Polarization aberrations. 1. Rotationally symmetric optical systems

James P. McGuire; Russell A. Chipman

doi:10.1364/AO.33.005080

Applied Optics
Vol. 33,
Issue 22,
pp. 5080-5100
(1994)
•https://doi.org/10.1364/AO.33.005080

Polarization aberrations. 1. Rotationally symmetric optical systems

James P. McGuire and Russell A. Chipman

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James P. McGuire and Russell A. Chipman, "Polarization aberrations. 1. Rotationally symmetric optical systems," Appl. Opt. 33, 5080-5100 (1994)

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Abstract

The polarization in isotropic radially symmetric lens and mirror systems in the paraxial approximation is examined. Polarized aberrations are variations in the phase, amplitude, and polarization state of the electromagnetic field across the exit pupil. Some are dependent on the incident polarization state and some are not. Expressions through fourth order for phase, amplitude, linear diattenuation, and linear retardance aberrations are derived in terms of the chief and marginal ray angles of incidence and the Taylor series expansion coefficients of the Fresnel equations for reflection and transmission at uncoated and thin-film-coated interfaces. Applications to polarization ray tracing are discussed.

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Term	Definition
Diattenuation	The property of having an intensity transmittance that is dependent on the incident polarization state.
Polarization	The property of altering the polarization state of light. Polarization includes the subsets of diattenuation and retardance.
Polarizera	An optical element that transmits a fixed polarization state independent of the incident polarization state. Examples include dichroic sheets (Polaroid) and Glan–Thompson prisms.
Polarization elementa	Any optical element showing polarization. Examples include retarders, polarizers, and metallic interfaces.
Retardance	The property of having a phase or optical path length that is dependent on the incident polarization state.

Matrix	Eigenstatesa	Names of Eigenstatesa
$σ_{0} = (\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix})$	Degenerate	Not applicable
$σ_{1} = (\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix})$	$∣ 1^{+} 〉 = (\begin{matrix} 1 \\ 0 \end{matrix})$	Horizontally polarized
	$∣ 1^{-} 〉 = (\begin{matrix} 0 \\ 1 \end{matrix})$	Vertically polarized
$σ_{2} = (\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix})$	$∣ 2^{+} 〉 = \frac{1}{\sqrt{2}} (\begin{matrix} 1 \\ 1 \end{matrix})$	Linearly polarized at 45°
	$∣ 2^{-} 〉 = \frac{1}{\sqrt{2}} (\begin{matrix} - 1 \\ 1 \end{matrix})$	Linearly polarized at −45°
$σ_{3} = (\begin{matrix} 0 & - j \\ j & 0 \end{matrix})$	$∣ 3^{+} 〉 = \frac{1}{\sqrt{2}} (\begin{matrix} 1 \\ j \end{matrix})$	Right circularly polarized
	$∣ 3^{-} 〉 = \frac{1}{\sqrt{2}} (\begin{matrix} j \\ 1 \end{matrix})$	Left circularly polarized

Coefficient	Matrix	Physical Significance
a ₀	σ₀	Re(a ₀) Polarization-independent amplitude
		Im(a ₀) Polarization-independent phase
a ₁	σ₁	Re(a ₁) Linear diattenuation along the coordinate axes
		Im(a ₁) Linear retardance along the coordinate axes
a ₂	σ₂	Re(a ₂) Linear diattenuation at 45° to the coordinate axes
		Im(a ₂) Linear retardance at 45° to the coordinate axes
a ₃	σ₃	Re(a ₃) Circular diattenuation
		Im(a ₃) Circular retardance

Commutator	Condition
[σ₀, σ _k ] = 0	k ∈{0, 1, 2, 3}
[σ _k , σ _k ] = 0	k ∈ {0, 1, 2, 3}
[σ _k , σ _l ] = 2jσ _m	k, l, m Cyclic permutations of 1, 2, 3

Aberration Coefficient	Physical Significance
Im(P ₀ _uvwx )	Phase aberrations
Re(P ₀ _uvwx )	Amplitude aberrations
Re(P ₁ _uvwx )	Linear diattenuation aberrations
Im(P ₁ _uvwx )	Linear retardance aberrations
Re(P ₃ _uvwx )	Circular diattenuation aberrations
Im(P ₃ _uvwx )	Circular retardance aberrations

Aberration Property	On-Axis Objects	Off-Axis Objects

		Tangential	Sagittal
Polarization	Linear	Linear	Elliptical
Orientation	Radial or tangential	Horizontal or vertical	Arbitrary
Symmetry	Rotational	None	Odd about the y axis

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Applied Optics