Abstract

Characterizing real-life optical surfaces usually involves finding the best-fit of an appropriate surface model to a set of discrete measurement data. This process can be greatly simplified by choosing orthonormal polynomials for the surface description. In case of rotationally symmetric aspherical surfaces, new sets of orthogonal polynomials were introduced by Forbes to replace the numerical unstable standard description. From these, for the application of surface retrieval using experimental ray tracing, the sag orthogonal Qcon-polynomials are of particular interest. However, these are by definition orthogonal over continuous data and may not be orthogonal for discrete data. In this case, the simplified solution is not valid. Hence, a Gram-Schmidt orthonormalization of these polynomials over the discrete data set is proposed to solve this problem. The resulting difference will be presented by a performance analysis and comparison to the direct matrix inversion method.

© 2015 Optical Society of America

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References

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  1. U. Ceyhan, T. Henning, F. Fleischmann, D. Hilbig, and D. Knipp, “Measurements of aberrations of aspherical lenses using experimental ray tracing,” Proc. SPIE 8082, 80821K (2011).
    [Crossref]
  2. Z. Hosseinimakarem, H. Aryan, A. Davies, and C. Evans, “Considering a Zernike polynomial representation for spatial frequency content of optical surfaces,” in Imaging and Applied Optics, Vol. 1 of 2015, OSA Technical Digest Series (OSA, 2015), paper FT2B.2.
  3. E. Goodwin, U. Fuchs, S. Gangadhara, S. Kiontke, V. Smagley, and A. Yates, Design and implementation of a new freeform surface based on Chebyshev Polynomials,” in Imaging and Applied Optics, Vol. 1 of 2015, OSA Technical Digest Series (OSA, 2015), paper FT2B.3.
  4. P. Jester, C. Menke, and K. Urban, “B-spline representation of optical surfaces and its accuracy in a ray trace algorithm,” Appl. Opt. 50(6), 822–828 (2011).
    [Crossref] [PubMed]
  5. G. E. Forsythe, “Generation and use of orthogonal polynomials for data-fitting with digital computers,” J. Soc. Ind. Appl. Math. 5(2), 74–88 (1957).
    [Crossref]
  6. C. B. Braunecker, R. Hentschel, and H. J. Tiziani, Advanced Optics Using Aspherical Elements (SPIE, 2008), Ch. 2.
  7. G. W. Forbes and C. P. Brophy, “Asphere, o asphere, how shall we describe thee?” Proc. SPIE 7100, 710002 (2008).
    [Crossref]
  8. G. W. Forbes, “Shape specification for axially symmetric optical surfaces,” Opt. Express 15(8), 5218–5226 (2007).
    [Crossref] [PubMed]
  9. G. W. Forbes, “Fitting freeform shapes with orthogonal bases,” Opt. Express 21(16), 19061–19081 (2013).
    [Crossref] [PubMed]
  10. D. Ochse, K. Uhlendorf, and L. Reichmann, “Describing freeform surfaces with orthogonal functions,” in Imaging and Applied Optics, Vol. 1 of 2015, OSA Technical Digest Series (OSA, 2015), paper FT2B.4.
  11. D. Malacara and S. L. DeVore, “Interferogram Evaluation and Wavefront Fitting,” in Optical Shop Testing, 2nd ed., D. Malacara ed. (Wiley, 1992).
  12. V. N. Mahajan, “Zernike Polynomials and Wavefront Fitting,” in Optical Shop Testing, 3rd ed., D. Malacara ed. (Wiley, 2007).
  13. G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers, (McGraw-Hill, 1968).
  14. G. B. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, 1985).
  15. J. Ye, X. Li, Z. Gao, S. Wang, W. Sun, W. Wang, and Q. Yuan, “Modal wavefront reconstruction over general shaped aperture by numerical orthogonal polynomials,” Opt. Eng. 54(3), 034105 (2015).
  16. J. E. Gentle, Numerical Linear Algebra in Statistics (Springer, 1998).
  17. J. Y. Wang and D. E. Silva, “Wave-front interpretation with Zernike polynomials,” Appl. Opt. 19(9), 1510–1518 (1980).
    [Crossref] [PubMed]
  18. E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK User’s Guide, 3rd ed. (SIAM, 1999).
  19. M. P. Rimmer, C. M. King, and D. G. Fox, “Computer program for the analysis of interferometric test data,” Appl. Opt. 11(12), 2790–2796 (1972).
    [Crossref] [PubMed]
  20. U. W. Hochstrasser, “Orthogonal Polynomials,” in Handbook of Mathematical Functions, M. Abramowitz and I. A. Stegun ed. (Dover, 1978).
  21. G. W. Forbes, “Robust and fast computation for the polynomials of optics,” Opt. Express 18(13), 13851–13862 (2010).
    [Crossref] [PubMed]
  22. R. Upton and B. Ellerbroek, “Gram-Schmidt orthogonalization of the Zernike polynomials on apertures of arbitrary shape,” Opt. Lett. 29(24), 2840–2842 (2004).
    [Crossref] [PubMed]
  23. V. N. Mahajan and G. M. Dai, “Orthonormal polynomials in wavefront analysis: analytical solution,” J. Opt. Soc. Am. A 24(9), 2994–3016 (2007).
    [Crossref] [PubMed]

