Abstract

With the recent emergence of slow-servo diamond turning, optical designs with surfaces that are not intrinsically rotationally symmetric can be manufactured. In this paper, we demonstrate some important limitations to Zernike polynomial representation of optical surfaces in describing the evolving freeform surface descriptions that are effective for optical design and encountered during optical fabrication. Specifically, we show that the ray grids commonly used in sampling a freeform surface to form a database from which to perform a φ-polynomial fit is limiting the efficacy of computation. We show an edge-clustered fitting grid that effectively suppresses the edge ringing that arises as the polynomial adapts to the fully nonsymmetric features of the surface.

© 2011 OSA

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References

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  1. E. Abbe, Lens system. U.S. Patent No. 697,959, (April, 1902).
  2. G. W. Forbes, “Shape specification for axially symmetric optical surfaces,” Opt. Express 15(8), 5218–5226 (2007).
    [CrossRef] [PubMed]
  3. B. Ma, L. Li, K. P. Thompson, and J. P. Rolland, “Applying slope constrained Q-type aspheres to develop higher performance lenses,” Opt. Express 19(22), 21174–21179 (2011).
    [CrossRef] [PubMed]
  4. Y. Tohme and R. Murray, “Principles and Applications of the Slow Slide Servo,” Moore Nanotechnology Systems White Paper (2005).
  5. F. Zernike, “Beugungstheorie des schneidenver-fahrens und seiner verbesserten form, der phasenkontrastmethode,” Physica 1(7-12), 689–704 (1934).
    [CrossRef]
  6. J. P. Rolland, C. Dunn, and K. P. Thompson, “An Analytic Expression for the Field Dependence of FRINGE Zernike Polynomial Coefficients in Rotationally Symmetric Optical Systems,” Proc. SPIE 7790, (2010).
  7. K. Fuerschbach, J. P. Rolland, and K. P. Thompson, “A new family of optical systems employing φ-polynomial surfaces,” Opt. Express 19(22), 21919–21928 (2011).
    [CrossRef] [PubMed]
  8. O. Cakmakci, B. Moore, H. Foroosh, and J. P. Rolland, “Optimal local shape description for rotationally non-symmetric optical surface design and analysis,” Opt. Express 16(3), 1583–1589 (2008).
    [CrossRef] [PubMed]
  9. G. W. Forbes, “Robust and fast computation for the polynomials of optics,” Opt. Express 18(13), 13851–13862 (2010).
    [CrossRef] [PubMed]
  10. M. Born and E. Wolf, Principles of Optics, (Cambridge, 1999).
  11. A. B. Bhatia, E. Wolf, and M. Born, “On the circle polynomials of Zernike and related orthogonal sets,” Proc. Camb. Philos. Soc. 50(1), 40–48 (1954).
    [CrossRef]
  12. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, chap. 22, (Dover, 1972).
  13. G. E. Fasshauer, Meshfree Approximation Methods with MATLAB (World Scientific Publishing, Singapore, 2007).
  14. B. Fornberg, E. Larsson, and N. Flyer, “Stable computations with Gaussian radial basis functions,” SIAM J. Sci. Comput. 33(2), 869–892 (2011).
    [CrossRef]
  15. R. Platte, Accuracy and Stability of Global Radial Basis Function Methods for the Numerical Solution of Partial Differential Equations, Ph.D. Thesis, (University of Delaware, 2005).

2011 (3)

2010 (2)

J. P. Rolland, C. Dunn, and K. P. Thompson, “An Analytic Expression for the Field Dependence of FRINGE Zernike Polynomial Coefficients in Rotationally Symmetric Optical Systems,” Proc. SPIE 7790, (2010).

G. W. Forbes, “Robust and fast computation for the polynomials of optics,” Opt. Express 18(13), 13851–13862 (2010).
[CrossRef] [PubMed]

2008 (1)

2007 (1)

1954 (1)

A. B. Bhatia, E. Wolf, and M. Born, “On the circle polynomials of Zernike and related orthogonal sets,” Proc. Camb. Philos. Soc. 50(1), 40–48 (1954).
[CrossRef]

1934 (1)

F. Zernike, “Beugungstheorie des schneidenver-fahrens und seiner verbesserten form, der phasenkontrastmethode,” Physica 1(7-12), 689–704 (1934).
[CrossRef]

Bhatia, A. B.

