Abstract

Zernike polynomials are an orthonormal set of scalar functions over a circular domain, and are commonly used to represent wavefront phase or surface irregularity. In optical testing, slope or curvature of a surface or wavefront is sometimes measured instead, from which the surface or wavefront map is obtained. Previously we derived an orthonormal set of vector polynomials that fit to slope measurement data and yield the surface or wavefront map represented by Zernike polynomials. Here we define a 3-element curvature vector used to represent the second derivatives of a continuous surface, and derive a set of orthonormal curvature basis functions that are written in terms of Zernike polynomials. We call the new curvature functions the C polynomials. Closed form relations for the complete basis set are provided, and we show how to determine Zernike surface coefficients from the curvature data as represented by the C polynomials.

© 2013 Optical Society of America

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    [Crossref] [PubMed]
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  13. F. Roddier, “Wavefront sensing and the irradiance transport equation,” Appl. Opt. 29, 1402–1403 (1990).
  14. M. Schulz, “Topography measurement by a reliable large-area curvature sensor,” Optik (Stuttg.) 112, 86–90 (2001).
  15. B. C. Kim, T. Saiag, Q. Wang, J. Soons, R. S. Polvani, and U. Griesmann, “The Geometry Measuring Machine(GEMM) Project at NIST,” in Proceedings of ASPE 2004 Winter Topical Meeting on Free-Form Optics: Design, Fabrication, Metrology, Assembly, 108 (2004).
  16. B. C. Kim, M. C. Kwon, B. U. Choo, and I. J. Yoon, “3-D Shape easurement Using Curvature Data,” Proc. SPIE 7389, 73892H (2009).
  17. C. Elstner, J. Gerhardt, P. Thomsen-Schmidt, M. Schulz, and I. Weingärtner, “Reconstructing surface profiles from curvature measurements,” Optik (Stuttg.) 113, 154–158 (2002).
  18. W. Han, “New method for estimating wavefront from curvature signal by curve fitting,” Opt. Eng. 34(4), 1232–1237 (1995).
  19. S. Ríos, E. Acosta, and S. Bara, “Modal phase estimation from wavefront curvature sensing,” Opt. Commun. 123, 453–456 (1996).
  20. T. M. Apostol, Linear Algebra: A First Course, with Applications to Differential Equations (John Wiley & Sons, 1997), pp. 111–114.
  21. R. Upton and B. Ellerbroek, “Gram–Schmidt orthogonalization of the Zernike polynomials on apertures of arbitrary shape,” Opt. Lett. 29, 2840–2842 (2004).
  22. D. W. Kim, B. C. Kim, C. Zhao, C. J. Oh, and J. H. Burge, “Algorithms for surface reconstruction from curvature data for freeform aspherics,” Proc. SPIE 8838, 88380B (2013).
  23. J. E. Sheedy and R. F. Hardy, “The optics of occupational progressive lenses,” Optometry 76(8), 432–441 (2005).

2013 (1)

D. W. Kim, B. C. Kim, C. Zhao, C. J. Oh, and J. H. Burge, “Algorithms for surface reconstruction from curvature data for freeform aspherics,” Proc. SPIE 8838, 88380B (2013).

2009 (1)

B. C. Kim, M. C. Kwon, B. U. Choo, and I. J. Yoon, “3-D Shape easurement Using Curvature Data,” Proc. SPIE 7389, 73892H (2009).

2008 (1)

2007 (2)

P. C. V. Mallik, C. Zhao, and J. H. Burge, “Measurement of a 2-meter flat using a pentaprism scanning system,” Opt. Eng. 46, 023602 (2007).

C. Zhao and J. H. Burge, “Orthonormal vector polynomials in a unit circle, Part I: basis set derived from gradients of Zernike polynomials,” Opt. Express 15, 18014–18024 (2007).

2005 (1)

J. E. Sheedy and R. F. Hardy, “The optics of occupational progressive lenses,” Optometry 76(8), 432–441 (2005).

2004 (1)

2002 (1)

C. Elstner, J. Gerhardt, P. Thomsen-Schmidt, M. Schulz, and I. Weingärtner, “Reconstructing surface profiles from curvature measurements,” Optik (Stuttg.) 113, 154–158 (2002).

2001 (1)

M. Schulz, “Topography measurement by a reliable large-area curvature sensor,” Optik (Stuttg.) 112, 86–90 (2001).

1996 (2)

S. Ríos, E. Acosta, and S. Bara, “Modal phase estimation from wavefront curvature sensing,” Opt. Commun. 123, 453–456 (1996).

