Introduction
The Stokes parameters describe the polarization state of electromagnetic radiation and can be easily calculated from the measured transmission and reflection coefficients of the system. In this example, we show how to use the Stokes parameters to analyze the polarization properties of the transmission and reflection of a multi-stack circular polarizer.
The Stokes vector is defined as:
$$\vec{S} = \begin{pmatrix} I\\Q\\U\\V\end{pmatrix}$$
Some Stokes vectors for common states of polarization of light [1] are shown below:
The Stokes parameters are related to the measured transmission and reflection coefficients (\(t_{ss}\), \(t_{sp}\), \(t_{pp}\), \(t_{ps}\), \(r_{ss}\), \(r_{sp}\), \(r_{pp}\), \(r_{ps}\)), and consequently to the basis electric fields (Jones vector) (\(E_{ts}^{out}\), \(E_{tp}^{out}\), \(E_{rs}^{out}\), \(E_{rp}^{out}\)). For transmission the following expressions can be used:
\[ \left[\begin{array}{ccc} E_{ts}^{out}\\ E_{tp}^{out} \\ \end{array}\right] = \left[\begin{array}{ccc} t_{ss} & t_{ps} \\ t_{sp} & t_{pp}\\ \end{array}\right] \left[\begin{array}{ccc} E_{s}^{in}\\ E_{p}^{in} \\ \end{array}\right] \]
$$I = |E_{ts}^{out}|^2 + |E_{tp}^{out}|^2$$
$$Q = |E_{ts}^{out}|^2 - |E_{tp}^{out}|^2$$
$$U = 2Re(E_{ts}^{out}E_{tp}^{*out})$$
$$V = -2Im(E_{ts}^{out}E_{tp}^{*out})$$
For reflection the following expressions can be used:
\[ \left[\begin{array}{ccc} E_{rs}^{out}\\ E_{rp}^{out} \\ \end{array}\right] = \left[\begin{array}{ccc} r_{ss} & r_{ps} \\ r_{sp} & r_{pp}\\ \end{array}\right] \left[\begin{array}{ccc} E_{s}^{in}\\ E_{p}^{in} \\ \end{array}\right] \]
$$I = |E_{rs}^{out}|^2 + |E_{rp}^{out}|^2$$
$$Q = |E_{rs}^{out}|^2 - |E_{rp}^{out}|^2$$
$$U = 2Re(E_{rs}^{out}E_{rp}^{*out})$$
$$V = -2Im(E_{rs}^{out}E_{rp}^{*out})$$
Run and results
In this example, we use the stackrt command to calculate the Stokes parameters from the transmission of the circular polarizer by replacing the top PEC material with air.
Open and run the script file [[circular_polarizer.lsf]] to calculate the complex reflection and transmission coefficients. Then open and run the [[Stokes_parameters.lsf]] script file. The Stokes parameters for both transmission and reflection will be displayed in the script prompt. For the transmission, the calculated Stokes parameters are:
$$\begin{pmatrix} 1\\-0.000348488\\0.00287026\\-0.999996\end{pmatrix}$$
The calculated parameters are approximately (1,0,0,-1), implying that the transmitted light is indeed circularly polarized (left-handed) as expected.
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Notes:
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[1] https://en.wikipedia.org/wiki/Stokes_parameters