Abstract

Zernike functions are orthogonal within the unit circle, but they are not over the discrete points such as CCD arrays or finite element grids. This will result in reconstruction errors for loss of orthogonality. By using roots of Legendre polynomials, a set of points within the unit circle can be constructed so that Zernike functions over the set are discretely orthogonal. Besides that, the location tolerances of the points are studied by perturbation analysis, and the requirements of the positioning precision are not very strict. Computer simulations show that this approach provides a very accurate wavefront reconstruction with the proposed sampling set.

© 2012 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. G. Dai and V. N. Mahajan, “Orthonormal polynomials in wavefront analysis: error analysis,” Appl. Opt. 47, 3433–3445(2008).
    [CrossRef]
  2. V. N. Mahajan, “Zernike polynomial and wavefront fitting,” in Optical Shop Testing, D. Malacara, ed. (Wiley, 2007), pp. 498–546.
  3. P. de Groot, “Design of error-compensating algorithms for sinusoidal phase shifting interferometry,” Appl. Opt. 48, 6788–6796 (2009).
    [CrossRef]
  4. Z. Shi, J. Zhang, Y. Sui, J. Peng, F. Yan, and H. Jiang, “Design of algorithms for phase shifting interferometry using self-convolution of the rectangle window,” Opt. Express 19, 14671–14681 (2011).
    [CrossRef]
  5. K. B. Doyle, V. L. Genberg, G. J. Michels, and G. R. Bisson, “Optical modeling of finite element surface displacements using commercial software,” Proc. SPIE 5867, 58670I (2005).
    [CrossRef]
  6. SigFit Reference Manual, Sigmadyne Inc. (2011).
  7. J. C. Wyant and K. Creath, “Basic wavefront aberration theory for optical metrology,” in Applied Optics and Optical Engineering, R. R. Shannon and J. C. Wyant, eds. (Academic, 1972), pp. 1–53.
  8. M. Pap and F. Schipp, “Discrete orthogonality of Zernike functions,” Mathematica Pannonica 16, 137–144 (2005).

2011 (1)

2009 (1)

2008 (1)

2005 (2)

K. B. Doyle, V. L. Genberg, G. J. Michels, and G. R. Bisson, “Optical modeling of finite element surface displacements using commercial software,” Proc. SPIE 5867, 58670I (2005).
[CrossRef]

M. Pap and F. Schipp, “Discrete orthogonality of Zernike functions,” Mathematica Pannonica 16, 137–144 (2005).

Bisson, G. R.

K. B. Doyle, V. L. Genberg, G. J. Michels, and G. R. Bisson, “Optical modeling of finite element surface displacements using commercial software,” Proc. SPIE 5867, 58670I (2005).
[CrossRef]

Creath, K.

J. C. Wyant and K. Creath, “Basic wavefront aberration theory for optical metrology,” in Applied Optics and Optical Engineering, R. R. Shannon and J. C. Wyant, eds. (Academic, 1972), pp. 1–53.

Dai, G.

de Groot, P.

Doyle, K. B.

K. B. Doyle, V. L. Genberg, G. J. Michels, and G. R. Bisson, “Optical modeling of finite element surface displacements using commercial software,” Proc. SPIE 5867, 58670I (2005).
[CrossRef]

Genberg, V. L.

K. B. Doyle, V. L. Genberg, G. J. Michels, and G. R. Bisson, “Optical modeling of finite element surface displacements using commercial software,” Proc. SPIE 5867, 58670I (2005).
[CrossRef]

Jiang, H.

Mahajan, V. N.

G. Dai and V. N. Mahajan, “Orthonormal polynomials in wavefront analysis: error analysis,” Appl. Opt. 47, 3433–3445(2008).
[CrossRef]

V. N. Mahajan, “Zernike polynomial and wavefront fitting,” in Optical Shop Testing, D. Malacara, ed. (Wiley, 2007), pp. 498–546.

Michels, G. J.

K. B. Doyle, V. L. Genberg, G. J. Michels, and G. R. Bisson, “Optical modeling of finite element surface displacements using commercial software,” Proc. SPIE 5867, 58670I (2005).
[CrossRef]

Pap, M.

M. Pap and F. Schipp, “Discrete orthogonality of Zernike functions,” Mathematica Pannonica 16, 137–144 (2005).

Peng, J.

Schipp, F.

M. Pap and F. Schipp, “Discrete orthogonality of Zernike functions,” Mathematica Pannonica 16, 137–144 (2005).

Shi, Z.

Sui, Y.

