1051-232: Imaging Systems Laboratory II
Laboratory 2: Snell’s Law, Dispersion and the Prism
March 19 & 21, 2002
Abstract. This laboratory exercise will demonstrate two basic properties of the way light interacts
with matter. The first is the law of refraction (Snell’s Law), which tells us that light rays will suffer
a change in direction when they cross a boundary between two indices of refraction. The second is
the phenomenon of dispersion, which is the variation of the index of refraction with wavelength. To
investigate both effects, we will use prisms made of three different materials: glass, water and
mineral oil. Lab write-ups are due March 26, 2002 for the Tuesday lab section and March 28, 2002
for the Thursday section.
I. Theory
In the lecture part of the course, we learned that a light ray incident on an interface between
two transparent materials is refracted at an angle determined by the incident angle, θ1, (measured
from the normal to the interface) and indices of refraction, n1 and n2, of the two materials. This
relationship is known as Snell’s Law, and can be written in the form
n1 sin θ1 = n 2 sin θ 2 ,
(1)
where n1 and n2 are the indices of refraction for the two media and θ1 and θ2 are the angles that the
incident and transmitted rays make with the normal to the surface, respectively. The situation is
shown in Figure 1 below.
n1
n2
θ2
θ1
Figure 1: A ray traversing the boundary of two indices of refraction.
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We also saw that when one applies Snell’s law to the case of a prism, it can be shown by
way of geometrical arguments that for a ray that passes through one face of the prism and emerges
from another face without reflections, the total deviation of the ray from its original path is given
by the formula
(
)
δ = θ i1 + arcsin sin α n 2 − sin 2 θ i1 − cos α sin θ i1 − α ,
(2)
where δ is the total deviation angle of the ray, α is the apex angle between the entrance and exit
face, n is the index of refraction of the prism material, and θi1 is the angle of incidence on the
entrance face. Figure 2 illustrates this situation.
α
δ
θi1
α
α
Figure 2: A triangular prism with apex angle α.
Figure 3: Typical dependence of the deviation angle, δ, as a function of the angle of incidence, θI1. In this case, n=1.5
and α=60 degrees.
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For a given prism, α and n are fixed, so the above expression for δ is a function only of the
angle of incidence on the entrance face, i.e. δ = δ(θi1). This function generally has a minimum at
some incident angle that depends on the index of refraction for the prism material, so that for
example, a curve of δ as a function of θi1 typically looks like Figure 3 above. This minimum should
occur when the ray’s path is parallel to the base of the prism, for an isosceles-triangular prism (like
the one shown in Figure 2). If the minimum angle of deviation δmin can be measured for a given
prism, it follows from Snell’s law and the above equation that
sin (δ min2+α )
n=
.
sin (α2 )
(3)
In the above discussion, it is assumed that the index of refraction is constant. This
assumption is perfectly valid for considering what happens to a single wavelength (i.e. color) as it
travels through the prism, but we have also learned that the index of refraction is a function of
wavelength. This effect has to do with how individual atoms respond as electromagnetic waves
pass by, a process known as non-resonant scattering, but the end result is that most materials used
in optical systems show a slight increase in the index of refraction for light of lower wavelength.
Figure 4 shows the wavelength dependence of the index of refraction for several different
substances. The unit of wavelength used is the Angstrom (Å), where 1 Å = 10-10 m.
Figure 4: Dependence of the index of refraction on wavelength for several different transparent substances.
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II. Experimental Set-Up
To get ready to do the experiments below, you will need (1) a glass prism, (2) one of the
prisms made of glass slides that can be filled with water or mineral oil, (3) a rotary table or angle
sheet on which to place the prisms to change the orientation easily, and (4) a laser. Once you have
all of these things, set the prism on the rotary table and arrange the laser so you can point the laser
beam through the prism easily, as in Figure 5. DO NOT LOOK DIRECTLY AT THE LASER
BEAM. The prism should be able to rotate through a large range in the angle of incidence on the
input face. Use a book with a white piece of paper taped to it or a piece of white paper taped to the
wall as a screen. Make sure before taking any data that the laser spot remains on your screen as you
rotate the prism.
C
laser
prism
A
δ
B
Rotary table
screen
Figure 5: Experimental set-up for this laboratory exercise.
III. Procedure
Start with the glass prism, and complete the following steps:
1) Measure the distance from the center of the rotary table to the screen as best you can. Try
taking 3 to 5 independent measurements and averaging the result. What is the standard
deviation, σ, of the result? Use σ / N as an estimate of the uncertainty in your distance
measure.
2) Mark the position of the laser spot on the screen in the case where no prism is present. This will
give you the position of the undeviated beam.
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3) Measure the apex angle of the prism. You can draw lines parallel to the faces of the prism, and
extend these lines with a ruler to make the measurement easier to do with a protractor.
4) Place the prism in the center of the rotary table or angle sheet. Mark the position of the laser
spot for different incident angles for as large a range as you can measure. Take at least 8 data
points, recording the value of the incident angle (as judged by the position of the rotary table)
and the distance from the undeviated spot position.
5) Convert the distances from the previous step to deviation angles δ(θi1). This can be done by
noting that
dist(AB)/dist(AC) = tan δ,
(4)
where dist(AB) is the distance from A to B in Figure 5 and dist(AC) is the distance from A to
C, both of which you have measured.
6) Now remove your glass prism and fill the hollow prism made from the microscope slides with
the distilled water provided. Repeat steps 1 through 5.
7) Next try steps 1 through 5 with after filling the microscope-slide prism with mineral oil.
8) Finally, go back to the glass prism and use one of the fiber-fed light sources and a slit instead of
the laser. Put the fiber source at least a couple of feet away from the prism. Make a slit that has
width of a couple of millimeters and length of about 1 cm. Measure the angle of incidence as
best you can, and then measure the deviation δ for three different colors: red, green or yellow,
and blue. If you have trouble, you may want to consider holding filters from the optics kit in
front of the slit to cut out the unwanted colors. In any case, you will probably have to turn the
lights off and shield your screen from the ambient light to be able to see the slit on your screen.
IV. Analysis
In your lab write-up, be sure to complete the following:
1) For each of the three prisms, graph the deviation angle δ as a function of the incident angle on
the input face of the prism, θi1. Find the minimum deviation angle, δmin and estimate the
uncertainty in this quantity.
2) Using Formula 3 on page 3, calculate the index of refraction for the glass, water, and mineral
oil used, based on your value for δmin and α in each case.
3) From your measurements in Step 8 above, derive the index of refraction for red (λ ≈ 700nm),
green (λ ≈ 550nm) and blue (λ ≈ 450nm) light. Do your data show an increase in n as the
wavelength is reduced, as in Figure 4? The laser is also red (λ = 632.8nm). How well does your
value of n agree between the laser measurements and the red filter measurement?
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