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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 27 — Dec. 19, 2011
  • pp: 26962–26974
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Edge clustered fitting grids for φ-polynomial characterization of freeform optical surfaces

Ilhan Kaya, Kevin P. Thompson, and Jannick P. Rolland  »View Author Affiliations


Optics Express, Vol. 19, Issue 27, pp. 26962-26974 (2011)
http://dx.doi.org/10.1364/OE.19.026962


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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

There is a revolution occurring in the field of optical design that is being driven by two concurrent, but unrelated major developments that have forced the community to retool the fundamental surface shapes that are and will be used in optical design. The first development resulted from discovering that the power series definition of rotationally symmetric aspheric surfaces, which was introduced by Abbe over 100 years ago [1

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

], is failing in part because of the lack of orthonormalization. This development has become apparent as new polishing methods, including small tool polishing, ion beam polishing, and MRF polishing come to be adopted throughout the industry resulting in the economic factors favoring, for the first time, the introduction of rotationally symmetric aspheres in many optical designs. The mathematical proposition of Q-type surfaces in place of power series to describe rotationally symmetric aspheres was made in the recent work of Forbes [2

2. G. W. Forbes, “Shape specification for axially symmetric optical surfaces,” Opt. Express 15(8), 5218–5226 (2007). [CrossRef] [PubMed]

]. The fabrication community that requires minimization of the maximum slope on any specified asphere drives these surface types. Of key importance in the development of these new surface descriptors is the fact that one of the preferred Q-type mathematical surface descriptions, Qbfs, yields normalizing coefficients that, when squared and summed, provide a direct computation of the RMS slope error. As a result of this computationally effective metric, it becomes possible, for the first time, to design optical systems with aspheres that comply with the fabricators guidelines on maximum testable slope. This effective slope constraint metric that arises directly from the Q-type surface descriptions has now been fully integrated in optical system design software with dramatic outcomes. As an example, Ma recently showed that the design of a 28 element lithographic lens with Q-type surfaces and an optimization integrated RMS slope constraint resulted in an order of magnitude decrease in overall sensitivity to tilts and decenters [3

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]

].

The second development is the introduction in the diamond turning community of the slow-servo axis [4

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

], which allows, again for the first time, the controlled manufacture of optical quality surfaces that are not rotationally symmetric. As a result of this development, the optical design community is working to broaden the definition of optical surface shapes to leverage these new capabilities. The initial impact of this development has been to broaden the shape of optical surfaces from a conic surface plus power series terms, and occasionally an anamorphic factor to model astigmatic surfaces, to a conic surface plus the terms of the FRINGE Zernike polynomial, a specific surface definition in the φ-polynomial family that is traceable to current interferometric optical testing. Indeed, since their introduction by F. Zernike while developing the theory of phase-contrast in the 1930s [5

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

], Zernike polynomials have emerged as a pervasive means of describing as-fabricated optical surface deformations and more recently to illustrate the field dependence of the wavefront aberrations in optical systems without rotational symmetry [6

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).

]. In optical design and manufacturing, Zernike polynomial representations of surface departure, placed as an added layer on top of a conic surface, form an enabling fundamental basis. These polynomials are complete and orthogonal over the unit circle, whether the image forming optical system is rotationally symmetric or not. In addition, the lower-order terms are readily identified with the Seidel aberrations. These properties suggest that any departure could be represented in terms of Zernike polynomials. This property has led to their early adoption as a surface representation for the new nonsymmetric diamond-turning process enabled by slow-servo technology.

φ-polynomials, such as Zernike polynomials, play an important role in optical modeling, simulation, and design given their listed above properties. As a consequence, the optical testing industry has chosen to present the departure of a wavefront under test from the reference sphere created by the interferometer in terms of Zernike polynomials. This new level of surface complexity is consistent with evolving innovative optical designs and diamond turning and computer controlled fabrication processes and this formulation in particular has been shown to dramatically extend the performance space of certain classes of optical design (e.g. head worn displays and missile seekers). Fuerschbach demonstrated, in an unobscured three-mirror design that presents the largest planar entrance aperture in the smallest spherical volume, dramatic advances in the diffraction-limited field of view (from 3 to 10) and the operational f/number (from ~f/10 to ~f/2) of an existing production form as a result of extending the surface shapes to the φ-polynomial family [7

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]

]. Even though the optical design community is just beginning to leverage these new opportunities for optical design with φ-polynomial surface shapes, it is not too soon to be looking ahead to their potential limitations when further extending surface shape descriptions to fully freeform surfaces. In this paper, the term freeform describes a conic surface with additive (or subtractive) polynomial departure that may not be rotationally symmetric (e.g. φ-polynomial) and in addition, which is a key point, we anticipate additive localized multi-centric functions that are described as radial basis functions [8

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]

].

