Abstract

We propose an improved double freeform-optical-surface design method for shaping a prescribed irradiance distribution whilst forming a desired wavefront from a given incident beam. This method generalizes our previous work [Opt. Exp. 21, 14728-14735 (2013)] to tackle non-separable beam irradiances. We firstly compute a proper ray mapping using an adaptive mesh method in the framework of the L2 Monge-Kantorovich mass transfer problem. Then, we construct the two freeform optical surfaces according to this mapping using a modified simultaneous point-by-point procedure which is aimed to minimize the surface errors. For the first surface, the modified procedure works by firstly approximating a value to the next point by only using the slope of the current point and then improving it by utilizing both slopes of the two points based on Snell’s law. Its corresponding point on the second surface can be computed using the constant optical path length condition. A design example of producing a challenging irradiance distribution and a non-ideal wavefront demonstrates the effectiveness of the method.

© 2013 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
  4. W. A. Parkyn and D. G. Pelka, “Free-form lenses for rectangular illumination zones,” Anthony, Inc., US Patent 7674019 (2010).
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    [CrossRef]
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    [CrossRef]
  16. V. Oliker, “Designing freeform lenses for intensity and phase control of coherent light with help from geometry and mass transport,” Arch. Ration. Mech. Anal.201(3), 1013–1045 (2011).
    [CrossRef]
  17. J. Rubinstein and G. Wolansky, “Intensity control with a free-form lens,” J. Opt. Soc. Am. A24(2), 463–469 (2007).
    [CrossRef] [PubMed]
  18. V. Oliker, J. Rubinstein, and G. Wolansky, “Ray mapping and illumination control,” J. Photonics Energy3(1), 035599 (2013).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  21. J. D. Benamou and Y. Brenier, “A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem,” Numer. Math.84(3), 375–393 (2000).
    [CrossRef]
  22. S. Haker, L. Zhu, A. Tannenbaum, and S. Angenent, “Optimal mass transport for registration and warping,” Int. J. Comput. Vis.60(3), 225–240 (2004).
    [CrossRef]
  23. M. M. Sulman, J. F. Williams, and R. D. Russell, “An efficient approach for the numerical solution of Monge-Ampère equation,” Appl. Numer. Math.61(3), 298–307 (2011).
    [CrossRef]
  24. R. T. Rockafellar, “Characterization of the subdifferentials of convex functions,” Pac. J. Math.17(3), 497–510 (1966).
    [CrossRef]
  25. R. J. McCann, “Existence and uniqueness of monotone measure-preserving maps,” Duke Math. J.80(2), 309–323 (1995).
    [CrossRef]
  26. W. B. Elmer, The Optical Design of Reflectors, 2nd ed. (Wiley, 1980), Chap.4.
  27. J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts and Company, 2005).
  28. M. Gdeisat and F. Lilley, “Two-dimensional phase unwrapping problem,” http://www.ljmu.ac.uk/GERI/CEORG_Docs/Two_Dimensional_Phase_Unwrapping_Final.pdf .

2013

2012

2011

D. Michaelis, P. Schreiber, and A. Bräuer, “Cartesian oval representation of freeform optics in illumination systems,” Opt. Lett.36(6), 918–920 (2011).
[CrossRef] [PubMed]

V. Oliker, “Designing freeform lenses for intensity and phase control of coherent light with help from geometry and mass transport,” Arch. Ration. Mech. Anal.201(3), 1013–1045 (2011).
[CrossRef]

M. M. Sulman, J. F. Williams, and R. D. Russell, “An efficient approach for the numerical solution of Monge-Ampère equation,” Appl. Numer. Math.61(3), 298–307 (2011).
[CrossRef]

2010

2008

2007

2006

B. Parkyn and D. Pelka, “Free form lenses designed by a pseudo-rectangular lawnmower algorithm,” Proc. SPIE6338, 633808 (2006).
[CrossRef]

2005

V. Oliker, “Geometric and variational methods in optical design of reflecting surfaces with prescribed illuminance properties,” Proc. SPIE5942, 594207 (2005).
[CrossRef]

H. Ries, “Laser beam shaping by double tailoring,” Proc. SPIE5876, 587607 (2005).
[CrossRef]

2004

T. Glimm and V. Oliker, “Optical design of two-reflector systems, the Monge-Kantorovich mass transfer problem and Fermat’s principle,” Indiana Univ. Math. J.53(5), 1255–1277 (2004).
[CrossRef]

