Three-dimensional polarization ray-tracing calculus I: definition and diattenuation

Garam Yun; Karlton Crabtree; Russell A. Chipman

doi:10.1364/AO.50.002855

Applied Optics
Vol. 50,
Issue 18,
pp. 2855-2865
(2011)
•https://doi.org/10.1364/AO.50.002855

Three-dimensional polarization ray-tracing calculus I: definition and diattenuation

Garam Yun, Karlton Crabtree, and Russell A. Chipman

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Garam Yun, Karlton Crabtree, and Russell A. Chipman, "Three-dimensional polarization ray-tracing calculus I: definition and diattenuation," Appl. Opt. 50, 2855-2865 (2011)

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Abstract

A three-by-three polarization ray-tracing matrix method for polarization ray tracing in optical systems is presented for calculating the polarization transformations associated with ray paths through optical systems. The method is a three-dimensional generalization of the Jones calculus. Reflection and refraction algorithms are provided. Diattenuation of the optical system is calculated via singular value decomposition. Two numerical examples, a three fold-mirror system and a hollow corner cube, demonstrate the method.

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$J_{q}$	${\hat{k}}_{q - 1}$	${\hat{k}}_{q}$	$P_{q}$

$(\begin{matrix} e^{- i π / 4} & 0 \\ 0 & e^{i π / 4} \end{matrix})$	$(\begin{matrix} 0 \\ 0 \\ 1 \end{matrix})$	$(\begin{array}{l} 0 \\ 0 \\ 1 \end{array})$	$(\begin{matrix} e^{- i π / 4} & 0 & 0 \\ 0 & e^{i π / 4} & 0 \\ 0 & 0 & 1 \end{matrix})$


$(\begin{matrix} e^{- i π / 4} & 0 \\ 0 & e^{i π / 4} \end{matrix})$	$(\begin{array}{l} 0 \\ 1 \\ 0 \end{array})$	$(\begin{array}{l} 0 \\ 1 \\ 0 \end{array})$	$(\begin{matrix} e^{- i π / 4} & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & e^{i π / 4} \end{matrix})$


$(\begin{matrix} e^{- i π / 4} & 0 \\ 0 & e^{i π / 4} \end{matrix})$	$(\begin{array}{l} 1 \\ 0 \\ 0 \end{array})$	$(\begin{array}{l} 1 \\ 0 \\ 0 \end{array})$	$(\begin{matrix} 1 & 0 & 0 \\ 0 & e^{- i π / 4} & 0 \\ 0 & 0 & e^{i π / 4} \end{matrix})$

q	${\hat{k}}_{q - 1}$	${\hat{k}}_{q}$	$P_{q}$

1	$(\begin{array}{l} 0 \\ 0 \\ 1 \end{array})$	$(\begin{array}{l} 1 \\ 0 \\ 0 \end{array})$	$(\begin{matrix} 0 & 0 & 1 \\ 0 & - 1 & 0 \\ - 1 & 0 & 0 \end{matrix})$


2	$(\begin{array}{l} 1 \\ 0 \\ 0 \end{array})$	$(\begin{array}{l} 0 \\ 1 \\ 0 \end{array})$	$(\begin{matrix} 0 & - 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & - 1 \end{matrix})$


3	$(\begin{array}{l} 0 \\ 1 \\ 0 \end{array})$	$(\begin{array}{l} 0 \\ 0 \\ 1 \end{array})$	$(\begin{matrix} - 1 & 0 & 0 \\ 0 & 0 & - 1 \\ 0 & 1 & 0 \end{matrix})$

q	${\hat{k}}_{q - 1}$	${\hat{k}}_{q}$	${\hat{P}}_{q}$	${\hat{P}}_{q}^{'}$	${\hat{s}}_{q}$	Surface Normal	$P_{q}$

1	$(\begin{matrix} 0 \\ 0 \\ - 1 \end{matrix})$	$(\begin{array}{l} \frac{- 2 \sqrt{2}}{3} \\ 0 \\ \frac{- 1}{3} \end{array})$	$(\begin{array}{l} - 1 \\ 0 \\ 0 \end{array})$	$(\begin{array}{l} \frac{- 1}{3} \\ 0 \\ \frac{2 \sqrt{2}}{3} \end{array})$	$(\begin{array}{l} 0 \\ - 1 \\ 0 \end{array})$	$(\begin{array}{l} \sqrt{\frac{2}{3}} \\ 0 \\ \frac{- 1}{\sqrt{3}} \end{array})$	$(\begin{matrix} 0.27 + 0.16 i & 0 & 0.94 \\ 0 & - 0.96 - 0.18 i & 0 \\ - 0.75 - 0.46 i & 0 & 0.33 \end{matrix})$



