Abstract

We consider the diffraction of a time-harmonic wave incident upon a grating (or periodic) structure. We study mathematical issues that arise in the direct modeling, inverse, and optimal design problems. Particular attention is paid to the variational approach and to finite-element methods. For the direct problem various results on existence, uniqueness, and numerical approximations of solutions are presented. Convergence properties of the variational method and sensitivity to TM polarization are examined. Our recent research on inverse diffraction problems and optimal design problems is also discussed.

© 1995 Optical Society of America

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  1. R. Petit, ed., Electromagnetic Theory of Gratings, Vol. 22 of Topics in Current Physics (Springer-Verlag, Berlin, 1980).
    [CrossRef]
  2. T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
    [CrossRef]
  3. L. Li, “A model analysis of lamellar diffraction gratings in conical mountings,” J. Mod. Opt. 40, 553–573 (1993).
    [CrossRef]
  4. O. Bruno, F. Reitich, “Numerical solution of diffraction problems: a method of variation of boundaries,” J. Opt. Soc. Am. A 10, 1168–1175 (1993).
    [CrossRef]
  5. O. Bruno, F. Reitich, “Numerical solution of diffraction problems: a method of variation of boundaries. II. Finitely conducting gratings, Padé approximants, and singularities,” J. Opt. Soc. Am. A 10, 2307–2316 (1993).
    [CrossRef]
  6. O. Bruno, F. Reitich, “Accurate calculation of diffractive grating efficiencies,” in Smart Structures and Materials 1993: Mathematics in Smart Structures, H. T. Banks, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1919, 236–247 (1993).
    [CrossRef]
  7. O. Bruno, F. Reitich, “Numerical solution of diffraction problems: a method of variation of boundaries. III. Doubly periodic gratings,” J. Opt. Soc. Am. A 10, 2551–2562 (1993).
    [CrossRef]
  8. O. Bruno, F. Reitich, “Approximation of analytic functions: a method of enhanced convergence,” Math. Comp. 63, 195–213 (1994).
    [CrossRef]
  9. D. Dobson, “A variational method for electromagnetic diffraction in biperiodic structures,” Model. Math. Anal. Numer. 28, 419–439 (1994).
  10. T. Abboud, “Étude mathématique et numérique de quelques problémes de diffraction d’ondes électromagnétiques,” Ph.D. dissertation (Ecole Polytechnique, Palaiseau, France1991).
  11. M. Cadilhac, “Some mathematical aspects of the grating theory,” in Electromagnetic Theory of Gratings, R. Petit, ed., Vol. 22 of Topics in Current Physics (Springer-Verlag, Berlin, 1980), pp. 53–62.
    [CrossRef]
  12. M. Cadilhac, “Rigorous vector theories of diffraction gratings,” in Progress in Optics, E. Wolf, ed. (North-Holland, New York, 1984), Vol. 21.
  13. J. A. Cox, “Inverse and optimal design problems for imaging and diffractive optical systems,” in Inverse Problems and Optimal Design in Industry, H. Engl, J. McLaughlin, eds. (B. G. Teubner, Stuttgart, 1994), pp. 29–36.
  14. R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).
  15. M. W. Farn, “New iterative algorithm for the design of phase-only gratings,” in Computer and Optically Generated Holographic Optics, I. N. Cindrich, S. Lee, eds., Proc. Soc. Photo-Opt. Intrum. Eng.1555, 34–42 (1991).
    [CrossRef]
  16. S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
    [CrossRef] [PubMed]
  17. A. Friedman, Mathematics in Industrial Problems, Vol. 16 of the IMA Volumes in Mathematics and Its Applications (Springer-Verlag, New York, 1988).
    [CrossRef]
  18. A. Friedman, Mathematics in Industrial Problems. Part 4, Vol. 38 of the IMA Volumes in Mathematics and Its Applications (Springer-Verlag, New York, 1991).
    [CrossRef]
  19. A. Friedman, Mathematics in Industrial Problems. Part 7, Vol. 67 of the IMA Volumes in Mathematics and Its Applications (Springer-Verlag, New York, to be published).
  20. G. Bao, D. Dobson, J. A. Cox, “Mathematical issues in the electromagnetic theory of gratings,” in Diffractive Optics: Design, Fabrication and Applications, Vol. 11 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 8–11.
  21. B. Engquist, A. Majda, “Absorbing boundary conditions for the numerical simulation of waves,” Math. Comp. 31, 629–655 (1971).
    [CrossRef]
  22. B. Engquist, A. Majda, “Radiation boundary conditions for acoustic and elastic wave calculations,” Commun. Pure Appl. Math. 32, 313–375 (1979).
    [CrossRef]
  23. D. Givoli, “Non-reflecting boundary conditions: a review,” J. Comp. Phys. 94, 1–29 (1991).
    [CrossRef]
  24. X. Chen, A. Friedman, “Maxwell’s equations in a periodic structure,” Trans. Am. Math. Soc. 323, 465–507 (1991).
  25. D. Dobson, A. Friedman, “The time-harmonic Maxwell equations in a doubly periodic structure,” J. Math. Anal. Appl. 166, 507–528 (1992).
    [CrossRef]