P. Benítez, J. C. Miñano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng.43(7), 1489–1502 (2004).
[CrossRef]

S. Haker, L. Zhu, A. Tannenbaum, and S. Angenent, “Optimal mass transport for registration and warping,” Int. J. Comput. Vis.60(3), 225–240 (2004).
[CrossRef]

2002

2000

J. D. Benamou and Y. Brenier, “A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem,” Numer. Math.84(3), 375–393 (2000).
[CrossRef]

1998

W. A. Parkyn, “Illumination lenses designed by extrinsic differential geometry,” Proc. SPIE3482, 389–396 (1998).
[CrossRef]

1995

R. J. McCann, “Existence and uniqueness of monotone measure-preserving maps,” Duke Math. J.80(2), 309–323 (1995).
[CrossRef]

1966

R. T. Rockafellar, “Characterization of the subdifferentials of convex functions,” Pac. J. Math.17(3), 497–510 (1966).
[CrossRef]

Angenent, S.

S. Haker, L. Zhu, A. Tannenbaum, and S. Angenent, “Optimal mass transport for registration and warping,” Int. J. Comput. Vis.60(3), 225–240 (2004).
[CrossRef]

Bäuerle, A.

Benamou, J. D.

J. D. Benamou and Y. Brenier, “A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem,” Numer. Math.84(3), 375–393 (2000).
[CrossRef]

Benítez, P.

P. Benítez, J. C. Miñano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng.43(7), 1489–1502 (2004).
[CrossRef]

Blen, J.

P. Benítez, J. C. Miñano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng.43(7), 1489–1502 (2004).
[CrossRef]

Bräuer, A.

Brenier, Y.

J. D. Benamou and Y. Brenier, “A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem,” Numer. Math.84(3), 375–393 (2000).
[CrossRef]

Bruneton, A.

Cassarly, W. J.

Chaves, J.

P. Benítez, J. C. Miñano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng.43(7), 1489–1502 (2004).
[CrossRef]

Ding, Y.

Dross, O.

P. Benítez, J. C. Miñano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng.43(7), 1489–1502 (2004).
[CrossRef]

Falicoff, W.

P. Benítez, J. C. Miñano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng.43(7), 1489–1502 (2004).
[CrossRef]

Feng, Z.

Fournier, F. R.

Glimm, T.

T. Glimm and V. Oliker, “Optical design of two-reflector systems, the Monge-Kantorovich mass transfer problem and Fermat’s principle,” Indiana Univ. Math. J.53(5), 1255–1277 (2004).
[CrossRef]

Gong, M.

Gu, P. F.

Haker, S.

S. Haker, L. Zhu, A. Tannenbaum, and S. Angenent, “Optimal mass transport for registration and warping,” Int. J. Comput. Vis.60(3), 225–240 (2004).
[CrossRef]

Han, Y.

Hernández, M.

P. Benítez, J. C. Miñano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng.43(7), 1489–1502 (2004).
[CrossRef]

Huang, L.

Jin, G.

Li, H.

Liu, P.

Liu, X.

Loosen, P.

Luo, Y.

McCann, R. J.

R. J. McCann, “Existence and uniqueness of monotone measure-preserving maps,” Duke Math. J.80(2), 309–323 (1995).
[CrossRef]

Michaelis, D.

Miñano, J. C.

P. Benítez, J. C. Miñano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng.43(7), 1489–1502 (2004).
[CrossRef]

Mohedano, R.

P. Benítez, J. C. Miñano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng.43(7), 1489–1502 (2004).
[CrossRef]

Muschaweck, J.

Oliker, V.

V. Oliker, J. Rubinstein, and G. Wolansky, “Ray mapping and illumination control,” J. Photonics Energy3(1), 035599 (2013).
[CrossRef]

V. Oliker, “Designing freeform lenses for intensity and phase control of coherent light with help from geometry and mass transport,” Arch. Ration. Mech. Anal.201(3), 1013–1045 (2011).
[CrossRef]

V. Oliker, “Geometric and variational methods in optical design of reflecting surfaces with prescribed illuminance properties,” Proc. SPIE5942, 594207 (2005).
[CrossRef]

T. Glimm and V. Oliker, “Optical design of two-reflector systems, the Monge-Kantorovich mass transfer problem and Fermat’s principle,” Indiana Univ. Math. J.53(5), 1255–1277 (2004).
[CrossRef]

Parkyn, B.

