LIGHT CALCULATIONS
AND MEASUREMENTS
PHILIPS
TECHNICAL
LIBRARY
LIGHT CALCULATIONS
AND
MEASUREMENTS
An introduction to the system of
quantities and units in light-technology
and to photometry
H.A.E. KEITZ
SECOND REVISED EDITION
MACMILLAN
© N.V. Philips' Gloeilampenfabrieken, Eindhoven (The Netherlands), 1971
Softcover reprint of the hardcover 2nd edition 1971
All rights reserved. No part of this publication may be reproduced or transmitted, in any
form or by any means, without permission.
Published by
MACMILLAN AND CO LTD
London and Basingstoke
Associated companies in New York, Toronto, Melbourne, Dublin, Johannesburg and Madras
ISBN 978-1-349-00012-8 (eBook)
ISBN 978-1-349-00014-2
DOI 10.1007/978-1-349-00012-8
PHILIPS
Trademarks of N.V. Philips' Gloeilampenfabrieken
No representation or warranty is given that the matter treated in this book is free from
patent rights; nothing herein should be interpreted as granting, by implication or otherwise, a licence under any patent rights
FOREWORD
Since the first edition of this book appeared 14 years ago, it has been widely
used in lighting practice and as a textbook, and appears to have received an
enthusiastic response from users. It was therefore possible to leave the form
and content of the book basically unchanged in this second edition.
A new section on Dourgnon and Fleury's quasi-central projection has been
added to the chapter on the representation of light distributions. A section on
the determination of the reflection properties of road surfaces in the chapter
on reflection, absorption and transmission has completely been revised. Finally,
a new section on the special measures to be taken when measuring the properties of gas-discharge lamps, with or without fittings, appears in the chapter
on the measurement of luminous flux. The absorption of colour filters has also
been dealt with in somewhat greater detail than in the first edition.
Measuring techniques have become much more mechanized and automated
of recent years; electronics, and in particular digital techniques, are widely used
in this connection. Such methods, which are also widely used for lighting
measurements, belong to the field of electronics rather than that of lighting
proper. They have therefore not been discussed in detail in this book, which
is mainly concerned with basic photometric methods.
The author has made grateful use of the remarks and suggestions received
from many readers since the appearance of the first edition. The whole text
and the illustrations have also been carefully revised, and it is hoped that this
second edition will also be of service to its readers in the practical and educational fields.
Thanks are due to all who have helped in the production of this book, and
in particular to Prof. Dr. H. W. Bodmann of the University of Karlsruhe (until
recently attached to the Philips' Lighting Laboratory in Aachen) who read
the entire revised second edition through, and to whom many improvements
in the text are due.
H. A. E. Keitz
CONTENTS
Page
PART I
LIGHT CALCULATIONS
Chapter I
Introduction
I-1.
2.
3.
4.
5.
6.
Chapter II
11-1.
2.
3.
4.
5.
Chapter III
III-I.
2.
3.
4.
5.
6.
The nature of light . . . .
Light as a wave-phenomenon
Polarisation. . . . . . . . .
Photometry; the photometric system of Lambert
The development of lighting engineering
Subjects dealt with in this book .
I
3
7
8
9
16
Solid Angle
Solid angle, steradian . . .
Some special solid angles .
Significance of the conception "solid angle" in illuruinating engineering
Small solid angles
Table of solid angles
18
19
21
21
24
Luminous Flux, Luminous Intensity, Quantity
of Light
Luminous flux; luminous intensity . . . . . . .
Units of luminous intensity and luminous flux
Formulae giving the relationship between luminous
intensity and luminous flux . . . . . .
Horizontal, spherical luminous intensity. . .
Luminous efficiency . . . . . . . . . . . .
Quantity of light, lumen-second, lumen-hour
25
28
32
33
35
35
Chapter IV
Light Distribution, Rousseau and Zonal Luminous Flux Diagrams
IV-I.
Light distribution, the Rousseau diagram . . .
Construction of the Rousseau diagram . .
39
45
of the infinitesimal calculus . . .
. . . . .
Determination of the efficiency of a lighting fitting
from the Rousseau diagram . . .
. . . . . . .
45
2.
3. Derivation of the Rousseau diagram with the aid
4.
47
VIII
CONTENTS
5.
6.
7.
8.
9.
10.
11.
Chapter V
The zonal luminous flux diagram
The area enclosed within the luminous intensity
curve is not a measure of the luminous flux emitted
Average luminous intensity calculated from the
Rousseau diagram
Russel angles.
The long-base Rousseau diagram for narrow beams
of light
Detailed example: determination of a 1000-lm curve
for a symmetrical lighting fitting .
Classification of lighting fittings according to their
light distibution
Luminous intensity table; polar and rectangular light
distribution diagrams
2. The isocandela diagram .
3. Spherical co-ordinates .
4. Sinusoidal projection
5. Transformation of spherical co-ordinates
6. The azimuthal projection
7. Comparison of the sinusoidal and the azimuthal
projections . . . . . .
8. Quasi central projection . •
9. The application of preferred numbers in isocandela
diagrams .
. ..
10. Examples of isocandela diagrams of a fitting ·. .
VI-1.
2.
3.
4.
5.
50
51
51
53
54
58
!If ethods of Representing Light Dt:stribution
V-1.
Chapter VI
48
60
60
62
62
64
67
70
71
74
.76
Illumination
Illumination; foot-candle; lux
The inverse square law . . .
Illumination with oblique incidence .
Horizontal and vertical illumination
Other units of illumination . , . .
Chapter VII
Illumination Calculations and Diagrams
VII-1.
2.
3.
Introduction . . . . . . . . . .
The lumen method; coefficient of utilization
The point-by-point method . . .
78
79
81
85
86
88
88
90
CONTENTS
IX
Illumination diagrams . . . . . . . .
92
93
The rectangular illumination diagram.
95
The solid of illumination.
95
7. The plane isolux diagram . . . . . .
8. The polar isolux diagram . . . . . .
96
9. Determination of the luminous flux from the plane
99
isolux diagram . . . . . . . . . . . . . .
10. The isolux diagram in quasi central projection
100
4.
5.
6.
Chapter VIII
Luminance and Luminous Emittance
104
Luminance; the stilb, candelas per sq. in. .
105
Lambert's law . . . . . . . . . . . . .
106
The luminous flux of uniform diffusers . .
108
Luminous flux of a uniformly diffuse cylinder.
111
.
.
.
.
.
.
.
.
.
Luminous emittance
Emittance and luminance of uniformly diffusing
surfaces . . . . . . . . . . . . . . . . . . . . 112
7. Units of luminance, based ~m the emittance of
uniform diffusers . .
112
VIII-I.
2.
3.
4.
5.
6.
Chapter IX
Non-Faint Sources
IX-1. Luminous intensity of non-point sources . . . . . 115
2. The inverse square law in another form . . . . . 120
3. Universal formula for the illumination produced by
uniformly diffuse circular light sources . . . . . . 121
4. Alternative derivations of the formula E = :rcL sin2 8 124
5. Luminance of light beams. . . . . . . . . . . . 128
6. Comparison between the illumination values obtained from equations E = Lw and E = :rcL sin2 8 128
7. Some special applications of the formula
E = :rcL sin2 8 = Ija 2 • • • • • • • • • • • • • • 130
8. Illumination produced by a uniformly diffuse circular light source in planes parallel to the source . 131
9. Illumination produced by linear light sources . . . 136
10. Comparison of illumination values of linear light
sources as obtained by exact calculation with those
obtained by means of the inverse square law . . . 140
11. Other kinds of non-point source . . . . . . . . . 143
12. The significance of the foregoing considerations as
applied to practical forms of light source . . . . . 143
X
CONTENTS
Chapter X
Reflection, Absorption, Transmission
146
Reflection, absorption, transmission. . . .
147
Regular reflection . . . . . . . : . . . .
3. Reflection factors of non-conductive materials
149
4. Reflection factor of conductive materials
·152
152
5. Diffuse reflection . . . .
. 15 7
6. Uniform diffuse reflection . . . . . . .
. 158
7. Luminance factor . . . . . . . . . . .
8. The luminance factor as a constant of the material 162
167
9. Gloss . . . . . . . . . . .
168
10. Reflection from road surfaces
II. Transmission . .
174
12. Density . . . . .
176
13. Diffusion factor . .
178
14. Mvltiple reflection
180
15. Absorption . . . .
183
16. Absorption of coloured filters
187
17. Colour of reflected light . .
188
X-1.
2.
0
Chapter XI
XI-I.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
Properties of Optical Systems
Introduction . . . . . . . . . . . . .
190
Relationship between luminance and refractive index;
Abbe's law . . . . . . . . . . . . . . . . . . . 191
Luminance of images formed by lenses and mirrors,
and of the lenses and mirrors themselves . . . . . 196
Optical systems as light sources; the exit pupil of
optical systems . . . . . . . . . . . . . . . . . 200
Determination of the exit pupil of an optical system;
entrance pupil . . . . . . . . . . . . . . . .
203
Object at the focus of a lens; angle of divergence . 205
The exit pupil of lens systems; vignetting . . . . 207
The luminous flux of optical systems. Aperture of
lens and mirror systems . . . . .
. · 212
Some remarks on mirror systems .
214
Drum lenses . . .
214
Cylindrical mirrors . . . . . . .
216
Facetted mirrors . . . . . . . .
·217
Optical systems for the projection of images
218
Diascopic projection. . . . . . .
219
CONTENTS
15.
16.
17.
18.
19.
Chapter XII
XII-I.
Slide projection. . . . .
Standard-film projection .
Sub-standard film projection.
The luminous flux emitted by projection systems
Episcopic projection. . . . . . . . . . . . . .
5.
6.
7.
8.
9.
221
225
227
228
230
The Photometric Measuring-units System
Introduction . . . .
2. The Luminance criterion . . .
3.
4.
XI
233
234
236
239
242
Luminance of coloured light .
The relative luminous efficiency of radiation
The summation law . . . .
Definitions of photometric quantities and units
based on v,\ . . . . . . . . . . . . . . . . . . 243
Vision at high and low luminance levels; Purkinje
effect . . . . . . . . . . . . . . . . . . . .
248
Equivalent luminance . . . . . . . . . . . . . . 251
Minimum perceptible luminance difference and
sensitivity to luminance difference . . . . . . . . 254
MEASUREMENT OF LIGHT
PART II
Chapter XIII
XIII-I.
2.
3.
4.
Chapter XIV
XIV-I.
2.
3.
4.
General C on.siderations
Introduction. Principles of visual and physical photometry . . . . . . . . . . . . . . . .
263
Standard light sources
. . . . .
264
Simultaneous and substitution methods .
268
Some practical hints . . . . . . . . .
269
Visual Photometry and Photometers
Principle of the visual photometer .
. . . . . . 273
Forms of photometer field . . .
. . . . . . 276
Methods used in photometers for obtaining the
variation of luminance . . . . . . . . . . .
282
Photometer bench. Some examples of visual photometers . . . . . . . . . . . . . . . . . . . . . 286
XII
CONTENTS
5.
6.
Homochromatic and heterochromatic photometry . 291
Choice of observer . . . . . . . . . . . . . . . 292
Chapter XV
XV-I.
2.
3.
4.
5.
6.
Physical Photometers and Photometry
Principles of physical photometers and photometry 295
Photo-emissive cells . . . . . .
296
Photo-voltaic cells . . . . . .
302
313
Bolometers and thermo-couples
Photographic photometry . . .
314
Physical photometers for heterochromatic photometry 315
Chapter XVI
XVI-I.
2.
3.
4.
5.
Measttrement of Luminous Intensity
Principle of luminous intensity measurement
The photometer bench . . . . . . . . . .
Apparatus for measurement of light distribution
Measurements on projectors . . .
Photometer with Maxwellian view . . . . . .
Chapter XVII
XVII-I.
2.
3.
4.
5.
6.
7.
8.
2.
3.
Measurement of Luminous Flux, Quantity of Light
and Luminous Emittance
The Ulbricht sphere photometer . . .
. . . . . 335
Illumination of the sphere window in the "ideal"
integrating photometer . . . . . . . . . . . . . 336
Illumination of the window of a non-ideal integrating
photometer . . . . . . . . . . . . . . . . . . . 338
Measures to be taken with non-ideal integrating
photometers to approximate to the ideal sphere 340
The measurement of the luminous flux of fluorescent
lamps and the appropriate fittings . . . . . . . . 351
Determination of luminous flu~ from the light
distribution or from an isolux diagram .
. 355
Measurement of quantity of light .
. 355
Measurement of luminous emittance
. 357
Chapter XVIII
XVIII-I.
. 321
. 322
324
329
332
Jfeasttrement of Illumination
Introduction . . . . . . . . . . .
. . . . . . 359
Measurement by means of laboratory photometers 360
Visual illumination photometers . . . . . . . . . 36 0
XIII
CONTENTS
4.
5.
Chapter XIX
Physical illumination photometers
Calibration of illumination photometers .
362
366
AIeasurement of Luminance
Direct visual measurement of luminance
. . 367
2. Visual and physical luminance measurements obtained from measurement of illumination . . .
372
3. Measurement of luminance distribution . . . .
374
4. Determination of the size of the exit pupil of lenses 380
XIX-I.
Chapter XX
]I,Jeasurement
of
Reflection, Transmission
Absorption
XX-I. Measurement of reflection factor
2. Measurement of transmission factor.
3. Measurement of absorption . . . .
4. Measurement of luminance factors and glos~
and
381
388
390
391
APPENDIX
International co-operation in Illuminating Engineering
403
Table I
Table II
- Table of solid angles . . . . . . . . .
40 7
- Values of cos31X cos3 p for a number of values of the
angles IX and p . . . . . • . • . . . . . . . . 408
Table III - Values of tanp /cos IX for a number of values of the
angles IX and P . . . . . . . . . . . . . . . . . 410
Table IV - Units of illumination and their mutual conversion
factors • . . . . . . . . . . . . . . . . . . . 4 12
Table V - The units of luminance and their mutual conversion
factors . • . . . . • . . . . . . • . . . . . . 4113
Table VI - International relative luminous efficiency of radiation
for photopic vision . • . . . . . . . • . . . . . 414
Table VII - International relative luminous efficiency of radiation
for scotopic vision . . . . . . . . . . • . . . . 415
Light distribution and zonal luminous flux diagrams for a number
oflighting fittings· •
417
Index
426
TABLE OF MOST IMPORTANT SYMBOLS USED IN THE TEXT
ex
~
y
8
£
7J
>.
1-'
v
7T
p
a
T
'I'
'P
w
J
e
E
<P
1
b
c
d
I
g
h
i
l
n
r
r, R
t
u
v
E
H
I
K>.
L
L
M
Q
s
T
V>.
V>,'
alpha
beta
gamma
delta
epsilon
eta
lambda
mu
nu
pi
rho
sigma
tau
phi
psi
omega
delta
theta
s1gma
phi
Plane angle, absorption factor
Plane angle, luminance factor
Plane angle
Plane angle
Emission factor
Efficiency of fittings, coefficient of ultilization
Wavelength
Micro
Frequency
3.1415 ....
Reflection factor
Diffusion factor, transmissive exponent of diffusing
media
Transmission factor
Plane angle
Plane angle
Solid angle
Small part. of a quantity
Half-aperture angle of light beams
Sum of a number of quantities
Luminous flux
Transmissive exponent of transparent media
Geographical latitude
Velocity of light
Distance
Focal distance
Optical limit distance of photometry, gloss number
of Harrison
Height
Angle of incidence
Geometrical longitude, length
Refractive index
Angle of refraction
Radius
Time
Object distance
Image distance
Illumination, energy (power)
Luminous emittance
Luminous intensity
Luminous efficiency of radiation at wavelength >.
Luminance
Equivalent luminance
Mechanical equivalent of light, linear magnification
Quantity of light
Area, density
Periodic time
International relative luminous efficiency of radiation
for photopic vision at wavelength >.
International relative luminous efficiency of radiation
for scotopic vision at wavelength >.
LIGHT CALCULATIONS
AND
MEASUREMENTS
CHAPTER I
INTRODUCTION
I-1. The nature of light
The natural phenomenon which we know as light governs to a large
extent all our activities; it is therefore not surprising that ever since
the earliest times man has endeavoured to produce artificial light.
Sources of such light have in the course of time been evolved, from the
primitive wood-fire by way of oil lamps, candles, paraffin and gas lamps
in numerous varieties, to the present-day electrical sources of light,
the electric filament and gas discharge lamps.
Naturally enough, the efforts of scientists have from a very early
date been directed towards finding an answer to the question: what
is light?
The ancient Greeks held several theories which for a long time amounted
to the assumption that the eyes emitted radiations which located the
objects around us. In this, they overlooked the fact that the sun is the primary source of light. When, amongst others, A r i s t o t l e demonstrated
the fallacy of this idea the learned men of the day came to the conclusion
that light consisted of a current of very small particles emitted by
incandescent substances; the particles, on entering the eyes, were supposed to be responsible for the sensation of light.
Of those theories which were developed in more recent times those of
Christiaan Huygens (1629-1695) and Isaac Newton
(1642-1727) are the most well-known. Their views on the nature of
light are to a large extent compatible with present-day conceptions.
In 1678 H u y g e n s put forward the opinion (published in 1690) that
light is a wave phenomenon (the wave theory); according to this theory
light consists of vibrations in a hypothetical medium, the light-ether
or, in short, the ether. Such vibrations would be propagated in straight
lines.
Newton had quite a different conception of the nature of light;
he held the theory that light sources emit particles which, on entering
the eye, produce the sensation of light, and this was actually an extension of the theory propounded by the ancient Greeks. Newton's
theory is known as the corpuscular theory.
2
INTRODUCTION
rr
Both H u y g e n s and N e w t o n had their supporters for a great
many years. One of the followers of the wave theory was F r e s n e 1
(1788-1827) who made certain elaborations on it. Many phenomena
could be explained in terms of both of these theories, but in the long
run it proved to be impossible to explain certain newly discovered
characteristics of light with the aid of the corpuscular theory, whereas
these could be fitted in with H u y gens' wave theory which thus
gradually gained supremacy. None the less, it was not until the middle
of the 19th century that the wave theory came to be universally accepted.
At the same time there were some very valid objections to the properties
of the ether as postulated by the wave theory and on which that theory
was based. These objections were overcome by the electromagnetic theory
of light formulated by Max we 11 (1831-1879): according to this
theory light is compounded of electrical and magnetic vibrations of
the same kind as electromagnetic waves such as are produced . by an
oscillating electric current, e.g. in a spark discharge.
In the course of time it has been found that a large number of radiations
are of the same nature as light, amongst these being x-rays and radio
waves. All these consist of electromagnetic waves, amongst which light
waves differ from the others in that, when they enter the eye, they
constitute a stimulus which conveys to the brain the sensation of "light".
This difference, from the physical point of view, will be discussed in the
next section.
In its turn the electromagnetic theory, too, failed to supply an explanation of certain light characteristics later discovered; for example
it did not explain the photo-electric effect, a subject which we shall mention
in connection with photo-electric cells.
A new principle was next introduced to the science of physics, namely
the quantum theory, as developed by P 1 a n c k. This theory states
that all radiations consist in the emission of energy, not progressively
as in the case of electromagnetic waves, but in certain very small
discrete quantities at a time. These quantities are called light quanta,
or photons.
Although the photons are emitted at irregular intervals, there is still
this much regularity that a constant-burning light source emits the
same average number of photons in the same interval of time. So much
evidence has. been brought forward in support of the photon that it
is now a generally accepted convention.
It may be noted here that the quantum theory really re-introduces
Newton's corpuscular theory, albeit in a modified form.
I-2]
LIGHT AS A WAVE-PHENOMENON
3
What then, of the wave-like character which light is assumed to possess
and which, with one or two exceptions, has been the means of explaining
all the properties of light? The answer to this was given in 1924 by
d e B r o g 1 i e and again in 1925 by S c h r 6 d i n g e r in his wave
mechanics; agreement was thus reached on the dual character of radiation.
The modern conception of light may be expressed as follows.
As far as materials are concerned (photo-electric cell, the eye, photographic plate) light - and radiation in general - behaves as though
it were composed of quanta, but results, i.e. the average number of
quanta which reach a surface per unit of time can in every case be
computed accurately by means of the wave theory.
We may therefore regard light (and radiation in general) as a wavemotion propagated in straight lines and conveying energy with it. Light
and all radiations are therefore energy and, when absorbed, this energy
can be converted to other forms of energy such as heat or electrical
energy. We shall refer to this again later.
For so far as may be necessary in the discussions in this book to refer
to the physical character of light, we shall regard light as a wavephenomenon.
I-2. Light as a wave-phenomenon
Before proceeding to a discussion of light as a wave-phenomenon, let
us consider wave-motion in general. Fig. I shows a line AB. Assume
that every point on this line from A to B is set in motion successively,
i.e. that they move rapidly up and down. Since each point commences
to vibrate later than the preceding one, a wave-like motion is set up,
such as can be observed when a stone is thrown into the water. The
wave-motion is thus propagated along the line AB and this is called
the direction of propagation of the wave. If we now draw in the deflections
exhibited by the points on AB at a given moment and join these points
by a line, we obtain the wave-shaped curve shown in Fig. I.
The direction of vibration of this curve is perpendicular to the direction
of propagation, and waves of this kind are called transverse. When the
direction of vibration coincides with the direction of propagation, the
wave is said to be longitudinal.
If the curve, of the kind depicted in Fig. I, is sinusoidal, we speak of a
harmonic vibration or wave.
It is now necessary to define some dimensions and conventions relating
to wave motion.
4
ll
INTRODUCTION
The greatest deflection from the position of rest is known as the amplitude
of the vibration (a in Fig. I). Of those points which, from the point
of view of distance and mobile conditions, are situated similarly with
respect to AB, it is said that they are in phase (e.g. points P and Q,
Fig. 1).
-Qv \
Av \
1/
\
v \
I
~
v
~
1\ v
I
1-
v~
A
/' ~
I
I
~ .,.1
1\~ ,..1
Fig. I. Transverse harmonic vibration. AB = direction of
propagation. a = amplitude. .\ = wavelength. The points
P & Q, R & S are in phase.
The distance between two successive points having the same phase
is termed the wavelength (A.); this is accordingly also the distance between
two successive peaks in the wave (e.g. R and S).
The time during which a point describes a complete oscillation, that
is, the time taken by it to travel from the position of rest first in the
one direction and then back in the other direction, through the point
of rest, to return finally to the starting condition, i:s the periodic time (T).
From Fig. I it will be seen that one wavelength is just completed during
the periodic time, for, when the wave has travelled the distance A from
R to S, the point R, in completing one vibration, has again acquired
the same phase. Hence the speed of propagation, o:r the velocity (c)
of the wave-motion is found to be
A
c = T'
(I-1)
By the frequency (v) of the vibration is meant the number of vibrations
per second. If the periodic time of one vibration be denoted by T, the
frequency is
, =
-y·
(1-2)
1-2]
LIGHT AS A WAVE-PHENOMENON
5
From equations (I -1) and (I -2) it follows that
c =A.. v.
(1-3)
EVf~ry wave-motion can thus be characterised by the dimensions wavelength, velocity and frequency, and the relationship between these quantities is expressed by equation (1-3).
In electromagnetic waves (and therefore also in light) we are concerned
with an electrical field, the strength of which varies with the time, the
value differing from one point to another. The electrical field strength
is perpendicular to the direction of propagation, and electromagnetic
waves are therefore transverse waves. The instantaneous values of the
field strength along a ray of light may be represented by lines (vectors)
in the manner shown in Fig. I.
The electrical field strength is associated with a magnetic field which
varies simultaneously with the electrical field; the magnetic field strength
is also perpendicular to the direction of propagation and is at right
angles to the electrical field.
If we represent both of the field strengths by vectors, an electromagnetic
wave-motion may be illustrated in the manner shown in Fig. 3; here,
the wavelength is equal to the distance PQ.
The waves are sinusoidal waves and it may accordingly be said that
the "vibration" of the electrical and magnetic field strengths is harmonic.
Of the characteristic dimensions v, A and c, the velocity for all electromagnetic waves in a vacuum is the same, viz. 2.99792 x 1010 cmfsec,
or, rounded off, 2.998 X 1010 cmfsec, which is almost 300.000 kmfsec.
In all other media, (air, glass etc) the velocity is lower, but, whereas
the velocity in vacuum is the same at all wavelengths, it is different
at every wavelength in other media. In air the velocity is only slightly
lower than in vacuum (only 3 per 10.000 for light in air at a pressure
of 760 mm, and oo C), and the differences for the various wavelengths
are only small.
When the value of the velocity in equation (I-3) is varied, the question
arises which of the two quantities in the second term (A and v) will
vary with it. It is found that the frequency of a given radiation is a
constant, i.e. it is independent of the medium through which the radiation
is passing. From (I-3) it follows that the wavelength of a radiation
varies proportionately with the velocity of propagation.
It is customary to designate the different kinds of radiation by their
wavelength, although, in view of the fact that the wavelength is dependent on the medium, it would be more logical to indicate the fre-
6
[I
INTRODUCTION
quency. We shall follow the established practice and name the wavelength, the values being understood to be in respect of vacuum. These
values differ only slightly from those in air.
Radio waves are usually specified in metres. However, for the range
of wavelengths of interest in lighting engineering, this unit is much
too large to yield convenient values. Centimetres and millimetres are
also too large for the purpose and various smaller units are employed, viz:
the micron (1 p.m = 10-3 mm)
the millimicron or nanometre (1 mp. or lnm = 10 - 6 mm)
the Angstrom unit (LA = t0- 7 mrr1 = 0.1 m,u)
The symbols used throughout this book for the miGron and the millimicron
will be p.m and mp. respectively.
It is generally assumed that the human eye is capable of perceiving
radiations of wavelengths between 0.40 and 0.70p.m 400-700 m,u or
4000-7000 A). The range of_ wavelengths to which the eye is sensitive
actually extends from 313 m,u to 1050 m,u, but it is only the range
from 400 to 700 m,u that is of general interest in lighting technology.
The difference between wavelengths in the visible range is seen by the
human eye as colour. Radiation at 0.4 .urn is perceived as violet, while
that in the range above 0.6 .urn is seen as red, with the colours visible in
the rainbow lying between these two. The sequence of colours in the
visible spectrum is violet, blue, green, yellow and red, with the intermediate colours like turquoise, yellow-green and orange between them.
Fig. 2 gives the electromagnetic spectrum and shows the various wavelength ranges with the names by which they are known.
wsrble lrghl
cosm;c
rod1otion
ll-rays
1
X-rays
T
UV-rad Infrared rod.
t
TOO lj.J
10 100 lmm 10
1
;o ··· 10 '. 10 ' 10·' 10 ' TO 1 10 · 5 TO· IL ·l 10··' 10·'
!
:
Imp 10
Fig. 2.
Radrowaves
;;
1- ~ ~
EHFSHFUHFVHF SW MW LW
I
I
T
I
•
I
TO' 10 -' 10' 10" 10·· 10' 10 7 mm
100 lm 10 100 TKm /0
The electromagnetic spectrum.
I-3]
POLARISATION
7
The wavelength zones which border on the range of light w;~ves are
the ultra-violet (extending from 0.2.um to 0.4pm approx.) and the infrared or heat rays (from 0 7 f..lm to 100 .um approx.).
The characteristics of ultra-violet radiations are of interest mainly in
chemistry and biology (tanning and reddening· of the skin, therapeutic
action e.g. in rickets, germicidal properties).
Infra-red radiations are known for the pronounced sensation of heat
which they produce; for this reason they are often referred to as heat
rays, although this is not strictly correct, seeing that visible. radiations
(and ultra-violet) also produce a sensation of heat. In such cases the
energy of the light wave is converted to heat energy.
Now, the light emitted by temperature or incandescent radiators such
as the sun or the electric filament lamp is always accompanied by a
large amount of infra-red radiations; the light from such sources gives
a pronounced sensation of heat which is sometimes pleasant, but sometimes a distinct source of discomfort, for which reason cold light is often
asked for. From the foregoing it will be seen tha:t there is really no such
thing as "cold" light in the absolute sense·; when we speak of such light
we therefore merely mean light that is accompanied by little or no
invisible radiation such as infta-red *).
Radiations consisting of light of only one single wavelength are known
as monochromatic, and monochromatic kinds of light of different
wavelengths are distinguished from each other by the colour sensation
which they produce (spectral colottrs). Combinations of different wavelengths, with which we are almost always concerned in practical work,
also produce colour sensations; these are to be regarded as mixed colours
of spectral colours. Certain combinations will give an impression of
white light. Colour perception, however, falls outside the scope of this
book and will not form part of our discussion 1 )**).
I-3. Polarisation
It is stated in the previous section that light can be regarded as an
electromagnetic, transverse, wave-motion. From the aspect of the plane~
in which the el~ctrical and magnetic field strengths "vibrate", various
possibilities exist. For example the "vibration" of the electrical field
*) The meaning of "cold" light intended above should not be confused with
another meaning. often attributed to this term, viz. light of a certain colour such
as blue which produces an unpleasant "cold" sensation.
**) The numerals refer to the bibliography at the end of each chapter.
8
INTRODUCTION
[I
E
Fig. 3. Electromagnetic wave. The electrical field strength E
"vibrates" in the plane W, the magnetic field strength H in
the plane V, perpendicular to W.
strength may lie in one plane only (with the magnetic field strength
in a plane perpendicular to it), this being the in~tance depicted in Fig. '3.
That plane which is at right angles to the electrical field (V, Fig. 3) is
then known as the plane of polarisation and we say that the light polarised
is in this plane.
Possibly there may be no definite direction of polarisation, in which
case the electrical "vibration" takes place in all directions without
any preference for one or the oth.;:r; we then have natural, or non-polarised,
light.
The light from sources such as the sun, electric incandescent lamps
and gas discharge lamps may be regarded as non-polarised light.
1-4. Photometry; the photometric system of Lambert
Until the 18th century the study of light was limited almost entirely
to ·geometrical optics which deals with the behaviour of rays in lenses,
prisms etc. Little or nothing had thus far been done in the quantitative
measurement of radiation.
The first to announce a more or less successful C~;ttempt to measure light
(photometry) was the Frenchman B o u g u e r (1729), who dealt with
only some of the quantities and conceptions which we now employ
in photometry and illuminating engineering. There is very little evidence
of any mathematical treatment of the problems, or satisfactory definitions of the conceptions in Bouguer's work.
1-5]
THE DEVELOPMENT OF LIGHTING ENGINEERING
9
In 1760 L a m b e r t *) published his "Photometria sive de mensura
et gradibus luminus, colorum et umbrae", i.e. "Photometry, or the measurement and classification of light, colour and shadow" 2 ).
In this, L a m b e r t developed a system of conceptions (photometric
system), the principle of which is still in use unchanged today.
Mathematically he established a large number of relationships between
the different concepts and, although many of these were found. to
be of little practical interest, it is surprising to note when reading his
book that so many of his formulae have been adopted in publications
on light and photometry of the last decade or two. Here one senses
the genius of this founder of photometry who built up his system
unaided.
The practical methods of photometry described in Lambert's work
were primitive in the extreme and there is no record of any photometer
in the form in which it is known today. It was not until the second half
of the 19th century, when lighting technology came to be developed,
that justice was done to the work of Lambert. We shall have
something to say about this development in the next section.
I-5. The develo~ment of lighting engineenng
The term lighting engineering is understood to be the technique which
embraces everything relating to the production and application of
light •*).
For that branch of lighting technique which deals with the actual
production of light we have no specific term; we might refer to it as the
technique of light production.
The branch of the technique that relates to the applications of light,
i.e. the illumination, falls under the heading of illuminating engineering.
The technique of light production covers the development and manufacture of the primary sources of Jight, that is, the equipment which converts
the energy supplied to it into visible radiation.
Illuminating engineering includes not only the design and execution
*) J. H. L am b e r t, born at Mulhausen in Alsace in 1728; died 1777 in Berlin.
Ht: ~as self-educated, a .fact that pr?bably accounts for the originality of his
wntings. J:Ie was versed m many subJects and also wrote works on philosophy,
mathematics, heat, sound and astronomy.
• *)The term "technique" is taken to cover the entire equipment and methods
employed in the execution of one of the arts.
10
INTRODUCTION
[I
of lighting installations, but also the development of lighting fittings,
which refers to the equipment in which the primary light source is
contained and which serves to throw the light from the source in those
directions where it is required by the lighting engineer. Or again, the
lighting fitting may be such as will mask the light partially or wholly
in directions where it would otherwise be found a hindrance (glare),
alternatively the fitting may fulfil a decorative function, or it may
merely protect the lamp.
When we speak of the development of lighting technique we should
first make it clear that, until the latter half of the 19th century, developments related almost entirely to techniques in the production
of light. By the second half of the 18th century there had been little
question of any development, for the firebrand or torch, the candle and
oil lamp were until then the ordinary sources of light which, in the
technical sense, had not risen above the level of the primitive wood
fire. Such light sources were too weak for the execution of more than
the simplest domestic activities and therefore served mainly to maintain
domestic and social life after sunset. There was little demand to extend
the working day with the aid of increased or improved artificial lighting.
The second half of the 18th century marked, particularly in England,
the commencement of the industrial era which came about mainly as
a result of the increased demand for merchandise, especially in the
European colonies and America. In order to meet this demand production
had to be increased constantly and the machine made its entry into
the factories (spinning machines and looms, driven by a steam engine).
The daily hours of work, too, had to be increased to keep pace with the
demand, and the need for better sources of light arose. It is therefore
only natural that at this time numerous improvements to existing light
sources were introduced, amongst which we may mention the cylindrical
lamp chimney (Quinque t, 1765) and the centre-draught oil burner
which took the place of the solid wick (A r g and, 1786).
At the beginning of the 19th century the technique of light production
was much improved by the introduction of the coal gas jet (batwing
burners).
By the middle of the 19th century the replacement by paraffin of the
oil used in oil lamps marked another important step forward.
The greatest impetus to the production of artificial light was given by
E d i s o n in 1879 when he succeeded in making a serviceable electric
filament lamp suitable for manufacturing as an industrial product.
Up to that time only the carbon arc was known as an electrical sourcP
1-5]
11
THE DEVELOPMENT OF LIGHTING ENGINEERING
of light, but this was a powerful source of light and therefore unsuitable
for the small rooms of private residences or offices.
It was possible to manufacture Edison's lamp in units of relatively
small power and these accordingly promoted the use of electric light
in almost all lighting installations.
The whole impact of the development of the electric lamp would have
teen lost, however, if it were not for the fact that at the same period
a development in electrical technology, namely the invention of the
dynamo and suitable means of distribution of electric current made it
possible to generate electrical energy on a large scale and to supply it
~~~~~~.
.
Edison's electric lamp contained a carbon filament which was
heated to incandescence, thus making it a source of light, by passing
electric current through it. To prevent the carbon filament from being
burnt it was mounted in a glass bulb from which the air was exhausted.
The carbon filament evaporates rather quickly, which is why it cannot
be taken to very high temperatures. This means that the useful output of
these lamps was relatively low. It was possible to iinprove this somewhat
by metallising the carbon filament, but no basic increase in the useful
output was achieved thereby. Such an increase was successfully brought
about only when the carbon filament was replaced by metal filaments.
Higher temperatures could be .used first with osmium ( 1902) and later
with tantalum ( 1905), 'which therefore gave more light for less power.
Even better results were obtained with tungsten (tungsten filament lamps,
1906).
To this day tungsten is the material from which electric lamp filaments are
made. Originally these lamps had a straight filament and evacuated
bulb (vacuum lamps).
Later still, in 1913, Langmuir introduced lamps filled with an inert
gas. The gas filling reduces the rate of evaporation of the tungsten, b"yt
also lowers the temperature of the filament because it dissipates the
heat more quickly. Langmuir remedied this drawback by coiling the
filament. Since the heat dissipation is proportional to the length of the
incandescent body and depends only slightly on its diameter, the losses
due to the gas were more than compensated by the coiling of the filament.
Coiled coil lamps were first made in 1934. The purpose of this arrangement is to shorten the incandescent body even further and to reduce the
losses through the gas even more.
Development of incandescent lamps has been continued over the past few
years and is still in progress. The life of an incandescent lamp, and its
12
INTRODUCTION
[I
efficiency, are determined to a considerable degree by the evaporation
of tungsten from the filaments. The evaporated tungsten is deposited on
the envelope of the lamp in the form of a grey or black coating, gradually
reducing the light output during the life of the lamp. If this evaporation
process could be reduced, therefore, the lamp would last much longer and
produce a greater useful output.
It has been found that the effects of evaporation can be gr~atly reduced
by the addition of a small quantity of iodine to the gas filling. As the
tungsten evaporates, it combines with the iodine to form tungsten iodide
at temperatures up to some 800 °C. At temperatures above 2000 °C,
however, the tungsten iodide decomposes again to form iodine and
metallic tungsten. The latter process, therefore, can take place only at the
filament itself or in its immediate vicinity.
The temperature of the glass bulb of the ordinary incandescent lamp
remains below that which favours the combination of tungsten and
iodine to tungsten iodide. If, therefore, the temperature of the bulb
could not be raised, there would be no point in introducing iodine into it.
It is, however, possible in quite a simple manner to attain the desired
increase in the temperature of the bulb- it only has to be made smaller.
Nevertheless, the temperatures of up to 800 oc required at the envelope
do generally mean that the latter must no longer be made of glass but of
quartz. An increase in the efficiency of about 25% has been obtained
for the same life.
Recently, other halogens* and mixtures of different halogens have been
used in these lamps, too, which is why they are referred to as halogen
lamps. This development has, in the initial stages, been directed towards
special-purpose lamps.
While on the subject of electrical light sources we should also mention
gas discharge lamps, the development of which has taken place during
the last 30 years 3). In these lamps, e.g. mercury vapour and sodium
lamps, a pilot discharge in an auxiliary gas renders the vapour conductive
by splitting it into ions and electrons. When current flows, processes
occurring between the electrons and atoms of the vapour result in the
emission of light. The colour of the light depends upon the type of
vapour or gas and also upon the gas pressure. In these lamps the light
is not produced as a result of incandescence due to high temperatures
as in filament lamps; gas discharge lamps are therefore not temperature
radiators.
*) The halogens are the clements chlonne, bromine, iodine and fluorine.
1-5]
THE DEVELOPMENT OF LIGHTING ENGINEERING
13
Whereas temperature radiators emit all the visible wavelengths (as well
as infra-red and some ultra-violet), thus producing a contznuous spectrum,
gas discharge lamps radiate only those wavelengths which are characteristic of the vapour or gas in the lamp (line spectrum).
The latest development of the gas discharge lamp embodies an entirely
new principle in the form of the tubular fluorescent lamp 4 ). Here, the
inside of the tube is coated with a powder which has the property of
converting radiation of the shorter wavelengths into radiation of longer
wavelenghts. By carefully blending these fluorescent substances it is
possible to modify the colour of the light emitted and in this way lamps
can be manufactured to give light of the same colour as that of a
temperature radiator, or the colour of daylight. In these lamps, too,
the spectrum is continuous.
The main advantage of gas discharge la~ps as compared with filament
lamps is their higher efficiency; on the other hand the fact that a currentlimiting device is essential is a drawback.
Improvements effected in the efficiency of light sources during the
course of time are illustrated in graphical form in Fig. 4. In this diagraJ;n
the efficiency of the light source is plotted vertically in lmfW, against
time on the horizontal axis. The unit lm/W is explained in section
111-5; at this point it is sufficient to say that it is a measure of the amount
of light emitted per second by the source, per unit of power in watts.
In the case of combustion or flame sources of light such as the candle
or paraffin lamp, the energy is supplied by the heat of combustion.
It
be seen that the efficiency of the candle and paraffin lamp, which
is 0.1 to 0.3 lmfW, is improved upon by the carbon filament lamp to
the extent of a factor of 10. T4e introduction of the tungsten lamp
yielded further improvement by a factor of 5 and the gas discharge
lamps increase efficiency by another 2 to 10 times.
A new method of producing light, known as electroluminescence, has
recently been developed. Basically, an electroluminescence plate is a
capacitor with one of its electrodes made of a translucent material and
with a certain luminescent material as the dielectric. If an alternating
electric field is applied to this device, light will be produced in the
luminescent layer.
Because the efficiency of these "elu plates" is very low, they are not
suitable for lighting purposes. For the moment, their use is restricted
to warning notices, house numbers, radio receiver scales, instrument
dials, etc.
No discussion of techniques of light production would be complete
will
14
[I
INTRODUCTION
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Fig. 4. Progressive increase in the efficiency of artificial light sources
a) Wax candle
b) Paraffin lamp
c) Edison carbon filament lamp
d) Carbon filament lamp
e) Lamp with metallised carbon filament
f) Tungsten filament lamp (straight filament, vacuum)
g-)
(single spiral, gas-filled)
h)
( ..
..
vacuum)
i)
..
..
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(coiled coil,. gas-filled)
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k) Sodium lamp
l) Fluorescent lamp
m) Blended light lamp (fila ment and mercury vapour light)
n) Electroluminescence
1-5]
THE DEVELOPMENT OF LIGHTING ENGINEERING
15
without mention of the. important improvement in gas lighting introduced by Au e r von We 1 s bach in 1886, in the shape of the
incandescent gas light. V o n W e 1 s b a c h made use of the peculiar
radiating properties of the oxides of certain metals in the rare-earth
group, chiefly cerium and thorium. When heated to incandescence these
oxides radiate strongly those wavelengths which lie within the visible
spectrum, with relatively little infra-red. The "incandescent mantle"
may consequently be heated to a high temperature to give radiations
in the zone of wavelengths to which the eye is particularly sensitive.
The radiations of such oxides are accordingly said to be selective.
This improvement in gas lighting arrived at a time when the invention
of the electric lamp made it look as though electric lighting would take
the world by storm and rapidly replace gas. Owing to v o n W e 1 sb a c h's invention, however, the need for electric light was not by
any means so strongly felt; gas lighting therefore held its ground for
many years and is still used.
Acetylene lighting may be mentioned as being the third great step forwarq
in techniques of light production. For isolated buildings it was particularly useful, but nowadays the carbide-lamp is employed almost
exclusively for signal and portable lamps.
Development of the E d i s o n lamp and systems for the distribution
of electricity since 1880 made it comparatively easy to meet the demand
for improved lighting, especially in industry, in not too costly a manner.
With the increases in illumination levels - which should not be overestimated, however - came the realisation that the new light sources
could not be used in the same way as the old flame sources, that is,
free to radiate light in all directions. Glare was a new factor to be reckoned
with and, this being harmful to the eyes, lamps were fitted with shades;
these constituted virtually the first lighting fittings as such and in this
way illuminating engineering came into being.
The next step was to make use of reflecting materials to direct the light
towards those points where it was required, i.e. towards the working
plane.
At that time the associated problems were purely physical and technological, development being aimed mainly at higher and higher illumination
levels. Later (ca. 1905) it was seen that this in itself was not enough,
but that effective lighting was more an optico-physiological problem.
Subsequently (ca. 1930) it was realised that vision also involves a
psychological factor and that illuminating engineering would have to
take this into account.
16
INTRODUCTION
[I
So far we have only touched upon artificial light and its sources in
lighting engineering, but it must not be overlooked that this subject
also includes natural daylighting. Particularly within the last few decades
numerous methods have been evolved for computing indoor daylighting
from the geometrical proportioning of windows and other boundaries
of the natural light from the sky. A discussion of this would take us
beyond the scope of this work, however 5 ).
1-6. Subjects dealt with in this book
The first part of the book is devoted to a review of the system of dimensions and nomenclature which we have so far referred to as the
photometric system. A number of the relationships existing between
these quantities will be developed, these being essential for a
thorough understanding of the photometric system, as well as for the
various lighting computations which the lighting engineer will usually
encounter.
The reader may possibly say: all this is sufficiently clear as applied to
white light (e.g. from an incandescent lamp), but what happens in the
case of coloured light such as that produced by the sodium lamp?
To this we reply that in many books the answer to this question is
given at the start, but that in the author's opinion a discussion of the
subject is more simply followed if the reader is conversant with the
photometric system. Preference is therefore given to a postponement
of an explanation of the significance of lighting nomenclature as applicable to coloured light until the last chapter of the first part of the
book. Until then the reader is asked to assume that the discussions
refer to white light, bearing in mind that in chapter XII we shall explain how the various considerations and definitions may be extended
to cover coloured light.
The second part of the book deals with methods of photometry and
makes frequent use of the theory contained in the first part.
The author considered that it would be useful to precede the section
on lighting computations by a chapter covering the mathematical
conception of solid angle which plays such an important part in illuminating engineering. In this it is assumed that the significance of the
term is not always sufficiently understood amongst those who may wish
to know something more about lighting technology.
I-6J
SUBJECTS DEALT WITH IN THIS BOOK
17
REFERENCES
See for instance: P. ]. Bouma: "Physical Aspects of Colour". Philips Technical Library, Eindhoven 1948
J. Bergmans, "Seeing Colours", Philips Technical Library, Eindhoven, 1960.
2 ) In 1892 an abridged German translation of
Lambert's "Photometria"
was published in the series "0 s twa 1 d's Klassiker der exakten Wissenschaften" (nos 31, 32 and 33)
3 ) J. Funke and P. ]. 0 ran j e: "Gas Discharge Lamps". Philips Technical
Library, Eindhoven 1951
4) W. W. Elenbaas, "Fluorescem Lamps and Lighting", Philips Technical Library
Eindhoven, 2nd edition, 1963.
'
") Some publications on daylightmg:
"The Lighting of Buildings", Post-war Building Studies no. 12. H. M. Stationery
Office, London 1944
W. A. A 11 en, Trans. I.E.S. London, 11, 1946, 205-218. "The Basis of Daylighting Calculations"
A. F. D u f ton: "Protractors for the Computation of Daylight Factors"
Building Research Technical Paper no. 28. H. M. Stationery Office, London
1946
P. ]. W a 1 dram, ]. Jun. Inst. Eng. 54, 1943, 27. "Daylight Illumination
in Factories and Workshops"
Prof. Dr. Ing. W. Arndt: "Praktische Lichttechnik". Berlin 1938
In Dutch: R. Swier s t r a: "Licht en Zicht" ("Light and Vision"), Tome II:
"Bezonning en Beschaduwing" ("Sun-lighting and Shadow") Haarlem 1954
J. W. T. Walsh, "The science of day-light", London, 1961.
1)
CHAPTER II
SOLID ANGLE
Il-l. Solid angle, steradian
In plane geometry, that part c f a plane area lying between two lines
meeting at a point (the apex) is called an angle. The size of an angle
can be indicated in two different ways, viz:
a) the whole plane area may be divided (by lines passing through the
apex of the angle) into 360 equal parts (degrees), the angle being
then expressed in degrees.
b) with the apex of the angle as centre a circle of any radius may be
drawn. Use is then made of the fact that the arc of the circle enclosed
by the two lines forming the angle is proportional to the angle. The
length of the arc, expressed in terms of the radius is employed as a
measure of the angle. The unit is the angle subtended by an arc of
the same length as the radius of the circle, this unit being known as
the radian: 1 radian= 57°17'44.8". A plane angle is therefore
arc length
---::-,------ radians.
radius
Since the circumference of a circle is 2n times the radius, an angle
of 360° contains 2n radians, an angle of 180° contains :rc rad. and
a right angle ~ rad.
In the geometry of solids (stereometry) the term solid angle is employed
by analogy with the idea of angle in plane geometry; instead of the two
lines enclosing a plane angle, however, there will be a conical surface,
and the space enclosed within this surface is the solid angle, usually
denoted by the Greek letter w.
Such conical surfaces can assume quite arbitrary and irregular forms;
in practice they will often consist of a regular conical surface or the sides of
a pyramid and here we have the means of computing their solid angles.
The size of a solid angle is expressed in a similar way to the measurement
of a plane angle in radians. To do this we imagine a sphere of any radius
r from the apex of the solid angle (see Fig. 5); that part of the spheriCJ.l
surhce which is enclosed by the conical boundary surface of the solid
11-2]
19
SOME SPECIAL SOLID ANGLES
angle is then proportional to the solid angle. When the size of the portion
of the spherical surface is equal to r 2 , we
say that the associated solid angle is I steradian
(abbrev. sterad). If the subtended part of the
spherical surface is not equal to r 2, but may
be denoted by 5, then
5
w=-.
r2
(II-I )
The surface of a sphere of radius r is 4nr2 ;
the solid angle enclosed by the whole sphere
is thus
4nr2
- - = 4n sterad.
r2
Hence the half sphere contains 2n sterad.
Example:
Fig. 5. The solid angle w
subtends at the surface
of the sphere an areaS.
When the radius of the
sphere is denoted by Y
and the subtended area
is ,s, the solid angle is
equa l to 1 steradian
A solid angle w subtends an area of 12 sq. ft. at the surface of a sphere
of radius 3ft. From equation (11-1) it will be found that this solid angle is
w =
s
12
;z = 3i = 1.33 sterad .
II- 2. Some special solid angles
Fig. 6 shows the cross-section of a sphere with centre M and radius r.
A solid angle w, enclosed by a right cone of
which the half-apex angle is ex, subtends a
circular area whose cross-section is AB. The
solid angle may now be expressed in terms
of ex.
5
r2
According to formula (II-I) w = -.
Fig. 6. The value of the
conical solid angle of
which the half-apex angle
is ex rna y be expressed in
terms of ex.
( w = 211 (1- cos ex) =
=
4" sin1
i)
The area subtended is a circular spherical
segment; hence 5 = 2n rh, where his the height
of the spherical segment (i.e. h = CD).
We may now write:
5
w = -
=
r2
2n
=-
r
2nr . CD
r2
2n
= -(MD- MC) =
r
(r- r cos ex) = 2n {1 -cos ex).
20
[II
SOLID ANGLE
A conical solid angle may thus be expressed by
w = 2n (I -cos ex).
(11-2)
For logarithmic applications this may be put in the form:
(II-2a)
Example:
A conical solid angle w has a half-apex angle IX = 6.5". The solid
angle then contains
•
w = 47T sm
2 IX
•
2
6.5o
2 = 47T sm 2
=
0.0404 sterad.
In the above we have defined the idea of "solid angle" as the space
enclosed by a conical surface. Space can also be enclosed by hyo conical
surfaces having a common apex; the space thus enclosed is also called
a solid angle and the definition given in section II-I should be extended
accordingly.
Let us take the case of a solid angle w as
depicted in Fig. 7, enclosed by two right
cones having a common axis and half-apex
angles cx 1 and cx 2 ; the solid angle can then be
G
expressed in terms of cx 1 and cx 2 •
Hence the solid angle encloses an area equal
to the round surface of a disc of the sphere.
Equation (II-I) once more tells us that
w · Sfr 2 , in which S is now the spherical
Fig. 7. When a solid
surface of the disc ABDC; this is equal to
angle w is enclosed by 2
2nrh;
where h = EF.
coaxial conical surfaces
having half-apex angles
Now, EF = MF- ME= r cos cx 1 - r cos cx 2 =
of IX 1 and IX 2 :
= r (cos cx 1 - cos cx 2);
w = 21r (cos IX1 -cos 1X2 )
therefore
S
w = - =
r2
2nr X r (cos cx 1 -
r
2
cos cx 2)
w = 2n (cos cx1 -
(
= 2n cos cx 1 -
cos cx 2)
)
cos cx 2 .
(II-3)
or, for logarithmic treatment:
(II-3a)
11-4]
SMALL SOLID ANGLES
21
If oc 1 = oo, that is cos oc 1 = 1, we have a solid angle bounded by a single
conical surface, and equations (11-3) and (II-3a) revert to (II-2) and
(II-2a).
The problem dealt with at the commencement of this section is thus
a special instance of the case discussed above.
Equation (11-3) can also be derived by regarding the solid angle MABDC
as the difference between solid angles MAGB and MCGD. It then follows
that in order to apply equations (11-3) and (ll-3a) it is not necessary
for the conical surfaces to be coaxial. However, the circle CD must not
cut circle AB, or lie completely outside it.
Example:
A solid angle w is enclosed by conical surfaces having half-apex angles
of 28° and 12° (ex 1 = 12°, ex 2 = 28°). This solid angle would contain
w =
47T
sin ex 1 ;
ex 2 sin ex 2 2 ex 1 = 47T sin 20° sin 8° = 0.0598 sterad.
11-3. Significance of the conception "solid angle" in illuminating
engineering
What are the applications of the idea "solid angle"?
Anticipating somewhat the uses to which this conception will later
be put, we may mention the following. When a certain surface appears
at a given distance from the eye, that surface is seen because a beam
of light rays reaches the eye. Now, if the surface is sufficiently far away
from the eye, we may regard the eye as a point; the beam of light then
constitutes a solid angle, the size of which will depend on the size of
the surface and the distance from the eye. We say that this surface
subtends a certain solid angle.
Similarly a beam of light emitted by a light source which may be regarded as a point source occupies a solid angle.
Another example is to be found in a beam of light brought to a focus
by a lens.
It will thus be seen that the idea "solid angle" figures very largely in
lighting technology: it will accordingly be encountered frequently in
our computations.
II-4. Small solid angles
Let us now take the case of a fiat circular surface of radius a and at a
di~tance d from the eye with the line joining the centre of the eye to the
22
[II
SOLID ANGLE
surface perpendicular to the latter. What is now the size of the solid
angle w subtended by this surface? Let AB be the cross-section of
the surface (Fig. 8) and let M be the place
occupied by the eye; then AC = CB = a, and
MC =d. The angle AMC is denoted by IX.
In order to compute the solid angle we take
M as the centre of a sphere, the circumference
of the surface being a small circle on this; in
the cross-section the sphere is then a circle
having M as centre and passing through
A and B.
The
solid angle to be computed is conical and
Fig. B. The circular surequation (ll-2a) therefore applies:
face AB, as seen from a
point M subtending a
solid angle of which the
half-apex angle is ex.
When ex is small, the
solid angle can be ob-
7Tex2
tained by means of ---;J,2
instead of from the
quotient of the spherical
surfaceAB divided by the
square of the radius MA
of the sphere
•
IX
w = 4n s1n 2 -
2"
We now calculate the value that would be
obtained for the solid angle with respect to
the flat surface instead of to the spherical
surface, and for the perpendicular distance (d)
from the eyes to the flat surface instead of
the radius of the sphere. Then
nAC 2
na2
w = MC2 = d2
(11-4)
in place of which we may write, since ajd = tan IX:
(ll-4a)
w = n tan 2 IX.
What would be the difference if we employed equations (11-2) and
(ll-2a) or (11-4) and (ll-4a)? For angles IX which are so small that
the angle, sine and tangent are interchangeable, the results show no
difference, as will be seen from the following:
Equation (ll-2a) reads:
•
IX
2
w- 4nsm -2"
For small angles we may put this in the form
w = 4n
(~r = niX
2•
(11-5)
11-4]
23
SMALL SOLID ANGLES
Equation (II-4a) for small angles becomes
(II-6)
Using (II-5) and (II-6) we therefore obtain the same result for w with
angles IX for which IX = sin IX = tan IX.
With larger angles IX there is a difference in the results; the variations
as percentages are given for a number of instances in Table A.
Apart from the angle IX, this table includes the relationship between
the diameter AB" ( = 2a) of the circular surface to the distance d from
the eyes to that surface. The last column contains the amount of error
as a percentage, involved when equations (II--4) and (II-4a) are employed instead of (II-2) and (II-2a).
TABLE A
For the solid angle {half-apex angle a:, column 1) within which a circular disc
of diameter 2a is seen from a distance d, column 2 gives the ration 2a : d appropriate to each value ohx. Columns 3 and 4 show the relative solid angles in
steradians computed from formulae (II-2) and (II-4a) respectively. Column 5
indicates the difference as a percentage between the values in columns 3 and 4.
rx
I 2a: d I
w1 =
47T sin 2
i I
w2 =
1T
tan2 rx
I
w2 :
w1
X 100%
-----~--------~--------------c--------------T---
1 : 7.2
1 : 5. 7
1: 4.7
I : 4.1
1 : 3.5
1 : 3.2
1 : 2.8
0.004872 X 11
0.007611 X 11
0.01096 X 11
0.01491 X 11
0.01946 X 11
0.02462 X 11
0.03038 X 11
0.004890 X 11
0.007654 X 11
0.01105 X 1r
0.01507 X 11
0.01'975 X 11
0.02509 X 11
0.03109 X 11
0.4%
0.6%
0.8%
1.1%
1.5%
1.9%
2.3%
From the table it will be seen that with a ratio of 2a : d of about I : 4
the error attendant on the use of the equation
na 2
w = -
d2
= n tan 2 IX
is only about I%In practice, therefore, there will be very many cases where the solid
angle can be computed by dividing the flat surface (5v) by the square
of the distance from the eye (d); thus
51)
w = d2'
24
SOLID ANGLE
[II
II-S. Table of solid angles
In Table I (see p. 407) will be found the values of conical solid angles
with half-apex angles of from oo to 180° in steps of S0 , together with the
difference in solid angle with respect to the angle ex which is smaller
by S0 •
From this table it will be noted that up to goo the increase in w becomes
more and more pronounced, but that from that point up to 180° it
decreases. Annular solid angles for the same arc length are thus much
greater at goo than at 180° and it is important to bear this in mind,
as will appear later.
CHAPTER III
LUMINOUS FLUX, LUMINOUS INTENSITY, QUANTITY
OF LIGHT
III-I. Luminous flux; luminous intensity
Light sources emit energy in the form of electromagnetic waves which
spread out in all directions. The amount of energy radiated during a
unit of time (the power) may be expressed in physical units as ergs per
second, or in watts. Apart from the total energy, we may also consider
the amount of energy passing through a certain part of a plane, or
within a certain solid angle, and express this in the same units.
Generally speaking, only a part of the energy entering the eye produces
an impression of light, viz. that part of which the wavelengths lie between
about 0.4 and 0.7 ~tm. As we shall see later (Chapter XII), the sensation
of light induced in the eyes by a certain amount of energy in the form
of electromagnetic waves is not the same at all wavelengths. Because
of this fact the energy emitted by a light source is not expressed in
watts or ergs/sec, even for so far as it lies between 0.4 and 0. 7 ~tm; instead,
the energy is evaluated in terms of the sensation of light produced in
the eyes (Cf. definition of· photometry: the measurement of radiation
evaluated in accordance with the visual impression).
Radiated energy thus evaluated on the basis of the impression of light which
it induces in the eyes is termed. the luminous flux (symbol f/J). The unit
of luminous :flux is the lumen, the definition of which follows later.
The quantity lumindus flux is frequently compared with analogous
conceptions employed in other branches of physical science. It is the
most conveniently comparable with a flow of liquid or electrical current.
The first instance would refer to the quantity of water passing a certain
point in a pipe within a unit of time; the second would represent the
quantity of electric current (coulombs) flowing at a given point in a
conductor within the unit of time.
If we regard a light source as a point from which light is emitted, we
can imagine a sphere round the source, with the source as centre; the
whole of the luminous flux will then pass through this sphere. Now,
if measurements be taken of the luminous :flux passing through areas
of a certain size at different points on the surface of the sphere, it will
be found that the :flux differs more or less between one point and another.
26
LUMINOUS FLUX, LUMINOUS INTENSITY, Q.UANTITY OF LIGHT
[III
The radiated luminous flux is in effect not uniformly distributed in
space, but varies with the direction.
When the size of the sphere is increased or decreased and the size of
the small area from which the measurement is taken is altered in proportion, we find exactly the same values of luminous flux, seeing that
light is propagated in straight lines. We thus measure the luminous
flux radiated in different solid angles of equal size.
The manner in which luminous flux from a light source is distributed
in space is, of course, of considerable importance to the illuminating
engineer, since a knowledge of this enables him to direct the light in.
the most effective and economical manner towards the objects .which
he wishes to illuminate. It is necessary, therefore, to measure values of
luminous flux in the different directions and to express these in a certain
unit. We might decide to measure the flux radiated in a certain solid
angle (e.g. with an apex-angle of 10°), but this would not be very practical, if for no other reason than that the energy in such a solid angle
might be unevenly distributed.
The solution to this problem can be solved along the lines of the definition of velocity; in mechanics the velocity of a moving body is the
quotient of the distance travelled, divided by the time. For example, the
distance covered in I minute is measured and this, divided by the time
(in this case 1 minute) gives the velocity in, say metres per minute. It
is quite possible, however, that the speed of the moving body is not
constant during that particular minute. In that case we would have
measured the average speed in that minute, but we should know nothing
about any variation that may have occurred in the speed during that
time. In order to arrive at this we must reduce the period of the measurement to so small a value that the speed in that space of time can
be regarded as uniform.
Accordingly, we approximate the time to zero, so that it becomes infinitely short, and express this mathematically in such a way the velocity
(v) is the quotient of the distance travelled (s) divided by the time (t)
with respect to t -> 0. As a formula this is
. s
v = hm -.
t-o t
Needless to say, practical measurements of velocity based on measurements of distance and time have to be effected within a finite period,
but this should be so short that the relevant velocity can be regarded
as constant.
111-1]
LUMINOUS FLUX; LUMINOUS INTENSITY
27
To return to the problem of the distribution of luminous flux in space,
we require to know what luminous flux is radiated within a certain
distance)
solid angle and, by analogy with the idea of velocity, ( .
, we
tzme
now employ the quotient of luminous flux divided by the solid angle (~).
In the same way that the time is made to approximate to zero for the
velocity, we now approximate the solid angle to zero.
, . . .
.
. . .
luminous flux
fh1s s1mllarly prov1des a hm1tmg value of the quotient
.
soltd angle .
with the solid angle approaching zero; this is known as the luminous
intensity (symbol I): formula:
I= lim
if>
w-*0 W
Formerly, the name candle-power was generally given to this ratio.
In the consideration that has led to this definition the light source is
assumed to be a point source, i.e. infinitely small. All practical forms
of light source are naturally of finite dimensions, hut in the following
we shall nevertheless consider practical light sources to be point sources,
assuming thereby that all the light emitted emanates from one single
point at the source. In Chapter IX it will be shown that in most cases
this assumption is permissible, as it involves no serious error. The
conditions under which the source may be considered to be a point
source are also dealt with in that chapter.
In the same way that measurement of velocity in terms of distance
covered and time must be effected in a finite period of time, in the measurement of luminous intensity the luminous flux must be measured within a
finite solid angle which should be so small that the distribution of the
luminous flux in that angle can be regarded as uniform. In place of the
unit of time we employ the unit of solid angle, i.e. the steradian.
It is possible to express the luminous intensity in the unit lumens per
steradian, but this term is never employed. Formerly, the unit
was known as the candle. The standardised unit at present is the candela,
to which reference will be made in the next section where the units of
luminous intensity are defined.
When the luminous intensity, that is, the luminous flux per steradian,
is constant within a certain solid angle, the luminous flux in that
28
LUMINOUS FLUX, LUMINOUS INTENSI~Y, QUANTITY OF LIGHT
LIII
solid angle is obtained by multiplying the luminous intensity by the
solid angle, v1z.
(III-I)
C/J = w. I.
The average luminous intensity in a solid angle in which a luminous
flux C/J is radiated is the quotient of luminous flux divided by the solid
angle; thus
(III-2)
Vsing the symbols employed in the infinitesimal calculus we express
the luminous intensity in accordance with what has been said above
as the differential ~ :. Hence
(111-3)
In general, therefore
<P = JI.dw.
(III-4)
111-2. Units of luminous intensity and luminous flux
The unit of luminous intensity
In the preceding section it was stated that the definition of the lumen
as the unit of luminous flux would be given later. The reason for this
is that the unit of luminous flux is derived from the unit of luminous
intensity, it being therefore necessary to define the latter first.
There have been numerous units of luminous intensity during the course
of time. They have all had this much in common that they represented
the luminous intensity of certain light sources, defined as accurately
as then possible.
It is a requirement of such light sources that they must not vary with
the time and that it must be possible to reproduce the source at any
time and place. Moreover, the luminous intensity must not vary when
the source is in use, that is, when measurements are taking place.
In those days when the first endeavours were being made to measure
light (the middle of the 18th century) the wax candle of all the sources
in use at that time came closest to meeting the conditions of constant
intensity and reproduceability; quite naturally, therefore, the luminous
intensity (in the horizontal direction) of a wax candle was taken to be
the unit of luminous intensity. This unit thus became known as the
"candle".
III-2]
UNITS OF LUMINOUS INTENSITY AND LUMINOUS FLUX
29
Needless to say, the luminous intensity of a flickering candle, which
was not invariably of the same composition and of which the flame
was dependent on the condition of the wick, was not particularly constant,
and it was not long before a more constant source of light was sought.
Of the many standard sources devised and used since that time there
arc two which have remained in use for a considerable period.
These are the Hefner candle (HK) employed in Germany and certain
other European countries, and the International candle (ic) which was
favoured in the English-speaking countries and France.
As there were objections to both the Hefner a~d the international
candle, these were replaced as standards on 1st January 1948 by a new
unit, originally called the New Candle, but now known as the Candela (cd).
The Hefner candle represents the luminous intensity radiated horizontally
by the Hefner lamp, designed by von Hefner A 1 ten e c k in
1884 and consisting of a kind of oil lamp which, however, burned amyl
acetate instead of oil.
When the flame height is properly adjusted, all Hefner lamps will, under
specified conditions of humidity, carbon dioxide content and atmospheric
pressure, yield an intensity of I Hefner candle. For other humidity and
pressure conditions the correct intensity is calculated with the aid of a
correction. formula.
Notwithstanding all this, the Hefner lamp has serious disadvantages;
not only is the flame size difficult to control, but the fact that the intensity of the flame is not more than one candle results in difficulties
in measurement. There are also objections to the yellow colour of the
flame.
In 1909 the United Stq.tes, Great Britain and France decided to standardise the unit of luminous intensity and this gave rise to the International
candle, established with the aid of the intensity of a number of carbon
filament electric lamps. A certain number of international candles was
attributed to the luminous intensity of these lamps in the horizontal
direction, and from these lamps, which served as primary standard,
secondary standards were prepared.
The primary standards were maintained in the State Laboratories in
the United States, Great Britain and France and were very seldom
used, so as to limit as far as possible the blackening of the bulb which
evaporation of the carbon filament produces. At the same time, however
slight the utilisation and consequent blackening, there must ultimately
be a point where the blackening results in a~ appreciable deterioration
30
LUMINOUS FLUX, LUMINOUS INTENSITY, Q.UANTITY OF LIGHT
[III
of the luminous intensity, with a change in the value of the unit of
intensity. This was duly foreseen and another method of standardising
the unit has been adopted.
It was agreed internationally that the new unit should be known as the
Candela (abbrev. cd). This represents one sixtieth of the luminous intensity
of 1 sq. em of the surface of the black body at the temperature of solidifying
platinum, radiated perpendicular to that surface 1 ).
The black body or full radiator is a body that absorbs all radiations
falling upon it. The radiation characteristics of such bodies are accurately
known and the radiations at all wavelengths and temperatures can be
very precisely calculated by means of a formula*). No existing freely
radiating materials will actually meet such absorption requirements,
but an artificial black body can be made in the following manner. A
hollow body of any material, in the wall of which
there is an aperture that is small compared with
the diameter of the body, will absorb all the light
entering through the aperture. If the inside of
I
I
I
the body be rendered incandescent, the radiations
emitted through the aperture will be those of the
black body, and such equipment can be employed
for the purposes of measurement.
For the practical realization of the primary standard
to be used for calibrating secondary standards
in candelas the Bureau of Standards at Washington
have designed an apparatus shown schematically
in Fig. 9. A crucible K is filled with pure platinum;
Fig. 9. Appara tus into this is placed a tube of thorium oxide (B)
employed for the
primary standard of containing powdered thorium oxide. To ensure
the Candela.
effective thermal insulation the whole is placed
B = thorium oxide
tube (black body), in a larger container, also filled with thorium
K = thorium oxide oxide. The platinum is heated by generating H.F.
crucible,
eddy currents in the metal. When the metal has
Pt = platinum
been heated to a temperature above the melting
point and then allowed to cool, the temperature, and with it the radiation,
remains constant during the time required by the platinum to solidify;
this time is utilised for taking the measurements.
The new unit was originally to have been introduced in all countries
on 1st January 1940, but, as the various government laboratories had
------th
I
*) This calculation is dealt with further in Section XII-6.
111-21
UNITS OF LUMINOUS INTENSITY AND LUMINOUS FLUX
31
not all completed the necessary arrangements for the introduction
and also because of hostilities, the introduction was postponed until
1st January, I948.
In concluding this section we would like to add a few words on the
relationship between the Hefner and international candles and the
candela. Originally, it was generally accepted on the basis of comparative
tests that
Hefner candle = 0.9 intern. candles, or
Intern. candle = I.II Hefner candles.
In the course of time, however, more accurate measurements have
shown that, owing to various circumstances, these conversions are
only valid for light sources having an incandescent temperature equal
to that of carbon filaments, that is to say, for lamps of the same kind
as the primary standards of the international candle. Lamps operating
at higher temperatures yielded other ratios, these being l.I4 for tungsten
filament vacuum lamps and I.I65 for gas-filled tungsten lamps.
This fact obviously served to increase the need for a new unit.
The candela lies in value between the Hefner and international candles;
it is slightly smaller than the latter, the difference for vacuum lamps being 0.5% and for
Pas-filled lamps 1.5 to 2%.
The unit of luminous flux
The unit of luminous flux is obtained by
means of equation (III-I): ifJ = w. I, I= I
(c or cd) and w = I (sterad.). Then ifJ = I
lumen, which may be defined in the following
manner: the lumen (abbrev. lm) is the luminous Fig. 10. Illustrating the
flux radiated by a light source of uniform definition of the lumen.
the light source L has
luminous intensity, at unit intensity in 1 aIf uniform
himinous insteradian. (Fig.lO).
tensity of 1 cd within the
If we express the luminous intensity in Hefner solid angle depicted, the
luminous flux in this
candles, the luminous flux will be in Hefner solid angle will be 1 lm
lumens (Him): if we employ the international
candle the result will be in international lumens (ilm). The unit thus
derived from the candela is known merely as the lumen.
For a number of years in some countries the dekalumen (= 10 lm) was
used as well as the lumen, but as it has been agreed internationally that
preference should be given to units differing from the basic unit only
32
LUMINOUS FLUX, LUMINOUS INTENSITY, QUANTITY OF LIGHT
[Ill
by 103 , 106 etc (i.e. micro, milli, kilo, mega etc), the dekalumen is now
no longer employed.
111-3. Formulae giving the relationship between luminous intensity and
luminous flux
Let us now illustrate the application of formula (111-1), viz.
W = w. I.
Example:
A light source has a luminous intensity of 280 cd within a conical solid
angle with a half-apex angle of 5°. From equation (111-l). then
.
ex
<I>= w. I = 41T sm 2 2 X I = 0.0239 X 280 lm = 6.7 lm.
In section 11-1 we have seen that space with respect to a point contains
4n steradians. A light source having a luminous intensity in all directions
of 1 cd (i.e.a. uniform intensity of 1 cd) thus gives in accordance with
equation (111-1) a total luminous flux of 4n lumens. Let I denote the
uniform luminous intensity; the luminous flux is then 4ni lm. Hence
we may write:
(/> = 4ni.
Example:
A light source having a luminous intensity of 25 cd in all directions
gives a total luminous flux of
<I> = 41TI = 41T X 25 lm = 314 lm.
In place of the uniform intensity we may also take the average intensity
in all directions (I 0 ), in which case we put:
(/> =
(/>
4ni0 or I 0 = -.
4n
(111-5)
Example:
An incandescent lamp gives a luminous flux of 1400 lumens. 1'he average
luminous intensity of this lamp wilr be
<I>
1400
I 0 = - = - - 1:::::! I l l cd.
41T
41T
Before it became standard practice to qualify a light source by its
luminous flux, the average luminous intensity was known as the mean
spherical candle-power (M.S.C.P.), expressed in spherical candles. The
TII-4]
HORIZONTAL, SPHERICAL LUMINOUS INTENSITY
33
conceptions of mean lower (I 0 ) and mean upper (I0 ) hemispherical
candle-power were also employed, these being, as the terms imply, the
mean luminous intensities of the radiation in the lower and upper
hemispheres respectively.
If we denote the luminous flux emitted in the lower and upper hemispheres by t/J 0 and t/J0 respectively, it may be said that, as each hemisphere
contains 2:n: steradians,
111-4. Horizontal, spherical luminous intensity
In many instances it will interest us to know the luminous intensity
of a source in a horizontal direction. Take for example projector lamps,
the light from which reaches the lens mainly in a horizontal direction.
The mean horizontal luminous intensity (Ih) may also be important,
e.g. in the case of the old straight filament vacuum lamps, which give
very little light downwards, but emit most of it horizontally. Before
quick methods of measuring luminous flux were known, the mean horizontal luminous intensity of such lamps was measured by rotating the
lamp rapidly about its axis. Owing to the uniform arrangement of the
filament of these lamps, the light emission could be sufficiently accurately
indicated by the average horizontal luminous intensity, in accordance
with which the lamps were typified. This system resulted in lamps of,
say, 10, 16 and 25 candle power.
All this was changed when the coiled filament was introduced. The
greatest intensity of lamps having this type of filament lies in a downward direction; furthermore, the horizontal luminous intensity is subject
to considerable variation, since it depends on the extent to which the
filament sags between the pigtails. The greater the number of coil
sections screened by the other sections, the lower the horizontal intensity.
In the meantime simple methods of effecting measurements of luminous
flux had been evolved and these lamps were accordingly photometered
spherically. It would have been possible to rate the lamps in spherical
candles, but this would yield lower values for spirallized filaments than
the "candles" of straight filament lamps of the same luminous flux.
To overcome this difficulty, lamps with spiral filaments were rated
for some time in accordance with the nominal candle-power *), defined
*) This term was used only for a time and is merely of historical interest.
34
LUMINOUS FLUX, LUMINOUS INTENSITY, QUANTITY OF LIGHT
[III
as the horizontal luminous intensity which a straight filament lamp
would yield for the same luminous flux, (and hence also spherical luminous intensity), as the lamp with spiral filament.
As we shall show later, the theoretical ratio of the spherical to the
horizontal luminous intensity of straight filament lamps should be n/4,
and practical values do not in fact deviate much from this. For spiral
filament lamps the nominal candle-power was thus found by multiplying the spherical candle-power by 4/n.
This was obviously not very satisfactory, and a system of standardisation
of lamp wattage was resorted to, the rated power being shown on the
bulb. This has its disadvantages, too, seeing that for the same amount
of power, the luminous flux of lamps of different manufacture may vary
considerably and may be increased by technical improvements in
the luminous efficiency *).
The next step, adopted in various countries, was to standardise the
luminous flux of the lamps, the unit Dekalumen being introduced for
this purpose. Consequently the luminous flux, as far as the numerical
value was concerned, was almost identical with the horizontal and
nominal luminous intensity previously employed.
n
_As already stated, the equation I 0 = 4 I,. is valid for straight filament
lamps. Now
where lP is expressed in lumens. Putting this into Dekalumens, we
have
In the case of coil-filament lamps I,. can be substituted £or the nominal
luminous intensity in this equation, making it possible to tell the customer
that instead of the lamp which he formerly purchased to give n candlepower, he could ask for a lamp of n Dim and receive one that would
give practically the same luminous flux.
The rating of electric lamps according to their luminous flux was adopted
by a limited number of countries only and in these it was not found
possible to accustom users to the idea of the Dekalumen.
*) See section III-5.
III-6]
Q.UANTITY OF LIGHT,
LUMEN-SECOND, LUMEN-HOUR
35
In recent years, therefore, electric lamp manufacturers have reverted
to the standardisation of wattages, the lamps now being branded with
the wattage as well as the luminous flux in lumens.
In the case of gas discharge lamps it is impossible to standardise the
luminous flux, seeing that the ballasts are manufactured to suit a
certain lamp wattage; the wattage of a given lamp type must therefore
always be the same, so that the same ballast can be used each time.
Any technical improvement resulting in an increase in the luminous
efficiency will accordingly represent an improvement in the luminous
flux for the same wattage.
Ill-5. Luminous efficiency
Electrical light sources consume a certain amount of electrical power
per second (watts) and yield a certain luminous flux. The efficiency of a
light source is higher according as the output of light per watt, i.e. the
quotient of luminous flux divided by the power consumed, is greater.
This quotient is known as the luminous efficiency of the light source
and it is expressed in lmfW.
Example:
1. The luminous efficiency of a 100 W-filament lamp giving a luminous
flux of 1425 lm is
1425
LOO = 14.25 lmfW.
2. The efficiency of a 40 W fluorescent lamp (lamp only) giving a luminous
flux of "2800"lm is
2800
40=
70 lmjW.
Suppose that the losses in the ballast amount to 10 W and that these
are included in the calculation; the total consumption of lamp and
ballast will then be 50 W and the efficiency is
2800 = 56 l /W
50
m
·
III-6. Quantity of light, lumen-second, lumen-hour
In section III-I we have seen that luminous flux is the quantity of
light radiated per second. When the luminous flux of a source is known,
the quantity of light (symbol Q) emitted by the source within a certain
time is found by multiplying the flux by the time. Thus
Q = cp. t.
(III-6)
36
LUMINOUS FLUX, LUMINOUS INTENSITY, QUANTITY OF LIGHT
[III
If the luminous flux is not constant with time we must write:
Q = 1:<1>. At, or preferably Q = j<~>. dt.
(III-6a)
When the time in equation (111-6) is in seconds, Q is in lumen-seconds
(lmsec); if it is in hours, Q is expressed in lumen-hours (lmh).
The idea of light quantity is of practical importance in two special connections, viz:
a) for flash bulbs (e.g. Philips Photoflux);
b) when an appraisal of the total quantity of light emitted by a source
during the working period is required.
a) The density *) of a photographic plate or film is' in the first instance
dependent on the quantity of light reaching the emulsion. When the
:3'.5
:3'.0
!J.. .lr
70:
v ""'
I
7
I
I
2.5
2.0
1.5
1.0
0.5
v
i\.
~
)
10
20
:30
""
-millisec
40
............
~ 1-50
Fig. ll. Curve of the luminous flux v. time (flash curve) of the
Philips PF 60 Photoflux flash bulb. The area included between
the curve and the horizontal axis is a measure of the quantity
of light emitted
plate or film is exposed to the light from a flash bulb for the whole of
the available flash, the density is proportional to the quantity of light
*) See section X-12 for further details.
III-6]
QUANTITY OF LIGHT,
LUMEN-SECOND, LUMEN-HOUR
37
emitted by the bulb. The manner in which the luminous flux waxes and
wanes during the flash does not affect the density.
Fig. 11 shows the complete flash curve of Philips Photoflux flash bulb
type PF 60 plotted with the time in milliseconds horizontally and
the luminous flux (i.e. the quantity of light per second) vertically, this
latter being the light emitted at every moment during combustion of
the wire in the flash bulb. It is seen that after ignition has taken
place the luminous flux increases up to a maximum value and then
drops to the point where the bulb finally extinguishes. The quantity
of light is obtained by multiplying the luminous flux at every
moment by the (infinitely) short time dt during which light is emitted,
and subsequently totalling these values; it is thereby assumed that
the luminous flux is constant for this infinitely short period. This can
also be expressed graphically by determining the size of the rectangles
formed by the infinitely short times (abscissa) and the luminous flux
values relating to those times (ordinate). In Fig. 11 a number of these
rectangles are shown, obviously for finite periods. When the periods are
made smaller and smaller until infinitely small, the sum of all the
small rectangles will be equal to the area enclosed between the curve
and the horizontal ordinate.
Expressed mathematically, this area is Jq, . dt and is, in accordance
with equation (III-6a), equal to the quantity of light Q.
It will be seen from the curve that a maximum of 3 X 106 lm is emitted.
The average quantity of light given by the PF 60 is 62.000 lmsec, this
being emitted in 52 milliseconds. Using equation (111-6) we can now
find the average luminous flux emitted in this 52 msec: from (111-6)
it follows that (/) = Qjt, hence
(/Jm =
62 000
0 _052 = 1 200 000 lm = 1.2 million lm.
A comparison of such values of luminous flux with those of ordinary
100 W gas-filled electric lamps for 220-230 V which yield about
1400 lm, will give some idea of the powerful source of light which
the Photoflux flash bulb places in the hands of the photographer
and explain how it is that instantaneous exposures can be made with
this very small light source.
b) The luminous flux of all known sources of artificial light decreases during the life of the source. The amount of the depreci-
38
LUMINOUS FLUX, LUMINOUS INTENSITY, Q.UANTITY OF LIGHT
--r--
¢(225v)
/500
lm
f
/000
a
r-- r-e- r-I
500
0
0
500
-
h
/OCX)
Fig. 12. Luminous flux as a
function of time, for two
filament lamps of the same
rating (100 W)
a) coiled coil lamp
b) single coil lamp
[III
ation differs between lamps of different
make and type. In order to make a comparison between two lamps of the same
rating it is necessary to consider not only
the initial value of the luminous flux, but
also the total quantity of light produced
by the lamps during their life. These
factors can be obtained from a graph
showing the luminous flux as a function of
the number of hours of burning. As an
example Fig. 12 shows the lines representing the average depreciation of the
luminous flux of a 220-230 V Philips
100 W coiled coil "Biarlita" lamp and also
of the earlier 100 W single coil "Arga"
lamp for the same voltage. The areas
representing the quantities of light of the
two lamp types for an average life of 1000
hours on 225 V is found to be
Coiled coil
Single coil
1.37 X 1()6 lmh
1.14 X 106 ,
Whereas the initial luminous flux of the coiled coil averages 14% more
than that of the single coil type, the quantity of light is 20% higher.
The coiled coil lamp therefore not only has the advantage of the higher
initial flux, but the decline in output during its life is less than in the
single coil lamp.
REFERENCES
1) G. He II e r, Ph. T. R. *) 5, 1940, 1-5. "The new luminous Standard"
W. de Groot, Ph. T. R. 10, 1948, 150-153. "The New Candle"
J. W. T. Walsh, W. Barnett, R. G. Berry andJ. S. Preston, Proc. Inst. Electr.
Eng. JOB, 1961, 173-181. "Units and standards oflight maintained at the National
Physical Laboratory, 1915-1960."
*) Ph. T. R. = Philips Technical R<view.
CHAPTER IV
LIGHT DISTRIBUTION, ROUSSEAU AND ZONAL LUMINOUS
FLUX DIAGRAMS
IV-1. Light distribution, the Rousseau diagram
In order to be able to evaluate the luminous characteristics of light
sources it is necessary to know, amongst other things, the radiated
luminous flux in the various directions within the space around the
source, i.e. the distribution of the luminous intensity, termed the light
distribution of the source. It must also be possible to express this numerically or graphically.
Once the luminous intensity in a number of directions in a plane passing
through the source has been established, the light distribution in that
plane can be represented graphically by plotting the intensity values
as radius vectors in the direction of measurement and subsequently
joining the extremities of these vectors to form a curve. The curve thus
obtained is the light distribution curve to which the designation polar
may be added, since the values represented are plotted from a single
point known as the pole.
If the light distribution of a source is axially or rotationally symmetrical,
the whole distribution may be represented by plotting it for one half of a
plane through the axis of symmetry.
When the distribution is for all practical
purposes rotationally symmetrical, the
average light distribution around the
axis is determined and plotted as the
average light distribution diagram.
Fig. 13 is an example of a light distribution diagram for an axially symmetrical light source. The latter should be
regarded as being suspended with the
axis of symmetry vertical.
The (average) luminous intensities are
Fig. f3. Polar light distridetermined for a number of angles
bution diagram of a light
around the axis of symmetry (e.g. every
source
40
LIGHT DISTRIBUTION AND THE ROUSSEAU DIAGRAM
[IV
10 degrees) and these are plotted from the pole in the diagram, in the
direction of measurement. A smooth curve is drawn through the ends
of the radius vectors as shown in Fig. 13. As a general rule the direction
vertically downwards is taken to be oo.
In cases where the light distribution is not axially or nearly axially
symmetrical it may be represented by curves plotted for different
planes.
If we plot the luminous intensities from a single point in the direction
of- measurement we can imagine a curved surface passing through
the ends of the radius vectors, and the solid thus obtained is called
a solid of light distribution. In the following chapter we shall show
how this solid may be represented as a plane figure. When the light
distribution is axially symmetrical, the solid of light distribution is a
solid of revolution whose axis of rotation coincides with the axis of
symmetry.
It is usual to convert and plot the distribution of lamps and lighting
fittings for a luminous flux of 1000 lumens. For lighting fittings this
luminous flux is taken to be that of the bare lamp. This conversion to a
fixed value of luminous flux has the advantage over the use of the
actual luminous intensity of the lighting fitting that the curves can be
employed for lamps of different luminous flux. Accordingly it is not
necessary to plot separate curves for lamps of different rated voltages,
and variations in the luminous flux due to modifications in lamps do
not make the curves invalid. Moreover, the curves can be used for lamps
of different power, provided that the light distribution of the fitting is
not thereby affected. Such is the case with opal glass globes or vitreous
enamelled and lacquered reflectors in which the lampholder is adjustable
so that the filament or gas discharge of the different lamp types can be
made to assume a constant position.
When a 1000-lm curve is available, the values on the curve can be
multiplied in every case by the number of times 1000 lm at which
the lamp is rated, in order to find the actual luminous intensity in a
given case.
From the foregoing it would seem obvious that the light distribution
curves for 1000 lm are derived by measuring the actual intensity of a
lighting fitting with a calibrated lamp (i.e. of known luminous flux)
placed in it. The values in respect of I 000 lm for the lamp could then
be found simply by dividing the actual luminous intensities as obtained,
by the number of times 1000 lm by which the reference lamp is the
greater.
IV-1]
LIGHT DISTRIBUTION, THE ROUSSEAU DIAGRAM
41
There are, however, various practical objections to such a procedure in
that it would entail the use of numerous calibrated measuring instruments
(and lamps), whilst it would also be necessary carefully to maintain
correct calibration (i.e. of intensity meter, voltmeter andfor ammeters,
and lamps).
Preference is usually given to another method which, although apparently,
more complicated, is more reliable and, since it involves no calibration
fulfils its purpose just as quickly as the seemingly shorter method
suggested above.
The method is as follows. The luminous intensity of the naked lamp
is measured with suitable apparatus*) in different directions on a
relative scale; the fitting with the same lamp in it and operated at the
same current or voltage, is then measured on the same scale; the rest
is purely a matter of computation and plotting curves.
In this method it should be noted that all calibration, even of the meter
used for checking the lamp voltage or current, is quite unnecessary;
the only condition is that all the instruments, as well as the luminous
flux of the lamp, shall remain constant.
Having thus obtained the light distribution of lamp and fitting according
to the same relative scale, we convert these to values in respect of 1000 lm
and, to do this, we must connect the average light distribution curve
with the luminous flux of the lamp.
Obviously, this 1s established by means of equation (III-1) which
states that
t:P = w. I.
In section III-3 it was shown that when I is constant throughout the
whole of the solid angle 4n, ([>is equal to 4ni; in practical forms of light
source, however, I is not constant; it varies between one direction and
another and ([> can be obtained from the luminous intensity only by
dividing the entire solid angle 4n into a large number of solid angles
Llw, small enough to justify the assumption that the luminous intensity
within them is constant, or that the mean intensity can be sufficiently
accurately approximated to without difficulty.
The total luminous flux can then be represented by the sum of all the
values for these small solid angles, in which case
t:P = EiiJw. Llw,
*) See section XVI-3.
42
LIGHT DISTRIBUTION AND THE ROUSSEAU Dl'I\GRAM
[IV
where lt:lw denotes the average luminous intensity in each of the solid
angles L1 w concerned.
From the measurements we know the average relative luminous intensity
for all the directions making the same angle with the axis. As solid
angles L1 w we now consider disc-shaped solid angles of so small a height
that the mean intensity within them
can be taken as being equal to the
intensity towards the centre of the
arc subtended by the solid angle.
Let us now turn to Fig. 14a which
~---t---15'UQ----dl-----i---l shows a polar light distribution
curve L drawn on an arbitrary
scale. The cross-section of a hemisphere of radius r is shown with the
-+---....6-----' pole 0 as centre, with also one of
L
the small solid angles L1 w corresFig. 14'l. Polar light distribution ponding to a height CD = h. The
curve from which the Rousseau
solid angle subtends a disc on the
diagram Fig. 14b is derived
surface of the sphere, corresponding
with the arc AB. The luminous intensity in this solid angle varies from
IA. to IB, and we take as average intensity It:lw the intensity in the
direction of the point mid-way between A and B. The luminous flux
in the disc-shaped solid angle L1 w is then
Lll/> = lt:lw. Llw.
(IV-1)
The solid angle is determined as in section Il-l, by dividing the area
subtended at the surface of the sphere by the square of the radius. Now,
the area Lla of the disc is
hence
Lla = 2nrh
Lla
r2
2nrh
r2
2nh
r
L1w=-=--=-
For (IV-I) we may therefore write:
2nh
Lll/> = ft:lwr
and for the total luminous flmc of the source:
(/> =
2nh
1: lt:lwr
IV-1]
LIGHT DISTRIBUTION, THE ROUSSEAU DIAGRAM
43
.
2n.
Smce - 1s constant, we may also put
r
2n
cp = -
r
E I Jw . h.
(IV-2)
If we now represent I Jw and h as the sides of a rectangle, the area
of that rectangle will represent the product of IJw and h. Proceeding
thus for each Llw and combining the areas so obtained, we then have
the means of determining the luminous flux graphically from the I Jw
values in the luminous intensity difl,gram.
Now let us look at Fig. 14b. This shows IJw projected horizontally and
h vertically. The area EFGH is therefore equal to IJw. hand represents
the luminous flux in the solid angle Ll w divided by 2nfr. Conversely,
the luminous flux Ll(f> is obtained from EFGH by multiplying this
by 2nfr.
The same procedure can be followed for the whole of the solid angl~
4n, in which case all the h's placed together will give us KL, or 2r.
Clearly, then, instead of constructing rectangles such as EFGH, we
may join the ends of the lines representing IJw by means of a smootr
curve. In this way we obtain as sum of all the areas IJw. h the figure
KLMN, whose area may be denoted by 5. This area can be measured
with a planimeter, but we can also measure IJw at regular intervals
and multiply the average of all the values of I.dw by the base*) KL = 2r
of the figure KLMN.
The construction of diagrams such as that shown in Fig. 14b was originated in 1882 by the Belgian engineer Rousseau, after whom these
diagrams are named 1).
How can we now use the Rousseau diagram to convert luminous intensity
curves known on a purely relative basis, to a certain luminous flux of
the source?
Let us assume that r = 12 em, so that the base of the Rousseau diagram
KL = 24 em. The scale on which I is represented is as yet unknown
and we assume tentatively that 1 em = a cd; I is then represented
I
by- em.
a
*) Although the line LK is usually drawn vertically with the luminous intensity
shown horizontally, we still refer to LK as the base of the diagram.
44
[IV
LIGHT DISTRIBUTION AND THE ROUSSEAU DIAGRAM
According to (IV-2):
2n
(/) = -
r
E ILJw . h,
where E IiJw. h is represented by the area 5. So, if 1 em = a cd,
E IiJw. h =a. 5
since, as each horizontal centimetre of KLMN represents a cd, the area 5,
measured in sq. em, must be multiplied by a in order to yield E ILJw. h.
We may therefore write:
2n
(/)=-a. 5,
r
from which it follows that
(/)r
a=-.
2n5
(IV--3)
When (/) = 1000 lm we may write
1000 >< r
2nS
a=----
When r = 12 em the value of a is
a=
1000 X 12
1910
2n5
=s·
In this case, then, the luminous intensities for 1000 lm are found by
. .
.
.
1910
mulbplymg the relative values by the conversiOn factor
s·
Conversely we can compute the area represented by 1000 lm, assuming
a luminous intensity scale and radius r.
If a= 1 cdjcm, then with r = 12 em, 5 = 1910 sq. em and 1000 lumen
will be represented by an area of 1910 sq. em.
If a= 20 cdjcm, then 5 = 1910: 20 = 95.5 sq. em and 1000 lm will
be represented by an area of 95.5 sq. em.
The conversion factor 1910/5 is valid for all values of the luminous
intensity obtained from the same arbitrary scale, and therefore also
for that of the lighting fitting.
This operation having been carried out, the light distribution curves
of lamp and fitting can be drawn.
IV-3]
45
DERIVATION OF THE ROUSSEAU DIAGRAM
IV-2. Construction of the Rousseau diagram
Fig. 14b shows the Rousseau diagram as derived from the light distribution curve; it is also possible to construct the Rousseau diagram
independently.
.aor=
Let ex. denote the angle AOD
tsd'
19~
in Fig. 14a; then
14d 181-
=
=
~
t-
171-
OD = rcos ex..
In Fig. 14b, therefore we also
have
1-
15f--
15~
f--
14~
13 1-
/30
~,;;d
no•
-
=
~
1-
l.dl=
Jod f-f--
Taking L as the origin of
the Rousseau diagram, we
find for the angle oc along
the base LK
/0~
9<f 1-
7 5 ::::;
eif 1-
LH = O'L-O'H =
= r - r cos oc = r (I- cos oc).
~5=
f--
O'H = rcoscx..
Il l -
9~
8-
"'4=
03_
s:
~ t.d 1' I ft-
111-
~
f--
70°
60°
5if
40°
30
~
f--
r=
~
~
~
1-
f-In Fig. 15 a Rousseau
1- '-''-A
diagram is depicted in which
61191
the lines associated with the
Fig. 15. Rousseau diagram
various angles are shown.
These lines lie at a distance r (I - cos IX) from the abscissa. The base
of the diagram is 7.5 em (3 in.) long.
In order to ensure sufficient accuracy when calculating the area of
the Rousseau diagram it is desirable to employ a diagram with a longer
base, and for practical purposes a base of, say, 24 em (about I 0") is
convenient.
With a base of 24 em and a scale of 20 cd = I em, an area of
95.5 sq. em will correspond to 1000 lm luminous flux at the source
(see previous section) .
IV- 3. Derivation of the Rousseau diagram with the aid of the infinitesimal calculus
With the aid of the infinitesimal calculus, the relationship b etween
light distribution and luminous flux is expressed in the following
manner.
Fig. l6a shows the light distribution curve of a light source within the
half cross section of a sphere of radius r. In the solid angle d w between
angles ot and ot+dot the luminous flux emitte:l is equal to
d <l> = lot. dw.
46
[IV
LIGHT DISTRIBUTION AND THE ROUSSEAU DIAGRAM
Now
= 21T1'2 sinadiX 2 . d
d --~ = da
= 1rSin1X IX
r•
r•
d~ = 21rl1X sin ada.
The luminous flux between IX1 and IX2 is thus found to be
and
(IV-4)
Fig. 16. Derivation of the , Rousseau
diagram from the light distribution by
means of the infinitesimal calculus
C!>IX
"
.
IX•
= 21rJ I IX sin IX diX
IX
(IV-5)
and the total luminous flux between IX = 0° and 180° = 1r rad., as
1T
C1>
= 21r J I IX sin IX diX.
(IV-6)
0
This integration is carried out graphically in the Rousseau diagram
in the following manner (fig. 16b):
The surface element dS in the Rousseau diagram corresponding to dOt
is equal to
dS = /IX r sin IX diX.
The area between the luminous intensity lines IX 1 and a 2 is
a.
(IV-7)
sa,, Ots = r I IX sin IX diX
f
Ott
and the whole area enclosed within the curve is
J
1T
S = r I a sin IX diX.
