Abstract

When leveraging orthogonal polynomials for describing freeform optics, designers typically focus on the computational efficiency of convergence and the optical performance of the resulting designs. However, to physically realize these designs, the freeform surfaces need to be fabricated and tested. An optimization constraint is described that allows on-the-fly calculation and constraint of manufacturability estimates for freeform surfaces, namely peak-to-valley sag departure and maximum gradient normal departure. This constraint’s construction is demonstrated in general for orthogonal polynomials, and in particular for both Zernike polynomials and Forbes 2D-Q polynomials. Lastly, this optimization constraint’s impact during design is shown via two design studies: a redesign of a published unobscured three-mirror telescope in the ball geometry for use in LWIR imaging and a freeform prism combiner for use in AR/VR applications. It is shown that using the optimization penalty with a fixed number of coefficients enables an improvement in manufacturability in exchange for a tradeoff in optical performance. It is further shown that, when the number of coefficients is increased in conjunction with the optimization penalty, manufacturability estimates can be improved without sacrificing optical performance.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article
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References

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    [Crossref]
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    [Crossref] [PubMed]
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2018 (4)

2017 (4)

B. Chen and A. M. Herkommer, “Alternate optical designs for head-mounted displays with a wide field of view,” Appl. Opt. 56(4), 901–906 (2017).
[Crossref] [PubMed]

B. G. Crowther and J. R. Rogers, “Desensitization in aspheric and freeform optical designs,” Proc. SPIE 10590, 1059010 (2017).

T. Blalock, B. Myer, I. Ferralli, M. Brunelle, and T. Lynch, “Metrology for the manufacturing of freeform optics,” Proc. SPIE 10448, 46 (2017).
[Crossref]

A. Broemel, U. Lippman, and H. Gross, “Freeform surface descriptions - part I: Mathematical representations,” Adv. Opt. Technol. 6, 327–336 (2017).

2016 (1)

U. Fuchs and S. R. Kiontke, “Discussing design for manufacturability for two freeform imaging systems,” Proc. SPIE 9948, 99480L (2016).

2015 (3)

2014 (1)

2013 (3)

B. Ma, K. Sharma, K. P. Thompson, and J. P. Rolland, “Mobile device camera design with Q-type polynomials to achieve higher production yield,” Opt. Express 21(15), 17454–17463 (2013).
[Crossref] [PubMed]

C. Menke and G. W. Forbes, “Optical design with orthogonal representations of rotationally symmetric and freeform aspheres,” Advanced Optical Testing 2(1), 97–109 (2013).
[Crossref]

F. Z. Fang, X. D. Zhang, A. Weckenmann, G. X. Zhang, and C. Evans, “Manufacturing and measurement of freeform optics,” CIRP Ann. 62(2), 823–846 (2013).
[Crossref]

2012 (1)

2011 (3)

2010 (1)

2008 (1)

Z. Zheng, X. Sun, X. Liu, and P. Gu, “Design of reflective projection lens with Zernike polynomials surfaces,” Displays 29(4), 412–417 (2008).
[Crossref]

2006 (1)

J. R. Rogers, “Using Global Synthesis to find tolerance-insensitive design,” Proc. SPIE 6342, 63420M (2006).
[Crossref]

2005 (1)

2004 (1)

M. Bray, “Orthogonal Polynomials: A set for square areas,” Proc. SPIE 5252, 314–321 (2004).
[Crossref]

1994 (1)

1981 (1)

1980 (1)

D. S. Grey, “Orthogonal Polynomials As Lens-Aberration Coefficients,” Proc. SPIE 0237, 85–90 (1980).
[Crossref]

1970 (1)

1934 (1)

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

Bauer, A.

A. Bauer, E. M. Schiesser, and J. P. Rolland, “Starting geometry creation and design method for freeform optics,” Nat. Commun. 9(1), 1756 (2018).
[Crossref] [PubMed]

N. Takaki, A. Bauer, and J. P. Rolland, “Degeneracy in freeform surfaces described with orthogonal polynomials,” Appl. Opt. 57(35), 10348–10354 (2018).
[Crossref] [PubMed]

Bauman, B. J.

Blalock, T.

