1. (A) Find the thicknesses of a particular birefringent crystal (n1 = 1.4737 and n2 = 1.4714)
needed to produce /4, /2, and  retardation plates, respectively, for the Argon laser line (
= 488 nm).
Retardation  = d·(n1 – n2) 

d = (n – n )
1
2
where (n1–n2)=0.0023

For the /4-retardation, d(/4) = 4  0.0023 = 53 m

For the /2-retardation, d(/2) = 2  0.0023 = 106 m

For the -retardation, d() = 4  0.0023 = 212 m
(B) Explain, including a sketch of the polarization states, how you can use these retarders (in
combination with a linear polarizer) to produce:
Since n1 > n2, n1 is the slow axis and n2 the fast axis.
(a) right circular polarized light
(b) left circular polarized light
(c) 90° rotation of plane-polarized light
fast
/2 plate
slow
Light
Polarization
90° Rotation
of Linear
Polarization
Light
Propagation
(C) How do you produce with the help of the above mentioned optical elements elliptically
polarized light? (Explain and make a sketch.)
2. You are working with a linearly polarized laser which emits a beam of light both at 1 = 700
nm and 2 = 450 nm wavelengths, and you have available material of linear birefringence
with nx = 1.7735 and ny = 1.7719. You want to design and cut it with a proper thickness
which can be used simultaneously as a circular polarizer for 1, and which changes the
linear polarization of 2 into 90° crossed linear polarization.
(A) What is the thickness of your chosen plate?
We need a retarder which can be a quater-plate for 1 and a half-plate for 2
simultaneously.
m1
1
1
Retardation  = d· (nx – ny) = 2 1 + 4 1 = m2 2 + 2 2
where m1 and m2 must be integers.
1
1

(2m
+
1)
=
1
1
4
2 2 (m2 + 1)
2m1 + 1
2 / 2
225
9
=
=
=
2m2 + 1
175
7
1 / 4
Therefore, (m1, m2) = (4,3) or (13,10) or (22,17) or (31,24) or .... (4+9n,3+7n) ..
We choose the set of the smallest numbers, m1 = 4 and m2 = 3. Then,
9
7
1.575 m
 = 4 1 = 2 2 = 1.575 m  d = 0.0016
= 984 m
(B) Show in a sketch how you arrange the laser and its polarization relative to the retardation
plate, and indicate the achieved polarizations for the two wave lengths behind the plate.
1
fast
Right
Circular
Polarization
2
Light
Polarization
slow
90° Rotation
of Linear
Polarization
Light
Propagation
3. You want to distinguish experimentally unpolarized from circularly polarized light of the
same wavelength . Describe the procedures in words and sketches.
(A) What physical tools will you need to do this and how will you arrange and use them?
We need two tools which are one /4-retarder and one linear polarizer.
When the circular polarized light goes through the /4-retarder, it will become a
linear polarized light which can be easily detected by a linear polarizer. When
the unpolarized light goes through the /4-retarder, however, it is still the
unpolarized light. Its intensity will not be changed under the rotation of the
linear polarizer.
Unpolarized
Light
/4
retarder
Unpolarized
Light
fast
Linear
Polarizer
Under
Turning
No
Intensity
Change
slow
Intensity
Change
Circular Polarized
Light
Linear Polarized
Light
(B) How will you determine if a circularly polarized light is left-circular or right-circular?
We have to know the fast- and slow-axis of the /4-retarder, and the transmissionaxis of linear polarizer. See the following sketch.
/4
retarder
Linear
Polarizer
f
fast
Right
Circular
Polarized
s
slow
Left
Circular
Polalrized
f
s
Linear Polarized
Light
Testing the
linear
polarized
direction
relative to the
fast- or
slow-axis as
shown in this
figure.
Problem 4:
(a) If you have two polarizers in crossed position and put one in between you will get some light
through again. Why ?
(b) How would you need to position 8 additional polarizers between two crossed polarizers to
obtain maximum transmission.
(a) just use Malus Law. The additional polarizer will reduce the intensity according to Malus
law and turn the polarization into its axis. This polarization is only reducd but not
completely blocked by the final polarizers
(b) For 8 additional polarizers you need to position one of an other rotated by an angle of
90/(8+1) relative to the previous polarizer