(IV-8)
0
Equations (IV-7) and (IV-8) differ only in the factor, from equations
(IV-5) and (IV-6) for the luminous flux. The luminous flux is therefore obtained from the area enclosed within the curve in the Rousseau
diagram by multiplying this by 21rjr.
IV-4]
EFFICIENCY OF A FITTING FROl\1 THE ROUSSEAU DIAGRAM
47
IV-4. Determination of the efficiency of a lighting fitting from the
Rousseau diagram
The Rousseau diagram also gives us the means· of ascertaining the
efficiency of a lighting fitting, by which is meant the ratio of the total
luminous flux emitted by the fitting to that of the bare lamp. Efficiency is
usually denoted by the Greek letter 'YJ and is expressed as a percentage.
It was showp. in section IV-I that the luminous flux values of the lamp
and fitting can both be represented by areas in the Rousseau diagram;
division of the one area by the other therefore gives us the efficiency.
It is a condition, however, that it must be possible to plot an average
light distribution curve for the fitting, which can usually be done only
in the case of axially symmetrical fittings.
The efficiency of a lighting fitting can also be determined by direct
measurement with a luminous-flux meter, this being a method of verifying
the result obtained from the Rousseau diagram.
This direct method of efficiency measurement also provides a simple
means of ascertaining the light distribution of symmetrical fittings,
since, when the efficiency is known from the direct measurement, measurement of the relative light distribution of the lighting fitting is all
that is needed to determine the light distribution for I 000 lm. The
luminous intensity values thus determined on an arbitrary scale are
set out in the Rousseau diagram and the area S thus obtained is measured
with the planimeter. In order to make the conversion for 1000 lm of
the bare lamp, the scale ( 1 em = a cd) is found for the luminous intensity
with r = 12 em, as
'YJ
1910
a= 100 X S'
.
1910
for, instead of the 1000 lm to w1th the factor S
applies, we now
have _!}_ x 1000 lm, and the relative luminous intensity values
100
obtained for the fitting must also be multiplied by this factor a.
As a rule the graphical method of obtaining the efficiency of a fitting,
with a check by means of direct measurement will be preferred to the
system just described.
For asymmetric fittings the method is not employed, since determination
of the average light distribution is as a rule too cumbersome.
48
LIGHT DISTRIBUTION AND THE ROUSSEAU DIAGRAM
[IV
If the light distribution from a fitting is uniform in every plane passing
through one axis of the fitting, the luminous flux can easily be found
by a Moerman method 2) for which the light distribution in only two
planes need be known.
This method is particularly useful for fluorescent lamp fittings. Here, the
light distribution curves are generally uniform in the planes through the
longitudinal axis, i.e. in every plane the luminous intensity at every
angle p to the axis bears the same proportion to the luminous intensity
vertical to this axis.
If the light distribution in the vertical plane containing the longitudinal
axis of the fitting and that in the plane perpendicular to the longitudinal
axis are known, the light distribution in all other planes can readily be
found.
If, now, the mean luminous intensity in the plane perpendicular to
the axis is calculated and it is found that a proportion a of the mean
luminous intensity is directed vertically downwards, it then becomes
necessary merely to multiply all luminous intensities in the vertical
plane through the axis by a to find the mean light distribution. This can
then be plotted on the Rousseau diagram in order to find the luminous
flux or the efficiency of the fitting. In practice this is made simpler by
transferring the light distribution in the vertical plane through the axis
to the Rousseau diagram, measuring with a planimeter and subsequently
multiplying by a.
IV-5. The zonal luminous flux diagram
It is often necessary to know what part of the luminous flux radiated
by a lighting fitting occurs within certain solid angles, i.e. in certain
zones and, in the case of
(/) (%)
symmetrical fittings, the
100
""
....
zonal luminous flux diagram
80
"
1-- 1is plotted for this purpose
, "
60
(Fig. 17). In this diagram
,
I'
,
half-apex angles of the
the
v
40
1-solid angles which
conical
1/ ,
axis of the fitting
the
have
-~
_... "
are plotted horizonooo ~20 40° 60 00 KXP f2(J" HCI' 160 IBCfl as axis
tally and the luminous flux
Fig. 17. Zonal luminous flux diagram
radiated within those solid
fitting
- - - - - - - lamp,
angles vertically.
-
""'
v
IV-5]
49
THE ZONAL LUMINOUS FLUX DIAGRAM
Usually the luminous flux is expressed as a percentage of the flux of
the bare lamp, so that the over-all efficiency of
the fitting may be read off at 180°.
From the example shown in Fig. 17 it will be seen
that the efficiency is 74%, also that between oo
and goo (i.e. in the lower hemisphere), 33% is
radiated, and between goo and 180° (upper hemisphere) 41%. The ratio of the flux radiated downwards (direct) to that radiated upwards (indirect) is
thus as 1 : 1.25.
A
Between 0° and 10° the emission is 4%, between
oo and 20° 11%, hence between 10° and 20° it is 7%
of the luminous flux of the lamp.
The luminous flux values of the zonal luminous flux
Fig. 18. The lumidiagram can be obtained from the Rousseau nous flux of a light
diagram; the area between the oo and 10°-lines cor- source can be determined from the
responds to the flux radiated between oo and 10°, polar distribution
and the planimeter reading, divided by the area curve by measurement of the length
corresponding to 1000 lm, gives the percentage of of the lines
luminous flux emitted between 0° and 10°. Re- AB = lex sin ex for
peating this process for 0°-20° and 0°-30° etc we ex = 5°, 15°, 25° etc.
The sum of these
obtain the values needed for the zonal flux diagram. lengths, multiplied
This is not a quick method, however; another by 1.095 R;j l.l gives
the luminous flux
method will be found simpler but not quite so accurate, although sufficiently accurate for ordinary practical purposes.
Let us turn to Fig. 18. This shows the polar light distribution curve
of a light source. We now compute the luminous flux between the angles
1X 1 and IX 2 , assuming (and this is the source of the error) that the average
luminous intensity I ex within the particular solid angle is equal to the
intensity radiated in the direction of the bisector of the angle 1X2 - IX1 ,
i.e. at the angle
(II-3a))
IX
1
+ 2 • The solid angle between 1 and 2 is (Equation
2
w =
IX
1X
IX
+
IX2
4 . IX2 - IXl • IXl
sm
nsm
2
2 .
In this solid angle the luminous flux is therefore
lf> ex,ex, =
•
4nl ex sm
IX2 -
2
IX1
•
sm
ot1
+
2
ot2
.
50
LIGHT DISTRIBUTION AND THE ROUSSEAU DIAGRAM
[IV
Assuming a constant difference between ot1 and ot 2 of, say, 10°, then
otl
ot2 ---,-----=
2
so an d otl + ot2 =ot
2
'
so we may write:
<PIX, IX• = 4n sin so I IX sin ot = 1.09S I IX sin ot ~ I. I I IX sin ot.
In Fig. 18 IIX sin ot is denoted by the line AB, and the value of IIX sin ot
can be measured directly from the light distribution curve with a rule.
For the calculation of </> in lumens it is of course necessary to take
into account the scale. If I em represents a candelas, the length of IIX sin ot
as measured must be multiplied by a as well as by the factor 1.09S.
The luminous flux between oo and I oo is founft by measuring I IX sin -ot
for ot = S0 , and that between 10° and 20° by taking the measurement
at ot = IS 0 and so on. The luminous flux between oo and ot1 is found
by adding up the values of ot lying between those limits, and the over-all
efficiency of a lighting fitting can be obtained by totalling the values
of ot for all the angles between 0 and I80°, as expressed by
</> = 1.09S E IIX sin ot.
(IV-9)
IV-6. The area enclosed within the luminous intensity curve is not
a measure of the luminous flux emitted
This is demonstrated by Fig. 19a which shows the luminous intensity curves
for two light sources of the same luminous flux. Fig. 19b is the Rousseau
diagram of the two curves; the areas enclosed are of equal size, whereas
the area for the curve of source A in Fig. 19a is much larger than that of
source B. The reason for this is that, as we have seen in section 11-5
(Table I), the solid angles
within the same plane angle
ot 1) increase as ot ap( ot2 proaches 90°. This is not exo
pressed in the luminous inI"
~
'\
I
I
0
\s
'J
!/
g
....A
..
Fig. 19. The area enclosed by the
7rf' polar light distribution curve is not a
measure of the luminous flux.
600 Whereas the area within the full-line
curve in Fig. 19a is considerably
500 smaller than that of the dotted-line
0
40
curve. the Rousseau dia~ram in Fi~ .
:30 0 19b shows that the lummous flux lS
oo the same in each case
IV-8]
RUSSEL ANGLES
51
tensity diagram, which contains only plane angles, but it is inherent in the
Rousseau diagram in which, as ex approaches 90°, the parts of the base
corresponding to the same plane angle (cx2 - a 1) increase progressively.
IV-7. Average luminous intensity calculated from the Rousseau diagram
Equation (IV-2) tells us that the luminous flux of a source may be
represented by
2n
J: I IJw • h.
r
t/J = -
The spherical luminous intensity I 0 is derived by means of equation
(III-5a) from the luminous flux:
tP
Io=-.
4n
Substitution of the expression in (IV-2) for tP in equation (III-5a)
gives
2n
- J: I11w. h
r
Io=--4-n--
We have seen that 1: I 11 w. h is represented by area S in the Rousseau
diagram, and that 2r is the base of this diagram; from the Rousseau
diagram, therefore, I 0 is found to be
s
Io = Tr·
(IV-10)
In other words the spherical luminous intensity of a light source is
found by dividing the area of the Rousseau diagram by its base.
It will be clear that, in order to find the average luminous intensity in
solid angles smaller than 4n we must divide the area of the diagram
corresponding to the luminous flux in the solid angle by that part of
the base which also corresponds to the solid angle.
Hence the lower and upper hemispherical luminous intensities (I 0 and I 0 )
are evaluated by dividing respectively the areas between oo and goo,
and goo and 180° by the half base r of the Rousseau diagram.
IV-8. Russel angles
In the absence of a planimeter for measuring the area of a Rousseau
52
LIGHT DISTRIBUTION AND THE ROUSSEAU DIAGRAM
[IV
diagram, an ordinary rule can be used to determine the area fairly
accurately (as pointed out in section IV-1), by measuring the distances
representing lAw at regular distances. The average of these measurements
is then multiplied by the length of the base of the diagram.
This can be facilitated by. drawing blocks at each side of the Rousseau
diagram in the manner shown in Fig. 15. Here, the diagram is thus
divided into 20 parts, each with a centre line. The rule is laid along
these centre lines and the value of I Aw in em or inches is measured
at the centre of the block, this being taken as the average luminous
intensity in that block.
The average of the measured values of lAw represents the average
for the whole Rousseau diagram and this, multiplied by the base, gives
the area enclosed by the curve.
Comparing this method with the system for ascertaining the spherical
luminous intensity from the Rousseau diagram mentioned in section
IV-7, we see that the average for the measurement of IiJw is at the
same time the spherical intensity.
The centre of each block in the diagram corresponds to a certain
direction of measurement, which can be calculated as follows.
In the diagram in Fig. -15:
AB = r (1 -cos oc).
At the centre of the lowest block:
AB = 0.05r.
From 0.05 r = r (I -cos oc) it follows that cos oc = 0.95.
At the centre of the second block:
AB = 0.15 r;
hence cos oc = 0.85.
The value of oc is thus calculated for the centre of each block and the
results are tabulated as in Table B.
Instead of measuring the intensities at every 5 or 10 degrees and plotting
these in the form of a Rousseau diagram to obtain an area corresponding
to the luminous flux, which may be measured by the method just
described, we may also measure the luminous intensity in those directions
which correspond to the centres of the blocks, i.e. at angles as given
in Table B. The average of these 20 luminous intensity values then
gives us 10 (see section IV-7) and multiplication by 4n yields the
luminous flux.
IV-9]
LONG-BASE ROUSSEAU DIAGRAM FOR NARROW BEAMS OF LIGHT
53
In English publications this is referred to as the method of R u s s e 1,
in Germany that of B 1 o c h and the angles in Table B are known
as Russel, or Russel-Bloch angles.
TABLE B
Russel-Bloch angles
1
2
3
4
5
6
7
8
9
10
18°12'
31°47'
4e25'
49°28'
56°38'
63°15'
69°31'
75°31'
81°22'
87° 8'
ll
12
13
14
15
16
17
18
19
20
9~52'
98°38'
104°29'
ll0°29'
ll6°45'
123°22'
130°32'
138°35'
148°13'
161°48'
This method appears at first sight to be very attractive, but it does
suffer from the disadvantage that the angles are not round values.
They can of course be inscribed on the protractor on the test instrument,
but it is difficult to plot the luminous intensity in the form of a polar
diagram with proportional angular divisions. The Russel method is
not suitable for narrow-angle fittings or for fittings with a freakish form
of the light distribution curve unless the number of measuring directions
is considerably increased.
IV-9. The long-base Rousseau diagram for narrow beams of light
When the light distribution is of the kind produced by a projector;
for instance, - in which case the luminous flux is radiated wholly
or for the greater part within a very narrow solid angle, there is an
objection to ascertaining the luminous flux of the beam from a Rousseau
diagram having a base of 24 em (about 10"). The greater part of the
area corresponding to the luminous flux would thus be compressed into
a long narrow space which would be very difficult or impossible to
measure at all accurately with a planimeter. In such an attenuated
diagram the thickness of the lines themselves would affect the result
and, for this reason, the diagram in such cases is extended vertically
so as to provide sufficient height. In effect this consists in enlarging
the base of the Rousseau diagram, to the extent, say, of r = 24, l 00 or
1000 em (10", 40" or 400").
54
[IV
LIGHT DISTRIBUTION AND THE ROUSSEAU DIAGRAM
When r =
r=
r =
24 em, a =
3820/S (Eq. IV-3)
100 em, a= 15920/S, and
1000 em, a= 159200/S.
With so large a base it is not practicable to construct the complete
diagram (i.e. with the whole base of 2r), nor is this even necessary. It
is sufficient to show that part which corresponds to the solid angle
containing the luminous flux to be determined.
If desired, the luminous flux can be split, one part being determined
from a diagram in which r = 1000 em for instance, and another with
r = 100 em. The division and bases of the diagrams should be such
that the form of the area will lend itself to accurate measurement with
the planimeter.
IV-10. Detailed example: determination of a 1000-lm curve for a
symmetrical lighting fitting
Let us now work out in detail the light distribution curve and zonal
luminous flux curve of a lighting fitting. This fitting contains a 300 W
incandescent lamp and is, for practical purposes, axially symmetrical.
Measurements taken from the lamp and fitting give the average values
of luminous intensity listed in Table C on an arbitrary scale.
TABLE C
Relative luminous intensity values of the lamp and fitting of Figs. 20 and 21
oo
100
20°
30°
40°
50°
60°
70°
80°
90°
I
Lamp
11.9
11.75
11.55
11.2
10.9
10.85
10.25
10.1
9.7
8.1
I
Fitting
13.3
12.75
ll.35
9.2
7.55
6.7
6.1
5.7
5.55
6.1
I
I
100°
ll0°
120°
130°
140°
150°
160°
170°
180°
Lamp
9.2
9.75
10.25
10.8
ll.25
ll.5
10.05
6.1
0.2
I
Fitting
8.3
ll.O
12.25
12.95
12.75
11.35
8.3
5.25
4.75
The results with respect to the lamp are plotted as a Rousseau diagram
with a base of 24 em, and a curve is drawn through the points as plotted.
IV-10]
55
DETERMINATION OF A 1000-LM CURVE FOR A FITT~G
15 20
The resultant Rousseau diagram
/SG
is depicted in Fig. 20, reduced
1"'>.
18
in size by a factor of 3. The area
1I
enclosed by the curve, and as
i/ r-- 1- /15
I
measured with the planimeter,
v I /10 14
was found to be 243 sq. em.
I
- - -1To obtain the luminous intensity
?'
/2
in cd from the readings, that is,
/ /
-r- I
/
the intensity scale (1 em= a cd),
90
KJ
\:the values in Table C are multiI BO
plied by 1910j243 = 7.86, and
I
A I
8
the results are as given in
?0
Table D.
~
These data now enable us to
,__ \ \
11\
50 4
plot the required light distri1'40
bution curves for lamp and fitting,
11- ' 0 2
and these are shown in Fig. ·21.
1I I-'0
It now remains only to ascertain
0
24
68KJ/214
the efficiency and plot the zonal
Fig. 20. Rousseau diagram with curves
luminous flux curve.
for an m candescent la mp (dotted line)
The efficiency can be determined and for a lighting fitting with the same
lamp (full line)
in accordance with the method
described in section IV- 5 for the
zonal diagram, by measurement of I or. sin oc. The values of I oc sin oc between
~
\
..
TABLED
Luminous intensity values in cd of t he lamp and fitting of Figs. 20 and 21, referred
to a lamp luminous flux of 1000 lm
oo
10°
20°
30°
40°
50°
60°
70°
80°
90°
I
Lamp
93.5
92
90.5
88
85.5
85
S0.5
79.5
76
63.5
I
Fitting
104.5
100
89
72.5
59
52.5
48
45
43.5
48
II
100°
110°
120°
130°
140°
150°
160°
170°
180°
Lamp
Fitting
72
76.5
80.5
85
88.5
90
79
48
1.5
65
86.5
98
101.5
100
89
65
41
34.5
56
LIGHT DISTRIBUTION AND THE ROUSSEAU DIAGRAM
[IV
0 and 180° when totalled and multiplied by 4n sin 5° (= 1.095) give
the luminous flux of the fitting, and from this the efficiency is computed.
The totalling of so many values each of which is subject to a certain
amount of error in measurement necessarily yields a less accurate
result than that obtained by planimeter measuremen.t of the area in
the Rousseau diagram.
To ascertain the efficiency of the lighting fitting we therefore utilise
the Rousseau diagram to check the result obtained from Irx sin ex.
For a luminous flux of 1000 lm we obtain from equation (IV-9) the
value which should be found for the sum of all the measured lines
Irx sin ex.
Equation (IV-9) states that
(/) = 1.095 E Irx sin ex.
For (/) = 1000 lm,
.
1000
E Irx sm ex = 1. 095 = 913.
Now, if 1 em= 1 cd, E Irx sin ex con•esponds to 913 em.
In the original light distribution curve from which I rx sin ex was measured,
100 cd represented 6.25 em, so that 1 em = 100/6.25 = 16 cd. Hence
.E Irx sin ex would be 913/16 = 57.1 em= 571 mm.
Irx sin ex as measured from the luminous intensity curves yields the
values shown in Table E.
It will be seen that the value of E I rx sin ex for 1000 lm is not the same
as the calculated value of 571 mm and this is due to inaccuracies in
plotting and measurement. The measured value of 569 is taken as
basis for the calculation of the percentages.
"''
The efficiency is also obtained .from the Rousseau diagram. The readings
in respect of the lighting fitting are shown plotted in Fig. 20, and
planimetry of the area within the curve gives us 208.5 sq. em; the
efficiency is thus 2~:~5 X 100% = 86%, which is the same value as
that obtained by measuring Irx sin ex.
Instead of employing the readings we could also plot as Rousseau diagram
the luminous intensity of the fitting, calculated for 1000 lm, and divide
the area thus obtained by the area corresponding to 1000 lm.
Let us now determine the areas in the Rousseau diagram by measuring
the luminous intensities in 20 blocks of equal height; lines are included
at the side of the diagram in Fig. 20 to represent the centres of the
20 blocks, and the result of this measurement is reproduced in Table F.
IV-10]
57
DETERMINATION OF A 1000-LM CURVE FOR A FITTING
TABLE E
Actual measured values of I a. sin a in the case of Fig. 21, together with totals
for the construction of the zonal luminous flux diagram. In the original instance
100 cd corresponded to 6.25 em.
Lamp
ZONE
0°- 10°
10°- 20°
20°- 30°
30°- 40°
40°- 50°
50°- 60°
60°- 70°
70°- 80°
80°- 90°
90°-100°
100°-110°
110°-120°
120°-130°
130°-140°
140°-150°
150°-160°
160°-170°
170°-180°
Lighting fitting
I a. sin a;' L'l a; sin a;
(mm)
(mm)
a.' L'l(mm)
a; sin a.' 1000
% of
lm
I a. sin
(mm)
so
15°
25°
35°
45°
55°
65°
75°
85°
95°
105°
115°
125°
135°
145°
155°
165°
175°
5
14.5
23
30.5
37
42
45
47
46
41.25
45
44.5
42
38
32.5
23
11.5
1.25
5
19.5
42.5
73
110
152
197
244
290
331.25
376.25
420.75
462.75
500.75
533.25
556.25
567.75
569
1
3.5
7.5
13
19.5
26.5
34.5
43
51
58
66
74
81
88
93.5
97.5
99.5
100
5.5
15.5
21
23
24
25
25.75
26
27.25
34.5
46.5
52.75
51
44.75
34.5
20.5
9
2
,
50
"
40
cd
20
°o
>0
80
-
,
"
40 0
5.5
21
42
65
89
114
139.75
165.75
193
227.5
274
326.75
377.75
422.5
457
477.5
486.5
488,5
120
w
Fig. 21. Light distribution curve and zonal luminous flux
diagram derived from the Rousseau diagram Fig. 20
I%
of
1000 lm
1
3.5
7.5
11.5
15.5
20
24.5
29
34
40
48
57.5
66.5
74
80
84
85.5
86
58
[IV
LIGHT DISTRIBUTION AND THE ROUSSEAU DIAGRAM
TABLE F
Relative luminous intensity values measured at the centre of each of the 20 bands
in the Rousseau diagram Fig. 20, in order to ascertain the areas enclosed by the
Rousseau curves.
I
1
2
3
4
5
6
7
8
9
10
Lamp
Fitting
11.6
11.1
10.85
10.8
10.45
10.15
10.1
10.0
9.8
8.6
11.6
8.9
7.5
6.8
6.3
6.0
8.7
5.6
5.6
5.8
I
II
11
12
13
14
15
16
17
18
19
20
Lamp
8.2
9.0
9.75
9.8
10.1
10.4
10.8
11.2
11.55
10.9
I
Fitting
6.4
7.8
9.6
11.05
11.9
12.6
12.95
12.85
12.0
10.8
In this way we find the areas to be 10.25 em X 24 em= 246 sq. em
and 8.89 em X 24 em= 213.5 sq. em, both of which are slightly higher
than the results obtained with the planimeter. The efficiency as obtained
in this manner is 86.5%.
IV-II. Classification of lighting fittings according to their light
distribution
There are many different kinds of lighting· fittings, of which the light
distribution differs very considerably between the one kind and another.
It is therefore not very surprising that many endeavours have been
made to classify these. We shall not discuss all the suggestions put
forward to this end, but will confine ourselves to the proposal of the
C.I.E., made in 1935.
This classification is based on the ratio of upper to lower hemispherical
flux of the fitting; this ratio is important to the lighting engineer in that
it largely determines the system on which a lighting installation may
be based.
Let us imagine a lighting fitting suspended with its axis vertical above the
horizontal surface to be illuminated (working plane). If the fitting radiates
the whole of its luminous flux in the lower hemisphere, the flux will fall
directly upon the working plane; this method of lighting is termed
direct lighting and the fitting is a direct-lighting fitting.
If the fitting emits the whole of the flux in the upper hemisphere, the
working plane receives only light reflected from the ceiling and walls,
IV-11] CLASSIFICATION OF FITTINGS ACCORDING TO LIGHT DISTRIBUTION
59
and none direct from the fitting. This is called indirect lighting and the
kind of fitting which produces it is an indirect-lighting fitting.
Between these two extremes the working plane may be lighted in
part directly and in part indirectly and, according as the mode of
radiation is either more direct or more indirect, we speak of semi-direct
and semi-indirect lighting and lighting fittings.
When the distribution of the luminous flux between the upper and
lower hemispheres is roughly half and half, the lighting is known as
general diffuse.
Limits have been laid down for these different kinds of lighting, the
C.I.E. classification being as follows:
Percentage of luminous flux
DIRECT
SEMI-DIRECT
GENERAL DIFFUSE
SEMI-INDIRECT
INDIRECT
Lower hemisphere
I Upper hemisphere
100-90
90-60
60-40
40-10
10-0
0-10
10-40
40-60
60-90
90-100
The illustrations in the Appendix (A to U) (p. 417-4 25) are examples
of different kinds of lighting fittings with their light distribution,
classified in accordance with the above.
Since much less of the luminous flux radiated upwards reaches the
working plane (i.e. with lower efficiency) than of that emitted direct
towards the working plane, the direct luminous flux of general diffuse
lighting fittings and in many cases also of semi-indirect fittings (i.e.
radiated straight towards the working plane) will be greater than the
flux derived from the indirect or upward radiations. The terms employed
in the classification, of "general diffuse" and "semi-indirect", therefore,
convey nothing of the ratio of the direct to the indirect luminous flux
in the working plane, or of the proportion in which the flux reaching the
working plane is direct or indirect; they refer only to the distribution
of the flux leaving the lighting fitting. In this respect the two terms are
rather misleading.
REFERENCES
In addition to the Rousseau method there are a number of other graphical
methods for the determination of luminous flux which are dealt with fully
in the book by E. L. J. M a t t h e w s : ''Etude critique des procedes graphiques
ou analytiques pour la determination du flux lumineux des sources de lumiere".
Ghent 1929
2 ) J. J. B. Moerman, Elektrotechniek, 34, 1956, 407-412, "Bepaling van het lichtstroomverdelingsdiagram van armaturen" (Determination of the luminous flux
distribution diagram of lighting fittings).
1)
CHAPTER V
METHODS OF REPRESENTING LIGHT DISTRIBUTION
V-I. Luminous intensity table; polar and rectangular light distribution
diagrams
In section IV-I it was stated that, if the distribution of the luminous
intensity of a light source is known, it can be reproduced either numerically or graphically. The numerical method is very simple, entailing only
a table specifying the different directions and associated intensities; this
is obviously a very convenient method when the data are required for
computations, as it is then unnecessary to read the required values from
graphs. If a table compiled from measurements of the light distribution
is available, the curve is therefore superfluous, but, should it be necessary
to interpolate from the values given, a curve will be necessary. On the
other hand, a table is seldom used to characterise a light source as
it does not present a clear picture of the light distribution.
The following methods of graphical representation are employed:
I) the polar diagram,
2) the rectangular diagram (cartesian co-ordinates),
3) the isocandela diagram.
Polar and rectangular diagrams have already been discussed in Chapter
IV, so that on the subject of these we need only be brief. The polar
diagram is the more generally used of the two, but in some cases the
rectangular is preferred, for example when it is necessary to plot the
curves of narrow beams of light.
If the light distribution to be plotted covers a wide range of values,
it is advisable to employ logarithmic graph paper. Both kinds of
diagram can be plotted for symmetrical as well as asymmetric lighting
fittings. Fully symmetrical sources can be characterised by showing
the distribution in only one half of the plane of symmetry. The distribution of asymmetric sources is plotted in several planes all of which
usually pass through a single line. In the case of bilateral symmetry
it is sufficient to show the curves for the half-plane only. For pendant
fittings this reference line is generally taken vertically through the
fitting, but, as will be seen later, there are instances in which it is better
to place the line elsewhere.
V-2. The Isocandela diagram
When the light source is imagined as being at the centre of a sphere,
V-2]
THE ISOCANDELA DIAGRAM
61
the luminous intensities can be shown at the points of intersection of
the radius vectors for the different directions of measurement with
the surface of the sphere. Points of equal luminous intensity on the
surface of the sphere can be found by interpolation, and these points
are joined together to form lines; such lines are known as isocandela
lines and the diagram thus obtained is called an isocandela diagram.
The isocandela diagram for the 3-dimensional spherical surface has now
to be represented in a flat plane. This is not a new problem, as it is
substantially the same as that which confronts the cartographer with
his maps. E u 1 e r has already demonstrated that it is not possible
to depict spherical objects in the flat in such a way that both the area
and the form are faithfully reproduced; stated more simply this means
that a spherical projection system can be accurate (or proportionate)
either from the area or from the angular point of view. Numerous systems
of projection are in use in which either the one or the other of these
two methods is adopted.
For representing light distribution, a spherical projection system is
employed which is proportional in area since the spherical surfaces to
be represented correspond to solid angles. In order to determine the
luminous flux within a certain solid angle we can measure the area
corresponding to this solid angle
Meridian
from the plane diagram (by
Pole
planimetry), and multiply this
area (converted to steradians) by
the luminous intensity (or average
lum. intensity) in that solid angle.
Two of the systems employed in
cartography, both of which are
area-proportional, have been
adopted in illuminating engineering, namely the sinusoidal
Pole
projection of Sanson-Flamsteed Zero meridian
and Lambert's azimuthal
Fig. 21. Fixing the bearing of a
projection.
point P on a sphere by means of
spherical co-ordinates. The latitude
For some years now a projection
b is. measured along the zero
described in a publication by
meridian or along the P-meridian,
and the longitude Z along the
Dourgnon and Fleury I) has
equator or the parallel
been used to represent isocandela
curves. This projection, which we shall call "quasi central projection", is
not area or angle-proportional. It does, however, have other advantages
about which we shall learn later.
62
METHODS OF REPRESENTING LIGHT DISTRIBUTION
z
XI-+++-Ir++++-lr+++-HH-t-+1X
V-3. Spherical co-ordinates
[V
Fig. 23. Development of the sinusoidal projection. The
surface of a hemisphere (a) is divided
into plane and infinitely narro'Y segments (b). When
these segments are
closed up horizontally, whilst maintaining the original
width of each one,
the spherical surface
assumes the form of
an onion-shaped figure in which the
meridians are sinusoidal (c)
z
Before discussing these systems of
spherical projection let us see how
the relative position of a point on a
spherical surface is determined. We
use for this purpose the method and
terminology employed in geography.
In geography the position of a point on
the earth's surface is fixed by means
of two co-ordinates, the latitude and
the longitude (see Fig. 22). Of these, the
c
z
latitude b is the distance from the
equator, this being measured as an angle in degrees or arc-length, along
the meridian (the great circle passing through the poles) on which the
point lies. The longitude l is the distance of the point from a given
meridian (the zero meridian); it is an angular measure along the small
circle or parallel (parallel to the equator) on which the point lies.
Latitude can of course also be measured along the zero meridian, and
is then the distance from the equator to the parallel on which the
point lies.
Longitude can also be measured along the equator, and is then the
distance from the zero meridian to the meridian on which the point lies.
V-4. Sinusoidal projection
We imagine the surface of the sphere, with its parallels, cut into sections
by a number of meridian planes to form semi-circular wedges (like
segments of an orange) (Fig. 23a).
The spherical surfaces of these segments are then laid out fiat (Fig. 23b) ;
V-4]
SINUSOIDAL PROJECTION
63
these surfaces- as is the whole surface of the sphet:e- are double curved
and cannot therefore be developed as plane surfaces, but we assume that
the segments are so narrow that they may be regarded as curved in one
direction only, and that the included portions of the parallels are similarly
narrow enough to be considered to be straight.
The strips representing the segments are thus divided by the straight
parallel lines into figures bounded by two parallel straight lines and
two curved lines.
If we now take the spacing of the parallels to be sufficiently small, these
curved lines may be regarded as straight and the strip can then be
said to consist of equilateral trapezoids.
The parallel sides, that is, the parts of the parallels in these trapezoids
are now displaced horizontally until they meet, so that all the parts
of each parallel form a straight line of the same length as the original
parallel. This results in distortion of the trapezoids, but the distortion
does not alter the area, seeing that the height and the parallel sides
remain the same. The figure thus obtained, which is reproduced for a
hemisphere in Fig. 23c, will then have the same area as that of the
strips of which it is .composed, i.e. of the hemispherical surface from
which the strips were obtained 1).
z
If the radius of the sphere be
denoted by R, the area of the
onion-shaped figure as shown in
Fig. 23c will be 2nR 2 • Let the
radius of any of the parallels be
denoted by rand the arc of the Xr------7rl'-v-'-t...-=-=:::..::::...--Jx
portion of the parallel by L1l;
then, since L1l is assumed to be
so small that the angle, sine and
tangent are interchangeable, the
length of this part of the parallel
will be r X L1l. If the latitude of
z
the parallel (in the geographical Fig. 24. Derivation of the plane co-ordinates x andy of a point P from the latitude
sense) is b; then
and longitude
r = R cos b
and the length of the section of the parallel will be
R cos b L1l.
The joint height of the segments and therefore also of the ultimate
64
METHODS OF REPRESENTING LIGHT DISTRIBUTION
[V
figure (ZZ in Fig. 23c) is equal to half the circumference of the sphere,
i.e. nR. The length of the half-equator (XX) is also nR and the length
of the half-parallel is nR cos b. In the projection the distance OA from
the parallel to the equator (X-axis) (Fig. 24) 1s
b
OA = 90 X !nR
with b in degrees, or
b X R
with b in radians.
On the equator, the distance OB from the meridian to the polar axis (Zaxis) is
l
OB = 90 X !nR
with l in degrees, or
l X R
with l in radians.
The location of a point on the sphere being fixed by the spherical co-
ordinates longitude l (rad) and latitude b (rad) the point is characterised
in the projection by the plane co-ordinates:
x = AP = OB cos b = R . l cos b, and
y = OA = R. b.
(V-1)
V-5. Transformation of spherical co-ordinates
In Fig. 23a the Z-axis is the polar axis of the meridians and parallels.
Obviously, any diameter of the sphere can be taken as polar axis and
the associated meridian and parallels drawn, and the latitude and
longitude of a point on the sphere can be indicated in accordance with
any such system. Among all the possible polar axes apart from the
Z-axis shown in the diagram, however, two are of interest. these being
the X- and Y-axes in Fig. 23a.
The two systems of meridians and parallels can be drawn in the sinusoidal
projection as in Fig. 23c and the co-ordinates of a point, indicated
as latitude and longitude in the X- or Y-system can be converted to
co-ordinates in the Z-system; by means of this conversion of co-ordinates
the meridians and parallels in the projection of Fig. 23c can be calculated.
V-5]
TRANSFORMATION OF SPHERICAL CO-ORDINATES
65
z
Fig. 25 shows the projection of
a number of meridians and
parallels drawn in the three
systems.
This conversion will now be
illustrated with reference to a
particular example; in the other
cases only the formulae will be
given.
Fig. 26 shows a hemisphere
represented by its sinusoidal
projection. The meridian of the
point P relative to the X-system
z
is given. The latitude of P can
Fig. _25i" Sinusoidal representation of a
hemisphere, showing a number of mebe represented by the curve P A
ridians and parallels corresponding to
(= bz) and the longitude by
the X-, Y- and Z-systems
- - - - - - X-system
the straight line AY (= l z). The
- - - - Y-system
meridian of P relating to the
- - Z-system
Z-axis is also included, the
latitude of P in this system being given by the curve PB (= bz) and
the longitude by the line BY (= lz).
Required: to calculate bz and lz from bz and lz. The figure PBX represents
a right-angled spherical triangle with the right angle at B. The sides are:
z
X ~~--+..~.--4~----------~X
z
Fig. 26. Derivation of formulae for the
conversion of the co-ordinates in one
system (e.g. the X-system) to those of
another (e.g. Z-system)
PB = bz,
XB = 90°-lz,
XP = 90°-bz.
Further, the angle PXB is equal
to the longitude of P in the
X-system: lz.
Employing two formulae used in
spherical trigonometry, it is a
simple matter to compute bz and
l. from bz and lz.
In the spherical triangle PBX:
sin bz = sin lz sin (90° - b.,) =
=sin lz cos bz;
further tan (90° - lz) =
= cos lz tan (90°- bz),
66
[V
METHODS OF REPRESENTING LIGHT DISTRIBUTION
from which it follows that
cot
z. = cos ltz cot b~.
The formulae for any transposition can be similarly derived, viz:
X--+ Y: sin b11 = cos l., cos b.,
(V-2a)
~~=~~~~
~~
Y --+ Z: sin b. = sin l 11 cos b11
tan = cos l 11 cot b11
z.
(V-2c)
(V-2d)
Z --+ X: sin b., = sin z. cos b.
cot ltz = cos z. cot b.
(V-2e)
(V-2/)
Y --+ X: sin b., = cos l 11 cos b11
tan l., = sin l 11 cot b11
(V-2g)
(V-2h)
Z--+ Y: sin b11 =cos z. cos b.
cot l 11 = sin z. cot b.
(V-2i)
(V-2j)
X --+ Z: sin b. = sin l., cos b.,
cot l. = cos l., cot b.,
(V-2k)
(V-2l)
To draw the projections of meridians and parallels in the X- and Ysystems, the co-ordinates of a number of the relevant points can be
referred to co-ordinates in the Z-system, after which equations (V-1)
can be used to compute the plane co-ordinates for plotting the points
on the circles.
There is a much simpler method, however, which will now be demonstrated by means of an example. Let us draw the meridian in the Xsystem, having a longitude l.,. For calculation purposes we select those
points on the meridian whose positions are determined by the latitude
b. in the Z-system. l., and b. in respect of the point are then known.
We can compute z. from equation (V-2/):
cot l., = cos z. cot b•.
From this it follows that
cot l.,
cos l, = - - = cot l~ tan b•.
cot b,
Having thus obtained z., we find from equation (V-1) the plane co-
V-6]
67
THE AZIMUTHAL PROJECTION
ordinates of the points required for drawing in the projection of the
meridian desired. In practice it amounts to this, that lz may be laid
out on the straight horizontal lines representing the parallels, since in
Fig. 26, RQ is the projection of a quarter-parallel and PQ is proportional
to lz.
The equations employed for computing the projections of the meridians
and parallels are as follows:
X-system
Y-system
l
l
meridians:
cos lz = cot l., tan bz
(V-3a)
parallels:
sin b.,
sin lz = - cos b%
(V-3b)
meridians:
sin lz = cot !11 tan bz
(V-3c)
parallels:
sin b11
coslz = - cos bz
(V-3d)
V-6. The azimuthal projection
In this system the surface of a hemisphere is reproduced within a circle.
The meridians are drawn as they appear when the sphere is regarded
in the direction of the polar axis, thus as diameters of the outermost
circle which is in this case the equator. The parallels are represented
as circles, concentric with the outermost circle.
The parallels are drawn in such a way that the areas enclosed by them
are equal to the areas bounded by the parallels on the· sphere; this gives
the projection of the meridians and parallels, with the Y-axis as polar
axis.
Let us denote the radius of the sphere by R, the radius of the outermost
circle in the projection by r, the latitude of the circle (i.e. the distance
in degrees of the parallel from the equator or outermost circle in the
figure) by b11 , and the radius of the projected parallel by r 1; the area S
of the spherical disc bounded by the parallel is then (see Fig. 27)
S = 2nRh = 2nR 2 ( I - sin b,.).
In the projection, the area enclosed by this parallel is (see Fig. 28)
Putting now S' = S, then
nr1 2 = 2nR 2 (1- sin b11 )
68
(V
METHODS OF REPRESENTING LIGHT DISTRIBUTION
from which it follows that
r 1 = Rv2 (1 -sin b11 ) = 2R sin (45°-=- }b11 ).
(V-4)
y
Fig. 28.
Fig. '17.
Fig. 27-28: Derivation of the plane co-ordinates of a point on a
sphere from the spherical co-ordinates in the azimuthal projection
At the equator, b11 = 0° and r 1 = r, so that we may write:
r = Rv'2.
Solving for R, we obtain
r
R= -v2·
Substitution of this value for R in (V-4) yields:
r 1 = rVz. sin (45° -}b 11).
(V-5)
The plane co-ordinates of a point P (l 11 , b11) on the sphere are thus
x = r 1 cos l = rVz. sin (45° -!bti) cos 1
11
and
11 ,
(V-6)
Taking successive values of 0°, 10°, 20° etc for b11 in equation (V-5).
we can now compile table G.
V-6]
69
THE AZIMUTHAL PROJECTION
Fig. 29 depicts the azimuthal representation of a hemisphere with
a number of parallels and meridians (full lines). As stated above, the
axis of these is the Y -axis (see Fig. 23a). In the same way as in sinusoidal
TABLE G
Hadii of the parallels of latitude b in the azimuthal projection in terms of the
radius of the projected equator. Llr1 is the increment of r 1 for increases of 10° in b.
b
oo
100
20°
30°
40°
50°
60°
70°
80°
90°
I
rl
r
r X 0.909
r x 0.811
r X 0.707
r X 0.598
r X 0.484
r X 0.366
r X 0.246
r x 0.123
0
I
Llr1
r X 0.091
r X 0.098
r X 0.104
r X 0.109
r X 0.114
r X 0.118
r X 0.120
r X 0.123
r X 0.123
projection, it is again possible to draw the projection of the meridians
and parallels in the other (X- and Z-systems). In this case the latter
must be converted to the
Y -system. For this purpose
equations (V-2) can be employed and the peculiarity is
then observed that the Zsystem projection can be
obtained from the X-system
by rotating the diagram
through 90°. It is thus only
necessary to compute the
plane co-ordinates of the
points in the X-system, the
Z-system co-ordinates being
then found by substituting
y for x and vice versa.
For the conversion from the
Fig. 29. Azimuthal representation of a hemisX-system to the Y -system phere
with a number of meridians and
equations (V-2a and b) are
parallels
70
[V
METHODS OF REPRESENTlNG LIGHT DISTRIBUTION
sin b11 = cos l., cos b.,, and
used, viz:
tan l 11 = sin l., cot b.,.
In equations (V-6) b11 and l 11 must now be expressed in terms of l., and b.,;
this gives for the plane co-ordinates of the points in the X-system:
x=r
sin b.,
VI +cos l., cos b.,
and
sin l., cos b.,
y=r---;====:====
VI +cos l., cos b.,·
(V-7)
A number of parallels and meridians of the Z-system are shown by the
dotted lines in Fig. 29.
As the azimuthal method of representation is also area-proportional,
we are once more in a position to ascertain luminous flux values by
planimetering the area between 2 isocandela lines and, after conversion
to the corresponding solid angle, to multiply it by the average luminous
intensity in the measured area.
The solid angle can be computed from the measured area in the following
manner. Let S denote this area; then the ratio of this to the area of the
hemisphere is as S : :n:r2• The hemisphere contains 2:n: steradians; hence
S corresponds to
-
s X 2:n: sterad. = 25
sterad.
:n:r2
r2
V-7. Comparison of the sinusoidal and the azimuthal projections
When a comparison is made between the sinusoidal and azimuthal
projections it is found that the latter has certain advantages to offer:
the diagram as obtained by Lambert's method shows much less
distortion and, moreover, in the azimuthal projection, the amount
of distortion for the same distance from the centre of the figure is the
same in all directions, whereas the nature of the distortion in the sinusoidal system, for the same angular distance, depends on the particular
position with respect to the centre of the figure.
The relatively slight distortion in Lambert's projection is due
to the fact that the scale on which one degree is plotted radially differs
only slightly over the different sections of the diagram. Within the
circle of radius 30° about the axis (i.e. between b = 60° and b = 90°)
V-8]
Q.UASI CENTRAL PROJECTION
71
the scale is practically constant (Cf table G). It is also an important
advantage that interpolation from the diagram is accordingly much
simplified. The various meridians and parallels in the Z-system in
Fig. 29 do not intersect each other precisely at right angles, but the
discrepancies at 90° are not so great as in the sinusoidal projection
(Fig. 25).
In the sinusoidal projection it is not possible to see whether a given
light distribution is rotationally symmetrical, as is possible in the
Lambert diagram. This is not important for practical purposes,
seeing that symmetrical light distributions are seldom represented by
means of isocandela diagrams.
V-8.
Quasi central projection
1 his method of plotting a light distribution on one plane was published
by Dourgnon and Fleury under the title "Universal Diagram".
It is our opinion, however, that the method is better described by the
name "quasi central projection".
A point P on a sphere (Fig. 30) can be centrally projected on a horizontal
Fig. 30. Point P on : sphere is centrally projected on to the plane H in point P'
(coordinates on the Z-system).
72
METHODS OF REPRESENTING LIGHT DISTRIBUTION
[V
plane li with the centre M of the sphere as the centre of projection.
Point P' is centrally projected as the point of intersection of the produced
straight line MP and plane H.
The position of projection P' can be established with the aid of the same
coordinates used to find the position of point P on the sphere, i.e. the
geographic longitude and latitude. All points or lines on the sphere
can be projected on to plane H in this way. If, with a number of points P',
the luminous intensities of the appropriate points P on the sphere were to
be written, isocandela curves could be determined by interpolation in
plane H. The great disadvantage of this metod is that projections P' move
further and further away from the projection of pole Z as the latitude
decreases, i.e. the closer points P are to the equator. The points on the
equator would be projected further and further away from Z' ad infinitum.
According to Dourgnon and Fleury it is better with this projection
not to use the latitude b but the distance b' from the pole which is measured from the pole and is the complement of the latitude b.
Distance Z'P' is equal to MZ' tan b' and this expression becomes infinitely great where b' = goo.
In order to remedy this drawback, Dourgnon and Fleury do not
locate point P' on plane H at its natural position obtained by central
projection but they use a system of coordinates in which the angles
are shown proportionally (Fig. 31). They take the longitude l 11 and the
distance b'y of theY system from the pole as Cartesian coordinates. This
has the advantage that the important lines parallel to the road axis
correspond in the system of coordinates to lines parallel to one another
and to the Y-axis.
It must now be asked how the systems of meridians and parallels are
represented on this chart if XX, YY and ZZ are taken as polar axes.
The simplest case is provided by theY-system, since here the meridians
and parallels give a system of straight lines intersecting one another at
right angles (Fig. 31). A point P on the sphere with the geographic
coordinates b' 11 and l 11 corresponds to a point P' on the chart with
Cartesian coordinates b' 11 and j 11 •
With polar axis ZZ the meridians and parallels on the chart are represented by curves, which can be calculated with the aid of spherical
trigonometrical formulae (Fig. 32).
The X-system gives a projection of meridians and parallels in curves
as shown in Fig. 33. This curve network can be derived from the foregoing
by drawing the grid in Fig. 32 symmetrically and then effecting a goo
shift parallel to the axis of l 11 •
V-8]
Q.UASI CENTRAL PROJECTION
73
0
Fig. 31. Plotting point P' on a diagram in which the angular magnitudes of b'y und /y
are entered proportionally.
y
0
30°
45°
ly
60°
75°
F1g. 32. Plotting the meridians and parallels of the Z-system on the diagram with
b'y and /y in Fig. 31.
74
METHODS OF REPRESENTING LIGHT DISTRIBUTION
[V
Fig. 33 .. Plotting the meridians and parallels of the X-system on the diagram with
bjy and 7& in Fig. 31.
As we shall see in VII .1 0, this system has considerable advantages in the
calculation and plotting of illumination diagrams.
The diagram with b' y and ly shown in perspective in Figs. 31, 32 and 33
is, of course, transferred to the drawing plane for practi~al use (Fig. 34).
The isocandela curves can then be plotted on such a diagram.
V-9. The application of preferred numbers in isocandela diagrams
Isocandela diagrams can be constructed, say, for 100, 50, 25, 10, 5, 2.5,
1 cd and so on, but it is much better so to arrange the isocandela lines
that the luminous intensity of each successive line is greater than that
of the preceding line by a constant factor.
To this end the use of preferred numbers is recommended and we shall
now devote a little space to this subject 3).
Preferred numbers are the terms of certain geometrical progressions,
every nth term of which is a whole power of 10; n may be 5, 10, 20 or 40.
When we taken = 5, we have the so-called R5 series of which the ratio
(i.e. the ratio of one term to the next) is ~TO ~ 1.6. In other words,
V-9]
PREFERRED NUMBERS IN ISOCANDELA DIAGRAMS
75
25
Fig. 34. lsocandela magram of the lighting fitting of Fig. 35, on Dourgnon and
Fleury's quasi.central projection.
the terms increase in steps of about 60%; the terms between I and IO
in this series become 1.6-2.5-4--6.3. If the steps of the R5 series
are too large for a given purpose, the RIO series (n = 10) may be used,
the ratio then being -\o/ 10 R; 1.25 and the increment about 25%. The
terms from 1 to 10 are then I.25-I.6-2-2.5-3.I5-4-5-6.3-8. The
R I0 series can be derived from the R5 series by interpolating one term
between each pair in the latter series. Interpolation between the terms
of the R I 0 series gives the R20 series, in which n = 20 and the ratio
20-
is ylO R; 1.12 (increment approx. I2%). Similarly an R40 series can
40-
be set up with n = 40, ratio viO R; 1.06 (increment approx. 6%).
For the luminous intensities in isocandela diagrams we shall in general
employ the R5 series. The lines are then for 1-1.6-2.5-4--6.3-1016-25-40-63-1 00-160_:_250-400-630-1 000 cd. According to our
requirements we shall utilise a part of these or extend the series upwards
or downwards.
76
METHODS OF REPRESENTING LIGHT DISTRIBUTION
[V
When an isocandela diagram is plotted from luminous intensity values
according to a preferred number system, the concentration of the
isocandela lines yields more information than with intensities of, say,
10, 20, 50 and 100, since
the intensities of the
successive lines then bear
a constant ratio to each
other.
Examples of isocandela diagrams of
a fitting
Figs. 35a and b show the
light distribution of a
highly asymmetric street
lightingunitin bothsinusoidal and azimuthal projections; the values of the
luminous intensities for
the isocandela lines are
based on the R5 series
of numbers. If the fitting
is suspended at a height
of I 0 m, overhanging the
roadway at a distance of
I m from the curb of a
street I 0 metres in width,
the light between the
meridians + 42° and -5! 0
as shown, will fall on the
•V -10.
Fig. 35a.
Fig. ;351b.
Fig. 35. lsocandela diagram
of an asymmetric lighting
fitting (a) sinusoidal projection; (b) azimuthal projection
roadway. In these figures only the meridians and parallels of the Ysystem are shown. We have here an instance, then, in which instead
of the projection on the Z-system, the Y-system as computed from
the Z-system is preferred. Both figures are isocandela diagrams from
V-10]
EXAMPLES OF ISOCANDELA DIAGRAMS
77
the point of view of the horizontal line through the lighting unit,
parallel to the axis of the roadway. The meridians are cross-sections
of planes passing through this line. These planes are accordingly represented by straight lines in the azimuthal projection, and curved
lines in the sinusoidal system. The azimuthal representation is clearer
than the other.
Fig. 34 shows the same light distribution on Dourgnon and Fleury's
quasi central projection. Because the perspective diagram in Fig. 31
represents only an eighth of the surface of the sphere and because a
quarter of the sphere ought to be shown with this asymmetric fitting,
the diagram of Fig. 31 has been duplicated here. The shape of the isocandela curves in the upper hemisphere cannot be shown here on the
same diagram. This is a disadvantage of this method, but one which is not
generally obtrusive, because the light radiated in the upper hemisphere
does not contribute to the lighting of the street. There is another advantage to this method of depicting a light distribution. If the fitting is moved
through, say, 10° in a plane perpendicular to the axis of the street, the
new isocandela diagram is obtained in a simple way by shifting the locus
of the curve in the diagram horizontally through 10°.
If light then falls on the surface to be illuminated from what was the
upper hemisphere in the initial position, the abovementioned disadvantage becomes evident.
REFERENCES
This derivation of the sinusoidal projection is taken from a publication by
R. Swier s t r a, Euclides 12, 1935, 56 (in Dutch)
2) J. Dourgnon and D. Fleury, Lux 28, 1Y60, 53-68, ,Diagramme univenel pour-la
representation des repartitions lumineuses des sources dissymetriques."
3) I.S.A. Bulletin 11, "Preferred numbers". Dec. 1935
1)
CHAPTER VI
ILLUMINATION
VI-1. Illumination; foot-candle; lux
Obviously, the amount of luminous flux falling on an object emitting
no light of its own is very important from the aspect of observation
of that object. We wish to know, therefore, how strongly an object or a
surface is illuminated and the natural course is to express the intensity
of the illumination, or illumination (symbol E) in terms of the luminotts
flux per unit area of the surface thus illuminated.
Denoting the luminous flux by f/J and the area by 5, we may say that
f/J
E=s-·
(VI-I)
f/J =E. S.
(VI-la)
Hence
The luminous flux falling on a surface is thus ascertained by multiplying
the illumination by the area.
Generally speaking, the illumination on a surface will differ between
one point and another, so that division of the incident luminous flux
by the total area of the surface in accordance with equation (VI-I)
yields the average illumination of that surface.
If we now consider a part of the surface only, it may be said that the
smaller the area, the less the variation in the illumination in that area
and, if this process be continued to produce an area that is in fact only
a point, the illumination of that point will be uniform. We may then
refer to the actual illumination of that point in a plane. It must be
remembered, however, that in speaking of such a point we really mean
a very small area, so that, when referring to the illumination of a point,
it is necessary to state the plane in which that point lies.
In terms of the infinitesimal calculus, illumination can be expressed as
E
= dtl>
dS
the incident luminous flux being obtained from
<P = jE. dS.
VI-2]
79
THE INVERSE SQ,UARE LAW
If the luminous flux is expressed in lumens and the area in square feet
in equation (VI-I), the illumination is obtained in foot-candles (fc).
Hence, 1 fc is the illumination produced by 1 lumen falling on a surface
1 sq. ft in area.
The foot-candle unit is employed only in English-speaking countries;
the internationally standardized unit of illumination is the lux (abbrev.lx).
One lux is the illumination of an area of 1 sq. m. produced by a luminous
flux of 1lm. In equation (VI-I), with S =1m 2 and if>= 11m, E = I lux.
Since I foot is 0.305 m,
1 fc =
(o.~o5r lx = 10.76lx ~ 10 lx, and
0.305) 2
llx = (- - fc = 0.0929 fc ~ 0.1 fc.
. I
Examples:
1. If the incident luminous flux on a surface of 4.5 sq. ft is 400 lm, the
average illumination on that surface is
tf>
400
E = S = 4.5 = 89 fc = 89 X 10.76lx = 958 lx.
2. An area of 12 m 2 receives an average illumination of 150 lx. In accordance with equation (VI-la) the incident luminous flux must
then be
tf>
= E X 5 = 12 X 150 lm = 1800 lm.
VI-2. The inverse square law
In Fig. 36, L represents a light source, the luminous intensity of which,
in all directions within a solid angle w, is denoted by I. The luminous
flux if> in this solid angle is then if> = w . I.
If we now draw a sphere with L as centre
and with radius r, the spherical surface
corresponding to w will receive the luminous
flux if>.
The illumination of this surface is asL
certained by dividing the incident luminous flux by the area; this area is
S = wr 2 and the illumination is therefore
if>
wi
I
E=-=-=-.
S
wr 2
r2
{VI-2) Fig. 36.
Derivation of the
inverse square law
80
ILLUMINATION
[VI
Hence the illumination on the spherical surface is equal to the luminous
intensity in the direction of that surface, divided by the square of the
radius.
·
Now, if we imagine the surface diminished in size to a mere point, that
point can be regarded as representing a plane surface perpendicular to
the line passing between the point and the light source.
We may therefore state that the illumination at a point in a plane perpendicular to the line joining the point and the source is equal to the
luminous intensity of the source in the direction of the point, divided by
the square of the distance between point and source. If we denote this
distance by d, equation (VI-2) becomes
I
E = d2 •
(VI-3)
In equation (VI-3), when
E = I fc
and I= I cd, d = I foot, or
E = I lux and I= I cd, d = I metre.
when
The lux and the' foot-candle may thus also be defined as the illumination
produced on a surface 1 metre or 1 foot from a light source of1 cd luminous
intensity, measured in a plane perpendicular to the direction of radiation .
.Formula (VI-3) can be expressed in words as follows. The illumination
at a point in a plane is proportional to the luminous intensity, and inversely
proportional to the square of the distance from the light source. For this
reason equation (VI-3) is generally known as the inverse square law.
In our consider~tion of this law, as well as that of luminous intensity
and luminous flux, the light source has each time been assumed to be
a point source, although in practice all light sources have finite dimensions, these being sometimes even quite large. The extent to which
the assumption of a point source is justifiable in practical applications
of the inverse square law is discussed in Chapter IX (Non-point sources).
Example:
A light source is suspended 10 feet above a horizontal surface; the luminous intensity of the source vertically downwards is 420 cd.
The illumination of the plane, at a point vertically below the source is
I
420
E = (j2 = W fc = 4.2 fc.
VI-3]
81
ILLUMINATION WITH OBLIQ.UE INCIDENCE
VI-3. Illumination with oblique incidence
Equation (VI-3) applies only when the plane of which the illumination
is to be determined is perpendicular to the line connecting the light
source and point, i.e. with normal incidence of the light at that point
in the plane.
Let us now ascertain the illumination when the plane lies at an angle
-90°-oc to the ray of light, in other words, when the light is incident at an
angle oc to the normal to the plane.
In Fig. 37 L is a light source with a luminous intensity of I cd in the
direction of the point P at which we
wish to know the illumination E. The
point P lies in a plane V and the line
LP is at an angle oc to the normal to· the
\
plane. The illumination to be computed
d
is that of this plane.
It is once more necessary to imagine that
we wish to know the illumination of a
very small area in the plane V, instead
of that of the actual point P. If we
provide · a plane V', perpendicular to Fig. 37. When light from a
source L falls at a point P on
LP at P, the illumination E' at P in a plane V at an angle ex to the
normal at P, the illumination
that plane will be
of V at P is cos ex times the
illumination at P nerpendicular
to LP.
E' = !__
(E =
d2
~2 cos ex)
The luminous flux thus reaches a small
area of V' which may be represented in cross-section by the line CD.
Now, if the same luminous flux falls on the plane V, the area concerned
may be represented in cross-section by AB; then
CD
AB=--.
cos oc
Since the dimensions of the illuminated areas of V and V' are the same
in the direction perpendicular to the plane of the drawing, the relevant
I
.
area of V will be - - times as large as that of V'. The illumination
COS IX
82
[VI
ILLUMINATION
of V is therefore cos ot times that of V' and we may accordingly write:
E = E' cos ot, or
I
E = d2 cos IX.
(VI-4)
Example:
A light source L is 12 feet from a point P in a plane V. The normal to
V lies at an angle rx = 25° to the line LP. The luminous intensity of L
in the direction of Pis 200 cd. The illumination on plane Vat the point P
is then
I
200
o
E = (ji" cos rx = W cos 25 = 1.27 fc.
In practice it is often necessary to determine the illumination at different
points in a plane lighted from above by a
t
source at a known height above the plane.
This height, which we will denote by h, is
thus the perpendicular distance of the source
L from the plane H (see Fig. 38 . Formula
(VI-4) gives the illumination of the plane H
at the point P, viz:
I
E = d2 cos IX.
Fig. 38. If a light source L
is suspended at a height h
above a plane H and the
light falls on a point P on H
at an angle rx, the luminous
intensity of Lin the direction
of P being denoted by I, the
illumination of H at P is
I
E = JiB cos sa;
h
d=--,
Now
COS IX
hence
I
I
E = -;-(---::-----,.)-::- X cos ot = hs cos 3ot.
2
CO: IX
I
E = - cos 3ot
h2
(VI-5)
If we now apply equation (VI-5) for various points in respect of which ot
is constant, we at once obtain the illumination values for circles of
radius AP = h tan ot. If the luminous intensity for all such points of
the same angle otis constant, we thus obtain in one operation the illumination value for the whole of the circle whose radius is h tan ot. Equation
(VI-5) is therefore very useful for ascertaining the illumination of
VI-3]
ILLUMINATION WITH OBLIQ.UE INCIDENCE
83
surfaces lighted by a source the distribution of which is wholly or nearly
symmetrical around the axis.
Examples:
I. A light source L is suspended 15ft above a point A in a horizontal
plane. A point Plying in the plane is so positioned that L ALP= 35°.
If the luminous intensity of L in the direction of P is 275 cd, the
illumination on the horizontal plane at point P will be
I COSa IX = 27 5 COSa 35o = 0.6 7 fC.
E = Jii'
151
2. If the point Pin the first example is 8 feet from point A, it is necessary first to calculate the angle IX:
tan IX = ~: = 185 = 0.533; hence IX 1'1::1 30°.
The illumination of the plane at Pis then, taking I = 275 cd,
I
275
E = - 1 cos3 IX = cos 3 30° = 0.8 fc.
h
151
When illumination calculations are to be effected for elongated areas,
another formula will in many
cases be found more useful,
especially when the light distribution is asymmetric. A good
example of this is a roadway
with lighting standards. In
Fig. 39, L represents a light
source at a mounting height
LA = h above the plane H.
H may thus be a roadway
Fig. 39. The illumination of a plane H at
a point P is
bounded by the parallel lines
I
QQ' and RR'. The illuminaE = }il cos1 IX cos3 {3
tion at a point P on the
plane H can be calculated from equation (VI-5):
I
E = hz coslly.
The point P lies in the plane LBP which makes an angle at with the
normal to plane H and the line LP makes an angle P with LB. LB lies
in a plane through L which is at right angles to H. The illumination
at P can be computed when at and P are known.
LA
In tJ. LAP, cos y = LP
84
[VI
ILLUMINATION
LB
In 6 LAB, LA = LB cos oc, and in 6 LBP, LP = --.
cos{J
LA
LBcosoc
LB = cos oc cos {J.
Hence cos y = LP =
cos fJ
Substituting this for cosy in the equation for E, we have
I
E = };2 cos 3oc cos 3/J.
(VI-6)
The distance AB is found to be
AB = h tan oc
and
tan fJ
BP = LB tan{J =h--.
cos (X
For calculations of illumination values by means of equation (VI-6),
Table II on p. 408/9 provides the values of cos3 oc cos3 fJ for a number of
angles oc and f3.
tan{J
Table Ilion p. 410/1 I gives the values of h tan oc and h - - for a number
cos oc
of angles oc and {J.
If the plane H represents a roadway, as suggested, the plane LAB
will be taken perpendicular to the direction of the road. BP is then
parallel to the centre-line of the roadway. In this way we calculate
the illumination along different lines, all of which are parallel to the
road axis.
It is essential for the purposes of the calculation that the luminous
intensity at the various angles fJ in planes at various angles oc to the
vertical LA shall be known, in other words the light distribution in these
planes must be ascertained. All these planes pass through the line TT'
lying parallel to the roadway and passing through the light source L.
There is another advantage in this method of selecting the planes of
measurement, for the same family of light distribution curves can
be utilised for different positions of the lighting fitting. Suppose the
unit to be canted so from the position to which the angles oc refer; this
will mean that, in order to compute the illumination for the line for
which oc = S0 , we must take the values for the plane where oc = oo
from the set of light distribution curves relating to the original position.
VI-4]
HORIZONTAL AND VERTICAL ILLUMINATION
85
For the line whereby ot = 25°, the values of the plane where ot = 20°
are taken, and so on.
Usually, the calculation will be confined to those planes which are
essential for determining the illumination of a road of a given width
and with a certain mounting height.
Example:
Suppose the mounting height of the light source L in Fig. 39 to be 25 ft,
with ot = 30°, {J = 40° and I = 540 cd.
The illumination of the plane H at the point P is then
540
I
E = }jl cos3 ot cos3 {J = 25, cos 3 30° cos 3 40° = 0.25 fc.
VI-4. Horizontal and vertical illumination
The illumination at a point, i.e. that of a very small area, in a horizontal
plane, is referred to as the horizontal illumination at, or of, that point
(EH). If, on the other hand, the point occurs in a vertical plane, we
speak of the vertical illumination of the point (Ev)·
In this connection it must be borne in mind that only one horizontal
plane, but an infinite number of vertical planes, can pass through. a
given point. When stating and computing the vertical illumination of
a point, the vertical plane in which the point lies must be denoted.
In the case illustrated in Fig. 38 we calculate the illumination of the
horizontal plane H, at the point P, that is, the horizontal illumination
at P. We therefore put:
(VI-7)
If we regard Pas a point in the vertical plane V j_ AP, it is readily seen
that the illumination of V at the point P, i.e. the vertical illumination
at P, is
I
Ev = hi cos 2ot sin ot.
It follows from (VI-7) and (VI-8) that
Ev
-
EH
sin ot
cos ot
= - - = tan ot.
(VI-8)
86
ILLUMINATION
[VI
In the example shown in Fig. 39 we have calculated the illumination
at the point Pin the horizontal plane H, i.e. the horizontal illumination
at P; hence we can state:
(VI-9)
For the vertical illumination at P in a plane perpendicular to BP (i.e.
perpendicular to the road axis if we assume that H represents a road),
we put:
I
.
Ev = h2 cos 21X sm fJ cos 2/J.
(VI-10)
From (VI-9) and (VI-10) it then follows that
Ev
En
tanfJ
cos IX
BP
LA
Now, tan fJ = LB and cos IX = LB.
hence
Ev
En
tanfJ
cos IX
BP
LA
As will be seen in Part II (Light Measurements) it is easier and more
accurate- if the incidence of the light is very oblique- to measure the
vertical instead of the horizontal illumination. When the vertical illumination Ev is measured at a point perpendicular to the road-axis,
En can be determined by multiplying Ev by the quotient of the mounting
height h divided by the distance BP from the measuring point to the
vertical plane through the lighting fitting.
VI-5. Other units of illumination
The foot-candle and the lux are not the only units of illumination in use.
Low illumination levels are often expressed in millilux (mlx) (= 0.001 lux).
We also sometimes come across the phot, this being the illumination
VI-5]
87
OTHER UNITS OF ILLUMINATION
produced by 1 lm on 1 sq. em, or by 1 cd or candle from a distance of
1 em. This unit fits into the C.G.S. system (1 phot = 10,000 lux).
English publications on the subject of coast lighting mention the seamile candle, which is the illumination resulting from 1 cd at a distance
of l sea-mile. As the sea-mile is 6080 feet
sea-mile candle=
(60~0r fc = 2.70 X 10- fc
8
(= 2.9 X t0- 7 lux).
In table IV (p. 412) the various units of illumination are listed together
with the relevant conversion factors.
CHAPTER VII
ILLUMINATION CALCULATIONS AND DIAGRAMS
VII-1. Introduction
Of the various factors which determine the quality of a lighting installation as an aid to effective vision, the illumination is one of the
most important.
In the design of a lighting installation the lighting engineer will ascertain
roughly the illumination level which the installation will have to produce
and, in some cases the degree of non-uniformity that may be tolerated.
He chooses a type of lighting fitting to suit the particular method of
illumination and, on the basis of the required illumination level and
permissible inequalities therein, he calculates the number of fittings
required and the power of the lamp to be installed in each fitting. For
this purpose details of the lighting fittings are required and these must
be provided by the manufacturer.
Calculations for complete lighting installations are outside the scope
of this book, but are dealt with extensively in most manuals on illumination engineering 1 ). In the following sections, however, we shall
describe briefly the more usual methods employed, with the emphasis
on those calculations which the lighting engineer has to make from the
distribution characteristics of the lighting fitting and which usually have
to precede any calculations on the complete installation, for which they
serve as basis.
There is a choice of two methods of calculation for lighting installations.
viz:
1) the lumen method,
2) the point-by-point method.
VII-2. The lumen method; coefficient of utilization
Lighting installations usually comprise a number of lighting fittings.
The engineer knows from experience based on measurement in practice
that a given type of fitting will ensure a certain degree of uniformity
in the illumination when given a certain height in relation to the spacing.
Once the mounting height has been decided upon to suit local conditions,
the most satisfactory spacing is thus determined and so also the number
of fittings required.
If it is also known what part of the luminous flux of each lamp with
its appropriate fitting reaches the plane to be illuminated (the working
plane), the luminous flux which will have to be supplied by each lamp
can be calculated, and this in turn indicates the lamp rating.
Vll-21
THE
LUMEN
METHOD;
COEFFICIENT
OF
UTILIZATION
89
Let the size of the working plane be denoted by 5 sq. ft (or sq. m)
and the required average horizontal illumination by E fc (or lux); the
luminous flux required in that area will then be E X 5 lm. If we now
denote that portion of the light from the lamps which reaches the working
plane by 'YJ, the total luminous flux of the lamps will have to be
(/)tot
Ex5
= - - lm.
'YJ
With n fittings, the luminous flux per lamp will be
(/Jia
Ex5
= - - lm.
nxn
n=
11 X
(VII-IJ
{[Jza
where {[Jza is the total luminous flux of the lamps in every lighting fitting.
The values of E, 5 and n can be determined at once by the lighting
engineer, but the factor rJ is a different matter. This is the coefficient
of utilization of the room and the lighting installation therein, this being
mainly determined by the following:
a. The light distribution of the fittings. When all the light from a fitting
is directed downwards (direct lighting), the degree to which the
ceiling is capable of reflecting light hardly affects the illumination
in the working plane at all. On the other hand, when the fitting
radiates all the light upwards (indirect lighting), the condition of
the ceiling is the all-important factor. If the fitting distributes its
light in all directions, the ceiling and walls are important, but to a
lesser degree than in indirect lighting.
b. The efficiency of the lighting fitting.
c. The amount of incident light reflected by walls and ceiling, of which
a part reaches the working plane. Obviously a whitewashed ceiling
will reflect more light downwards than a grey one, and light-coloured
walls similarly contribute more towards the illumination in the
working plane than dark ones.
d. The ratio of the over-all dimensions of the area to be illuminated to the
height above this area at which the lamps are mounted. The influence
of this on the illumination is included in the data relating to 'YJ under
the name of room index and methods of computing this index may
be found in any existing book on illuminating engineering.
90
ILLUMINATION CALCULATIONS AND DIAGRAMS
[VII
Curves and tables are in existence from which approximate coefficient
of utilization values may be read or interpolated in relation to all practical
forms of lighting installation 2 ).
It is usual to make allowance in the coefficient of utilization for a certain
depreciation in the efficiency of the lighting equipment during use,
such being due to:
1. drop in the luminous flux of the lamps,
2. reduction in the efficiency of lighting fittings due to dust and dirt;
even with regular maintenance a certain average "dirty condition"
must be taken into account.
3. deterioration in the reflection factors of walls and ceiling, to which
the same remarks apply as in 2. regarding average working conditions.
The desired illumination must be produced as an average during the
working period of the equipment and, owing to the three factors mentioned above, this average is below the initial value for the new equipment.
It is necessary, therefore, to plan an installation for a higher illumination
level than the average output. A suitable correction is usually incorporated in the tables or charts of illumination values.
The factor by which the initial illumination level should be divided
or, conversely, that by which the desired average should be multiplied
to allow for depreciation in output during use is known as the depreciation
factor. This may therefore be defined as the ratio of initial to average
efficiency (or illumination) of the equipment during use; hence it is
greater than unity.
The method described above is based only on the luminous flux of the
light sources as installed and is appropriately referred to as the lumen
method.
VII-3. The point-by-point method
The lumen method is adopted for calculations of lighting installations
in cases where a given average illumination is envisaged in which some
latitude is allowed in the uniformity of the lighting, or in which a certain,
degree of uniformity is specified. Such uniformity is generally ensured
by installing a number of light sources working in co-operation; the
illumination at any given point in the working plane is then produced
by several light sources in the vicinity of that point.
In most cases the lumen method is suitable for interior lighting schemes;
the lighting of streets, squares, in fact all outdoor lighting presents a
rather different problem. As will be seen later (Ch. X), the illumination
at the different points of the road, i.e. the distribution of the illumina-
VII-3]
THE POINT-BY-POINT METHOD
91
tion, is our main concern in street lighting. In order to know something
of this distribution we ascertain the illumination values at a number
of points and if necessary interpolate the levels at intermediate
points.
The distribution of the illumination in any street lighting project will
have to be derived from the light distribution of the fitting on the
basis of a given mounting height and spacing, and in conjunction with
the width of the street or roadway to be illuminated. Here again, thus,
it is essential to have details of the distribution characteristics of the
lighting fittings to be used. We can here, however, go a step further
and say that the makers could be expected to provide the data in a
form that would render the application of such data universal, that
is, suitable for obtaining the distribution for any combination of
mounting height, spacing and road width. Whether or not such data
are determined by calculation from the light distribution or by direct
measurement is not important.
We shall discuss various forms in which such illumination data might
be presented. Of the two methods of originating such data we shall
concern ourselves here only with the calculation method. The direct
method is described in Part II, Light Measurements (section XVIII-4).
Since the illumination is computed for a number of points in the plane
to be illuminated, the method under review is known as the point-bypoint method.
This system is not employed exclusively for exterior lighting; it is also
used for interior illumination. In section VII-2 we have said that the
lighting engineer usually knows from experience how much uniformity
of illumination may be expected from a given project and that such
experience is backed by practical measurement; this is only half the
truth, however, for the expert can also calculate the degree of uniformity
from the distribution of illumination in the working plane produced
by one fitting. The illumination in this plane is two-fold; one part
comes from the source itself (direct) and the other is reflected from
walls and ceiling (indirect). The illumination due to the direct light can
be computed from the characteristic distribution of the lighting fitting
by the point-by-point method.
Thus, also in the case of indoor lighting fittings the illuminating engineer
must have data concerning the illumination at his disposal, especially in
the case of direct lighting, and here too the data supplied should be
universal in their application.
92
ILLUMINATION CALCULATIONS AND DIAGRAMS
[VII
VII-4. Illumination diagrams
Illumination diagrams are graphical representations of illumination
patterns in a plane or along a line. A diagram of this kind will illustrate
either the illumination distribution of a complete lighting installation,
or local differences in the illumination produced by a single light source.
In the first instance the number of light sources, the mounting height
and the spacing are all specified, and also the condition, and therefore
the influence, of walls and ceiling is similarly fixed. It is difficult and
sometimes impossible to base a diagram of this kind on calculation,
and recourse is usually taken to direct measurement of the illumination,
this being, as it were, proof by results.
For estimating the effect of a
lighting installation in advance,
various forms of diagram are employed, showing the illumination
resulting from the use of particular lighting fittings. These do
not, of course, take into account
the effect of the condition of the
walls or ceiling. The diagrams are
usually so arranged that they can
be used in relation to any mounting
height by interpolation or simple
conversion.
We shall now describe the more
current types of illumination distribution diagram and the method of
working them out from the light
distribution of the fitting. There
are other forms than those which
we shall review, but the reader will
have no difficulty in understanding
Fig. 40. Polar light distribution diathese on encountering them in the
gram of a lighting fitting suspended
at a height h above a horizontal plane.
literature, once the following has
The isolux diagrams in Figs. 42, 45
been studied.
and 47 were constructed with the aid
of this Cl!rve
For the derivation of the diagrams
relating to a single lighting fitting
let us assume an axially symmetrical and also a bilaterally symmetrical
fitting. In the first type we take as starting point the light distribution
of a fitting the axis of symmetry of which is vertical and which illuminates
VII-5]
THE RECTANGULAR ILLUMINATION DIAGRAM
93
a horizontal surface; the situation is as shown in Fig. 40. The bilaterally
symmetrical arrangement is depicted in Fig. 4la which is similar to
that in Fig. 39. The light distribution of the fitting is given in planes
through a line parallel to the illuminated plane (Fig ..41 b).
In principle, a distinction can be made between two types of diagram, viz:
1. those in which the illumination is plotted as a function of its location.
2. isolux diagrams in which
--.T-·-.
L
Ill
-lines are drawn joining
It I
It I
points of equal illumina;71
I I I
I
I I
1/ I
tion. These lines are called
I 1
I
II I
I I
I
I1 I
isolux curves. It is useful
I I :
to employ the preferred
/ I I
number system for the
values in isolux diagrams
and the R5 series will
generally be found the
most suitable.
The illumination can of course
be expressed in foot-candles
or lux. In the following we
I
0
shall use the foot-candle and
I
the foot, but the values can
be read as they stand in lux
cd
by substituting metres for
feet. In the figures both
notations are shown.
The following are the most
usual forms which the illuFig. 't-1. The light distribution curves in
Fig. 4lb refer to planes making an angle Ot
mination diagrams assume:
with the vertical line passing through the
1. rectangular diagram,
lighting fitting. The assumed position of the
fitting with respect to the illuminated plane
2. solid of illumination,
is seen in Fig.4la. The curves in Fig. 4lb
3. plane isolux diagram,
serve as basis of calculation for the isolux
diagrams in Figs. 43, 44 and 46
4. polar isolux diagram.
·-·-·-·r
~
~0
~
VII-5. The rectangular illumination diagram
In rectangular illumination diagrams the illumination is plotted with
respect to various points on a line, the horizontal axis representing
the distance of the points from a reference point and the vertical axis
the illumination values.
For the axially symmetrical fitting (Fig. 40) the illumination produced
94
ILLUMINATION CALCULATIONS AND DIAGRAMS
can be wholly characterised by a
single line (Fig. 42), but if the fitting
is not axially symmetrical, more
than one line is needed (Fig. 43).
The diagram in Fig. 42 can be
easily constructed with the aid
of equation (VI-5):
Fe {lx)
250
h=lft(m)
cfi= 1000/m
200
150
100-
I
E = - cos 3 :x.
ft2
50
u
[VII
0
0.5
1.5
~~-~
2
2.5 ft(m)
.. a- tan a.
Fig. 42. Illumination diagram for the
lighting fitting shown in Fig. 40 ;_
the illumination is shown as a function
of the distance from the vertical
through the fitting
The distance a from the axis, at
which the computed illumination
is produced at an angle oc, is found
from: a = h tan oc. In a universal
diagram h may be I foot or I 0
feet; when h = I foot,
E =Ia. cos 3 oc and
a= tan oc.
Fig. 42 is based on the light distribution illustrated in Fig. 40 for a
height h = I foot and a luminous flux of the bare lamp used in the
fitting, of 1000 lumens. For other mounting heights the illumination
values are divided by h2 , and the distances a arc multiplied by h.
Illumination curves derived Fe (lx)
from the ligh~ distribution
200
a.=O.b=O
shown in Fig. 41b are depicted
/ a.=l5~b=h fan15"=2.68ff(mJ
.....
,
/
in Fig.43;here the illumination
h=70ft{mJ
~
is plotted as produced along
</>=1000/m
the lines of intersection QQ', 100
/ ct.=30~ b= h fan .30° ="5.77ff (mJ
SS' etc of the illuminated plane
a=45?b=hfan 45°=10ftrmJ
and the planes for which the
50
light distribution curves are
shown in Fig. 41b.
--.a
The illumination values are
calculated by means
of Fig. 43. Illumination diagram for the
lighting fitting shown in Fig. 41. The
equation (VII-6):
'
I
E =-cos 3oc cos 3/J.
ft2
illumination along the lines such as H.H.'
;md SS' in Fig. 41 a, is plotted as a function
of the distance from the vertical plane
through the fitting
VII-7]
95
THE PLANE ISOLUX DIAGRAM
The distances along the lines SS' etc. (see Fig. 4la) are obtained from
tan ,8
a= h - - ,
cos ex
the positions of these lines being in accordance with the distances
b = h tan ex.
The curves in Fig. 43 refer to a height h = I 0 feet and a luminous
flux of the bare lamp of I 000 lm. Conversion for other mounting heights
I0 2
and the
is carried out by multiplying the illumination value by h2
h
.
distances by 1o·
VII-6. The solid of illumination
The illumination at any point in a plane can also be plotted along
perpendiculars to that plane to form a solid figure; when the upper
~
I
I
01'tm,J
qSI'ttn,.l
-<7"tr;,l'tr~
Fig. 44. Solid of illumination for the lighting fitting shown in Fig. 41.
ends of the lines are joined, a curved surface is obtained and this threedimensional figure gives a very good idea of the illumination in the
plane concerned. Fig. 44 shows such a figure in relation to the lighting
fitting illustrated in Fig. 41 and as derived from Fig. 43.
This type of figure is not used for calculations on lighting installations,
but merely provides a pictorial representation of the illumination levels.
Plaster models are sometimes constructed in the same manner for
demonstration purposes.
VII-7. The plane isolux diagram
Plane isohtx diagrams consist of isolux curves drawn in the illuminated
plane.
96
ILLUMINATION CALCULATIONS AND DIAGRAMS
[VII
Clearly, the isolux curves for an axially symmetrical lighting fitting
must be circles (Fig. 45b) .. In such case the isolux diagram is best constructed by derivation from the
rectangular diagram, and the
H=lft(m)
example given in Fig. 45a-b
~=IOOO!m needs no further explanation.
Isolux diagrams of fittings which
a
are nol axially symmetrical can
similarly be derived from the
rectangular diagram, although
it is more usual to compute or
measure the illumination for a
number of points, to record the
values obtained in respect of
those points and interpolate
p0ints of equal illumination.
1,6 Smooth curves are then drawn
10 joining up these final points in
76 the
manner shown in the
25
40 example in Fig. 46 which refers
63 to the fitting in Fig. 41.
700 The same method can· be em160
ployed when the light distriFig. 45. Construction of the plane isolux
diagram (b) for the axially symmetncal bution of the light source is given
fitting shown in Fig. 40, from the illumination in planes cutting a vertical line
curve (a)
through the source; in this case
the points for which the illumination values are calculated lie on lines,
all of which pass through the foot of the vertical line through the source.
VII-8. The polar isolux diagram
This is described in reference to Fig. 47a-b. In Fig. 47a, A denotes the
Fig. 46. Plane isolux diagram for the bilaterally
symmetrical fitting in Fig. 41. Mounting height l ft
(m). Luminous flux of the lamp 1000 lm.
VII-8]
THE POLAR ISOLUX DIAGRAM
97
position of the lighting fitting shown in Fig. 40. The plane of the drawing
represents a vertical plane through the axis of the fitting. We now
have to determine a number of points in this plane where a certain
value of horizontal illumination occurs (say I fc). The manner in which
this takes place will be described later. The points thus
obtained are joined by a
smooth curve which then
represents the )-foot-candle
line. Other lines are drawn
for other illumination values.
Concerning the point P on
the I-fc line it may be said
that the horizontal illumination at this point is I fc;
the location of this point P
is determined by the vertical
and horizontal distances AQ
and PQ, of which AQ is the
height of the source above
the horizontal plane through
P, i.e. the mounting height;
5
PQ is the distance of P from
the vertical through the
source.
Conversely, gi\·en a certain
mounting height, say 5 feet, Fig . .47. a) Polar isolux diagram for the
we might wish to know axially symmetrical fitting in Fig. 40;
curve of the same fitting, as
something about the illumi- b) Illumination
derived from the diagram in Fig. 47a
nation in the plane 5 feet
below the fitting. \\re then draw a horizontal line RS at 5 feet below A.
This line intersects the isolux curves for 6.3, 4, 2.5 and I fc; hence at
these points of intersection the horizontal illumination is equal to the
values of the isolux curves. The distance from these points to the axis
of the fitting can be read from the chart and from this it is possible
to construct a rectangular illumination diagram. Thus, for a height
It = 5 feet we obtain from Fig. 47a the curve shown in Fig. 47b.
The polar isolux diagram is accordingly always a universal diagram,
that is, it can be used for different mounting heights .. Although this
applies in principle also to asymmetric lighting fittings, the use of the
98
ILLUMINATION CALCULATIONS AND DIAGRAMS
[VII
diagram is straightforward only in the case of symmetrical ones and,
in practice, such diagrams are used only for the last mentioned type
of fitting.
The method of calculation for a polar isolux diagram may be explained
with reference to Fig. 48 in which A represents the light source. The
polar light distribution curve is that of
Fig. 40. In a direction that makes an
A
angle oc with the axis of symmetry we
can now calculate the distance r along the
radius vector to a point P at which the
fitting produces a certain horizontal illumination E fc. According to equation
(IV-4) this illumination is
I a.
E =-cos oc,
r2
hence
r=
1/I a. .
f E cos oc.
(VII-2)
Using the light distribution curve of Fig. 40,
let
us now calculate the distance r almig
diagram
the radius vector in respect of oc = 40°,
withE= 2.5, 4, 6.3 and 10 fc. From the figure we find that 1 41p = 198 cd.
In equation (VII-2) therefore, I a.= 198 cd, cos oc =cos 40° = 0.766,
and E is successively 2.5, 4, 6.3 and 10 fc. For E = 2.5 we then have
Fig. 48. Illustrating the calculations for a polar isolux
r=
Similarly,
r-
l/198
X 0.766 ft = 7.8 feet.
2.5
E = 4
fc ; r = 6.2 ft,
E = 6.3 fc ; r = 4.9 ft,
E = 10 fc ; r = 3.9 ft.
The values of r are plotted along the radius vectors. All the points of
equal illumination on the various vectors are then connected, producing
a diagram as shown in fig. 47a. Since we are interested only in the
horizontal and vertical distances, the radius vectors themselves arc
omitted from the final diagram, these being replaced by a system of
horizontal and vertical lines.
VII-9]
THE LUMINOUS FLUX FROM THE PLANE ISOLUX DIAGRAM
99
VII-9. Determination of the luminous ftux from the plane isolux diagram
The plane isolux diagram provides a means of ascertaining the luminous
flux radiated in the direction of a plane or part of a plane. According
to (VI-la):
<P =EX S.
If we measure the area between two isolux curves with a planimeter
and multiply this by the average illumination in that" area, we obtain
the luminous flux incident to that area between the two curves. In
some cases it may be found difficult to determine the average illumination,
but, generally speaking, this can be sufficiently accurately estimated
for all ordinary purposes. If the isolux curves are more or less concentric
it must be remembered that, assuming the distribution of the illumination
to be linear from the centre outwards, the arithmetical mean of the
values of the isolux curves will yield luminous flux values which will
be slightly too high. The reason for this is that the area on the "high"
side of the line representing the average is smaller than that on the
"low" side; accordingly a somewhat lower value than the arithmetical
mean should be employed.
As an example let us turn once more to the isolux diagram in Fig. 46
and assume that we wish to know the luminous flux falling on the roadway
depicted. We accordingly measure the areas between the isolux curves
with the planimeter *), convert these to square feet and multiply the
values by the appropriate (estimated) average illumination; in this
way Table H is obtained.
Since the lighting fitting is bilaterally symmetrical and Fig. 46 only
shows one half of the isolux diagram for the whole roadway, the total
luminous flux intercepted by the road is 2 X 142 = 284 lm. As the
diagram refers to a luminous flux of 1000 lm for the bare lamp, about
28.5% of the flux reaches the roadway.
From the table it is seen that the light falling beyond the 2.5 fc line
is very little indeed and that, although the figure does not go beyond
4 feet, we can safely say that the total flux on the roadway is equal to
that as ascertained for the part shown (and even for a smaller section).
It will be obvious from this example that when the foot-candle is replaced
by the lux and the square foot by the square metre, the values remain
the same.
The fact that as an example for this method of determining the amount of
*) For accurate planimetry the diagram should be drawn on a larger scale than
that employed in the figure in order to save space.
100
[VII
ILLUMINATION CALCULATIONS AND DIAGRAMS
TABLE H
Calculation of luminous flux reaching the road surface in the isolux diagram
Fig. 46
Between isolux
curves
160 fc
and 100
and 63
and 40
and 25
and 16
and 10
and 6.3
and 4
and 2.5
2.5 and 1.6
1.6 and 1
and 0.63
1
0.63 and 0.4
160
100
63
40
25
16
10
6.3
4
Total
I
Area in
sq.ft
Estimated illumination
in fc
Luminous
flux in
lumens
0.13
0.33
0.37
0.36
0.36
0.35
0.34
0.35
0.31
0.27
0.36
0.21
0.19
0.07
175
125
80
52
32
20
13
8.2
5.2
3.1
2.0
1.3
0.8
0.45
22.8
41.3
29.6
18.7
11.5
7.0
4.4
2.9
1.6
0.84
0.72
0.27
0.15
0.03
4.00 sq.ft
I
I - 142 lm
luminous flux the case of an illuminated roadway has been chosen, does
not imply that it is of any great importance in street lighting calculations.
The object of the example is merely to demonstrate the manner in which
the luminous flux may be evaluated from the isolux chart.
Vll-10. The isolux diagram. in quasi central projection
In Section V-8 we learnt about the Dourgnon and Fleury quasi
central projection method of representing the light distribution and
pointed out that this method has particular advantages in the calculation
and plotting of illumination diagrams. Fig. 49 shows the quasi central
projection of the isocandela diagram of the fitting in Fig. 41.
To calculate the illumination at point P' of this diagram, we must multiply the luminous intensity in the direction of P', i.e. 100 cd, by the third
power of the cosine of the angle formed by the direction towards P' and
the Z-axis, i.e. by cos3 b' z, and then divide it by the square of the suspension height. If we standardise the height at 1 metre, we need then only
multiply by cos3 b' z.
VII-10
THE ISOLUX DIAGRAM IN Q.UASI CENTRAL PROJECTION
101
Fig. 49. lsocandela diagram of Fig. 34. For the luminous intensity values their logarithms
are also indicated.
Fig. 50. Diagrams with curves of log cos3 b' z with the same coordinates as m Fig. 49
(only half of the symmetrical diagram is represented).
This multiplication is a simple matter if we draw curves for cos3 b'~n a
diagram with the same coordinates as in Fig. 49 (Fig. 50). If this is drawn
on transparent paper and laid over the diagram in Fig. 49, the appropriate value of the illumination can be calculated by multiplication
at each point of intersection of the curves in Figs. 49 and 50.
102
ILLUMINATION CALCULATIONS AND DIAGRAMS
[VII
_,..
F1g. 51. Isolux diagram in angular coordinates converted from Figs. 49 and 50.
Fig. 52. Isolux diagram in Cartesian coordinates converted from Fig. 51.
The cos3 b' z curves can now. be calculated for round numbers. If the
cos3 b' z curves and the isocandela curves for preferred numbers are
plotted, the products at the points of intersection once more provide
preferred numbers, so that multiplication can be quickly done without
any arithmetic with the aid of a table.
If, the illuminations at the points of intersection of the curves are given
on the transparent diagram in preferred numbers, the isolux curves
required are found by joining the points with the same preferred numbers.
The result of graphical calculation with the help of Figs. 49 and 50 is
given in Fig. 51. This isolux diagram is plotted on angular coordinates.
They can be converted into Cartesian coordinates by a simple process of
calculation. The result of this conversion for Fig. 51 is given in Fig. 52.
VII-10 J
THE ISOLUX DIAGRAM IN Q.UASI CENTRAL PROJECTION
103
REFERENCES
J o h. Jansen: "Beleuchtungstechnik", Philips Technical Library, Eindhoven 1954
J. W. Favie, "Lighting", Philips Technical Library, Eindhoven, 1962.
W. B. Boast: "Illuminating Engineering", 2nd edition. New York 195:l
"Illumination Design for Interiors", Published by ELMA Lighting Service
Bureau, London 1948
H. H. Higbie: "Lighting Calculations", New York 1934
2 ) W.
Harrison and E. A. Andersen, Trans. I.E.S. Am. 11, 1916,
67-91, "Illumination efficiencies as determined in an experimental room"
and 15, 1920, 97-123, ~·coefficients of Utilization"
H. Z ij 1, Ph. T. R. 7, 1942, 97-103. "Efficiencies of Lighting Installations".
H. Z ij 1 : "Manual for the Illuminating Engineer on Large Size Perfect Diffusors". Philips Technical Library, Eindhoven, 1951
W. B. Boast: "Illuminating Engineering", 2nd edition p. 182 et. seq.
New York 1953
"Illumination Design for Interiors" p. 81 et. seq. (see under 1 )
"I.E.S. Lighting Handbook", 4th edttion, section 9, New York 1966.
3 ) J. van H u 1 sen, Electrotechniek 19, 1941, 130-131. .,Een weinig bekende
methode voor het aangeven van verlichtingssterkten" ("A little-known Method
for the Indication of Illumination Values")
4) j. t5ourgnon and D. Fleury, Lux 28, 1960, 53-68. "Diagrarnme umversel pour 1a
representation des repartitions lumineuses des sources dissymetriques".
1)
CHAPTER VIII
LUMINANCE AND LUMINOUS EMITTANCE
VII I -I. Luminance; the stilb, candelas per sq.in.
When two light sources of the same luminous intensity, one of which
has a larger area than the other, are regarded successively, the smaller
appears to the eye to be brighter than the other.
In the smaller of two such light sources the luminous intensity per unit
area is higher than in the larger and we accordingly say that the
luminance of the one is higher than that of the other.
Luminance (symbol L) is defined as the luminous intensity radiated
per unit area:
I
L
- -s.
(VIII-I)
\\"ith I in candelas and S in sq. em, I is expressed as cdjsq. em, or in
stilbs (sb). Hence I stilb is I cdjsq. em.
The stilb is the internationally standardized unit of luminance; in
English-speaking countries the candela per sq. in. (cdjsq. in.), or the
cdjsq.ft is employed. The luminance is also expressed in cdjm2. This unit
is known under the name nit (symbol nt). The choice of unit usually depends on which of the two is the more convenient under given conditions
In formulae this alternative is not always open, but may depend on
the units employed for the other quantities.
The term luminance was introduced by the C.I.E. in 1951 to take the
place of brightness (Symbol JJ) which, used exclusively in older literature
and to a certain extent in modern publications, is synonymous with
luminance, although no longer the universal term.
Generally speaking, the luminance of an area is not uniform, so use
is made of the average luminance, this being the quotient of the total
It.iminous intensity divided by the total area. In many cases this average
luminance can be used in calculations without the need for knowing
the distribution of the luminance throughout the area.
VIII-I]
LUMINANCE; THE STILB, CANDELAS PER SQ.. IN.
105
An example of this will be found in the filaments of projector and
cinema lamps which usually comprise a number of parallel coil sections.
The luminance of the open spaces between the coils is zero, but for the
purposes of calculations on optical systems the luminance of the area
enclosing the entire filament is the important factor; this is the average
luminance or, in short, the luminance of the filament.
Example:
The luminous intensity of an area 8 mm in width and 10 nun in length
is 800 cd. The (average) luminance is then
I
L = 5
800
= 8 X 10 cdfmm 2 =
1000 sb (cdfsq.cm) =
J07 cdfrn2,
VIII-2. Lambert's law
In the same way that the luminous intensity of a light source is stated
with respect to the direction of radiation, it is
s
also necessary to indicate the direction from which
luminance is observed. When the direction of
observation is perpendicular to the luminous
surface, and provided that this surface is plane,
the actual area must of course be taken when
use is made of formula (VIII-I) (see Fig. 52).
If the direction concerned is at an angle ex from
Fig. 52. If the luthe normal to the surface of the light source, the
minous intensity perpendicular to a unisurface appears to ce smaller, to the extent of
formly diffusing sur5 cos ex; the luminance at angle ex is then
face S be denoted
by I., the intensity
I rx at an angle rx from
the normal to that
surface is I 0 cos rx
(L a m b e r t's law)
L =
a
Irx
5 cos ex
(VIII-2)
and the luminous intensity is
I rx =--= Lrx . 5 . cos ex.
(VIII-2a)
The more precise definition of luminance is thus the quotient of the luminous intensity divided by the apparent surface of the light source. Cnder
this definition it is not necessary for the source to be a plane source;
it may be of any cross-section.
When the luminance of a source is the same in all directions, L is independent of ex, the index may ce omitted and equation (VIII-2a)
then reads:
I rx = L . 5 . cos ex,
106
LUMINANCE AND LUMINOUS EMITTANCE
[VIII
where L. S = I 0 (the luminous intensity perpendicular to the plane,
1.e. direction of radiation 0°). Hence we may also write:
I= Io cos IX
(VIII-3)
which is the mathematical expression of Lambert's law.
This law states that if a surface has the same luminance in all directions,
its luminous intensity in a given direction is equal to the luminous intensity
perpendicular to the plane (Io), multiplied by the cosine of the angle between
that direction and the normal to the plane.