A. B. Bhatia, E. Wolf, and M. Born, “On the circle polynomials of Zernike and related orthogonal sets,” Proc. Camb. Philos. Soc. 50(1), 40–48 (1954).
[CrossRef]

Born, M.

A. B. Bhatia, E. Wolf, and M. Born, “On the circle polynomials of Zernike and related orthogonal sets,” Proc. Camb. Philos. Soc. 50(1), 40–48 (1954).
[CrossRef]

Cakmakci, O.

Dunn, C.

J. P. Rolland, C. Dunn, and K. P. Thompson, “An Analytic Expression for the Field Dependence of FRINGE Zernike Polynomial Coefficients in Rotationally Symmetric Optical Systems,” Proc. SPIE 7790, (2010).

Flyer, N.

B. Fornberg, E. Larsson, and N. Flyer, “Stable computations with Gaussian radial basis functions,” SIAM J. Sci. Comput. 33(2), 869–892 (2011).
[CrossRef]

Forbes, G. W.

Fornberg, B.

B. Fornberg, E. Larsson, and N. Flyer, “Stable computations with Gaussian radial basis functions,” SIAM J. Sci. Comput. 33(2), 869–892 (2011).
[CrossRef]

Foroosh, H.

Fuerschbach, K.

Larsson, E.

B. Fornberg, E. Larsson, and N. Flyer, “Stable computations with Gaussian radial basis functions,” SIAM J. Sci. Comput. 33(2), 869–892 (2011).
[CrossRef]

Li, L.

Ma, B.

Moore, B.

Rolland, J. P.

Thompson, K. P.

Wolf, E.

A. B. Bhatia, E. Wolf, and M. Born, “On the circle polynomials of Zernike and related orthogonal sets,” Proc. Camb. Philos. Soc. 50(1), 40–48 (1954).
[CrossRef]

Zernike, F.

F. Zernike, “Beugungstheorie des schneidenver-fahrens und seiner verbesserten form, der phasenkontrastmethode,” Physica 1(7-12), 689–704 (1934).
[CrossRef]

Opt. Express (5)

Physica (1)

F. Zernike, “Beugungstheorie des schneidenver-fahrens und seiner verbesserten form, der phasenkontrastmethode,” Physica 1(7-12), 689–704 (1934).
[CrossRef]

Proc. Camb. Philos. Soc. (1)

A. B. Bhatia, E. Wolf, and M. Born, “On the circle polynomials of Zernike and related orthogonal sets,” Proc. Camb. Philos. Soc. 50(1), 40–48 (1954).
[CrossRef]

Proc. SPIE (1)

J. P. Rolland, C. Dunn, and K. P. Thompson, “An Analytic Expression for the Field Dependence of FRINGE Zernike Polynomial Coefficients in Rotationally Symmetric Optical Systems,” Proc. SPIE 7790, (2010).

SIAM J. Sci. Comput. (1)

B. Fornberg, E. Larsson, and N. Flyer, “Stable computations with Gaussian radial basis functions,” SIAM J. Sci. Comput. 33(2), 869–892 (2011).
[CrossRef]

Other (6)

R. Platte, Accuracy and Stability of Global Radial Basis Function Methods for the Numerical Solution of Partial Differential Equations, Ph.D. Thesis, (University of Delaware, 2005).

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, chap. 22, (Dover, 1972).

G. E. Fasshauer, Meshfree Approximation Methods with MATLAB (World Scientific Publishing, Singapore, 2007).

E. Abbe, Lens system. U.S. Patent No. 697,959, (April, 1902).

M. Born and E. Wolf, Principles of Optics, (Cambridge, 1999).

Y. Tohme and R. Murray, “Principles and Applications of the Slow Slide Servo,” Moore Nanotechnology Systems White Paper (2005).

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Figures (9)

Fig. 1
Fig. 1

Illustration of a F/1 Parabola with a Gaussian bump height of 200 µm with a shape parameter of 800 µm−1 within a normalized unit circle.

Fig. 2
Fig. 2

Computation of high-order Born and Wolf Zernike term, Z 50 4 : (a) with Eq. (2). (b) with Eq. (5) and (6).