T. Blalock, B. Myer, I. Ferralli, M. Brunelle, and T. Lynch, “Metrology for the manufacturing of freeform optics,” Proc. SPIE 10448, 46 (2017).
[Crossref]

Bray, M.

M. Bray, “Orthogonal Polynomials: A set for square areas,” Proc. SPIE 5252, 314–321 (2004).
[Crossref]

Broemel, A.

A. Broemel, U. Lippman, and H. Gross, “Freeform surface descriptions - part I: Mathematical representations,” Adv. Opt. Technol. 6, 327–336 (2017).

Brunelle, M.

T. Blalock, B. Myer, I. Ferralli, M. Brunelle, and T. Lynch, “Metrology for the manufacturing of freeform optics,” Proc. SPIE 10448, 46 (2017).
[Crossref]

Chen, B.

Chow, W. W.

Crowther, B. G.

B. G. Crowther and J. R. Rogers, “Desensitization in aspheric and freeform optical designs,” Proc. SPIE 10590, 1059010 (2017).

Evans, C.

F. Z. Fang, X. D. Zhang, A. Weckenmann, G. X. Zhang, and C. Evans, “Manufacturing and measurement of freeform optics,” CIRP Ann. 62(2), 823–846 (2013).
[Crossref]

Fang, F. Z.

F. Z. Fang, X. D. Zhang, A. Weckenmann, G. X. Zhang, and C. Evans, “Manufacturing and measurement of freeform optics,” CIRP Ann. 62(2), 823–846 (2013).
[Crossref]

Ferralli, I.

T. Blalock, B. Myer, I. Ferralli, M. Brunelle, and T. Lynch, “Metrology for the manufacturing of freeform optics,” Proc. SPIE 10448, 46 (2017).
[Crossref]

Ferreira, C.

Forbes, G. W.

Fuchs, U.

U. Fuchs and S. R. Kiontke, “Discussing design for manufacturability for two freeform imaging systems,” Proc. SPIE 9948, 99480L (2016).

Fuerschbach, K.

Grey, D. S.

D. S. Grey, “Orthogonal Polynomials As Lens-Aberration Coefficients,” Proc. SPIE 0237, 85–90 (1980).
[Crossref]

D. S. Grey, “Tolerance sensitivity and optimization,” Appl. Opt. 9(3), 523–526 (1970).
[Crossref] [PubMed]

Gross, H.

Y. Zhong and H. Gross, “Vectorial aberrations of biconic surfaces,” J. Opt. Soc. Am. A 35(8), 1385–1392 (2018).
[Crossref] [PubMed]

A. Broemel, U. Lippman, and H. Gross, “Freeform surface descriptions - part I: Mathematical representations,” Adv. Opt. Technol. 6, 327–336 (2017).

Gu, P.

Z. Zheng, X. Sun, X. Liu, and P. Gu, “Design of reflective projection lens with Zernike polynomials surfaces,” Displays 29(4), 412–417 (2008).
[Crossref]

Herkommer, A. M.

Jin, G.

Kiontke, S. R.

U. Fuchs and S. R. Kiontke, “Discussing design for manufacturability for two freeform imaging systems,” Proc. SPIE 9948, 99480L (2016).

Li, L.

Lippman, U.

A. Broemel, U. Lippman, and H. Gross, “Freeform surface descriptions - part I: Mathematical representations,” Adv. Opt. Technol. 6, 327–336 (2017).

Liu, X.

Z. Zheng, X. Sun, X. Liu, and P. Gu, “Design of reflective projection lens with Zernike polynomials surfaces,” Displays 29(4), 412–417 (2008).
[Crossref]

López, J. L.

Lynch, T.

T. Blalock, B. Myer, I. Ferralli, M. Brunelle, and T. Lynch, “Metrology for the manufacturing of freeform optics,” Proc. SPIE 10448, 46 (2017).
[Crossref]

Ma, B.

Mahajan, V.

Menke, C.

C. Menke and G. W. Forbes, “Optical design with orthogonal representations of rotationally symmetric and freeform aspheres,” Advanced Optical Testing 2(1), 97–109 (2013).
[Crossref]

Myer, B.

