Optoelectronics: Photonic Materials and Devices
DT086/DT085
Polarization of light (Part 1, theory)
Exercise #3
Suppose that you have a pair of ideal polarizers located along an optical axis.
The polarization axis of the first one is vertical and the second horizontal. You
place a third ideal polarizer between this pair. This polarizer can have a
polarization axis at angle  with respect to the vertical.
a) Show that the transmittance T = Iout/Iin for unpolarized (Ex=Ey=E)
light is given by
T ( ) 

1
1  cos 2 (2 )
8

b) For which angles is the transmission maximum, and for which is it
minimum. Analyse this dependence using your favourite graphing
software. Are you surprised that you can get a transmitted intensity,
despite the original polarizers being oriented orthogonal?
c) Consider the case when the horizontal and vertical polarizers can be
considered ideal, while the 3rd polarizer (in between) is a dichroic
sheet polarizer with k1=0.8 and with extinction ratio (k2/k1) of 0.01.
Derive new formula for the transmittance through such a system.
d) Compare dependencies of transmission versus the rotation angle of
the middle polarizer for ideal and non-ideal cases.
The Jones Matrix of a Polarizer
The quality of a polarizer is characterized by two parameters: the major
principal transmittance k1 and the minor principal transmittance k2. k1 is the
ratio of the intensity of the transmitted light to that of the incident light when
the polarizer is oriented to maximize the transmission of linearly polarized
incident light. k2 is the same ratio but when the polarizer is oriented to
minimize transmission. The values of k1 and k2 are defined when the incident
light direction is perpendicular to the surface of the polarizer.
The Jones matrix of a polarizer whose major principal transmission axis is
along the x axis is
k
P 1
0
0
k 2 
For an ideal polarizer, k1 = 1 and k2 = 0.
Next, the case when the polarizer is rotated in its plane will be considered. Let
the direction of the transmission axis be rotated by  from the x axis.
cos 
P  
 sin 
 sin   k1
cos    0
k cos 2   k 2 sin 2 
P   1
 (k1  k 2 ) sin  cos 
0   cos 
k 2   sin 
sin  
cos  
(k1  k 2 ) sin  cos  

k1 sin 2   k 2 cos 2  
If the polarizer is ideal, the Jones matrix becomes:
 cos 2  sin  cos  
P  

sin 2  
sin  cos
Intensity of Light
Intensity of the electromagnetic wave, represented by the Jones vector J:
Ex 
J  
E y 
can be found as follows:

I  J * J  E x*

Ex 
E *y    E x* E x  E *y E y
E y 