# Polarization of Light

```Polarization of Light
References
Halliday/Resnick/Walker Fundamentals of Physics, Chapter 33, 7th ed. Wiley 2005
PASCO EX9917A and EX9919 guide sheets (written by Ann Hanks)
1 weight – Exercises 1 and 2
2 weights - all
Introduction
Light, viewed classically, is a transverse electromagnetic wave. Namely, the underlying
oscillation (in this case oscillating electric and magnetic fields) is along directions perpendicular
to the direction of propagation. This is in contrast to longitudinal waves, such as sound waves,
in which the oscillation is confined to the direction of propagation. Light is said to be linearly
polarized if its oscillation is confined to one direction (the direction of the oscillation of the
electric field is defined as the direction of polarization). Most light sources in nature emit
unpolarized light i.e., light consists of many wave trains whose directions of oscillation are
completely random.
Light may be polarized by passing it through a sheet of commercial material called Polaroid,
invented by E.H. Land in 1938. A sheet of Polaroid transmits only the component of light
polarized along a particular direction and absorbs the component perpendicular to that direction.
Consider a light beam in the z direction incident on a Polaroid which has its transmission axis in
the y direction. On the average, half of the incident light has its polarization axis in the y
direction and half in the x direction. Thus half the intensity is transmitted, and the transmitted
light is linearly polarized in the y direction.
Malus’ Law
Suppose we have a second piece of Polaroid whose transmission axis makes an angle  with
that of the first one The E vector of the light between the Polaroids can be resolved into two
components, one parallel and one perpendicular to the transmission axis of the second Polaroid
(see Figure 1).
If we call the direction of transmission of the second Polaroid y ' ,
Ex '  E sin  and E y '  E cos
Only the E y ' component is transmitted by the second Polaroid.
The intensity of light is proportional to the square of the electric field amplitude. Thus the
intensity transmitted by both Polaroids can be expressed as:
I ( )  E y2'  E 2 cos2  .
If I 0  E 2 is the intensity between the two Polaroids, the intensity transmitted by both of them
would be:
I ( )  I 0 cos 2 
(1)
This equation is known as Malus’ Law after its discoverer, E.L. Malus (1775-1812). It applies to
any two polarizing elements whose transmission directions make an angle  with each other.
When two polarizing elements are placed in succession in a beam of light as described here,
the first is called polarizer and the second is called analyzer. As you will see, no light reaches
the photocell when the polarizer and analyzer are crossed ( = 90).
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Figure 1: Two Polaroids whose transmission directions make an angle  with each other.
Theory for three polarizers
Unpolarized light passes through 3 polarizers (see Figure 2):
Figure 2: Electric Field Transmitted through Three Polarizers
The first and last polarizers are oriented at 90o with respect to each other. The second polarizer
has its polarization axis rotated an angle Φ from the first polarizer. Therefore, the third polarizer
2


   from the second polarizer. The intensity after passing through the
2

is rotated an angle 
first polarizer is I1 and the intensity after passing through the second polarizer, I2, is given by:
I 2  I1 cos 2  .
The intensity after the third polarizer, I3, is given by:


I 3  I 2 cos 2      I1 cos 2 
2


 cos
2



    

2
(2)
Rearranging Equation 2, we obtain:
I3 
I1
sin 2 ( 2 )
4
(3)
Because the data acquisition begins when the transmitted intensity through Polarizer 3 is a
maximum, the angle (  ) measured in the experiment is zero when the second polarizer is 45o
from the angle Φ. Thus the angle Φ is related to the measured angle  by:
  45o  
(4)
Reflectance
Consider unpolarized light incident to a surface separating air and glass or air and water. Define
the plane of incidence, containing the incident, reflected and refracted rays as well as the
normal to the surface.
When light is reflected from a flat surface, the reflected light is partially polarized. This is due to
the fact that the reflectance of light R= (Reflected Intensity)/(Incident Intensity) depends on the
polarization itself. The degree of polarization depends on the angle of incidence and the indices
of refraction of the two media.
For reflection at an air-glass interface (indices of refraction n1 for air and n2 for glass), Fresnel
equations give the reflection coefficients r|| , r :
r 
n1 cos 1  n2 cos  2
n1 cos 1  n2 cos  2
(5a)
r// 
n1 cos 2 n2 cos 1
n1 cos  2  n2 cos 1
(5b)
1 is the angle of incidence, and 2 is the angle of refraction.
r|| and r refer to the reflection coefficients for polarized light whose direction of polarization lie
in the plane of incidence and perpendicular to the plane of incidence, respectively.
Reflectance R (parallel or perpendicular) is defined as the square of the corresponding
reflection coefficient: R//  r//2 ; R  r2
Note that 2 is not measured in this experiment and must be inferred from Snell’s law of
refraction:
sin  2 n1
(6)

sin 1
n2
3
Figure 3 shows initially unpolarized light incident at the ‘polarizing’ angle P, for which the
reflected light is completely polarized with its electric field vector perpendicular to the plane of
incidence. The electric field vector E of the incident wave can be resolved into two components:
parallel to the plane of incidence and perpendicular to the plane of incidence.
