Abstract

We analyze an alternative to classical Zernike fitting based on the cubic B-spline model, and compare the strengths and weaknesses of each representation over a set of different wavefronts that cover a wide range of shape complexity. The results obtained show that a Zernike low-degree polynomial expansion or a cubic B-spline with a low number of breakpoints are the best choices for fitting simple wavefronts, whereas the cubic B-spline approach performs much better when more complex wavefronts are involved. The effect of noise level in the fit quality for the different wavefronts is also studied.

© 2006 Optical Society of America

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References

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  1. R. R. Rammage, D. R. Neal, and R. J. Copland, "Application of Shack-Hartmann wavefront sensing technology to transmissive optic metrology," in Advanced Characterization Techniques for Optical, Semiconductor, and Data Storage Components, A. Duparré and B. Singh, eds., Proc. SPIE 4779, 161-172 (2002).
    [CrossRef]
  2. W. H. Southwell, "Wave-front estimation from wave-front slope measurements," J. Opt. Soc. Am. 70, 998-1006 (1980).
  3. K. W. Farmer, "Scattered data interpolation by C quintic splines using energy minimization," M. A. thesis (University of Georgia, 1997).
  4. G. Vdovin and P. M. Sarro, "Flexible mirror micromachined in silicon," Appl. Opt. 34, 2968-2972 (1995).
  5. P. M. Prieto, E. J. Fernández, S. Manzanera, and P. Artal, "Adaptive optics with a programmable phase modulator: applications in the human eye," Opt. Express 12, 4059-4071 (2004).
    [CrossRef]
  6. L. Seifert, J. Liesener, and H. J. Tiziani, "The adaptive Shack-Hartmann sensor," Opt. Commun. 216, 313-319 (2003).
    [CrossRef]
  7. X. Liu and Y. Gao, "B-spline based wavefront reconstruction for lateral shearing interferometric measurement of engineering surfaces," Advances in Abrasive Technology V (Trans Tech Publications, 2003), Vol. 238-239, pp. 169-174.
  8. L. Seifert, H. J. Tiziani, and W. Osten, "Wavefront reconstruction with the adaptive Shack-Hartmann sensor," Opt. Commun. 245, 255-269 (2005).
    [CrossRef]
  9. D. Malacara and S. L. DeVore, "Interferogram evaluation and wavefront fitting," in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, 1992).
  10. C. de Boor, A Practical Guide to Splines, revised edition (Springer-Verlag, 2001).
  11. J. Arasa, S. Royo, and N. Tomás, "Simple method for improving the sampling in profile measurements by use of the Ronchi test," Appl. Opt. 39, 4529-4534 (2000).
  12. S. Royo, J. Arasa, and C. Pizarro, "Profilometry of toroidal surfaces with an improved Ronchi test," Appl. Opt. 39, 5721-5731 (2000).
  13. C. L. Lawson and R. J. Hanson, Solving Least Square Problems (Prentice Hall, 1974).
  14. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, 1997).

2005 (1)

L. Seifert, H. J. Tiziani, and W. Osten, "Wavefront reconstruction with the adaptive Shack-Hartmann sensor," Opt. Commun. 245, 255-269 (2005).
[CrossRef]

2004 (1)

2003 (1)

L. Seifert, J. Liesener, and H. J. Tiziani, "The adaptive Shack-Hartmann sensor," Opt. Commun. 216, 313-319 (2003).
[CrossRef]

2002 (1)

R. R. Rammage, D. R. Neal, and R. J. Copland, "Application of Shack-Hartmann wavefront sensing technology to transmissive optic metrology," in Advanced Characterization Techniques for Optical, Semiconductor, and Data Storage Components, A. Duparré and B. Singh, eds., Proc. SPIE 4779, 161-172 (2002).
[CrossRef]

2000 (2)

1995 (1)

1980 (1)

Arasa, J.

Artal, P.

Copland, R. J.

R. R. Rammage, D. R. Neal, and R. J. Copland, "Application of Shack-Hartmann wavefront sensing technology to transmissive optic metrology," in Advanced Characterization Techniques for Optical, Semiconductor, and Data Storage Components, A. Duparré and B. Singh, eds., Proc. SPIE 4779, 161-172 (2002).
[CrossRef]

de Boor, C.

C. de Boor, A Practical Guide to Splines, revised edition (Springer-Verlag, 2001).

DeVore, S. L.

