Abstract

We propose a method to design double smooth freeform surfaces applied in beam shaping with a ray mapping method in the paper. We couple the calculation of ray mapping and the construction of freeform surfaces to approach the surface normal field integrability condition based on the symplectic flow mapping scheme. In this paper, the incident beam wavefront is not limited to be planar or spherical. Several challenging design examples are presented that include transforming a circular Gaussian beam to an unconventional beam with variously shaped contour, and transforming an elliptic beam to a convergent beam with complex irradiance distribution in non-paraxial regime. The results show the high efficiency and feasibility of the proposed method in designing freeform optics for beam shaping applications.

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

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References

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  1. F. M. Dickey and S. C. Holswade, Laser Beam Shaping: Theory and Techniques (Marcel Dekker, 2000).
  2. F. M. Dickey, S. C. Holswade, T. E. Lizotte, and D. L. Shealy, Laser Beam Shaping Applications (Taylor Francis Group, 2006).
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  8. H. Ma, Z. Liu, P. Jiang, X. Xu, and S. Du, “Improvement of galilean refractive beam shaping system for accurately generating near-diffraction-limited flattop beam with arbitrary beam size,” Opt. Express 19(14), 13105–13117 (2011).
    [Crossref]
  9. D. L. Shealy and S.-H. Chao, “Geometric optics-based design of laser beam shapers,” Opt. Eng. 42(11), 3123–3139 (2003).
    [Crossref]
  10. Y. Zhang, R. Wu, P. Liu, Z. Zheng, H. Li, and X. Liu, “Double freeform surfaces design for laser beam shaping with monge–ampère equation method,” Opt. Commun. 331, 297–305 (2014).
    [Crossref]
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    [Crossref]
  12. V. Oliker, L. L. Doskolovich, and D. A. Bykov, “Beam shaping with a plano-freeform lens pair,” Opt. Express 26(15), 19406–19419 (2018).
    [Crossref]
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    [Crossref]
  16. Z. Feng, L. Huang, G. Jin, and M. Gong, “Designing double freeform optical surfaces for controlling both irradiance and wavefront,” Opt. Express 21(23), 28693–28701 (2013).
    [Crossref]
  17. Z. Feng, B. D. Froese, C.-Y. Huang, D. Ma, and R. Liang, “Creating unconventional geometric beams with large depth of field using double freeform-surface optics,” Appl. Opt. 54(20), 6277–6281 (2015).
    [Crossref]
  18. C. Bösel and H. Gross, “Ray mapping approach for the efficient design of continuous freeform surfaces,” Opt. Express 24(13), 14271–14282 (2016).
    [Crossref]
  19. C. Bösel, N. G. Worku, and H. Gross, “Ray-mapping approach in double freeform surface design for collimated beam shaping beyond the paraxial approximation,” Appl. Opt. 56(13), 3679–3688 (2017).
    [Crossref]
  20. L. L. Doskolovich, A. A. Mingazov, D. A. Bykov, E. S. Andreev, and E. A. Bezus, “Variational approach to calculation of light field eikonal function for illuminating a prescribed region,” Opt. Express 25(22), 26378–26392 (2017).
    [Crossref]
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    [Crossref]
  22. D. A. Bykov, L. L. Doskolovich, A. A. Mingazov, E. A. Bezus, and N. L. Kazanskiy, “Linear assignment problem in the design of freeform refractive optical elements generating prescribed irradiance distributions,” Opt. Express 26(21), 27812–27825 (2018).
    [Crossref]
  23. K. Desnijder, P. Hanselaer, and Y. Meuret, “Ray mapping method for off-axis and non-paraxial freeform illumination lens design,” Opt. Lett. 44(4), 771–774 (2019).
    [Crossref]
  24. Z. Feng, B. D. Froese, R. Liang, D. Cheng, and Y. Wang, “Simplified freeform optics design for complicated laser beam shaping,” Appl. Opt. 56(33), 9308–9314 (2017).
    [Crossref]
  25. C. Gannon and R. Liang, “Ray mapping with surface information for freeform illumination design,” Opt. Express 25(8), 9426–9434 (2017).
    [Crossref]
  26. Z. Feng, D. Cheng, and Y. Wang, “Iterative wavefront tailoring to simplify freeform optical design for prescribed irradiance,” Opt. Lett. 44(9), 2274–2277 (2019).
    [Crossref]
  27. K. Desnijder, P. Hanselaer, and Y. Meuret, “Flexible design method for freeform lenses with an arbitrary lens contour,” Opt. Lett. 42(24), 5238–5241 (2017).
    [Crossref]
  28. R. Wu, Y. Zhang, M. M. Sulman, Z. Zheng, P. Benítez, and J. C. Miñano, “Initial design with l 2 monge-kantorovich theory for the monge–ampère equation method in freeform surface illumination design,” Opt. Express 22(13), 16161–16177 (2014).
    [Crossref]
  29. B. D. Froese and A. M. Oberman, “Convergent filtered schemes for the monge–ampére partial differential equation,” SIAM J. on Numer. Analysis 51(1), 423–444 (2013).
    [Crossref]
  30. Z. Feng, B. D. Froese, and R. Liang, “Freeform illumination optics construction following an optimal transport map,” Appl. Opt. 55(16), 4301–4306 (2016).
    [Crossref]
  31. S. Haker, L. Zhu, A. Tannenbaum, and S. Angenent, “Optimal mass transport for registration and warping,” Int. J. computer vision 60(3), 225–240 (2004).
  32. M. M. Sulman, J. Williams, and R. D. Russell, “An efficient approach for the numerical solution of the monge–ampère equation,” Appl. Numer. Math. 61(3), 298–307 (2011).
    [Crossref]
  33. C. Prins, R. Beltman, J. ten Thije Boonkkamp, W. IJzerman, and T. W. Tukker, “A least-squares method for optimal transport using the monge-ampere equation,” SIAM J. on Sci. Comput. 37(6), B937–B961 (2015).
    [Crossref]
  34. S. Wei, Z. Zhu, Z. Fan, Y. Yan, and D. Ma, “Multi-surface catadioptric freeform lens design for ultra-efficient off-axis road illumination,” Opt. Express 27(12), A779–A789 (2019).
    [Crossref]
  35. L. Casetti, “Efficient symplectic algorithms for numerical simulations of hamiltonian flows,” Phys. Scr. 51(1), 29–34 (1995).
    [Crossref]

