## Abstract

Chebyshev and Legendre polynomials are frequently used in rectangular pupils for wavefront approximation. Ideally, the dataset completely fits with the polynomial basis, which provides the full-pupil approximation coefficients and the corresponding geometric aberrations. However, if there are horizontal translation and scaling, the terms in the original polynomials will become the linear combinations of the coefficients of the other terms. This paper introduces analytical expressions for two typical situations after translation and scaling. With a small translation, first-order Taylor expansion could be used to simplify the computation. Several representative terms could be selected as inputs to compute the coefficient changes before and after translation and scaling. Results show that the outcomes of the analytical solutions and the approximated values under discrete sampling are consistent. With the computation of a group of randomly generated coefficients, we contrasted the changes under different translation and scaling conditions. The larger ratios correlate the larger deviation from the approximated values to the original ones. Finally, we analyzed the peak-to-valley (PV) and root mean square (RMS) deviations from the uses of the first-order approximation and the direct expansion under different translation values. The results show that when the translation is less than 4%, the most deviated 5th term in the first-order 1D-Legendre expansion has a PV deviation less than 7% and an RMS deviation less than 2%. The analytical expressions and the computed results under discrete sampling given in this paper for the multiple typical function basis during translation and scaling in the rectangular areas could be applied in wavefront approximation and analysis.

© 2013 Optical Society of America

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### Equations (25)

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(1)
$${F}_{i}(x+u,y+v)={F}_{i}+\sum _{k=1}^{K}\frac{1}{k!}{(u\frac{\partial}{\partial x}+v\frac{\partial}{\partial y})}^{(k)}{F}_{i}+{R}_{K},$$
(2)
$${\tilde{F}}_{i}(x,y)={F}_{i}(x+u,y+v)\approx {F}_{i}+(u\frac{\partial}{\partial x}+v\frac{\partial}{\partial y}){F}_{i}.$$
(3)
$${\tilde{F}}_{i}(x,y)=\sum _{k=1}^{\infty}{a}_{ik}{F}_{k}(x,y),$$
(4)
$${a}_{ik}={\iint}_{S}{\tilde{F}}_{i}(x,y)\xb7{F}_{k}(x,y)\mathrm{d}S.$$
(5)
$${F}_{i}(px,qy)=\sum _{k=1}^{\infty}{s}_{ik}{F}_{k}(x,y),$$
(6)
$${s}_{ik}={\iint}_{S}{F}_{i}(px,qy)\xb7{F}_{k}(x,y)\mathrm{d}S.$$
(7)
$${T}_{1}(x)=1;\phantom{\rule[-0.0ex]{1em}{0.0ex}}{T}_{2}(x)=x;\phantom{\rule[-0.0ex]{1em}{0.0ex}}\dots {T}_{n+1}(x)=2x{T}_{n}(x)-{T}_{n-1}(x).$$
(8)
$${\int}_{-1}^{1}{T}_{t}(x){T}_{r}(x)\frac{\mathrm{d}x}{\sqrt{1-{x}^{2}}}=\{\begin{array}{c}\begin{array}{cc}0:& t\ne r\end{array}\\ \begin{array}{cc}\pi :& t=r=1\end{array}\\ \begin{array}{cc}\pi /2:& t=r\ne 1\end{array}\end{array}.$$
(9)
$${C}_{i}(x,y)={T}_{m}(x){T}_{n}(y).$$
(10)
$${c}_{ij}=\frac{1}{K}{\int}_{-1}^{1}{\int}_{-1}^{1}\frac{[{C}_{j}+(u\xb7\frac{\partial {C}_{j}}{\partial x}+v\xb7\frac{\partial {C}_{j}}{\partial y})]\xb7{C}_{i}(x,y)}{\sqrt{1-{x}^{2}}\sqrt{1-{y}^{2}}}\mathrm{d}x\mathrm{d}y,$$
(11)
$$K=\{\begin{array}{cc}{\pi}^{2},& i=j=1\\ \frac{{\pi}^{2}}{4},& i\ne 1,j\ne 1\\ \frac{{\pi}^{2}}{2},& \text{else}\end{array}.$$
(12)
$${C}_{j}(px,qy)={T}_{m}(px)\xb7{T}_{n}(qy).$$
(13)
$${c}_{ij}={\int}_{-1}^{1}{\int}_{-1}^{1}\frac{{C}_{j}(px,qy){C}_{i}(x,y)}{K\sqrt{1-{x}^{2}}\sqrt{1-{y}^{2}}}\mathrm{d}x\mathrm{d}y.$$
(14)
$${L}_{n}(x)={2}^{n}\sum _{k=0}^{n}{x}^{k}\left(\begin{array}{c}n\\ k\end{array}\right)\left(\begin{array}{c}\frac{n+k-1}{2}\\ n\end{array}\right).$$
(15)
$${\int}_{-1}^{1}{L}_{m}(x){L}_{n}(x)\mathrm{d}x=\frac{2}{2n+1}{\delta}_{mn}\phantom{\rule{0ex}{0ex}}({\delta}_{mn}\text{\hspace{0.17em}}\text{is the Kronecker notation}).$$
(16)
$${q}_{ij}=\frac{1}{4}{\int}_{-1}^{1}{\int}_{-1}^{1}[{Q}_{j}+(u\xb7\frac{\partial {Q}_{j}}{\partial x}+v\xb7\frac{\partial {Q}_{j}}{\partial y})]\xb7{Q}_{i}(x,y)\mathrm{d}x\mathrm{d}y.$$
(17)
$${Q}_{j}(px,qy)={L}_{m}(px)\xb7{L}_{n}(qy),$$
(18)
$${q}_{ij}={\int}_{-1}^{1}{\int}_{-1}^{1}{Q}_{j}(px,qy){Q}_{i}(x,y)\mathrm{d}x\mathrm{d}y\mathrm{.}$$
(19)
$${\tilde{L}}_{1}=3{L}_{1};$$
(20)
$${\tilde{L}}_{2}=3u{L}_{1}+\frac{5}{3}{L}_{2};$$
(21)
$${\tilde{L}}_{3}=\frac{9{u}^{2}}{2}{L}_{1}+5u{L}_{2}+\frac{7}{5}{L}_{3};$$
(22)
$${\tilde{L}}_{4}=(3u+\frac{15{u}^{3}}{2}){L}_{1}+\frac{25{u}^{2}}{2}{L}_{2}+7u{L}_{3}+\frac{9}{7}{L}_{4};$$
(23)
$${\tilde{L}}_{5}=(15{u}^{2}+\frac{105{u}^{4}}{8}){L}_{1}+(5u+\frac{175{u}^{3}}{6}){L}_{2}+\frac{49{u}^{2}}{2}{L}_{3}+9u{L}_{4}+\frac{11}{9}{L}_{5}.$$
(24)
$$\text{Deviation}=\frac{\text{Direct Expansion}-1\text{st Order Expansion}}{\text{Direct Expansion}}.$$
(25)
$${L}_{1{\mathrm{RMS}}^{2}}=1;\phantom{\rule[-0.0ex]{2em}{0.0ex}}{L}_{2{\mathrm{RMS}}^{2}}=1/3;\phantom{\rule[-0.0ex]{2em}{0.0ex}}{L}_{3{\mathrm{RMS}}^{2}}=1/5;\phantom{\rule[-0.0ex]{2em}{0.0ex}}{L}_{4{\mathrm{RMS}}^{2}}=1/7;\phantom{\rule[-0.0ex]{2em}{0.0ex}}{L}_{4{\mathrm{RMS}}^{2}}=1/9.$$