Mueller matrix roots depolarization parameters

Hannah D. Noble; Stephen C. McClain; Russell A. Chipman

doi:10.1364/AO.51.000735

Applied Optics
Vol. 51,
Issue 6,
pp. 735-744
(2012)
•https://doi.org/10.1364/AO.51.000735

Mueller matrix roots depolarization parameters

Hannah D. Noble, Stephen C. McClain, and Russell A. Chipman

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Hannah D. Noble, Stephen C. McClain, and Russell A. Chipman, "Mueller matrix roots depolarization parameters," Appl. Opt. 51, 735-744 (2012)

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Abstract

The Mueller matrix roots decomposition recently proposed by Chipman in [1] and its three associated families of depolarization (amplitude depolarization, phase depolarization, and diagonal depolarization) are explored. Degree of polarization maps are used to differentiate among the three families and demonstrate the unity between phase and diagonal depolarization, while amplitude depolarization remains a distinct class. Three families of depolarization are generated via the averaging of different forms of two nondepolarizing Mueller matrices. The orientation of the resulting depolarization follows the cyclic permutations of the Pauli spin matrices. The depolarization forms of Mueller matrices from two scattering measurements are analyzed with the matrix roots decomposition—a sample of ground glass and a graphite and wood pencil tip.

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Number	Generator	First-Order Form
1	$(\begin{matrix} 1 & d_{1} & 0 & 0 \\ d_{1} & 1 & 0 & 0 \\ 0 & 0 & \sqrt{1 - d_{1}^{2}} & 0 \\ 0 & 0 & 0 & \sqrt{1 - d_{1}^{2}} \end{matrix})$	$(\begin{matrix} 1 & d_{1} & 0 & 0 \\ d_{1} & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix})$
2	$(\begin{matrix} 1 & 0 & d_{2} & 0 \\ 0 & \sqrt{1 - d_{2}^{2}} & 0 & 0 \\ d_{2} & 0 & 1 & 0 \\ 0 & 0 & 0 & \sqrt{1 - d_{2}^{2}} \end{matrix})$	$(\begin{matrix} 1 & 0 & d_{2} & 0 \\ 0 & 1 & 0 & 0 \\ d_{2} & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix})$
3	$(\begin{matrix} 1 & 0 & 0 & d_{3} \\ 0 & \sqrt{1 - d_{3}^{2}} & 0 & 0 \\ 0 & 0 & \sqrt{1 - d_{3}^{2}} & 0 \\ d_{3} & 0 & 0 & 1 \end{matrix})$	$(\begin{matrix} 1 & 0 & 0 & d_{3} \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ d_{3} & 0 & 0 & 1 \end{matrix})$
4	$(\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & \cos d_{4} & \sin d_{4} \\ 0 & 0 & - \sin d_{4} & \cos d_{4} \end{matrix})$	$(\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & d_{4} \\ 0 & 0 & - d_{4} & 1 \end{matrix})$
5	$(\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & \cos d_{5} & 0 & - \sin d_{5} \\ 0 & 0 & 1 & 0 \\ 0 & \sin d_{5} & 0 & \cos d_{5} \end{matrix})$	$(\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & - d_{5} \\ 0 & 0 & 1 & 0 \\ 0 & d_{5} & 0 & 1 \end{matrix})$
6	$(\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & \cos d_{6} & \sin d_{6} & 0 \\ 0 & - \sin d_{6} & \cos d_{6} & 0 \\ 0 & 0 & 0 & 1 \end{matrix})$	$(\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & d_{6} & 0 \\ 0 & - d_{6} & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix})$