T. Blalock, B. Myer, I. Ferralli, M. Brunelle, and T. Lynch, “Metrology for the manufacturing of freeform optics,” Proc. SPIE 10448, 46 (2017).
[Crossref]

Nakano, T.

Navarro, R.

Ochse, D.

D. Ochse, K. Uhlendorf, and L. Reichmann, “Describing freeform surfaces with orthogonal functions,” Proc. SPIE 9626, 962612 (2015).
[Crossref]

Reichmann, L.

D. Ochse, K. Uhlendorf, and L. Reichmann, “Describing freeform surfaces with orthogonal functions,” Proc. SPIE 9626, 962612 (2015).
[Crossref]

Rogers, J. R.

B. G. Crowther and J. R. Rogers, “Desensitization in aspheric and freeform optical designs,” Proc. SPIE 10590, 1059010 (2017).

J. R. Rogers, “Using Global Synthesis to find tolerance-insensitive design,” Proc. SPIE 6342, 63420M (2006).
[Crossref]

Rolland, J. P.

Schiesser, E. M.

A. Bauer, E. M. Schiesser, and J. P. Rolland, “Starting geometry creation and design method for freeform optics,” Nat. Commun. 9(1), 1756 (2018).
[Crossref] [PubMed]

Schneider, M. D.

Sharma, K.

Sinusía, E. P.

Sun, X.

Z. Zheng, X. Sun, X. Liu, and P. Gu, “Design of reflective projection lens with Zernike polynomials surfaces,” Displays 29(4), 412–417 (2008).
[Crossref]

Swantner, W.

Takaki, N.

Tamagawa, Y.

Thompson, K. P.

Uhlendorf, K.

D. Ochse, K. Uhlendorf, and L. Reichmann, “Describing freeform surfaces with orthogonal functions,” Proc. SPIE 9626, 962612 (2015).
[Crossref]

Weckenmann, A.

F. Z. Fang, X. D. Zhang, A. Weckenmann, G. X. Zhang, and C. Evans, “Manufacturing and measurement of freeform optics,” CIRP Ann. 62(2), 823–846 (2013).
[Crossref]

Yang, T.

Zernike, F.

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

Zhang, G. X.

F. Z. Fang, X. D. Zhang, A. Weckenmann, G. X. Zhang, and C. Evans, “Manufacturing and measurement of freeform optics,” CIRP Ann. 62(2), 823–846 (2013).
[Crossref]

Zhang, X. D.

F. Z. Fang, X. D. Zhang, A. Weckenmann, G. X. Zhang, and C. Evans, “Manufacturing and measurement of freeform optics,” CIRP Ann. 62(2), 823–846 (2013).
[Crossref]

Zheng, Z.

Z. Zheng, X. Sun, X. Liu, and P. Gu, “Design of reflective projection lens with Zernike polynomials surfaces,” Displays 29(4), 412–417 (2008).
[Crossref]

Zhong, Y.

Zhu, J.

Adv. Opt. Technol. (1)

A. Broemel, U. Lippman, and H. Gross, “Freeform surface descriptions - part I: Mathematical representations,” Adv. Opt. Technol. 6, 327–336 (2017).

Advanced Optical Testing (1)

C. Menke and G. W. Forbes, “Optical design with orthogonal representations of rotationally symmetric and freeform aspheres,” Advanced Optical Testing 2(1), 97–109 (2013).
[Crossref]

Appl. Opt. (6)

CIRP Ann. (1)

F. Z. Fang, X. D. Zhang, A. Weckenmann, G. X. Zhang, and C. Evans, “Manufacturing and measurement of freeform optics,” CIRP Ann. 62(2), 823–846 (2013).
[Crossref]

Displays (1)

Z. Zheng, X. Sun, X. Liu, and P. Gu, “Design of reflective projection lens with Zernike polynomials surfaces,” Displays 29(4), 412–417 (2008).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Nat. Commun. (1)

A. Bauer, E. M. Schiesser, and J. P. Rolland, “Starting geometry creation and design method for freeform optics,” Nat. Commun. 9(1), 1756 (2018).
[Crossref] [PubMed]

Opt. Express (8)

