Physics 3340
Spring 2010
Polarized Light and verification of the Fresnel Equations
Purpose
This experiment will test the electromagnetic theory of transmission and reflection of polarized
light from a dielectric surface, as expressed in Fresnel's equations. The lab begins with an
introduction to methods for producing and analyzing linear and circularly polarized light. You
will go on to measure the angular dependence of reflection and transmission for both p- and spolarizations of a He-Ne laser beam incident on a lucite surface.
Introduction
1. Description of polarized light

The plane of polarization of an electromagnetic wave is taken to be the plane containing
the electric field vector, E , and the direction of propagation, k . Coherent linearly
polarized light is described by the equation:
E  E0, X cos  kz  t  xˆ
This equation represents a wave of amplitude, E0, X , wavelength   2 k , propagating
in the z-direction, polarized in the (x,z) plane. See Figure 8.1(a) below.

Unpolarized light can be visualized as a stream of photons, whose individual
polarizations are randomly oriented. Any on photon behaves like an electromagnetic
wave packet up to ten-thousand wavelengths long, whose polarization is unrelated to that
of its companions. See Figure 8.1 (b) below.

For incoherent linearly polarized light, the photon E-fields are parallel to a fixed plane.

In circularly polarized light, the vector, E , rotates about the direction of propagation at
the frequency of the light wave, i.e., one revolution per period of the wave. A circular
polarized wave can be represented by superposing two perpendicular waves of equal
amplitude, whose phases differ by 90 degrees. For example:
E  E0 cos  kz  t  xˆ  cos  kz  t   2  yˆ 
The case with the positive phase is right-hand circularly polarized light, while the
negative phase represents the case of left-hand circular polarization. See Figure 8.1 (c)
below.
Polarized Light
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Figure 8.1. (a) Coherent linearly polarized light. (b) Unpolarized light. (c) Right-handed
circularly polarized light.
2. Production of polarized light

Linear polarization. A linear polarizer, such as polaroid, transmits only that component
of the incident E-field that is parallel to the transmission axis of the polarizer. The
component of the E-field perpendicular to this axis is either absorbed, or reflected in a
different direction. Such a material is known as a dichroic filter. If the emerging light is
passed through a second filter, known as the analyzer, only that component of the field
parallel to the transmission axis of the analyzer is passed. If the axes of the polarizer and
analyzer are at relative angle,  , the E-field of radiation emerging from the analyzer is
related to that emerging from the polarizer as E2  analyzer   E1  polarizer  cos . Since
light intensity is proportional to the square of the E-field amplitude, we have Malus' Law:
I 2  I1 cos 2 