Fig. 3
Fig. 3

Zernike coefficients resulting from the least squares fitting of a conventional asphere with 136 Zernike polynomials using the Born and Wolf ordering of terms.

Fig. 4
Fig. 4

The test surfaces described in Eq. (9). (a) A five-term conventional aspheric mirror; (b) A F/1 parabola with 600, 50, and 30 µm bumps; (c) Franke surface. Note that in this illustration the apertures were normalized to 1 in radius.

Fig. 5
Fig. 5

Fitting grids used to demonstrate efficacy for data sampling: (a) hex grid, (b) Cheby-polar grid (c) uni-random grid (Halton points) (d) e_clust-random grid that clusters points towards the boundary over the unit circle.

Fig. 6
Fig. 6

(a) Rotationally symmetric analytic five-term asphere departure from best sphere based on minimum RMS; (b) rotationally symmetric analytic five-term asphere; (c) RMS error in Zernike polynomials approximant performance relative to the analytic function expressed in meters for an increasing number of Zernike coefficients with hex, uni-random, Cheby-polar, and e-clust-random sampling grids for the asphere shown in (b).

Fig. 7
Fig. 7

(b) RMS error in Zernike polynomials approximant performance relative to the analytic function expressed in meters for an increasing number of Zernike coefficients with hex, uni-random, Cheby-polar, and e-clust-random sampling grids for the F/1 parabola with 3 bumps, shown in (a).

Fig. 8
Fig. 8

Comparison of the approximants obtained with two different fitting grids for the F/1 parabola with bumps; Top row: Approximant with uni-hex grid sampling with (a) 25 samples, (b) 204 samples, (c) 1990 samples, (d) 4980 samples; Bottom row: Approximant with e_clust-random sampling with (e) 25 samples, (f) 204 samples, (g) 1990 samples, and (h) 4980 samples.

Fig. 9
Fig. 9

(b) Zernike polynomial fit RMS error expressed in meters as a function of coefficient order with hex, uni-random, Cheby-polar, and e_clust-random fitting grids for the freeform Franke surface (Eq. (9)c) shown in (a).

Tables (1)

Tables Icon

Table 1 Zernike Polynomials with significant coefficients for a rotationally symmetric conventional asphere

Equations (12)

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Z n m (ρ,φ)= R n ±m (ρ){ cosmφ sinmφ },
R n ±m (ρ)= q=0 nm 2 ( 1 ) q ( nq )! q!( n+m 2 q )!( nm 2 +q )! ρ n2q .
R nw m (ρ)= ρ m Z nf m ( ρ 2 ),
Z nf m ( ρ 2 )= P nf (0,m) (2 ρ 2 1),
P n+1 (ρ)=( a n + b n ρ) P n (ρ) c n P n1 (ρ).
a n = (s+1)[ ( sn ) 2 + n 2 +s ] ( n+1 )( sn+1 )s , b n = (s+2)(s+1) ( n+1 )( sn+1 )s , c n = (s+2)(sn)n ( n+1 )( sn+1 )s ,
f(ρ,φ)= k=1 N c k Z n m (ρ,φ),
Ac=f.
f aspherics (ρ)= ( ρ 2 )/200 1+ 1+1/ (200) 2 ( ρ 2 ) +1.876× 10 -5 ρ 2 0.5× 10 -7 ρ 4 +0.545× 10 -10 ρ 6 0.25× 10 -13 ρ 8 +0.4× 10 -17 ρ 10 ,
f threebumps (x,y)= ( ρ 2 ) 80 +0.05 e 0.25[ (x7) 2 + (y+6) 2 ] +0.6 e 0.49[ (x+3) 2 + (y2) 2 ] +0.03 e 0.81[ (x5) 2 + (y7) 2 ] ,
f Franke (x,y)=0.75 e 0.25[ (9x2) 2 + (9y2) 2 ] +0.75 e [ (9x+1) 2 /49+ (9y+1) 2 /10] +0.5 e 0.25[ (9x7) 2 + (9y3) 2 ] 0.2 e [ (9x4) 2 + (9y7) 2 ] .
ρ k =cos( (2k+1)π 2N ),k=0,1,...,N1.

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