2019 (3)

2018 (3)

2017 (5)

2016 (3)

Z. Feng, B. D. Froese, and R. Liang, “Freeform illumination optics construction following an optimal transport map,” Appl. Opt. 55(16), 4301–4306 (2016).
[Crossref]

C. Bösel and H. Gross, “Ray mapping approach for the efficient design of continuous freeform surfaces,” Opt. Express 24(13), 14271–14282 (2016).
[Crossref]

S. Chang, R. Wu, L. An, and Z. Zheng, “Design beam shapers with double freeform surfaces to form a desired wavefront with prescribed illumination pattern by solving a monge-ampère type equation,” J. Opt. 18(12), 125602 (2016).
[Crossref]

2015 (2)

Z. Feng, B. D. Froese, C.-Y. Huang, D. Ma, and R. Liang, “Creating unconventional geometric beams with large depth of field using double freeform-surface optics,” Appl. Opt. 54(20), 6277–6281 (2015).
[Crossref]

C. Prins, R. Beltman, J. ten Thije Boonkkamp, W. IJzerman, and T. W. Tukker, “A least-squares method for optimal transport using the monge-ampere equation,” SIAM J. on Sci. Comput. 37(6), B937–B961 (2015).
[Crossref]

2014 (2)

R. Wu, Y. Zhang, M. M. Sulman, Z. Zheng, P. Benítez, and J. C. Miñano, “Initial design with l 2 monge-kantorovich theory for the monge–ampère equation method in freeform surface illumination design,” Opt. Express 22(13), 16161–16177 (2014).
[Crossref]

Y. Zhang, R. Wu, P. Liu, Z. Zheng, H. Li, and X. Liu, “Double freeform surfaces design for laser beam shaping with monge–ampère equation method,” Opt. Commun. 331, 297–305 (2014).
[Crossref]

2013 (2)

Z. Feng, L. Huang, G. Jin, and M. Gong, “Designing double freeform optical surfaces for controlling both irradiance and wavefront,” Opt. Express 21(23), 28693–28701 (2013).
[Crossref]

B. D. Froese and A. M. Oberman, “Convergent filtered schemes for the monge–ampére partial differential equation,” SIAM J. on Numer. Analysis 51(1), 423–444 (2013).
[Crossref]

2011 (3)

