Abstract

Numerous applications require the simultaneous redistribution of the irradiance and phase of a laser beam. The beam shape is thereby determined by the respective application. An elegant way to control the irradiance and phase at the same time is from double freeform surfaces. In this work, the numerical design of continuous double freeform surfaces from ray-mapping methods for collimated beam shaping with arbitrary irradiances is considered. These methods consist of the calculation of a proper ray mapping between the source and the target irradiance and the subsequent construction of the freeform surfaces. By combining the law of refraction, the constant optical path length, and the surface continuity condition, a partial differential equation (PDE) for the ray mapping is derived. It is shown that the PDE can be fulfilled in a small-angle approximation by a mapping derived from optimal mass transport with a quadratic cost function. To overcome the restriction to the paraxial regime, we use this mapping as an initial iterate for the simultaneous solution of the Jacobian equation and the ray mapping PDE by a root-finding algorithm. The presented approach enables the efficient calculation of double freeform lenses with small distances between the freeform surfaces for complex target irradiances. This is demonstrated by applying it to the design of a single-lens and a two-lens system.

© 2017 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Single freeform surface design for prescribed input wavefront and target irradiance

Christoph Bösel and Herbert Gross
J. Opt. Soc. Am. A 34(9) 1490-1499 (2017)

Ray mapping approach for the efficient design of continuous freeform surfaces

Christoph Bösel and Herbert Gross
Opt. Express 24(13) 14271-14282 (2016)

A mathematical model of the single freeform surface design for collimated beam shaping

Rengmao Wu, Peng Liu, Yaqin Zhang, Zhenrong Zheng, Haifeng Li, and Xu Liu
Opt. Express 21(18) 20974-20989 (2013)

