Operator Formalism in the Theory of Partial Polarization*

A. S. Marathy

doi:10.1364/JOSA.55.000969

Journal of the Optical Society of America
Vol. 55,
Issue 8,
pp. 969-980
(1965)
•https://doi.org/10.1364/JOSA.55.000969

Operator Formalism in the Theory of Partial Polarization

A. S. Marathy

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A. S. Marathy, "Operator Formalism in the Theory of Partial Polarization*," J. Opt. Soc. Am. 55, 969-980 (1965)

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Abstract

In the coherency-matrix formalism (CMF), the state of polarization of the field as well as the physical devices are represented in terms of 2×2 matrixes. The possibility of being able to exchange the role played by these matrices in the CMF is considered and a versatile operator formalism is developed. The physical meaning of the various operator relations is given and the possibility of obtaining the form of the most general operator that commutes with a given physical device is discussed. The operator formalism is applied to the study of certain physical devices whose matrix operator representations have been constructed through the use of Pauli spin matrices. Emphasis is placed on the identification of all the operators (including the Pauli spin matrices) in terms of combinations of known physical devices. In this way, the physical meanings of the various operator relations are discussed together with the meanings of the commutation relations between the various well-known physical devices. Finally, the forms of the most general operators that commute with the respective operators considered in this paper are obtained.

