Abstract

We generalize the analytical form of the orthonormal elliptical polynomials for any arbitrary aspect ratio to arbitrary orientation and give expression for them up to the 4th order. The utility of the polynomials is demonstrated by obtaining the expansion up to the 8th order in two examples of an off-axis wavefront exiting from an optical system with a vignetted pupil.

© 2014 Optical Society of America

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References

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    [CrossRef]
  2. J. Mazzaferri and R. Navarro, “Wide two-dimensional field laser ray-tracing aberrometer,” J. Vis. 12(2), 1–14 (2012).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  7. G.-m. Dai and V. N. Mahajan, “Nonrecursive orthonormal polynomials with matrix formulation,” Opt. Lett. 32, 74–76 (2007).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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  17. R. Navarro, “Refractive error sensing from wavefront slopes,” J. Vis. 10(13), 1–15 (2010).

2013 (3)

2012 (2)

V. N. Mahajan, “Orthonormal polynomials in wavefront analysis: analytical solution: errata,” J. Opt. Soc. Am. A 29, 1673–1674 (2012).
[CrossRef]

J. Mazzaferri and R. Navarro, “Wide two-dimensional field laser ray-tracing aberrometer,” J. Vis. 12(2), 1–14 (2012).

2010 (1)

R. Navarro, “Refractive error sensing from wavefront slopes,” J. Vis. 10(13), 1–15 (2010).

2007 (2)

2002 (1)

1996 (1)

1994 (2)

W. Swantner and W. W. Chow, “Gram–Schmidt orthogonalization of Zernike polynomials for general aperture shapes,” Appl. Opt. 33, 1832–1837 (1994).
[CrossRef]

W. Gander, G. H. Golub, and R. Strebel, “Least-squares fitting of circles and ellipses,” BIT Num. Math. 34, 558–578 (1994).

1976 (1)

1968 (1)

Bradley, A.

Cheng, X.

Chow, W. W.

Dai, G.-m.

Díaz, J. A.

Gander, W.

W. Gander, G. H. Golub, and R. Strebel, “Least-squares fitting of circles and ellipses,” BIT Num. Math. 34, 558–578 (1994).

Golub, G. H.

W. Gander, G. H. Golub, and R. Strebel, “Least-squares fitting of circles and ellipses,” BIT Num. Math. 34, 558–578 (1994).

Harbers, G.

Hong, X.

Horn, R. A.

R. A. Horn and C. R. Johnson, Matrix Analysis, 2nd ed. (Cambridge, 2013), p. 441.

Johnson, C. R.

R. A. Horn and C. R. Johnson, Matrix Analysis, 2nd ed. (Cambridge, 2013), p. 441.

King, W. B.

Kocaoglu, O. P.

Kunst, P. J.

Leibbrandt, G. W. R.

Liu, Z.

Mahajan, V. N.

Mazzaferri, J.

J. Mazzaferri and R. Navarro, “Wide two-dimensional field laser ray-tracing aberrometer,” J. Vis. 12(2), 1–14 (2012).

Miller, D. T.

Navarro, R.

J. Mazzaferri and R. Navarro, “Wide two-dimensional field laser ray-tracing aberrometer,” J. Vis. 12(2), 1–14 (2012).

R. Navarro, “Refractive error sensing from wavefront slopes,” J. Vis. 10(13), 1–15 (2010).

Noll, R. J.

Rudolf, P.

P. Rudolf, “Photographic objective,” U.S. patent721,240 (July15, 1902).

Strebel, R.

W. Gander, G. H. Golub, and R. Strebel, “Least-squares fitting of circles and ellipses,” BIT Num. Math. 34, 558–578 (1994).

Swantner, W.

Thibos, L. N.

Appl. Opt. (5)

Biomed. Opt. Express (1)

BIT Num. Math. (1)

W. Gander, G. H. Golub, and R. Strebel, “Least-squares fitting of circles and ellipses,” BIT Num. Math. 34, 558–578 (1994).

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

J. Vis. (2)

R. Navarro, “Refractive error sensing from wavefront slopes,” J. Vis. 10(13), 1–15 (2010).

J. Mazzaferri and R. Navarro, “Wide two-dimensional field laser ray-tracing aberrometer,” J. Vis. 12(2), 1–14 (2012).

Opt. Lett. (1)

Other (3)

V. N. Mahajan, Optical Imaging and Aberrations, Part III: Wavefront Analysis (SPIE, 2013), pp. 71–72, 222.

R. A. Horn and C. R. Johnson, Matrix Analysis, 2nd ed. (Cambridge, 2013), p. 441.

P. Rudolf, “Photographic objective,” U.S. patent721,240 (July15, 1902).

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Figures (4)

Fig. 1.
Fig. 1.

Elliptical pupil with unit major axis, aspect ratio 0ϵ<1, and rotated an angle 0α<π, from the x-axis.

Fig. 2.
Fig. 2.

Matrix plots of (a) C; (b) M. These plots show the nonzero elements for both matrices in the general case. Matrix plots shown in (c) and (d) are the numerical values of the elements for both matrices considering an example in which ϵ=0.85 and α=1rd. Zero values are white, negative values are bluish, and positive values reddish.

Fig. 3.
Fig. 3.

Simulated interferogram (in a.u. of intensity, left column) and wavefront (in λ units, right column) of the Tessar lens US Patent 721240 for the field positions: (a) (0°, 25°), (b) (20°, 25°). The fitted ellipses obtained for the application of the elliptical polynomials are also drawn (dotted red lines).

Fig. 4.
Fig. 4.

Interferograms obtained by using the reconstructed wavefront (left column) and difference between the elliptical wavefront provided by ZEMAX program and that by using orthonormal elliptical polynomials, up to 8th order (right column), for the field positions as in Fig. 3: (a) (0°, 25°); (b) (20°, 25°).

Tables (3)

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Table 1. Orthonormal Zernike Polynomials Zj up to the 4th Order in Cartesian Coordinates (x,y)

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Table 2. Orthonormal Elliptical Polynomials Ej up to the 4th order in Terms of the Circle Zernike Polynomials

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Table 3. Expansion Coefficients up to the 8th Order in Terms of Zernike Circle and Elliptical Polynomials, as well as Sigma and P–V Values

Equations (13)

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

W(x,y)=jejEj(x,y;ϵ,α),
Ej(x,y;ϵ,α)=i=1JMij(ϵ,α)Zi(x,y),
M=(QT)1,
QTQ=C.
Cij(ϵ,α)=1ϵπxminxmaxdxy()y(+)Zi(x,y)Zj(x,y)dy.
(xcosα+ysinα)2+(ycosαxsinα)2ϵ2=1,
y()=x(1ϵ2)sin2αϵηx2η
xmin(),xmax(+)=η,
η=ϵ2+(1ϵ2)cos2α
E2=2(xcosαysinα);E3=2ϵ(ycosα+xsinα).
1ϵπxminxmaxdxy()y(+)E2(x,y)E3(x,y)dy=(1ϵ2)sin4α2ϵ
ej=1ϵπxminxmaxdxy()y(+)W(x,y)Ej(x,y;ϵ,α)dy.
σ=j=245cj2.

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