References

  • View by:
  • |
  • |
  • |

  1. P. Benítez and J. C. Miñano, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43, 1489–1502 (2004).
    [Crossref]
  2. J. Chaves, Introduction to Nonimaging Optics, 2nd ed. (CRC Press, 2015), pp. 321–406.
  3. J. C. Miñano, P. Benítez, and A. Santamaría, “Free-form optics for illumination,” Opt. Rev. 16, 99–102 (2009).
    [Crossref]
  4. J. Chaves, J. C. Miñano, and P. Benítez, “Afocal video-pixel lens for tricolor LEDs,” Proc. SPIE 5942, 594203 (2005).
    [Crossref]
  5. H. Ries, “Laser beam shaping by double tailoring,” Proc. SPIE 5876, 587607 (2005).
    [Crossref]
  6. 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. Comm. 331, 297–305 (2014).
    [Crossref]
  7. 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–Ampère type equation,” J. Opt. 18, 125602 (2016).
    [Crossref]
  8. V. I. Oliker, J. Rubinstein, and G. Wolansky, “Ray mapping and illumination control,” J. Photon. Energy 3, 035599 (2013).
  9. A. Bäuerle, A. Bruneton, P. Loosen, and R. Wester, “Algorithm for irradiance tailoring using multiple freeform optical surfaces,” Opt. Express 20, 14477–14485 (2012).
    [Crossref]
  10. A. Bruneton, A. Bäuerle, P. Loosen, and R. Wester, “High resolution irradiance tailoring using multiple freeform surfaces,” Opt. Express 21, 10563–10571 (2013).
    [Crossref]
  11. Z. Feng, L. Huang, G. Jin, and M. Gong, “Beam shaping system design using double freeform optical surfaces,” Opt. Express 21, 14728–14735 (2013).
    [Crossref]
  12. Z. Feng, L. Huang, G. Jin, and M. Gong, “Designing double freeform optical surfaces for controlling both irradiance and wavefront,” Opt. Express 21, 28693–28701 (2013).
    [Crossref]
  13. Z. Feng, B. D. Froese, C. Huang, D. Ma, and R. Liang, “Creating unconventional geometric beams with large depth of field using double freeform-surface optics,” Appl. Opt. 54, 6277–6281 (2015).
    [Crossref]
  14. Z. Feng, B. D. Froese, and R. Liang, “Freeform illumination optics construction following an optimal transport map,” Appl. Opt. 55, 4301–4306 (2016).
    [Crossref]
  15. L. L. Doskolovich, E. S. Andreev, S. I. Kharitonov, and N. L. Kazansky, “Reconstruction of an optical surface from a given source-target map,” J. Opt. Soc. Am. A 33, 1504–1508 (2016).
    [Crossref]
  16. L. L. Doskolovich, E. A. Bezus, M. A. Moiseev, D. A. Bykov, and N. L. Kazanskiy, “Analytical source-target mapping method for the design of freeform mirrors generating prescribed 2D intensity distributions,” Opt. Express 24, 10962–10971 (2016).
    [Crossref]
  17. C. Bösel and H. Gross, “Ray mapping approach for the efficient design of continuous freeform surfaces,” Opt. Express 24, 14271–14282 (2016).
    [Crossref]
  18. C. Bösel and H. Gross, “Ray mapping approach in double freeform surface design for collimated beam shaping,” Proc. SPIE 9950, 995004 (2016).
    [Crossref]
  19. N. K. Yadav, J. H. M. ten Thije Boonkkamp, and W. L. IJzerman, A Least-Squares Method for the Design of Two-Reflector Optical Systems, CASA-report 1619 (2016).
  20. T. Glimm and V. I. Oliker, “Optical design of single reflector systems and the Monge–Kantorovich mass transfer problem,” J. Math. Sci. 117, 4096–4108 (2003).
    [Crossref]
  21. T. Glimm and V. I. Oliker, “Optical design of two-reflector systems, the Monge–Kantorovich mass transfer problem and Fermat’s principle,” Indiana University Math. J. 53, 1255–1278 (2004).
    [Crossref]
  22. X.-J. Wang, “On the design of a reflector antenna II,” Calc. Var. 20, 329–341 (2004).
  23. J. Rubinstein and G. Wolansky, “Intensity control with a free-form lens,” J. Opt. Soc. Am. A 24, 463–469 (2007).
    [Crossref]
  24. T. Glimm, “A rigorous analysis using optimal transport theory for a two-reflector design problem with a point source,” Inverse Probl. 26, 045001 (2010).
    [Crossref]
  25. V. I. 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]
  26. S. Haker, L. Zhu, A. Tannenbaum, and S. Angenent, “Optimal mass transport for registration and warping,” Int. J. Comp. Vis. 60, 225–240 (2004).
    [Crossref]
  27. J. Rubinstein and G. Wolansky, “Reconstruction of optical surfaces from ray data,” Opt. Rev. 8, 281–283 (2001).
    [Crossref]
  28. M. M. Sulman, J. F. Williams, and R. D. Russel, “An efficient approach for the numerical solution of Monge–Ampère equation,” Appl. Numer. Math. 61, 298–307 (2011).
    [Crossref]
  29. J.-D. Benamou, B. D. Froese, and A. M. Oberman, “Numerical solution of the optimal transportation problem using the Monge–Ampère equation,” J. Comp. Physiol. 260, 107–126 (2014).
    [Crossref]

2016 (6)

2015 (1)

2014 (2)

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. Comm. 331, 297–305 (2014).
[Crossref]

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

2013 (4)

2012 (1)

2011 (2)

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

V. I. 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]

2010 (1)

T. Glimm, “A rigorous analysis using optimal transport theory for a two-reflector design problem with a point source,” Inverse Probl. 26, 045001 (2010).
[Crossref]

2009 (1)

J. C. Miñano, P. Benítez, and A. Santamaría, “Free-form optics for illumination,” Opt. Rev. 16, 99–102 (2009).
[Crossref]

2007 (1)

2005 (2)

J. Chaves, J. C. Miñano, and P. Benítez, “Afocal video-pixel lens for tricolor LEDs,” Proc. SPIE 5942, 594203 (2005).
[Crossref]

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

2004 (4)

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

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

T. Glimm and V. I. Oliker, “Optical design of two-reflector systems, the Monge–Kantorovich mass transfer problem and Fermat’s principle,” Indiana University Math. J. 53, 1255–1278 (2004).
[Crossref]

X.-J. Wang, “On the design of a reflector antenna II,” Calc. Var. 20, 329–341 (2004).