L a m b e r t arrived at this law by observing the luminance of the
sun's disc. This he found to be uniform over the whole surface, although
the surface at the periphery is perpendicular to the centre part. From
this uniformity of luminance at the. periphery and centre L a m b e r t
concluded that the luminous intensity is proportional to the apparent
area as given by the product of the actual area and the cosine of the
angle of radiation.
This law, which we nowadays have little difficulty in understanding,
was not regarded as being quite so apparent in Lambert's day,
as illustrated by the fact that, according to so sagacious a man as E u I e r,
luminous intensity was held to be independent of the position of the
radiating surface and, hence, that the sun as a sphere produces just
as much illumination on the earth as it would do if the hemisphere
which we see were flattened out.
There are no light sources which conform wholly to Lambert's
law; such hypothetical sources do, however, provide a very usefu1
basis for theoretical arguments, and we shall make frequent use ot
them in this book. Light sources of this kind are called uniformly diffuse
sources or uniform diffusers.
Practical light sources radiating approximately in accordance with
Lambert's la~ are called diffuse radiating or, briefly, diffuse light
sources.
VIII-3. The luminous flux of uniform diffusers
We have seen in the previous section that when a surface radiates in
accordance with Lambert's law and has, therefore, the same' luminance
in all directions, the luminous intensity I a. at an angle IX from the
normal, is given by equation (VIII-3):
I a.= 10 cos ot.
By means of this formula we can now construct the light distribution
curve of a uniform diffuser, which is found to be a circle, touching
the radiating surface (see Fig. 53a).
VIII-3]
THE LUMINOUS FLUX OF UNIFORM DIFFUSERS
107
From the right-angled £::.01)1Xin Fig.53a it will be seen that IIX = Io cos ex.
If we plot the luminous intensity
in
a Rousseau diagram in order
.------,tad'
1-----1/Sd'
to be able to calculate the luminous flux from the light distri1-----11200
bution curve (Fig. 53b), we find
that the line joining the extremities of the intensity lines
is straight; this may be verified in the following manner.
Draw a line from the end of the
line representing I IX to P, and
let e be the angle between this
Fig. 53. Light distribution (a) and line and the Y -axis; then:
Rousseau diagram (b) of a uniformly
diffusing surface
IIX
I 0 cosex
I0
tan€J=--=--=-.; -.
r cos ex
r cos ex
r
Since I 0 fr is constant, the value of e is the same for all lines joining
the ends of each of the I IX lines to P; in other words all these lines coincide, or, there is only one (straight) line that can be drawn through
the ends of the I IX lines.
The area enclosed by this line can be calculated as being that of the
right-angled triangle of height r and base I 0 •
This area is
!ri0 •
In accordance with equation (IV-2), this area must be multiplied by
2nfr to give the value of the luminous flux. Hence
2n
(/) = -
r
X !ri0 = ni0 •
(VIII-4)
In the case of uniform diffusers, the value of the radiated luminous flux
is n times the luminous intensity perpendicular to the surface.
Conversely
(VIII-4a)
The luminous flux ((/}IX) between the angles of emission 0 and ex corresponds
to the area enclosed by the lines I 0 and I IX in the Rousseau diagram
108
[VIII
LUMINANCE AND LUMINOUS EMITTANCE
This area is a trapezoid, the parallel sides of which are I 0 and I IX and
whose height is equal to r (I -cos tX). Hence this area is
!r (I -cos tX) (I0 +IIX) = !r ( I - cos tX) (I 0 + I 0 cos tX) =
= !r ( I - cos tX) I 0 (I +cos tX) =!rio (I--- cos2tX) = !ri0 sin 2tX.
In accordance with section IV-I we now obtain the luminous flux
by multiplying by 2nfr the area as computed above:
2
I . 2
</J I X- - -2n
r X 21 ri o sin tX- n o sm tX •
(VIII-5)
Equations (VIII-4) and (VIII-5) can also be derived from the integral
of equation (IV-4) which states that
IX,
.P = 27T
JI sin diX.
IX
IX
IX,
In the case m point I IX = I. cos IX, so that
.P = 27T
IX,
JI cos sin diX.
0
IX
IX
Solution gives us:
IX,
tP = 1TI0 sin 2 IX J·
7T
If 1X 1 = 0 and IX 2 = 9 , we
tP = 1Tl0 sin 2 IX.
obtain
.P = 7Tl0 ,
if
IX 1
=
0 and
IX 2
=
IX,
-
Examples:
l. A surface radiating in accordance with Lambert's law and having
a luminous intensity of I 0 = 60 cd in the direction of the normal,
will have a luminous flux of
tP = 7TI0 = 7T X 60-lm = 188.5 lm.
2. If the luminous flux of such a surface is 445 lm, the luminous intensity I 0 in the direction of the normal will be
tP
445
I 0 = - = - cd = 141.5 cd.
7T
1T
VIII-4. Luminous flux of a uniformly diffuse cylinder
The light distribution of a luminous cylinder, e.g. a wire, whose luminance
is the same in all directions, can be expressed by means of a simple
formula if we may assume that the diameter of the cylinder is very
small in comparison with the length of the cylinder; here, the axis of
symmetry coincides with the longitudinal axis of the cylinder.
Fig. 54a depicts the light distribution of a cylinder of this kind. The
luminous intensity perpendicular to the wire is denoted by I 0 • At angle
VIII-4]
109
UNIFORMLY DIFFUSE CYLINDER
oc from the normal to the wire, or at an angle of 90° -oc from the wire
itself, the luminous intensity is (VIII-3)
I ex= Io cos oc.
(VIII-6)
In this we are departing from the usual practice of indicating directions
of radiation by the angles which these make with the axis of symmetry;
instead, we state the angle from the normal to the uniform diffuser.
A glance at Fig. 54a at once
r-=-----.;:g~
shows that the corresponding
light distribution curve is a
1--------',-------1+30°
circle touching the axis of
symmetry at 0. The solid of
tk----.-+t--JOa
light distribution is toroidal
.
~
in form.
_
-:30_o_~A l--=-~---tL___-----l-30° !~x d~~:rm~:~ue~he0/u~in~~:
1----::7""------J-600
~
plotted to give the Rousseau
diagram shown in Fig. 54h; for
convenience a luminous inFig. 54. Light distribution (a) and Rousseau
tensity scale is used such that
diagram (b) of a uniformly diffusing cylinder
Io = r.
The Rousseau curve is thus found to be a semi-circle and this is easily
proved in. the following manner.
In the right-angled !:c. PAB:
'-=-----'-90°
PB 2 = PA2
+ AB
2•
Here, PA = r sin oc and AB = I ex = ! 0 cos oc; hence PB 2 = r 2 sin 2 oc +
+ ! 0 2 cos 2oc.
Since r = ! 0 we may write:
PB 2 . r 2 sin 2oc + r 2 cos 2oc = r 2 , and PB = r.
The line joining P to the end of each line representing lex is thus equal
tor and the curve will be a semi-circle of radius r. The area of this semicircle is i:nr 2 and this, multiplied by 2nfr gives a luminous flux of
2n
C/J = - X !nr 2 = n 2r.
r
Since we have said that ! 0 = r, we may also put:
(/J =
or
n2Jo
(/J
Io= -.
n2
(VIII-7)
(VIII-7a)
110
LUMINANCE AND LUMINOUS EMITTANCE
[VIII
Provided that the distance at which 10 is used is great enough, this
formula may also be employed for illumination calculations as applied
to a number of luminous cylinders parallel to each other, but naturally
only when such cylinders are sufficiently widely spaced to ensure that
they will not mask any appreciable portion of each other's light.
A practical example of this situation is found in a straight filament
vacuum lamp in which a number of thin filament sections are mounted
almost parallel to each other. The light distribution curve of such lamps
conforms sufficiently well to equation (VIII-6).
In this 10 is the horizontal luminous intensity 1h of the lamp, and the
luminous flux of such lamps can be calculated quite accurately from
(/) = n21h.
The spherical luminous intensity is
(/)
1o=-
4n
and the ratio of spherical to horizontal luminous intensity is therefore
(/)
1o
I,,
4n
n
cp
4"
-n2
This is the proof of what has already been said in section III-4.
Equation (\'III-7): <P = TT 2 l 0 can also be obtained from the integral
of equation (IV-5) which states that
a,
<P = 2'" I a sin IX da.
a,
J
It must be remembered that for this formula a is measured from the
axis of symmetry, whereas, in the case of the luminous cylinder, a is
measured from the normal to the cylinder; equation (IV -5) must therefore be adapted to this method of indication of the angle.
To avoid confusion, let us refer to angle f3 in (IV -5} instead of angle a.
Then
{3,
<P = 2'" j I f3 sin f3 df3,
{3,
(IV-5a)
in which, in the present instance:
I f3 = I o cos a, f3 = 90'~- a, with a positive in an upward direction
and negative in the downward direction; thus sin f3 = cos a and df3 = da
Equation (I\'-5a)· then becomes
<P = 2'"
JI. cos sin (90°- drx = 2'" JI. cos
~2
IX,
~2
IX
IX}
2 IX drx.
VIli-S]
111
LUMINANCE EMITTANCE
For the total luminous flux we have to integrate from
1X 1 = - 90° = -
i and 2
then
IX
+90° = +
=
i:
+:::2
<P = 211
JI. cos2
IX'
diX,
7T
2
the solution of which is:
<P = 11!0 {IX +
+:::2
t sin 21X) / = 11 !
2
0•
7T
2
Examples:
l. A straight filament lamp of 25 cd (i.e. horizontal luminous intensity
I 11 = 25 cd) gives an approximate luminous flux of
cp = 7T2J 11 = 11 2 x 251m= 247 lm.
2. A similar lamp, having a measured luminous flux of 155 lm will
have a horizontal luminous intensity of
<P
155
~
! 11 = 2 = - 2 cd = 15.7o cd.
7T
7T
VIII-5. Luminous emittance
The luminous flux emitted or transmitted by, or incident to a surface
per unit area is known as the luminous emittance or, briefly, the emittance
of that surface (Symbol H).*)
We have already seen that the luminous flux per unit area incident
to a surface is termed the illumination; illumination, therefore, is another
term for the emittance of an illuminated surface.
Mathematically, luminous emittance can be defined along the same
lines as illumination:
(/>
(VIII-8)
H=s·
(H = ~;)
(VIII-Sa)
The considerations mentioned in section VI-I concerning average
illumination as well as the illumination of a point in a plane apply
equally to the more general conception of emittance and need not be
repeated here.
Emittance is evaluated by means of equation (VIII-8) in lmfm 2 , lmfcm 2
or lmfsq. ft; it is only in the case of the emittance of an illuminated surface
that these units have been given a specific name, viz. lux, phot, footcandle.
*} The quantity luminous emittance was formerly called radiance or radiancy. In
1951 the C.l.E. recommended the discontinuance of the use of the term radiancy
and to use in the future the term luminous emittance or emittance.
112
LUMINANCE AND LUMINOUS EMITTANCE
[VIII
VIII-6. Emittance and luminance of uniformly diffusing surfaces
A surface of area 5, radiating in accordance with Lambert's law
with a luminance L will have a luminous intensity in the direction
of the normal of I o = L . S candelas.
According to equation (VIII-4) the luminous flux of that area would be
$
= nl = nL . S.
0
(/J
Hence
s=nL,
(/J
\\'here S is the luminous flux per unit of area, that is, the emittance H.
\Ve may therefore write:
H=nL,
(VIII-9)
H
L=-.
(VIII-9a)
n
This means that the luminous emittance of a uniformly diffusing surface
is found from the luminance by multiplying it by n. Conversely, the
luminance is the quotient of the emittance divided by n.
When applying equations (VIII-9) and (VIII-9a) we have to take
due care of the units employed. If L is expressed in sb (cdfcm 2 ), H will
be in lmfcm 2 • L expressed in cdfm2 gives H in lmfm 2 , and, if expressed in
cdfsq. ft, lmfsq. ft.
.
Examples:
l. A surface radiating
in accordance with Lambert's law, with a
luminance of 16 sb will have an emittance of H = 1r X 16 = 50.2 1mjcm2 •
2. Should the emittance of such a surface be 700 lmfm 2 , the luminance
700
of that surface would be L = = 223 cdjm 2 = 0.0223 sb.
1T
VIII-7. Units ofluminance, based on the emittance of uniform diffusers
From the above remarks it follows that the luminance of a uniform
diffuser is fully determined once the emittance is known. A luminance
unit can therefore be established as being the luminance of a surface
which, radiating in accordance with Lambert's law, has a certain
emittance.
Units of luminance are derived from three emittance units, viz:
from 1 lmfm 2 :
from I lmfcm 2 :
from 1 lmfsq. ft:
the. apostilb (asb),
the lambert (L or la),
the foot-lambert (ftla).
VIII-7]
113
UNITS OF LUMINANCE
The apostilb, the lambert and the foot-lambert can thus be defined
as the luminance of a surface radiating in accordance with Lambert's
law and having respectively an emittance of I lmfm 2, I lmfcm 2 and
I lmfsq. ft.
What is the relationship between these and the stilb?
H
According to equation (VIII-9a): L = -.
;'1;
Substituting su,ccessively I lmfm 2, I lmfcm 2 and I lmfsq. ft, we then
have
I
~-- sb = 3.183 X I0- 5 sb,
1 asb = - cdfm2
;'1;
n X 104
1 la
1
= - cdfcm 2
;'1;
1
1 ftla = - cdfsq. ft
;r
1
= - sb =
;r
3.28 2
0.3183 sb,
~~cc-: sb
;r
X 10 4
= 3.426 x I0-4 sb.
If L be expressed in these units and H in the corresponding units lmfm 2 ,
lmfcm 2 and lmfsq. ft, the luminance is numerically equal to the emittance
so we write:
L=H
(e.g. L in cdfsq. ft and H in foot-lamberts).
In Germany the apostilb is used as the unit of luminance, but the C.I.E.
does not accept this and its use is therefore not recommended.
The lambert is employed in the United States; the millilambert (mL or mla)
is also used (1 L = 1000 mL).
In Great Britain the foot-lambert is the more current unit, other expressions for the same unit being the equivalent or apparent foot-candle. For
the origi_n of these terms see section X -6 (p. 15 7).
On p. 413 the current luminance units are listed in Table V together
with their conversion factors.
In Fig. 228 on p. 412 the various units in order of their size are shown
on a logarithmic scale; the different ratios are indicated alongside the
arrows which show the relative positions of the units in the system.
Example:
If a luminance of 25 mL is mentioned in a publication, this luminance
is (see Table V or Fig. 228).
25 X 3.183 X I0-4 sb = 7.96 X I0- 3 sb
or 25 X 3.183 cd(m2 = 79.6 cd(m2
(or 25 X 10 asb = 250 asb).
114
LUMINANCE AND LUMINOUS EMITTANCE
[VIII
The association of the luminance units discussed above with the
luminance of a surface radiating in accordance with Lambert's law
does not imply that only the luminance of surfaces radiating in this
manner, or nearly enough in this manner, may be expressed in the
units given.
The luminance of any radiating surface in any direction can be expressed in these units, proceeding from the following argument. If the
luminance of a given radiating surface in a certain direction is known,
say n lamberts, this means that the luminance of that surface in that
direction is just the same as that of a surface radiating in accordance
with Lambert's law and having a luminance of n lamberts, i. e. with
an emittance of n lmfcm 2 •
CHAPTER IX
NON-POINT SOURCES
IX-I. Luminous intensity of non-point sources
The formulae developed in the preceding chapters all refer to light
sources which can be regarded as a point constituting the apex of a solid
angle*). The derivations have each time been based on the cone-shaped
propagation of the luminous radiation, i.e. on the inverse square law.
The definition of luminous intensity is also founded on the conical
propagation of luminous flux from a point. In Ch. III (section III-I)
this is defined as the flux per unit solid angle, the light source being
considered to be a point source.
In practice, however, all light sources have finite dimensions. When
dealing with sources of very small dimensions we can at once accept
the approximation to a point source, but it is not quite so simple when
the definition recalled above is applied to large sources such as searchlights or tubular fluorescent lamps with their appropriate lighting
fittings.
As always, the terms large and small are only relative conceptions and
it is necessary to indicate the standard against which light sources
can be considered large or small. In our case we have to apply these
terms in reference to the distance from the source to the observer, or
to the illuminated surface. Such relativity will often be intuitive; we
have only to think of the sun or the stars, which we so readily accept
as point sources, i.e. small sources, whereas they are, in fact, of very
large dimensions compared with artificial light sources.
In the case of large sources such as the fluorescent lamps already
mentioned, it is difficult to look upon these as being point sources in the
sense implied in the definition of luminous intensity given in Ch. III,
viz. as representing the apex of a solid angle. The question then arises
whether this definition is acceptable for such light sources.
*) We have thereby strictly assumed, and also treated, the light source as a
mathematical point (which has no area), not as a physical point (a plane element
having an infinitely small area). In our considerations we should of course justify
this inaccuracy, but it will be seen upon closer investigation of light sources of
finite dimensions that the discrepancy has no practical significance.
116
[IX
NON-POINT SOURCES
The answer to this can be given in reference to Fig. 55, showing a light
source LL' which can be imagined as
disc- or line-shaped. It is assumed that
LL' radiates uniformly diffusely with
a luminance L. Let us denote the
area of the light source by S; the
luminous intensity perpendicular to
L'
the surface of the source will then
Fig. 55. Demonstrating the fact
be according to the considerations given
that the inverse square law does
not apply when the light source
in Ch. VIII:
is large compared with its dist10 =LX S.
ance from the illuminated point
Now, is it permissible to employ this
same 10 when applying the inverse square law to compute the illumination at the point P in Fig. 55? A glance at the figure will show whether
this is permissible or not. A surface element at L, of a size LIS, will
have a luminous intensity at right angles to the source equal to:
LII0 = L. LIS. But, in the direction of P, the luminous intensity is only
LI/IX =Lifo COS at= L. LIS COS at.
Hence the illumination produced at P by LIS is
L1JIX
L1/o
(.
d )
E p = - - cos APL = - - cos4 oc since LP=-- .
LP 2
d2
cos oc
That part of the illumination at P which is contributed by the surface
element LIS at L is therefore really smaller by a factor of cos4 oc than
the value that we should iQclude when applying the inverse square law,
using ! 0 and d. The closer the elements of LL' approach A, the smaller
this error becomes, and it follows, therefore, that the error in the total
illumination at P will be smaller according as the angle at is reduced,
that is as the distance d becomes greater in comparison with LL'.
It will be seen, therefore, that for computations of illumination values
for distances which are not great compared with the area of the light
source, the conception of luminous intensity as defined in an earlier
chapter, in conjunction with the inverse square law, no longer holds.
In our further discussions it will be found that this is not such a very
great drawback, but it is nevertheless necessary to determine just how
far the inverse square law is valid in practice.
It is also possible to demonstrate along other lines the peculiarity of
the conception of a point source assumed in the inverse square law and
the definition of luminous intensity as applied to large light sources.
Using the inverse square law as based on a cone-shaped propagation
IX-1]
LUMINOUS INTENSITY OF NON-POINT SOURCES
117
of luminous flux we can compute the illumination at a point, i.e. a
surface element at a certain distance from the light source. Such a
surface element should be regarded as the base of a cone with the source
at its apex. Actually the "apex" of the "cone" may be a light source
possibly several feet across, the "base" being infinitely small. The
conception and the representation of a light source as being the apex
of a solid angle for the purpose of defining luminous intensity, is thus
forced. Fortunately, however, there is a way out of the difficulty, by
taking into account the manner in which the luminous intensity is
measured; this is never actually carried out by measuring the luminous
flux and the solid angle within which it is radiated, and then dividing
the one by the other according to the definition.
All luminous intensity measurements are based on measurement of the
illumination, the inverse square law then being applied to compute
the luminous intensity, by multiplying the illumination by the square
of the distance from the point of measurement to the source, i.e. I = Ed 2 .
We then employ the value of I thus obtained to compute E for other
distances, e.g. £ 1 = Ijd 1 2 •
If the result is to be accurate we must write Ed 2 = E 1 d1 2 , in other
words the product of Ed 2 must be constant; but, if some error is permissible, Ed 2 must be constant within certain limits. The point is, when
is Ed 2 constant?
From Fig. 55 we have seen that the error entailed by the use of the
inverse square law is smaller according as the angle ot is made smaller,
i.e. as d is increased with respect to the light source LL'; this means
that the product Ed 2 will approximate more closely to a constant value
when d is increased with respect to LL'.
This is demonstrated in Fig. 56 which shows the product Ed 2 plotted
as a function of d (full line curve A) for a white diffuse lighting fitting
(depicted in Fig. 40), containing a 200 W incandescent lamp. Ed 2 is
thus obtained from the illumination as measured on the axis of the
fitting, and the distances. It is seen that when d R::i 2 m, the product
Ed 2 reaches the constant value 690; this is obtained with a ratio of
lighting fitting diameter to d of about 1 : 5. Thus, when applying the
inverse square law to this fitting we may use for I the value of Ed 2
(expressed in candelas), as measured at d >2m, whilst, for the same
distances, the inverse square law can .be employed to calculate the
illumination values. If an error of at most 5% is permissible in a computation of the illumination, Fig. 56 shows that d may then be
2 0.85 m.
118
[IX
NON-POINT SOURCES
The lighting fitting, of which the Ed 2 (d) curve is reproduced in Fig. 56,
consists of a reflector
cd
~00,.---·
containing an incan@Q_ - - - A
descent lamp whose
~
/
7
filament is an inch
-8 f-·
/
or two above the
./'"
5 ""'
··plane through the
/
/
/
rim of the reflector.
625
The light-emitting
I
I
I
parts of the reflector
600
0
2
4m
3--d
and lamp are there2
Fig. 56. Curves showing the product Ed as a
fore displaced in the
function of the distance d. The two curves A and B
direction of the axis
refer to distances measured from different points of
to the extent of some
the light source
inches in respect of each other, and this gives rise to the question
as to what part of the reflector should be taken for the measurement
of d. The solution lies in a comparison of the dotted curve (B) in Fig. 5&
with the full curve (A). Curve A refers to a distance d measured from
a point 4 em from the plane of the reflector rim; for curve B, d was
measured from the actual plane of the rim. The illumination measured
at every point in curve A is thus multiplied by the square of a "d"
which is 4 em larger than in curve B. At every distance, therefore, the
values of Ed 2 are lower in B than in A, as will be seen from the diagram.
When d = 4 m the difference is about 2%, corresponding to a difference
in d of I% (400 and 404 em).
It can also be seen from the. diagram that, when d = 4 m, curve B con-·
tinues to rise and Ed 2 has not by then become constant, whereas curve A
is horizontal from d = 2 m, and Ed2 is already contant from that distance.
The greater the distance d, the closer will B approach A until, when d
is infinitely great, the curves coincide and assume the same value of Ed2.
If d were measured from a plane more than 4 em within the reflector,
the calculated curve would lie above A and, after first yielding higher
values than 690 for Ed 2, would drop gradually to reach that value at
a certain distance.
The point from which d is measured should thus be the one that gives
the shortest distance at which Ed 2 becomes constant.
As a rule the location of this point is not of much practical importance,
but it is, nevertheless, useful to realise the effect of the particular choice.
The fact that the practical importance is not so great may be demonstrated in reference to Fig. 56. If E be measured at d = 4 m from the
I
I
v
--
IX-1]
LUMINOUS INTENSITY OF NON-POINT SOURCES
119
plane of the reflector rim, curve B will yield a luminous intensity of
I= Ed 2 = 675 cd. Using this intensity to compute the illumination
at 2 m, we obtain a value that is I.5% too high (we should, of course,
have taken the value 664 from curve B, for d = 2 m). When E is computed with d > 4 m, the resultant value is slightly too low, since with
this distance a higher value of Ed 2 should really be employed. At the
very most the error will be 2%, that is, the difference between 675 and
690 cd; an error of this order of size can generally be accepted in lighting
engineering.
When smaller distances d are taken for tl)e measurement, the error
is increased, and it is important, therefore, that this distance should
not be too small.
The tendency of Ed 2 to approach a fixed value when the distance d
is increased may now be utilised to formulate a definition of luminous
intensity which will apply to all light sources, whatever _their dimensions. This must be expressed in a mathematical manner. We can do
this by saying that, whem d is increased, it will approach infinity, and
at the same time the product Ed 2 acquires a limiting value. The definition
would thus read: The luminous intensity of a light source is the limiting
value of the product Ed 2 with an infinitely high value of d; as a formula
this will take the form
I= lim Ed 2 •
d-+ 00
(IX-I)
As shown in the above example (Fig. 56), the expression infinitely long
may be interpreted as long compared with the light source. In practice
it is often sufficient to accept a value which differs from the theoretical
limit, depending on the degree of accuracy required.
Our new definition of luminous intensity is of the greatest utility when
applied to "large" light sources, generally known as non-point sources,
as distinct from point sources (for which a better term might be quasi
point sources).
In equation (IX-I) we may write for E:
E = $fS
where (/J is the luminous flux incident on an area of size 5. Hence
d2
1 = lim (/J .5 .
d-~ 00
120
[IX
NON-POINT SOURCES
Now, ~ is a solid angle (w), of which the apex is at the light source,
d2
so we may put:
I= lim~.
d--'?-00 w
(IX-la)
The luminous intensity of a light source can accordingly be further
defined as the limit of the luminous flux per unit solid angle to which
that quantity approximates with increasing distance from the source, the
apex of the solid angle coinciding with an arbitrary point in the light source.
Equation (IX-la) can be more suitably written in the form of a differential, viz:
L1 tJ>
dt;l>
d-+00 L1Sfd2 .d(l)--+0 Lfw
dw
.
..dt/>
.
I= hm - - = hm-- = --.
(IX-lb)
Since the definition of luminance is bound up with the luminous intensity, this definition should also be revised to agree with equation
(IX-1). Instead of (VIII-2) we therefore write:
La. =
I
a.
= lim
Ed2
s cos Ot d--'?-00 s cos Ot
.
(IX-2)
For the remainder of this chapter we shall concern ourselves with the
derivation of formulae for the illumination produced by non-point
sources that will be valid for any distance fFom source to illuminated
area. In this we shall confine ourselves to uniformly diffuse circular
and linear light sources and, having derived the formulae, consider
under which conditions the inverse square law can be used. We shall
then also discuss in how far these considerations are of value when
applied to sources which are not perfectly uniformly diffuse.
To arrive at our formulae we shall make use of the inverse square law
when the light source is very small compared with the distance and,
to this end, non-point sources will be regarded as consisting of a large
number of point sources.
IX-2. The inverse square law in another form
If we are going to use the inverse square law E = I fd 2 for diffuse nonpoint sources, we may substitute L . S for I, where L is the luminance
and S the area of the light source; thus
s
E =LX d2 •
IX-3]
UNIVERSAL FORMULA FOR THE ILLUMINATION
121
In this, Sfd2 is the solid angle w under which the source is observed
from a distance d, assuming d great comL
pared with S; hence
0
(IX-3)
E = Lw.
This, then, is the inverse square law in
another form, which leads us to an important conclusion. Let us consider Fig. 57.
This shows a plane L radiating uniformly
diffusely with a luminance L. At a distance
Fig. 57. The deciding elements in calculation of the from L there is an opaque screen D having
illumination at the point P
in it an aperture S. Suppose that we wish
are the aperture of the
diaphragm D, the distance to know the illumination at P, at a distance
of the diaphragm from P
d from D.
and the luminance of the
equation (IX-3) is used for this purpose:
If
light-emitting surface L behind it, but not the distance
from L to P
E = Lw ( = L X ; 2).
It will be seen that, apart from the luminance L, the illumination depends
only on the aperture in D and the distance from D to P. The distance
between L and D is quite immaterial, provided only that L is so large
that, from the point of view of P, the aperture will be completely "filled"
by the luminance of L (the aperture must be completely "flashed").
The location of the limiting element in the space is quite unimportant;
the only factor that determines the illumination is the solid angle
subtended by the flashed aperture.
With diffuse sources the illumination is unaffected even by the form
in depth of the light-emitting surface; hence the illumination at P
from the three sources depicted in Fig. 58
L
is the same in each case provided that
the luminance is the same.
It may be superfluous to add once more
that the equation E = Lw can only be
used when the inverse square law is ap- L'
Fig. 58. The shape in the
plicable. The conditions under which this
depth of the light source LL'
does not affect the illuis permissable will be investigated in the
mination at the point P as
next section.
long as the apparent size is
the same
IX-3. Universal formula for the illuinination produced by uniformly diffuse circular light sources
For the construction of this formula we shall make use of the law of
122
[IX
NON-POINT SOURCES
reciprocity, the derivation of which is as follows. Suppose the luminance
of two diffuse sources S1 and S2 of any given shape (Fig. 59) to be L 1
and La respectively. S1 then receives a certain luminous flux
d
from S2 and so also does Sa from S1 •
We now compute the luminous
10409
flux received by one from the
Fig. 59. The law of reciprocity
other, for which purpose we first
determine the amount of luminous flux which: an element Lf5a of the
surface Sa receives from a similar element Lf51 in S1 .
Let us denote the distance between these elements by d, and the angles
between the normals to the elements and d by oc1 and oca respectively.
The luminous intensity of .151 in the direction of L15a is then
L 1 • Lf51 cos oc1 ;
the illumination of L15a will be
L 1 • Lf51 • cos oc1 • cos oca
da
and the luminous flux received by Lf5 2 from Lf51
_ L 1 • .151 • .152 • cos oc1 • cos oc2
d2
•
tP
.ds, _.. .ds. -
Similarly, the luminous flux received by .151 from Lf52 is found to be
tP.ds•.- .ds, =
L 2 • .151 • Lf5 2 • cos oc1 • cos !Xz
d2
.
Division then gives
tP.ds,- .ds.
L1
tP.ds.- .ds, = L2 ·
The two luminous flux values are thus proportional to the luminance
values of the radiating surfaces.
As the two sources S1 and S2 can be imagined as composed of a large
number of surface elements of which each and every combination possesses the property just mentioned, this rule applies to all surfaces;
it can be expressed as
tPs,-+- s.
EtP.ds,-+ .ds.
Ll
tPs.-+-s, = EtP.ds,-+.ds, = L2.
IX-3]
UNIVERSAL FORMULA FOR THE ILLUMINATION
123
This, then, is the equation for the law of reciprocity, which states that
if any two diffuse light sources illuminate each other, they deliver luminous
fluxes to each other in the same proportion as their luminance (or emittance)
values.
When the luminance (or emittance) values are the same, the luminous
flux values are also the same.
Using this law of reciprocity we can now very easily derive a formula
for the illumination produced by uniformly
A
diffuse circular light sources.
In Fig. 60 AB represents a light source of this
kind, emitting light with a luminance L. Let us
now compute the illumination perpendicular
to CP at the point P; L APC =e. The point P
must once more be regarded as a surface element,
80410
B
the size of which we shall denote by L1 S. Now,
Fig. 60. Illustrating the AB emits just as much luminous flux towards
derivation of the forL1S as L1S would emit towards AB if L1S were
mula E = 1rL sin1 8
of the same luminance, viz. L.
The last mentioned luminous flux can be very easily computed, for
the luminous flux in the solid angle of half-apex angle e, produced
by a diffusely radiating surface L1S of luminance L is (VIII-5):
fP = nL . L1S . sinz e.
This is therefore also the luminous flux radiated to L1S by AB. The
illumination of L1S is now obtained as the quotient of the incident
luminous flux divided by the area L1S; thus
E = nL sin 2 e.
(IX-4)
. e = AC
In Fig. 60, sm
AP' so t h at we may put f or (X
I -4):
AC 2
AP2
E=nL - -
where n AC 2 is the area of AB; hence n L . AC 2 is the luminous intensity I
of AB according to the definition given in section IX-I. If we now
denote AP by a, we can write in place of (IX-4):
I
az
E=-.
This is another inverse square law, which differs from that relating
124
NON-POINT SOURCES
[IX
to point sources in so far that the distance is not measured perpendicular to the light source, but along a line drawn to the periphery of
the circular source.
If the source is small compared with the distance, the difference between
the perpendicular and oblique distances will also be small, and the
former distance may be used in the formula, in which case E = Ifa 2
becomes E = lfd 2 • The size of the error thereby introduced is discussed in section IX-6.
IX-4. Alternative derivations of the formula E = nL sin2 e
We shall now describe another way of deriving the formula: E = nL sin2€J.
Our object in so doing is that this method makes use of certain very
interesting introductory considerations.
Let us consider a sphere of radius R (see
Fig. 61). It will be assumed that the sphere
transmits light without loss. In the surface
of this sphere, at A, there is a very small
light-emitting area L1S radiating according
to Lambert's law, with luminance L.
We will now calculate the illumination on
the surface of the sphere at any given
point, say P.
As L1S is assumed to be quite small compared with AP, the inverse square law may Fig. 61. A point in a unibe applied. The luminous intensity of L1S formly diffuse light-emitting
sphere uniformly illuminates
in the direction of the centre M of the
the wall of the sphere
sphere is L . L1S. If L MAP = oc., the luminous intensity Ia. in the direction of P will be Ia. = L. L1S. cos oc..
Since AB = R cos oc. and AP = 2 AB, we may write AP = 2R cos oc..
The illumination E'a. at P, measured perpendicular to AP is therefore
,
Ia.
L.L1S.cosoc.
L.L1S
Ea.=--=
----AP2
4R 2 cos2 oc.
4R 2 cos oc..
Since AP makes an angle oc. with the normal at P (i.e. the radius PM),
E' a. must be multiplied by cos oc. to give the illumination E' p at P on
the surface of the sphere:
E'p=
L . L1S . cos oc.
L . L1S
---4R2 cos oc. - 4R2 .
(IX-5)
Since the angle oc. no longer occurs in this equation, it follows that the
lX-4]
ALTERNATIVE DERIVATIONS OF THE FORMULA E
= 1rL sins 8
125
illumination produced by a uniformly diffuse light-emitting part of a
sphere is the same at every point on the sphere.
If we now introduce the emittance H to take the place of the luminance,
so that L = Hfn, we may express equation (IX-5) in the following form:
H. LIS
E'p = 4nR 2 •
(IX-6)
The product H . LIS represents the luminous flux emitted by the surface
element LIS, and 4nR2 the surface of the sphere. Equation (IX-6) may
therefore be expressed in words as follows:
The illumination on a sphere produced by a uniformly diffuse light-emitting
element of the surface of the sphere is equal to the quotient of the luminous
flux emitted by that element, divided by the surface of the sphere.
It may also be said that the luminous flux emitted is uniformly distributed
over the whole of the sphere.
If we now consider a part of the sphere, consisting of a number of small
elements, formula (IX-6) will apply to each such element. The total
illumination produced by the particular radiating portion of the sphere's
surface is thus the sum of the illumination values produced by the
individual elements. If the area of the
s
radiating portion be denoted by S = ELlS,
then H. S = EH. LIS, and
(IX-7)
What has been said above regarding the
uniformity of distribution of the luminous
flux of an element over the whole surface
of a sphere thus applies equally to the
80412
luminous flux of a part of the sphere's
surface of any given size.
Fig. 62. A sphere at the
periphery of a circular, uni- Let us now look at Fig. 62. This again
formly dfffuse light source represents a perfectly transparent sphere
(AC) is uniformly illuminated
throughout
of radius R. The upper segment ABC emits
(M c A 11 is t e r's equilux uniformly diffuse light with an emittance H.
spheres)
The area of that part of the sphere is
S = 'b&R. BE and this area is bounded by the circle AEC (radius r).
The illumination EABC of the wall of the sphere produced by the segment
ABC according to formula (IX-7) is
126
[IX
NON-POINT SOURCES
H .S
EABC
= 4nR2 =
H . 2nR . BE
. 4nR2
BE
= H 2R"
(IX-8)
Let us now compute the illumination at the wall of the sphere below
the circle AEC, supposing the spherical segment ABC to be replaced
by a circular light source of the same size and in the same location as
the circle AEC. This light source is assumed to have the same emittance
H as the segment ABC.
We wiU. now calculate the luminous flux passing through the circle AEC
from the segment ABC. This is the luminous flux f/> ADC received by
the remainder of the sphere ADC and distributed uniformly over it.
Let the area of this part be S'; then
(/>ADC
= EABC. S',
BE
= H 2 R and S' = 2nR. ED.
where
EABC
Hence
f/> ADC = H 2R . 2nR . ED = H . n . BE . ED.
BE
In the right-angled .6ABD, AE 2 = BE . ED, so that we may write
for (/> ADC:
(/> ADC = H . 'J'l • AE2 = H . 'J'l1'2.
This, however, is also the luminous flux emitted by AEC, assuming this
to be a uniformly diffuse light source of emittance H.
We see, therefore, that the illumination of the part of the sphere ADC,
when illuminated by the circle AEC, is the same as when illuminated by
the portion ABC. This result confirms the statement made at the end of
section IX-2, viz. that the determining factor for the illumination is the
solid angle subtended by the flashed uperture which is placed in front of a
diffuse radiating light source. It is also clear that a uniformly diffuse
circular light source uniformly illuminates any spherical surface passing
through its circular bourtdary. The illumination of such spherical surfaces
is the ·same at every point.
M c A 11 i s t e r, who published these considerations in 1911, referred
to such spherical surfaces as equilux spheres l).
Equation (IX-8) for EABC can be written in another form by introducing the angle e (B = L ADB = L BAE).
IX-4]
ALTERNATIVE DERIVATIONS OF THE FORMULA E = '"L sin1 8
127
a
AD
Then BE= AEtan 8 = rtan8, and 2R = BD = - - = - - .
cos e
cos e
Since the illumination produced by the circular light source AEC is
identical with that produced by the segment ABC, the index ABC
can be omitted, and we may then write:
r
rtan 8
BE
E = H 2 R = H --a- =H.-;; sin 8.
cos e
Now, rfa = sin 8; hence
E = Hsin2 8 = H a2·
A
AsH= nL,
E = nL sin2 8
or:
Fig. 63. Illustrating the derivation of the formula
E = '"L sin2 9
by means of the infinitesimal
calculus
r2
E = nL -
a2
Since nr2 is the area of the circular light,
sourc~ AEC, L. nr 2 is the luminous intensity of this disc, so that it is permissible to write:
I
E=-.
a2
The formula E ·= '"L sin 2 9 can also be derived with the aid of the
infinitesimal calculus, as follows.
The uniformly diffuse circular light source AB (see Fig. 63) illuminates
a point P on the normal to the centre of the circle. We now want to know
the illumination at point P perpendicular to the normal, as produced
by the spherical segment enclosed within two circles of radii rand r + dr.
All points on this ring are therefore similarly oriented with respect to P
(we shall waive the calculation for a surface element and integration
through 360° around the axis CP).
The area of this ring is
dS = '" { (r + dr) 2 -
Now
r 2 } = '" (r 2 + 2r dr + d 2r - r 2 ) = 2~ dr.
,
dr = _;___ da. - 1 - = . r da.
sm a. cos a.
cos a.
sm a.
so that the area can also be expressed as
dS= .2'"r2da. .
stn a. cos a.
128
NON-POINT SOURCES
[IX
The apparent area of the ring as seen from P is therefore
dS' = . 2..,.• dr~. cos r~. = 2":"• dr~.
Sln fl. '
Sln fl. COS fl.
and the luminous intensity in the direction of P is dl = !_"~ ,.. dr~..
sm r~.
This luminous intensity produces an illumination at P at a distance
rjsin r~., the light being incident at an angle r~. to the normal at P; the
illumination at P produced by the ring is thus
2"L r• dr~.
dE =
,..
sin r~.
cos r~. = 2"L sin r~. cos r~. dr~..
sin1 r~.
The total illumination at P produced by the surface AB as enclosed
within the angle 9 is then
9
21rL sin r~. cos r~. dr~.,
E =
J
the solution of which is:
0
E = 1rL sin• 9.
IX-5. Luminance of light beams
In section IX-2 we have seen that when a diaphragm is placed in front
of a light source the diaphragm must be regarded as a source of light,
having a luminance equal to that of the actual light source. It is a
condition, however, that, in the direction in which the luminous intensity
of the diaphragm is measured or computed, the diaphragm shall be
completely filled with light from the source (completely flashed).
This leads us to tbe conclusion that the quantity luminance need not
be limited to an actual light source, but that it can be determined and
stated in respect of any point or plane in a beam of light.
In the example in Fig. 57, seeing that a layer of air is tacitly assumed
to exist between Land n; in which practically no light is absorbed, the
luminance of D. is equal to that of L. Should a light-absorbing medium
occur between L and D, it would be necessary in equations E = Lw
and E = nL sin2 e not to employ the luminance at L, but that at D.
IX-6. Comparison between the illumination values obtained from
equations E = Lw and E = nL sin11 e
In the foregoing we have shown that the formula E = nL sin2 8 is
valid for all circular light sources and all values of the angle e, that
is to say, it yields the exact value of E for all ratios of light source
diameter to distance, whereas formula E = Lw may only be used
when w is small, i.e. when the light source diameter is small compared
with the distance.
IX-6]
COMPARISON BETWEEN DIFFERENT ILLUMINATION VALUES
129
In order to ascertain the extent of the
error entailed when formula
E = Ifd 2 = Lw
is applied instead of the exact expression
E = nL sin2 e
we shall now compute the results for a
range of ratios of light source diameter
to distance, using the two forms of
equation.
For E = lfd 2 = L. w we may write
E = nL tan 2 e, so that we obtain for
comparison E = nb sin2 e and
oo~o~~~--~~~o~~ffi~o~--a~aoo
1:20f.IO 1:5 1:4 1:3
1:2
1' 1•5
.3f The comparative results are illustrated
82544
Fig. 64.' Demonstrating the
amount of the error when the
illumination produced by a
uniformly diffuse circular light
source is calculated by means
of the inverse square law
instead of the exact formula
E =TTL sin2 8
tan 2 e- sin2 e
sin2 e
E = nL tan 2 e.
--:--::---=--- X
in Fig. 64 for angles e of 0 to 20°;
below the diagram a number of ratios
of light source diameter 2r to the
distance d are also given. In this figure
the error as a percentage, involved when
the inverse square law is used in place
of the exact formula, is plotted vertically;
this error is
100% (= tan 2 @ X 100%).
Since tan e = rfd, we may compute the error as (rfd) 2 X 100%, or, if
we denote the ratio 2r: d with q, as !q2 X 100%.
It will be seen from the figure that a ratio of 1 : 7 entails an error of
l% when the inverse square law is employed. A ratio of 1 : 5 shows an
error of about 1%. Should an error of 5% be permissible, and this is
usually the case in rough practical calculations, the use of the inverse
square law can be extended to distances corresponding to rather more
than twice the diameter of the light source.
This comparison demonstrates once more the significance of non-point,
or large light sources, viz. sources of which the dimensions are large
(or preferably "not small") compared with the distance from the surface
which they illuminate. The "largeness" is therefore relative and does
not refer to the absolute dimensions of the source.
130
NON-POINT SOURCES
[IX
The term point source *) is applied to a source whose dimensions are so
small in relation to the distance from the illuminated surface that
the product Ed 2 approximates to its limiting value.
"Large" light sources can be regarded as point sources in conjunction
with large distances, whereas a "small" source may be looked upon
as "large" when the resultant illumination is to be measured only a
short distance from the source.
IX-7. Some special applications of the formula E = nL sin2 8 = Ifa 2
a. When 8 = 90° as applied to equation IX-4, in which case sin 8 = 1,
the formula reverts to E = nL. This happens when a point in a plane
is enclosed by a light source of luminance L
(see Fig. ~5).
The illumination on the plane AB is at all
points nL. For L we may put Hfn, so that
E = H (E in lux, H in lmfm 2 , or E in fc and H
in lmfsq. ft). It may be said that in this case A+----...I...J::l----:!:
80414
the emittance in the plane AB is equal to that
of the dome-shaped light source (illumination
is, after all, a particular case of emittance). But
Fig. 65. The illumination produced by a
uniformly diffuse domeshaped light source is
equal to the emittance
of this source
it is not only in the plane AB that E, and hence
also the emittance, are at all points the same
and equal to the emittance of the dome; this
applies in every horizontal plane above AB.
Accordingly it may be stated that, within the space between the light-emitting
dome and the plane A B, the emittance of the light passing downwards through
every horizonta~ plane is constant.
Taking the luminance of the sky to be uniform, we can accordingly
compute the illumination in any open space by means of the formula
E = nL = H or, conversely, compute the luminance of the sky from
the measured illumination. From the considerations mentioned in
section IX-5 it follows that we need not ask ourselves how the sky
should be represented as an actual light. source.
*) In lighting technology another meaning of point source is also recognised,
namely a source that is perceived by the eye as a point. The image of such light
sources as formed on the retina is so small that it occupies only one of the lightsensitive elements of the retina. A reduction in the size of the source, or an increase in the distance from which it is observed, does not therefore alter the apparent size of the (point) source. When the si?=e of the source is increased, or the
distance reduced, the point as seen by the eye only becomes larger when the
image on the retina covers more than one of the light-sensitive elements.
IX-7]
SOME SPECIAL APPLICATIONS OF THE FORMULA E = nL sin 1 8
131
b. Fig. 66 shows a spherical light source, the luminance L of which
is uniform. Let the radius· of the sphere be r. Using the exact formula
we shall now compute the illumination perpendicular to MP at a
point P located a small distance
p
d from M.
10415
Now, E = nL sin2 8 = nL d2 ,
m
which nr 2 is the apparent area of
the sphere as seen from infinity or,
in more practical terms, from a
point at a great distance from the
sphere. nr2L therefore represents the
luminous intensity of the sphere, in
accordance with the definition in section IX-1.
Hence we may write:
I
Fig. 66. The illumination from a
uniformly diffuse spherical light source
may be computed by means of the
inverse square law in respect of any
distance, when the distance from the
illuminated point is measured from
the centre of the sphere
E=d2'
the distance being measurea trom the centre of the sphere.
With spherical light sources of uniform luminance, then, the inverse
square law applies for all ratios of light source diameter to distance,
when the distance is measured from the centre of the sphere.
IX-8. Illumination produced by a uniformly diffuse circular light source
in planes parallel to the source
Before working out the illumination in
such cases, let us first consider one or
two further points in connection with
M c A 11 is t e r's equilux spheres.
In Fig. 67 AC represents a uniformly
Vmax
diffuse circular light source. We have
already shown in section IX-4 that the
illumination over the surface of all perfectly transparent spheres coinciding with
the circle AC is constant. The illumination
is therefore computed for a plane tangent- Fig. 67. Demonstrating that
the horizontal illumination on
ial to the sphere at P.
an equilux sphere is equal to
What is now the horizontal illumination that on the wall of the sphere
at P? To obtain this we make use of the following characteristic of
illumination (see Fig. 68 ).
132
[IX
NON-POINT SOURCES
If we introduce a very small surface .15 in any beam of light and then
rotate this surface so that the illumination
on it is as great as possible (Emaz) (full
line, Fig. 68), the illumination Ea. in
any other position of the surface is equal
... _..
to Ema., times the cosine of the angle (ex}
through which the surface is rotated from
~s
the optimum position.
--~
~---
Ea. = Ema,. COS ex.
Thus
(IX-9)
Fig. 68. If at any point in
a beam of light optimum
illumination occurs in the
plane shown by the elliptical
full line, the illumination in
a plane at angle a. to the
optimum plane is
Ea. = Emaz COS a.
The illumination Eex follows this rule up to
the direction in which part of the light beam
falls no longer on the surface under consideration but on the back of it. This theorem
can be proved in the following way:
The luminous flux incident to the surface AS may be regarded as consisting of an infinitely large number of beams with an infinitely small solid
angle, which can be looked upon as emanating from an equal number of
point sources.
If such a small beam, or a number of them with their axes parallel, be
allowed to fall on a surface which is rotated about any line in its own
plane, the illumination will follow a cosine curve. This can be seen
from Fig. 69, in which the line LP represents a beam of light coming
from a light source L and illuminating the point P in the
J; L
plane V 1 . If we denote by E
the illumination at P as measured perpendicular to LP, the
illumination E 1 of V1 at P
will be E cos y 1 , where y 1 is the
angle between LP and the
normal LA 1 to V1 .
If we now rotate the plane vl
about the line QR through an
angle b and denote the plane
'::...'--'/
in the new position by V 2 , the
Fig. 69. Illustrating the derivation of the illumination E 2 of V 2 at P will
formula Ea. = Emaz cos a.
be: E cos Y2, where Y2 is the
angle between LP and the normal LA 2 to V2 •
---
- --
IX-8]
UNIFORMLY DIFFUSE CIRCULAR LIGHT SOURCE
E cos y 2
Hence
E cos y 1
cos 1'2
cos 'Yl
133
(IX-10)
We now draw lines A1 B and A2B perpendicular to QR and denote
the angles PLB, A 1 LB and A 2LB by {3, cx1 and cx2 ; according to section
VI-3 we may then write:
cos y 1 = cos cx1 cos {3, and
cos y 2 = cos cx 2 cos {3.
Substituting these expressions for cos y 1 and cos y 2 in equation (IX-I 0)
we obtain:
cos cx 2 cos {3
cos cx1 cos {3
cos cx 2
cos cx1
(IX-II)
The angle of rotation is here <5 = cx2 - cx1 •
Suppose that cx1 = 0, in which case the plane is perpendicular to LB;
then cos cx1 = 1 and <5 = cx 2 - 0 = cx 2 • In this position the illumination
at P is at its maximum when the plane is rotated about QR. Let us
denote this value by Emaz; then equation (IX-11) becomes
E2
cos cx2
1
- - = - - = cos <5
Emaz
or
'
E = Emaz COS <5.
Fig. 70. Since the component parts of
the illumination elements at a point
describe cosine curves (C 1 , C2 , C3} according to the position of the illuminated
surfaces, the total illumination also
follows a cosine curve (C)
The variation in the illumination
produced by the elementary
beams of light accordingly follows a cosine curve. For beams
of different directions the position at which E = Emax also
differs, and the curves for beams
in three different directions may
be represented as the cosine
curves C1 , C2 and C3 in Fig. 70.
The total illumination is then
represented by adding together
the cosines of the individual
bf!ams. It can be shown mathematically that the resultant
curve C is also a cosine curve;
134
NON-POINT SOURCES
[IX
in accordance with equation (IX-9) the total illumination is thus
E
= Ema:x: COS a;,
It should also be pointed out that the cosine can become negative but
illumination cannot.
The validity of expression (IX-9) is therefore restricted to the angular
range in which the rays of the beam fall only on the front of the surface
under consideration. In the case of Fig. 70, the total illumination thus
follows this formula only between the directions indicated by A and B.
Let us now turn once more to Fig. 67. If we can determine the position
of the plane for Ema:x: in respect of P, we can compute the horizontal
illumination at P from the illumination at the wall of the sphere at P.
Now, the beam which illuminates P is symmetrical with respect to
the plane of the drawing and to a plane through BP and perpendicular
to the plane of the drawing (arc AB = arc BC, and accordingly
L APB = L BPC). The maximum illumination Ema:x: at P therefore occurs
in the plane Vma:x: perpendicular to BP, so that BP is the normal to Vma:x:
at P. The normal to the surface of the sphere at P is the radius MP
and the normal to the horizontal plane H at P is FP.
Further, L MPB = L MBP = a: and L FPB = L MBP = a:, so that
the horizontal plane at P makes the same angle with Vma:x: as the tangential plane to the sphere. Hence the horizontal illumination at P is equal
to that at the wall of the sphere, and the characteristic of M c A ll i st e r's equilux spheres is just as applicable to the horizontal illumination
at all points on the surface of the sphere as to the illumination of the
wall itself.
This characteristic is of value in computing the illumination in planes
parallel to uniformly diffuse circular light
sources.
In Fig. 7 i, AB represents a circular source
of radius r and luminance L. We wish to
calculate the illumination E p at the point
P in a plane parallel to the source, at a
distance CD = d from AB. The distance
from P to CD is p.
A sphere is imagined, passing through P
and the periphery of AB, this being shown
f
in Fig. 71 as a circle having M as centre Fig. 71. Illustrating the deriand R as radius. This sphere is then an vation of the formula for
calculating the illumination
equilux sphere of which the illumination produced by a uniformly diffuse circular light source in
is E = nL sin 2 8 (8 = L CFB).
planes parallel to the source
IX-8]
135
UNIFORMLY DIFFUSE CIRCULAR LIGHT SOURCE
Since the horizontal illumination at P is equal to that on the wall
of the sphere at P, it can also be said that
Ep = nL sin2 e.
To express sin 2 8 in terms of d, r and p we draw the following lines
in the diagram: BP, HGJJDP, so that BG = GP, MG = Rand BIJJCF.
sin 2 8 = t ( I - cos 28).
Now
(IX-12)
The angle 28 is obtained from !::::,. CMB:
CM
cos28 =If·
In the similar triangles HGM and IBP:
HM : IP = GH : BI,
where
(IX-13)
r+P and BI =d.
IP = p-r, GH = - 2 -
Inserting these in equation (IX-13) we find on solving for HM that
p2-y2
HM= -2d-.
CM is obtained from
CM = CH
P
+ HM = J.d
2 + --2d =
2
y2
J2 + p2
y2
----2~d~--
and the radius R can be calculated from !::::,. CMB:
It is now possible to compute cos 28 ( =
c:), insert this value m
equation (IX-12) and then multiply by nL, which finally gives
E p = !nL [ 1 -
J2 + p2- y2
]
_
_.
V (d2 + p2 _ r2)2 + 4d2r2
(IX-14)
136
NON-POINT SOURCES
[IX
By means of equation (IX-14) the variation of the illumination in a
plane parallel to the light source can be computed, and the results
for a number of different ratios pjd and rjd are shown graphically in
Fig. 72. The illumination is here given as a function of the ratio pjd;
it is expressed as a percentage of the illumination in the centre below
the light source.
E
fOOr--.~=r------.------.------.------.
0/o
60o~----,a~,,----,a~,2~--~a~.3~--~o~~~----~as
-Pfd
Fig. 7~2. Illumination produced by a uniformly diffuse
circular light source in planes parallel to the source, as
a function of the ratio of distance (p) of the point from
the axis of the light source, tu the height (d) of the
source above the plane. The resultant illumination is
illustrated for various ratius of the radius (r) to the height
(d) of the source. The illumination with pjd = 0 is taken
to be 100% for each ratio of r: d. The values for rjd = l/00
correspond to those as computed by means of the inverse
square law
The line for the ratio rfd = I joo represents the illumination produced
by a point source, the light distribution of which is that of a plane
light source radiating in accordance with Lambert's law.
IX-9. Illumination produced by linear light sources
Among other forms of non-point sources, linear sources have become
particularly interesting since the introduction of the tubular fluorescent
lamp. Let us now see how the illumination produced by such sources
can be computed.
As the diameter of the tube is small compared with the length, such
light sources may be regarded as linear, i.e. in .the computations it
may be assumed that the light is concentrated along a line, viz. the
axis of the tube.
In Fig. 73, AB represents a linear light source suspended parallel to
IX-9]
LINEAR LIGHT SOURCES
137
a plane H; we require to know the illumination at a point P in this
plane.
The radiation from AB is uniformly diffuse anu the solid of light distribution will accordingly be toroidal (see section VIII-4). Let the length
B
Fig. 73. Calculation of the illumination produced by a
linear light source (AB) at any given point (P) in a plane
of AB be l and the luminous flux f/J; the luminous flux per unit length
is then f/J 1 = f/Jjl. The maximum luminous intensity 1 1 of a part of AB
one unit in length will then be 1 1 = fPJn 2 = f/Jjln 2 •
A convenient formula for the illumination at P is obtained if this is
expressed in terms of the height h of the source above H, the perpendicular distance PC = a of P from the source, and angles ex1 and ex 2 •
These are the angles subtended by the parts of AB on each side of C,
with respect to P (i.e. L L APC and CPB in Fig. 73).
We shall compute E by calculating first the contribution dEa. of a
(cylindrical) surface element, dl in length, towards the illumination at P,
and then integrating dEa. between the angles ex1 and ex2 • The element
under consideration is shown at D in the figure. The distance DP is
denoted by r and L QDP by fJ. In the length of the element, dl, an
angle dex is subtended with respect to P.
The luminous intensity of dl in the direction of P is
dla. = 1 1 dl cos ex,
(since DP is at an angle ex from the normal to AB at D).
The contribution dEa. made by dl towards the total illumination at
P in the plane H may be represented by
dla.
I
dE a. = -cos fJ = _~: dl cos oc cos {J,
y2
y2
(IX-15)
138
[IX
NON-POINT SOURCES
in which r, dl and cos fJ can be expressed in terms of h, a and oc:
a
r
a
h
h
r = - - , dl = - - doc= ---doc, cos fJ =-=-cos oc.
cos oc
cos oc
cos 2 oc
r
a
Substitution in equation (IX-J 5) then yields:
h
2
dErx -- I 1 -cos
oc doc.
a2
(IX-I6)
The total illumination at th~ point P is evaluated by integration between
the limits oc1 and oc 2 , taking into account the fact that in Fig. 73 the
angle oc 2 is positive, so that oc1 must be negative. (For the perpendicular
PC, angle oc = 0). This gives the equation:
I + sin 2oc !oc,
E = I 1 h- -I 2oc
a2 4
(IX-I7)
IXt
in place of which we may write:
I [ 2 (oc2 E = I 1 ah 2 4
. 2oc oc1 ) + sm
2
. 2oc ] •
sm
1
(IX-I7a)
If h and a are in metres and I 1 in cdfm, E will be in lux; with h and a
in em and ! 1 in cdfcm, a factor of I ()4 must be included to give E in lux.
h and a in feet and I 1 in cdjft will give E in foot-candles, oc must be
expressed in radians.
Example:
Suppose that six 40 W tubular fluorescent lamps are mounted in a
horizontal line. Let the luminous flux per lamp be 3000 lumen. We will
assume that the lamps emit uniformly diffuse light. The length of each
lamp is 4 feet and the space between them 2n, so that the luminous flux
per foot of tube, including the gaps is <P 1 =
~~~8° = 720
lmjft.
The
luminous intensity per foot is then / 1 = 7 ~0 ~ 73 cdjft. These lamps
are assumed to be suspended 12 feet above'"a horizontal surface.
The arrangement is shown in Fig. 73; P will b.e so located that PR = 10ft
and AC = 15ft. In the figure, then, AB = 6 x 4If8 ft = 25ft (BC =10ft),
h = 12 ft.
In order to apply equation (IX-17a) we must first compute PC= a,
and the angles lXI and ot 2 :
a= PC= Vh 2 + PR 2 = Vl2 2 + 10 2 = 15.62 ft.
BC
-10
tan lXI =
= 15 .62 = -0.64; OC1 = -0.57 rad.
ta:1 OC2
a
AC
15
=a=
15 .62 = 0.96;
1X2
= 0.765 rad.
IX-9]
139
LINEAR LIGHT SOURCES
All the values can now be filled in in equation (IX-17a) and this gives:
12
E = 73 X 15.621 X
x t { 2 (0. 765 + 0.57) + sin 2 X 0. 765- sin 2 (-0.57)} = 4.12 fc.
Let us now consider the universal formula (IX-17) in the special case
where the point P at which the illumination is to be computed is located
at a distance a from the light source that is small compared with the
length l of the source. Here
oc1 = -
n
n
2 and oc2 = 2, so that
h
n
E = Iz. a2. 2"
Since ~ = cosy (see Fig. 73), this may be written as
a
n 11
E = - . - . cos y.
2 a
For a surface element perpendicular to the shortest line between it
and the light source, y = 0, and we then find that the illumination is
inversely proportional to the distance from the source:
!J = 1.57 !.J.
a
E = ~.
2 a
(/ 1
(IX-18)
in cdfm and a in metres gives E direct in lux; if / 1 is in cdfcm and
a in em, the result must be multiplied by 1()4 to give E in lux. 1 1 in
cdjft and a in feet gives E in fc).
Equation (IX-18) can also be derived in a more simple manner. The
tubular light source of length l gives a luminous flux of
We now imagine a cylinder of radius a around the light source and
concentric with it. If l is very long compared with a, it may be said
that the whole of the luminous flux from the source will fall on the
cylinder. The illumination of the cylinder wall is therefore found by
dividing the luminous flux f/J of the source by the area 2nal of the
cylinder, viz:
140
NON-POINT SOURCES
[IX
E = _!!__ = n 2I 1 X l
n !.J
2nal
2nal
2 a
which is identical with equation (IX-18).
A diagram can be constructed for the practical application of equation
(IX-17), in which the value of the expression ! 120t + sin 2at 1::
is
plotted as a function of Ot. Values read from this diagram are then
multiplied by I Na 2 to give the value of the illumination.
A further simplification can be effected in the application of this formula
by calculating values of at 2 - at1 which will represent the same contribution towards the value of the expression ! 120t + sin 2at 1:: 2).
IX-10. Comparison of illumination values of linear light sources as
obtained by exact calculation with those obtained by means
of the inverse square law
In the same way as we have done for large circular light sources, we
shall now ascertain the extent of the error involved when the illumination of linear light sources is computed with the aid of the inverse
square law instead of accurately by means of equation (IX-17).
When the inverse square law is employed, the luminous intensity is
regarded as being concentrated at the centre of the line (in tubular
lamps mid-way along the tube). This point is considered to be the
point whence the light is emitted and it
is therefore the point to which the inverse
square law refers.
Calculations will be made for fwo different
sets of conditions, and these are now explained in reference to Figs. 74 and 75.
Case 1 (see Fig. 74).
A horizontal light source AB, the length
of which is l, illuminates a horizontal surFig. 74. Calculation of the
face H.
illumination produced by a
The height of the source above H is denoted linear light source (AB} in
its projection on a plane
by h. The luminous intensity per unit
parallel to it
length is I 1•
We wish to compute the illumination at a point P, vertically below the
centre of the source AB on the plane H. AB subtends an angle of 2at at P.
IX-10]
141
LINEAR LIGHT SOURCES
In order to keep the argument general, calculations will not be made
for specific dimensions of h and l, but for a range of ratios h : l.
Applying equation (IX-17) to Fig. 74 we have
E = I 1 X : 2 X tl2oc + sin 2oc
1:: ;
a= h and oc1 and oc2 are numerically equal, so that equation (IX-17)
becomes
E =
t l
X
(2oc + sin 2oc).
(IX-19)
l
Angle oc is found from tan oc = 2h. The factor l (2oc + sin 2oc) can now
be evaluated for various ratios of h to l.
If AB be regarded as a point source at C, the formula for the inverse
square law E = Ifd 2 will contain I= I 1 X l and d = h, in which
case
0_--
I, . l I 1 l
E=J;2=;;·~t·
(IX-20)
There is a third method of computing E, that is by means of
equation (IX-18). The light
source is here regarded as being
infinitely long compared with
the height h. Formula (IX-18):
Fig. 75 Calculation of the illumination
produced by a light source AB at a point P
in a plane parallel to the source, where
PQ = QR = CR = h
E =
n
I1
2 . ;- then assumes the
.
_ n I1
form. E - 2 . },;
(IX-21)
Equations (IX-19), (IX-20) and (IX-21) differ only in the factor by
which I)h is to be multiplied to yield E, and these factors accordingly
determine the difference in the results as obtained by these three methods
of calculation.
If we now work out these factors for a range of ratios h : l they can be
plotted graphically as a function of hfl. With hfl on the horizontal axis,
the vertical axis will show the illumination E divided by I 1fh. Figure 76
shows this for ratios hfl of from 0.1 to 10 (curves I), which means a
142
[IX
NON-POINT SOURCES
variation in the height of the light source of from 1f10th of the length
to 10 times its own lengt.h.
From the figure it will be seen
that the results are quite ac- E/f!.
ceptable for technical purposes 10
as regards their accuracy when
5
the inverse square law is used
"1'-. .... 'lj
for distances greater than twice
2a
I 2a.+sin
/
the length of the source (with
2
--- - - -~
..--- r-:: Fhfl = 2 the error is about 4%).
-c~
With distances less than roughly
one quarter of the light source
n•
".).;
""'
length it can be said with suffi.....
kcosacr c6s4f II
cient accuracy that the illumination is inversely proportional qa
][
to the distance, in which case
0,1 18(.aa,
equation (IX-18) can be used.
,'(
--
'
"
.aa,+sL.aa.-sin~
Case 2 (see Fig. 75).
Here the situation as regards
0,0
''
"I\
-·
light source and illuminated plane a,a
is the same as in the above
a,Of
10
instance, but we now wish to
0,1
o,5
G718J
know the illumination at the
point P of which the location Fig. 76. Illumination values for different
ratios of h : l, relating to t~e conditions
in the plane H is determined by depicted in figs. 74 and 75. The values
on the ordinate are not those of the
RQ = PQ =h.
In order to apply formula.(IX-17) actual illumination, but of
(1 1 =
we determine oc1 and oc 2 from
luminous intensity per unit length of the
Eft
BD
h-0.5l
tan oc1 = DP =
hy' 2 and
+
AD
h
0.5l
tan oc 2 = DP = hy' 2
light source)
a= DP = hy'2.
For (IX-17) we can therefore put:
E = I 1 X _h2 X ! [2oc +sin 2oc]ex• = I 1 X~ l2oc +sin 2oc]ex'.
2h
ex, h
8
ex,
(IX-22)
If AB is regarded as a point source, equation (VI-5) can be used, viz.
IX-10]
LINEAR LIGHT SOURCES
143
I
E = - cos3 oc cos3 {J in which in the present instance
h2
'
z
I = I 1 • cos
e, oc = <5, fJ = e,
thus giving us
I .l
I l
E = - 1- cos3 <5 cos1 8 = __! • - cos3 <5 cos4 8.
h2
h h
(IX-23)
Equations (IX-22) and (IX-23) thus again differ in the factor by which
Izlh must be multiplied to give the value of E.
These factors have also been calculated: for a series of ratios of h : l
(between hfl = 0.1 and hfl = 10), and the results are depicted in Fig. 76
as a function of hfl (curves II).
This diagram shows that the error involved in the approximating method
of calculation is smaller than in the first instance.
In general it may be said that the inverse square law yields values
which approximate sufficiently closely for technical purposes to the
exact values when the mounting height is more than twice the length
of the source.
IX-II. Other kinds of non-point source
Since the wa11s and ceiling of a room reflect light falling upon them,
thus adding to the illumination in the working plane, it is also of interest
to find formulae for the evaluation of this illumination as well as for
the luminous flux radiated towards the working plane.
For this purpose the reflecting surfaces are regarded as uniformly diffuse
sources of light and, in practice, this is a fair approximation. Such
sources of light are usually rectangular or square in shape; the plane
in respect of .which the illumination is to be computed, when illuminated
by a ceiling, is parallel to the plane of the light source, or, when i11uminated by the walls, perpendicular to the plane of the source. The method
of arriving at the relevant formulae lies outside the scope of this book,
however 3 ).
I X -12. The significance of the foregoing considerations as applied
to practical forms of light source
a. The exact formula for the illumination produced by circular light
sources (E = nL sin 2 8) is not often used in practical work. This also
applies to the form E = I Ja 2 •
The reason for this is that the luminance of practical forms of circular
144
NON-POINT SOURCES
[IX
light source are hardly ever uniform, moreover their radiation is often
not uniformly diffuse. A contributory reason is that the degree of error
permissible in lighting calculations is usually so generous that for most
distances the inverse square law yields sufficiently good results.
It will be seen later (Chapter XI) that in optical systems (lenses etc.),
the exact formula is used to give more accurate results.
For calculations of illumination as produced by reflecting walls and
ceilings, in which case the source of light is large compared with the
distance, the inverse square law is not accurate enough, and the exact
formula€ have to be employed. The shape of such reflecting surfaces
is nearly always rectangular however, for which reason the exact formula for circular light sources is very rarely applicable in such cases
as well. 1he significance of the formula and the considerations regarding
its use, as far as the usual circular primary light sources are concerned,
are centred mainly in the fact that the distance at which the inverse
square law is applicable can be ascertained as an approximation, so
that discretion must be exercised in the use of this law.
Should the lighting engineer wish to calculate illumination values with
an error of not more than 5% he will know that he can make use of
the inverse square law when the distance is, roughly, 'not less than
twice the overall dimensions of the light source. If data are needed
in respect of shorter distances it will usually not be possible to calculate
these, since the exact formula holds good only for perfectly uniform
diffusers with uniform luminance; details can in such cases be obtained
only by measurement.
If we need to measure the luminous intensity of a light source and
accordingly obtain this from the product Ed 2 , we must know roughly
from what distance the measurement of the illumination must be carried
out in order to remain within a certain percentage of the limiting value
of Ed 2 •
If the permissible error is 1% we know that the distance at which the
measurement is to be effected must be at least 5 times the diameter
of the light source; for !% error the distance is lO times the diameter.
As shown in section IX-1, when deciding upon the measuring distance
we must take into account the particular point on the lighting fitting
from which the measurement is to be made.
b. The practical point of view with regard to linear light sources is
rather different from that adopted for circular sources. The emission
of light from many present-day kinds of light source approximates
IX-12]
PRACTICAL FORMS OF LIGHT SOURCE
145
so closely to uniformly diffuse radiation that it is worthwhile using
the exact formula for short distances.
On the other hand, when considering the lighting fittings in which
such light sources are mounted, we cannot apply the exact formula
and direct measurement must be resorted to. Here again, the value
of the theoretical considerations lies more in the possibility of ascertaining roughly the distance at which the inverse square law is still
valid, with a view to the required accuracy of the results.
c. As already mentioned in a) above, the usefulness of the exact formulae
for rectangular light sources lies in their application for calculating
the illumination produced by walls and ceilings. In these cases the
ratios of the dimensions of the light source to the distance are never
such that the inverse square law will yield a sufficiently accurate result.
The formulae can be used successfully for computing the coefficient of
utilisation of lighting equipment in enclosed spaces.
REFERENCES
E. D. M c A IIi s t e r, Trans. I.E.S. Am. 9, 1911, 703-721. "The Law of
Conservation as applied to Illumination Calculations". See also: N. A. H a 1b e r t s m a: "Der Lichtstrombegriff und seine Anwendungen", Berlin, 1921,
p. 34 e.s.
2 ) H.
Z ij I, Ph. T. R. 6, 1941, 147-152. "The Calculation of Lighting Installations with Linear Sources of Light"
3 ) H. Z ij I : "Manual for the Illuminating Engineer on Large Size Perfect Diffusors". Philips Technical Library, Eindhoven, 1959
1)
CHAPTER X
REFLECTION, ABSORPTION, TRANSMISSION
X-1. Reflection, absorption, transmission
When light passing from one homogeneous medium falls upon the
interface between this and another medium, part of it is reflected and
the rest passes into the second medium; the light entering the second
medium is then partly or wholly absorbed. Any light that is not absorbed
may pass through the medium, and this is referred to as transmitted
light.
These phenomena are accordingly known as reflection, absorption and
transmission, respectively.
The ratio of the reflected luminous flux (<Pe) to the incident flux is
.
<Pe
called the reflect~on factor (e); hence e = q;·
The ratio of the absorbed luminous flux (<Pa.) to the incident flux (<P)
<P
is the absorption factor (oc); oc = ;.
The ratio of the transmitted luminous flux (<P-r) to the incident flux (<P)
<P
is the transmission factor (-r); -r = <PT
Since <PP, <Pa. and <P.T are together equal to <P:
For opaque substances, i.e. substances which do not transmit light, -r = 0;
hence
e+oc= I.
Almost all substances which are opaque in the thicknesses in which
they are normally used will transmit light when they occur in very
thin layers; this can be observed in incandescent lamps which have
a silver or aluminium mirror on the inside of the bulb; the filament
is o.rten visible through this layer.
X-2]
REGULAR REFLECTION
147
X-2. Regular reflection
A parallel beam of light falling on to a polished plane surface of glass or
metal is reflected as a parallel beam. Before and after reflection the
rays lie in the same plane through the normal at the point of incidence,
and the angles made by the incident and reflected rays with the normal
are equal (angle of incidence = angle of reflection).
This kind of reflection conforms to the optical laws and is known as
specular or regular reflection.
What is now the relationship beA
tween the luminous intensity and
luminance of the reflected light and
the characteristics of the light before
reflection?
Fig. 77 shows a mirror M which gives
- ~- -- ~- -- -- ~ M'
==---~~~p~"""""";;.;-----regular reflection. A beam of light
1 l
~f
-M
from a source A of luminance L
l ! /;1
falls on a point P (infinitely small
I ,,!/ /
area of M); the solid angle of the
beam is w.
A'
In studying the reflection of the
beam, P must be regarded as a small
Fig. 77. The luminance of an image
plane mirror lying in the plane M'
A', and of a point P on a mirror M,
is equal to that of the reflected light
which can be drawn tangentially to
source, multiplied by the reflection
the convex mirror at P. The normal
factor of the mirror
at P is PN. The light rays are
reflected symmetrically with respect to the normal, so that the reflected
beam subtends the same solid angle as that of the incident beam. What,
then, is the luminance as observed by the eye when looking into the
reflected beam in the direction of P?
The emittance at P (i.e. in this case the illumination) is H = Lw cos Cl.
If the observed luminance be denoted by L', then the emittance at P
after reflection will be H' = L' w cos Cl.
Now, if we denote the reflection factor of the mirror by e, the luminous
flux, and therefore also the emittance of the reflected light at P, will
be equal to (! times that before reflection, i.e.
tV
H' = (!H, or L' w cos Cl = (!Lw cos Cl.
Hence
L' = eL.
(X-1)
148
REFLECTION, ABSORPTION, TRANSMISSION
[X
This can be represented in the following manner. The flat mirror
M' produces an image A' of A, and the luminance of the image is reduced
by the mirror to a value QL. The plane element P may be regarded as
a diaphragm which is flashed with the luminance of A'.
The luminance of the image of the light source and that of the mirror
are accordingly equal to the luminance before reflection has taken place,
multiplied by the reflection factor of the mirror. This luminance is thus
found to be independent of the emittance, i.e. the illumination, of the mirror.
As this rule applies to every point or surface element of a specular
surface, it is valid also for the whole surface, irrespective of the form
of the mirror.
With the aid of this law it is possible to compute the luminous intensity
of curved mirrors; these can be regarded as light sources, the luminance
of which is L' = eL. When the area S of the mirror is completely flashed
with the reflected luminance of the light source, the luminous intensity is
I = SL' = SeL.
Exactly what is meant by mirrors and lenses being "flashed" is dealt
with in chapter XI (section XI-6).
Let us now see how the luminous intensity of a light source is changed
by regular reflection. In Fig. 78 a point
A
source A is depicted as being reflected
by a plane mirror M. The optical laws
tell us that the plane mirror forms an
image A' of the light source A, whereby
A' and A are symmetrically located with
respect to the reflecting surface.
What we wish to know is the luminous
intensity of the image A' in the direction
of A'R. The line A'R passes through the
reflecting surface M at the point P, and
the reflection in the direction of A'R thus
A'
takes place at the point P.
Let I denote the luminous intensity of Fig. 78. The luminous intensity
A in the direction of P; then the emittance of an image A' of a light source
A as formed by a plane mirror
H (the illumination) at P IS
M is equal to that of A, mulI
H = AP2 cos IX.
tiplied by the reflection factor
of the mirror
If we denote the luminous intensity of A' in the direction of P by I',
X-3]
REFLECTION FACTORS OF NON-CONDUCTIVE MATERIALS
149
the emittance at P will be
I'
H' = A'P 2 cos rx,
or, smce A'P = AP,
I'
H' = AP 2 cos rx.
As the emittance after reflection is: H' = eH, we may write:
I'
I
AP2 cos rx = 12 AP2 cos rx,
so that
I'= ei.
(X-2)
In the case of plane mirrors the luminous intensity of the image of a light
source is equal to that of the source itself multiplied by the reflection factor
of the reflecting surface.
The most well known surfaces which produce regular reflection are
those of polished metals, and glass with silvering on the back. Unsilvered
glass will also produce specular reflection if the surface is quite smooth;
so also will opal glass, glazed porcelain, vitreousenamel etc.
Examples:
l. If in Fig. 77 L = 800 sb and
p =
0.9:
L' = pL = 0.9 X 800 sb = 720 sb.
If a curved mirror of circular cross-section 10 em in diameter, i.e.
of which the apparent surface perpendicular to the boundary plane
is ~ x· 102 em 2 , is flashed with a luminance L' = 720 sb, the luminous
intensity of this mirror (perpendicular to the boundary plane) will be
I =
i X 10 X 720 cd
2
= 56,500 cd.
2. If a light source of 250 cd luminous intensity is reflected by a plane
mirror of which p = 0. 75, the luminous intensity of the image of
the source will be
I' = pl = 0.75 x 250 cd = 187.5 cd.
X-3. Reflection factors of non-conductive materials
The reflection factor of a regularly reflecting surface is dependent on
the angle· of incidence of the light. Now, there is a characteristic difference
in this relationship as between electrical conductors and non-conductors.
150
REFLECTION, ABSORPTION, TRANSMISSION
[X
In non-conductors which do not absorb light or of which the absorption
is only slight*), the reflection factor at
different angles of incidence is governed
by the refractive index, formulated by F r e sn,
n e I as
(!;
sin 2 ( i - r)
[
tan 2 (i- r)]
= ! sin2(T-=i--~) + tan 2 (i + r)
(X-3)
where e; is the refleGtion factor for an angle
of incidence i, and r is the angle of refraction.
The refractive indices determine the relationship between i and r according to Snell's
law (see Fig. 79), viz.
Fig. 79. Snell's Jaw of
refraction:
sin i
n'!
sin r
n1
sin i
sin r
(X-4)
where n 1 and n 2 are respectively the refractive indices of the media
through which the light travels before and after refraction. If the first
medium is a1r, n 1 = I and equation (X-4) reads:
sin i
. .
.
- - = n or sin t = n sin r.
sin r
'
In the process of reflection the light is polarised. The first term of (X-3)
represents that part of the reflected light for which the electrical field is
perpendicular to the plane of incidence, the second term to that part
whereby this field is parallel to that plane. This polarisation can only
be demonstrated by means of special apparatus such as the Nicol prism or
polarising filter; the human eye is not capable of appraising the polarised
condition of light, for which reason the two terms in equation (X-3)
are made additive.
With perpendicular incidence equation (X-3) becomes
eo= (::~::r
(X-5)
which, for n1 = I (air) and n 2 = n, gives
eo=(:+ ~r
(X-5a)
*) By "slight" is meant that the light absorbed in passing through a layer equal
to the wavelength of the light is relatively only very little, e.g. at most 2 or 3
per cent.
X-3]
REFLEXION
FACTORS
OF
NON-CONDUCTIVE
MATERIALS
151
Formula (X-5) can be derived from (X-3) by using the angles themselves instead of the sines and tangents; for very small angles this is
permissible.
When i = goo, i.e. with glancing incidence, e90 = I, in which case the
reflection is complete.
The curve representing the reflection factor of glass with a refractive
__
index of 1.5 in air is shown in Fig. 80
(the fact that the refractive index
1
1
1
13"11 25°22' 34"51 40°5"3
38oa9'
6°39' d2B' 00'22
41°'49' differs according to the wavelength
of the light is here disregarded).
It is seen that, when i = 0°, glass
0,8
gives about 4% reflection; when the
incidence of light to a glass pane is
0,6
perpendicular, 4% of the light is
j_
reflected from the front. Some of
I
it is absorbed (very little, e.g. about
0,4
1
2% for a given thickness), so that
about g4% falls on the rear surface,
I
0,2
where 4% is again reflected, i.e.
/
l----'
almost 4% of the incident light. In
0
300
600
0
.soo all, then, the glass reflects some 8%.
Up to an angle of about 60° there is
hardly any increase in the reflection
Fig. 80. The reflection factor P; as
a fm1ction of the angle of incidence
factor, but thereafter the factor rises
i on reflection of light from air to
quickly to unity at goo. When light
glass with n = 1.5. \\"hen light in
the glass falls on the interface
is reflected in glass or water by a
glass-air, p; is obtained by reversing
surface
backing on to air, the values
r and i in the chart
of i and r are reversed in equation
(X-3). In this case the curve in Fig. 80 will also give values of e if,
instead of the angles i, the corresponding values of the angles r are
plotted horizontally. This has been done in the figure, where the angle r
is shown at the top of the diagram. (! then attains unity at the critical
angle, which is determined by
,.
1
.
sin goo
stn r = ---- = -.
n
n
If the angle of incidence exceeds the critical angle, the reflection factor
remains unity, such being encountered in what is known in optics as the
totally reflecting prism in which the angle of incidence at the hypotenuse
face is 45° and thus greater than the critical angle.
152
REFLECTION, ABSORPTION, TRANSMISSION
[X
With non-conducting substances, therefore, the reflection factor must be
stated in reference to the angle of incidence at which it is measured. If this
is measured with light incident from all directions, i.e. with diffuse
light, the reflection factor will be greater than with light incident at
an angle of from 0 to 45°.
X-4. Reflection factor of conductive materials
The reflection factor of electrically conductive materials, i.e. metals, too,
is dependent on the angle of incidence. The region within which the reflection factor is practically constant ranges from the perpendicular almost
up to glancing incidence, in the neighbourhood of which direction it rises
rapidly to unity when incidence is fully glancing, as in non-conductive
materials.
For metals of sufficient thickness (e.g. > 1 Jlm) cc = l - p. Now the
absorption factor is equal to the emission factor (<),that is, the emittance
of an incandescent substance in a certain direction, divided by that
of the black-body at the same true temperature and in the same direction.
Black-body radiation is quite diffuse, but that of all metals is not wholly
diffuse, and the extent of the departure from the diffuse is manifested
in the glancing directions of the radiation. Whereas at radiating angles
of 0° to nearly 90° the emission factor is fairly constant, it decreases
as 90° is approached more closely and is zero at 90° (for, if p -+ I,
l - p = cc = < -+ 0). This means that at 90° the absorption factor
is also zero and the reflection factor accordingly unity.
X -5. Diffuse reflection
Most surfaces found in practice do not give regular reflection; they
spread or diffuse the light in all directions, and we accordingly speak of
diffuse reflection. For the moment it will be sufficient merely to mention
this fact; later we shall consider more closely the manner in which
diffuse reflection occurs.
Similarly, we speak of the diffuse reflection factor, by which is meant
the ratio of the diffuse reflected luminous flux to the incident flux*).
The ratio of regularly reflected luminous flux to the incident flux is
known as the regul~r reflection factor.
Whereas a surface producing regular reflection appears bright only
in the direction of the (regular) reflected light, a diffuse reflecting surface reveals a certain luminance in all directions.
The distribution of the light reflected by a diffuse reflecting surface
*) Older publications employ the term albedo, i.e. whiteness in the same sense
as diffuse reflection factor. This word is still current in astronomy, but should
not be employed in illuminating engineering.
X-5]
DIFFUSE REFLECTION
153
element can be determined. Usually only one cross-section of the solid
of light distribution (which is generally axially symmetrical only in the
case of perpendicular incidence) is reproduced, i.e. the most characteristic, this being in the plane passing through the direction of incidence
and the normal to the surface.
When dealing with reflection and transmission the term light distribution
Fig. 81. Luminous intensity iridicatrices. a and b are examples of
spread reflection; c is the indicatrix
of a uniformly diffusely reflecting
surface
Fig. 82.
Luminance indicatrices
derived from the luminous intensity
indicatrices in fig. 7a, b and c
curve is not used; in its place we use the term characteristic diffusion curve
or indicatrix of diffusion. This can be plotted on either polar or cartesian
co-ordinates.
This curve can give the distribution of the luminous intensity of the
reflected light, but we can also plot the luminance. In the latter case we will
plot the relative luminance in respect to a certain illumination, preferably per lux, or foot-candle. We accordingly determine the quotient of
luminance divided by illumination for a certain direction of observation
154
REFLECTION, ABSORPTION, TRANSMISSION
[X
and incidence. This quotient is called the luminance factor ({3); further
reference will be made to this concept in section X-7.
Two luminous intensity indicatrices are shown in Figs. 8la and b and
these also indicate the direction in which the light is assumed to be
incident. Figs. 82a and b depict the corresponding luminance indica trices.
It is seen that the maximum luminance of the reflected light occurs in
directions which make roughly the same angle with the normal to the
surface as the incident light. The form of the diagram becomes less
and less dependent on the direction of incidence according as the extent
of the diff~sion is increased, and we can therefore visualise the case where
the luminous intensity indicatrix is a circle, being entirely independent
of the angle of incidence. We are then concerned with a uniform diffuser
(Figs. 81c and 82c); the surface accordingly emits light in accordance
with Lambert's law.
In his book "Photometria" mentioned earlier (see p. 8), Lambert
formulated the law that diffuse reflecting surfaces (and also radiating
surfaces such as the sun and incandescent solid bodies) have the same
luminance in all directions. This is still sometimes referred to as Lamb e r t's law. Lambert discovered it experimentally. Owing to
the crudity of the apparatus with which he worked, he did not observe
the departures from this law, particularly at glancing directions of
observation. Furthermore, L a m b e r t worked with directions of
incidence not far removed from the normal, for which this law certainly
holds good in fair approximation.
·
The two extreme types of reflection are accordingly the regular and the
uniformly diffuse, but neither occurs in perfect form with any known
kind of surface. Even the most perfectly ground and polished mirror
produces some, albeit a very small amount, of diffuse reflection in
adqition to regular reflection.
Between these two extreme, hypothetical kinds of reflection there are
the more practical forms, and also combinations of more or less regular
and diffuse reflection.
If the luminous intensity indicatrix of diffusion differs to any great
extent from a circle we use the term spread reflection; where a surface
produces not only diffuse or spread reflection but also regular reflection,
we refer to it as being mixed reflecting.
The luminous intensity indicatrices of many surfaces are, for perpendicular incidence of the light, practically circular, the reflection being
then almost uniformly diffuse. As the direction of incidence departs
from the perpendicular, the form of the indicatrix differs more and more
from the circular. With oblique incidence, therefore, all diffusely reflecting
surfaces show spread reflection. This is in many cases of considerable
importance. The high luminance observed in certain directions on
X-5]
DIFFUSE REFLECTION
155
lighted roadways is due to the fact that the road surfaces which with
perpendicular incidence give almost uniformly diffuse reflection, are
strongly spread reflecting when the incidence is oblique.
With· perpendicular incidence, surfaces of frosted opal glass, powders,
matt paint etc all produce approximately uniformly diffuse reflection,
but in every case spread reflection occurs with oblique incidence.
Other surfaces, such as the matt side of frosted glass, give spread reflection at all angles of incidence.
All such surfaces giving diffuse reflection are known as matt.
Mixed reflection is produced by the smooth side of frosted glass, unfrosted opal glass, glazed porcelain, vitreous enamel, glossy paper etc.
Diffuse reflection can arise in two different ways:
1. As a consequence of the fact that a surface is not smooth. Such
surfaces may be regarded as consisting
of a great number of small planes, all
at different angles to each other, so
that their normals all assume different
positions. Now, if these small planes
themselves reflect regularly, parallel
rays falling on them are reflected in
different directions and are thus difFig. 83. Heflection from uneven
fused. This is illustrated schematically
surfaces
in Fig. 83. Such conditions are found
in metals with a matt surface.
2. With inhomogeneous media at the surface of which diffuse reflection
occurs, but which nevertheless transmit light to a certain .extent, so
that some of the light penetrates the medium. That part of the light
is then refracted and again reflected by the crystals or small homogeneous areas within the medium, and some of it is thrown back; such
light undergoes many changes in direction in its devious path through
the medium and it is diffuse when it finally leaves the medium.
This occurs with frosted opal glass, matt paint, powders and so on.
In mixed reflection we are concerned with regular reflection at the
surface, combined with diffuse reflection of the kind just described
in 2) above.
We shall now look a little more closely into the reflection characteristics
of diffusely reflecting surfaces as a whole*).
*) Later, in section X-8, we shall revert to the manner in which diffuse reflection
takes place at a surface-element.
156
REFLECTION, ABSORPTION, TRANSMISSION
[X
The form of the reflection indicatrix of a matt surface depends on the
structure of the surface, as well as on the angle of incidence of the light.
Such surfaces exhibit small crests and troughs. With perpendicular
incidence both the crests and troughs arc illuminated (Fig. 84a); .the
former produce no masking effects and no shadows are formed that
will reduce the luminance in any direction. The reflection indicatrix is
thus axially symmetrical.
When incidence is oblique, however, only those sides of the crests which
face the light source are illuminated, the far sides remaining dark (Fig. 84b). This
does not mean, of course, that
the surface as observed against
the light appears to be dark,
for the tops of the crests,
which are always more or less
b
a
rounded, reflect the light fairly
(a)
incidence
perpendicular
Fig. 84. With
both.· the crests and the troughs are ilregularly (Fig. 84b). Such
luminated. With oblique incidence (b) only
reflection can be quite conone side of the prominence is illuminated,
and that side which faces the light source
siderable, since the reflection
is seen to have a certain luminance. At the
factor of non-conductors is
tops of the crests regular reflection takes
place, so that the surface a lso appears
high with glancing incidence.
light when regarded from the side remote
form which the reflection
The
from the light source
indicatrix assumes depends
upon the depth of the troughs and the rounding of the crests. Deep
troughs and sharp crests result in much light in the direction of the
light source itself, whereas shallow troughs and flat crests give more light
in the other direction. The more oblique the incidence of the light, the
more these differences are accentuated and the lower the luminance
of the reflected light in the direction of the normal to the surface.
We can differentiate between micro- and macro-structures; the first refers
to details that cannot be discerned with the naked eye, and the second
to discernable irregularities. Surfaces exhibiting micro-structure without
any macro-structure can be termed smooth; those with macro-structure
are rough. Usually, rough surfaces also reveal micro-structure.
In general, the troughs in smooth surfaces are shallow and the crests
fairly flat, so that, when 1ight falls on them obliquely, the reflection
in the direction away from the light source predominates. Only with
glancing incidence is an appreciable quantity of light reflected in the
direction of the source.
X-6]
UNIFORM DIFFUSE REFLECTION
157
With rough surfaces the troughs may be so deep that the luminance
as seen from the light source is greater than in the opposite direction,
even when the angle of
1
incidence is not so great.
+
This is illustrated in
Fig. 85 which is taken
from an article by
J an sen 1 ).
-60°
From the foregoing it
follows that it is possible to influence the
reflection characteristics
of a material with miFig. 85. Luminance indicatrices of a rough surface
cro-structure by modelwhich; with oblique incidence of the light reflects
more light in the direction of the light source than
ling its surface.
in the opposite direction.
- - - - perpendicular incidence
- - - - - - angle of incidence 30°
- · - · - · - angle of incidence 60°
Incidence in the direction of the arrows
X -6. Uniform diffuse
reflection
This mode of reflection
is mathematically defined and it may therefore be used as a basis for
calculations.
Perfectly uniformly diffuse reflection does not occur as such in practice, but
many matt-surfaced materials approximate to the hypothetical conditions
very closely, at all events when the direction of incidence is not too
oblique. In many instances, therefore, we can base our computations
with sufficient accuracy on the assumption of uniform diffuse reflection.
As mentioned in the previous section, a surface that constitutes a
uniformly diffuse reflector is to be regarded as a secondary light source
radiating in accordance with Lambert's law*).
The characteristics of such light sources have been dealt with in Chapter
VIII, so that we can now employ the equations already derived.
When a luminous flux f/> falls on a uniformly diffusely reflecting surface
of reflection factor e, the reflected luminous flux is ecf>. According to
equation (VIII-4a) the luminous intensity in the direction of the normal
to the plane is then ! 0 = ecf> and in the direction including an angle IX
:n;
with the normal
e. cJ>
Ia. =--cos IX.
:n;
*) As applied to secondary light sources we speak of remission of the light, in
contrast with emission from the primary sources.
158
REFLECTION, ABSORPTION, TRANSMISSION
[X
If we compute these quantities per unit of area, the illumination E
is substituted for <P and the luminance L for I. Then, per unit area,
the reflected luminous flux will be eE, and the luminance, which is the
same in all directions. is
L = eE_
(X-6)
n
If E is stated in lux, L will be in cdfm 2 • To obtain L in stilb (cdfcm 2)
a factor of 10-4 must be added. E in fc gives L in cdjft 2•
If E is expressed in lux and L in asb, n disappears from the equation,
according to section VIII-6; hence L = eE. The same applies with L
in ftla and E in fc.
In the case of a surface for which e = 1, L = E, that is, the luminance
in asb is numerically equal to the illumination in lux.
Reflecting surfaces which are perfect diffusers, i.e. which reflect 100%
of the light, are sometimes known as perfectly white, for which reason
the asb was formerly known in Germany as Lux auf Weiss.
An analogy with the linking of the luminance unit to the illumination
unit through the concept of the 100% diffuse reflecting surface is to be
found in the expression equivalent, or apparent footcandle for the footlambert.
X-7. Luminance factor
Let us now once more consider the concept luminance factor, the definition of which is given in section X-5 as the quotient of the luminance
divided by the illumination for a certain direction of observation and
a certain direction of incidence. This definition may be stated in the
form of an equation as
L
(X-7)
Po,I = E
where the indices 0 and I indicate that {3 relates to the directions of
observation 0 and incidence I.
If we express L in cdfm 2 and E in lux, {3 will be in (cdfm 2)/lux; L in
cdjft 2 and E in fc give {3 in (cdfft 2)/fc. Once f3o,I has been determined,
the luminance in the direction 0 with the direction of incidence I can
be obtained from the illumination:
L = {30 ,1 X E
(X-7a)
in which {3 is a factor.
The luminance factor fJ is also defined as the ratio between the luminance
X-7]
LUMINANCE FACTOR
159
L of a sample for a given direction of observation and the luminance L of
the perfect diffuser (Lw = canst.; Pw = 1) for the same illumination. Thus
p = LfLw.
L = Efn (Lw in cdfm2, E in lux)
Because
applies to the perfect diffuser, we can also write
P = 1t X LfE (cdfm2flx)
The luminance factor thus defined, therefore, differs from the one first
defined by the factor 7t. Furthermore, P = LfLw is an abstract number,
while with P = LfE the units (e.g. cdfm2flx) must be given.
The luminance factor is primarily used in the reflection from road surfaces, and it is easier in this case to work with P = LfE (cdfm2flx).
In the foregoing we have repeatedly spoken of one certain direction
of incidence and one given direction of observation. It should be borne
in mind, however, that the light falling on a point in a plane is always
a pencil of rays with a finite solid angle, i.e. that it embraces a large
number of different directions of incidence. Again, the acceptance
surface of a photometer as used for the measurement of luminance, or
the eye with which such luminance is observed, always intercepts a
pencil of rays having a finite solid angle containing thus a great number
of different directions of observation. Of these numerous directions of
incidence and observation, one is taken as the direction of incidence,
and one as the direction of observation, these being characterised by
angular co-ordinates. It is obviously convenient to choose the axes
of the solid angles of both the incident light and the light entering the
eye or photometer for this purpose.
In the hypothetical case of one single direction each for incidence and
observation these directions
L
might be defined as the axes
of infinitely small solid angles.
In practice, however~ we invariably employ various directions of incidence, and various
directions of observation, at
the same time.
There is a third factor to be
taken into account, namely
Fig. 86. Principle of measurement of lu- the size of the surface involved
minance factor. LL' light source. FF' in the measurement. For difacceptance surface of the photometer.
8 ac,ceptance angle of the photometer ferent points in this surface
160
REFLECTION, ABSORPTION, TRANSMISSION
[X
the directions of observation differ and therefore also the luminance
factors may differ.
