Abstract

We propose a method for designing refractive optical elements for collimated beam shaping in the geometrical optics approximation. In this method, the problem of finding a ray mapping is formulated as a linear assignment problem, which is a discrete version of the corresponding mass transportation problem. A method for reconstructing optical surfaces from a computed discrete ray mapping is proposed. The method is suitable for designing continuous piecewise-smooth optical surfaces. The design of refractive optical elements transforming beams with circular cross-section to variously shaped (rectangular, triangular, and cross-shaped) beams with plane wavefront is discussed. The presented numerical simulation results confirm high efficiency of the designed optical elements.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article
OSA Recommended Articles
Optimal mass transportation and linear assignment problems in the design of freeform refractive optical elements generating far-field irradiance distributions

Leonid L. Doskolovich, Dmitry A. Bykov, Albert A. Mingazov, and Evgeni A. Bezus
Opt. Express 27(9) 13083-13097 (2019)

Linear assignment problem in the design of freeform refractive optical elements generating prescribed irradiance distributions

Dmitry A. Bykov, Leonid L. Doskolovich, Albert A. Mingazov, Evgeni A. Bezus, and Nikolay L. Kazanskiy
Opt. Express 26(21) 27812-27825 (2018)

Ray-mapping approach in double freeform surface design for collimated beam shaping beyond the paraxial approximation

Christoph Bösel, Norman G. Worku, and Herbert Gross
Appl. Opt. 56(13) 3679-3688 (2017)