2003 (1)

T. Glimm and V. I. Oliker, “Optical design of single reflector systems and the Monge–Kantorovich mass transfer problem,” J. Math. Sci. 117, 4096–4108 (2003).
[Crossref]

2001 (1)

J. Rubinstein and G. Wolansky, “Reconstruction of optical surfaces from ray data,” Opt. Rev. 8, 281–283 (2001).
[Crossref]

Andreev, E. S.

Angenent, S.

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

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–Ampère equation,” J. Comp. Physiol. 260, 107–126 (2014).
[Crossref]

Benítez, P.

J. C. Miñano, P. Benítez, and A. Santamaría, “Free-form optics for illumination,” Opt. Rev. 16, 99–102 (2009).
[Crossref]

J. Chaves, J. C. Miñano, and P. Benítez, “Afocal video-pixel lens for tricolor LEDs,” Proc. SPIE 5942, 594203 (2005).
[Crossref]

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

Bezus, E. A.

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]

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

Bruneton, A.

Bykov, D. A.

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–Ampère type equation,” J. Opt. 18, 125602 (2016).
[Crossref]

Chaves, J.

J. Chaves, J. C. Miñano, and P. Benítez, “Afocal video-pixel lens for tricolor LEDs,” Proc. SPIE 5942, 594203 (2005).
[Crossref]

J. Chaves, Introduction to Nonimaging Optics, 2nd ed. (CRC Press, 2015), pp. 321–406.

Doskolovich, L. L.

Feng, Z.

Froese, B. D.

Glimm, T.

T. Glimm, “A rigorous analysis using optimal transport theory for a two-reflector design problem with a point source,” Inverse Probl. 26, 045001 (2010).
[Crossref]

T. Glimm and V. I. Oliker, “Optical design of two-reflector systems, the Monge–Kantorovich mass transfer problem and Fermat’s principle,” Indiana University Math. J. 53, 1255–1278 (2004).
[Crossref]

T. Glimm and V. I. Oliker, “Optical design of single reflector systems and the Monge–Kantorovich mass transfer problem,” J. Math. Sci. 117, 4096–4108 (2003).
[Crossref]

Gong, M.

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]

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

Haker, S.

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

Huang, C.

Huang, L.

IJzerman, W. L.

N. K. Yadav, J. H. M. ten Thije Boonkkamp, and W. L. IJzerman, A Least-Squares Method for the Design of Two-Reflector Optical Systems, CASA-report 1619 (2016).

Jin, G.

Kazanskiy, N. L.

Kazansky, N. L.

Kharitonov, S. I.

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–Ampère type equation,” J. Opt. 18, 125602 (2016).
[Crossref]

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. Comm. 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. Comm. 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. Comm. 331, 297–305 (2014).
[Crossref]

Loosen, P.

Ma, D.

Miñano, J. C.

J. C. Miñano, P. Benítez, and A. Santamaría, “Free-form optics for illumination,” Opt. Rev. 16, 99–102 (2009).
[Crossref]

J. Chaves, J. C. Miñano, and P. Benítez, “Afocal video-pixel lens for tricolor LEDs,” Proc. SPIE 5942, 594203 (2005).
[Crossref]

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

Moiseev, M. A.

Oberman, A. M.

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

Oliker, V. I.

V. I. Oliker, J. Rubinstein, and G. Wolansky, “Ray mapping and illumination control,” J. Photon. Energy 3, 035599 (2013).

V. I. 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. I. Oliker, “Optical design of two-reflector systems, the Monge–Kantorovich mass transfer problem and Fermat’s principle,” Indiana University Math. J. 53, 1255–1278 (2004).
[Crossref]

T. Glimm and V. I. Oliker, “Optical design of single reflector systems and the Monge–Kantorovich mass transfer problem,” J. Math. Sci. 117, 4096–4108 (2003).
[Crossref]

Ries, H.

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

Rubinstein, J.

V. I. Oliker, J. Rubinstein, and G. Wolansky, “Ray mapping and illumination control,” J. Photon. Energy 3, 035599 (2013).