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2×2 representation	eigenvalues	eigenstate	J^ij (i, j = 1, 2)	Mueller matrix	eigenvalues	eigenstate S^i,j = (i, j = 1, 2)
$\begin{array}{l} Ĉ (δ) & = {\hat{σ}}_{0} cos δ + {i \hat{σ}}_{1} sin δ \\ = [\begin{array}{l} e^{i δ} & 0 \\ 0 & e^{- i δ} \end{array}] \end{array}$	e⁺ⁱ^δ	$E_{1}^{C} = [\begin{matrix} 1 \\ 0 \end{matrix}]$	$J_{C}^{11} = [\begin{array}{l} 1 & 0 \\ 0 & 0 \end{array}]$	$Ĉ_{M} (δ) = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & cos 2 δ & - sin 2 δ \\ 0 & 0 & sin 2 δ & cos 2 δ \end{matrix}]$	+1	$S_{C}^{11} = [\begin{matrix} 1 \\ 1 \\ 0 \\ 0 \end{matrix}]$
	e⁺ⁱ^δ	$E_{1}^{C} = [\begin{matrix} 1 \\ 0 \end{matrix}]$	$J_{C}^{22} = [\begin{array}{l} 0 & 0 \\ 0 & 1 \end{array}]$		+1	$S_{C}^{22} = [\begin{matrix} 1 \\ - 1 \\ 0 \\ 0 \end{matrix}]$
	e^{−i δ}	$E_{2}^{C} = [\begin{matrix} 0 \\ 1 \end{matrix}]$	$J_{C}^{12} = [\begin{array}{l} 0 & 1 \\ 0 & 0 \end{array}]$		eⁱ²^δ	$S_{C}^{12} = [\begin{matrix} 0 \\ 0 \\ 1 \\ - i \end{matrix}]$
	e^{−i δ}	$E_{2}^{C} = [\begin{matrix} 0 \\ 1 \end{matrix}]$	$J_{C}^{21} = [\begin{array}{l} 0 & 0 \\ 1 & 0 \end{array}]$		e⁻ⁱ²^δ	$S_{C}^{21} = [\begin{matrix} 0 \\ 0 \\ 1 \\ + i \end{matrix}]$
$\begin{array}{l} \hat{K} (β) & = {\hat{σ}}_{0} cos β + {i \hat{σ}}_{2} sin β \\ = [\begin{array}{l} cos β & i sin β \\ i sin β & cos β \end{array}] \end{array}$	e^j^β	$E_{1}^{K} = [\begin{matrix} 1 \\ 1 \end{matrix}]$	$J_{K}^{11} = [\begin{array}{l} 1 & 1 \\ 1 & 1 \end{array}]$	${\hat{K}}_{M} (β) = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & cos 2 β & 0 & + sin 2 β \\ 0 & 0 & 1 & 0 \\ 0 & - sin 2 β & 0 & cos 2 β \end{matrix}]$	+1	$S_{K}^{11} = [\begin{matrix} 1 \\ 0 \\ 1 \\ 0 \end{matrix}]$
	e^j^β	$E_{1}^{K} = [\begin{matrix} 1 \\ 1 \end{matrix}]$	$J_{K}^{22} = [\begin{matrix} 1 & - 1 \\ - 1 & 1 \end{matrix}]$		+1	$S_{K}^{22} = [\begin{matrix} 1 \\ 0 \\ - 1 \\ 0 \end{matrix}]$
	e⁻ⁱ^β	$E_{2}^{K} = [\begin{matrix} 1 \\ - 1 \end{matrix}]$	$J_{K}^{12} = [\begin{array}{l} 1 & - 1 \\ 1 & - 1 \end{array}]$		eⁱ²^β	$S_{K}^{12} = [\begin{matrix} 0 \\ 1 \\ 0 \\ + i \end{matrix}]$
	e⁻ⁱ^β	$E_{2}^{K} = [\begin{matrix} 1 \\ - 1 \end{matrix}]$	$J_{K}^{21} = [\begin{array}{r} 1 & 1 \\ - 1 & - 1 \end{array}]$		e⁻ⁱ²^β	$S_{K}^{21} = [\begin{matrix} 0 \\ 1 \\ 0 \\ - i \end{matrix}]$
$\begin{array}{l} \hat{R} (θ) & = {\hat{σ}}_{0} cos θ + {i \hat{σ}}_{3} sin θ \\ = [\begin{array}{l} cos θ & - sin θ \\ sin θ & cos θ \end{array}] \end{array}$	eⁱ^θ	$E_{1}^{R} = [\begin{matrix} 1 \\ - i \end{matrix}]$	$J_{R}^{11} = [\begin{array}{l} 1 & i \\ - i & 1 \end{array}]$	${\hat{R}}_{M} (θ) = [\begin{matrix} 1 & 0 & 1 & 0 \\ 0 & cos 2 θ & - sin 2 θ & 0 \\ 0 & sin 2 θ & cos 2 θ & 0 \\ 0 & 0 & 0 & 1 \end{matrix}]$	+1	$S_{R}^{11} = [\begin{matrix} 1 \\ 0 \\ 0 \\ 1 \end{matrix}]$
	eⁱ^θ	$E_{1}^{R} = [\begin{matrix} 1 \\ - i \end{matrix}]$	$J_{R}^{22} = [\begin{array}{l} 1 & - i \\ i & 1 \end{array}]$		+1	$S_{R}^{22} = [\begin{matrix} 1 \\ 0 \\ 0 \\ - 1 \end{matrix}]$
	e⁻ⁱ^θ	$E_{2}^{R} = [\begin{matrix} 1 \\ i \end{matrix}]$	$J_{R}^{12} = [\begin{array}{l} 1 & - i \\ - i & - 1 \end{array}]$		eⁱ²^θ	$S_{R}^{12} = [\begin{matrix} 0 \\ 1 \\ - i \\ 0 \end{matrix}]$
	e⁻ⁱ^θ	$E_{2}^{R} = [\begin{matrix} 1 \\ i \end{matrix}]$	$J_{R}^{21} = [\begin{array}{r} 1 & i \\ i & - 1 \end{array}]$		e⁻ⁱ²^θ	$S_{R}^{21} = [\begin{matrix} 0 \\ 1 \\ i \\ 0 \end{matrix}]$