  26. G. Bao, “Diffractive optics in periodic structures: TM polarization,” preprint Ser. 1241 (Institute for Mathematics and Its Applications, University of Minnesota, Minneapolis, Minn., 1994).
  27. G. Bao, D. Dobson, “Nonlinear optics in periodic diffraction structures,” in Second International Conference on Mathematical and Numerical Aspects of Wave Propagation, R. Kleinman, T. Angell, D. Colton, F. Santosa, I. Stakgold, eds. (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1993), pp. 30–38.
  28. G. Bao, D. Dobson, “Diffractive optics in nonlinear media with periodic structure,” preprint Ser. 1124 (Institute for Mathematics and Its Applications, University of Minnesota, Minneapolis, Minn., 1993).
  29. J. C. Nédélec, F. Starling, “Integral equation methods in a quasi-periodic diffraction problem for the time-harmonic Maxwell’s equations,” SIAM J. Math. Anal. 22, 1679–1701 (1991).
    [CrossRef]
  30. J. A. Cox, D. Dobson, “An integral equation method for biperiodic diffraction structures,” in International Conference on the Applications and Theory of Periodic Structures, J. Lerner, W. McKinney, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1545, 106–113 (1991).
    [CrossRef]
  31. D. Dobson, “Optimal design of periodic antireflective structures for the Helmholtz equation,” Eur. J. Appl. Math. 4, 321–340 (1993).
    [CrossRef]
  32. T. Abboud, J. C. Nédélec, “Electromagnetic waves in an inhomogeneous medium,” J. Math. Anal. Appl. 164, 40–58 (1992).
    [CrossRef]
  33. T. Abboud, “Electromagnetic waves in periodic media,” in Second International Conference on Mathematical and Numerical Aspects of Wave Propagation, R. Kleinman, T. Angell, D. Colton, F. Santosa, I. Stakgold, eds. (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1993), pp. 1–9.
  34. A. Bonnet-Bendhia, F. Starling, “Guided waves by electromagnetic gratings and nonuniqueness examples for the diffraction problem,” Vol. 17 of Mathematical Methods in the Applied Sciences (B. G. Teubner, Stuttgart, 1994), pp. 305–338.
    [CrossRef]
  35. A. Kirsch, “Diffraction by periodic structures,” in Proceedings of the Lapland Conference on Inverse Problems, L. Pävarinta, E. Somersalo, eds. (Springer-Verlag, Berlin, 1993), pp. 87–102.
  36. P. G. Ciarlet, The Finite Element Method for Elliptic Problems (North-Holland, Amsterdam, 1978).
  37. G. Bao, “Finite elements approximation of time harmonic waves in periodic structures,” SIAM J. Numer. Anal. (to be published).
  38. G. Bao, A. Friedman, “Inverse problems for scattering by periodic structures,” Arch. Ration. Mech. Anal. (to be published).
  39. D. Dobson, “Phase reconstruction via nonlinear least-squares,” Inverse Probl. 8, 541–557 (1992).
    [CrossRef]
  40. D. Dobson, “Designing periodic structure with specified low frequency scattered far field data,” in Advances in Computer Methods for Partial Differential Equations VII, R. Vichnevetsky, D. Knight, G. Richter, eds. (International Association for Mathematics and Computers in Simulation, New Brunswick, N.J., 1992), pp. 224–230.
  41. Y. Achdou, “Numerical optimization of a photocell,” Opt. Comput. Methods Appl. Mech. Eng. 102, 89–106 (1993).
    [CrossRef]
  42. Y. Achdou, O. Pironneau, “Optimization of a photocell,” Optimal Control Appl. Methods 12, 221–246 (1991).
    [CrossRef]
  43. R. Kohn, G. Strang, “Optimal design and relaxation of variational problems. I, II, III,” Commun. Pure Appl. Math. 39, 113–137, 139–182, 353–377 (1986).
    [CrossRef]
  44. G. Bao, “A uniqueness theorem for an inverse problem in periodic diffractive optics,” Inverse Probl. 10, 335–340 (1994).
    [CrossRef]
  45. D. Dobson, “A boundary determination problem from the design of diffractive periodic structures,” in Free Boundary Problems: Theory and Applications, Pitman Research Notes in Mathematics (Pitman, New York, 1994).
  46. G. Bao, “An inverse diffraction problem in periodic structures,” in Third International Conference on Mathematical and Numerical Aspects of Wave Propagation, (Society for Industrial and Applied Mathematics, Philadelphia, Pa., to be published).
  47. A. Kirsch, “Uniqueness theorems in inverse scattering theory for periodic structures,” Inverse Probl. 10, 145–152 (1994).
    [CrossRef]
  48. J. A. Cox, D. Dobson, “Mathematical modeling for diffractive optics,” in Diffractive and Miniaturized Optics, S. Lee, ed., Proc. Soc. Photo-Opt. Instrum. Eng.CR49, 32–53 (1994).
  49. D. Dobson, “Exploiting ill-posedness in the design of diffractive optical structures,” in Smart Structures and Materials 1993: Mathematics in Smart Structures, H. T. Banks, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1919, 248–257 (1993).
    [CrossRef]
  50. G. Bao, D. Dobson, “Second harmonic generation in nonlinear optical films,” J. Math. Phys. 35, 1622–1633 (1994).
    [CrossRef]
  51. G. A. Kriegsmann, C. L. Scandrett, “Large membrane array scattering,” J. Acoust. Soc. Am. 93, 3043–3048 (1993).
    [CrossRef]

