Abstract

Slow-servo single-point diamond turning as well as advances in computer controlled small lap polishing enables the fabrication of freeform optics, or more specifically, optical surfaces for imaging applications that are not rotationally symmetric. Various forms of polynomials for describing freeform optical surfaces exist in optical design and to support fabrication. A popular method is to add orthogonal polynomials onto a conic section. In this paper, recently introduced gradient-orthogonal polynomials are investigated in a comparative manner with the widely known Zernike polynomials. In order to achieve numerical robustness when higher-order polynomials are required to describe freeform surfaces, recurrence relations are a key enabler. Results in this paper establish the equivalence of both polynomial sets in accurately describing freeform surfaces under stringent conditions. Quantifying the accuracy of these two freeform surface descriptions is a critical step in the future application of these tools in both advanced optical system design and optical fabrication.

© 2012 OSA

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References

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    [CrossRef]
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2012

2011

2010

2009

J. C. Miñano, P. Benitez, and A. Santamaria, “Freeform optics for illumination,” Opt. Rev. 16(2), 99–102 (2009).
[CrossRef]

2008

O. Cakmakci, S. Vo, K. P. Thompson, and J. P. Rolland, “Application of radial basis functions to shape description in a dual-element off-axis eyewear display: Field-of-view limit,” Inf. Display - J. Soc. I 16, 1089–1098 (2008).

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]

2007

1934

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

Benitez, P.

J. C. Miñano, P. Benitez, and A. Santamaria, “Freeform optics for illumination,” Opt. Rev. 16(2), 99–102 (2009).
[CrossRef]

Cakmakci, O.

O. Cakmakci, S. Vo, K. P. Thompson, and J. P. Rolland, “Application of radial basis functions to shape description in a dual-element off-axis eyewear display: Field-of-view limit,” Inf. Display - J. Soc. I 16, 1089–1098 (2008).

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]

Forbes, G. W.

Foroosh, H.

Fuerschbach, K.

Jester, P.

Kaya, I.

Menke, C.

Miñano, J. C.

J. C. Miñano, P. Benitez, and A. Santamaria, “Freeform optics for illumination,” Opt. Rev. 16(2), 99–102 (2009).
[CrossRef]

Moore, B.

Rolland, J. P.

Santamaria, A.

J. C. Miñano, P. Benitez, and A. Santamaria, “Freeform optics for illumination,” Opt. Rev. 16(2), 99–102 (2009).
[CrossRef]

Thompson, K. P.

Urban, K.

Vo, S.

O. Cakmakci, S. Vo, K. P. Thompson, and J. P. Rolland, “Application of radial basis functions to shape description in a dual-element off-axis eyewear display: Field-of-view limit,” Inf. Display - J. Soc. I 16, 1089–1098 (2008).

Zernike, F.

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

Appl. Opt.

Inf. Display - J. Soc. I

O. Cakmakci, S. Vo, K. P. Thompson, and J. P. Rolland, “Application of radial basis functions to shape description in a dual-element off-axis eyewear display: Field-of-view limit,” Inf. Display - J. Soc. I 16, 1089–1098 (2008).

Opt. Express

Opt. Rev.

J. C. Miñano, P. Benitez, and A. Santamaria, “Freeform optics for illumination,” Opt. Rev. 16(2), 99–102 (2009).
[CrossRef]

Physica

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

Other

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

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

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Figures (6)

Fig. 1
Fig. 1

The two types of ray-grids used for creating data sites for polynomial fitting with about 900 rays in this figure: (a) Hexagonal uniform (b) Edge clustered.

Fig. 2
Fig. 2

Sag departure from the best fit sphere (bfs): (a) f1-bfs, F/1 parabola with the Gaussian bump away from the edge (b) f2-bfs, F/1 parabola with the Gaussian bump near the edge of the aperture.

Fig. 3
Fig. 3

Sag fit residual profiles for f1 ; the F/1 parabola with a Gaussian bump away from the edge of the aperture with T = 80; (a) fit residual with hexagonal uniform sampling, (b) fit residual with edge clustered sampling. The gradient-orthogonal Q-polynomial and the Zernike polynomial representations give indistinguishable results, so only one is shown.

Fig. 4
Fig. 4

Sag fit residual profiles for f2 ; the F/1 parabola with Gaussian bump at the aperture edge with T = 80; (a) fit residual with hexagonal uniform sampling, (b) fit residual with edge clustered sampling. Zernike and gradient-orthogonal Q-polynomials perform very similarly, so only one is shown.

Fig. 5
Fig. 5

Comparing Zernike polynomials and gradient-orthogonal Q-polynomials as freeform surface representations. The fidelity is investigated with both edge clustered and hexagonal uniform data site samples in the case of fitting analytical functions with these surface descriptions; the evolution of the RMS fit residual vs. the number of coefficients for the test case (a) f1, F1 parabola with the bump away from the edge, (b) f2, F1 parabola with the bump at the edge.

Fig. 6
Fig. 6

Zernike (solid lines) and gradient-orthogonal Q-polynomials (dash lines) surface approximation performance over a range of heights of the rotationally nonsymmetric bump with hexagonal uniform and edge clustered sampling for the test cases (a) f1, (b) f2.

Equations (7)

Equations on this page are rendered with MathJax. Learn more.

f(ρ,θ)= c ρ 2 1+ 1 c 2 ρ 2 + 1 1 c 2 ρ 2 u 2 ( 1 u 2 ) n=0 N a n 0 Q n 0 ( u 2 ) + 1 1 c 2 ρ 2 m=1 M u m n=0 N [ a n m cos(mθ)+ b n m sin(mθ) ] Q n m ( u 2 ),
f(ρ,θ)= c ρ 2 1+ 1 c 2 ρ 2 + m=1 M u m n=0 N [ a n m cos(mθ)+ b n m sin(mθ) ] Z n m ( u 2 ),
P n+1 ( ρ )=( a n + b n ρ ) P n ( ρ ) c n P n1 ( ρ ),
Q n m ( u 2 )= A n m ( u 2 ) g n1 m Q n1 m ( u 2 ) f n m ,
f 1 (x,y)= x 2 + y 2 320 +0.0125 e 0.09[ ( x20 ) 2 + ( y5.5 ) 2 ] ,
f 2 (x,y)= x 2 + y 2 320 +0.0125 e 0.09[ ( x+26 ) 2 + ( y26 ) 2 ] ,
c bfs = 2 f( ρ max ,θ ) ρ max 2 + f( ρ max ,θ ) 2 ,

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