Note that R||= 0 when n1 cos  2  n 2 cos  1 which leads to the definition of the Brewster angle (or
‘polarizing’ angle):
n
(7)
tan  P  2
n1
ΘP is the angle of incidence of unpolarized light which makes the reflected light completely
polarized in the perpendicular direction to the plane of incidence (Sir David Brewster, 1812).
When the angle of incidence of the initially unpolarized light is ΘP, the reflected and refracted
rays are perpendicular to each other.
Normal
Incident ray
(unpolarized)
n1
Reflected ray
(polarized,  )
θP
θP
n2
θ2
Refracted ray
(polarized)
Figure 3 Unpolarized light incident at the polarizing angle.
Note: If the incident light has no component of E perpendicular to the plane of incidence, there is no
reflected light.
Malus’ Law: Apparatus notes
Be careful not to leave your fingerprints on optical surfaces.
- Rotate the aperture disk so the translucent mask covers the opening to the light sensor.
- Verify that the Rotary motion sensor is mounted on the polarizer bracket and connected to the
polarizer pulley with the plastic belt.
- Verify that the Rotary motion sensor is plugged into the National Instruments (NI) interface.
- Place all the components on the Malus’ Law Optics Track in the order shown in Fig. 4. Space
components apart along the optics track (largest distance should be between detector and
polarizer).
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Figure 4: Experiment Components
SOFTWARE SET UP
Click on the desktop shortcut “Polarization of Light”. Click on the little arrow located in the upper
left corner of the screen to start the acquisition. The program is self-explanatory.
Exercise 1: two polarizers, verify Malus’ Law
In the first two procedure steps, polarizers are aligned to allow the maximum amount of light
through.
- Since the laser electromagnetic wave is already polarized, the first polarizer must be aligned
with the polarization axis of light. Remove the holder with the polarizer and Rotary motion
sensor from the track. Slide all the other components on the track close together and dim the
room lights. Click ON/OFF and rotate the polarizer that does not have the Rotary motion sensor
until the light intensity on the graph is at its maximum.
- To allow the maximum intensity of light through both polarizers, bring back the holder with the
polarizer and Rotary motion sensor on the track, and rotate the polarizer until light intensity on
the graph is maximum. Before you begin a new scan, you may clear previous data.
Note: If the maximum exceeds 4.5 V, decrease the gain on the light sensor. If the maximum is less than 0.5 V,
increase it.
- To scan the light intensity versus angle press ON/OFF and rotate the polarizer which has the
Rotary motion sensor through 180 degrees. Rotation should be constant. Do not rotate back!
Acquisition stops by itself at the end of 180 degrees. Practise until you get the best (smooth)
recording of Intensity of light vs. angle.
Analysis
Students not from PHY224/324: Export data, analyze dependencies: Intensity vs. cosθ, and
Intensity vs. cos2θ
 PHY224/324 students: Python Requirement 1: write a fitting program for Intensity vs.
cos2θ. You have to include experimental errors for both light intensity and angle readings. The
data set includes about 300 values (Intensity, θ). Compare with Malus’ Law prediction, Eq. 1.
Exercise 2: three polarizers
- Repeat the experiment with 3 polarizers (see setup in Figure 6). Place Polarizer 1 on the track
and rotate it until the transmitted light is a maximum.
- Place Polarizer 2 on the track and rotate it until the light transmitted through both polarizers is
a minimum.
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- Place a Polarizer 3 on the track between the first and second polarizers. Rotate it until the
light transmitted through all three polarizers is a maximum (see Fig. 9).
Figure 6: Setup with three polarizers
- Press ON/OFF and scan Intensity vs. Angle for 360 degrees as you rotate the third polarizer
that has the Rotary motion sensor. Try the qualitative fitter.
- Select your data from 2 polarizers and from 3 polarizers. What two things are different for the
Intensity vs. Angle graph for 3 polarizers compared to 2 polarizers?
Export data and find the best fit matching Equation (3). Include experimental errors.
 Python Requirement 2 (PHY224/324 students only): write a fitting program using your
data and the intensity function from Equation 3
Questions
1) For 3 polarizers, what is the angle between the middle polarizer and the first polarizer to
get the maximum transmission through all 3 polarizers? Remember: in the experiment,
the angle of the middle polarizer automatically reads zero when you start taking data but
that doesn't mean the middle polarizer is aligned with the first polarizer.
2) For 3 polarizers, what is the angle between the middle polarizer and the first polarizer to
get the minimum transmission through all 3 polarizers?