There are two major bottlenecks in representing an optical surface sag or wavefront aberration function over the unit circle with φ-polynomials to the accuracies demanded for most stringent optics applications. One is the numerical ill-conditioning associated with computation of the higher-order polynomials that might be required to achieve an accurate representation of an as-fabricated optical surface. Forbes addressed this bottleneck with the development of three-term recurrence relations for Zernike polynomials [9

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

]. This recurrence relation also reduces the computational cost significantly. Second is the substantial number of Zernike terms required, sometimes thousands. Prior to arriving at the proper number of terms in the representation of the optical surface, intermediate results with an insufficient number of coefficients exhibit high-departure errors at the edges. Importantly, as will be shown in this paper, the rate of convergence to an adequate number of terms in the representation is sensitive to the surface sampling.

The major contribution of this paper is the application of an edge-clustered fitting grid to create the database for input to the least squares fitting process resulting in a Zernike polynomial approximant to the freeform surface. The impact of this fitting grid on the reduction of edge ringing and the improvement of surface-fit quality by several orders of magnitude compared to current fitting grids will be demonstrated. We compare the effectiveness of fit with commonly used grid types such as 1) a set of uniform size hexagonal subgrids centered on a uniform rectangular grid, 2) a polar grid with a Chebyshev-based radial sample points, and 3) a uniformly random point fitting grid to show the significance of the edge-clustered random fitting grid on the accuracy of surface approximation.

The remainder of this paper is organized as follows; Section 2 briefly summarizes the three-term recurrence relation developed by Forbes for Zernike polynomials, which will be used as a representative from the family of more general φ-polynomials. A detailed explanation of the least-squares procedure to map the freeform surface based on the sampled database created on the fitting grids to a Zernike polynomial representation is presented in Section 3. In Section 4, the surfaces that were created to be the benchmark cases for freeform surfaces are described. In Section 5, four fitting grids that supply the database for the least-square fitting process are detailed. Finally, numerical results that show RMS errors as a function of the number of Zernike polynomial coefficients for each test surface with different fitting grids fitting are reported, before concluding the paper.

2. The three-term recurrence relation for Zernike polynomials

As recently brought forth to the optics literature, recurrence formula must be applied in computing any polynomial from mild to higher orders to ensure an accurate representation of the polynomial. In this section, the recurrence relation of Zernike polynomials developed by Forbes [9

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

] is briefly revisited. The recurrence relation for Zernike polynomials removes the numerical round-off errors that might lead to a numerical catastrophe in the computation of Zernike polynomials if the explicit definitions are followed in the computation of very high-order Zernike terms. More and more high-order Zernike polynomials might become necessary as more complex analysis is performed that goes beyond simply the resolution of the optics and expands to include the impact of a midspatial-frequency fabrication residual, as an example. A Zernike polynomial definition is given in Born and Wolf [10

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

] as follows
Znm(ρ,φ)=Rn±m(ρ){cosmφsinmφ},
(1)
where n>m and the difference n-m is even. The radial polynomial is expressed as

Rn±m(ρ)=q=0nm2(1)q(nq)!q!(n+m2q)!(nm2+q)!ρn2q.
(2)

As demonstrated by Fig. 2a, a computation based on an explicit expression gives peak values caused by round-off errors on the order of 100, where the value of the radial polynomial should be 1. The recursively computed result for Z504(ρ,φ) shown in Fig. 2b is free of these debilitating computational artifacts.

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

A freeform surface over a circular aperture may be represented as a functionf(ρ,φ) that is a weighted sum of Zernike basis functions that form a complete and orthogonal set over the unit circle as
f(ρ,φ)=k=1NckZnm(ρ,φ),
(7)
where ck represents the coefficients associated with each Zernike polynomial. If the function is known beforehand, the coefficients, ck 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, ck, 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.

