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Mathematics Science Partnership (MSP) 2011
Polarization of Light
(Color online if you have a black and white printout)
Pengqian Wang
About the instructor:
Pengqian Wang, Associate Professor, Department of Physics, Western Illinois University
Phone: 309-298-2541
E-mail: [email protected]
http://faculty.wiu.edu/P-Wang
Contents:
1. Electromagnetic Waves
2. Polarization of Light
3. Polarizers
4. Polarization by Reflection
5. Birefringence
6. Optical Activity
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1. Electromagnetic Waves
Lights are electromagnetic waves.
A light has an oscillating electric field and an oscillating magnetic field.
The directions of the electric field (E), the magnetic field (B), and the propagation of the
light (v) are perpendicular to each other. They are related by the right hand rule:
E×B//v.
Waves are categorized into transverse waves and longitudinal waves.
Light is a transverse wave, because its oscillation directions (E and B) are perpendicular
to its propagation direction (v).
Assignment 1.
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2. Polarization of Light
2.1 Definition of the polarization direction of light
The polarization direction of light is defined as in the direction of its electric field.
1) Since Fe = qE, the polarization determines the force direction. Most light phenomena
are governed by the fact that the electrons in the molecules receive forces from the
electric field of the light waves.
2) An electron usually receives a much stronger force from the electric field than from
the magnetic field of the light. Fe = qE, Fm = qv×B.
2.2 Polarization states of light
Suppose the light is propagating in the z-direction. Because light is a transverse wave, the
electric field (E-field) direction, which has been defined as the polarization of the light,
must be in the x-y plane. Its two components on the x and y axes are algebraically
E x ( z, t )  ˆiE0 x cos( kz  t )
E0 x , E0 y  Amplitude
E y ( z, t )  ˆjE0 y cos( kz  t   ) k  Wave vector
  Frequency
E( z, t )  E x ( z, t )  E y ( z, t )
  Phase lag
y
Ey
x
Ex
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1) Linear polarization
The E-field is always oscillating in one direction. This corresponds to phase lag  = 0 or
 = ±p in the algebraic description.
y
y
E
E
Ey
Ex
x
Ex
x
Ey
 = ±p
=0
2) Circular polarization
The direction of the E-field rotates. The end point of the E vector traces out a circle.
This corresponds to E0x=E0y=E0,  = ± p /2 in the algebraic description.
y
y
E0
E0
t
E0
x
E
t
E0
x
E
 = -p /2
 = p /2
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3) Elliptical polarization
The E vector rotates and changes its magnitude as well. The end point of the E vector
traces out an ellipse.
Elliptical polarization is the most general form of the polarization state of light.
E0y
E
E0x
4) Random polarization
The direction of the E-field is not predictable. Also called unpolarized light.
Nature light (sun light) is randomly polarized.
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3. Polarizers
A polarizer is an optical device whose output is a certain form of polarized light.
A linear polarizer is the most common type of polarizer. It has a transmission axis and
an extinction axis, which are perpendicular to each other. Light polarized in the
direction of the transmission axis passes through the polarizer freely. Light polarized
in the direction of the extinction axis is totally absorbed or deflected. Therefore the
transmitted light is polarized in the direction of the transmission axis of the polarizer.
Malus’s law
The energy transmitivity of a linear polarizer is given by the Malus’s law. Suppose a
linearly polarized light with intensity I0 is impinging upon a linear polarizer, the
intensity of the transmitted light is given by
I ( )  I 0 cos 2 
Here  is the angle between the polarization
direction of the input light and the
transmission axis of the polarizer.
Assignment 2.
For randomly polarized light, the transmission
through a linear polarizer is always ½.
Input
polarization

Output
polarization
Light
propagation
direction
Transmission
axis
Polarizer
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Activities:
1. Preparing the polarizer.
Tear off the protection film. Mark the two shorter sides with transparent tape. They are
the transmission axes. Cut the polarizer into halves. You now have two linear
polarizers.
2. Observation of light sources.
1) Ceiling lights. Please hold a polarizer and look through it at a ceiling light. Rotate the
polarizer through 360º. What happens to the intensity of the transmitted light? What
is the polarization state of the light emitted by the ceiling lights?
2) Two polarizers. Please hold one polarizer in each hand, one in front of the other, and
look through them at a ceiling light. Now rotate one of the polarizers through 360º.
What happens to the intensity of the light transmitted through the two polarizers?
What is the angular relationship between the transmission axes of the two polarizers
when the light is blocked?
3) Laser pointer. Please observe laser safety.
4) Flashlight.
5) LCD screens. E.g., a computer monitor, a cell phone, a camera, a calculator, etc.
Assignment 3.
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4. Polarization by Reflection
4.1 s-polarization and p-polarization
Plane-of-incidence is the plane formed by the incident light beam and the normal of the
interface between the two media.
The s-polarization is defined for those light whose E-field is perpendicular to the planeof-incidence. The p-polarization is defined for those light whose E-field is parallel
to the plane-of-incidence.
Ei
i r
t
Er
Ei
Er
i r
ni
nt
Et
s-polarization
t
ni
nt
Et
p-polarization
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4.2 Reflectance
The energy reflectance of a light beam incident on an optical medium depends on the
polarization of the incident light, i.e., whether it is s-polarized or p-polarized.
 n cos  i  nt cos  t
Rs   i
 ni cos  i  nt cos  t
2
 sin 2 ( i   t )
 
