EXPERIMENT – III
(2 sessions)
MEASUREMENT OF THE REFRACTIVE INDEX OF A PRISM:
DISPERSION RELATION [n()] IN THE GLASS
1- OBJECTIVES AND THEORETICAL BASIS
This experiment is organized to be carried out in two session:
1st Session: Experimental measurement of the refractive index of a transparent material.
· We first need the material to be cut
in the shape of a prism, i.e. at least two
planar and polished faces forming an angle
. The result is a triangular prism, in which
the third face takes no part in the refraction
process (Fig.1). When a beam of light is
incident on one face it is refracted inside the
prism, travels to the other where is refracted
again, producing an overall deviation of the
beam, . This angle depends on the angle ,
on the angle of incidence on the first face,
and on the refractive index of the media, n
(n=c/v, a velocity ratio that shall be
considered a constant in this part).
n 

´
´


Fig.1: Refraction of a collimated beam (represented by a
single ray) at the entrance and exit faces of a prism.  is the
dihedron angle, characteristic of the prism, and is the
deviation angle.
· When calculated, it is found that  has a minimum, m, for a certain angle of incidence, this
magnitude depending only on the value of  and n. This angle –the smallest value found for - is
not a difficult magnitude to measure, and, together with the measurement of the angle , will allow
us to calculate the value of the refractive index.
The aforementioned dependence of the index can be summarized in the following expression:
sin 

