## Abstract

The classical aberrations of an anamorphic optical imaging system, representing the terms of a power-series expansion of its aberration function, are separable in the Cartesian coordinates of a point on its pupil. We discuss the balancing of a classical aberration of a certain order with one or more such aberrations of lower order to minimize its variance across a rectangular pupil of such a system. We show that the balanced aberrations are the products of two Legendre polynomials, one for each of the two Cartesian coordinates of the pupil point. The compound Legendre polynomials are orthogonal across a rectangular pupil and, like the classical aberrations, are inherently separable in the Cartesian coordinates of the pupil point. They are different from the balanced aberrations and the corresponding orthogonal polynomials for a system with rotational symmetry but a rectangular pupil.

© 2010 Optical Society of America

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### Equations (28)

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(1)
$${p}^{2},\text{}{x}^{2},\text{}px,\text{}{q}^{2},\text{}{y}^{2},\text{and}qy.$$
(2)
$$W(p,q;x,y)=\sum _{i,j,k,l,m,n}{C}_{i,j,k,l,m,n}({p}^{2}{)}^{i}({q}^{2}{)}^{j}({x}^{2}{)}^{k}({y}^{2}{)}^{l}\phantom{\rule{0ex}{0ex}}(px{)}^{m}(qy{)}^{n},$$
(3)
$$\text{degree}=2(i+j+k+l+m+n).$$
(4)
$$W(p,q;x,y)=({C}_{1}{p}^{3}+{C}_{2}p{q}^{2})x+({C}_{3}{p}^{2}q+{C}_{4}{q}^{3})y\phantom{\rule{0ex}{0ex}}+({C}_{5}{p}^{2}+{C}_{6}{q}^{2}){x}^{2}+{C}_{7}pqxy\phantom{\rule{0ex}{0ex}}+({C}_{8}{p}^{2}+{C}_{9}{q}^{2}){y}^{2}+{C}_{10}px{y}^{2}+{C}_{11}qy{x}^{2}\phantom{\rule{0ex}{0ex}}+{C}_{12}p{x}^{3}+{C}_{13}q{y}^{3}+{C}_{14}{x}^{2}{y}^{2}+{C}_{15}{x}^{4}\phantom{\rule{0ex}{0ex}}+{C}_{16}{y}^{4},$$
(5)
$${W}_{cx}(x,y)={x}^{3}.$$
(6)
$${\sigma}_{cx}^{2}=\u3008[{W}_{cx}(x,y){]}^{2}\u3009-\u3008{W}_{cx}(x,y){\u3009}^{2},$$
(7)
$$\u3008g(x,y)\u3009=\frac{\underset{-1}{\overset{1}{\int}}\underset{-1}{\overset{1}{\int}}g(x,y)\mathrm{d}x\mathrm{d}y}{\underset{-1}{\overset{1}{\int}}\underset{-1}{\overset{1}{\int}}\mathrm{d}x\mathrm{d}y}=(1/4)\underset{-1}{\overset{1}{\int}}\underset{-1}{\overset{1}{\int}}g(x,y)\mathrm{d}x\mathrm{d}y.$$
(8)
$${W}_{bcx}(x,y)={x}^{3}+bx.$$
(9)
$${\sigma}_{bcx}^{2}=\frac{1}{7}+\frac{2b}{5}+\frac{{b}^{2}}{3}.$$
(10)
$${W}_{bcx}(x,y)={x}^{3}-(3/5)x.$$
(11)
$$\frac{1}{2}\underset{-1}{\overset{1}{\int}}{P}_{n}(x){P}_{{n}^{\prime}}(x)\mathrm{d}x=\frac{1}{2n+1}{\delta}_{n{n}^{\prime}},$$
(12)
$${P}_{n}(-x)=(-1{)}^{n}{P}_{n}(x).$$
(14)
$${P}_{n}(-1)=\{\begin{array}{cc}1& \text{for even}n\\ -1& \text{for odd}n\end{array},$$
(15)
$$\begin{array}{c}{P}_{n}(0)=0\phantom{\rule[-0.0ex]{1em}{0.0ex}}\text{for odd}n\mathrm{.}\\ {P}_{n}(0)\text{}\mathrm{is}\text{}\mathrm{positive}\text{}\mathrm{or}\text{}\mathrm{negative}\text{}\mathrm{depending}\text{}\mathrm{on}\text{}\mathrm{whether}\text{}n/2\text{}\mathrm{is}\text{}\mathrm{even}\text{}\mathrm{or}\phantom{\rule[-0.0ex]{0.10em}{0.0ex}}\mathrm{odd}\mathrm{.}\end{array}$$
(16)
$$(n+1){P}_{n+1}(x)=(2n+1)x{P}_{n}(x)-n{P}_{n-1}(x).$$
(17)
$${L}_{n}(x)=\sqrt{2n+1}{P}_{n}(x).$$
(18)
$$\frac{1}{2}\underset{-1}{\overset{1}{\int}}{L}_{n}(x){L}_{{n}^{\prime}}(x)\mathrm{d}x={\delta}_{n{n}^{\prime}}.$$
(19)
$${Q}_{j}(x,y)={L}_{l}(x){L}_{m}(y),$$
(20)
$${Q}_{1}(x,y)={L}_{0}(x){L}_{0}(y)=1.$$
(21)
$$\frac{1}{4}{\int}_{-1}^{1}{\int}_{-1}^{1}{Q}_{j}(x,y){Q}_{{j}^{\prime}}(x,y)\mathrm{d}x\mathrm{d}y={\delta}_{j{j}^{\prime}}.$$
(22)
$${N}_{n}=(1/2)(n+1)(n+2).$$
(23)
$${Q}_{32}(x,y)={L}_{4}(x){L}_{3}(y).$$
(24)
$$W(x,y)=\sum _{j}{A}_{j}{Q}_{j}(x,y),$$
(25)
$${A}_{j}=(1/4){\int}_{-1}^{1}{\int}_{-1}^{1}W(x,y){Q}_{j}(x,y)\mathrm{d}x\mathrm{d}y.$$
(26)
$$\u3008W(x,y)\u3009={A}_{1}.$$
(27)
$$\u3008[W(x,y){]}^{2}\u3009=\sum _{j}{A}_{j}^{2}.$$
(28)
$${\sigma}_{W}^{2}=\u3008[W(x,y){]}^{2}\u3009-\u3008W(x,y){\u3009}^{2}=\sum _{j\ne 1}{A}_{j}^{2}.$$