References

  • View by:
  • |
  • |
  • |

  1. B. R. Frieden, “Lossless conversion of a plane laser wave to a plane wave of uniform irradiance,” Appl. Opt. 4(11), 1400–1403 (1965).
    [Crossref]
  2. J. L. Kreuzer, “Coherent light optical system yielding an output beam of desired intensity distribution at a desired equiphase surface,” U. S. Patent No. 3,476,463 (4Nov., 1969).
  3. 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] [PubMed]
  4. J. A. Hoffnagle and C. M. Jefferson, “Design and performance of a refractive optical system that converts a Gaussian to a flattop beam,” Appl. Opt. 39(30), 5488–5499 (2000).
    [Crossref]
  5. 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] [PubMed]
  6. X. Hui, J. Liu, Y. Wan, and H. Lin, “Realization of uniform and collimated light distribution in a single freeform-Fresnel double surface LED lens,” Appl. Opt. 56(15), 4561–4565 (2017).
    [Crossref] [PubMed]
  7. 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]
  8. 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(12), 125602 (2016).
    [Crossref]
  9. 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] [PubMed]
  10. Z. Feng, L. Huang, M. Gong, and G. Jin, “Beam shaping system design using double freeform optical surfaces,” Opt. Express 21(12), 14728–14735 (2013).
    [Crossref] [PubMed]
  11. 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]
  12. 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] [PubMed]
  13. 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] [PubMed]
  14. J. Rubinstein and G. Wolansky, “Intensity control with a free-form lens,” J. Opt. Soc. Am. A 24(2), 463–469 (2007).
    [Crossref]
  15. V. I. Oliker, “Designing freeform lenses for intensity and phase control of coherent light with help from geometry and mass transport,” Arch. Ration. Mech. Analysis 201(3), 1013–1045 (2011).
    [Crossref]
  16. J. Munkres, “Algorithms for the assignment and transportation problems,” SIAM J. Appl. Math. 5(1), 32–38 (1957).
    [Crossref]
  17. X. Mao, S. Xu, X. Hu, and Y. Xie, “Design of a smooth freeform illumination system for a point light source based on polar-type optimal transport mapping,” Appl. Opt. 56(22), 6324–6331 (2017).
    [Crossref] [PubMed]
  18. X. Mao, H. Li, Y. Han, and Y. Luo, “Polar-grids based source-target mapping construction method for designing freeform illumination system for a lighting target with arbitrary shape,” Opt. Express 23(4), 4313–4328 (2015).
    [Crossref] [PubMed]
  19. Y. Ding, X. Liu, Z.-R. Zheng, and P.-F. Gu, “Freeform LED lens for uniform illumination,” Opt. Express 16(17), 12958–12966 (2008).
    [Crossref] [PubMed]
  20. L. L. Doskolovich, N. L. Kazansky, S. I. Kharitonov, and V. A. Soifer, “A method of designing diffractive optical elements focusing into plane areas,” J. Mod. Opt. 43(7), 1423–1433 (1996).
    [Crossref]
  21. R. Jonker and A. Volgenant, “A shortest augmenting path algorithm for dense and sparse linear assignment problems,” Computing 38(4), 325–340 (1987).
    [Crossref]
  22. D. P. Bertsekas, “The auction algorithm: A distributed relaxation method for the assignment problem,” Ann. Oper. Res. 14(1), 105–123 (1988).
    [Crossref]
  23. C. de Boor, A Practical Guide to Splines (Springer-Verlag, 2001).
  24. 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(8), 1504–1508 (2016).
    [Crossref]
  25. J. Rubinstein and G. Wolansky, “Reconstruction of optical surfaces from ray data,” Opt. Rev. 8(4), 281–283 (2001).
    [Crossref]
  26. V. Oliker, “Mathematical aspects of design of beam shaping surfaces in geometrical optics,” Trends in Nonlinear Analysis, M. Kirkilionis, S. Krömker, R. Rannacher, and F. Tomi, eds. (Springer, 2003).
    [Crossref]
  27. V. Oliker, “Controlling light with freeform multifocal lens designed with supporting quadric method (SQM),” Opt. Express 25(4), A58–A72 (2017).
    [Crossref] [PubMed]
  28. L. L. Doskolovich, M. A. Moiseev, E. A. Bezus, and V. Oliker, “On the use of the supporting quadric method in the problem of the light field eikonal calculation,” Opt. Express 23(15), 19605–19617 (2015).
    [Crossref] [PubMed]
  29. L. L. Doskolovich, K. V. Borisova, M. A. Moiseev, and N. L. Kazanskiy, “Design of mirrors for generating prescribed continuous illuminance distributions on the basis of the supporting quadric method,” Appl. Opt. 55(4), 687–695 (2016).
    [Crossref] [PubMed]
  30. R. K. Luneburg, Mathematical Theory of Optics (University of California, 1964).
  31. L. L. Doskolovich, A. Y. Dmitriev, E. A. Bezus, and M. A. Moiseev, “Analytical design of freeform optical elements generating an arbitrary-shape curve,” Appl. Opt. 52(12), 2521–2526 (2013).
    [Crossref] [PubMed]
  32. Fast linear assignment problem using auction algorithm (mex). http://www.mathworks.com/matlabcentral/fileexchange/48448
  33. Opto-mechanical software TracePro. https://www.lambdares.com/tracepro
  34. Computer-aided design software Rhinoceros. http://www.rhino3d.com

2017 (5)

2016 (3)

2015 (3)

2014 (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]

2013 (3)

2011 (2)

V. I. Oliker, “Designing freeform lenses for intensity and phase control of coherent light with help from geometry and mass transport,” Arch. Ration. Mech. Analysis 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] [PubMed]

2008 (1)

2007 (1)

2001 (1)

J. Rubinstein and G. Wolansky, “Reconstruction of optical surfaces from ray data,” Opt. Rev. 8(4), 281–283 (2001).
[Crossref]

2000 (1)

1996 (1)

L. L. Doskolovich, N. L. Kazansky, S. I. Kharitonov, and V. A. Soifer, “A method of designing diffractive optical elements focusing into plane areas,” J. Mod. Opt. 43(7), 1423–1433 (1996).
[Crossref]

1988 (1)

D. P. Bertsekas, “The auction algorithm: A distributed relaxation method for the assignment problem,” Ann. Oper. Res. 14(1), 105–123 (1988).
[Crossref]

1987 (1)

R. Jonker and A. Volgenant, “A shortest augmenting path algorithm for dense and sparse linear assignment problems,” Computing 38(4), 325–340 (1987).
[Crossref]

1980 (1)

1965 (1)

1957 (1)

J. Munkres, “Algorithms for the assignment and transportation problems,” SIAM J. Appl. Math. 5(1), 32–38 (1957).
[Crossref]

Andreev, E. S.

Bertsekas, D. P.