n
 m   
 2
sin 

2 
(1)
· The value of  can be either known beforehand from the manufacturer or measured with the
same set-up used in the measurement of m. The measurement of m is done for an incident beam of
a wavelength corresponding to the yellowish-orange doublet of Na, o  5893Å, located in an
intermediate zone of the visible spectrum.
2nd Session: There are two objectives in this session: We must carry out an estimate of the
dispersion curve associated to the material of which the prism is constituted. Secondly, and for the
sake of completion, we shall measure the index of some unknown liquid with the help of a
conventional instrument known as the Abbe refractometer.
· The dependence of the refractive index (i.e. the dependence of the speed of light in the
material), with  is known as dispersion relation. Then, in n depends on , m depends as well. The
dispersion curve is characteristic of the material, and for a typical glass in the visible range it has
the shape shown in Fig.2 (corresponding to the so-called normal dispersion). An accurate
knowledge of the shape of these curves requires that additional values of n are obtained for several
wavelengths. In our case we shall use one towards the red and another towards the blue.
· In the Na lamp used in this
experiment, we find, among others, the
following emission lines:
r =6161Å (red)
v =5685 Å (average value of a
green doublet)
We shall measure the values of
m corresponding to these wavelengths
(obviously  remains the same), and
from Eq. (1) we obtain the values of
the refractive index n(r) and n(v).
This pair of values constitute two
points in the dispersion curve of the
material, n(). It is expected that n(v)
> n(r) (normal dispersion).
Fig.2: Dispersion curves for some transparent materials
· There are analytical expressions
for the dependence n(). The simplest
expression is the reduced Cauchy Equation, a development in which we take only two terms,
enough for an approximate description of the behaviour of meny materials in the visible:
n() = A + B / 2
(2)
With the values obtained for n(v) and n(r) we can fit the coefficients A and B in Eq.2 and
obtained an approximate dispersion curve for the glass of our experiment. With this formula, we
can substitute for the value o=5893Å of the first session and obtain approximately n(o). But there
is another practical application of this Cauchy Equation: We can use it to obtain a parameter used
by manufactures to characterize the dispersion power of a glass: the Abbe Number. It is defined as:
n 1
V= d
(3)
n F  nC
where nd, nF and nC are by definition the indices of the glass for the following wavelengths:
d → d  5876Å (transition of an He atom)
F→
C→
F  4861Å
C  6563Å
(transition of a H atom
(transition of a He atom)
· Measurement of the refractive index of a liquid, by means of Abbe’s refractometer. [This is
the third objective, and it can be carried out either at the end of the first or the second session].
The refractometer is an instrument devised by Abbe, and based in the phenomenon of total
internal reflection (TIR) experienced by rays when passing from a medium to another of higher
refractive index (n and n’ respectively). In Abbe’s apparatus, the first index corresponds to a well
known high index prism (around 1.7), and the second to some unknown liquid, usually of a much
lower index. The critic angle takes the value:
sinC = [n’/ n]
Abbe’s refractometer (see Fig.9 for a
picture of the apparatus) is prepared and
calibrated so that it can assign a value n’ from
a situation where a particular L is used for the
alignment of a system. Looking through the
right eyepiece we have the scheme represented
in Fig.3. Rays from the source enter the first
prism with a variety of angles of incidence.
Some will be refracted, but others arriving
over the critic angle will be totally reflected.
Those passing to the unknown liquid will be
refracted again recovering their original
direction in the second prism. Looking through
the optical system we observe a line separating
a brilliant zone from a dark one. By rotating
the pair of prism with an external wheel, we
can place that line in the center of a reference
cross seen from the eyepiece. The wheel
moves an internal calibrated scale that can be
observed through the left eyepiece. The result
is a quite straightforward procedure for
refractive index measurement.
Further details on chromatic aberration
are commented in Section 11.
Eye
Eyepiece (right)
Intermedium
Focal plane
Collecting
lens
High index
prisms
Liquid
(index
unknown)
Light Source
Fig.3: Two adjacent zones are formed with very
different illumination. Below the critic angle there
is refraction (high index prism--> liquid) and the
light reaches the second prism. Over the critic
angle there is TIR, giving rise to a dark zone in
the intermedium plane.
2- MATERIALS (Sessions 1 and 2)
• A Sodium (Na) lamp.
• A glass prism of unknown n and Abbe number.
• A goniometer.
• An auxiliary external collimator.
• A magnifying glass.
C
G
P
T1
T2
T3
T
Fig.4: Scheme of a goniometer.
A goniometer is an instrument devised to
measure the angle formed by two light beams (incident
and emerging. It consist of (Fig 4): a rotating platform
P (or platina) with three levelling screws (T1, T2 y T3)
on which we shall place the glass prism; a telescope
arm A, mounted on a rotating disk containing a vernier that, when the arm rotates, moves on a 360º
graded circle G. The scale of G has a 0,5º precision, and with the 30 parts of the vernier produces an
