Light scattering/reflection from surface,
the Saunderson correction
Anna Lundberg
2004-2005
Supervisor: Nils Pauler
Examiner: Mats Nyhlén
Umeå University
Department of physics
Thesis project
M-real TC
ÖRNSKÖLDSVIK
Umeå University
Department of physics
Thesis project
M-real TC
ÖRNSKÖLDSVIK
Abstract
To be able to optimise the printing quality of an image, it is important to
be familiar with the optical properties in both the printing media and the
colour. The general problem for describing illumination inside a turbid
medium is known as the radiative transfer problem. The Kubelka Munk
theory is a two-flux version of this problem. This method is frequently
used within the paper industry, mostly because of its ease of use. Not
surprisingly the approximations made within this method affects the
result. Sometimes more and sometimes less depending of the material
used.
This thesis work has included studies on one of the approximations
made within the Kubelka Munk equation, namely surface scattering. The
correction is called the Saunderson corrections. The corrections has
been studied and calculated both numerically and with a Monte-Carlo
simulation program named GRACE 2.4. The light-scattering and
absorption parameters has been calculated with the variable Rg-method,
which is a special version of the Kubelka Munk equation. This has been
performed film within the visible spectrum (400-700 nm) for different ink
printed on Mylar, with and without corrections. One important thing in the
determination of the surface reflectance corrections has been the optical
geometry of the device. Another parameter that affects the result seems
to be the surface roughness.
The result has been tested by using the calculated values for the
scattering and absorption parameters for calculating the reflectance and
the printing density in the Kubelka Munk theory and compare with
measured values . The result from this method indicates that the light
scattering and absorption coefficient calculated with the corrected
Kubelka Munk equation can be used at some wavelengths. The
corrections has also been investigated in a pragmatic way for Ecolith
cyan at 500, 570 and 620 nm.
The last part of this work has been to study the optical dot gain. This is
an illusion for the human eye, meaning that the printed object seems to
be larger than it actually is. The magnitude of this phenomenon depends
on the optical and mechanical properties of the background of the print
and it has been found that printing on a medium with a high scattering
coefficient reduces the dot gain. GRACE 2.4 has been used to do
simulations.
The largest source of errors affecting the result are probably the
refractive indices. These constants are wavelength dependent and have
been measured on paper, not on Mylar film. This affects the
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determination of the Saunderson correction parameters since these
parameters are dependent of refractive index. One of the Saunderson
parameters representing the internal surface reflectance can be
regarded as “un-determinative“ because it is impossible to describe the
distribution for the light when entering the surface from below.
Acknowledgements
I would like to express my gratitude to people at M-real in Ö-vik for their
support during my thesis project, Lisettie Gidlund, Marie Tjärnström,
Birgitta Sjögren for help during experimental measurements, Jerker
Wågberg and Niklas Johansson for their support to my project and
report. I would give a special thanks to my supervisor Nils Pauler for
introducing me to the paper industry and Ludovic Coppel at ACREO for
the support to simulations with GRACE and for the help with calculations.
Table of Contents
1
INTRODUCTION .............................................................................................................. 1
1.1
1.2
1.3
1.4
2
BACKGROUND .................................................................................................................. 1
DIFFERENT APPROACHES .................................................................................................. 1
PURPOSE........................................................................................................................... 2
TASK ................................................................................................................................ 3
THEORY............................................................................................................................. 3
2.1 LIGHT ............................................................................................................................... 3
2.1.1
Light, general aspects ............................................................................................ 3
2.1.2
Radiometry ............................................................................................................. 6
2.1.3
Light scattering from a surface .............................................................................. 9
2.1.4
Body reflection - light scattering from bulk. ........................................................ 13
2.2 OPTICAL DOT GAIN ......................................................................................................... 15
2.3 KUBELKA MUNK ............................................................................................................ 15
2.3.1
Derivation of the Kubelka Munk theory ............................................................... 16
2.3.2
Variable Rg-method .............................................................................................. 18
2.3.3
Errors in Kubelka-Munk ...................................................................................... 18
2.4 D/0° GEOMETRY AND ELREPHO GEOMETRY .................................................................... 20
2.4.1
Elrepho light source and detector ........................................................................ 20
2.4.2
The Saunderson corrections ................................................................................. 21
3
MEASUREMENTS .......................................................................................................... 24
4
THE GRACE MODEL .................................................................................................... 25
4.1 GENERAL........................................................................................................................ 25
4.2 LIGHT ............................................................................................................................. 26
4.2.1
Scattering from a surface ..................................................................................... 26
4.2.2
Bulk scattering ..................................................................................................... 27
4.2.3
The light sources .................................................................................................. 28
4.3 DETECTORS .................................................................................................................... 29
4.3.1
Simulating Elrepho detector................................................................................. 29
4.3.2
Angle resolved detector (ARS) ............................................................................. 29
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4.4
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HOW MANY WAVE PACKETS? ......................................................................................... 29
METHODS ....................................................................................................................... 30
5.1 THE SAUNDERSON CORRECTIONS WITH FRESNEL ........................................................... 30
5.1.1
External reflectance k1 ......................................................................................... 30
5.1.2
Internal reflectance k2 .......................................................................................... 32
5.2 SAUNDERSON CORRECTIONS WITH GRACE 2.4 ............................................................. 33
5.3 THE SCATTERING AND ABSORPTION COEFFICIENT FOR MYLAR FILM .............................. 33
5.4 THE SCATTERING AND ABSORPTION COEFFICIENT FOR INK PRINTED ON MYLAR FILM .... 34
5.4.1
Ecolith cyan and Ecolith black............................................................................. 34
5.5 KUBELKA MUNK REFLECTANCE AND THE PRINT DENSITY .............................................. 34
5.6 OPTICAL DOT GAIN ......................................................................................................... 35
6
RESULTS .......................................................................................................................... 36
6.1 KUBELKA MUNK WITH AND WITHOUT SAUNDERSON USING RO AND ROO SIMULATED WITH
GRACE ................................................................................................................................... 36
6.1.1
Rink with different value of s, k and the Saunderson corrections .......................... 38
6.2 SAUNDERSON WITH FRESNEL’S LAW .............................................................................. 39
6.2.1
Ecolith black ........................................................................................................ 39
6.2.2
Ecolith cyan.......................................................................................................... 40
6.3 SAUNDERSON WITH GRACE .......................................................................................... 41
6.3.1
Mylar .................................................................................................................... 41
6.3.2
Ecolith black ........................................................................................................ 42
6.3.3
Ecolith Cyan ......................................................................................................... 42
6.3.4
k0 .......................................................................................................................... 43
6.3.5
Test of variabel Rg for different k0,k1,k2 ( Mylar) ................................................. 45
6.3.6
Test of different k0,k1,k2 of Mylar and ink for a printed construction .................. 46
6.4 SKM(INK) AND KKM(INK) WITH THE VARIABLE RG-METHOD ............................................. 48
6.4.1
Ecolith black ........................................................................................................ 48
6.4.2
Ecolith Cyan ......................................................................................................... 51
6.5 SMYLAR WITH VARIABLE RGV-METHOD ............................................................................. 55
6.6 THE PRINT DENSITY WITH AND WITHOUT SAUNDERSON CORRECTIONS .......................... 56
6.7 TEST OF PARAMETERS TO FIT THE KUBELKA MUNK MODEL ........................................... 60
6.8 OPTICAL DOT GAIN ......................................................................................................... 64
7
ANALYSIS AND DISCUSSION ..................................................................................... 66
7.1 GENERAL........................................................................................................................ 66
7.2 THE SAUNDERSON CORRECTIONS ................................................................................... 68
7.2.1
k1 .......................................................................................................................... 68
7.2.2
k2 .......................................................................................................................... 68
7.2.3
k0 .......................................................................................................................... 69
7.2.4
Variable Rg- method ............................................................................................. 70
7.2.5
sKM and kKM with and without corrections ............................................................ 70
7.3 SIMULATIONS FOR INK RASTER OBJECT .......................................................................... 70
8
CONCLUSIONS ............................................................................................................... 71
9
REFERENCES ................................................................................................................. 72
10
APPENDIX ....................................................................................................................... 74
10.1
REFLECTANCE MEASUREMENTS................................................................................. 74
10.1.1 Mylar .................................................................................................................... 74
10.1.2 Ecolith cyan.......................................................................................................... 74
10.1.3 Ecolith black ........................................................................................................ 77
Umeå University
Department of physics
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M-real TC
ÖRNSKÖLDSVIK
Umeå University
Department of physics
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1
INTRODUCTION
1.1
Background
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The first impression that we receive from a printed paper is the colour
and gloss of the print, the brightness of the paper and maybe also the
gloss of the paper. The colour of a print is an interplay between the light,
the ink and the printed medium and to be able to optimise the printing
quality it is important to be familiar with the optical properties of the ink
and the material.
This report is a thesis work in Master of Science in engineering physics
for the department of physics at Umeå University. It has been performed
at M-real Technology Centre in Örnsköldsvik and is a part of the ongoing
project to study the optical properties of paper and ink. The aim of the
optical project is to find ways to determine optical constants of offset ink
using traditional optical theories and evaluate if proposed corrections are
in line with known theories.
1.2
Different approaches
Usually when describing light as a wave, Maxwell’s electromagnetic
wave theory is applied. For a complicated structure such as paper and
ink, it would be impossible to solve Maxwell’s equations. This problem
has led to different types of theories for linking the optical properties of
radiation to the bulk properties of the material.
The general problem for calculating the light intensity inside an
illuminated turbid medium is known as the radiative transfer problem.
There are several methods for deriving practical solutions to this
problem.
The first approximate solution to the radiative transfer problem was
presented by Schuster [9], who considered radiation only in a forward
and a backward direction. Clearly influenced by this, Kubelka and Munk
[11] developed their model for applications in plastics and paint. This was
further refined by Kubelka [6,10].
These two models, by Schuster and Kubelka, are known as two flux
models of the radiative transfer problem [4].
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The Kubelka Munk theory has turned out to be the most commonly used
theory for describing light scattering and absorption in paper, paint and
plastic, because of its simple form and ease of use.
.
Figure 1. Kubelka Munk is a two-flux version of the multi-flux problem
The theory assumes that all incident light is diffuse and surface
scattering are not included.
This theory does not include surface scattering. The Saunderson
corrections are proposed to correct the Kubelka Munk equation for this.
Another method for solving the radiative transfer problem is to use the
Discrete Ordinate Radiative Transfer method, (DORT), which makes it
possible to solve the problem in more than two channels (two-flux
calculations). This is called multi-flux calculations. Wick [12] was the first
to introduce a solution to the radiative transfer problem in discrete
ordinary methods. [4] The DORT theory is now adapted to paper optics
and initial evaluation show that light scattering and light absorption are
influenced by the degree of diffusion in the bulk.
The radiative transfer problem can also be solved by use of the Monte
Carlo method. GRACE 2.4 is a Monte Carlo simulation program,
developed by the Acreo institute in Stockholm, for calculating light
scattering and absorption in paper. The program treats the incident light
as indivisible wave packets and uses Monte Carlo methods, using
stochastic processes, for describing light scattering in a turbid medium.
1.3
Purpose
The aim of this thesis work is to understand the physical background of
the derivation of the Saunderson corrections and investigate how these
corrections affects calculations made with the Kubelka Munk equation.
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Task
The main task was to determine the optical constants of these
corrections for substrates with different refractive index and test if they
could make it possible to determine the light scattering and light
absorption coefficients of offset ink printed on Mylar films. The Monte
Carlo simulation program GRACE was used to test the calculated result.
If the result turned out to be satisfactory, optical constants of the ink
should be used in Monte Carlo simulation and halftoning of prints.
2
THEORY
2.1
Light
2.1.1
Light, general aspects
By the term "light", we often mean the part of electromagnetic spectrum
that is visible to the human eye (380-770 nm), but it can also refer to
other forms of electromagnetic radiation, such as infrared and ultraviolet
light.
It is characterised by its velocity, frequency and wavelength. Short
wavelengths (high frequency) correspond to high energy and vice versa.
White light has the same intensity for all wavelengths and light with
different wavelengths are perceived as different colours for a human eye.
Light can be described both as a particle and as a wave; this is called the
wave/particle duality.
2.1.1.1
Light as an electromagnetic wave motion
An electromagnetic wave (EM-wave) is built up of oscillating electric and
magnetic fields. The waves oscillate in space and carries energy from
one place to another. The electric and magnetic field can be described
as a harmonic wave in the form [2]


E  E0  e i k r  t 


B  B0  e i k r  t 
(2.1)
(2.2)
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


The electric field E , magnetic field, B and propagation vector k , are
always mutually perpendicular (fig 1). E0 and B0 represent their

amplitudes, r the position and  the angular frequency.
Figure 2. Plane electromagnetic wave.
If the wave is viewed at a fixed time (fig. 3) the relation between the
spatial wavelength,  and the propagation constant k can be found [2].
k
2

(2.3)
When viewing the wave at a fixed position (fig 4), it is periodic in time
with a period, T. The relation between the propagation constant k, period
T and the wave velocity, is:
kvT  2
(2.4)
Figure 3. A plane electromagnetic Figure 4. A plane electromagnetic
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wave viewed in fixed time.
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wave viewed at a fixed point.


Both E and B satisfies a differential equation of the form:
2
 1  E
2 E   2  2
 c  t
2.1.1.2
(2.5)
Light as wave packets - photons
Light can also be described as wave packets, known as photons.
Photons carries an energy quanta, frequency and wavelength. The
energy is related to its frequency according to
E  h 
(2.6)
where h is Planck's constant. h  4.1356692  10 15 eVs
2.1.1.3
Refractive indices
Refractive indices describe the ratio of velocity c of an electromagnetic
wave to its velocity in vacuum c0. It describes the optical response of the
material to the incident EM-wave and is a property of the medium.
n
c
c0
(2.7)
If the reflecting surface is metallic, the refraction index becomes a
complex number.
2.1.1.3.1
Pseudo refractive indices
If the material has an conductivity , the E-field creates a current density
J ( A / m 2 ). This is described by Ohm’s law.