J. Rubinstein and G. Wolansky, “Intensity control with a free-form lens,” J. Opt. Soc. Am. A 24, 463–469 (2007).
[Crossref]

J. Rubinstein and G. Wolansky, “Reconstruction of optical surfaces from ray data,” Opt. Rev. 8, 281–283 (2001).
[Crossref]

Russel, R. D.

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

Santamaría, A.

J. C. Miñano, P. Benítez, and A. Santamaría, “Free-form optics for illumination,” Opt. Rev. 16, 99–102 (2009).
[Crossref]

Sulman, M. M.

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

Tannenbaum, A.

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

ten Thije Boonkkamp, J. H. M.

N. K. Yadav, J. H. M. ten Thije Boonkkamp, and W. L. IJzerman, A Least-Squares Method for the Design of Two-Reflector Optical Systems, CASA-report 1619 (2016).

Wang, X.-J.

X.-J. Wang, “On the design of a reflector antenna II,” Calc. Var. 20, 329–341 (2004).

Wester, R.

Williams, J. F.

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

Wolansky, G.

V. I. Oliker, J. Rubinstein, and G. Wolansky, “Ray mapping and illumination control,” J. Photon. Energy 3, 035599 (2013).

J. Rubinstein and G. Wolansky, “Intensity control with a free-form lens,” J. Opt. Soc. Am. A 24, 463–469 (2007).
[Crossref]

J. Rubinstein and G. Wolansky, “Reconstruction of optical surfaces from ray data,” Opt. Rev. 8, 281–283 (2001).
[Crossref]

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–Ampère 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–Ampère equation method,” Opt. Comm. 331, 297–305 (2014).
[Crossref]

Yadav, N. K.

N. K. Yadav, J. H. M. ten Thije Boonkkamp, and W. L. IJzerman, A Least-Squares Method for the Design of Two-Reflector Optical Systems, CASA-report 1619 (2016).

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–Ampère equation method,” Opt. Comm. 331, 297–305 (2014).
[Crossref]

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–Ampère 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–Ampère equation method,” Opt. Comm. 331, 297–305 (2014).
[Crossref]

Zhu, L.

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

Appl. Numer. Math. (1)

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

Appl. Opt. (2)

Arch. Ration. Mech. Anal. (1)

V. I. 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]

Calc. Var. (1)

X.-J. Wang, “On the design of a reflector antenna II,” Calc. Var. 20, 329–341 (2004).

Indiana University Math. J. (1)

T. Glimm and V. I. Oliker, “Optical design of two-reflector systems, the Monge–Kantorovich mass transfer problem and Fermat’s principle,” Indiana University Math. J. 53, 1255–1278 (2004).
[Crossref]

Int. J. Comp. Vis. (1)

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

Inverse Probl. (1)

T. Glimm, “A rigorous analysis using optimal transport theory for a two-reflector design problem with a point source,” Inverse Probl. 26, 045001 (2010).
[Crossref]

J. Comp. Physiol. (1)

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

J. Math. Sci. (1)

T. Glimm and V. I. Oliker, “Optical design of single reflector systems and the Monge–Kantorovich mass transfer problem,” J. Math. Sci. 117, 4096–4108 (2003).
[Crossref]

J. Opt. (1)

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–Ampère type equation,” J. Opt. 18, 125602 (2016).
[Crossref]

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

J. Photon. Energy (1)

V. I. Oliker, J. Rubinstein, and G. Wolansky, “Ray mapping and illumination control,” J. Photon. Energy 3, 035599 (2013).

Opt. Comm. (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. Comm. 331, 297–305 (2014).
[Crossref]

Opt. Eng. (1)

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

Opt. Express (6)

Opt. Rev. (2)

J. Rubinstein and G. Wolansky, “Reconstruction of optical surfaces from ray data,” Opt. Rev. 8, 281–283 (2001).
[Crossref]

J. C. Miñano, P. Benítez, and A. Santamaría, “Free-form optics for illumination,” Opt. Rev. 16, 99–102 (2009).
[Crossref]

Proc. SPIE (3)

J. Chaves, J. C. Miñano, and P. Benítez, “Afocal video-pixel lens for tricolor LEDs,” Proc. SPIE 5942, 594203 (2005).
[Crossref]

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

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

Other (2)

N. K. Yadav, J. H. M. ten Thije Boonkkamp, and W. L. IJzerman, A Least-Squares Method for the Design of Two-Reflector Optical Systems, CASA-report 1619 (2016).