Physica (1)

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

Proc. SPIE (7)

D. S. Grey, “Orthogonal Polynomials As Lens-Aberration Coefficients,” Proc. SPIE 0237, 85–90 (1980).
[Crossref]

D. Ochse, K. Uhlendorf, and L. Reichmann, “Describing freeform surfaces with orthogonal functions,” Proc. SPIE 9626, 962612 (2015).
[Crossref]

M. Bray, “Orthogonal Polynomials: A set for square areas,” Proc. SPIE 5252, 314–321 (2004).
[Crossref]

U. Fuchs and S. R. Kiontke, “Discussing design for manufacturability for two freeform imaging systems,” Proc. SPIE 9948, 99480L (2016).

T. Blalock, B. Myer, I. Ferralli, M. Brunelle, and T. Lynch, “Metrology for the manufacturing of freeform optics,” Proc. SPIE 10448, 46 (2017).
[Crossref]

J. R. Rogers, “Using Global Synthesis to find tolerance-insensitive design,” Proc. SPIE 6342, 63420M (2006).
[Crossref]

B. G. Crowther and J. R. Rogers, “Desensitization in aspheric and freeform optical designs,” Proc. SPIE 10590, 1059010 (2017).

Other (10)

M. Davies, Mechanical Engineering and Engineering Science, The University of North Carolina at Charlotte, (personal communication, October 2, 2018).

J. Renze, C. Stover, and E. W. Weisstein, “Inner Product” (from MathWorld - A Wolfram Web Resource), retrieved January 25, 2019, http://mathworld.wolfram.com/InnerProduct.html .

E. W. Weisstein, “Jacobi Polynomials” (from MathWorld - A Wolfram Web Resource), retrieved October 30, 2018, http://mathworld.wolfram.com/JacobiPolynomial.html .

“Appendix A Zernike Polynomials,” in CODEV Lens System Setup Reference Manual (2015).

M. Nikolic, P. Benítez, B. Narasimhan, D. Grabovickic, J. Liu, and J. Minano, “Optical design through optimization for rectangular apertures using freeform orthogonal polynomials: a case study,” Optical Engineering 55, 071204(071201–071206) (2016).

Synopsys, “CODE V Q-Freeform UD1 Release Notes,” (Synopsys, Inc, 2017).

L. Cook, G. Perron, B. Zellers, and B. Cohn, “Display system having coma-control plate in relay lens,” U.S. Patent 4,826,287 (May 2 1989).

“ISO 10110-19 Optics and Photonics - Preparation of drawings for optical elements and systems - Part 19,” (ISO, Geneva, Switzerland, 2015).

E. W. Weisstein, “Chebyshev Polynomial of the First Kind” (from MathWorld - A Wolfram Web Resource), retrieved September 1, 2018, http://mathworld.wolfram.com/ChebyshevPolynomialoftheFirstKind.html .

E. W. Weisstein, “Legendre Polynomial” (from MathWorld - A Wolfram Web Resource), retrieved September 1, 2018, http://mathworld.wolfram.com/LegendrePolynomial.html .

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Figures (9)