J  E
(2.8)
With Maxwell’s relations, it can be shown that the conductivity leads to a
modification of equation (2.5) [2]. This leads to a differential equation
described by
2
 1   E    E
(2.9)
 2 E   2   2  
2 
 c  t
  0 c  t
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By combining the E-field as a harmonic wave (eq.2.1 and eq. 2.2 ) with
the differential equation (2.9), it can be seen that the propagation vector,
k, must be complex:
 
  

k  1  i
c
  0 
1
2
(2.10)
From equation (2,1), (2.2), (2.3) and (2.9), the refractive index on the
complex form can be derived as
  
 

n  1  i
  0 

1
2
(2.11)
The refractive index can be written in general form:
n  nr  ni
(2.12)
where the real part, Re( n)  nr behave as the ordinary refractive index
and the imaginary part, Im(n)=ni determines the rate of absorption, , in
the conductive medium.

4ni

(2.13)
The imaginary part, also known as extinction coefficient, is related to the
damping of the oscillation amplitude of the incident field.
A non-metallic material (dielectric) is a low absorbing material and the
imaginary part is small and often neglected. If the material is conductive,
the imaginary part is larger and must be taken into account. [2]
2.1.2
Radiometry
Radiometry is the science of measurement of electromagnetic radiation.
To be able to do reflectance calculations in a correct way, it is important
to take the energy distributions for incident and reflected light into
account.
The energy flowing through a surface per unit time is called radiation.
The radiant flux emitted per unit of solid angle that is incident on, passing
through or emerging from a point source in a given direction is called the
radiation intensity, I. The radiation intensity for an element of radiant flux
d through an element of solid angle d is illustrated in figure 5, and is
given by
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I 
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d
d
(2.14)
where d is the differential element of solid angle measured in spherical
co-ordinates [3].
d  sin dd
(2.15)
The irradiance, E (brightness), is the radiant flux per unit area that is
incident on, passing through or emerging from a specified surface.
E
d
dA
(2.16)
The irradiance decreases inversely with the square of the distance from
the light source.
E
d I

dA r 2
(2.17)
The irradiance is illustrated in figure 6.
The radiance L describes the radiation intensity per unit of projected area
perpendicular to the specified direction and is defined as
dI
d 2
(2.18)
L

dA cos d (dA cos )
for an element of radiant flux d2. This is illustrated in figure 7.
Depending on the optical device used, the measurements are either
radiance or irradiance[3]. When measurements are made with an
integrating sphere geometry, the irradiance is measured and when a
detector of fixed area is used, the radiance is measured.
The relation between the irradiance E and the radiance L is given by
combining equation 2.17 and equation 2.18 and integrating over the
hemisphere .
dE  L  cosd
(2.19)
2 / 2
E
  L  cos sin dd
0
(2.20)
0
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Figure 5. The radiation intensity is Figure 6. The flux leaving a point
the flux through the cross section source within any solid angle is
dA per unit of solid angle.
distributed over increasingly larger
areas, producing an irradiance that
decreases inversely with the
square of the distance.
Figure 7. The radiance L, describes the radiant intensity per unit of
projected area, perpendicular to the specified direction
2.1.2.1
Lambertian surface
When a radiating or reflecting surface has a radiance that is independent
of the viewing angle, the surface is said to be perfectly diffuse, or a
Lambertian surface. The radiance for a Lambertian surface is constant
for all viewing angles . Therefore, for Lambertian surfaces, the term I()
in equation 2.18 can be substituted with I(0)cos. This gives the following
relation for a fixed surface A.
L
dI ( )
dI (0) cos dI (0)


 const
dA cos
dA cos
dA
(2.21)
The Lambert cosine law [2] describes the angle dependence of intensity.
I ( )  I (0) cos 
(2.22)
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2.1.3
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Light scattering from a surface
If a parallel beam of light strikes a surface of a dielectric material, it can
be both transmitted into the material and reflected at the boundary. Light
that is reflected at the boundary is said to be specularly reflected and is
described by Snell’s law of reflection and Fresnel’s law.
The reflectance is the dimensionless ratio of the reflected flux d to the
incident flux d i.
R
2.1.3.1
d r
d i
(2.23)
Snell’s law
Light incident on a flat surface can be described by Snell’s law (fig 8).
Figure 8. Light incident on a flat surface can be described by Snell’s law
The relation between the angle of incident light and the angle of refracted
light is
n1 sin 1  n2 sin  2
(2.24)
where n1 and n2 are the refractive indices for the media at both sides of
the surface. 1 is the incident angle, and 2 the angle of refraction.
According to the law of reflection, the reflected wave will be reflected at
the same angle as the incident wave. This phenomena is called specular
reflection
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Fresnel’s law
2.1.3.2
Figure 9. Incident, reflected and refracted wave
The fraction of incident energy transmitted or reflected from a plane
surface can be described by Fresnel’s equations [2]. The subscript 
describes the polarisation direction normal to the plane (TM-mode) and ||
denotes the polarisation direction parallel to the plane of incident (TEmode).
r 
n1 cos1  n2 cos 2
n1 cos1  n2 cos 2
(2.25)
t 
2n1 cos1
n1 cos1  n2 cos 2
(2.26)
r|| 
n2 cos1  n1 cos 2
n1 cos 2  n2 cos1
(2.27)
t|| 
2n1 cos1
n1 cos 2  n2 cos1
(2.28)
 is the incident angle, and can be found by Snell’s law of reflectance.
Reflectance for both polarisation directions are plotted as a function of
the angle of incidence for internal reflectance and external reflectance in
figure 10.
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Figure 10. Fresnel reflectance
In case of internal reflection both r and r║ reaches values of unity at
angles less than 90°. This is the phenomenon of total internal reflection.
It occurs at the critical angle,  c .
 n1 

n
2
 
 c  sin 1 
(2.29)
No wave energy is reflected for TM-mode at the so-called Brewster
angle,  B . This occurs when
 n1 

 n2 
 B  tan 1 
(2.30)
The reflection coefficients r and r║ sometimes becomes complex
numbers and since instrument cannot measure complex quantities the
measured reflectance is defined as
R r
2
(2.31)
for both polarisation coefficients [13]. Natural light can be viewed as an
equal mixture of both polarisation components.
1
r  r ||
2
1
T  t  t ||
2
R
(2.32)
(2.33)
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Reflectance calculations for a Lambertian distribution
As earlier mentioned in the theory chapter, radiometry is an important
part in reflectance calculations. If the light has a totally diffuse
(isotrophic) distribution, i.e. has the same intensity in all directions when
it enters the surface or emerges from the surface, then the incident light
will have an angular distribution as follows. [14].
f ( i )  sin  i .
(2.34)
This is a result from equation 2.15. The angular distribution is
independent of the azimuthal angle and therefore only dependent of the
polar angle .
A Lambertian surface is said to have the same radiance in all directions,
( Equation 2.21). Lambert’s cosine law (equation 2.22) describes how
light intensity depends on viewing angle. This gives the angular
distribution for a Lambertian surface
f  i   cos i 
(2.35)
The energy distribution in the solid angle d for incident light can be
written as
F  i   2  f  i   cos i   2  sin  i  cos i  sin 2 i
(2.36)
f(isin is the isotropic distribution for the polar angle and the number
2 is a normalization constant. When the distributions are taken into
account, the Fresnel reflectance can be written as
R  F  r d r 
1
R  R || F  i   1 R  R || sin 2 i d r
2
2
(2.37)
Figure 11 a-b illustrates the distribution for incident and reflected light,
where n1=1 and n2=1.5 for both internal reflectance and external
reflectance. The distributions are plotted versus incident angle in
degrees.
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Figure 11a. The distribution of Figure 11 b. The distribution of
internal reflectance. The red line external reflectance. The red line
represents the distribution of represents the distribution of
incident
light
and the
blue incident light and the blue
represents the distribution of represents the distribution of
reflected light.
reflected light
The distributions for the internal reflectance is illustrated schematic in
Figure 12 a.
Figure 12 A schematic illustration of the internal reflectance.
Figure 12 illustrates the internal reflectance. No light is refracted at the
critical angle, c in the figure. Light is totally reflected back into the
medium at angles larger the critical angle, this light that never escapes
the medium, is called the internal reflectance.
2.1.4
Body reflection - light scattering from bulk.
When light has entered the bulk, it can be reflected or refracted between
particles inside the bulk and multiple reflections are created. Another
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word for this phenomenon is scattering. Light can also be absorbed by
the particles.
2.1.4.1
Scattering
Reflection, refraction and diffraction are summarised in one concept,
namely light scattering. From here on the Kubelka Munk scattering are
represented with the subscipt sKM and the scattering coefficient
calculated with GRACE are subscripted sGRACE.
Reflection
When light hits particles inside the bulk material it can be reflected by
objects inside the bulk, (figure 13)
Refraction
As light passes from one transparent medium to another, from air to filler
for example, it changes speed, and is refracted (figure 14). How much
depends on the refractive index of the mediums and the angle between
the light ray and the line perpendicular (normal) to the surface separating
the two mediums.
Diffraction
When a wave interacts with a single particle that is as large as or smaller
than the wavelength of light, the wave is diffracted. The particle scatters
the incident beam uniformly in all directions, (figure 15).
Figure 13. Reflection
2.1.4.2
Figure 14. Refraction
Figure 15. Diffraction
Absorption
Molecules inside the bulk can absorb and emit packets of
electromagnetic radiation. Different molecules absorb radiation of
different wavelengths due to the discrete energies dictated by the
detailed atomic structure of the atoms, this is the mechanism of how
colour is created.
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2.2
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Optical dot gain
To be able to optimise the printing quality of halftone prints, it is of great
importance to take the phenomena called optical dot gain into account.
The magnitude of this phenomenon depends on the optical and
mechanical properties of the printing substrate and it has been found out
that printing on a medium with a high scattering coefficient reduces the
dot gain. Figure 16 shows an optical dot gain for black colour.
Figure 16. The optical dot gain is an illusion for the human eye. A dot
with a certain size seems to be larger for the eye than it actually is.
The tone value for a printed sample can be calculated with equation 2.38
where Rpaper is the reflectance of the background, Rrast is the reflectance
of ink raster object printed on a surface and Rfulltone is the reflectance of a
full tone surface.
tonv 
R paper  Rrast
R paper  R fulltone
(2.38)
The optical dot gain, odg, can be calculated with equation 2.39.
odg  tonv  Rrast
2.3
(2.39)
Kubelka Munk
Kubelka Munk is a two-flux version of the radiative transfer function
where the illumination and scattering is completely isotropic.
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2.3.1
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Derivation of the Kubelka Munk theory
Figure 17. Two fluxes which are completely diffuse. One in the positive xdirection, J, and one in the negative x-direction, I.
During passage trough the layer dx, some of the light will be scattered
and absorbed so that i and j will be reduced at the same time. The part
that was reduced from I will be added to J and vice versa. The total
change in light intensity can therefore be written for both directions as.
 dI  k KM Idx  s KM Jdx  s KM Idx
dJ  k KM Jdx  s KM Jdx  s KM Idx
(2.40)
(2.41)
sKM is the light scattering coefficient in m2/kg and kKM is the absorption
coefficient in m2/kg.
The mean path length for diffuse flux is 2 times the linear path length,
[15]. The Kubelka Munk equations for diffuse light is
 dI  k KM 2Idx  s KM 2Jdx  s KM 2Idx
dJ  k KM 2Jdx  s KM 2Jdx  s KM 2Idx
(2.42)
(2.43)
Setting K=2 kKM and S=2 sKM and re-arranging terms gives
dI
 k  s I  sJ
dx
dJ
 K  S J  SI
dx

Let
SJ
a
S
(2.44)
(2.45)
(2.46)
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then

dI
  aI  J
Sdx
dJ
 aJ  I
Sdx
(2.47)
(2.48)
The approximation that light intensity is diffuse in both directions gives
the relation for reflectance over the differential layer dx.
J
r
I
(2.49)
 J  idj  jdi
dr  d   
i2
I
(2.50)
Let Eq (2.47) and (2.48) into (2.50)
R
w
dr
dr
 r 2  2ar  1   2
 S  dx
Sdx
Rg r  2ar  1
0
(2.51)
After integration between 0<x<d and Rg<r<R
ln
( R  a  a 2  1)( Rg  a  a 2  1)
( Rg  a  a  1)( R  a  a  1)
2
2
 2Sd a 2  1
(2.52)
The reflectance for an opaque bulk can be determined by considering an
infinitely thick layer, d   .
R 
1
a  a2 1
 a  a2 1  1
K
K2
K

