Mathematical modeling in diffractive optics

  • Avner Friedman
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 67)


Diffractive optics is an emerging technology with many practical applications. Some of the applications and the need for mathematical modeling underlying them were discussed in two previous presentations by Allen Cox (from Honeywell Technology Center) as reported in [1; Chap. 22] and [2; Chap. 5]. One of the most elementary applications is replacing conventional lens by diffractive gratings which are produced by interference fringes on holographic plates. Other applications include nonreflective interface (called moth-eye grating), beam splitters, sensors, and variety of corrective lenses.


Helmholtz Equation Direct Problem Diffractive Optic Nonlinear Optical Material Fredholm Alternative 
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  1. [1]
    A. Friedman, Mathematics in Industrial Problems, IMA Volume 16, Springer—Verlag, New York (1988).zbMATHGoogle Scholar
  2. [2]
    A. Friedman, Mathematics in Industrial Problems, Part 3, IMA Volume 38, Springer—Verlag, New York (1991).zbMATHGoogle Scholar
  3. [3]
    Electromagnetic theory of Gratings,R. Petit ed., Topics in Physics, Vol. 22, Springer—Verlag, Heidelberg (1980).Google Scholar
  4. [4]
    X. Chen and A. Friedman, Maxwell’s equations in a periodic structure, Trans. Amer. Math. Soc., 323 (1991), 465–507.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    D.C. Dobson and A. Friedman, The time harmonic Maxwell equations in doubly periodic structure,J. Math. Anal. Appl., 166 (1992), 507528.Google Scholar
  6. [6]
    D.C. Dobson, Optimal design of periodic antireflective structures for the Helmholtz equation, European J. Appl. Math., 4 (1993), 321–340.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [7]
    Y. Achdou and O. Pironneau, Optimization of a photocell, Opt. Control Appl. Math., 12 (1991), 221–246.zbMATHCrossRefGoogle Scholar
  8. [8]
    G. Bao and D.C. Dobson, Diffractive optics in nonlinear media with periodic structures, IMA Preprint #1124, March 1993.Google Scholar
  9. [9]
    T. Abboud and J.C. Nédélec, Electromagnetic waves in an inhomogeneous medium, J. Math. Anal. Appl., 164 (1992), 40–58.MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    T. Abboud, Electromagnetic waves in periodic media, in “Second International Conference on Mathematical and Numerical Aspects of Wave Propagation,” edited by R. Kleinman et al, SIAM, Philadelphia (1993), pp. 1–9.Google Scholar
  11. [11]
    D.C. Dobson, A variational method for electromagnetic diffraction in biperiodic structures,RAIRO, Modél, Math. Anal. Numér., to appear.Google Scholar
  12. [12]
    O.P. Bruno and F. Reitich, Solution of a boundary value problem for Helmholtz equation of the boundary into the complex domain, Proc. Roy. Soc. Edinburg, 122A (1992), 317–340.MathSciNetzbMATHGoogle Scholar
  13. [13]
    G. Bao, Finite element approximation of time harmonic waves in periodic structures,SIAM J. Numerical Analysis, to appear.Google Scholar
  14. [14]
    O.P. Bruno and F. Reitich, Numerical solution of diffraction problems: A method of variation of boundaries, J. Optical Soc. Amer. A., 10 (1993), 1168–1175.CrossRefGoogle Scholar
  15. [15]
    O.P. Bruno and F. Reitich, Numerical solution of diffractive problems: A method of variation of boundaries II, Dielectric gratings, Pade approximants and singularities, J. Optical Soc. Amer. A, 10 (1993), 2307–2316.CrossRefGoogle Scholar
  16. [16]
    O.P. Bruno and F. Reitich, Accurate calculation of diffractive grating efficiencies, SPIE, 1919, Mathematics in Smart Structures (1993), 236–247.Google Scholar
  17. [17]
    O.P. Bruno and F. Reitich, Numerical solution of diffraction problems: A method of variation of boundaries III. Doubly-periodic gratings, J. Optical Soc. Amer., 10 (1993), 2551–2562.CrossRefGoogle Scholar
  18. [18]
    O. Bruno and F. Reitich, Approximation of analytic functions: a method of enhanced convergence,Math. Comp., to appear.Google Scholar
  19. [19]
    D.C. Dobson, Designing periodic structure with specified low frequency scattered far field data, in “Advances in Computer Methods for Partial Differential Equations VII,” edited by R. Vichnevetsky, D. Knight and G. Richter, IMACS (1992), 224–230.Google Scholar
  20. [20]
    M. Born and E. Wolf, Principles of Optics, Sixth edition, Pergamon Press, Oxford (1980).Google Scholar
  21. [21]
    M.T. Eismann, A.M. Tai and J.N. Cederquist, Holographic beam-former designed by an iterative technique, SPIE Proc., vol. 1052 (1989), 10–18.Google Scholar
  22. [22]
    F. Wyrowski and O. Bryndahl, Iterative Fourier-transform algorithms applied to computer holography, J. Optical Soc. Amer., A 5 (1988), 1058–1065.CrossRefGoogle Scholar
  23. [23]
    N.E. Hurt, Phase Retrieval and Zero Crossings, Kluwer, Dordrecht (1989).zbMATHGoogle Scholar
  24. [24]
    D.C. Dobson, Phase reconstruction via nonlinear least-squares, Inverse Problems, 8 (1992), 541–557.MathSciNetzbMATHCrossRefGoogle Scholar
  25. [25]
    G. Bao, A uniqueness theorem for an inverse problem in periodic diffractive optics, Inverse Problems, to appear.Google Scholar
  26. [26]
    G. Bao and D.C. Dobson, Second harmonic generation in nonlinear optical films, J. Math. Physics, 35 (1994), 1622–1633.MathSciNetzbMATHCrossRefGoogle Scholar
  27. [27]
    G.A. Kriegsman and C.L. Scandrett, Large membrane array scattering, J. Acoust. Soc. Am., to appear.Google Scholar
  28. [28]
    B. Lichtenberg and N.C. Gallagher, Finite element approach for the numerical analysis and modeling of diffractive and scattering objects, Purdue University, Department of Electrical Engineering, 1994. Preprint.Google Scholar

Copyright information

© Springer-Verlag New York, Inc. 1995

Authors and Affiliations

  • Avner Friedman
    • 1
  1. 1.Institute for Mathematics and its ApplicationsUniversity of MinnesotaMinneapolisUSA

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