Abstract

Control of the optical fields of laser beams, i.e., laser beam shaping, is of great importance to many laser applications. Freeform optics offers plenty of advantages for complex beam shaping requirements, including precise beam control, energy efficiency, compact structure, and relatively low cost. We present a modified ray-mapping method to simplify the freeform optics design for complicated optical field control and achieve a challenging task of producing two prescribed beam profiles on two successive target planes. This method begins by calculating an approximate output ray sequence that connects the two prescribed beam profiles and a corresponding input ray sequence. After setting an initial profile of the first freeform optical surface on the input ray sequence, we can obtain the second freeform optical surface based on the optical path length constancy between the given input wavefront and the computed output wavefront. Then, we can acquire all the normal vectors of the first freeform optical surface using Snell’s law and approximately reconstruct the first freeform optical surface by solving a relationship between the surface points and normal vectors using a fast least squares method. The surface construction process is repeated until the stop criterion is satisfied. We design three freeform lenses, and Monte Carlo simulations demonstrate their abilities of simultaneously producing two challenging beam profiles from a divergent Gaussian beam.

© 2017 Optical Society of America

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References

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    [Crossref]

2017 (2)

T. Zhou, X. Liu, Z. Liang, Y. Liu, J. Xie, and X. Wang, “Recent advancements in optical microstructure fabrication through glass molding process,” Front. Mech. Eng. 12, 46–65 (2017).
[Crossref]

X. Wu, G. Jin, and J. Zhu, “Freeform illumination design model for multiple light sources simultaneously,” Appl. Opt. 56, 2405–2411 (2017).
[Crossref]

2016 (6)

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

Y. Jin, A. Hassan, and Y. Jiang, “Freeform microlens array homogenizer for excimer laser beam shaping,” Opt. Express 24, 24846–24858 (2016).
[Crossref]

T. Gissibl, S. Thiele, A. Herkommer, and H. Giessen, “Sub-micrometre accurate free-form optics by three-dimensional printing on single-mode fibres,” Nat. Commun. 7, 11763 (2016).
[Crossref]

H. Li, N. J. Naples, X. Zhao, and A. Y. Yi, “An integrated approach to design and fabrication of a miniature endoscope using freeform optics,” Adv. Opt. Technol. 5, 335–342 (2016).

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

C. Bösel and H. Gross, “Ray mapping approach in double freeform surface design for collimated beam shaping,” Proc. SPIE 9950, 995004 (2016).
[Crossref]

2015 (2)

2014 (4)

J. Zhu, X. Wu, T. Yang, and G. Jin, “Generating optical freeform surfaces considering both coordinates and normals of discrete data points,” J. Opt. Soc. Am. A 31, 2401–2408 (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-Ampére equation method,” Opt. Commun. 331, 297–305 (2014).
[Crossref]

M. Maksimovic, “Design and optimization of compact freeform lens array for laser beam splitting: a case study in optimal surface representation,” Proc. SPIE 9131, 913107 (2014).
[Crossref]

J. D. Benamou, B. D. Froese, and A. M. Oberman, “Numerical solution of the optimal transportation problem using the Monge-Ampére equation,” J. Comput. Phys. 260, 107–126 (2014).
[Crossref]

2013 (7)

2012 (2)

2011 (2)

A. Bruneton, A. Bäuerle, P. Loosen, and R. Wester, “Freeform lens for an efficient wall washer,” Proc. SPIE 8167, 816707 (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, 1013–1045 (2011).
[Crossref]

2008 (1)

2007 (1)

2005 (2)

R. A. Hicks, “Designing a mirror to realize a given projection,” J. Opt. Soc. Am. 22, 323–329 (2005).
[Crossref]

H. Ries, “Laser beam shaping by double tailoring,” Proc. SPIE 5876, 587607 (2005).
[Crossref]

2004 (3)

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, 1255–1278 (2004).
[Crossref]

P. Benítez and J. C. Miñano, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43, 1489–1502 (2004).
[Crossref]

J. Rubinstein and G. Wolansky, “A variational principle in optics,” J. Opt. Soc. Am. A 21, 2164–2172 (2004).
[Crossref]

2003 (1)

D. L. Shealy and S. Chao, “Geometric optics-based design of laser beam shapers,” Opt. Eng. 42, 3123–3138 (2003).
[Crossref]

2002 (1)

2000 (1)

1998 (1)

1994 (1)

K. Nemoto, T. Fujii, and N. Goto, “Laser beam-forming by deformable mirror,” Proc. SPIE 2119, 155–161 (1994).
[Crossref]

1980 (2)

1974 (1)

1972 (1)

Barsky, B.