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

H. Ma, Z. Liu, P. Jiang, X. Xu, and S. Du, “Improvement of galilean refractive beam shaping system for accurately generating near-diffraction-limited flattop beam with arbitrary beam size,” Opt. Express 19(14), 13105–13117 (2011).
[Crossref]

2008 (1)

2007 (1)

2004 (1)

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

2003 (1)

D. L. Shealy and S.-H. Chao, “Geometric optics-based design of laser beam shapers,” Opt. Eng. 42(11), 3123–3139 (2003).
[Crossref]

2000 (1)

1995 (1)

L. Casetti, “Efficient symplectic algorithms for numerical simulations of hamiltonian flows,” Phys. Scr. 51(1), 29–34 (1995).
[Crossref]

1980 (1)

1965 (1)

1964 (1)

V. Galindo, “Design of dual-reflector antennas with arbitrary phase and amplitude distributions,” IEEE Trans. Antennas Propag. 12(4), 403–408 (1964).
[Crossref]

An, L.

S. Chang, R. Wu, L. An, and Z. Zheng, “Design beam shapers with double freeform surfaces to form a desired wavefront with prescribed illumination pattern by solving a monge-ampère type equation,” J. Opt. 18(12), 125602 (2016).
[Crossref]

Andreev, E. S.

Angenent, S.

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

Beltman, R.

C. Prins, R. Beltman, J. ten Thije Boonkkamp, W. IJzerman, and T. W. Tukker, “A least-squares method for optimal transport using the monge-ampere equation,” SIAM J. on Sci. Comput. 37(6), B937–B961 (2015).
[Crossref]

Benítez, P.

Bezus, E. A.

Bösel, C.

Bykov, D. A.

Casetti, L.

L. Casetti, “Efficient symplectic algorithms for numerical simulations of hamiltonian flows,” Phys. Scr. 51(1), 29–34 (1995).
[Crossref]

Chang, S.

S. Chang, R. Wu, L. An, and Z. Zheng, “Design beam shapers with double freeform surfaces to form a desired wavefront with prescribed illumination pattern by solving a monge-ampère type equation,” J. Opt. 18(12), 125602 (2016).
[Crossref]

Chao, S.-H.

D. L. Shealy and S.-H. Chao, “Geometric optics-based design of laser beam shapers,” Opt. Eng. 42(11), 3123–3139 (2003).
[Crossref]

Cheng, D.

Desnijder, K.

Dickey, F. M.

F. M. Dickey, S. C. Holswade, T. E. Lizotte, and D. L. Shealy, Laser Beam Shaping Applications (Taylor Francis Group, 2006).

F. M. Dickey and S. C. Holswade, Laser Beam Shaping: Theory and Techniques (Marcel Dekker, 2000).

Doskolovich, L. L.

Du, S.

Fan, Z.

Feng, Z.

Frieden, B. R.

Froese, B. D.

Galindo, V.

V. Galindo, “Design of dual-reflector antennas with arbitrary phase and amplitude distributions,” IEEE Trans. Antennas Propag. 12(4), 403–408 (1964).
[Crossref]

Gannon, C.

Gong, M.

Gross, H.

Haker, S.

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

Hanselaer, P.

Hoffnagle, J. A.

Holswade, S. C.

F. M. Dickey and S. C. Holswade, Laser Beam Shaping: Theory and Techniques (Marcel Dekker, 2000).

F. M. Dickey, S. C. Holswade, T. E. Lizotte, and D. L. Shealy, Laser Beam Shaping Applications (Taylor Francis Group, 2006).

Huang, C.-Y.

Huang, L.

IJzerman, W.

C. Prins, R. Beltman, J. ten Thije Boonkkamp, W. IJzerman, and T. W. Tukker, “A least-squares method for optimal transport using the monge-ampere equation,” SIAM J. on Sci. Comput. 37(6), B937–B961 (2015).
[Crossref]

Jefferson, C. M.

Jiang, P.

Jin, G.

Kazanskiy, N. L.

Kreuzer, J. L.

J. L. Kreuzer, “Coherent light optical system yielding an output beam of desired intensity distribution at a desired equiphase surface,” (1969). US Patent 3,476,463.

Li, H.

Y. Zhang, R. Wu, P. Liu, Z. Zheng, H. Li, and X. Liu, “Double freeform surfaces design for laser beam shaping with monge–ampère equation method,” Opt. Commun. 331, 297–305 (2014).
[Crossref]

Liang, R.

