Invariant aberrations. I. Preliminaries

H. A. Buchdahl

doi:10.1364/JOSAA.5.001957

Journal of the Optical Society of America A
Vol. 5,
Issue 11,
pp. 1957-1962
(1988)
•https://doi.org/10.1364/JOSAA.5.001957

Invariant aberrations. I. Preliminaries

H. A. Buchdahl

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H. A. Buchdahl, "Invariant aberrations. I. Preliminaries," J. Opt. Soc. Am. A 5, 1957-1962 (1988)

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Abstract

In this paper I begin the investigation, in the context of Lagrangian aberration theory, of the existence and the explicit forms of aberrations of any order that are invariant, possibly only when ancillary conditions are satisfied, under displacements of the plane of the entrance pupil or of that of the object or both simultaneously, i.e., under p-variations, l-variations, or s-variations. Much of the formal work revolves about the effects of infinitesimal linear transformations of coordinates on linear combinations of homologous aberration coefficients. Generically, any invariant is the product of a scale factor with such a linear combination, and recurrence relations are found for the numerical multipliers that enter into it.

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Equations (52)

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NC	OE−	OT−	ÔE−	ÔT−	O^*E−
σ	1	j	1	j	1
$\bar{σ}$	−p	−jp	−p	−jp	−p
τ	−1	−1	$- l$	$- l$	$- j l$
$\bar{τ}$	l	l	1	1	J
g	l/j	l	1/j	1	1
δπ	$j l d p$	$- j p d l$	$j l d p$	$- j p d l$	$j l d p$
$δ \bar{π}$	$j l d p$	$j^{2} l d p$	jdp	j²dp	dp
δρ	$j l d l$	$l d l$	$j d l$	$d l$	$j^{2} d l$
$δ \bar{ρ}$	$j l d l$	$j l d l$	$j p d l$	$j p d l$	$- j l d p$
δg	−dp + dl	dl	$- l d p - p d l$	0	0

Coefficient	Dimension
$G_{μ, ν \| a}^{(n)}$	−(2n − μ − ν + 2)d_s − (μ + ν)d_t + 1
${\bar{G}}_{μ, ν \| a}^{(n)}$	−(2n − μ − ν + 1)d_s − (μ + ν + 1)d_t + 1
$G_{μ, ν \| b}^{(n)}$	−(2n − μ − ν + 1)d_s − (μ + ν + 1)d_t + 1
${\bar{G}}_{μ, ν \| b}^{(n)}$	−(2n − μ − ν)d_s − (μ + ν + 2)d_t + 1

Case	λ^′	λ^′	λ^″
x_λ	0(0 ≤ k ≤ n)	k − n(n + 1 ≤ k ≤ 2n)	[½k]
${\bar{α}}_{λ}$	0(1 ≤ k ≤ n + 1)	k − n − 1(n + 2 ≤ k ≤ 2n + 1)	[½(k − 1)]
β_λ	0(1 ≤ k ≤ n + 1)	k − n − 1(n + 2 ≤ k ≤ 2n + 1)	[½(k − 1)]
${\bar{β}}_{λ}$	1(2 ≤ k ≤ n + 2)	k − n − 1(n + 3 ≤ k ≤ 2n + 2)	[½k]
(12.1)	0(1 ≤ k ≤ n + 1)	k − n − 1(n + 2 ≤ k ≤ 2n + 1)	[½(k − 1)]
(12.2)	0(2 ≤ k ≤ n + 2)	k − n − 2(n + 3 ≤ k ≤ 2n + 2)	[½k − 1]
(12.3)	0(2 ≤ k ≤ n + 2)	k − n − 2(n + 3 ≤ k ≤ 2n + 2)	[½k − 1]
(12.4)	1(3 ≤ k ≤ n + 3)	k − n − 2(n + 4 ≤ k ≤ 2n + 3)	[½(k − 1)]
(12.5)	0(1 ≤ k ≤ n + 1)	k − n − 1(n + 2 ≤ k ≤ 2n + 1)	[½(k − 1)]
(12.6)	0(2 ≤ k ≤ n + 1)	k − n − 1(n + 2 ≤ k ≤ 2n)	[½k − 1]
(12.7)	0(2 ≤ k ≤ n + 1)	k − n − 1(n + 2 ≤ k ≤ 2n)	[½k − 1]
(12.8)	0(1 ≤ k ≤ n)	k − n(n + 1 ≤ k ≤ 2n − 1)	[½(k − 1)]

NC	OE−	OT−	ÔE−	ÔT−	O^*E−
σ	1	j	1	j	1
$\bar{σ}$	−p	−jp	−p	−jp	−p
τ	−1	−1	$- l$	$- l$	$- j l$
$\bar{τ}$	l	l	1	1	J
g	l/j	l	1/j	1	1
δπ	$j l d p$	$- j p d l$	$j l d p$	$- j p d l$	$j l d p$
$δ \bar{π}$	$j l d p$	$j^{2} l d p$	jdp	j²dp	dp
δρ	$j l d l$	$l d l$	$j d l$	$d l$	$j^{2} d l$
$δ \bar{ρ}$	$j l d l$	$j l d l$	$j p d l$	$j p d l$	$- j l d p$
δg	−dp + dl	dl	$- l d p - p d l$	0	0

Coefficient	Dimension
$G_{μ, ν \| a}^{(n)}$	−(2n − μ − ν + 2)d_s − (μ + ν)d_t + 1
${\bar{G}}_{μ, ν \| a}^{(n)}$	−(2n − μ − ν + 1)d_s − (μ + ν + 1)d_t + 1
$G_{μ, ν \| b}^{(n)}$	−(2n − μ − ν + 1)d_s − (μ + ν + 1)d_t + 1
${\bar{G}}_{μ, ν \| b}^{(n)}$	−(2n − μ − ν)d_s − (μ + ν + 2)d_t + 1

Abstract

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Equations (52)

Journal of the Optical Society of America A