1994 (5)

O. Bruno, F. Reitich, “Approximation of analytic functions: a method of enhanced convergence,” Math. Comp. 63, 195–213 (1994).
[CrossRef]

D. Dobson, “A variational method for electromagnetic diffraction in biperiodic structures,” Model. Math. Anal. Numer. 28, 419–439 (1994).

G. Bao, “A uniqueness theorem for an inverse problem in periodic diffractive optics,” Inverse Probl. 10, 335–340 (1994).
[CrossRef]

A. Kirsch, “Uniqueness theorems in inverse scattering theory for periodic structures,” Inverse Probl. 10, 145–152 (1994).
[CrossRef]

G. Bao, D. Dobson, “Second harmonic generation in nonlinear optical films,” J. Math. Phys. 35, 1622–1633 (1994).
[CrossRef]

1993 (7)

G. A. Kriegsmann, C. L. Scandrett, “Large membrane array scattering,” J. Acoust. Soc. Am. 93, 3043–3048 (1993).
[CrossRef]

D. Dobson, “Optimal design of periodic antireflective structures for the Helmholtz equation,” Eur. J. Appl. Math. 4, 321–340 (1993).
[CrossRef]

Y. Achdou, “Numerical optimization of a photocell,” Opt. Comput. Methods Appl. Mech. Eng. 102, 89–106 (1993).
[CrossRef]

L. Li, “A model analysis of lamellar diffraction gratings in conical mountings,” J. Mod. Opt. 40, 553–573 (1993).
[CrossRef]

O. Bruno, F. Reitich, “Numerical solution of diffraction problems: a method of variation of boundaries,” J. Opt. Soc. Am. A 10, 1168–1175 (1993).
[CrossRef]

O. Bruno, F. Reitich, “Numerical solution of diffraction problems: a method of variation of boundaries. II. Finitely conducting gratings, Padé approximants, and singularities,” J. Opt. Soc. Am. A 10, 2307–2316 (1993).
[CrossRef]

O. Bruno, F. Reitich, “Numerical solution of diffraction problems: a method of variation of boundaries. III. Doubly periodic gratings,” J. Opt. Soc. Am. A 10, 2551–2562 (1993).
[CrossRef]