2
 sin ( i   t )
2
 nt cos  i  ni cos  t 
tan 2 ( i   t )
 
R p  
2
n
cos


n
cos

tan
( i   t )
t
t
i 
 i
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4.3 Brewster angle
When i  t  90,  i  arctan
nt
  B is called the Brewster angle.
ni
At the Brewster angle, the reflectance of the p-polarization is zero, therefore the reflected
nature light is totally s-polarized, i.e., it is linearly polarized in the direction
perpendicular to the plane-of-incidence.
We can then reduce the reflected light using a polarizer. This is how a sunglass work.
It is especially helpful when you are fishing.
For a glass with n=1.5, the Brewster angle is  B  arctan 1.5  56.
Assignment 4.
p
s
B
Rp = 0
E
Polarizer
Brewster
angle
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Activities:
Glare from the floor:
Please find a place on the floor in the classroom or hallway where you can see “glare”
(reflection) from a ceiling light (carpet won’t work). You may need to turn off other
lights. Look at the glare through a single polarizer and rotate the polarizer through
360º. At the minimum transmission, please further reduce the transmitted light by
adjusting the angle of reflection. This is done by moving yourself a little forward or
backward. What is the orientation of the transmission axis of the polarizer when the
glare is minimized? What is the polarization direction of the light being reflected
from the floor?
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5. Birefringence
5.1 Birefringence
Birefringence refers to the phenomenon that when light is propagating inside an
optically anisotropic crystal, mostly a uniaxial crystal, the refractive index of the light
depends on its polarization direction.
The crystal has an optic axis. Light whose polarization is perpendicular to the optic axis
feels a refractive index of no and propagates with a speed of vo=c/ no. Light whose
polarization is parallel to the optic axis feels a refractive index of ne and propagates
with a speed of ve=c/ ne.
Inside the crystal, a light beam is in general split into an ordinary light and an
extraordinary light, each with different speed and somehow different propagation
direction.
An anisotropic crystal can be used to separate light into two perpendicularly polarized
light beams.
Extraordinary-light
Ordinary-light
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5.2 Wave plates
Wave plates (retardation plates, or retarders) are optical elements used to transform
the polarization states of light. They are made from one or more pieces of birefringent
crystals. Each wave plate has a fast axis and a slow axis. Inside the crystal, the incident
light is separated into two lights, each has refractive indices no and ne, and is polarized
parallel to the fast or slow axes. The two light have different speed, therefore at the
output surface there is a phase lag between the two lights. The phase lag determines the
polarization state of the output light.
1) A half-wave plate introduces a phase lag of p. It converts a linearly polarized light
into another linearly polarized light, mirrored by the fast or slow axis.
2) A quarter-wave plate introduces a phase lag of p/2. It converts a linearly polarized
light into a circularly polarized light, when the input polarization is 45° to the fast and
slow axes.
Assignment 5.
f
s
Half-wave plate
f
f
f
s
s
s
Quarter-wave plate
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Activities:
1. Calcite crystal
1) Please put the crystal on a page of paper and see the transmission of some letters
without polarizers. Rotate the crystals a little so that the two images of the letters are
well separated. Then look at the letters with one polarizer. Rotate the polarizer and
explain what you observe. Please notice the relation between the polarization
directions of the two images of the letters.
Assignment 6.
2) Please put the crystal on a picture of a bird, and see the transmission of the picture
when you are rotating the polarizer back and forth. I hope you have fun.
2. Crossed polarizers.
Please fix the flashlight on your table with light shooting up, using a double-sided tape.
Adjust the two polarizers until they are exactly crossed and the light coming from the
flashlight is extinguished. Place each of the following objects between the polarizers
and rotate them. You will see that light is transmitted in very interesting ways.
1) A plastic protractor.
2) A U-shape plastic ― squeeze it and see the stress-induced bands.
3) A small plastic box.
4) Pieces of cellophane tape on glass.
5) A sheet of mica.
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6. Optical Activity
8.10 Optical activity
Optical activity refer to the phenomena that when a linearly polarized light is passing
through a medium, the polarization plane is rotated. It is thus also called optical rotation.
When this happens, the medium is said to be optically active. The rotation of the
polarization plane is proportional to the path length of the light, and can be measured by
degree/centimeter. Common optically active media include quartz, sugar and syrup. It can
be used to measure blood sugar concentration in diabetic people.
Activities:
Corn syrup
Please fill a beaker with corn syrup to about 1.5 inch high. Put the beaker between the
crossed polarizers above our flashlight station.
Please rotate the upper polarizer in very fine step and observe the color changes of
the syrup. Please take photos of your favorite colored liquid when possible.
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