D. P. Bertsekas, “The auction algorithm: A distributed relaxation method for the assignment problem,” Ann. Oper. Res. 14(1), 105–123 (1988).
[Crossref]

Bezus, E. A.

Borisova, K. V.

Bösel, C.

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(12), 125602 (2016).
[Crossref]

de Boor, C.

C. de Boor, A Practical Guide to Splines (Springer-Verlag, 2001).

Ding, Y.

Dmitriev, A. Y.

Doskolovich, L. L.

Du, S.

Feng, Z.

Frieden, B. R.

Froese, B. D.

Gong, M.

Gross, H.

Gu, P.-F.

Han, Y.

Hoffnagle, J. A.

Hu, X.

Huang, C.-Y.

Huang, L.

Hui, X.

Jefferson, C. M.

Jiang, P.

Jin, G.

Jonker, R.

R. Jonker and A. Volgenant, “A shortest augmenting path algorithm for dense and sparse linear assignment problems,” Computing 38(4), 325–340 (1987).
[Crossref]

Kazanskiy, N. L.

Kazansky, N. L.

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(8), 1504–1508 (2016).
[Crossref]

L. L. Doskolovich, N. L. Kazansky, S. I. Kharitonov, and V. A. Soifer, “A method of designing diffractive optical elements focusing into plane areas,” J. Mod. Opt. 43(7), 1423–1433 (1996).
[Crossref]

Kharitonov, S. I.

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(8), 1504–1508 (2016).
[Crossref]

L. L. Doskolovich, N. L. Kazansky, S. I. Kharitonov, and V. A. Soifer, “A method of designing diffractive optical elements focusing into plane areas,” J. Mod. Opt. 43(7), 1423–1433 (1996).
[Crossref]

Kreuzer, J. L.

J. L. Kreuzer, “Coherent light optical system yielding an output beam of desired intensity distribution at a desired equiphase surface,” U. S. Patent No. 3,476,463 (4Nov., 1969).

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(12), 125602 (2016).
[Crossref]

Li, H.

X. Mao, H. Li, Y. Han, and Y. Luo, “Polar-grids based source-target mapping construction method for designing freeform illumination system for a lighting target with arbitrary shape,” Opt. Express 23(4), 4313–4328 (2015).
[Crossref] [PubMed]

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.

Lin, H.

Liu, J.

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]

Y. Ding, X. Liu, Z.-R. Zheng, and P.-F. Gu, “Freeform LED lens for uniform illumination,” Opt. Express 16(17), 12958–12966 (2008).
[Crossref] [PubMed]

Liu, Z.

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (University of California, 1964).

Luo, Y.

Ma, D.

Ma, H.

Mao, X.

Mingazov, A. A.

Moiseev, M. A.

Munkres, J.

J. Munkres, “Algorithms for the assignment and transportation problems,” SIAM J. Appl. Math. 5(1), 32–38 (1957).
[Crossref]

Oliker, V.

Oliker, V. I.

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

Rhodes, P. W.

Rubinstein, J.

J. Rubinstein and G. Wolansky, “Intensity control with a free-form lens,” J. Opt. Soc. Am. A 24(2), 463–469 (2007).
[Crossref]

J. Rubinstein and G. Wolansky, “Reconstruction of optical surfaces from ray data,” Opt. Rev. 8(4), 281–283 (2001).
[Crossref]

Shealy, D. L.

Soifer, V. A.

L. L. Doskolovich, N. L. Kazansky, S. I. Kharitonov, and V. A. Soifer, “A method of designing diffractive optical elements focusing into plane areas,” J. Mod. Opt. 43(7), 1423–1433 (1996).
[Crossref]

Volgenant, A.

R. Jonker and A. Volgenant, “A shortest augmenting path algorithm for dense and sparse linear assignment problems,” Computing 38(4), 325–340 (1987).
[Crossref]

Wan, Y.

Wolansky, G.

J. Rubinstein and G. Wolansky, “Intensity control with a free-form lens,” J. Opt. Soc. Am. A 24(2), 463–469 (2007).
[Crossref]

J. Rubinstein and G. Wolansky, “Reconstruction of optical surfaces from ray data,” Opt. Rev. 8(4), 281–283 (2001).
[Crossref]

Worku, N. 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–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]

Xie, Y.