Fig. 1
Fig. 1 Layout and optical performance (λ = 10 µm) for unpenalized designs used as benchmarks with (a,b) Zernikes and (c,d) 2D-Qs.
Fig. 2
Fig. 2 For the three-mirror telescope, optical performance and manufacturability estimates versus square-sum penalty weight with (top) Zernikes and (bottom) 2D-Qs. All values of the lower-bound designs are within 5% of the benchmark.
Fig. 3
Fig. 3 Layout for (left) the benchmark design and square-sum penalized designs with 2D-Qs for weights (middle-left) 10−2, (middle-right) 100, and (right) 102. The layout changes from the benchmark to the weight 10−2 and weight 100 designs, but not from the weight 100 design to the weight 102 design. The Zernike designs (not shown) have a similar trend.
Fig. 4
Fig. 4 Layout and optical performance (λ = 10 µm) for 36-term square-sum penalized designs with (a,b) Zernikes and (c,d) 2D-Qs.
Fig. 5
Fig. 5 Location of the centered and outermost 3 mm sub-pupils used for evaluating optical performance within the 8 mm eyebox. Performance was also evaluated over sub-pupils located halfway between the outermost and center sub-pupils (not shown).
Fig. 6
Fig. 6 Layout and optical performance for unpenalized benchmark reflective prism designs with (top) Zernikes and (bottom) 2D-Qs. FFDs of sagittal and tangential MTF at 50 cyc/mm are shown for best-case and worst-case sub-pupils.
Fig. 7
Fig. 7 For the reflective prism, optical performance versus manufacturability estimates as square-sum penalty changes with (top) Zernikes and (bottom) 2D-Qs. All values of the lower-bound design are within 5% of the benchmark.
Fig. 8
Fig. 8 Layout of Zernike prism designs for (left) the unpenalized benchmark and (right) the highest-weight square-sum penalized designs. Notice the change in layout. The 2D-Q designs (not shown) show a similar trend.
Fig. 9
Fig. 9 Layout and optical performance for 36-term square-sum penalized prism designs with (top) Zernikes and (bottom) 2D-Qs. FFDs of sagittal and tangential MTF at 50 cyc/mm are shown for best-case and worst-case sub-pupils.

Tables (6)

Tables Icon

Table 1 Three-Mirror Telescope Specifications

Tables Icon

Table 2 For the three-mirror telescope with Zernikes, optical performance and manufacturability estimates of (left) the 16-term unpenalized benchmark design, (middle) the 36-term unpenalized design, and (right) the 36-term square-sum penalized design.

Tables Icon

Table 3 For the three-mirror telescope with 2D-Qs, optical performance and manufacturability estimates of (left) the 16-term unpenalized benchmark design, (middle) the 36-term unpenalized design, and (right) the 36-term square-sum penalized design.

Tables Icon

Table 4 Specifications of Reflective Prism

Tables Icon

Table 5 For the freeform prism with Zernikes, optical performance and manufacturability estimates of (left) the 16-term unpenalized benchmark design, (middle) the 36-term unpenalized design, and (right) the 36-term square-sum penalized design.

Tables Icon

Table 6 For the freeform prism with 2D-Qs, optical performance and manufacturability estimates of (left) the 16-term unpenalized benchmark design, (middle) the 36-term unpenalized design, and (right) the 36-term square-sum penalized design.

Equations (16)

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

z=f(ρ,θ)= c ρ 2 1+ 1 c 2 ρ 2 + n=0 N s n P n (u,θ).
P n , P n' =0whennn'.
S, P n P n , P n = n' s n' P n' , P n P n , P n = s n .
S,S = n S, s n P n = n P n , P n s n 2 .
Err SS = Err Aber +ζ n P n , P n s n 2 ,
z=f(ρ,θ)= c ρ 2 1+ 1 c 2 ρ 2 +D(u,θ), D(u,θ):= n=0 N m=n n C n m Z n m (u,θ) .
fg = 1 π π π 0 1 f(u,θ)g(u,θ)ududθ .
| D(u,θ) | 2 = ( n,m C n m Z n m )( n',m' C n' m' Z n' m' ) = n,m ( C n m ) 2 ε m 2n+2 ,
Err SS = Err Aber +ζ n,m ( C n m ) 2 ε m 2n+2 .
z=f(ρ,θ)= c ρ 2 1+ 1 c 2 ρ 2 + δ(u,θ) 1 c 2 ρ 2 , δ(u,θ):= u 2 (1 u 2 ) n=0 N a n 0 Q n 0 ( u 2 )+ m=1 M u m n=0 N [ a n m cos(mθ)+ b n m sin(mθ)] Q n m ( u 2 ).
fg = π π 0 1 f(u,θ)g(u,θ) ududθ u 2 (1 u 2 ) / π π 0 1 ududθ u 2 (1 u 2 ) .
| δ(u,θ) | 2 = ( δ u ) 2 + 1 u 2 ( δ θ ) 2 = m,n [ ( a n m ) 2 + ( b n m ) 2 ].
Err SS = Err Aber +ζ 1 ρ max 2 m,n [ ( a n m ) 2 + ( b n m ) 2 ].
1 ρ max 2
u
ρ.

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