This document describes an experiment to verify Fresnel's equations for reflection at a dielectric surface. The experiment involves measuring the reflection coefficients and rotation of the polarization plane for light reflected at various angles of incidence off a glass prism. The reflection coefficients will be measured for perpendicular and parallel polarization and plotted against angle of incidence. The refractive index of the glass will be determined. The measured reflection coefficients and polarization rotation will be compared to values calculated using Fresnel's equations to test the equations. Key concepts covered include Brewster's angle, reflection and transmission coefficients, polarization, and Fresnel's equations.

1. Polarization by Reflection at a Dielectric and verifying Fresnel’s
Equations
Principle
Plane-polarized light is reflected at a glass surface. Both the rotation of the plane
of polarization and the intensity of the reflected light are to be determined and
compared with Fresnel's formulae for reflection.
Tasks
i. The reflection coefficients for light polarized perpendicular and parallel to
the plane of incidence are to be determined as a function of the angle of
incidence and plotted graphically.
ii. The refractive index of the flint glass prism is to be found.
iii. The reflection coefficients are to be calculated using Fresnel's formulae and
compared with the measured curves.
iv. The reflection factor for the flint glass prism is to be calculated.
v. The rotation of the polarization plane for plane polarized light when
reflected is to be determined as a function of the angle of incidence and
presented graphically. It is then to be compared with values calculated
using Fresnel's formulas.
What you can learn about
▪ Electromagnetic theory of light
▪ Reflection coefficient, Reflection factor
▪ Brewster's law
▪ Law of refraction
▪ Polarization, Polarization level
Theory:
When unpolarized light is incident on a dielectric medium such as a glass, the
reflected rays are partially plane – polarized, i.e., it is a mixture of a plane
polarized and unpolarized (ordinary) light.
Let a plane light wave strike a boundary between two media having refractive
indexes n1 and n2. The angles , , and are the angles of incidence, reflection,
and refraction, respectively. In this case:
Optics Lab – Department of Physics – Baghdad University 2012 – Exp. No. 10 1

2. | | | | (the law of reflection)
and
(the law of refraction)
The fraction of light reflected or refracted depends on:
i. The refractive indices of the media
ii. The angles of incidence
iii. The state of polarization of the incident light
At a certain angle of incidence , the reflected light is totally plane – polarized,
(Figure 1). This angle is called the “Brewster’s Angle” or “Polarizing Angle”, and it
is related to the refractive index of the Normal
Reflected ray
medium by: Incident ray
(polarized, 𝐸 ) ⊥
(Unpolarized)
𝜑 𝜑
This relation is called “Brewster’s law”. 𝑛
𝐵 𝐵
Also, for this angle the reflected electric 90
𝑛
vector E is normal to the plane of 𝜑
incidence, i.e., the plane containing the
incident ray and the normal at the point Refracted ray
(polarized)
of incidence.
Let the plane of vibration makes an angle
with the normal to the plane of
incidence (figure 2). This angle is called Figure 1: Unpolarized light incident at the polarizing angle
the orientation angle whether it refers to
light vibration in the incident, reflected or refracted light. The vector E in the
incident, reflected or refracted light can be resolved into two amplitude
components:
Normal
𝐸𝑝
𝐸 𝑅𝑠
𝜓
𝐸𝑠
𝜑
𝑅
𝑅𝑝
𝜑 80
Figure 2: Plane - polarized light externally reflected from a glass surface at 80 degrees angle of
incidence.
Optics Lab – Department of Physics – Baghdad University 2012 – Exp. No. 10 2

3. i. Components parallel to the plane of incidence Ep, Rp, . These amplitude
components are called the p – components.
ii. Components normal to the plane of incidence Es, Rs, . They are called the
s – components.
For unpolarized light, Ep=Es.
According to the electromagnetic theory, the amplitudes, phase, and state of
polarization of the reflected and refracted waves for the p – vibrations and s –
vibrations are related to the corresponding characteristics of the incident wave by
Fresnel’s equations:
for the reflected amplitudes.
for the refracted amplitudes.
It follows from equations (2 - 5) that for any value of the angles the
signs of and and the signs of and coincide. This means that the
phases also coincide; that is, in all cases the refracted wave retains the phase of
the incident wave. For the components of the reflected wave (Rp and Rs), the
phase relations depend on , n1, and n2. For example, if = 0, then when n2 > n1
the phase of the reflected wave will be shifted by π.
From 2 and 3, one gets:
This is the tangent of the angle , i.e.,
In experiments, rather than measuring the amplitude, scientists usually measure
the intensity of a light wave, that is, the energy flux carried by it, which is
proportional to the square of the amplitude. The ratios of the average energy
Optics Lab – Department of Physics – Baghdad University 2012 – Exp. No. 10 3

4. fluxes over a period of time in the reflected and refracted waves to the average
energy flux in the incident wave are called the reflection coefficient r and the
transmission coefficient t. We obtain from equations (2 - 5) the Fresnel equations
that define the reflection and transmission coefficients for the s- and p-
components of the incident wave:
( )
In the absence of light absorption, rs + ts = 1 and rp + tp = 1, in accordance with the
law of the conservation of energy. If natural light is incident on the boundary,
that is, if all directions of oscillations of the electric vector are equally probable,
then, one half of the wave’s energy is accounted for by p – oscillations and the
other half by s – oscillations. In this case, the total reflection coefficient is:
( ) [ ]
If 90 and , then rp = 0; that is, light that is polarized
such that its electric vector lies in the plane of incidence under these conditions
will not be reflected at all from the boundary. The reflected light, when natural
light is incident at this angle, will be completely polarized.
For normal incidence of light on the boundary between two media ( = 0), the
Fresnel equations for the amplitudes of reflected and refracted waves can be
reduced to the form:
( ) ( )
( ) ( )
Optics Lab – Department of Physics – Baghdad University 2012 – Exp. No. 10 4