Number	Generator	First-Order Form
7	$(\begin{matrix} 1 & d_{7} & 0 & 0 \\ - d_{7} & 1 & 0 & 0 \\ 0 & 0 & \sqrt{1 - d_{7}^{2}} & 0 \\ 0 & 0 & 0 & \sqrt{1 - d_{7}^{2}} \end{matrix})$	$(\begin{matrix} 1 & d_{7} & 0 & 0 \\ - d_{7} & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix})$
8	$(\begin{matrix} 1 & 0 & d_{8} & 0 \\ 0 & \sqrt{1 - d_{8}^{2}} & 0 & 0 \\ - d_{8} & 0 & 1 & 0 \\ 0 & 0 & 0 & \sqrt{1 - d_{8}^{2}} \end{matrix})$	$(\begin{matrix} 1 & 0 & d_{8} & 0 \\ 0 & 1 & 0 & 0 \\ - d_{8} & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix})$
9	$(\begin{matrix} 1 & 0 & 0 & d_{9} \\ 0 & \sqrt{1 - d_{9}^{2}} & 0 & 0 \\ 0 & 0 & \sqrt{1 - d_{9}^{2}} & 0 \\ - d_{9} & 0 & 0 & 1 \end{matrix})$	$(\begin{matrix} 1 & 0 & 0 & d_{9} \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ - d_{9} & 0 & 0 & 1 \end{matrix})$
10	$(\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & \sqrt{1 - d_{10}^{2}} & d_{10} \\ 0 & 0 & d_{10} & \sqrt{1 - d_{10}^{2}} \end{matrix})$	$(\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & d_{10} \\ 0 & 0 & d_{10} & 1 \end{matrix})$
11	$(\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & \sqrt{1 - d_{11}^{2}} & 0 & d_{11} \\ 0 & 0 & 1 & 0 \\ 0 & d_{11} & 0 & \sqrt{1 - d_{11}^{2}} \end{matrix})$	$(\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & d_{11} \\ 0 & 0 & 1 & 0 \\ 0 & d_{11} & 0 & 1 \end{matrix})$
12	$(\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & \sqrt{1 - d_{12}^{2}} & d_{12} & 0 \\ 0 & d_{12} & \sqrt{1 - d_{12}^{2}} & 0 \\ 0 & 0 & 0 & 1 \end{matrix})$	$(\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & d_{12} & 0 \\ 0 & d_{12} & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix})$
13	$(\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & - d_{13} + \sqrt{1 - d_{13}^{2} / 2} & 0 & 0 \\ 0 & 0 & d_{13} + \sqrt{1 - d_{13}^{2} / 2} & 0 \\ 0 & 0 & 0 & \sqrt{1 - d_{13}^{2}} \end{matrix})$	$(\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 - d_{13} & 0 & 0 \\ 0 & 0 & 1 + d_{13} & 0 \\ 0 & 0 & 0 & 1 \end{matrix})$
14	$(\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & \frac{d_{14}}{\sqrt{3}} + \sqrt{1 - 2 d_{14}^{2} / 3} & 0 & 0 \\ 0 & 0 & \frac{d_{14}}{\sqrt{3}} + \sqrt{1 - 2 d_{14}^{2} / 3} & 0 \\ 0 & 0 & 0 & - \frac{2 d_{14}}{\sqrt{3}} + \sqrt{1 - 2 d_{14}^{2} / 3} \end{matrix})$	$(\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 + d_{14} / \sqrt{3} & 0 & 0 \\ 0 & 0 & 1 + d_{14} / \sqrt{3} & 0 \\ 0 & 0 & 0 & 1 - 2 d_{14} / \sqrt{3} \end{matrix})$
15	$(\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 - \sqrt{\frac{2}{3}} d_{15} & 0 & 0 \\ 0 & 0 & 1 - \sqrt{\frac{2}{3}} d_{15} & 0 \\ 0 & 0 & 0 & 1 - \sqrt{\frac{2}{3}} d_{15} \end{matrix})$	$(\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 - \sqrt{\frac{2}{3}} d_{15} & 0 & 0 \\ 0 & 0 & 1 - \sqrt{\frac{2}{3}} d_{15} & 0 \\ 0 & 0 & 0 & 1 - \sqrt{\frac{2}{3}} d_{15} \end{matrix})$

$α I$	Attenuator
$D_{H / V} (q, r)$	Horizontal/Vertical Linear Diattenuator
$D_{45 ° / 135 °} (q, r)$	$45 ° / 135 °$ Linear Diattenuator
$D_{R / L} (q, r)$	Circular Diattenuator
$R_{H / V} (δ)$	Horizontal/Vertical Linear Retarder
$R_{45 ° / 135 °} (δ)$	$45 ° / 135 °$ Linear Retarder
$R_{R / L} (δ)$	Circular Retarder

