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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References

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  1. P. Reardon, F. Liu, and J. Geary, “Schmidt-like corrector plate for cylindrical optics,” Opt. Eng. 49, 053002 (2010).
    [CrossRef]
  2. F. Liu, B. M. Robinson, P. J. Reardon, and J. M. Geary, “Analyzing optics test data on rectangular apertures using 2-D Chebyshev polynomials,” Opt. Eng. 50, 043609 (2011).
    [CrossRef]
  3. V. N. Mahajan, “Orthonormal aberration polynomials for anamorphic optical imaging systems with rectangular pupils,” Appl. Opt. 49, 6924–6929 (2010).
    [CrossRef]
  4. K. A. Goldberg and K. Geary, “Wave-front measurement errors from restricted concentric subdomains,” J. Opt. Soc. Am. A 18, 2146–2152 (2001).
    [CrossRef]
  5. A. Guirao, D. R. Williams, and I. G. Cox, “Effect of rotation and translation on the expected benefit of an ideal method to correct the eye’s higher-order aberrations,” J. Opt. Soc. Am. A 18, 1003–1015 (2001).
    [CrossRef]
  6. S. Bará, J. Arines, J. Ares, and P. Prado, “Direct transformation of Zernike eye aberration coefficients between scaled, rotated, and/or displaced pupils,” J. Opt. Soc. Am. A 23, 2061–2066 (2006).
    [CrossRef]
  7. L. Lundström and P. Unsbo, “Transformation of Zernike coefficients: scaled, translated, and rotated wave-fronts with circular and elliptical pupils,” J. Opt. Soc. Am. A 24, 569–577 (2007).
    [CrossRef]
  8. J. Schwiegerling, “Scaling Zernike expansion coefficients to different pupil sizes,” J. Opt. Soc. Am. A 19, 1937–1945 (2002).
    [CrossRef]
  9. C. E. Campbell, “Matrix method to find a new set of Zernike coefficients from an original set when the aperture radius is changed,” J. Opt. Soc. Am. A 20, 209–217 (2003).
    [CrossRef]
  10. H. Shu, L. Luo, G. Han, and J.-L. Coatrieux, “General method to derive the relationship between two sets of Zernike coefficients corresponding to different aperture sizes,” J. Opt. Soc. Am. A 23, 1960–1966 (2006).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  15. G.-M. Dai and V. N. Mahajan, “Orthonormal polynomials in wave-front analysis: error analysis,” Appl. Opt. 47, 3433–3445 (2008).
    [CrossRef]

2011 (2)

F. Liu, B. M. Robinson, P. J. Reardon, and J. M. Geary, “Analyzing optics test data on rectangular apertures using 2-D Chebyshev polynomials,” Opt. Eng. 50, 043609 (2011).
[CrossRef]

J. Schwiegerling, “Scaling pseudo-Zernike expansion coefficients to different pupil sizes,” Opt. Lett. 36, 3076–3078 (2011).
[CrossRef]

2010 (3)

2008 (1)

2007 (2)

2006 (2)

2003 (1)

2002 (1)

2001 (2)

Ares, J.

Arines, J.

Bará, S.

Campbell, C. E.

Coatrieux, J.-L.

Cox, I. G.

Dai, G.-M.

Geary, J.

P. Reardon, F. Liu, and J. Geary, “Schmidt-like corrector plate for cylindrical optics,” Opt. Eng. 49, 053002 (2010).
[CrossRef]

Geary, J. M.

F. Liu, B. M. Robinson, P. J. Reardon, and J. M. Geary, “Analyzing optics test data on rectangular apertures using 2-D Chebyshev polynomials,” Opt. Eng. 50, 043609 (2011).
[CrossRef]

Geary, K.

Goldberg, K. A.

Guirao, A.

Han, G.

Liu, F.

F. Liu, B. M. Robinson, P. J. Reardon, and J. M. Geary, “Analyzing optics test data on rectangular apertures using 2-D Chebyshev polynomials,” Opt. Eng. 50, 043609 (2011).
[CrossRef]

P. Reardon, F. Liu, and J. Geary, “Schmidt-like corrector plate for cylindrical optics,” Opt. Eng. 49, 053002 (2010).
[CrossRef]

Lundström, L.

Luo, L.

Mahajan, V. N.

Prado, P.

Reardon, P.

P. Reardon, F. Liu, and J. Geary, “Schmidt-like corrector plate for cylindrical optics,” Opt. Eng. 49, 053002 (2010).
[CrossRef]

Reardon, P. J.

F. Liu, B. M. Robinson, P. J. Reardon, and J. M. Geary, “Analyzing optics test data on rectangular apertures using 2-D Chebyshev polynomials,” Opt. Eng. 50, 043609 (2011).
[CrossRef]

Robinson, B. M.

F. Liu, B. M. Robinson, P. J. Reardon, and J. M. Geary, “Analyzing optics test data on rectangular apertures using 2-D Chebyshev polynomials,” Opt. Eng. 50, 043609 (2011).
[CrossRef]

Schwiegerling, J.

Shu, H.

Unsbo, P.

Williams, D. R.