2015 (1)

J. Ye, X. Li, Z. Gao, S. Wang, W. Sun, W. Wang, and Q. Yuan, “Modal wavefront reconstruction over general shaped aperture by numerical orthogonal polynomials,” Opt. Eng. 54(3), 034105 (2015).

2013 (1)

2011 (2)

U. Ceyhan, T. Henning, F. Fleischmann, D. Hilbig, and D. Knipp, “Measurements of aberrations of aspherical lenses using experimental ray tracing,” Proc. SPIE 8082, 80821K (2011).
[Crossref]

P. Jester, C. Menke, and K. Urban, “B-spline representation of optical surfaces and its accuracy in a ray trace algorithm,” Appl. Opt. 50(6), 822–828 (2011).
[Crossref] [PubMed]

2010 (1)

2008 (1)

G. W. Forbes and C. P. Brophy, “Asphere, o asphere, how shall we describe thee?” Proc. SPIE 7100, 710002 (2008).
[Crossref]

2007 (2)

2004 (1)

1980 (1)

1972 (1)

1957 (1)

G. E. Forsythe, “Generation and use of orthogonal polynomials for data-fitting with digital computers,” J. Soc. Ind. Appl. Math. 5(2), 74–88 (1957).
[Crossref]

Brophy, C. P.

G. W. Forbes and C. P. Brophy, “Asphere, o asphere, how shall we describe thee?” Proc. SPIE 7100, 710002 (2008).
[Crossref]

Ceyhan, U.

U. Ceyhan, T. Henning, F. Fleischmann, D. Hilbig, and D. Knipp, “Measurements of aberrations of aspherical lenses using experimental ray tracing,” Proc. SPIE 8082, 80821K (2011).
[Crossref]

Dai, G. M.

Ellerbroek, B.

Fleischmann, F.

U. Ceyhan, T. Henning, F. Fleischmann, D. Hilbig, and D. Knipp, “Measurements of aberrations of aspherical lenses using experimental ray tracing,” Proc. SPIE 8082, 80821K (2011).
[Crossref]

Forbes, G. W.

Forsythe, G. E.

G. E. Forsythe, “Generation and use of orthogonal polynomials for data-fitting with digital computers,” J. Soc. Ind. Appl. Math. 5(2), 74–88 (1957).
[Crossref]

Fox, D. G.

Gao, Z.

J. Ye, X. Li, Z. Gao, S. Wang, W. Sun, W. Wang, and Q. Yuan, “Modal wavefront reconstruction over general shaped aperture by numerical orthogonal polynomials,” Opt. Eng. 54(3), 034105 (2015).

Henning, T.

U. Ceyhan, T. Henning, F. Fleischmann, D. Hilbig, and D. Knipp, “Measurements of aberrations of aspherical lenses using experimental ray tracing,” Proc. SPIE 8082, 80821K (2011).
[Crossref]

Hilbig, D.

U. Ceyhan, T. Henning, F. Fleischmann, D. Hilbig, and D. Knipp, “Measurements of aberrations of aspherical lenses using experimental ray tracing,” Proc. SPIE 8082, 80821K (2011).
[Crossref]

Jester, P.

King, C. M.

Knipp, D.

U. Ceyhan, T. Henning, F. Fleischmann, D. Hilbig, and D. Knipp, “Measurements of aberrations of aspherical lenses using experimental ray tracing,” Proc. SPIE 8082, 80821K (2011).
[Crossref]

Li, X.

