Three-dimensional polarization algebra

Colin J. R. Sheppard; Marco Castello; Alberto Diaspro

doi:10.1364/JOSAA.33.001938

Journal of the Optical Society of America A
Vol. 33,
Issue 10,
pp. 1938-1947
(2016)
•https://doi.org/10.1364/JOSAA.33.001938

Three-dimensional polarization algebra

Colin J. R. Sheppard, Marco Castello, and Alberto Diaspro

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Colin J. R. Sheppard, Marco Castello, and Alberto Diaspro, "Three-dimensional polarization algebra," J. Opt. Soc. Am. A 33, 1938-1947 (2016)

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Abstract

If light is focused or collected with a high numerical aperture lens, as may occur in imaging and optical encryption applications, polarization should be considered in three dimensions (3D). The matrix algebra of polarization behavior in 3D is discussed. It is useful to convert between the Mueller matrix and two different Hermitian matrices, representing an optical material or system, which are in the literature. Explicit transformation matrices for converting the column vector form of these different matrices are extended to the 3D case, where they are large $(81 \times 81)$ but can be generated using simple rules. It is found that there is some advantage in using a generalization of the Chandrasekhar phase matrix treatment, rather than that based on Gell–Mann matrices, as the resultant matrices are of simpler form and reduce to the two-dimensional case more easily. Explicit expressions are given for 3D complex field components in terms of Chandrasekhar–Stokes parameters.

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Equations (75)

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Standard Basis	Pauli Basis	Number × Size (3D)
$ρ$ Pauli to standard	$σ$ Pauli matrices	$9 \times (3 \times 3)$
$I$	$Λ$	$9 \times 9$
$Λ^{- 1}$	$I$	$9 \times 9$
$e$ Electric field vector		$3 \times 1$ complex [5]
$P$ (Hermitian) Polarization matrix	$S$ (real) Stokes matrix	$3 \times 3$ [9]
$P$ Polarization vector	$S$ (real) Stokes vector	$9 \times 1$
$J, (e^{'} = Je)$ Jones matrix	$C$ “C-matrix”	$3 \times 3$ complex [17]
$j$	$c$ “c-vector”	$9 \times 1$ complex [17]
$N, (p^{'} = Np)$ Mueller matrix in standard basis	$M$ (real), ( $s^{'} = Ms$ ) Mueller matrix	$9 \times 9$ [81]
$n$	$m$	$81 \times 1$
$H$ Standard Hermitian matrix	$G$ Pauli Hermitian matrix	$9 \times 9$
$h$	$g$	$81 \times 1$
$R, (h = Rn)$ row operator	$K, (n = Km)$	$81 \times 81$
$V$ matrix of eigenvectors	$U$ matrix of eigenvectors	$9 \times 9$
${Ψ_{μ v}}$	${Γ_{μ v}}$	$81 \times (9 \times 9)$
${Ψ_{μ v}}$	${γ_{μ v}}$	$81 \times (81 \times 1)$
$Ψ, (h = Ψ m)$	$Γ, (g = Γ m)$	$81 \times 81$

	$h$	$g$	$n$	$m$
$h$	$I$	$K$	$R$	$RK = Ψ$
$g$	$K^{†}$	$I$	$K^{†} R = Ψ^{†}$	$K^{†} RK = Γ$
$n$	$R$	$RK = Ψ$	$I$	$K$
$m$	$K^{†} R = Ψ^{†}$	$K^{†} RK = Γ$	$K^{†}$	$I$

Standard Basis	Pauli Basis	Number × Size (3D)
$ρ$ Pauli to standard	$σ$ Pauli matrices	$9 \times (3 \times 3)$
$I$	$Λ$	$9 \times 9$
$Λ^{- 1}$	$I$	$9 \times 9$
$e$ Electric field vector		$3 \times 1$ complex [5]
$P$ (Hermitian) Polarization matrix	$S$ (real) Stokes matrix	$3 \times 3$ [9]
$P$ Polarization vector	$S$ (real) Stokes vector	$9 \times 1$
$J, (e^{'} = Je)$ Jones matrix	$C$ “C-matrix”	$3 \times 3$ complex [17]
$j$	$c$ “c-vector”	$9 \times 1$ complex [17]
$N, (p^{'} = Np)$ Mueller matrix in standard basis	$M$ (real), ( $s^{'} = Ms$ ) Mueller matrix	$9 \times 9$ [81]
$n$	$m$	$81 \times 1$
$H$ Standard Hermitian matrix	$G$ Pauli Hermitian matrix	$9 \times 9$
$h$	$g$	$81 \times 1$
$R, (h = Rn)$ row operator	$K, (n = Km)$	$81 \times 81$
$V$ matrix of eigenvectors	$U$ matrix of eigenvectors	$9 \times 9$
${Ψ_{μ v}}$	${Γ_{μ v}}$	$81 \times (9 \times 9)$
${Ψ_{μ v}}$	${γ_{μ v}}$	$81 \times (81 \times 1)$
$Ψ, (h = Ψ m)$	$Γ, (g = Γ m)$	$81 \times 81$

	$h$	$g$	$n$	$m$
$h$	$I$	$K$	$R$	$RK = Ψ$
$g$	$K^{†}$	$I$	$K^{†} R = Ψ^{†}$	$K^{†} RK = Γ$
$n$	$R$	$RK = Ψ$	$I$	$K$
$m$	$K^{†} R = Ψ^{†}$	$K^{†} RK = Γ$	$K^{†}$	$I$

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Equations (75)

Journal of the Optical Society of America A