2×2 representation	eigenvulues	eigenstates	J^ij (i, j = 1, 2)	Mueller matrix	eigenvalues	eigenstate S^ij (i, j = 1,2)
$\begin{array}{l} Ĉ^{'} (δ^{'}) & = {\hat{σ}}_{0} cos δ^{'} - {\hat{σ}}_{1} sin δ^{'} \\ = [\begin{array}{l} e^{- δ^{'}} & 0 \\ 0 & e^{+ δ^{'}} \end{array}] \end{array}$	e⁻^δ′	$E_{1}^{C^{'}} = [\begin{matrix} 1 \\ 0 \end{matrix}]$	$J_{C^{'}}^{11} = [\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}]$	$Ĉ_{M}^{'} (δ^{'}) = [\begin{matrix} cosh 2 δ^{'} & - sinh 2 δ^{'} & 0 & 0 \\ - sinh 2 δ^{'} & cosh 2 δ^{'} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}]$	e⁻²^δ′	$S_{C^{'}}^{11} = [\begin{matrix} 1 \\ 1 \\ 0 \\ 0 \end{matrix}]$
	e⁻^δ′	$E_{1}^{C^{'}} = [\begin{matrix} 1 \\ 0 \end{matrix}]$	$J_{C^{'}}^{22} = [\begin{matrix} 0 & 0 \\ 0 & 1 \end{matrix}]$		e⁺²^δ′	$S_{C^{'}}^{22} = [\begin{matrix} 1 \\ - 1 \\ 0 \\ 0 \end{matrix}]$
	e⁺^δ′	$E_{2}^{C^{'}} = [\begin{matrix} 0 \\ 1 \end{matrix}]$	$J_{C^{'}}^{12} = [\begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix}]$		+1	$S_{C^{'}}^{12} = [\begin{matrix} 0 \\ 0 \\ 1 \\ - i \end{matrix}]$
	e⁺^δ′	$E_{2}^{C^{'}} = [\begin{matrix} 0 \\ 1 \end{matrix}]$	$J_{C^{'}}^{21} = [\begin{matrix} 0 & 0 \\ 1 & 0 \end{matrix}]$		+1	$S_{C^{'}}^{21} = [\begin{matrix} 0 \\ 0 \\ 1 \\ i \end{matrix}]$
$\begin{array}{l} {\hat{K}}^{'} (β^{'}) & = {\hat{σ}}_{0} cosh β^{'} - {\hat{σ}}_{2} sinh β^{'} \\ = [\begin{array}{l} cosh β^{'} & - sinh β^{'} \\ - sinh β^{'} & cosh β^{'} \end{array}] \end{array}$	e⁻^β′	$E_{1}^{K^{'}} = [\begin{matrix} 1 \\ 1 \end{matrix}]$	$J_{K^{'}}^{11} = [\begin{matrix} 1 & 1 \\ 1 & 1 \end{matrix}]$	${\hat{K}}_{M}^{'} (β^{'}) = [\begin{matrix} cosh 2 β^{'} & 0 & - sinh 2 β^{'} & 0 \\ 0 & 1 & 0 & 0 \\ - sinh 2 β^{'} & 0 & cosh 2 β^{'} & 0 \\ 0 & 0 & 0 & 1 \end{matrix}]$	e⁻²^β′	$S_{K^{'}}^{11} = [\begin{matrix} 1 \\ 0 \\ 1 \\ 0 \end{matrix}]$