G. Harbers, P. J. Kunst, and G. W. R. Leibbrandt, “Analysis of lateral shearing interferograms by use of Zernike polynomials,” Appl. Opt. 35, 6162–6172 (1996).

1995 (1)

W. Han, “New method for estimating wavefront from curvature signal by curve fitting,” Opt. Eng. 34(4), 1232–1237 (1995).

1992 (2)

R. G. Lane and M. Tallon, “Wave-front reconstruction using a Shack-Hartmann sensor,” Appl. Opt. 31, 6902–6907 (1992).

P. E. Glenn, “Lambda-over-one-thousand metrology results for steep aspheres using a curvature profiling technique,” Proc. SPIE 1531, 61–64 (1992).

1990 (2)

P. E. Glenn, “Robust, sub-angstrom-level midspatial-frequency profilometry,” Proc. SPIE 1333, 175 (1990).

F. Roddier, “Wavefront sensing and the irradiance transport equation,” Appl. Opt. 29, 1402–1403 (1990).

1982 (1)

1980 (1)

1976 (1)

Acosta, E.

S. Ríos, E. Acosta, and S. Bara, “Modal phase estimation from wavefront curvature sensing,” Opt. Commun. 123, 453–456 (1996).

Bara, S.

S. Ríos, E. Acosta, and S. Bara, “Modal phase estimation from wavefront curvature sensing,” Opt. Commun. 123, 453–456 (1996).

Burge, J. H.

D. W. Kim, B. C. Kim, C. Zhao, C. J. Oh, and J. H. Burge, “Algorithms for surface reconstruction from curvature data for freeform aspherics,” Proc. SPIE 8838, 88380B (2013).

C. Zhao and J. H. Burge, “Orthonormal vector polynomials in a unit circle, Part II: completing the basis set,” Opt. Express 16, 6586–6591 (2008).

C. Zhao and J. H. Burge, “Orthonormal vector polynomials in a unit circle, Part I: basis set derived from gradients of Zernike polynomials,” Opt. Express 15, 18014–18024 (2007).

P. C. V. Mallik, C. Zhao, and J. H. Burge, “Measurement of a 2-meter flat using a pentaprism scanning system,” Opt. Eng. 46, 023602 (2007).

Choo, B. U.

B. C. Kim, M. C. Kwon, B. U. Choo, and I. J. Yoon, “3-D Shape easurement Using Curvature Data,” Proc. SPIE 7389, 73892H (2009).

Ellerbroek, B.

Elstner, C.

C. Elstner, J. Gerhardt, P. Thomsen-Schmidt, M. Schulz, and I. Weingärtner, “Reconstructing surface profiles from curvature measurements,” Optik (Stuttg.) 113, 154–158 (2002).

Gavrielides, A.

Gerhardt, J.

C. Elstner, J. Gerhardt, P. Thomsen-Schmidt, M. Schulz, and I. Weingärtner, “Reconstructing surface profiles from curvature measurements,” Optik (Stuttg.) 113, 154–158 (2002).

Glenn, P. E.

P. E. Glenn, “Lambda-over-one-thousand metrology results for steep aspheres using a curvature profiling technique,” Proc. SPIE 1531, 61–64 (1992).

P. E. Glenn, “Robust, sub-angstrom-level midspatial-frequency profilometry,” Proc. SPIE 1333, 175 (1990).

Han, W.

W. Han, “New method for estimating wavefront from curvature signal by curve fitting,” Opt. Eng. 34(4), 1232–1237 (1995).

Harbers, G.

Hardy, R. F.

J. E. Sheedy and R. F. Hardy, “The optics of occupational progressive lenses,” Optometry 76(8), 432–441 (2005).

Kim, B. C.

D. W. Kim, B. C. Kim, C. Zhao, C. J. Oh, and J. H. Burge, “Algorithms for surface reconstruction from curvature data for freeform aspherics,” Proc. SPIE 8838, 88380B (2013).

B. C. Kim, M. C. Kwon, B. U. Choo, and I. J. Yoon, “3-D Shape easurement Using Curvature Data,” Proc. SPIE 7389, 73892H (2009).

Kim, D. W.

D. W. Kim, B. C. Kim, C. Zhao, C. J. Oh, and J. H. Burge, “Algorithms for surface reconstruction from curvature data for freeform aspherics,” Proc. SPIE 8838, 88380B (2013).

Kunst, P. J.

Kwon, M. C.

