Homogeneous and inhomogeneous Jones matrices

Shih-Yau Lu; Russell A. Chipman

doi:10.1364/JOSAA.11.000766

Journal of the Optical Society of America A
Vol. 11,
Issue 2,
pp. 766-773
(1994)
•https://doi.org/10.1364/JOSAA.11.000766

Homogeneous and inhomogeneous Jones matrices

Shih-Yau Lu and Russell A. Chipman

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Shih-Yau Lu and Russell A. Chipman, "Homogeneous and inhomogeneous Jones matrices," J. Opt. Soc. Am. A 11, 766-773 (1994)

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Abstract

The classification of polarization properties of polarization elements is studied to derive data-reduction equations for extracting the diattenuation, retardance, and other polarization properties from their Jones matrices. Polarization elements, and Jones matrices as well, are divided into two classes: homogeneous, with orthogonal eigenpolarizations, and inhomogeneous, with nonorthogonal eigenpolarizations. The basic polarization properties, diattenuation and retardance, of homogeneous polarization elements are straightforward and well known; these elements are characterized by their eigenvalues and eigenpolarizations. Polarization properties of inhomogeneous polarization elements are not so evident. By applying polar decomposition, the definitions of diattenuation and retardance are generalized to inhomogeneous polarization elements, providing an understanding of their polarization characteristics. Furthermore, an inhomogeneity parameter is introduced to describe the degree of inhomogeneity in a polarization element. These results are then adapted to degenerate polarization elements, which have only one linearly independent eigenpolarization.

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Equations (81)

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Matrix	$[\begin{matrix} cos θ & 0 \\ sin θ & 0 \end{matrix}]$	$cos θ [\begin{matrix} cos θ & 0 \\ sin θ & 0 \end{matrix}]$	$\frac{1}{\sqrt{2}} [\begin{matrix} 1 + i cos 2 θ & 0 \\ i sin 2 θ & 0 \end{matrix}]$	$[\begin{matrix} cos θ & - sin θ \\ 0 & 0 \end{matrix}]$	$cos θ [\begin{matrix} cos θ & - sin θ \\ 0 & 0 \end{matrix}]$	$\frac{1}{\sqrt{2}} [\begin{matrix} 1 + i cos 2 θ & - i sin 2 θ \\ 0 & 0 \end{matrix}]$
ξ_q	cos θ	cos²θ	$(1 + i cos 2 θ) / \sqrt{2}$	cos θ	cos²θ	$(1 + i cos 2 θ) / \sqrt{2}$
E_q	$(\begin{matrix} cos θ \\ sin θ \end{matrix})$	$(\begin{matrix} cos θ \\ sin θ \end{matrix})$	$\frac{1}{\sqrt{2}} (\begin{matrix} 1 + i cos 2 θ \\ i sin 2 θ \end{matrix})$	$(\begin{matrix} 1 \\ 0 \end{matrix})$	$(\begin{matrix} 1 \\ 0 \end{matrix})$	$(\begin{matrix} 1 \\ 0 \end{matrix})$
ξ_r	0	0	0	0	0	0
E_r	$(\begin{matrix} 0 \\ 1 \end{matrix})$	$(\begin{matrix} 0 \\ 1 \end{matrix})$	$(\begin{matrix} 0 \\ 1 \end{matrix})$	$(\begin{matrix} sin θ \\ cos θ \end{matrix})$	$(\begin{matrix} sin θ \\ cos θ \end{matrix})$	$\frac{1}{\sqrt{2}} (\begin{matrix} i sin 2 θ \\ 1 + i cos 2 θ \end{matrix})$
η	\|sin θ\|	\|sin θ\|	$∣ sin 2 θ ∣ / \sqrt{2}$	\|sin θ\|	\|sin θ\|	$∣ sin 2 θ ∣ / \sqrt{2}$
E_max	$(\begin{matrix} 1 \\ 0 \end{matrix})$	$(\begin{matrix} 1 \\ 0 \end{matrix})$	$(\begin{matrix} 1 \\ 0 \end{matrix})$	$(\begin{matrix} cos θ \\ - sin θ \end{matrix})$	$(\begin{matrix} cos θ \\ - sin θ \end{matrix})$	$\frac{1}{\sqrt{2}} (\begin{matrix} 1 - i cos 2 θ \\ i sin 2 θ \end{matrix})$
JE_max	$(\begin{matrix} cos θ \\ sin θ \end{matrix})$	$cos θ (\begin{matrix} cos θ \\ sin θ \end{matrix})$	$\frac{1}{\sqrt{2}} (\begin{matrix} 1 + i cos 2 θ \\ i sin 2 θ \end{matrix})$	$(\begin{matrix} 1 \\ 0 \end{matrix})$	$(\begin{matrix} cos θ \\ 0 \end{matrix})$	$(\begin{matrix} 1 \\ 0 \end{matrix})$
T_max	1	cos²θ	1	1	cos²θ	1
E_min	$(\begin{matrix} 0 \\ 1 \end{matrix})$	$(\begin{matrix} 0 \\ 1 \end{matrix})$	$(\begin{matrix} 0 \\ 1 \end{matrix})$	$(\begin{matrix} sin θ \\ cos θ \end{matrix})$	$(\begin{matrix} sin θ \\ cos θ \end{matrix})$	$\frac{1}{\sqrt{2}} (\begin{matrix} i sin 2 θ \\ 1 + i cos 2 θ \end{matrix})$
JE_min	$(\begin{matrix} 0 \\ 0 \end{matrix})$	$(\begin{matrix} 0 \\ 0 \end{matrix})$	$(\begin{matrix} 0 \\ 0 \end{matrix})$	$(\begin{matrix} 0 \\ 0 \end{matrix})$	$(\begin{matrix} 0 \\ 0 \end{matrix})$	$(\begin{matrix} 0 \\ 0 \end{matrix})$
T_min	0	0	0	0	0	0
$D$	1	1	1	1	1	1
ℛ	2\|θ\|	2\|θ\|	2 cos⁻¹[(1 + cos² 2θ)/2]^1/2	2\|θ\|	2\|θ\|	2 cos⁻¹[(1 + cos² 2θ)/2]^1/2

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Equations (81)

Journal of the Optical Society of America A