# 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:

$$
(r/t)*\text{total}=\prod*{n=1}^{N\_\text{max}} \begin{bmatrix}
e^{\phi\_n} & (r/t)*{n,n+1} e^{\phi\_n} \\
(r/t)*{n,n+1} e^{-\phi\_n} & e^{-\phi\_n} \end{bmatrix}
$$

#### References

1. E. Collett, *Field Guide to Polarized Light*, SPIE - the International Society for Optical Engineering, Bellingham, WA (2005).
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)


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