Ac=f.
(8)

The linear system given in Eq. (8) can be solved with backward substitution. The matrix 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 ck are the weights multiplying the columns of the matrix 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
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.
, 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
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.
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

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

table-icon
View This Table
provides a list of the nonzero Zernike polynomial coefficients of Fig. 3.

4. The test surfaces

faspherics(ρ)=(ρ2)/2001+1+1/(200)2(ρ2)+1.876×10-5ρ20.5×10-7ρ4+0.545×10-10ρ60.25×10-13ρ8+0.4×10-17ρ10,
(9a)
fthreebumps(x,y)=(ρ2)80+0.05e0.25[(x7)2+(y+6)2]+0.6e0.49[(x+3)2+(y2)2]+0.03e0.81[(x5)2+(y7)2],
(9b)
fFranke(x,y)=0.75e0.25[(9x2)2+(9y2)2]+0.75e[(9x+1)2/49+(9y+1)2/10]+0.5e0.25[(9x7)2+(9y3)2]0.2e[(9x4)2+(9y7)2].
(9c)

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

In this section, we present four different fitting grid patterns that form the database content for the least squares fit to yield the Zernike polynomial coefficients that then are used to describe the surface sag at any point on the optical surface. The resulting Zernike polynomial definition of the surface are re-sampled during either an analysis of the optical system performance or during optical system optimization. When optical surfaces are specified by a polynomial function in an optical system design and analysis simulation environment, the predicted performance of the optical system is computed by an often sparse sample of the wavefront obtained by ray tracing. Typically, in a commercial raytrace code, uniform rectangular grids are used to form a sampled database for the wavefront at an individual wavelength and at an individual field point where the grid density is often in a user specified range between 64 x 64 and 1024 x 1024.

In this paper four different sampling grids are considered to create the database that is applied to transform the specially selected set of analytic test surfaces defined by a small number of coefficients to an equivalent surface, but now computed from a Zernike polynomial description in all cases. The four types of grids applied to the transformation process are 1) a set of uniform size hexagonal subgrids centered on a uniform rectangular grid (hex grid), 2) a polar grid with Chebyshev-based radial weighting (Cheby-polar grid), 3) a uniformly random-point grid (uni-random grid), and 4) an edge-clustered, random-point grid (e_clust-random grid). Each is illustrated in 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.
.

The purpose of compiling and illustrating the evolution of the fit accuracy is to highlight the relationship between the spatial-frequency content of the test surface and the number of Zernike terms required. Moving forward, one of the features of the broader class of freeform surfaces will be an ability to introduce surface features with higher spatial-frequency content. With the hex grid setting, the unit circle is divided into regular hexagonal cells. The center point for each hexagonal cell is also included as a sample point. We have chosen a hexagonal grid structure for the uniform setting since a circular aperture is more uniformly covered with hexagons rather than rectangular cells. In the Cheby-polar grid, the sample points are placed along an expanding set of circles at the roots of the Chebyshev polynomial of degree 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

ρk=cos((2k+1)π2N),k=0,1,...,N1.
(10)

The third grid type, uni-random, consists of simply randomly placed sample points. A point sampling function generates uniformly distributed random points across the unit square (see [13

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

], Halton points). The fourth sampling method is termed e_clust-random point sampling. Here the radial coordinate of the random Halton points is modified with a sine function weighting to move them towards the boundary of the unit circle. If a point has coordinates (ρ,φ) then the corresponding clustered point has(sin(πρ/2),φ)as its coordinates. This type of edge-clustered sampling was previously used in the context of an RBF-QR method by Fornberg et al. [14

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]

]. As will be shown, it is the fourth fitting grid that has clearly demonstrated the best efficacy. Figure 5 displays each of these grid types using approximately 450 points over the unit circle.

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

To study the efficacy of each grid type for determining the coefficients of a Zernike polynomial fit to each of the test surfaces, over an increasing set of coefficients, we have sampled each surface with the four different fitting grids, carried out the least squares approximation to create the Zernike polynomial fit and then evaluated the approximant at around 7000 points that are on a uniform grid as would be the case in any commercial lens design program. Then, RMS errors in the resulting approximant to the test surface, which quantifies the difference between the height of the test surface computed using Eq. (9) and the Zernike polynomial fit determined for each fitting grid type, are recorded in meters in order for 10−9 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 ≈1.5N. 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

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).

].