Wyant, J. C.

J. C. Wyant and K. Creath, “Basic wavefront aberration theory for optical metrology,” in Applied Optics and Optical Engineering, R. R. Shannon and J. C. Wyant, eds. (Academic, 1972), pp. 1–53.

Yan, F.

Zhang, J.

Appl. Opt. (2)

Mathematica Pannonica (1)

M. Pap and F. Schipp, “Discrete orthogonality of Zernike functions,” Mathematica Pannonica 16, 137–144 (2005).

Opt. Express (1)

Proc. SPIE (1)

K. B. Doyle, V. L. Genberg, G. J. Michels, and G. R. Bisson, “Optical modeling of finite element surface displacements using commercial software,” Proc. SPIE 5867, 58670I (2005).
[CrossRef]

Other (3)

SigFit Reference Manual, Sigmadyne Inc. (2011).

J. C. Wyant and K. Creath, “Basic wavefront aberration theory for optical metrology,” in Applied Optics and Optical Engineering, R. R. Shannon and J. C. Wyant, eds. (Academic, 1972), pp. 1–53.

V. N. Mahajan, “Zernike polynomial and wavefront fitting,” in Optical Shop Testing, D. Malacara, ed. (Wiley, 2007), pp. 498–546.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (3)

Fig. 1.
Fig. 1.

Sampling patterns and orthogonality deviations comparisons for 4N+1 and 2N1 azimuthal points. (a) 4N+1 azimuthal sampling, (b) orthogonality deviations of 4N+1 sampling, (c) 2N1 azimuthal sampling, and (d) orthogonality deviations of 2N1 sampling.

Fig. 2.
Fig. 2.

Sampling patterns and orthogonality deviation comparisons for radial and azimuthal perturbation about ideal points. (a) 2N1 sampling patterns, (b) deviations of 104 levels in radial direction, (c) deviations of 104 levels in azimuthal direction, and (d) deviations of 104 levels in both directions.

Fig. 3.
Fig. 3.

Zernike surface and reconstruction errors. (a) Constructed Zernike surface, and (b) differences of reconstructed and original coefficients.

Equations (22)

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

Znm(ρ,θ)=n+1Rnm(ρ)eimθ,
01Rnm(ρ)Rnm(ρ)ρdp=δnn2(n+1),
02πeimθeimθ¯dθ=2πδmm.
1π02π01Znm(ρ,θ)Znm(ρ,θ)¯ρdρdθ=δnnδmm.
lk(x)=j=1Njkxλjλkλj,
Ak=11lk(x)dx,
11f(x)dx=k=1Nf(λk)Ak.
ρk=1+λk2(0,1),k=1,,N,
n+|m|2+n|m|22N1,n|m|2+n+|m|22N1,
δnn2(n+1)=01Rnm(ρ)Rnm(ρ)ρdρ=14k=1NRnm(ρk)Rnm(ρk)Ak.
θj=2πj4N+1,j=0,,4N,
X={zjk=(ρk,θj),k=1,,N,j=0,,4N}.
Gpp=k=1Nj=04NZnm(ρk,θj)Znm(ρk,θj)¯Ak2(4N+1)=1π02π01Znm(ρ,θ)Znm(ρ,θ)¯ρdρdθ=δnnδmm.
θj=2πj2N1,j=0,,2N2.
Cmn=k=1Nj=0N1T(ρk,θj)Znm(ρk,θj)¯Ak2N(N=2N1&4N+1)
l1(x)=j=2Nxλjλ1λj=k=2Npk·l1(x),lk(x)=xλ1λkλ1·j=2Njkxλjλkλj=pklk(x)Δλ1pklk(x)xλ1,
pk=λ1λkλ1λk,
A1=11l1(x)dx=k=2Npk·A1,Ak=11lk(x)dx=pkAkΔλ1pkBk,
Bk=11lk(x)xλ1dx,
k=1Nf(λk1)Ak=f(λ1)A1+k=2Nf(λk)AK(f(λ1)+Δλ1f(λ1))A1+k=2Nf(λk)AK=(f(λ1)A1+Δλ1f(λ1)A1)·k=2Npk+k=2Nf(λk)pkAkΔλ1·k=2Nf(λk)pkBk.
ΔG=k=1Nf(λk)Akk=1Nf(λk)AkΔλ1(k=2N(f(λ1)A1+f(λk)Akλ1λk+f(λk)Bk)f(λ1)A1).
ΔCmn=O(Δλ1)(Δλ1=λ1λ10).

Metrics