Xu, S.

Xu, X.

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. Commun. 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(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]

Zheng, Z.-R.

Ann. Oper. Res. (1)

D. P. Bertsekas, “The auction algorithm: A distributed relaxation method for the assignment problem,” Ann. Oper. Res. 14(1), 105–123 (1988).
[Crossref]

Appl. Opt. (9)

L. L. Doskolovich, A. Y. Dmitriev, E. A. Bezus, and M. A. Moiseev, “Analytical design of freeform optical elements generating an arbitrary-shape curve,” Appl. Opt. 52(12), 2521–2526 (2013).
[Crossref] [PubMed]

L. L. Doskolovich, K. V. Borisova, M. A. Moiseev, and N. L. Kazanskiy, “Design of mirrors for generating prescribed continuous illuminance distributions on the basis of the supporting quadric method,” Appl. Opt. 55(4), 687–695 (2016).
[Crossref] [PubMed]

B. R. Frieden, “Lossless conversion of a plane laser wave to a plane wave of uniform irradiance,” Appl. Opt. 4(11), 1400–1403 (1965).
[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] [PubMed]

J. A. Hoffnagle and C. M. Jefferson, “Design and performance of a refractive optical system that converts a Gaussian to a flattop beam,” Appl. Opt. 39(30), 5488–5499 (2000).
[Crossref]

X. Hui, J. Liu, Y. Wan, and H. Lin, “Realization of uniform and collimated light distribution in a single freeform-Fresnel double surface LED lens,” Appl. Opt. 56(15), 4561–4565 (2017).
[Crossref] [PubMed]

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] [PubMed]

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] [PubMed]

X. Mao, S. Xu, X. Hu, and Y. Xie, “Design of a smooth freeform illumination system for a point light source based on polar-type optimal transport mapping,” Appl. Opt. 56(22), 6324–6331 (2017).
[Crossref] [PubMed]

Arch. Ration. Mech. Analysis (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. Analysis 201(3), 1013–1045 (2011).
[Crossref]

Computing (1)

R. Jonker and A. Volgenant, “A shortest augmenting path algorithm for dense and sparse linear assignment problems,” Computing 38(4), 325–340 (1987).
[Crossref]

J. Mod. Opt. (1)

L. L. Doskolovich, N. L. Kazansky, S. I. Kharitonov, and V. A. Soifer, “A method of designing diffractive optical elements focusing into plane areas,” J. Mod. Opt. 43(7), 1423–1433 (1996).
[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(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. Express (8)

Z. Feng, L. Huang, M. Gong, and G. Jin, “Beam shaping system design using double freeform optical surfaces,” Opt. Express 21(12), 14728–14735 (2013).
[Crossref] [PubMed]

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]

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] [PubMed]

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] [PubMed]

X. Mao, H. Li, Y. Han, and Y. Luo, “Polar-grids based source-target mapping construction method for designing freeform illumination system for a lighting target with arbitrary shape,” Opt. Express 23(4), 4313–4328 (2015).
[Crossref] [PubMed]

Y. Ding, X. Liu, Z.-R. Zheng, and P.-F. Gu, “Freeform LED lens for uniform illumination,” Opt. Express 16(17), 12958–12966 (2008).
[Crossref] [PubMed]

V. Oliker, “Controlling light with freeform multifocal lens designed with supporting quadric method (SQM),” Opt. Express 25(4), A58–A72 (2017).
[Crossref] [PubMed]

L. L. Doskolovich, M. A. Moiseev, E. A. Bezus, and V. Oliker, “On the use of the supporting quadric method in the problem of the light field eikonal calculation,” Opt. Express 23(15), 19605–19617 (2015).
[Crossref] [PubMed]

Opt. Rev. (1)

J. Rubinstein and G. Wolansky, “Reconstruction of optical surfaces from ray data,” Opt. Rev. 8(4), 281–283 (2001).
[Crossref]

SIAM J. Appl. Math. (1)

J. Munkres, “Algorithms for the assignment and transportation problems,” SIAM J. Appl. Math. 5(1), 32–38 (1957).
[Crossref]

Other (7)

J. L. Kreuzer, “Coherent light optical system yielding an output beam of desired intensity distribution at a desired equiphase surface,” U. S. Patent No. 3,476,463 (4Nov., 1969).

