Polarization elements: a group-theoretical study

Sudha; A. V. Gopala Rao

doi:10.1364/JOSAA.18.003130

Journal of the Optical Society of America A
Vol. 18,
Issue 12,
pp. 3130-3134
(2001)
•https://doi.org/10.1364/JOSAA.18.003130

Polarization elements: a group-theoretical study

Sudha and A. V. Gopala Rao

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Sudha and A. V. Gopala Rao, "Polarization elements: a group-theoretical study," J. Opt. Soc. Am. A 18, 3130-3134 (2001)

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Abstract

The classification of polarization elements, the polarization affecting optical devices that have a Jones-matrix representation according to the type of eigenvectors they possess, is given a new visit through the group-theoretical connection of polarization elements. The diattenuators and retarders are recognized as the elements corresponding to boosts and rotations, respectively. The structure of homogeneous elements other than diattenuators and retarders are identified by giving the quaternion corresponding to these elements. The set of degenerate polarization elements is identified with the so-called null elements of the Lorentz group. Singular polarization elements are examined in their more illustrative Mueller-matrix representation, and, finally the eigenstructure of a special class of singular Mueller matrices is studied.

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Type of Polarization Element	Example 1	Example 2
Homogeneous	$\frac{1}{\sqrt{2}} [\begin{matrix} \sqrt{2} - i & 1 - \sqrt{2} i \\ 1 - \sqrt{2} i & \sqrt{2} - i \end{matrix}]$	$\frac{1}{2} [\begin{matrix} 2 - 3 i & - 2 \sqrt{3} - \sqrt{3} i \\ 2 \sqrt{3} + \sqrt{3} i & 2 - 3 i \end{matrix}]$
	$X_{1} = \frac{1}{\sqrt{2}} {1, - 1}; X_{2} = \frac{1}{\sqrt{2}} {1, 1}$	$X_{1} = \frac{1}{\sqrt{2}} {1, - i}; X_{2} = \frac{1}{\sqrt{2}} {1, i}$
Inhomogeneous	$[\begin{matrix} - i & - 1 - 2 i \\ 1 & 2 + i \end{matrix}]$	$[\begin{matrix} 2 + \sqrt{3} & 0 \\ 2 (1 - i) & 2 - \sqrt{3} \end{matrix}]$
	$X_{1} = \frac{1}{\sqrt{2}} {1, - 1}; X_{2} = \frac{\sqrt{5}}{\sqrt{6}} \{1, \frac{- 1}{1 + 2 i}\}$	$X_{1} = \frac{\sqrt{3}}{\sqrt{5}} \{1, \frac{1 - i}{\sqrt{3}}\}; X_{2} = {0, 1}$
Degenerate	$[\begin{matrix} - i & 1 - i \\ 1 - i & 2 + i \end{matrix}]$	$[\begin{matrix} 1 & 0 \\ 2 (1 - i) & 1 \end{matrix}]$
	$X_{1} = X_{2} = \frac{1}{\sqrt{2}} {1, i}$	$X_{1} = X_{2} = {0, 1}$

Type of Polarization Element	Example 1	Example 2
Homogeneous	$[\begin{matrix} 1 & 1 \\ 1 & 1 \end{matrix}]$	$[\begin{matrix} 0 & 0 \\ 0 & 1 \end{matrix}]$
	$X_{1} = \frac{1}{\sqrt{2}} {1, 1}; X_{2} = \frac{1}{\sqrt{2}} {1, - 1}$	$X_{1} = {0, 1}; X_{2} = {1, 0}$
Inhomogeneous	$[\begin{matrix} 1 & i \\ 1 & i \end{matrix}]$	$[\begin{matrix} 1 & 0 \\ 1 & 0 \end{matrix}]$
	$X_{1} = \frac{1}{\sqrt{2}} {1, 1}; X_{2} = \frac{1}{\sqrt{2}} {1, i}$	$X_{1} = \frac{1}{\sqrt{2}} {1, 1}; X_{2} = {0, 1}$
Degenerate	$[\begin{matrix} 1 & - 1 \\ 1 & - 1 \end{matrix}]$	$[\begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix}]$
	$X_{1} = X_{2} = \frac{1}{\sqrt{2}} {1, 1}$	$X_{1} = X_{2} = {1, 0}$

Type of Polarization Element	Example 1	Example 2
Homogeneous	$\frac{1}{\sqrt{2}} [\begin{matrix} \sqrt{2} - i & 1 - \sqrt{2} i \\ 1 - \sqrt{2} i & \sqrt{2} - i \end{matrix}]$	$\frac{1}{2} [\begin{matrix} 2 - 3 i & - 2 \sqrt{3} - \sqrt{3} i \\ 2 \sqrt{3} + \sqrt{3} i & 2 - 3 i \end{matrix}]$
	$X_{1} = \frac{1}{\sqrt{2}} {1, - 1}; X_{2} = \frac{1}{\sqrt{2}} {1, 1}$	$X_{1} = \frac{1}{\sqrt{2}} {1, - i}; X_{2} = \frac{1}{\sqrt{2}} {1, i}$
Inhomogeneous	$[\begin{matrix} - i & - 1 - 2 i \\ 1 & 2 + i \end{matrix}]$	$[\begin{matrix} 2 + \sqrt{3} & 0 \\ 2 (1 - i) & 2 - \sqrt{3} \end{matrix}]$
	$X_{1} = \frac{1}{\sqrt{2}} {1, - 1}; X_{2} = \frac{\sqrt{5}}{\sqrt{6}} \{1, \frac{- 1}{1 + 2 i}\}$	$X_{1} = \frac{\sqrt{3}}{\sqrt{5}} \{1, \frac{1 - i}{\sqrt{3}}\}; X_{2} = {0, 1}$
Degenerate	$[\begin{matrix} - i & 1 - i \\ 1 - i & 2 + i \end{matrix}]$	$[\begin{matrix} 1 & 0 \\ 2 (1 - i) & 1 \end{matrix}]$
	$X_{1} = X_{2} = \frac{1}{\sqrt{2}} {1, i}$	$X_{1} = X_{2} = {0, 1}$

Type of Polarization Element	Example 1	Example 2
Homogeneous	$[\begin{matrix} 1 & 1 \\ 1 & 1 \end{matrix}]$	$[\begin{matrix} 0 & 0 \\ 0 & 1 \end{matrix}]$
	$X_{1} = \frac{1}{\sqrt{2}} {1, 1}; X_{2} = \frac{1}{\sqrt{2}} {1, - 1}$	$X_{1} = {0, 1}; X_{2} = {1, 0}$
Inhomogeneous	$[\begin{matrix} 1 & i \\ 1 & i \end{matrix}]$	$[\begin{matrix} 1 & 0 \\ 1 & 0 \end{matrix}]$
	$X_{1} = \frac{1}{\sqrt{2}} {1, 1}; X_{2} = \frac{1}{\sqrt{2}} {1, i}$	$X_{1} = \frac{1}{\sqrt{2}} {1, 1}; X_{2} = {0, 1}$
Degenerate	$[\begin{matrix} 1 & - 1 \\ 1 & - 1 \end{matrix}]$	$[\begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix}]$
	$X_{1} = X_{2} = \frac{1}{\sqrt{2}} {1, 1}$	$X_{1} = X_{2} = {1, 0}$

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Journal of the Optical Society of America A