Optical systems with translational invariance

Dmitry Zhuridov

doi:10.1364/AO.511270

Applied Optics
Vol. 63,
Issue 6,
pp. 1506-1516
(2024)
•https://doi.org/10.1364/AO.511270

Optical systems with translational invariance

Dmitry Zhuridov

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Dmitry Zhuridov, "Optical systems with translational invariance," Appl. Opt. 63, 1506-1516 (2024)

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Abstract

The leading order monochromatic aberrations are investigated for optical systems, which obey single-plane symmetry and translational invariance. These aberrations are classified from the symmetry principles for the wave aberration function. Fermat’s principle is applied for a system made of generic cylindrical surfaces, and a complete set of the aberration coefficients required for calculation of the aberrations for any paraxial ray is obtained. To demonstrate the applied value of the analytical results obtained, the criteria of compensation of the leading aberrations for the cylindrical analog of the Cassegrain telescope were found.

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Corrections

Dmitry Zhuridov, "Optical systems with translational invariance: erratum," Appl. Opt. 63, 2750-2751 (2024)
https://opg.optica.org/ao/abstract.cfm?uri=ao-63-10-2750

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$α$	Refracting Surface of Eq. (38)	Reflecting Surface, Eq. (80)
$α_{e 1}$	$- \frac{1}{2} [\frac{κ}{R^{3}} (n_{1} - n_{0}) + b + n_{0} Γ Γ_{R}^{2}]$	$- \frac{n_{0}}{R^{3}} [κ + {(\frac{μ + 1}{μ - 1})}^{2}] - \frac{b}{2}$
$α_{e 2}$	$- \frac{n_{0}}{2 s_{0}^{3}} (1 - \frac{s_{0}^{2}}{s_{1}^{2}} \frac{1 + 4 β + 6 β^{2} + 4 β^{3}}{β})$	$- \frac{n_{0}}{2 s_{0}^{3}} \frac{1 - 4 μ + 6 μ^{2} - 3 μ^{3}}{μ^{3}}$
$α_{e 3}$	$- \frac{n_{0}}{2 s_{0}^{2}} (1 - \frac{n_{0}^{2}}{n_{1}^{2}}) Γ_{R}$	$- \frac{n_{0}}{s_{0}^{2} R} \frac{μ + 1}{μ - 1}$
$α_{c 1}$	$- \frac{3}{2} n_{0} θ_{0} Γ Γ_{R}$	$- 3 \frac{n_{0} θ_{0}}{R^{2}} \frac{μ + 1}{μ - 1}$
$α_{c 2}$	$- \frac{n_{0} θ_{0}}{2 s_{0}^{2}} (1 - \frac{n_{0}^{2}}{n_{1}^{2}})$	0
$α_{a 1}$	$- n_{0} θ_{0}^{2} (Γ + \frac{n_{1} - n_{0}}{2 n_{1} R})$	$- \frac{n_{0} θ_{0}^{2}}{R}$
$α_{a 2}$	$- \frac{n_{1} - n_{0}}{R} - \frac{n_{0} θ_{0}^{2}}{s_{0}} (1 - \frac{n_{0}^{2}}{n_{1}^{2}})$	$- \frac{n_{0}}{s_{0}} \frac{μ - 1}{μ}$
$α_{d}$	$- \frac{n_{0} θ_{0}^{3}}{2} (1 - \frac{n_{0}^{2}}{n_{1}^{2}})$	0