In principle the conditions under which the measurements arc taken
are as shown in Fig. 8_6. The surface involved in the measurement is AB.
LL' is the light source and FF' the acceptance surface of the photometer. By means of this arrangement we now wish to measure the
luminance as observed by the eye in the "main direction" CE, of a
surface element (point) C, illuminated in the "main direction" DC.
Now, the illuminating directions vary, from L'A to LB, and the observation directions from AF' to BF; in effect, then, we measure in this
way the average of various luminance factors, and only if this average
is equal to the luminance factor in respect of the main directions DC
and CE will the result be correct.
The fewer the rays deviating from the main directions that are included
in the measurement, the more closely will the average {J approximate
to the true {J relating to the main directions.
In practice the main illuminating direction DC can be approximated
by lighting the surface under measurement with a so-called parallel
beam, this being obtained by placing a "point source" at the focus
of a lens. It is true that the rays from the lens are not all parallel, but
the axes of the solid angles of all the beams reaching the surface are
parallel*). This method of illuminating an area may be represented
diagrammatically in the manner shown in Fig. 87. Every point on
the surface is thus illuminated
by beams having the same solid
angle and whose axes are parallel. We shall call these solid
angles the solid angles of illumination (wr)· The angle included
by those rays which deviate most
from the main direction during
measurement of the luminance we
shall call the acceptance angle (8). Fig. 87. In a "parallel" beam of light
By reducing WI and e the true the axes of the solid angles of illumination w 1 are parallel, and the solid
value of {J for the main directions
angles are equal
DC and CE will be more nearly
approximated.
When the luminance indicatrices for a giveri surface are not too pointed,
*) This is referred to again in Chapter XI, (section Xl-6).
X-7]
LUMINANCE FACTOR
161
i.e. when they include an area in which {3 is fairly constant, it will be
noticed that if wr and f9 be reduced, a point occurs at which any further
reduction yields no further change in the value of {3.
In reducing wr and I or 8, therefore, we have a means of ascertaining
whether the measuring arrangement used will yield values of {3 which
can be used to compute the luminance as will be observed by the eye.
It is not necessary, however, to reduce wr beyond the solid angles of
illumination that occur in practice.
If the luminance factor of a surface is constant for all directions within
a certain acceptance angle, the luminance in that angle will be constant
for a certain illumination, and the surface can be regarded as a light
source radiating light of that luminance in the direction of observation.
We can therefore apply the inverse square law to that surface as a
light source, taking into account, of course, the question as to whether
it may be regarded as a point source or a non-point source.
When a surface, however, gives regular reflection, the inverse square
law applies with the image as light source; in this case no fixed value
of the luminance factor can be attributed to the surface.
Hence, if the reflecting surface is a light source to which the inverse
square law may be applied, i.e. if it can be regarded as the surface from
which the rays originate, it will also have a luminance factor.
If the inverse square law cannot be applied to the surface as a light
source, i.e. the image behind the surface must be regarded as the origin
of the rays, the luminance factor will have no fixed value.
When the reflecting surface is a perfectly plane mirror then the position
of the image is clearly defined; if, however, the surface is specular but
gives a distorted image owing to irregularities in the surface, the point
from which the rays originate is not fixed but may vary; here again, no
fixed value of the luminance factor will be found.
With mixed reflection, the reflecting surface being a true plane, the
diffuse and specular reflections can be determined separately by making
use of the difference between the points of origin of the diffuse and
regular reflections. If the regular reflection is not completely regular,
however, this cannot be done.
162
REFLECTION, ABSORPTION, TRANSMISSION
[X
X-8. The luminance factor as a constant of the material*)
If we wish to regard the luminance factor as a constant of the material,
it is necessary to recall the fact, already mentioned, that surfaces which
reflect specularly have no luminance factor.
No matter how far the values of Wz and e are reduced, no constant
value of {J will be obtained, from which it may be deduced that a luminance factor can exist as a constant of the material only if reductions
in wz andfor e will at a certain point yield a constant value of {J.
Naturally, there are limits, imposed by the technique, beyond which
"'I and B cannot be reduced.
Now, between (hypothetical) uniformly diffusely reflecting surfaces and
specular surfaces there are very many intermediate forms, and we
shall now see for which of these there is, and for which there is not,
a luminance factor.
With uniformly diffuse reflecting surfaces the luminance is independent
of the directions of incidence and observation; hence {J is independent
Of (JJI and e.
If the reflection factor be denoted by (!, we know from equation (X-6)
that
e
L
eE
L = - and {J = - = - (cdfm 2)/lux
:n:
E
:n:
for all directions of illumination and observation.
It will be clear that for surfaces whose reflective properties do not
differ much from the uniformly diffuse, the values of Wz and fJ are not
critical in a determination of {J.
The position is different, however, in relation to surfaces giving a decidedly spread reflection and, when the incidence is very oblique, with
surfaces giving almost diffuse reflection.
In order to see what happens here let us consider the mechanism of
reflection from a small area (i.e. the influence of the micro-structure).
In their simplest form, the surfaces of materials consist of particles
(granules) of various sizes. In general, the surfaces of these particles
are curved, but they may also have flat facets with random orientation.
Not all the tops of the particles will be on the same level; they exhibit
miniature crests and troughs. Such surfaces are matt.
*) The term "constant of the material" is used here in another meaning from
the usual one; the luminance factors are not constants of the material whatever
its form, but only in relation to its actual surface.
X-8]
163
LUMINANCE FACTOR
An example of this is the type of surface obtained by smoothing out
the surface of a powder.
The situation is entirely changed, however, if a binding medium is introduced between and over the grains. When light falls on a surface
of this kind:
I) it is reflected regularly from the surfaces of the grains and binder;
2) it penetrates both the grains and the binder; in the former repeated
refraction and reflection diffuse the light, part of which re-emerges
from the surface.
This is what takes place in paints in which a resin or oil, which is in
itself transparent and usuc~.lly nearly colourless, lies over and between
the grains that give the paint its colour (the pigment). The surface of
the binder will more or less follow the contours of the grains of pigment;
the less it follows the outline of the grains of pigment, the more glossy
it is, up to the point where the binder itself, with a perfectly smooth
surface, will give regular reflection.
In the case of particles of the same order ot size as the waYclength ot
the light, what is known as Rayleigh diffusion occurs instead of reflection
and refraction. Each particle diffuses the light in all directions, and
the laws of regular reflection no longer apply.
Of the light that emerges after penetration and diffusion in the particles
it is clear that the points of origin lie in or on the particles, i.e. that
L
A
Fig. 8H. Diagram illustrating the mechanism
of diffuse reflection:
a) when the radii of the particles are of the
same order of size as the particles
themselves·
b) when the ;adii are large compared with
the particles
L
L
A
B
164
REFLECTION, ABSORPTION, TRANSMISSION
[X
the surface functions as a light source to which the inverse square !aw
applies and for which a luminance factor can be established.
In reference to the diagrams in Figs. 88a and b let us now consider in
outline how reflection takes place at the surface under the different circumstances. Fig. 88a shows a few particles adjacent to each other, these
being drawn as spheres for the sake of simplicity. In practice they would
not of course be true spheres nor exactly next to each other; the situation
as depicted should be regarded as a model to which our considerations
will apply. For our purpose, which is merely to explain qualitatively
those facts already learned in practice, the situation represented in the
figure is quite justifiable.
Let us take the case of reflection from the centre sphere in Fig. 88a.
Light falls on the sphere in the direction LM, M being the centre of the
sphere. Owing to spherical aberration the spherical surface does not
produce a sharply defined image of the light source. It may be said
that the sphere forms an image of the light source on the line LM for
every point of incidence. Since the exact positions of the images do
not really affect our arguments, which are only qualitative, the composite image as a whole may- be denoted by L'. L' is thus the point from
which a pencil of rays emanates, the size of this being determined by
the amount of masking by the neighbouring particles. The average
limiting rays in the cross-section are thus L' A and L'B, corresponding to
the incident rays LP and LQ respectively.
It is seen from the figure that the beam of rays at the side which is
remote from the light source proper (the right hand side) runs further
towards the horizontal than that facing the light source (left hand side).
The angles of incidence with the normal to the sphere are greater on
the right hand side than those on the left; consequently, the reflection
factor of the surface of the sphere (and hence also the luminous intensity
of the image) will be higher on the right hand side than on the other.
These two facts together make it seem plausible that the luminance
factor, with respect to the oblique directions of incidence shown in the
figure will be higher in directions more remote from the light source
than nearer to it. This also explains why the maximum luminance factor
of many surfaces does not lie in the direction of specular reflection, but
in directions that make greater angles with the normal.
Let us now consider the effect of the solid angle of incidence (wi).
If we vary wi, this means that the images L' are varied in size in proportion with wi. For a given luminance of the light source, the luminous
intensities of the images will vary in proportion to the variation in wi
X-8]
LUMINANCE FACTOR
165
(and hence with the illumination on the surface). The observed luminance
of the surface accordingly also varies in proportion with wi, provided
that the masking effect of the neighbouring particles does not alter the
situation.
Since the images in this case lie so close beneath the surface, the luminance of the surface remains proportional to the illumination, and
the measured luminance factor will be constant provided that w I does
not change too much.
We will now consider the acceptance angle of the photometer.
The light accepted by the photometer depends on the light distribution
of the image L', as well as on the masking conditions. The light distribution curve will be very gradual and, except for glancing directions
of observation, the masking conditions will not vary much with the
direction of observation, so that the effect of the acceptance angle on
the final result will only be small under the conditions shown in the
figure.
In practice, divergences from the idealised situation as outlined will
have an equalizing effect on the light distribution, so we may say in
general that, with surfaces consisting of particles whose radii of curvature
~re of the order of size of the dimensions of the particles themselves,
the luminance factors are to a fairly large extent independent of wi
and e.
Let us now turn to the situation as depicted in Fig. 88b. Here the radii
are large compared with the size of the particles.
The images of the light source formed by the spherical surfaces lie
relatively far below the surface; the greater the radii of curvature, the
deeper the images, up to the point at which, with infinitely large radius
(perfeCtly plane surface), the light source and its image will be symmetrical with respect to the surface.
The effect of variations in wro that is, in the size of the images, upon
the masking of the reflected beam will be the more marked according
as the beams become narrower and hence the radii of curvature greater
compared with the size of the particles.
From the form and direction of the beam AL'B it is seen that the spread
in the reflected light is much less than in the case of the corresponding
beam in Fig. 88a, and that the reflected light is then much more directional. The indicatrix, therefore, is very much more pointed, which
means that the effect of the size of the acceptance angle is much greater
than that in Fig. 88a.
The size of w r and 8 thus both have a pronounced effect on the measured
166
REFLECTION, ABSORPTION, TRANSMISSION
[X
luminance factor and, when Wr and €J are reduced, the likelihood of
finding a constant value of {3 becomes less according as the radii of
curvature becomes greater compared with the particle size.
Since the reflection factors of the convex surfaces of the particles as
shown in Fig. 88b will not differ so much on average in the specular
direction from those in a more horizontal direction, such surfaces will
give a maximum luminance factor in or near the specular direction.
This is in contrast with the surfaces shown in Fig. 88a, where the maximum lies on the whole in more horizontal directions. This is borne
out in practice.
In paint coatings the particles as represented in Fig. 88a are covered
with a clear layer which more or less follows the surfaces of the particles;
in principle, such coatings assume the form shown in Fig. 88b and
possess the reflection properties described.
The light penetrating the outer layer is reflected in the manner shown
in Fig. 88a, but the reflection factors of the particles are only low, seeing
that the difference between the refraction indices of the outer layer and
the particles of pigment is small. The diffuse reflection from the particles
of pigment is thus of less significance than without the outer layer.
The transmission of the outer layer is greater. This can be clearly demonstrated by moistening a powder (e.g. glass powder) with water.
The dry powder is white because it diffuses the. light in all directions,
but, when the powder is wetted, the reflection factor is greatly reduced;
the transmission, on the other hand, is considerably increased. The
"whiteness", i.e. the relatively high luminance in all directions, almost
entirely disappears.
Recapitulating, then, we may say that luminance factors as constants
of the material will be encountered only when the radii of curvature
of the component parts of the surface are of the same order of size as
these component parts themselves.
Proceeding from the concept of uniformly diffuse, better: approximately
uniformly diffuse, through spread to regular and mixed reflection, it is
not possible to say just where the luminance factor ceases to be a constant
of the material. In the doubtful intermediate zones, measurements with
different solid angles of illumination and for photometer acceptance angles
will have to prove the existence or otherwise of a constant value for {3.
X-9]
GLOSS
167
X-9. Gloss
A problem closely related to the considerations outlined in the previous
s~ction is that of fixing a numerical scale for the effect generally referred
to as gloss.
Surfaces are described as glossy when, apart from some diffusion, they
show more or less perfect specular reflection. Measurements of gloss
are of considerable practical importance in the appraisal of paper and
of paint finishes.
To be effective, measurements of gloss must satisfy two requirements, viz.
I) the results of measurement of two surfaces giving the same im,pression of gloss to a normal observer must be equal;
2) higher and lower values as the results of measurement must correspond to impressions of higher and lower gloss.
We have seen in the previous section that surfaces which are practically
specular have no luminance factors as constants of the material. The
relationship between the measured luminance and the illumination is
wholly dependent on the measuring conditions, viz. the solid angle of
the illumination and the acceptance angle of the photometer. The
many investigators who have published definitions and results of measurement of gloss have all employed very widely divergent measuring
conditions (usually a fairly large wr and 0), and it is therefore not
surprising that their definitions and results show so little agreement.
It would be going outside the scope of this book to examine this subject
in greater detail. Readers wishing to know more about it are recommended to read Harrison's work "Definition and measurement of gloss"2).
This book contains an extensive discussion of almost all the literature on
the subject of the definition and measurement of gloss. It finishes with
a survey of exerything published on this problem up to 1945. The book
gives a clear impression of the confusion which reigns in this field.
Since I945 Harrison has carried out further extensive work on the
subject of the appraisal and measurement of gloss. His investigations
\vere concerned with samples of paper with various degrees of gloss and
of different colours. On the one hand he evaluated the gloss of different
kinds of paper by visual assessment, and on the other hand tried to
find a system of photometric measurements which gives results that
grade the papers in the same order as does the visual assessment.
Harrison takes three measurements: the paper being illuminated at
45o he measures (I) the maximum intensity at or ncar the specular
angle, 45°, (2) the intensity 5o nearer the normal, i.e. approx. 40°, (3)
168
REFLECTION, ABSORPTION, TRANSMISSION
[X
the intensity normal to the surface. The results are combined in a rather
complicated equation which yields a gloss number G. This gloss number
gives a fairly good correlation with the visual assessment under mean
viewing conditions but, according to Harrison's opinicn, the final
solution of the problem is herewith not yet found, since the gloss number thus established does not give an objective evaluation for all observers and for all viewing conditions,
X- I0. Reflection from road surfaces
Amongst reflecting surfaces in general, roadways present a problem
of their own; because of the importance of this in street lighting, therefore, we shall devote a separate section to the subject.
On a lighted roadway we see obstacles usually as silhouettes against
a brighter background, the latter being the roadway itself. The luminance,
or rather the distribution of luminance on the roadway is therefore the
important factor. It is useful to be able to compute this distribution
in respect of given lighting systems and roadways, and it is also important for the designer of lighting fittings to be in a position to ascertain the particular light distribution that his fittings will have to
provide in order to ensure the distribution of luminance on the roadway
that will promote optimum seeing conditions along the road.
Given the isolux diagram of the roadway, that is, the illumination at any
point, we can compute the luminance at every point (with formula (X-7 a)),
provided that we also know the luminance factor for each such point in
respect of the associated directions of illumination and observation.
To assist our considerations we depict the conditions relating to illumination and observation on
L
a roadway in Fig. 89. A light
source L is suspended above
the kerb of the roadway AB-CD.
0 is the eye of the observer.
In respect of a point P the
directions of illumination and
observation are determined by
the angles rp and J, and If/ and r
respectively. rp and If/ are measured from the normal at P, J and
r from a line passing through P
Fig. 89. Street lighting. AB-CD roadway,
parallel to the kerbs.
L = light source, 0 = observer's eye
The part of the road surface of
X-10]
REFLECTION FROM ROAD SURF ACES
169
importance to the direction of observation of the car driver is fairly far,
e.g. 30 metres, in front of him. This means that only values of 'I' greater
than 87.5° are of significance to our considerations. Measurements have
shown that the angle 'If, provided that it is greater than 87.5°, has little
effect on the luminance factor, and this, in turn,means that we can always
use the value at 'I' = 89° for p. Angle r is rather small for points on that
part of the road important as a background for the driver's observation,
and the effect of angle r on p can be ignored between zero and these
small values.
If we introduce these two approximations (ignoring the effects of 'I' and
ron p), the luminance becomes a function only of qJ and J. This simplification enables us to calculate the luminance of the road surface from the
light distribution of the fittings and data on p for the values found for
qJ and J, on the condition that the fittings can be regarded as point
sources.
This assumption means that P is constant for every part of a fitting,
although qJ and J differ slightly because of the finite dimensions. Fortunately it has been found that this is sufficiently accurate in the case of dry
road surfaces.
However, there are varying degrees of dampness for which the luminance
distribution is equally important.
As is known from experience, the reflecting properties of a roadway
change appreciably when the surface becomes wet; they tend towards
specular reflection and, if the rainfall is heavy enough to fill up all the
irregularities in the surface and thus produce a more or less continuous
layer of water, the reflection becomes almost completely regular and
the luminance factor ceases to have a fixed value.
With such regular reflection no other light reaches the eyes than that
which is reflected in the vertical plane through the light source and
the eye, so that, in very wet weather, narrow bright streaks appear
on the roadway, the latter appearing almost completely dark outside
the bright streaks.
Between the dry and flooded conditions many different modes of reflection occur, including those for which there is no fixed value of {J.
B e r g m a n s 3) has shown that under conditions such as those that
occur just after rainfall, that is, when most of the water has run off,
all types of road surface yield a fixed value of {J for directions of illumination whereby q; < 80°, i.e. with lighting fittings of the cut-off type.
This is important, because the wet concJitions persist, sometimes for a
considerable time. The distribution of luminance in the wet condition
170
[X
REFLECTION, ABSORPTION, TRANSMISSION
may therefore be satisfactorily characterised by the distribution in the
condition described.
We are giving h.ere a brief description of the method of de Boer, Onate
and Oos trij ck4) as an example of a method of calculating the luminance
of road surfaces.
Let us refer to Fig. 90 which represents the plan of a road. The observer's
f.ll--.1
4---tF-4~<<~~
·-
Fig. 90. Diagram of the reflection properties of a road surface in which the
curves represent projections of lamp positions for which point P has the same
luminance factor for the observer above 0'. The plane of the drawing represents
the road surface. The diagram can be used for distances between 0' and P greater
than 40 times the eye-level of the observer
eye 0 is 1.50 m above 0', and the observed point P is, say, 150 m from
the observer.
At some point on the road a light source is assumed to be at a height
hand to have a luminous intensity I cd in the direction of P. The illuI
mination at P is thus E = h2 cos3 rp. Under these conditions we now
measure the luminance L at P in the direction PO, which will enable
us to compute the luminance factor from
L
h2
{3=- = L - - - .
E
I cos3 rp
We imagine the light source displaced .along the road at the same height,
and measure in the above described manner the luminance factor at
the point P in the direction PO for a large number of positions of the
light source. At each position of the light source in which f3 is measured,
we note the value of {3, and, from the values thus obtained we derive
the locus by interpolation from the positions of the light source that
give the same values f3 at P. In this way a diagram such as that shown
in Fig. 90 is obtained.
In order to render the diagram universal the mounting height of the
light source is taken as unit of length (indicated in the diagram by the
line h). The effect of the variation in "P upon the values of f3 is negligible
at distances OP > 60 m.
X-10]
REFLECTION FROM ROAD SURFACES
171
it is clear that the curves in Fig. 90 are symmetrical with respect to
the line O'P provided that the structure of the road surface introduces
no directional effects.
The diagram in Fig. 90 does not give a spontaneous impression of
the contribution by the various light sources in the system towards
the luminance at P in the direction of the observer. A diagram can be
derived from Fig. 90, however, that will do this.
For positions of the light source which are far from the observed point
the values of {3 will be high. The horizontal illumination at P produced
in such positions, however, will be low; hence the contribution by these
sources towards the total luminance of P will be small, in spite of the
high values of {3.
h2
. From this the
As we have seen above, we may write {3 = L
I I cos3 q;
luminance can be computed as L = {3 cos3 q; h2 • The value of {3 can be
read from the diagram Fig. 90. For each position of the light source
in the diagram there is a corresponding value of q;. This leads to another
diagram in which the curves are the loci of the light source positions
yielding the same value of {3 cos3 q; at the point P in the direction PO.
The new diagram is shown in Fig. 91, in which the unit of length is once
more the mounting height h of the light source.
0.00'3
o'
~-
0.002
0.005
Fig. 91. Representation of the reflection properties of a road surface by means
of an E.P. diagram. The observer is over point 0' and looking at point P on the
road surface. The curves represent projections of lamp positions for which the
product {3 cos 3 qJ for point P as seen by the observer has the value indicated at
each curve. The diagram can be used for distances O'P greater than 40 times
the eye-level of the observer
The authors call this diagram an E.P. diagram in which "E.P." stands
for "equivalent positions", i.e. light source positions which are equivalent
172
[X
REFLECTION, ABSORPTION, TRANSMISSlON
in so far as the value of fJ cos3 qJ at P is concerned.
In order to compute the contributions towards the luminance at P
we read the values of fJ cos3 qJ for different light source positions from
a diagram of the kind shown in Fig. 91 and multiply these values by
the luminous intensity I in the direction of P, dividing the result by h2 •
For a lighting fitting that has the same luminous intensity in all directions, Fig. 9 immediately gives the luminance contribution of the
fittings in different positions.
For any given lighting fitting, the light distribution is usually known
and, if this be represented as an iso-candela diagram, projected on to
the plan of the road, the combination of E.P. diagram and iso-candela
diagram enables us to compute the luminance contributed by the fittings.
Fig. 92 illustrates the manner in which the E.P. diagram is used to
compute the luminance values for a number of selected points on a road
surface, more than 60 m from the observer.
I
I
0.002 I
0.005
/
/
I
I
I
I
---1
I "'
I
I /
-----O.o;
- - - ...... ..........,..:,9~
o
...
I
1/ ,--......
11
-.Q.o3 ............
_....L.-1-- -1-I U.--!..Lfo4aos;£;;"> _--:.....
a
0~
'
\
I
\
I
1 /~--
1
'
\ \
\ \
\
\
\
'
o'1o3' ' '
--
, ~00
-
... ..-
_..- - - - -
· - -·-7- _L!._
.......
L3
I
\ I \
·~\
\ ' '
/VV ' \
\
\
gg;•
\ - -Ls
c
/
I
o /
• I
I
I
I
/
h
1-----l
"' ,
, .. "'
..,--roo---... '
o........ .,.,. ....
..-
-"2-:::r' " \
I'.,~.. ' ' ) \I
\
I I
\
® J
\ \, L '.J I I
\
, ___ .....
,_.,...
/
.,...,.
.,.....-..--__ ........ -oQf> - - - - -
h
o.o.92,--
...... - -
640 • 400
2500
I rl'---,
I
--------,
\
I I I ~....
150
'
\
I~
__ ..,J 1 p / '
I
b
.......... ,..
' ...... _.....
.- ..............
\ \ l5\ \ ' ,......
'
--
,,___
640 400\ \
, .... - - , \ \ \
, •
J \.L.l
"'?-'~
\ , __ . ,.
\
....
'~
~
Jo
I I
1,11
"
'Lilt-' I I
1 ~ I
Fig. 92a. and b. Explaining the use of·the E.P. diagram.
a) To calculate the contributions of the light sources L 1 to L 5 towards the lu-
minance of the point P, a diagram of the lighting system on transparent
paper is placed over the E.P. diagram so that P coincides with point P in
the E.P. diagram, with the direction PO' lying on the axis of symmetry of
the E.P. diagr<).m. The values of fJ cos 3 rp can then be read off and multiplied
by the luminous intensities of the light sources in the direction of P to give
the contribution of each source to the luminance of P.
b) !so-candela diagram of the light sources projected on the road surface, from
which the luminous intensity values in the direction of P can bt> read
X-10]
REFLECTION FROM ROAD SURFACES
173
A plan view of the road with its lighting system (lamps L 1 to L 5) is
drawn on a sheet of transparent paper. The mounting height of the
lighting fittings is used as unit of length, as in the E.P. diagram. Let
P be one of the points in the plan, as seen by the observer. The transparent drawing is now placed over the E.P. diagram in such a way
that P coincides with P in the diagram. The situation is then as shown
in Fig. 92a. The values of {J cos3 cp for positions L 1 to L 5 of the light
sources are now read off and tabulated (column II in Table 1), after
which the luminous intensity in the direction of P is determined for
each light source. The simplest way of doing this is to project an isocandela diagram of one of the lighting fittings on to a plan of the roadway.
TABLE I
Lamp
Ll
L2
La
L,
L;
I{Jcos rp in P I I-+ p I Jt2.iJL
3
0.008
0.0045
0.02
0.015
0.005
40cd
640cd
325cd
205cd
190cd
0.32 cd
2.88 cd
6.5 cd
3.08 cd
0.95cd
1: 112.AL = 13.73 cd
h=8m
L = 0.214 cdfm2
This iso-candela diagram is in turn covered with the transparent plan
of the road so that fittings L 1 to L 5 in the plan successively coincide
with L in the iso-candela diagram. Care must be taken, that, if the
light distribution is not .axially symmetrical, the iso-candela diagram
is correctly positioned with respect to the roadway. In this way the
luminous intensity in the direction of P can be ascertained for any
given lighting fitting.
Fig. 92b shows the iso-candela diagram for L 3 . The third column of
Table I gives the luminous intensity values, and we have now to multiply the values in column II by those in column III. The result, entered
in column IV, represents the contribution L1L of each lighting fitting to the
luminance at the observed point P, multiplied by h2 • Addition of the
values in column IV, and division of this sum by h2 then gives the total
luminance of the road surface at P, as seen by the observer at 0.
This is repeated for each point, and, when this has been done for
a sufficiently large number of points, iso-luminance curves can be
plotted for an observer 0 on a road having a number of light sources
Lt .... Ln.
174
REFLECTION, ABSORPTION, TRANSMISSION
[X
This gives us the iso-luminance diagram in the plan of the road. For
an evaluation of the distribution of luminance as seen by the observer,
this plan should really be drawn in perspective, but, as the iso-luminance
lines then fall very close to each other, the diagram becomes difficult
to read. Conclusions can also easily be drawn from the plan view with
regard to the uniformity, or points of highest and lowest luminance, etc.
The method described and others known the literature 5) are much too
complicated for general use by lighting engineers. It is doubtful whether
a method of calculation can be found that is not too time-consuming
and contains no simplifications that can adversely affect the required
accuracy.
In public lighting practice a satisfactory result can often be attained if
research is restricted to the directions of observation of significance to
road traffic 6 ). Here a number of simplifications become possible,
especially in giving the average luminance of the road surface.
X-11. Transmission
The same distinctions can be made for transmission, in accordance with
the distribution of the transmitted light, as for reflection.
Hence we speak of regular, diffuse, uniformly diffuse, spread and mixed
transmission.
As the corresponding terms as applied to reflection have already been
defined, it is not necessary to give the relevant definitions with respect
to transmission.
Transmission can be characterised in the same way as reflection, by
means of luminous-intensity and luminance indicatrices. It is also
possible to determine luminance factors. With regard to the occurrence
or otherwise of fixed values, the influence of the solid angle of the illuminating beam of light and the acceptance angle of the photometer,
similar consid&ations may be said to apply as those outlined in
section X-5.
We shall now give a few examples of materials displaying the different
kinds of transmission:
a. Regular transmission takes place in clear glass and similar materials
which are called transparent.
b. Uniformly diffuse transmission is closely approximated by perpendicular incidence of light on opal glass. Here again, deviations from
uniformly diffuse transmission occur in the more oblique directions, up
to glancing inCidence.
X-11]
TRANSMISSION
175
With oblique incidence, opal glass also shows spread transmission.
Owing to the deviation which opal glass shows from uniformly diffuse
reflection and uniformly diffuse transmission, the luminance of an
opal glass bulb such as that of
some kinds of incandescent lamp
is not the same over the whole of
the apparent surface. Although
the luminance of such bulbs appears to the eye to be uniform,
it is, at the periphery, only 50
to 60% of the value at the centre
of the apparent surface. This also
applies to both tubular opal lamps
and tubular fluorescent lamps.
c. Spread transmission is producd Fig. 93. Luminance indicatrices of
by frosted glass. It is interesting to glass frosted on one side, illuminated
at an angle of 45°;
note that glass frosted on one side - - - - Light incident to smooth
side
yields different luminance indi- - - --- - Light incident to frosted
catrices according to the side from
side
which it is illuminated.
Fig. 93 depicts the luminance indicatrices of glass frosted on one side,
with· light incident at 45o *). It will be seen that when the smooth side
faces the light source the emergent rays are deflected towards the normal
and that, conversely, with the frosted side illuminated, the emergent
beam bends away from the normal.
This is important from the point of view of lighting fitting design, since
the illumination can be modified by frosting either the inside or the
outside of the glass.
The following may be noted in this connection.
When light falls on the smooth side of the glass it is deflected towards
the normal. If the other side were also smooth, refraction of the beam
on emerging from the glass would restore it to its original direction
(i.e. in Fig. 93, to 45°). If the side remote from the light source is frosted,
that is, if it consists of facets all oriented differently, it is found that
the principal direction of the transmitted beam is at an angle of 35°
with the normal. For the principal direction of the transmitted light
to show no variation when light falls on the frosted side, this direction
would have to be at an angle of 28° (with n = 1.5). On balance,
*) Measurements by the author.
176
REFLECTION, ABSORPTION, TRANSMISSION
[X
therefore, the light is deflected less from the normal than when the
reverse side is smooth.
When the light is incident on the frosted side of the glass it is not
deflected so much on average (in this case towards the normal) as
when it enters from the smooth side. At the smooth reverse side it is
deflected away from the normal, so that the principal direction of the
transmitted rays includes a larger angle with the normal than if the
glass were plain on both sides (according to Fig. 93 about 55°).
d. Mixed transmission. This occurs only rarely. With frosted glass it is
very rare; the glass would have to be so lightly frosted that it would
hardly be possible to speak of frosting at all.
Some opal glasses give regular transmission of part of the light (opalin
glass). When a clear incandescent lamp is placed behind an opalin glass,
the filament can be seen through it; apart from a marked reduction
in the luminance of the filament, a difference in the colour is to be seen,
this being much redder than otherwise~ The reason for this is that in the
process of diffusion by particles of the same order of size as the wavelength of light (Rayleigh diffusion), the shortwave rays are diffused
more strongly than those of longer wavelengths.
This phenomenon explains the blue colour of a cloudless sky the sunlight
is diffused by the small particles suspended in the atmosphere and,
since short-wave radiations are diffused to a greater extent than those
of longer wavelength, the diffuse light is blue.
X-12. Density
In photography it is useful to know the transmission factor of developed
negatives and diapositives, as a function of the exposure i.e. the quantity
of light thrown onto the light-sensitive layer (sensitometry).
This is also important in cases where plates or films are used for the
measurement of light, e.g. in spectro-photometry or in measurements
of luminance (photographic photometry; section XV-5).
Instead of transmission factor the conception density is employed
(symbol 5), the definition of density being the logarithm of the reciprocal
of the transmission factor. We accordingly write:
S =-log T.
Suppose that the density is 3. Then T = I0- 3 = 0.001.
When it is remembered that the transmission factor is defined as the
quotient of the total transmitted luminous flux divided by .the incident
X-12]
DENSITY
177
luminous flux, it will be seen that the above definition of density does
not correspond in most cases with photographic practice.
It is only when prints are taken from negatives, the sensitive paper
being in contact with the negative, that the total transmitted luminous
flux is used. When an image is projected by means of a lens, (e.g. in
enlarging a negative, in diapositive projection or in the examination
of an X-ray film), only that light which falls on the projection lens, or
on the lens of the eye is utilised, and light diffused beyond this is of no
consequence.
In such cases the total luminous flux falling on the image is of no consequence either; we are interested only in the luminous flux reaching
the projection lens without the photographic image in front of it.
The total incident luminous flux is important, however, in the case
of contact prints.
In accordance with the conception of density as actually employed
in photography and as measvred by means of densitometers, this should
be stated as the logarithm of the attenuation which the luminous flux of a
given system of illumination fa!Ung on a "receiver" undergoes when a
photographic plate or film is placed between the lighting system and the
receiver. The receiver is understood to be a projection lens, the eye, or
the sensitized paper used for making contact prints.
if the attenuation is, say 100 times, the density will be equal to log 100 = 2.
Owing to the diffusing properties of the silver granules in the photographic material, one and the same object will yield different values
according to the solid angles of the incident light and of the receiver as
seen from the plate or film 7 ). The density measured, therefore, depends
on the design of the densitometer employed.
If we measure the density of a plate or film in accordance with our
second definition, using a small acceptance angle, first with perpendicularly incident and then with diffuse incident light, we obtain a higher
density with the former than with the latter. This is obvious when it
is remembered that, owing to the diffusing properties of the silver
granules, a certain amount of light is included in the measurement
which originates from light falling obliquely on the plate and which,
when the lighting system is measured without the plate, does not enter
the receiver.
C a 11 i e r ( 1909) found that the two densities measured thus differ
by a factor which, in general, is a constant for any kind of plate or film.
This is known as the Callier coefficient and the effect itself as the Callier
effect.
178
REFLECTION, ABSORPTION, TRANSMISSION
[X
Let us now demonstrate the practical importance of the Callier effect
by means of a numerical example.
We will suppose that we enlarge a plate or film, once with diffuse light
and once with light from a condenser, which can be regarded as directional
light. Let us also assume that two areas in the plate have densities of
I and 2, respectively, when measured with diffuse light. The light passing from the diffuse background of the plate to the projecting lens
is thus attenuated down to 10% and I%, respectively, and the luminous
flux values of the pencils of light in the two areas are thus in the ratio
of I 0 : I.
If the Callier coefficient of the plate is 1.2, the densities with directional
light will be 1.2 and 2.4, corresponding to an attenuation of the beams
to 7.6% and 0.575%. The ratio of the luminous flux of the beams is
then 13.2 : I, i.e. the contrast between the two areas, which was I 0 : I
in diffuse light is 13.2: I with light from the condenser.
This explains why negatives enlarged in an enlarger with condenser,
and thus receiving light from directions which differ but little, arc
"harder", i.e. show much greater contrast than when enlarged with
diffuse light.
Apart from the method of illumination and the acceptance angle of the
photometer, density values must be given in reference to the kind of
light used for the measurement, since the diffusing properties of silve;
granules are dependent on the wavelength of the incident light.
X -13. Diffusion factor
In the course of time numerous suggestions have been made for the
characterisation, by means of one value, of the diffusing properties
of diffusely reflecting or transmitting materials. For this purpose the
obvious thing to do is to take the case where the incident light falls
as a parallel beam perpendicular to the surface, since this yields a symmetrical indicatrix.
Fig. 94. Explanation of the concept of diffusion factor according
to H a I b c r t s m a. The surface
of which curve A is the luminous
intensity indicatrix, has a diffusion
factor <;f
1,.
a=-1 ma.r
I max
X-13]
DIFFUSION FACTOR
179
The total average illumination is thus found to be
if>
Em= 5 (I + e + e2 + .... ) =
if>
5 I_ e
(X-9)
1
Suppose that e is, say, 0.75, so that - - = 4. The average illumi1-e
nation owing to the multiple reflection is then 4 times as much as without
1
reflection at the boundary surfaces. If e = 0.2, i.e. - - - = 1.25, the
1-e
increment is 25%. As a result of multiple reflection it is possible for
the coefficient of utilisation to be more than 100%, which represents the
curious situation in which more light falls on the working area than is
emitted by the light source. However, this obviously conflicts with the
law of the conservation of energy. It must be remembered that the
light falls on the working surface, but is only partially absorbed; the
energy is not dissipated at the working plane, but is in turn returned
into space.
Equation (X-9) can also be derived along other lines. The luminous
flux absorbed by the boundary faces at any moment (if>abs) is equal to the
luminous flux if> emitted by the light source at that moment. If this
were not so, the luminous flux in space W(}u]d increase steadily; hence
if> = if>abs"
The absorbed luminous flux is also equal to the total incident luminous
flux if>; multiplied by the absorption factor oc (= 1 - e), so that we
may write
if>a bs =if>= if>. (1- e),
l
from which it follows that
1
if>.= i f > - - .
1- e
•
To·obtain the average illumination Em we divide the incident luminous
flux by the illuminated area S, so that
if>
1
Em=-.--.
s 1-e
If the boundary surfaces transmit light (transmission factor -r), it is
possible to calculate what part of the luminous flux is transmitted.
Suppose, for example, that in the case of a lighting fitting consisting
180
[X
REFLECTION, ABSORPTION, TRANSMISSION
For a long time the diffusion factor propos<:;d by H a I bert s m a 8 )
has been in use. This may be defined in reference to Fig. 94. Let curve A
represent the luminous intensity indicatrix of a surface element with
perpendicular incidence of the light. Further, let I max denote the maximum luminous intensity and C/J the luminous flux after diffusion. If
this luminous flux were uniformly diffused, the indicatrix would be the
circle B, for which the maximum luminous intensity is C/Jfn.
According to Halbertsma the diffusion factor (a) is the ratio of
maximum luminous intensity with uniform diffusion C/Jfn to the actual
maximum luminous intensity (/max); hence
<P
n
<P
(X-8)
a=--=--.
!max
nlmax
Now, nlmax is the luminous flux of a uniform diffuser with 1 = [max
(indicatrix is the circle in C Fig. 94). We can thus also define a as the
ratio of the luminous flux actually diffused to that which the surface
would radiate if it were a uniform diffuser having the same maximum
luminous intensity. If the surface is actually a uniform diffuser, a = I.
In 1939 the C.I.E. gave a different definition of diffusion factor, viz.
the ratio of the average luminance at angles of 20° and 70° to that
at 5° with perpendicular incidence.
0
X-14. Multiple reflection
In section VII-2 we have stated that the coefficient of utilisation (viz.
the illumination) within a given space depends on the reflecting properties of the walls and ceiling. If we assume that both these and the
floor, i.e. all the boundary planes have the same reflection factor (!,
the effect of the reflection factor can be demonstrated by simple calculation.
Suppose that a light source of luminous flux <Pis suspended in the room;
this luminous flux will then produce an average illumination <PfS on the
boundary surfaces, where S is the total area of these surfaces. Of this
luminous flux a part e<P is reflected and distributed over the total space,
thus producing an average illumination of
e;.
Again, part of this
secondary incident flux f! 2<P is reflected, resulting in an illumination of
e2<P
.
S' and so on, ad infinitum.
X-14]
MULTIPLE REFLECTION
181
of a glass globe, e and o of the glass are known; we can ascertain the
efficiency, assuming that the whole of the globe surrounds the light
source, thus disregarding any cover plates for lamp entry-holes. Of
the incident luminous flux f/>, a part of!> is transmitted, and another
part ef!> is reflected; the reflected light again falls on the wall, causing
o(!f!> to be transmitted and e2f/> to be reflected. Of this, oe 2f/> is then
transmitted and e3rP is reflected and the total transmitted light is equal to
of!> (I
+ e + e2 + ea + ....) = f/> -I --og.
The efficiency of the lighting fitting is therefore
o
1J = I - e·
(X-IO)
A globe made of clear glass, with o = 0.90 and e = 0.08 thus has an
efficiency of
o
0.90
1J = - - = - - - - R; 0.98, or 98%.
I - e I - 0.08
If the luminous flux were incident to a flat pane of the same glass,
only 90% would be transmitted; the repeated reflections within the
hollow globe increase this to 98%.
Frosted glass gives various values of (!, ex and o according to whether
the light falls on the plain or frosted side, and this explains why the
efficiency of inside frosted lamps is higher than that of lamps with the
same frosting on the outside of the bulb. Measurements of the reflection,
absorption and transmission factors of some frosted glasses have been
carried out by Pi rani and S c h 6 n born 9 ), who have come to
the conclusion that when light falls on the frosted side (as with inside
frosting)! the reflection factor is lower, the absorption factor also lower
and, hence, the transmission factor higher than with incidence from the
other side (as in outside frosting).
Let us now take as an example one of the kinds of glass investigated
by Pi rani and S c h 6 n born, in order to show that if lamp
bulbs were to be made of such glass, the efficiency of the inside frosted
bulb would be higher than that of the outside frosted one. They find
that for a particular kind of glass with perpendicular incidence:
light on frosted side: e = 0.087, ex = 0.055, o = 0.865;
light on plain side: e = 0.123, ex= 0.09, o = 0.78.
182
REFLECTION, ABSORPTION, TRANSMISSION
[X
Calculation of the efficiency *) using these valu~s gives:
0.865
0.087 = 0 ·95 •
Inside frosted:
17 = I -
outside frosted:
17 = I - O.I23 = 0 ·88 ·
0.78
The other samples of glass investigated by P i r a n i and S c h o nb o r n yielded similar differences.
The calculation thus explains the fact that the efficiency of inside
frosted lamps is a few per cent higher than that of the outside frosted
type.
P i r a n i and S c h 6 n b o r n suggest a very acceptable explanation
of the fact that the absorption is higher when the light is incident on
the plain side of the glass. In this case the likelihood of total reflection
from a number of facets when the light emerges from the other (frosted)
side, is much greater than when the light enters from the frosted side.
If, after total reflection, the light leaves nevertheless at the frosted side,
the greater absorption takes place because of the longer path that the
light has to travel. The total and almost total reflection that occurs with
smaller incidental angles when light passes from glass to air than when
it enters the glass from the air is thus the reason for the higher reflection
factor with incidence of the light on the smooth side.
In practice, of course, there is no such thing as a glass bulb consisting
of a complete hollow globe in one piece. There is always an opening
through which the lamp is inserted, and also some suspension device
usually in the form of a cover plate, on which a lampholder is mounted
with a bush and ring. If we wish to know the efficiencies of a number
of different globes, t_he effect of reflection from the suspension device
upon the efficiency must be eliminated as far as possible; this can be
done by blackeqing the inside of the cover plate. The luminous flux
of the lamp with suspension device, but without globe is measured
first, then that of the complete lighting fitting, obviously with the lamp
in the same position with regard to the suspension device. The ratio
qf the one luminous flux value to the other is then a measure of the
efficiency of the globe, or the glass efficiency.
*) For this calculation we should not actually use the values for perpendicular
incidence since, owing to the diffusion, other angles of incidence occur as well, to
which other values of p, ex and T relate. There is no objection, however, to the
use of these values merely to show the tendency to a difference in efficiency.
X-15]
183
ABSORPTION
X-15. Absorption
We shall be looking iqto the way in which absorption is governed by the
thickness of the absorbing layer (see Fig. 95).
We assume that the absorption is a for a layer thickness of, say, d mm.
a is the percentage of the penetrating luminous flux absorbed. The
penetrating luminous flux is equal to the incident luminous flux minus
the reflected luminous flux. 1 - a, therefore, is transmitted.
I
I
d
#;
..,..
I
I
I
I
I
I
I
I
I
I
I
I
d
I
•'•
I
I
I
I
I
I
I
I
I
I
I
I
d .. I
I
I
I
I
#u
I
I
I
I
I
I
Fig. 95. If the absorption in the layer thickness dis a, <1>11>11 = <Pt (1 - a)n will be
transmitted through n layers of thickness d, <Pt being the penetrating luminous flux
If we add a layer of the same thickness to this layer, so that the total
thickness of both layers becomes 2d, a luminous flux of 1 - a enters the
second layer. Here again, a portion a is absorbed, i.e. a ( 1-a). In both
layers together, then, a + a(l-a) is absorbed, so that part 1 - (a +
a (1- a)) = 1 - a - a + a2 ~ (1- a)2 emerges from the second layer.
This luminous flux enters a (possible) subsequent layer of the same
thickness (the total thickness thus becoming 3d mm), in which a part
a ( l - a)2 is absorbed. The total quantity, therefore, absorbed in the three
layers is a + a(l -a) + a(l - a)2 and that transmitted is
1 -{a + a (l -a) + a (l - a)2} = (1 - a)3.
The survey below shows the ratios on absorption.
in lst layer
in 2nd layer
in 3rd layer
in nth layer
absorbed
transmitted
a
a (1 -a)
a (1 - a)2
a (1 - a)n-1
1-a
(l - a)2
(l - a)3
(l - a)n
If we designate the penetrating luminous flux by fiJi and the luminous
flux transmitted through n layers of thickness d by f!Ju,n, we can write
f!Ju,n =fiJi (1- a)n
184
[X
·REFLECTION, ABSORPTION, TRANSMISSION
or, in logarithms,
log cPu,n =log cPi
+ n log (1- a),
where a represents the part of the penetrating luminous flux absorbed in
one layer.
As can be seen, we are using the transmitted light here because this makes
for simpler calculations than the absorbed light.*
Once the absorption in a Ia yer of thickness d mm is known (e.g. a!),
we can calculate the absorption (a 2 ) in a layer of d2 mm in the following
way.
If we take the absorption in a layer 1 mm thick as a, with a thickness d 1 :
cPu,d! =
and at thickness d2 :
cPt (1 -
a)di
Now
and
We can therefore write
and
a
1 - a = (1-a 1 ) 1/di
From the first equation we obtain for 1 If we insert this value for 1 -
a in the second equation, we obtain
1 - a 2 = (1- aJ)di/d2
(X-11)
The absorption in the layer d 2 mm thick is thus
1-
(1 -
a 1)d2fd1
(X-12)
Examples
l. A layer I mm thick absorbs 10% (a = 0.1) of the penetrating light.
A layer 5 mm thick will then transmit
C/Ju =
C/11 (1-a) 5
= cJ>1 (1-0.1) 5 = 0.595 C/Ji
The absorption of 5 mm is therefore 40.5% of the penetrating luminous
flux.
2. A pane of glass reflects 8% of the incident luminous flux. At a thickness
of I mm, 20% (a = 0.2) of the penetrating light is absorbed. How much
*) To simplify we shall not take into acount the reflexion on the exit face of the absorbing
medium
X-15]
185
ABSORPTION
light will a 3 mm thick pane of this type of glass transmit? At a reflection
of 8%, 92% enters the pane.
Of this,
tP,. - tP1 (1-a)3 = tP1 (1~0.2)3 = 0.83 tP1 = 0.512 tP,
will be transmitted through a thickness of 3 mm. Therefore,
0.512 X 0.92 = 0.471 = 47.1%
of the incident luminous flux will be transmitted.
3. A 4 mm pane absorbs 15% of the penetrating luminous flux. How
much light will a 6 mm pane of the same material absorb?
According to equation (X-12), the absorption in the 6 mm layer is
1- (l-a 1)d2fdt = 1- (1-0.15) &f4 = 1-0.85 &f4 = 1-0.784 = 0.216 = 21.6%
of the penetrating luminous flux.
We shall now derive two equations for the absorption in diffuse and nondiffuse media, using infinitesimal calculus.
a. Absorption in a non-diffusing medium
When a luminous flux fP 0 is incident at right angles to the interface
of a non-diffusing medium, part of this flux is
reflected and the rest enters the medium; in the
latter, absorption takes place. When the light
travels through a distance dx in the medium
(see Fig. 96), that part which is absorbed is:
df/J = -a . f/J . dx
Fig. 96. Derivation of
the absorption formula IIi,. = lfi; e-•d
(X-13)
where ~ is a constant of the medium. The second
term is given the negative sign to indicate that
df/J is a reduction in the luminous flux.
Equation (X-13) can also be written as:
-=-a
rp
. dx .
df/J
....
Integration gives the absorption for a finite distance of travel, viz.
f !rp J=
a . dx
+C
lnf/J =-ax+ C.
The constant C can be computed for x 1= 0, in which case C = In f/J;
f/J is the luminous flux entering the medium, which we shall denote
by f/J;. Hence:
In f/J = -ax
In f/J;
+
186
REFLECTION, ABSORPTION, T,RANSMISSION
[X
from which it follows that:
If we imagine a medium of thickness d (e.g. a filter) and denote the
luminous flux perpendicular to the exit face by f/J,.:
(X-14)
The coefficient a is known as the transmissive exponent.
Formula (X-14) is valid only for perpendicular incidence and emergence
of the light at the interfaces; for other directions the path travelled is
not equal to d, but is greater.
b. Absorption in a diffusing medittm
If the medium represented in Fig. 96 is diffuse, the luminous flux passing through a layer dx in thickness will be partly absorbed. In contrast
with transparent media, however, some of the light is diffused in all
directions in the layer dx, i.e. also rearwards. This luminous flux is again
in part absorbed by layers lying more towards the rear, and in part is
diffused, the latter component being thus added to the light falling
on the layer dx. This absorption and diffusion is repeated ad infinitum,
so that ultimately a part of the light falling on the layer dx reaches the
other side and another part does not. Let the last mentioned part be
denoted by df/> = -af/>dx; we can then derive a formula for the light
transmitted by a layer of finite thickness d, which will be analogous
to equation (X-14), viz.
(X-15)
This equation is based on the assumption that the light entering all
the layers dx is similarly diffused. If the incident light is diffuse, the
distribution of the light entering the layer is the same for all layers
dx from the moment of entering the diffusing medium, and equation
(X-15) is then valid for any layer thickness.
When light enters the diffusing medium from one direction only, the
distribution of the transmitted light becomes uniform only after the
light has penetrated to a certain depth; in the process of diffusion in
the initial layers the light remains to a certain extent directional. Formula (X-15) can be employed for directional light only when the medium
is sufficient! y thick.
X-16]
ABSORPTION OF COLOURED FILTERS
187
X-16 Absorption of coloured filters
Coloured filters absorb differently in the different parts of the spectrum.
Thus they alter the spectral composition of the light falling on the filter
from a light source, if the light is not monochromatic, i.e. does not contain one wavelength only. This gives the visual impression of a change in
the colour of the light.
If a filter of a certain composition and thickness produces a certain
colour in the light transmitt~d, the colour will change if the thickness
of the filter is altered. This can be explained with reference to Fig. 97.
o.a
('
I
'To.&
o.s
11.4
Q;l
0,2
Ol
0.7
~m)
Fig. 97. Change in the relative spectral composition, and thus in the colour of light
passing through a coloured filter, when the thickness of the filter is altered.
In Fig. 97, curve I shows the luminous flux transmitted through
a filter of thickness d, i.e. the value of 1 -a in the previous considerations.
If the thickness of the filter is doubled, the transmission at each wavelength can be calculated from (1 - a) 2 • The result is shown in curve II
in Fig. 97. The ratios of the intensities of the various wavelengths have
now changed, e.g.
with 0.55 p.m, ( 1 - a) in curve I is 0.6
and (1 - a)2 in curve II is 0.36,
with 0.5 p.m,(l -a) in curve I is 0.22
an.d (1 - a)2 in curve II is 0.045.
188
REFLECTION, ABSORPTION, TRANSMISSION
[X
Whereas the ratio for the wavelengths 0.55 and 0.5 pm in curve I is
0.6 : 0.22 = 2. 7, in curve II it has become 0.36 : 0.045 = 8.
This change in the transmission ratio occurs at all wavelengths. Thus
the colour of the light transmitted changes with the thickness of the
filter.
X-17
Colour of reflected light
We have seen that with media which transmit light fairly well, a large
part of the reflected diffuse luminous flux consists of light that has
penetrated the granules and has ultimately been re-emitted after repeated
reflection and refraction. If the medium is colourless, i.e. if light of all
wavelengths is transmitted and absorbed to the same degree, the spectral
composition (the colour) of the reflected light will be the same as that
of the incident light.
On the other hand, if the medium is coloured, that is, if more light of a
certain wavelength is transmitted, or if less is absorbed than for other
wavelengths, the spectral composition i.e. the colour of the transmitted
light will differ from that of the incident light.
For example, if transmission in the green is high, the diffuse light transmitted after repeated reflection and refraction of white light penetrating
the medium will also be green. In such media there is relatively little
reflection compared with transmission, so that, when there is not so much
absorption, we have the curious fact that the colour of the diffusely
reflected light is the same as that of the transmitted light. Thus the
leaves of trees are green because the chlorophyll in them transmits
more green than red or violet.
A very different effect is produced by materials which absorb much
light and transmit little, as do metals; the light is then absorbed on
penetrating to a depth equal to only a part of the wavelength. At the
same time, the penetration is so restricted that light entering is not
attenuated, but is only reflected. In such cases, therefore, the colour
of the reflected light is again determined wholly by the absorption
factor, but in the sense that it is just those wavelengths for which absorption is highest that are reflected best. An example of this is found
in copper, which in very thin layers gives green transmitted light, but
which when thicker gives a red reflection. The same thing occurs in
some non-metallic substances such as methyl violet, which as a dilute
solution absorbs green light and in the form of crystals also gives a green
reflection.
X-17]
COLOUR OF REFLECTED LIGHT
189
REFERENCES
1) J o h. J an sen, Ph. T. R 5, 1940, 125-130. "The Distribution of the Light
:reflected by different Ceiling and Wall Materials"
2 ) V. G. N.
H a r rison : "Definition and Measurement of Gloss". Published
by the "Printing and Allied Trade Research Association", London 1945
V. G. N. Harrison and S. R C. Po u 1 t e r, Res. Sci. Appl. Ind. 7, 1954,
128-136. "Gloss Measurement of High-glass Papers"
3 ) J.
B e r g m a n s : "Lichtreflectie door wegdekken" ("Light Reflection by
Road Surfaces"). Thesis, Delft 1938; see also: J. Bergman s, Ph. T. R 3,
1938, 313-321. "The Brightness of Road Surfaces under Artificial Illumination"
4) J. B. d e B o e r, V. 0 ii a t e and A. 0 o s t r ij c k, Philips Res. Rep. 7,
1952, 54-76. "Practical Methods for Measuring and Calculating the Luminance
of Road Surfaces"
J. B. d e B o e r and A. 0 o s t r ij c k, Philips Res. Rep. 9, 1954, 209-2'24.
"Reflection Properties of Dry and Wet Road Surfaces and a simple Method
for their Measurement"
S) J. M. Waldram, Ill. Eng. 27, 1934, 305, "Road surface reflection characteristics and
their influence on street lighting practice"
A. Bloch, Tmns. I.E.S.. (London), 8, 1939, 113-128, "Light scattering by road
surfaces. A theoretical treatment"
A. J. Harris and A. W. Christie, Public Lighting 19, 1954, 553-569 "Relative
importance of the variables controlling street lighting performance"
H. R. Ruff and G. K. Lambert, Public Lighting 22, 1957, 177-190, "Relative
importance of the variables controlling street lighting performance"
J. B. de Boer, Monograph no. 4 of the I.E.S. London (1962), "The concept road
surface luminance and its application to public, lighting"
6 ) J.
Bergman s and W. L. Verve s t, Ph. T. R 5, 1940, 222-230. "A
New Fitting for Road Lighting"
7) G. A.
Bout r y: "Mesure de densites photographiques par la methode
photo-electrique" Publ. scient. et techn. du minist. de l'air, Paris 1934
J. E. de G r a a f, Zts. f. Wiss. Photogr. 37, 1938, 147-159. "Zur Densitometrie von Rontgenfilmen und ihrer Normung"
8 ) N. A. H a 1 bert s m a, E.T.Z. 39, 1918, 207-209. "Die Streuung (Diffusion)
des Lichtes als Mittel zur Verringerung der Flachenhelle kiinstlicher Lichtquellen''
9 ) M. Pi rani and H. Schon born, Li. u. La. 15, 1926, 458-460, "LJber
den Lichtverlust in mattierten Glasern"
CHAPTER XI
PROPERTIES OF OPTICAL SYSTEMS
XI -I. Introduction
In optics, an optical system is understood to be a system of refracting
andjor specular reflecting surfaces, i.e. lenses or mirrors, as used in
combination with an object which can be reproduced by the optical
system. Here, the object may be a primary or secondary light source.
The paths travelled by light rays through optical systems are dealt with
in geometrical optics, but this largely falls outside the scope of this
book 1). Here we are interested in the characteristics of the light emerging from optical systems, i.e. the luminous intensity, the luminous
flux and its distribution, the luminance of the light and also the illumination produced by such systems. Nevertheless, in discussing these,
we shall in many cases be unable to avoid taking into consideration the
paths of the rays through the optical system.
In the following, therefore, a certain knowledge of the behaviour of
light rays in optical systems is essential and, as this work is not intended
as a textbook on geometrical optics, we shall have to assume that the
reader possesses the necessary knowledge.
The luminous characteristics of optical systems will, in general, have
to be considered in conjunction with the object, i.e. the light source.
At the same time, certain characteristics can also be attributed to
optical systems without an object which are important from the point
of view of lighting calculations.
In order to avoid ambiguity we shall speak of optical systems when we
mean systems of refracting andjor reflecting surfaces combined with a
light-emitting object, and of the optics where only the system of refracting
or reflecting surfaces is concerned.
In the following we shall show how the luminous characteristics of
optical systems can be computed and, conversely, how such systems
can be designed to meet given requirements as regards their luminous
properties.
The luminous characteristics of optical systems are governed by, apart
from the geometrical-optical rules, Abbe's law, which we shall now
explain.
XI-2]
LUMINANCE AND REFRACTIVE INDEX
191
XI-2. RelationsJrlp between luminance and refractive' index; A b b e•s
law
In the previous chapter we have seen that when light falls on the interface
between two media, a part of it is reflected from the face and another
part passes through it into the second medium; we have also seen what
happens to the luminous flux; this is partly or even wholly absorbed.
Let us now look into the question as to how the luminance and luminous
intensity are modified when light passes from one medium into another.
Obviously, there is no object in considering anything but a medium
the absorption of which is low (clear glass etc).
It is not possible a priori to say anything definite about what happens
to the luminous intensity, since the refraction modifies the apparent
area of the light source. It may be enlarged or reduced, dependent upon
the shape of the refracting face, the ratios of one index of refraction
to another and also upon the distance of the light source from the
refracting face.
As to the luminance, this quantity varies with the refraction, and the
law to which this obeys can be ascertained; in doing this the reflection
and absorption losses will for the moment be disregarded.
In Fig. 98, L is a light-emitting surface radiating in a medium I, the
index of refraction of which is n1 • The light from L falls on the interface G
of a medium II whose index of refraction is n 2 • At a point P on G we
imagine a very small area of size LIS which alone is transparent; around
LIS, therefore, no light is transmitted. NP is the normal to G at P. It is
assumed that LIS is so small that all the normals at LIS are in the same
direction, that is, parallel to NP. The luminance of L in the direction
of P is denoted by L 1 .
The rays are refracted at P and, after this refraction, the luminous flux
incident at P again forms a beam.
The aperture LIS is virtually a diaphragm situated within the luminous
flux emitted by L. As shown in section IX-2, a diaphragm of this kind
functions as a light source with respect to the space in front of it. The
luminous intensity of the diaphragm is thus equal to the luminance
of the light source that can be imagined behind it, multiplied by the
area of the diaphragm. It is a condition, however, that the whole of the
diaphragm be flashed.
Since the solid angle after refraction (w2) corresponds to that before
refraction (w 1), LIS as seen from w 2 is completely flashed with the
luminance produced by the radiation in medium II; this we shall denote
by L 2 •
192
PROPERTIES OF OPTICAL SYSTEMS
[XI
Accordingly, LIS, for medium II, may be regarded as a light source
emitting light in the solid angle w 2 with a luminance L 2 •
L 2 can be computed in the following manner.
Fig. 99 shows a narrow beam of light originating from a part of the light
source Lin Fig. 98. This beam is incident on the interface of medium II
at P, which lies in the plane G' as drawn and representing the tangent
plane at P to the interface G in Fig. 98: Hence the normal NP and all
the normals to the small area LIS· are perpendicular to G'.
Fig. 98.
Figs. 98 and 99. Illustrating
A b b e's law. The luminance values in media I and
II, having indices of refraction of n 1 amd n 2 respectively, are proportional to
the squares of the indices of
refraction
Fig. 99.
For the purposes of our calculation we introduce in medium I a plane
V1 at any point between P and L, and parallel to G'. The normal NP
intersects V1 at M1 .
The narrow beam of light from L is defined in the following manner.
Two circles are drawn in V1 with M1 as centre; if points on these circles
are now connected to P, two cones are produced, the half-apex angles
of which are i 1 and i 2 • From the area between the two circles we now
select a part L' bounded by two radii of the concentric circles; these
radii enclose an angle ex.
The beam formed by connecting the periphery of L' to P is then that
XI-2]
193
LUMINANCE AND REFRACTIVE INDEX
which we assume to be emitted by a part of L. L' is thus the equivalent
light source for that part of L.
L 2 is now determined by successively:
a) calculating the luminous flux emitted by L' towards L1S with a
luminance L 1 ;
b) calculating the luminous flux emitted by L1S in the beam after
refraction, with an unknown luminance L 2 ;
c) equating the luminous flux values as calculated in a) and b) and
evaluating L 2 from the equation. The reflection at P and absorption
in the media are thereby disregarded.
a) Calculation of the luminous flux emitted by L' towards L1S
According to equation (IX-4) the illumination of G at P as produced
by that part of V1 which is enclosed by the inner circle is: E 1 = nL 1 sin 2 i 1 ;
that produced by the part of V 1 which is enclosed by the outer circle is:
E 2 = nL 1 sin 2 i 2 •
The illumination of G1 at P, produced by that part of V1 which lies
between the two circles, is therefore
Ep = E 2 -
E 1 = nL1 (sin 2 i 2 -
sin 2 i 1 }.
(XI-I)
Since the circles are symmetrical with respect to the normal NP, each
sector contributes in similar fashion to the illum~nation at P, and it
may thus be said that the illumination at Pis proportional to the central
angle of the sectors of the circles, i.e. also to the centre angle of the
parts of the sectors lying between the two circles.
The illumination E P in equation (XI -I) refers to the zone between the
complete circles, that is, to the central angle 2n rad. For L', being the
difference between two sectors of central angle IX rad, the illumination
E L'-+ p at p is
IX
E L' _.. p = 2n n L 1 (sin 2 i 2 -
sin 2 i 1 ) = "2 L 1 (sin 2 i 2 IX
sin 2 i 1 }.
The luminous flux (/JL' ._..LIS incident to L1S from L' is then
</JL'
-..LIS=~ L1S. L 1 (sin 2 i 2 - sin 2 i 1 ).
(XI-2)
b) Calculation of the luminous fhtx from L', emitted by L1S in medium II
after refraction
To make this calculation we first see .what happens to the rays emanating
from L' in the process of refraction, usi~g Snell's law.
194
PROPERTIES OF OPTICAL SYSTEMS
[XI
Air the rays following th~ generators of the cone whose half-apex angle
is i 1 (of the ihnex: circle), and which therefore make an angle i 1 with the
normal at P, <;~.re refracted at P in such a way that the angle from the
normal changes to rl according to the equation:
Similarly, for the rays from· the outer circle on V1 , we write:
--=-
sin: r 2
n1
These rays thus once more form conical surfaces, the half-apex angles
of which are r 1 and r 2 respectively.
The rays of light from the radii of the two circles bounding the area L'
occur in two planes passing through the normal, and these rays remain
in the same planes after refraction.
In Fig. 99 a plane V2 is shown in medium II, parallel to V1 and G'.
The conical surfaces o,n which lie the rays emanating from the two circles
. on vl intersect this plane to produce two circles; the rays from the lines
M1 A1 B 1 and M1J) 1 C1 thus pass through lines M2 A2B 2 and M2D 2C2 respectively. The angle formed by these lines is again oc and the lines are
parallel to the corresponding lines in plane V1 • This is illustrated in the
figure by two reference lines M1E 1 and M2E 2 , both of which are horizontal
in the figure. Angles E 1 M1 C1 and E 2M2C2 are then equal ({3).
Now, i.f the luminance of L1S within the cone having a half-apex angle
r 1 be denoted by L 2 , L1S will be a light source, the luminous intensity
of which in the direction of the normal is ! 0 = L1S. L 2 • According to
equation (VIII-5) the luminous flux within this cone will be
f/>1 = nl0 sin 2 r 1 = n. L1S. L 2 sin2 r 1 .
Within the cone with half-apex angle r 2 the luminous flux is
f/>2 = n . L1S. L 2 sin2 r 2 .
The flux if>.J.s m the solid angle between the two conical surfaces is
therefore
if>.J.s = n . L1S . L 2 (sin 2 r 2 -
sin2 r 1 ).
Within these solid angles the luminous flux is uniformly distributed
around the axis i.e. the normal PN, and a sector with a central angle
xt-2]
LUMINANCE AND, REFRACTIVE .INDEX
of oc rad thus contains a part of the total luminous flu~ f/J within the cones
01:
equal to 2:n; x tP.
The luminous flux in the beam of light emitted by AS is equal to the
flux emitted by L' to LIS and can also be denoted by f/JL'...,. ., 5 :
f/JL' ...... As =
~ :n; • LIS . L 2 (sin2 r 2 - sin2 r 1 ) =
(XI-3)
c) Equating the luminous flux to and from LIS and evaluating L 2 from
the equation
The expressions found for ·f/JL' ...... .::ls (XI-2) and (XI-3) a.re now equated
to give the following equation
From this it follows that
L1
-=
sin 2 r 2 sin 2 i 2 -
sin2 r 1
sin2 i 1 •
According to Snell's law:
so that
L2 =
or
n2
1
= n22
(XI-4)
Since this equation applies to every part of L, and because the beam
of light from L can be regarded as consisting of a number of beams
of the assumed form, formula (XI-4) is universally valid. Put into
words, this formula means that the luminance of a light source is proportional to the square of the index of refraction o1 the medium in ·which
it is measured, disregarding losses due to reflection and absorption.
This is known as Abbe's law.
196
PROPERTIES OF OPTICAL SYSTEMS
[XI
XI-3. Luminance of images formed by lenses and mirrors, and of the
lenses and mirrors themselves
The formula derived in the preceding section for the relationship between
luminance and index of refraction implies that the luminance of the
image of an object emitting light in air, formed by a lens or mirror in air,
is the same as that of the object, disregarding reflection and absorption
losses in the lens or mirror.
We notice that the luminance of the image of an object produced in the
air may not be interchanged with that of an image formed on a screen of,
e.g. paper or ground glass.
The luminance produced on the screen depends on the illuminatio~
on the screen and the reflective or transmissive properties of the screen
itself. The illumination and thus the luminance of it are governed in this
case by the aperture of the image forming system, as shown in XI-8.
Concentration af the light from a source, say the filament of a projection
lamp, by means of a lens or mirror does not therefore increase the
luminance. Concentration refers only to the luminous flux which, in a
lens, is "compressed" into a smaller solid angle. This is accompanied
by an enlarged image of the light source; the luminance remains the
same as before.
Since light from sources in general is emitted in air and is returned
to the air through other media, it may be said that luminance cannot
be increased by optical means.
Owing to the fact that when
p'
L
lenses and mirrors form images
they themselves function as light
sources for these images, the
relationship between luminance
Fig.lOO.
and refractive index also means
that the luminance of .lenses and
mirrors is equal to that of the
H H'
I
object of which they form an
: :----._
P'
image*).
P
9
i '~e
Figs.lOO and 101. Demonstrating that the
luminance of a lens and that of the image
formed by it (P') is equal to the luminance
of the subject P (disregarding reflection
and absorption losses in the lens)
- •--·-
TT--- -- --·
1
Fig. 101.
*) This has already been shown in another manner as applied to mirrors in
section X-2.
Xl-3]
LUMINANCE OF IMAGES, LENSES AND MIRRORS
197
In relation to both lenses and mirrors the above statement can be proved
in a different manner from that used in the previous section; as far as
lenses are concerned, this is as follows.
Fig. 100 shows an infinitely thin lens L producing an image P'
of an object P. Now, if the object emits light with a luminance L, what
we wish to know is the luminance L' of the image P'. We shall disregard
reflection and absorption by the lens.
The luminous flux emitted by P towards L is equal to the flux emitted
by L towards P'. Now, since the rays in an image-forming system of
this kind are reversible, we may say that the luminous flux which P'
would emit towards L_- if the luminance were L' - is equal to the
flux which P emits towards L.
Let us first consider the case of image-forming in the paraxial zone
(Fig. 100). This is the zone, close to the axis, in which the sine, the tangent
_and the arc of the angles between the rays and the optical axis may be
regarded as equal.
The luminous flux f/J passing from P to L is then
f/J = LSOJ,
where Sis the area of P and OJ the solid angle at P, subtended by L.
The luminous flux f/J' that would be emitted by P' to L if the luminance
of P' were L' is
f/J' = L'S'OJ',
where S' is the area of P' and OJ' is the solid angle at P' subtended by L.
Since f/J = f/J',
LSOJ = L'S'OJ';
hence
L
S'OJ'
L' = SOJ.
(XI-5)
Now, OJ= SLfu2 and OJ'= SLfv2 , where SL denotes the area of the
lens, u the object distance and v the image distance. The values of S'
and S are proportional to the square of the linear magnification, i.e.
to v2fu 2• For (XI-5) we can therefore write:
SL
L
v2
v2
L' = uz X S L = I.
uz
Hence the luminance of the image is indeed equal to that of the object.
If the angles included by the image-forming rays of light are so wide
198
PROPERTIES OF OPTICAL SYSTEMS
[XI
that the sine, tangent and angle between the rays and the axis cannot
be regarded as equal, the formula M = vfu, which corresponds to the
I
1
I
simplified lens formula - + - = -, cannot be used. It is then necessary
u
v
t
to take into account the sine condition, which may be expressed as
sin 8
sin 8''
M=--
in which 8 and 8' are the angles included in the object and image
spaces by the optical axis and the rays emerging from a point on the
axis.
In the case of an infinitely thin lens it is permissible with small angles
8 and 8' to say that sin 8 =tan 8 and sin 8' =tan 8'. Then
sin 8
sin 8'
tan 8
tan 8'
M = - - becomes M = - - .
If we denote the radius of the lens by r, tan 8 = r fu and tan 8' = r fv,
so that
tan 8
tan8'
M = - - = vfu.
In Fig. 101, H and H' represent the two positive principal planes of a
lens system. The half-aperture angles are 8 and 8'. P, of which the
luminance is L, is reproduced as an image at P' with a luminance of L'.
The luminous flux emitted by P towards the lens is then
l/J = nLS sin2 8
(XI-6)
and the luminous flux l/J' which P' would emit to the lens if its luminance
were L' is
l/J' = nL'S' sin 2 8'.
Equating l/J with l/J' we then obtain:
nLS sin 2 8 = nL'S' sin 2 8',
from which it follows that
L
S' sin2 8'
L' =. S sin2 8 .
According to the sine condition, the linear magnification is equal to
sin 8/sin 8', so that S' JS = sin 28fsin2 8', and substitution in the equation
L
for L' thus gives
XI-3]
LUMINANCE OF IMAGES, LENSES AND MIRRORS
199
L
L' =I.
That lenses have the same luminance as the iiilage of the light source
can be shown in the following manner.
The luminous flux emitted by the lens system in the direction of P'
is equal to that radiated by P towards the system. In the case of images
p:roduced by paraxial rays (see Fig.lO.Ol,it may be said that the luminous
intensity of L is 5LLL, LL being the luminance of the lens. The illu5L
mination at P' is therefore ~; thus the luminous flux incident at
v2
P' is 5 LLL x 5', and this must be equal to the luminous flux if> emitted
v2
by P towards L, which, as we have seen above, is L. 5. w.
Hence
L5
so that
L
LL
(J.)
=5LLL5'
'
v2
5L
1
5'
;2 X-;;; X S'
where w · 5Lfu 2 and 5'/5 = v2 ju 2 •
Substitution then yields:
L
5L
u2
v2
- = -2x - x - - 2- 1
LL
SL
v
u
-
·
In the formation of images produced by non-paraxial rays the reproducing
system consists of a .circular light source of luminance LL which illuminates P' (see Fig .I 01). According to formula (IX-4) the illumination
at P' is: nLL sin2 @', and the luminous flux emitted towards P' is:
if>' = nLL5' sin 2 @'.
The luminous flux if> emitted by P to the reproducing system is calculated above as
if>= nL5 sin 2 e.
Equating if> and if>' we thus have
nL5 sin 2 6J = nLL5' sin 2 6J';
from which follows, once again,
L
- = 1.
LL
200
PROPERTIES OF OPTICAL SYSTEMS
[XI
The relationship between the luminances can be similarly derived for
optical systems consisting of mirrors.
Let us denote the transmission factor of a lens system by -r; then
L' = -rL, and LL = -rL.
If we similarly denote the reflection factor of a reflecting system by e,
L' = eL, and LM = eL,
where LM is the luminance of the mirror.
XI-4. Optical systems as light sources; the exit pupil of optical systems
We have just seen that optical systems serve as light sources, not only
for the image itself, but also for every point within the image space
that receives light from the system.
Apart from the consideration of the illumination in the image space
(whereby we may regard the system as a point source or a non-point
source), the luminous flux emitted by an optical system may also be
of interest. This must also be discussed in the following sections.
The considerations put forward are theoretical and refer in general
to the ideal, perfectly formed, image; in practice, of course, it is often
necessary to make allowances for imperfections. None the less, these
theoretical considerations have their uses, since they serve for estimating
the behaviour of optical systems and, conversely, for the approximate
design of such systems to meet given requirements from a light-technical
point of view.
In this respect lenses generally have the advantage over mirrors in
that they provide a much nearer approach to the ideal image than
mirrors. In principle, the same considerations apply to lenses and mirrors
alike.
In the following we shall deal in general with lenses, taking these to be
axially symmetrical, the axis of symmetry of the system coinciding with
that of the beams of light. In so far as it is necessary we shall then
proceed to mirrors and certain types of system which are not axially
symmetrical (cylindrical parabolic mirrors), or of which the axes of
symmetry of the beams and the optics do not coincide (drum lenses).
Let us commence by ascertaining what we should regard as the light
source with respect to points on the optical axis of an optical system
XI-4]
OPTICAL SYSTEMS AS LIGHT SOURCES
201
comprising a simple lens and an object. This may be termed the equivalent
light source of the system.
In Fig. 102 L represents a thin lens. An image of an object Pis produced
at P', this being brought
about by reason of the
fact that every point on
L emits a pencil of rays
G
to P'.
The observer's eye is supposed to be at A and
this point A is then
only
Fig. 102. A lens L forms a real image P' of an
object P. Since P' is smaller than L, P' is the
seen as a bright spot. If
exit pupil of the system for points on the axis
the eye is moved along
beyond Gthe optical axis within
the image space, i.e. to the right in the figure, it will be among beams
of light ceming from points in the vicinity of A, so that flashed circles
are then seen.
For instance, when the eye arrives at D it will be just on the edge of
beams coming from the points C (C 1 and C2 and all other points on the
circle of which the radius is AC). What we then see is the face of the
lens, limited by the circle CcA--C 2 , flashed.
AtE, the eye is just at the inner edge of the beams proceeding from the
periphery of the lens, and the whole lens is flashed. From that point
onwards we can therefore compute the illumination E on the optical
axis, by regarding the whole lens as a light source. It depends on the
accuracy with we wish to evaluate E whether we employ the inverse
square law or the formula E = nL sin 2 e.
If the eye be now moved further, we continue to see the whole lens
flashed, until the point G is reached, as we are then at the outer edge
of the beams emitted from the periphery of the lens. Beyond that point
we see that flashed rings at the edge of the lens disappear, i.e. they
become dark. When the eye is removed still further from P', the outer
edges of the beams wherein the eye is still just stationed approximate
to a cylinder and at infinity do form a cylinder. It will now be seen that
the beam of light falling on the eye is limited beyond G by the periphery
of P'; this means that the image P' may be substituted for the flashed
area of the lens, or that, in other words, as far as points beyond G are
concerned, P' serves as equivalent light source of the optical system; it
also means that, at a sufficiently large distance, the illumination can be
computed from the luminous inten~ity of the image P'.
202
PROPERTlES. OF t>PTICAL SYSTEMS
[XI
Let us now turn to the optical system depicted diagrammatically in
Fig. 103.
Pis once more the object, an image P' of which is produced by a lens L:
P' is greater than the area: of the lens. If the observer's eye be now
imagined to be tnoving along the optical axis towards the right, the
lens, commencing from the point A at which the lens is completely
flashed, will continue to serve as light source for all points on the axis.
Since the beam illuminating the eye is always sma1ler at P' in crosssection than P' itself,
P' can never take over
the function of light
source in its entirety;
from the point D for
instance only the part
C1C2 of P' can be seen.
Fig. 103. A lens L forms a real image P' of an object P.
For all points on the Since
L is smaller than P', L is the exit pupil of the
axis beyond A, theresystem for points on the axis beyond A
fore, the lens functions
as light source with the same luminance as that of the object. Accordingly it may be said that the image P', with respect to points beyond
Gin Fig. 102, and the area of the lens for points beyond A in Fig. 103,
ta.ke the place of the object P and the lens L.
In the first instance the image P', and in the second the lens L is known
as the exit pupil of the optical system under consideration.
In general, the exit pupil of an optical system may be defined as the
flashed area as observed from any point in the image space of the system,
and accordingly serving as equivalent light source for that point.
When the exit pupil is formed by the image of a light source of which
the luminous intensity in the direction of the axis is known, the luminous
intensity of the image can be quite easily determined without reference
to the luminance of the light source.
If we denote the linear magnification by M, the area of the image will
be M 2 times that of the light source. Since the luminance of the image
is the same as that of the source, the luminous intensity of the image
will therefore be M 2 times the intensity of the source.
When an image of the filament of an incandescent lamp of 500 cd
luminous intensity is obtained with a linear magnification of 3 x , the
luminous intensity of the image in the direction of the optical axis is
32 x 500 = 4500 cd (This value should of course be reduced by ·the
extent of the losses in the optics).
XI-5]
THE EXIT PUPIL OF AN OPTICAL SYSTEM
203
In closing this section we may mention the fact that the highest illumination on the axis occurs at point E in Fig.l02 and at point A in
Fig.l03, for it is at these points that the solid angles of the illumination
attain their highest values. HenGe the maximum illumination is not
produced at the points where the images occur, as is sometimes thought.
XI-5. Determination ofthe exit pupil of an optical system; entrance pupil
We now come to the problem of determining the exit pupil of optical
systems. In this we shall in general limit ourselves to the exit pupil in
respect of points on the optical axis,although the same considerations
will in most cases apply to points in the near vicinity of the optical axis.
To determine the exit pupil for points ftuther removed from the optical
axis it is necessary to invoke geometrical-optical considerations going
beyond the scope of this book. Certain qualitative considerations,
nevertheless, will be gone into.
According to our definition, the optical system includes not only the
optics, but also the object. It is possible to determine the exit pupil
of the optics by assuming that the object would be located at one of
the foci of the optics.
In Figs.l02 and 103 the object and lens constitute boundaries for the
rays of light, these boundaries being of course physical. Apart from
these the figures show certain boundaries at the images P' of the objects
P, as formed by the lenses, and these boundaries may also consist of
virtual images which have to be considered as lying within the image
space.
All the areas within these boundaries emit light of the same luminance as
that of the object. From within the image space we observe only those
boundaries which lie within the image space. If we direct our eyes from
within this space, towards the optical system, we see only the smallest
boundaries of the rays in the image space flashed.
Let us illustrate this with reference to Fig.l04. In this figure an image
P' of an object Pis produced by a lens L 1 . This image is then reproduced
by a lens L 2 , giving an image P".
In this system there are three physical boundaries for the rays, viz. the
object and the two lenses. Further there are three non-physical boundaries, these being the images P' and P" and image L1 ' of the lens L1
as formed by L 2 . Three of these boundaries lie within the image space,
viz. the lens L 2 and images P" and L1 '; one of these is the exit pupil.
From the point A, L 1 ' is seen to be the smallest boundary, and this is
therefore the exit pupil of this system for A.
204
[XI
PROPERTIES OF OPTICAL SYSTEMS
A
Fig. 104. Determination of the exit pupil of an optical
system. Of the boundaries of the rays in the image space
(P•, L 1 ' and L 2 ) the image L 1 ' is seen from point A the
smallest. Thus the image L 1 ' is the exit pupil of the
system, for point A
The above shows how the exit pupil can be ascertained.
The images of all the boundaries to the rays are determined within the image
space; among the boundaries thus found, as well as among any physical
boundaries within the space, that one which is seen to be the smallest is the
exit pupil.
A demonstration of this may be given in the following manner (see
Fig. 105).
In the image space a number of diaphragms of different diameter are
depicted (i.e. boundaries for the rays). Behind these there is a large
area whose luminance is that
A
p
·of the light source. Now, suppose that we are looking towards
the diaphragms from a point in
the image space; only the smallest of these is seen flashed and
this one is the exit pupil. .
If the observation is made from
a point removed from the optical axis, the exit pupil may Fig. 105. For the point P some distance
be bounded by portions of two from the axis of diaphragms A, B and C
the exit pupil is bounded from above
diaphragms; in Fig. 105 this is
by B and from below by A
illustrated at the point P, in
respect of which the exit pupil is formed by the upper edge of diaphragm
B and the lower edge of A.
In the object space, the exit pupil corresponds to an object which in
the words of geometrical optics is conjugate to it, and this is called the
entrance pupil of an optical system. The entrance pupil is accordingly
determined by the image of the exit pupil in the object space. In Fig.l02
P is the entrance pupil and in Fig. 103 it is L.
The significance which can be attributed to the entrance pupil is
XI-6]
OBJECT AT THE FOCUS OF A LENS
205
demonstrated in Fig. 106, in which an image P' of an object Pis produced
by a lens L 1 • At P' there is a
p'
second
lens 'L 2 which is smaller
L,
~-- than the image P' itself, and
the exit pupil of the whole
system is thus the lens L 2 •
The conjugate entrance pupil
occurs at P, but it is smaller
Fig. 106. An image of an object P is produced at P' by a lens L 1 • At P' there is than P.
another lens L 2 • L 2 is then the exit pupil. It will be seen, then, that P
The entrance pupil of this system is that
part of the object P which corresponds to can be reduced to the actual
the size of L 2 • P can therefore be reduced
size of the entrance pupil
to the size of this part without decreasing
(this being the image of L 2
the luminous intensity of L 2
in the object space) without
varying the exit pupil, i.e. without varying the luminous intensity of
the system in the direction of the optical axis.
--
XI-6. Object at the focus of a lens; angle of divergence
When an object is placed at the focus of a lens an image of it is formed
at infinity. In such cases the exit pupil is the lens itself, and it is important to ascertain the point from which the whole area of the lens
actually functions as exit pupil. Let us turn to Fig. 107.
An object P is situated at the focus of a lens L; we can then say that
every point on L emits a beam of light. The apex angles of all these
Fig. 107. A parallel beam formed by the lens L of the
object P located at the focus of the lens. It is only beyond
G that the lens is wholly flashed, G is called the beam
cross-over point
beams are the same (cp). We again trace the development of the illumination along the optical axis. Starting frorp A, from where only the
point A is seen flashed, and moving along the axis, we observe larger
and larger areas of the lens flashed, until at G the whole lens becomes
206
PROPERTIES OF OPTICAL SYSTEMS
[XI
flashed. Beyond G, then, we may consider L as a light source whose
luminance is that of P. Lis therefore the exit pupil of the system. When
the distance AG is sufficiently long compared with the size of L, so that
L may be regarded as a point source, it is permissible to speak of the
luminous intensity of L.
Should we wish to ascertain the luminous intensity of L by measuring
the illumination at the axis, we should have to be careful to locate the
point of measurement beyond G. In photometry this point G is known
as the beam cross-over point, which is further referred to in Part II,
(section XVI-3).
The distance AG can be calculated by means of the following for a
circular object P.
AG = R cot }rp, where R is the radius of the lens.
Cot tiP= ffr, where f is the focus of the lens and r the radius of the
object. We thus see that
AG = R .f.
r
Let us now also see what the illumination is for points not on the axis,
at a large distance from L, i.e. the luminous intensity of Lin directions
other than that of the axis. In other words, let us examine the light
distribution of the system.
Within the cone-shaped space BGC the lens is seen completely flashed;
L is therefore the exit pupil for this space. The apex angle of this cone
is equal to rp.
From Fig.l07 it is seen that, outside the cone BGC the lens will not be
flashed when at relatively short distances from L. At an infinite distance,
however, the parallel lines DB and FH, as also FC and DE, will coincide
and it may be said that the light beam is bounded by a conical surface
of apex angle rp, and that the lens, which is then infinitely small compared
with the distance, is the apex of this cone.
In practice we may regard the beam as tapering at an angle of rp even
at finite distance, provided only that the distance is great compared
with the size of the lens.
The luminous intensity of the beam in a certain direction is then equal
to 1he apparent area of the lens in that direction, multiplied by the
luminance of the objec!. If the angle between the particular direction
and the axis be denoted by IX, the apparent area will be S cos IX, where
S is the actual area of th~ lens. The luminous intensity is thus:
I = -rLS cos IX,
XI-7]
THE EXtT PUPIL OF LENS SYSTEMS
207
where L denotes the luminance of P, and l' the transmission factor of
the lens.
Apart from a certain amount of diffused light from the lens, no light
falls beyond the lines FH and DE, and the apex angle of the beam is qJ.
The half-apex angle of a beam is known as the angle of divergence of
the beam, but, as this name is sometimes given to the whole apex angle,
it is advisable to indicate the angle of divergence as twice the half-apex
angle, e.g. 2 X 5°, or 2 X 8.5°.
In our example we have assumed that an ideal image is formed, that is,
one without aberration and, in so doing, we have regarded the beams
of light as sharply defined. In practice, however, there will often be
errors in the formation of the image (in reflecting systems invariably),
in consequence of which the beams of light are not sharply P,efined.
In such systems the luminous intensity will exhibit a maximum,
falling off towards the periphery. Angles of divergence are given for the
beam in such cases, but these are then the angles at which the luminous
intensity"(or average luminous intensity) is 50%. 25% or 10% of the
maximum.
Reference is often made to parallel beams and these are illustrated by
drawing rays from the focus which emerge from the lens parallel to the
axis; in such cases qJ = 0. The angle of divergence is determined by
tan t(/J = rff. Since all light sources have finite dimensions and the
value of r is accordingly always finite, tan l(/J must also always have
a finite value.
There is, then, really no such thing as a "parallel" beam in the literal
sense of the word. A certain amount of spread must always be taken into
account, however small this may be.
XI-7. The exit pupil of lens systems; vignetting
When an object is placed at the focus of a simple lens the exit pupil
is the lens itself. However, if the system comprises a number of lenses
and possibly one or more diaphragms, if must not at once be assumed
that the area of the last lens is the exit pupil of the system; it is necessary to ascertain the exit pupil in the manner outlined above;
We shall now proceed to do this as applied to two widely used kinds of
lens system, i.e. two objectives. These are systems of lenses used for
projection purposes. Errors in the formation of the image, such as occur
with single lenses, are in such cases more or less fully corrected by combining a number of single lenses having different characteristics.
208
PROPERTIES OF OPTICAL SYSTEMS
[XI
Amongst other applications, objectives are employed in picture projectors, photographic apparatus and microscopes.
Objectives may consist of separate single lenses, i.e. with air spaces
between them, or of lenses cemented together, or combinations of these.
Diaphragms, or stops, may be interposed between the separate components.
Let us now ascertain the exit pupil of:
a) an objective comprising two components separated by an air space, and
b) an objective having a stop between the components.
The components of such objectives usually consist of one or more single
lenses cemented together, but for our purpose such compound lenses
can be regarded as single lenses.
a) Exit pupil of objectives consisting of lenses with space between them
(see Fig. 108)
H'H
I
I
I
I
·- -· ---,,-----1;:Fi'
f
: l fxit.,
I
!--------''----....;
-------S.a
pup!
Fig. 108. Exit pupil of an objective without internal stop.
L 1 and L 2 are the rear and front lenses respectively. H and H'
are the positive principal planes in the object and image spaces
respectively. L 1 ' is the image of L 1 , formed by L 2 • L 2 is the
exit pupil
L 1 and L 2 are the component lenses of an objective the foci of which
are F and F'; the foci of the components L1 and L 2 are respectively
F 1 , F 1 ' and F 2 , F 2 '.
We now have to define the image L1 ' of Lv as formed by L 2 • Since L 1
lies between F 2 and L 2 , this image is virtual. If this image is greater
than L 2 , L 2 will be the exit pupil, but, if it is smaller than L 2 , the exit
pupil will be the image L1 '.
Now, in most practical objectives, L 2 is always smaller than L 1 ' so that
in practice the exit pupil of an objective without built-in stop is always
the front lens.
When the exit pupil is the front lens, this means that the other boundaries
to the rays are at least so large that they will pass the light emerging
from the exit pupil in the axial direction. The only other boundary
present in the system depicted in Fig.l08 is the rear lens. Let us now
see how large this lens must at least be in. order to meet this condition.
This can be done in two ways, viz: -
XI-7]
THE EXIT PUPIL OF LENS SYSTEMS
209
I) we can construct the size of L 1 such that the image L 1' is just as large
as L 2 , or
2) we can construct the paths of the rays emerging from L 2 as a beam
parallel to the axis.
The latter procedure has been followed in Fig.l08 in accordance with
the method employed in geometrical optics; it is thus found that L1
may be appreciably smaller than L 2 •
In actual practice, however, L1 is usually made larger, the reason for
this being that parallel rays emerging obliquely from L 2 require a larger
rear lens in order to permit the whole of lens L 2 to function as exit pupil.
This is seen from the beam S1S2 in Fig. 108, which converges at the
point S in the focal plane. In order to be able to pass all oblique rays
such as 5 11 5 2 , the lens should have a cross-section equal to 2 X MA.
b) Exit pupil of objectives consisting of 2 lenses with a stop between
them (see Fig. 109)
If a stop D be placed within the objective of Fig.l08 in the manner
shown in Fig. 109, we must, in
order to determine the exit pupil,
Li
also
construct the image of this
t
stop as produced by the front lens.
·-·-·-·- In the case depicted in Fig. 109
it will be seen that this image
D' is smaller than L 2 , so that
this virtual image is the exit
Fig. 109 Exit pupil of an objective pupil, not L 2 • The practical method
with internal stop D. The image D'
of ascertaining the size of the
of D formed by L 2 is the exit pupil of
the objective. Other references as in image D', i.e. of the exit pupil,
Fig. 100
is discussed in Part II, Light
measurements (section XIX-4).
Let us now ascertain by means of an example the exit pupil of an
objective for directions other than the axial. Fig. 110 shows the same
objective as that depicted in Fig. 108.
According to what has been said in section XI-5 in regard to the boundaries of the rays, the objective shown in Fig. 110 has two flashed stops,
viz. the lens L 2 and the image L1 ' of L1 as produced by L 2•
If the objective be observed in the direction of the axis, L 2 is the smaller
diaphragm and is thus the exit pupil. However, if we look at the objective from a direction deviating from the axial, L 2 will be the exit
210
PROPERTIES OF OPTICAL SYSTEMS
[XI'
pupil for those directions that fall within the conical surface which
can be drawn through the
peripheries of L1 ' and L 2 •
The extreme directions are
represented in the figure
by the line E'C.
Geometrical-optical construction shows that the s
point on the object which r~=::-:~==__!,~~~~~~====--corresponds to the extreme direction in which
Fig. 110. Vignetting in an objective. The rays
L 2 is still the exit pupil, from
point T on the object at the focus do
is the point S. The emer- not quite fill the front lens L 2 because the image
to serve as background
gent beam from this L 1 ' is not large enough
for the whole area of L 2 in the· direction of the
point is bounded by AS 1
parallel emergent rays T (T1 and T 2 )
and CS 2 , where CS 2 is the
production of E'C.
For points outside the circle of radius FS, for example T, at which
the emergent beam (ATcBT2) makes a larger angle with the axis than
that from S, L 1' does not offer a completely flashed background, so that L 2
will not be completely flashed. If we draw a line E'B from E' in the
direction of the beam from T, this line will intersect the lens L 2 at B.
In this manner the lower boundary of the emergent beam from S is
found for the plane of the drawing*).
If L 2 should be the exit pupil for T, the
radius of L 1 would have to be extended
to the point G, and L 1' would be correspondingly larger.
What we actually see of the exit pupil
is depicted in Fig. 111, in which both the
diaphragms L1 ' and L 2 are shown as we
see them displaced with respect to each
other. The points A, B, C and D' in this
f
figure correspond to those in Fig. ll 0.
Fig. 111. Front view of an
The hatched area of L 2 remains dark and
objective in which vignetting
occurs. The points A, B, C
the exit pupil is therefore smaller than L 2•
and D' are the same as those
L
1 ' and L 2 are not drawn as circles, but
similarly indicated in Fig. 110
as ellipses. Due to the oblique viewing
*) The same result can be obtained with the aid of another geometrical-optical
cons ruction.
XI-7]
211
THE EXIT PUPIL OF LENS SYSTEMS
direction we see that the circles are deformed to ellipses, whose minor
axes are equal to the diameters of the circles, multiplied by the cosine
of the angle of emergence (angle T 1 AF1 ).
The above described phenomenon is known as masking or vignetting.
This may be further elucidated with reference to Fig. 112a and b. On
the left hand side of each of the diagrams, two stops D 1 and D 2 are shown
in front of a surface L emitting diffuse light. The stops are of equal size,
but the spacing is not so great in Fig. 112b as in 112a.
a
,.,--,,\
I
I
\
I
\
I
' .... __
I
_.,.//
,,..---....
I
~
'\
I
',~'
''
'
I
',..,
p
I
/
'
__ ... ,-' '
R
I
....
/
''
\~)
b
~
........ __ ,,
\
p
Q
I
R
Fig. , 12. Vignetting by two diaphragms D 1 and D 2 filled with
lighf oy a diffuse light-emitting surface L. With the longer distance
between D 1 and D 2 (a), the vignetting is more pronounced than
with the shorter distance (b). The figures P, Q and R shown on
the right illustrate the flashed portions as seen from the directions
of P, Q and Ron the left
Now, if we look in the direction of the axis (from P) towards L, we
see D 2 wholly flashed with light from L, as shown by P on the right
hand side of the figures a and b.
When the direction of vision is removed from the axial, the stop D 2
is seen to be fully flashed up to a certain direction Q. The direction
in which D 2 can still just be seen to be fully flashed is closer to the axis
in Fig. 112a, with its greater spacing DcD2, than in Fig. 112b.
At wider angles of observation with respect to the axis, vignetting
occurs because a part of the background against which D 2 is observed
212
PROPERTIES OF OPTICAL SYSTEMS
[XI
is now formed by the side of D1 remote from L. This vignetting effect
sets in more quickly with the wide separation of D 1 and D 2 than with
the closer spacing.
In the right hand section of Fig. ll2a and b, R shows what is seen in
the two cases when the observation is made in the direction of R. The
direction of R in Fig. 112a is the same as in 112b. This clearly demonstrates the difference in the vignetting effect, which is more pronounced
in the case of the more widely separated stops D 1 and D 2 •
Objectives intended to cover a wide angle of the field of view, such as
most photographic objectives, are therefore made as short as possible.