1992 (3)

T. Abboud, J. C. Nédélec, “Electromagnetic waves in an inhomogeneous medium,” J. Math. Anal. Appl. 164, 40–58 (1992).
[CrossRef]

D. Dobson, “Phase reconstruction via nonlinear least-squares,” Inverse Probl. 8, 541–557 (1992).
[CrossRef]

D. Dobson, A. Friedman, “The time-harmonic Maxwell equations in a doubly periodic structure,” J. Math. Anal. Appl. 166, 507–528 (1992).
[CrossRef]

1991 (4)

J. C. Nédélec, F. Starling, “Integral equation methods in a quasi-periodic diffraction problem for the time-harmonic Maxwell’s equations,” SIAM J. Math. Anal. 22, 1679–1701 (1991).
[CrossRef]

D. Givoli, “Non-reflecting boundary conditions: a review,” J. Comp. Phys. 94, 1–29 (1991).
[CrossRef]

X. Chen, A. Friedman, “Maxwell’s equations in a periodic structure,” Trans. Am. Math. Soc. 323, 465–507 (1991).

Y. Achdou, O. Pironneau, “Optimization of a photocell,” Optimal Control Appl. Methods 12, 221–246 (1991).
[CrossRef]

1986 (1)

R. Kohn, G. Strang, “Optimal design and relaxation of variational problems. I, II, III,” Commun. Pure Appl. Math. 39, 113–137, 139–182, 353–377 (1986).
[CrossRef]

1985 (1)

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

1983 (1)

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

1979 (1)

B. Engquist, A. Majda, “Radiation boundary conditions for acoustic and elastic wave calculations,” Commun. Pure Appl. Math. 32, 313–375 (1979).
[CrossRef]

1972 (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

1971 (1)

B. Engquist, A. Majda, “Absorbing boundary conditions for the numerical simulation of waves,” Math. Comp. 31, 629–655 (1971).
[CrossRef]

Abboud, T.

T. Abboud, J. C. Nédélec, “Electromagnetic waves in an inhomogeneous medium,” J. Math. Anal. Appl. 164, 40–58 (1992).
[CrossRef]

T. Abboud, “Étude mathématique et numérique de quelques problémes de diffraction d’ondes électromagnétiques,” Ph.D. dissertation (Ecole Polytechnique, Palaiseau, France1991).

T. Abboud, “Electromagnetic waves in periodic media,” in Second International Conference on Mathematical and Numerical Aspects of Wave Propagation, R. Kleinman, T. Angell, D. Colton, F. Santosa, I. Stakgold, eds. (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1993), pp. 1–9.

Achdou, Y.

Y. Achdou, “Numerical optimization of a photocell,” Opt. Comput. Methods Appl. Mech. Eng. 102, 89–106 (1993).
[CrossRef]

Y. Achdou, O. Pironneau, “Optimization of a photocell,” Optimal Control Appl. Methods 12, 221–246 (1991).
[CrossRef]

Bao, G.

G. Bao, “A uniqueness theorem for an inverse problem in periodic diffractive optics,” Inverse Probl. 10, 335–340 (1994).
[CrossRef]

G. Bao, D. Dobson, “Second harmonic generation in nonlinear optical films,” J. Math. Phys. 35, 1622–1633 (1994).
[CrossRef]

G. Bao, D. Dobson, “Nonlinear optics in periodic diffraction structures,” in Second International Conference on Mathematical and Numerical Aspects of Wave Propagation, R. Kleinman, T. Angell, D. Colton, F. Santosa, I. Stakgold, eds. (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1993), pp. 30–38.

G. Bao, D. Dobson, J. A. Cox, “Mathematical issues in the electromagnetic theory of gratings,” in Diffractive Optics: Design, Fabrication and Applications, Vol. 11 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 8–11.

G. Bao, “Finite elements approximation of time harmonic waves in periodic structures,” SIAM J. Numer. Anal. (to be published).

G. Bao, “Diffractive optics in periodic structures: TM polarization,” preprint Ser. 1241 (Institute for Mathematics and Its Applications, University of Minnesota, Minneapolis, Minn., 1994).

