Question

(a) Show that the normal modes of the linear polarizer are linearly polarized waves

(b) Show that the normal modes of the wave retarder are linearly polarized waves

(c) Show that the normal modes of the polarization rotator are right and left circularly polarized waves

What are the eigenvalues of the systems described above?

Answer 1

We know that

$J_1=\begin{bmatrix} E_{1x}\\ E_{1x} \end{bmatrix}$

$J_2=\begin{bmatrix} E_{2x}\\ E_{2x} \end{bmatrix}$

$J_3=\begin{bmatrix} E_{3x}\\ E_{3x} \end{bmatrix}$

$J_1=tJ_1;$ $2\times2$ jhon metrics for transmission

$J_2=rJ_1$ $2\times2$ jhon metrics for reflection

$t=\begin{bmatrix} t_x& 0 \\ 0 & t_y \end{bmatrix}$

$r=\begin{bmatrix} r_x& 0 \\ 0 & r_y \end{bmatrix}$

$E_{2x}=t_xE_{1x};E_{2y}=t_yE_{1y};$

$E_{3x}=r_xE_{1x};E_{3x}=r_yE_{1y};$

TE mode polerization

$r_x=\frac{n_1cos\theta_1-n_2cos\theta_2}{n_1cos\theta_1+n_2cos\theta_2}$

$t_x=1+r_x$

TM mode polerization

$r_y=\frac{n_2cos\theta_1-n_1cos\theta_2}{n_2cos\theta_1+n_1cos\theta_2}$

$t_y=\frac{n_1}{n_2}(1+r_y)$

Now

$Y=rel(Aexp(j(w(t-\frac{z}{c}))))$

Then

$A=A_x\hat{x}+A_y\hat{y}$

Then

$Y_x=a_xcos(w(t-\frac{z}{c})+\phi_x)$

Now

$Y_y=a_ycos(w(t-\frac{z}{c})+\phi_y)$

Now

$\frac{\xi^2}{a_x^2}+\frac{\xi^2}{a_y^2}-2cos\phi\frac{\xi_x\xi_y}{a_xa_y}=sin^2\phi$

$\phi=0°$ linear polarization

$\phi_y>\phi_x$

Clock wise

$\phi_y<\phi_x$

Anti clock wise