5. In such a case, the difference between the components s and p disappears, since
the concept of plane of incidence loses meaning. In this case, we obtain, in
particular,
( )
It follows from the above two equations that the reflection of light at the
boundary is greater the larger the absolute value of the difference n2 – n1; the
coefficients r and t do not depend on which side of the boundary the incident
light wave arrives from.
A condition for applicability of the Fresnel equations is that the refractive index of
the medium be independent of the amplitude of the electric field strength of the
light wave. This condition, which is trivial in classical (linear) optics, is not satisfied
for high – power radiant fluxes, such as those radiated by lasers. In these cases,
the Fresnel equations do not provide a satisfactory description of the observed
phenomena, and consequently the methods and concepts of nonlinear optics
must be used.
Procedure
 Brewster’s angle and the index of refraction n.
1. Arrange all the optical components on the optical bench as in figure 3.
0
360 - graduated
rotatable stage Polarizer
Prism or
glass plate
Laser
Analyzer
Intensity meter
Figure 3 – Experimental setup for reflectance measurements as a function of angle of incidence.
Optics Lab – Department of Physics – Baghdad University 2012 – Exp. No. 10 5

6. 2. Make the transmission axis of the polarizer in the horizontal orientation,
i.e., the transmitted laser beam will have horizontal polarization with the
vibration of the electric field vector component E parallel to the plane of
incidence ( ). Remove the analyzer from the optical bench.
3. Align the incident laser beam spot with the 0 mark on the graduated
rotating stage.
4. Align the photocell with the incident polarized laser spot and read the
initial intensity (I0) as registered by the attached multimeter.
5. Put the prism on the rotating stage which must be level and make the laser
beam incident normally on the prism. This is done whenever the reflected
and incident laser spots coincide with each other. If not, simply rotate the
prism (not the stage!) clockwise or anticlockwise direction around the
vertical axis until the reflected laser spot precisely becomes aligned with
the incident beam.
6. Slowly turn the graduated rotatable stage clockwise in 10 – degree steps.
At each orientation, rotate the Photocell arm slightly to get into the beam.
Read the intensity (I) as registered by the attached multimeter. Record the
measured data in a table 1 below.
I0= mV at =0 degree
Ip Is
rp=Ip/I0. rs=Is/I0
degrees mV mV
10
20
30
40
.
.
90
7. Repeat step 3 by rotating the stage of the prism in 10 – degree angle
increments until reaching the value 80 0 (or even 90 0, if possible).
8. Using the same procedure as before, take measurements with the
transmission axis of the polarizer in the vertical orientation, i.e., the
Optics Lab – Department of Physics – Baghdad University 2012 – Exp. No. 10 6

7. transmitted laser beam will have vertical polarization: the electric field
component vibrating perpendicularly to the plane of incidence ( ⊥ ).
Analysis
1. Take the ratio Ip/Io, graph it versus angle . Determine the Brewster’s
angle.
2. Use Brewster's angle to calculate the index of refraction of glass prism
using equation 1. Use n1 = 1.000308
3. Calculate the parallel and perpendicular reflectances using Fresnel
equations.
 Verification of Fresnel’s equations
1. Record the value of the analyzer scale at which the laser beam has
totally been extinct. Notice that there are always two analyzer positions
to give the extinction. Use the one closest to the zero reading of the
analyzer.
2. Rotate the analyzer exactly 450 from its initial reading Remove the
0
prism and return the analyzer arm to 180 . Insert the polarizer between
the laser and the prism (on the optical bench) and rotate it until the
light is extinguished. The transmission axis of the polarizer is now 45 0 to
the horizontal and vertical directions. Hence, Ep=Es. Don’t change this
polarizer setting during the remaining part of the experiment.
3. Turn the analyzer arm to position 170 0 ( 8 . Place the prism on
the stage. Rotate this stage until the laser beam falls on the analyzer.
4. Rotate the transmission axis of the analyzer until the light is
extinguished. The difference between the current analyzer reading
and the zeroth reading recorded previously at the Brewster’s angle
gives the angle . Be sure that is closest to the zeroth reading of the
analyzer scale.
5. Repeat steps 6 and 7 for rotating stage angles of 160, 150, 140, 130,
120, 110, and 100. Tabulate your readings as in table 2 below.
Optics Lab – Department of Physics – Baghdad University 2012 – Exp. No. 10 7

8. 170 85
160 80
. .
. .
100 50
Analysis
1. Use Snell’s law and the evaluated refractive index n to calculate the
angle of refraction for each angle of incidence
2. Calculate the value of using equation 7.
3. Plot a graph of vs. Compare it with that of Jenkins and White.
Questions
1. Would Brewster's angle be more or less for light in air reflecting off water?
2. How would data look like for an arrangement with vertical square
polarizer?
3. How do polarized sunglasses reduce glare? Which direction is the axis of
polarization in a pair of polarized sunglasses? How could you check this?
Optics Lab – Department of Physics – Baghdad University 2012 – Exp. No. 10 8