2
2
S
S
S
(2.53)
Solving the equation for a finite layer gives [6]
R 
(1 / R )( R g  R )  R ( R g  1 / R ) exp( Sd (1 / R  R))
(2.54)
( R g  R )  ( R g  1 / R ) exp( Sd (1 / R  R ))
R0 is the reflectance of the sample over an ideally black background,
(Rg=0).
If one assumes that Rg=0.
R0 
exp( Sd (1 / R  R))  1
(1 / R ) exp( Sd (1 / R  R ))  R
(2.55)
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Solving for S and by using the relation K=2 kKM and S=2 sKM.
s KM 
 ( R  Rg )( R  Rg ) 
R

ln  
2
d (1  R )  (1  RgR  )( R  R) 
kKM can then be calculated using the relationship
k KM (1  R ) 2

s KM
2 R
(2.56)
(2.57)
If the material is completely homogeneous, it is possible to use
grammage, w, instead of d.
2.3.2
Variable Rg-method
An alternative method is to measure the reflectance over two different
backgrounds. The reflectance factors Rv and Rg are measured for a
sample over backgrounds with reflectance factors Rgv and Rgs, [17]. The
calculation for s is as follows.
a
1( Rgv  Rgs )(1  RvRs )  ( Rv  Rs )(1  RgvRgs )
2( RsRgv  RvRgs )
R  a  a 2  1
(2.58)
(2.59)
and finally
s KM 
2.3.3
1
 1

w
 R 
 R

ln
(1  RsR  )( R  Rgs )
(1  R Rgs )( R  Rs )
(2.60)
Errors in Kubelka-Munk
There are some assumptions made in the Kubelka Munk theory. These
are:
1. Light fluxes are completely diffuse.
2. Light fluxes in forward and reverse directions have the same angular
distribution.
3. The background is in complete optical contact with the paper.
4. There are no surfaces in the model i.e. specular reflections are not
being taken into account. It is only the light scattering and the light
absorption of the bulk that is described by the theory.
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5. The material is considered as completely homogeneous.
In general, the forward and reverse fluxes do not have the same angular
distribution.[15] This implies that eq. 2.40-2.41 fails. As a result, the
absorption coefficients in the two directions are not strictly equal to each
other. Usually this difference can be neglected but not always. The
Kubelka Munk theory fails for strong absorbing materials.
The value of the absorption coefficient depends on the angular
distribution of the light in the scattering material [15].
If collimated light is passing perpendicularly through a thin layer, the
Kubelka Munk absorption coefficient will be equal to the linear absorption
coefficient for collimated light. If the incident light is diffuse, then some of
the light will have a longer path through the layer.
Figure 18. This picture illustrates a layer in the material. Ray A, which
has a longer path through the layer than Ray B, is much more strongly
absorbed.
Figure 18 shows layer in the material. Ray B is perpendicular to the
layer. If the incident light is diffuse, then some of the light will have a
longer path through the layer, this is represented by Ray A in the picture
If the light is perfectly diffuse, then the relation between the Kubelka
Munk absorption coefficients and linear absorption coefficient is [15]
k KM  2k Grace
(2.61)
If light travels in directions far from the normal in a strong absorbing
material, kKM can be much larger than 2kGRACE and the Kubelka Munk
theory therefore fails to describe light scattering and absorption in a
strongly absorbing material.
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The relation between Kubelka Munk and radiative linear scattering
coefficients is
s KM 
3
s Grace
4
(2.62)
where sKM is the Kubelka Munk scattering coefficient and sGRACE is the
radiative transfer coefficient for scattering. This is valid when the
radiation is nearly isotropic. [15]
The error caused by paragraph 4 above is the focus in this report, and
can be corrected by the Saunderson corrections, which is a method for
correcting the Kubelka Munk theory for surface reflectance. Studies in
this experiment has been made on high absorbent material, e.g. ink
layers. This implies that the problems with strong absorbing materials will
affect the result .
2.4
D/0° geometry and Elrepho geometry
An ELREPHO spectrophotometer is a standard instrument for
reflectance measurements used in the paper and pulp industry. This
instrument has a d/0° geometry, which means that the sample is
illuminated with diffuse light and the reflectance is measured from a fixed
point perpendicular to the sample (only in one direction). There are also
other instruments with other optical geometry, for example 45°/0° and
8°/0° and angle resolved scatterometer (ARS).
In this work, measurements have been made with an Elrepho instrument
and calculations with this geometry are therefore in focus in this report.
2.4.1
Elrepho light source and detector
An Elrepho instrument (ISO 2469) is build up by a sphere coated with
BaSO4 inside. The Barium sulphate pigment is used to ensure that the
illuminating light is diffuse. Figure 23 illustrates a model of an Elrepho
instrument .
There are three openings in the sphere. A xenon lamp is placed in
opening number 1. This lamp is screened so that neither the sample nor
the detector is directly illuminated by the lamp. A detector is located at
number 2 and finally, the sample is placed in opening number 3.
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To eliminate the specular reflection, a gloss trap is used, (no.4 ). The
extent for the gloss trap is 15.5° (calculated from Snell’s law of
reflection).
Figure 23. The Elrepho instument has a D/0 geometry which means that
the sample is illuminated with diffuse light and the reflected light is
measured in a direction normal to the sample.
The output from the Elrepho instrument is the reflectance factor, defined
as the ratio of the scattered power from the sample to the scattered
power from an ideal Lambertian surface.
STFI, the Swedish Pulp and Paper Research Institute, is authorised by
ISO/TC6, to supply reference standards which are distributed once a
month to calibrate the instruments. This reference standards represents
“the perfect reflecting diffuser” [17].
2.4.2
The Saunderson corrections
J.L Saunderson [16] derived an equation for correcting Kubelka-Munk for
surface reflectance in 1942. To derive these corrections, he assumed
that collimated light strikes the film and a fraction k1 is reflected from the
surface. The rest of the light is transmitted into the film and is reflected
upwards. When the reflected light enters the surface from below, a
fraction k2 is reflected back into the film.
The internal surface reflection and the multiple scattering are linked
together, and changes in one will influence the other, (Figure 19-21).
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Figure 19. Cycle 1
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Figure 20. Cycle 2
Figure 21. Cycle 3….
The sum of all light leaving the top boundary and going up, can be
expressed as
2
Rmeasured  k1  (1  k1 )(1  k 2 ) Rcorr (1  k 2 Rcorr  k 22 Rcorr
 ...)
(2.63)
Where the term Rcorr represents the Kubelka Munk reflectance and
Rmeasured is the measured reflectance.
This can be rewritten by using the relations for geometric series.
a
q
 1  a  a 2  ...  a n  (1  a) 1
(2.64)
q0
The measured reflectance can then be written as
Rmeasured  k1 
(1  k1 )(1  k 2 ) Rcorr
(1  k 2 Rcorr )
(2.65)
Or inversely
Rcorr 
Rmeasured  k1
1  k1  k 2 (1  Rmeasured )
(2.66)
The Saunderson correction was derived for spectrophotometer
measurements made with integrating sphere geometry. In this report, it is
also necessary to take the Elrepho geometry into account.
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2.4.2.1
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External reflectance k1
k1 is the external reflectance, and can be calculated by considering the
incident light as diffuse with a Lambertian distribution and integrate
Fresnel’s equation over the surface.
The Elrepho geometry for the light source and for the detector must be
taken into account.
Elrepho instrument has a gloss trap. This means that no light is incident
on the surface for angles lower than 15.5°. Calculations for k1 should be
done by integrating Fresnel’s equations in the interval 15.5°- 90° for
incident angle.
sin 2 1
2
15.5
90
k1 
2.4.2.2

 n1 cos1  n2 cos 2   n2 cos1  n1 cos 2 

 
d 2
 n1 cos1  n2 cos 2   n1 cos 2  n2 cos1 
(2.66)
Internal reflectance k2
k2 describes the fractional reflectance when the light entering the sample
is partially reflected at the air-sample interface. This correction term is a
little bit more complicated to determine than k1 because it is difficult to
describe the distribution for the light entering the upper surface from
below.
If the background has a refractive index lower than the sample, then light
incident on the upper surface will have a more narrow distribution
compared to the light incident on the lower surface and can be calculated
with Snell’s law and Fresnel’s law. This is illustrated in figure 22.
Figure 22. k2 is the internal reflectance and depends on the refractive
index of the background.
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The incident angle on the upper surface i is same as the refracted angle
at the lower surface t.
The refracted angle at the lower surface can be calculated directly by
Snell's law (equation(2.24). This means that the reflected part at the
upper surface will have a Lambertian distribution with angles lower than
maximum angle calculated with Snell’s law.
2.4.2.3
k0
The first term in equation 2.65 is the fraction of the surface reflectance
that actually reaches the detector. This term depends on the optical
device used and makes it necessary to modify (equation 2.65 ).
Rmeasured  k 0 
(1  k1 )(1  k 2) Rcorr
(1  k 2 Rcorr )
(2.67)
For an integrating sphere geometry with the specular component
included k0 equals k1. Because of the gloss trap in the Elrepho detector,
no specular reflection should be measured, i.e. k0=0 and equation 2.67
would be written as.
Rmeasured  0 
(1  k1 )(1  k 2) Rcorr
(1  k 2 Rcorr )
(2.68)
This equation fails if a fraction of light reaches the detector. This can
occur if the surface has a micro surface roughness, or if the sample
decline.
Another factor that can affect k0 is the geometry of the detector. The
geometry of the detector implies that a part of the light can be reflected
from the lens, and is therefore not collected by the gloss trap.
k0 can be described as a part of k1 and depends on the optical geometry
for the device, the surface roughness and the accuracy in
measurements.
3
MEASUREMENTS
The reflectance factor was measured with a LW Elrepho instrument for
ink printed on Mylar film at 400-700 nm. The ink studied was Ecolith
black and cyan. Each measurement was repeated three times on each
side of the sample to get a reliable value of the reflectance. Figure 24
illustrates the printed sample.
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Figure 24. Reflectance measurements on ink printed on Mylar film
The reflectance of the Mylar film was measured with black and white
background over all wavelengths. This was also made for ink printed on
Mylar film. The results from the measurements can be found in appendix.
4
THE GRACE MODEL
GRACE is the name of a computer software for light scattering
simulations developed within the Light and Paper project. The
simulations are based on Monte-Carlo techniques.
The project Light and Paper started in 1995 and its purpose was to get a
description of the interaction between light and paper [19]. The two main
goals of the project were to develop software for describing light
scattering simulations and to study the optical properties in paper and its
components. GRACE was used in this project to verify the shortcomings
of Kubelka Munk theory and for calculation of optical dot gain.
4.1
General
In the program, a paper is described as a three dimensional structure,
including rough surfaces, coating layers, transparent layers, ink layers
and base sheet layers containing fibres, fillers, pores and fines.
The Monte-Carlo model, using stochastic processes, is used to do light
scattering calculations in three dimensions. The input parameters to the
light scattering model include a detailed description of the paper
structure and its scattering and absorption parameters.
A short overview of the GRACE program is given in the following section.
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Simulations for base sheet layers containing fibres and pores have not
been done in this work and are therefore not described.
4.2
Light
In GRACE, the wave particle duality is applied. For some processes, e.g.
reflection, refraction and polarisation, the program considers the wave
nature of light. For propagation and interaction, light is represented as
indivisible wave packets.
The light is defined by wave packets. These retain a number of
properties and each wave packet is described by a 10-element column
vector containing the co-ordinates (x,y,z) of the current position of the
wave packet, the direction cosines (l,m,n) and the direction cosines of the
polarisation vector multiplied by the relative electromagnetic energy.
GRACE only allows linearly polarised wave packets and only one
wavelength can be simulated at a time.
In order to simulate the whole spectrum, it is necessary to run several
simulations at different wavelengths. It is possible to choose the number
of wave packets when running the program, all the column vectors
representing the wave packets are then put together in a matrix with ten
rows
4.2.1
Scattering from a surface
In GRACE, the surface scattering is treated as a combination of two
effects. The surface is spatially filtered to separate long-range
topographic structure from the micro-roughness containing shorter
spatial wavelengths. The long-range topographic structure, called the
surface waviness, is the part that deflects incident wave packets
according to Snell’s law and Fresnel equations. In addition, there is the
short spatial wavelength roughness that scatters the light diffusely.
The surface micro roughness is described by the root mean square (rms)
roughness measured with an atomic force microscope.
Simulation of scattering by a surface is done as follows. First, the
intercept of each wave packet with the surface is calculated. At each
intercept, the normal to the surface is calculated, and the Fresnel
equations and Snell’s law is applied to determine the direction,
magnitude and polarisation state of the reflected and refracted wave.
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The scattering by the micro roughness is introduced by letting a fraction
of the reflected and refracted light be distributed in a Lambertian manner
according to eq. 4.1 and 4.2
 4Rq n1 cos 



2

diffusely reflected power
 1 e 
totally reflected power
diffusely refracted power
 1 e
totally reflected power
(4.1)
 2Rq ( n1 cos 1  n2 cos  2 ) 





2
(4.2)
where n1 and n2 are the indices of refraction for the two media.  and 
are the angles from the surface normal.  is the wavelength of the
incident wave and Rq is the micro-roughness, rms. for the surface. These
formulae are only valid for smooth surfaces (Rq<<) but are used as
approximations.
The polarisation state for specular reflected light is determined by
Fresnel’s equation. For diffuse light, the polarisation state is determined
by the the polarisation factors Q, where  and  are the polarisation
states before and after scattering has occurred.