$A_{31}$	$- \frac{n_{0}}{8} (\frac{κ}{R^{3}} \frac{n_{1} - n_{0}}{n_{0}} + b + Γ Γ_{R}^{2})$
$A_{32}$	$- \frac{n_{0}}{8 s_{0}^{3}} [1 - \frac{s_{0}^{2}}{s_{1}^{2}} \frac{{(1 + β)}^{4} - β^{4}}{β}]$
$A_{33}$	$- \frac{n_{0}}{4 s_{0}^{2}} (1 - \frac{n_{0}^{2}}{n_{1}^{2}}) Γ_{R}$
$A_{21}$	$- \frac{n_{0}}{2} (W ψ) [\pm \frac{κ}{R^{3}} \frac{n_{1} - n_{0}}{n_{0}} \pm b + (\frac{1}{W} \pm \frac{1}{R}) Γ Γ_{R}]$
$A_{22}$	$- \frac{n_{0}}{2 s_{0}^{2}} (W ψ) (\frac{1}{W} \pm \frac{1}{R}) (1 - \frac{n_{0}^{2}}{n_{1}^{2}})$
$A_{11}$	$- \frac{n_{0}}{4} (W ψ)^{2} [3 {(\frac{1}{W} \pm \frac{1}{R})}^{2} - {(\frac{1}{W} \mp \frac{1}{s_{0}})}^{2}] Γ$
	$- \frac{n_{0}}{4} (W ψ)^{2} [{(\frac{1}{W} \mp \frac{1}{s_{0}})}^{2} \frac{n_{1} - n_{0}}{n_{1} R} + \frac{3 κ}{R^{3}} \frac{n_{1} - n_{0}}{n_{0}} + \frac{3 b}{n_{0}}]$
$A_{12}$	$- \frac{n_{1} - n_{0}}{2 R} - \frac{n_{0}}{2 s_{0}} (W ψ)^{2} (\frac{1}{W} \mp \frac{1}{s_{0}})^{2} (1 - \frac{n_{0}^{2}}{n_{1}^{2}})$
	$- \frac{n_{0}}{2 s_{0}^{2}} (W ψ)^{2} (\frac{1}{2 R} \pm \frac{1}{W} - \frac{1}{2 s_{0}}) (1 - \frac{n_{0}^{2}}{n_{1}^{2}})$
$A_{01}$	$- \frac{n_{0}}{2} (W ψ)^{3} {(\frac{1}{W} \pm \frac{1}{R}) (\frac{1}{R} \pm \frac{2}{W} - \frac{1}{s_{0}}) Γ$
	$+ (\frac{1}{W} \mp \frac{1}{s_{0}})^{2} [(\frac{1}{W} \mp \frac{1}{s_{0}}) (1 - \frac{n_{0}^{2}}{n_{1}^{2}}) \pm \frac{n_{1} - n_{0}}{n_{1} R}]$
	$\pm (\frac{κ}{R^{3}} \frac{n_{1} - n_{0}}{n_{0}} + \frac{b}{n_{0}})}$

$A_{31}$	$- \frac{n_{0}}{4 R^{3}} [κ + {(\frac{μ + 1}{μ - 1})}^{2}] - \frac{b}{8}$
$A_{32}$	$- \frac{n_{0}}{8 s_{0}^{3}} \frac{1 - 4 μ + 6 μ^{2} - 3 μ^{3}}{μ^{3}}$
$A_{33}$	$- \frac{n_{0}}{2 s_{0}^{2} R} \frac{μ + 1}{μ - 1}$
$A_{21}$	$- \frac{n_{0}}{R^{2}} (W ψ) [(\frac{1}{W} \mp \frac{1}{R}) \frac{μ + 1}{μ - 1} \pm \frac{κ}{R}] \mp \frac{b}{2} (W ψ)$
$A_{22}$	$\mp \frac{n_{0}}{2 s_{0}^{2} R} (W ψ) \frac{μ + 1}{μ - 1}$
$A_{11}$	$- \frac{n_{0}}{2 R} (W ψ)^{2} [3 {(\frac{1}{W} \mp \frac{1}{R})}^{2} - 2 {(\frac{1}{W} \mp \frac{1}{s_{0}})}^{2} + \frac{3 κ}{R^{2}}]$
	$- \frac{3 b}{4} (W ψ)^{2}$
$A_{12}$	$- \frac{n_{0}}{2 s_{0}} \frac{μ - 1}{μ} - \frac{n_{0}}{2 s_{0}^{2} R} (W ψ)^{2} \frac{μ + 1}{μ - 1}$
$A_{01}$	$\mp \frac{n_{0}}{2 R} (W ψ)^{3} [3 {(\frac{1}{W} \mp \frac{1}{R})}^{2} - {(\frac{1}{W} \mp \frac{1}{s_{0}})}^{2}]$
	$\mp \frac{n_{0}}{2 R^{3}} (W ψ)^{3} [κ - {(\frac{μ + 1}{μ - 1})}^{2}] \mp \frac{b}{2} (W ψ)^{3}$