G. Bao, D. Dobson, “Diffractive optics in nonlinear media with periodic structure,” preprint Ser. 1124 (Institute for Mathematics and Its Applications, University of Minnesota, Minneapolis, Minn., 1993).

G. Bao, “An inverse diffraction problem in periodic structures,” in Third International Conference on Mathematical and Numerical Aspects of Wave Propagation, (Society for Industrial and Applied Mathematics, Philadelphia, Pa., to be published).

G. Bao, A. Friedman, “Inverse problems for scattering by periodic structures,” Arch. Ration. Mech. Anal. (to be published).

Bonnet-Bendhia, A.

A. Bonnet-Bendhia, F. Starling, “Guided waves by electromagnetic gratings and nonuniqueness examples for the diffraction problem,” Vol. 17 of Mathematical Methods in the Applied Sciences (B. G. Teubner, Stuttgart, 1994), pp. 305–338.
[CrossRef]

Bruno, O.

Cadilhac, M.

M. Cadilhac, “Some mathematical aspects of the grating theory,” in Electromagnetic Theory of Gratings, R. Petit, ed., Vol. 22 of Topics in Current Physics (Springer-Verlag, Berlin, 1980), pp. 53–62.
[CrossRef]

M. Cadilhac, “Rigorous vector theories of diffraction gratings,” in Progress in Optics, E. Wolf, ed. (North-Holland, New York, 1984), Vol. 21.

Chen, X.

X. Chen, A. Friedman, “Maxwell’s equations in a periodic structure,” Trans. Am. Math. Soc. 323, 465–507 (1991).

Ciarlet, P. G.

P. G. Ciarlet, The Finite Element Method for Elliptic Problems (North-Holland, Amsterdam, 1978).

Cox, J. A.

J. A. Cox, “Inverse and optimal design problems for imaging and diffractive optical systems,” in Inverse Problems and Optimal Design in Industry, H. Engl, J. McLaughlin, eds. (B. G. Teubner, Stuttgart, 1994), pp. 29–36.

J. A. Cox, D. Dobson, “An integral equation method for biperiodic diffraction structures,” in International Conference on the Applications and Theory of Periodic Structures, J. Lerner, W. McKinney, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1545, 106–113 (1991).
[CrossRef]

G. Bao, D. Dobson, J. A. Cox, “Mathematical issues in the electromagnetic theory of gratings,” in Diffractive Optics: Design, Fabrication and Applications, Vol. 11 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 8–11.

J. A. Cox, D. Dobson, “Mathematical modeling for diffractive optics,” in Diffractive and Miniaturized Optics, S. Lee, ed., Proc. Soc. Photo-Opt. Instrum. Eng.CR49, 32–53 (1994).

Dobson, D.

G. Bao, D. Dobson, “Second harmonic generation in nonlinear optical films,” J. Math. Phys. 35, 1622–1633 (1994).
[CrossRef]

D. Dobson, “A variational method for electromagnetic diffraction in biperiodic structures,” Model. Math. Anal. Numer. 28, 419–439 (1994).

D. Dobson, “Optimal design of periodic antireflective structures for the Helmholtz equation,” Eur. J. Appl. Math. 4, 321–340 (1993).
[CrossRef]

D. Dobson, A. Friedman, “The time-harmonic Maxwell equations in a doubly periodic structure,” J. Math. Anal. Appl. 166, 507–528 (1992).
[CrossRef]

D. Dobson, “Phase reconstruction via nonlinear least-squares,” Inverse Probl. 8, 541–557 (1992).
[CrossRef]

G. Bao, D. Dobson, “Diffractive optics in nonlinear media with periodic structure,” preprint Ser. 1124 (Institute for Mathematics and Its Applications, University of Minnesota, Minneapolis, Minn., 1993).

D. Dobson, “A boundary determination problem from the design of diffractive periodic structures,” in Free Boundary Problems: Theory and Applications, Pitman Research Notes in Mathematics (Pitman, New York, 1994).

J. A. Cox, D. Dobson, “An integral equation method for biperiodic diffraction structures,” in International Conference on the Applications and Theory of Periodic Structures, J. Lerner, W. McKinney, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1545, 106–113 (1991).
[CrossRef]

D. Dobson, “Exploiting ill-posedness in the design of diffractive optical structures,” in Smart Structures and Materials 1993: Mathematics in Smart Structures, H. T. Banks, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1919, 248–257 (1993).
[CrossRef]

J. A. Cox, D. Dobson, “Mathematical modeling for diffractive optics,” in Diffractive and Miniaturized Optics, S. Lee, ed., Proc. Soc. Photo-Opt. Instrum. Eng.CR49, 32–53 (1994).