cos 
QSS 
  sin 2  i cos s    sin 2  s
i
QSP 
  1
cos 


  1
(4.3)
2
  sin 2  S sin  s
  sin 2  i  cos s    sin 2  s
i

(4.4)
2
  sin 2  i sin  s
 cos    sin  cos    sin  
  1   sin    sin  cos    sin  sin  

cos    sin   cos    sin  
QPS 
2
i
i
s
s
2
i
S
s
2
i
(4.4)
2
2
QSP
2
  1 cos  s
i
s
2
i
s
2
(4.5)
s
where  is the relative dielectric constant of the transmitting medium, 
and 2 are the polar angles and  is the azimuthal angle.
4.2.2
Bulk scattering
In this report, the simulated bulk is considered as homogenous. There
are two processes that take place in bulk scattering. Light is either
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scattered or absorbed. The scattering and absorption parameters are
determined by the user. The thickness is also determined. The extinction
coefficient is the sum of the scattering and absorption coefficient. A wave
has a mean free path, given by the extinction coefficient. When a packet
has travelled its free path in the layer, it is either absorbed or scattered
according to the ratio between the respective coefficients.
The path length, or travel distance of a wave packet in the coating layer
decays exponentially with the mean free path.
The mean free path, ℓ, is the inverse of the volumetric extinction
coefficient, (cross section per unit volume)
 ext  C ext / V
(4.6)
The extinction coefficient is the sum of the volumetric scattering
coefficient and the volumetric absorption coefficient. The path length
probability distribution is
P( x)   ext exp(  ext x)
(4.7)
where x is the path length.
By integrating eq. 5.7 and using the fundamental transformation law of
probabilities with P(x) from a uniformly distributed random number, an
expression for the path length is obtained.
x(a)   log( 1  a) /  ext
(4.8)
When a wave packet has travelled its path length, the ratio between the
extinction coefficients determines the probability for absorption or
scattering. Each absorbed wave packet is removed from the simulation.
For each scattered wave packet the new direction is determined.
4.2.3
The light sources
There are three light sources implemented in GRACE: a beam source,
an Lambert source and an Elrepho source. In this section only two of
them has been used, Elrepho and Lambert. Beam light source is
therefore not described.
4.2.3.1
Lambert
This function is used to simulate light from diffuse light sources. The light
intensity is proportional to cos(  ) according to Lambert law of cosine.
28
Umeå University
Department of physics
Thesis project
4.2.3.2
M-real TC
ÖRNSKÖLDSVIK
Simulating Elrepho light source
This function generates wave packets according to the specification of
an Elrepho light source (Lambertian distribution with all angles less than
15.5° removed). This geometry has been described in the theory section
and follows the ISO-2469 standard.
4.3
Detectors
4.3.1
Simulating Elrepho detector
The Elrepho detector returns the reflectance factor that would be
measured by an Elrepho instrument. The energy reflected into the
Elrepho detector aperture is normalised by the power that would be
measured from a perfectly diffuse sample.
The exact design rules for an Elrepho instrument are defined in the ISO2469 standard.
4.3.2
Angle resolved detector (ARS)
The angle resolved detector (ARS) returns the total power of the wave
packets passing through the defined aperture where the measured
power is transformed into a Bi-directional Scatter Distribution Function
(BSDF). The instrument is defined according to ASTM E1392.
4.4
How many wave packets?
Looking at simulations made with the ARS, it can be seen that the
simulations are noisy with a million wave packets. The reflectance
measured with an ARS and an Elrepho light source is plotted vs. incident
angle in figure 25.
29
Umeå University
Department of physics
Thesis project
M-real TC
ÖRNSKÖLDSVIK
Figure 25. Reflectance from an Elrepho light source measured with an
ARS.
In order to estimate the number of wave packets needed for the values
of reflectance and transmittance, ten simulations were made and the
mean value and standard deviation were calculated. Results from this
indicated that simulations could be done with 1000000 wave packets
since the standard deviation of ten simulations with 1000000 wave
packets was low (approximately 0.1).
5
METHODS
5.1
The Saunderson corrections with Fresnel
The Saunderson correction parameters were calculated directly by use
of Fresnel ‘s equation. Matlab 6.5 was used to do the calculations.
5.1.1
External reflectance k1
k1 was calculated for Lambertian distributed and Elrepho distributed light.
The equation for modelling Fresnel reflectance with lambertian
distributed light can be found in the theory section.
This is a complicated integral and is very difficult to solve. Therefore the
numerical trapezoid-method was used.
30
Umeå University
Department of physics
Thesis project
M-real TC
ÖRNSKÖLDSVIK
By dividing the area under the graph into small area elements, the total
area can be found by adding the sub areas.
90
R   F ( r ) p  p
(5.1)
p 1
MATLAB 6.5 was used to calculate the reflectance. The accuracy of the
calculations depends on the step size for the incident angle.
Figure 26. The reflectance for Lambertian distributed light was calculated
by trapezoid method.
Elrepho distribution has a Lambertian distribution with all
lower than 15.5° removed. k1 was also calculated for
distributed light source because this geometry has been
experiment. Figure 27 shows the Elrepho distribution for
reflected light.
polar angles
an Elrepho
used in this
incident and
Figure 27. The reflectance for Elrepho distributed light was calculated by
trapezoid method.
31
Umeå University
Department of physics
Thesis project
5.1.2
M-real TC
ÖRNSKÖLDSVIK
Internal reflectance k2
Snell’s law of reflectance was used to calculate the maximum angle of
incidence for the internal reflectance, and k2 was thereafter calculated
directly by integrating Fresnel’s law of reflectance, with a Lambertian
distribution. This would give a value of k2 near 0.6 for a refractive index
1.5. The result from this method would be correct if the light inside the
film were completely diffuse. This is however not in this case.
Earlier experiments has shown that the Kubelka Munk equations give
better results if a value near 0.4 is used for k2 instead of 0.6 [5]. Mudgett
and Richards presented in 1973 [15 ] a theoretical explanation to this.
They compared the ratio between the absorbing coefficient and
scattering coefficient for 16-flux calculations with the two flux Kubelka
Munk coefficients. If the two ratios were in completely agreement, then
the results (curves ) would agree, but they did not.
To fit the results from 16-flux calculations, they calculated the ratio
kKM / sKM in the Kubelka Munk equation with different values for the
correction parameter k2. After that, they made a polynomial expression
for k2 in terms of refractive index to correct the Kubelka Munk equation.
The polynomial expression were then determined by the least-squares
criterion.
For diffuse illumination the expression for k2 was found to be
k 2  0.3527 + 3.6311/n - 8.0405/n 2 + 4.0405/n 3
(5.2)
In this report the reflectance measurements were made on a sample
laying on a paper. This set-up allows an airgap between the paper and
the film, caused by the surface roughness of the paper. The difference in
refractive index causes a more narrow distribution for the internal
reflectance than the external reflectance. This has been introduced
earlier in the theory section. This would give a lower value for k2, near 0.4
for a refractive index 1.5.
In this report k2 was calculated directly with Fresnel’s equation and with
the polynomial expression from Mudgett and Richards.
32
Umeå University
Department of physics
Thesis project
5.2
M-real TC
ÖRNSKÖLDSVIK
Saunderson corrections with GRACE 2.4
External reflection, k1
k1 was simulated in GRACE 2.4. with an Elrepho light source and ARSdetector. The number of wave packets was set to 1 000 000. Incident
light was assumed to have a uniform distribution. This was made for
different values of refractive indexes over all wavelengths.
Internal reflection, k2
k2 was simulated in GRACE 2.4. with an Lambertian light source and
ARS- detector. The number of wave packets was set to 1 000 000.
Incident light was assumed to have a uniform distribution. This was
made for different values of refractive indexes
External reflection, k0
The Saunderson correction parameter k0 for the Mylar film was calculated
by simulating a model of a surface with a micro-roughness in GRACE
2.4. The surface roughness was measured with a Parker Print Surface
(PPS). In the simulation, the optical geometry used for the device was an
Elrepho geometry.
5.3
The scattering and absorption coefficient for Mylar film
The reflectance from the reference paper was measured and used as Rgv
(white paper) and Rgs ( black paper), in equation 2.60. Rv is the measured
reflectance from the film with white background and Rs is the measured
reflectance from the film with black background.
The correction parameters found for the Mylar film was used to subtract
the surface reflectance from the measured reflectance from the film (Rv
and Rs) and the background (Rgv and Rgs) with equation 2.66.
sKM and kKM was calculated according to the equations 2.56-2.57, with
and without correction parameters.
33
Umeå University
Department of physics
Thesis project
M-real TC
ÖRNSKÖLDSVIK
5.4
The scattering and absorption coefficient for ink printed
on Mylar film
5.4.1
Ecolith cyan and Ecolith black
The background reflectance of the film was corrected with k0,k1,k2 Mylar
by using equation 2.66 and the measured reflectance of the sample, with
black and white background was corrected with k0 ,k1, k2 ink by using
equation 2.66 and the measured reflectance of the ink.
The obtained values for R (equation 2.59) was studied. This should be
constant over grammage, but it did not turned out to be completely
constant. Table 1 shows the values for R calculated in equation 2.62 for
cyan at 400 and 500 nm. The values for R differs more at low and high
grammage. A representative value for R would be somewhere in the
middle of this. The values for Roo at the grammage, 1.3 g/m 2 was
chosen.
Grammage [g/m2] Roo, =400nm Roo, =500nm
0.375
0.788
1.075
1.300
1.713
2.288
2.600
3.463
4.800
0.2093
0.1986
0.1423
0.1153
0.0906
0.0763
0.0695
0.0572
0.0493
0.3768
0.3464
0.2695
0.2309
0.1888
0.1571
0.1315
0.0969
0.0707
Table 1. R at different grammage for Ecolith cyan.
Equation 2.59 was used to find sKM and kKM for ink
5.5
Kubelka Munk reflectance and the print density
The calculated scattering and absorption coefficients with and without
corrections was used to calculate the Kubelka Munk reflectance. The
print density was also studied, and is given by
 Rbackground 