M. Halstead, B. Barsky, S. Klein, and R. Mandell, “Reconstructing curved surfaces from specular reflection patterns using spline surface fitting of normals,” in 23rd Annual Conference on Computer Graphics and Interactive Techniques (Association of Computing Machinery, 1996), pp. 335–342.

Bäuerle, A.

Benamou, J. D.

J. D. Benamou, B. D. Froese, and A. M. Oberman, “Numerical solution of the optimal transportation problem using the Monge-Ampére equation,” J. Comput. Phys. 260, 107–126 (2014).
[Crossref]

Benítez, P.

P. Benítez and J. C. Miñano, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43, 1489–1502 (2004).
[Crossref]

R. Winston, J. C. Miñano, P. Benítez, N. Shatz, and J. C. Bortz, “Concentrators for prescribed irradiance optics,” in Nonimaging Optics (Elsevier, 2005), pp. 159–180.

Bortz, J. C.

R. Winston, J. C. Miñano, P. Benítez, N. Shatz, and J. C. Bortz, “Concentrators for prescribed irradiance optics,” in Nonimaging Optics (Elsevier, 2005), pp. 159–180.

Bösel, C.

C. Bösel and H. Gross, “Ray mapping approach in double freeform surface design for collimated beam shaping,” Proc. SPIE 9950, 995004 (2016).
[Crossref]

Brenner, K.-H.

K.-H. Brenner, “General solution of two-dimensional beam-shaping with two surfaces,” in Information Optics and Photonics: Algorithms, Systems, and Applications, B. Javidi, ed. (Springer, 2010), pp. 3–11.

Bruneton, A.

Bryngdahl, O.

Chang, S.

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

Chao, S.

D. L. Shealy and S. Chao, “Geometric optics-based design of laser beam shapers,” Opt. Eng. 42, 3123–3138 (2003).
[Crossref]

Dickey, F. M.

L. A. Romero and F. M. Dickey, “The mathematical and physical theory of lossless beam shaping,” in Laser Beam Shaping: Theory and Techniques, F. M. Dickey and S. C. Holswade, eds. (Marcel Dekker, 2000), pp. 90–92.

Evans, C.

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

Evans, N. C.

Fang, F. Z.

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

Feng, Z.

Froese, B. D.

Fujii, T.

K. Nemoto, T. Fujii, and N. Goto, “Laser beam-forming by deformable mirror,” Proc. SPIE 2119, 155–161 (1994).
[Crossref]

Giessen, H.

T. Gissibl, S. Thiele, A. Herkommer, and H. Giessen, “Sub-micrometre accurate free-form optics by three-dimensional printing on single-mode fibres,” Nat. Commun. 7, 11763 (2016).
[Crossref]

Gissibl, T.

T. Gissibl, S. Thiele, A. Herkommer, and H. Giessen, “Sub-micrometre accurate free-form optics by three-dimensional printing on single-mode fibres,” Nat. Commun. 7, 11763 (2016).
[Crossref]

Glimm, T.

T. Glimm and N. Henscheid, “Iterative scheme for solving optimal transportation problems arising in reflector design,” ISRN Appl. Math. 2013, 1–12 (2013).
[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, 1255–1278 (2004).
[Crossref]

Gong, M.

Goto, N.

K. Nemoto, T. Fujii, and N. Goto, “Laser beam-forming by deformable mirror,” Proc. SPIE 2119, 155–161 (1994).
[Crossref]

Gross, H.

C. Bösel and H. Gross, “Ray mapping approach in double freeform surface design for collimated beam shaping,” Proc. SPIE 9950, 995004 (2016).
[Crossref]

Grossberg, M. D.

R. Swaminathan, S. K. Nayar, and M. D. Grossberg, “Framework to design catadioptric imaging and projection systems,” in IEEE Conference on ICCV-PROCAMS (2003).

Halstead, M.

M. Halstead, B. Barsky, S. Klein, and R. Mandell, “Reconstructing curved surfaces from specular reflection patterns using spline surface fitting of normals,” in 23rd Annual Conference on Computer Graphics and Interactive Techniques (Association of Computing Machinery, 1996), pp. 335–342.

Hassan, A.

Henscheid, N.

T. Glimm and N. Henscheid, “Iterative scheme for solving optimal transportation problems arising in reflector design,” ISRN Appl. Math. 2013, 1–12 (2013).
[Crossref]

Herkommer, A.