Liu, P.

Y. Zhang, R. Wu, P. Liu, Z. Zheng, H. Li, and X. Liu, “Double freeform surfaces design for laser beam shaping with monge–ampère equation method,” Opt. Commun. 331, 297–305 (2014).
[Crossref]

Liu, X.

Y. Zhang, R. Wu, P. Liu, Z. Zheng, H. Li, and X. Liu, “Double freeform surfaces design for laser beam shaping with monge–ampère equation method,” Opt. Commun. 331, 297–305 (2014).
[Crossref]

Liu, Z.

Lizotte, T. E.

F. M. Dickey, S. C. Holswade, T. E. Lizotte, and D. L. Shealy, Laser Beam Shaping Applications (Taylor Francis Group, 2006).

Ma, D.

Ma, H.

Meuret, Y.

Miñano, J. C.

Mingazov, A. A.

Oberman, A. M.

B. D. Froese and A. M. Oberman, “Convergent filtered schemes for the monge–ampére partial differential equation,” SIAM J. on Numer. Analysis 51(1), 423–444 (2013).
[Crossref]

Oliker, V.

Prins, C.

C. Prins, R. Beltman, J. ten Thije Boonkkamp, W. IJzerman, and T. W. Tukker, “A least-squares method for optimal transport using the monge-ampere equation,” SIAM J. on Sci. Comput. 37(6), B937–B961 (2015).
[Crossref]

Rhodes, P. W.

Russell, R. D.

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

Shealy, D. L.

D. L. Shealy and S.-H. Chao, “Geometric optics-based design of laser beam shapers,” Opt. Eng. 42(11), 3123–3139 (2003).
[Crossref]

P. W. Rhodes and D. L. Shealy, “Refractive optical systems for irradiance redistribution of collimated radiation: their design and analysis,” Appl. Opt. 19(20), 3545–3553 (1980).
[Crossref]

F. M. Dickey, S. C. Holswade, T. E. Lizotte, and D. L. Shealy, Laser Beam Shaping Applications (Taylor Francis Group, 2006).

Sulman, M. M.

Tannenbaum, A.

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

ten Thije Boonkkamp, J.

C. Prins, R. Beltman, J. ten Thije Boonkkamp, W. IJzerman, and T. W. Tukker, “A least-squares method for optimal transport using the monge-ampere equation,” SIAM J. on Sci. Comput. 37(6), B937–B961 (2015).
[Crossref]

Tukker, T. W.

C. Prins, R. Beltman, J. ten Thije Boonkkamp, W. IJzerman, and T. W. Tukker, “A least-squares method for optimal transport using the monge-ampere equation,” SIAM J. on Sci. Comput. 37(6), B937–B961 (2015).
[Crossref]

Wang, Y.

Wei, S.

Williams, J.

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

Worku, N. G.

Wu, R.

S. Chang, R. Wu, L. An, and Z. Zheng, “Design beam shapers with double freeform surfaces to form a desired wavefront with prescribed illumination pattern by solving a monge-ampère type equation,” J. Opt. 18(12), 125602 (2016).
[Crossref]

Y. Zhang, R. Wu, P. Liu, Z. Zheng, H. Li, and X. Liu, “Double freeform surfaces design for laser beam shaping with monge–ampère equation method,” Opt. Commun. 331, 297–305 (2014).
[Crossref]

R. Wu, Y. Zhang, M. M. Sulman, Z. Zheng, P. Benítez, and J. C. Miñano, “Initial design with l 2 monge-kantorovich theory for the monge–ampère equation method in freeform surface illumination design,” Opt. Express 22(13), 16161–16177 (2014).
[Crossref]

Xu, X.

Yan, Y.

Zhang, Y.

R. Wu, Y. Zhang, M. M. Sulman, Z. Zheng, P. Benítez, and J. C. Miñano, “Initial design with l 2 monge-kantorovich theory for the monge–ampère equation method in freeform surface illumination design,” Opt. Express 22(13), 16161–16177 (2014).
[Crossref]

Y. Zhang, R. Wu, P. Liu, Z. Zheng, H. Li, and X. Liu, “Double freeform surfaces design for laser beam shaping with monge–ampère equation method,” Opt. Commun. 331, 297–305 (2014).
[Crossref]

Zheng, Z.