overall accuracy of the measurement of 1 arcminute (1’); and finally a collimator arm C (a fixed
arm). The goniometer has two small screws under the telescope arm for fixing it of for producing a
fine movement. Another pair an be found for producing the same effect on the central platform,
though they are not so practical, because the scale G is part of it and this may change the origin with
respect to some former measurements.
3- ALIGNMENT AND PREPARISON (Sessions 1 and 2)
1.- Place the spectral lamp in such a way that it illuminates uniformly the entrance slit in the
collimator arm.
2.- Focus the eyepiece of the telescope arm, by moving it within its frame untill a sharp
image of the cross contained in it is observed. Turn the cross in order to have a vertical line. Then,
tighten the external screw while keeping the focus and the vertical. The eyepiece is prepared.
3.- Now, with the lateral wheel of the telescope, we focus an object in the infinity (an
external cross in an auxiliary collimator, for instance). The telescope arm is prepared.
4.- Align both arms of the goniometer, so that we can observe the slit through the telescope.
5.- With the lateral wheel of the collimator arm, focus the slit. (Do not touch the wheel in
the telescope to do this). Now the collimator arm is prepared. We can close the slit to gain
resolution, but we must be sure that we have enough luminosity.
6.- Check that the vertical line of the cross is aligned with the slit.
7.- At this point, approximately the same length of slit should be visible over and under the
horizontal line of the reference cross (centered slit). Then, when the vertical reference overlaps the
slit we read the position of the telescope arm on the goniometer scale: xo.
8.- Now place the prism on the platform as shown in Fig. 5.The dihedron angle of the prism
should point to the source, approximately. Check that the height of the platform is adequate for the
beam to reach the full prism and illuminate both faces.
9.- Observe the light reflected by one face of
the prism through the telescope arm, so that the slit
overlaps the vertical line of the reference cross.
Then, check that the horizontal line crosses the slit
approximately in the middle. The same should
happen for the second face of the prism. If not, the
Colimador
Anteojo
Fig.5: Initial position of the prism on the
goniometer.
three screws under the platform should be adjusted
in small steps (you may call the person supervising
the experiment if you don’t reach a levelled position.
4- MEASUREMENT 1 OF  (Session 1)
1.- Once the goniometer is ready and the prism is in the position indicated in Fig. 5, observe
the slit reflected in one of the faces of the prism through the telescope arm, well centered with the
reference line. Take the angular position x1 in the scale (with arcminute precision).
2.- Repeat the same operation with the slit reflected in the other face of the prism (take x2).
3.- Calculate the value of  from the expression: 2 = x1- x2.
4.- The former 3 steps should be repeated 5 times, therefore obtaining the mean value of 
and the variance and standard deviation 2 and  , respectively.
2
1 N
 =   i   
N i 1
It is important to have the values of 2 in rad2 units.
2
5- MEASUREMENT OF m (PART 1)
1.- Place the prism as shown in Fig 6, so that
the light may go across the prism through the main
surfaces of the prism. Observe the output beam with
the telescope arm: some spectral lines can be easily
seen. Centre the orange-yellowish line with the
vertical reference of the eyepiece. We shall refer to
the wavelength of this line as o. The refractive
index of the medium for this wavelength will be
referred to as no.
Collimating
Arm
Telescope
Arm
Fig.6: Prism on the plattform of the goniometer
correctly placed for the measurement of m.
2.- Turn the prism with your fingers (try to touch it only on the upper surface, no on any of
the polished ones). At the same time, look at the desired line through the telescope arm (‘following’
it) until the line stops approaching the collimating arm direction and turns backwards. If you
manage to overlap the vertical reference line of your eyepiece with the exact point where the line
‘turns’ then the position at which a minimum-deviation phenomenon is occurring (=m) can be
characterized (this is a delicate process). The reading of the angular position of the telescope arm
gives us the value 1 (with a precision of arcmin).
3.- Put the prism in a position so that an identical, though symmetric, experiment can be
performed (if we were observing with our telescope arm in the right side, we should be observing in
This part ( measurement) may be delayed, just in case the experimenter runs out of time while doing the rest of the
tasks. For the moment, we can take the value =60º, and later perform this measurement if there is enough time. Once it
is measured, this value is valid for parts 1 and 2 (there is no need to repeat it again when doing part 2).
1
the left one now). The light is entering the prism through a different face, but the process can be
repeated in an identical way, finding again a turning point, which angular position is 2 (read on the
same scale-window)
4.- Calculate m by making : m = (1-2)/2.
5.- Repeat these measurements 5 times changing slightly the position of the prism on the
platform., and calculate the average, variance and standard deviation values:  m , m2 y m. It is
important to have the value of m2 in radian units (rad2)
6. CALCULATION OF no (PART 1)
1.- As explained in the first section, the refractive index can be calculated from Eq(1):