	e⁻^β′	$E_{1}^{K^{'}} = [\begin{matrix} 1 \\ 1 \end{matrix}]$	$J_{K^{'}}^{22} = [\begin{array}{r} 1 & - 1 \\ - 1 & 1 \end{array}]$		e⁺²^β′	$S_{K^{'}}^{22} = [\begin{matrix} 1 \\ 0 \\ - 1 \\ 0 \end{matrix}]$
	e⁺^β′	$E_{2}^{K^{'}} = [\begin{matrix} 1 \\ - 1 \end{matrix}]$	$J_{K^{'}}^{12} = [\begin{matrix} 1 & - 1 \\ 1 & - 1 \end{matrix}]$		+1	$S_{K^{'}}^{12} = [\begin{matrix} 0 \\ 1 \\ 0 \\ + i \end{matrix}]$
	e⁺^β′	$E_{2}^{K^{'}} = [\begin{matrix} 1 \\ - 1 \end{matrix}]$	$J_{K^{'}}^{21} = [\begin{array}{r} 1 & 1 \\ - 1 & - 1 \end{array}]$		+1	$S_{K^{'}}^{21} = [\begin{matrix} 0 \\ 1 \\ 0 \\ - i \end{matrix}]$
$\begin{array}{l} \hat{R} (θ^{'}) & = {\hat{σ}}_{0} cosh θ^{'} - {\hat{σ}}_{3} sinh θ^{'} \\ = [\begin{array}{l} cosh θ^{'} & - i sinh θ^{'} \\ i sinh θ^{'} & cosh θ^{'} \end{array}] \end{array}$	e⁻^θ′	$E_{1}^{R^{'}} = [\begin{matrix} 1 \\ - i \end{matrix}]$	$J_{R^{'}}^{11} = [\begin{matrix} 1 & i \\ - i & 1 \end{matrix}]$	${\hat{R}}_{M} (θ^{'}) = [\begin{matrix} cosh 2 θ^{'} & 0 & 0 & - sinh 2 θ^{'} \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ - sinh 2 θ^{'} & 0 & 0 & cosh 2 θ^{'} \end{matrix}]$	e⁻²^θ′	$S_{R^{'}}^{11} = [\begin{matrix} 1 \\ 0 \\ 0 \\ 1 \end{matrix}]$
	e⁻^θ′	$E_{1}^{R^{'}} = [\begin{matrix} 1 \\ - i \end{matrix}]$	$J_{R^{'}}^{22} = [\begin{matrix} 1 & - i \\ i & 1 \end{matrix}]$		e⁺²^θ′	$S_{R^{'}}^{22} = [\begin{array}{r} 1 \\ 0 \\ 0 \\ - 1 \end{array}]$
	e⁺^θ′	$E_{2}^{R^{'}} = [\begin{matrix} 1 \\ i \end{matrix}]$	$J_{R^{'}}^{12} = [\begin{matrix} 1 & - i \\ - i & - 1 \end{matrix}]$		+1	$S_{R^{'}}^{12} = [\begin{matrix} 0 \\ 1 \\ - i \\ 0 \end{matrix}]$
	e⁺^θ′	$E_{2}^{R^{'}} = [\begin{matrix} 1 \\ i \end{matrix}]$	$J_{R^{'}}^{21} = [\begin{array}{r} 1 & i \\ i & - 1 \end{array}]$		+1	$S_{R^{'}}^{21} = [\begin{matrix} 0 \\ 1 \\ i \\ 0 \end{matrix}]$

