Help with polarization and half-wave plates

[benoy]

I am taking a course in Electro-Optics for Electrical Engineering and I am having some problem solving the following question.

Take two ideal polarizers (the first with its axis vertical and the second, horizontal) and insert between them a stack of 10 half-wave plates, the first with its fast axis rotated pi/20 rad. from the vertical, and each subsequent one rotated pi/20 rad. from the previous one. Determine the ratio of intensity after the second polarizer to that after the first polarizer.

Does anyone know how to solve this problem?

 

[Aaron Barnard]

The answer to this question can be found by using the Jones Calculus. This method involves expressing the polarization state of the light as a matrix which is then multiplied with the matrices of each of the polarizers and half-wave plates in order to find the transmission coefficient for each element. The intensity ratio is then simply the square of this coefficient.

In this case, the matrix for the first polarizer (with its axis vertical) is given by:
P1 = [1 0; 0 0]

The matrix for the first half-wave plate is given by:
HWP1 = [cos(pi/20) sin(pi/20); -sin(pi/20) cos(pi/20)]

The matrix for the second polarizer (with its axis horizontal) is given by:
P2 = [0 0; 0 1]

The product of these three matrices gives the transmission coefficient:
T = P2*HWP1*P1 = [0 0; 0 (cos(pi/20))^2]

Therefore, the intensity ratio is given by the square of the transmission coefficient:
I_ratio = (cos(pi/20))^4

This calculation can be repeated for each successive half-wave plate in the stack in order