   
sin  m

2 

no 
sin 
2
 
2.-In order to assess the error involved in that calculation we must propagate the
experimental errors: Lets have a magnitude represented by f(x,y); its variance is given by:

2
 f 
 f 
 
 x2     y2  2
 x  xo , yo
 y  xo , yo
2
2
f ( x, y )
 f f 


 x2, y
 x y  xo , yo
x and y representing the experimental magnitudes,  and m. Because these two
measurements are independent,  x2, y  0 . In order to obtain the error f , that is no, we must operate
introducing  2 and  2m in rad2 units. (Remember that both no and no are non-dimensional
magnitudes. Concerning  y m, obviously it doesn’t matter which units are used, provided that the
functions sin and cos are properly calculated. In your hand calculator DEG means degrees, RAD
means radians). In case we took =60º with zero error, then we have  2 =0 )
RESULTS CORRESPONDING TO PART 1
 Mean value and error of angle  (if measured)
 Mean value and error of m
 Value obtained for no  no
7. MEASUREMENT OF m(r) and m(v).
(PART 2)
1.- Repeat 3 times steps 5.1 to 5.4 for the spectral lines r y v.
r =6161Å (red)
v =5685.5 Å (yellowish green)
We obtain the average values  m (r ) y  m (v ) .
8. CALCULATION OF n(r) and n(a).
(PART 2)
1.- Substituting the values obtained for  and  m (r ) in Eq.1, calculate n(r).
2.- Substituting the values obtained for  and  m (v ) in Eq.1, calculate n(v).
9. CALCULATION OF AN APPROXIMATE CAUCHY EQUATION. CALCULATION OF
THE ABBE NUMBEROF THE GLASS, ‘V’.
(PART 2)
1.- From the values obtained for n(r) and n(v) calculate coefficients A and B in Eq(2).
This provides us with an approximate version of the Cauchy dispersion equation.
2.- By substituting the value o in our equation, we must obtain a value for n(o).
3.- This approximate value n(o) should be very close to the value no experimentally
obtained in Section 6. Compare n(o) and no  no and check the agreement.
4.- Finally, by using our dispersion Cauchy equation we are going to (approximately)
calculate the Abbe number, V. The values for the standard wavelengths d, C y F must be
introduced in the dispersion equation. (these values are given in Section 1), therefore obtaining
n(d), n(C) and n(F). These values must be taken to Eq (3) and an approximate value of V is
obtained.
RESULTS CORESPONDING TO PART 2
 Mean values  m (r ) and  m (v ) , and errors.
 Values obtained for n(r) and n(v)
 Values of coefficients A and B: Approximate Cauchy equation.
 Value of n(o) obtained from Cauchy equation. Comparison with no.
 Values of n(d), n(C) and n(F) : Calculation of the Abbe’s number of
the material.
10.- DESCRIPTION OF ABBE’S REFRACTOMETER
1- Eyepiece with X-shape reference lines.
2- Chromatic compensator.
3- Chromatic compensator wheel.
4- Thermometer holder
5- Ouktput hole for the excess-liquid.
6- Two-prisms body.
7- Button for closing/opening the
prisms system.
8- Mirror to drive light into the system.
9- Main wheel: Turns the prisms (right
side) AND the refractive index
scale (left side).
10- Inside: Refractive index scale.
11- Eyepiece of the small microscope
used to read the scale.
(PART 3)
1
11
2
3
10
5
4
9
7
6
8
Fig.7: Abbe’s Refractometer
11.- PROCEDURE FOR THE MEASUREMENT OF THE REFRACTIVE INDEX OF A
LIQUID.
We shall use either a standard liquid (an index matching liquid) provided by the supervisor, or
water.
1.- After separating the two high index prisms, some drops of the liquid are put on the
surface of one of them, and the prisms are again put together. The space between them is now filled
with the liquid of unknown index, instead of air. It is important that this space is properly filled in,
so that the apparatus can work correctly. When observing through the eyepiece, two zones must be
found with very different illumination.
2.- By turning the prisms set (wheel #9 in Fig.7) the line should cut the reference cross of the
eyepiece (as indicated in Gig.8). At this point the line will possible look like a blurred and coloured
stripe, more than like a sharp line. This is due to the wavelength dependence of the refractive index,
that makes the critic angle to be wavelength-dependent as well (a border line is formed for each
colour).
3.- By rotating the chromatic compensator (wheel #3 in Fig 7) we manage to make the line
much sharper, so that a single index value is reached. The compensator artificially takes all colour
lines to the same position, therefore making the instrument insensitive to the person operating it.
4.- Finally, the reading of the index is directly made by looking through the left eyepiece
(the instrument is calibrated and an internal wheel turns when rotating wheell #9 in Fig.7). Being
the scale internal, it needs to be illuminated with the torch through a small lateral mirror. The
reading gives the refractive index with four digits (precision of one thousandth). The scale looks as
shown in Fig.8 (left).
4
20C
1,3
3
2
Fig.8: Aspect of the view obtained, respectively, through
the left and right eyepieces of Abbe’s refractometer.
RESULTS CORRESPONDING TO PART 3
 Mean value of the refractive index for a liquid (four digit precision).
12.- QUESTIONS:
1.- Why do we affirm that our measurement of the dihedron angle  does not depend on the exact
position of the prism, and that it is enough for it to point “approximately” to the source?
2.- Do you think possible to find the same value of m for all colours (i.e. all lines “bouncing” back
in the same angular position?)
3.- What would happen if we used a white lamp in order to measure the dispersion curve of the
prism? Identify the practical problem in doing so, and try to devise a practical solution.
4.- What do you think it would happen if you try to repeat this experiment with a coloured glass?
(Think that coloured implies the presence of some absorption line within the visible).