$B_{22}$	$- \frac{n_{0} (W χ)}{2 s_{0}^{3}} [1 - \frac{s_{0}^{2}}{s_{1}^{2}} \frac{{(1 + β)}^{4} - β^{4}}{β}]$
$B_{23}$	$- \frac{n_{0} (W χ)}{2 s_{0}^{2}} (1 - \frac{n_{0}^{2}}{n_{1}^{2}}) Γ_{R}$
$B_{11}$	$- \frac{n_{0} {(W χ)}^{2}}{4 s_{0}^{2}} (1 - \frac{n_{0}^{2}}{n_{1}^{2}}) Γ_{R}$
$B_{12}$	$- \frac{3 n_{0} {(W χ)}^{2}}{4 s_{0}^{3}} [1 - \frac{s_{0}^{2}}{s_{1}^{2}} \frac{{(1 + β)}^{4} - β^{4}}{β}]$
$B_{13}$	$- \frac{n_{0} W^{2}}{s_{0}^{2}} ψ χ (1 - \frac{n_{0}^{2}}{n_{1}^{2}}) (\frac{1}{W} \pm \frac{1}{R})$
$B_{01}$	$- \frac{n_{0}}{2 s_{0}^{2}} W^{3} ψ χ^{2} (1 - \frac{n_{0}^{2}}{n_{1}^{2}}) (\frac{1}{W} \pm \frac{1}{R})$
$B_{02}$	$- \frac{n_{1} - n_{0}}{R} (W χ) (1 + \frac{χ^{2}}{3}) - \frac{n_{0} {(W χ)}^{3}}{2 s_{0}^{3}} [1 - \frac{s_{0}^{2}}{s_{1}^{2}} \frac{{(1 + β)}^{4} - β^{4}}{β}]$
	$- \frac{n_{0} W^{3}}{2 s_{0}^{2}} ψ^{2} χ (1 - \frac{n_{0}^{2}}{n_{1}^{2}}) [2 \frac{s_{0}}{W} (\frac{1}{W} \mp \frac{1}{s_{0}}) + Γ_{R}]$

$B_{22}$	$- \frac{n_{0} (W χ)}{2 s_{0}^{3}} \frac{1 - 4 μ + 6 μ^{2} - 3 μ^{3}}{μ^{3}}$
$B_{23}$	$- \frac{n_{0} (W χ)}{s_{0}^{2} R} \frac{μ + 1}{μ - 1}$
$B_{11}$	$- \frac{n_{0} {(W χ)}^{2}}{2 s_{0}^{2} R} \frac{μ + 1}{μ - 1}$
$B_{12}$	$- \frac{3 n_{0} {(W χ)}^{2}}{4 s_{0}^{3}} \frac{1 - 4 μ + 6 μ^{2} - 3 μ^{3}}{μ^{3}}$
$B_{13}$	$\mp \frac{2 n_{0}}{s_{0}^{2} R} W^{2} ψ χ \frac{μ + 1}{μ - 1}$
$B_{01}$	$\mp \frac{n_{0}}{s_{0}^{2} R} W^{3} ψ χ^{2} \frac{μ + 1}{μ - 1}$
$B_{02}$	$- n_{0} \frac{(W χ)}{s_{0}} (1 + \frac{χ^{2}}{3}) \frac{μ - 1}{μ} - n_{0} \frac{W^{3}}{s_{0}^{2} R} ψ^{2} χ \frac{μ + 1}{μ - 1}$
	$- \frac{n_{0} {(W χ)}^{3}}{2 s_{0}^{3}} \frac{1 - 4 μ + 6 μ^{2} - 3 μ^{3}}{μ^{3}}$

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Tables (5)

Equations (144)

Applied Optics