# Muller Calculus & Transfer Matrix Method (TMM)

Polarization effects in KostaCLOUD via Muller Calculus & Transfer Matrix Method (TMM)
In KostaCLOUD every ray keeps track of a Polarization Stokes Vector which specifies a mixed polarization state of light which enters into an optical system.
Many people are accustomed to the standard Jones Vector, which is a complex valued vector which can be used to describe solely pure polarization states. A Jones vector can be represented in the following way:
$\bf{E} = \begin{bmatrix} E_x e^{i\delta_x}\\ E_y e^{i\delta_y} \end{bmatrix}$
We can convert a Jones Vectors into Stokes Vectors in a fixed cartesian basis (as defined previously) as follows:
$\begin{bmatrix} I\\ Q \\ U \\ V \end{bmatrix} =\begin{bmatrix} I_\text{Unpolarized}\\ 0 \\ 0 \\ 0 \end{bmatrix} +\begin{bmatrix} E_x^2+E_y^2\\ E_x^2-E_y^2 \\ 2 E_x E_y \cos(\delta_x -\delta_y) \\ -2 E_x E_y \sin(\delta_x-\delta_y) \end{bmatrix}$
You can specify any simple mixed or pure polarization state of light utilizing Stokes parameters. A mixed state is defined as an unknown state, which translates to a random polarization. As light propagates through a system, it generally turns into a more pure state, as we have a defined system with known interactions. A few Examples are as follows
$\begin{bmatrix} I_0\\ Q_0 \\ U_0 \\ V_0 \end{bmatrix} =\begin{bmatrix} 1\\ 0 \\ 0 \\ 0 \end{bmatrix}$
-> Unpolarized Light
$\begin{bmatrix} I_0\\ Q_0 \\ U_0 \\ V_0 \end{bmatrix} =\begin{bmatrix} 1\\ 1 \\ 0 \\ 0 \end{bmatrix}$
-> Linear Horizontally Polarized Light,
$-Q_0$
is Linear Vertical Polarized Light
$\begin{bmatrix} I_0\\ Q_0 \\ U_0 \\ V_0 \end{bmatrix} =\begin{bmatrix} 1\\ 0 \\ 1 \\ 0 \end{bmatrix}$
-> Linear
$45^{\circ}$
Polarized Light,
$-U_0$
is Linear
$-45^{\circ}$
Polarized Light
$\begin{bmatrix} I_0\\ Q_0 \\ U_0 \\ V_0 \end{bmatrix} =\begin{bmatrix} 1\\ 0 \\ 0 \\ 1 \end{bmatrix}$
-> Right Hand Circularly Polarized Light,
$-V_0$
is Left Hand Circularly Polarized Light
In Reflection of an interface we use the following matrix:
$\begin{bmatrix} I'\\ Q' \\ U' \\ V' \end{bmatrix} =\begin{bmatrix} \frac{1}{2} (\rho_s^2+\rho_p^2) & \frac{1}{2} (\rho_s^2-\rho_p^2) & 0 & 0\\ \frac{1}{2} (\rho_s^2-\rho_p^2) & \frac{1}{2} (\rho_s^2+\rho_p^2) & 0 & 0 \\ 0 & 0 & \rho_s \rho_p \cos(\Delta_\rho) & -\rho_s \rho_p \sin(\Delta_\rho) \\ 0 & 0 & -\rho_s \rho_p \sin(\Delta_\rho) & \rho_s \rho_p \cos(\Delta_\rho) \end{bmatrix} \begin{bmatrix} I_0\\ Q_0 \\ U_0 \\ V_0 \end{bmatrix}$
Where we can relate local quantities to our reflection Fresnel coefficients:
$\rho_s = |r_s| \\ \rho_p = |r_p| \\ \Delta_\rho = \measuredangle r_s - \measuredangle r_p$
In Transmission of an interface we use the following matrix:
$\begin{bmatrix} I'\\ Q' \\ U' \\ V' \end{bmatrix} =\begin{bmatrix} \frac{1}{2} (\tau_s^2+\tau_p^2) & \frac{1}{2} (\tau_s^2-\tau_p^2) & 0 & 0\\ \frac{1}{2} (\tau_s^2-\tau_p^2) & \frac{1}{2} (\tau_s^2+\tau_p^2) & 0 & 0 \\ 0 & 0 & \tau_s \tau_p \cos(\Delta_\tau) & -\tau_s \tau_p \sin(\Delta_\tau) \\ 0 & 0 & -\tau_s \tau_p \sin(\Delta_\tau) & \tau_s \tau_p \cos(\Delta_\tau) \end{bmatrix} \begin{bmatrix} I_0\\ Q_0 \\ U_0 \\ V_0 \end{bmatrix}$
Where we can relate local quantities to our transmission Fresnel coefficients:
$\tau_s = |t_s| \\ \tau_p = |t_p| \\ \Delta_\tau = \measuredangle t_s - \measuredangle t_p \\$
In calculating Dielectric Stacks KostaCLOUD utilizes the Transfer Matrix Method (TMM). TMM is projected into the Stokes Vector Space to include effects such as layer thickness, which plays directly into the transmission/reflection Fresnel coefficients. We can calculate an induced phase factor due to each matrix as follows:
$\phi_n=i k_\parallel \Delta t_n$
, where
$k_\parallel$
is the projected wave-vector,
$\Delta t_n$
is our layer thickness. We can then include these in a transfer matrix:
$(r/t)_\text{total}=\prod_{n=1}^{N_\text{max}} \begin{bmatrix} e^{\phi_n} & (r/t)_{n,n+1} e^{\phi_n} \\ (r/t)_{n,n+1} e^{-\phi_n} & e^{-\phi_n} \end{bmatrix}$
1. 1.
E. Collett, Field Guide to Polarized Light, SPIE - the International Society for Optical Engineering, Bellingham, WA (2005).
2. 2.
Charalambos C. Katsidis and Dimitrios I. Siapkas, "General transfer-matrix method for optical multilayer systems with coherent, partially coherent, and incoherent interference," Appl. Opt. 41, 3978-3987 (2002)