T. Gissibl, S. Thiele, A. Herkommer, and H. Giessen, “Sub-micrometre accurate free-form optics by three-dimensional printing on single-mode fibres,” Nat. Commun. 7, 11763 (2016).
[Crossref]

Hermann, J.

Hicks, R. A.

R. A. Hicks, “Designing a mirror to realize a given projection,” J. Opt. Soc. Am. 22, 323–329 (2005).
[Crossref]

R. A. Hicks and R. K. Perline, “Geometric distributions for catadioptric sensor design,” in IEEE Conference on CVPR (2001), pp. 584–589.

Huang, C.

Huang, L.

Jiang, Y.

Jin, G.

Jin, Y.

Kirkici, H.

Klein, S.

M. Halstead, B. Barsky, S. Klein, and R. Mandell, “Reconstructing curved surfaces from specular reflection patterns using spline surface fitting of normals,” in 23rd Annual Conference on Computer Graphics and Interactive Techniques (Association of Computing Machinery, 1996), pp. 335–342.

Li, A.

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

Li, G.

Li, H.

H. Li, N. J. Naples, X. Zhao, and A. Y. Yi, “An integrated approach to design and fabrication of a miniature endoscope using freeform optics,” Adv. Opt. Technol. 5, 335–342 (2016).

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

R. Wu, L. Xu, P. Liu, Y. Zhang, Z. Zheng, H. Li, and X. Liu, “Freeform illumination design: a nonlinear boundary problem for the elliptic Monge-Ampére equation,” Opt. Lett. 38, 229–231 (2013).
[Crossref]

Li, T.

Li, Y.

Liang, R.

Liang, Z.

T. Zhou, X. Liu, Z. Liang, Y. Liu, J. Xie, and X. Wang, “Recent advancements in optical microstructure fabrication through glass molding process,” Front. Mech. Eng. 12, 46–65 (2017).
[Crossref]

Liu, K.

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-Ampére equation method,” Opt. Commun. 331, 297–305 (2014).
[Crossref]

R. Wu, L. Xu, P. Liu, Y. Zhang, Z. Zheng, H. Li, and X. Liu, “Freeform illumination design: a nonlinear boundary problem for the elliptic Monge-Ampére equation,” Opt. Lett. 38, 229–231 (2013).
[Crossref]

Liu, X.

T. Zhou, X. Liu, Z. Liang, Y. Liu, J. Xie, and X. Wang, “Recent advancements in optical microstructure fabrication through glass molding process,” Front. Mech. Eng. 12, 46–65 (2017).
[Crossref]

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

R. Wu, L. Xu, P. Liu, Y. Zhang, Z. Zheng, H. Li, and X. Liu, “Freeform illumination design: a nonlinear boundary problem for the elliptic Monge-Ampére equation,” Opt. Lett. 38, 229–231 (2013).
[Crossref]

Liu, Y.

T. Zhou, X. Liu, Z. Liang, Y. Liu, J. Xie, and X. Wang, “Recent advancements in optical microstructure fabrication through glass molding process,” Front. Mech. Eng. 12, 46–65 (2017).
[Crossref]

Loosen, P.

Ma, D.

Ma, X.

Maksimovic, M.

M. Maksimovic, “Design and optimization of compact freeform lens array for laser beam splitting: a case study in optimal surface representation,” Proc. SPIE 9131, 913107 (2014).
[Crossref]

Mandell, R.

M. Halstead, B. Barsky, S. Klein, and R. Mandell, “Reconstructing curved surfaces from specular reflection patterns using spline surface fitting of normals,” in 23rd Annual Conference on Computer Graphics and Interactive Techniques (Association of Computing Machinery, 1996), pp. 335–342.

Miñano, J. C.

P. Benítez and J. C. Miñano, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43, 1489–1502 (2004).
[Crossref]

R. Winston, J. C. Miñano, P. Benítez, N. Shatz, and J. C. Bortz, “Concentrators for prescribed irradiance optics,” in Nonimaging Optics (Elsevier, 2005), pp. 159–180.

Muschaweck, J.

Naples, N. J.

H. Li, N. J. Naples, X. Zhao, and A. Y. Yi, “An integrated approach to design and fabrication of a miniature endoscope using freeform optics,” Adv. Opt. Technol. 5, 335–342 (2016).

Nayar, S. K.

R. Swaminathan, S. K. Nayar, and M. D. Grossberg, “Framework to design catadioptric imaging and projection systems,” in IEEE Conference on ICCV-PROCAMS (2003).