J. Ye, X. Li, Z. Gao, S. Wang, W. Sun, W. Wang, and Q. Yuan, “Modal wavefront reconstruction over general shaped aperture by numerical orthogonal polynomials,” Opt. Eng. 54(3), 034105 (2015).

Mahajan, V. N.

Menke, C.

Rimmer, M. P.

Silva, D. E.

Sun, W.

J. Ye, X. Li, Z. Gao, S. Wang, W. Sun, W. Wang, and Q. Yuan, “Modal wavefront reconstruction over general shaped aperture by numerical orthogonal polynomials,” Opt. Eng. 54(3), 034105 (2015).

Upton, R.

Urban, K.

Wang, J. Y.

Wang, S.

J. Ye, X. Li, Z. Gao, S. Wang, W. Sun, W. Wang, and Q. Yuan, “Modal wavefront reconstruction over general shaped aperture by numerical orthogonal polynomials,” Opt. Eng. 54(3), 034105 (2015).

Wang, W.

J. Ye, X. Li, Z. Gao, S. Wang, W. Sun, W. Wang, and Q. Yuan, “Modal wavefront reconstruction over general shaped aperture by numerical orthogonal polynomials,” Opt. Eng. 54(3), 034105 (2015).

Ye, J.

J. Ye, X. Li, Z. Gao, S. Wang, W. Sun, W. Wang, and Q. Yuan, “Modal wavefront reconstruction over general shaped aperture by numerical orthogonal polynomials,” Opt. Eng. 54(3), 034105 (2015).

Yuan, Q.

J. Ye, X. Li, Z. Gao, S. Wang, W. Sun, W. Wang, and Q. Yuan, “Modal wavefront reconstruction over general shaped aperture by numerical orthogonal polynomials,” Opt. Eng. 54(3), 034105 (2015).

Appl. Opt. (3)

J. Opt. Soc. Am. A (1)

J. Soc. Ind. Appl. Math. (1)

G. E. Forsythe, “Generation and use of orthogonal polynomials for data-fitting with digital computers,” J. Soc. Ind. Appl. Math. 5(2), 74–88 (1957).
[Crossref]

Opt. Eng. (1)

J. Ye, X. Li, Z. Gao, S. Wang, W. Sun, W. Wang, and Q. Yuan, “Modal wavefront reconstruction over general shaped aperture by numerical orthogonal polynomials,” Opt. Eng. 54(3), 034105 (2015).

Opt. Express (3)

Opt. Lett. (1)

Proc. SPIE (2)

U. Ceyhan, T. Henning, F. Fleischmann, D. Hilbig, and D. Knipp, “Measurements of aberrations of aspherical lenses using experimental ray tracing,” Proc. SPIE 8082, 80821K (2011).
[Crossref]

G. W. Forbes and C. P. Brophy, “Asphere, o asphere, how shall we describe thee?” Proc. SPIE 7100, 710002 (2008).
[Crossref]

Other (11)

D. Ochse, K. Uhlendorf, and L. Reichmann, “Describing freeform surfaces with orthogonal functions,” in Imaging and Applied Optics, Vol. 1 of 2015, OSA Technical Digest Series (OSA, 2015), paper FT2B.4.

D. Malacara and S. L. DeVore, “Interferogram Evaluation and Wavefront Fitting,” in Optical Shop Testing, 2nd ed., D. Malacara ed. (Wiley, 1992).

V. N. Mahajan, “Zernike Polynomials and Wavefront Fitting,” in Optical Shop Testing, 3rd ed., D. Malacara ed. (Wiley, 2007).

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers, (McGraw-Hill, 1968).

G. B. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, 1985).

Z. Hosseinimakarem, H. Aryan, A. Davies, and C. Evans, “Considering a Zernike polynomial representation for spatial frequency content of optical surfaces,” in Imaging and Applied Optics, Vol. 1 of 2015, OSA Technical Digest Series (OSA, 2015), paper FT2B.2.

E. Goodwin, U. Fuchs, S. Gangadhara, S. Kiontke, V. Smagley, and A. Yates, Design and implementation of a new freeform surface based on Chebyshev Polynomials,” in Imaging and Applied Optics, Vol. 1 of 2015, OSA Technical Digest Series (OSA, 2015), paper FT2B.3.

C. B. Braunecker, R. Hentschel, and H. J. Tiziani, Advanced Optics Using Aspherical Elements (SPIE, 2008), Ch. 2.

J. E. Gentle, Numerical Linear Algebra in Statistics (Springer, 1998).

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK User’s Guide, 3rd ed. (SIAM, 1999).

