## Edge clustered fitting grids for φ-polynomial characterization of freeform optical surfaces |

Optics Express, Vol. 19, Issue 27, pp. 26962-26974 (2011)

http://dx.doi.org/10.1364/OE.19.026962

Acrobat PDF (2955 KB)

### 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

## 1. Introduction

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

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]

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]

## 2. The three-term recurrence relation for Zernike polynomials

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

*n>m*and the difference

*n-m*is even. The radial polynomial is expressed as

*(ρ,φ)*shown in Fig. 2b is free of these debilitating computational artifacts.

## 3. Least squares data fitting to create a Zernike polynomial surface

*c*represents the coefficients associated with each Zernike polynomial. If the function is known beforehand, the coefficients,

_{k}*c*could be determined by taking a double integral of the function multiplied with the corresponding Zernike polynomial over the circle. In optical system design, these surfaces are to be determined from optimization of the user selected variable coefficients, based on wavefronts that are initiated at different field points, by a real ray trace of a grid (typically uniform, rectangular) of rays in the pupil at one wavelength through a model of a complete optical system. In this scenario, the least squares approach given in Eq. (8) is used to find the coefficients,

_{k}*c*, associated with each Zernike polynomial at each wavelength and field point. Here, the goal is to represent the sag of a surface to be used as an exact representation of a freeform surface, to the extent possible.

_{k}**A**is constructed as follows; the matrix

**A**is

*M by N*, where

*N*is the number of Zernike polynomial coefficients to be fit, and

*M*is the number of sample points throughout the aperture. Each row corresponds to a sample point, where each Zernike polynomial is computed. Each column corresponds to a Zernike polynomial coefficient evaluated over all the points in the circular aperture. The coefficients

*c*are the weights multiplying the columns of the matrix

_{k}**A**to match the surface sag vector,

**f**. Once the coefficients are determined, the approximant can be evaluated at any point across the aperture as would routinely occur in evaluating the impact of the surface in an overall optical system design. In Fig. 3 , as an example, we have shown the 136 Zernike polynomials coefficients that are computed as the result of least squares fitting of a conventional rotationally symmetric asphere, which is shown in Fig. 4a and whose description is given in Eq. (9)a. As can be seen in Fig. 3, most of the coefficients are almost zero up to the working double precision limit, as expected. The non-zero valued coefficients are aligned with increasing orders of the Zernike polynomial terms that are affiliated with spherical aberration. Table 1 provides a list of the nonzero Zernike polynomial coefficients of Fig. 3.

## 4. The test surfaces

## 5. Hexagonal Grid, Chebyshev-Polar Grid, Uniform-Random Grid, and Edge-Clustered Grid fitting

*N*in the

*ρ*direction. The points along the

*ρ*direction are called Chebyshev abscissas. Therefore we call these circles Chebyshev circles. The radii of the circles are given as

*(ρ,φ)*then the corresponding clustered point has

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]

## 6. Results of efficacy of fitting the test surfaces with four different fitting grids

^{−9}on the vertical scale to correspond to nanometers. For each approximation, the size of the approximation matrix is

*M*by

*N*, where

*N*is the number of Zernike polynomials coefficients, and

*M*is the number of samples over the circular aperture, where

*M*≈1.5

*N.*We have chosen this ratio such that the computational cost is not too high in the approximation; meanwhile the number of samples is sufficient to avoid working within an interpolation setting (i.e.

*M*=

*N)*. In Chebyshev-based clustering methods, this ratio has been studied in 1D, and optimum results are obtained when this ratio is around 1.5 for different bases [15].

^{−5}m to 1m, which is huge compared to the sizes of the bumps on the F/1 parabola spanning 30 to 600 µm. This is caused by the dominant effect of the oscillations on the circular boundary when there is no edge weighting. On the other hand, applying either of the edge clustered sampling grids, Cheby-polar or e_clust-random, produces

*exponentially decaying errors*as the number of sampling points increases. However, it is only with the e_clust-random fitting grid that we can obtain an approximant described with machine precision accuracy, as illustrated in Fig. 7b. The Cheby-polar grid produces a similar RMS error trend as that of the e_clust-random grid until the number of point samples reaches around 3000, after which it yields less accuracy. Only the edge clustered sampling grids achieved sub-nanometer accuracies, while the unclustered fitting grids could not even describe the surface with micron accuracies. Hex grid and uni-random grid point sampling produced consistently poor surface approximants when fitting Zernike polynomials to intrinsically nonsymmetric surfaces.