D  log 10 
 R

pr int


(5.3)
34
Umeå University
Department of physics
Thesis project
5.6
M-real TC
ÖRNSKÖLDSVIK
Optical dot gain
Equation 2.38-39 was used to calculate the value of the optical dot gain
for ink raster objects printed on a paper with a gel coating layer between
the paper and ink. GRACE 2.4 was used to do the simulations.
Following models was used in the simulations.
1. Rpaper
An infinitely thick paper with refractive index=1.5 and a grammage of 1
000 g/m2 was placed at the bottom of the structure with a gel coating
layer at the top. The scattering and absorption parameters for the paper
was 60 m2/kg and 0.2 m2/kg respectively. The values for sKM for the gel
coating layer varied between 10 and 100 m 2/kg. This is illustrated in
figure 28.
2. Rrast
An ink raster object was placed at the top of the gel coating layer. The
values for the scattering and absorption parameters was determined
from the results from the calculations for sKM and kKM with the variable Rgmethod in this report and the equation 2.61 and 2.62. The test was
performed at the wavelength 560 nm. This model is illustrated in figure
29.
3. Rfulltone
In this model a fulltone layer was placed at the top of the gel coating
layer. The scattering parameters, absorption parameters and grammage
was the same as for the ink scattering object. This is illustrated in figure
30.
Figure 28. A paper with Figure 29. Ink raster Figure 30. Fulltone
a gel coating layer on object on the gel print on top of a gel
top
coating layer
coating layer.
35
Umeå University
Department of physics
Thesis project
M-real TC
ÖRNSKÖLDSVIK
6
RESULTS
6.1
Kubelka Munk with and without Saunderson using Ro
and Roo simulated with GRACE
The purpose of this exercise was to study the influence of Saunderson
corrections on the Kubelka Munk reflectance in a theoretical experiment.
R0 and R∞ were simulated in GRACE 2.4 at different grammages at the
wavelength 560 nm and without surface. In this calculation the density of
the material was assumed to be 1 g/cm3
The reflectance, R0 and R∞ was determined by simulating a material
without surface. This was made by setting the refractive indexes on each
side of the bulk equal to the refractive index inside the bulk. The
scattering and absorption coefficient in GRACE was calculated according
to equation 2.6.
Figure 36. R0 simulations.
The Saunderson corrections were used to theoretically add a surface to
the bulk. (Equation 2.65). The values of the constants, k1 and k2 was 0.9
and 0.6 respectively.
In figure 37, the simulated Ro is plotted vs. grammage with and without
surface. The blue line represents the simulated values for Ro for all
grammage and the red line represents the values for Ro after the surface
has been added.
36
Umeå University
Department of physics
Thesis project
M-real TC
ÖRNSKÖLDSVIK
Figure 37. The simulated R0 in GRACE with and without surface.
The Kubelka Munk equation (2.56-2.57) was then used to calculate the
light scattering and absorption coefficients inside the bulk.
The scattering coefficient without surface is illustrated in figure 38.
Figure 38. s calculated with Kubelka Munk without surface.
s was calculated in the same way with a surface as without surface. The
results can be seen in figure 39.
37
Umeå University
Department of physics
Thesis project
M-real TC
ÖRNSKÖLDSVIK
Figure 39. s calculated with Kubelka Munk with and without surface.
It is well known that s is a property within the material, and should not
change with grammage. Figure 39 indicates what happens to s Kubelka
when a surface is added to the bulk material. The light scattering with
surface included is far from being constant for different grammages.
6.1.1
Rink with different value of s, k and the Saunderson
corrections
A test was performed to study the influence of the Saunderson correction
parameters and different values of the scattering and absorption
coefficients on measured reflectance. The test was performed for a
layered construction with optical constants corresponding to mylar film
with a ink layer printed on it.
The reflectance was calculated with equation 2.53 and 2.55. Equation
2.66 was used to subtract the surface from the measured value of
reflectance both for the Mylar film and for the sample surface.
In this experiment the background reflectance, Rg was set to 0.8 and the
correction parameters for the Mylar film k0, k1 and k2 was set to 003, 0.12
and 0.65 respectively. The values of the parameters can be found for the
different cases in table 2 and the result is plotted in figure 40.
case
sink
kink
1
2
3
4
5
6
7
10
1
50
10
50
10
10
2000
200
10000
200
2000
2000
2000
k0ink
0.003
0.003
0.003
0.003
0.003
0.001
0.1
k1ink
k2ink
0.11
0.11
0.11
0.11
0.11
0.11
0.11
0.6
0.6
0.6
0.6
0.6
0.6
0.6
Table 2. Different values of test parameters
38
Umeå University
Department of physics
Thesis project
M-real TC
ÖRNSKÖLDSVIK
Figure 40. Kubelka Munk reflectance calculated with different values of
parameters
The effect of different values of the Saunderson corrections, sKM and kKM
can be studied in the figure above. A big value on the correction
parameter ko increases the reflectance, this is represented by curve no 7
in the figure. Line no 3 shows that the Kubelka Munk reflectance can be
reduced by increasing the absorption coefficient.
6.2
Saunderson with Fresnel’s law
The Saunderson corrections was calculated with Fresnel’s law and is
represented in table 3-4. Calculations made for Ecolith black is
represented in table 3 and for Ecolith cyan in table 4. The values for the
refractive indexes used in this project was handed out from ACREO and
is obtained from ellipsometry measurements.
6.2.1
Wavelength
400
410
420
Ecolith black
Refractive index
1.52
1.52
1.53
k1
0.09
0.09
0.09
k2 Fresnel
0.61
0.61
0.61
k2-polynom
k2-Fresnel with
0.42
0.42
0.42
snell’s law for
maximum angle
0.42
0.42
0.42
39
Umeå University
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Thesis project
430
440
450
460
470
480
490
500
510
520
530
540
550
560
570
580
590
600
610
620
630
640
650
660
670
680
690
700
1.53
1.54
1.54
1.55
1.54
1.56
1.56
1.56
1.57
1.57
1.58
1.58
1.58
1.59
1.59
1.59
1.60
1.60
1.60
1.61
1.61
1.61
1.62
1.62
1.62
1.63
1.63
1.63
M-real TC
ÖRNSKÖLDSVIK
0.09
0.09
0.09
0.09
0.09
0.09
0.09
0.09
0.09
0.09
0.09
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.62
0.62
0.62
0.62
0.62
0.63
0.63
0.63
0.64
0.64
0.64
0.64
0.64
0.64
0.64
0.64
0.65
0.65
0.65
0.65
0.66
0.66
0.66
0.66
0.66
0.66
0.66
0.66
0.43
0.42
0.43
0.43
0.44
0.44
0.44
0.45
0.45
0.45
0.45
0.45
0.46
0.46
0.46
0.47
0.47
0.47
0.47
0.47
0.48
0.48
0.48
0.49
0.49
0.49
0.49
0.49
0.42
0.42
0.42
0.42
0.40
0.40
0.40
0.40
0.40
0.40
0.40
0.40
0.40
0.39
0.39
0.39
0.39
0.39
0.39
0.39
0.39
0.39
0.39
0.39
0.37
0.37
0.37
0.37
Table 3. The Saunderson corrections calculated with Fresnel’s law for
Ecolith black
6.2.2
Wavelength
400
410
420
430
440
450
460
470
480
490
500
510
520
530
540
550
560
570
580
590
600
610
620
630
640
650
Ecolith cyan
Refractive index
1.64
1.63
1.62
1.61
1.61
1.60
1.59
1.58
1.58
1.57
1.56
1.54
1.53
1.51
1.49
1.46
1.44
1.43
1.43
1.43
1.44
1.46
1.49
1.52
1.55
1.58
k1
0.11
0.11
0.11
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.09
0.09
0.09
0.08
0.08
0.07
0.07
0.07
0.08
0.08
0.09
0.09
0.10
0.10
k2 Fresnel
0.67
0.66
0.66
0.66
0.65
0.64
0.64
0.64
0.64
0.63
0.62
0.62
0.61
0.60
0.59
0.57
0.55
0.54
0.55
0.55
0.55
0.57
0.59
0.60
0.62
0.64
k2-polynom
k2-Fresnel with
0.50
0.49
0.48
0.48
0.47
0.46
0.46
0.46
0.45
0.44
0.44
0.43
0.42
0.40
0.39
0.37
0.35
0.34
0.34
0.34
0.35
0.37
0.39
0.41
0.43
0.45
snell’s law for
maximum angle
0.37
0.37
0.39
0.39
0.39
0.39
0.39
0.40
0.40
0.40
0.42
0.42
0.42
0.44
0.46
0.47
0.49
0.49
0.49
0.47
0.47
0.45
0.44
0.42
0.40
0.39
40
Umeå University
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660
670
680
690
700
1.59
1.58
1.57
1.57
1.58
M-real TC
ÖRNSKÖLDSVIK
0.10
0.10
0.10
0.10
0.10
0.64
0.63
0.64
0.64
0.64
0.46
0.46
0.44
0.45
0.46
0.39
0.39
0.40
0.39
0.39
Table 4. The Saunderson corrections calculated with Fresnel’s law for
Ecolith cyan.
6.3
Saunderson with GRACE
The Saunderson corrections were simulated in GRACE and is
represented in table 5-7.
6.3.1
Mylar
The Saunderson correction k0 for the mylar film was simulated in
GRACE at different wavelengths. The refractive index was set to 1.6 for
the Mylar film and the surface roughness was set to 0.463 m.
Wavelength
400
410
420
430
440
450
460
470
480
490
500
510
520
530
540
550
560
570
580
590
600
610
620
630
640
650
660
670
680
690
700
k0, r0=0.463
0.09
0.09
0.09
0.08
0.08
0.08
0.08
0.08
0.09
0.08
0.08
0.08
0.08
0.09
0.08
0.09
0.08
0.08
0.07
0.08
0.08
0.07
0.08
0.08
0.07
0.08
0.07
0.06
0.07
0.08
0.07
Table 5. The Saunderson corrections, k0 simulated in GRACE for the
Mylar film.
41
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6.3.2
M-real TC
ÖRNSKÖLDSVIK
Ecolith black
The simulated values for k2 and k1 can be found in table 6 for Ecolith
black.
Wavelength
Refractive index
k2
k1
400
410
420
430
440
450
460
470
480
490
500
510
520
530
540
550
560
570
580
590
600
610
620
630
640
650
660
670
680
690
700
1.64
1.63
1.62
1.61
1.61
1.60
1.60
1.59
1.57
1.57
1.56
1.54
1.53
1.51
1.49
1.46
1.44
1.42
1.43
1.43
1.44
1.46
1.49
1.52
1.55
1.58
1.59
1.58
1.57
1.57
1.58
0.61
0.61
0.61
0.62
0.62
0.62
0.62
0.63
0.63
0.63
0.64
0.64
0.64
0.64
0.64
0.65
0.65
0.65
0.65
0.65
0.65
0.65
0.66
0.66
0.66
0.66
0.66
0.67
0.66
0.67
0.67
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.11
0.11
0.11
0.11
0.11
0.11
0.11
0.11
0.11
0.11
0.11
0.11
0.11
0.11
0.11
0.11
0.11
0.11
0.11
Table 6. The Saunderson correction, k1 and k2 in GRACE for Ecolith
black
6.3.3
Ecolith Cyan
The simulated values for k2 and k1 can be found in table 6 for Ecolith
cyan.
Wave-length
Refractive index
k2
k1
400
410
420
430
440
450
460
470
480
490
500
510
520
530
540
1.64
1.63
1.62
1.61
1.61
1.60
1.59
1.58
1.57
1.57
1.56
1.54
1.53
1.51
1.49
0.66
0.66
0.66
0.66
0.66
0.65
0.65
0.64
0.64
0.64
0.63
0.62
0.61
0.60
0.59
0.12
0.11
0.11
0.11
0.11
0.11
0.11
0.11
0.11
0.10
0.10
0.10
0.10
0.10
0.10
42
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550
560
570
580
590
600
610
620
630
640
650
660
670
680
690
700
M-real TC
ÖRNSKÖLDSVIK
1.46
1.44
1.42
1.43
1.43
1.44
1.46
1.49
1.52
1.55
1.58
1.59
1.58
1.57
1.57
1.58
0.577
0.55
0.55
0.55
0.554
0.56
0.57
0.59
0.60
0.62
0.64
0.65
0.64
0.64
0.64
0.64
0.09
0.09
0.09
0.08
0.08
0.08
0.09
0.09
0.10
0.10
0.11
0.11
0.11
0.11
0.11
0.11
Table 7. The Saunderson correction, k1 and k2 in GRACE for Ecolith cyan
6.3.4
k0
The simulations for k0 in GRACE did not turn out to give satisfactory
results for all wavelengths. Calculations of sKM and kKM for the Mylar film
with this constant gave a complex or negative value for some
wavelengths.
Conclusions was found that k0 could be determined experimentally. One
way to assess the magnitude of k0 is to measure the Mylar film over a
cavity (which absorbs all radiation) with and without ink. By studying the
measured reflectance in the absorption region, the magnitude of max k0
can be found.
The measured reflectance was studied for both cyan, black and for the
Mylar film (figure 31-33).
43
Umeå University
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ÖRNSKÖLDSVIK
Figure 31. Measurements for cyan with 9 different ink weights printed on
Mylar film.
Figure 32. Measurements for black with 8 different ink weights printed
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on Mylar film.
Figure 33. Measured reflectance for the Mylar film measured over a
cavity.
The figures above illustrates the reflectance measured over a cavity
within the visible spectrum (400-700nm). Each line corresponds to a
specific grammage.
6.3.5
Test of variabel Rg for different k0,k1,k2 ( Mylar)
The variable Rg-method (Equation2.61) was used to calculate the
scattering coefficient, sKM for the Mylar film with and without corrections.
k1-Mylar was determined to 0.1080 with Fresnel’s equation (equation
2.70). The refractive index for the Mylar film was 1.64. k2-Mylar was
determined with equation 2.74 to 0.4933.
Different values for k0 was tested. It would be natural to think that this
constant should be close to zero , because the film has a very glossy
surface, but k0=0 into equation 2.63 with these values of k1 and k2 gave a
complex or a negative value for sKM.
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Umeå University
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Measurements of the surface roughness, r0, gave a reading of 0.4633
μm. Simulations in GRACE with this value gave k0 near 0.07. This can be
compared to the measured reflectance at 400-480 nm.
The measured values of k0 at all wavelengths were used to calculate the
scattering and absorption parameters for the Mylar film. Unfortunately
these values for k0 did not give a successful result. sKM. and kKM.
calculated with the variable Rg-method turned out to be negative at some
wavelengths,0.8* k0 seemed to be a better value for k0.The correction
parameters for the background was set to 0.0960 and 0.4336 for k1 and
k2. k1was calculated with Fresnel’s equation and k2 was determined with
equation 2.74.
An important part of this section is to study the corrected reflectance.
The corrected reflectance should not exceed the measured reflectance
from the background. It is also important to study the R-infinity. The
value of R-infinity should be within the region 0-1.
6.3.6
Test of different
construction
k0,k1,k2 of Mylar and ink for a printed