D. Malacara and S. L. DeVore, "Interferogram evaluation and wavefront fitting," in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, 1992).

Farmer, K. W.

K. W. Farmer, "Scattered data interpolation by C quintic splines using energy minimization," M. A. thesis (University of Georgia, 1997).

Fernández, E. J.

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, 1997).

Gao, Y.

X. Liu and Y. Gao, "B-spline based wavefront reconstruction for lateral shearing interferometric measurement of engineering surfaces," Advances in Abrasive Technology V (Trans Tech Publications, 2003), Vol. 238-239, pp. 169-174.

Hanson, R. J.

C. L. Lawson and R. J. Hanson, Solving Least Square Problems (Prentice Hall, 1974).

Lawson, C. L.

C. L. Lawson and R. J. Hanson, Solving Least Square Problems (Prentice Hall, 1974).

Liesener, J.

L. Seifert, J. Liesener, and H. J. Tiziani, "The adaptive Shack-Hartmann sensor," Opt. Commun. 216, 313-319 (2003).
[CrossRef]

Liu, X.

X. Liu and Y. Gao, "B-spline based wavefront reconstruction for lateral shearing interferometric measurement of engineering surfaces," Advances in Abrasive Technology V (Trans Tech Publications, 2003), Vol. 238-239, pp. 169-174.

Malacara, D.

D. Malacara and S. L. DeVore, "Interferogram evaluation and wavefront fitting," in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, 1992).

Manzanera, S.

Neal, D. R.

R. R. Rammage, D. R. Neal, and R. J. Copland, "Application of Shack-Hartmann wavefront sensing technology to transmissive optic metrology," in Advanced Characterization Techniques for Optical, Semiconductor, and Data Storage Components, A. Duparré and B. Singh, eds., Proc. SPIE 4779, 161-172 (2002).
[CrossRef]

Osten, W.

L. Seifert, H. J. Tiziani, and W. Osten, "Wavefront reconstruction with the adaptive Shack-Hartmann sensor," Opt. Commun. 245, 255-269 (2005).
[CrossRef]

Pizarro, C.

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, 1997).

Prieto, P. M.

Rammage, R. R.

R. R. Rammage, D. R. Neal, and R. J. Copland, "Application of Shack-Hartmann wavefront sensing technology to transmissive optic metrology," in Advanced Characterization Techniques for Optical, Semiconductor, and Data Storage Components, A. Duparré and B. Singh, eds., Proc. SPIE 4779, 161-172 (2002).
[CrossRef]

Royo, S.

Sarro, P. M.

Seifert, L.

L. Seifert, H. J. Tiziani, and W. Osten, "Wavefront reconstruction with the adaptive Shack-Hartmann sensor," Opt. Commun. 245, 255-269 (2005).
[CrossRef]

L. Seifert, J. Liesener, and H. J. Tiziani, "The adaptive Shack-Hartmann sensor," Opt. Commun. 216, 313-319 (2003).
[CrossRef]

Southwell, W. H.

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, 1997).

Tiziani, H. J.

L. Seifert, H. J. Tiziani, and W. Osten, "Wavefront reconstruction with the adaptive Shack-Hartmann sensor," Opt. Commun. 245, 255-269 (2005).
[CrossRef]

L. Seifert, J. Liesener, and H. J. Tiziani, "The adaptive Shack-Hartmann sensor," Opt. Commun. 216, 313-319 (2003).
[CrossRef]

Tomás, N.

Vdovin, G.

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, 1997).

Appl. Opt. (3)

J. Opt. Soc. Am. (1)

Opt. Commun. (2)

L. Seifert, J. Liesener, and H. J. Tiziani, "The adaptive Shack-Hartmann sensor," Opt. Commun. 216, 313-319 (2003).
[CrossRef]

L. Seifert, H. J. Tiziani, and W. Osten, "Wavefront reconstruction with the adaptive Shack-Hartmann sensor," Opt. Commun. 245, 255-269 (2005).
[CrossRef]

Opt. Express (1)

Proc. SPIE (1)

R. R. Rammage, D. R. Neal, and R. J. Copland, "Application of Shack-Hartmann wavefront sensing technology to transmissive optic metrology," in Advanced Characterization Techniques for Optical, Semiconductor, and Data Storage Components, A. Duparré and B. Singh, eds., Proc. SPIE 4779, 161-172 (2002).
[CrossRef]

Other (6)

K. W. Farmer, "Scattered data interpolation by C quintic splines using energy minimization," M. A. thesis (University of Georgia, 1997).