2×2 representation	eigenvalues	eigenstate	J^ij (i, j = 1, 2)	Mueller matrix	eigenvalues	eigenstate S^i,j = (i, j = 1, 2)
$\begin{array}{l} Ĉ (δ) & = {\hat{σ}}_{0} cos δ + {i \hat{σ}}_{1} sin δ \\ = [\begin{array}{l} e^{i δ} & 0 \\ 0 & e^{- i δ} \end{array}] \end{array}$	e⁺ⁱ^δ	$E_{1}^{C} = [\begin{matrix} 1 \\ 0 \end{matrix}]$	$J_{C}^{11} = [\begin{array}{l} 1 & 0 \\ 0 & 0 \end{array}]$	$Ĉ_{M} (δ) = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & cos 2 δ & - sin 2 δ \\ 0 & 0 & sin 2 δ & cos 2 δ \end{matrix}]$	+1	$S_{C}^{11} = [\begin{matrix} 1 \\ 1 \\ 0 \\ 0 \end{matrix}]$
	e⁺ⁱ^δ	$E_{1}^{C} = [\begin{matrix} 1 \\ 0 \end{matrix}]$	$J_{C}^{22} = [\begin{array}{l} 0 & 0 \\ 0 & 1 \end{array}]$		+1	$S_{C}^{22} = [\begin{matrix} 1 \\ - 1 \\ 0 \\ 0 \end{matrix}]$
	e^{−i δ}	$E_{2}^{C} = [\begin{matrix} 0 \\ 1 \end{matrix}]$	$J_{C}^{12} = [\begin{array}{l} 0 & 1 \\ 0 & 0 \end{array}]$		eⁱ²^δ	$S_{C}^{12} = [\begin{matrix} 0 \\ 0 \\ 1 \\ - i \end{matrix}]$
	e^{−i δ}	$E_{2}^{C} = [\begin{matrix} 0 \\ 1 \end{matrix}]$	$J_{C}^{21} = [\begin{array}{l} 0 & 0 \\ 1 & 0 \end{array}]$		e⁻ⁱ²^δ	$S_{C}^{21} = [\begin{matrix} 0 \\ 0 \\ 1 \\ + i \end{matrix}]$
$\begin{array}{l} \hat{K} (β) & = {\hat{σ}}_{0} cos β + {i \hat{σ}}_{2} sin β \\ = [\begin{array}{l} cos β & i sin β \\ i sin β & cos β \end{array}] \end{array}$	e^j^β	$E_{1}^{K} = [\begin{matrix} 1 \\ 1 \end{matrix}]$	$J_{K}^{11} = [\begin{array}{l} 1 & 1 \\ 1 & 1 \end{array}]$	${\hat{K}}_{M} (β) = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & cos 2 β & 0 & + sin 2 β \\ 0 & 0 & 1 & 0 \\ 0 & - sin 2 β & 0 & cos 2 β \end{matrix}]$	+1	$S_{K}^{11} = [\begin{matrix} 1 \\ 0 \\ 1 \\ 0 \end{matrix}]$
	e^j^β	$E_{1}^{K} = [\begin{matrix} 1 \\ 1 \end{matrix}]$	$J_{K}^{22} = [\begin{matrix} 1 & - 1 \\ - 1 & 1 \end{matrix}]$		+1	$S_{K}^{22} = [\begin{matrix} 1 \\ 0 \\ - 1 \\ 0 \end{matrix}]$
	e⁻ⁱ^β	$E_{2}^{K} = [\begin{matrix} 1 \\ - 1 \end{matrix}]$	$J_{K}^{12} = [\begin{array}{l} 1 & - 1 \\ 1 & - 1 \end{array}]$		eⁱ²^β	$S_{K}^{12} = [\begin{matrix} 0 \\ 1 \\ 0 \\ + i \end{matrix}]$
	e⁻ⁱ^β	$E_{2}^{K} = [\begin{matrix} 1 \\ - 1 \end{matrix}]$	$J_{K}^{21} = [\begin{array}{r} 1 & 1 \\ - 1 & - 1 \end{array}]$		e⁻ⁱ²^β	$S_{K}^{21} = [\begin{matrix} 0 \\ 1 \\ 0 \\ - i \end{matrix}]$
$\begin{array}{l} \hat{R} (θ) & = {\hat{σ}}_{0} cos θ + {i \hat{σ}}_{3} sin θ \\ = [\begin{array}{l} cos θ & - sin θ \\ sin θ & cos θ \end{array}] \end{array}$	eⁱ^θ	$E_{1}^{R} = [\begin{matrix} 1 \\ - i \end{matrix}]$	$J_{R}^{11} = [\begin{array}{l} 1 & i \\ - i & 1 \end{array}]$	${\hat{R}}_{M} (θ) = [\begin{matrix} 1 & 0 & 1 & 0 \\ 0 & cos 2 θ & - sin 2 θ & 0 \\ 0 & sin 2 θ & cos 2 θ & 0 \\ 0 & 0 & 0 & 1 \end{matrix}]$	+1	$S_{R}^{11} = [\begin{matrix} 1 \\ 0 \\ 0 \\ 1 \end{matrix}]$
	eⁱ^θ	$E_{1}^{R} = [\begin{matrix} 1 \\ - i \end{matrix}]$	$J_{R}^{22} = [\begin{array}{l} 1 & - i \\ i & 1 \end{array}]$		+1	$S_{R}^{22} = [\begin{matrix} 1 \\ 0 \\ 0 \\ - 1 \end{matrix}]$
	e⁻ⁱ^θ	$E_{2}^{R} = [\begin{matrix} 1 \\ i \end{matrix}]$	$J_{R}^{12} = [\begin{array}{l} 1 & - i \\ - i & - 1 \end{array}]$		eⁱ²^θ	$S_{R}^{12} = [\begin{matrix} 0 \\ 1 \\ - i \\ 0 \end{matrix}]$
	e⁻ⁱ^θ	$E_{2}^{R} = [\begin{matrix} 1 \\ i \end{matrix}]$	$J_{R}^{21} = [\begin{array}{r} 1 & i \\ i & - 1 \end{array}]$		e⁻ⁱ²^θ	$S_{R}^{21} = [\begin{matrix} 0 \\ 1 \\ i \\ 0 \end{matrix}]$