XI-8. The luminous flux of optical systems. Aperture of lens and mirror
systems
We shall now show how the luminous flux emerging from an optical
system may be ascertained.
Let us commence with a single, thin, lens with an object at the focus.
If we once more disregard the
loss.es in the lens, the luminous
flux emitted by the lens will
be equal to that reaching it from
the object, this being calculated
in the following manner.
Fig. 113 shows the object P at
the focus of the lens L. A small Fig. 113 Illustrating the method of
computing the luminous flux emitted
area LJS p of P lying on the axis
by a lens
has a luminous intensity m the
direction of the axis of ! 0 = LJS p X L (L =luminance of P). The
luminous flux ifJ LISP emitted by LJS p towards the lens is
ifJLisp = L15p X nL sin 2 e.
As long as P is not too large compared with the focal length, we may
assume as an approximation that each small part of P emits the same
luminous flux towards L, so that the total flux reaching L is
(XI-6)
This could be, for example, the luminous flux that reaches a projection
screen (ignoring losses in the lens), when an image of the light source
is produced a great distance away (which may be put equal to infinity).
If the light travels through the lens in the reverse direction from infinity,
XI-8]
THE LUMINOUS FLUX OF OPTICAL SYSTEMS
213
an image is formed at the focus and the illumination E of the image
is E = nL sin2 e, L being orice more the luminance of the object which
is reproduced.
This is, for instance, the illumination received at the centre of a photographic film or plate with the lens at a great distance from the object to
be photographed; this illumination determines the exposure time.
In the first instance the luminous flux, and in the second the illumination,
is determined by the angle e. Now, cote= 2/fD, where I is the focal
length andD the diameter of the lens. The luminous flux and illumination
thus appear to be dependent on the ratio of the focal length to the lens
diameter, but independent of their absolute values. This ratio is known
as the aperture, or /-number, of a lens.
An /-number of n is designated as ffn, or f: n, e.g. //1.5, or f: 8.
Having dealt so far with simple lenses, let us now see what takes place
when the lens system comprises a number of components, as is the case
with most objectives.
In the case· of the objectives depicted in Figs. 108 and 109 it will be
seen from the geometrical arrangement that the angle e is determined
by the exit pupil EP and the focal length, viz. that cote= 2ffEP. Here,
then, the /-number is the ratio of the focal length to the diameter of the
exit pupil.
If the directions of the rays be reversed, the exit pupil shown in the
diagram functions as entrance pupil. In photographic objectives, therefore, the /-number has to be defined as the ratio of focal length to en~
trance pupil. With a degree of accuracy that is sufficienffor photographic
purposes we can say that in practice sin2 e is equal to tan 2 e. Since
the /-number is proportional to cote, we may also say that the illumination on the sensitized material, which is proportional to sin 2 e,
is inversely proportional to the square of the /-number. Hence, as an
approximation, the exposure time is directly· proportional to the square
of the /-number.
In photographic objectives it is .usual to mark the different stops in
such a way that each setting corresponds roughly to one half, or twice,
the .exposure time for the adjacent stop. The /-numbers thus constitute
~ 1.4 (//4.5, 6.3, 9, 12.5, 18,
geometrical series with a ratio of roughly
or // 5.6, 8, II, 16, 22).
The above remarks relate to the condition whereby the object or image
is at the focus of the lens or objective, i.e. the lens is "focussed to infinity".
In the case of a camera objective it will be seen that when the camera
y2
214
PROPERTIES OF OPTICAL SYSTEMS
[XI
is focused on a nearby object, the image moves back, producing a smaller
.angle e than when the focussing is set at infinity and, hence, that the
exposure time needs to be longer. In extreme cases this must be taken
into account. Suppose that under-exposure to the extent of 25% is
permissible; no correction should then be necessary for distances more
than 10 times the focal length of the lens.
The foregoing remarks apply to all objectives in general, but if a comparison is to be made between one objective and another, it must not
be forgotten that the illumination is proportional to the transmission
factor of the objective and that this may differ considerably. The transmission factor of a simple lens is about 0.9; that of an objective comprising a number of lenses might be as low as 0.5.
XI-9. Some remarks on mirror systems
The considerations outlined in the preceding sections apply in principle
to all types of optical system. The examples given relate to lens systems,
but mirror or reflecting systems, as well as combinations of lenses and
mirrors, can be dealt with in the same way. Whilst not pursuing this
further it will nevertheless be useful to note one or two points peculiar
to mirror systems only.
Whereas with lenses (with corrected lenses at least) one obtains exact
images of objects, free from spherical aberration, mirrors give an exact
image only within certain limits. \Vith concave mirrors of the three
different types most in use, spherical aberration is absent only in the
following cases:
a) in spherical mirrors: when rays are emitted from the centre. These
are returned to the centre.
b) in parabolic mirrors: when rays are emitted from the focus (or pass
through the focus; these are reflected parallel to the axis. In this
case one can speak of an image at infinity.
c) in ellipsoidal mirrors: when rays are emitted from a focus (or pass
through a focus): these converge at the other focus.
When the /-number is high (i.e. relatively small mirror diameter), it
is in practice usually possible to anticipate a perfect image of objects
within the paraxial zone.
XI-10. Drum lenses
In lighthouses use is made of cylindrical lenses of the type shown in
the diagram in Fig. 114, in which b is the horizontal cross-section through
Xl-10]
215
DRUM LENSES
the line AB in the vertical section a. These lenses are known as drum
lenses.
c
The vertical cross-section is the
same as that of an ordinary
spherical lens; this rotated about
a vertical axis through the focus
F gives the cylindrical form.
When a light source is placed
at F the rays are concentrated
in the vertical direction; horizontally no concentration occurs and
the beam emerging from the lens
is accordingly fan-shaped, this
being particularly suitable for
lighthouses.
The luminous. intensity of an
optical system of this kind can
be computed in the following
manner.
Fig. 114. Drum lens
If we look at the lens from a
considerable distance in the direction of the line AB, we see a part
of the lens flashed. Vertically this area extends from top to bottom,
but in the horizontal direction it is no wider than the width of the light
source, since there is no concentrating effect in that direction and the
light source is neither magnified nor reduced.
We shall denote the luminous intensity of the light source in the direction
of the lens by I, the height by hand the width by b. Let H be the height
of the lens. The area of the flashed part is then b X H. The luminance
is that of the light source multiplied by the transmission factor -r: of
. -r:.I
the lens, that 1s: --.
b.h
The luminous intensity of the lens is then
-r:.I
I
h = b . h X b . H = -r: . h . H.
In this expression it is seen that the width b of the light source does
not occur; hence that the luminous intensity of the lens is not dependent
on this width. The luminous intensity is ascertained by multiplying
the height of the lens by the quotient of the luminous intensity of the
216
PROPERTIES OF OPTICAL SYSTEMS
[XI
source divided by the height of the source, so the candelas per em or
inch of height. It is evident that the important thing for lighthouse
lamps. is not the luminance but the luminous intensity per em or inch
of height.
XI -11. Cylindrical mirrors
Here we shall deal only with cylindrical-parabolic mirrors. Used in
conjunction with linear light sources such as the high-pressure mercury vapour lamp and the halogen lamp, such mirrors have assumed
practical importance, in particular for flood-lighting.
Fig. ll5a is a front view of a mirror of the kind used with a linear light
source; Fig. 115h shows the
cross-section.
The form of the mirror is
developed by displacing a parabola parallel to itself; the
focus is thus moved along a
line perpendicular to the plane
of the parabola. This line is
Fig. 115. Cylindrical-parabolic reflector
known as the focal line of the
with linear light source
mirror.
Here again, concentration takes place in a vertical direction, but not
horizontally. The height of the area seen to be flashed in the horizontal
direction, perpendicular t'o the axis, is equal to the height of the mirror,
and the width is equal to that of the light source. Directly in front of
the mirror, therefore, the luminous intensity is equal to this area,
multiplied by the luminance of the source (and the reflection factor of
the mirror).
Since the luminance of the arc in the mercury vapour lamps employed
with these mirrors is not uniform over the width of the tube, the luminous
intensity per c:m of height can not be used here, so that we have to
ascertain what luminance should be taken for the purposes of calculation.
The light that we see comes from the direction of the focal line of the
mirror and, hence, from the centre of the apparent area of the arc.
At this point the luminance is at its maximum, and this is therefore
taken as basis for our calculation.
This is valid of course only when the form of the mirror is a true parabola
in cross-section, but in practice this is never actually the case. Some
divergence from the true form must accordingly be taken into account~
this resulting in the fact the light reflected horizontally emanates partly
XI-12]
FACETTED MIRRORS
217
from areas in the discharge which are outside the centre and consequently of lower luminance. Owing to such divergences, parts of the
mirror may ev·en be entirely dark, since the reflected rays in these parts
are emitted away from the horizontal.
Even though the above may reveal a similarity between drum lenses
and cylindrical-parabolic reflectors, there is also an important difference.
In drum lenses the luminous intensity is the same on all sides in the
horizontal direction, provided that the width of the light source is also
the same all round, as is the case with cylindrical incandescent sources.
With the cylindrical-parabolic mirror,' however, •the luminous intensity
decreases as we move from directly in front to towards the sides; in
principle, this decrease is proportional to the cosine of the angle between
the direction of observation and the direction straight ahead. It continues to the point at which the intensity suddenly falls off to that of
the light source alone; the angle at which this takes place is governed by
the lengths of the mirror and the light source.
XI-12. Facetted mirrors
A third kind of mirror will now be discussed, this being of special
constr!lction, known as facetted. Such mirror-reflectors consist of a
number of small contiguous mirrors; they may be plane or curved,
and are usually so shaped that the basic form is that of a paraboloid
of revolution, for example. The facets will then constitute tangent
planes to the paraboloid.
Optically, facetted mirrors differ from mirrors having the same basic
shape but having a continuous curve in the greater spread which they
produce; this is accompanied by a lower luminous intensity of the
beam.
Whereas continuously curved mirror reflectors often produce patchy
or streaked beams, this is not the case with the facetted mirror, since
the individual mirrors of which it consists each function as a separate
source of light, the solid angle of the emitted light being bounded by
the facets.
Facetted mirrors are employed in place of the usually more costly,
continuously curved, mirrors in all applications where their increased
spread is not a drawback, or may even be desirable.
The luminous intensity of facetted mirrors is equal to the sum of the
luminous intensities of the facets and, if these can be calculated, the
intensity of the whole reflector can of course be determined. Such a
calculation will be simple and reasonably accurate only when the facets
218
PROPERTIES OF OPTICAL SYSTEMS
[XI
are plane and the whole of the reflected image of the light source can be
seen from the direction in which the luminous intensity is to be ascertained.
F
Such conditions are illustrated in Fig. 116. The
three facets of the mirror F
produce 3 images L 1 , L 2
and L3 of the light source L.
In the direction of the axis
Fig. 116. Facetted mirror
these images are all fully
visible and their luminous intensity can be computed.
If we denote the reflection factor of the facets of the mirror by (}, the
luminous intensity I in the direction of the axis will be
I= n X (} X I 1
+I
1,
if I 1 is the luminous intensity of L in all directions.
In order to secure the effect of the greater beam spread andjor higher
uniformity of the beam, reflectors are also made of which the continuously curved surface is more or less uniformly indented (display
reflect{)rs).
XI-13. Optical systems for the projection of images
Optical systems for producing images of h; ->cts or figures are known
to all; these include the old-time magic ~.tl ern, projectors for stills or
moving pictures, and microscopes. The last-mentioned will not be dealt
with here.
Let us now consider one or two aspects of projection apparatus, mainly
from the point of view of the lighting technology involved; the geometrical optics will be discussed only in so far as is essential to the discussion.
Projection systems for both still and moving pictures comprise the
following components:
a) the reproducing element which projects the image on to a screen.
This can be a simple lens, i.e. a bi-convex or meniscus lens, but, as
an image of greater sharpness is usually required than that which
a single lens is capable of producing, corrected lens systems (objectives), are generally employed.
b) means of illuminating the object of which an image is required.
When the object is illuminated from the rear (transmitted light) we
speak of diascopic projection and the apparatus may be referred to as
XI-14]
DIASCOPIC PROJECTION
219
a diascope. If the object is illuminated from the front, i.e. with incident
light, the projection is episcopic and the apparatus concerned is known
as an episcope. Units designed for both systems of projection are called
epidiascopes.
In diascopic projection the objective projects a shadow-image of the
object, so that it is necessary to provide a bright background for the
object; in episcopic projection the light projected is reflected from the
object.
XI-14. Diascopic projection
Fig. 117 depicts an object P, which may be a lantern slide or film strip,
of which an image is projected by a lens system; for the sake of simplicity the lens system (the obo,
jective) is represented by an
infinitely thin lens 0.
Although projection is effected
at a finite distance, this is in
general so large compared with
the focal length of the lens that
Fig. 117. In order to flash a projection
we may assume for our purpose
lens 0 which is to project an image
of an object P, a background is required
that P is at the focus of the lens.
that will be at least equal to C1C2 at A,
Hence, of every point on P,
or D 1D 2 at B
0 forms an image at infinity. Jhe
rays emitted from each point on P leave the lens as parallel rays, of
which it may be said that they meet at a point at infinity.
From each point on P a beam travels towards 0 as shown in Fig. 117
for points P 1 and P 2 , from which lines are drawn to Q1 and Q2 on the
periphery of 0. P 1 Q1 and P 1 Q2 are then the peripheral rays of the
beam from P 1 • If we now project these lines in the reverse direction
we have the peripheral rays of the pencil of light that has to travel
to Pv and, if the same be done for P 2, we have the boundary of the
whole beam that has to reach the object in order that every point on P
will fill the whole lens with light.
The figure thus shows what size the background of P has to be in relation to the point at which this background is placed. If this be at A,
the diameter must be at least equal to C1 C2 ; if at B it must be at least
DtD2.
It will be seen at once from the figure that this background is much
too large for it to consist of the filament of an incandescent lamp or the
crater of a carbon arc. Such would be uneconomical, moreover, as the
220
PROPERTIES OF OPTICAL SYSTEMS
[XI
light from the background is required only within a limited solid angle.
Now, lenses and mirrors provide the means of concentrating the light
from a source within a wide solid angle into a smaller angle, whereby
the luminance, apart from losses due to reflection and absorption,
remains the same.
Concentrating systems of this kind are called condensers, and they may
consist of lenses, mirrors, or a
combination of both. Fig. 118a,
b, c illustrates the principles on
L
which projection systems are
--tP-1<t=::::C ,...----..:::,:r..-based in conjunction with a
condenser, a mirror and a comc
bination of lens and mirror,
respectively.
The luminous intensity of such
systems can be computed for the
direction of the optical axis, and,
for this purpose, according to
section XI-4, it is necessary to
know the size of the exit pupil,
the luminance of the light source
and the losses due to reflection
three main systems of
and absorption in the lenses and Fig. 118. Theprojection
mirrors.
for projecting lantern slides.
In most cases it is difficult to a) system
The diapositive P is large compared
with the objective 0. The condenser
calculate the luminous intensity
C is placed just behind P. The narin any other direction than that
rowest point in the path of the rays
is the objective
of the axis, since, as we have
already seen in section XI-7 b) and c). Systems for projecting standard
film. The film gate Pis small compared
on the subject of objectives, the
with the objective 0. The narrowest
dependence of the intensity on
point in the path of the rays is the
film gate. In b) a mirror condenser
the geometrical-optical characis shown; in c) a condenser comteristics of the optical system is
prising a parabolic mirror and a lens
much more complex.
Calculations can also be made to determine the luminous flux emitted
by the objective. Such calculations of both the luminous intensity
and the luminous flux of projection systems enables us to determine
to a fairly high degree of accuracy whether a system as designed will
meet the requirements imposed on it.
The size and position of the exit pupil are ascertained from the geo-
XI-15]
SLIDE PROJECTION
221
metrical-optical characteristics of the system. These depend on the
ratio of the size of the object to that of the objective, whilst the dimensions of the objective, condenser and light source are in turn governed
by limits set by the technique, or by economic factors, in other words
by the cost of the equipment.
In the following sections we shall consider two examples, viz.
1) object large compared with diameter of the objective;
2) object small compared with diameter of the objective.
The first of these conditions is found in the projection of lantern slides
(e.g. 3tx3t" and 24x36 mm), and we shall accordingly classify this
kind of projection as slide projection. The second category includes the
projection of standard 35 mm film, and this may be referred to as
standard-film projection.
The projection of sub-standard film (such as 8 and 16 mm) will be
discussed thereafter.
XI-15. Slide projection
We shall be dealing with this using an example with given dimensions.
Fig. 119 shows a slide projection system. The pictures to be projected
measure 23 x 35 mm2, their diagonal being 42 mm.
~-&-~~-·-·-·-·-·~·Fig. 119. Illustrating the method of calculating data for the optical elements of a
system for lantern slide projection.
The condenser lens, located immediately behind the slide, has a free
aperture 50 mm in diameter. The condenser system consists of two
lenses, an aspherical one close to the lamp and a spherical one close to the
slide. This condenser system forms an image of the filament of the lamp
222
[XI
PROPERTIES OF OPTICAL SYSTEMS
in the exit pupil of the objective. The focal length of the objective is
madej3.5 so that exit pupil diameter is 100 : 3.5 ~ 29 mm. The linear
magnification of the image of the filament in the object lens is 3.6.
With these data we now want to calculate the luminous intensity at the axis
of the beam emerging from the objective using two different projection
lamps, assuming that the total transmittance r of the system is 0.5.
The lamps we use are two Philips projection lamps, the data on them
that are important to us being listed in the table below.
llOV-lOOW
220V-500W
filament
bxh
mm
lum. flux
mean luminance
lm
horiz. lum.
intensity
cd
6x5
lOX 9.5
1730
11400
182
1650
600
1740
bs
It is noticeable in a comparison of the luminance that that of the 500 W
lamp is much greater than that of the I 00 W lamp. The main reason
for this is the arrangement of the coils in the I 00 W lamp in one plane
r-0
~rD
~
a
r;:
vb
Fig. 120. In slide projection, the objective can be filled by the image of the filament
of the projection lamp in two ways, viz:
a) the image falls entirely within the exit pupil of the objective;
b) the image pr~jects beyond the exit pupil.
XI-15]
SLIDE PROJECTION
223
next to each other, involving, for reasons of lamp manufacture, rather
large gaps between them. (These can partially be compensated by the
use of a hemispherical auxiliary reflector by images of the filaments.)
In the 500 W lamp the coils are arranged one close behind the other in
two planes. The gaps between the parts of the filaments in the foremost
plane are filled by the filaments in the rearmost plane. This means that
the "concentration" of the filaments in the 500 W lamp is much greater
than in the 100 W type, resulting in a much greater average luminance.
We shall now be calculating the luminous intensity of the objective with
the two lamps in succession.
1. 110 V-100 W lamp
The image of the filament in the exit pupil of the objective measures
3.6 X 6 = 21.6 mm (breadth) and
3.6 X 5 = 18 mm (height).
The diameter of the exit pupil of the objective is 29 mm, so that the
image of the filament falls wholly within it (Fig. 120a).
In this case the image of the filament is the exit pupil of the entire system
and the luminous intensity at the centre of the beam is the luminous
intensity of this image.
We calculate this in the following way:
The linear magnification is 3.6, so that the luminous intensity of the
image is 3.62 x the luminous intensity of the lamp. We still have to
take the transmission factor of the optic (0.5) into consideration, and the
luminous intensity in the light beam is
l = 3.6 2 X 182 X 0.5 cd R:i 1180 cd.
2. 220 V-500 W lamp
The image of the filament in the exit pupil here measures
3.6 X 10 = 36 mm (breadth) and
3.6 X 9.5 = 34.2 mm (height).
Because the dtameter of the exit pupil of the object lens is 29 mm,
the image of the filament projects beyond the exit pupil of the object
lens on all sides. The lens is thus completely flashed by the luminance
from the lamp, and the luminous intensity of the beam can be calculated
by multiplying the luminance of the lamp by the exit pupil (bearing in
mind the transmission factor of the optic, of course).
224
PROPERTIES OF OPTICAL SYSTEMS
[XI
The size of the exit pupil is t n x 2.9 cm 2 = 6.6 cm2. The luminous
intensity of the beam is then
1 = 6.6 X 1740 X 0.5 cd ~ 5750 cd.
If we now compare the luminous intensities of the beams with the horizontal luminous intensities of the lamps, we find that, while the luminous
intensity ratio of the lamps is 1 : 9, the luminous intensities of the beams
are in the ratio 1 : 5. If we take an objective with an aperture off: 2.5,
so that the exit pupil diameter is 100 : 2.5 = 40 mm, the image of the
filament of the 500 W lamp also falls entirely within the exit pupil of the
objective, and we can calculate the luminous intensity of the beam in the
same way as with the 100 W lamp:
I= 3.62 X 1650 X 0.5 cd ~ 10600 cd.
Of course, the ratio between tile luminous intensities of the beams is then
exactly the same as that between the horizontal luminous intensities
of the lamps.
From this table it will be seen how important it is to have the image
of the light source wholly or almost wholly within the objective, in order
to make the most effective use of the luminous flux from the lamp
(and also of the wattage consumed by the lamp).
We can now calculate from the luminous intensity along the axis of
the system the illumination at the centre of the projection screen. If
a certain illumination is required at a given distance from the projector,
the aperture of the objective and the light source itself provide the
means of approximating closely the required value. There is, however,
a limit to which such requirements can be carried, as imposed by technical and economic considerations.
As regards the screen illumination elsewhere than on the axis of the
system, the following may be noted.
In principle, the screen illumination is directly proportional to the 4th
power of the cosine of the angle between the axis and the direction
in which the illumination is to be evaluated, but in many cases vignetting
will occur at oblique directions (see section XI-7). This effect will be
all the more pronounced according as the image of the light source fills
the objective to a greater extent. In practice, however, such vignetting
rarely needs to be taken into account in lantern slide projection, since
the rays do not travel through the objective very obliquely. If the image
of the light source is not symmetrically disposed in the objective, i.e.
Xl-16]
STANDARD-FILM PROJECTION
225
if the source is not properly centred, different degrees of vignetting will
occur at the sides, or top and bottom, resulting in irregular illumination on the screen (e.g. higher at the top than at the bottom).
XI- I 6. Standard-film projection
In standard-film projection the projected object,
bounded-by the film gate,
is usually smaller than
~::f::::~1 ~1 -l"""""==::::=:=:=~ the exit pupil of the
objective; an example of
this arrangement is deFig. 121. In order to fill the exit pupil of the
picted in Fig. 121.
objective 0 with light from every point in the film
In this diagram AB is
gate AB in standard film projection, the solid
angles w 1 , w 2 , w 3 must be filled with light. The
the diagonal of the film
rearward projections of the boundary lines of
gate (24.8 mm for the
the solid angles determine the size of the condenser necessary to ensure the optimum flashing
standard sound-film size
of the objective (e.g. C)
of 15.2 X 20.9 mm 2 , with
rounded corners). The objective 0 has a focal length of 12.5 em and
an angular aperture of f/2, so that the diameter of the exit pupil is
62.5 mm.
Proceeding from the exit pupil of the objective, the figure shows the
path of the rays from a few points in the film gate. If the exit pupil is
to be fully flashed, the solid angles w 1 , w 2 , w 3 etc must be filled with
light.
Now, if we try to achieve this in the same way as for the slide projector
l;>y placing a condenser just behind the film gate, say at C, to produce
an image of the light source in the exit pupil of the objective, it is found
when a standard type of projector lamp or carbon arc is used, that
the source has to be magnified to such an extent - i.e. that the focal
length must be so short - that it is not practicable to make a condenser
of the required diameter.
On the other hand, for a condenser having the smallest /-number that
is technically possible, the light source would have to be so large that
the whole unit would become uneconomical and therefore unacceptable.
There is a solution, however, that entails neither of these objections,
and this consists in producing an image of the light source at the point
where the cross-section of the beam passing through the optical system
is smailest. According to Fig. 121 this is the film gate. If, however,
---r
226
PROPERTIES OF OPTICAL SYSTEMS
[XI
an image of the filament of an incandescent cinema lamp is formed in
the film gate, this image is projected on the screen by the objective,
with consequent unpleasant irregularity in the screen illumination.
But, if the image of the spiral be made to lie between the film gate and
the objective, the image of the spiral in the film gate itself is so blurred
as to ensure reasonable uniformity in the luminance. The spherical
aberration present in most condenser systems also has a blurring effect
on the image. In any case the luminance of the light source must be as
uniform as possible.
The magnification required to fill the film gate with the (unsharp) image
of the light source presents no difficulty as far as the dim~nsions of the
condenser are concerned.
The condenser usually takes the form of a spherical mirror even though,
owing to spherical aberration, this is not the ideal form for a mirror
condenser*).
The arrangement is now as shown in Fig. 122, in which M is the mirror.
AB the film gate, and 0
M
the objective. Let us
determine the exit pupil
of this system.
In the image space there
are three boundaries of
Fig. 122. To determine the exit pupil of a pro- the rays, L 2 , L 1 ' and the
jector for standard film with mirror condenser M.
In the image space there are three boundaries image M' of the mirror
to the rays, viz. M', that is the image of the M. The last mentioned
mirror M formed by the objective 0, L 2 , the front
lens of the objective, and L 1 ', the image of L 1 lies in front of the obas formed by L 2 • Under the conditions as shown,
jective. L 1 ' is always
L 2 is the smallest and is therefore the exit pupil
greater than L 2, as alof the system for points on the axis
ready mentioned, so that
either L 2 or M' will be the exit pupil.
Two possibilities then arise, viz.:
1. M' smaller than L 2• In this case M' is the exit pupil.
2. M' larger than or equal to L 2 • L 2 is then the exit pupil.
To calculate the luminous intensity of the system, the luminance of
the light source must be known in either case. If a 15 V, 50 A Philips
cinema lamp is used, the luminance is about 3000 sb.
For the luminance of the exit pupil we take about half this value, viz.
*) V.ie shall not go into the geometrical optics of projection. For this the reader
is referred to a series of articles by Naumann published in "Die Kinotechnik" 2 ).
XI-17]
SUB-STANDARD-FILM PROJECTION
227
1500 sb; the loss is due in the first place to reflection and absorption
in the system, and in the second to the fact that, owing to the spherical
aberration of the mirror, the exit pupil is not entirely·flashed with the
luminance of the light source.
If we now suppose that (Fig. 122) the exit pupil of the system is that
of the objective (diam. 6.25 em), the luminous intensity of the system
in the direction of the axis will be
I= in X 6.252 X 1500 cd ~ 46,000 cd.
The illumination of a screen 20m from the objective is then 46,000/202 =
115 lux. At points not lying on the axis of the system the illumination
is lower, and the amount of the reduction as against the axial direction
depends on the degree of vignetting by the lens mountings, as well as
on the relative position and size of M'.
If the exit pupil ~f the system is smaller than that of the objective, the
axial illumination on the screen is correspondingly lower.
XI-17. Sub-standard film projection
The leading sizes of sub-standard film are the 16 mm (gate size 7.16 X
9.6 mmz; diagonal 11.6 mm) and the 8 mm (3.3 x 4.4 mm 2 ; diagonal
5.3 mm).
In general the ratios of film gate dimensions to objective diameter
follow the same principle as for standard film, and the smallest crosssection of the beam therefore mostly occurs in the film gate.
At first sight, then, the obvious thing to do would be to adopt the same
system of illumination as that used for standard film, but there is an
important difference.
As pointed out in the previous section, the use of a condenser just behind
the film gate would necessitate prohibitively large lamp filaments in
order to fill the exit pupil of the objective with light, but in sub-standard
film projection this method can be employed; the filaments of substandard-film projector lamps usually require a linear magnification of
3 X in order to fill the objective, and this can be obtained with condensers
of standard !-numbers. In sub-standard film projectors the condenser
is therefore usually placed just behind the film gate. In principle, this
is the same arrangement as employed for the projection of slides, and
the calculations are accordingly wholly analogous to those applicable
to slide projectors, for which reason we shall not give any further
examples.
Sometimes the condenser is placed slightly further away from the film
228
PROPERTIES OF OPTICAL SYSTEMS
[XI
gate, so that the image of the filament lies somewhere between the
gate and the objective; in certain cases this gives a higher luminous
flux from the objective, but, if the image is brought too close to the
film gate the gain in luminous flux is obtained at the expense of the
uniformity of the screen illumination.
When this system is employed, another factor, the non-uniform distribution of luminance of the lamp filament, is involved. If the image of
the filament occurs in or near the film gate, an image of the parallel
spirals of which the filament usually consists is projected more or less
clearly on to the screen, as we have already seen in the preceding section;
the illumination of the screen is then noticeably irregular. Owing to the
fact that the geometrical-optical conditions differ somewhat from those
relating to standard-film projection (this difference will not be enlarged
upon here), the effect referred to is more pronounced than in 35 mm
film projection.
All this does not imply, however, that when the condenser is placed
just behind the film gate the non-uniformity of the luminance of the
filament is unimportant. On the contrary, if the filament is too irregular
the screen illumination will also be irregular, since the vignetting effect
reduces the luminous intensity of the exit pupil in oblique directions
more or less in stages, the more so according as the image of the filament
more completely fills the objective. In the projection of slides, too,
uniformity of filament luminance has its importance.
XI-18. The luminous ftux emitted by projection systems
Equation (XI-6) enables us to calculate the maximum luminous flux
that will reach a projection screen from a projection system with a given
objective and lamp. In this formula we have the means of verifying the
quality of the illuminating system. Formula (XI-6) states that:
(/) = SnL sin2 8;
in this case S is the area of the gate, L the luminance with which the
objective illuminates, i.e. the luminance of the light source multiplied
by the transmission factor of the whole optical system, and 8 the semiaperture angle of the objective. The latter is computed from the /-number
of the objective.
When making use of equation (XI-6) we must ensure that the whole
solid angle, of which the semi-apex angle is 8, is filled with luminous
XI-18]
THE LUMINOUS FLUX EMITTED BY PROJECTION SYSTEMS
229
flux with respect to every point in the gate (we thus assume that, for
points beyond the axis, the solid angles are equal to those of the point
on the axis, which is approximately true).
Let us now compute the maximum obtainable luminous flux in a given
case and compare this with that of the light source. In practice, in order
to check the quality of the projection system, this is compared with
the actual luminous flux from the objective.
Take the case of a standard-film projector having as light source a
15 V, 50 A cinema projector lamp with spherical mirror, and an objective
with an /-number of f : 2. (It will be seen from formula (XI-6) that
the focal length of the objective is immaterial). The transmission factor
of the system may be 0.6, a value which is frequently met with in practice.
The luminance of the projector lamp is 3000 sb; therefore the luminance
of the objective is 1800 sb. f: 2 corresponds to an angle e = 14°2', so
that sin2 e = 0.059. The area of the gate in the case of standard film
is 3.18 cm2.
These values in equation (XI-6) give a luminous flux of
f/> = 3.18 X n X 1800 X 0.059 ~ 1050 lumens.
Of the 20,000 lrn (approx.) of the luminous flux of the lamp, the objective
under consideration thus passes at most 1050 lm, i.e. about 5% reaches
the projection screen. In practice, the actual luminous flux reaching the
screen is somewhat less than that.
In sub-standard film projection the efficiency is generally even lower,
and a luminous flux of 2% from the objective is by no means a rarity.
When the objective is completely filled with light, the luminous flux
from the lens is proportional to the luminance of the light source. An
increase in the size of the source does not affect the luminous flux emitted
by the objective; only an increase in the luminance will do this. For
this reason the lamps used in sub-standard film projectors have often
much larger filaments than are required according to the geometricaloptical characteristics, the luminance being considerably higher, however,
than that of lamps with smaller filaments, consuming less power.
To increase the luminance, spherical mirrors are employed, the filament
being placed at the centre of curvature so that the image of it will
coincide with the filament itself. The mirror should preferably be so
adjusted that the images of the separate sections of the filament fall
between the sections themselves, thus ensuring greater uniformity
of the luminance of the luminous area. Another means of increasing
230
[XI
PROPERTIES OF OPTICAL SYSTEMS
and equalising the luminance consists in making the filament in two
sections, each comprising a row of
vertical spirals, these being so
assembled that the spirals in the
one lie behind the gaps in the other
(Bi-plane lamps).
M
XI -19. Episcopic projection
The principle of the episcope is
illustrated in Fig. 123. The subject
to be projected, e.g. a book or
drawing, is illuminated by a number of lamps L, with or without
--------L--------V
the use of reflectors, and projection Fig. 123. Diagram illustrating the
is effected by means of an objective principle of the episcope. The object
V is illuminated by lamps L and an
0. An image of the horizontal image
is formed by the objective 0.
subject is projected on to a vertical A mirror M throws the vertical rays
on to a projection screen
screen with the aid of a mirror M
placed at an angle of 45° from
the vertical. If a glass mirror is used, the specular layer should be applied
to the front surface of the glass in order to avoid double contours in
the image.
To calculate the illumination of the screen it is necessary for us to know
the luminance of the subject. If we assume for the sake of simplicity
that the reflection is uniformly diffuse, we must know the illumination
of the subject, but it is not usually possible to calculate this from the
illumination system; we shall therefore assume that the illumination
is known and denote this by E v· The luminance of the subject is then
L
v
= e X Ev sb
:rt X 1()4
(e =reflection factor of the subject, Ev expressed in lux).
If we know the area (S cm2) of the exit pupil of the objective, we can
compute the luminous intensity from the following equation:
e. Ev
I = S X -- X T
:7l • 1()4
{Xl-7)
(-r =transmission factor of the objective).
Let us suppose that we wish to project a drawing on paper of which
XI-19]
EPISCOPIC PROJECTION
231
{! = 0.7, using an episcope, the exit pupil of the objective being 14 em
diameter and the transmission factor -r = 0.8. The reflection of the
plane mirror might be em= 0.9. Suppose that a screen illumination of
25 lux is required at a range of 4 metres.
From these particulars we can calculate the required illumination of
the drawing. The range, the screen illumination and the reflection
factor of the mirror enable us to find the required luminous intensity
of the objective:
25 X 42
Ed 2
~ 440 cd.
0. 9
=
I = -
f!m
Ev is evaluated from equation (XI-7), viz:
__ I X n X 1()4
E vSe-r:
here
n
I= 440cd, S = 4 x 142 = 154 cm2 , e = 0.7 and -r = 0.8,
hence
.
Ev =
440xnx 104
1ux = 160,000 lux.
I 54 X 0 .7 X 0 .8
This example will serve to demonstrate the order of the illumination
required to ensure reasonable illumination on the screen. It is obvious
that the objectives used for episcopes must have a small /-number,
i.e. large diameter. A practical objective size, on which the above calculation was based, has a focal length f of 50 em and an aperture of
f/3.5, which means a diameter of 14 em.
Apart from a concentrating- type of optics, light sources of very high
luminous flux must be used, i.e. taking a great deal of power. This
constitutes a problem in view of the heat generated in the episcope,
but with some forms of light source the excess energy can be dispersed
by means of water-cooling. A light source having outstanding qualities
from the point of view of its use in episcopes is the water-cooled superhigh-pressure mercury vapour lamp (such as the Philips types.SP), which
combines a high luminous flux with small physical dimensions, the
water-cooling for the dispersal of the surplus energy being an integral
part of the design.
232
PROPERTIES OF OPTICAL SYSTEMS
[XI
REFERENCES
Some books on geometrical ootics:
B. K. Johnson: "Optics and optical instruments", 3rd edition, London 1960
B. K. Johnson: "Optical Design and Lens Computation". London 1948
D. H. Jacobs: "Fundamentals of Optical Engineering". New York 1943
R. W. Ditch burn:· "Light", 2nd edition, 1963
M ii 11 e r-P o u i 11 e t s: "Lehrbuch der Physik", 2er Band, 1e Halfte, Brunswick 1926
M. Be reck: "Grundlagen der praktischen Optik". Berlin 1930
2 ) H.
N au m.a n n: die Kinotechnik, 11, 1929, 311-315. "Zusammenhange
zwischen Spiegel, Bildfensterbeleuchtung und Lichtleistung bei der Kinoprojektion"
Idem. 11, 1929, 651-657. "Ober den Durchmesser von Projektionsobjektiven"
Idem. 12, 1930, 10-13. "Zur Kinoprojektion mit kurzbrennweitigen Objektiven"
1)
CHAPTER XII
THE PHOTOMETRIC MEASURING-UNITS SYSTEM
XII-I. Introduction
In chapter I we said that when we had dealt with the photometric
system and all that this embodies, we should see in how far the definitions of the conceptions and units employed in illuminating engineering
would have to be extended in scope to include coloured light, and that,
until then, the reader should regard the considerations dealt with as
referring to white light only.
Before going into this problem we will review.. what we have so far
considered in the field of photometric units, and see how far this fulfils
our purpose, viz. the measurement of radiation evaluated according to the
visual sensation produced.
In so doing we may confine ourselves to the basic unit, the candela,
from which the other photometric units are derived. The candela has
been defined as I /60th of the luminous intensity of_ I sq. em of the
black body radiating at the temperature of solidifying platinum, measured perpendicularly to the surface.
This means that a certain luminous intensity is attributed to a given
power (W) per steradian of a radiation having a given spectral distribution. Neither here, nor in the definitions of the other photometric
quantities and units, however, has the visual sensation produced by
the radiation been introduced. In fact, we have gone no further than
to arrive at a physical conception in which the eye plays no part.
In addition to defining the candela, it was tacitly assumed that luminous
intensity and the other photometric quantities are proportional to the
power of the radiation (e.g. we have defined luminous efficiency as the
luminous flux in lumens divided by the power in watts and it was thus
assumed that each watt yields an equal number of lumens).
If there were only one kind of light, with a standard spectral distribution,
it would be possible to express luminous intensity equally well in W Jsterad
instead of in candelas (= lmfsterad) and luminous flux in watts instead
of in lumens, and the measurement could be carried out in a purely
physical manner.
There are, however, different kinds of light and here enters the problem
234
THE PHOTOMTERIC MEASURING-UNITS SYSTEM
[XII
of evaluating all these kinds "according to the visual sensation they
produce". We must, therefore, relate the physical phenomenon "radiation" to the physiological effect "light".
The expression "visual sensation" is vague and indeterminate, and
requires to be further· defined; hence we have to ascertain what characteristic or criterion is the deciding factor in our visual evaluation of radiation.
As will be shown in the next section, the sensation of light is a sensation
produced by the luminance of the objects observed, not by the luminous
intensity or the luminous flux. This physiological sensation must not
be identified with the physical quantity which produces the sensation.
It is, therefore, necessary to distinguish the two conceptions by different
terms. As we have seen, the physical quantity is called luminance.
The visual effect is denoted by the terms subfective brightness or lum,·nosity.
Thus we may say that luminance produces luminosity.
We will call the criterion which enables us to evaluate radiation according to its visual effect the luminance criterion.
XII-2. The luminance criterion
Illuminated objects, or light sources, are perceived by reason of their
form, colour and luminance. The fact that we perceive a surface by its
luminance and not by its luminous intensity or luminous flux will be
understood if we know something of the principle on which vision is
based (see Fig. 124).
As far as the optical construction is concerned, the
eye may be compared to a
PL----t
c<.mera; the lens L forms an
image of objects within the
84219
field of vision (e.g. an image
P'Q' of an object PQ) on Fig. 124. Cross-section of the human eye
a "screen" which corresponds in diagram. R retina; L lens; I pupil; C cornea;
A anterior chamber; V vitreous humour;
to the frosted glass or film N optic nerve.
in the camera. This "screen" An object PQ directly in front of the eye
produces an image P'Q' on the fovea
lies against the rear wall of
centralis F
the eye and is called the
retina (R, Fig. 124). As in the camera, too, the eye has a variable stop,
the iris or pupil (1).
The retina is made up of light-sensitive elements which transmit stimuli
a
XII-2]
THE LUMINANCE CRITERION
235
when light falls on them and these stimuli are· passed through the
optic nerve to the brain where we are made aware of a sensation of light.
An image is formed on the retina, then, of a certain size and illumination;
the size depends on the dimensions and distance of the object observed,
and the illumination, according to section XI-4, is proportional to the
luminance of the object as well as to the entrance pupil of the lens,
i.e. the iris. Now, the pupil of the eye is not constant in size; it varies
between 2 and roughly 8 mm in diameter, and it adjusts itself automatically to the luminance and distribution of luminance in the field of
yision. When the luminance of the field is low the pupil is fully dilated,
and it closes gradually as the luminance is increased.
Investigations by S t i 1 e s and C r a yv ford 1 ) have shown that the
more the light rays entering the eye are directed towards the periphery
of the pupil, the less they contribute to the subjective brightness (StilesCrawford effect). No explanation of this effect has yet been found.
Before the effect was discovered it was held that the luminosity was
goverrted entirely by the illumination on the retina, and retinal illumination units based on this premise are mentioned in the literature.
One such unit is the luxon, defined as the retinal illumination produced
by a surface having a luminance of I cdfm2 when the pupil area is lsq.mm.
This unit as such should not be used.
The illumination of the retina sets up a stimulus which by way of the
optic nerve produce& a mental sensation of light. This effect, related
only to the luminance and not the luminous flux or intensity, is named
the subjective brightness or luminosity. In short, we may say that apart
from form and colour, we perceive luminance.
Now, the luminosity depends not only on the retinal illumination, but
also on the sensitivity conditiot'\ of the retina. When we undergo a change
from a visual field of high luminance to one of low luminance, the
luminosity is at first not constant; this is partly due to the slow dilation
of the pupil, as a result of which the retinal illumination is increased,
and partly to a variation in the sensitivity of the retina. In other words
the eye adjusts itself to the luminance of the new field, and this adjustment in the sensitivity is known as the adaptation of the eye. Once the
eye has completed the process of adaptation we say that it is fully
adapted. Use is made of the terms light and dark adaptation when the
eye has adapted itself to visual fields of high and low luminance
respectively. Light adaptation is accomplished quickly, but dark adaptation is a much slower process, as everyone will have noticed at one
time or another when passing from light to dark places, and vice versa.
In photometry adaptation is an important phenomenon, since reliable
visual measurements are obtained only when the eyes are fully adapted.
236
THE PHOTOMETRIC MEASUKING-UNITS SYSTEM
LXII
The eye is unable to tell us anything about the intensity of the luminosity
in terms of dimensions or quantities and no means are at our disposal
for effecting such measurement. It is also not capable of giving any
information regarding degrees of luminance, other than that luminance
is either higli or low; if two fields of different luminance are observed
simultaneously we can only say that one is brighter than the other.
In any appraisal of quantitative differences in luminance the eye cannot
help us; it can do no more than inform us that two fields observed
simultaneously give an equal or different luminosity.
And this brings us to an important definition for our measuring system,
viz. that when two fields simultaneously produce the same luminosity,
the luminances of the fields are held to be equal.
To this must be added that we regard the luminance as being proportional
to the power; by definition, n times as much power yields n times as
much luminance.
Given a luminance A (proportional to power a), if this luminance be
increased to a luminance B of the same kind of light (proportional to
power b), the sum of these luminances A and B (proportional to a+ b)
will give the same luminosity as the luminance (of the same kind of
light) that is proportional to c, if a
b = c. We may therefore say that
the eye adds up the luminances.
As we have seen above, this does not refer to the luminosities, but to
the luminances themselves, as based on the equality of the luminosities.
From the fact that the eyes are capable only of a comparison of luminances it follows that, if we wish to take visual measurements of other
quantities used in illuminating engineering, these must be converted to
luminances by one means or another.
+
XII-3. Luminance of coloured light
In the preceding section we have said briefly that our measuring system
is based on an equality of luminosity thus assuming that such equality,
even among the most widely diverging colours of light, can always be
perceived. If luminances. being compared differ only slightly in colour
or not at all (homochromatic comparison) this is readily acceptable,
but it is not so acceptable when the kinds of light differ considerably
(heterochromatic comparison). As this may require some explanation
let us turn to Fig. 125 which shows diagramatically a form of apparatus
for the comparison of two luminances. The lines AB and BC represent two
white surfaces having equal reflective properties, and both are observed
by the eye 0. Let us suppose that AB receives blue light and BC red
XII-3]
LUMINANCE OF COLOURED LIGHT
237
light, the lummance of AB being kept constant whereas that of BC
can be varied. Comparison between the luminances of AB and BC is
facilitated by a dividing line
at B, between the two fields.
The observer is asked to adjust
the luminance of the red field
so that the luminosity produced equals that of the blue;
according to the definition
given in the previous section,
the luminances of the red
Fig. 125. Arrangement for comparison of
two luminances. AB and BC are diffusely surface will then be the same
reflecting surfaces, separated from each
as that of the blue surface.
other at B. Both AB and BC are observed
So much red light can now be
from 0.
allowed by the observer to
fall on BC that he can clearly judge the luminosity produced to be
greater than that of the blue. If he then decreases the quantity of red
he arrives at a point where he cannot say with any certainty whether
the luminosities and thus the luminances are equal or different. With
still less red he will quickly be able to see quite clearly that the luminance
of the red is lower than that of the blue. Notwithstanding the great
difference in colour he will thus have observed a difference in luminance,
first in favour of the red, and subsequently in favour of the blue. Within
the region of uncertainty, therefore, the difference in luminance must
have been reversed.
If the observer adjusts his apparr;~.tus a large number of times to the
point where in his opinion there is no difference in luminosity and thus
in luminance, a statistical analysis will show that it can be said of one
certain adjustment that, according to the observations, the luminance
of the red is equal to that of the blue.
In comparisons of markedly heterochromatic light it is difficult and
sometimes impossible for an unskilled observer to effect such an adjustment for equal luminosity, and the adjustments whereby he balances
the luminosities, and thus the luminances themselves, may vary widely.
Nevertheless, it has been found that observers can become practised
in "forgetting" the colours, so that they succeed fairly well in responding
only to the luminance components of the fields compared, as distinct
from colour components.
Fortunately, however, there are other methods of heterochromatic
photometry which demand less of the observer than the method of
238
THE PHOTOMETRIC MEASURING-UNITS SYSTEM
[XII
direct comparison and which make possible much greater accuracy in
measurement.
It has been said above that adjustments for equal luminosity are not
difficult when the differences in colour are sufficiently small, and use
can be made of this fact also to compare luminances· between widely
differing colours. Instead of making a direct comparison between the
red and blue as in our example, we now compare the blue with another
blue colour which is not so very different from the first, but contains
also a little red; this colour is then compared with another containing
slightly more red. The colours can be so selected that they constitute
a gradual transition from blue to red, e.g. by increasing the red content
each time and reducing the blue, thus giving a change over from blue
to red in stages. In the last stage the red is compared with another red
having a small blue component, but so little that it is not disturbing
when the luminosities are being balanced.
This method of comparison is useful only if it appears that the final
result is independent of the series of intermediate colours used (S c h r 6d in g e r 2 )). This has been investigated and verified by K 6 n i g and
others.
By means of this step-by-step method, then, it is possible accurately to
determine luminance ratios of all kinds of coloured light. If in place of
the blue in our example we use light of the standard spectral distribution
from which the candela is derived, we can thus in the same way compare
the luminance of red light with the standard light.
The luminosity produced by the red light being matched with that
of the light of standard distribution, the same value (e.g. in stilbs) is
attrihuted to the red as to the white luminance.
Another method of heterochromatic photometry is the flickn method,
whereby a direct comparison is made. Suppose once more that a red
field is to be compared with a blue one; if the two fields are alternated
rapidly the effect is a flickering of the field of view and, if a start is made
once more at the point where the luminosity produced by the red is
clearly greater than that of the blue, the flicker will be very marked.
If the luminance of the red be then gradually reduced, the flicker will
become less pronounced, or even cease altogether; a further reduction
of the red luminance then again increases the flicker.
The flicker method is based on the property of the eye to react more
slowly to colour stimuli than to luminance stimuli. It is possible so to
adjust the speed of the alternations that the eye perceives changes of
luminance, but not changes of colour. It may also be said that there
is a colour flicker (1), and a luminance flicker (2). At a certain frequency
(1) disappears and (2) remains.
XII-4]
THE RELATIVE LUMINOUS EFFICIENCY OF RADIATION
239
If the luminances of the red and blue fields at the moment of minimum
flicker should be measured by the above described step-by-step method,
the luminances should prove to be the same. The accuracy with which
the point of minimum flicker, and therefore also equality of luminance,
can be adjusted, is very much higher than can be obtained by direct
comparison of stationary fields.
Here, then, we have a method of measuring coloured light by direct
comparison, this being the one employed with the flicker photometer
to be discussed in Part II (section XIV-4). ~he cumbersome step-bystep method is therefore not essential in procuring sufficiently accurate
measurements of coloured light.
XII--4. The relative luminous efficiency of radiation
From the preceding section it would appear that the problem of measurring coloured light, i.e. of expressing the various quantities as applied
to coloured light in the conventional units, has been effectively solved.
Using a photometer, we merely compare the luminance of the coloured
light with that of light of the standard composition and, by balancing
the luminosities, we should be able to attribute to the coloured light
a certain value in stilbs, candelas, lumens etc.
Unfortunately, however, it is not quite as easy as that, for, if a number
of different observers are asked to measure in this manner the same
quantity of coloured light (i.e. a certain quantity of radiant energy of
a given, but arbitrarily selected, spectral distribution), the results will
generally differ among the various observers. Luminosities produced
by the same amount of energy at different wavelengths of the spectrum
differ between one person and another, and are not even constant for
any one observer in the course of time.
As it is not acceptable that the results of measurement should depend
on accidental peculiarities of the eyes of the observer, efforts have been
made to lay down characteristics for the "average eye" with respect
to the evaluation of light. The means to this end are based on a determination of the sensitivity of the eye to radiant energy of different
wavelengths, the relative luminous efficiency of radiation.
To define this let us turn once more to Fig. 125. We will assume that
the left hand ·field is illuminated with light of wavelength A0 , and
that the energy *) reaching the eye and inducing a certain luminosity
is E 0 • This can be expressed in watts. The right hand field may be
*) Where in this and following sections the term energy is used, it is intended to
represent energy per second (power).
240
[XII
THE PHOTOMETRIC MEASURING-UNITS SYSTEM
illuminated with light of wavelength A, this being so adiusted as to produce
the same luminosity as the other. The energy of the light of that
wavelength A that will then have to enter the eye may be denoted byE>...
Now suppose that, in order to obtain the same luminosity, three times
as much energy of wavelength A is needed as that of wavelength A0 ;
we then say that the relative luminous efficiency of radiation of the wavelength A is lf3rd of that of wavelength A0 • Hence the relative luminous
efficiency of radiation is inversely proportional to the energy required to
produce equal luminosities.
In the experiment described, the wavelength A can be successively
replaced by a number of others and, by ascertaining each time the
power required to induce the same luminosity as that induced by the
reference wavelength A0 of energy E 0 , we can determine the ratios of
the relative luminous efficiency of radiation in relation to the wavelength A0 for different wavelengths. The values thus obtained can then
be plotted as a function of A. As it is here a question of ratios, it is
customary to multiply these ratios by a factor such that the maximum
point in the curve represents unity. They are denoted· by the symbol V>...
The relative luminous efficiency of radiation of a large number of persons
has been measured by various workers and, in 1924, the C.I.E. standardized the average of these VA measurements. Formerly these standardized
values of VA were called the International Luminosity Factors.
Later, others have carried out further measurements of VA but, although
many of these were found to
diverge from the internationally 1.0
accepted values, the C.I.E. have o,a
:I
up to now not considered it
necessary to modify the curve 0.6
originally adopted.
ot4 - 1-·
The values of the relative luJ
2
minous efficiency of radiation
'\
I-).~
have been plotted in Fig. 126, o
400
500
600
m~ ?00
and the respective values of the
wavelengths, in stages of 5 mt-t Fig .. 126. The. internation;~J !elative lu.
. T bl VI (
)
mmous effic1ency of rad1atlon ( V>..).
are g1ven m a e
p. 414 .
It will be seen from the curve
that V.\ reaches a maximum at 555 mt-t (or more precisely at 554 mt-t)
and falls off towards each end of the spectrum. For red or blue light,
therefore, much more power is required to induce a given luminosity
than for green or yellow light.
lfl
°·
( \
I
-v
I
\
'\
\
XII-4]
THE RELATIVE LUMINOUS EFFICIENCY OF RADIATION
241
It should be pointed out here that the curve in Fig. 126 is valid only
for luminances above 3 cdfm2 (approx.); what happens to the relative
luminous efficiency of radiation at lower luminance levels will be seen
in section XII-7.
Using VA we can now derive the conditions to be met by the radiant
energy in order to induce balanced luminosities.
Let us denote the relative luminous efficiency of radiation at wavelengths A0 and A1 by V 0 and V1 ; then
(XII-I)
So far we have considered only those kinds of light which consist of a
single wavelength, but in practice we are nearly always concerned
with 1ight ·that comprises numerous components, of different wavelengths. In such cases it would of course be possible to proceed along
the lines given in section XII-3, but this would not be practical in view
of the very large number of possible combinations.
There is another course open to us, however. It is found that it is permissible to assume that the eye adds up the luminosities of the different
components (summation law*), and we can accordingly lay down the
conditions for balanced luminosities for heterochromatic kinds of 1ight
in this way.
We substitute for the light of wavelength ..t1 another consisting of a
combination of wavelengths A2 and A3 whose energies are respectively
E 2 and E 3 ; these energies are so adjusted that the resultant luminosity
is equal to that of the light of wavelength A0 •
Then, assuming from the summation law that the luminances due to
the components ..t2 and ..t3 are added up by the eye, the expression
V 0 E 0 = V1 E 1 is now replaced by
(XII-2)
If the light has numerous components ..t2, ..t3 , A4 , Ali .... whose energies
are E 2 , E 3 , E 4 , E 5 •••• , the conditions of balanced luminosities will be:
V 0 E 0 = V2 E 2
+ V E + V4 E4 + V 5E 5 •••.
3
3
or
If individual differences in relative luminous efficiency of radiation are
*) referred to again in section XII-5.
242
THE PHOTOMETRIC MEASURING-UNITS SYSTEM
[XII
to be eliminated, the internationally adopted values of V>. must be
used in place of V2, V3 etc., so we usually write:
(XII-3)
Where the spectrum of the light is continuous it may, without incurring any appreciable error, be regarded as being composed of small
bands of wavelengths e.g. 10 mJ-t
in width, and the radiant power £'~~)
r aA.
at each wavelength within these
bands can be taken as being that
of the mean wavelength of each
band.
Fig. 127 depicts such a continuous
spectrum with radiant energy plotted against wavelength. In this
figure the total energy is represented by the area enclosed within
0,6
0.5
the curve and the abscissa; this
area can be divided into strips
LIA. in width (e.g. 10 mJ-t) and the Fig. 127. Distribution of spectral
energy of a light source. The area
area of each strip will then re- between the curve, the abscissa and
present the energy radiated the ordinates of 0.4 andO. 7 ~represents
the energy radiated in the visib1e part
between the wavelengths A.1 and
of the spectrum.
A.l + LIA..
In applying equation (XII-3) we assume this energy to be radiated
at a wavelength of A(= A.1 + !LIA.).
t
This can be stated with more mathematical precision by taking in place
of strips of finite width .H, strips of infinitely small width d>.. The expression .EV;..E>. is then replaced by J V;..E';..d~, in which E'>. is the energy
per wavelength unit (~~).
In practical calculations, however, the
integral is replaced by the summation as above.
XII-5. The summation law
The validity of the summation law in the measurement and evaluation
of luminance is of great importance for photometry and illuminating engineering, since it enables us to add up the values of luminance, luminous flux etc. of kinds of light which differ from one another.
For example, given an incandescent lamp of 1000 lm and a mercury
vapour lamp of 500 lm, we may accordingly assume a total luminous
flux of 1500 lm for the blended light.
XII-6]
PHOTOMETRIC QUANTITIES AND UNITS BASED ON VA.
243
This law may also be formulated in another manner; when a luminous
flux t;/J1 (spectral distribution 1) is equivalent to a luminous flux t/J2
(spectral distribution 2), a blended luminous flux comprising
is also equivalent to t/J1 and t;/J2 {otis an arbitrary value between 0 and 1).
The same formulation will also apply to luminous intensity, luminance etc.
It is not usually realised when the quantities employed in photometry
are added up that use is being made of an important law, the validity
of which was established only after considerable investigation (in certain
circumstances it is not actually valid at all).
This becomes all the more apparent if we look for a moment at the
consequences of employing another criterion than the luminance criterion, e.g. the amount of light that might be required for the execution
of a certain 'visual task, e.g. the visual acuity.
By visual acuity is meant, expressed in simple terms, the capacity of
the eye to distinguish fine details in the field of view.
The summation law does not hold good when based on such criteria,
however, and a system of measurement based on them would lead
to great practical difficulties. This means, then, that quantities of light
which are equivalent according to the system of measurement founded
on the luminance criterion may in other respects not be equivalent.
A striking example of this is to be found in the far higher visual acuity
experienced with sodium light than with the same quantity of light
from tungsten lamps.
For further information regarding the requirements to be met by
measuring systems, and the extent to which our photometric measuringunits system satisfies these requirements, the reader is referred to a
very interesting work by H. Konig: "Der Begriff der Helligkeit",
p. 40 et seq. a)
XII-6. Definitions of photometric quantities and units based on V,\
In section XII-4 we have expressed equality of luminosity and, by
definition, also equality of luminance as equation (XII-3):
VOEO = 1: V,\E,\.
By analogy we can express equality of luminous flux as: V 0 G0 = 1: V,\G.\,
G being the power of the luminous flux. In this the terms V 0 G0 and
1: V ,\G,\ represent the luminous flux, but in a different unit from the
244
THE PHOTOMETRIC MEASURING-UNITS SYSTEM
[XII
lumen, which is the unit so far employed for luminous flux. If G be
expressed in watts, the unit in which E V,xG,x is expressed is called
the light-watt; we therefore write:
l/J = E V,xG,x (light-watts).
(XII-4)
To convert the value of the luminous flux in light-watts to the usual
unit employed in photometry, viz. the lumen, it must be multiplied
by a - for the moment unknown- factor, C. Equation (XII-4) then
becomes
l/J = C E V,xG,x (lumens).
(XII-5)
Analogous formulae can be constructed for the other photometric
quantities in which the constant C will appear as the ratio of the candela
(lmfsterad) to light-watt/sterad, the lux (lmfm 2) to light-wattfm2 and
lm
)
light-watt
the stilb ( --fcm2 to
fcm 2 •
,stj:!rad
sterad
watts
For example, if E in equation (XII-3) is in--dfcm 2 , the terms V 0 E 0
stera
light-watts
and E V,xE,x will represent the luminance in
fcm 2 • The lusterad
minance L in stilbs can then be written:
(XII-6)
L = C E V"E".
In this way the photometric quantities and units are based on the international
relative luminous efficiency of radiation V ..\·
It now remains to evaluate C, and this can be done experimentally
or by calculation. Experimentally, C may be determined by measuring
on the one hand the luminous intensity of a light source as compared
with a source calibrated in candelas and, by ascertaining on the other
hand the absolute values of the spectral distribution of energy.
By calculation, C may be evaluated with the aid of a formula introduced
by P 1 a n c k, by computing the spectral distribution of the black
body at the melting point of platinum, that is, the number of W fcm 2
radiated at each wavelength. This, divided by 7£, gives values in Wfcm2
per steradian perpendicular to the surface of the black body.
The values thus obtained are inserted in equation (XII-6), in which
L = 60 sb (cdfcm 2); in this equation V,x is known and the value of C
can easily be computed.
The two methods of evaluating C should of course yield the same results,
but so far the extent to which the empirical result agrees with the
"theoretical" value of C has not proved very satisfactory.
XII-6]
PHOTOMETRIC QUANTITIES AND UNITS BASED ON
VA.
245
The discrepancy may be put down firstly to errors inherent in the
experiment, and secondly to uncertainty regarding the values of the
constants occurring in P I an c k's formula, in consequence of which
the calculation tends to yield unreliable results.
So far, all the experiments carried out in order to determine the value
of C have given a value in the region of 630. Calculations based on
P 1 an c k's formula, applying therein the values "of the constants which
are commonly considered to be the most accurate, yield a value of 680.
In a paper read before the CIE meeting in Vienna in 1963 4 ), Preston
put forward a new proposal for the basic unit of light. He suggested
that the original standard of the candela and its definition be replaced
by a definition of C. With the known absolute spectral energy distribution
of a light source, the luminous flux can then be calculated by means of
this agreed value. This method presupposes accurate measurement of the
spectral energy distribution, which, however, is very difficult to do.
The difficulties with the accurate spectral energy distribution can be
circumvented by measuring a light source with a calibrated radiation
meter (a radiometer) with a filter interposed, the spectral transmission of
which is similar to VA,.
The paper describes the method and a few tests made with their results.
Preston found the mean value of C to be 680 lmfW using standard
lamps from the National Physiral Laboratory at Teddington, Britain,
the photometric data on which were, of course, known.
Further tests and measurements in other laboratories will be necessary
to find any differences in the measurement of energy and to eliminate
them. If this were to be successful, the original standard of the candela as
the basic photometric unit could be abandoned.
In P 1 an c k's formula, E (,\, T) = c1
,\-5
,\T
, th~re are two constants,
ec,; -1
c1 and c2 , which must be known. At the same time, the exact temperature
for which the spectral distribution of the black body is to be computed
must also be known. In our case this is the melting point of platinum
(T, 1 = 2043 °K). Now, there is some uncertainty in regard to the value
of c1 , as well as to the melting point of platinum; T , 1 is calculated on the
basis of the melting point of gold (Tau).
Until recently it was thought that the accepted value of Tau (1336 °K)
was quite accurate, but latterly some doubt has been cast upon its
accuracy and some investigators are inclined to the view that, of the
three constants c1, c2 and T au• the last mentioned is the least reliable 5).
Further experiments will have to show whether or not Tau (and therefore also T 211 ) is some degrees higher than the generally accepted value,
thus eliminating the discrepancy between the experimental and calculated
values.
The argument may also be reversed, and from the discrepancy between
Ctheor- = 680 lmjW and C•xv· = 630 lmjW the conclusion might be
drawn that Tau is equal to 1342° K (this being perhaps the most reliable
246
THE PHOTOMETRIC MEASURING-UNITS SYSTEM
[XII
determmation of Tau)· The adoption of th1s value for Tau would at
the same time eliminate another discrepancy, viz. that between the
"theoretical" value of the Stephan-Boltzmann constant u (5.67 X
IQ-18 Wfm 2 °K 4 ) and the "experimental" value of the same constant
(5. 74- 5. 79 X lQ-1&), the experimental value then being 5.65- 5. 70 X
IQ-18Wfm2 oK4.
New measurements in Germany, whose results however are not published,
seem to show that the melting point of gold lies indeed some degrees higher
than it was accepted up to now. The temperature found, however, is not so
much higher that the above-mentioned discrepancy can be completely
explained.
Experimental photometry is effected by comparison with the candela
(and units derived from it). Let the luminous intensity of a light source
be n candelas; we can then write:
(cd)
(XII-7)
where .E V~E~ and .EV~E~,ca are the luminous intensities in lightwatts/sterad of the light source, and 1 cm 2 of the black body at the
freezing point of platinum respectively.
C can be eliminated from both sides of the equation (XII-7) and need
not therefore be known; this is not necessary, since we have created
a light source for which the value of 1 (cd) has been attributed to the
expression C .E V~E~,ca·
Reverting to formula (XII-5):
cJ> = C .E V~G~
this may also be put in the form:
cJ> = .E CV~G~.
We will now consider one of the terms of the summation, e.g. for a
wavelength which we shall denote by 1, then
cJ>1 = CV1G1 .
We can now calculate the value of CV1 for this wavelength, for example,
and thus obtain a constant for the latter (usually denoted by K~), indicating the ratio of the luminous flux (lm) to the power (W) radiated
as monochromatic light at that wavelength. K>.. is thus given in lmfW,
and is known as the luminous efficiency of radiation at wavelength A..
For A.= 555 mfl (V~ = 1) K~ reaches its maximum (Km). This is then
numerically equal to the constant C in equation (XII-5), and, in this
formula, Km (or· K) is usually employed instead of C. We shall accordingly
employ K in place of C in the following.
K or Km (680 lmfW) is thus theoretically the highest possible value
of the luminous efficiency of a light source. If we regard K as the ratio
XII-6]
PHOTOMETRIC QUANTITIES AND UNITS BASED OV ·VA.
221-7
of lumens to light-watts (both units of luminous flux), K will be an
abstract value. But, if K (= Km) be regarded as one of the values of KJ...,
so that K becomes the quotient of the luminous flux (lm) by the energy
(W) emitted at a wavelength of A.= 555 mp, where KJ... = Km and
V" = I, then K must be expressed in lmjW.
The reciprocal of K or Km is known as the mechanical equivalent of
light and is denoted by M. This is expressed in Wjlm. Assuming that
K = Km = 680 lmjW, M = 0.00147 Wjlm.
The expression "mechanical equivalent of light" is more or less obsolete and its use is not to be recommended.
The definitions of the photometric units, and with them the quantities
used in photometry, as based on the international values of VJ..., have
completely ohanged the face of photometry.
For many decades those interested in photometry were concerned with
the question which method of photometry was the best, that is, the
best suited to the characteristics of the average eye. The answer to
this is now: "a method of photometry is correct ff it yields results which are
compatible with formula (XII-6) L = K E EJ... VJ...".
Since there are few observers whose spectral sensitivity correspond
exactly to the international values of V J.. , it is a fortunate circumstance
that in the last decades we have found in the photo-electric cell a means
of determining the value of E E" VJ... along purely physical lines, thus
eliminating the human eye as an element in photometry. This is possible
because the spectral sensitivity of these cells can be matched with the
international relative luminous efficiency of radiation; further reference
is made to this point in Part II.
In section XII-3 we have already mentioned the term employed for
photometry by comparison of light sources of different colours, viz.
heterochromatic photometry. The spectral distributions of the kinds
of light to be compared are thus necessarily different.
When the kinds of light to be compared have the same spectral comP9Sition and are of the same colour we speak of homochromatic photometry; in this case the results of measurement are not dependent on
the spectral sensitivity of the observer's eye or of the photo-electric
cell.
Where the colours of the two kinds of light are similar but the spectral
composition is different (and this is quite possible), the results of measurement are then indeed dependent on the spectral sensitivity of the
observer or photo-electric cell. This is known as subjective homochromatic
photometry.
248
THE PHOTOMETRIC MEASURING-UNITS SYSTEM
[XII
The term pseudo- homochromatic photometry is sometimes employed
when the difference in colour is so slight that it does not interfere with
the accuracy of the balance and when the difference in spectral composition is so small that the measurement is practically independent
of the spectral sensitivity of the observer or photo-electric cell.
Every photometric value expressed in units oflight corresponds to a radiation oxpressed in units of energy, and, indeed, every photometric value is
the corresponding radiation value assessed in relation to VA..
The radiation quantities have also been given appropriate terms. The
values that correspond to one another are given in table VIII (page 416)
for the purposes of comparison.
XII-7. Vision at high and low luminance levels; Purkinje effect
We have mentioned in section XII-3 that the internationally adopted
values of v,\ are valid only for luminance levels above 3 cdfm2 (approx.). We shall now see how spectral sensitivity is affected at lower
luminance levels.
Let us first consider the construction of the eye and in particular that
of the retina.
The latter consists of light-sensitive elements of two kinds, between
which a clear distinction in form can be made, viz. rods and cones; in
addition to this difference in form there is a very much greater difference,
viz. in the manner in which they function.
Practically speaking, at very low luminance levels only the rods operate,
whereas at luminance levels above about 3 cdjm2 only the cones are
called into play. Between these two regions there is a zone in which
the luminosity is due to both rods and cones.
A second difference is that the cones enable us to perceive colours,
whereas the rods are not capable of distinguishing differences in colour.
With the rods alone, everything would appear to us as in a black and
white photograph, that is, we should perceive only differences in luminance. The transition from cone-vision to rod-vision when the luminance is reduced is the reason for the fact that the perception of
colours of everything around us fades away when darkness is falling.
The cause of the difference in colour sensitivity between the rods and
cones· must be sought in the light-sensitive (photo-chemical) substances
present in the rods and cones. In the rods this is the visual purple, of
which the photo-chemic.al properties are f~.i~ly accurately_ k!lown. The
substances responsible for the photo-sens1t1ve charactenstlcs of the
cones have so far not been identified with any certainty.
A third important difference between the rods and cones is to be found
in the local distribution of the two different kinds of photo-sensitive
elements in the retina. Around that point in the retina where the image F
(Fig. 124) is formed of an object viewed directly by the eye i.e. which
we see by looking straight ahead, there is a small practically circular
XII-7]
VISION AT HIGH AND LOW LUMINANCE LEVELS
249
area about 0.25 mm in diameter, which contains only cones. This is
the fovea centralis and the point where it occurs reveals a slight depression in the retina. The fovea centralis lies in the centre of an area
of the retina about 2 mm in diameter known as the macula lutea, or
yellow spot, so called because it is of a slightly yellow tint. The number
of cones per unit area gradually decreases from the fovea towards the
edge of the yellow spot and, at the same time, there is an increase in the
number of rods. This gradual transition from cones to rods continues
beyond the yellow spot so that, away from the centre (at the periphery),
the concentration of cones is only low and that of the rods relatively
high.
One of the many peculiarities of vision that can be explained by this
distribution of the rods and cones is the following: if an effort is made
to fix the gaze on a small object which, owing to its low luminance, is
barely visible (e.g. a very faint star), it seems to disappear. The image
is then formed on the fovea, which contains no rods, but, if the direction
of vision is shifted through a small angle, say 10°, the object again
becomes visible because the image is then formed at a point on the
retina where rods do occur.
The fourth difference between rods and cones lies in their respective
spectral sensitivities. In the same way that the relative luminous efficiency
of radiation above 3 cdfm2 has been determined (region of photopic
vision), the spectral sensitivity has also been measured in that range
of luminances where only the rods function (region of scotopic vision).
The curve thus obtained is reproduced in Fig. 128 (curve V,x'), in which
1,0
'
the international curve for pho-
topic vision has been included
for comparison (V,x). The V,x'
0,8
\
1
'~
curve has been adopted by the
I
0,5
I I \
C.I.K in 1951 (Stockholm). The
\
I
1/ I \
\
0,4
values of V ,x' for values of
v,:
I
%
wavelengths in stages of 5 mp.
\
I
I
L
\
I
v
\.
are given in Table VII (p. 415).
I
......
./_ 1---'
.......
A
noticeable difference is to
0 400
500
mp 700
500
-).
be seen between the two curves.
Fig. 128. V,x'. The international relative
In cone vision the maximum
luminous efficiency of radiation of the
occurs
at 555 mp. (yellow-green),
dark-adapted eye (Scotopic vision).
V,x. The international relative luminous whereas for rods the maximum
efficiency of radiation of the light-adapted
is at 507 mit (blue green); the
eye (Photopic vision).
curve for scotopic vision is
displaced with respect to that for photopic vision in the direction of
the shorter wavelengths. There is a pronounced difference particularly
v
l
.
J
v
'l
250
THE PHOTOMETRIC MEASURING-UNITS SYSTEM
[XII
at the red end of the spectrum; the photopic vision curve (VA) extends
much further into the red than the scotopic vision curve (VA').
In the region between pure cone vision and pure rod vision the relative
luminous efficiency of radiation lies between V ;~ and VA' dependent
on the luminance level. In this intermediate region the curve shifts
from V;~ to VA'·
In consequence of this difference in spectral sensitivity the ratio of
the luminosity of one coloured object to that of another varies when
the radiant power is considerably reduced, but is maintained at the
same relative levels for the different colours. Let us illustr.ate this by
means of an example.
Suppose that one half of the field of view is a surface radiating yellow
light of 581 mp, with high luminance, and that the other half radiates
green light of 530 mp,. From Fig. 128 it will be seen that VA for both
wavelengths is the same. If we now make the luminosities equal we
shall also have balanced the respective energies. If we then reduce the
luminance levels by reducing both of the energy values by ·a factor of,
say, 1000, the energy at each side will still be the same, but the relative
luminous efficiencies which we can read from curve V >.' in Fig. 128
will now be different, viz. 0.114 and 0.81. This means that, after reducing
the luminance level we should have to make the energy of the yellow
light seven times as much as that of the green to achieve similar luminosities. Hence, for the same amount of energy, the green light gives a
much greater luminosity than the yellow. The effect when the experiment
is carried out with red and blue spectral colours is even more marked.
Something very similar is experienced in nature. When we look at red
flowers among green foliage by twilight, the flowers appear almost
black, because the rods in the retina are so insensitive to red light
(curve VA', Fig. 128). This phenomenon is clearly perceptible, although
not quite so prominent as in monochromatic light, seeing that the
objects also reflect other rays which usually tend to lessen the effect.
In general, this peculiarity of the eye is known as the Pur kin j e effect,
after the Czech physiologist P u r k i n j e who was the first to describe
it in the literature. The cause, that is, the displacement in the VA curve,
whictl in Pur kin j e's time was not yet known, is often alluded to
by the same name.
The occurrence of the Purkinje effect is the reason why in visual
photometry on the basis of the international VA curve we have to
ensure that the luminance of the surface perceived by the eye is in the
zone of cone-vision, i.e. above 3 cdfm2 (approx.).
XII-8]
EQ.UIVALENT LUMINANCE
251
After extensive investigations B o u m a') has ascertained that the
summation law applies with all accuracy in the region of pure rod
vision.
XII-8. Equivalent luminance
Equality of luminosity has been formulated in (XII-3) as
VOEO = E V>.E>..
By definition, it is held that wherr the luminosities are balanced, the
luminances are also equal.
V 0 E 0 and E V>.E>. are expressions for the luminance, the unit being
light-watt
---d~fcm2 ; the luminance can be expressed in stilbs (XII-6) as
stera
L = K EV>.E>..
This definition is based on the international V>. curve; it applies also
to the luminance region (below about 3 nt) where the rods function
either partially or exclusively and regarding which we have seen in the
previous section that the luminous efficiency of radiation is not the
same as it. is above 3 nt. This means that, in that particular region,
equal luminances in accordance with expression (XII-6) are no longer
accompanied by equal luminosities, as already shown in the previous
section, in which a comparison was made between yellow light of
581 m.u and green light of 530 m,u. At the higher luminance level the
luminosities were balanced, whereas at the lower level the balance no
longer exists.
Although it is not possible, outside the photopic region, to relate subjective brightness to luminance in a unique manner, it is of interest
to have some scale in order to be able to evaluate subjective brightness.
To this end the conception of equivalent luminance has been introduced.
In 1891 K 6 n i g formulated the definition of equivalent luminance
(named by him Helligkeit) as follows:
Put for a light of a certain spectral distribution the equivalent luminance
proportional to the energy (and thus to the luminance) for all values of
luminance; in order to ascertain the equivalent luminance of another
kind of light this is compared with that of the reference spectral distribution and the equivalent luminances are said to be equal when the
two luminosities are balanced.
The choice of the reference spectral distribution i!> arbitrary. K 6 n i g
chose, as reference, light of 535 m,u and determined the equivalent
luminance for the other wavelengths over a large range of luminances.
252
THE PHOTOMETRIC MEASURING-UNITS SYSTEM
[XII
After K 6 n i g several other kinds of reference sources have been
proposed. In 1951 the C.I.E. agreed upon standardizing as the reference
spectral distribution that of the light emitted by a full radiator at the
freezing point of platinum (2042° K).
The results of measurements of equivalent luminance with light of the
C.I.E. spectral distribution as reference light are shown in fig. 129.')
Here the luminance (L) and the wavelengths have been plotted on the
ordinates. Each point of the area of the diagram thus represents a
physically defined starting point (defined by L and A.), and at each of
these points the relevant result of the measurement of the equivalent
luminance can be indicated. If, thereafter, the points with equal equivalent luminance are connected by curves, then fig. 129 is obtained.
The dotted line connects the wavelengths for which the luminance is
equal to that of an apparently equally bright field with light with the
spectral distribution of the black body at 2042° K.
The Purkinje effect can clearly be edt,.
IO•r---t:--,...---.,..----r----!-.,0
seen in the figure. If, for instance,
Q
blue light of 475 mp, is compared with red light of 625 mp,,
1---+---+--.;-+--....."1.~,.
then it is to be seen that in
order to obtain an equivalent
luminance of 10, the same
10"'1------+----b_,."""""+---::::l
luminances are required (points
P and Q). If, however, an equivalent luminance of 10-3 must
be produced, the luminance
required is for the red light
(Q') about 100 times as great
as for the blue light (P'). Points
P and Q lie in the region of
pure cone vision whereas at
P' and Q' the luminosity is
produced· by nearly pure rod
vision.
If we denote the relative
luminous efficiencies of radiation at a certain (low) lu650mp
minance level by Vo' and V1 '
for wavelengths A0 and A. 1
Fig. 129. The equivalent luminance L' as a
function of wavelength ,\ and luminance
L. Points P, Q and P', Q' demonstrate the
Purkinje effect.
XII-8]
respectively,
EQUIVALENT LUMINANCE
then
253
balanced luminosities can again be expressed by:
Here V 0 ' E 0 and V1 ' E 1 are expressions for equivalent luminance, analogous to the expressions for luminance. To bring these equivalent
luminances on the above adopted scale and to express them in e.g.
cdfm2 , we have to match them with the reference light, for which the
equivalent luminance is equal to the luminance or K' .EV>.'E.\ = K .EV>.E>.·
For the equivalent luminance of the light A.0 we have the equation
K'V 0 'Eo = K' .EV>.'E>. = K.EV>.E>..
Here K' is again a luminous efficiency of radiation. Its value changes
with decreasing brightness levels till in the region of pure rod vision,
where V>.' become~ constant, it also takes a constant value. It is only
in this region that the equivalent luminance is related by a constant
factor to the luminance which factor, however, depends on the spectral
distribution of the light considered.
The value of K' depends on the choice of the reference spectral distribution. This can be seen from the following argument:
If V 0 ' E 0 = V1 ' E 1 and A.1 is the wavelength of the reference light, then
the equivalent luminance of the light with wavelength A0 can be expressed by
K'Vo'Eo = K'V1'E1 = KV1E1
in which V 1 is the relative luminous efficiency of radiation of A. 1 in the
photopic region.
If the equivalent luminance is defined as above by
K' must be given another value since V1 'E1 and .EV/E>. are equal
but, consequently, V1 E 1 and .EV>.E>. are different.
In the region intermediate between the pure cone vision and pure rod
vision, doubling K' V 0 ' E 0 means by definition doubling the equivalent
luminance but not of the energy since the energy governs the luminance
level and thus also V >.'.
From the point of view of the lighting engineer this means that at
low luminance levels twice as many lamps will give to an object twice
the luminance, but not twice the equivalent luminance.
254
[XII
THE PHOTOMETRIC MEASURING-UNITS SYSTEM
XII-9. Minimum perceptible luminance difference and sensitivity to
luminance difference
We have shown in sections XII-2 and 3 that the only quantitative
assessment of luminosity that the eye is capable of making is one of
equality in respect to two fields. If the experiment described in section
XII-3 be carried out with the wedge depicted in Fig. 125 with the
same kind of light on each side, and we then balance the luminosities,
the luminance on one side can be varied slightly without producing a
visual sensation that the balance has been disturbed. The threshold
value of the differences in luminance, i.e. the lowest value of the difference
in luminance, that is only just perceptible, is called the minimum perceptible luminance difference.
If we denote this difference with respect to a luminance L by LJL, the
variation which is only just perceptible will be LJLfL.
Hence high sensitivity to luminance differences is accompanied by a
low value of LJLfL. In order to be able to indicate greater sensitivity
by means of a higher numerical
LfAL
value, it is more convenient, J
however, to employ the reI
1
fib
ciprocal LfLJL, and this is
known as the sensitivity to 120
luminance difference. The con100
0
ception is often called contrast
v
~
:7
sensitivity but since different B
v
kinds of contrast occur, it is
/
60
1/
I/
preferable to use a more ex~
j
/
plicit term.
l......,....v
Values obtained for sensitivity
20
,
to luminance difference depend
0
very largely on the conditions
Q01 0,1
1
10 1001 fOJ 104 J
-:a -1
0
a 3 4
under which measurement is
carried out, and in particular on Fig. 130. Sensitivity to luminance difference
Lf.dL.
the luminance distribution in a) dark background,
b) and c) bright
background.
the field of view. It is therefore
not surprising that the values
published by different workers vary considerably among themselves.
Fig. 130 illustrates the results of three measurements; as the range
of luminances over which the measurements extend is so very great,
the luminance L has been plotted on a logarithmic scale.
Curve a in this figure represents the average results of measurements
I
I
I
1'\
I\
XII-9]
SENSITIVITY TO LUMINANCE DIFFERENCE
255
by Konig, B Ian chard and Bouma, and all of them are in
fairly close agreement; curve b shows measurements by S t i I e s B),
and curve c was obtained from a large number of tests carried out by
Schuhmacher 9).
Curve a is based on comparison fields covering only a small part of the
field of view, the rest of the field being dark. The conditions under
which curves b and c were produced were roughly the same; the field
of view, filled with light of uniform luminance L subtended an angle
of about 40°, and the luminance of a small part of the field was varied
to an extent LfL.
The difference between curves band c, which were nevertheless obtained
under very similar conditions, may probably be put down to slight
differences in the measuring conditions on the one hand and individual
differences among the observers on the other.
The curves should not be looked upon as average for a large number
of observers, seeing that S t i I e s employed two observers and S c h u hm ache r only one *).
In view of the position of the results of S t i 1 e s' last two measurements
the form of the end of curve b is somewhat uncertain. The last point
of all indicates a bend in the curve, but the measurements do not reveal
exactly where· the bend commences.
Curve a is particulary important in photometry, since the conditions
under which it was obtained are entirely similar to those under which
most visual photometers are operated. From curve a we see that with
luminance values between 10 and 5000 cdfm 2 the sensitivity to luminance
difference is between 50 and 60, which means that in vimal photometry,
in the range of luminance concerned, the difference .between any two
levels may be approximately 2% without implying that one of .the
adjustments is incorrect. If a precise result is required, this can be
obtained by readjusting the luminance a number of times in succession
and taking the average of the results.
The curve refers to homochromatic comparison. With heterochromatic
comparison the sensitivity is lower and the accuracy of the measurement
becomes lower according as the luminous fields in the photometer differ
more widely in colour.
Curves b and c are also of interest in connection with photometry, not
that it is likely that measurements would ever be carried out under
*) The results obtained by both investigators are the outcome of a series of
comparative tests in which the absolute values of the measured sensitivities were
of minor importance.
256
THE PHOTOMETRIC MEASURING-UNITS SYSTEM
[XII
precisely the same conditions as those under which these curves were
taken and thus yield the degree of accuracy inherent in these curves.
The curves do teach us, however, how the accuracy of the balance can
be increased by the conditions under which measurement is carried
out. It may be deduced from the diagram that the sensitivity to
luminance difference, and therefore also the accuracy of the matching,
may be increased by giving the surroundings of the photometer fields
a certain luminance, instead of leaving them dark.
The curves shown in Fig. 130 are also of interest to the lighting engineer,
since conditions will arise in his field of activity which are the same as,
or in any case very similar to those under which the curves were obtained.
For example, the conditions met with in street lighting are quite comparable with those under which curve a was determined; conditions in
rooms with light-coloured walls will often be similar to those relating
to curves b and c, or somewhere between curve a and curves b and c.
Apart from the effect of other physiological and psychological elements
in vision, it is the influence of the luminance distribution in the field
of vision and the sensitivity to luminance difference which have been
largely responsible for prompting the lighting engineer to concentrate
more on the distribution of luminance in his projects. In this connection
the term brightness engineering, or better luminance engineering has
become current.
When the luminance of the surroundings, partly or wholly, is higher
than that of a field in which small differences in luminance are to be
observed, the sensitivity to luminance difference is greatly reduced;
such circumstances give rise to what is known as glare. The effect of
a light of great luminance upon our powers of observation in general
and not only of differences in luminance, is only too well known.
Measurements of sensitivity to luminance difference were carried out
even in the earliest days of lighting technology, for in 1729 B o u g u e r
stated that it was possible to distinguish one field of luminance from
another only when the difference in luminance is as I : 64.
Others including Weber (1834) came to the conclusion that sensitivity to luminance difference in the (relatively small) luminance
range within which they made their observations, was roughly constant,
and from this arose Weber's law, which states that Ljt1L is constant
for all values of L.
Fig. 130 shows how much truth there is in this. It is only under the
conditions relating to curve a (dark surroundings) that the law holds,
for a certain range of luminance and then only as a rough approximation.
XII-9]
SENSITIVITY TO LUMINANCE DIFFERENCE
257
Curve b shows that under other conditions there is no question of any
constancy in Lf AL.
Weber's law is often referred to as the law of Fe c h n e r-W e be r
but this is not accurate, for the law was formulated by W e b e r in
1834. Fechner (1851) drew from Weber's law the conclusion
that the luminosity is proportional to the logarithm of the luminance L.
but the validity of this has been the subject of much controversy. The
proof is as follows. The minimum perceptible contrast Cmtn is expressed
as Cmtn = ALJL (The fraction iJLfL (the minimum perceptible luminance
difference) is also known as the Fechner fraction).
If AL/L is constant a number of luminance values, which form a geometrical progression, correspond to a number of just distinguishable
levels of luminance. This suggests that the luminosity (h) is proportional
to the number of steps between the level considered and the lowest
perceptible level •(or another standard level) giving
h = C1 log L + C2
which is the mathematical equivalent of Fe c h n e r's law.
There are objections to this equation, however, particularly from
the point of view of practical illuminating engineering. For example,
the mathematical interpretation and treatment of the expression ALJL
is inaccurate as it implies that AL would have this meaning that, if the
luminance of a given part of the field of view be increased from L to
(L + AL) the luminosity with respect to that part would be just perceptibly higher. This would in turn imply that the variation in the
luminosity refers to a certain part of the retina, and this is not the
meaning to be attached to AL in Weber's law. Under this law AL
is the difference in luminance between two adjacent parts of the field
of view that will produce a perceptible variation in the luminosity. AL
thus relates to the sensation arising from two adjacent parts of the retina,
which is very different from the assumption in Fechner's law.
For, if the luminance of a part of the field of view is varied, the adaptation
of the eye varies with it and so also does the luminosity of the whole
field, including that part of which the luminance has not actually been
altered.
Wlien the luminance of part of the field of view is increased, the luminosity resulting from the rest of the field is reduced accordingly.
Cmtn should "therefore properly be expressed as the difference between
luminance levels, e.g. L 1 - L 2 •
Even though Fe c h n e r's law must be discarded in face of indisputable
proof, we must nevertheless concJude from the results of a test as described below that, for one and the same adaptation of the eye, the
luminosity is approximately proportional to the logarithm of the luminance.
An observer is given three contiguous fields, the outer two of which are
of constant but different luminance (L 1 and La). The luminance of the
central field (L 2) is variable. He is then asked to vary L 2 so that the
difference in the luminosity between L 1 and L 2 will be the same as that
between L 2 and La. The adjustment is not made so that L 2 = ! (L 1 +La),
but that, roughly, L 2 = V L 1 X La; in other words, whereas the luminosity induced by L 2 is the arithmetical mean of the sensations in respect
to L 1 and La. luminance L 2 appears to be the geometrical Mean of L 1
and La. From L 2 = VL 1 x-La it follows that
log L 2 = l (log L 1
+ log La).
258
THE PHOTOMETRIC MEASURING-UNITS SYSTEM
[XII
Whereas we measure the luminance on a linear scale, the luminosity
follows a more or less logarithmic scale, and this can be verified roughly
by comparing two rows of grey fields arranged in order of increasing
luminance. The luminance values of the successive gradations in the one
row differ by a certain factor, and those of the other row differ each time
by a certain value of the luminance, so that for the one we write:
r = rLa = .... = constant
r = L2
LI
2
3
4
and for the other:
L 2 - L 1 = L 3 - L 2 = L, - L 3 = . . . . = constant.
The variations in luminance in the first row are therefore logarithmic,
and those in the second arithmetic.
The logarithmic row will give an impression of a uniform variation in
luminance; the other one does not, the sensation being that the steps
are too great at the darker end of the row and too small at the lighter
end to suggest uniform variation.
A more precise idea of visual evaluation of luminance or, more properly
of luminance ratios, is found in the work of M u n s e II in his system
of colour ranging, and also in N e w h ill's investigations.
M u n s e II painted a scale in 10 steps of luminance, inducing equal
hlminosity gradation, from the "blackest black" to the "whitest white"
that he could produce.
Photometry has shown that the relative values of the luminances of
these steps V are as follows:
10
9
8
7
6
5
4
3
2
v= 1
100
76.7
19.3 29.3 42.0 57.6
11.7
6.4
3.05
1.18
1.30
1.52 1.43 1.37 1.33
1.65
1.83
2.1
2.6
Below the relative luminance values are shown the ratios of each step
to the previous one , and from this it will be seen that the ratios become
smaller as the luminance values increase.
If the variations were logarithmic the terms would be:
100
37.2 61.0
5.25 8.45 13.85 22.7
3.2
1.94
1.18
with a constant ratio of 1.64.
Although there is an obvious depatture from a logarithmic evaluation
of the luminance ratios, the gradual decrease in the ratios among the
steps of the M u n s e II chart seems to explain the fact that when a
variable luminance located between two constant luminances is adjusted
for equal luminance differences, the ratios are found to be roughly
logarithmic. The divergence from the logarithmic variation will be
greater according as the relative difference between the two constant
luminances is increased.
The results of M u n s ell's investijations are confirmed by investigations carried out by New hi IJlu),who submitted to a number of
observers a series of surfaces of which the reflection factors were tak;m
at random. The observers were then asked to give the luminance of
each surface a relative value between 0 and 10 in tenths of unity. By
plotting the results in a graph, the reflection factors corresponding
to the steps 0 to 10 could then be determined.
The scale thus obtained was found to agree very closely with the M u ns e II scale.
XII-9J
SENSITIVITY TO LUMINANCE DIFFERENCE
259
REFERENCES
W. S. S t i I e s and B. W. Crawford, Proc. Roy. Soc. B 112, 1933, 428.
"The l~minous Efficiency of Rays entering the Eye Pupil at different Points"
See also: W. D z i o be k, Das Licht 4, 1934, 150-153. "Der Stiles-CrawfordEffekt und seine Bedeutung fiir die Photometrie"
2 ) E.
S c h rod in g e r, Ann. Phys. 63, 1920, 481-520. "Grundlinien einer
Theorie der Farbenmetrik im Tagessehen III"
3 ) H.
Konig: "Der Begriff der Helligkeit". Neuchatel 1947
4) J. S. Preston, "A radiometric basis for the unit of light", Paper P.63.13 for the
C.I.E. meeting in Vienna (1963)
S) H. Moser, l.J. Stille and<..:. Tingwaldt, Optik4, 1948/49,463-471.
"Zweite Strahlungskonstante und Goldpunkt"
W. de Groot, Physica 16, 1950, 419-420. "The Radiation Constants and
the Light Equivalent of Energy"
W. de Groot, Ph. Res. Rep. 8, 1953, 401-410. "Some Remarks about the
so-called Crova Wavelength"
6) P. J. Bouma, Proc. Kon. Akad. Wet. Amsterdam 38, 1935, 35-45, 148-161.
"Grundlinien einer allgemeinen Theorie der Farbenmetrik", I und II
') W. de Groot, Ph. T. R. 15, 1953, 182-187. "Photometry at Low Luminance
Levels"
8) W. S. S t i I e s, Proc. Roy. Soc. London, B 104, 1929, 322-351. "The Effect
of Glare on the Brightness Difference Threshold"
9) R. 0. Schumacher: "Die Unterschiedsempfi.ndlichkeit des helladaptierten menschlichen Auges". Thesis Berlin 1940. See also: Das Licht 11, 1941,
134-135
IO) S.M. New hi 11, Amer. Journ. of Psych. Austin 63, 1950, 221-228. A Method
of Evaluating the Spacing of Visual Scales
1)
PART II
MEASUREMENT OF LIGHT
CHAPTER XIII
GENERAL CONSIDERATIONS
XIII-I. Introduction. Principles of visual and physical photometry
In Part I (Ch. XII) we have already defined photometlry and the principles upon which it is founded. Let us briefly review these once again.
Photometry is the measurement of radiation on the basis of the visztal
sensation induced by it.
This visual sensation is the luminosity or subjective brightness; the
eye receives visual sensations of colours foo, but the evaluation of colours
lies beyond the scope of photometry; it is dealt with in colorimetry.
Qualitatively, we can only judge one luminosity to be equal to another
when the two are observed at the same time. Of the actual intensity
of a luminosity or of the luminance of the object that produces the
sensation, the eye is unable to tell us anything.
By definition, the luminance is proportional to the energy. Comparisons
of different kinds of light, i.e. light of which the spectral compositions
differ, must be carried out with the aid of the international relative
luminous efficiency of radiation V .\·
For measurements of luminous intensity, for example, this is expressed
by equation (XII-7):
K E V.\E.\ = nK E V.\EA,ca cd.
As the value of 60 candelas has been attributed to a certain light source
(1 sq. em of the black body at the freezing point of platinum), it is
not necessary in photometry to know the value of K in the expression
K E V.\E.\,ca·
Photometers must be so designed as to be capable of yielding results
in conformity with equation (XII-7) and with the corresponding
equations for the other quantities used in photometry.
There are two kinds of photometer, based on different principles, viz.
those in which the eye as a light-sensitive element forms an integral
part of the measuring system (visual photometers), and those which
make use of the physical or chemical action of the light, whereby the
eye has no other task than the reading of electrical instruments (physical
and photographic photometers).
264
GENERAL CONSIDERATIONS
[XIII
When dealing with the two main types of photometer, viz. visual
and physical, the question immediately arises as to which of the two
is to be preferred? In general it is the physical photometer, the
advantages of this system as compared with the visual being as
follows.
I. The elimination of the visual element and differences among the
spectral sensitivities of individuals, this being particularly important
in heterochromatic photometry.
2. The possibility of obtaining greater accuracy.
3. The possibility of effecting technical measurements (i.e. those which
need not conform to such lligh standards of accuracy), using only
simple instruments requiring no great skill in operation and maintenance.
For many types of measurement, therefore, visual methods of photometry are no longer employed; apart from in special instances, photometry has now reached the stage where all homochromatic and technical
heterochromatic measurements may be effected with physical photometers. Visual measurement has long been supreme in the field of heterochromatic precision photometry, but it has nonetheless had to give way
here, too, largely to the objective methods. The instruments required
for this are rather expensive, which has delayed their general introduction into photometric laboratories.
In the meantime, since visual photometers are still used, even in homochromatic photometry, we must devote some considerable space to this
subject.
In the meantime, since visual photometers are still so widely used,
even in homochromatic photometry, we must devote some considerable
space to this subject.
We shall not deal with the special branch of photometry known as
spectro-photometry, by which we understand the step-wise comparison
of the spectra of two light sources, over a range of wavelengths, and
the plotting of energy/wavelength curves. This includes, for example,
measurements of the spectral transmission of coloured filters 1).
XIII-2. Standard light sources
By "measurement" is meant comparison with a standard to which a
certain value, expressed in a particular unit, has been attributed. Just
as length is measured against a standard such as a rule, standard lamps
of which the value is known, and expressed in one of the units employed
in photometry, are used in photometry.
Such standards are all derived from a primary standard.