X. Liu and Y. Gao, "B-spline based wavefront reconstruction for lateral shearing interferometric measurement of engineering surfaces," Advances in Abrasive Technology V (Trans Tech Publications, 2003), Vol. 238-239, pp. 169-174.

D. Malacara and S. L. DeVore, "Interferogram evaluation and wavefront fitting," in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, 1992).

C. de Boor, A Practical Guide to Splines, revised edition (Springer-Verlag, 2001).

C. L. Lawson and R. J. Hanson, Solving Least Square Problems (Prentice Hall, 1974).

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, 1997).

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Figures (13)

Fig. 1
Fig. 1

Basic functions of the cubic B-spline representation along the fictitious knot domain.

Fig. 2
Fig. 2

(Color online) Zernike ill conditioning at high polynomial degrees (solid curve) for the ideal spherical wavefront, which is not present in any cubic B-spline scheme (dashed curve): (a) From the 13th degree, the Zernike matrix becomes singular, (b) which degrades the least-squares fitting solution obtained.

Fig. 3
Fig. 3

(Color online) Number of computational operations involved in the SVD matrix inversion method for the different Zernike degrees (solid curve) and cubic B-spline NBP (dashed curve). An operation is a multiplication or a division plus an addition.

Fig. 4
Fig. 4

(Color online) rms wavefront deviation (bottom, circles) and rms fit deviation (top, triangles) depending on the Zernike fit degree (filled symbols) and on the cubic B-spline NBP (nonfilled symbols), for the simulated noisy spherical wavefront. The upper horizontal reference line reflects the calculated rms deviation between the ideal and noisy wavefronts. (a) Noise amplitude between ± 0.1 % wavefront PV, (b) ± 0.5 % PV, and (c) ± 1 % PV .

Fig. 5
Fig. 5

(Color online) rms fit deviation for a spherical wavefront measured with a digital Ronchi test setup, under the different Zernike (solid curve) and cubic B-spline conditions (dashed curve).

Fig. 6
Fig. 6

(Color online) Zernike (solid curve) and cubic B-spline (dashed curve) rms wavefront deviation for the ideal toroidal wavefront rotated 30°.

Fig. 7
Fig. 7

(Color online) rms wavefront deviation (bottom, circles) and rms fit deviation (top, triangles) depending on the Zernike fit degree (filled symbols) and on the cubic B-spline NBP (nonfilled symbols), for the simulated noisy toroidal wavefront. The upper horizontal reference line reflects the calculated rms deviation between the ideal and noisy wavefronts. (a) Noise amplitude between ± 0.1 % wavefront PV, (b) ± 0.5 % PV, and (c) ± 1 % PV.

Fig. 8
Fig. 8

(Color online) rms fit deviation of the Zernike (solid curve) and cubic B-spline (dashed curve) representations for the experimental rotated toroidal wavefront.

Fig. 9
Fig. 9

Highly complex wavefront selected.

Fig. 10
Fig. 10

(Color online) rms wavefront deviation for the ideal highly complex wavefront under different Zernike degree (solid curve) and cubic B-spline NBP (dashed curve) cases.

Fig. 11
Fig. 11

(Color online) Comparison of Zernike (top) and cubic B-spline (bottom) rms wavefront deviation (circles) and rms fit deviation (triangles) for the highly complex wavefront with random noise added between ± 0.1 % wavefront PV. The horizontal reference line reflects the calculated rms deviation between the ideal and noisy wavefronts. (a) Low-degree–NBP schemes do not reproduce the complex shape at all, and (b) the appropriate working zone, in which the cubic B-spline performs better than the Zernike ( NBP = 16 × 16 is the optimum fit).

Fig. 12
Fig. 12

(Color online) Comparison of Zernike (top) and cubic B-spline (bottom) rms wavefront deviation (circles) and rms fit deviation (triangles) for the highly complex wavefront with random noise added between ± 0 .5% wavefront PV. The horizontal reference line reflects the calculated rms deviation between the ideal and noisy wavefronts. (a) Low-degree–NBP fits do not follow the wavefront shape and (b) the appropriate working zone, which gives the NBP = 11 × 11 cubic B-spline as the best fit.