2×2 representation	eigenvulues	eigenstates	J^ij (i, j = 1, 2)	Mueller matrix	eigenvalues	eigenstate S^ij (i, j = 1,2)
$\begin{array}{l} Ĉ^{'} (δ^{'}) & = {\hat{σ}}_{0} cos δ^{'} - {\hat{σ}}_{1} sin δ^{'} \\ = [\begin{array}{l} e^{- δ^{'}} & 0 \\ 0 & e^{+ δ^{'}} \end{array}] \end{array}$	e⁻^δ′	$E_{1}^{C^{'}} = [\begin{matrix} 1 \\ 0 \end{matrix}]$	$J_{C^{'}}^{11} = [\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}]$	$Ĉ_{M}^{'} (δ^{'}) = [\begin{matrix} cosh 2 δ^{'} & - sinh 2 δ^{'} & 0 & 0 \\ - sinh 2 δ^{'} & cosh 2 δ^{'} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}]$	e⁻²^δ′	$S_{C^{'}}^{11} = [\begin{matrix} 1 \\ 1 \\ 0 \\ 0 \end{matrix}]$
	e⁻^δ′	$E_{1}^{C^{'}} = [\begin{matrix} 1 \\ 0 \end{matrix}]$	$J_{C^{'}}^{22} = [\begin{matrix} 0 & 0 \\ 0 & 1 \end{matrix}]$		e⁺²^δ′	$S_{C^{'}}^{22} = [\begin{matrix} 1 \\ - 1 \\ 0 \\ 0 \end{matrix}]$
	e⁺^δ′	$E_{2}^{C^{'}} = [\begin{matrix} 0 \\ 1 \end{matrix}]$	$J_{C^{'}}^{12} = [\begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix}]$		+1	$S_{C^{'}}^{12} = [\begin{matrix} 0 \\ 0 \\ 1 \\ - i \end{matrix}]$
	e⁺^δ′	$E_{2}^{C^{'}} = [\begin{matrix} 0 \\ 1 \end{matrix}]$	$J_{C^{'}}^{21} = [\begin{matrix} 0 & 0 \\ 1 & 0 \end{matrix}]$		+1	$S_{C^{'}}^{21} = [\begin{matrix} 0 \\ 0 \\ 1 \\ i \end{matrix}]$
$\begin{array}{l} {\hat{K}}^{'} (β^{'}) & = {\hat{σ}}_{0} cosh β^{'} - {\hat{σ}}_{2} sinh β^{'} \\ = [\begin{array}{l} cosh β^{'} & - sinh β^{'} \\ - sinh β^{'} & cosh β^{'} \end{array}] \end{array}$	e⁻^β′	$E_{1}^{K^{'}} = [\begin{matrix} 1 \\ 1 \end{matrix}]$	$J_{K^{'}}^{11} = [\begin{matrix} 1 & 1 \\ 1 & 1 \end{matrix}]$	${\hat{K}}_{M}^{'} (β^{'}) = [\begin{matrix} cosh 2 β^{'} & 0 & - sinh 2 β^{'} & 0 \\ 0 & 1 & 0 & 0 \\ - sinh 2 β^{'} & 0 & cosh 2 β^{'} & 0 \\ 0 & 0 & 0 & 1 \end{matrix}]$	e⁻²^β′	$S_{K^{'}}^{11} = [\begin{matrix} 1 \\ 0 \\ 1 \\ 0 \end{matrix}]$