XIII-2]
STANDARD LIGHT SOURCES
265
As standard of measurement of length we have the international metre,
is maintained in Paris in the form of a platinum rod; for the
standard light source we use the black body at the freezing point of
platinum, as described in section 111-2.
This last mentioned standard can be reproduced at any given time or
place and satisfies the requirement that it is always constant, both in
luminous intensity and spectral composition.
At the same time, it is not quite such a simple matter to reproduce the
actual apparatus required; owing to its form and size, and the limited
solidifying time of platinum, this standard is not suitable for use in
routine photometry. Moreover, it is only a luminous intensity standard
(also used as a luminance standard), whereas standards of luminous
flux are also required.
Only a limited number of laboratories possess the primary standard of
luminous intensity. They include:
the National Bureau of Standards (N.B.S.), Washington, U.S.A.;
the National Physical Laboratory (N.P.L.), Teddington, England;
Laboratoire Central d'Electricite, Paris;
Physikalisch-TechnischeBundesanstalt(P.T.B.),Brunswick,Germany,and
the National Research Council (N.R.C.), Ottawa, Canada.
For daily use, technical light sources are employed as standards, although
these do not satisfy the requirement that they shall be constant for an
indefinite period. The luminous intensity and flux of all such standards
depreciate during their life, and also the spectral composition of the
light may change in consequence. For this reason the following system
is applied: some of the lamps are calibrated by comparison with the
primary standard. This is generally done in one of the Standard Laboratories. These . sub-standards (also called secondary standards) are
in tum used for the calibration of working standards; since little use
need be made of the sub-standards, which can therefore be regarded
as constant for a considerable time, they can be kept in service longer
and it is not necessary to invoke the somewhat costly services of the
standard laboratory so often for re-calibration purposes.
The working standards are intended for every-day use, and are regularly
re-calibrated against the sub-standards.
Incandescent and gas-discharge lamps both display a fairly rapid change
in luminous flux during. the initial part of their life (in incandescent
lamps an increase, in discharge lamps a decrease). After this a very
gradual deterioration in the luminous flux sets in, and it is only then
that we can speak of a "constant" luminous flux (and constant electrical
characteristics).
whic~
266
GENERAL CONSIDERATIONS
[XIII
All incandescent and gas discharge lamps on which light measurements
must be taken must therefore be allowed to bum until they became
reasonably constant, this being known as ageing the lamps; filament
lamps with a normal life of 1000 hours are thus aged for about 24 hours
at rated voltage, and gas discharge lamps for about 100 hours.
The construction of standard lamps must be such that during the time
that they are used no variation shall occur in their characteristic photometric values as a result of changes in the geometrical conditions, in
connection with which the following may be noted.
a) Incandescent lamps
For standards of luminous flux, vacuum lamps with straight or coiled
filament may be used, or, in the higher ratings, commercial gas-filled,
single coil lamps. The use of coiled coil lamps as standards is not to be
recommended since the risk that at different times varying numbers
of turns of the coil may be short-circuited, and the luminous flux thus
modified, is greater than with single coils.
Gas-filled lamps must always be burned in the same position. Changes
in the burning position alter the flow of the gas in the lamp and hence
also the cooling conditions of the filament, and this necessarily affects
the luminous flux of the lamp.
The construction of standard lamps for the measurement' of luminous
intensity must conform to higher requirements than that of those intended for luminous flux measurement. In the latter only a constant
total luminous flux is required; the distribution of the flux in space
is less critical.
In the case of luminous intensity standards, however, not only must
the total luminous flux be constant, but so also must be the luminous
intensity within a certain solid angle. This is necessary firstly because
the size of the acceptance surface of the photometer is finite and therefore
subtends a finite solid angle with the lamp, and secondly because - of
even greater importance - any slight deviation from the correct direction
of measurement in the set-up must otherwise be taken into account.
For these reasons the following conditions must be avoided, or steps
taken to eliminate the undesirable effects which they produce:
a. cords or blisters in clear bulbs; these function as small lenses and,
especially when the filament is small, tend to produce irregularities
in the light distribution;
b. images of the filament formed by the rear wall of a (clear) bulb and
coinciding partially or wholly with the filament itself; if small differ-
XIII-2]
STANDARD LIGHT SOURCES
267
ences in the direction of measurement occur, the masking effect of
the filament on the image may vary considerably and, with it, the
luminous intensity;
c. straight coils; the projection of the luminous area of a spiral increases
when the line of observation is moved from the direction perpendicular
to the axis of the spiral in slightly divergent directions in a plane
through the axis;
d. ring-shaped filaments; owing to gradual sagging of the spiral sections
between the "pigtails", the form of the luminous area, and hence
also the light distribution, varies.
It is not always possible to avoid such contingencies in the design of
standard lamps for luminous intensity. For example, in the higher
ratings the use of a coiled filament is unavoidable; the adverse consequences must then be avoided by taking suitable precautionary
measures. As examples of standard lamps for which this has been done
we may mention the following, which have proved reliable in practice.
1. Straight filament vacuum
lamps (for low ratings
up to about 50 cd)
a. With the filament
suspended in the form
of a cylinder (Fig.
131) ;such lamps must
be rotated when in use
06467
to eliminate inconstancy due to differences in the mutual
Fig. 132. Straight filFig. 131. Straight
ament vacuum lamp
various
the
of
masking
filament vacuum
with filament sections
sections of the filalamp. Can be used
mounted in a single plane.
as luminous intenCan be used as luminous
imatheir
and
ment
sity standard when
intensity standard withges. Since there is no
rotated.
out rotation.
gas in the lamp there
can be no eddies to disturb the constancy of the luminous intensity
at different speeds of rotation.
b. With filament suspended in a single plane (Fig. 132).
2. Gas-filled lamps with single-coil filament and opal glass bulb.
Such lamps are operated horizontally; the luminous intensity is measured
in the axial direction of the lamp. The opal glass bulb ensures that the
268
GENERAL CONSIDERATIONS
[XIII
luminous intensity is practically independent of possible minor changes.
in the geometrical conditions of the filament.
In many cases standard lamps for luminance measurements take the
form of tungsten ribbon lamps 9) (see Fig. 133), the light emitting part
consisting of tungsten rolled into a thin strip. This results in a relatively
large area of almost completely uniform luminance.
b) Gas discharge lamps
Although the performance of gas discharge lamps in
general is not as constant as that of tungsten filament
lamps, their use as luminous flux standards sometimes has so many advantages compared with incandescent lamps that the inconstancy is accepted.
It is essential always to use these lamps in the same
burning position and, in the case of sodium lamps,
it is important to leave the lamp in the photometer
until it is quite cold, since displacement of the metal
on the wall of the bulb, which may occur if the lamp
66461
in the molten
Fig. 138. Tungsten is moved whilst the sodium is still
in
state, results in appreciable differences the luminous
ribbon lamp
flux. Even when these precautions are taken it is
advisable to re-calibrate the standards much more frequently than is
done with tungsten lamp standards.
Since the wattage consumed by gas discharge lamps burning on a certain
of the ballast, there is no point in quoting the luminous flux of a gas
discharge lamp for a given supply voltage. Another reason why it is
useless to state the luminous flux for a certain lamp voltage or current
is that the wattage .consumed at such values depends on the form of
the current, which determines the apparent lamp power factor. The
luminous flux of gas discharge lamps can therefore only be given in
respect of the wattage consumed by the lamp; for a given wattage the
luminous flux is independent of the voltage, current and power factor
that determine the actual wattage consumed, within fairly wide limits.
XIII-3. Simultaneous and substitution methods
Comparisons of quantities for measurement purposes can be effected
by the following methods.
a. by direct, i.e. simultaneous comparison with a standard in the photometer (simultaneous method),
XIII-4]
269
SOME PRACTICAL HINTS
b. by comparison in the photometer of the value to be measured with
that of any constant light source (comparison lamp), after which
the light to be measured is replaced by a standard which is in turn
compared with the comparison lamp (substitution method).
The substitution method is always employed when the photometer
used is calibrated.
In photometry preference is usually given to this method, since those
parts of the photometer which correspond to the two comparative
fields need not be identical. With the simultaneous method a second
measurement with the photometer fields reversed is usually necessary
in order to render the final result independent of the optical asymmetry
of the photometer.
XIII-4. Some practical hints
For the purposes of measurements in general - and this of course
includes photometry- it is essential that we do in fact measure only the
quantity that we are interested in, neither more, nor less. This appears
to be stating the obvious, but it is not so obvious that this always does
in fact take place. Unless care is taken, it many be found that inaccuracies result due to more light entering the photometer than should
be there, owing to the
presence of stray light, i.e.
D:s
D2
0,
light reaching the photo-
P
t
-
---+----=.=-.·-1----
--L
meter other than directly ., t-i:::-::.=-:-.=-.=:--.-:--. - - - .::::::~:....~.~;:;;;_;~,..,~~
from the sourc.e being meas,. ~
~
--ured.
~Such stray light may be Fig. 134. Arrangement for screening-off stray
radiated from objects, walls
light
or the ceiling in the laboratory, or direct from other light sources, but may also be caused
by scattering in compound lens systems (objectives). More often than
not such light can be excluded in the manner shown in Fig. 134. A
number of screens with diaphragms (stops) is placed between the light
source and the photometer in such a way that the acceptance surface of
the photometer is completely filled by the light of the source to be
measured and the sides of the stops facing the photometer. In a later
section we shall see how stray light may be avoided in certain special
instances.
Stray light is quite the most prolific source of error in photometry and
1----------r--=.---
270
GENERAL CONSIDERATIONS
[XIII
it would be useful if all photometric laboratories were to display notices
reading:
"LOOK OUT FOR STRAY LfGHT''.
In most cases the presence or absence of stray light can be detected
by screening the light to be.measured at one or two suitable points
(usually close to the light source); any light then measured in the photometer will be stray light. It is even possible in this way to ascertain
roughly the degree of error entailed when stray light is admitted and,
from this, to decide whether further screening against stray light is
necessary, or whether conditions are compatible with the required
degree of accuracy.
It is also possible to obtain less light in the photometer than that corresponding to the quantity to be measured. This may be due to accidental
screening of the light, or it may be thought that an exit pupil is completely flashed with light whereas in fact it is not. This should be
checked.
Mention has already been made of the necessity of ageing standard
light sources, and this applies equally to other light sources which have
to be measured, unless a value relating to the source prior to ageing
is ex-pressly required.
It is generally advisable to effect calibration with at least two standards;
should one of them have undergone a change for some reason, this will
become apparent if a second standard is used. Should the differences
revealed by two standards exceed the anticipated degree of inaccuracy,
the calibration can be checked against a third standard to show which
of the other two is faulty.
In order to ensure the highest attainable accuracy, the value of the
standard should correspond as closely as possible to the value to be
measured. If the values differ by too mnch for comparison on one and
the same scale on the photometer, one of them must be reduced, and
this inevitably introduces an element of inaccuracy in the measurement.
In view of the fact that the luminous flux of almost all electric light
sources is largely dependent on the supply voltage and current, it is
important to make sure when measuring lamps that this is carried
out at the correct values of current and/or voltage. In the first place
this necessitates sufficiently accurate instruments, and in the second
place a mains supply that is sufficiently constant, so that the lamp
can be operated long enough at the specified voltage or current to
permit the necessary readings to be taken. In the third place, it is
XUI-4]
SOME PRACTICAL HINTS
271
essential that the voltage or current as read from the instruments is
actually that at which the lamp is operating, and that no error is introduced by contact resistance. To
ensure a correct measurement of the
voltage at the lamp terminals, a
special lampholder having four contacts is recommended. Such a lampholder is shown diagrammatically in
Fig. 135. Current is supplied by
means of contacts I and 3, this being
measured by an ammeter A; contacts
2 and 4 connect the lamp to a voltmeter V. If contacts I and 3, which
6fJ47o
carry the current, were also connected
Fig. 135. Diagram of lampholder
to the voltmeter, any contact resistance for use in photometry. Two special
between the larrip cap and the contacts contacts (2 and 4) are provided
for the voltmeter leads; contacts
would result in a reading higher than 1 and 3 are for the current supplyc
the actual voltage; the difference
consists of the voltage drop across the contact resistance, which is
proportional to the current flowing through this resistance. If the
voltage is measured between contacts I and 3, the voltage drop is
proportional to the lamp current, but if we measure the voltage
across contacts 2 and 4, it is proportional to the voltmeter current and
is consequently negligibly small, provided that the contacts have been
L
well designed .
..---(::::--..;;..--The use of separate voltmeter contacts is particularly necessary when
the lamps to be measured carry a
heavy current or are for use on low
voltage. It is advisable, when using
standard lamps, to measure the
current after adjustment of the voltage and to compare this with the
calibration current of the lamp, in
order to have a check on the effect
Fig. 136. Circuit for measurement of of any possible contact resistance in
the power consumed by gas discharge
the circuit.
lamps.
It should be remembered that when
the current is being measured, the voltmeter current is also flowing
through the ammeter; to obtain the exact value of the lamp current,
272
GENERAL CONSIDERATIONS
[XIII
therefore, we deduct the voltmeter current from the reading of the
lamp current.
To eliminate errors due to contact resistance, incandescent lamp standards taking large currents andjor for use on low voltage (e.g. tungsten
ribbon lamps) should preferably be calibrated and employed at a certain
current.
The best circuit that can be employed for adjusting a gas discharge
lamp to a required power consumption is shown in Fig. 136. Here, the
voltage coil of a wattmeter W is connected in parallel with the lamp L
and curren.t coil A. The wattmeter thus indicates the wattage taken by
the lamp, plus that absorbed by the current coil of the wattmeter. If
the latter amount of power cannot be disregarded it must be deducted
from the meter reading.
In order to avoid too much distortion of the alternating lamp voltage,
the resistance of the voltage coil should be such that the current flowing
in the coil is not more than 5% of the lamp current.
Details of suitable electrical measuring instruments for use in the photometry of gas discharge lamps may be found in a publication by Mart erst o c k s).
Further details on the measurement of the electrical data on fluorescent
lamps will be found in I.E.C. Publication No 81 4 ). The recommendations
in this Publication should also be observed in the photometry of these
lamps. See also XVII-5.
REFERENCES
See for instance:
P. J. Bouma 1 ) Ch. I, p. 144 et seq.
J. W. T. W a 1 s h: "Photometry", 2nd edition, Chapter XI, London 1952
~) Ph. T. R. 5, 1940, 82-87. "Tungsten Ribbon Lamps for Optical Measurements".
Compiled by J. V o o g d
..
3) J. Marterstock, Lichttechnik 2, 1950, 177-181. "Uber die Eignung
elektrischer Messinstrumente fiir die Photometrie von Gasentladungslampen"
4) "International specification for tubular fluorescent lamps for general lighting
service", I.E.C. Publication 81
1)
CHAPTER XIV
VISUAL PHOTOMETRY AND PHOTOMETERS
XIV- I . Principle of the visual photometer
As we have seen in section XII-2, for the visual measurement of other
quantities than luminance, these must be converted to luminance. The
manner in which this is done, however, will be only an incidental point
in our discussion. In this section we will deal with photometers in the
limited sense of the word, viz. those parts of the system by means of
which the comparison and adjustment of the luminances are effected.
Seeing that in visual photometry two luminances have to be compared
in every case, the visual photometer must in the first place provide
two photometer fields (usually referred to as the photometer field), one
part of which derives its luminance from the quantity to be measured,
the other from a standard or comparison source.
An example of this has already been mentioned in section XII-3, viz.
the wedge depicted in Fig. 125.
Once the photometer has been adjusted for equal luminosities, that
is, in the photopic region, also for equal luminances, we say that photometric balance has been obtained.
In order to ensure optimum accuracy of adjustment the two parts of
the photometric field must be contiguous.
It is generally accepted that the dividing line between the two parts
of the field should be as sharply defined as possible for maximum
accuracy of adjustment, although investigations by K r u it h of 1),
together with those of Middleton have shown that, under certain
circumstances, some lack of distinctness of the dividing line does not
affect sensitivity to luminance differences unfavourably. In the design
of photcmeter fields the makers usually endeavour to make the dividing
line as fine as possible. A small space between the two sections will
easily reduce the sensitivity to luminance difference by a factor
of 5.
In the second place, the photometer must permit of adjustment for
equal luminosity. To this end, at least one of the parts of the photometer field should be variable in luminance to a measurable degree.
In order to give a more general picture of the process and to assist
274
VISUAL PHOTOMETRY AND PHOTOMETERS
[XIV
in the further description of photometers let us first consider a particular
photometric system. This is depicted in Fig. 137, and is of a type that
is suitable for the measuring of luminous intensity.
For measurement of the
X
luminous intensity I., of
a light source X the
a
source is placed at a
distance of a metres
Fig. 137. Equipment for the measurement of from the photometer
luminous intensity. PF Photometer field; X light field PF, which in this
source to be measured; CL comparison lamp;
case is the wedge shown
EP eyepiece; T tube; 0 observer's eye.
in Fig. 125.
The illumination produced on the right hand side of PF results in a
luminance that is proportional to the illumination. The wedge reflects
diffusely and the luminance is accordingly observed by the eye 0.
The left hand side of PF is illuminated by a comparison light source
CL (luminous intensity ICL); the resultant luminance of this side of
PF is also observed by the eye. The distance between CL and PF is
variable and is now adjusted so that the two halves of the field induce
the same luminosity. Let the distance of CL as read from a scale be d.,
metres. The light source being measured is now replaced by a standard
lamp of known luminous intensity I, and the distance between CL
and PF is again adjusted to obtain photometric balance. Let this new
distance be d metres.
From these values we can now compute the luminous intensity of X
in the following manner.
With the initial adjustment the luminances and also the illumination
values of the two halves of the field were equal, so that
_[__~__;lf __:_ ____l __
JJ7
For the second adjustment we have:
I
a2 =
IcL
-;p:·
Dividing the first term of the first equation by that of the second, and
proc ~eding similarly for the second terms we then have:
XIV-1]
PRINCIPLE OF THE VISUAL PHOTOMETER
275
I.,
I
from which it follows that
I
d2
"'
= - X I.
d.,2
Hence the luminous intensity I CL of the comparison lamp and the
distance a are immaterial provided that the luminance of the photometric field remains above the Purkinje region, i.e. above 3 cdfm2.
In the type of photometer just described the method used for obtaining
the variation of luminance is based on the inverse square law; this is
the method generally employed in photometry, but there are others
which will be dealt with later.
·
Between the photometer field and the eye of the observer a tube is
usually provided (Tin Fig. 137) for the purpose of excluding extraneous
light; the end of the tube is covered with a disc (eye-piece, EP) provided
with an 8 mm aperture, the size of the pupil of the dark-adapted eye.
That part of the photometer which contains the photometer field and
the eye-piece is known as the photometer head.
In the arrangement shown in Fig. 137 the light to be measured is collected
by a photometer field which in this case is simultaneously the comparison
surface; in most photometers the photometer field is distinct from the
comparison surface and consists of diaphragms mounted in front of the
comparison surface. In such cases the tube usually includes a lens
which is adjustable with the tube itself so that each and every observer
can adjust the field to the distance at which his eye is best accommodated,
thus avoiding fatigue. In this way, too, the field of view is limited,
the eye is not distracted and measurement is unaffected by the surroundings of the photometer field. The adjustable lens, moreover,
provides a means of varying the angle under which the photometer
field is observed.
In the photometer field in Fig. 137 the distances from the light source
to the various points on the field are not equal; hence, especially at
small distances, the illumination and luminance of the two halves will
not be perfectly uniform. For a precision photometer, however, this
must be so, and the light is then accordingly made to fall perpendicularly
on to the comparison surface. If the comparison surfaces are not
sufficiently diffuse, so that the luminance factor varies with the direction
of observation, this direction must be fixed.
276
VISUAL PHOTOMETRY AND PHOTOMETERS
[XIV
For accuracy in a photometer we can therefore lay down three conditions, viz.
I. The photometer fields must be as close together as possible; the
dividing line must not display more then a limited lack of sharpness.
2. The light to be measured must fall on the comparison surface at
right angles.
3. The comparison surface must be sufficiently diffuse, or the direction
of observation must be fixed.
In the following sections we shall deal with the most common forms
of photometer fields and the most common methods used for obtaining
the variation of luminance, thereby restricting our review to a few
of the many photometer designs that have appeared in the course of
time. Once the form and functions of the more important components
and of a few different kinds of photometer are known, the method of
functioning of other types will present no difficulty. Moreover, many
publications on the subject are available. 2 )
XIV-2. Forms of photometer field
Although the oldest forms of photometer field went out of use long
ago, it may nevertheless be instructive to say something about them,
if only to demonstrate the wrong
L2
X
way of doing things.
A
I
f
The
oldest form of photometer
f
L,
~
field
is depicted in Fig. 138.
i
I
f
I
The
apertures
in the screen S
f
I
I
I
are
covered
with
translucent
I
I
I
I
paper and are illuminated by
the lamp to be measured and
5
by the standard light source
Fig. 138. Bouguer photometer
(L 1 ~nd L 2 respectively). Screen
A serves to ensure that each
aperture receives light only from its own respective source. This arrangement was designed by B o u g u e r (published in 1760). It does
not in any way meet the requirement that the fields should be contiguous,
and the attainable accuracy was accordingly very low.
In this respect the design of Lambert (1760) and Rumford
(1792) as illustrated in Fig. 139 was an improvement. Here the two
light sources L 1 and L 2 illuminate a plane F. To prevent the light from
one source overlapping with that from the other, two screens sl and s2
are arranged so as to be capable of rotation about a vertical axis; they
XIV-2]
FORMS OF PHOTOMETER FIELD
277
are so adjusted that the edges of the shadows which they throw on
the surface F are contiguous. The eye of the observer is at W and accordingly receives reflected light; (B o u g u e r
--....,.....,..........---F
employed transmitted light). An objection
I /I\ \
to this photometer is that the light to be
/~//~,
measured is not incident at right angles
I "'\
~/1
\ \
i
/1
and so does not uniformly illuminate the
\\
:klw
/
\\
comparison surface (in this case simultane- x
\~
I
ously the photometer field). Furthermore, L,
~
I
\
I
the demarcation of the shadows is not
66474 \
i
L2
sharply defined, a fact that has been regarded
as a serious defect in the design. This sup- Fig. 139. The Lambert
f 0 r d photoPosed shortcoming has led to various and R u m
meter
other designs which went a long. way
towards meeting the objections. All these were based on the principle
of Ritchie's wedge (1826) (Fig. 140). The sides of the wedge consist
of mirrors M1 and M2 which receive light from the two sources under
comparison respectively, and reflect it on to transparent paper P. At
a later stage the mirrors were replaced by matt white paper, the
luminance of which was observed directly. This gave the photometer
fields depicted in Figs. 125 and 137.
A subsequent improvement was introduced by Buns en (1843) in the
form of his greasespot photometer; the
field
photometer
opaque
of
was
p --.1.---...J-Fig. 14l. Photowhite paper, a small meter field in the
u n s e n greaseFig. 140. Ritchie wedge
spot on which was Bspot
photometer
rendered translucent by impregnation with oil or wax (Fig. 141).
The Bunsen photometer head can be employed in several different
ways, viz.
1. The illumination on one side of the field can be kept constant by a
comparison lamp, whilst the other side is illuminated successively
by the lamp on test and the standard source (i.e. by the substitution
method). In both cases the lamp distances are so adjusted that the
luminance of the translucent spot is seen to be the same as that of
278
VISUAL PHOTOMETRY AND PHOTOMETERS
[XIV
its surroundings. The simple proof that the luminous intensities
are then in inverse proportion to the square of the distance to the
translucent spot need not be given here.
2. The light sources to be compared can be placed on both sides of
the screen in tum and the distances determined at which the spot
disappears in both instances. This gives four positions (two for each
source), from which the required luminous intensity can be computed.
3. The lamps can be placed as before on both sides of the screen and
so positioned that the contrast between the grease-spot and its surroundings is the same as seen from both sides, the arrangement
then being as shown in Fig. 142. Mirrors M1 and M2 enable the
screen to be observed from both sides simultaneously. Here again
the luminous intensities are inversely proportional to the square
of the distances.
It should be noted that it is not correct to effect the adjustment for
disappearance of contrast when only
one side of the field is observed.
In the case of the third method
it must be noted that the adjustment
is not carried out for equal luminance,
but for equal contrast. It has been
found that under favourable conditions the eye is capable of registering
equality of contrast more accurately
than equality of luminance. This
Fig. 142. Photometer head in which
the translucent spot can be observed principle is encountered again in one
from both sides simultaneously. of the most accurate types of modem
photometer head.
A disadvantag.e of the grease-spot photometer is that, since the spot
receives light from both sides, sensitivity of adjustment is lower than
with photometer h~ads in which each half of the field receives light
from only one of the sources.
This is to say that a displacement of, for example, 1 em of one of the
light sources involves a smaller variation in the ratios of the luminances
of the comparison surfaces with the Bunsen photometer than with
the photometer heads where each half of the field is illuminated by
one of the light sources only.
We come now to the types of photometer field which go the farthest
towards satisfying the requirements formulated above, viz. the prisms
of Lummer and Brodhun (1889).
XlV-2]
279
FORMS OF PHOTOMETER FIELD
One construction is depicted in Fig. 143; this comprises two 90-degree
glass prisms. The plane hypotenuse AC of one of them (ABC) makes
angles of 45° with the sides of the
right angle, and light perpendicular
to one of the sides of the right
angle is totally reflected by the PS2
I
hypotenuse and emerges at right
I
I
angles from the other side.
I
I
The hypotenuse of the other prism
--r--DEF is convex, with a part of the
IH
centre ground flat; this part includes angles of 45° with the sides
of the right angle. The flat part
lies against the hypotenuse of prism
ABC in such a way that it makes
p
optical contact with it, i.e. there
is no intervening air space, and
light passes through the pair as
6647(}
though they were a single piece
Fig. 143. L u m m e r B r o d h u n
of glass. Hence light falling on GH
equality-of-luminance prism
is not reflected by AC, but passes
straight through it.
If a comparison surface PSI be placed behind DE, light passing from it
to GH goes straight through. Light from a comparison surface PS2
placed behind AB, falling on parts AG and HC of the
hypotenuse AC is totally reflected and emerges from BC;
light falling on GH. passes through towards the right.
When the prism is viewed from the direction P a figure
is seen as shown in Fig. 144. The luminance of the figure
in the centre is derived from PSI and that of the outer
ring from PS2 • If the prism is carefully prepared, the
line of division between the two figures is so fine and
66479
Fig. 144.
sharp that it can be made to disappear entirely when
Image of the
photometer the luminances are balanced.
field in a
An example of a complete photometer head with LumLummer
mer-Brodhun
equality-of-luminance prism is depicted
Brodhun
prism
in Fig. 145. In this, PS is a matt white screen (consisting
of,. say, a Plaster of Paris plate a few millimetres in
thickness), of which the two sides serve as comparison surfaces. MI
and M2 are mirrors to reflect the light from both sides of PS into the
f
280
[XIV
VISUAL PHOTOMETRY AND PHOTOMETERS
Lummer-Brodhun prism P. The tube T contains a lens as described
in the previous section. EP is the eye-piece.
The photometer head is usually capable of rotation
through 180° about an axis
I
I
XY, so that the comparison
surfaces can be reversed with
respect to the light sources
under comparison. In this
way any differences in the
reflecting properties of the
two sides of the Plaster of
"' T
Paris plate can be checked
''"''"
corrected.
and
:-..'
,~,
The other kind of Lum~
EP'
mer-Brodhun prism is ilFig. 145. Photometer head with L u m merB r o d h u n prism
lustrated in Fig. 146; this
admits of adjustment for
equal contrast.
In this case the hypotenuses of both of the prisms ABC and BCD are
plane, these being placed together along BC. In the hypotenuse of
BCD a figure is etched or ground so that its face is not in optical contact
with the hypotenuse of ABC. The
form of this figure is shown by the
hatched portion of Fig. 147.
When the line of vision is m the
PS2
direction of P, light from the comparison surface PS 1 (Fig. 146) is
only transmitted by those parts of
BC which are in optical contact,
whilst only that light from PS 2 is
seen which is reflected in front of the
c
recessed figure.
I
I
The figure as observed is depicted in
I
I
Fig. 14 7; the luminance of the hatched
I
portion is derived from PS 2 and that
I
!PI I I
of the non-hatched part from PS 1 •
66481
As so far described, the prism is still
an equality-of-luminance prism, but Fig. 146. L u m me r-B rod hun
prism for measurement by equality
it may easily be converted to a contrast
of luminance or contrast
--------~~+~
----------
p~~,7~
'"
t
XIV-2]
FORMS OF PHOTOMETER FIELD
281
prism by placing two plane pieces of glass
at E and F. These lie in the paths of the
rays transmitted and reflected, respectively,
by the two trapezoids in Fig. 147, and the
luminance of these trapezoids is then 8%
lower than that of the surrounding area.
Photometric balance is established when
the contrast of the left-hand trapezoid with
respect to the left-hand semi-circle is equal
66482
to that of the right-hand trapezoid and Fig. 147. Image of the
photometer field in a L u rocorresponding semi-circle.
me r-B rod hun contrast
In some photometer heads the glass plates
prism
are hinged so that the prism can be employed as an equality-of-luminance or an equality-of-contrast prism.
It has already been mentioned in our description of the Bunsen greasespot photometer that balancing of contrast has the advantage over
balancing of luminance in that the eye can better appraise the former
than the latter.
PSt
With contrast photometers the adjustment for balanced contrast sometimes presents difficulties, but these can often ~e
avoided if the photometrist is made aware of the fact that 1t
is easy to induce the impression that the trapezoids lie above
the semi-circles when they are darker, and below them when
brighter than the semi-circles themselves. The contrast on
the two sides is balanced when the two trapezoids give the
impression that they are at equal distance above the semi-circles.
Today, Lummer-Brodhun prisms have superseded the
Bunsen grease-spot photometer for laboratory work.
A~+-+---+-+-'78 Of the remaining types of photometer prism we shall
mention one other, as this will be encountered in one
of the types of photometer to be discussed, viz. the
I
Fresnel bi-prism, depicted in Fig. 14.8. This is
1\
I \
an isosceles prism ABC. Two comparison surfaces PS 1
I
\
and PS 2 are placed in front of it and, of the light from
I I \
I
+ \
PS 1 only that part is refracted in the direction of the
observer
which passes through face BC; of the light
\
66483
from PS 2 only that part is seen which passes through
Fig. 148.
F res n e 1 bi- AC. The two fields AC and BC are divided by a sharp
prism
line of demarcation at C.
282
[XIV
VISUAL PHOTOMETRY AND PHOTOMETERS
XIV-3. Methods used in photometers for obtaining the variation of
luminance
In order to enable the luminance of photometer fielgs to be balanced,
the luminance of one of them is usually made variable. This generally
applies to the light from the comparison lamp.
A distinction is made between step-wise variation and continuous systems.
T~e first kind is used for the approximate levelling out of pronounced
differences in the luminance of the fields, the second serving for the
more accurate adjustment.
It is almost always essential that the. method of variation, in its operation,
shall not modify the spectral composition of the light. Let us now
discuss some of the methods commonly employed.
1. Step-wise variation of luminance
Rotating shutter. This is a rotating disc from which a sector has been
cut (Fig. 149). It is placed in the path of
the rays illuminating the comparison surface or between the comparison surface and
the photometer field.
If the disc is rotated slowly, a flickering of
the field is observed, but if the speed is
sufficiently high the flicker disappears and
the luminance appears constant.
The central angle of the sector being denoted
by ex, the observed luminance according to
Fig. 149. Rotating sector
disc
T a 1 bot's law (1834)
ex
is 360 times that
without the shutter. The attenuation factor
ex
is thus 360.
The shutter may have either a fixed or a variable attenuation factor,
and some shutters are even continuously variable during rotation.
It is better, especially for low attenuation factors (small central angle)
to determine the actual factor by calibration and not by computing
it from the dimension of the angle.
Rotating shutters naturally satisfy the condition that the spectral composition of the light must not be modified.
XIV-3]
283
METHODS FOR OBTAINING THE VARIATION OF LUMINANCE
Absorption filters. It is very difficult to produce a perfectly neutral
filter, for which reason it is not advisable to use neutral filters for attenuation factors of less than 0.1. Filters of greater density usually
cause an alteration in the colour which is in most cases not acceptable.
Good neutral filters are supplied by various manufacturers. An exposed
photographic plate will in some cases prove useful, but care must be
taken that it is uniformly exposed, and allowance should be made for
the Callier effect (See Section X-12) that such plates exhibit.
A variable attenuation factor can be obtained by means of a neutral wedge, which
is a filter of graded density, usually obtained
by tapering the material (Fig. 150). In order
to compensate the refraction, this wedge
66485
is combined with a clear wedge having as
Fig.
150.
Neutral
wedge
nearly as possible the same refractive index, with transparent wedge to
or with a second neutral wedge. Neutral compensate the refraction
of the neutral one
filters of which the density varies in stages
are also on sale.
Diffusing glasses. Sometimes an opal glass is inserted in the path of
the rays to serve as secondary light source for illuminating the comparison surface of the photometer. In such cases care should be taken
that the glass is sufficiently non-selective, since many kinds of opal
glass transmit appreciably more in the green than in other parts of
the spectrum; others, again, transmit more red.
Reflecting glasses. Attenuation can also be obtained through reflection
from an optically fiat glass plate. With perpendicular incidence and
e = 8% the attenuation factor is then 0.125. High requirements are
put on the flatness of the glass surface.
When polarised light is attenuated, and when variable adjusting systems
employing polarised light are adopted (see 2 below) it should be remembered that light falling on the glass obliquely will be polarised.
Diaphragms. When lenses are included in the path of the rays from
the light source to be measured, diaphragms can also be incorporated.
They can also be employed in conjunction with diffuse glass for controlling the luminous intensity of the glass as secondary light source.
l
!
~I
2. Continuously variable systems for luminance variation
Based on the inverse square law. An example of this kind of system has
already been given in section XIV-I (Fig. 137). The principle of this
and the actual design are so simple that little need be said about them;
284
VISUAL PHOTOMETRY AND PHOTOMETERS
[XIV
it is only necessary to note that in some photometers the comparison
lamp is not movable as in Fig. 137, but :fixed, the comparison
surface instead being adjusted during
measurement.
Based on the use o.f polarised light. When
a ray of natural light (I in Fig. 151)
falls on a prism cut in a certain manner
from calcite or quartz it is split by the
prism into two rays 2 and 3, the polarisation planes of which (see Section
I -3) are perpendicular to each other; in
other words the two rays of light are
polarised at right angles to each other.
As the polarised rays follow different
directions in the prism, such prisms are
Fig. 151. Double-refracting
known as double-refracting prisms. In
.Prism
these, then, we have the means of splitting a beam of natural light into two rays which are polarised in planes
perpendicular to each other. Prisms of the kind shown in Fig. 151 are
not so suitable for this purpose, in that the emergent rays are parallel.
A prism that is quite suitable in this
respect is the Wollaston prism (Fig.
152), which consists of a double-refracting
or a glass prism I and a .double-refracting
prism II, so proportioned and combined
that the emergent rays, which are polarised
I
perpendicular to each other, are symmetrical.
For photometric purposes the most widely
][
used prism is the Nicol, as depicted in
Fig. 153. It is so constructed that one of
the polarised beams is transmitted whereas
the other is disposed of laterally, to be
absorbed by the black housing of the
prism.
'2
The Nicol comprises two half-prisms separated by a layer of cement. An incident
ray I is split into two rays 2 and 3 which Fig. 152. W o 11 aston
prism
are polarised at right angles to each other.
At the interface between the two half-prisms ray 3 is totally reflected,
passes laterally through the prism and is absorbed in the manner
XIV-3 J
METHODS FOR OBTAINING THE VARIATION OF LUMINANCE
285
described. Ray 2 passes unchanged through the interface and emerges
from the underside, parallel to the direction of
incidence.
To serve as a means for luminance variation in a
photometer two identical Nicols can be placed one
behind the other (Fig. 154). Nicol I alters the ray
of natural light A into polarised light. Now, if Nicol II
is positioned exactly as Nicol I, the ray B will pass
through it (parallel Nicols); in Nicol II there is no
longer a ray which, as ray 3 in Fig. 153, is absorbed.
But, if the second Nicol is rotated goo on its axis,
no light at all passes it (crossed Nicols); all the light
in ray B is reflected and absorbed in the manner
of 3 in Fig. 153.
If the second Nicol be rotated to an intermediate
position, the rays are split as in the first Nicol, so
that a part of the light emerges from II, the rest
being suppressed. When the second Nicol is slowly
rotated on its axis from the parallel to the crossed
position, the amount of transmitted light is gradually Fig. 153. Nicol
prism
reduced from maximum to zero. Between the luminance of the incident light (L;) and that of the transmitted light (L,.)
the following relationship exists:
(XIV-I)
where oc. is the angle between the polarisation planes of the two Nicols.
When high attenuation is employed (oc. ~goo) great accuracy is
needed in the calibration and reading of the angle oc., for which reason
it is in such cases often preferable to employ three Nicols, one behind
the other. Initial attenuation is obtained by rotating the first of these
to a giYen position, for which the attenuation factor can be determined.
Fig. 154. Two Nicols in combination
The exact adjustment is then effected by turning the third continuously
rotatable Nicol as required.
In some photometers use is made of polarised light with Nicols in the
286
VISUAL PHOTOMETRY AND PHOTOMETERS
[XIV
following manner. The kinds of light to be compared are split in a doublerefracting prism, such as a Wollaston prism, into beams polarised
at right angles to each other. Only one beam of each kind of light is
utilised for the measurement, these being beams which are polarised
at right angles to each other. Each fills one half of the photometer
field, which is observed through a Nicol. The zero position of the Nicol
is taken to be that in which one of the fields (e.g. field I) is completely
dark, the luminance of the other being then at a maximum. When
the Nicol is rotated through 90° the situation is reversed. In practice
the Nicol is actually adjusted for equal luminance of the two sections
of the field. If the luminance of fields I and II be denoted by L 1 and L 2 ,
and if with balanced luminance the Nicol has been rotated through
an angle oc from the zero position, the following relation ship applies:
(XIV-2)
Variable diaphragms. Continuously variable diaphragms can also
be employed for variation of luminance, these being usually operated
by means of a screw movement. An example is given in Fig. 155.
664{)()
Fig. 155. Diaphragm with screw adjustment
Other systems for luminance variation than those described above
are also in use, some of which will be mentioned in our discussion of
photometers.
XIV-4. Photometer bench. Some examples of visual photometers
Photometer bench
One of the most useful and important pieces of equipment in a photometric laboratory is the photometer bench. It is particularly useful because
XIV-4]
PHOTO)'dETER BENCH
287
it may be used for so many different kinds of measurement and is thus
a universal instrument. At the same time photometer benches are
restricted in their use in that they are not transportable-.
A photometer bench (Fig. 156) consists of supports with rails A (usually
12 to 20 feet in length) on which the various accessories required for
the measurement, such as the photometer head B, lamps C, and
screens D, can be moved backward and forward. Often these are mounted
on carriages, if necessary with pointers, and graduations are provided
along the length of the bench so that the relative positions of the lamps
Fig. 156. Photometer bench
and photometer head can be noted. Different photometer heads can
be used on the bench. Together with the head and the screens, the
bench then comprises a complete photometer.
To overcome the drawback of non-transportability of the photometer
bench various transportable photometers have been designed, the
system for luminance variation and the comparison lamp being contained
within the unit.
The Weber photometer
Among transportable photometers the W e b e r photometer is one of
the most convenient and widely used. It consists of two tubes (A and B,
Fig. 157) mounted at right angles to each other. P is a LummerBrodhun (equality or contrast) prism whose fields are observed
through a lens L and eye-piece EP. The end of the tube B carries a
detachable lamphouse with incandescent comparison lamp CL. The
comparison lamp illuminates a movable opal glass PS 1 inside the tube B,
288
VISUAL PHOTOMETRY AND PHOTOJ',fETERS
[XIV
this being the comparison surface for the light from the comparison
lamp. The luminance of the opal glass is inversely proportional to the
distance d from the lamp, this distance being measured off on a scale.
The scale is frequently graduated with the inverse squares of the
distances, thus facilitating the computation involved in the measurement.
The comparison surface for the light to be measured may be an opal glass
PS 2 placed in the square holder C, or a Plaster of Paris plate mounted
at an angle of 45° in front of the tube A. Filters can also be included in
the holder C.
c
A
1.:.
Fig. 157. Weber photometer
The Weber photometer is used on a stand or, if mounted on a carriage,
can be included on a photometer bench.
Martens Polarisation photometer
In this photometer the system for luminance variation makes use of
polarised light. The principle is illustrated in Fig. i58. S is the comparison surface (e.g.a Plaster of Paris plate) for the light to be measured.
By way of the totally reflecting prisms P 1 and P 2 an image of this surface is produced at A by lens L, and the luminance at A is thus proportional to that of the surface S.
The comparison lamp CL is mounted in a fixed position and illuminates
an opal glass M (luminance LB). Behind M there is a diaphragm B.
The comparison is made between A and B, the light from which passes.
through a lens with a Wollaston prism W which splits each beam into
XIV-4]
PHOTOMETER BENCH
289
two, polarised at right angles to each other. A Fresnel hi-prism is mounted
against the Wollaston prism, and
this again divides each of the four
beams into two others. In the plane
i
of the eye-piece EP, therefore, 4
images each of A and B occur. The
optical system is so designed that
of the eight images 6 are screened
off by the eye-piece and only two
reach the eye, one each from A and B,
their polarisation planes being at
I .
right angles to each other. Between
the hi-prism and the eye-piece there
is a rotatable Nicol N; if that position
1/
,f
of the Nicol be taken as zero whereby the luminance of B is at a maximum whilst A is totally dark, the
luminance LA of A when the Nicol
EP
is rotated through an angle oc to
Fig. 158.. Martens polarisation
establish photometric balance, in
photometer
accordance with (XIV- 2), is
I
I
I
I
tt
,;
The Be c h stein Flicker photometer
In Chapter XII (section XII-3) we have already seen that direct
comparison of kinds of light of widely differing colours cannot be carried
out with any great degree of accuracy, but that accurate results can
be obtained by illuminating the fields in rapid alternation with the
two kinds of light.
This is the principle on which the various flicker photometers are based.
Of the many types available we shall deal only with the instrument
devised by Be c h stein, details of which are depicted in Fig. 159.
A and B are the comparison surfaces of the photometer. The light
passes through prisms P 1 and P 2 and lenses L 1 and L 2 , and falls on
prism P. This prism, which . forms the photometer field, rotates on its
axis XX'. It is in two parts, the outer, annular section C (see Fig. 160)
and central section D being tapered in the manner shown. The photometer is also equipped with the usual tube fitted with lens and
eye-piece.
290
VISUAL PHOTOMETRY J\ND PHOTOMETERS
[XIV
When prism P assumes the position shown in Fig.
160a, the light from the comparison surface A,
shown here as a Ritchie wedge, is refracted by the
annular section of the prism C so that it travels
to the eye parallel to the axis. The light from B
reaches the eye after refraction in the central part
of the prism D. The field of view may be represented
by the figure E in Fig. 160a, in which the horizontally
hatched part denotes the light from A and the
part with the vertical hatching the light from B.
When the prism is rotated through 180° it will be
in the position shown ih Fig. 160b; refraction by
the central part D now directs the light from A
towards the eye, and the ring C the light from B;
the field of view then assumes the form shown by
6M04
diagram F. A comparison of the diagrams E and F
Fig. 159. Be c h- shows that the position of the two light sorts have
s t e i n flicker
now been reversed.
photometer
Rapid rotation of prism P produces flickering, which
is at a minimum when the luminances of the two kinds of light are
equal, hence the instrument is adjusted for m .nimum flicker. At high
speeds of the prism the flicker disappears
entirely and the photometer field exhibits a
mixed colour; if the speed be increased still
I II
II I
further the flicker disappears even when the
I If
IfI II
luminances are not the same. To ensure the
highest possible accuracy, therefore, it is necessary to adjust the flicker frequency so that a
I II
II I
certain amount of flicker remains, or so that it
just disappears, but reappears with the slightest
10 1 I
IDI I
I I 1
I I I
alteration in either of the luminances to be.
I I I
I I I
compared.
The optimum flicker frequency is dependent on
the luminance of the photometer field, of the
colours of the two kinds of light being compared, E• •F
and on the observer himself. It has to be found
b
each time afresh, for which reason flicker photo66495
meters are equipped with means of adjusting
the frequency. If a D.C. mptor is used for the Fig. 160. The prism
in the Bechstein
drive, the speed can be controlled with a rheostat.
flicker photometer
M~
\~
), I
c~·c
XJV-5]
HOMOCHROMATIC AND HETEROCHROMATIC PHOTOMETRY
291
XIV-5. Homochromatic and heterochromatic photometry
All the photometers described in the foregoing can of course be employed for homochromatic and quasi-homochromatic photometry. In
homochromatic photometry a photometer field of 6° to 7° is recommended
for the most accurate results.
Flicker photometers were designed especially for heterochromatic
photometry; other systems for the latter have already been mentioned
in Chapter XII (section XII-3), viz. the direct comparison and stepby-step methods.
The direct comparison method is not employed because of its inaccuracy,
except possibly for fundamental research into the peculiarities inherent
in the method of comparison itself.
The step-by-step method is reasonably accurate, but, as it is timeconsuming, it is not used for routine photometry.
As fourth method we must now mention the filter method, which is
indeed used for routine work. By this method the colour difference
between the two kinds of light is reduced by means of filters to such
an extent that the measurement becomes homochromatic, or at least
quasi-homochromatic, and can accordingly be effected by the use of
normal photometers as for homochromatic comparison.
If the light from a coloured light source such as a sodium or mercury
vapour lamp is to be measured, the comparison lamp used is an incandescent lamp burning at a certain temperature. The spectral distribution of the light of the latter can be measured or computed; the
photometer is calibrated with a standard incandescent lamp.
To measure the coloured light a filter is placed in the path of the rays
from the comparison lamp so as to match the light with that of the
lamp to be measured. The transmission factor can be determined for
each wavelength by means of a spectrophotometer, and these data,
together with the spectral distribution of the light from the comparison
lamp, are used to compute the total transmission factor of the filter
for the light from the comparison lamp. This transmission factor can
also be ascertained by measurement in a flicker photometer.
The filter method can also be employed in flicker photometry; the
matching of the colours of the light increases the accuracy of the photometric balance obtainable.
To ensure a high degree of accuracy it is essential always to us.! the
filter at the same temperature, since the spectral transmission of the
filter varies with the temperature.
In heterochromatic photometry, selectivity of the comparison surfaces
292
VISUAL PHOTOMETRY AND PHOTOMETERS
[XIV
and photometer fields may prove a source of error; this should be borne
in mind when making a choice of these components.
If all the literature ever written on the question as to which of the
above mentioned four systems of photometry should be employed to
obtain the best results were collected together, it would fill volumes,
and we should find ourselves with a collection of exchanges of opinion
and friendly controversy extending over many decades.
Now that the relative luminous efficiency of radiation has been fixed
in the form of an objective standard, this, in conjunction with the
definitions of photometric units, has made the above question superfluous and we can say that any method is correct which gives results
in conformity with the definitions mentioned, i.e. in accordance with
the V A-curve.
Two conditions may be laid down that must be satisfied by any visual
system of heterochromatic photometry likely to give satisfactory
results, viz.
1. The luminance of the photometer field must be above the zone in
which the Purkinje effect occurs (> 3 cdjm2);
2. The photometer field should be of such a size that only the area of
cone vision in the retina is involved in the observations, i.e. that
the field of vision subtends an angle of not more than 2°. Some
investigators prefer not to exceed 1.5°, but an objection to this is
that the balancing, if made at smaller angles than 2°, is very fatiguing.
For the rest, the luminous efficiency of radiation of the eye of the
observer and consequently the choice of observer is very important.
More will be said about this in the next section.
XIV-6. Choice of observer
If we apply the name standard observer to one whose spectral sensitivity
curve agrees exactly with the International V A-curve, it can be said
at once that no such person exists. This fact constitutes a serious handicap
in photometry, but fortunately we have the means of limiting the
drawbacks of having to work with non-standard observers, and in many
cases to overcome them altogether.
Only perpons with normal trichromatic vision are suitable as observer,
i.e. only those who have a normal colour sense, without any form of
colour-blindness.
Whether or not a person is a normal trichromat can be determined by
means of colour tables, for example that of S t i 11 in g. This consists
of a number of rectangular panels containing rows of coloured spots
XIV-6]
CHOICE OF OBSERVER
293
or circles, some of which together form a letter or figure. This latter
and the surrounding spots are of different colours, a;nd the differences
between the colours of the rectangles and figures ar~ sufficiently
pronounced for persons with normal colour vision to distinguish all
the latter without difficulty. To persons showing some form of colourblindness, some of the cyphers appear to be of the same colour as that
of the surroundings, so that they are unable to read them.
Even though an observer may thus prove to have normal colour vision,
his spectral sensitivity will differ from the internationally accepted
values, in some instances considerably, but to some extent invariably.
In order to be able to ascertain the correct values from results obtained
by non-standard observers I v e s and K i n g s b u r y 3 ) have devised
a simple system, although this is limited to the photometry of light
sources having a continuous spectrum such as incandescent lamps
combined with filters. Buck 1 e y 4 ) has shown that this system is
not suitable for use with light sources having a line spectrum, such
as gas discharge lamps.
I v e s and K i n g s b u r y employ two liquid filters consisting of a
yellow and a blue solution in plane-parallel-sided glass envelopes giving
a liquid thickness of exactly 10 mm. The yellow liquid .is a solution
of 72 g potassium bichromate in 1 litre of distilled water, and the blue
liquid 57 g copper sulphate in 1 litre of water.
These solutions have been selected such that for a standard observer
their transmission factors at a temperature of 20° C are exactly the
same for light from a vacuum incandescent lamp burning at a colour
temperature of 2077°· K. We say then that, to the standard observer,
the yellow-to-blue ratio (YIB) is unity.
An observer whose YIB ratio is > 1 has higher luminous efficiency of
radiation at the long-wave end of the spectrum, and is less sensitive
at the short-wave end than the standard observer. The reverse holds
for an observer when his YIB ratio is < 1.
It has been found empirically with heterochromatic measurements
on light sources, the spectrum of which is continuous, that there is a
linear relationship between the results obtained by different observers,
and their Y j B ratios. It is therefore possible to ascertain the correct
values from only a small number of observers whose YjB ratios are
known.
The spectral sensitivity of the individual is not constant; it changes
slowly in the course of time and it is also subject to periodic variation.
Dr e s 1 e r 5 ) has ascertained that for some observers the spectral
294
VISUAL PHOTOMETRY AND PHOTOMETERS
[XIV
sensitivity varies according to the season, and he suggests as a possible
explanation that this is related to seasonal dietary conditions.
For this reason it is necessary to check the Y IB ratio from time to time.
The selection of observers for measuring light with line spectrum is
rather more of a problem; in this case the Y IB ratio of the observer is
of little assistance. This may be explained by the fact that the YIB ratio
is a measure of the spectral sensitivity over two wide spectral zones
as a whole, whereas sensitivity within smaller zones in the spectrum
tends to vary considerably among persons having the same YIB ratio.
For example, the results obtained by two such observers in measuring
sodium lamps, of which the light consists almost exclusively of radiation
of 589 mp, may be found to differ by an appreciable percentage.
The following system gives satisfactory results in the measurement
of light with a line spectrum. Observers are selected whose YIB ratio
is in the region of 1 (say between 0.98 and 1.02). There will then be
every chance that these observers will correctly evaluate certain kinds
of monochromatic light. Whether this is so, and which of the observers
is the most suitable, is ascertained experimentally by having them
take measurements from standard lamps with line spectra (sodium and
mercury vapour lamps) previously calibrated at one of the standard
laboratories. It is quite likely that one or more of the observers will
give a value that agrees with the calibration. The results obtained by
the others will usually exhibit a definite deviation from the standard
and, once this has been ascertained for light from different kinds of
lamp, their measurements can be corrected accordingly.
REFERENCES
A. M. K r u it h of, Ph. T. R. 11, 1950, 333-339. "Perception of Contrasts
when the Contours of Details are blurred"
2) For the study of the different types of photometer the reader is referred to
the various handbooks in existence, of which we will mention here the books
of W a I s h and S e w i g listed on page 388
3 ) H. E. I v e s and E. F. Kingsbury, Trans. I.E.S. New York 10, 1915,
203-208. "On the Choice of a Group of Observers for Heterochromatic Measurements"
The same authors, Trans. I.E.S. New York 10, 1915, 259-270. "A method of
Correcting Abnormal Color Vision and its Application to the Flicker Photometer"
H. E. I v e s, J. Frankl. Inst. 188, 1915, 217-235. "The Photometric Scale"
See also: K. S. Gibson, J.O.S.A. 9, 1924, 113-121. "Spectal characteristics
of Test Solutions used in Heterochromatic Photometry"
4 ) H.
Buck 1 e y, Ill. Eng. London 27, 1934, 118-122 and 148-157. "Heterochromatic Photometry with Particular Reference to the Photometry of Luminous Discharge Tubes"
..
5 ) A. DressIer, Das Licht 10, 1940, 79-82. "Uber eine jahreszeitliche Schwankung der spektralen Hellempfindlichkeit"
1)
CHAPTER XV
PHYSICAL PHOTOMETERS AND PHOTOMETRY
XV--1. Principles of physical photometers and photometry
In physical photometers the energy of the light rays falling upon the
surface of a light-receptor is transformed into another form of energy.
Since "energy", or better "energy per second" i.e. "power", in terms
of lighting technology refers to "luminous flux", the flux falling on the
light-receptor, i.e. the illumination is measured.
Whereas in visual photometry all the photometric quantities are converted to luminance, in physical photometry they are converted to
illumination.
The principal types of physical photometer are the following:
1. Photo-emissive cells
2. Photo-voltaic cells (or barrier-layer cells)
Both of these convert the radiant energy to electrical energy, which
can be measured with a voltmPter or ammeter.
3. Bolometers
Here a wire is heated by the radiant energy; the variation in the
resistance of the wire is a measure of its temperature and therefore
also of the incident radiation.
4. Thermo-couples
In these the temperature of the junctions of a number of thermocouples is raised by the radiant energy, and the e.m.f. thus generated
is a measure of the incident radiation.
To the above types of photometer, must be added a type of photometer in which use is made of the chemical action of light:
5. Photographic photometers
Here the radiant energy is utilised to expose a photographic plate.
The density of the negative is a criterion of the exposure (illumination
X exposure time).
Of these five types, photo-emissive and photo-voltaic cells are by far
the most important, and the only ones employed in routine photometry.
The other types arc used for special purposes only.
In general, physical photometers are calibrated with a standard light
*) Besides the light and radiation sensitive types of receiver 1 to 5, there are also
nowadays "photoresistors", generally made of cadmium sulphide, the electrical resistance of which drops as the illumination increases. We should also mention photodiodes
and phototransistors, made of semiconductive silicon or germanium. The feature of
these is that, on being. illuminated, they pass, amplify or supply an electric current.
These photoelectric components are at present mainly used in electronic switching and
control, since their stability as receivers in photometry is usually inadequate or can be
achieved only under special conditions.
296
PHYSICAL PHOTOMETERS AND PHOTOMETRY
[XV
source in order to provide the relationship between the reading of an
electrical measuring instrument and the illumination. This calibration
then serves in tum to determine the illumination produced by other
light sources. In some cases a balance method is used, the photometer
then being illuminated alternately by the light source under examination
and a standard or comparison light source. An electrical measuring instrument then indicates when the illumination produced by the two
sources is equal.
XV- 2. Photo-emissive cells
When exposed to light or other electromagnetic radiation, many substances have the property that they then emit electrons. Under certain
circumstances these electrons can be measured as electric current. Such
effects are grouped under the heading of photo-electric phenomena.
When the electrons are emitted from the irradiated surface we speak of external photo-electric
effect; this takes place with many metals. Two
other kinds of photo-electric effect will be introduced in the next section, which deals with
K
photo-voltaic cells.
Use is made of the external photo-electric effect
in photo-emissive cells 1) to measure light. The
cell consists of a bulb with a thin metallic layer
on the inside (the photo-cathode K, Fig. 161).
The bulb may be either exhausted (vacuum
cells), or filled with a rare gas (gas-filled cells). A
part of the bulb is left uncoated so that light can
enter, and when the light falls on the photocathode electrons are emitted in all d~rections
within the bulb. To cause an electric current to
66531
flow it is n~cessary to include a second electrode
(the anode, A, in Fig. 161); the electric circuit
Fig. 161. Photo-emissive cell. A anode,
outside the cell must be closed with a resistance,
K cathode
for example that of a milliammeter. The current
then flowing is very small indeed, however, since only a small part of
the electrons liberated reach the anode, so that, in order to "capture"
more or, if possible, all the free electrons, the anode must be given a
positive potential with respect to the photo-cathode.
A serviceable photo-current is thus produced only when the cell is
incorporated in a circuit such as that shown in Fig. 162. The battery B
XV-2]
PHOTO-EMISSIVE CELLS
297
supplies the positive voltage for the anode; a meter (M) is included
for measuring the photo-current.
If at a given illumination level the anode voltage be raised slowly from
zero, the current is seen to increase. The
higher the voltage the greater the number
K
of emitted electrons drawn towards the
anode; vacuum and gas-filled cells behave
A
quite differently, however, with increasing
anode voltage.
In the case of vacuum cells a condition is
reached at a certain value of the voltage
whereby all the electrons liberated from Fig. 162. Simple circuit for
the measurement of photothe cathode are caught by the anode. A current. A and K are resfurther increase in voltage beyond that pect vely the anode and
cathode of the photo-emispoint is not accompanied by a consequent sive cell. B battery for
rise in the photo-current, the current having anode supply, M meter
reached a saturation point (see curve V
in Fig. 163). When the anode voltage is maintained above the saturation
level the photo-current, for a given illumination, is constant.
In gas-filled cells ionisation occurs when a certain voltage (the ionisation
voltage of the gas) is reached, which means that electrons leaving the
cathode and colliding with the gas molecules, liberate electrons from
the latter in far greater numbers than those of the colliding electrons
themselves. In this way the photo-current is amplified (see curve G,
Fig. 163).
When the voltage on a gas-filled cell is raised still further, increased
ionisation finally leads to breakl
down, that is to say a luminous
gas
discharge occurs, the current
tl---t----------f------1
flowing through the gas being
then independent of the illuv
mination.
There are also other differences
between vacuum and gas-filled
cells. The photo-current in a
XJO
150V
i6531
vacuum cell is strictly proporFig. 163. Photo-current as a function
tional to the luminous flux enof the anode voltage: V in vacuum
cell, G in gas-filled cell. In contrast with tering the cell, whereas with gasthe gas-filled cell no further rise in filled cells the relationship between
photo-current takes place in the vacuum
the two quantities is not linear;
cell beyond a certain voltage
l-
298
PHYSICAL PHOTOMETERS AND PHOTOMETRY
[XV
the photo-current rises more steeply than does the luminous flux.
Again, in vacuum cells the photo-current follows all :fluctuations in
the luminous :flux without any -measurable delay, whereas the current
:flowing in gas-filled cells shows a certain inertia with respect to fluctuations due to the time taken by the ionisation process.
Another important point of difference between the two types is the
stability; the photo-current of vacuum cells is very constant, not only
during the short time in which measurements are carried out, but also
over longer periods. Gas-filled cells are not so stable.
Comparing the peculiarities of the two kinds of photo-emissive cell as
outlined above we see that the vacuum cell is obviously the more suitable
for photometric purposes, and in the following, therefore, we shall
consider this kind of cell exclusively.
The composition of the photo-cathode is important from the point of
view not only of the manufacturer, but also of the user of the cell in
photometers, seeing that upon this factor depends the magnitude of
the photo-current and also the spectral sensitivity.
It is possible by means of quantum theory to compute for photo-cathodes
of pure metals the extreme wavelength at which the cell will be capable
of delivering current. For some metals the upper limit lies in the ultraviolet region, with others, namely the alkali metals (lithium, sodium,
potassium, rubidium and caesium), ~his limit (known as the red limit)
occurs in the visible spectrum, or :tnay even be in the infra-red. It has
been found possible to displace the red limit towards higher wavelengths
by making use of special cathode compositions. Here, the most important
elements are antimony and caesium. Photo-cathodes with different
proportions of these materials give a large number of different spectral
sensitivity curves.
The spectral sensitivity curves of a few photocells are given in Fig. 164.
Besides those for a caesium and a caesium-antimony cell is that for a
potassium type developed as the first photocell thirty years ago and long
used primarily for photometric purposes. For purposes of comparison,
the international spectral sensitivity (curve 4) is also shown in the figure.
The scale has been so chosen that the curves pass through the same point
at 555 nm. Further reference is made to the spectral sensitivity of photoemissive cells in section XV-6.
The construction of photocells has changed in the course of time. The
first potassium and caesium cells looked roughly like the one shown
in Fig. 161. The whole of the inside of the glass bulbs was coated with
a thin layer of metal except for the part through which the light was to
be allowed to enter the cell.
XV-'2]
299
PHOTO-EMISSIVE CELLS
Modern photocells consist of a cylindrical glass bulb in which the photocathode is arranged in one of a variety of ways. It may be located
directly on the cylindrical wall, on a bent metal plate or on the flat top of
the bulb (frontal cathodes).
In the latter case, the photocathode must be semi-transparent. There are
photocells with quartz tops, so as to permit the measurement of ultraviolet radiation.
Before proceeding to a discussion of the methods employed in measuring
the photo-current it is necessary to consider one or two other properties
of photo-emissive cells, which are of interest in photometry.
~~--~--~----~--~--~
i
300
1500
Fig. 164. Spectral sensitivity. 1 potassium vacuum cell, 2 caesium vacuum cell.
3 ceasium-antimony cell. Curve 4 is the V"A curve. The scales are such that the curves
pass through the same point at the wavelength at which visual sensitivity is highest,
(555 mJ.t)
First of all comes a warning. Photo-emissive cells must never be operated
at temperatures exceeding 50° C.
The sensitivity varies a little from one point to another in a photocathode and it is accordingly advisable to carry out measurements in
such a way that the same area of the cathode is utilised each time,
e.g. by placing a diffusing glass in front of the window.
In frontal cathode photocells, of course, a shutter can be placed directly
in front of the cathode to provide constant limitation.
Photo-emissive cells may exhibit a dark current, that is, a current which
can flow even in the absence of light in the cell. This may be due to
el~ctrons leaving the cathode under the influence of temperature, or to
leakage as a result of inadequate insulation between anode and cathode.
Such defects in the insulation may occur either on the inside or the
300
PHYSICAL PHOTOMETERS AND PHOTOMETRY
[XV
outside of the bulb and, in order to avoid them, the space between
the anode and the cathode is usually made as large as possible. For
example, the cathode is taken upwards out of the bulb .and the anode
downwards. External leakage current is prevented by applying a
conductive band or ring round the bulb and connecting this to earth.
If the cathode is also earthed the leakage will then be to earth and thus
does not give rise to errors.
At constant temperatures dark current of thermal origin does not usually
give rise to any difficulties, especially when the current is small compared with the photo-current. If it is of the same order as the photocurrent itself it may present some difficulty. This may be avoided by
reducing the temperature of the cell. Alternatively, with the cell at a
constant temperature, compensation may be provided for the dark
current.
Measurement of the photocurrent
Formerly the photo-voltaic cells were not as stable as they are to-day.
This is why photo-emissive cells were almost exclusively used for all
really precision light measurements, i.e. for measuring both stationary
and rapidly changing light phenomena.
Nowadays, the photo-voltaic cell has largely replaced the photoemissive cell i~ the field of the measurement of stationary light phenomena, that is to say,. those especially that relate to lamps that burn steadily.
Furthermore, photo-voltaic cells have the advantage over the photoemissive cells that it is easier to adapt' them to V.~. with the aid of filters.
The current generated by the photo-voltaic cell cannot, because of its
slow response speed, follow fast changes in illumination, like the lighting
up and extinguishing of ·a "Photoflux" lamp. The field of the photoemissive cell is therefore almost totally restricted to the measurement
of such phenomena. We thus propose here to ignore photo-emissive cell
circuits for the measurement of constant luminous flux. A few circuits for
the measurement of very rapidly developing light phenomena are discussed in connection with the measurement of quantityoflight (XVII-7).
Photocells with secondary electron amplification
Finally, we must deal with another kind of photocell developed over the
last twenty years which, because of its special properties, has its own
place among the above mentioned types. We are referring here to photocells with secondary emission, also known as secondary electron multipliers or
SEM's2). Here, the electrons emitted from the photocathode are attracted
by an auxiliary anode capable of emitting electrons itself under the effect
of the electron bombardment. The auxiliary anode emits five to ten
XV-2]
PHOTO-EMISSIVE CELLS
301
electrons for every incident one, thus amplifying the original photocurrent
five to ten times. If this stream of electrons is allowed to fall on another
auxiliary anode, once again at a higher positive potential, the same cycle
is repeated. If, therefore, several auxiliary anodes are connected or
arranged one behind the other, the photocurrent will be considerably
amplified. These cells, then, permit the measurement directly with
a pointer instrument of photocurrents that would otherwise require an
amplifier
However, the greatest illumination that can be measured with SEM's is
still relatively small. The limit is set by the heaviest current permissible
between the final auxiliary anode and the final stage anode. In its turn,
this current is determined by the greatest loadability permissible in view
of the heat developed between these two anodes. Furthermore, it must
be borne in mind that any emission of heat (dark current) from the
photocathode is also amplified, and finally the sensitivity of this cathode
changes, making the accuracy of measurement no greater than with a
vacuum cell. A better range should therefore be used in measurement.
Because the degree of amplification rises steeply with the voltage, the
latter must be maintained constant within at least 1% and, for greater
precision, to within 1Ofoo.
Initially, the stability of SEM cdls was not very high either during a
series of measurements or over a longer term. It was also governed
by periods of load and rest, and thus where rather scrupulous measurement was required, repeated calibration was necessary. These cells have,
however, been improved to such a great degree now that they are quite
usable for higher-precision measurements, provided that the calibration
of the instruments is regularly checked.
SEM cells are used where ordinary photocells or photo-voltaic cells
supply photocurrents that are too weak, e.g. in spectral photometry, in
astronomy to measure the luminous intensity of stars and, lately, for
measurements dealing with street lighting.
There are no serious problems involved in the measurement of the
photocurrents of SEM cells. If the photocurrent is passed through a large
resistor, the voltage generated across the resistor can be measured
with an amplifier, a sensitive diode voltmeter or an oscillograph. A
sensitive current measuring instrument can also be inserted in the circuit
between the anode and the final· auxiliary anode, where, in this case,
there should also be a protective resistor. The greatest difficulty is the
current supply because, as already stated, high requirements are made of
the stabilisation of the stage voltages.
302
[XV
PHYSICAL PHOTOMETERS AND PHOTOMETRY
XV-3. Photo-voltaic cells
This type of cell is capable of converting incident light into electrical
energy without the aid of the auxiliary voltage required by the photoemissive cell. The property responsible for this is the barrier-layer photoeffect, by which light causes electrons liberated from a semi-conductor to
migrate via a banier layer towards an adjacent metallic layer where
they can return to the semi-conductor by way of an external resistance
e.g. a milliameter *). Without any other apparatus, then, a photocurrent of measurable magnitude can be obtained, a fact that has gone
far towards promoting the use of photo-voltaic cells in photometry.
On account of the presence of the barrier layer, the photo-voltaic cell's
are often called barrier-layer cells.
~"'"""'""""",.,.,.Jl"".,..,l'!l"" .,
1
Fig. 165. Diagram showing construction of a) front-wall cell, b) rear-wall cell.
1 semi-conductor, 2 barrier layer, 3 metal layer, 4 metal layer in contact with
semi-conductor
Photo-voltaic cells 3 ) thus consist in effect of a metallic layer and a
semi-conductor separated from each other by a thin insulating layer,
the barrier layer. The layer of metal or counter-electrode must be so
thin that it will transmit light which can then penetrate into the semiconductor and there liberate electrons. A support (base) of metal, in
contact with the semi-conductor must also be provided.
There are two kinds of barrier-layer cell, viz. those in which the barrier
layer is either in front of, or behind, the semi-conductive layer, the
electrons leaving the latter from that side on which the light enters
(front-wall cells), or from the other side (rear-wall cells). The two systems
are illustrated diagrammatically in Fig. 165. In these diagrams 1 is
*) In these cells another phenomenon occurs besides the barrier layer effect,
i.e. the internal photo-effect, which consists in an increase in the conductivity of
certain semi-conductors (including selenium) due to the liberation of electrons
by the incident light. Use is made of this fact for technical purposes. A selenium
cell consisting in principle of a piece of selenium with two electrodes is placed in
a closed circuit under tension. When light is allowed to fall on the cell the resistance
drops and the current can be used to operate a relay. Photo-electric cells of this
kind are sometimes referred to as photo-conductive cells; due to their inaccuracy
however, they are not suitable for the measurement of light.
XV-3]
PHOTO-VOLTAIC CELLS
303
the semi-conductor and 2 the barrier layer; 3 is the metal electrode
to which the photo-electrons migrate from the semi-conductor, through
the barrier layer. The metallic layer, in contact with the semi-conductor
to provide a return path to the latter for the electrons, is denoted by 4.
Light enters in the direction of the arrows, so that in front-wall cells
the metal counter-electrode 3 and the barrier layer 2 have to be transparent, or in rear-wall cells the metal base 4. As will be seen from Fig.
165 the :flow of electrons in front-wall cells is against the direction of
the incident light, and in rear-wall cells in the same direction as that
of the light.
Two different kinds of barrier-layer cell are made, namely cuprous
oxide, and selenium cells, of which the former are produced as either
front or rear-wall cells, whereas the latter are front-wall cells only.
For photometric purposes the selenium cell is employed almost exclusively, and we shall accordingly concern ourselves with this type
only. In any case, the characteristics of the two kinds of cell are roughly
the same. As mentioned above, the selenium cell is a front-wall cell, and
the order of the layers is as shown in Fig. 165a.
From the point of view of the current and voltage delivered, photovoltaic cells behave very differently from photo-emissive cells. In the
latter, at any rate in vacuum cells, the photo-current is strictly proportional to the incident luminous :flux. In photo-voltaic cells the current
delivered is only in one special case proportional to the incident luminous flux.
Apart from its functioJ? as photo-electric cell the system also works as a
rectifier, which is a good conductor when the metallic layer 3 (Fig. 165)
is negative, and a poor conductor
when the semi-conductor 1 is negative. Illumination renders the metallic layer 3 negative; hence an
electric current will :flow from 3
to 1, i.e. in the opposite direction
to that of the photo-current. The
greater the external resistance the
higher the voltage across the barrier
layer (between 1 and 3), and the
stronger the current opposed to the
Fig. 166. Equivalent circuit of a
barrier-layer cell. The cell can be photo-current.
regarded as consisting of a barrier
layer S, a capacitance C and a The photo-voltaic cell may be reresistance R
presented by the equivalent circuit
304
PHYSICAL PHOTOMETERS AND PHOTOMETRY
depicted in Fig. 166. S is the
barrier layer, functioning as rectifier, R is the resistance of the
semi-conductor in which the current flows in the opposite direction to that in the barrier
layer; R is dependent on the
incident light. Re is the external
resistance, e.g. a milliammeter.
The transparent metal film and
the selenium with metal baseplate,
separated by the thin barrier
layer (th·e resistance of which
is high), together constitute a
capacitor as denoted by C in
Fig. 16B.
Whereas the current generated
by the barrier layer is proportional to the luminous flux (curve
a in Fig. 167), the curve of the
inverse current arising from the
voltage produced by the photocurrent across the external re-
700
600
66539
Fig. f67. Photo-current i as a function
of the illumination E on a photo-voltaic
cell. a) photo-current flowing in the
absence of external resistance; b) counter
current produced by the voltage across
the barrier layer. c is the total current
resulting from a and b
to.n.
100.11.
500
300
[XV
soo.n.
2500Lux
665(0
Fig. 168. Cell-current i as function of the
illumination E for different values of the
e:x;ternal resistance
sistance is more or less as
shown by curve bin Fig. 167.
The total current flowing in
the external circuit is as shown
by curve c.
The photo-current characteristic depends upon the value
of the resistance in the external current circuit in the
manner illustrated in Fig. 168.
With a small resistance the
current is very nearly proportional to the illumination
on the cell; with shorted
resistance it is perfectly linear.
With increasing resistance, the
characteristics become more
XV-3]
PHOTO-VOLTAIC CELLS
305
and more curved at high illumination values. If the ext.ernal resistance
be made sufficiently high the current will show no further rise at a
certain illumination
c
c
owing to the opposing
current. In practice this
means that the meaR
suring range of seleniumcell photometers should
not be increased by connecting a resistor in series
with
the meter, but by
!2
shunting the meter with
Fig. 169. The measuring range of a photovoltaic cell with meter should not be increased
a resistor (Fig. 169).
with resistance in series (a), but with resistance
Fig. 170 shows the voltin parallel (b)
age developed by a given
cell for different values of the external resistance.
The current characteristic is accordingly only linear with low external
resistance and low illumination. As a general rule it may be taken that
there will be sufficient linearity as long as the cell voltage remains below
15 mV. For accurate measurements, however, calibration is essential
in all measuring ranges.
Photo-voltaic cells are not quite so stable as vacuum photo-emissive
cells. They need rather frequent re-calibration, the frequency of this
operation depending on
the desired degree of
V (volt)
accuracy. These cells,
ff000.11.
2000.11.
moreover, are not so
1000.11.
500.11.
constant over short periods as photo-emissive
cells and they are there0,1
fore less suitable for precision measurements. At
the same time, selenium
cells are now being marketed, the properties of
I
~E,
500
1000
1500
2000
2500
3000 Lux
which are so good that
o&S42
these cells can be used Fig. 170. The voltage
V developed across different
for all routine photoresistances as a function of the illumination E
metry.
In a humid atmosphere barrier-layer cells tend to absorb the moisture,
., I
306
PHYSICAL PHOTOMETERS AND PHOTOMETRY
[XV
to the possible destruction of the cell, as manifested by a considerable
deterioration, or even a complete absence of photo;-sensitivity; these
cells should accordingly never be used or stored in too damp an atmosphere. Barrier-layer cells are now available which are completely
sealed in an enclosure consisting partly of glass and partly of metal.
This enclosure is filled with an inert gas. In this manner a perfect protection from moisture is afforded which considerably extends the life of
the cells.
Another peculiarity of photo-voltaic cells is that they may be subject
to fatigue, that is, the current drops when once the illumination is
applied, and takes some minutes to become constant; the drop in value
amounts to some few per cent.
A typical spectral sensitivity curve for a selenium cell is reproduced in
Fig. 171 together with
the VA curve. The various makes of cell differ
in their sensitivities, and
the cells also tend to
75
vary in this respect individually. From Fig. 171
it is seen that the spectral sensitivity of the
selenium cell extends
beyond the visible region
of the spectrum, so that
it is possible by means
of coloured filters to
match the sensitivity Fig. 171. Spectral sensitivity of a selenium cell
(full line) compared with the VA curve (dotted
more or less perfectly
line).
with the VA curve. We
shall have more to say about this in Section XV-6.
The spectral sensitivity of selenium cells also varies somewhat with the
illumination, that is to say the current versus illumination curves differ
slightly according to the wavelength of the light. For technical measurements this is of no great importance, but for precision work allowance
should be made.
The current generated by a selenium cell is dependent on the temperature.
In cases where measurement and calibration are effected at a constant
temperature this has no significance, but, if after prior calibration a
sele• tium cell photometer is to be used at different temperatures, this
XV-3]
307
PHOTO-VOLTAIC CELLS
fact must be taken into account if the degree of accuracy required
justifies it.
This variation in the current may amount to 0.5% per degree centigrade.
It is dependent on the particular kind of cell and the external resistance.
In measuring the extent to which the cell is dependent on temperature it
should not be forgotten that the resistance of the meter has also a
temperature coefficient, which is, however, much lower than that of the
cell itself. For precision measurements the temperature coefficient of the
cell and meter should be determined together.
Very high illumination levels (> 10,000 lux) should be avoided, as they
......_
1, 0
0, 9
8
"~
0 ,8
\ !\.
0 ,7
6
5
,~
0, 6
\
\
o.5
\
0 ,4
0,..J
0, 2~ - ·
0, 1
0
10
~
d'
Fig.
I
4
\
,,. K2J
1\'1'\
I
~1
~3
2
1,5
~
' \900
5o"
66544
172. Angular response of
barrier-layer cells
I
1,25
1,1
0
0
20°
v
40°
/
v
60°
2
.,//
80°
66545
Fig. 173. Curves for correction of
the reading error due to dependence
on angle of incidence of the cell
responses shown in Fig. I 71
tend to raise the temperature of the cell too much and thus reduce its
life. It is better not to exceed 1000 to 2000 lux (100-200 fc).
The sensitivity of the working area of the cell is not usually the same
at all points, and the illumination should therefore be distributed as
evenly as possible. If illun1ination of only a part of the area is unavoidable,
the masked cell should be calibrated for accurate results.
An important feature of barrier-layer cells is the dependence of the
photo-current on the incident angle of the light, in other words: their
angular response. When a cell is illuminated by a light source of constant
luminous intensity and at a constant distance from it, the actual illumination varies with rotation of the cell in accordance with the cosine of
the angle turned through (curve I in Fig. 172). The variation in the meter
reading follows a rather different curve, i.e. curve 2 or 3, Fig. 172.
308
[XV
PHYSICAL PHOTOMETERS AND PHOTOMETRY
The construction of the cell has a great bearing on the shape of the
curve; a cell with a flat rim will give a curve similar to curve 2, but one
with a rather high rim which begins to introduce a masking effect at
relatively small angles of incidence will produce a characteristic like
curve 3.
If a protective glass window or coloured glass filter is fitted in front of
the cell the effect of variations in the reflection factor of the glass will
also have to be taken into account.
Readings in respect of directions of incidence other than normal must
thus be multiplied by a factor which can be derived from Fig. 172, from
which charts of the kind depicted in Fig. I 73 can be plotted. These
charts show more clearly than Fig. 172 the amount of error in the
uncorrected reading.
Without a correction, therefore, the cell is not suitable for measuring
light falling at larger angles of incidence than about 45°. The necessary
correction can be computed from curves such as those shown in Fig.l72,
but this is usually not very convenient, seeing that the angle of incidence
of the light to be measured must then be known. When light enters
from numerous directions at the same time it is quite impossible to
calculate the appropriate correction.
:::~
2
66547
I
Fig. 174. 1 cell, 2 internally blackened
tube, 3 spherical opal glass
Fig. 175. Two 'cells placed one above
the other. With perpendicular incidence of the light the upper cell
completely masks the lower one.
With oblique incidence the lower
cell is also partly or wholly illuminated
and thus makes good the deficiency
in the current supplied by the upper
cell
Needless to say many investigators have sought means of counteracting
this dependence of barrier-layer cells on the direction of incidence; of
the rather numerous suggestions put forward we may mention the
following, which most nearly answer the purpose.
I. The arrangement shown in Fig. 174, comprising a tube 2 mounted on
the cell I and supporting a spherical opal glass 3. The height of the
XV-3]
PHOTO-VOLTAIC CELLS
309
tube and the radius of curvature of the glass depend on the transmission factor and absorption of the opal glass and must be determined empirically*).
2. Two cells, one above the other (Fig. 175). With perpendicularly
incident light the upper cell only is illuminated; the wider the
angle of incidence the more the lower cell takes part in the measurement. Hence the lower cell supplies that part of the current which
the upper cell fails to deliver. The distance between the two cells
can be calculated approximately from the angular response of the
two cells, but should, however, be determined empirically.
3. The device shown in Fig. 176 which is a cover with a central hole,
fitted over the cell. This cover is painted white on the inside. With
oblique angles of incidence, light reflected by the cell is reflected
back, in part, from the inside of the cover. The correction thus
6&549
66548
Fig. 176. Cover with central
hole placed over cell
Fig. 177. Cell with lens
obtained is incomplete and, moreover, results in an appreciable
reduction in the measuring sensitivity (roughly proportional to the
ratio of cell area to size of hole). For results which need not be too
accurate, however, it -is useful and fairly simple to construct.
4. A flat opal glass plate, frosted on the upper face, placed over the
cell. The correction obtained is scarcely complete, but the method
is simple and serviceable where only a rough correction need be made.
5. A thin lens cemented to the cell (Fig. 177), permitting almost complete correction, with negligible reduction in sensitivity. 4 )
6. Another suggestion is to use a sphere with white interior as shown
in Fig. 178a. Since the wall of the sphere will never reflect uniformly
diffusely, the cell receives relatively more light with the direction
of incidence shown in the figure than with other directions, and
over-compensation is thus obtained in this and adjoining directions.
This drawback can be overcome by using the arrangement of Larche
*) This description is taken from an unpublished report of the Illuminating
Engineering Laboratory of the N.V. Kema, Arnhem.
310
PHYSICAL PHOTOMETERS AND PHOTOMETRY
[XV
and S c h u 1z e 5) designed for the measurement of ultra-violet radiation,
but which can be used also for measuring visible radiation. L a r c h e
and S c h u 1 z e provided the sphere of Fig. 178a with a cone-shaped
screen (Fig. 178b).
The diameter of the screen must be somewhat
greater than that of the photo-cell, whereas
the entrance opening must be much greater
than the photo-cell. With this arrangement
a practically perfect correction for the effect
of the angle of incidence can be obtained.
Measurement of the photo-current
The current from a photo-voltaic cell can
usually be measured with a moving-coil in- Fig.· 178a. Sphere painted matt white, as sugstrument. If the current is too small, however, gested
for counteracting
a mirror galvanometer may be used. With the dependence on angle
incidence of photovery small currents (< 1o-s A) a sort of of
electric cells. Owing to
creeping effect occurs, that is, the current inadequate diffuse reflecfrom the white paint
takes some time to assume a constant value, tion
the counteracting effect
owing to the fact that the internal resistance is not likely to be sufficient
of the cell is then very high compared with
the resistance of the galvanometer.
Since it is not possible with photo-voltaic
cells to increase, as with photo-emissive cells,
the voltage obtained across a high resistance,
there is nothing to be gained by using an
amplifier with the former type of cell.
Since the short-circuit current is proportional
to the illumination, a circuit for measuring
this current is important. A suitable circuit,
suggested by S e 1 en y i, is shown in Fig. 179.
The cell current I. is taken through resistors b. Sphere of fig. 178a,
corrected according to
R 1 and R 2• Current (I) from the battery B Larche and Schulze.
S conical screen .
flows through R 2 in the opposite direction
to that of the cell current I.. By means of
a rheostat R3 current I is so adjusted that there is no potential difference
between the points P and Q, and no current then flows through the
galvanometer (centre zero) G. This instrument is connected across the
terminals of the cell and: when no current is passing, the cell voltage
is zero and I • is the short-circuit current of the cell.
XV-3]
PHOTO-VOLTAIC CELLS
311
According to Kirchhoff's law:
from which:
Ic.R 1 = (I-Ic)R 2
R2
I=I--c
R1
R2
+
Hence, if R 1 be made many times larger than R 2 , I will be many times
higher than I c and the current I can thus be made so large that it can
be measured with a simple meter, e.g. a milliammeter.
p
-
Q
Fig. 179. Circuit for measuring the shortcircuit current of photo-voltaic cells (according to S e l e n y i)
Ftg. 180 is a diagram of a perfectly practicable circuit for measuring the
short-circuit current of a photo-voltaic cell. The short-circuit current
flows through resistors R 7, R 8 and R 10 , R 7 + R 10 corresponding to R 1 in
Fig. 179 and Rs to Rz.
R 10 is a decade resistor chain with which the factor by which the shortcircuit current of the cell is amplified can be altered. That is to say that
R 10 can be used to regulate the range of measurement of the arrangement.
Points P and Q correspond to points P and Q in Fig. 179.
The current compensating the short-circuit current in R 8 is faken from
the battery B via the potentiometer Rt and Rz. Rt is for coarse and R2
for fine adjustment. A is the ammeter for measuring the battery current.
The potentiometer R 4 + R 5 can be used to obtain a whole-number ratio
between the indication of A and the value of the calibration lamp to be
measured during calibration.
R 6 serves to compensate the temperature effect on the internal resistance
of ammeter A.
R3 is a limiting resistor to prevent undesirably heavy currents from the
battery.
312
PHYSICAL PHOTOMETERS AND PHOTOMETRY
[XV
+
Fig. 180. Circuit diagram of an amplifier for the measurement of the short-circuit
current of photo-voltaic cells
= milliammeter, class 0.5, 0-l rnA
A
B
= 4.5 V flashlamp battery
G
= 0 - l pA light-beam galvanometer, R = 1000 n
R 1 = 5 k!l, 3 W wire-wound potentiometer
R 2 = 500 n, 3 W wire-wound potentiometer
R3
3.7 kn, l W carbon resistor
l kn, l W wire-wound resistor
R4
R5
= 1 k!l, 3 W wire-wound potentiometer
R6
= l 00!1, l W wire-wound resistor
R 7_ 8 = 1000!1 wire-wound resistors, R1 = 900 nand Rs = 100 n
= 25 kn, 3 W wire-wound potentiometer
R9
R 10 = Decade resistor, divided into thousands and tens of thousands and possibly
also hundreds of ohms
Sk 1 = 6 A switch
Sk2 = telephone switch
The sensitivity of the galvanometer G can be controlled with the potentiometer R9.
Sk 2 is a "telephone key" with which the photo-voltaic cell and the galvanometer G can be short-circuited in the inoperative state.
The compensation of the short-circuit current of the cell can be automized and the current proportional to the current of the cell can be
read-off directly. This automation can be improved so that the measuring
results can be read-off with an digital device, or be recorded on paper
with an electric type-writer, directly in numbers. An example, an automatic photometer for measuring the luminous flux of incandescent lamps,
is mentioned in XVII-5.
XV-4]
BOLOMETERS AND THERMO-COUPLES
313
XV-4. Bolometers and thermo-couples
In bolometers and thermo-couples the surfaces rece1vmg the radiation
are blackened to render them non-selective. For the comparison of
light of different spectral compositions it is accordingly necessary to
match the spectral sensitivity more or less with VA (see Section XV-6).
The essential part of a bolometer is the receptor, which consists of a
metal wire or strip, this being very thin in order to limit its thermal
capacity. This element is heated by absorption of the radiation falling
on it, ana the electrical resistance thereby increases. This variation in
resistance can be measured, and is a direct indication of the amount
of the radiation absorbed. In order to avoid difficulties due to irregular
cooling of the element by air currents, the element is mounted in a
bulb which is preferably exhausted.
To render the measurement independent of variations in the ambient
temperature, bolometers are made which have two identical receptors,
one of which is screened from the incident radiation. Each of the receptors
is connected to one arm of a Wheatstone bridge, and the variation in the
resistance of the irradiated receptor is thus quite simply measured.
With thermo-couples use is made of the property that two different
metals placed in electrical contact with each other develop an e.m.f.
when the temperature at the
junction is higher than that at
the extremities. The junction is
blackened and thereby absorbs
the incident radiation, so raising
the temperature.
In photometry, and measurement
of radiation in general, theM o ll
thermo-pile 6 ) (see Fig. 181.) is the
most widely used. This consists
of a number of thermo-couples
connected in series. The thermocouples themselves are blackened
66552
strips (300 x 6,u) of the alloys
constantan and manganin, joined
Fig. HH. The M o 11 thermo-pile
with silver solder; these strips
are made by soldering together
fairly thick plates of the two metals and then rolling them out to a
thickness of 6 microns. Here, again, the strips are made as thin as possible in order to keep the thermal capacity low. The ends of the elements
314
PHYSICAL PHOTOMETERS AND PHOTOMETRY
[XV
are attached to a copper plate, electrically insulated but making effective
thermal contact; these ends thus have a high thermal capacity and
remain cool during the measurement, whereas the soldered ends which
remain free from the copper plate are heated. The working area is
limited, by means of a plate with a hole in it, to the extent of a circle
6 mm in diameter.
The voltage from the thermo-pile is measured directly by means of a
galvanometer or millivoltmeter.
At low illumination levels the e.m.f. generated is proportional to the
illumination, but at higher levels this no longer holds, because the
blackened strips lose a portion of their absorbed energy by radiation;
this loss is relativelygreaterat higher temperatures, and the rise in temperature is then no longer directly proportional to the energy absorbed.
Sensitive instruments (galvanometers) must be employed with bolometers and thermo-couples, which are accordingly suitable for laboratory
use only. Because of their fairly high sensitivity, bolometers and thermocouples are particularly useful where the radiation intensities are only
low, e.g. in the spectral analysis of light (spectrophotometry).
XV-5. Photographic photometry
The density of an exposed photographic plate or film is a function of
a great variety of factors, such as the illumination, the e:)\posure time,
the method of development, the spectral composition of the light and,
of course, the characteristics of the plate or film; the latter differ widely
between one kind and another, and may even vary- albeit not to a
considerable extent - from batch to batch of a given kind.
In first approximation, the density of most films and plates is a function
of the product of the illumination and the exposure time, that is, the
exposure.
If the material is exposed in a camera with objective, the illumination,
as we have seen in Section XI-8, is proportional to the luminance of
the object photographed. It can accordingly be said that the plate or
film measures luminance.
If it can be made possible during measurement for all the variables
apart from the luminance to be held constant, the density will be a
function only of the luminance, and we thus have a means of measuring
luminance photographically. This can actually be done by making an
exposure, not only of the luminance to be measured, but also of a series
of known luminances of the same spectral composition as that of the
XV-6)
PHOTOMETERS FOR HETEROCHROMATIC PHOTOMETRY
315
first-mentioned luminance, preferably using the same exposure time
and lens stop. A comparison can then be made between the densities
in respect of the light under test and the reference densities, by means
of a densitometer.
In so doing, allowance must be made for the effects of possible vignetting of the objective, of the decrease in illumination towards the
edges of the plate due to reduction in the apparent area of the exit pupil
of the objective, and to the fact that the beams do not strike the plate
at right angles. These effects can be determined for every aperture of
the particular objective by using a photo-emissive cell or photo-voltaic
cell to measure the relative illumination values at various points on
the plate or film with constant luminance of the object.
Allowance must also be made for the characteristics peculiar to photographic material in general, viz. the different "effects" known in photography (e.g. Callier effect (section X-12), Eberhard effect).
Since the density increases with the exposure time (the effect on plate
or film is cumulative), it is possible photographically to measure much
smaller luminances than by physical methods.
The photographic method of photometry is employed for the measurement of the intensity of spectral lines, i.e. in spectrophotometry,
and also of the distribution of luminance on artificially illuminated
roads (see Section XIX-3).
XV-6. Physical photometers for heterochromatic photometry
We have seen from the previous section that none of the physical photometers described possesses an inherent spectral sensitivity corresponding
to VA· For the comparison of kinds of light of different spectral composition the following procedures can be followed:
a) In front of the light receptor one or more filters can be placed whose
spectral transmission is such that the spectral sensitivity of the
combination more or less equals VA (filter method).
b) Correction factors can be ascertained for certain kinds of light.
c) The light under test can be broken down into its spectrum by means
of a monochromator; in the plane of the spectrum a template is
placed which has the form of the V .\ curve (geometrical method).
d) The energy v. wavelength curve (spectral energy distribution) of
the light can be plotted with the aid of a spectrophotometer, the
energy value being then multiplied by the appropriate value of V .\
for each wavelength. In effect, the value of the expression .EV.\E~
is computed (algebraic method).
316
PHYSICAL PHOTOMETERS AND PHOTOMETRY
[XV
This last method is difficult and cumbersome, and is adopted only in
fundamental research; in photometry it is of little importance. For
routine measurements the methods based on the use of filters and
correction factors are both used; for precise results the filter and geometrical methods are mainly employed.
The more the two kinds of light differ in composition, and the higher
the required accuracy of the results, the better must be the agreement
with V "; this determines whether and how many filters should be used.
It should be borne in mind, however, that the correction for measurement of light the spectrum of which is continuous need not be so precise
as for light with line spectrum.
With light of a single wavelength, any difference with respect to V"
at that wavelength is carried through in its entirety into the final
result. Should the divergence, say, at 590 mtl (roughly the wavelength
of the light from a sodium lamp) be for example 10%, the error in the
measurement of sodium lamps will be 10%.
If in a certain region of the spectrum the spectral sensitivity of the
cell diverges from V -'• then in measuring light having a continuous
spectrum, only that part of the light comprising wavelengths occuring
in that region will be wrongly measured. Since the remainder is measured
correctly, the divergence from V" is only partly reflected in the overall
measurement.
Since each and every filter used represents a source of absorption of
light and hence a depreciation in the sensitivity, as few filters must
be used as possible, commensurate with the purpose in hand.
Filters are made of glass or gelatine, or they may be of liquid. As the
spectral transmission of such filters is to some extent dependent on
temperature, it is necessary, for precision measurements, to maintain
a temperature that is constant within narrow limits. Amongst gelatine
filters there are some that gradually decolorize when exposed to
light, a fact that should not be overlooked. Compared with the other
kinds, liquid filters have the advantage that the spectral transmission
can be modified by varying the concentration of the solution. These
filters of course lend themselves only to laboratory work, and not to
transportable photometers.
In using the filter method it should be remembered that the spectral
sensitivities of individual cells vary; when precise results are required,
therefore, the appropriate filter should be found for each cell and, as
the characteristics of both cells and filters are liable to change in time,
the combination should be subjected to a regular check.
X. V-6]
PHOTOMETERS FOR HETEROCHROMATIC PHC'T'OMETRY
317
It is also to be borne in mind when using filters that light p<'.ssing obliquely
through a filter has a longer path to travel than perpendicularly incident
rays and that the absorption thus varies with the direction of incidence.
It is necessary, therefore, in precision measurement, to carry out the
calibration and the measurements with the same direction of incidence.
Let us now consider the adaptation by filters of the different receptors
discussed in the foregoing.
1. Photo-emissive cells
There are special filters on the market for caesium-antimony cells providing good adaptation to VA,. For exact matching to VA, the correct
filter, or rather the correct combination of filters, must be found for
each individual cell.
Caesium cells can be corrected for use with the usual temperature
radiators by means of a blue filter; for more effective correction K 6 n i g 7 )
has prepared a combined filter consisting of eight glasses.
2. Photo-voltaic cells
Cell-filter combinations, the sp~ctral sensitivities of which approximate
to V" are marketed by a number of firms. For technical purposes these
can usually be employed without a correction factor, but for precision
work the necessary factor should be determined for each cell and each
kind of light.
When this is done it should be remembered, however, that the spectral
sensitivity of photo-voltaic cells tends to vary slightly with the illumination. When filter cells are used, the variation due to the effect of
the direction of incidence on the filter has also to be taken into account.
3. Bolometers and thermo-couples
For matching non-selective thermal receptors with V" Gibson,
Tee 1 e and Keegan s) have developed a filter composed of one
glass and one liquid filter, the spectral transmissivity of which is for
all practical purposes equal to V". However, as this combination transmits some infra-red radiation, T e e 1 e 9) used it in conjunction with
a red filter that transmits no visible radiation but which has a high
transmission in the infra-red. T e e 1 e takes two measurements, one
with, and one without this extra filter; the difference between the two
results is then the required value.