The calculated correction parameters for the Mylar film, k0 k1 and k2,
should be used to correct the background reflectance, before using the
variable Rg-method to calculate s-ink. k1 and k2 was calculated according
to section 5.2.1.1-5.2.1.3.
As mentioned earlier in section 2.4.2.2, a value for k2 near 0.4 would be
better than a value near 0.6. To illustrate the influence of surface
reflectance on the Kubelka Munk equation, the scattering coefficient, sKM,
was calculated with and without the two different correction parameters
with the Kubelka Munk equation. The calculations were based upon the
measured reflectance values. sKM was calculated at 690 nm for cyan.
This is illustrated in figure 34.
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Umeå University
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M-real TC
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Figure 34. sKM for cyan calculated in Kubelka Munk with and without
Saunderson corrections at 690 nm.
The same was made for Ecolith black at the wavelength 690 nm.
Figure 35. sKM for black calculated in Kubelka Munk with and without
corrections at 690 nm
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Umeå University
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M-real TC
ÖRNSKÖLDSVIK
k0 was determined experimentally. By studying the reflectance in the
absorbing region in figure 31-32 it can be seen that k0 would be about
0.6% for both colours. This reflectance value is the measured reflectance
in the absorbing region, and should therefore correspond to the surface
reflectance, but it also contains the bulk reflectance as well. Calculations
of sKM and kKM with this value of k0 did not give satisfactory result; kKM
became very big and sKM became negative at some wavelengths.
With k0=0.6% as a starting point, the constant could be set by calculating
sKM and kKM for different values of k0 . The best value of k0 for Ecolith
black was set to 0.32%, and k0=0.55% for Ecolith cyan.
6.4
sKM(ink) and kKM(ink) with the variable Rg-method
The calculated values for the light scattering and absorption coefficients
calculated with the Kubelka Munk equation with and without corrections
for the ink is represented in this section. The Saunderson corrections
used in this section is the result from calculations made with Fresnel’s
equation for k1 and the polynomial expression for k2
6.4.1
Ecolith black
The light scattering and absorption coefficient for Ecolith black calculated
with the Kubelka Munk equation with and without corrections is illustrated
in figure 41. The figure shows how these parameters changes with
wavelength. Each grammage is representing a line with a specific colour.
Figure 41 The calculated values for sKM and kKM with the variable Rg48
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method in the Kubelka Munk equation plotted vs. wavelength.
The result has also been illustrated in a three dimensional plot (figure 4245).
In figure 42 the light scattering coefficient calculated with the Kubelka
Munk without correction is represented and plotted vs. grammage and
wavelength. Figure 43 shows the light scattering coefficient calculated
with the Kubelka Munk with corrections .
Figure 42 The calculated values for sKM with the variable Rg-method in
the Kubelka Munk equation without corrections plotted vs. wavelength,
and grammage
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Figure 43 The calculated values for s with the variable Rg-method in the
Kubelka Munk equation with corrections plotted vs. wavelength, and
grammage
Figure 44 The calculated values for kKM with the variable Rg-method in
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the Kubelka Munk equation without corrections plotted to wavelength,
and grammage
Figure 45 The calculated values for k with the variable Rg-method in the
Kubelka Munk equation with corrections plotted to wavelength, and
grammage
In figure 44 the absorption coefficient calculated with the Kubelka Munk
without correction is represented and plotted vs. grammage and
wavelength. Figure 45 shows the light abosorption coefficient calculated
with the Kubelka Munk and with corrections.
It can be seen from these figures that the Saunderson corrections does
not affect the result so much. There is still a variation in sKM even after
the Saunderson correction has been applied. It can be noticed that the
variations are bigger for wavelengths <500 nm.
6.4.2
Ecolith Cyan
The light scattering and absorption coefficient for Ecolith cyan calculated
with the Kubelka Munk equation with and without corrections is illustrated
in figure 46. The figure shows how these parameters changes with
wavelength. Each grammage is representing a line with a specific colour.
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Figure 46 The calculated values for s and k with the variable Rgv-method
in the Kubelka Munk equation plotted to wavelength.
This has also been illustrated in a three dimensional plot where the light
scattering and absorption coefficient is plotted vs. wavelength and
grammage. Figure 47 shows the result from Kubelka Munk without
corrections and figure 48 shows the result from Kubelka Munk with
corrections.
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Figure 47 The calculated values for sKM with the variable Rg-method in
the Kubelka Munk equation without corrections, plotted vs. wavelength
and grammage.
Figure 48 The calculated values for sKM with the variable Rg-method in
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the Kubelka Munk equation with corrections, plotted to wavelength, and
grammage
Figure 49 The calculated values for kKM with the variable Rg-method in
the Kubelka Munk equation without corrections plotted to wavelength,
and grammage
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Figure 50 The calculated values for kKM with the variable Rg-method in
the Kubelka Munk equation with corrections plotted to wavelength, and
grammage
It can be seen in the figures that there is also a variation in the
parameters for different grammage even after the Saunderson
corrections has been applied for cyan, but the variations decreases after
the corrections has been applied. An interesting result is that the
Saunderson corrections reduces the light scattering within the region
550-700 nm. The absorbing region for cyan is for wavelengths larger
than 500 nm so the results after the corrections has been made should
be reliable . The absorption coefficient is not affected as much as the
scattering coefficient of the Saunderson corrections.
6.5
sMYLAR with variable Rgv-method
The scattering coefficient and absorbing coefficient for the Mylar film
calculated with variable Rg-method at wavelengths 400-700nm is
presented in the following graph.
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Figure 51. s and k calculated in Kubelka Munk with and without
corrections for the Mylar film.
The figures that shows the corrected value for the absorption in figure 51
agrees with the values from the manufacturer [8].
6.6
The print density
corrections
with
and
without
Saunderson
The print density calculated with and without corrections are presented
for both Ecolith black and Ecolith cyan in this section.Figure 52 illustrates
the print density for Ecolith black calculated with the Kubelka Munk
equation compared to measured values. The blue “smooth” surface
represents the measured values for the print density over the
wavelengths and grammage and the meshed surface represents the
print density calculated with the Kubelka Munk equation without
corrections. The decrease in print density for high ink grammage was not
expected, and the reason not fully understood.
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Figure 52. The print density vs. wavelength and grammage for Ecolith
black .
Figure 53 illustrates the print density for Ecolith black calculated with the
Kubelka Munk equation with corrections compared to measured values.
The blue, “smooth” surface represents the measured values for the print
density over all wavelengths and grammage and the checked surface
represents the print density calculated with the Kubelka Munk equation
with corrections.
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Figure 53 The print density vs. wavelength and grammage for Ecolith
black.
When studying the both pictures (Figure 53 and 54) it can be seen that
the Saunderson corrections makes the calculations for printing density to
agree better with measured values over all wavelengths and grammage
than calculations made without the Saunderson corrections. It can be
noticed that the print density is not in total agreement with measured
values in either figure. The results from the print density for Ecolith cyan
can be studied below.
A three dimensional plot for the print density vs. wavelength and
grammage has been made for Ecolith cyan in figure 54-55. In figure 54,
the print density has been plotted to wavelength and grammage. The
blue, “smooth ” surface represents the measured value and the meshed
surface represents the values calculated with the Kubelka Munk equation
without corrections.
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Figure 54 The print density vs. wavelength and grammage for Ecolith
cyan.
Figure 55 shows, the print density plotted vs wavelength and grammage.
The blue, “smooth ” surface represents the measured value and the
meshed surface represents the values calculated with the Kubelka Munk
equation with corrections.
Figure 55 The print density vs. wavelength and grammage for Ecolith
cyan
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By studying figure 54 and 55, it can be seen that the Saunderson
correction make the Kubelka Munk print density to fit the measured
values for wavelengths >550 nm. For wavelength<500 nm the Kubelka
Munk print density agrees better with measured values without than with
corrections.
6.7
Test of parameters to fit the Kubelka Munk model
The result indicated that the theoretical calculated values of the
Saunderson correction did not make the Kubelka Munk reflectance and
print density curve fit the measured reflectance exactly for all
wavelengths.
To be able to determine the scattering and absorption coefficients for ink
printed on Mylar film, a pragmatic method was used. Different values of
the correction parameters k0, k1 and k2-ink respectively was tested to
make the Kubelka Munk reflectance to fit the curve including the
measured values. Cyan was studied in this section at the wavelength
500 nm, 570 nm and 620 nm.
An important part of this section is to study the influence of the
Saunderson corrections on both the Mylar film and the ink. i.e. the
factors that should be investigated is R, the corrected reflectance and
the scattering and absorption parameters sKM and kKM. The corrected
reflectance should not exceed the background reflectance. Negative and
complex or extremely large values are not accepted as result of the R or
of the optical properties in the Mylar film or the ink.
The result from the pragmatic approach is represented in this section.
cyan 500 nm
The Saunderson correction parameters that make the Kubelka Munk
model to fit the measured reflectance at 500 nm was k0, k1 and k2 =0 for
both the Mylar film and the ink. The result is presented in figure 56 and
57. The print density is plotted to grammage in figure 56.The red line
represents the corrected values. In this case, when the correction
parameters are zero, the results are identical with the uncorrected
results. The scattering and absorption parameters calculated with the
Kubelka Munk equation with corrections is plotted to grammage in figure
57.
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Figure 56. The Kubelka Munk print density for cyan calculated at 500 nm
with the Saunderson corrections.
Figure 57. The scattering and absorption parameter calculated at 500
nm.
It can be seen from the figure above that there is still a variance with
grammage after the Saunderson corrections has been applied.
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cyan 570 nm
The Saunderson correction parameters that make the Kubelka Munk
model to fit the measured reflectance at 570 nm was k0, k1 and k2
ink=0,0055, 0.1031 and 0,4 respectively. k0, k1 and k2 for the Mylar film
was set to 0.05, 0.1080 and 0.27. The print density is plotted vs.
grammage in figure 59.The red line represents the corrected values. The
scattering and absorption parameters calculated with the Kubelka Munk
equation with the correction is plotted to grammage in figure 58.
Figure 58. The scattering and absorption parameter calculated at 570
nm.
The corrected values for the light scattering parameter in the figure
above can be compared to the uncorrected parameter in the same
figure. The variation is much smaller after the corrections has been
made.
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Figure 59. The Kubelka Munk print density for cyan, calculated at 570
nm with the Saunderson corrections.
cyan 620 nm
The Saunderson correction parameters that make the Kubelka Munk
model to fit the measured reflectance at 620 nm was k0, k1 and k2
ink=0,00608, 0.1031 and 0,4 respectively. k0, k1 and k2 for the Mylar film
was set to 0.03, 0.1080 and 0.4933. The result is presented below.
Figure 60. The scattering and absorption parameter calculated at 620 nm
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Figure 61.The Kubelka Munk print density for cyan calculated at 620 nm
