Image analysis of curved gradient-index rods

Mary J. Nadeau; Joseph J. Miceli; Duncan T. Moore

doi:10.1364/AO.25.001780

Applied Optics
Vol. 25,
Issue 11,
pp. 1780-1790
(1986)
•https://doi.org/10.1364/AO.25.001780

Image analysis of curved gradient-index rods

Mary J. Nadeau, Joseph J. Miceli, and Duncan T. Moore

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Mary J. Nadeau, Joseph J. Miceli, and Duncan T. Moore, "Image analysis of curved gradient-index rods," Appl. Opt. 25, 1780-1790 (1986)

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Abstract

A method of ray tracing through curved gradient-index imaging rods is developed. A study of the imaging ability of a radial gradient-index relay rod, with end-to-end imaging of an object in the meridional plane, bent along some simple curves is performed. The study includes an analysis of changes in the first order of properties and the transverse aberrations in the image plane. Approximations of the first-, second-, and third-order transverse aberration coefficients are made.

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$ɛ_{1 Y} = ψ_{1} \bar{ρ} cos θ + l \bar{ρ} cos θ + ψ_{3} \bar{h},$
$ɛ_{1 Z} = ψ_{2} \bar{ρ} sin θ + l \bar{ρ} sin θ,$
where ψ₁,ψ₂ =	first-order linear astigmatism
l =	defocus
ψ₃ =	linear distortion
$ɛ_{2 Y} = (ω_{1} + ω_{2} cos 2 θ) {\bar{ρ}}^{2} + ω_{4} \bar{ρ} \bar{h} cos θ + ω_{6} {\bar{h}}^{2}$
$ɛ_{2 Z} = ω_{3} {\bar{ρ}}^{2} sin 2 θ + ω_{5} \bar{ρ} \bar{h} sin θ,$
where ω₁,ω₂,ω₃ =	second-order zeroth degree coma
ω₄,ω₅ =	second-order linear astigmatism
ω₆ =	second-order distortion
$\begin{array}{l} ɛ_{3 Y} = & τ_{1} {\bar{ρ}}^{3} cos θ + τ_{2} {\bar{ρ}}^{3} cos θ sin θ + (τ_{5} + τ_{6} cos 2 θ) {\bar{ρ}}^{2} \bar{h} \\ + τ_{8} \bar{ρ} {\bar{h}}^{2} cos θ + τ_{10} {\bar{h}}^{3}, \end{array}$
$\begin{array}{l} ɛ_{3 Z} = & τ_{3} {\bar{ρ}}^{3} sin θ + τ_{4} {\bar{ρ}}^{3} {cos}^{2} θ sin θ + τ_{7} {\bar{ρ}}^{2} \bar{h} sin 2 θ \\ + τ_{9} \bar{ρ} {\bar{h}}^{2} sin θ, \end{array}$
where τ₁,τ₂,τ₃,τ₄ =	third-order third degree astigmatism
τ₅,τ₆,τ₇ =	Third-order linear coma
τ₈,τ₉ =	Third-order linear astigmatism
τ₁₀ =	Third-order distortion

Order	Coefficient	Half-period	1 Period	1.5 Periods	2 Periods
First	ψ₁	0	0	–	–
	ψ₂	−0.077 μm	0.156 μm	−0.230 μm	0.325 μm
	ψ₃	0.018	−0.006	0.027	−0.028
Second	ω₁	−0.032	0	−0.032	0
	ω₂	−0.016	0	−0.016	0
	ω₃	0	0	–	–
	ω₄	0.082	−0.167	–	–
	ω₅	−0.282	0.567	–	–
	ω₆	−0.156	0	−0.154	0
Third	τ_l	0.143	−0.290	0.448	−0.620
	τ₂	0	0	0	0
	τ₃	0.143	−0.289	0.446	−0.617
	τ₄	0	0	0	0
	τ₅	0	0	–	–
	τ₆	0	0	–	–
	τ₇	0	0	–	–
	τ₈	0.119	−0.237	–	–
	τ₉	−0.395	0.790	–	–
	τ₁₀	0	0	0	0

