Decomposition of a depolarizing Mueller matrix into its nondepolarizing components by using symmetry conditions

Ertan Kuntman; Oriol Arteaga

doi:10.1364/AO.55.002543

Applied Optics
Vol. 55,
Issue 10,
pp. 2543-2550
(2016)
•https://doi.org/10.1364/AO.55.002543

Decomposition of a depolarizing Mueller matrix into its nondepolarizing components by using symmetry conditions

Ertan Kuntman and Oriol Arteaga

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Ertan Kuntman and Oriol Arteaga, "Decomposition of a depolarizing Mueller matrix into its nondepolarizing components by using symmetry conditions," Appl. Opt. 55, 2543-2550 (2016)

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Abstract

A procedure for the parallel decomposition of a depolarizing Mueller matrix with an associated rank 2 covariance matrix into its two nondepolarizing components is presented. We show that, if one of the components agrees with certain symmetry conditions, the arbitrary decomposition becomes unique, and its calculation is straightforward. Solutions for six different symmetries, which are relevant for the physical interpretation of polarimetric measurements, are provided. With this procedure, a single polarimetric measurement is sufficient to fully disclose the complete polarimetric response of two different systems and evaluate their weights in the overall response. The decomposition method we propose is illustrated by obtaining the ellipsometric responses of a silicon wafer and a holographic grating from a single measurement in which the light spot illuminates sectors of both materials. In a second example, we use the decomposition to analyze an optical system in which a polarizing film is partially covered by another misaligned film.

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Corrections

23 March 2016: Corrections were made to Refs. 19 and 20.

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Case	Mueller Matrix $M$	Covariance Matrix $H$ ^a
Type 1	$[\begin{matrix} 1 & m_{01} & 0 & 0 \\ m_{01} & 1 & 0 & 0 \\ 0 & 0 & m_{22} & m_{23} \\ 0 & 0 & - m_{23} & m_{22} \end{matrix}]$	$[\begin{matrix} P & 0 & 0 & W \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ W^{} & 0 & 0 & \bar{P} \end{matrix}] \begin{array}{l} P = (1 + m_{01}) / 2 \\ W = (m_{22} + i m_{23}) / 2 \\ \bar{P} = 1 - P \\ P \bar{P} = W^{} W \end{array}$
Type 2	$[\begin{matrix} 1 & 0 & m_{02} & 0 \\ 0 & m_{11} & 0 & m_{13} \\ m_{02} & 0 & 1 & 0 \\ 0 & - m_{13} & 0 & m_{11} \end{matrix}]$	$[\begin{matrix} K & N & N & K \\ N^{} & \bar{K} & \bar{K} & N^{} \\ N^{} & \bar{K} & \bar{K} & N^{} \\ K & N & N & K \end{matrix}] \begin{array}{l} K = (1 + m_{11}) / 4 \\ N = (m_{02} + i m_{13}) / 4 \\ \bar{K} = 0.5 - K \\ K \bar{K} = N^{*} N \end{array}$
Type 3	$[\begin{matrix} 1 & 0 & 0 & m_{03} \\ 0 & m_{11} & m_{12} & 0 \\ 0 & - m_{12} & m_{11} & 0 \\ m_{03} & 0 & 0 & 1 \end{matrix}]$	$[\begin{matrix} E & V & - V & E \\ V^{} & \bar{E} & - \bar{E} & V^{} \\ - V^{} & - \bar{E} & \bar{E} & - V^{} \\ E & V & - V & E \end{matrix}] \begin{array}{l} E = (1 + m_{11}) / 4 \\ V = (m_{12} + i m_{03}) / 4 \\ \bar{E} = 0.5 - E \\ E \bar{E} = V^{*} V \end{array}$