B. Parkyn and D. Pelka, “Free form lenses designed by a pseudo-rectangular lawnmower algorithm,” Proc. SPIE6338, 633808 (2006).
[CrossRef]

Parkyn, W. A.

W. A. Parkyn, “Illumination lenses designed by extrinsic differential geometry,” Proc. SPIE3482, 389–396 (1998).
[CrossRef]

Pelka, D.

B. Parkyn and D. Pelka, “Free form lenses designed by a pseudo-rectangular lawnmower algorithm,” Proc. SPIE6338, 633808 (2006).
[CrossRef]

Qian, K. Y.

Ries, H.

Rockafellar, R. T.

R. T. Rockafellar, “Characterization of the subdifferentials of convex functions,” Pac. J. Math.17(3), 497–510 (1966).
[CrossRef]

Rolland, J. P.

Rubinstein, J.

V. Oliker, J. Rubinstein, and G. Wolansky, “Ray mapping and illumination control,” J. Photonics Energy3(1), 035599 (2013).
[CrossRef]

J. Rubinstein and G. Wolansky, “Intensity control with a free-form lens,” J. Opt. Soc. Am. A24(2), 463–469 (2007).
[CrossRef] [PubMed]

Russell, R. D.

M. M. Sulman, J. F. Williams, and R. D. Russell, “An efficient approach for the numerical solution of Monge-Ampère equation,” Appl. Numer. Math.61(3), 298–307 (2011).
[CrossRef]

Schreiber, P.

Stollenwerk, J.

Sulman, M. M.

M. M. Sulman, J. F. Williams, and R. D. Russell, “An efficient approach for the numerical solution of Monge-Ampère equation,” Appl. Numer. Math.61(3), 298–307 (2011).
[CrossRef]

Tannenbaum, A.

S. Haker, L. Zhu, A. Tannenbaum, and S. Angenent, “Optimal mass transport for registration and warping,” Int. J. Comput. Vis.60(3), 225–240 (2004).
[CrossRef]

Wang, L.

Wester, R.

Williams, J. F.

M. M. Sulman, J. F. Williams, and R. D. Russell, “An efficient approach for the numerical solution of Monge-Ampère equation,” Appl. Numer. Math.61(3), 298–307 (2011).
[CrossRef]

Wolansky, G.

V. Oliker, J. Rubinstein, and G. Wolansky, “Ray mapping and illumination control,” J. Photonics Energy3(1), 035599 (2013).
[CrossRef]

J. Rubinstein and G. Wolansky, “Intensity control with a free-form lens,” J. Opt. Soc. Am. A24(2), 463–469 (2007).
[CrossRef] [PubMed]

Wu, R.

Xu, L.

Zhang, Y.

Zheng, Z.

Zheng, Z. R.

Zhu, L.

S. Haker, L. Zhu, A. Tannenbaum, and S. Angenent, “Optimal mass transport for registration and warping,” Int. J. Comput. Vis.60(3), 225–240 (2004).
[CrossRef]

Appl. Numer. Math.

M. M. Sulman, J. F. Williams, and R. D. Russell, “An efficient approach for the numerical solution of Monge-Ampère equation,” Appl. Numer. Math.61(3), 298–307 (2011).
[CrossRef]

Appl. Opt.

Arch. Ration. Mech. Anal.

V. Oliker, “Designing freeform lenses for intensity and phase control of coherent light with help from geometry and mass transport,” Arch. Ration. Mech. Anal.201(3), 1013–1045 (2011).
[CrossRef]

Duke Math. J.

R. J. McCann, “Existence and uniqueness of monotone measure-preserving maps,” Duke Math. J.80(2), 309–323 (1995).
[CrossRef]

Indiana Univ. Math. J.

T. Glimm and V. Oliker, “Optical design of two-reflector systems, the Monge-Kantorovich mass transfer problem and Fermat’s principle,” Indiana Univ. Math. J.53(5), 1255–1277 (2004).
[CrossRef]

Int. J. Comput. Vis.