2	$(\begin{array}{l} \frac{- 2 \sqrt{2}}{3} \\ 0 \\ \frac{- 1}{3} \end{array})$	$(\begin{array}{l} \frac{- \sqrt{2}}{3} \\ \sqrt{\frac{2}{3}} \\ \frac{1}{3} \end{array})$	$(\begin{array}{l} \frac{- 1}{6} \\ \frac{\sqrt{3}}{2} \\ \frac{\sqrt{2}}{3} \end{array})$	$(\begin{array}{l} \frac{5}{6} \\ \frac{1}{2 \sqrt{3}} \\ \frac{\sqrt{2}}{3} \end{array})$	$(\begin{array}{l} \frac{- 1}{2 \sqrt{3}} \\ \frac{- 1}{2} \\ \sqrt{\frac{2}{3}} \end{array})$	$(\begin{array}{l} \frac{- 1}{\sqrt{6}} \\ \frac{- 1}{\sqrt{2}} \\ \frac{- 1}{\sqrt{3}} \end{array})$	$(\begin{matrix} 0.25 - 0.08 i & 0.44 + 0.33 i & 0.7 + 0.23 i \\ - 0.95 - 0.05 i & - 0.04 + 0.08 i & 0.23 + 0.14 i \\ - 0.15 & 0.72 + 0.27 i & - 0.57 - 0.01 i \end{matrix})$



3	$(\begin{array}{l} \frac{- \sqrt{2}}{3} \\ \sqrt{\frac{2}{3}} \\ \frac{1}{3} \end{array})$	$(\begin{array}{l} 0 \\ 0 \\ 1 \end{array})$	$(\begin{array}{l} \frac{1}{6} \\ \frac{- 1}{2 \sqrt{3}} \\ \frac{2 \sqrt{2}}{3} \end{array})$	$(\begin{array}{l} \frac{1}{2} \\ \frac{- \sqrt{3}}{2} \\ 0 \end{array})$	$(\begin{matrix} \frac{- \sqrt{3}}{2} \\ \frac{- 1}{2} \\ 0 \end{matrix})$	$(\begin{array}{l} \frac{- 1}{\sqrt{6}} \\ \frac{1}{\sqrt{2}} \\ \frac{- 1}{\sqrt{3}} \end{array})$	$(\begin{matrix} - 0.65 - 0.09 i & - 0.53 - 0.15 i & 0.38 + 0.23 i \\ - 0.53 - 0.15 i & - 0.04 + 0.08 i & - 0.65 - 0.4 i \\ - 0.47 & 0.82 & 0.33 \end{matrix})$

$I_{\max}$	$v_{1}$	$I_{\min}$	$v_{2}$	D Diattenuation

0.774	$(\begin{array}{l} 0.65 e^{- 0.85 i} \\ 0.76 e^{- 2.15 i} \\ 0 \end{array}) = e^{- 2.15 i} (\begin{array}{l} 0.65 e^{1.30 i} \\ 0.76 \\ 0 \end{array})$	0.757	$(\begin{array}{l} 0.76 e^{- 0.91 i} \\ 0.65 e^{- 0.94 i} \\ 0 \end{array}) = e^{- 0.91 i} (\begin{array}{l} 0.76 \\ 0.65 e^{1.85 i} \\ 0 \end{array})$	0.014

$J_{q}$	${\hat{k}}_{q - 1}$	${\hat{k}}_{q}$	$P_{q}$

$(\begin{matrix} e^{- i π / 4} & 0 \\ 0 & e^{i π / 4} \end{matrix})$	$(\begin{matrix} 0 \\ 0 \\ 1 \end{matrix})$	$(\begin{array}{l} 0 \\ 0 \\ 1 \end{array})$	$(\begin{matrix} e^{- i π / 4} & 0 & 0 \\ 0 & e^{i π / 4} & 0 \\ 0 & 0 & 1 \end{matrix})$


$(\begin{matrix} e^{- i π / 4} & 0 \\ 0 & e^{i π / 4} \end{matrix})$	$(\begin{array}{l} 0 \\ 1 \\ 0 \end{array})$	$(\begin{array}{l} 0 \\ 1 \\ 0 \end{array})$	$(\begin{matrix} e^{- i π / 4} & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & e^{i π / 4} \end{matrix})$


$(\begin{matrix} e^{- i π / 4} & 0 \\ 0 & e^{i π / 4} \end{matrix})$	$(\begin{array}{l} 1 \\ 0 \\ 0 \end{array})$	$(\begin{array}{l} 1 \\ 0 \\ 0 \end{array})$	$(\begin{matrix} 1 & 0 & 0 \\ 0 & e^{- i π / 4} & 0 \\ 0 & 0 & e^{i π / 4} \end{matrix})$

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Applied Optics