S. Chang, R. Wu, L. An, and Z. Zheng, “Design beam shapers with double freeform surfaces to form a desired wavefront with prescribed illumination pattern by solving a monge-ampère type equation,” J. Opt. 18(12), 125602 (2016).
[Crossref]

Y. Zhang, R. Wu, P. Liu, Z. Zheng, H. Li, and X. Liu, “Double freeform surfaces design for laser beam shaping with monge–ampère equation method,” Opt. Commun. 331, 297–305 (2014).
[Crossref]

R. Wu, Y. Zhang, M. M. Sulman, Z. Zheng, P. Benítez, and J. C. Miñano, “Initial design with l 2 monge-kantorovich theory for the monge–ampère equation method in freeform surface illumination design,” Opt. Express 22(13), 16161–16177 (2014).
[Crossref]

Zhu, L.

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

Zhu, Z.

Appl. Numer. Math. (1)

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

Appl. Opt. (7)

Arch. Ration. Mech. Anal. (1)

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]

IEEE Trans. Antennas Propag. (1)

V. Galindo, “Design of dual-reflector antennas with arbitrary phase and amplitude distributions,” IEEE Trans. Antennas Propag. 12(4), 403–408 (1964).
[Crossref]

Int. J. computer vision (1)

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

J. Opt. (1)

S. Chang, R. Wu, L. An, and Z. Zheng, “Design beam shapers with double freeform surfaces to form a desired wavefront with prescribed illumination pattern by solving a monge-ampère type equation,” J. Opt. 18(12), 125602 (2016).
[Crossref]

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

Opt. Commun. (1)

Y. Zhang, R. Wu, P. Liu, Z. Zheng, H. Li, and X. Liu, “Double freeform surfaces design for laser beam shaping with monge–ampère equation method,” Opt. Commun. 331, 297–305 (2014).
[Crossref]

Opt. Eng. (1)

D. L. Shealy and S.-H. Chao, “Geometric optics-based design of laser beam shapers,” Opt. Eng. 42(11), 3123–3139 (2003).
[Crossref]

Opt. Express (10)

H. Ma, Z. Liu, P. Jiang, X. Xu, and S. Du, “Improvement of galilean refractive beam shaping system for accurately generating near-diffraction-limited flattop beam with arbitrary beam size,” Opt. Express 19(14), 13105–13117 (2011).
[Crossref]

V. Oliker, L. L. Doskolovich, and D. A. Bykov, “Beam shaping with a plano-freeform lens pair,” Opt. Express 26(15), 19406–19419 (2018).
[Crossref]

Z. Feng, L. Huang, G. Jin, and M. Gong, “Designing double freeform optical surfaces for controlling both irradiance and wavefront,” Opt. Express 21(23), 28693–28701 (2013).
[Crossref]

C. Bösel and H. Gross, “Ray mapping approach for the efficient design of continuous freeform surfaces,” Opt. Express 24(13), 14271–14282 (2016).
[Crossref]

R. Wu, Y. Zhang, M. M. Sulman, Z. Zheng, P. Benítez, and J. C. Miñano, “Initial design with l 2 monge-kantorovich theory for the monge–ampère equation method in freeform surface illumination design,” Opt. Express 22(13), 16161–16177 (2014).
[Crossref]

S. Wei, Z. Zhu, Z. Fan, Y. Yan, and D. Ma, “Multi-surface catadioptric freeform lens design for ultra-efficient off-axis road illumination,” Opt. Express 27(12), A779–A789 (2019).
[Crossref]

C. Gannon and R. Liang, “Ray mapping with surface information for freeform illumination design,” Opt. Express 25(8), 9426–9434 (2017).
[Crossref]

L. L. Doskolovich, A. A. Mingazov, D. A. Bykov, E. S. Andreev, and E. A. Bezus, “Variational approach to calculation of light field eikonal function for illuminating a prescribed region,” Opt. Express 25(22), 26378–26392 (2017).
[Crossref]

L. L. Doskolovich, D. A. Bykov, E. S. Andreev, E. A. Bezus, and V. Oliker, “Designing double freeform surfaces for collimated beam shaping with optimal mass transportation and linear assignment problems,” Opt. Express 26(19), 24602–24613 (2018).
[Crossref]