J. Chaves, Introduction to Nonimaging Optics, 2nd ed. (CRC Press, 2015), pp. 321–406.

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (9)

Fig. 1.
Fig. 1.

(a) Irradiance distributions IS(x,y) and IT(x,y) with the boundaries ΩS and ΩT are given on the planes z=0 and z=zT, respectively. In the first step an integrable ray mapping u(x,y)=(ux(x,y),uy(x,y)) is calculated between the distributions, which defines the vector field s4 between source and target plane. (b) In the second step the freeform surfaces zI(x,y) and zII(x,y) are calculated from the law of refraction and the constant OPL condition. The collimated input and output beams are represented by the vector fields s1 and s3, and the refracted beam by the vector field s2 [18].

Fig. 2.
Fig. 2.

(a) Input irradiance IS(x,y); (b) output irradiance IT(x,y) of the example “IAP;” (c) output irradiance IT(x,y) of the example “house.” The irradiances are normalized to ensure energy conservation.

Fig. 3.
Fig. 3.

To compare the quality of the initial mapping u(x,y) and optimized mapping u(x,y), they are plugged into the Jacobian Eq. (15). The Jacobian equation for “IAP” (a) before the optimization with u(x,y) (rms=1.1730·106) and (b) after the optimization with u(x,y) (rms=1.2387·106). According to the rms values, there is a minimal decrease of quality of the local energy conservation property. The ray-mapping condition [Eq. (16)] (c) before the root finding with u(x,y) (rms=0.0164) and (d) after the root finding with u(x,y) (rms=1.5199·108). The decrease of the rms and peak-to-valley leads to an approximate path-independent integration of the map.

Fig. 4.
Fig. 4.

To compare the quality of the initial mapping u(x,y) and optimized mapping u(x,y), they are plugged into the Jacobian Eq. (15). The Jacobian equation for “house” (a) before the optimization with u(x,y) (rms=5.6635·107) and (b) after the optimization with u(x,y) (rms=6.7198·107). There is a minimal decrease of quality of the local energy-conservation property. The ray-mapping condition [Eq. (16)] (c) before the root-finding with u(x,y) (rms=0.0206) and (d) after the root finding with u(x,y) (rms=7.4189·108).

Fig. 5.
Fig. 5.

Layout of the lens system with freeform surfaces zI(x,y) and zII(x,y) mapping a collimated Gaussian beam onto a collimated beam (a) with the “IAP” irradiance distribution by a two-lens system with plane surfaces as the first and last surface and (b) with the “house” target irradiance distribution by a single optical element with two freeform surfaces.

Fig. 6.
Fig. 6.

Results from the ray-tracing evaluation for the predefined irradiance distribution “IAP.” Output irradiance distribution from the ray tracing using surfaces from (a) the initial map u(x,y) and (b) the optimized map u(x,y). Absolute difference ΔIT(x,y) between predefined [Fig. 2(b)] and irradiance distribution from the ray tracing using surfaces from (c) the initial map u(x,y) and (d) the optimized map u(x,y). Optical path difference from the ray tracing with a reference wavelength of λ=550  nm using surfaces from (e) the initial map u(x,y) and (f) the optimized map u(x,y). Since the maps are integrated first from (0.5,0.5) along the x direction and then along the y direction, the deviations from the plane wavefront in (e) are consistent with Fig. 3(c).

Fig. 7.
Fig. 7.

Results from the ray-tracing evaluation for the predefined irradiance distribution “house.” Output irradiance distribution from the ray tracing using surfaces from (a) the initial map u(x,y) and (b) the optimized map u(x,y). Absolute difference ΔIT(x,y) between predefined [Fig. 2(c)] and irradiance distribution from the ray tracing with a reference wavelength of λ=550  nm using surfaces from (c) the initial map u(x,y) and (d) the optimized map u(x,y). Optical path difference from the ray tracing using surfaces from (e) the initial map u(x,y) and (f) the optimized map u(x,y).