In Fig. 6c
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).
, we have plotted the RMS errors in meters versus the number of point samples on a uniform grid for the four different fitting grids. This analytic test surface is a conventional rotationally symmetric test surface described in Eq. (9)a and shown in Fig. 6a-b, which is well approximated with all fitting strategies by using around 200 points. We see that initially, all the fitting grids achieve good RMS errors, and the geometry of the fitting grid has little influence on the quality of the approximation. As shown in Fig. 6c, all fitting grids quickly achieved sub-nanometer accuracies with less than 100 points. However, as the number of Zernike coefficients increases, the RMS error grows with hex grid, uni-random, and Cheby-polar grids, whereas the edge clustered random point set produced the most stable results with errors kept on the order of machine precision. As this surface is a conventional rotationally symmetric surface rather than a freeform surface, the number of Zernike polynomials is minimal, i.e. 66 terms, to reach the high accuracy levels demanded by precision or lithography applications (also see Fig. 3).

In Fig. 7b
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).
, we illustrate the RMS errors in meters versus the total number of point samples sampled on a uniform grid for the four fitting grids applied to fitting a Zernike polynomial to the F/1 parabola with three bumps described in Eq. (9)b. Results show that the hex grid and the uni-random grid produce large errors in the order of 10−5 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.

The reason that all of the fitting grids initially produce RMS errors around 5x10−5 m for the F/1 parabola with bumps is best captured in 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.
., 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.

The reason the edge clustered fitting grids produce better surface approximants can be explained by its success in reducing the boundary errors over the unit circle. In the same way that adjusting an equispaced grid to Chebyshev points are the remedy for the Runge type oscillations that results in using equispaced points in 1D, the edge clustered fitting grids act as a remedy for φ-polynomials edge ringing for surface shapes with offset localized structured such as will be seen with multi-centric radial basis functions.

In the process of optical design, due to the number of parameters, fields, wavelengths, etc. the number of sample points within any one ray set used in evaluating the metric for optimization is minimal, often less than 100. However, once a solution is established, the analysis of the performance is typically conducted with hundreds of thousands if not millions of ray samples per surface. In the context of the process being pursued here, the Zernike coefficients that characterize the surface are computed with the relatively sparse optimization grid. The result is that there is a spatial frequency content on the wavefront that will not typically be sampled during the optimization. This gap will result in a poor match to the wavefront by the surface model. It is necessary in fact, for the surfaces shown here, especially the parabola with a series of three bumps and the Franke surface, to greatly exceed the number of samples that can be applied in a typical lens design software if the distinguishing features are to be picked up in the surface models as revealed by the high resolution analysis that is readily conducted in a ray trace code. Additionally, as compiled here using a standalone code, the number of Zernike terms that needs to be available in the optimization procedure exceeds by thousands the parameterization sets available during an optical design, which currently rarely exceeds 100.

7. Conclusion

In this paper, we have investigated the effect of edge clustering points towards the boundary of the unit circle when fitting Zernike polynomials to a general surface shape that represents a family of freeform optical surfaces. We demonstrated that these grids are effective and that the edge-clustered random-fitting grid is particularly effective. We have also compared this fitting grid with a hexagonal sub-grid spaced on uniform centers and a simple random fitting grid for optical surface approximation with Zernike polynomials. We have observed that edge-clustered fitting grids produced very good approximants, and improved the approximation performance by several orders of magnitude compared to that of fitting grids without edge clustering. For highly varying freeform surfaces (e.g. surfaces with no specific symmetry) like Franke, only the edge-clustered sampling method achieved sub-nanometer accuracies. The RMS errors produced by edge clustered fitting grids stably and exponentially decreased as the number of point samples was gradually increased for freeform optical surfaces. In the test cases considered, edge clustered random sampling has achieved in 2-D what a Chebyshev-radial spaced sample achieves in 1-D in removing the impact of edge ringing.

Acknowledgements

The work was supported by the National Science Foundation GOALI grant ECCS-1002179, the II-VI Foundation, and the NYSTAR Foundation C050070.

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 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).

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

  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. Express15(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. Express19(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,” Physica1(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. SPIE7790, (2010).
  7. K. Fuerschbach, J. P. Rolland, and K. P. Thompson, “A new family of optical systems employing φ-polynomial surfaces,” Opt. Express19(22), 21919–21928 (2011). [CrossRef] [PubMed]
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