Order	Coefficient	Half-period	1 Period
First	ψ₁	0	0
	ψ₂	−0.135 μm	0.508 μm
	ψ₃	−0.012	0.019
Second	ω₁	−0.190	0.381
	ω₂	−0.095	0.190
	ω₃	0	0
	ω₄	−1.547	0.168
	ω₅	−1.525	−0.551
	ω₆	0.034	−0.067
Third	τ₁	0.141	−0.279
	τ₂	0	0
	τ₃	0.141	−0.279
	τ₄	0	0
	τ₅	0	0
	τ₆	0	0
	τ₇	0	0
	τ₈	0.118	−0.238
	τ₉	−0.394	0.791
	τ₁₀	0	0

$ɛ_{1 Y} = ψ_{1} \bar{ρ} cos θ + l \bar{ρ} cos θ + ψ_{3} \bar{h},$
$ɛ_{1 Z} = ψ_{2} \bar{ρ} sin θ + l \bar{ρ} sin θ,$
where ψ₁,ψ₂ =	first-order linear astigmatism
l =	defocus
ψ₃ =	linear distortion
$ɛ_{2 Y} = (ω_{1} + ω_{2} cos 2 θ) {\bar{ρ}}^{2} + ω_{4} \bar{ρ} \bar{h} cos θ + ω_{6} {\bar{h}}^{2}$
$ɛ_{2 Z} = ω_{3} {\bar{ρ}}^{2} sin 2 θ + ω_{5} \bar{ρ} \bar{h} sin θ,$
where ω₁,ω₂,ω₃ =	second-order zeroth degree coma
ω₄,ω₅ =	second-order linear astigmatism
ω₆ =	second-order distortion
$\begin{array}{l} ɛ_{3 Y} = & τ_{1} {\bar{ρ}}^{3} cos θ + τ_{2} {\bar{ρ}}^{3} cos θ sin θ + (τ_{5} + τ_{6} cos 2 θ) {\bar{ρ}}^{2} \bar{h} \\ + τ_{8} \bar{ρ} {\bar{h}}^{2} cos θ + τ_{10} {\bar{h}}^{3}, \end{array}$
$\begin{array}{l} ɛ_{3 Z} = & τ_{3} {\bar{ρ}}^{3} sin θ + τ_{4} {\bar{ρ}}^{3} {cos}^{2} θ sin θ + τ_{7} {\bar{ρ}}^{2} \bar{h} sin 2 θ \\ + τ_{9} \bar{ρ} {\bar{h}}^{2} sin θ, \end{array}$
where τ₁,τ₂,τ₃,τ₄ =	third-order third degree astigmatism
τ₅,τ₆,τ₇ =	Third-order linear coma
τ₈,τ₉ =	Third-order linear astigmatism
τ₁₀ =	Third-order distortion

Order	Coefficient	Half-period	1 Period	1.5 Periods	2 Periods
First	ψ₁	0	0	–	–
	ψ₂	−0.077 μm	0.156 μm	−0.230 μm	0.325 μm
	ψ₃	0.018	−0.006	0.027	−0.028
Second	ω₁	−0.032	0	−0.032	0
	ω₂	−0.016	0	−0.016	0
	ω₃	0	0	–	–
	ω₄	0.082	−0.167	–	–
	ω₅	−0.282	0.567	–	–
	ω₆	−0.156	0	−0.154	0
Third	τ_l	0.143	−0.290	0.448	−0.620
	τ₂	0	0	0	0
	τ₃	0.143	−0.289	0.446	−0.617
	τ₄	0	0	0	0
	τ₅	0	0	–	–
	τ₆	0	0	–	–
	τ₇	0	0	–	–
	τ₈	0.119	−0.237	–	–
	τ₉	−0.395	0.790	–	–
	τ₁₀	0	0	0	0

Abstract

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Equations (7)

Applied Optics