Appl. Opt. (3)

J. Opt. Soc. Am. A (8)

V. N. Mahajan and G.-M. Dai, “Orthonormal polynomials in wave-front analysis: analytical solution,” J. Opt. Soc. Am. A 24, 2994–3016 (2007).
[CrossRef]

K. A. Goldberg and K. Geary, “Wave-front measurement errors from restricted concentric subdomains,” J. Opt. Soc. Am. A 18, 2146–2152 (2001).
[CrossRef]

A. Guirao, D. R. Williams, and I. G. Cox, “Effect of rotation and translation on the expected benefit of an ideal method to correct the eye’s higher-order aberrations,” J. Opt. Soc. Am. A 18, 1003–1015 (2001).
[CrossRef]

S. Bará, J. Arines, J. Ares, and P. Prado, “Direct transformation of Zernike eye aberration coefficients between scaled, rotated, and/or displaced pupils,” J. Opt. Soc. Am. A 23, 2061–2066 (2006).
[CrossRef]

L. Lundström and P. Unsbo, “Transformation of Zernike coefficients: scaled, translated, and rotated wave-fronts with circular and elliptical pupils,” J. Opt. Soc. Am. A 24, 569–577 (2007).
[CrossRef]

J. Schwiegerling, “Scaling Zernike expansion coefficients to different pupil sizes,” J. Opt. Soc. Am. A 19, 1937–1945 (2002).
[CrossRef]

C. E. Campbell, “Matrix method to find a new set of Zernike coefficients from an original set when the aperture radius is changed,” J. Opt. Soc. Am. A 20, 209–217 (2003).
[CrossRef]

H. Shu, L. Luo, G. Han, and J.-L. Coatrieux, “General method to derive the relationship between two sets of Zernike coefficients corresponding to different aperture sizes,” J. Opt. Soc. Am. A 23, 1960–1966 (2006).
[CrossRef]

Opt. Eng. (2)

P. Reardon, F. Liu, and J. Geary, “Schmidt-like corrector plate for cylindrical optics,” Opt. Eng. 49, 053002 (2010).
[CrossRef]

F. Liu, B. M. Robinson, P. J. Reardon, and J. M. Geary, “Analyzing optics test data on rectangular apertures using 2-D Chebyshev polynomials,” Opt. Eng. 50, 043609 (2011).
[CrossRef]

Opt. Lett. (1)

Other (1)

Standard (2005).

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Figures (7)

Fig. 1.
Fig. 1.

Horizontal translation and scaling.

Fig. 2.
Fig. 2.

First 15th-order wavefront distribution and coefficients generated by 2D-Chebyshev polynomials.

Fig. 3.
Fig. 3.

First 15th-order wavefront distribution and coefficients generated by 2D-Legendre polynomials.

Fig. 4.
Fig. 4.

2D-Chebyshev coefficients change after different scaling.

Fig. 5.
Fig. 5.

2D-Legendre coefficients change after different scaling.

Fig. 6.
Fig. 6.

Comparison of PV deviation with 1D-Legendre polynomials.

Fig. 7.
Fig. 7.

Comparison of RMS deviation with 1D-Legendre polynomials.

Tables (8)

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Table 1. First Five Terms in a 1D-Chebyshev Polynomial

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Table 2. Expressions for 2D-Chebyshev Polynomials after Translation

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Table 3. Expressions for 2D-Chebyshev Polynomials after Scaling

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Table 4. First Five Terms in 1D-Legendre

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Table 5. First 15 Terms and the Corresponding Aberration Changes in the 2D-Legendre Polynomial after Translation

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Table 6. Coefficient Changes of the First 15 Terms in a 2D-Legendre Polynomial after Scaling, and Their Corresponding Aberration

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Table 7. Expressions for Discrete Chebyshev after Translation

Tables Icon

Table 8. Expressions for Discrete Chebyshev after Scaling

Equations (25)

Equations on this page are rendered with MathJax. Learn more.

Fi(x+u,y+v)=Fi+k=1K1k!(ux+vy)(k)Fi+RK,
F˜i(x,y)=Fi(x+u,y+v)Fi+(ux+vy)Fi.
F˜i(x,y)=k=1aikFk(x,y),
aik=SF˜i(x,y)·Fk(x,y)dS.
Fi(px,qy)=k=1sikFk(x,y),
sik=SFi(px,qy)·Fk(x,y)dS.
T1(x)=1;T2(x)=x;Tn+1(x)=2xTn(x)Tn1(x).
11Tt(x)Tr(x)dx1x2={0:trπ:t=r=1π/2:t=r1.
Ci(x,y)=Tm(x)Tn(y).
cij=1K1111[Cj+(u·Cjx+v·Cjy)]·Ci(x,y)1x21y2dxdy,
K={π2,i=j=1π24,i1,j1π22,else.
Cj(px,qy)=Tm(px)·Tn(qy).
cij=1111Cj(px,qy)Ci(x,y)K1x21y2dxdy.
Ln(x)=2nk=0nxk(nk)(n+k12n).
11Lm(x)Ln(x)dx=22n+1δmn(δmnis the Kronecker notation).
qij=141111[Qj+(u·Qjx+v·Qjy)]·Qi(x,y)dxdy.
Qj(px,qy)=Lm(px)·Ln(qy),
qij=1111Qj(px,qy)Qi(x,y)dxdy.
L˜1=3L1;
L˜2=3uL1+53L2;
L˜3=9u22L1+5uL2+75L3;
L˜4=(3u+15u32)L1+25u22L2+7uL3+97L4;
L˜5=(15u2+105u48)L1+(5u+175u36)L2+49u22L3+9uL4+119L5.
Deviation=Direct Expansion1st Order ExpansionDirect Expansion.
L1RMS2=1;L2RMS2=1/3;L3RMS2=1/5;L4RMS2=1/7;L4RMS2=1/9.

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