Nemoto, K.

K. Nemoto, T. Fujii, and N. Goto, “Laser beam-forming by deformable mirror,” Proc. SPIE 2119, 155–161 (1994).
[Crossref]

Oberman, A. M.

J. D. Benamou, B. D. Froese, and A. M. Oberman, “Numerical solution of the optimal transportation problem using the Monge-Ampére equation,” J. Comput. Phys. 260, 107–126 (2014).
[Crossref]

Oliker, V.

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, 1013–1045 (2011).
[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, 1255–1278 (2004).
[Crossref]

Perline, R. K.

R. A. Hicks and R. K. Perline, “Geometric distributions for catadioptric sensor design,” in IEEE Conference on CVPR (2001), pp. 584–589.

Qiu, Y.

Ries, H.

H. Ries, “Laser beam shaping by double tailoring,” Proc. SPIE 5876, 587607 (2005).
[Crossref]

H. Ries and J. Muschaweck, “Tailored freeform optical surfaces,” J. Opt. Soc. Am. A 19, 590–595 (2002).
[Crossref]

Romero, L. A.

L. A. Romero and F. M. Dickey, “The mathematical and physical theory of lossless beam shaping,” in Laser Beam Shaping: Theory and Techniques, F. M. Dickey and S. C. Holswade, eds. (Marcel Dekker, 2000), pp. 90–92.

Rubinstein, J.

Schruben, J. S.

Serkan, M.

Shatz, N.

R. Winston, J. C. Miñano, P. Benítez, N. Shatz, and J. C. Bortz, “Concentrators for prescribed irradiance optics,” in Nonimaging Optics (Elsevier, 2005), pp. 159–180.

Shealy, D. L.

Southwell, W. H.

Stollenwerk, J.

Swaminathan, R.

R. Swaminathan, S. K. Nayar, and M. D. Grossberg, “Framework to design catadioptric imaging and projection systems,” in IEEE Conference on ICCV-PROCAMS (2003).

Thiele, S.

T. Gissibl, S. Thiele, A. Herkommer, and H. Giessen, “Sub-micrometre accurate free-form optics by three-dimensional printing on single-mode fibres,” Nat. Commun. 7, 11763 (2016).
[Crossref]

Wang, H.

Wang, X.

T. Zhou, X. Liu, Z. Liang, Y. Liu, J. Xie, and X. Wang, “Recent advancements in optical microstructure fabrication through glass molding process,” Front. Mech. Eng. 12, 46–65 (2017).
[Crossref]

Weckenmann, A.

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

Wester, R.

Winston, R.

R. Winston, J. C. Miñano, P. Benítez, N. Shatz, and J. C. Bortz, “Concentrators for prescribed irradiance optics,” in Nonimaging Optics (Elsevier, 2005), pp. 159–180.

Wolansky, G.

Wu, R.

S. Chang, R. Wu, A. Li, and Z. Zheng, “Design beam shapers with double freeform surfaces to form a desired wavefront with prescribed illumination pattern by solving a Monge-Ampére type equation,” J. Opt. 18, 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-Ampére equation method,” Opt. Commun. 331, 297–305 (2014).
[Crossref]

R. Wu, L. Xu, P. Liu, Y. Zhang, Z. Zheng, H. Li, and X. Liu, “Freeform illumination design: a nonlinear boundary problem for the elliptic Monge-Ampére equation,” Opt. Lett. 38, 229–231 (2013).
[Crossref]

Wu, X.

Xie, J.

T. Zhou, X. Liu, Z. Liang, Y. Liu, J. Xie, and X. Wang, “Recent advancements in optical microstructure fabrication through glass molding process,” Front. Mech. Eng. 12, 46–65 (2017).
[Crossref]

Xu, L.

Xue, Q.

Yang, T.

Yi, A. Y.

H. Li, N. J. Naples, X. Zhao, and A. Y. Yi, “An integrated approach to design and fabrication of a miniature endoscope using freeform optics,” Adv. Opt. Technol. 5, 335–342 (2016).

Zhang, G.

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

Zhang, X. D.

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

Zhang, Y.

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

R. Wu, L. Xu, P. Liu, Y. Zhang, Z. Zheng, H. Li, and X. Liu, “Freeform illumination design: a nonlinear boundary problem for the elliptic Monge-Ampére equation,” Opt. Lett. 38, 229–231 (2013).
[Crossref]

Zhang, Z.

Zhao, X.