Fig. 13
Fig. 13

(Color online) Comparison of Zernike (top) and cubic B-spline (bottom) rms wavefront deviation (circles) and rms fit deviation (triangles) for the highly complex wavefront with random noise added between ± 1 % wavefront PV. The horizontal reference line reflects the calculated rms deviation between the ideal and noisy wavefronts. (a) Low-degree–NBP fits do not follow the wavefront shape and (b) the appropriate working zone, in which NBP = 11 × 11 is the optimum fitting scheme.

Tables (1)

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Table 1 Spherical Wavefront Fitting Results a

Equations (19)

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W ( ρ , ϑ ) = n = 0 k m = 0 n a n m Z n l ( ρ , ϑ ) = i = 1 L a i Z i ( ρ , ϑ ) ,
l = n 2 m ,
Z n l ( ρ , ϑ ) = { R n l ( ρ ) cos ( l ϑ ) for   l 0, R n l ( ρ ) sin ( l ϑ ) for   l > 0 ,
R n     l ( ρ ) = R n     n 2 m ( ρ ) = s = 0 m ( 1 ) s ( n s ) ! s ! ( m s ) ! ( n m s ) ! ρ n 2 s ,
L = ( k + 1 ) ( k + 2 ) 2 .
Z a = W ,
[ Z 1 ( ρ 1 , ϑ 1 ) Z 2 ( ρ 1 , ϑ 1 ) Z L ( ρ 1 , ϑ 1 ) Z 1 ( ρ 2 , ϑ 2 ) Z 2 ( ρ 2 , ϑ 2 ) Z L ( ρ 2 , ϑ 2 ) Z 1 ( ρ N , ϑ N ) Z 2 ( ρ N , ϑ N ) Z L ( ρ N , ϑ N ) ] [ a 1 a 2 a L ] = [ W ( ρ 1 , ϑ 1 ) W ( ρ 2 , ϑ 2 ) W ( ρ N , ϑ N ) ] ,
W ( x , y ) = i = 0 n j = 0 m a i j B i , k ( x ) B j , l ( y ) ,
knot   multiplicity s = k μ s .
n + 1 = ( s = 1 NBP x - 2 k μ s ) + 2 k k .
B i , k ( x ) = x t i t i + k 1 t i B i , k 1 ( x ) + t i + k x t i + k t i + 1 B i + 1 , k 1 ( x ) ,
B i ,1 ( x ) = { 1, t i x t i + 1 , 0, otherwise .
t x r = x r t i t i + 1 t i ,   x r ( t i , t i + 1 ) .
B i , 4 ( t x ) = B i , 1 ( t x ) B i + 1 , 1 ( t x ) B i + 2 , 1 ( t x ) B i + 3 , 1 ( t x ) ,
      B i , 1 ( t x ) = t x 3 6 , B i + 1 , 1 ( t x ) = 1 6 ( 1 + 3 t x + 3 t x 2 3 t x 3 ) , B i + 2 , 1 ( t x ) = 1 6 ( 4 6 t x 2 + 3 t x 3 ) , B i + 3 , 1 ( t x ) = 1 6 ( 1 t x ) 3 , t x [ 0 , 1 ] .
[ B 0 , 4 ( t x 1 ) B 0 , 4 ( t y 1 ) B 0 , 4 ( t x 1 ) B 1 , 4 ( t y 1 ) B n , 4 ( t x 1 ) B m , 4 ( t y 1 ) B 0 , 4 ( t x 2 ) B 0 , 4 ( t y 2 ) B 0 , 4 ( t x 2 ) B 1 , 4 ( t y 2 ) B n , 4 ( t x 2 ) B m , 4 ( t y 2 ) B 0 , 4 ( t x N ) B 0 , 4 ( t y N ) B 0 , 4 ( t x N ) B 1 , 4 ( t y N ) B n , 4 ( t x N ) B m , 4 ( t y N ) ] [ a 00 a 01 a n m ] = [ W ( t x 1 , t y 1 ) W ( t x 2 , t y 2 ) W ( t x N , t y N ) ] .
x x 0 = x r   cos   ϑ + y r   sin   ϑ ,
y y 0 = x r   sin   ϑ + y r   cos   ϑ ,
z = ( x x 0 ) 2 / R 1 + ( y y 0 ) 2 / R 2 1 + { 1 [ ( x x 0 ) 2 / R 1 + ( y y 0 ) 2 / R 2 ] 2 ( x x 0 ) 2 + ( y y 0 ) 2 } 1 / 2 .

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