318
PHYSICAL PHOTOMrTERS AND PHOTOMETRY
[XV
It is possible to determine and employ correction factors for kinds of
light of known spectral distribution e.g. from sodium lamps, mercury
vapour lamps working at different vapour pressures, neon lamps,
fluorescent lamps etc. If the composition of the light to be measured
is not known, however, use must be made of receptors whose spectral
sensitivity is matched with V ,~..
Wide use is made of correction factors for simple illumination photometers
with barrier-layer cells. The makers of such photometers usually supply
correction factors for the more generally used technical light sources.
Whereas in principle the filter method is an approximation method for
matching receptors with V ,~.,the geometrical method is, in principle, exact.
Since we are dependent on the physical constants (spectral transmission)
of materials, in the manufacture of filters, it is not possible to produce
a filter the transmissivity of which, wavelength for wavelength, will
guarantee a perfect match with V ,~..
At the same time this is not such an obstacle as it would seem, for it
has been found possible by means of combinations of filters to secure
a match with V >.that is sufficiently accurate for precision measurements.
In the geometrical method, agreement with V >. is not d~pendent on
the physical constants of the material employed, but on the degree of
accuracy with which we are able to make a template of a given form,
and on the accuracy with which the spectral sensitivity of the receptor
can be measured.
Fig. 182. Schematic diagram of photometer based on
the geometric principle
We shall now describe the geometriCal method in reference to Fig. 182
which is taken from an article by V o o g d 10), in which be describes
a physical photometer working on the geometrical principle.
The light to be measured, passing through a slit S, is concentrated as
a ,parallel" beam by a lens L 1 and is then analysed into its various
XV-6]
PHOTOMETERS FOR HETEROCHROMATIC PHOTOMETRY
319
wavelengths by a prism Pr. The rays of different wavelength, which
are parallel also after refraction by the prism, are brought together
by another lens L 2 in its focal plane, to form a spectrum. The height
of the spectrum is the same at all points and corresponds to the height
of the slit S. The light from the spectrum is passed through a lens L3
to ·the receptor R. ·
Whereas in the filter method each wavelength is so absorbed by the
filter that the spectral sensitivity of the filter-receptor combination
as nearly as possible equals V .\. this is achieved in the geometrical
method by screening off a part of the radiation of each of the projected
adjacent wavelengths, a template D being used for this purpose, in
the plane of the spectrum.
The shape of the template can be computed from the spectral sensitivity
of the receptor (G.\) and V .\· At every point on the template the height
must be proportional to V .\/G.\.
Nevertheless, the method is not so simple in realisation as would appear
at first sight. In the first place it is not easy to make a template exactly
of the required form. Further, owing to the finite width of the slit, the
wavelength is not reproduced in the spectrum as a line, but as a band,
the width of which equals that of the image of the slit. These bands
therefore overlap, in consequence of which the transmission of a wavelength A. is not determined by the height of the template at the position D, but by the average height of the template corresponding to
the image of the slit.
The smaller the slit the greater the accuracy, but the lower the sensitivity of the photometer. The average transmission for a given wavelength will the more precisely agree with the computed transmission
according as the form of the template more nearly approaches the linear.
Again, the more the spectral sensitivity of the receptor approximates
to V .\. the more linear the template will be; for this reason K 6 n i g 11 )
has suggested that the sensitivity of the cell be roughly matched with
V.\ by using one or more filters (F, Fig. 182).
The first photometer based on the geometrical principle was constructed
in 1915 by I v e s, the receptor being in this case a thermo-couple
(photo-electric cells were then unknown).
320
PHYSICAL PHOTOMETERS AND PHOTOMETRY
[XV
REFERENCES
1} M. C. Teves, Ph. T. R. 2, 1937, 13-17. "The Photo-electric Effect and
its Application in Photo-electric Cells"
A. S o m m e r: "Photoelectric Cells". London 1946
R. Sew i g: "Objektive Photometrie". Berlin 1935
V.·K. Zworikin and E. G. Ramberg "Photoelectricity and its application,"
l~ew York 1949
H. Carter A. M. I.E.E. and M. Donker, "Photo-electric devices in theory and
practice", Philips Technical Library, 1963.
2) M. C. Teves, Ph. T. R. 5, 1940, 253-257. "A Photocell with Amplification
by Means of Secondary Emission"
3) W. C h. van Gee I, Ph. T. R. 8,. 1946, 65-71. "Blocking-Layer Photocells"
4~ G. B. Buck, Ill. Eng. 44, 1949, 293-302. "The Correction of Light-sensitive
Cells for Angle of Incidence and Spectral Quality of Light"
5) K. Larche and R. S c h u I z e, Zts. f. Techn. Phys. 23, 1942, 114-117.
"Uber ein Ultraviolett Messgerat mit Vorsatzkugel fiir Strahlungseinfall
unter grossem Winkel"
6) L. S. 0 r n s t e i n, W. ] . H. M o 11 and H. C. B u r g e r: "Objektive
Spektralphotometrie". P. 4 et seq. Brunswick 1932
7) H. K 6 n i g, Helv. Phys. Acta 16, 1943, 421-422. '"Praz1sions-Photometrie
mit Caesiumzelle und Kombinationsfilter"
S) K. S. Gibson, R. P. Tee 1 e and H. ]. Keegan, J.O.S.A. 29, 1939, 144.
"An Improved Luminosity Filter"
9) R. P. Tee 1 e, J.O.S.A. 31, 1941, 696-704. "A Physical Photometer"
10) ]. V o o g d, Ph. T. R. 4, 1939, 260-266. "Physical Photometry"
11) H. K 6 n i g, Helv. Phys. Acta 7, 1934, 433-453. "Beitrage zum Problem
des Vergleiches verschiedenfarbiger Lichtquellen"
CHAPTER XVI
MEASUREMENT OF LUMINOUS INTENSITY
XVI-I. Principle of luminous intensity measurement
In Part I two definitions of luminons intensity were introduced. According to the first of these, luminous intensity is the quotient of luminous flux divided by the solid angle, with infinitely small solid angle,
that is, the luminous flux per steradian (see Section III-1), and according to the second it is the limit of the product Ed 2 when d --+ oo
(Section IX-1).
The second definition can be employed for all practical forms of light
source, i.e. of which the dimensions are finite. In the case of the ideal
point source the second definition gives way to the first.
It has already been stated in Section IX-1 that luminous intensity is
always measured as a quantity answering to the second definition, i.e.,
the product of the illumination and the square of the distance from the
light source to the photometer*).
Sufficient has already been said about the measuring distance for diffuse
light sources in Section IX-12, and the reader is therefore referred to
that section for details.
In effect, luminous intensity measurements amount to measurement
of illumination. As mentioned in Section XII-2 this is converted to
luminance in visual photometry very simply by using a diffusely
transmitting or reflecting screen as comparison surface in the photometer.
In our description of a luminous intensity measurement in Section
XIV-1 we have already indicated the general method of carrying out
such measurements. Instead of the primitive photometer, however, a
well-constructed instrument will usually be employed, of the kind
described in Section XIV-4.
With physical photometers for the measurement of illumination, luminous intensities are measured in the same way, a light-receptor being
used instead of the comparison surface of the visual photometer.
*) The method of measuring luminous intensity discussed in Section X\'I-4 is
an exception.
322
MEASUREMENT OF LUMINOUS INTENSITY
[XVI
XVI-2. The photometer bench
A description of a photometer bench has already been given in Section
XIV-4, so that here we need only deal with its use for the measurement
of luminous intensities.
Figures 183a, b and c show the
OL
X
different ways in whifh the bench
can be used with visual photog
meters.
In Fig. 183a the positions of the
lamp on test X and comparison CL
X
lamp CL are fixed, the photo~p
meter P being adjusted on the
bench to establish photometric
!2
balance.
Fig. 183b shows the lamp on CL
-x
test X as fixed; the photometer
~p
P and comparison lamp CL are
at a fixed distance from each
other and can be moved together Fig .. 183. Possibilities of the photometer
on the bench for photqmetric bench for the measurement of luminous
intensity
balance.
CL = comparison lamp. P = photometer
The arrangement shown in Fig. head. X = light source to be measured
a) CL and X fixed, P adjustable
183c differs from b in so far b)
X fixed, CL and P adjustable
that P and CL-are fixed and X c) CL and P fixed, X adjustable
is adjustable.
If a physical receptor is to be employed, the substitution method is
usually adopted. It is then possible to dispense with calibration of the
cell and hence to disregard its characteristics (important in the case
of photo-voltaic cells) by so adjusting the distance of the standard
lamp, after measurement of the lamp on test, that the cell delivers the
same current for the same external resistance, i.e. that it receives the
same illumination. The luminous intensities will then be proportional
to the square of the distance from lamps to cell. In this case the cell
functions, as it were, as a null instrument.
Other null methods can also be employed, for example a photo-emissive
cell may be illuminated rapidly in turn by the lamp on test and standard
lamp. The distance from the cell to one of the lamps is adjustable. If
the illumination on the cell differs between one lamp and the other the
cell current varies, and this results in a pulsating current. For equal
1
-~p
--
1
1
1
1
-
l
XVI-2]
THE PHOTOMETER BENCH
323
illumination values and cell currents the total cell current is constant.
If the cell current be measured with the aid of an A.C. amplifier the
66555
=-
Fig. 184. Null method of
luminous intensity measurement with two photovoltaic cells. When the
illumination is the same
on each cell the meter
does not deflect
Fig. 185. Luminous intensity measurement with the aid of two lenses.
Lens L 1 produces an image of the
light source in L 2• The adjustable
diaphragm D enables the image to be
isolated almost completely from its
background, thus effectively screening
off any stray light
output meter will indicate no current when photometric balance is
established.
In another null method 2 barrier-layer cells are used, these being placed
back to back on the photometer bench with a meter in series with
them in the manner shown in Fig. 184. If the sensitivities of the cells
are equal (or matched by placing a filter or stop in front of one of them)
the meter will not deflect when photometric balance is obtained.
Fig. 185 shows an arrangement which offers two advantages, viz. an
increase in sensitivity of the cell employed, and the fact that it is
practically independent of extraneous light.
The light to be measured falls on a lens L 1 which forms an image of
the source in lens L 2 • The focal iengths of L 1 and L 2 are such that L 2
produces an image of L 1 in the celi C. A variable stop placed in front
of L 2 makes it possible to isolate the image of the light source from the
image of its background, so that only light from the source enters the
cell. The light collected by L 1 is distributed evenly over the image of L 1
in the cell, and the same part of the cell accordingly always receives
uniform illumination. If the diameter of L 1 is made considerably larger
than that of the cell, the photometer is rendered more sensitive than when measurements
are carried out with the naked
cell.
D
D
D
66557
D
Fig. 186. Tube K with screens D for ex- Another device for excluding
cluding stray light
stray light, which is, however,
:: }
324
MEASUREMENT OF LUMINOUS INTENSITY
[XVI
a little less effective than the system of lenses, consists of a long tube
placed before the cell (Fig. 186). This tube is painted matt black on the
inside and is provided with diaphragms as photometer screens (D).
XVI-3. Apparatus for measurement of light distribution
The preceding seGtion deals in particular with the measurement of the
luminous intensity of a light source in only one direction. To measure
the light distribution of a source it must be possible to ascertain the
intensity in various directions, and some form of equipment is required
by means of which this can be done quickly and easily.
The simplest method, at any rate one that can be carried out using the
simplest apparatus, consists in measuring the illumination at a number
of points on a plane surface upon which the light from the source is
allowed to fall.
Formula (VI-5) (E =I cos3 r~../h 2 ) can be used to determine the luminous
intensity, in the different direction5, from the illumination. Needless to
say this method is suitable only where the measuring surface need not
be too large, i.e., when the beam is quite narrow or the measuring
distance not too great (e.g. car and cycle lamps).
In all other instances specially designed apparatus is generally used.
The construction of apparatus for measuring light distribution is governed
by the following factors:
a) the minimum distance between photometer and light source, compatible with the desired accuracy;
b) the dimensions of the light source;
c) whether or not the burning positions of the light sources may be varied.
a) In Chapter IX we have seen that the minimum measuring distance,
in order to obtain a reasonable accuracy, for circular diffuse light sources
is five times the diameter of the light source, and that this distance for
linear sources is twice the length of the source. This means that with
conventional fittings the measuring distance will be 4 to 6 metres
(15-20 ft).
As applied to mirrors and lenses it is shown in Section XI-6 that the
point of measurement of the illumination must lie beyond the beam
cross-over point. The distance from the beam cross-over point to the
projector varies.according to the type of optical system and light source
used in it *) and, for large searchlights, may be several miles. The distance
for a motor-car headlamp will be as much as 7 to 10 metres (25-35 ft),
*) Further details in Section XVI-4.
XVI-3]
MEASUREMENT OF LIGHT DISTRIBUTION
325
but two or three metres (7-10 ft) is sufficient for cycle lamps owing to
their small dimensions.
In any determination of the minimum measuring distance for a beam
type source the beam cross-over point is not the only factor to be taken
into account, however; it should be remembered that at points beyond
the beam cross-over point a projector functions as a luminous disc (the
limiting area of a projector is the exit pupil). Hence the same considerations apply to the minimum measuring distance required to yield
a certain degree of accuracy of Ed 2 as for diffusely radiating sources.
In practice the distance from projector to beam cross-over point is
generally much greater than the minimum distance necessary for a
sufficiently close approximation to the limit value of Ed 2 •
R e e b 1) makes a distinction between the optical limit distance and the
photometric limit distance. By the former is understood the minimum
measuring distance as determined by the geometrical characteristics
of the projector, that is, the distance from projector to beam cross-over
point. Reeb's definition of photometric limit distance is the minimum
measuring distance necessary to ensure a sufficiently close approximation
to the limiting value of Ed 2 •
b) This calls for little comment, since it is obvious that large, heavy
fittings, searchlights etc. will necessitate more robust constructions
than small, light, sources.
c) With many light sources, e.g. gas-filled incandescent lamps and
sodium lamps, the luminous flux is to some extent dependent on the
burning position.
The above considerations have led to the design of two types of apparatus for the measurement of light distribution, based on different
principles. Both types are made in a large number of variations.
Stated briefly, the difference in principle amounts to this, that in the
one type the light source is stationary and the photometer is moved,
whereas in the other the reverse is the case.
The first kind is employed for light sources which can be measured
at short range, and in cases where the burning position of the source
may not be changed during measurement. The second is more suitable
for the measurement of sources requiring such a long measuring range
that it is not possible to move the photometer around the source.
In practical forms of the first mentioned type of apparatus the photometer is, nevertheless, usually stationary; a system of mirrors arranged
to rotate in a meridian plane around the source so as to r.eflect the
326
[XVI
MEASUREMENT OF LUMINOUS INTENSITY
light into the photometer produces the same effect as a photometer
moving round the source. For measurements in other meridian planes
the source is rotated on its axis, leaving the burning position as before.
Fig. 187 depicts an arrangement of this kind. 6 )
The mirrors M1 and M2 , which are mounted on an arm, are rotated
together about the axis AB. Light from the source L travels via M1
and M 2 to the photometer P. For measurements
in other meridian planes the source is rotated
about axis CD. Facilities for rotation are an
c
advantage in measuring the average distribution
p
(Section IV-1). The speed of rotation, de.
B
pendent on the asymmetry of the light distri- A
·
'
L
·
\ I? ' .
bution, is from 100 to 200 r.p.m. and, where
. I
large lighting fittings are concerned, this im.
poses high requirements on the robustness of
the apparatus.
The mirrors "fold up" the measuring distance,
and the supporting arm can thus be very
\ .
much shorter than if the photometer itself
were moved.
\·I
Visual photometers must, of course, be kept
stationary, but with cell-photometers the cell
mounted on a bracket, can be moved round the
light source, in which case the mirrors are Fig. 187. Apparatus for
measurement of light
unnecessary. This works quite well with photo- distribution.
The light
voltaic cells, but not so well with photo- from a source L is re,
towards
the
emissive cells, since movement of the leads fleeted
photometer by mirrors
may cause interference with the measurements. M1 and M 2 mounted on
which pivots
Types of equipment used for the measurement a bracket
about AB.
of projectors are depicted in Figs. 188 and 189 2).
Fig. 188 shows a frame capable of rotation about the fixed horizontal
axis AB. The projector is mounted in the frame in such a way as to
rotate about axis YY, which follows the movement about AB.
In the system shown in Fig. 189 the fixed axis CD is vertical and the
axis of rotation XX is in the initial position perpendicular to the direction
of measurement. The motion is applied through reduction gearing, the
position of the projector being read from a graduated arc fitted, if
desired, with a vernier.
A different kind of instrument is depicted in Fig. 1903). This consists
of two gimballed rings A and B, in the inner one of which the projector
\
\ I
\. II
\I
XVI-3]
327
MEASUREMENT OF LIGHT DISTRIBUTION
is mounted. The inner ring also carries an arm C with slide D which
can be moved along an arc E. This arc can be rotated about the horizontal axis ZZ (through worm gearing), and the inner ring with the
projector is able to follow this motion by reason of the gimbals.
In the three types of apparatus described the measuring direction is
in each case adjusted in a different manner, that is, in each the direction
is determined by different systems of co-ordinates. If the results of
measurement are to be plotted as an isocandela diagram it is necessary
to ascertain what spherical co-ordinates are to be included in the chart
(see Ch. V). In the latter we plot the angular displacement from the
axis of the projector in respect of which the measurement is taken.
For the measurement, however, the photometer is stationary and the
y
c
'R
A
B
R
66559
Fig. 188. The frame R rotates
on the fixed horizontal axis
AB; the projector P rotates
about axis YY which follows
the motion of R about AB
Fig. 189. The frame R rotates
on the fixed vertical axis CD;
the projector is pivoted on
axis XX which follows the
movement of R about CD
projector is rotated, so that when it is turned to the right and upwards
the measuring direction with respect to the axis of the beam is displaced
to the left and downwards.
If in Fig.l88 the projector is rotated through a certain angle IX about the
fixed axis AB, and is then rotated about YY so the the measuring
direction maintains an a,ngle IX with the normal to the axis YY, the
measuring direction will describe a parallel (line of latitude).
For the direction of measurement, the rotation about YY is thus rotation
in longitude, and that about AB one in latitude. The system of co-
328
MEASUREMENT OF LUMINOUS INTENSITY
[XVI
ordinates for this apparatus accordingly has a vertical pole axis, and
is therefore the Y-system, whilst that for the apparatus depicted in
Fig. 189 is the X-system.
With measuring directions which do not diverge more than 10 to 15°
from the axis there is no need to make any distinction between the one
system of co-ordinates
A
or apparatus, and the
other, seeing that up
to such angles the meridians of the one run
very close to the parallels of the other.
y.;e now come to the
apparatus depicted in
Fig. 190.
Here, on rotation about
axis ZZ the directions
of measurement describe conical surfaces
about the axis, ZZ being
then the pole axis. Displacement along the
circle E is displacement
in latitude and rotation
about ZZ is the motion
in longitude, so that Fig. 190. The projector is mounted in the inner
measurements are ef- gimballed ring of A and B. The axis of the projector
fected with co-ordinates is deflected from the initial position a long a graduated
arc E. Rotation of the arc E enables the luminous
of the Z-system.
intensity of the projector to be measured in directions
This instrument is used
lying on the surface of a cone.
to advantage for the azimuthal spherical projection, since the parallels
of latitude described when the instrument is rotated about ZZ are
reproduced as circles. Furthermore, this apparatus facilitates the determination of the average luminous intensity of a source at similar angles
from the axis in cases where measurement in four directions is not
enough to ensure an accurate average. The four directions can be easily
reproduced with the systems shown in Figs. 188 and 189 by rotating
upwards, downwards, to left and right.
XVI-4]
MEASUREMENTS ON PROJECTORS
329
XVI-4. Measurements on projectors
A few supplementary remarks must be added to what has been said in
the previous section about measurements on projectors 4).
In section Xl-6 the beam cross-over point is defined as the point at
which the exit pupil of an optical system is seen to be completely flashed.
Only as from this point may the inverse square law be applied.
If we are to determine the luminous intensity of projectors as being
the product Ed2 , E must be measured for a value of d that at least
equals the distance g from the beam cross-over point to the projector,
in addition to which this distance must be great enough to permit of
a sufficiently close approximation to the limiting value of Ed2 • In
general, this condition will be met at
the beam cross-over
point.
In section XI-6 the
distance g is computed for circular
diffuse light sources
placed at the focus
&&562
of a lens. Let us
now calculate this
Fig. 191. Figure illustrating the calculation of the
location of the beam cross-over point (distance g)
for a parabolic mirof a parabolic mirror with spherical light source
ror with spherical difat the focus
fuse light source at
the focus, with reference to Fig. 191. If the beams reflected from individual points on the mirror be drawn, it is found that g is determined
by the inner boundary rays of the beam as reflected at the edges of the
paraboloid.
From Fig. 191 it is found that:
g = yjtan <5.
If r ~ y, we may write for tan <5: tan <5 = rjv, where v is the vector
radius of the edge of the parabola. Now, v =I+ x, where I is the focal
length and x the depth of the paraboloid; hence:
+
y (f x)
g=--r
(XVI-I)
Similar calculations can be made for other .forms of light source, but
there will not usually be much object in so doing, seeing that in practice
330
MEASUREMENT OF LUMINOUS INTENSITY
[XVI
numerous divergencies from the theoretical conditions occur. The
significance of the derived formula for g lies thus mainly in the fact
that it tells us upon what elements the position of the beam cross-over
point depends. It may be said that g is greater according as the light
source is made smaller and the mirror larger.
In practice the beam cross-over point is found to be nearer to the
projector itself than is indicated by theory, and to an increasing extent
as the mirror departs more from the theoretical form.
If there is any uncertainty whether the measuring distance is sufficiently
large, or if it has to be determined in advance, the best method is to
measure the luminous intensity at different distances to find the distance
as from which Ed2 retains the same value.
Some typical measuring distances occurring in practice have already
been given in Section XVI-3.
In the case of search lights the distance from projector to beam crossover point is often so large (sometimes several miles) that absorption
and scattering of the light by the particles of solid matter suspended
in the atmosphere between the projector and photometer must be taken
into account.
This is particularly important where measurements are taken in the
open. Such atmospheric effects can at all times be ascertained by setting
up next to the projector a diffuse light source whose luminous intensity
can be measured at such close range that the effects of the atmosphere
on the result can be disregarded e.g. in the photometric laboratory.
Each time that measurements are taken from the projector - or less
frequently if stability of the atmospheric conditions permits - the intensity of the diffuse source is then also measured and the results obtained from the projector are corrected with the aid of those taken
from the diffuse source. n necessary the latter source can be placed
closer to the photometer. In this case the corrections of the projector
measurements by means of the measurements from the diffuse source
must be carried out with the aid of the absorption formula (X-12).
The light spot produced by a beam on a screen can be projected at a
short distance by placing a positive lens in front of the projector (Fig.
192) 5).
If we imagine a screen at an infinite distance from the projector B
(in practice this would simply mean a long distance), the image on
the screen would be reproduced (obviously reduced in size) in the focal
plane of the lens. If the screen be placed in the focal plane F of L, this will
show on a reduced scale what the beam looks like at infinity. To project
XVI-4]
MEASUREMENTS ON PROJECTORS
331
the cross-section of the beam at other distances by means of the lens,
the lens formula enables us to compute the distance from the lens at
which the screen should be mounted.
This method can be employed for measuring purposes as well, but
two factors should then be taken into consideration. In the first place
the lens reflects light back to the projector and this is once more thrown
back by the latter. This light is superimposed on the actual beam, but
its light distribution
F
is entirely different, so
that the distribution
L
of the projection of
the beam differs from
that of the beam itself.
Carried out in this
way, measurements
would yield faulty
Fig. 192.. , Measurement of the light distribution of results, and a means
a projector at short range. The lens L produces at
correcting this
the focal plane F an image of the beam projected of
to infinity by the projector. The effect of light consists in introducing
reflected by L can be almost completely eliminated
a filter (Fi in Fig. 192)
by a filter Fi
at an angle of 45°
with the optical axis, between the projector and the lens. The light to
be measured than passes once through this filter and is attenuated
by a factor r. The light reflected by the lens passes through the filter
three times and is accordingly attenuated to the extent of r. The filter
is mounted at an angle of 45° so that light reflected from it is defleeted
from the direction of the axis and does not enter the projector.
The second point concerned is the deficiencies inherent in single lenses
(mainly spherical aberration). For the degree of accuracy usually
demanded in such measurements the aperture of the lens should not
be greater than f : 8. This system can be successfully employed for
motor-car headlamps for which a lens 25 em (10") in diameter, with
a focal length of 2 metres (7') is suitable.
There is another method that can be used in certain cases to attain
a measuring distance that is less than the distance between projector
and beam cross-over point. If it is possible to split the light-emitting
surface of the projector into precisely defined elements (e.g. by means
of screens), the distribution of the light-elements can be measured
separately. The luminous intensity of the whole projector in a given
332
MEASUREMENT OF LUMINOUS INTENSITY
[XVI
direction is then obtained by adding up the intensities of the elements
in that direction.
From the formula (XVI-I) for the distance from projector to beam
cross-over point it is seen that g is proportional toy, that is, the distance
from the edge of the projector to the axis. When elements of the projector
are measured, y is the distance from the edge of the element to the
axis of the element and is accordingly much smaller than the y that
relates to the whole projector. Therefore, g can also be very much
smaller.
This method is quite suitable for the measurement of cylindrical-parabolic mirrors and drum lenses.
XVI-5. Photometer with Maxwellian view
In the visual photometers described in Section XIV-4 the pupil of
the observer's eye is not reduced by an artificial pupil and is thus wholly
flashed with the luminance of the comparison surface.
Sometimes, however, measurements of luminous intensity can be effected
by means of a photometer in which the
Lz R
pupil of the eye is only ~Lar~::::::::::.:::==::==:=~(~
partially flashed with · - · - · - - · - · - ·
the light by producing
in it a reduced image
v
of the light source to
be measured (MalCwel- Fig. 193. 'Diagrammatic representation of a photolian view).
meter with Maxwellian view
The principle of such
photometers will now be explained with reference to Fig. 193, which
shows the paths of the rays of light measured. For the luminance of
the photometer field of the comparison source the usual system with
completely flashed pupil can be used. This section is not included in
the figure.
La is the light source the intensity of which is to be measured. A reduced
image of La is formed by the lens L 1 in the lens of the eye L 2 • PF
is the photometer field, e.g. a Lummer-Brodhun prism. The eye is
accommodated to PF, hence an image of PF is produced on the
retina R.
Let us now compute the illumination E R on the retina as produced
by La, disregarding the losses in the lens L 1 and in the eye. The following
notation will be used:
XVI-5]
I
u
v
f
v0
5
333
PHOTOMETER WITH MAXWELLIAN VIEW
= luminous intensity of La in the direction of the photometer,
= distance from La to Lv
= distance from L 1 to L 2 (PF is considered as coincident with L 1 ),
=
focal length of Lv
= distance from L 2 to retina R,
area of L 1 effective in forming the image of La in L 2 ( ~-= area
of PF),
<PR = luminous flux falling on the retina,
5' = area of retinal image of S.
=
Then
<J>R
ER = 51 .
The luminous flux <PR falling on the retina is equal to that which 1s
received from La by area 5 of L 1 ; hence
I
<J>R = - 5.
uz
The area 5' of the retinal image of 5 is reduced by a factor equal to
the square of the quotient of image and object distances of 5 and 5'
with respect to the lens L 2 , so that
v2
5' =-i-S.
v
We can therefore write:
I
-S
<P
u2
v2
I
ER=~=--=- X - .
2
5'
v....!!_5
v02
u2
v2
1
1
Introducing the lens formula -:- = 1
u
uz f2
-·
+-v1 we find that v = (u- /)2'
2
I
f2
ER = - X
.
vo2
(u- /)2
The retinal illumination is thus inversely proportional to (u- /) 2 and
we therefore again have a square law, in which, however, the distances
are measured up to the front focus of L 1• Usually the distance u is so
great compared with the focal length that it can be measured right
up to L 1•
With such photometers only the substitution method is employed;
for the comparison luminance, as already mentioned, the normal system
with wholly flashed pupil is used.
hence
334
MEASUREMENT OF LUMINOUS INTENSITY
[XVI
If the retinal illumination be computed for a particular light source
and distance as found by the two methods, it is found that with the
Maxwellian view E R is generally thousands of times greater than that
with completely flashed pupil. The lower measuring limit by the firstmentioned is therefore very much lower than by the second.
Nevertheless the method whereby the pupil is completely flashed is
to be rreferred wherever possible, and the other method is only used
when the first is impracticable owing to the luminance of the photometer
field being too low.
The method with Maxwellian view has several disadvantages. Since
the dividing line between the photometer fields never disappears
entirely, but remains visible as a light or dark strip, the accuracy of the
balance is lower than when the pupil is wholly filled.
Again, the image of the light source must be exactly adjusted in the
centre of the eye-piece. Owing to the Stiles-Crawford effect (see Section
XII-2) any displacement of the eye during measurement results in a
variation in the retinal illumination and therefore also of the luminosity.
The use of a chin-rest is therefore recommended. In order to minimize
the effects of inevitable small displacements, care should be taken that
the image of the lamp on the pupil of the eye is small, preferably not
larger than about 1 mm in diameter.
Maxwellian view is employed in photometers when light sources of
relatively low luminous intensity have to be measured at large distances,
and also for the measurement of very low intensity sources at normal
distances, e.g. cycle rear lamps and reflectors.
REFERENCES
I) 0. R e e b, Optik 9, 1952, 254-273. "Zur Frage der photometrischen Grenzentfernung"
2) T h. H. Pro j ector, Ill.,Eng. 48, 1953, 189-191. "The Use of Zonal Constants in the Calculation of Beam Flux"
The same author, Ill. Eng. 48, 1953, 192-196. "Versatile Goniometer for Projection Photometry"
a) J. Bergman s and H. A.
Keitz, Ph. T. R. 9, 1947, 114-122. "Determining the Light Distribution and Luminous Flux of Projectors"
') J. M. W a 1 dram, Trans. Ill. Eng. London 16, 1951, 187-207. !'The Photometry of Projected Light"
") A. B 1 on de 1, Comptt..:- Rendus 188, 1929, 1464-1467. "Sur une methode
nouvelle pour !'etude en laboratoire des faisceaux des appareils optiques"
P. C i b i e, Comptes Rendus 200, 1936, 2136-2138. "Methode de contr6le en
laboratoire des projecteurs de lumiere des automobiles"
=·
CHAPTER XVII
MEASUREMENT OF LUMINOUS FLUX, QUANTITY OF LIGHT, AND
LUMINOUS EMITTANCE
XVII-I. The Ulbricht sphere photometer
Towards the end of the last century it was gradually realised that the
correct quantity characterising a light source is not the luminous intensity, but the total luminous flux. Until then no other method of
determining luminous flux was known than that of computing it from
the light distribution. This method is not straight forward, however,
and efforts have quite naturally been made to devise an instrument
by means of which luminous flux could be measured in a simple manner.
Various suggestions were put forward, but none of these gave satisfactory results until in 1900 U 1 b r i c h t 1) solved the problem by
introducing his sphe1'e photometer, or integrating photometer.
According to (III-4) q, = Jldw, and it can be said that this photometer
determines the integral Jldw.
The photometer consists of a hollow sphere, the interior of which is
painted matt white. The light source is suspended inside the sphere.
Now, if a part of the interior be screened from the direct rays from the
source, the illumination of that part will be proportional to the luminous
flux of the source, provided certain conditions are fulfilled. These conditions will be discussed later.
In this way it is possible to measure an unknown luminous flux, once
the photometer has been calibrated with a standard source. The substitution method is almost invariably employed, the source under
examination being suspended at the same point as the standard source.
It is only necessary to measure the illumination at the measuring point
on a relative scale and, as a rule, we do not measure the illumination
as such, but another quantity that is proportional to the illumination.
This can be done either visually or physically.
Figs. 194a and b depict two methods of effecting the measurement. In
both fig11res K is the sphere, L the light source, M the measuring point,
and S the screen that masks the direct rays from L in the direction
of M. The difference between the two diagrams lies in the manner in
336
[XVII
MEASUREMENT OF LUMINOUS FLUX
which the illumination of M, or more properly the quantity proportional
thereto, is measured. In effect, in the first case (Fig. 194a) M is measured
from the rear, and in the second (Fig. 194b) from the front.
With the arrangement shown in Fig. 194a a hole is provided in the wall
of the sphere at M, into which a light-receptor (photo-electric cell or
opal glass) is inserted. The luminance or luminous intensity of the opal
glass is measured, this being proportional to the illumination on the
inside face of the glass. Alternately a photo-electric cell is placed immediately behind the opal glass.
11
---o
------------,------- -
------
,- .......
Fig. 194. Principle of the U 1 b rich t sphere. K sphere painted matt white;
L light source to be measured; S screen; M measuring position. With the arrangement shown in a) a quantity that is proportional to the illumination at M
is measured behind M (the sphere window); in b) a similar quantity (the luminance)
is measured from the front through aperture 0 in the sphere
Since an opal glass is often fitted at the measuring point (invariably,
during the period when no other form of photometry than the visual
was known), the measuring point is referred to as the sphere window,
or, for short, the window.
In Fig. 194b the luminance of M - which is proportional to the illumination - is measured through an aperture 0 in the wall of the
sphere. This measurement is generally effected by the visual method.
It is only rarely that this arrangement of the photometer is employed.
XVII-2. Illumination of the sphere window in the "ideal" integrating
photometer
Let us now compute the illumination E of the sphere window *) as a
*) We shall also refer to the sphere window as applied to Fig. 194b, as the following
considerations apply to both arrangements.
XVII-2]
IDEAL INTEGRATING PHOTOMETER
337
function of the luminous flux in the "ideal" sphere photometer.
By this is meant a sphere photometer the interior of which is uniformly
diffuse, it being assumed, moreover, that the screen, light source and
reflective properties of the window do not disturb the distribution
paths of the rays in the sphere, and that the reflection factor is independent of the wavelength of the light, i.e. that reflection in the
sphere is not selective.
Let us denote the reflection factor of the sphere wall by (!, the luminous
flux by (/> and the radius of the sphere by R *).
A small area L15 of the sphere wall receives a luminous flux equal to L1$.
Hence L1S reflects a luminous flux of eL1$. This area is a uniformly
diffuse light source which illuminates the whole interior of the sphere.
According to section IX-4 the sphere is a McAllister equilux sphere,
that is to say the illumination of the sphere as produced by L1S is
the same at all points. The luminous flux incident on the sphere wall
being eL1$ and the area of the sphere 4nR2, the illumination L1E 1 of the
sphere wall, and hence also that of the sphere window, produced after
reflection by L1S, will be:
Every part L1$ of the luminous flux falling on the sphere wall contributes an element L1E 1 towards the total illumination E 1 of the sphere
as a result of the initial reflection so that
The total reflected luminous flux e$ is again reflected by the sphere
wall, and the illumination E 2 of the window arising from this second
reflection is found in the same way as Ev viz:
(!2(/J
E2 = 4nR2.
Proceeding thus for the 3rd, 4th, 5th ... nth reflection, we find for the
total illumination E of the window, resulting from the 1st ... nth
reflections together:
*) The following is taken largely from an unpublished article by the Lighting
Engineering Laboratory of N.V. Kema, Arnhem (1944), to whom the author's
acknowledgements are due.
338
MEASUREMENT OF LUMINOUS FLUX
[XVII
E = Et + £2 + Ea + ... +En=
er/J
= 4nR2
=
e2r/J
earp
+ 4nR2 + 4nR2 + · · · +
enrp
4nR2 =
(/)( + e2 + ea + ... + en ) = 4nR
(/) X t -e e.
4nR 2 e
2
(XVII-I)
Let k be the quotient of E divided by r/J; then:
E = kr/J.
(XVII-I a)
If there were no screen in the sphere, light direct from the source would
fall on the window; the illumination on the window would then depend
on the luminous intensity of the light source in the direction of the
window and thus on the light distribution of the source and the position
of the source with respect to the window. It is therefore dependent
on variable and casual factors, and is not proportional to the total
luminous flux, as is the case with the illumination produced by the
reflected light. The screen is therefore provided betw~en the source
and the sphere window in order to eliminate the effect of this illumination
which is not proportional to the luminous flux.
From the above derivation of equation (XVII-I) it appears that in the
case of the ideal, uniformly diffuse reflecting sphere the illumination
of the window, produced after the luminous flux has been reflected at
least once (the indirect illumination), is independent of the distribution
of the flux over the sphere wall, i.e. is independent of the light distribution
and position of the light source.
XVII-3. Illumination of the window of a non-ideal integrating photometer
For a hollow body of arbitrary form, of which the reflection factor is
the same at all points, we have already derived a formula that will give
the mean illumination at the wall (section X-14, equation X-9). This is:
1
Em= S. 1 - e'
(/)
where S is the area of the hollow body. In deriving this equation we
proceeded from the assumption that the light from the source reaching
the wall is not screened. In integrating photometers, however, the
direct rays in the direction of the window are screened off, so that we
have now to consider the illumination produced by light that is reflected
XVII-3]
NON-IDEAL INTEGRATING PHOTOMETER
339
at least once. For the mean indirect illumination Eim , then, equation
(X-9) becomes
E.
'm
=~.-e-.
S
1- (}
(XVII-2)
If the illumination of the window be related to Eim by a factor p, we
may write:
(XVII-3)
Now, if we put:
we find -
p_ _e_ =k
s ·1-e '
as for the ideal sphere -
that
E = kf/>.
In the derivation of equation (XVII-3) we have made no assumptions
regarding the shape of the hollow body. What we have assumed is
that e is independent of the place at the sphere wall, and is the same
for all reflections, i.e. that e is independent of the angle of incidence, the
factor p being independent of the place on the sphere wall and of the
light distribution of the light source~
In practice, however, there are certain departures from these ideal
conditions, viz.:
1. The reflection from the sphere wall is never uniformly diffuse.
2. The reflection factor of the sphere wall is not exactly the same at
all points.
3. The screen masks not only the light source, but also a part of the
sphere wall.
4. A part of the reflected light falls on the light source, the suspension
device and the screen, and is partly absorbed by these.
5. The window, or the receptor, is not subject to the cosine law in
transmitting or absorbing light. Glancing rays are not evaluated
by the photometer to the same extent as rays in directions more
closely approaching the normal.
6. The reflection factor may differ slightly at different wavelenghts,
i.e. the sphere wall may reflect selectively.
In consequence of these deviations from the ideal conditions k = E (f/>
is in practice dependent on the position, the light distribution, size
340
MEASUREMENT OF LUMINOUS FLUX
[XVII
and absorption factor of the light source, as well as on the spectral
distribution of the light.
XVII-4. Measures to be taken with non-ideal integrating photometers
to approximate to the ideal sphere
Let us now see what limitations are imposed by the above divergences
in the use of the integrating photometer, and what steps can be taken
- or in some cases must be taken - to correct the effects of these deviations.
The following cases may occur in practice:
1 (a) equal relative light distribution of the light sources to be compared,
(b) light distributions not equal,
2 (a) absorption of the light travelling to and fro within the sphere
the same for both light sources,
(b) absorption not the same,
3 (a) spectral distributions of the light sources identical,
(b) spectral distributions not the same.
Combinations of Ia orb, 2a orb and 3a orb are, of course, always met
with in practice.
Ia and b. Light distribution of sources to be compared
If a narrow beam of light were to be thrown successively towards different
points in the ideal integrating photometer with receptor answering
the cosine law, exactly the same reading would be obtained for each
direction of the beam. The photometer reading would thus be independent of the distribution of the luminous flux in space (the light
distribution) and it would be possible to compare light sources of widely
differing distribution patterns without the slightest error.
If this test be carried out in a practical form of integrating photometer,
different readings are obtained for different directions of the beam.
Fig. 195a shows the result of such a test in an Ulbricht sphere; the
divergence from the average value of k as a percentage is here plotted
as a function of the point where the beam, coming from the centre of
the sphere, strikes the wall. The average value of k is the average value
as computed over all the solid angles with apex at the centre of the
sphere.
The curve exhibits two dips, one at A as produced by the presence
of the screen, the light falling on the window being thus reflected at
least twice (screen error), and another at B, this being due to the fact
that the luminous flux reflected by that part of the wall which lies
XVII-4]
APPROXIMATION TO THE IDEAL SPHERE
341
opposite the window is partly masked by the screen (screen shadow
error).
The presence of the screen is not the only source of divergence from the
ideal conditions, however; departures from uniformly diffuse reflection
also play a part.
These differences between the ideal and practical forms of sphere result
in an error in the comparison of light sources of which the light distribution differs.
s
p
®
-
-
Fig. 195. Evaluation by the
photometer P of the luminous
flux as a function of the part
of the sphere on which it falls.
The curves show as a percentage the divergence from
the average in integrators a) of
pure spherical form, b) semiregular 14-sided body, c) of
the form shown. S indicates
the 'location of the screen
342
MEASUREMENT OF LUMINOUS FLUX
[XVII
The measures that can be adopted to minimize this error as much as
possible will be discussed later.
Seeing that a true sphere, with its doubly curved surface, is more
difficult and more expensive to make than a body with plane boundary
faces, many integrating photometers are made in the form of more or
less regular polyhedrons; a much favoured pattern is a semi-regular
tetradecahedron or fourteen-sided body which is, in effect, a cube with
flattened corners.
When curves are plotted for an integrating photometer of this type
along the same lines as those in Fig. 195a they will appear as shown
in Fig. 195b (two curves are given, because measurements in two planes
of symmetry have been taken.
Comparison of Fig. 195awith 195b shows that, in the sphere, k is practically constant over a large range, whereas in the polyhedron k is not
constant in any ranges; the error involved when the light distributions
are not the same is thus greater than in the case of the sphere.
It is possible, however, to construct a polyhedron that is very much
more satisfactory in this respect than the conventional patterns, and
the results obtained from a model are illustrated in Fig. 195c, together
with the form of the polyhedron itself.
Compared with figures 195a and b, the curve in Fig. 195c is seen to
correspond much more closely to those of the sphere.
From the above it will be clear that the true ~;phere is preferable to
polyhedrons when the light distribution of the sources shows a marked
difference.
It will be seen from Figs. 195a, b and c that the evaluation of the luminous flux radiated in various directions by a source will vary from
one type of photometer to the other. The illumination on the window,
or in the case of the non-ideal receptor the reading obtained from the
photometer, is a function of the location on the wall of the sphere of
the luminous flux to be measured.
If we write:
E = k<P,
it follows that k = f (oc, {3), where oc and {3 indicate the position on the
wall of the sphere in angular co-ordinates. If the light is distributed
over a number of points:
(XVII-4)
E = L:k. LI<P
if LI<P denotes the luminous fluxes incident at the various points of the
sphere wall.
XVII-4]
APPROXIMATION TO THE IDEAL SPHERE
343
In case Ia (light distributions equal) in every term k . Lf«P the value of
Lf«P varies in the same proportion (say: a) when the standard light
source is replaced by the light source to be measured, so that we obtain
for the illumination or the reading of the photometer for a light source X:
E.,= 1: k. a. Lf«P =a. E.
In this case the dependence of k on the position in the sphere is immaterial which means that, with equal relative distribution of the sources
to be compared, the geometry of the photometer and uniformity of the
reflection factor are no longer sources of error.
From (XVII-4) it follows, moreover, as already discussed above, that
if the relative light distribution of the sources is not the same, the
variation in k may have a pronounced effect on the results obtained.
In order to reduce this effect as much as possible the following measures
can be taken:
a .. The reflection factor of the paint applied to the sphere must be made
as high and as uniform as possible. This implies that the inevitable
soiling of the bottom of the sphere will necessitate frequent repainting.
b. The sphere should be large and the screen as small as possible.
c. The light source must be so arranged in the sphere that the greatest
possible part of the luminous flux falls on the
areas of the sphere for which the value of k
varies the least from the average value in equation
(XVII-4), i.e. on those parts which lie outside
the zones of screen and screen shadow error.
Linear light sources such as tubular fluorescent
lamps and fittings should accordingly be suspended in the sphere with the axis at right angles
to the screen (see Fig. 196). Beams of light should
always be made to fall on areas outside the zones
Fig. 196. The light of screen and screen shadow error.
sources placed in the
d. The screen should be diffusely translucent,
integrator should be
so arranged that the
with a transmission factor of about 3%.
greatest possible part
of the luminous flux
e. A receptor, or window, should be used which,
falls in the areas of
as far as is possible, also follows the cosine
constant k, which imlaw at glancing incidence.
plies that linear sources should lie per- To the above may be added the following expendicular to both
screen and window
planatory remarks.
a. The fact that e should be as high as possible
344
MEASUREMENT OF LUMINOUS FLUX
[XVII
may be explained as follows. The value of k as a function of the position
on the sphere wall depends in the first place on the contribution, after
initial reflection, as made by each point towards the illumination of
the window. As photometer paints are reasonably matt, the luminous
flux is nearly enough uniformly distributed over the sphere wall after
the initial reflection, and the contributions of the 2nd and 3rd, and
subsequent reflections, towards E are thus dependent almost exclusively
on the geometry of the integrator.
The contribution to E arising from the first reflection from a certain
point in the sphere may be said to be proportional to k1f/J.
The sum of the contributions due to subsequent reflection is proportional to (e 2
e3 e4
e") = e2/(I -e).
When e is increased the term e2I(1 - e) increases in value more
steeply than the term k 1 e, so that, when e is high, the effect of k1 and its
variations is reduced.
In practice, a lack of uniformity in the reflection factor will be mainly
the result of the heavier soiling at the bottom than at other parts of
the integrator, in consequence of which k is lower at the bottom than
elsewhere. The effect of this on the measurement is the more marked
according as a larger part of the flux is allowed to fall on the bottom
part. Particularly in the measurement of light beams radiating vertically
downwards should care be taken to maintain a sufficiently high degree
of uniformity; this can be done by accurately ascertaining (e.g. from the
light distribution) the luminous flux of a concentrated type of source
with constant luminous flux (e.g. a lamp with silvered bowl), and subsequently taking measurements from this source in the integrator by
comparing it with a standard lamp, which radiates in all directions.
If the value of the luminous flux thus obtained is found to be below a
certain limit (say 97% of the original value), it will be time to repaint
the integrator.
In order, in the case of narrow beams, to be less dependent on variations
in e, the directly illuminated part of the integrator can be enlarged by
mounting the projector in the wall of the sphere instead of at the centre,
the position and size of the screen being adapted to suit.
b. Zones in which the screen and screen-shadow errors occur should
be made as small as possible. To this end the screen itself should be
made as small as possible compared with the sphere. At the same time,
this is determined mainly by the size of the light source, which means
that the sphere should be as large, and the screen as small, as the dimensions of the light source will permit.
+ + + ...
XVII-4]
APPROXIMATION TO THE IDEAL SPHERE
345
c. Requires no further explanation.
d. The screen error can be almost fully corrected by employing a diffuse
translucent screen. The necessary transmission factor can be calculated;
it is found that for all practical purposes -c = 3% is satisfactory. It is
not possible by such means to eliminate error due to the screen shadow.
e. Light entering the receptor or sphere window at glancing incidence
comes from the zone of the screen error, so that, if the cosine law is
not sufficiently satisfied by the receptor, this results in an increase in
the screen error.
2a and b. Absorption by the light sources under comparison
If the light travelling to and fro in the sphere is absorbed to the same
degree by each of the sources under comparison, the illuminations of
the window and thus the reading taken from the photometer are affected
to the same extent, hence no correction is necessary.
However, if the light-absorbing properties are dissimilar, an endeavour
must be made to ascertain the effect on the final result. The absorption
which affects the accuracy of the result is that of the light which is
reflected at least once.
According to H e 1 w i g 2) the influence of absorption of the indirect
light by the light source under examination can be measured by placing
in the sphere a constant indirect light
source, which emits a constant indirect
luminous flux into the sphere. If the illumination on the window produced by this
indirect flux be measured successively with
the two source to be compared extinguished
in turn, and the difference is found to be
say, no/o, this means that the one source
absorbs n% more of the indirect light than
the other. This correction of n% is applied
to the readings obtained from the sources
Fig. 197. Integrator with
auxiliary lamp H procompared.
posed by H e 1 w i g to
A constant indirect luminous flux can be
eliminate error arising
in the comparison of light
obtained by placing in the sphere an auxisources whose absorption
liary lamp screened in such a way that only
differs
indirect light, i.e. reflected at least once,
falls on the sources under comparison (see Fig. 197). The whole procedure
of the luminous flux measurement is then carried out as follows:
346
MEASUREMENT OF LUMINOUS FLUX
[XVII
Standard source
Source
"
on, auxiliary lamp out: reading a5
out,
h5
on:
to be measured on,
out:
ax
out,
on:
hx
Disregarding the indirect light absorption we find that the unknown
luminous flux is
ax
cpx = - X cps•
as
where cps is the luminous flux of the standard.
This is corrected for absorption of the indirect light by multiplying
it by the quotient of the readings taken from the auxiliary lamp (hsfhx)·
Thus the required luminous flux is
ax
hs
<l>x = - X - X cps·
as
hx
Particularly for the measurement of lighting fittings, the absorption
of which can be anything up to 40% more than that of the naked lamp,
He 1 wig's method employing an auxiliary lamp is a very useful
means of ensuring accurate results.
An objection to the method, which does not, however, weigh heavily
against its advantages, is that the result is obtained from four readings
instead of oniy two, so that the probable error is v2 times greater.
3a and b. Spectral distribution of the light sottrces under comparison
When the spectral distributions of the light sources to be compared are
the same the spectral reflection from the sphere wall has no effect on
the results of the measurement.
If the composition is not the same, however, even relatively slightly
selective reflection may have a pronounced effect on the results. The
illumination of the window is proportional to e/(1 -e) and, if the
values of (! for two wavelengths are equal to 80 and 81%, respectively,
this difference in (! will give a relative difference of about 6% in the
illumination of the window.
In consequence, thus, the photometer paint should be as little selective
as possible and, of equal importance, should remain so in use. Paints
which tend to turn yellow are therefore to be avoided at all costs, for
which reason oil-bound paints are never used; size-bound paints are
employed instead, these being applied as aqueous suspensions.
Various recipes for such paints are to be found in the literature, but we
XVII-4]
APPROXIMATION TO THE IDEAL SPHERE
347
append two, one of which has been adopted by the National Physical
Laboratory, corresponding to the specification in British Standard
354: 1961 3), and another, taken from the German D ..I.N. 5032 4 ).
~pecification to B.S. 354: 1961
1. Pre-treatment and primer on the wall will not generally affect the
optical properties of the whole of the coating system, but should ensure
proper keying to the basic material and provide protection against
cor~osion and peeling.
2. The recommended top-coating consists of two layers:
2.1 a white primer and
2.2 a water-soluble, matt layer that can be washed off when soiled and
replaced by a fresh coating.
2.1. It is desirable for the white primer to have a matt surface and to be
proof against yellowing in time. The colouring material should therefore
contain .a sufficient proportion of a strong covering pigment, e.g. titanium
oxide, dispersed in a non-yellowing medium. Two-component-coatings
of the polyvinyl acetate emulsion type are to be recommended.
2.2. The washable, matt white topcoat can be made from fine.precipitated barium sulphate ("blanc-fixe") dispersed in a water-soluble, yellowing-resistant medium, e.g. sodium carboxymethyl cellulose that is as
free from fibres as possible. There are other colloids with similar properties.
A recommended mixture consists of
1000 parts by weight of "blanc-fixe" (barium sulphate);
25 parts by weight of sodium carboxymethyl cellulose (low viscosity);
1000 parts by weight ofwater.
The mixture can be further improved to give the most non-selective
reflection possible. The slight decrease in the spectral range of the
white pigment (barium sulphate) in the blue region can be compensated
by the addition to the mixture of an aqueous paste of a carbon pigment
that emphasises the blue, e.g. ivory black.
For further details on the preparation of the mixture, the British Standard
should be consulted.
2. Recipe to DIN 5032, July 1957
For the undercoating of the integrator it is recommended:
On the primer of Nitro-binder and zinc-oxyde a thick magnesium-
348
MEASUREMENT OF LUMINOUS FLUX
[XVII
oxyde solution with 0.5 w% gelatine (for 200 g MgO 1 g gelatine)
diluted with water to a sprayable, or paintable thickness must be
applied at least two times. Attention should be paid that no drip occurs.
Such coating is almost aselective and its reflection factor is independent
of the temperature of the ball. The renewal of the integrator coating
should take place regularly, at least once a year.
The results of measurement depend not only on the spectral reflectivity
of the sphere wall, but also on the spectral sensitivity of the operator
in visual photometry, or of the photo-electric cell in physical photometry. In the latter instance matching of the spectral sensitivity of the
cell with V.\ gives us the means of correcting any selective reflection in
the integrator.
The influence of the spectral reflectivity of the sphere wall and the
spectral transmission of the window can be determined spectrometrically
by suspending a lamp in the integrator, then ascertaining the spectral
composition of the light transmitted by the window, and comparing this
with the spectral composition of the source itself 5).
An incandescent lamp can be employed for the long-wave region of the
spectrum. In order to ensure sufficient light in the blue, it is useful to
carry out the test also with an incandescent lamp with blue bulb, or
with a "daylight" (U.S.A.) or "colour matching" (G.B.) fluorescent lamp.
A curve is then drawn giving the calibration of the integrator as a
function of the wavelength. This curve can be combined with the spectral
sensitivity curve of the photo-electric cell (by multiplying the values
of the two curves at each wavelength) to produce a curve representing
the sensitivity of the integrator and cell at every wavelength.
From this it is possible to derive the spectral transmission curve to
which a filter placed before the cell would have to conform in order
to ensure readings from the cell that would be wholly in accordance
with V.\. Such an ideal filter can be approximated as closely as possible
by a filter or combination of filters as obtainable on the market.
The system of using correction factors can be employed for integrators,
just as in the case of photo-electric cells; as applied to integrators the
correction factors are determined for the combined integrator and photoelectric cell and, even when filters are used in the manner described
above to approximate to the ideal measurement (to match V.\), it may
be necessary to determine the correction factor.
The difficulties arising from selective reflection from the wall of the
integrator when luminous flux measurements of coloured sources are
XVII-4]
APPROXIMATION TO THE IDEAL SPHERE
349
made by comparison with standard incande~cent lamps can be surmounted by making a homochromatic instead of a heterochromatic
comparison; to this end the photometer is calibrated with standard
light sources of the same kind as the source under test. This restricts the
difficulty to one of measuring the standards. In the routine photometry
of, say, gas discharge lamps, the difficulty can be entrusted to a standard
laboratory.
A method of routine photometry for gas discharge lamps by which
calibration is effected with lamps of the same kind as that under
test has been given by 0 ran j e 6 ), who describes integrators of
cubic form with barrier-layer cells roughly matched with VA as light
receptors. For a rapid check on the integrator, and also in order to save
the standard gas discharge lamps, the standard lamp in the integrator
is replaced by an incandescent lamp suspended at a fixed point. The
reading thus obtained ftom a meter indicates a certain number of lumens,
which represent an "apparent" lumen value of the filament lamp, since
the light distribution of the latter may differ considerably from that of
the discharge lamp, and also because the whole integrator does not
give a measurement that is strictly in accordance with the V ,~ curve.
The incandescent lamp is then used as sui--standard for the measurement
of other gas discharge lamps of the same type and need only be compared with the standard lamp periodically.
Although the luminous flux of gas discharge lamps is not as constant
as that of standard incandescent lamps, this method nevertheless gives
more reliable results than direct comparison with standard incandescent
lamps, a fact that is of considerable importance in routine photometry.
We have now dealt with the influence of the light distribution, absorption
by the light source, and the spectral distribution. It remains for us to
consider the effect of the location of the source in the integrating photometer.
The curve in Fig. 195a (k as a function of the point where the light
strikes the sphere wall) was plotted for perpendicular, or almost perpendicular, incidence, that is, with the source at the centre of the sphere.
The more obliquely the light strikes, the more pronounced the change
in the form of the curve, seeing that the sphere is not a uniformly diffuse
reflector. When the light distribution is the same for each light source,
the result of the measurement is independent of the variation in k,
and it makes no difference at what point the sources are suspended,
provided that this is the same point in each case.
When the light distribution of the sources differs, the two sources are
350
MEASUREMENT OF LUMINOUS FLUX
[XVII
suspended in such a way that the curve k (ex) shows the largest possible
area in which k is constant, or nearly so; that is to say in the centre of
the sphere (projectors as we have already seen, can be mounted outside
the sphere so that the beam falls on the opposite wall).
All this holds good as long as the sources to be compared are the same
size or can be regarded as such. But, if a comparison is to be made
between a large and a small source and it is not possible to regard the
large one as small compared with the dimensions of the integrator, it
must be remembered that the various parts of the larger source occupy
different parts of the sphere. Such is the case with tubular fluorescent
lamps which are made in lengths of several feet. In order to avoid having
to make integrators which would be very large in all directions, th"s
kind of lamp can be measured in cylindrical integrators. A number of
photo-electric cells are mounted in the wall of the cylinder, these being
connected in parallel so that the readings will be independent of the
part of the lamp whence the luminous flux is received. The number
and positions of the cells are determined experimentally by moving a
small tubular lamp to and fro within the integrator. Calibration can
be effected with a standard tubular lamp having the same light distribution as an element of the fluorescent lamp.
If it is possible to calibrate with a lamp of the same type as that under
test, any integrator can of course be used.
The luminous flux of an image-projection objective can be measured in an
integrator in the same way as that of a beam projector, through an
opening in the integrator. In the case of such narrow beams, however,
it is advisable to calibrate the integrator with a similar beam of known
luminous flux.
The luminous flux from objectives can also be determined from the
screen illumination measured at uniformly distributed points in the
image. This will give the average screen illumination, and the total
luminous flux is found by multiplying this average by the area of the
image. The luminous flux of the beam used for calibrating the integrator
can be ascertained in the same way.
To conclude this section, we should also like to mention an automatic
photometer for measuring the luminous flux of incandescent lamps,
as described by Van Gorcum and van der WaaP). With this photometer,
all that need be done once the lamp is inserted is the pressing of a button.
The photometric and electrical data on the lamp are then automatically
recorded.
XVII-5]
LUMINOUS FLUX OF FLtrORESCENT LAMPS
351
It has been considered useful, in view of the special properties of fluorescent lamps, to add some remarks on their measurement and that of
fittings using them.
These considerations apply not only to measurements of the luminous
flux but to all photometric measurements made on gas discharge
lamps.
First of all, a few remarks pertaining to all gas discharge lamps. The
power consumed by the gas discharge lamp is determined by the voltage
applied and the impedance connected to the lamp in series (the ballast).
A choke or a choke in series with a capacitor is generally used as a ballast.
In order to obtain comparable and reproducible values for the luminous
flux of the lamp, we can use two methods:
1. We can supply the lamp with its rated power with the aid of some
choke by adjusting the voltage across the lamp and the choke.
2. We can connect the lamp in series with a choke with standardised
electrical characteristics The lamp and choke must then be connected to
a fixed voltage.
The International Electrotechnical Commission (I.E.C.) has chosen the
second method and has standardised, in its Publication 81, chokes for
a series of fluorescent lamps, these chokes being referred to as reference
ballasts. This series is being supplemented by specifications for reference
ballasts for other gas discharge lamps.
The actual luminous flux of the lamp(s) does not affect the measurement
of the efficiency of a fitting. Nevertheless, the lamp must provide the
same luminous flux when both the naked lamp and the fitting are
measured. This condition can be met by the use of the same, arbitrary
choke in both cases.
The most important remarks on the measurement of mercury vapour
and sodium vapour lamps have now been made. We have not yet finished
with tubular fluorescent lamps, however. In contradistinction to mercury
and sodium vapour lamps, the luminous flux and the electrical properties
of tubular fluorescent lamps depend greatly on the temperature of the
surroundings in which the lamp is burning. To put it more precisely, the
luminous flux is determined by the temperature of the coldest point on the
lamp (which is generally the centre of the underside of the tube). This
means that not only the ambient temperature, but also the movement
of the air in the measuring room determine the coldest point. An addi-
352
t
[XVII
MEASUREMENT OF LUMINOUS FLUX
100
100
t
;t%)
; (o/o)
90
90
80
80
70
70
60
60
50
so
40 ~~~~~~~~~~~~~~
10 15 20 25 30 35 40 45 50 55 60 65 70 75 80
---rube wall temperature(•c)
-8
3,5 ~ 14; 2p ~ 3,2 38 4fo Sf' ~ 62 6( 7,3
40
-¥
-25-Ambient temperature ("C)
a
+---;!n;';::;;;~-=:n:;t-;c:;-+;:~;-;::\;~";;!::-;;t:-:"1=-:;f=10 15 20 25 30 35 40 45 50 55 60 65 70 75 80
-13-'7,5 -i
---Tube wall temperature(°C)
4 10 1,6 21 I~ 34J!l,S~55}.557,563,5fi9.5
-25-.Ambient temperature(oc)
b
Fig. 198 Dependence of the luminous flux of Philips fluorescent lamps on the ambient
temperature. a) TL40W, b) TL65W.
tional condition for measurement, therefore, is a draught-proof measuring
chamber. The I.E.C. has specified an ambient temperature of 25 °C.
The conditions for measurement in the case of fluorescent lamps are therefore clearly specified: reference ballast, voltage, ambient temperature
(25 aq and a draught-proof measuring chamber. Figs. 198a and b are
the typical tube wall temperature/luminous flux curves for Philips TL
fluorescent lamps of 40 and 65 Win colour 33 (white).
In the case of the 40-watt lamp (Fig. 198a) the optimum for the luminous
flux lies at 39 ac; this is the temperature attained by the lamp in a
draught-proof room at an ambient temperature of 25 °C. The luminous
flux decreases at both higher and lower temperatures.
With the 65-watt lamp (Fig. 198b) the optimum is also at 39 °C. This
temperature is already attained with a room temperature of some 20 ac
because, compared to the forty-watt lamp, the output rises more rapidly
than the temperature of the surface of the fluorescent lamp (for the same
tube diameter, the output is about 60% greater, while the tube is only
some 25% longer).
At the prescribed ambient temperature of 25 ac, the luminous flux of the
65-watt lamp is about 1% lower than at the optimum temperature.
The measurement of high-power fluorescent lamps, e.g. the Philips
TL120W, is particularly fraught with difficulties. Because the power
in this case is almost double that of a 65-watt lamp, while the diameter
is slightly smaller and the length is the same, the ambient temperature for
XVII-5]
LUMINOUS FLUX OF FI,.UORESCENT LAMPS
353
the optimum luminous flux is about -5 oc unless special measures are
tp.ken. However, artificial aids can raise the ambient temperature at
which the best possible luminous flux is obtained. In the case of the Philips
TL120W lamp, a bulge is fashioned at the centre of the underside of
the tube. This locally increases the cooling surface and lowers the
temperature at the coldest point on the tube. The difficulties in measurement are caused by the fact that these lamps require a very long stabilisation time. If such a lamp is measured after having been moved, it can
take up to eight to twenty-four hours before the luminous flux and electrical data are constant. If, however, the lamp is then left in position, and
this is usual in a lighting installation, although hardly possible for the
purposes of photometry, two to three hours will generally be sufficient
for re-stabilisation. *)
In the measurement of such a lamp in a fitting where adjustment has
to be made to an entirely different point on the characteristic, stabilisation
takes a great deal of time. If measurements are to be made in the morning,
the lamp should be switched on the previous evening.
The curve of the luminous flux as a function of temperature means that,
in fittings and particularly in closed fittings in which the temperature
is higher than room temperature, the luminous flux of fluorescent lamps
is reduced. The efficiency of such a fitting is determined not only by the
absorption of the reflective and transmissive parts of the fitting, but also
by the temperature increase brought about by the enclosing of the
lamp(s). We now have to find out how to define the efficiency of a
fluorescent lamp fitting, bearing the above remarks in mind.
Because a lighting installation must be calculated from the efficiency
and the luminous flux data provided by the manufacturer for the lamps,
the obvious thing to do is to take for the luminous flux of the naked
lamps that found from a measurement of the luminous flux made in the
draught-proof room at 25 °C. If the efficiency is to be properly determined, the naked lamp must be operated with the same ballast as that in
the fitting.
*) The difference in the stabilisation time between, for instance, the 40 and 65-watt
lamps and the high-power lamps may be explained in the following way: While the
lamp is not burning, there are droplets of mercury in the tube, which can move about in
the tube during transport. When the lamp is ignited, the mercury droplets begin to evaporate. The vapour must diffuse inside the tube. As soon as it has reached the coldest part
of the tube and the vapour pressure at this point is established, the luminous flux
i_s stabilised. We have spoken of the coldest point in 40 and 65-watt lamps, but, in fact,
the temperature along the length of the tube is almost the same and the diffusing
mercury vapour reaches a point oflowest temperature almost immediately. In the 120W
lamp the lowest temperature prevails only in the bulge in the tube. It can therefore
take a very long time before the diffusing mercury vapour reaches this point, and the
stabilisation time can be very long.
354
MEASUREMENT OF LUMINOUS FLUX
[XVII
The essential condiuon here is that the power provided by the ballast for
the lamp shall not be too far from the rated value. If, for instance, the
ballast supplies a 40 W lamp with only 30 W, calculation of a lighting
installation on the measured efficiency with the rated luminous flux of
the lamp as 100% would provide only 75% of the calculated illuminations in the installation. The condition that the power of the lamp be not
too far from the rated value may be considered satisfied if the ballast
meets the specifications of I.E.C. Publication 82 9 ).
Of course, 25 °C should be taken as the ambient temperature during
measurement.
The definition of the efficiency of a fluorescent lamp, which we refer to as
the operating efficiency, can, then, be given in the following form:
the quotient of the luminous flux of the fitting and that of the lamp measured in a draught-proof room at 25 oc, with each lamp operated naked
and with its appropriate ballast.
When using the operating efficiency thus defined, the lighting engineer
must remember that the efficiency will change if the temperature is very
different from 25 °C. By and large, the efficiency initially increases as the
room temperature drops, but subsequently decreases. This will be of no
great significance for indoor lighting, but very important to lignting
installations in the open. In some cases the effect of low temperatures
must be included in calculations: The cooling effect of wind must also be
taken into account.
There are two ways of determining the losses occurring in the fitting
for purely optical reasons (by absorption) when stating the efficiency,
viz:
1. A thermoelement can be attached to the centre of the underside
of the lamp(s) before the luminous flux of the fitting is measured, and,
after the final temperature lias been attained, the temperature of the
lamps can be measured. The percentage decrease in the luminous flux of
the lamp due to the temperature increase in the fitting can then be
read off from a graph like the one in Fig. 198. The "optical" efficiency
can easily be calculated from this value and the operating efficiency.
2. If the lamp in the fitting is switched on after attaining the ambient
temperature of 25 oc, the temperature of the tube wall will attain the
value that it assumes on burning naked at 25 °C within two to three
minutes. The enclosure by the fitting then has no effect. If the l11minous
flux is now measured, the value with the lamps at 25 oc ambient temperature, and thus the "optical" efficiency is measured.
The moment at which the value must be read off is fairly sharply
XVII-6]
LUMINOUS FLUX FROM LIGHT DISTRIBUTION
355
defined. After being switched on, the lamp heats up rapidly to the
temperature that it attains at an ambient temperature of 25 °C. This is
accompanied by an increase in the luminous .flux. This process finishes
after two to three minutes. Because the rise in temperature due to the
heating of the fitting is a much slower process, the luminous flux of the
lamp after stabilisation at 25 oc remains constant for sufficiently long
for measurement. This method cannot be used for lamps with a long
stabilisation time, like the overloaded "TL" 120 W.
XVII-6. Determination of luminous flux from the light distribution
or from an isolux diagram
The luminous flux of axially symmetrical light sources can be determined
by means of the Rousseau diagram in the manner described in Section
IV-1.
When the light source is asymmetric the luminous flux can be ascertained
from the average distribution with the aid of a Rousseau diagram or
from an isocandela diagram, provided this is drawn on a spherical
projection proportional to the area (Section V-6). The method of deriving
the luminous flux from the isolux diagram has already been given in
Section VII-9.
Determination of the luminous flux from the light distribution has
its uses in some cases, because, owing to selective reflection from the
interior of the integrator, direct measurement of the luminous flux
of coloured light is more difficult than a luminous intensity measurement.
To determine the luminous flux of standard coloured light sources,
therefore, the light distribution is sometimes measured. If the luminous
flux only is required, irrespective of the light aistribution, it is not
necessary to take into account the rules for the measuring distance.
The illumination E on a spherical surface surrounding the light source
can be measured at any desired or suitable distance, and multiplication
of E by the area of the sphere then gives the luminous flux. For this
purpose, use may be made of a Rousseau or an isolux diagram plotted
on a spherical projection proportional to the area.
This amounts in fact to what is done when the luminous flux is ascertained
from the "actual" light distribution, since the luminous intensity is
always computed from the illumination.
XVII-7. Measurement of quantity of light
The quantity of light emitted by long-life light sources is determined
graphically from the kind of chart shown in Fig. 12, the points as plotted
356
MEASUREMENT OF LUMINOUS FLUX
[XVII
being obtained by measuring the light source at different times during
the burning period.
With short-lived sources such as flash bulbs the quantity of light produced
can be measured in the manner
described by v a n L i e m p t
and d e V r i en d 10) (see Fig.
199). The flash bulb is suspended
c
in the integrator K with matt
glass window M in front of which
66572
a vacuum photo-emissive cell P
is placed. Included in the photocurrent circuit is a capacitor C
Fig. 199. Arrangement used by van
L i e m p t and d e V r i e n d for the
which is charged up -- with
measurement of the quantity of light
cell in darkness -- to some
produced by flash bulbs. The capacitor C,
charged from an external source, is
300 V D.C. as read from an
partially discharged by the photoelectrostatic voltmeter V. When
current from a cell P, induced by the
flash bulb in the sphere. The voltage
the flash bulb is fired a photodrop across the capacitor is proportional
current flows in the circuit and
to the quantity of light from the flash
bulb
the capacitor voltage functions
as anode voltage for the cell. The photo-current discharges the capacitor.
Provided that, when the flash has taken place the voltage on the capacitor
is still higher than the saturation voltage of the cell, the voltage drop
across the capacitor will be proportional to the quantity of light delivered
by the flash bulb. A final voltage in excess of the saturation voltage
of the photo-electric cell can be ensured by a suitable choice of capacity.
p
l
~--------------------------~+
Fig. 200. Circuit for measuring the quantity oflight in lighting phenomena that change
abruptly.
XVII-8]
MEASUREMENT OF LUMINOUS EMITTANCE
357
Another circuit permitting the determination of the luminous flux
cycle of the flash bulb while the content is burning is shown in Fig. 200.
While there is no light falling on photocell P, capacitor C is charged to
a certain voltage, e.g. 200 V. When the light from the ignited flash-bulb
falls on the photocell, the photocurrent partially discharges the capacitor.
Once burning is over, the residual voltage can be read off. The voltage
drop across the capacitor is proportional to the quantity oflight that has
fallen on the photocell.
If a resistor R with an oscillograph in parallel with it is inserted in the
circuit, the luminous flux cycle can be displayed on th.e oscillograph
and recorded by photog•aphy.
XVII-8. Measurement of luminous emittance
As we have seen in Section VIII-5, luminous emittance, in the case
of a surface that does not radiate light of itself, is identical with illumination, and its measurement is accordingly a measurement of illumination. Methods of carrying out the measurement are described in
the following chapter.
In principle it should be possible to measure the luminous emittance of
radiating surfaces by placing an illumination photometer on the surface
to be measured, if it were not for the fact that reflection from the
comparison surface of the photometer increases the emittance (repeated
reflection). As a rule the luminous flux must be determined by means
of an integrator or from the light distribution, the emittance being then
arrived at by dividing the value of the luminous flux thus obtained by the
area of the radiating surface.
With near-uniformly diffuse radiating surfaces it may be sufficient to
measure the luminous intensity perpendicular to the surface, and to
calculate from this the luminance and multiply the result by n.
REFERENCES
1) R U 1 b rich t, E.T.Z. 21, 1900, 595-597. "Die Bestimmung der mittleren
raumlichen Lichtintensitat durch nur eine Messung"
R. U 1 b r i c h t : "D~s Kugelphotometer". Munich and Berlin 1920
2) H.- J. H e 1 wig: "Uber lichttechnische Messungen mit Hilfe von Hohlraumen". Thesis Berlin 1933. See also: das Licht 4, 1934, 115-118, 135-140 and
156-159. "Uber li<;:.httechnische Messungen mit Hilfe von Hohlraumen" and
5, 1935, 33-34. "Uber praktische Erfahrungen mit der neuen Messmethode
fur die Ulbrichtsche Kugel"
3) British Standard 354: 1961. Recommendations for photometric integrators
358
MEASUREMENT OF LUMINOUS FLUX
[XVII
DIN 5032. "Photometrische Bewertung und Messung von Lampen und Beleuchtung". Juli 1957
See also: W. E. K. Mid dIet on and C. L. Sanders, Ill. Eng. 48, 1953,
254-256. "An Improved Sphere Paint"
5 ) M. H. Gab r i e 1, C. F. Koenig and E. S. Steeb, Gen. El. Rev. 54,
1951, no. 9, 30-37 and no. 10, 23-29. "Photometry". This article gives a survey
of the use of the integrator for measuring luminous flux of lamps.
6 ) P. J. 0 ran j e,
Ph. T. R. 5, 1940, 166-170. "Technical Photometry of GasDischarge Lamps"
7) A. H. van Gorcum and B. v.d. Waal, Electrotechniek, 41, 1963, (335-342),
"Automatische gloeilampenfotometer met digitale registratie'' (in Dutch) (Eng.:
Automatic incandescent lamps photometer with digital recording)
The same authors: "Automatic photometer with digital recording for checking
quality of incandescent lamps", Control, 11, 1967, 158-162 and 236-240.
8) H. J. J. van Boort and D. Kolkman, Philips Techn. Review, 19, (1957 /58)
333-337, "The double-flux "TL"-lamp, a fluoresent lamp of high output per unit
length".
9) I.E. C. Publication 82. "Recommendations for ballasts for fluorescent lamps".
10) J. A . .M. van Lie m p t and J. A.~ de V r i end, Rec. Trav. Chim. Pays
Bas 52, 1933, 163. "Studien tiber die Explosion von Schwefelkohlenstoff-Stickstoffmonoxyd -Gemischen''
4)
CHAPTER XVIII
MEASUREMENT OF ILLUMINATION
XVIII-1. Introduction
Illumination is measured either in order to know the illumination itself,
i.e. to obtain a value of this quantity expressed in lux or footcandles,
or as an intermediate stage in ascertaining values of other photometric
quantities, in which case it is usually sufficient to know the illumination
as a relative value only. In the foregoing sections on the measurement
of luminous flux and luminous intensity, examples of this procedure
have already been given.
In principle there is no difference between the one kind of measurement
of illumination and the other, but the distinction has to be made seeing
that special instruments have been designed for use where the objective
is the measurement of the illumination, these being adapted to the
necessity of measuring this quantity in lighting installations.
Portable illumination photometers, from which a direct reading of the
illumination in lux or footcandles can be taken, need not, in general,
be highly accurate, but reliable and compact in size; the cost should
also be reasonable, so that this need not form an obstacle to the wide
use of these very handy meters among lighting contractors, architects
and so on.
Apart from such "pocket" illumination photometers, more accurate
meters are also obtainable which are larger in size and can be used down
to lower levels of illumination. Illumination photometers can of course
be employed for other measurements as well, but the question then
arises as to whether the instrument is accurate enough for the purpose
envisaged.
Naturally, illumination can be measured either visually or physically.
In visual meters the illumination to be measured must once again be
converted to luminance, but this can be done quite simply by introducing
a diffuse reflecting or translucent screen in the plane in which the
measurement is to be taken. This screen serves as comparison surface
for the photometer.
With physical illumination photometers the field of the photometer
360
MEASUREMENT OF ILLUMINATION
[XVIII
(photo-electric cell) is in itself light-sensitive and it is only necessary
to place the cell at the point where the illumination is to be measured.
XVIII-2. Measurement by means of laboratory photometers
In our example of a luminous intensity measurement (Fig. 137) it has
already been seen that the measurement is in fact one of illumination.
In that case the measurement gives the luminous intensity as the product
Ed 2 , although the numerical value of E does not appear in the ultimate
calculation.
All conventional visual photometers which work with a comparison
surface are suitable for the measurement of illumination; if an absolute
value is required the photometer must of course be calibrated, this being
done the most simply on the photometer bench with the aid of standard
lamps of known luminous inten.sity. The value of the illumination is
obtained from the inverse square law E = I jd 2 when a value in lux
or footcandles is desired.
After what has been said in the previous section and in Chapter XV
regarding photo-electric cells there is no need here to enlarge on the
use of physical photometers. Calibration is effected in the same way
as for visual photometers.
XVIII-3. Visual illumination photometers
When illuminating engineering ultimately arrived at the stage in its
growth where it was evident that illumination levels and uniformity
of illumination were important factors for effective seeing conditions,
the need arose for simple, portable and not too costly instruments
capable of measuring the illumination produced by installations reasonably accurately.
At that time photo-electric cells were not known, and visual illumination
photometers were accordingly designed and marketed in a variety
of types.
Because of the lower requirements to be met by such instruments
compared with those imposed on laboratory equipment it was not
necessary to employ costly optical devices such as the Lummer-Brodhun
prism, nor would it have been possible to reconcile such designs with
the need for cheapness.
Again, from the point of view of size and weight and also the price, large
voltmeters or ammeters could not be employed in the instruments, and
accuracy was accordingly not so high.
Since the importance of such illumination photometers has declined
XVIII-3]
VISUAL ILLUMINATION PHOTOMETERS
361
owing to the appearance of physical meters, only one specimen of the
former will be described; this is the Osram luxmeter 1), which was one of
the last designs to be produced.
A sketch of this instrument is shown in Fig. 201, partly cut away to
reveal the interior.
14
12
2-f¥WWi~~~~
9--ff--------"'......"""1
7
5
3
4
If
Fig. 201. A visual illumination photometer (Osram luxmeter)
The receptor for the light on test is section l of an opal glass sphere
2 which projects slightly above the top cover of the instrument. This
arrangement ensures that only a small error is involved when the light
enters at glancing incidence (this being more or less the same device
that is used for barrier-layer cells (Fig. 174). The sphere works on the
same lines as the Ulbricht sphere and functions as comparison surface
for the photometer.
A second opal glass sphere 3, which is illuminated by the comparison
lamp 4, serves as comparison field. The photometer field comprises a
glass plate 5 mounted at an angle of 45° between the spheres; this plate
is divided into narrow bands which are alternately silvered and clear.
To balance the luminance the observer looks through a window 6 in the
cover plate, towards this glass plate. He then sees the non-silvered bands
with sphere 3 as background, and, reflected in the silvered bands the
luminance of sphere 2.
Seen thus close together, the alternate bands of luminance from the
two sources greatly facilitate observation and ensure accuracy of balance.
The variation of luminance is obtained by rotating the cylinder 7,
the wall of which is cut away obliquely. On rotation, the wall of the
362
MEASUREMENT OF ILLUMINATION
[XVIII
cylinder thus cuts off a larger or smaller portion of the aperture 8, through
which the light from the comparison lamp passes. The cylinder is adjusted by means of a knurled flange 9 attached to it and projecting
through the case of the instrument.
The scale, calibrated in lux, is read through a window in the cover (10).
A pocket lamp battery II supplies the lamp, and the current is adjusted
to the correct value by means of a rheostat (operated by knob 12),
as read from an ammeter 13.
This illumination photometer has two measuring ranges, one of which
runs from 0.4 to 400 lux. The other is obtained by sliding a filter having
an attenuation factor of I 0 in front of the comparison lamp; the maximum
illumination that can then be measured is thus increased to 4000 lux.
XVIII-4. Physical illumination photometers
Visual illumination photometers have now been rendered almost completely obsolete by the introduction of the photo-voltaic cell which
permits of designs that do not need any external sources of voltage
and which can be used by any layman in photometry. The photoemissive cells manufactured before the appearance of barrier-layer
cells did not lend themselves well to portable illumination photometers,
seeing that they require an auxiliary voltage; only one model appears
to have been marketed, and this has been superseded by the barrierlayer cell type.
The introduction of physical illumination photometers brought with
it not only the advantage of simplicity, but also a further advantage
over the visual type. The latter were of course designed only for direct
comparison, but when gas discharge lamps with their coloured light made
their appearance measurements by direct comparison became difficult
and unreliable. By employing correction factors
or by .placing correcting filters in front of the
<>ells even the layman will have no difficulty in
taking reliable measurements of coloured light.
For technical measurements in the illumination
range from 50 to 1000 lux (5-100 fc) commercial illumination photometers are available
in a size of about 3" x 2" x If' complete with
cell and meter, these being typical pocket instruments (see Fig. 202).
Fig. 202. Pocket size
In this type of instrument the cell is usually physical
illumination
photometer
recess-mounted and is often · protected from
XVIII-4]
PHYSICAL ILLUMINATION PHOTOMETERS
363
damage by a glass window. This system involves considerable risk of
the instrument being wrongly used, since even at fairly steep directions
of incidence the meter begins to depart from the cosine law, this becoming the more marked as the direction is made more oblique. An
example of this has already been given in Section XV-3 (Figs. 172
and 173). 'The use of these illumination photometers for modern lighting
installations with their numerous large size light sources (particularly
fluorescent lamps) may well result in errors.
To illustrate this point we have calculated the error in the case of the
two cells whose angular response is shown in Fig. 172, in respect of
measurement under two different conditions of illumination. For the
one example we have assumed an infinitely large, uniformly-diffusely
radiating surface, and for the other an infinitely long linear source,
also giving uniformly diffuse radiation. It is further assumed that the
luminance of each source is uniform at all points.
We assume also that the cells (Fig. 172) are calibrated with light entering with perpendicular incidence; with the aid of the curves shown
in the figure it is then possible to compute the error made when measurements are taken of illumination produced by the sources on a surface
parallel to them. The illumination produced by a part of the source
radiating towards the cell at an angle of, say, 40° appears to be evaluated
4% too low by cell 1 and 36% too low by cell 2. To find the total error
in the reading from the cell in respect of the total illumination we must
first calculate the relative illumination values Ea. produced in the cell
for every value of the angle ex; added together, these elements will give
the total illumination.
The illumination elements for all the angles of incidence must be multiplied by the evaluation factor to be derived from Fig. 172 to give the
cell-reading for each such element; these values added together then
give the cell-reading for the total illumination.
It can be computed that for the diffuse surface, Ea. is proportional
to sin ex cos ex, and, with a linear light source, to cos 2 ex.
The proof of this is to be found in the derivation of equations (IX-4)
and (IX-17), respectively, by means of the infinitesimal calculus.
The next stage is to effect a graphical integration of the expressions
E = J sin a cos rxda and E = J cos 2 ada. If we denote the evaluation
factor of the cell by Pa. and the cell-reading by A, the second summation
must be the graphical integration of the equation A = f Pry, sin a cos rxda
and A = J Pa. cos 2 a.da. respectively. Division of A by E then gives the
error in the cell-reading.
364
[XVIII
MEASUREMENT OF ILLUMINATION
The elements of the illu- E«
mination and cell-readings fo.SJ--t--+-~~t--+---+---1
can be added by plotting Ea.
and A a. as a function of ex
(Figs. 203a and b); in these o,3l----t---,~-+-+--+~&-....lftl.:--~-l
figures the curves 1 relate to
Ea. and curves 2 and 3 to 0 •
cells 1 and 2, respectively.
The sum of the values of
Ea. and Aa. is obtained from
the areas enclosed between
!l
the curves and the abscissae.
I
The total areas indicate the
1
values of the infinitely large I
~ '-..
or long light source, i.e. for o.8
fcc
a= goo.
-
~
~
"
_.-I
In the case of light sources q 5
of finite dimensions, if the 0
~ ft
point.of measurement lies on
1((3
the central perpendicular to q "'
~~
the source, the error in the 0
0° 10° 20° 30° 40° 50° 50'" 70° 80° 90°
reading can be ascertained
-oc
by measuring the areas in
66966
Figs. 203a and b lying beFig. 203. Illumination elements Ea and
their evaluation by the barrier-layer cell
tween the ordinates of 0°
shown in Fig. 172 as a function of the
and ex, where ex is the angle
angle of incidence a. a) with uniformly
diffuse radiating surface, b) with uniformly
tween the rays emitted from
diffu~e linear so~rce. In a) E~ is proportional
the boundaries of the light
to sm a. cos a.; m b) to cos a.. The values
of sin a. cos a. and cos 1 0c are plotted a!'
sources and the normal at
ordinates
the measuring point.
The errors for ex= 45° and ex= goo have been computed along these
lines from Figs. 203a and b, and the results are given in the table on
page 365.
Although under practical conditions of illumination the situation will
not usually be such as to entail quite so much error as is shown in our
examples, these will nevertheless serve as a warning against the use
under certain conditions of barrier-layer cells having an angular response
which diverges too much from the cosine law.
If the cell is fitted with a flat correcting filter, oblique incidence of the
light introduces another source of error in that the path travelled by
'~
XVIII-4]
I
Cell I
Cell 2
365
PHYSICAL ILLUMINATION PHOTOMETERS
0°-45°
I
Circular
light source
\
Linear
light source
I%
<I%
0°-90°
4.5%
2%
0°-45°
5.5%
3%
0°-90°
24%
9%
the light is longer than with perpendicular incidence, with the result
that the absorption is increased.
Larger portable illumination photometers are also made, usually with
a separate cell connected to the meter by a flexible lead; these provide
for a lower measuring limit and greater accuracy. The better instruments
of this type can also be used for laboratory work; they can be obtained
with multi-range provision, achieved by connecting shunts across the
meter; some models contain two or even four cells in parallel in order
to secure a lower measuring threshold.
The lower measuring ranges are required not only for laboratory purposes, but also for measurements on street lighting.
Even though the cells in this type of photometer are not usually covered
with a glass, and have only a low rim which does not produce a screening
effect, care should be taken when they are used for lighting installations
comprising large size sources, or in street lighting. For measurements
on roadways at points some distance from the actual light source, i.e.
where the light enters at wide angles to the normal, the difficulty of
the angular response can be overcome by measuring the vertical instead
of the horizontal illumination, in planes perpendicular to l"!l.e road
axis. As mentioned in Section VI-4 the horizontal illumination can be
computed from this quite easily.
Owing to the impression of greater accuracy produced by the design of
these instruments (which is certainly justified when they are used
correctly) there is nevertheless a tendency to attribute rather too much
accuracy to the results obtained under all conditions.
The problem of correction of the cells for use in illumination photometers,
both with respect to the angular response and to the absorption by
filters without too much reduction in sensitivity has been dealt with
in detail by B u c k 2), who suggests the use of filters which are thicker
at the edges than in the centre.
In conclusion we repeat here the precautions mentioned in Section
366
[XVIII
MEASUREMENT OF ILLUMINATION
XV-3 to be observed in the use of barrier-layer cells with a view to
preserving the cells and avoiding faulty measurement:
a) Have the photometers re-calibrated at regular intervals;
b) Keep the cells away from damp as much as possible.
XVIII-5. Calibration of illumination photometers
Illumination photometers can be calibrated on the photometer bench
by means of luminous intensity standards.
If the calibration is to be precise it should
be effected for various distances. With
the physical illumination photometers this
G
is essential in order to lay down the deviaI
tion from linearity which the cells extk
I
hibit.
In pocket photometers it is usually sufficient
I
to check the scale at a number of points.
I
d
If many such photometers are to be
I
calibrated it is useful to employ a specially
I
constructed apparatus of the kind shown
in the sketch in Fig. 204. Here L is a
I
j LH
lamp below which a frosted glass disc G
,..1
is mounted. Diaphragms (B) of different
'1!11'
aperture can be inserted under the glass G
LJ
in order to vary the illumination on the
l
photometer LM, which is placed on a
66577
table with vertical adjustment so that
Fig. 204. Sketch of appathe distance d from the diaphragm to the
ratus used for calibrating
illumination photometers
cell will always be the same. Calibration
is done with the aid of an accurate illumination photometer calibrated with standard lamp on the photometer
bench.
(l)L
'/
REFERENCES
1)
2)
Li.u.La. 1932, 183-184. "Der neue Osram-Beleuchtungsmesser"
G. B. Buck, see 5) Chapter XV
CHAPTER XIX
MEASUREMENT OF LUMINANCE
XIX-L Direct visual measurement of luminance
Since luminance, among the photometric quantities, is the only one
that can be perceived as such by the eye, only luminance can be measured
directly by comparison with another luminance. We shall term this
method the direct visual measurement of luminance, in contrast with
the indirect methods in which luminance is measured with the aid of
another quantity.
Since photometric units are all based on the unit of luminous intensity,
i.e. the candela, the value of the standard luminance with which the
photometer has to be calibrated must be determined by measuring the
luminous intensity of a certain area and dividing this by the area itself,
unless the comparison be effected with the luminance of the black body
at the temperature on which 'the candela is based.
By means of such a standard luminance a photometer can be calibrated
by flashing one of the photometer fields with this luminance, the other
field obtaining its luminance from the comparison lamp. When the
measurement has been made the standard luminance is replaced by the
unknown luminance, photometric balance being then obtained between
this and the comparison lamp.
The photometer field can be flashed with the luminance to be measured
in various ways, viz.
1. The photometer can be so positioned in front of the surface X of
which the luminance is to be measured that the photometer field PF
is wholly flashed (Fig.
X
205). The surface
PF
under examination
then functions as com
surface for the
f~~;:::=~=~~~~~~~~~~5P parison
photometer. As will
Fig. 205. Direct visual luminance measurement by
means of a photometer. X = surface the luminance
of which is to be measured. PF = photometer field.
L = photometer lens. P = pupil of photometer
be seen from the figure, the surface under
examination must be
of a certain size if this
method is to be used.
368
MEASUREMENT OF LUMINANCE
[XIX
The size of the surface involved in the measurement can either be
computed from the geometrical-optical details of the photometer, or determined experimentally by placing a diffuse light source before the pupil P
of the eye-piece and measuring the diameter of the projected light
spot on the surface under examination. In many cases, however, this
area will not be sufficiently large, in which case it can be enlarged
artificially, by:
2. reproducing the surface under examination in the photometer with
a lens. The exit pupil of this lens, flashed with the light of which the
luminance is to be measured, then functions as comparison surface;
it must be large enough to ensure that the photometer field and the
pupil of the eye-piece are wholly flashed.
According to Abbe's law (Section Xl-2) the observed luminance is then
equal to the luminance of the surface under examination, apart from
losses in the optical system *).
The point at which the image of the surface under examination occurs
is immaterial and may therefore be such as will suit the circumstances,
provided that the photometer field and the pupil of the eye-piece are
completely flashed.
This has. to be verified, in every case, by noting the path of the rays
in the photometer and lens. This can be done experimentally by placing
a diffuse light source in front of the pupil, measuring the diameter of the
projected light spot at the exit pupil of the lens and then measuring
the diameter of the light spot projected in the surface under examination,
with the lens interposed.
Two examples of this method of measuring luminance are illustrated
in Fig. 206a and b. In Fig. 206a a part of the surface under examination
X is reproduced by the lens L 1 and the magnifier L 2 in the photometer, the image being formed in the pupil P of the eye-piece. P is
in the focal plane of L 2 , so that X is at the focus of L 1 • From the path
of the rays coming from the edge of P as bounded by the photometer
field PF the minimum size required for the exit pupil of L 1 will be seen.
This also shows what part of X actually takes part in the measurement.
The linear magnification is equal to the quotient of the focal length
of L 2 divided by that of L 1, and from this, together with the diameter
of the pupil, the size of the measured portion of X can be computed.
Fig. 206b depicts the same photometric arrangement, but in this case
a part of X is projected by L 1 onto PF. (Since X and L 1 are shown in
*) The measurement of the transmission factor of lenses is discussed in Section
XX-2.
XIX-1]
DIRECT VISUAL MEASUREMENT OF LUMINANCE
369
the same position in relation to PF as in Fig. 206a, the focal length of
L 1 is different from that in Fig. 206a(shorter)).
The diameter of the measured part of X is now determined by the
focal length of L 1 and the distance LcPF.
Which of the three methods described (or another in which an image
of X is produced at a different point) should be used, depends on various
factors.
In the first place this choice depends on the size of the part of the surface
PF
PF
Fig. 206. Visual luminance measurement based on an image of the surface for
measurement produced by a lens L 1 : a) in the pupil of the photometer. b) in the
photometer field PF. L 2 =photometer lens
under examination that is available for the measurement. In this respect
the method shown in Fig. 205 is the least favorable. The possibility of
employing one of the other methods depends on the available measuring
distance, the focal length and aperture of the lenses available, (for
greater magnifications the method of Fig. 206b requires a larger
aperture of L 1 ).
Another factor affecting the choice of method is the degree to which
small irregularities in the luminance of the surface under examination.
are observed. In the method not using projection (Fig. 205) each point
of the photometer field receives its light from a large part of X and,
in Fig. 206a, even from every point of the measured part of X. Small
irregularities in the luminance of X are thus not observed. With the
method shown in Fig. 206b each irregularity is reproduced at PF and
is therefore observed.
370
MEASUREMENT OF LUMINANCE
[XIX
Since the luminance is independent of the path of the rays it is not
generally necessary to employ corrected lenses; the use of uncorrected
lenses may even be recommended for smoothing out irregularities.
It may be added here that in the measurement of relatively large surfaces care should be taken that the lens L 1 is not placed too close to the
surface under examination; otherwise some error may be involved in
the measurement owing to the fact that light reflected by the lens L 1
increases the luminance of the surface. The error can be reduced by
screening off that part of the surface which is not actually included
in the measurement.
1 he optical pyrometer as luminance meter
Optical pyrometers are sometimes used for the measurement of luminance.
These instruments also come
within the category of direct x
8
r·
visual luminance meters and
Lt
~L La
should therefore be men- (:::.....::==:;~._~&l--"""""'=::·---=-t=:: A ______ !P
tioned in this section.
~
\[
Primarily, optical pyrometers are intended for Fig. 207. Optical pyrometer used as luminance
meter. An image of the surface of which the
measuring the temperature luminance
is to be measured is produced in the
of temperature radiators, plane of the filament of an incandescent lampCL
by means of a lens L 1 • The luminances of the
but we shall now discuss image
and filament are compared through a
lens L 2 and pupil P
the principle and use of this
instrument as a luminance
meter in reference to Fig. 207. The optical pyrometer as a means of
measuring temperatures does not fall within the scope of this book.
An image of the surface under examination X, possibly enlarged, is
produced by a lens L 1 in a plane B in the optical pyrometer. In this
plane there is the filament of a comparison lamp CL, the temperature
and hence the luminance of which can be adjusted by varying the current.
Via another lens L 2 , the filament and the image of X are observed
simultaneously by the eye, which is located behind the pupil P of the
eye-piece. The current flowing in CL is so adjusted that the filament
and the image of X appear to be equally bright, the filament thus
"disappearing". The instrument is calibrated with standard luminances,
from which a lamp-current versus luminance curve is plotted. At high
luminance levels, which have to be reduced in order to avoid difficulty
in measurement due to glare, a filter can be placed in the eye-piece.
If 2 selective filter is used, the instrument must be calibrated with light
!