G. Bao, D. Dobson, J. A. Cox, “Mathematical issues in the electromagnetic theory of gratings,” in Diffractive Optics: Design, Fabrication and Applications, Vol. 11 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 8–11.

G. Bao, D. Dobson, “Nonlinear optics in periodic diffraction structures,” in Second International Conference on Mathematical and Numerical Aspects of Wave Propagation, R. Kleinman, T. Angell, D. Colton, F. Santosa, I. Stakgold, eds. (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1993), pp. 30–38.

D. Dobson, “Designing periodic structure with specified low frequency scattered far field data,” in Advances in Computer Methods for Partial Differential Equations VII, R. Vichnevetsky, D. Knight, G. Richter, eds. (International Association for Mathematics and Computers in Simulation, New Brunswick, N.J., 1992), pp. 224–230.

Engquist, B.

B. Engquist, A. Majda, “Radiation boundary conditions for acoustic and elastic wave calculations,” Commun. Pure Appl. Math. 32, 313–375 (1979).
[CrossRef]

B. Engquist, A. Majda, “Absorbing boundary conditions for the numerical simulation of waves,” Math. Comp. 31, 629–655 (1971).
[CrossRef]

Farn, M. W.

M. W. Farn, “New iterative algorithm for the design of phase-only gratings,” in Computer and Optically Generated Holographic Optics, I. N. Cindrich, S. Lee, eds., Proc. Soc. Photo-Opt. Intrum. Eng.1555, 34–42 (1991).
[CrossRef]

Friedman, A.

D. Dobson, A. Friedman, “The time-harmonic Maxwell equations in a doubly periodic structure,” J. Math. Anal. Appl. 166, 507–528 (1992).
[CrossRef]

X. Chen, A. Friedman, “Maxwell’s equations in a periodic structure,” Trans. Am. Math. Soc. 323, 465–507 (1991).

A. Friedman, Mathematics in Industrial Problems, Vol. 16 of the IMA Volumes in Mathematics and Its Applications (Springer-Verlag, New York, 1988).
[CrossRef]

G. Bao, A. Friedman, “Inverse problems for scattering by periodic structures,” Arch. Ration. Mech. Anal. (to be published).

A. Friedman, Mathematics in Industrial Problems. Part 7, Vol. 67 of the IMA Volumes in Mathematics and Its Applications (Springer-Verlag, New York, to be published).

A. Friedman, Mathematics in Industrial Problems. Part 4, Vol. 38 of the IMA Volumes in Mathematics and Its Applications (Springer-Verlag, New York, 1991).
[CrossRef]

Gaylord, T. K.

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

Gelatt, C. D.

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

Gerchberg, R. W.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

Givoli, D.

D. Givoli, “Non-reflecting boundary conditions: a review,” J. Comp. Phys. 94, 1–29 (1991).
[CrossRef]

Kirkpatrick, S.

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

Kirsch, A.

A. Kirsch, “Uniqueness theorems in inverse scattering theory for periodic structures,” Inverse Probl. 10, 145–152 (1994).
[CrossRef]

A. Kirsch, “Diffraction by periodic structures,” in Proceedings of the Lapland Conference on Inverse Problems, L. Pävarinta, E. Somersalo, eds. (Springer-Verlag, Berlin, 1993), pp. 87–102.

Kohn, R.

R. Kohn, G. Strang, “Optimal design and relaxation of variational problems. I, II, III,” Commun. Pure Appl. Math. 39, 113–137, 139–182, 353–377 (1986).
[CrossRef]

Kriegsmann, G. A.

G. A. Kriegsmann, C. L. Scandrett, “Large membrane array scattering,” J. Acoust. Soc. Am. 93, 3043–3048 (1993).
[CrossRef]

Li, L.

L. Li, “A model analysis of lamellar diffraction gratings in conical mountings,” J. Mod. Opt. 40, 553–573 (1993).
[CrossRef]

Majda, A.