Exercise 3 Polarization by reflection and Brewster’s angle
Light from a diode laser is reflected off the flat side of an acrylic semi-circular lens. The
reflected light passes through a polarizer and is detected by a light sensor. The angle of
reflection is measured by a rotary motion sensor mounted on the spectrophotometer table. The
intensity of the reflected polarized light versus reflected angle is graphed to determine the angle
at which the light intensity is a minimum. This is Brewster's Angle, which is used to calculate
the index of refraction of acrylic (Eq. 7).
Since the index of refraction depends on the wavelength of incident light, using a
monochromatic light source (diode laser) should render the interpretation of results relatively
simple.
Preliminary Setup
- Move the diode laser to the optics bench marked Brewster’s angle.
- Attach 2 polarizers to the same holder, and place the collimating slits on the track as shown in
Figure 7. Make sure the rotary motion sensor is mounted on the spectrophotometer table with
the bigger diameter of spindle against the table (see Figure 7). Attach the spectrophotometer
table base to ground as instructed by your TA.
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Figure 7: Complete setup for Exercise 3. Note that the computer interface is different in your
experiment
- Two round polarizers are used on the holder. To read the polarizer angle, use the brass
pointer at the top of the holder. Note that these angle readings differ from the real orientation of the
polarizer by 180 degrees.
- Rotate the second polarizer (second from laser) to 225 degrees and lock it in place by
tightening the brass screw. (In reality, this is a 45 degree polarizer, used to solve the problem
that the laser light is already polarized). The first polarizer (closest to the laser) is used
throughout the experiment to adjust the light level.
- The square analyzing polarizer (see Figure 8) has its transmission axis marked, and for
normal use the label should be on top with its axis horizontal and thus 90 degrees from the
polarization axis of the reflected light. In this way, it is used to find the variation in the parallel
component of the reflected light and to determine Brewster’s angle. By placing the analyzing
polarizer with its transmission axis vertical, you can also look at the variation in the
perpendicular component of the reflected light.
Figure 8: The analyzing segment of the setup
- The small metal Brewster's angle accessory disk should be zeroed so that the mark at the top
of the label above the N in the word ANGLE is aligned with the zero angle mark on the
spectrophotometer disk.
- The step base has two marks. For reflected light, use the mark on the higher step side. Align
it to zero on the disk. The D lens is placed on the lower surface, flush against the step when
data is being collected.
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- To align the laser beam, remove all polarizers, collimating slits, and D lens. Set the
spectrophotometer arm near 180 degrees. Use the x-y adjustment on the laser diode to get the
beam at the center of the light sensor slit. The light sensor bracket slit should be set on slit #4.
Place the collimating slits on track and adjust the slit position so the laser beam passes through
the #4 slit.
Note about reflection angle measurements: The angle is calculated by dividing the actual angle
(recorded by the computer) by two, with a correction with respect to the spindle diameter. The markings
on Brewster's disk are there only for convenience (in this experiment) and are not used directly. To get
the laser beam exactly on to the slit, you must make fine adjustments while watching the digits display on
computer for the maximum light intensity. You can adjust either the Brewster’s disk or the
spectrophotometer arm until the intensity is maximized.
Procedure
Dim the room lights. Click on the Brewster’s Angle application tab.
Place the double polarizer on the track and the D lens on the step. Rotate the Brewster’s disk to
the 120 mark located counter-clockwise on the spectrophotometer table. Click ON/OFF and,
while watching the digits display of light intensity, rotate the spectrophotometer arm to get into
the beam. Rotate the first polarizer (nearest to the laser) to adjust the intensity level to be as
high as possible. The Light Sensor should be on gain of 1 or 10. This will setup the starting point
of the rotational motion sensor.
Start the acquisition without the square polarizer: slowly rotate the spectrophotometer arm
clockwise in 10 degree steps, over ~100 degrees. At each step, rotate the Brewster’s disk to get
into the beam. Use the cursors to read the intensity (Io) and angle values of graph maxima.
Place the square analyzing polarizer (axis horizontal) on the spectrophotometer arm in front of
the slits. Note: The square analyzing polarizer must sit level, flat on the arm.
Do an acquisition with square polarizer (horizontal) using the same procedure as before. Read
intensity (I) and angle values of graph maxima.
Analysis
Take the ratio I/Io, graph it versus angle. Include errors in reading intensity and angle. Determine
the Brewster’s angle.
Use Brewster's angle to calculate the index of refraction of acrylic using Equation 7. Use n1 = 1.
Calculate the parallel and perpendicular reflectances using Fresnel equations (5a and 5b).
 Python Requirement 3 (PHY224/324 students only): do the three analysis steps above by
writing a program to fit the data and to output Brewster’s angle, index of refraction of acrylic and
the two reflectances from (5a) and (5b).
Questions
Would Brewster's angle be more or less for light in air reflecting off water?
How would data look like for an arrangement with vertical square polarizer?
How do polarized sunglasses reduce glare? Which direction is the axis of polarization in a pair
of polarized sunglasses? How could you check this?
This guide sheet was written by Ruxandra Serbanescu in 2009 (revised 2015). The acquisition DAQ
board was built and programmed by Larry Avramidis
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