B. C. Kim, M. C. Kwon, B. U. Choo, and I. J. Yoon, “3-D Shape easurement Using Curvature Data,” Proc. SPIE 7389, 73892H (2009).

Lane, R. G.

Leibbrandt, G. W. R.

Mallik, P. C. V.

P. C. V. Mallik, C. Zhao, and J. H. Burge, “Measurement of a 2-meter flat using a pentaprism scanning system,” Opt. Eng. 46, 023602 (2007).

Noll, R. J.

Oh, C. J.

D. W. Kim, B. C. Kim, C. Zhao, C. J. Oh, and J. H. Burge, “Algorithms for surface reconstruction from curvature data for freeform aspherics,” Proc. SPIE 8838, 88380B (2013).

Ríos, S.

S. Ríos, E. Acosta, and S. Bara, “Modal phase estimation from wavefront curvature sensing,” Opt. Commun. 123, 453–456 (1996).

Roddier, F.

Schulz, M.

C. Elstner, J. Gerhardt, P. Thomsen-Schmidt, M. Schulz, and I. Weingärtner, “Reconstructing surface profiles from curvature measurements,” Optik (Stuttg.) 113, 154–158 (2002).

M. Schulz, “Topography measurement by a reliable large-area curvature sensor,” Optik (Stuttg.) 112, 86–90 (2001).

Sheedy, J. E.

J. E. Sheedy and R. F. Hardy, “The optics of occupational progressive lenses,” Optometry 76(8), 432–441 (2005).

Southwell, W. H.

Tallon, M.

Thomsen-Schmidt, P.

C. Elstner, J. Gerhardt, P. Thomsen-Schmidt, M. Schulz, and I. Weingärtner, “Reconstructing surface profiles from curvature measurements,” Optik (Stuttg.) 113, 154–158 (2002).

Upton, R.

Weingärtner, I.

C. Elstner, J. Gerhardt, P. Thomsen-Schmidt, M. Schulz, and I. Weingärtner, “Reconstructing surface profiles from curvature measurements,” Optik (Stuttg.) 113, 154–158 (2002).

Yoon, I. J.

B. C. Kim, M. C. Kwon, B. U. Choo, and I. J. Yoon, “3-D Shape easurement Using Curvature Data,” Proc. SPIE 7389, 73892H (2009).

Zhao, C.

D. W. Kim, B. C. Kim, C. Zhao, C. J. Oh, and J. H. Burge, “Algorithms for surface reconstruction from curvature data for freeform aspherics,” Proc. SPIE 8838, 88380B (2013).

C. Zhao and J. H. Burge, “Orthonormal vector polynomials in a unit circle, Part II: completing the basis set,” Opt. Express 16, 6586–6591 (2008).

C. Zhao and J. H. Burge, “Orthonormal vector polynomials in a unit circle, Part I: basis set derived from gradients of Zernike polynomials,” Opt. Express 15, 18014–18024 (2007).

P. C. V. Mallik, C. Zhao, and J. H. Burge, “Measurement of a 2-meter flat using a pentaprism scanning system,” Opt. Eng. 46, 023602 (2007).

Appl. Opt. (3)

J. Opt. Soc. Am. (2)

Opt. Commun. (1)

S. Ríos, E. Acosta, and S. Bara, “Modal phase estimation from wavefront curvature sensing,” Opt. Commun. 123, 453–456 (1996).

Opt. Eng. (2)

P. C. V. Mallik, C. Zhao, and J. H. Burge, “Measurement of a 2-meter flat using a pentaprism scanning system,” Opt. Eng. 46, 023602 (2007).

W. Han, “New method for estimating wavefront from curvature signal by curve fitting,” Opt. Eng. 34(4), 1232–1237 (1995).

Opt. Express (2)

Opt. Lett. (2)

Optik (Stuttg.) (2)

C. Elstner, J. Gerhardt, P. Thomsen-Schmidt, M. Schulz, and I. Weingärtner, “Reconstructing surface profiles from curvature measurements,” Optik (Stuttg.) 113, 154–158 (2002).

M. Schulz, “Topography measurement by a reliable large-area curvature sensor,” Optik (Stuttg.) 112, 86–90 (2001).

Optometry (1)

J. E. Sheedy and R. F. Hardy, “The optics of occupational progressive lenses,” Optometry 76(8), 432–441 (2005).

Proc. SPIE (4)

D. W. Kim, B. C. Kim, C. Zhao, C. J. Oh, and J. H. Burge, “Algorithms for surface reconstruction from curvature data for freeform aspherics,” Proc. SPIE 8838, 88380B (2013).