D. A. Bykov, L. L. Doskolovich, A. A. Mingazov, E. A. Bezus, and N. L. Kazanskiy, “Linear assignment problem in the design of freeform refractive optical elements generating prescribed irradiance distributions,” Opt. Express 26(21), 27812–27825 (2018).
[Crossref]

Opt. Lett. (3)

Phys. Scr. (1)

L. Casetti, “Efficient symplectic algorithms for numerical simulations of hamiltonian flows,” Phys. Scr. 51(1), 29–34 (1995).
[Crossref]

SIAM J. on Numer. Analysis (1)

B. D. Froese and A. M. Oberman, “Convergent filtered schemes for the monge–ampére partial differential equation,” SIAM J. on Numer. Analysis 51(1), 423–444 (2013).
[Crossref]

SIAM J. on Sci. Comput. (1)

C. Prins, R. Beltman, J. ten Thije Boonkkamp, W. IJzerman, and T. W. Tukker, “A least-squares method for optimal transport using the monge-ampere equation,” SIAM J. on Sci. Comput. 37(6), B937–B961 (2015).
[Crossref]

Other (3)

J. L. Kreuzer, “Coherent light optical system yielding an output beam of desired intensity distribution at a desired equiphase surface,” (1969). US Patent 3,476,463.

F. M. Dickey and S. C. Holswade, Laser Beam Shaping: Theory and Techniques (Marcel Dekker, 2000).

F. M. Dickey, S. C. Holswade, T. E. Lizotte, and D. L. Shealy, Laser Beam Shaping Applications (Taylor Francis Group, 2006).

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Figures (13)

Fig. 1.
Fig. 1. Sketch of the freeform beam shaping system.
Fig. 2.
Fig. 2. The conceptual diagram of the initial mapping calculation.
Fig. 3.
Fig. 3. Freeform surface construction process.
Fig. 4.
Fig. 4. Iteration process via symplectic integration coupled with surface construction
Fig. 5.
Fig. 5. Design specifications for collimated beam shaping.
Fig. 6.
Fig. 6. (a) The designed beam shaping system using integrable mapping; (b,c) the comparison between OMT mapping and integrable mapping.
Fig. 7.
Fig. 7. (a) The irradiance distribution and row irradiance in the plane $z$=10 mm; (b) The irradiance distribution and row irradiance in the plane $z$=50 mm.
Fig. 8.
Fig. 8. The convergence of the algorithm and the running time.
Fig. 9.
Fig. 9. Irradiance distributions of uniform hexagonal beam, fan-shaped beam and rectangular beam with prescribed irradiance ”OEI” generated by lenses in the plane $z$=12 mm and $z$=30 mm.
Fig. 10.
Fig. 10. (a) The design sketch diagram of non-planar input wavefront beam; (b) the incident beam irradiance; (c) the incident wavefront.
Fig. 11.
Fig. 11. (a) The integrable mapping of source and target; (b) the simulated irradiance distribution at the target plane of the designed system; (c) the row and column irradiance distributions as well as the comparisons of the simulated results and the desired results.
Fig. 12.
Fig. 12. The convergence of the algorithm and the running time.
Fig. 13.
Fig. 13. The designed system and relative irradiance distributions at different distances.

Equations (15)

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d e t ( D m ) = h ( ξ , η ) g ( m ( ξ , η ) ) ,
m ( H ) = Y ,
J B ( m , b ) = 1 2 H | m b | 2 d s ,
J 1 ( m , P ) = 1 2 H | | D m P | | 2 d ξ d η ,
J ( m , P , b ) = ( 1 α ) J B ( m , b ) + α J 1 ( m , P ) .
N = n i R n o I ,
( × N ) I = 0 ,
( × N ) I = ( × n i R n o I n i R I n o ) I = ( × n i R n i R I n o ) I ( × n o I n i R I n o ) I .
( × R ) I = 0 ,
C ( ξ , η ) = R η W ξ R ξ W η , ,
ϕ ( ξ , η ) = cos ( π 2 ξ ) cos ( π 2 η ) .
H ( ξ , η ) = C ( ξ , η ) ϕ ( ξ , η ) .
ξ ˙ = η H ( ξ , η ) , η ˙ =       ξ H ( ξ , η ) ,
ξ n + 1 = ξ n h ξ H ( ξ n + 1 , η n ) , η n + 1 = η n + h η H ( ξ n + 1 , η n ) ,
M F = i = 1 m j = 1 n H ( ξ i , j , η i , j ) ,