H. Li, N. J. Naples, X. Zhao, and A. Y. Yi, “An integrated approach to design and fabrication of a miniature endoscope using freeform optics,” Adv. Opt. Technol. 5, 335–342 (2016).

Zheng, Z.

S. Chang, R. Wu, A. Li, and Z. Zheng, “Design beam shapers with double freeform surfaces to form a desired wavefront with prescribed illumination pattern by solving a Monge-Ampére type equation,” J. Opt. 18, 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-Ampére equation method,” Opt. Commun. 331, 297–305 (2014).
[Crossref]

R. Wu, L. Xu, P. Liu, Y. Zhang, Z. Zheng, H. Li, and X. Liu, “Freeform illumination design: a nonlinear boundary problem for the elliptic Monge-Ampére equation,” Opt. Lett. 38, 229–231 (2013).
[Crossref]

Zhou, T.

T. Zhou, X. Liu, Z. Liang, Y. Liu, J. Xie, and X. Wang, “Recent advancements in optical microstructure fabrication through glass molding process,” Front. Mech. Eng. 12, 46–65 (2017).
[Crossref]

Zhu, J.

Zou, W.

Adv. Opt. Technol. (1)

H. Li, N. J. Naples, X. Zhao, and A. Y. Yi, “An integrated approach to design and fabrication of a miniature endoscope using freeform optics,” Adv. Opt. Technol. 5, 335–342 (2016).

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Arch. Ration. Mech. Anal. (1)

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[Crossref]

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[Crossref]

Front. Mech. Eng. (1)

T. Zhou, X. Liu, Z. Liang, Y. Liu, J. Xie, and X. Wang, “Recent advancements in optical microstructure fabrication through glass molding process,” Front. Mech. Eng. 12, 46–65 (2017).
[Crossref]

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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, 1255–1278 (2004).
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T. Glimm and N. Henscheid, “Iterative scheme for solving optimal transportation problems arising in reflector design,” ISRN Appl. Math. 2013, 1–12 (2013).
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J. D. Benamou, B. D. Froese, and A. M. Oberman, “Numerical solution of the optimal transportation problem using the Monge-Ampére equation,” J. Comput. Phys. 260, 107–126 (2014).
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S. Chang, R. Wu, A. Li, and Z. Zheng, “Design beam shapers with double freeform surfaces to form a desired wavefront with prescribed illumination pattern by solving a Monge-Ampére type equation,” J. Opt. 18, 125602 (2016).
[Crossref]

J. Opt. Soc. Am. (5)

J. Opt. Soc. Am. A (5)

Nat. Commun. (1)

T. Gissibl, S. Thiele, A. Herkommer, and H. Giessen, “Sub-micrometre accurate free-form optics by three-dimensional printing on single-mode fibres,” Nat. Commun. 7, 11763 (2016).
[Crossref]

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-Ampére equation method,” Opt. Commun. 331, 297–305 (2014).
[Crossref]

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D. L. Shealy and S. Chao, “Geometric optics-based design of laser beam shapers,” Opt. Eng. 42, 3123–3138 (2003).
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P. Benítez and J. C. Miñano, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43, 1489–1502 (2004).
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M. Maksimovic, “Design and optimization of compact freeform lens array for laser beam splitting: a case study in optimal surface representation,” Proc. SPIE 9131, 913107 (2014).
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R. A. Hicks and R. K. Perline, “Geometric distributions for catadioptric sensor design,” in IEEE Conference on CVPR (2001), pp. 584–589.

R. Swaminathan, S. K. Nayar, and M. D. Grossberg, “Framework to design catadioptric imaging and projection systems,” in IEEE Conference on ICCV-PROCAMS (2003).

R. Winston, J. C. Miñano, P. Benítez, N. Shatz, and J. C. Bortz, “Concentrators for prescribed irradiance optics,” in Nonimaging Optics (Elsevier, 2005), pp. 159–180.

K.-H. Brenner, “General solution of two-dimensional beam-shaping with two surfaces,” in Information Optics and Photonics: Algorithms, Systems, and Applications, B. Javidi, ed. (Springer, 2010), pp. 3–11.

L. A. Romero and F. M. Dickey, “The mathematical and physical theory of lossless beam shaping,” in Laser Beam Shaping: Theory and Techniques, F. M. Dickey and S. C. Holswade, eds. (Marcel Dekker, 2000), pp. 90–92.

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Figures (7)

Fig. 1.
Fig. 1.

Schematic of a single lens with double freeform optical surfaces for producing two prescribed irradiance distributions from a given input beam. The input beam from the light source can be collimated or divergent with or without aberrations. The ray bending should be monotonic.