P. E. Glenn, “Lambda-over-one-thousand metrology results for steep aspheres using a curvature profiling technique,” Proc. SPIE 1531, 61–64 (1992).

P. E. Glenn, “Robust, sub-angstrom-level midspatial-frequency profilometry,” Proc. SPIE 1333, 175 (1990).

B. C. Kim, M. C. Kwon, B. U. Choo, and I. J. Yoon, “3-D Shape easurement Using Curvature Data,” Proc. SPIE 7389, 73892H (2009).

Other (4)

B. C. Kim, T. Saiag, Q. Wang, J. Soons, R. S. Polvani, and U. Griesmann, “The Geometry Measuring Machine(GEMM) Project at NIST,” in Proceedings of ASPE 2004 Winter Topical Meeting on Free-Form Optics: Design, Fabrication, Metrology, Assembly, 108 (2004).

R. E. Parks and D. S. Anderson, “Surface Profile Determination Using a Two-ball Spherometer,” Optical Fabrication and Testing Workshop Technical Notebook, OSA, Tucson, AZ meeting, Nov. 5–7, 1979.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, 1980), P464–468.

T. M. Apostol, Linear Algebra: A First Course, with Applications to Differential Equations (John Wiley & Sons, 1997), pp. 111–114.

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Figures (1)

Fig. 1
Fig. 1

Illustration of an interferometer measuring a spherical surface. What is measured is the surface deviation from a sphere along the normal direction, not the absolute sag.

Tables (6)

Tables Icon

Table 1 Curvatures of Zernike polynomials: ZCj = CURV(Zj)

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Table 2 List of the inner products of the first 13 Zernike curvatures excluding the trivial terms

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Table 3 List of first 33 orthonormal curvature polynomials Cj as functions of curvatures of Zernike polynomials.

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Table 4 Summary of the recursion relations between the C polynomials and the Zernike polynomials.

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Table 5 List of the first 15 terms of the non-trivial C polynomials written in terms of the combinations of Zernike polynomials.

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Table 6 List of ZC terms for calculating C226 and the calculated C226 expressed in Zernike polynomials.

Equations (15)

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

j=226
c( x ) d 2 z d x 2
c( x,y )=( 1 2 ( d 2 z d x 2 + d 2 z d y 2 ) d 2 z dxdy 1 2 ( d 2 z d x 2 d 2 z d y 2 ) )=( c 1 c 2 c 3 )
CURV=( 1 2 ( d 2 d x 2 + d 2 d y 2 ) d 2 dxdy 1 2 ( d 2 d x 2 d 2 d y 2 ) )
Z even j = n+1 R n m (r) 2 cos(mθ) Z odd j = n+1 R n m (r) 2 sin(mθ) } m0 Z j = n+1 R n 0 (r), m=0
R n m (r)= s=0 (nm)/2 (1) s (ns)! s![(n+m)/2s]![(nm)/2s]! r n2s
Z C j =CURV( Z j )=( 1 2 2 x 2 + 1 2 2 y 2 2 xy 1 2 2 x 2 1 2 2 y 2 ) Z j ( x,y )
A B= 1 π ( A 1 B 1 + A 2 B 2 + A 3 B 3 )dxdy
C j( n,m ) = 1 k2( n 4 n 2 ) ( Z C j( n,m ) 4( n 2 1 ) ( n2 ) 2 Z C j'( n2,m ) + n 2 ( n+1 ) ( n2 ) 2 ( n3 ) Z C j"( n4,m ) )
C 226 = 1 4319200 ( Z C 226 1596 324 Z C 188 + 8400 5508 Z C 152 )
ϕ j( n,m ) = 1 k2( n 4 n 2 ) ( Z j( n,m ) 4( n 2 1 ) ( n2 ) 2 Z j'( n2,m ) + n 2 ( n+1 ) ( n2 ) 2 ( n3 ) Z j"( n4,m ) )
K( x,y )= α j C j ( x,y )
Φ( x,y )= α j ϕ j ( x,y )
Φ( x,y )= α j ϕ j ( x,y )= γ j Z j ( x,y )
γ j( n,m ) = α j( n,m ) k 1 2( n 4 n 2 ) 2 α j ( n+2,m ) k 2 2 n 2 ( n+2 ) 2 + α j ( n+4,m ) k 3 2( n+1 ) ( n+2 ) 2 ( n+3 )

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