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Optimal design of an optical lens

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Abstract

We wish to construct an optimal optical lens, in the sense that it focuses all rays from a given source onto a given focal point. The construction can be reduced mathematically to a global study of a system of two nonlinear functional differential equations. We prove that this system has precisely two analytic solutions.

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References

  1. J. Ockendon, S. Howison & E. Varley, The lens problem, R. P. I. Report, 1985.

  2. R. L. Sternberg, Successive approximation and expansion methods in the numerical design of microwave dielectric lenses, J. Math. Physics 34 (1955), 209–235.

    Google Scholar 

  3. R. L. Sternberg, On a multiple boundary value problem in generalized aplanatic lens design and its solution in series, Proceedings of the Physical Society (London) 81 (1963), 902–924.

    Google Scholar 

  4. R. L. Sternberg, Optimizing and extremizing nonlinear boundary value problems in lenticular antennas in oceanography, medicine and communications: Some solutions and some questions, in V. Lakshmikantham Ed., Nonlinear Systems and Applications, Academic Press, New York, 1977, 301–325.

    Google Scholar 

  5. R. L. Sternberg, A survey of numerical methods for a new class of nonlinear partial differential equations arising in nonspherical geometrical optics, Presented at the 11th IMACS World Congress in Oslo, Norway, August, 1985.

  6. R. L. Sternberg, M. J. Goldstein & D. Drinkard, On a new numerical method for a new class of nonlinear partial differential equations arising in nonspherical geometrical optics, in W. N. Everitt & B. D. Sleeman, Ordinary and Partial Differential Equations, Springer-Verlag, Heidelberg, 1982, 642–651.

    Google Scholar 

  7. R. L. Sternberg, R. B. Grafton & D. R. Childs, On a numerical minimization problem for a new class of nonlinear partial differential equations arising in nonspherical geometrical optics and beamforming lenticular antennas, J. Comput. and Appl. Math., to appear.

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Authors and Affiliations

  1. Institute for Mathematics and its Applications, University of Minnesota, Minneapolis

    Avner Friedman & Bryce McLeod

  2. Oxford University, Oxford, England

    Avner Friedman & Bryce McLeod

Authors
  1. Avner Friedman
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  2. Bryce McLeod
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Friedman, A., McLeod, B. Optimal design of an optical lens. Arch. Rational Mech. Anal. 99, 147–164 (1987). https://doi.org/10.1007/BF00275875

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  • DOI: https://doi.org/10.1007/BF00275875

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