U. W. Hochstrasser, “Orthogonal Polynomials,” in Handbook of Mathematical Functions, M. Abramowitz and I. A. Stegun ed. (Dover, 1978).

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Figures (4)

Fig. 1
Fig. 1 Selected coefficients of order m = 3 (left) and m = 4 (right) resulting from a best-fit using matrix inversion for different number of polynomials M.
Fig. 2
Fig. 2 Mean value of the individual polynomial terms of order m > 0 for the base set Qcon and its orthogonalized counterparts. Right hand side shows a comparison of the orthogonal terms only.
Fig. 3
Fig. 3 Difference between coefficients from simplified solution bm to coefficients from matrix inversion am. Right hand side focuses on the results from the orthogonal terms.
Fig. 4
Fig. 4 Reciprocal condition number of Gram matrix

Equations (36)

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z(ρ)= C ρ 2 1+ 1 C 2 ρ 2 (1+κ) + m=0 M a m ρ 2m+4
S= l=1 L b l P l .
E 2 ( b 1 , b 2 ,..., b L )= n=1 N [ S n l=1 L b l P l ( x n , y n ) ] 2 =minimum .
E 2 b l =0.
E 2 b l =2 n=1 N [ S n m=1 M b m P m ( x n , y n ) ] ( P l ( x n , y n ) )=0
E 2 b l = m=1 M b m n=1 N P m ( x n , y n ) P l ( x n , y n ) n=1 N S n P l ( x n , y n ) =0,
m=1 M b m [ n=1 N P m ( x n , y n ) P l ( x n , y n ) ] = n=1 N S n P l ( x n , y n ) .
c l = n=1 N S n P l ( x n , y n ) .
G m,l = n=1 N P m ( x n , y n ) P l ( x n , y n ) .
m=1 M b m G m,l = c l ,
S=bP.
P T Pb= P T S
b= ( P T P ) 1 P T S.
κ( G )= max| λ | min| λ | with λσ( G ),
G m,l = n=1 N J m ( x n , y n ) J l ( x n , y n ) = h m δ m,l ,
h m = n=1 N [ J m ( x n , y n ) ] 2
b m G m,m = c m
b m = c m G m,m = c m h m = n=1 N S n J m ( x n , y n ) n=1 N [ J m ( x n , y n ) ] 2 .
b m = c m = n=1 N S n H m ( x n , y n ) .
z=f(ρ, ρ max )= C ρ 2 1+ 1 C 2 ρ 2 (1+κ) + u 4 m=0 M a m con Q m con ( u 2 ),
E 2 ( a 0 con , a 1 con ,..., a M con )= [ g( u ρ max ) m=1 M a m con Q m con ( u 2 ) ] 2 =minimum,
p( u ) = 0 1 p( u )w( u 2 ) udu 0 1 w( u 2 ) udu ,
G m,l = u 8 Q m con ( u 2 ) Q l con ( u 2 ) = 0 1 x 4 Q m con ( x ) Q l con ( x )dx ,
0 1 Q m con (x) Q l con ( x ) x 4 dx = h m δ m,l ,
0 1 w(x) [ Q m con ( x ) ] 2 dx= h m .
Q m con (x)= P m ( 0,4 ) ( 2x1 )
a 1n f n+1 ( x )=( a 2n + a 3n x ) f n ( x ) a 4n f n1 ( x ).
f n+1 ( x )=( a n + b n x ) f n ( x ) c n f n1 ( x ),
a n = (2n+5)( n 2 +5n+10) (n+1)(n+2)(n+5) , b n = 2( n+3 )( 2n+5 ) ( n+1 )( n+5 ) , c n = n( n+4 )( n+3 ) ( n+1 )( n+2 )( n+5 ) .
J 1 = P 1
H m = J m J m w = J m J m , J m w = J m ( n=1 N J m 2 ( x n , y n )w( x n , y n ) ) 1 2
J m = P m k=1 m1 D m,k H k
D m,k = P m , H k = n=1 N w( x n , y n ) P m ( x n , y n ) H k ( x n , y n )
D m,k = P m , H k = n=1 N w( x n , y n ) P m ( x n , y n ) H k ( x n , y n )
C m,l = k=1 K=lm D l,lk C lk,m
a m ={4.36653e-7, -2.21714e-10, -1.70412e-13, -3.68093e-17, 8.94435e-21, 1.85012e-23, -6.27043e-27}

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