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Optimal design of periodic antireflective structures for the Helmholtz equation

  • David C. Dobson (a1)

We study the problem of designing a periodic interface between two homogeneous materials with different impedance properties, in such a way that time-harmonic waves incident on the interface over a given range of angles have minimal total reflected energy. It is shown that the problem can be ‘relaxed’ to include continuously varying profiles. A simple gradient descent minimization scheme is proposed and examples from several numerical calculations are given.

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[1] Y. Ono , Y. Kimura , Y. Ohta & N. Nishida 1987 Antireflection effect in ultrahigh spatial-frequency holographic relief gratings. Appl. Optics 26, 11421146.

[2] E. N. Glytsis & T. K. Gaylord 1988 Antireflection surface structure: dielectric layer(s) over a high spatial-frequency surface-relief grating on a lossy substrate. Appl. Optics 27, 42884304.

[5] M. Born & E. Wolf 1980 Principles of Optics, sixth edition. Pergamon.

[6] X. Chen & A. Friedman 1991 Maxwell's equations in a periodic structure. Trans. Amer. Math. Soc. 323, 465507.

[7] D. Dobson & A. Friedman 1992 The time-harmonic Maxwell equations in a doubly periodic structure. J. Math. Anal. Appl. 166, 507528.

[9] B. Engquist & A. Majda 1979 Radiation boundary conditions for acoustic and elastic wave calculations. Comm. Pure Appl. Math. 32, 313375.

[10] D. Givoli 1991 Non-reflecting boundary conditions. J. Comp. Phys. 94, 129.

[11] R. Kohn & G. Strang , 1986 Optimal design and relaxation of variational problems I, II, and III. Comm. Pure Appl. Math. 39, 113137, 139182, 353377.

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European Journal of Applied Mathematics
  • ISSN: 0956-7925
  • EISSN: 1469-4425
  • URL: /core/journals/european-journal-of-applied-mathematics
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