with the Saunderson corrections.
6.8
Optical dot gain
Following parameters was used for simulations
sgel / μm
0.0133-0.1333
kgel / μm
0.0001
sink / μm
0.02
kink / μm
1
dink
1
μm
μm
1,10,50
dgel
The result is presented below.
Figure 62. Optical dot-gain simulated with GRACE with different
scattering parameters for the gel coating layer.
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Figure 63. Optical dot-gain simulated with GRACE with different
scattering parameters for the gel coating layer.
Figure 64. Optical dot-gain simulated with GRACE with different
scattering parameters for the gel coating layer.
The optical dot gain has been plotted to the scattering coefficient for the
gel coating layer in figure 62-64. It can be seen that the dot gain can be
reduced by increasing the light scattering coefficient for the underlying
layer and that the dot gain can be reduced by increasing the thickness of
the layer.
It can be seen from figure 65 that the optical dot gain also depends on
the thickness of the ink.
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7
ANALYSIS AND DISCUSSION
7.1
General
M-real TC
ÖRNSKÖLDSVIK
During centuries, scientists and researcher has tried to describe nature
in a theoretical way. A theoretical description of nature always almost
includes approximations. One reason for this can be unknown physical
parameters but the most common reason is to be able to find a
mathematical solution to the problem. Light and light phenomena is very
complex and therefore difficult to describe.
The Kubelka Munk equation includes several approximations but is
frequently used within the paper and pulp industry for describing light
scattering and absorption in paper due to its simplicity and ease of use. It
is well known that this method is not optimal for describing light
scattering and absorption in material with strong absorption so it can be
discussed whether this method is reliable for describing light in ink. A
surface is represented by a difference in refractive index between two
materials. Saying that there is no surface would mean that there is no
difference in refractive index between the two materials. This is not
correct and should be observed. This thesis work has treated these
correction parameters and it has has been both exciting and informative.
Reality is, as mentioned earlier difficult to describe and this work
indicates that it still remains several physical properties in the printing
material and in the printing medium to be able to describe light scattering
and absorption in ink in a correct way. Surface roughness, the
composition of the ink at different grammage and more exact values on
the refractive index is parameters that increases the uncertainly in the
results.
The main purpose of this thesis work has been to investigate the
Saunderson correction and the influence of these corrections on
calculations made with the Kubelka Munk equation. The result has been
tested on real samples. Different samples with the colours Ecolith black
and cyan has been studied. It is important to notice that these
corrections are not general constants that can be used for any
measurement. They should be calculated for each specific sample. It has
also been found that the reflectance measurement varies between
measurements. Reflectance measurements of one sample varies from
one measurement to another. This can be explained by the calibrations
made for the detector and the ageing of the sample.
It can be seen from this experiments that the Saunderson corrections on
the Kubelka Munk equation works better for the wavelengths >550nm.
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One explanation to this can be that a difference in refractive index does
not affects radiation with high frequency and a surface correction would
therefore worse the result. Another parameter that can affect the result is
the insecurity of the refractive index measurements. The pragmatic part
of this work shows the parameters for cyan at a specific wavelength. By
comparing the different parameters at these wavelengths shows that the
constants received from this part differs from the theoretical part. It can
be discussed whether the pragmatic part is reliably or not — Are these
corrections the Saunderson correction parameters or not?
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7.2
The Saunderson corrections
7.2.1
k1
M-real TC
ÖRNSKÖLDSVIK
k1 is the correction parameter that has been the most easiest to describe
theoretically. This term represents the total surface reflectance at the
upper surface. Comparison between the value for k1 simulated in
GRACE and the directly calculated with the Fresnel’s equations shows
that these constants are not identical. The difference in the result can be
explained with the choice of numerical method and the resolution for the
detectors in the program. Applications of this term on real samples
assumes that the reflected light is diffuse, the light rays is assumed to be
collimated, and unpolarized and only dependent of the polar angle.
These approximations would lead to a difference between the theoretical
result and the true real values.
An important parameter that should be determined in the future is the
refractive index of the specific sample. Measurements should be
performed on Mylar film. The refractive index that is available today is
measured by ACREO with an ellipsometry measurement. It is well known
that the paper absorbs the mineral oil in the ink. The mineral oil and the
colour pigment has different refractive index and different composition
and it is therefore most likely that the refractive index for the ink printed
on paper and on a Mylar film is not the same.
7.2.2
k2
k2 is the term that has always been difficult to calculate analytically, since
it is impossible to describe the light distribution for light at the inside of
the upper surface. It would be most natural to describe the distribution as
a Lambertian distribution but with a smaller distribution in the way that
has been made in this report, but we still don’t know how the distribution
changes with increasing thickness of the colour. We would probably
never be able to get an answer to this problem.
Comparison between the values for k2 calculated with the polynom
obtained by Mudgett and Richards and the values for k2 calculated with
the Fresnel’s equation indicates that both is near 0.4. This is also
supported by other publications. k2 directly calculated with the Fresnel’s
equation and with a Lambertian distribution gave a value for k2 around
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0.6 and the result from this report indicates that this value for k2 is not
applicable in the calculations made with the Kubelka Munk equation.
7.2.3
k0
k0 should be around zero for a high glossy surface. Measurements of
surface roughness on both Mylar film and the printed surface indicate
that this is not true. k0 has been determined by studying the reflectance
within the absorbing area for cyan and within the whole spectrum for
black. This parameter has been regarded as a constant in the whole
spectrum and over all grammages for the ink. This introduces an error
because surface roughness measurements have shown that it varies
with grammage and wavelength , this variation can also be seen in the
figures from the reflectance measurements over a cavity .
k0 has appeared to be unexpectedly high for the Mylar film. It was set to
80% of the measured reflectance with black background since a tiny part
should come from the bulk scattering (about 20%).
The parameter k0 is described as a surface roughness parameter, but
another factors that should be included in this term is that a part of the
light can be reflected at the lens and an refractive index difference
between the reference and the sample could give a part that is not
collected by the gloss trap.
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7.2.4
M-real TC
ÖRNSKÖLDSVIK
Variable Rg- method
Calculations of s and k have been made in this report by using the
variable Rg-method. This is based upon the Kubelka Munk theory and the
derivation of this method is not investigated in this report.
This method assumes that the Mylar film is in complete optical contact
with the underlying reference paper. This is not completely correct,
because both the paper and the Mylar film has a certain surface
roughness. Another unknown factor is what happens at the surface
between the Mylar film and the ink. It is well known that these two
mediums has different refractive index. The electrical properties of the
Mylar film and the ink can also be discussed and investigated. Questions
like these are very difficult to find an answer to. Perhaps some of them
can be neglected and some not.
7.2.5
sKM and kKM with and without corrections
It can be seen in this report that the Saunderson corrections reduces the
dependence of grammage for sKM and kKM calculated with the Kubelka
Munk equation. It is known that these parameters should be constant
and not vary with grammage. There is still a variation after the
corrections has been applied but they are smaller than without
corrections.
7.3
Simulations for Ink raster object
Simulations in GRACE have been made for ink raster object. The optical
dot gain has been studied at different grammage and light scattering
coefficients for the gel-coating layer between the ink and the paper.
The purpose of this part was to investigate the possibility to simulate the
optical dot gain with the Monte Carlo simulation tool GRACE. The result
from this part shows that the optical dot gain can be reduced by
increasing the light scattering parameter for the gel coating layer or
increasing the thickness for the layer. More calculation and analyses
should done since optical dot gain normally has a value of about 10% for
coated papers.
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CONCLUSIONS
It can be seen from this experiment that the light scattering coefficient
sKM and kKM varies with grammage when using the Kubelka Munk
equation without the Saunderson corrections. The theoretical values of
the Saunderson corrections has turned out to correct the Kubelka Munk
equation in a satisfactory way for Ecolith cyan at wavelength >550nm.
The Saunderson correction depends on the optical geometry of the
device and the surface roughness for the sample. These parameters
depend on the refractive index for the material and in that way they are
also dependent of wavelength as well. k1 is the total external reflectance.
k2 is the internal reflectance. k0 is the part that is missed by the gloss trap.
It still remains several physical parameters for describing the reflectance
in a correct way and due to the fact that these methods used includes
several approximations. The Saunderson corrections is not the only
parameters that are needed to correct the Kubelka Munk equation.
It is now possible to simulate the optical dot gain with the Monte Carlo
simulation program GRACE 2.4. Simulations in GRACE has shown that
the optical dot gain can be reduced by increasing the light scattering
parameter for the underlying layer or by increasing the thickness of the
underlying layer.
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REFERENCES
[1] K. Johansson, P. Lundberg , R. Rydberg, “Grafisk kokbok 2.0”
second edition, ARENA, ISBN 91-7843-161-1, 2001,
[2] Frank L. Pedrotti, , Pedrotti , S. Leno, ”Introduction to optics”,
second edition, Prentice-Hall International,Inc, 1996, ISBN 0-13-0169730.
[3] C. Nordling, J.Österman, “Physics handbook for science and
engineering”, sixth edition, studentlitteratur, ISBN 91-44-00823-6.
[4] P. Edstrom, “Fast and stable solution method for angle-resolved light
scattering simulation”, 2002
[5] J, H. Nobbs, “Kubelka-Munk theory and the prediction of reflectance”,
Rev. Prog. Coloration vol. 15, 1985.
[6] P. Kubelka, “New contributions to the Optics of Intensely Lightscattering Materials”. Part I, J. Opt. Soc. Am 38 , 1948
[7] C.J. Bartleson, F. Grum,”optical radiation measurements” vol.2,
ACADEMIC PRESS INC,1980.
[8] DuPont Teijin films ,Product information Mylar polyester film
[9] A. Schuster, “Radiation trough a foggy atmosphere”, Astrophys. J.
21, 1905
[10] P. Kubelka, “New contributions to the Optics of Intensely Lightscattering Materials”. Part II, J. Opt. Soc. Am 44 , 1954.
[11] P.Kubelka Munk F, “Ein Beitrag zur Optik der Farbanstriche”, Z.
Tech. Phys, 1931.
[12] G.C. Wick, “Uber ebene Diffusionsprobleme”, Z. Phys 120, 1943.
[13] H. Wiklund, L University of Technology “Colour content in gloss”,
Thesis work M-real 2000,
[14] W.F. Sullivan, “Absolute Reflectances fron reflectometer readings ”,
APPLIED OPTICS, vol. 10, NO.7, (July 1971), 1550.
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[15] P.S. Mudgett, L.W. Richards, “Kubelka-Munk Scattering and
absorption Coefficients For Use with Glossy, Opaque Objects ”, Cabo
corporation, Concord Rd., Billerica, Mass 01821.
[16] J.L. Saunderson, “Calculation of the Colour of Pigmented Plastics”J.
Opt. Soc. Am. 32, 61, 1942.
[17] N. Pauler, “Paper optics” , AB LORENTZEN & WETTRE, ISBN, 91971-765-6-7, 2002.
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10
APPENDIX
10.1
Reflectance measurements
10.1.1
Mylar
Rgv (%)
Rgs
(%)
5.14
4.98
4.88
4.81
4.76
4.74
4.75
4.76
4.78
4.79
4.81
4.83
4.82
4.77
4.68
4.59
4.51
4.45
4.39
4.35
4.35
4.39
4.47
4.54
4.62
4.76
4.95
5.20
5.6
6.11
6.60
87.5
88.4
89.3
89.8