	e⁻^β′	$E_{1}^{K^{'}} = [\begin{matrix} 1 \\ 1 \end{matrix}]$	$J_{K^{'}}^{22} = [\begin{array}{r} 1 & - 1 \\ - 1 & 1 \end{array}]$		e⁺²^β′	$S_{K^{'}}^{22} = [\begin{matrix} 1 \\ 0 \\ - 1 \\ 0 \end{matrix}]$
	e⁺^β′	$E_{2}^{K^{'}} = [\begin{matrix} 1 \\ - 1 \end{matrix}]$	$J_{K^{'}}^{12} = [\begin{matrix} 1 & - 1 \\ 1 & - 1 \end{matrix}]$		+1	$S_{K^{'}}^{12} = [\begin{matrix} 0 \\ 1 \\ 0 \\ + i \end{matrix}]$
	e⁺^β′	$E_{2}^{K^{'}} = [\begin{matrix} 1 \\ - 1 \end{matrix}]$	$J_{K^{'}}^{21} = [\begin{array}{r} 1 & 1 \\ - 1 & - 1 \end{array}]$		+1	$S_{K^{'}}^{21} = [\begin{matrix} 0 \\ 1 \\ 0 \\ - i \end{matrix}]$
$\begin{array}{l} \hat{R} (θ^{'}) & = {\hat{σ}}_{0} cosh θ^{'} - {\hat{σ}}_{3} sinh θ^{'} \\ = [\begin{array}{l} cosh θ^{'} & - i sinh θ^{'} \\ i sinh θ^{'} & cosh θ^{'} \end{array}] \end{array}$	e⁻^θ′	$E_{1}^{R^{'}} = [\begin{matrix} 1 \\ - i \end{matrix}]$	$J_{R^{'}}^{11} = [\begin{matrix} 1 & i \\ - i & 1 \end{matrix}]$	${\hat{R}}_{M} (θ^{'}) = [\begin{matrix} cosh 2 θ^{'} & 0 & 0 & - sinh 2 θ^{'} \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ - sinh 2 θ^{'} & 0 & 0 & cosh 2 θ^{'} \end{matrix}]$	e⁻²^θ′	$S_{R^{'}}^{11} = [\begin{matrix} 1 \\ 0 \\ 0 \\ 1 \end{matrix}]$
	e⁻^θ′	$E_{1}^{R^{'}} = [\begin{matrix} 1 \\ - i \end{matrix}]$	$J_{R^{'}}^{22} = [\begin{matrix} 1 & - i \\ i & 1 \end{matrix}]$		e⁺²^θ′	$S_{R^{'}}^{22} = [\begin{array}{r} 1 \\ 0 \\ 0 \\ - 1 \end{array}]$
	e⁺^θ′	$E_{2}^{R^{'}} = [\begin{matrix} 1 \\ i \end{matrix}]$	$J_{R^{'}}^{12} = [\begin{matrix} 1 & - i \\ - i & - 1 \end{matrix}]$		+1	$S_{R^{'}}^{12} = [\begin{matrix} 0 \\ 1 \\ - i \\ 0 \end{matrix}]$
	e⁺^θ′	$E_{2}^{R^{'}} = [\begin{matrix} 1 \\ i \end{matrix}]$	$J_{R^{'}}^{21} = [\begin{array}{r} 1 & i \\ i & - 1 \end{array}]$		+1	$S_{R^{'}}^{21} = [\begin{matrix} 0 \\ 1 \\ i \\ 0 \end{matrix}]$

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Equations (180)

Journal of the Optical Society of America