Symmetry	Equations
Type 1	$α_{1} P = (h_{11} h_{00} - h_{10} h_{01}) / h_{11}$ $α_{1} W = [h_{10} h_{03} - h_{13} (h_{00} - α_{1} P)] / h_{10}$ $α_{1} \bar{P} = α_{1} W W^{*} / P$ alternatively, $α_{1} P = (h_{22} h_{00} - h_{20} h_{02}) / h_{22}$ $α_{1} W = [h_{20} h_{03} - h_{23} (h_{00} - α_{1} P)] / h_{20}$
Type 2a	$α_{1} K = \frac{h_{30} h_{03} - h_{00} h_{33}}{h_{30} + h_{03} - h_{33} - h_{00}}$ $α_{1} N = \frac{h_{30} h_{01} - h_{00} h_{31} + α_{1} K (h_{31} - h_{01})}{h_{30} - h_{00}}$ $α_{1} \bar{K} = α_{1} N N^{*} / K$
Type 2b	$α_{1} \bar{K} = \frac{h_{21} h_{12} - h_{11} h_{22}}{h_{21} + h_{12} - h_{22} - h_{11}}$ $α_{1} N = \frac{h_{11} h_{02} - h_{12} h_{01} + α_{1} \bar{K} (h_{01} - h_{02})}{h_{11} - h_{12}}$ $α_{1} K = α_{1} N N^{*} / \bar{K}$
Type 3a	$α_{1} E = \frac{h_{30} h_{03} - h_{00} h_{33}}{h_{30} + h_{03} - h_{33} - h_{00}}$ $α_{1} V^{} = \frac{h_{10} h_{03} - h_{00} h_{13} + α_{1} E (h_{13} - h_{10})}{h_{03} - h_{00}}$ $α_{1} \bar{E} = α_{1} V V^{} / E$
Type 3b	$α_{1} \bar{E} = \frac{h_{11} h_{22} - h_{21} h_{12}}{h_{21} + h_{12} + h_{22} + h_{11}}$ $α_{1} V = \frac{h_{01} h_{12} - h_{11} h_{02} + α_{1} \bar{E} (h_{01} + h_{02})}{h_{11} + h_{12}}$ $α_{1} E = α_{1} V V^{*} / \bar{E}$
In all cases $α_{1}$ is determined from $α_{1} = tr (α_{1} H_{1 S})$

Case	Mueller Matrix $M$	Covariance Matrix $H$ ^a
Type 1-2	$[\begin{matrix} 1 & m_{01} & m_{02} & m_{03} \\ m_{01} & m_{11} & m_{12} & m_{13} \\ m_{02} & m_{12} & m_{22} & m_{23} \\ - m_{03} & - m_{13} & - m_{23} & m_{33} \end{matrix}]$	$[\begin{matrix} A & G & G & M \\ G^{} & B & B & H^{} \\ G^{} & B & B & H^{} \\ M^{} & H & H & C \end{matrix}] \begin{array}{l} A B = G G^{} \\ G H^{} = M B \\ B C = H H^{} \end{array}$
Type 1-3	$[\begin{matrix} 1 & m_{01} & m_{02} & m_{03} \\ m_{01} & m_{11} & m_{12} & m_{13} \\ - m_{02} & - m_{12} & m_{22} & m_{23} \\ m_{03} & m_{13} & - m_{23} & m_{33} \end{matrix}]$	$[\begin{matrix} A & G & - G & M \\ G^{} & B & - B & H^{} \\ G^{} & - B & B & - H^{} \\ M^{} & H & - H & C \end{matrix}] \begin{array}{l} A B = G G^{} \\ G H^{} = M B \\ B C = H H^{} \end{array}$
Type 2-3	$[\begin{matrix} 1 & m_{01} & m_{02} & m_{03} \\ - m_{01} & m_{11} & m_{12} & m_{13} \\ m_{02} & - m_{12} & m_{22} & m_{23} \\ m_{03} & - m_{13} & m_{23} & m_{33} \end{matrix}]$	$[\begin{matrix} A & G & H & A \\ G^{} & B & Y & G^{} \\ H^{} & Y^{} & C & H^{} \\ A & G & H & A \end{matrix}] \begin{array}{l} A B = G G^{} \\ H G^{} = A Y \\ A C = H H^{} \end{array}$