Matrix	$α I$	$D_{H / V} (q, r)$	$D_{45 ° / 135 °} (q, r)$	$D_{R / L} (q, r)$	$R_{H / V} (δ)$	$R_{45 ° / 135 °} (δ)$	$R_{R / L} (δ)$
$α I$		( $D_{15}$ , $D_{14}$ , $D_{13}$ )	( $D_{15}$ , $D_{14}$ , $D_{13}$ )	$D_{15}$	( $D_{15}$ , $D_{14}$ , $D_{13}$ )	( $D_{15}$ , $D_{14}$ , $D_{13}$ )	$D_{15}$
$D_{H / V} (q, r)$	( $D_{15}$ , $D_{14}$ , $D_{13}$ )		$D_{15}$ , $D_{14}$ , $D_{12}$	( $D_{15}$ , $D_{14}$ , $D_{13}$ ), $D_{11}$	( $D_{15}$ , $D_{14}$ , $D_{13}$ )	( $D_{15}$ , $D_{14}$ , $D_{13}$ ), $D_{9}$	( $D_{15}$ , $D_{14}$ , $D_{13}$ ), $D_{8}$
$D_{45 ° / 135 °} (q, r)$	( $D_{15}$ , $D_{14}$ , $D_{13}$ )	$D_{15}$ , $D_{14}$ , $D_{12}$		( $D_{15}$ , $D_{14}$ , $D_{13}$ ), $D_{10}$	( $D_{15}$ , $D_{14}$ , $D_{13}$ ), $D_{9}$	( $D_{15}$ , $D_{14}$ , $D_{13}$ )	( $D_{15}$ , $D_{14}$ , $D_{13}$ ), $D_{7}$
$D_{R / L} (q, r)$	$D_{15}$	( $D_{15}$ , $D_{14}$ , $D_{13}$ ), $D_{11}$	( $D_{15}$ , $D_{14}$ , $D_{13}$ ), $D_{10}$		( $D_{15}$ , $D_{14}$ , $D_{13}$ ), $D_{8}$	( $D_{15}$ , $D_{14}$ , $D_{13}$ ), $D_{7}$	$D_{15}$
$R_{H / V} (δ)$	( $D_{15}$ , $D_{14}$ , $D_{13}$ )	( $D_{15}$ , $D_{14}$ , $D_{13}$ )	( $D_{15}$ , $D_{14}$ , $D_{13}$ ), $D_{9}$	( $D_{15}$ , $D_{14}$ , $D_{13}$ ), $D_{8}$		$D_{15}$ , $D_{14}$ , $D_{12}$	$D_{15}$ , $D_{14}$ , $D_{13}$ , $D_{11}$
$R_{45 ° / 135 °} (δ)$	( $D_{15}$ , $D_{14}$ , $D_{13}$ )	( $D_{15}$ , $D_{14}$ , $D_{13}$ ), $D_{9}$	( $D_{15}$ , $D_{14}$ , $D_{13}$ )	( $D_{15}$ , $D_{14}$ , $D_{13}$ ), $D_{7}$	$D_{15}$ , $D_{14}$ , $D_{12}$		( $D_{15}$ , $D_{14}$ , $D_{13}$ ), $D_{10}$
$R_{R / L} (δ)$	( $D_{15}$	$D_{15}$ , $D_{14}$ , $D_{13}$ ), $D_{8}$	( $D_{15}$ , $D_{14}$ , $D_{13}$ ), $D_{7}$	$D_{15}$	( $D_{15}$ , $D_{14}$ , $D_{13}$ ), $D_{11}$	( $D_{15}$ , $D_{14}$ , $D_{13}$ ), $D_{10}$

Roots Parameter	Specular Angle ${- 70 °, 70 °}$	Non-Specular Angle ${- 70 °, 10 °}$
$D_{0}$	0.029	0.096
$D_{1}$	$- 0.243$	$- 0.470$
$D_{2}$	0.002	0.006
$D_{3}$	$- 0.006$	$- 0.008$
$D_{4}$	2.612	0.785
$D_{5}$	0.007	0.005
$D_{6}$	$- 0.075$	$- 0.119$
$D_{7}$	$- 0.018$	$- 0.059$
$D_{8}$	0.001	0.016
$D_{9}$	0	$- 0.001$
$D_{10}$	0.108	$- 0.024$
$D_{11}$	0.029	$- 0.004$
$D_{12}$	0.007	$- 0.006$
$D_{13}$	$- 0.149$	$- 0.430$
$D_{14}$	0.065	0.459
$D_{15}$	0.292	1.169

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Equations (14)

Applied Optics