V. Oliker, “Mathematical aspects of design of beam shaping surfaces in geometrical optics,” Trends in Nonlinear Analysis, M. Kirkilionis, S. Krömker, R. Rannacher, and F. Tomi, eds. (Springer, 2003).
[Crossref]

C. de Boor, A Practical Guide to Splines (Springer-Verlag, 2001).

R. K. Luneburg, Mathematical Theory of Optics (University of California, 1964).

Fast linear assignment problem using auction algorithm (mex). http://www.mathworks.com/matlabcentral/fileexchange/48448

Opto-mechanical software TracePro. https://www.lambdares.com/tracepro

Computer-aided design software Rhinoceros. http://www.rhino3d.com

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 (6)

Fig. 1
Fig. 1 (a) Geometry of the problem of the design of an optical element. (b) Approximation of a circular domain G and a triangular domain D with equal-flux cells. Dots correspond to the centers of the cells.
Fig. 2
Fig. 2 (a) Surfaces of an optical element transforming a circular beam into a beam with rectangular cross-section. (b, c) Normalized irradiance distributions generated by the element in the planes z = 10 mm and z = 30 mm calculated using TracePro. The irradiance cross-sections along the coordinate axes are shown at the top and at the right of the distributions.
Fig. 3
Fig. 3 The mappings T: GD corresponding to the transformation of a circular beam (a) to a rectangular beam (b), a triangular beam (c), and a cross-shaped beam (d). Blue lines mark a straight line in the G domain and its images in the generated domains D.
Fig. 4
Fig. 4 (a) Surfaces of an optical element transforming a circular beam to a triangular one. (b, c) Normalized irradiance distributions generated by the element in the planes z = 10 mm and z = 30 mm calculated using TracePro. The irradiance cross-sections along the coordinate axes are shown at the top and at the right of the distributions.
Fig. 5
Fig. 5 (a) Surfaces of an optical element transforming a circular beam to a cross-shaped beam. A magnified fragment of the lower optical surface shown in Fig. 6 is highlighted with a rectangle. (b, c) Normalized irradiance distributions generated by the element in the planes z = 10 mm and z = 30 mm calculated using TracePro. The irradiance cross-sections along the coordinate axes are shown at the top and at the right of the distributions.
Fig. 6
Fig. 6 A magnified fragment of the lower surface [f(u)] of the designed optical element [see Fig. 5(a)]. The break is marked with a red line.

Equations (14)

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

G I 0 ( u ) d u = D I ( x ) d x ,
ω I 0 ( u ) d u = T ( ω ) I ( x ) d x ,
f ( u ) u i = n x i ( u ) u i δ 2 ( n 2 1 ) | x ( u ) u | 2 , i = 1 , 2 , g ( x ) x i = n x i u i ( x ) δ 2 ( n 2 1 ) | x u ( x ) | 2 , i = 1 , 2 ,
2 f ( u ) u 1 u 2 = 2 f ( u ) u 2 u 1 , 2 g ( x ) x 1 x 2 = 2 g ( x ) x 2 x 1 .
P ( T ) = G I 0 ( u ) C ( u T ( u ) ) d u ,
C ( s ) = { γ 2 | s | 2 , | s | < γ ; , | s | γ .
( M ) i , j = C ( u i x j ) , i , j = 1 , , N ,
P d ( j 1 , , j N ) = i C ( u i x j i ) min ,
g ( x ) = m , n p m , n B m ( x 1 ) P n ( x 2 ) ,
S ( p ) = i [ g ^ x 1 ( x i ) g x 1 ( x i ) ] 2 + i [ g ^ x 2 ( x i ) g x 2 ( x i ) ] 2 min ,
Φ i ( u ) + n | x i u | 2 + [ g ( x i ) Φ i ( u ) ] 2 = L i , i = 1 , , N g ,
Φ i ( u ) = g ( x i ) δ + n δ 2 ( n 2 1 ) | x i u | 2 n 2 1 .
f ( u ) = min i { 1 , , N g } Φ i ( u )
T ( u ) = x j j = argmin i { 1 , , N g } Φ i ( u ) .

Metrics