B. Engquist, A. Majda, “Radiation boundary conditions for acoustic and elastic wave calculations,” Commun. Pure Appl. Math. 32, 313–375 (1979).
[CrossRef]

B. Engquist, A. Majda, “Absorbing boundary conditions for the numerical simulation of waves,” Math. Comp. 31, 629–655 (1971).
[CrossRef]

Moharam, M. G.

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

Nédélec, J. C.

T. Abboud, J. C. Nédélec, “Electromagnetic waves in an inhomogeneous medium,” J. Math. Anal. Appl. 164, 40–58 (1992).
[CrossRef]

J. C. Nédélec, F. Starling, “Integral equation methods in a quasi-periodic diffraction problem for the time-harmonic Maxwell’s equations,” SIAM J. Math. Anal. 22, 1679–1701 (1991).
[CrossRef]

Pironneau, O.

Y. Achdou, O. Pironneau, “Optimization of a photocell,” Optimal Control Appl. Methods 12, 221–246 (1991).
[CrossRef]

Reitich, F.

Saxton, W. O.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

Scandrett, C. L.

G. A. Kriegsmann, C. L. Scandrett, “Large membrane array scattering,” J. Acoust. Soc. Am. 93, 3043–3048 (1993).
[CrossRef]

Starling, F.

J. C. Nédélec, F. Starling, “Integral equation methods in a quasi-periodic diffraction problem for the time-harmonic Maxwell’s equations,” SIAM J. Math. Anal. 22, 1679–1701 (1991).
[CrossRef]

A. Bonnet-Bendhia, F. Starling, “Guided waves by electromagnetic gratings and nonuniqueness examples for the diffraction problem,” Vol. 17 of Mathematical Methods in the Applied Sciences (B. G. Teubner, Stuttgart, 1994), pp. 305–338.
[CrossRef]

Strang, G.

R. Kohn, G. Strang, “Optimal design and relaxation of variational problems. I, II, III,” Commun. Pure Appl. Math. 39, 113–137, 139–182, 353–377 (1986).
[CrossRef]

Vecchi, M. P.

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

Commun. Pure Appl. Math. (2)

B. Engquist, A. Majda, “Radiation boundary conditions for acoustic and elastic wave calculations,” Commun. Pure Appl. Math. 32, 313–375 (1979).
[CrossRef]

R. Kohn, G. Strang, “Optimal design and relaxation of variational problems. I, II, III,” Commun. Pure Appl. Math. 39, 113–137, 139–182, 353–377 (1986).
[CrossRef]

Eur. J. Appl. Math. (1)

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G. Bao, D. Dobson, “Second harmonic generation in nonlinear optical films,” J. Math. Phys. 35, 1622–1633 (1994).
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Figures (9)

Fig. 1
Fig. 1

Geometry of the grating problem.

Fig. 2
Fig. 2

Linear grating in a silver substrate.

Fig. 3
Fig. 3

Execution time: the TM case.

Fig. 4
Fig. 4

Convergence of the MAXFELM code in TE and TM polarization.

Fig. 5
Fig. 5

Geometry of the conductor problem.

Fig. 6
Fig. 6

Geometrical configuration for TE design. For convenience the problem is scaled so that the grating period Λ = 2π.

Fig. 7
Fig. 7

Ideal array generator designs. (a) Relaxed mixture design: black, substrate; white, air; (b) equivalent interface profile: curve, a boundary between the air and the substrate.

Fig. 8
Fig. 8

(a) Optimal ramp profile and (b) gray-scale plot of the real part of the resulting diffracted field.

Fig. 9
Fig. 9

(a) Particular near-optimal profile and (b) gray-scale plot of the real part of the resulting diffracted field.