XIX-1]
DIRECT VISUAL MEASUREMENT OF LUMINANCE
371
of the same spectral composition as that of the light to be measured.
In some cases the red filter incorporated in the pyrometer for temperature
measurements can be employed.
Since the luminances are balanced at the boundaries of the filament,
only a very small part of the area to be measured is actually involved
in the measur·:ment.
Portable luminance meters for field-work
Just as for the measurement of illumination, instruments which are
easily portable are made for the measurement of luminance and are
therefore suitable for use other than in laboratories. Whereas the visual
type of illumination photometer has been almost completely superseded
by physical instruments, this is not the case with luminance meters
since the lowest values that generally have to be measured produce
an illumination on the cell that is too small to be measured with simple,
robust portable microammeters. In Section XIX-2 a portable luminance
meter will be mentioned in which a photo-emissive cell is used.
In the course of time
numerous visual luL
minance meters have
been devdoped. These
can usually also be
used as illumination
photometers by placing a matt white
screen at the point
where the illumination
R
664J/9
is to be measured and
Fig. 208. Portable visual luminance meter designed
measuring the luminby L u c k i e s h and T a y l o r
ance of this screen
with the meter. Calibration can be effected by means of a known illumination.
Of the various portable luminance meters we shall mention only one
viz. that designed by L u c k i e s h and T a y 1 or 1) (Fig. 208).
An image of the surface of which the luminance is to be measured is
produced by a lens L. The photometer field comprises this image and
two small silvered rectangles (depicted in Fig. 208b). These rectangles
reflect the light from a diffusing glass in front of the comparison lamp.
The photometer field is observed through the pupil P 1 and the current
of the comparison lamp is adjusted by means of a rheostat R, as read
372
[XIX
MEASUREMENT OF LUMINANCE
from an ammeter AM. Photometric balance is obtained by rotating
a disc A, and the luminance is read from a scale inside the instrument,
through a pupil P 2 • As reducing filters can be introduced in the path
of the rays of the light to be measured, or from the comparison lamp,
and, as the lamp can be adjusted for two different current values, the
measuring range of the instrument is quite wide, viz. from about 0.1
cdfm2 (0.03 mL) to about 180,000 cdfm2 (50 L). The size of the field
of view of the photometer is not more than 0.1 °.
XIX-2. Visual and physical luminance measurements obtained from
measurement of illumination
Since in this method the luminance measurement is converted to a
measurement of illumination, and as it is immaterial whether this is
measured visually or physically, it will not be necessary for us to deal
with the visual and physical methods separately.
One such method has already been mentioned in the preceding section;
the luminous intensity
is measured (through the x
illumination on the com- III
j
j
parison surface) of a part [====A~======~~-=====~==~c
of the surface under exa- rII
J
L
mination the size of
which can be exactly
determined, the lumin- Fig. 209. Physical luminance measurement. The
aperture A in the end cover of the tube is filled
ance value being ob- with the luminance to be measured from X and
tained by dividing the functions as light source for the photo-electric
cell C. The screens S check the passage of stray
luminous intensity by the
light in the tube
area. In so doing it is
obviously necessary to take into account the rules relating to the ratios
of measuring distance and the diameter of the light source (the area to
be measured), and the size of the acceptance surface of the photometer.
This method is suitable for use with a simple luminance meter with
barrier-layer cell of the kind depicted in Fig. 209. The meter consists
of a tube with matt black interior. The front cover has in it an aperture
A which is presented to the surface under examination; the barrier-layer
cell C is mounted at the other end. Screens S can be inserted in the tube
to prevent stray light from falling on the cell.
For the measurement of primary sources, the tube can be placed against
the surface to be measured; in the case of surfaces which derive their
luminance from incident light the tube is placed at a distance from
XIX-2]
VISUAL AND PHYSICAL LUMINANCE MEASUREMENTS
373
the surface that will ensure that the incident light is not screened by
the tube to any appreciable extent. Needless to say, when this method
is employed, the area available for measurement must be fairly large.
The method which we shall now
X
PF describe is particularly suitable
when the surface available for the
L
_ _ _ _ _ ~ measurement is only small.
An image of the surface X to be
measured (Fig. 210) is projected
by a lens L onto the comparison
Fig. 210.
Luminance measurement
surface
PF of a visual photometer
derived from measurement of illumination. An image of a part of the or onto a photo-electric cell C. The
surface X whose luminance is to be
measured is produced on the comparison luminance L of X is then computed
surface PF or photo-electric cell C. from the resultant illumination in
L functions as light source for the
the following manner.
photometer
The luminance of the lens is TL
(T =transmission factor of the lens). LetS be the size of the exit pupil
of L; then L is a light source of which the luminous intensity is T LS cd.
The illumination E produced by L on the comparison surface for a
distance L-PF equal to d is then
TLS
d2
E=-.
E can be measured in accordance with one of the methods outlined in
Chapter XVIII, and L can then be computed from
Ed 2
L=-.
Example:
TS
Let the measured illumination E be 8.5 lux; d is found to be 2 m, ,. =
0.75 and S = 30 cm 2 ; then:
Ed 2
8.5 X 2 2
L = -:;s = 0.75 X 30 = 1. 51 sb.
The size of the measured portion of the surface can be obtained from
the focal length of the lens, the distance d and the diameter of the light
receptor.
Suppose that the focal length of the lens in the above example is 125 mm
and the diameter of the photo-voltaic cell used, 40 mm; the diameter
of the measured area is then
125
.
2000 x 40 = 2.5 mm.
374
MEASUREMENT OF LUMINANCE
[XIX
Conversely, the appropriate focal length and measuring distance can
be computed from the size of the surface available for measurement
and that of the light-receiving area. Here, a further determining factor
is the threshold of measurement of the photometer. With visual photometers, therefore, it must be borne in mind that the luminance of the
photometer field should not be less than the minimum luminance of
3 cdfm2.
With this method of measurement, too, precautions should be taken
that the luminance to be measuretl is not appreciably increased by light
reflected from the lens.
A physical luminance meter suitable for field-work and having attractive
features is described by Freund 2 ). It contains a lens (objective f: 1.9)
in focussing mount which forms an image of the region to be measured
on a reticule in the optical housing. A circle in the center of this plate
indicates the exact extent of the area being measured. This image
is viewed magnified through a telescopic eye-piece. A partial reflecting
mirror just ahead of the reticule plate reflects a portion of the light
downward to the photo tube. An opaque aperture plate is located at
the focus of this light beam directly above the photo tube. This plate
excludes all light except the portion of the image that corresponds to
that within the circle of the reticule. The light emerging from this aperture passes through a filter wheel and then onto the cathode of a photoemissive cell. The photo-current is amplified, and the output current
can be read from a logarithmic-response microammeter built into the
luminance meter. The microammeter is calibrated to read directly the
footlambert luminance of the surface measured. The scale is calibrated
from 1 to 100 footlamberts. Range multipliers of 0.1, 1, 10, 100, 1000
and 10 000 are available on the range switch. The measuring range runs
from 0.1 ftla (about 0.3 cdfm2 ) up to 106 ftla (about 350 sb). The acceptance angle of the meter is 1.5°.
The batteries needed for the anode voltage of the cell and for the amplifier are contained in the meter. For continuous use over long periods
the meter can be connected to an external power supply to replace the
batteries.
XIX-3. Measurement of luminance distribution
It may be necessary to determine distributions of luminance in cases
where the luminance over a given area is not constant, but differs
between one point and another. Measurements are then taken at a
number of points over the surface and curves are plotted to show the
XIX-3]
375
MEASUREMENT OF LUMINANCE DISTRIBUTION
distribution along a line, or in the form of iso-luminance curves. It is
important that in the measurements at the individual points the area
included shall be so small that the variation in luminance over that
area does not make the final result more inaccurate than is compatible
with the requirements. The measuring method should be adapted
accordingly.
In its execution the method will depend on the nature of the surface
concerned, i.e. its size, and its possibility of movement. The distribution
of luminance of the bulb of an electric lamp or lighting fitting, for
example, can be measured with the aid of a lens and an illumination
·3-C
Fig. 211. Diagram illustrating the measurement of the distribution of luminance
of a lamp bulbK, (or lighting fitting). An image of a part of the bulb is projected
by lens L onto the photo-electric cell C. The bulb is moved in the direction of the
axis and perpendicular thereto, a sharp image of every measuring point being
obtained
photometer by moving the bulb in a direction perpendicular to the
axis of the lens (see Fig. 211). After each such movement the apparatus
is carefully re-focused, for which purpose lines drawn just above or
below the measuring point will be found useful.
In the case of small light sources (e.g. the arc of a mercury vapour
lamp) an image of the whole discharge can be projected onto a screen,
and the cell of the photometer, provided if necessary with a slit-shaped
diaphragm, is then moved across the image. Obviously, all such measurements are of the kind usually taken in the laboratory.
The measurement of the luminance of street lighting systems presents
quite a different problem.
The aim of luminance measurements in street lighting is mainly twofold:
the determination of the average road-surface luminance (as a measure
of the lighting level) and of the local variations in that luminance (as a
measure of the patchiness). The average value, as well as the local variations in luminance are only of interest as seen by the motorist and on that
part of the road which is of importance to him for his observations
(about 150 to 600 ft in front of the driver). Local variations in the luminance can only be measured if the field of vision of the luminance meter
is small enough; the solid angle it subtends should certainly not be
376
MEASUREMENT OF LUMINANCE
[XIX
greater than that corresponding to the smallest object of importance to
the driver (about 3 X 3 minutes of arc).
It is an obvious idea to carry out the measurements of the local luminance
from one point, which is representative of the positions normally occupied
by the eyes of the driver. Experience gained with a luminance meter3)
designed for making local measurements of the luminance from a single
point has shown, however, that very high demands must be made not
only on the accuracy of the means of adjusting the instrument but also
on the skill of the observer if sufficiently reproducible results are to be
obtained.
One important aim ofthe local measurements is the determination of the
average luminance of the road surface by suitable averaging of the local
values. This should be done either so that the points under measurement
are so distributed over the road surface that each one takes up an equal
area in the perspective picture, or, if another distribution is chosen
(normally a simpler one and thus easier to plot on the plan of the road)
each local value should be weighted in proportion to the corresponding
area taken up in the perspective picture.
An instrument for the measurement of road surface luminances which
avoids the difficulties involved in measurement of local values from one
point, i.e. from long distances and which, moreover, enables the average
luminance to be measured directly is described by Asmussen and de
Boer4). The instrument is provided with two identical objectives (0 1 and
0 2 in Fig. 212). Both produce an image of the part of the road under
measurement in the plane of an exchangeable slide S. The lowest piut
of the slide contains a diaphragm D. The light falling on the photomultiplier M then comes exclusively from a part of the road of shape
determined by the contours of the diaphragm D. These contours are
reproduced in the form of thin black lines on a transparent window A
in the top half of the slide S (taking the relative positions of the lenses
0 1 and 0 2 with respect to the part of the road in question into consideration). By aiming the luminance meter so that the black lines coincide with
the edges of the image of the part of the road on which it is desired to
carry out the measurements, one ensures that only light from this parit
of the road falls on the photomultiplier M.
·
If a local luminance has to be measured, a slide with a rectangular aperture D and a corresponding rectangle on the window A is placed in the
instrument, which is then placed flat on the road about 10 ft from the
spot under measurement (see Fig. 213). An image of the rectangle on the
window A is then projected via objective 0 2 (Fig. 212) onto a small screen
XIX-3]
MEASUREMENT OF LUMINANCE DISTRIBUTION
377
placed in the middle of the area under measurement. The light for this
purpose is provided by the lamp L, which is turned into the optical axis
of the objective 0 2 .The image of the rectangle is adjusted with the aid of
two setting screws so that half of it is. visible of the screen. In this position
the photomultiplier only receives light coming from the rectangle shown
in Fig. 213 about 4 inches broad and 31 inches long (0.1 X 0.8 m). When
the distance between the screen and the meter is 10 ft, the optical axis
of the objective makes an angle of nearly 1o with the surface of the road .
.Fig. :ll4: Luminance meter as described by Asmussen and de Boer. 0 1 and 0 2 = objectives; S = slide with diaphragm D and window A; E = eyepiece; L = lamp for
setting the meter for the measurement of the luminance of points on the road surface;
B = push-button switch for the lamp; M = photomultiplier; C = calibration de.vice;
K = hood for screening against light-scatter.
378
MEASUREMENT OF LUMINANCE
[XIX
It is permissible to carry out measurements of the local luminance exclusively at this angle, even though the angle at which the motorist
observes the part of the road of interest to him varies from about 0.5 to
about 1.5°, since the influence of a variation in the angle of observation
in this range on the observed luminance is negligible.
For measurements of the average luminance, the instrument is mounted
vertically on a stand (Fig. 214) or held vertically in the hand (Fig. 215).
A slide is then introduced with a trapeziumshaped aperture in the
diaphragm D and a corresponding trapezium on the window A. The
lamp L, which is used to determine the area under measurement for
local luminance measurements, is now turned out of the optical axis of
the objective 0 2 .The observer can then see through the eyepiece E the
outline of the trapezium on the window projected on to the surface of the
road, and can adjust the luminance meter until this outline coincides
with the part of the road he is interested in. A large number ofslides with
trapeziums of different shapes are provided with the instrument. The two
horizontal lines always correspond to distances of 200 and 530 ft in front
of the instrument, when it is held at a height of 5 ft (60 and 160m, when
at a height of 1.5 m). The slanting side lines correspond to various widths
of road, increasing in steps of about 1,5 ft (0.5 m) for narrow roads and
about 3 ft (1 m) for wide roads if the road is observed from a height of
5 ft ( 1.5 m) in the middle of the right-hand lane (or in the middle of the
left-hand lane if the slide is reversed). If it is desired to make measurements on a strip of road of another length, the meter should be held at a
different height. The length of the strip under measurement and its
distance from the meter are proportional to the height of the meter.
Since it is impossible simultaneously to point the meter accurately in the
right direction and to read the meter deflection, the meter is provided
with a cable release, which is pressed at the moment when the meter is
properly adjusted; this fixes the needle on the scale, so that the observer
can take the reading at his leisure and accurately.
The use of slides with diaphragms D of different areas means that the
instrument has a different sensitivity with each different slide, and must
thus be recalibrated each time the slide is changed. This is done with
the aid of a built-in calibration unit C (see fig. 212), which contains a
number of seasoned calibration lamps behind an opal-glass window,
which ensures that the luminance is distributed evenly over the whole
area of the window. This calibrating unit can be pushed up with the aid
of a button so that the opal-glass window comes in front of the objective
XIX-3]
MEASUREMENT OF LUMINANCE DISTRIBUTION
379
0 1• The voltage of the lamps in the calibration unit is stabilized by the
voltage-stabilizing unit.
The calibration proper consists in changing the amplification factor of the
amplifier so that with the given slide the instrument shows the known
luminance of the calibration unit, multiplied by a calibration factor
dependent on the sort of light under investigation.
The photographic method
By the photographic method, luminance measurements are not taken
from the road surface directly, but from a photographic reproduction;
the measuring points can therefore be precisely laid down in the photograph. The latter is of course in perspective and if necessary a plan
view can be derived from it by transposition.
An example of the photographic luminance measurement of a road
surface may be found in an article by Bouma 5). In this case a
photograph of the road was taken from eye-level on a rapid panchromatic
plate, a horizontal strip of which was shielded. Later, in the laboratt>ry,
an exposure was made on this strip of a light box having at the front
a glass plate divided into a number of bands blackened to different
densities, whose luminance values were known. This exposure was made
with the same kind of light as that of the street lighting system and
with the same exposure time. Comparison of the density of the different
parts of the road in the photograph, with the bands of which the luminance is known, enables the luminance of every point on the road
to be ascertained. The actual method of measuring the densities is
explained in Section XX-2.
With this method it must be remembered that for a given luminance
of the object the illumination and therefore also the density of the
photograph decreases towards the edges of the plate. As shown in
Section XI-14 the illumination varies as the fourth power of the cosine
of the angle of divergence from the axis, provided no vignetting occurs.
If this does occur the decrease in density in the direction of the edges
is even greater. In methods of measurement such as the one described,
this decrease in illumination has to be determined experimentally
by placing in front of the camera a surface of uniform luminance and
measuring the relative illumination by means of a photo-electric cell
having a very small acceptance surface.
380
MEASUREMENT OF LUMINANCE
[XIX
XIX-4. Determination of the size of the exit pupil of lenses
For the measurement of luminance as derived from measurement of
the illumination produced by lenses it is necessary to know the size
of the exit pupil of the lenses. Let us now see how this can be done.
The exit pupil of single lenses and of objectives without internal stops
(Section XI-7; example a) are defined by the mount of the lens (or of
the first lens respectively), and can be determined by measurement.
It is more difficult to ascertain the exit pupil of objectivfS having a
stop between the component lenses (Section XI-7; examP.le b). In most
cases the exit pupil is then represented by the image of the stop as
produced by the first lens. The size (and if necessary also the relative
position) can be found experimentally by projecting an enlarged image
of it through an objective of which the exact focal length is known.
The magnification is computed from the object and image distances
and the focal length; measurement of the image and the magnification
then enables us to determine the size of the exit pupil.
REFERENCES
M. Luckie s h and A. H. T a y 1 or, J.O.S.A. 27, 1937, 132. "A Brightness
Meter"
2 ) K. Freund, Ill. Eng. 48, 1953, 524-526. "Design Characteristics of a PhotoElectric Brightness Meter"
3) J. B. de Boer, Rev. Gen. des Routes et des Aerodromes 280, 1955, 83-S4 "Un
luminancemetre pour l'eclairage public".
4) E. Asmussen andJ. B. de Boer, Public Lighting 27, 1962, 136-140, "A luminance
meter for street lighting".
S) P. J. Bouma, Ph. T. R. 4, 1939, 292-301. "Measurements carried out on
Road Lighting Systems already installed"
1)
CHAPTER XX
MEASUREMENT OF REFLECTION, TRANSMISSION AND
ABSORPTION
XX-1. Measurement of reflection factor
The reflection factor of a surface depends on the angle of incidence and
it is therefore not strictly correct to speak of the reflection factor, as
such, of a material or surface. Nevertheless this is usually done, the
reflection factor being taken to refer to those angles of incidence at
which it is constant or nearly so. In Chapter X we have seen that the
reflection factor of glass of refractive index 1.5 is practically constant
up to angles of incidence from 45° to 50°, and in the case of metals, to
almost glancing incidence.
Since it is only angles such as these that are of interest in illuminating
engineering, in general, the unqualified reflection factor is accordingly
sufficient for most purposes.
In the design of instruments for the measurement of reflection factors
we have usually a choice, therefore, of angles of incidence between
oo and ca. 45°.
Besides the reflection factor at given angles of incidence there is of
course also the reflection factor in respect of light incident from all
sides, i.e. the diffuse reflection factor. The measurement of this necessitates special apparatus and will accordingly have to be included in
our discussion.
Measurement of the reflection factor of light incident from one direction only
Let us first consider the measurement of reflection factors with light
incident at angles between oo and 45°.
Since the reflection factor e is the ratio of two luminous fluxes, we can
obviously employ the two methods of luminous flux measurement,
i.e. by using an integrator or by determining the light distribution.
In the case of purely regular reflection by a perfectly smooth (polished)
surface (metal or transparent glass), e can also be evaluated by measuring
firstly the luminous intensity of a light source and secondly that of
the image of the light source.
A measurement obtained from the light distribution with the aid of a
382
REFLECTION, TRANSMISSION, ABSORPTION
[XX
Rousseau diagram is of course only appropriate for diffuse reflecting
surfaces, and then only when the distribution (i.e. the luminous intensity
or luminance indicatrix) is also to be ascertained. Otherwise this method
of measurement is far too cumbersome and it is much quicker to use
an integrator. Measurement of the reflection indicatrix will be discussed
in section XX-4.
For the measurement of reflection factors using integrators many
methods are given in the literature. Most of them employ a standard
reflecting surface, with which the surface to be tested is compared.
These methods can be described on general lines in reference to Fig. 216.
KisanUlbricht
sphere. A beam of
light enters through
an opening 0 1 and
falls on the plate
to be tested which
is mounted at 0 2
in the sphere. To
ensure a sharply
defined beam of
light the following Fig. 216. Principle of apparatus for measuring reflection
factors with the aid of a standard reflecting surface
arrangement
can
be used. A lens L 1 (condenser) produces an image of the light source in
another lens L 2 which throws an image of L 1 on the test plate and accordingly produces a clear-cut light spot. A diaphragm D of any desired
shape can be placed in front of L 1 • If the light does not fall perpendicularly
onto the plate to be measured a circular light spot can be ensured by
using an elliptical diaphragm. In order to exclude stray light, screens
can be interposed between L 1 and L 21 and it is important that L 2 be as
clean as possible.
ScreenS ensures that no light reaches the sphere window P directly from
the plate to be measured. When the photometer reading has been taken,
the test plate is replaced by a standard plate and the unknown reflection factor is then easily computed from the readings, which are
directly proportional to the respective reflection factors.
Either a visual or a physical photometer may be used. The precautions
to be taken in connection with the spectral sensitivity when measuring
luminous flux (section XVII-4) must of course also be observed here.
The standard surface usually takes the form of a plate coated with
magnesium oxide by deposition, i.e. the oxide produced by the com-
Fig. 213. Setting of the luminance meter for the measurement of the local luminance for a
part of the road surface of I 0 X 80 mm 2 (the rectangle indicated by the black lines in the
centre of the adjusting slide).
Fig. 214. Measurement of the average luminance of a road surface by means of the luminance meter mounted on a stand.
XX-1]
MEASUREMENT OF REFLECTION FACTOR
383
bustion of a piece of magnesium wire or tape is. made to form a deposit
on the plate.
The reflection factor of magnesium oxide is usually given in the literature
as 0.95, but if any given plate is coated with MgO without any particular
precautions the ,..esultant layer of MgO will never yield a reflection
factor of 0.95.
"Letter circular" LC-547 1) of the Bureau of Standards contains a
specification for the preparation of a standard MgO-coated plate. In
this the most important feature is the method to be employed to deposit
the magnesium oxide in a very finely divided condition on the plate.
This results in a thick layer (about 0.5 mm) which is not liable to flake
off and has a high reflection factor; for a plate prepared in accordance
with the specification a
reflection factor of 0.97
should be obtainable.
Magnesium
carbonate
blocks, supplied with data
8 on their reflection factor
are available on the market for use as standard
plates. When the surface
of such a block has beX
come dirty, a fresh clean
surface can be produced
Fig. 217. The Taylor method of measuring
reflection factors without the aid of a standard
simply by grinding off the
reflecting surface
soiled layer.
Another method of measurement, which does not necessitate the use of a
standard plate and which is accordingly very suitable for general purposes
outside the laboratory, has been devised by A. H. Taylor2).
The test plate X is placed in an Ulbricht sphere K (Fig. 217) and is
iliuminated by a beam of light. A screen S prevents direct light from X
from falling on the window P. Let the illumination at the window be Ex.
The beam is then rotated about the axis AA' so that it strikes a part of the
sphere wall, the window not being screened in this case. If the reading be
now Ek, the reflection factor is found from
The correctness of this method can be shown in the following manner.
If the luminous flux of the beam is 1/J lm, the plate will reflect e.,<P lm,
384
[XX
REFLECTION, TRANSMISSION, ABSORPTION
this being the luminous flux which we wish to measure. Further, denoting
the area of the sphere by S and the reflection factor of the sphere wall
by ek, it will be seen that, as in the derivation of formula (XVII-I),
the illumination at the window is
When the beam strikes the sphere wall directly, ek<l> lm is reflected.
Since in this instance the luminous flux falling on the sphere window for
the first time is not screened, E k is not proportional to ~.but to
1-
(!k
Division of the expressions for E., and E k then gives e.,. With this method
the light sources, i.e. the plate to be measured and the illuminated
area of the sphere wall, are not situated in the same place, for which
reason it is preferable to arrange for the illuminated areas to be located
symmetrically with respect to the sphere window.
If the plate to be measured is quite flat and gives regular reflection the
reflected light will also produce a sharply defined light spot in the sphere,
and the best arrangement is one in which the position of this light spot
with respect to the window is the same as for the other light spots. This
can be done along the lines shown in Fig. 217, in which the specularly
reflected beam from X strikes the sphere wall at B.
Since the distribution of the luminous flux from X to be measured may
occur in many different forms (owing to the reflection being diffused
to a lesser or greater extent) it is recommended that the shape of the
integrator be such that the factor k mentioned in section XVII-4 varies
as little as possible with ek, i.e. it should be a true sphere.
Seeing that the measurement is dependent on the value of ek at a
particular point on the sphere wall and, in the case of flat specular
plates on two such small areas, care should be taken that ek is uniform,
in other words that the sphere is repainted at the first sign of soiling.
Although this method does not necessitate a standard plate it is advisable
to check the photometer from time to time with a standard plate. If the
plates to be tested are specular it is also good practice to take a check
XX-1]
MEASUREMENT OF REFLECTION FACTOR
385
occasionally with a specular standard plate (e.g. a plate of well-silvered
plate glass) which can be calibrated from time to time by means of
luminous intensity measurements.
Measurement of diffuse reflection factor
For the measurement of diffuse reflection factors the
sphere is again the appropriate apparatus.
The sphere can then at the same time
supply the diffuse illumination for the plate
to be measured. The arrangement may be
as shown in Fig. 218a, b or c.
In Fig. 218a the sphere is illuminated by an
incandescent lamp, so screened that no
light from it falls directly on the plate
to be measured X. In Fig. 218b a beam
of light is projected into the sphere.
Ulbricht
p
.£
Fig. 218. Three methods of illuminating surfaces
for the measurement of the diffuse reflection factor
In both systems the illumination on X is from all sides, but not exactly
with the same luminance in all directions. The third method (Fig. 218c)
is better in this respect; here the light from a sphere containing three
or four screened lamps passes into a second sphere in which the test
plate is mounted.
Many methods of measuring diffuse reflection factors are to be found
in the literature, but they differ from one another only in detail and
all have this much in common that, although the measurement is straightforward, calculation of the required value of e is rather complicated.
We shall now take one of these methods as an example, and any one
of the systems of illumination shown in Figs. 218a, b and c will suit
our purpose.
386
REFLECTION, TRANSMISSION, ABSORPTION
[XX
An aperture QR is made in the wall of the sphere K (Fig. 219) and
three measurements are taken at the window P in order to determine
the reflection factor, viz:
1) with QR uncovered, or covered with a box lined with black velvet.
Let the illumination of the window be E 0 •
2) with QR covered with a plate having the same reflection factor as
the sphere wall (ek),. Illumination of window Ek.
3) with QR covered with the plate to be measured (reflection factor e.,).
Illumination of window E.,.
Now if the areas of the circle QR (A) and of the remainder of the
sphere (Ak) are known, the required reflection
factor e., can be computed from the three
measurements by making use of the fact that
the combined luminous flux absorbed by the
circle QR and by the rest of the sphere is
equal to the luminous flux entering the sphere
(4>). Let us denote the absorption factor of
the sphere by ock ( = 1 - ek) and that of the
Q
66.507 R
plate to be measured by oc., (= 1 -e.,). With
QR uncovered, the absorption factor of QR (ot
Fig. 219. Principle of
apparatus for measuris obviously unity.
ing diffuse reflection
Using the above data we can compute the
factor
luminous fluxes incident on the sphere wall
and the open circle in each of the three examples. Multiplication by the
respective absorption factors then gives the amount of the absorbed
luminous flux and, by equating the sum in each case with tP we obtain
three equations containing the unknown e.,, ek and 4>, from which ex
can be evaluated.
The luminous flux falling on the sphere wall and in the opening is equal
to the product of the illumination and the area.
Because, according to IX-4, the luminous flux of a point on a sphere
that is completely diffusely reflective is distributed evenly over the
surface of the sphere, i.e. the illumination is the same everywhere,
the illumination at the sphere window is equal to that of the aperture,
no matter what the reflectivity of the surface arranged in the aperture.
This also applies when the surface in the aperture is not spherical but
plane.
In the first case, with QR uncovered, the illumination of the wall of the
sphere (in which the sphere window is located) is E 0 , as is that of the
circle. There is a luminous flux (1 - p) S · E 0 lm incident on the wall of
0)
XX-1]
MEASUREMENT OF REFLECTION FACTOR
387
thesphere,partofwhich,rx.~c(l-p) S ·Eo= (1-e~c) (1-p) S ·Eo lm,
is absorbed. That incident on the circle is p · S · E 0 lm, which is completely absorbed. The sum of these absorbed luminous fluxes is equal
to the luminous flux rt> falling into the sphere. Thus the first equation is
In the second case, where QR is covered with a plate of the same reflection
factor as that of the wall of the sphere., we find the luminous flux absorbed
by the wall of the sphere in a similar way to the first case, with
(1- l?k) (1- p) S · E1c lm. The circle absorbs (1- ek) p · S · E1c 1m. In
total, therefore, the flux absorbed is ( 1 - ek) S · E 1c 1m. This luminous
flux is once again equal tort>, and therefore the second equation reads:
E~c (1 -
rt> =
e~c)S.
in the third case (QR covered by the plate to be measured) the luminous
flux absorbed by the wall of the sphere is Ex (1- e~c) (1- p) S lm and
that absorbed by the circle Ex ( 1 - ea:) p · S lm. For the third equation
we then have
ex) p · S + Ex (1 - e~c) (1-p) S
= Ex · S [(1- ex)P + (1 - e~c) (1 - p)].
rt> = Ex (1 -
rt> and S are eliminated by equating the right-hand sides of the three
equations, and Sneed not therefore be known, but only p, i.e. the fraction
of the total surface of the surface of the sphere corresponding to the
circle QR.
Solving the three equations to Px gives
ex= Ex- Eo . -E~c- .
E~c -
Eo
Ea:
e~c
In the latter expression, p .need not be known if the reflection factor of
the wall of the sphere is known.
It is possible to circumvent the determination of p (with ek unknown)
by making a fourth measurement in which the circle is covered with a
plate of known reflection factor. This then gives a fourth equation, and
thus p can also be eliminated.
388
REFLECTION, TRANSMISSION, ABSORPTION
[XX
XX-2. Measurement of transmission factor
Transmitted light is that part of the incident flux that is not reflected
or absorbed. Since the reflection and absorption factors depend on the
angle at which the light strikes, the transmission factor is also dependent
on incidence and must be measured at certain angles.
When we speak of a transmission factor, without further qualification,
this generally refers to perpendicular incidence. Whereas the reflection
factor is nearly always found to be constant for angles of incidence
of from oo tb about 45° and we accordingly have a certain amount of
freedom in regard to the angle at which measurements are taken, no
such latitude exists in the case of transmission, because at oblique
angles the absorption factor tends to increase steeply owing to the
longer path travelled by the light through the medium. The higher
the transmissive exponent and consequently the absorption factor, the
more sharply the latter increases with oblique incidence and the more
the transmission factor varies with the angle.
The Ulbricht sphere can be used for the measurement of transmission
factors, the arrangement then being as shown in Fig. 220.
The sphere has an opening in it at A through which a beam of light
is projected. A screen S is placed in the sphere to prevent light from A
from reaching the window P. Let us
assume that the photometer reading
in respect of the light entering at
A is a0 •
The plate to be measured X is placed
before the opening A and this produces
a reading a.,.
In the same way as for the measurement of reflection by Taylor's method
we find that the transmission factor
66508
is given by the quotient a.,ja 0 •
The same arrangement will serve for Fig. 220. Principle of apparatus
for measuring the transmission
the measurement of transmission facfactor
tors for diffuse incident light, but of
course with a different form of illumination; the arrangement shown
in Fig. 218c may be used instead.
The transmission factor of transparent materials can be measured by
placing them in front of a surface of uniform luminance, the luminance
of this area being measured first without and then with the plate in
front of it. Care must be taken, however, that the plate to be measured
==(-s
XX-2]
MEASUREMENT OF TRANSMISSION FACTOR
389
is placed sufficiently far from the luminous surface to ensure that the
luminance is not appreciably increased by light reflected from the plate.
This method is suitable for either plane or curved objects and is therefore suitable for determining the transmission factor of lenses and
lens systems, e.g. objectives. In such cases the area of uniform luminance
should not be any larger than is strictly necessary for the purposes of the
measurement, seeing that due to reflection from the faces of the lenses light
emanating from beyond the essential area produces a certain luminanc~
in the direction of the optical axis; if included in the measurement, this
light will falsify the result obtained for the transmission factor.
The transmission factor of lenses and objectives can also be measured
by producing an image of a surface of uniform luminance· and measuring
the illumination of this image. The luminance of the objective can then be
measured in the manner described in section XIX-2; this luminance
is of course less than that of the projected area, and the transmission
factor is the quotient of the luminance of the objective divided by the
known luminance of the projected surface.
Measurement of density
Density, as measured with a densitometer, may be defined as in section
X-12 as the logarithm of the attenuation in the luminous flux from a
given illumination system falling on a light receptor after passing through
a photographic film or plate.
In view of what has been said in Chapter XI we may now say that in
many cases the word luminance can be used instead of luminous flux.
Densitometers are employed in sensitometry, that is to say for determining
the photo-sensitivity of photographic materials. The apparatus consists
of a source of illumination and a receptor, i.e. a photometer, and may be
of the visual or physical type. The most obvious method is of course
to take measurements with and without the plate, from which the
density is then found by simple calculation, but the great objection
to this is the wide difference that may occur between the luminous flux
values with and without the plate (for a density of 3 the ratio of the
fluxes or luminances is 1 : 1000). For this reason many densitometers
compare the density of the test plate with that of a neutral wedge
(see section XIV-3).
Other types of densitometer work on the principle of the polarisation
photometer.
390
REFLECTION, TRANSMISSION, ABSORPTION
[XX
As a general rule, the area of the plate or film available for sensitometric
measurements is fairly large (at least I sq. em) and neither the construction of the instruments nor the measurement present any problem.
The illuminating and acceptance solid angles should correspond to the
<;onditions under which the plate or film on test is normally used, but it
should be noted here that many of the meters available on the market
are somewhat lacking in this respect. In publications on the subject,
moreover, this point is generally given far too little prominence.
Data regarding densities as measured with apparatus of differing optical
design cannot be compared one with the other.
For the measurement of the intensity of spectral lines (spectrophotometry), as represented by lines on a photographic plate or film, microphotometers are used. Since the spectral lines as produced on the photographic plate are often very narrow, only very small areas are available
for the measurement, and the precision of the instrument has to be
very high indeed. These instruments are usually self-recording.
The same remarks apply to photographic photometry, for which purpose
microphotometers can also be used. The areas available for measurement
are in this case generally so large, however, that it is not really necessary
to use a microphotometer.
For the measurement of densities in connection with the making of
contact prints, where it is a question of measuring the total incident
and transmitted fluxes, the method described at the commencement
of this section is of course quite suitable, viz. measurement of the transmission factor by means of an integrator.
Details of the many types of densitometers and microphotometers would
be beyond the scope of this book; the reader is referred to the many
publications already available on these subjects 4 ).
XX-3. Measurement of absorption
In view of the fact that a. = I - (e + T), the absorption factor can be
at once obtained if p and -r have already been measured in accordance
with one of the methods indicated in the previous sections. Ifthe material
to be tested transmits no light ( -r = 0), measurement of the reflection
factor is sufficient; with materials that do transmit light the absorption
factor can also be found by effecting one measurement only.
The plate under examination X is suspended in the centre of an Ulbricht
sphere (Fig. 221) fitted with a screen S to prevent light from X from
reaching the sphere window P directly.
XX-4]
391
LUMINANCE FACTORS AND GLOSS
L
A
Fig. 221. Principle of apparatus
for measuring the absorption
factor
66509
A beam of light L is projected through the opening A onto the plate
inside the sphere, and a reading is taken. The plate is then removed
from the beam (but is left inside the sphere), so that the beam strikes
the wall of the sphere, and another reading is taken.
In the same way as was shown in section XX-1, on the basis of Taylor's method of determining the reflection factor, we now find that
the quotient of the readings is the absorption factor.
If the reflection from the test plate is wholly or partially specular, care
should be taken that the plate is not set exactly perpendicular to the
beam of light, as an appreciable portion of the reflected light would then
be lost through the opening A.
The transmissive exponent of transparent or translucent plates (section
X-15) can be computed from the absorption and reflection factors
together with the thickness of the plate.
In formula (X-14) Wu = W;e-ad, W; denotes the luminous flux entering
the plate, i.e. the incident light less the reflected flux. W; is thus proportional to ( 1 - e).
(/Ju represents the emergent flux and is proportional to 1 - (e + ex).
Formula (X-14) can thus be written:
1 - (e +ex) = (1- e) e-a4.
One of the methods outlined in the preceding sections can be employed
to determine ex and e, and d can be measured with a micrometer, thus
enabling a to be computed.
Similarly, in formula (X-l5) Wu = W; e-ad the value of a in respect to
diffuse media can be ascertained.
XX-4. Measurement of luminance factors and gloss
The apparatus required for measuring luminance factors must be capable
of illuminating the surface under examination from different directions
392
REFLECTION, TRANSMISSION, ABSORPTION
[XX
of incidence, and of providing means of measurement of the luminance
in different directions. Apparatus of this kind are known as goniaphotometers.
Fig. 222 shows the principle of this apparatus. X is the plate under
examination, L a beam of light capable
of being moved in an arc about an
axis at X, and P is a photometer,
similarly mounted on a pivoted arm.
If luminance factors are to be measured
beyond the plane in which L can be
revolved, the axis on which the arm
of P rotates must be capable of rotation
on an axis at right angles to the first
axis.
I
By means of such apparatus luminance
Fig. 222. Principle of the goniaphotometer
factors can be measured with either
reflected or transmitted light.
As we have already seen in section X-7 the luminance factor is the
quotient of luminance and illumination. The latter can be measured
in accordance with one of the methods described in Chapter XVIII.
For the measurement of the luminance we have the choice between
direct measurement on the one hand and, on the other, determination
from the luminous intensity, this being divided by the (apparent) area.
Either a light source without optical system, or a "parallel" beam of
light may be used for the illumination, but in any case care must be
taken that the part of the plate involved in the measurement is sufficiently uniformly illuminated. With direct illumination from the
source this means that the source has to be placed sufficiently far from
the plate. This also ensures that· the angles of incidence in respect of
the plate do nat vary too much.
When a "parallel" beam of light is employed the reduction of the illumination towards the edges of the beam should be borne in mind
(see section XI-6). It is advisable, after first making a rough calculation
of that part of the beam in which the illumination on the plate is practically uniform, to check the variation in the illumination with a photoelectric cell having a small acceptance surface (using a diaphragm if
necessary).
If the luminanca is to be measured directly, thus not by way of a
luminous intensity measurement, it is necessary to check by calculation
that the area required by the photometer for the purposes of the mea-
'
''
XX-4]
LUMINANCE FACTORS AND GLOSS
393
surement is fully provided by the illuminated part of the plate (section
XIX-I). This is particularly important with glancing directions of
incidence, since the apparent foreshortening of the plate is then considerable.
When a luminance measurement is carried out by actually measuring
the intensity of the plate in the case of
transmitted light, the radiating area can
be precisely limited by a diaphragm. For
glancing directions of incidence, however,
the diaphragm should be extremely thin,
66511
as otherwise a part of the plate will be
screened (see Fig. 223).
Fig. 223. Reduction in the
For the measurement of reflected light apparent area of a surface acoblique incidence
by way of the luminous intensity the area companying
through a diaphragm of too
great a thickness
should be limited by confining the beam,
in view of the fact that diaphragms
always reflect a certain amount of light which is then included in the
measurement as stray tight. In this, any lack of uniformity in the
illumination is of little importance, as will be seen from the following.
From section X-7 we see that the luminance factor is
L
{J=E
(the indices are omitted for the sake of simplicity).
In this, L = IJS cos IX and E = C/>jS, where I is the luminous intensity,
cJ> the incident luminous flux and S the illuminated area. For (X-7)
we can therefore write:
I
s cos
IX
I
fJ = -cp- = cJ> cos IX.
s
Care must be taken that the beam is properly limited so as to avoid
stray light, and this can be done by using a clean lens and placing stops
between the light source and the lens to prevent light reflected from
the inside of the housing from leaving the lens in oblique directions.
In all measurements of luminance factors it is necessary first to consider
whether the result will really yield a luminance factor (as a constant
of the material), whether the acceptance angle of the photometer is
394
REFLECTION, TRANSMISSION, ABSORPTION
[XX
sufficiently small for the measurement of the luminance factors, and
whether it is necessary to take measurements at various acceptance
angles to ascertain if it is possible to measure the luminance factor (see
considerations in section X-8 in which a method of varying the acceptance
angle is given).
Luminance factors of road surfaces
As mentioned in section X-10 the luminance factors of road surfaces
play a very important part in street lighting.
Measurements of luminance factors can be taken over a small portion
of the road surface by means of a goniophotometer, either in situ, or
from a sample of the surface in the laboratory.
Alternatively the luminance distribution of the road surface can be
measured as illuminated by a certain light source, in conjunction with
measurement of the distribution of illumination. From the two distributions the luminance factor can be computed for any point on the
road surface (i.e. direction of illumination).
The photographic method of luminance measurement is also often
employed for road surfaces, either as applied to samples 4), or with the
aid of the luminance and illumination distribution in respect to the
lighting system of the whole road 5).
A simple method of measurement to find the value p cos3tp has been
worked out in connection with the method of calculation given in
X-10 for luminances of street lighting with the aid of this value.
It is possible with the luminance meter described in XIX-3 to find the
value p cos3. tp both in the open, i.e. in the. street, and in the laboratory.
Measurements in the open permit examinations of street and road
surfaces in their natural surroundings under varying weather conditions,
while laboratory measurements are suitable rather for systematic research, e.g. into the effect of different grain sizes or minerals. The principle of measurement is the same in both cases. A fitting with a constant
luminous intensity on to the surface to be measured is arranged at a
constant height. The luminances at the individual points are noted for the
actual angles of incidence tp and the angle t5 between the plane of incidence of the light and the plane of observation.
L= p
I cos3 tp
= L (rn,
h2
.,. t>).
Since the height and luminous intensity of the light source used are
XX-4]
LUMINANCE FACTORS AND GLOSS
395
kept constant, the scale of the luminance ,meter can be linearly transformed into a p cos3 rp scale with the factor h2fJ.
h2
p cos3 rp = I
. L = f (rp, J).
Since the p cos3 rp values are to be used for luminance measurements
as in X-10, the measurement conditions must correspond to the observation conditions. Objects on the road viewed at an angle of some 3
minutes of arc are just clearly distinguishable. For the most important
observation distances of the vehicle driver, this corresponds to an average
road surface area of0.2 X 2.0 m2. Because of the astigmatic character and
discrete light distribution of a fitting, different luminances will be produced at various points on this area, and they will be reflected to the
observer with widely varying luminance coefficients within this area.
Since differences in luminance within this area can no longer be perceived
separately, it is logical to speak of this area as a "point" and of the ayerage
reflective properties of the area as "point" reflection.
The measuring arrangement in all measurements should be restricted
to the smallest possible space, and, in laboratory measurements, too,
the field of measurement should be made as small as possible by the
size of the sample, which is always limited. In order to fully embrace
the surface structure properties of road surfacing materials, a measuring
field of 0.05 X 0.5 m2 is quite adequate. If this measuring field is to be
adapted to natural observation conditions, the true scale should be
converted to this model scale. In practical lighting installations, where the
size of the measuring field may be 0.2 X 2.0 m2, the suspension heights
are some 10 m and the area of the fittings about 0.2 m2. In the case of
the model scale, therefore, of 0.05 X 0.5 m2, the height of the fittings is
about 2.5 m with an illuminating area of 0.013 m2. The p cos3 rp values
can be measured in this arrangement with a luminance meter. Here,
however, the measuring aperture on the sample side of the meter may not.
be greater than three minutes of arc. W1th these methods, the reflective
properties of road surfaces are determined with the same integration
limits as with visual observation.
Fig. 224a is a diagram of the measuring arrangement for finding the
p cos3 rp values in the open. The P cos3 rp values of the test surface F for
different positions X of the fitting S are measured at an angle of 1° to the
plane of the road and a measuring aperture of 3' with an objective
luminance meter M (e.g. the design described in XIX-3). The values
measured are shown graphically in the form of "EP diagrams" (see
396
[XX
REFLECTION, TRANSMISSION, ABSORPTION
a
F
G
..,..---\ ,.
_..-t
.--·
b
Fig. 224: a. Finding p cos3 q~ values in the open.
b. The measurement in the laboratory.
X-10). In laboratorymeasurement (Fig. 224b), the principle ofmeasurement is unaltered, but here the p cos3 <p values can be continuously
recorded by a writer, dependently on the angle of rotation c5 with the aid
of the turntable T or dependently on the angle of incidence of the light rp
with the aid of the movable fitting S on the rail G. Fig. 225 shows the
practical design of such a measuring arran{{ement.
XX-4]
LUMINANCE FACTORS AND GLOSS
397
Measurement of gloss
The design of glossmeters has been adapted as far as possible to the
definition of gloss as formulated by the designers. In m<J,ny cases, however,
they do not fully meet the requirements, i.e. the acceptance angle of the
photometer may not be suitable for measuring correctly the quantities
on which the definition is based. It will be clear from what has been
said in section X-8 that where very glossy surfaces are encountered the
acceptance angle has a very pronounced effect on the results of measurement, and that different instruments will yield different values for
the gloss of one and the same surface even using the same definition of
gloss.
Descriptions of all the existing types of glossmeters would take us too
far afield and it will be sufficient to refer the reader to the work by
Harrison, of which mention is made in section X-9' 6 ). Here we
shall describe briefly only a few types, which are made in various
executions.
l) Meters based on the 0 s t w a l d K l u g h a r t method
The sample under examination AA' is illuminated from an angle of
45° to the normal and the luminance is measured in the direction of the
normal (Fig. 226a). The plate is then rotated through an angle of 22.SO
(Fig. 226b), the directions of observation and illumination remaining as
before, so that the luminance is measured in the specular direction.
In variations of this method the plate may be turned so that the maximum
luminance is reached.
~
I
""
t;
;;; A ~'?m=m'?m=:m-,A'
Q
Fig. 226. Principle of gloss measurement by the 0 s t w a 1 d
method
K 1u g h a r t
From the luminance values (measured in different ways according to
different methods) figures are computed which are taken to be a measure
of the gloss.
398
REFLECTION, TRANSMISSION, ABSORPTION
[XX
2) Polarisation glossmeters
In glossmeters of this type (e.g. the Ingersoll Glarimeter) the
percentage of polarised light is determined that occurs in the reflected
light. Such meters are used chiefly for the measurement of the gloss
of paper.
In section X-3 it has already been pointed out that the light reflected
specularly by electrically non-conductive media (water, glass) is polarised. At a certain angle practically all the reflected light is polarised
in one plane. The light reflected from uniformly diffusely reflecting
surfaces is not polarised.
On the assumption that the light reflected by paper is in part reflected
specularly and is therefore polarised~ the rest being diffusely reflected
(unpolarised), the percentage of polarised light is measured at the angle
of maximum polarisation in one plane, and this is taken to be a measure
of the gloss.
Since metals in general do not polarise light on reflection, these polarisation gloss meters are not suitable for the measurement of metallic
surfaces.
3) Objective gloss meters
With this type of glossmeter the surface to be tested is illuminated by a
beam of light from a given direction, and the reflected light is measured
in the specular direction, this being compared with the specularly
reflected light from a standard surface, usually of polished black glass.
Light reflected in other directions than the specular is thus ignored.
The value obtained by measurement is sometimes referred to as the
objective gloss, and the instruments as objective glossmeters. The measurement gives no indication, however slight, of the form of the reflection
indicatrix.
4. Method ~I H a r r i s o n
In section X-9 a method of gloss assessment by means of a gloss number,
as introduced by Harrison, has been mentioned. For carry~ng
out the photometric measurements needed to arrive at the gloss number,
only a goniophotometer is required. Since the principle of this kind of
apparatus has already been discussed in the beginning of this section
we need not give further details as to the method of measurement and
we may refer to what has been said in section X-9 on the subject.
Fig. 215. Measurement of the average luminance of a road surface by means of the luminance meter held in the hand.
Fig. 225. Arrangement for measuring the reflective properties of samples of road surfaces.
XX-4]
LUMINANCE FACTORS AND GLOSS
399
REFERENCES
National Bureau of Standards, Washington, Letter Circular LC-547, March 17,
1939. "Preparation and Colorimetric Properties of a Ma!fnesium-Oxide Reflectance Standard"
2 ) A. H. T a y I or, Sci. Pap. Bur. of Stand. 17, 1922, 1-6, no. 405. "A Simple
Portable Instrument for the Absolute Measurement of Reflection and Transmission Factors"
A. H. T a y I or, J.O.S.A. 25, 1935, 51-56. "Errors in Reflectometry"
3) See for instance: G. A. B o u t r y, 7 ) Chapter X and R. S e w i g: "Handbuch der Lichttechnik", P. 359 et seq.
4) J. B e r g m a n s, see 2) Chapter X
S) J. M. W a 1 dram, Ill. Eng. 27, 1934, 305-313 and 339-351. "Road Surface
Reflection Characteristics and their Influence on Street Lighting Practice"
Also: G.E.C. Journal 6, 1935, 67-86
6) V. G. N. Harrison: "Definition and Measurement of Gloss" London 1945
1)
APPENDIX
INTERNATIONAL CO-OPERATION IN
ILLUMINATING ENGINEERING 1 )
Anyone who is accustomed to employing the basic photometric quantities
and their corresponding units in his daily activities will find it hard
to realise that there was a time when all such matters were shrouded
in complete darkness.
It was at the international Congress of Electricians held at Geneva in
1896 that the system of photometric concepts still in use today was
first laid down on the proposals of A. B 1 on de 1. Not long afterwards, in 1900, the gas experts, at an international meeting in Paris
agreed upon the necessity for more accurate methods of determining the
light-giving power of lighting gas.
Arising from this the International Photometric Commission was inaugurated and met at Zurich in 1903, 1907 and 1911. This commission
had the greatest interest in establishing an internationally recognised
unit of luminous intensity. At that time there were fairly considerable
differences among the various countries as to the conception "candle".
In order to obtain legal backing, even though in one country only, the
scientific co-operation of a state laboratory was essential; the PhysikalischTechnische Reichsanstalt at Charlottenburg, Germany, fulfilled this
mission with the Hefner-unit lamp, upon which the "Hefner candle"
(HK) was based. By 1907 the National Physical Laboratory at Teddington, England, the Bureau of Standards at Washington and the
Conservatoire des Arts et Metiers, Paris, had agreed upon a unit of
luminous intensity, which was fixed by means of a set of carefully
calibrated carbon filament lamps, at a value corresponding to 1.11 HK.
The fact that this unit was somewhat prematurely called the International Candle was quite naturally an obstacle to all endeavours over
a number of years to establish a really universal unit of luminous
intensity.
The "New Candle", based on the radiation of the black body at the
freezing point of platinum, was introduced on I st January 1948
by the Bureau International de Poids et Mesures; in the summer of
1948 the Commission lnternationale d'Eclairage (C.I.E.) suggested
for this unit the international name "candela" (pronounced candela;
abbrev. cd), and this has since been adopted in various countries.
404
APPENDIX
The Commission Internationale d'Eclairage (C.I.E.), (Eng.: International
Commission on Illumination, Germ.: Internationale Beleuchtungskommission) was established in 1913 out of the old International Photometric Commission, as it was realised that illuminating engineering
covered a very much wider scope than merely the photometric measurements and calculations which can be regarded as the backbone of
illuminating engineering.
The C.I.E. has set up committees to deal with the following subjects
and to report their findings to the General Assembly of the C.I.E.:
Basic Quantities
Vocabulary
Photometry
Colorimetry
Colour Rendering
Signal Colours
Photopic, Mesopic and Scotopic Vision
Visual Performance
Sources of Visible Radiation
Sources of UV and IR Radiation, and Measurement
Operating Accessories
Characteristics of Illuminating Engineering
Materials
Photometric Requirements for Luminaires
Pre-determination of Illumination and Luminance
Causes of Discomfort in Lighting
Agreable Luminous Environment
Home and Hotel Lighting
School and Office Lighting
Industrial Lighting
Mine Lighting
Lighting of Public Buildings
Hospital Lighting
Lighting for Selling
Lighting for Stage and Studio
Lighting for Indoor Games
Lighting in Hazardous and Corrosive Situations
INTERNATIONAL CO-OPERATION IN ILLUMINATING ENGINEERING
405
Daylight
Street Lighting
Aviation Ground Lighting
Railway and Docks Lighting
Airborne Lighting and Signals
Lighting for Outdoor Sports
Automobile Headlights and Signal Lights
Floodlighting and Advertising Signs
Fundamentals on Traffic Signals
Education in Schools, etc.
Popular Education
Lighting Codes, Regulations and Legislation
At present (1967) 38 countries take part in the activities of the C.I.E.
Each country acts as the centre (secretariat committee) for one or more
of the abovementioned subjects. This function includes the presentation
of a summary report on the progress made since the last meeting at each
meeting of the C.I.E. These meetings take place every three to four years.
Sessions of the commissions are arranged in the intermediate periods too
where necessary.
The C.I.E. communicates with several international organisations, a
few of which we should like to mention here.
The work mentioned <;~-t the beginning of this book on the new unit of
luminous intensity was urged as early as 1931, together with the Bureau
International des Poids et Mesures.
For several years now the C.I.E. has been working successfully on the
illumination of airfields for the International Civil Aviation Organisation
(I.C.A.O.).
Moreover, together with, for example, the International Organisation
for Standardisation (I.S.O.) it has been seeking an agreement or. the
Uniformisation of Automobile Lighting.
One of the achievements of the C.I.E. has been the plotting of the curves
of the international relative luminous efficiency of radiation for the lightadapted and the dark-adapted eye.
The system proposed· by the C.I.E. known as the trichromatic system of
colour classification and the four-language list of quantities and terms,
with definitions, used in lighting engineering are also well known.
The meetings of the C.I.E. (Paris 1921, Geneva 1924, Saranac (U.S.A.)
406
AP;I'ENDIX
1928, Cambridge 1931, Berlin 1935, Scheveningen 1939, Paris 1948,
Stockholm 1951, Zurich 1955, Brussels 1959, Vienna 1963 and Washington 1967) bring experts in lighting engineering from all over the world
together. They thus promote the exchange ofviews and provide a survey
of the state of science and technology which is recorded each time in a
bulky volume, the "Comptes Rend us".
REFERENCES
1)
N. A. Halbertsma, Trans. Ill. Eng. Soc. London 12, 1947,97-107. "International
relations in illuminating engineering".
Folder C. I.E.: "What it is, what it is trying to do, how it can be helpful in many
countries where it is not now represented", published in 1954 by the Central Office
of the C. I.E:.
407
TABLE I
TABLE I
Table of solid angles
The size of the conical solid angle cu with half-apex angle ex is expressed in steradians. The increment of cu for each increment of 5° in ex is also given.
(X
oo
50
100
15°
20°
25°
30°
35°
40°
45°
50°
55°
60°
65°
70°
75°
80°
85°
90°
(sterad)
"'
Increment
of cu
0
0.0239
0.0955
0.214
0.379
0.589
0.842
1.136
1.470
1.840
2.24
2.68
3.14 H
3.63
4.13
4.66
5.19
5.74
6.28 (2")
0.0329
0.0716
0.1186
0.165
0.210
0.253
0.294
0.334
0.370
0.404
0.435
0.462
0.486
0.506
0.523
0.535
0.543
0.548
0.548
(X
"'
(sterad)
95°
100°
105°
110°
115°
120°
125°
130°
135°
140°
145°
150°
155°
160°
165°
170°
175°
180°
I
6.83
7,37
7.91
8.43
8.94
9.42 (3 ..)
9.89
10.31
10.72
11.10
11.43
11.72
11.98
12.19
12.35
12.47
12.54
12.57 (4 ..)
Increment
of cu
0.543
0.535
0.523
0.506
0.486
0.462
0.435
0.404
0.370
0.334
0.294
0.253
0.210
0.165
0.1186
0.0716
0.0239
APPENDIX
408
TABLE II
Values of coss ex cos 3 fJ for a number of values of the angles ex
and fJ (see Fig. 227).
cos 3 ex
cos 3 fJI
Olno
II 1.001[)11 0.955
I 0.901 1 0.830
0.650
0.550
0.450
0.353
5° 11 0.989
0.944
1 0.891
1 0.820
0.642
1 0.543
1 0.444
0.350
10° 11 0.955
11 0.912
1 0.861
1 0.793
1 0.620
1 0.525
1 0.429
1 0.338
15° 11 0.910
11
0.861
1 0.812
1 0.748
1 0.585
1 0.495
1 0.405
1 0.319
20° 11 0.830
11 0.793
1 0.748
1 0.689
1 0.539
1 0.456
1 0.373
1 0.293
0.671
1 0.618
1 0.484
1 0.409
1 0.334
1 0.263
1 0.585
1 0.539
1 0.422
1 0.357
1 0.292
1 0.230
0.495
1 0.456
1 0.357
1 0.302
1 0.247
1 0.194
25° II 0.744
, _ __!tic____ _
30° 11 0.650
I
0.711
ij 0.621
~II 0.550 11 0.525
1
1
40° 11 0.450
11 0.429
1 0.405
1 0.373
1 0.292
1 0.247
1 0.202
1 0.159
45°
11 0.338
1 0.319
1 0.293
1 0.230
1 0.194
1 0.159
1 0.125
~ 0.254 1 0.239 1 0.220 1 0.173 1 0.146
1 0.119
1 0.0940
1 0.0849
1 0.0667
11
0.353
50° ·1/ 0.266
55°
~ 0.189 11 0.180 1 0.170 1 0.157
1 0.123
1 0.104
~II 0.125 11 0.119 1 0.113 1 0.104 1 0.0812 1 0.0687 1 0.0562 1 0.0442
0.075511 0.0721 1 0.0680 1 0.0626 1 0.0490 1 0.0415
65°
11
70°
~ 0.0400 11 0.0382 1 0.0361
1
0.0332 1 0.0260 1 0.0220
1 0.0339
1 0.0267
1 0.0180
1 0.0141
75° 11 0.017311 0.0166 1 0.0156 1 0.0144 1 0.0113 1 0.009541 0.007791 0.00613
TABLE II
50°
I
55°
I
60°
65°
I
70°
1 72.5°
409
I
75°
1 76.5°
1 78.5°
80°
0.00524
0.266
1 0.189
0.125
0.0755
0.0400
0.0272
0.0173
0.0128
0.00792
0.263
1 0.187
1 0.124
1 0.0746
1 0.0396
1 0.0269
1 0.0171
0.0126
1 0.007831 0.00518
10.254
1 0.180
1 0.119
1 0.0721
1
0.0382
1 0.0260
1 0.0166
1 0.0122
1 0.007571 0.00500
0.239
1 0.170
1 0.113
II o.o680 -~ o.o361 1 0.0245 1 0.0156 1 0.0115 1 0.007141 0.00472
0.220
II o.157
1 0.104
1 0.0626
1 0.0332
1 0.0226
1 0.0144
1 0.0106
0.198
1 0.140
1 0.0931
1 0.0562
1 0.0298
1 0.0202
1 0.0129
1 0.009471 0.00590 1 0.00390
0.173
1 0.123
1 0.0812
1 0.0490
1 0.0260
1 0.0177
1 0.0113
1 0.008261 0.005151 0.00340
0.146
1 0.104
1 0.0687
1 0.0415
1 0.0220
1 0.0149
1 0.009531 0.006991 0.00436
0.119
1 0.0849
1 0.0562
1 0.0339
1 0.0180
1 0.0122
1 0.007791 0.005721 0.00356
0.0939
1 0.0667
1 0.0442
1 0.0267
1 0.0141
1 0.00961 1 0.006131 0.00450
1 0.0501
1 0.0332
1 0.0200
1 0.0106
1 0.007221 0.00460 1 0.00338
0.0236
1 0.0142
1 0.007551 0.005131 0.003271 0.00240
!
--
0.0705
1 0.006581 0.00434
L
a
..
0.0501
0.0332
0.0200
1
0.0356
1
{J
-- 1 0.0236
1 0.0156
,
Ip
4_ ___ - -
1 0.009441 0.00500 1 0.00340 1 0.00217
8
1 0.0142
1 0.009441 0.00570 1 0.003021 0.002051 0.00131
----------Fig. 227
0.0106
1 0.007551 0.00500 1 0.003021 0.00160 1 0.001091 0.000694
0.00460 1 0.003271 0.002171 0.00131 1 0.0006941 0.0004711 0.000301
p
17888
BP =h tan{J
cos a:
I
E = .2 cos 3 a: coss f3
"'
h2
410
APPENDIX
TABLE III
tanp
Values of - - for a number of values of the angles IX and p (see Fig. 214, page391)
COS IX
IX
p
oo
oo
II
·~ I
p
0
I
100
1
1 0.984
0
I
0
I
15°
I
0.966
I
0
20°
I
30°
0.866
0.940
35°
40°
45°
0.819
0.766
0.707
0
0
0
I
0
I
0
I
0.893
I
0.969
I
1.06
I
1.15
I
I
1.56
1.45
I
I
1-.86
1.74
I
I
2.26
2.11
I
I
2.62
I 2.80
I
I 3.35 I 3.59
4.87
4.56
I
I
I
0.124
0.114
0.107
0.101
0.0931
0.0906
50 I 0.0875 0.08751 0.0888
I
I
I
I
I
I
0.249
0.230
0.215
0.204
0.188
0.183
10°11 0.176 11 0.176 1 0.179
I
I
I
I
I
I
0.379
0.350
0.327
0.309
0.285
0.277
150 0.268 11 0.268 1 0.272
I
I
I
I
I
I
0.515
0.475
0.444
0.420
0.387
0.377
20°11 0.364
0.364 1 0.370
I
I
I
I
I
I
0.659
0.609
0:569
0.538
0.496
0.483
25°11 0.466
0.466 1 0.473
I
I
I
I
I
I
0.816
0.754
0.667
0.705
0.614
0.598
0.586
300 I 0.577
0.577
I
I
I
II
I
0.990
0.914
0.855
0.809
0.745
0.725
0.700 1 0.711
35°11 0.700
I
I
I
I
I
I
I
11
II
11
11
l
1
11
11
400 II 0.839
11
450 II 1.00
111.00
500 II 1.19
111.1 9
I
550 II 1.43
1\1.43
I 1.45
60°11 1.73
111. 73
I 1.76
65°11 2.14
~ 2.14 I 2.18
70°11 2.75
112.75
1
3.73
13.73
I
75°11
0.839
1
I
0.852
1.02
1.21
2.79
3.79
I
I
0.869
1.04
1.23
I
I
1.27
1.38
I
I
I
1.65
1.52
1.48
I
I
I
1.84
1.79
I 2.00
I
I
2.48
2.28
2.22
I
I
I
3.17
2.92
2.84
I
I
I
4.31
3.97
3.86
I
I
I
1.02
1.22
I
I
1.09
1.31
I
I
1.19
1.41
I
1.69
I
2.02
2.45
I
3.03
I
3.89
I
5.28
I
411
TABLE III
50°
55°
6QO
65°
70°
72.5°
75°
76.5°
78.5°
0.643
0.574
0.500
0.422
0.342
0.301
0.259
0.233
L199
I
80°
I
0.174
I~ I 0 I
I
I
I
I
0.375
0.136
I 0.439 I 0.504
I 0.153 I 0.175 I 0.207 I 0.256 I 0.291 I 0.338 I
0.755
0.274
I 0.884 I 1.02
I 0.307 I 0.353 I 0.417 I 0.516 I 0.586 I 0.681 I
0
i
I
0.417
0.566
I
I
I
0
0
0
0.467
l I
0.634
0.635
0.725
I
0.813
0.898
I
1.01
1.09
I
1.46
I
1.74
I
2.08
I
1.31
1.56
1.85
1.22
I
0.536
I
I
I
0.728
I
0.933
I
1.37
I
1.66
I
1.99
I
1.15
1.40
I
1.68
I
2.00
I
2.38
I
0.861
1.10
I
I
2.37
2.82
0
0
I
I
0.783
1.06
1.36
I
1.69
I
2.05
I
2.45
I
2.92
I
3.48
I
4.18
I
I
I
I
0.891
1.21
1.55
I
I
I
1.04
1.41
1.80
2.49
2.86
3.38
I
1.15
1.56
2.00
3.96
4.75
2.47
2.23
1.92
I
4.60
I
I
I
I
3.46
4.10
3.02
2.69
I 5.06 I 5.76 I 6.69
I
I
I
5.07
4.29
3.74
3.34
I
I 6.27 I 7.13 I 8.29
I
I
5.49
9.14
8.03
6.50
4.79
10.6
4.27
I
I
I
I
I
I
14.4
7.46
12.4
6.51
10.9
8.83
5.81
I
I
I
2.22
I
I
I
I
I
3.00
2.71
2.33
I
I
I
2.79
I 3.24 I 3.59
I
3.86
3.33
I 4.28
I
I
I
0
0
5.52
I
5.10
6.12
I
I
1.34
1.83
2.34
I
2.90
I
I
3.51
4.21
I
I
I
1.54
2.10
2.69
3.32
412
APPENDIX
TABLE IV
Units of illumination and their mutual conversion factors
I
---·
Lux= lx
Millilux = mlx
Phot =ph
I
II
I
F oat-candle = fc
II
lx
1
I0-3
I
I
I
mlx
103
1
104
I
107
10.76
I
10760
I
I
I
I
ph
I0-4
I0-7
1
11.076 X w-3
I
I
I
I
I
fc
9.29 X I0-2
9.29 X w-s
929
1
Fig. 228.
The relations between the various units of luminance. The units are plotted
on a logarithmic scale
I
footlambert =
equivalent footcandle =
apparent footcandle =
ftL or ftla
mL or mla
lj
3.43 X 10-4
I
II
I
II 3.18 X 10-4 1
0.318
Lambert = L or la
3.43
3.18
3183
I
I
0.318
0.296
296
I
I
12.21 X I0-3
2.05
1
I
I
I 2.05 X I0-3 1
1
I 2.96 X IQ- 2 12.05 X IQ- 4 1
I
0.318
113.18x 10-s1
Apostilb = asb
144
I
I
1550
0.155
16.94 X IQ-3 1
I
II
1
cdjin. 2
I
I 9.29 X IQ- 2 1 6.45 X IQ- 4 1
6.45
1
I
I
I
929
I
cdjin. 2
104
I
cdjft2
10.76
I
I
1
IQ-4
I
nt
111.076 X IQ- 3 1
II
II
I
sb
cdjft2
Nit = cdjm 2 = nt
Stilb = cdjcm2 = sb
TABLE V
10.76
10
104
1
4870
33.8
3.14
31400
asb
I
I
I
I
I
1.076 X I0- 3
IQ-3
1
IQ-4
0.487
IQ- 3
I
I
I
I
I
1
1.076
1
103
0.1
487
3.38
3140
I 3.38 X
I
I
mL
0.314
3.14
L
13.14xi0- 4 1
I
I
The units of luminance and their mutual conversion factors
452
3.14
0.292
2920
I
I
1
0.929
929
19.29 X IQ-2
I
I
I
I
I
ftL
I
t"'
~
V :l
-
<
t<l
~t:J:I
414
APPENDIX
TABLE VI
International Relative Luminous Efficiency of Radiation for Photopic Vision ( V~)
A
VA
A
A
(m,u)
v~
(m,u)
v~
390
395
0.0000
0.0001
0.0001
0.0002
525
530
535
540
545
0.7932
0.8620
0.9149
0.9540
0.9802
675
680
685
690
695
0.0232
0.0170
0.0119
0.0082
0.0057
400
405
410
415
420
0.0004
0.0006
0.0012
0.0022
0.0040
550
555
560
565
570
0.9950
1
0.9950
0.9786
0.9520
700
705
710
715
720
0.0041
0.0029
0.0021
0.0015
0.0010
425
430
435
440
445
0.0073
0.0116
0.0168
0.0230
0.0298
575
580
585
590
595
0·9154
0.8700
0.8162
0.7570
0.6949
725
730
735
740
745
0.0007
0.0005
0.0004
0.0003
0.0002
450
455
460
465
470
0.0380
0.0480
0.0600
0.0739
0.0910
600
605
610
615
620
0.6310
0.5668
0.5030
0.4412
0.3810
750
755
760
765
770
0.0001
0.0001
0.0001
0.0000
0.0000
475
480
485
490
495
0.1126
0.1390
0.1693
0.2080
0.2586
625
630
635
640
645
0.3210
0.2650
0.2170
0.1750
0.1382
775
0.0000
500
505
510
515
520
0.3230
0.4072
0.5030
0.6082
0.7100
650
655
660
665
670
0.1070
0.0816
0.0610
0.0446
0.0320
(m,u)
380
38~
415
TABLE VII
TABLE VII
International Relative Luminous Efficiency of Radiation for Scotopic Vision
(Young Eye) (VA')
VA' is equal to unity at 507 ffil£
).
(mp)
VA'
).
(mp)
VA'
).
(mp,)
I
VA'
I
0.0022
0.0045
525
530
535
540
545
0.880
0.8ll
0.733
0.650
0.564
400
405
410
415
420
0.0093
0.0185
0.0348
0.0604
0.0966
550
555
560
565
570
0.481
0.402
0.3288
0.2639
0.2076
425
430
435
440
445
0.1436
0.1998
0.2625
0.3281
0.3931
575
580
585
590
595
0.1602
0.1212
0.0899
0.0655
0.0469
450
455
460
465
470
0.455
0.513
0.567
0.620
0.676
600
505
610
615
620
0.0331
0.0231
0.0159
0.0109
0.0074
475
480
485
490
495
0.734
0.793
0.851
0.904
0.949
625
630
635
640
645
0.0050
0.0033
0.0022
0.0015
0.0010
500
505
510
515
520
0.982
0.998
0.997
0.975
0.935
650
655
660
665
670
0.0007
0.0005
0.0003
0.0002
0.0001
380
385
390
395
0.0006
O.OOII
675
680
685
690
695
0.0001
0.0001
0.0001
0.0000
0.0000
Luminous intensity
Luminous emittance
Luminance
Illumination
lP.
I. = w
lP.
M.=s
1.
lP.
Lo=---=
Scosa cosa·S·w
lP.
E.= S
Radiant intensity
Radiant emittance
Radiance
Irradiance
Luminous flux
Quantity of light
fPe =
~· = J lPoA d).
Q. = lP•. t
Radiant flux
Quantity of radiation
Km
f lPeA · VA · d).
lP
E=-g
I
lP
L=--=
Scosa cosa·S·w
lP
H=s
=
?
lP
I=w
lP =
Q=lP·t
TABLE VIII
The radiation quantities and the corresponding quantities in illuminating engineering
~
t:l
z
~
'1:1
t>l
a>
~
-
LIGHT DISTRIBUTION AND ZONAL LUMINOUS FLUX DIAGRAMS
41 7
LIGHT DISTRIBUTION AND ZONAL LUMINOUS FLUX DIAGRAMS
OF A NUMBER OF LIGHTING FITTINGS
The fittings are classified in accordance with the classification of the
C. I.E. (page 58).
f/J = total luminous flux of the fitting
f/J0 = luminous flux in upper hemisphere
f/J 0 = luminous flux in lower hemisphere
Fittings for Incandescent lamps
'lb
100
80
60
4()
J
:JO
1/
v
17111
0 0. /
0
60
0
120
0
White enamelled metal fitting
(direct)
f/J 0 = f/J
Fig. A
%
100
80
60
I
4()
20
0 ./
oo
v
v
1710
60
0
1200
Street lighting fitting; steel plate reflector,
inside white vitreous enamelled
(direct)
<Po=f/J
Fig. B
418
APPENDIX
%
TOO
80
60
/
40
20
0 ./
0
/
/
0
60 0
1200
White enamelled fitting, louvred
With bowl-silvered lamv
(direct)
(/> 0 = (/>
Fig. C
'*'
100
80
........
60
40
2
0
oo
v
v
/
&7153
Street lighting fitting; steel plate reflector white
enamelled, with opal glass ring
(direct)
= 0.91 (/>
(/> 0
Fig. D
,00
0,1,
80
/
40
0
0~
-
......... ~
60
..v
j
&765&
0
--"----=
180°
White enamelled metal fitting; open at the
upper side for illumination of the ceiling
(semi-direct)
0.85 (/>
(/> 0 =
Fig. E
419
LIGHT DISTRIBUTION AND ZONAL LUMINOUS FLUX DIAGRAMS
-
OAJ
100
80
60
4()
:lO
0 0
0
v
v
v
/
!--
v
...-
60 0
67&57
1800
120
Fitting with glass bowl; upper part opal
glass, bottom frosted clear glass
(semi-direct)
(/) 0 = 0.71 (/)
Fig. F
-
%
100
v
80
60
4()
:JO
0
o•
1--
/
v
/
6
67858
6(1.
Fitting with opal glass bowl
(semi-direct)
= 0.68 (/)
(/) 0
Fig. G
%
100
80
60
4()
y
0
:JO
-·
0
!7
./
-·
60
v
v
-·
1--
/800
6
67660
Fitting with glass bowl; upper part opal glas
with concave insert of frosted clear glass
(general diffuse)
(/) 0 = 0.57 (/)
Fig. H
420
APPENDIX
'II>
100
80
60
v
40
:;o
oo"
1.----
v~-""
v
L--
6
17111
I:Jo"
Fitting with opal glass bowl
(general diffuse)
f/> 0 = 0.55 f/J
100-
Fig. I
"
BO
60
40
10
0
so•
o•
tBO•
110"
Fitting of translucent plastic material; open
at the upper side
(semi-indirect)
f/>0 = 0.77 tP
Fig. J
'Yo
100
BO
v~"'
60
20
0
0
v 1--
J....-
_g
60
l/
I/
120
g
Fitting with reflector of translucent plastic
material, open at the upper side, louvred at
the bottom
(semi-indirect)
f/>0 = 0.75 f/J
Fig. K
421
LIGHT DISTRIBUTION AND ZONAL LUMINOUS FLUX DIAGRAMS
100
16)
v
~·
0
0
60
v
/,.,
/
17117
Metal fitting, closed at the bottom, open at the
upper side
(indirect)
t[>c = if>
Fig. L
F£ttings for two tubular fluorescent lamps
- - - Light distribution perpendicular to the axis of the fitting
- - - - - - - Light distribution in the vertical plane through the axis
180°15fl
0
o/o
100
1'-fl
900 80
60
I
40
100
600
20
0
v
j
....,~
00
60"
1:JO"
18
Trough fitting, reflector white enamelled, ends
open
{direct)
if> 0 = tP
300
Fig. M
422
APPENDIX
OJb
100
80
60
20
../
v
e
......
/
40
)
60
120 0
0
Fitting of Fig. M louvred
(direct)
(/J c =
(/J
Fig.. N
%
100
80
60 f----
_........v
I/
1/
~
,.... I
~-
1--
j
Fitting of Fig. M with apertures at the top,
ends open
(direct)
= 0.9 (/J
(/J c
Fig. 0
LIGHT DISTRIBUTION AND ZONAL LUMINOUS FLUX DIAGRAMS
ao
42 3
%
tOO
-- --
60
40
20
/
0
a•
v
e
//
'/
60°
t20°
Fitting of Fig. 0 louvred
tao•
(semi -direct)
= 0.87 ([>
([> 0
Fig. P
tao•
"
t5IJO
100 r-r-T""T""T"""T"-r-r-r-r...,.--r--,-,-,--,-,-,
ao·-
cd
60
40
hrrr.~++++++~~,hH
20hr~r+++++++~~~H
~~++++++~~goo
QWLLL~~~~~~~~
o•
60"
t2o•
tao•
60"
White, diffusing metal fitting with cover of opal plastic material
(direct)
q,o = q,
Fig. Q
424
APPENDIX
%
toor-T""'..-.-~~~~~,....-,..-,-,--,
cd
40
mm~m:tttttw
20~++~~~~~1-+-~+4
so•
t2o•
teo•
so•
Fitting with white ceiling plate and cover of opal plastic material
(semi-direct)
(/) CJ = 0.85 (/)
Fig. R
%
toor-T""'~~~~~~,....-,..-,-,....,
80~++++~+4+4~1-t-~H
so ttt=ttttttittm:$:1:~
40• ~++~+-J......,.~I-t-1-+-+-++-l
~~++~~~~1-t-t--+-+-++-l
0~~~~~~-LLLLL~
o•
so•
t2o•
teo•
Fitting with white ceiling plate and cover of corrugated clear plastic
material
(semi-direct)
(/)CJ = 0.8 (/)
Fig. S
LIGHT DISTRIBUTION AND ZONAL LUMINOUS FLUX DIAGRAMS
425
%
too
lUI
60
AO
20
!JO•
0
o•
60"
120"
teo•
Fitting with metal side-screens, open at the top, louvred
(general diffuse)
fP 0 = 0.55 fP
Fig. T
'Yo
100
-,
80
1200 60
40
20
k".l:=-lt"+-+-t---1-+-4 900
tJ
00
_v /
v'"'"
v
1.---
!..--
~
Fitting with side-screens of opalescent plastic
material, open at the top, louvred
(general diffuse)
(/> 0 = 0.50 (/>
Fig. U
INDEX
Where not otherwise indicated, the numbers refer to sections
. .XI-2
Abbe's law.
X-1, 15
Absorption
Measurement ofXX-3
. X-1
-factor . . . . .
Acceptance angle (of a photometer) . X-7
XII-2
Adaption . . . .
. X-5
Albedo . . . . .
XV-6
Algebraic method .
1-2
Amplitude . . . .
.XI-6
Angle of divergence .
1-2
Angstrom unit . . .
Angular response of photo-voltaic
XV-3
cells. . . . . . . . . .
. XI-8
Aperture of optical systems
Apparent foot-candle . . . XIII-7, X-6
Apparent surface (of a light source)
. VIII-2
. . . . . . . . . . .
. VIII-7
Apostilb . . . . . . . . .
. XVII-4
Auxiliary lamp of Helwig .
. VIII-I
Average luminance . . . .
Average luminous intensity . . III-3, IV-7
. V-6
Azimuthal projection .
Cone-vision . . . .
Continuous spectrum
Contrast prism . . .
Contrast sensitivity .
Corpuscular theory .
Correction factors of photo-cells
Critical angle of reflection
Cylindrical mirrors . . . . . .
XII-7
1-6
XIV-2
Xll-9
1-1
XV-6
. X-3
Xl-11
XV-3
Barrier-layer photo-cells
XV-3
Barrier-layer photo-effect
Beam cross-over point . . . XI-6, XVI--4
Bechstein flicker photometer . . . XIV-4
III-2, XII-6
Black body
XV-4
Bolometers. . . . .
. XIV-2
Bouguer photometer
. VIII-I
Brightness . . . . .
XII-I, 2
Subjective-. . .
XIV-2
Bunsen grease-spot photometer .
XV-2
Dark current .
. III-2
Dekalumen
XX-2
Densitometers
Density . . .
X-12, XV-5
XX-2
Measurement ofDepreciation factor .
. . VII-2
Diascope . . . . .
. . Xl-13
Diascopic projection
.XI-13, 14
Diffuse cylinder
VIII-4
luminous flux of. X-5
Diffuse reflection . .
. X-5
Diffuse reflection factor
.X-11
Diffuse transmission .
Diffuser
. X-6
Perfect - . . . .
VIII-2, X-5
. . .
Uniform VIII-3
Luminous flux of uniform - .
. X-13
Diffusion factor . . . . .
XII-3
Direct-comparison method . . .
IV-11
Direct lighting . . . . . . . .
Direct visual measurement of lumiXIX-I
nance . . . . . . .
XIV-3
Double-refracting prism
Xl-10
Drum lens . . . . . .
Calibration of illumination photoXVIII-5
meters . . . .
. X-12
Callier coefficient
. X-12
Callier effect . .
. 111-2
Candela . . . .
XI-10
Candelas per em or in. of height.
VIII-I
Candelas per sq. in. .
. 111-1
Candle . . . . .
. 111-2
. . .
Hefner . 111-2
International . 111-1
Candle-power
XV-2
Ceasium cell . . .
IX-3, 4, 8
Circular light sources
IV-II
Classification of lighting fittings.
Vll-2
Coefficient of utilization
1-2
Cold light . . . . .
XIV-1
Comparison lamp . .
XIV-I
Comparison surface.
Xl-14
Condenser . . . . .
1-5
Edison's lamp . . . . . .
. IV-4
Efficiency of lighting fittings
1-2
Electromagnetic . . . . .
1-1
Electromagnetic theory of light
Emittance (see luminous emittance)
. XI-5
Entrance pupil
. X-10
E.P. diagram.
Xl-13
Epidiascope .
. XI-13, 19
Episcope. . .
. XI-13, 19
Episcopic projection.
. . XIV-2
Equality-of-luminance prism
Equilux spheres of McAllister . . . IX-4
VIII-7, X-6
Equivalent foot-candle.
. . Xll-8
Equivalent luminance
1-1
Ether . .
XI--4, 5, 7
Exit pupil
X-12, XV-5
Exposure
. . . XIV-I
Eye-piece
INDEX
Facetted mirrors . . . . .
XI-I2
Filters
absorption of coloured. X-I6
Filter method . . . . . XIV-5, XV-6
Flicker photometer (Bechstein) . . XIV-4
Flicker photometry .
XII-3, XIV-4
Fluorescent lamps
I-6
Measurement of. XVII-5
f-number
. XI-8
Foot-candle .
. VI-I
Foot-lambert.
VIII-7
Fovea centralis
XII-7
Fresnel bi-prism
XIV-2
Frequency . .
I-2
Front-wall cell
XV-3
Full radiator .
Ill-2, XII-6
Gas-discharge lamps
Gas-filled cells . . .
Geometrical method
Gloss . . . . . . .
Objective - . . .
Measurement ofGloss number . . .
Goniophotometer . .
Grease-spot photometer
I-6
XV-3
XV-6
. X-9
XX-4
XX-4
. X-9
XX-4
XIV-2
Harmonic vibration .
I-2
Heat rays . . . . .
I-2
. III-2
Hefner candle . . .
Helwig (auxiliary lamp of-) .
.XVII-4
Hemispherical candle-power .
. . III-3
Heterochromatic photometry
. . . . . . . . . . . XII-3, XIV-5
Homochromatic photometry
. . . . . . . . . . . XII-3, XIV-5
Horizontal illumination . . .
. Vl-4
Horizontal luminous intensity
. III-4
Illuminating engineering
I-5
Ch. VI
Illumination . . . .
. VI-4
Horizontal Measurement ofCh. XVIII
Solid of-.
VII-6
Vertical-.
. VI-4
Illumination with oblique incidence . . . . . . . . . .
. VI-3
Illumination diagrams . . . . .
VII-4
Illumination diagram (rectangular) VII-5
Illumination photometers
XVIII-3, 4
XVIII-5
Calibration of- - .
Physical - . .
XVIII-4
Visual-- . . .
XVIII-3
Incandescent gas light
I-5
Incandescent lamps .
1-5
lndicatrix of diffusion
X-5
Indirect lighting
IV-11
Infra-red radiation .
I-2
Integrating photometer (see integrator)
427
Integrator . . . . . . . XVII-I, 2, 3, 4
. XVII-4
Approximation to ideal Ideal - . . . . .
. XVII-2
Non-ideal . . . .
. XVII-3
Internal photo-effect
XV-3
International candle .
. III-2
International luminosity factors.
XII-4
International relative luminous efficiency of radiation
Photopic vision .
XII-4
XII-7
Scotopic vision .
Inverse square Jaw
Vl-2, IX-I
Iris . . . . . . .
XII-2
.X-IO
Iso-/1-diagram . .
. V-2
Isocandela diagram
!so-luminance diagram
.X-IO
Isolux diagram
XII-7
Plane- . . . . . .
Polar- . . . . . .
VII-8
in quasi central projection
VII-IO
I-4
Lambert . . . . . . . . .
Photometric system of- . . I-4, Ch. XII
Lambert (unit of luminance)
VIII-7
Lambert photometer
XIV-2
XIII-2
Lambert's Jaw . .
Law of reciprocity
. IX-3
Light
Cold- . . . .
I-2
Light distribution .
Ch. IV
Polar - - curve .
. IV-I
Solid o f - - .
. IV-I
- - apparatus
XVI-3
Light ether . . . .
I-I
Lighting fittings
I-5
Classification of- - .
IV-11
Efficiency of - . IV-4
Light quanta . .
I-I
Light source
Circular-IX-3, 4, 8
Linear-- .
IX-9, IO
. Ch. IX
Non-point - Point-.IX-I, 6
Standard - - .
. XIII-2
Light-watt . . . .
. XII-6
Linear light source
IX-9, IO
1-6
Line spectrum .
. III-2
Lumen
Lumen method .
VII-2
Lumen-hour .
. III-6
. III-6
Lumen-second
Luminance . .
Ch. VIII, IX-I
Average-.
. VIII-I
Equivalent . . . . XII-8
Measurement ofCh. XIX
Measurement of- distribution . XIX-3
Measurement of - from illumination measurement . . . . . XIX-2
Measurement of - of road surfaces . . . . . . . . . . . XIX-3
428
INDEX
Luminance
. X-2,XI-3
-of images
. . . . IX-5
- of light beams
. X-2, XI-3
-of optics . .
. . XII-2
Luminance criterion .
. . X-5, 7, 8
Luminance factor . .
XX-4
Measurement of- - .
Measurement of - - of road
XX-4
surfaces . . . . . .
. X-5
Luminance indicatrix . .
XIX-I
Luminance meters (portable)
XII-I
Luminosity . . . . . .
XII-4
International - factors
111-5
Luminous efficiency. . .
XII-6
- - of radiation . . .
XII-4
Relative - - of radiation .
Vlll-5
Luminous emittance . . .
.XVII-7
Measurement of- - .
III-I
Luminous flux . . . . .
Ch. XVII
Measurement of- - .
XI-18
- - from projection systems
.XI-8
- - of optical systems . . .
- - of uniform diffuse cylinder VIII-4
VIII-3
- - of uniform diffusers
111-1, IX-I
Luminous intensity
Average-- . . . . . 111-3, IV-7
111-4
Horizontal - - . . . .
.111-3
Mean hemispherical-111-4
Spherical - - . . . . .
. IV-1
Polar curve of- - . . .
. X-5
Luminous influsity indicatrix
XIV-2
Lummer-Brodhun prism .
. VI-I
Lux . .
XII-2
Luxon . . . .
0
0
0
0
0
XII-7
Macula lutea.
Martens polarisation photometer XIV-4
XVI-5
Maxwellian view . . . . . . .
McAllister (Equilux spheres of-) . . IX-4
Mean lower and upper hemispheric111-3
al candle-power .
Measurement of
XX-3
absorption .
XX-2
density
. XVII-7
emittance
. XIX-4
exit pupil
. XX-4
gloss
. Ch. XVIII
illumination
Ch. XIX
luminance .
XIX-3
luminance distribution .
. XX-4
luminance factors. . .
. XIX-3
luminance of road surfaces .
. XVII-8
luminous emittance . . . .
Ch. XVII
luminous flux . . . . . .
luminous flux of fluorescent lamps
. XVII-5
and their fittings
. Ch. XVI
luminous intensity
XV-2, 3
photo-current
. XVI-4
projectors . . .
. XVII-7
quantity of light
0
Measurement of
reflection factor . . . . . .
XX-I
transmission factor . . . .
XX-2
Mechanical equivalent of light
XII-6
MgO-coated standard plate of reflection .
XX-I
Millilambert . . . . . . . . . . VIII-7
Millilux . . . . . . . . . . . . . Vl-5
Min. perceptible luminance differXII-9
ence . . . . .
Mirrors
Xl-11
Cylindrical FacettedXI-I2
Luminance of.XI-3
. X-5
Mixed reflection .
Mixed transmission
.X-I4
XV-4
Moll thermo-pile .
1-2
Monochromatic radiation
.X-14
Multiple reflection
XII-9
Munsell scale of luminances
Natural light .
New candle .
Neutral filter .
Neutral wedge
Nicol . . . .
Nit . . . . .
Non-point sources
1-3
. IV-2
XIV-3
XIV-3
XIV-3
VIII-I
Ch. IX
.XI-7
Objective . . . .
XX-4
Objective gloss . .
XX-4
--meters . .
XIX-I
Optical pyrometer
. XI-I
Optical system . .
Optical systems for image-projecXI-I3
tion.
.XI-I
Optics . . . . . . .
Parallel beam of light
Perfect diffuser
Periodic time .
Phase . . . .
Phot
Photo-cathode
Photo-cell
Barrier-layer - .
CaesiumCorrection factors for - .
Front-wall Gasfilled. . . .
Photo-voltaic Potassium - .
Rear-wallSeleniumVacuumPhoto-effect
Barrier-layer - .
ExternalInternal- . . .
.XI-6
X-6
1-2
1-2
. VI-5
XV-2
XV-3
XV-2
XV-6
XV-3
XV-2
XV-3
XV-2
XV-3
XV-3
XV-2
XV-3
XV-2
XV-3
429
INDEX
Photometer
Bechstein flicker
XIV-4
BouguerXIV-2
Bunsen- . .
XIV-2
Gonio. . . .
XX-4
Grease-spotXIV-2
Illumination - .
XVIII-3
LambertXIV-2
Martens polarisationXIV-4
Rumford- . . . .
XIV-2
Weber- . . . . .
XIV-4
with Maxwellian view
XVI-4
Photometer bench
XIV-4, XVI-2
Photometer field . .
XIV-I, 2
XIV-I
Photometer head . .
Photometric balance
. . . . XIV-I
Photometric system (of Lambert)
1-4, Ch. XII
. . . . .
Photometry
. . . XII-3
FlickerHeterochromatic - .
XII-3, XIV-5
XII-3, XIV-5
Homochromatic . XV-5
Photographic Physical-.
. Ch.XV
Visual.Ch. XIV
1-1
Photon . . . . .
Photo-multiplier cell
XV-2
XII-7
Photopic vision . . .
Photo-voltaic cell . .
XV-3
Physical illumination photometer XVIII-4
Physical photometry
Ch.XV
Planck's formula . .
XII-6
Plane isolux diagram
VII-7
1-3
Plane of polarisation
Point-by-point method
VII-3
Point source . . . . .
. IX-1, 6
Polar curve of luminous intensity
. IV-I
Polarisation . . . . .
I-3
Polarisation photometer
XIV-4
XX-4
Polarisation gloss meter
Polar isolux diagram .
VII-8
Potassium cell . . . .
XV-2
Preferred numbers
. V-9
in isocandela diagrams
in isolux diagrams
Vll-8
Primary standard . . . .
XIII-2
. III-2
- - of luminous intensity .
Projection
. V-6
Azimuthal - .
.XI-13, 14
Diascopic - .
. Xl-13, 19
Episcopic - .
. V-8
Quasi central . V-4
Sinusoidal - . .
Slide- . . . .
XI-15
Standard-film Xl-16
Sub-standard film XI-17
Projectors, measurement ofXVI-4
XII-7
Purkinje effect . . . . . . .
Quantity of light . . . . .
Measurement of- - Quantum theory . . .
Quasi central projection
. III-6
.XVII-7
Rear-wall cell
Reciprocity, law of- .
Red limit . . . . .
Reflection . . . . .
Critical angle ofDiffuse- .
Mixed- .
MultipleRegular-.
from Road surfaces
SpecularSpread- . . . .
Uniform diffuseRegular reflection . .
Regular reflection factor .
Regular transmission . .
Relative !urn. eff. of radiation
Retina . . . . . . . .
Reflection factor . . .
- - of Conductors
Diffuse-- . . . .
Measurement of- - .
- - of Non-conductors.
Regular-Ritchie wedge
Rod vision . . .
Room index . .
Rotating sector disc .
Rousseau diagram
Rumford photometer
Russel-Bloch angles .
XV-3
. IX-3
XV-2
X-1
X-3
X-5
X-5
.X-14
. X-2
.X-10
X-2
X-5
X-6
X-2
X-5
.X-ll
XII-4
XII-2
X-1
. X-4
. X-5
XX-1
. X-3
. X-5
XIV-2
XII-7
XII-2
XIV-3
Ch. IV
XIV-2
. IV-8
1-l
. V-8
Scotopic vision . . .
XII-7
Sea-mile candle
. VI-5
Secondary-emission cells .
XV-2
Secondary standards
XIII-2
Selenium cell. . . . . .
. XV-3
Sensitivity to luminance difference. Xll-9
Simultaneous method
. Xlll-3
Sine condition . . .
. XI-3
Sinusoidal projection
. V-4
Slide projection.
Xl-15
Snell's law . . . . .
. X-3
Solid angle . . . . .
. Ch. II
page 407
Table of solid angles
VII-6
Solid of illumination
Solid of light distribution
. IV-1
Spectrum
Continuous - .
1-6
Line- . . . .
1-6
Specular reflection
X-2
Sphere paints . .
.XVII-4
Sphere window . .
.XVII-I
Spherical candle .
. III-3, 4
Spherical co-ordinates .
. . V-3
430
. III-4
Spherical luminous intensity
. X-5
Spread reflection . . . .
. X-11
Spread transmission . . .
XI-I6
Standard-film projection .
XIII-2
Standard light sources.
XIV-6
Standard observer
XII-3
Step-by-step method
. II-I
Steradian . . . . .
VIII-I
Stilb . . . . . . .
XII-2
Stiles-Crawford effect
XIV-6
Stilling's colour tables .
XIII-4
Stray light . . . . . .
XII-I, 2
Subjective brightness .
XIII-2
Sub-standard . . . . .
XI-I7
Sub-standard film projection
XIII-3
Substitution method . . . .
XII-5
Summation law . . . . . .
Systems of obtaining variation in
luminance in photometers . . . · XIV-3
Talbot's law . . . . . . . . . . XIV-3
Taylor's method of measurement of
XX-I
reflection factor . . . . . .
I-5
Technique of light production
I-2
Temperature radiator .
XV-4
Thermo-couple . . . .
. X-3
Totally reflecting prism
X-I, II
Transmission .
.X-11
Diffuse- .
Mixed- .
.X-11
.X-11
Regular-.
.X-11
Spread- .
. X-11
Uniform diffuse. X-I
Transmission factor .
XX-2
Measurement of- - .
INDEX
Transmissive exponent .
Tungsten lamps . . . .
Tungsten ribbon lamp .
.X-I5
I-6
XIII-2
. . . XVII-I
Ulbricht sphere. . . . .
I-2
Ultra-violet radiation . .
VIII-2, X-5
Uniform diffuser . . . .
. X-6
Uniform diffuse reflection . .
Uniform diffuse transmission. . . . X-1I
Vacuum cell . . . .
. XV-2
Variable diaphragm .
XIV-3
Velocity of radiation
I-2
Vertical illumination
. VI-4
Vignetting . . . . . . . . . .
.XI-7
Visual illumination photometer . XVIII-3
Visual photometry
.Ch. XIV
Visual purple . .
XII-7
Wavelength . .
Wave-mechanics
Wave-motion. .
Wave theory . .
Weber photometer
Weber's law . . .
Weber-Fechner's law
Wollaston prism .
Working standards
XII-9
XIV-3
XIII-2
Yellow-blue ratio
Yellow spot . . .
XIV-6
XII-7
Zonal luminous flux diagram
IV-5
I-2
I-I
I-2
I-1
XIV-4
Xll-9