Fig. 8.
Fig. 8.

Layout of the lens system with freeform surfaces zI(x,y) and zII(x,y) mapping a collimated Gaussian beam onto a collimated beam with the “house” target irradiance distribution by a two-lens system.

Fig. 9.
Fig. 9.

Results from the ray-tracing evaluation for the predefined irradiance distribution “house.” The side length of the square input beam is 1a.u. and for the square output beam 2a.u. Output irradiance distribution from the ray tracing using surfaces from (a) the initial map u(x,y) and (b) the optimized map u(x,y). Absolute difference ΔIT(x,y) between predefined [Fig. 2(c)] and irradiance distribution from the ray tracing with a reference wavelength of λ=550  nm using surfaces from (c) the initial map u(x,y) and (d) the optimized map u(x,y). Optical path difference from the ray tracing using surfaces from (e) the initial map u(x,y) and (f) the optimized map u(x,y).

Tables (3)

Tables Icon

Table 1. Comparison of ΔIT and OPD for Example “IAP”

Tables Icon

Table 2. Comparison of ΔIT and OPD for Example “House”

Tables Icon

Table 3. Comparison of ΔIT and OPD for Example “House” with a Larger Output Beam

Equations (24)

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

det(u(x,y))IT(u(x,y))=IS(x,y).
d(IS,IT)2=infuM|u(x)x|2IS(x)dx,
yux(x,y)xuy(x,y)=0,
s1=(00zI(x,y)),s2=(ux(x,y)xuy(x,y)yzII(ux,uy)zI(x,y)),s3=(00zTzII(ux,uy)),s4=(ux(x,y)xuy(x,y)yzT).
n·(×n)=0
nI=n1s^1n2s^2,
s2(×s1)=n1{s2×[(s2)s2]}znI·s2s2(×s3)+s2(×s4),
vzI(x,y)=n1·v·[(v·)v]nI·s2+vzII(ux,uy)(zII(ux,uy)zI(x,y))v.
(zzI(x,y))=nI(x,y)(nI(x,y))z(xzI(x,y)yzI(x,y))=(n2·(uxx)|s2|·(nI)zn2·(uyy)|s2|·(nI)z)v,
vzI(x,y)=0,
n1v·[(v·)v]nI·s2+n2g(ux,uy)(nII)z·|s2|(zII(ux,uy)zI(x,y))v=0,
n1|s1|+n2|s2|+n1|s3|=constOPL.
zII(ux,uy)zI(x,y)=nn21OPLred1n21OPLred2+(n21)|u(x,y)Id|2,
(vx2+vy2)nI·s2v(zII(ux,uy)zI(x,y))v=0.
det(u(x,y))IT(u(x,y))IS(x,y)=0,
yuxxuyn21OPLred2+(n21)|u(x,y)Id|·[(uxx)2yux(uyy)2xuy+(uxx)(uyy)(yuyxux)]=0,
u(x,y)=u(x,y)+Δu(x,y)
Δux(0.5,y)=Δux(0.5,y)=0,y[0.5,0.5],Δuy(x,0.5)=Δuy(x,0.5)=0,x[0.5,0.5],
x(Δux)12Δx[(Δux)i;j+1(Δux)i;j1],y(Δux)12Δy[(Δux)i+1;j(Δux)i1;j]
x(Δux)12Δx[3(Δux)i;j+24(Δux)i;j+1+(Δux)i;j],y(Δux)12Δy[3(Δux)i+2;j4(Δux)i+1;j+(Δux)i;j]
det(u(x,y))IT(u(x,y))(1)IS(x,y)IT(u(x,y))IT(u(x,y)).
s^2=n·s^1+{n·n^·s^1+1n2·[1(n^·s^1)2]}n^
|s2|=OPLred+n·[zII(xm,ym)zI(xs,ys)].
s2(xs,ys,zII(xm,ym))=(xmxs,ymys,zII(xm,ym)zI(xs,ys))

Metrics