Advertisement

Ray Mappings and the Weighted Least Action Principle

  • Jacob Rubinstein
  • Yifat Weinberg
  • Gershon Wolansky
Conference paper
Part of the Mathematics in Industry book series (MATHINDUSTRY, volume 26)

Abstract

Two basic problems in optics are presented. The solutions to both problems are formulated in terms of the associated ray mappings. An alternative formulation based on a weighted sum of the actions along the rays is derived. Existence of solutions is established via the Weighted Least Action Principle. Numerical methods for computing the ray mappings are discussed. Finally, we demonstrate the theoretical considerations by presenting a complete solution to a specific beam shaping lens design.

Notes

Acknowledgements

This work is supported by grants from the Israel Science Foundation.

References

  1. 1.
    Angenent, S., Haker, S., Tannenbaum, A.: Minimizing flows for the Monge-Kantorovich problem. SIAM J. Math. Anal. 35, 61–97 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Benamou, J.D., Brenier, Y.: A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem. Numer. Math. 84, 375–393 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Brenier, Y.: Polar factorization and monotone rearrangement of vector-valued functions. Commun. Pure Appl. Math. 64, 375–417 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Dickey, F.M.: Laser beam shaping. Opt. Photonics News 14, 31–35 (2003)CrossRefGoogle Scholar
  5. 5.
    Kuhn, H.W.: The Hungarian Method for the assignment problem. Naval Res. Logist. Q. 2, 83–97 (1955)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Lee, C., Lowengrub, J., Rubinstein, J., Zheng, X.: Phase reconstruction by the weighted least action principle. J. Opt. A Pure Appl. Opt. 8, 279–289 (2006)CrossRefGoogle Scholar
  7. 7.
    Merigot, Q.: Multiscale approach to optimal transport. Comput. Graph. Forum 30, 1584–1592 (2011)CrossRefGoogle Scholar
  8. 8.
    Nam, J., Rubinstein, J., Thibos, L.: Wavelength adjustment using an eye model from aberrometry data. J. Opt. Soc. Am. A 27, 1561–1574 (2010)CrossRefGoogle Scholar
  9. 9.
    Roddier, F.: Curvature sensing and compensation: a new concept in adaptive optics. Appl. Opt. 27, 1223–1225 (1998)CrossRefGoogle Scholar
  10. 10.
    Rubinstein, J., Wolansky, G.: A variational principle in optics. J. Opt. Soc. Am. A 21, 2164–2172 (2004)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Rubinstein, J., Wolansky, G.: A weighted least action principle for dispersive waves. Ann. Phys. 316, 271–284 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Rubinstein, J., Wolansky, G.: Intensity control with a free-form lens. J. Opt. Soc. Am. A 24, 463–469 (2007)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Rubinstein, J., Wolansky, G.: Geometrical optics and optimal transport. J. Opt. Soc. Am. A (to appear)Google Scholar
  14. 14.
    Teague, M.R.: Deterministic phase retrieval: a Green’s function solution. J. Opt. Soc. Am. 73, 1434–1441 (1983)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2017

Authors and Affiliations

  • Jacob Rubinstein
    • 1
  • Yifat Weinberg
    • 1
  • Gershon Wolansky
    • 1
  1. 1.Department of MathematicsTechnionHaifaIsrael

Personalised recommendations