In circularly polarized light, the x- and y-amplitudes are equal, but 90 degrees out of
phase. The E-vector rotates at the frequency of the light about the direction of
propagation. It may be generated by passing linearly polarized light through a quarter
wave plate (see Welford, Chapter 4). The quarter wave plate is cut from a birefringent
material in which the velocity of light is different according to which of two
perpendicular directions the E-field is parallel. The thickness is such that the wave with E
in the 'fast' plane gets ahead in phase by  2 (quarter wave) relative to that in the 'slow'
plane. The orientation of the quarter wave plate must be such that the 'fast plane' is at 45
degrees relative to the plane of polarization of the incoming light. If the angle is not 45
degrees, then the resulting light is elliptically polarized.
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Figure 8.2.
3. At the boundary between two dielectrics, the reflected and transmitted intensities depend on
whether the E-vector lies parallel (p-polarization) or perpendicular (s-polarization) to the
plane of incidence (horizontal plane in this experiment). These ratios can be calculated from
the boundary conditions on the E and B-fields at the interface. The results are described by
the "Fresnel Equations". The ratios of intensities are given by:
sin 2 i  t 
 IR 
R    
2
 I 0  sin i  t 
tan 2 i  t 
 IR 
R   
tan 2 i  t 
 I0 
2
2
 IT 
nt 4sin t  cos i  cos t 
T    
cos i 
sin 2 i  t 
 I 0   ni
4sin 2 t  cos 2 i 
cos t 
 IT 
nt
T   
2
2
ni sin i  t  cos i  t  cos i 
 I0 
(s-pol)
(p-pol)
where the incident and transmitted angles,  i and  t are related by Snell's Law:
ni sin i  nt sin t
Note that at Brewster's angle, where i  t   2 , we have R  0 . All the reflected light
would be s-polarized even for un-polarized incident light. Polarization by reflection is a
second important method to produce polarized light. Furthermore, we find T  100% for ppolarized light. Glass windows tilted at the Brewster angle are known as Brewster windows.
They are used in producing polarized laser sources.
Figure 8.3. s- and p-polarization geometries.
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Apparatus
Figure 8.4. Apparatus to measure transmitted and reflected intensities of polarized light.
For the introductory experiments, the laser beam passes through the polarizer (polaroid sheet P1)
and the analyzer (polaroid sheet P3). The intermediate position, P2 is either empty or is occupied
by a third polariod sheet, a quarter-wave plate, or other sample of material. Each sheet can be
rotated in its own plane about the horizontal beam axis, and the angle of the polaroid
transmission axes measured relative to the vertical direction. The transmitted beam intensity is
measured with a photodiode or observed on a screen.
Figure 8.4 (b or c) shows the set up for measuring the transmission and reflection coefficients for
polarized light incident on a dielectric surface. The dielectric sample is a polished semicircular
slab of lucite (plexiglass) attached with wax to the upper member of a pair of rotation tables.
These can rotate independently about a vertical axis through the laser beam. The sample is
positioned so that the point of incidence of the laser beam is precisely at the center of the
diameter face and lies on the rotation axis of the tables. The photodiode detector is mounted on
the lower rotator, so it can move independently.
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The plane of incidence, containing the incident beam axis and the normal to the dielectric
surface, is the horizontal plane in this experiment. The angle of incidence on the plane face is
varied by rotating the upper table and is measured by viewing the beam (from above) against the
angular graph paper on the table surface. The transmitted ray exits along a radius of the curved
face, so it is not bent by refraction. The table can be rotated further, enabling the beam to enter
radially through the curved face and exit through the plane face. With the later configuration, you
can observe the critical angle,  C , and total internal reflection. The beam can be observed either
by its glow from scattering inside the lucite, or with a white card. Angles can be measured
relative to a piece of circular graph paper (xerox copies in the lab) which is stuck onto the top of
the table. Black paper just under the sample makes the beam inside it more visible.
The narrow laser beam is linearly polarized by passing through the polaroid polarizer, P1. The
polarization of the beam can be set either vertical (to give vector E perpendicular to the plane of
incidence) or horizontal (E parallel to the plane of incidence) by rotating P1 about the beam axis.
The intensity of incident light, I 0 , reflected light, I R , and transmitted light, I T , is measured with
a photodiode. You may need to change the resistor in your present metering circuit to
accommodate the different light intensity in this experiment. The photodiode is attached to the
lower of the pair of rotation tables, enabling it to be positioned independently in the beam it is to
measure. Complete collection of the beam at the photodiode aperture presents a problem. Verify
that the total of the reflected and transmitted powers DOES EQUAL the incident power
throughout the experiment in order to verify that the collection efficiency remains constant.
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Problems
1.
Polarization problems.
a) What is the fraction of light intensity transmitted by the second of two polaroids, whose
axis of transmission are 30 degrees apart?
b) Suppose that your laser produces light polarized vertically, and you need to make the
measurements with horizontally polarized light. Show that this can be accomplished by
placing an additional polaroid at 45 degrees between the laser and the final polaroid.
What fraction of the laser intensity is incident on the sample? Can you think of another
choice of polarizers and wave plates that also solves the problem?
2.
Assuming that the refractive index of lucite is n=1.50, calculate the critical angle and
Brewster's angle for comparison with your data. Prepare theoretic graphs of the four
Fresnel equations.
3.
Polarization analysis.
a) Devise a procedure to distinguish unambiguously between unpolarized light and
circularly polarized light..
b) Find a convincing explanation for the "one-way light valve". See page 8.7.
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Outline of the Experiment
1. Set up the laser and rotating table on an optic bench. Align the optic axis so that the laser
beam passes through the vertical axis of the rotator.
2. Determine whether or not the light from your laser is polarized using the polarization
analyzer, P1.
3. Test Malus' law quantitatively using the polarizer (P1) and analyzer (P3). Measure the
incident and transmitted beam powers with the photodiode. The intensity of a laser beam may
be controlled by inserting two polaroids and adjusting their relative orientation to give the
required beam power.
4. Set the polarizer (P1) and analyzer (P3) axes to be mutually perpendicular i.e., crossed.
Examine the effect of placing between them the following inserts.
a) A third polaroid sheet (P2). Observe how the intensity of light now emerging from the
analyzer (P3) changes as you rotate the inserted sheet (P2).
b) A quarter-wave plate. Demonstrate that light emerging from the wave-plate is circularly
polarized when the fast axis is turned 45 degrees relative to the polarization plane of light
from the polarizer. What happens at other angles?
c) Show that your polarizer and wave-plate jointly function as a one-way light valve. Try
returning the light with a mirror placed beyond P2.
d) Try various pieces of plastic and sheet and scotch tape.
5. For the lucite sample:
a) Using several angles of incidence between 10 and 60 degrees, determine the refractive
index using Snell's Law:.
ni sin i  nt sin t
b) Measure the critical angle and test the relation:
sin  C  1 n
c) Determine the Brewster angle and test the prediction:
tan  B  n
d) Measure the intensities of the incident, reflected, and transmitted beams as a function of
angle of incidence for both p- and s-polarized light. Plot the reflection and transmission
coefficients versus the angle of incidence on the theoretical graphs you produced for
Problem 2. Do you confirm the predictions of electromagnetic theory? Verify that the
reflected and transmitted powers do sum to the original incident power at each angle, as is
required by conservation of energy.
Polarized Light
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