^{−5}m for the F/1 parabola with bumps is best captured in Fig. 8 ., where we show the resulting approximants with the hex grid fitting (top row) and e_clust-random fitting (bottom row) grids while increasing the number of sample points and in proportion the number of Zernike polynomial coefficients for the F/1 parabola with bumps. Results shows that the number of sample points used in the evaluation of the approximation, in this case, is so small that the bumps on the F/1 parabola are under-sampled. However, as the number of Zernike polynomial coefficients increases to better fit the surface (and in conjunction the number of sampling points used in the evaluation), the least squares process tries to match the sag values at more points with a higher number of Zernike polynomial terms, causing severe oscillations at the edges when distributions that are not edge clustered are used as seen in the upper row of Fig. 8. As the lower row of displays in Fig. 8 shows, the e_clust-random fitting grid successfully describes the surface without significant edge ringing, therefore producing much better approximants than a sampling without edge clustering.

^{−10}m) demanded by lithography applications. Cheby-polar fitting grids produced similar results to e_clust-random fitting grids until around 2800 points. After this point, Cheby-polar fitting grids produce approximants with growing RMS errors. The stable and exponentially decaying errors produced by the edge clustered sampling make this method the method of choice for fitting Zernike polynomials to freeform optics surfaces and is the key result of this paper.

## 7. Conclusion

## Acknowledgements

## References and links

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 |

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 |

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 |

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 |

7. | K. Fuerschbach, J. P. Rolland, and K. P. Thompson, “A new family of optical systems employing φ-polynomial surfaces,” Opt. Express |

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 |

9. | G. W. Forbes, “Robust and fast computation for the polynomials of optics,” Opt. Express |

10. | M. Born and E. Wolf, |

11. | A. B. Bhatia, E. Wolf, and M. Born, “On the circle polynomials of Zernike and related orthogonal sets,” Proc. Camb. Philos. Soc. |

12. | M. Abramowitz and I. Stegun, |

13. | G. E. Fasshauer, |

14. | B. Fornberg, E. Larsson, and N. Flyer, “Stable computations with Gaussian radial basis functions,” SIAM J. Sci. Comput. |

15. | R. Platte, |

**OCIS Codes**

(000.3860) General : Mathematical methods in physics

(080.2740) Geometric optics : Geometric optical design

(220.0220) Optical design and fabrication : Optical design and fabrication

**ToC Category:**

Optical Design and Fabrication

**History**

Original Manuscript: October 17, 2011

Manuscript Accepted: November 28, 2011

Published: December 16, 2011

**Citation**

Ilhan Kaya, Kevin P. Thompson, and Jannick P. Rolland, "Edge clustered fitting grids for φ-polynomial characterization of freeform optical surfaces," Opt. Express **19**, 26962-26974 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-27-26962

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### References

- E. Abbe, Lens system. U.S. Patent No. 697,959, (April, 1902).
- G. W. Forbes, “Shape specification for axially symmetric optical surfaces,” Opt. Express 15(8), 5218–5226 (2007). [CrossRef] [PubMed]
- 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]
- Y. Tohme and R. Murray, “Principles and Applications of the Slow Slide Servo,” Moore Nanotechnology Systems White Paper (2005).
- F. Zernike, “Beugungstheorie des schneidenver-fahrens und seiner verbesserten form, der phasenkontrastmethode,” Physica 1(7-12), 689–704 (1934). [CrossRef]
- 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).
- 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]
- 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]
- G. W. Forbes, “Robust and fast computation for the polynomials of optics,” Opt. Express 18(13), 13851–13862 (2010). [CrossRef] [PubMed]
- M. Born and E. Wolf, Principles of Optics, (Cambridge, 1999).
- 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]
- 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).
- B. Fornberg, E. Larsson, and N. Flyer, “Stable computations with Gaussian radial basis functions,” SIAM J. Sci. Comput. 33(2), 869–892 (2011). [CrossRef]
- 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).

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