Fig. 2.
Fig. 2.

Flow diagram of the design method.

Fig. 3.
Fig. 3.

Sketch of the relationship between the wavefront surface points and unit output ray vectors.

Fig. 4.
Fig. 4.

(a) Model of Lens A and its simulation results; (b) computed output wavefront with the PV of 0.4798 mm; (c) first freeform optical surface with the PV of 0.8142 mm; (d) second freeform optical surface with the PV of 4.0632 mm. The ray tracing was implemented with 1 × 10 7 light rays, and the kernel size for noise reduction of the simulated irradiance distribution is set as 5.

Fig. 5.
Fig. 5.

(a) Model of Lens B and its simulation results; (b) computed output wavefront with the PV of 0.8089 mm; (c) first freeform optical surface with the PV of 0.3720 mm; (d) second freeform optical surface with the PV of 0.4809 mm. The ray tracing was implemented with 1 × 10 7 light rays, and the kernel size for noise reduction of the simulated irradiance distribution is set as 5.

Fig. 6.
Fig. 6.

(a) Model of Lens C and its simulation results; (b) computed output wavefront with the PV of 0.2993 mm; (c) first freeform optical surface with the PV of 0.7960 mm; (d) second freeform optical surface with the PV of 2.3079 mm. The ray tracing was implemented with 2 × 10 7 light rays, and the kernel size for noise reduction of the simulated irradiance distribution is set as 5.

Fig. 7.
Fig. 7.

Evolution of the error estimates for (a) Lens A, (b) Lens B, and (c) Lens C.

Equations (21)

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2 ϕ ξ 2 2 ϕ η 2 ( 2 ϕ ξ η ) 2 = I 1 ( ξ , η ) I 2 ( ϕ ( ξ , η ) ) ,
O ^ i , j = ( α i , j ξ j , β i , j η i , d 2 d 1 ) | ( α i , j ξ j , β i , j η i , d 2 d 1 ) | .
( W i + 1 , j W i , j ) · ( O ^ i + 1 , j + O ^ i , j ) = 0 ,
( W i , j W i , j + 1 ) · ( O ^ i , j + O ^ i , j + 1 ) = 0 ,
2 ψ ξ 2 2 ψ η 2 ( 2 ψ ξ η ) 2 = I 1 ( ξ , η ) I 0 ( ψ ( ξ , η ) ) ,
I ^ i , j = ( u i , j , v i , j , h ) | ( u i , j , v i , j , h ) | .
| P i , j W i , j | + n lens | P i , j P i , j | + | W i , j P i , j | = L const .
P i , j = W i , j + | P i , j W i , j | I ^ i , j ,
P i , j = W i , j | W i , j P i , j | O ^ i , j .
s i , j + | W i , j W i , j t i , j O ^ i , j s i , j I ^ i , j | + t i , j = L .
t i , j = b ± b 2 4 a c 2 a ,
a = n lens 2 1 , b = 2 [ n lens 2 O ^ i , j · ( W i , j W i , j ) + n lens 2 I ^ i , j · O ^ i , j s i , j L s i , j ] , c = n lens 2 | W W | + n lens 2 s i , j 2 2 n lens 2 I ^ i , j · ( W i , j W i , j ) s i , j L 2 s i , j 2 + 2 L s i , j .
R ^ i , j = ( P i , j P i , j ) / | P i , j P i , j | .
N ^ i , j = ( n lens R ^ i , j I ^ i , j ) / | n lens R ^ i , j I ^ i , j | .
( P i + 1 , j P i , j ) · ( N i + 1 , j + N i , j ) = 0 ,
( P i , j + 1 P i , j ) · ( N i , j + 1 + N i , j ) = 0 ,
i = 1 n j = 1 m | s i , j ( k + 1 ) s i , j ( k ) | i = 1 n j = 1 m | s i , j ( 1 ) s i , j ( 0 ) | < ε ,
I 0 ( u , v ) = A e 2 [ ( u ω ) 2 + ( v ω ) 2 ] , ( u , v ) Ω 0 ,
RRMSD = I s , 1 ( ξ , η ) I 1 ( ξ , η ) F I 1 ( ξ , η ) F , ( ξ , η ) Ω 1 ,
I 1 ( ξ , η ) = B e 2 [ ( ξ ω ) 30 + ( η ω ) 30 ] , ( ξ , η ) Ω 1 ,
I 2 ( α , β ) = C , ( α , β ) Ω 2 ,