Symmetry	Equations
Type 1-2	$α_{1} B = \frac{h_{21} h_{12} - h_{11} h_{22}}{h_{21} + h_{12} - h_{22} - h_{11}}$ $α_{1} G = \frac{h_{02} h_{11} - h_{01} h_{12} + α_{1} B (h_{01} - h_{02})}{h_{11} - h_{12}}$ $α_{1} H = \frac{h_{31} h_{22} - h_{21} h_{32} - α_{1} B (h_{31} - h_{32})}{h_{22} - h_{21}}$
Type 1-3	$α_{1} B = \frac{h_{11} h_{22} - h_{21} h_{12}}{h_{11} + h_{22} + h_{21} + h_{12}}$ $α_{1} G = \frac{h_{01} h_{12} - h_{11} h_{02} + α_{1} B (h_{01} + h_{02})}{h_{11} + h_{12}}$ $α_{1} H = \frac{h_{31} h_{22} - h_{21} h_{32} - α_{1} B (h_{31} + h_{32})}{h_{22} + h_{21}}$
Type 2-3	$α_{1} A = \frac{h_{30} h_{03} - h_{00} h_{33}}{h_{30} + h_{03} - h_{33} - h_{00}}$ $α_{1} G^{} = \frac{h_{10} h_{03} - h_{00} h_{13} - α_{1} A (h_{10} - h_{13})}{h_{03} - h_{00}}$ $α_{1} H^{} = \frac{h_{20} h_{03} - h_{00} h_{23} - α_{1} A (h_{20} - h_{23})}{h_{03} - h_{00}}$
In all cases $α_{1}$ is determined from $α_{1} = tr (α_{1} H_{1 S})$

Case	Mueller Matrix $M$	Covariance Matrix $H$ ^a
Type 1	$[\begin{matrix} 1 & m_{01} & 0 & 0 \\ m_{01} & 1 & 0 & 0 \\ 0 & 0 & m_{22} & m_{23} \\ 0 & 0 & - m_{23} & m_{22} \end{matrix}]$	$[\begin{matrix} P & 0 & 0 & W \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ W^{} & 0 & 0 & \bar{P} \end{matrix}] \begin{array}{l} P = (1 + m_{01}) / 2 \\ W = (m_{22} + i m_{23}) / 2 \\ \bar{P} = 1 - P \\ P \bar{P} = W^{} W \end{array}$
Type 2	$[\begin{matrix} 1 & 0 & m_{02} & 0 \\ 0 & m_{11} & 0 & m_{13} \\ m_{02} & 0 & 1 & 0 \\ 0 & - m_{13} & 0 & m_{11} \end{matrix}]$	$[\begin{matrix} K & N & N & K \\ N^{} & \bar{K} & \bar{K} & N^{} \\ N^{} & \bar{K} & \bar{K} & N^{} \\ K & N & N & K \end{matrix}] \begin{array}{l} K = (1 + m_{11}) / 4 \\ N = (m_{02} + i m_{13}) / 4 \\ \bar{K} = 0.5 - K \\ K \bar{K} = N^{*} N \end{array}$
Type 3	$[\begin{matrix} 1 & 0 & 0 & m_{03} \\ 0 & m_{11} & m_{12} & 0 \\ 0 & - m_{12} & m_{11} & 0 \\ m_{03} & 0 & 0 & 1 \end{matrix}]$	$[\begin{matrix} E & V & - V & E \\ V^{} & \bar{E} & - \bar{E} & V^{} \\ - V^{} & - \bar{E} & \bar{E} & - V^{} \\ E & V & - V & E \end{matrix}] \begin{array}{l} E = (1 + m_{11}) / 4 \\ V = (m_{12} + i m_{03}) / 4 \\ \bar{E} = 0.5 - E \\ E \bar{E} = V^{*} V \end{array}$

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Applied Optics