S. Haker, L. Zhu, A. Tannenbaum, and S. Angenent, “Optimal mass transport for registration and warping,” Int. J. Comput. Vis.60(3), 225–240 (2004).
[CrossRef]

J. Opt. Soc. Am. A

J. Photonics Energy

V. Oliker, J. Rubinstein, and G. Wolansky, “Ray mapping and illumination control,” J. Photonics Energy3(1), 035599 (2013).
[CrossRef]

Numer. Math.

J. D. Benamou and Y. Brenier, “A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem,” Numer. Math.84(3), 375–393 (2000).
[CrossRef]

Opt. Eng.

P. Benítez, J. C. Miñano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng.43(7), 1489–1502 (2004).
[CrossRef]

Opt. Express

Opt. Lett.

Pac. J. Math.

R. T. Rockafellar, “Characterization of the subdifferentials of convex functions,” Pac. J. Math.17(3), 497–510 (1966).
[CrossRef]

Proc. SPIE

H. Ries, “Laser beam shaping by double tailoring,” Proc. SPIE5876, 587607 (2005).
[CrossRef]

V. Oliker, “Geometric and variational methods in optical design of reflecting surfaces with prescribed illuminance properties,” Proc. SPIE5942, 594207 (2005).
[CrossRef]

W. A. Parkyn, “Illumination lenses designed by extrinsic differential geometry,” Proc. SPIE3482, 389–396 (1998).
[CrossRef]

B. Parkyn and D. Pelka, “Free form lenses designed by a pseudo-rectangular lawnmower algorithm,” Proc. SPIE6338, 633808 (2006).
[CrossRef]

Other

W. A. Parkyn and D. G. Pelka, “Free-form lenses for rectangular illumination zones,” Anthony, Inc., US Patent 7674019 (2010).

W. A. Parkyn, and Jr., “Illuminating lens designed by extrinsic differential geometry,” Teledyne Lighting and Display Products, Inc., US Patent 5924788 (1999).

W. B. Elmer, The Optical Design of Reflectors, 2nd ed. (Wiley, 1980), Chap.4.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts and Company, 2005).

M. Gdeisat and F. Lilley, “Two-dimensional phase unwrapping problem,” http://www.ljmu.ac.uk/GERI/CEORG_Docs/Two_Dimensional_Phase_Unwrapping_Final.pdf .

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Figures (8)

Fig. 1
Fig. 1

Sketch of the design problem.

Fig. 2
Fig. 2

The modified double freeform-surface construction method.

Fig. 3
Fig. 3

The flow diagram of the simultaneous point-by-point procedure for calculating the two desired freeform surfaces.

Fig. 4
Fig. 4

Sketch of the Galilean type dual-lens beam shaper setting.

Fig. 5
Fig. 5

(a) The desired target irradiance distribution, wherein the contrast is set as 4:1:0; (b) The prescribed output wavefront.

Fig. 6
Fig. 6

The resulting ray mapping for the given design, (a) The final adaptive mesh of the input beam; (b) A uniform Cartesian mesh of the output beam. For better visualization, the 256 × 256 grid was interpolated into an 32 × 32 one.

Fig. 7
Fig. 7

The (a) first and (b) second resulting freeform surfaces. They were uniformly interpolated into 32 × 32 grids for better visualization.

Fig. 8
Fig. 8

Simulation results: (a) the output irradiance and (b) the wavefront errors simulated from the beam shaper designed using the previous construction method; (c) the output irradiance and (d) wavefront errors simulated from the beam shaper designed using the modified construction method.

Tables (1)

Tables Icon

Table 1 Performance parameters of the simulated irradiance and wavefront

Equations (7)

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

I in (x,y)dxdy= I out (x',y')dx'dy'
I in (x,y)= I out (f,g)| J(f,g) |
( f(x,y),g(x,y) )=u(x,y)
I out ( u(x,y) )det 2 u(x,y)= I in (x,y)
u t =log( I out ( u(x,y) )det 2 u(x,y) I in (x,y) )
RRMSD= i,j=1 256 ( I s (i,j) I out (i,j) ) 2 / i,j 256 I out (i,j) 2
RMS= 1 256×256 i,j 256 ( z ' s (i,j)z'(i,j) ) 2

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