90.2
90.8
91.4
91.8
92.2
92.5
93.0
93.4
93.8
94.1
94.3
94.5
94.6
94.8
95.1
95.3
95.5
95.6
95.7
95.8
95.8
95.8
95.9
95.9
96.1
96.2
96.2
Rs (%)
Rv (%)
10.25
10.18
10.1
9.91
9.78
9.63
9.47
9.34
9.26
9.16
9.07
8.96
8.82
8.67
8.51
8.36
8.21
8.07
7.93
7.81
7.75
7.71
7.69
7.69
7.69
7.71
7.79
7.93
8.15
8.45
8.75
70.65
73.2
75.1
76.7
77.9
78.9
79.7
80.3
81.0
81.7
82.2
82.6
82.8
83.1
83.4
83.6
83.8
84.0
84.1
84.3
84.5
84.6
84.8
84.9
85.0
85.2
85.4
85.4
85.5
85.4
85.4
10.1.2
Ecolith cyan
10.1.2.1
Rs (%) at different wavelengths and grammage
Wavellength/
grammage
400
0.38
0.79
1.08
1.30
1.71
2.29
2.60
3.46
4.8
8.30
7.30
6.30
5.60
4.90
4.38
4.07
3.46
2.97
410
8.95
8.31
7.57
6.97
6.31
5.82
5.44
4.72
4.07
420
9.12
8.60
7.96
7.41
6.81
6.39
6.02
5.36
4.64
430
9.27
8.89
8.39
7.90
7.42
7.05
6.76
6.15
5.49
74
Umeå University
Department of physics
Thesis project
M-real TC
ÖRNSKÖLDSVIK
440
9.58
9.43
9.07
8.70
8.37
8.12
7.93
7.47
7.02
450
9.70
9.680
9.41
9.13
8.87
8.72
8.57
8.24
7.95
460
9.57
9.53
9.25
8.99
8.73
8.59
8.43
8.11
7.82
470
9.36
9.27
8.98
8.69
8.42
8.25
8.07
7.71
7.37
480
9.12
8.93
8.60
8.28
7.97
7.75
7.56
7.12
6.69
490
8.80
8.52
8.13
7.77
7.42
7.15
6.92
6.41
5.90
500
8.44
8.02
7.59
7.19
6.81
6.47
6.21
5.63
5.02
510
7.96
7.40
6.92
6.4600
6.01
5.62
5.31
4.67
3.97
520
530
7.36
6.65
6.63
5.73
6.06
5.05
5.54
4.46
5.01
3.87
4.55
3.36
4.22
3.02
3.52
2.35
2.79
1.72
540
5.86
4.73
3.94
3.31
2.71
2.23
1.93
1.42
1.01
550
4.95
3.62
2.78
2.19
1.69
1.35
1.15
0.88
0.71
560
3.97
2.53
1.77
1.33
1.02
0.84
0.76
0.67
0.62
570
3.14
1.75
1.16
0.87
0.72
0.66
0.64
0.61
0.59
580
2.61
1.35
0.90
0.71
0.64
0.63
0.62
0.60
0.58
590
2.32
1.15
0.79
0.65
0.62
0.61
0.61
0.59
0.58
600
2.10
1.03
0.74
0.63
0.61
0.62
0.62
0.61
0.59
610
1.98
0.96
0.72
0.63
0.62
0.64
0.63
0.62
0.60
620
1.95
0.95
0.72
0.64
0.63
0.65
0.65
0.64
0.61
630
1.98
0.96
0.72
0.64
0.63
0.65
0.65
0.64
0.61
640
2.05
0.99
0.73
0.64
0.63
0.65
0.64
0.63
0.61
650
2.23
1.06
0.76
0.64
0.62
0.63
0.63
0.62
0.59
660
2.48
1.21
0.82
0.66
0.62
0.62
0.62
0.61
0.58
670
2.63
1.29
0.85
0.68
0.62
0.61
0.61
0.60
0.57
680
2.63
1.27
0.83
0.66
0.62
0.61
0.61
0.60
0.57
690
2.57
1.19
0.79
0.64
0.60
0.61
0.61
0.60
0.57
700
2.51
1.13
0.76
0.64
0.61
0.62
0.62
0.61
0.58
10.1.2.2
Rv (%)at different wavelengths and grammage
Wavellength/
grammage
400
0.38
0.79
1.08
1.30
1.71
2.29
2.60
3.46
4.80
50.8
39.2
31.33
26.14
20.5
15.8
12.8
8.83
5.59
410
58.4
49.0
42.0
37.1
31.1
25.9
22.0
16.4
10.8
420
62.5
54.3
48.0
43.5
37.8
32.6
28.6
22.4
15.7
430
66.4
59.5
54.1
50.1
45.0
40.2
36.4
30.1
22.7
440
70.9
65.9
61.6
58.5
54.3
50.3
47.0
41.4
34.1
450
74.2
70.4
67.1
64.6
61.3
58.0
55.3
50.4
43.8
460
75.57
72.2
69.2
67.1
64.0
61.1
58.7
54.2
48..0
470
76.3
73.0
70.1
68.0
65.1
62.3
60.0
55.7
49.8
480
76.47
73.0
69.9
67.8
64.8
61.9
60.0
55.3
49.3
490
76.1
72.2
68.8
66.4
63.2
60.1
57.6
53.0
46.7
599
75.1
70.3
66.5
63.7
60.1
56.5
53.8
48.6
41.8
510
72.9
67.0
62.4
59.0
54.6
50.3
47.1
41.1
33.7
520
530
69.5
64.7
61.8
54.5
55.9
47.0
51.6
41.6
46.1
35.1
40.9
29.3
37.1
25.2
30.5
18.8
22.9
12.4
540
58.3
45.1
36.0
29.9
23.1
17.5
14.0
9.09
4.99
550
49.7
33.5
23.7
17.9
12.1
8.11
5.89
3.31
1.66
560
39.5
21.7
12.9
8.49
4.90
2.89
1.97
1.15
0.77
75
Umeå University
Department of physics
Thesis project
M-real TC
ÖRNSKÖLDSVIK
570
30.6
13.3
6.43
3.72
1.92
1.17
0.88
0.70
0.64
580
24.9
8.99
3.72
2.03
1.10
0.80
0.69
0.65
0.63
590
21.6
6.81
2.62
1.44
0.86
0.72
0.66
0.64
0.62
600
18.9
5.37
1.99
1.15
0.77
0.70
0.65
0.65
0.63
610
17.3
4.51
1.64
1.00
0.75
0.70
0.66
0.66
0.65
620
16.9
4.30
1.57
0.97
0.74
0.71
0.67
0.67
0.66
630
17.2
4.39
1.60
0.98
0.75
0.72
0.68
0.68
0.66
640
18.2
4.80
1.76
1.04
0.77
0.73
0.68
0.68
0.66
650
20.4
5.93
2.22
1.24
0.82
0.73
0.67
0.67
0.65
660
23.4
7.59
2.98
1.61
0.93
0.76
0.68
0.67
0.64
670
24.8
8.45
3.40
1.83
1.00
0.78
0.69
0.66
0.63
680
24.1
8.02
3.18
1.72
0.96
0.76
0.68
0.66
0.63
690
22.4
6.93
2.65
1.44
0.87
0.74
0.66
0.66
0.63
700
20.6
5.95
2.21
1.24
0.82
0.730
0.67
0.67
0.64
3.463
4.8
10.1.2.3
Rgs (%)at different wavelengths and grammage
Wavellength/
grammage
400
0.375
0.788
1.075
1.300
1.713
2.288
2.6
10.3
10.3
10.4
10.2
10.2
10.2
10.2
10.2
10.1
410
10.2
10.2
10.3
10.1
10.1
10.1
10.0
10.18
10.0
420
10.1
10.1
10.1
9.98
9.99
9.98
10.0
10.05
9.97
430
9.9
9.96
10.0
9.85
9.84
9.84
9.90
9.90
9.82
440
9.80
9.82
9.87
9.73
9.71
9.71
9.70
9.77
9.69
450
9.67
9.68
9.73
9.59
9.57
9.56
9.65
9.62
9.57
460
9.50
9.52
9.56
9.45
9.41
9.39
9.49
9.46
9.41
470
9.36
9.38
9.42
9.31
9.28
9.26
9.35
9.33
9.27
480
9.29
9.30
9.34
9.24
9.20
9.18
9.28
9.25
9.20
490
9.19
9.21
9.23
9.14
9.10
9.09
9.18
9.15
9.10
500
9.10
9.12
9.15
9.05
9.02
9.01
9.11
9.08
9.02
510
8.99
9.01
9.04
8.94
8.91
8.89
9.00
8.96
8.91
520
530
8.85
8.70
8.86
8.71
8.89
8.74
8.80
8.64
8.78
8.63
8.75
8.60
8.87
8.71
8.82
8.66
8.77
8.61
540
8.54
8.55
8.58
8.49
8.46
8.44
8.55
8.51
8.40
550
8.40
8.40
8.44
8.34
8.32
8.29
8.40
8.36
8.31
560
8.25
8.25
8.28
8.19
8.18
8.15
8.25
8.22
8.17
570
8.11
8.11
8.13
8.05
8.04
8.01
8.11
8.07
8.02
580
7.97
7.97
8.00
7.90
7.89
7.87
7.97
7.93
7.88
590
7.85
7.85
7.87
7.79
7.77
7.75
7.8
7.82
7.77
600
7.79
7.77
7.80
7.72
7.71
7.68
7.79
7.75
7.69
610
7.74
7.74
7.76
7.68
7.67
7.64
7.740
7.70
7.66
620
7.73
7.72
7.74
7.66
7.65
7.62
7.72
7.68
7.63
630
7.72
7.71
7.73
7.65
7.64
7.62
7.71
7.68
7.63
640
7.72
7.71
7.74
7.66
7.65
7.63
7.73
7.69
7.64
650
7.75
7.74
7.77
7.69
7.68
7.66
7.76
7.72
7.67
660
7.83
7.82
7.85
7.76
7.77
7.74
7.85
7.80
7.75
670
7.97
7.95
7.98
7.90
7.90
7.88
7.98
7.93
7.88
680
8.18
8.16
8.20
8.12
8.11
8.09
8.19
8.14
8.10
76
Umeå University
Department of physics
Thesis project
M-real TC
ÖRNSKÖLDSVIK
690
8.48
8.47
8.49
8.41
8.41
8.39
8.48
8.45
8.40
700
8.77
8.77
8.79
8.72
8.72
8.69
8.79
8.76
8.70
10.1.2.4
Rgv (%)at different wavelengths and grammage
Wavellength/
grammage
400
0.375
0.788
1.075
1.300
1.713
2.288
2.6
3.463
4.8
70.9
70.6
71.2
70.4
71.03
70.9
70.71
71.5
70.4
410
73.40
73.2
73.7
72.9
73.5
73.3
73.4
73.5
73.1
420
75.2
75.1
75.5
74.8
75.2
75.1
75.2
75.3
74.9
430
76.8
76.7
77.0
76.43
76.8
76.7
76.8
76.8
76.5
440
78.0
78.0
78.3
77.8
78.0
78.0
78.1
78.1
77.9
450
78.9
79.0
79.3
78.8
79.0
78.9
79.1
79.1
78.9
460
79.6
79.7
80.0
79.6
79.7
79.6
79.9
79.81
79.6
470
80.3
80.3
80.6
80.2
80.3
80.2
80.5
80.3
80.2
480
81.0
81.1
81.4
81.0
81.0
80.9
81.2
81.1
80.9
490
81.7
81.7
82.0
81.2
81.7
81.6
81.9
81.8
81.7
500
82.2
82.3
82.6
82.2
82.3
82.1
82.5
82.3
82.3
510
82.5
82.6
82.0
82.5
82.6
82.5
82.9
82.7
82.6
520
530
82.8
83.1
82.8
83.1
83.2
83.4
82.7
82.9
82.8
83.1
82.8
83.0
83.1
83.3
82.9
83.1
82.8
83.0
540
83.3
83.3
83.7
83.2
83.3
83.2
83.5
83.3
83.3
550
83.6
83.6
84.0
83.5
83.6
83.5
83.8
83.6
83.6
560
83.8
83.8
84.2
83.7
83.8
83.8
84.1
83.9
83.8
570
84.0
84.0
84.4
83.9
84.0
84.0
84.3
84.12
84.0
580
84.1
84.1
84.5
84.0
84.1
84.0
84.4
84.2
84.1
590
84.2
84.2
84.7
84.1
84.2
84.1
84.5
84.3
84.2
600
84.4
84.4
84.8
84.2
84.4
84.3
84.6
84.4
84.3
610
84.56
84.
84.9
84.4
84.5
84.4
84.7
84.5
84.4
620
84.7
84.7
85.1
84.5
84.7
84.6
84.8
84.7
84.6
630
84.8
84.8
85.2
84.7
84.8
84.7200
85.0
84.9
84.7
640
85.0
85.0
85.4
84.8
85.0
84.9
85.2
650
85.2
85.2
85.6
85.0
85.2
85.1
85.3
85.2
85.1
660
85.3
85.3
85.7
85.2
85.4
85.2
85.5
85.4
85.3
670
85.4
85.4
85.8
85.2
85.4
85.3
85.5
85.4
85.3
680
85.4
85.4
85.7
85.2
85.4
85.3
85.5
85.4
85.3
690
85.4
85.3
85.7
85.2
85.4
85.3
85.5
85.4
85.3
700
85.3
85.3
85.6
85.1
85.3
85.2
85.4
85.3
85.2
10.1.3
Ecolith black
10.1.3.1
Rs (%)at different wavelengths and grammage
Wavellength/
grammage
400
85.0
84.9
0.35
0.71
1.10
1.54
1.85
2.34
2.74
4.08
4.50
0.85
0.64
0.62
0.61
0.64
0.63
3.71
77
Umeå University
Department of physics
Thesis project
M-real TC
ÖRNSKÖLDSVIK
410
4.39
0.85
0.64
0.62
0.61
0.62
0.63
3.61
420
4.30
0.83
0.63
0.63
0.61
0.62
0.61
3.46
430
4.21
0.81
0.60
0.59
0.59
0.59
0.59
3.34
440
4.12
0.81
0.59
0.57
0.56
0.57
0.58
3.25
450
4.04
0.81
0.60
0.57
0.55
0.56
0.56
3.15
460
3.98
0.80
0.58
0.55
0.54
0.54
0.56
3.05
470
3.93
0.79
0.56
0.53
0.52
0.53
0.54
2.97
480
3.89
0.80
0.55
0.52
0.51
0.53
0.53
2.90
490
3.85
0.79
0.54
0.51
0.5
0.51
0.51
2.84
500
3.83
0.80
0.54
0.51
0.5
0.51
0.51
2.79
510
3.79
0.80
0.54
0.51
0.49
0.50
0.50
2.73
520
530
3.75
3.71
0.80
0.79
0.53
0.52
0.49
0.48
0.48
0.47
0.49
0.48
0.49
0.48
2.67
2.61
540
3.68
0.79
0.51
0.47
0.46
0.47
0.48
2.57
550
3.65
0.80
0.51
0.47
0.46
0.47
0.47
2.52
560
3.60
0.79
0.50
0.46
0.45
0.46
0.46
2.47
570
3.56
0.78
0.49
0.44
0.43
0.44
0.45
2.42
580
3.53
0.78
0.49
0.44
0.42
0.44
0.44
2.39
590
3.50
0.78
0.49
0.44
0.41
0.43
0.43
2.34
600
3.50
0.79
0.49
0.44
0.42
0.43
0.44
2.32
610
3.48
0.79
0.48
0.43
0.41
0.43
0.43
2.28
620
3.47
0.80
0.48
0.42
0.41
0.42
0.43
2.24
630
3.48
0.81
0.48
0.43
0.41
0.42
0.42
2.22
640
3.49
0.83
0.49
0.43
0.40
0.42
0.42
2.19
650
3.49
0.83
0.48
0.41
0.39
0.41
0.41
2.15
660
3.52
0.84
0.47
0.41
0.39
0.41
0.41
2.12
670
3.56
0.87
0.49
0.42
0.39
0.4
0.41
2.1
680
3.63
0.91
0.50
0.42
0.39
0.41
0.42
2.08
690
3.71
0.94
0.50
0.42
0.39
0.4
0.41
2.06
700
3.80
0.97
0.50
0.42
0.39
0.4
0.41
2.05
10.1.3.2
Rv (%) at different wavelengths and grammage
Wavellength/
grammage
400
0.35
0.71
1.10
1.54
1.85
2.34
2.74
4.80
10.2
1.61
0.72
0.63
0.62
0.63
0.65
3.78
410
10.6
1.73
0.74
0.67
0.64
0.64
0.66
3.67
420
11.0
1.82
0.74
0.69
0.64
0.64
0.64
3.56
430
11.3
1.9
0.73
0.65
0.62
0.62
0.61
3.41
440
11.7
2.01
0.73
0.63
0.60
0.61
0.61
3.30
450
12.1
2.14
0.75
0.63
0.59
0.59
0.60
3.22
460
12.5
2.26
0.76
0.62
0.58
0.58
0.58
3.12
470
12.9
2.39
0.77
0.61
0.57
0.57
0.57
3.03
480
13.3
2.52
0.79
0.61
0.56
0.56
0.57
2.98
490
13.7
2.66
0.80
0.61
0.54
0.54
0.55
2.91
500
14.2
2.82
0.83
0.62
0.55
0.54
0.56
2.86
510
14.6
2.98
0.86
0.62
0.54
0.54
0.55
2.81
520
530
15.0
15.4
3.13
3.29
0.88
0.91
0.62
0.62
0.53
0.52
0.5
0.52
0.54
0.52
2.74
2.68
78
Umeå University
Department of physics
Thesis project
M-real TC
ÖRNSKÖLDSVIK
540
15.8
3.45
0.94
0.61
0.52
0.51
0.52
2.64
550
16.2
3.6
0.97
0.62
0.52
0.51
0.52
2.59
560
16.5
3.74
1.00
0.63
0.51
0.50
0.51
2.54
570
16.8
3.86
1.02
0.62
0.50
0.48
0.50
2.50
580
17.2
4.01
1.05
0.63
0.50
0.49
0.49
2.46
590
17.5
4.15
1.09
0.63
0.49
0.48
0.48
2.42
600
17.9
4.31
1.13
0.65
0.50
0.48
0.49
2.39
610
18.2
4.47
1.17
0.65
0.50
0.47
0.48
2.36
620
18.6
4.63
1.21
0.67
0.50
0.47
0.48
2.32
630
19.0
4.81
1.26
0.69
0.50
0.47
0.47
2.29
640
19.4
5.01
1.31
0.71
0.51
0.47
0.48
2.27
650
19.8
5.22
1.37
0.72
0.50
0.46
0.47
2.23
660
20.2
5.43
1.45
0.74
0.50
0.46
0.46
2.20
670
20.6
5.64
1.52
0.77
0.52
0.46
0.47
2.18
680
20.9
5.80
1.58
0.79
0.53
0.47
0.48
2.16
690
21.1
5.94
1.62
0.81
0.53
0.47
0.47
2.14
700
21.3
6.03
1.65
0.82
0.53
0.47
0.47
2.13
10.1.3.3
Rgs (%) at different wavelengths and grammage
Wavellength/
grammage
400
0.35
0.71
1.10
1.54
1.85
2.34
2.74
4.8
10.2
10.4
10.3
10.19
10.2
10.3
10.3
10.2
410
10.1
10.4
10.2
10.14
10.1
10.2
10.2
10.1
420
10.0
10.1
10.1
10.01
10.0
10.1
10.1
9.98
430
9.89
9.98
9.97
9.87
9.87
9.90
9.96
9.84
440
9.76
9.83
9.84
9.74
9.74
9.77
9.81
9.72
450
9.61
9.68
9.69
9.60
9.59
9.63
9.68
9.58
460
9.45
9.52
9.53
9.43
9.43
9.46
9.52
9.42
470
9.32
9.38
9.39
9.31
9.31
9.33
9.38
9.30
480
9.24
9.30
9.32
9.24
9.22
9.25
9.29
9.22
490
9.14
9.18
9.21
9.13
9.12
9.15
9.19
9.12
500
9.05
9.11
9.12
9.04
9.04
9.06
9.11
9.03
510
8.95
8.99
9.01
8.93
8.93
8.95
8.99
8.92
520
530
8.81
8.66
8.85
8.71
8.88
8.73
8.79
8.64
8.79
8.64
8.81
8.66
8.86
8.71
8.78
8.62
540
8.50
8.54
8.57
8.48
8.48
8.50
8.55
8.47
550
8.35
8.39
8.42
8.33
8.34
8.35
8.40
8.32
560
8.20
8.24
8.26
8.18
8.19
8.20
8.25
8.18
570
8.06
8.09
8.12
8.03
8.05
8.06
8.11
8.04
580
7.92
7.96
7.98
7.89
7.91
7.92
7.97
7.90
590
7.81
7.84
7.86
7.78
7.80
7.80
7.85
7.78
600
7.75
7.77
7.80
7.73
7.73
7.74
7.78
7.71
610
7.71
7.73
7.76
7.68
7.70
7.70
7.74
7.68
620
7.69
7.72
7.74
7.66
7.68
7.68
7.72
7.66
630
7.68
7.71
7.73
7.66
7.67
7.67
7.72
7.65
640
7.68
7.71
7.74
7.66
7.68
7.68
7.73
7.66
650
7.71
7.74
7.76
7.68
7.71
7.71
7.75
7.69
660
7.79
7.81
7.84
7.76
7.79
7.79
7.83
7.77
79
Umeå University
Department of physics
Thesis project
M-real TC
ÖRNSKÖLDSVIK
670
7.93
7.95
7.98
7.90
7.92
7.92
7.97
7.90
680
8.14
8.17
8.19
8.12
8.13
8.1
8.18
8.10
690
8.46
8.48
8.49
8.42
8.43
8.44
8.48
8.40
700
8.75
8.78
8.79
8.73
8.73
8.74
8.77
8.72
10.1.3.4
Rgv (%)at different wavelengths and grammage
Wavellength/
grammage
400
0.35
0.71
1.10
1.54
1.85
2.34
2.74
4.80
70.7
70.9
70.5
70.6
70.8
70.8
70.5
70.4
410
73.2
73.4
73.1
73.1
73.3
73.3
73.1
73.0
420
75.2
75.3
75.0
75.1
75.3
75.2
75.0
74.9
430
76.8
76.9
76.6
76.7
76.8
76.7
76.6
76.5
440
78.0
78.1
77.9
78.0
78.1
78.0
77.9
77.9
450
78.0
79.1
78.8
79.0
79.1
78.9
78.87
78.9
460
79.7
79.8
79.6
79.7
79.8
79.6
79.6
79.6
470
80.4
80.4
80.2
80.3
80.4
80.2
80.2
80.2
480
81.1
81.2
81.0
81.1
81.2
80.9
81.0
81.0
490
81.7
81.8
81.6
81.7
81.8
81.6
81.6
81.6
500
82.2
82.3
82.1
82.3
82.3
82.1
82.2
82.1
510
82.6
82.7
82.5
82.6
82.6
82.5
82.5
82.5
520
530
82.9
83.1
82.9
83.2
82.8
83.1
82.9
83.1
82.9
83.2
82.8
83.0
82.7
83.0
82.7
82.9
540
83.5
83.5
83.3
83.4
83.4
83.3
83.3
83.2
550
83.7
83.8
83.6
83.7
83.7
83.6
83.6
83.5
560
83.9
84.0
83.8
83.9
83.9
83.8
83.8
83.7
570
84.1
84.2
84.0
84.1
84.1
84.0
83.9
83.9
580
84.2
84.3
84.1
84.2
84.2
84.1
84.1
84.0
590
84.4
84.4
84.2
84.3
84.3
84.2
84.2
84.1
600
84.6
84.6
84.4
84.5
84.5
84.4
84.4
84.2
610
84.7
84.8
84..
84.7
84.6
84.6
84.5
84.34
620
84.9
85.0
84.7
84.8
84.8
84.7
84.7
84.5
630
85.0
85.1
84.9
85.0
85.0
84.9
84.8
84.7
640
85.1
85.2
85.0
85.1
85.1
85.0
85.0
84.9
650
85.2
85.4
85.1
85.2
85.3
85.2
85.2
85.0
660
85.4
85.5
85.3
85.4
85.4
85.3
85.3
85.2
670
85.5
85.6
85.4
85.5
85.5
85.4
85.4
85.3
680
85.5
85.6
85.4
85.5
85.5
85.4
85.4
85.3
690
85.5
85.6
85.4
85.5
85.5
85.4
85.4
85.3
700
85.5
85.5
85.4
85.5
85.3
85.3
85.2
85.3
80