Equations (86)

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× E i ω μ H = 0 ,
× H + i ω E = 0 ,
( x 1 + n Λ , x 3 ) = ( x 1 , x 3 ) for all x 1 , x 3 ,
( x 1 , x 3 ) = 1 for x 3 b ,
( x 1 , x 3 ) = 2 for x 3 b .
( Δ + k 2 ) u = 0 in R 2 ,
( Δ α + k 2 ) u α = 0 in R 2 ,
u α ( x 1 , x 3 ) = n Z u α ( n ) ( x 3 ) exp ( i α n x 1 ) ,
u α ( n ) ( x 3 ) = 1 Λ 0 Λ u α ( x 1 , x 3 ) exp ( i α n x 1 ) d x 1 .
β j n ( α ) = exp ( i γ j n / 2 ) | k j 2 ( α n + α ) 2 | 1 / 2 n Z ,
γ j n = arg [ k j 2 ( α n + α ) 2 ] 0 γ j n < 2 π .
β j n ( α ) = { [ k j 2 ( α n + α ) 2 ] 1 / 2 k j 2 > ( α n + α ) 2 i [ ( α n + α ) 2 k j 2 ] 1 / 2 k j 2 < ( α n + α ) 2 .
u α Ω j = n Z a j n exp [ ± i β j n ( α ) x 3 + i α n x 1 ] j = 1 , 2 ,
u α ( n ) ( x 3 ) = { u α ( n ) ( b ) exp [ i β 1 n ( α ) ( x 3 b ) ] n 0 in Ω 1 u α ( 0 ) ( b ) exp [ i β 1 ( x 3 b ) ] + exp ( i β 1 x 3 ) exp [ i β 1 ( x 3 2 b ) ] n = 0 in Ω 1 u α ( n ) ( b ) exp [ i β 2 n ( α ) ( x 3 + b ) ] in Ω 2 .
u α ( n ) ν Γ j = { i β 1 n ( α ) u α ( n ) ( b ) n 0 on Γ 1 i β 1 u α 0 ( b ) 2 i β 1 exp ( i β 1 b ) n = 0 on Γ 1 i β 2 n ( α ) u α n ( b ) on Γ 2 .
u α ν | Γ 1 = n Z i β 1 n ( α ) u α ( n ) ( b ) exp ( i α n x 1 ) 2 i β 1 exp ( i β 1 b ) = T 1 ( u α Γ 1 ) 2 i β 1 exp ( i β 1 b ) ,
u α ν Γ 2 = n Z i β 2 n ( α ) u α ( n ) ( b ) exp ( i α n x 1 ) = T 2 ( u α Γ 2 ) ,
( T j f ) ( x 1 ) = n Z i β j n ( α ) f ( n ) exp ( i α n x 1 ) ,
( Δ α + k 2 ) u α = 0 in Ω
( Δ α + k 2 ) u α · ϕ ¯ = 0 .
Ω u α · ϕ ¯ + Ω ( k 2 α 2 ) u α ϕ ¯ + 2 i α Ω ( x 1 u α ) ϕ ¯ + Γ 1 ( T 1 u α ) ϕ ¯ + Γ 2 ( T 2 u α ) ϕ ¯ Γ 1 2 i β 1 exp ( i β 1 b ) ϕ ¯ = 0 ,
α · [ ( 1 / k 2 ) α u α ] + u α = 0 ,
Ω 1 k 2 u α · ϕ ¯ + Ω ( ω 2 α 2 k 2 ) u α ϕ ¯ + i α Ω 1 k 2 ( x 1 u α ) ϕ ¯ i α Ω 1 k 2 u α x 1 ϕ ¯ + Γ 1 1 k 1 2 ( T 1 u α ) ϕ ¯ + Γ 2 1 k 2 2 ( T 2 u α ) ϕ ¯ Γ 1 2 i β 1 1 k 1 2 exp ( i β 1 b ) ϕ ¯ = 0 ,
( x 1 + n 1 Λ 1 , x 2 + n 2 Λ 2 , x 3 ) = ( x 1 , x 2 , x 3 ) .
( x 1 , x 2 , x 3 ) = 1 for x 3 b ,
( x 1 , x 2 , x 3 ) = 2 for x 3 b .
E I = s exp ( i q · r ) , H I = p exp ( i q · r )
s = ( 1 / 1 ) ( p × q ) q · q = k 1 2 , p · q = 0 .
E α = exp [ i ( α 1 x 1 + α 2 x 2 ) ] E ( x 1 , x 2 , x 3 ) , H α = exp [ i ( α 1 x 1 + α 2 x 2 ) ] H ( x 1 , x 2 , x 3 ) ,
α = + i ( α 1 , α 2 , 0 ) .
α × E α i ω μ H α = 0 ,
α × H α + i ω E α = 0 .
α × [ ( 1 / μ ) α × H α ] ω 2 H α = 0 ,
α × H α + i ω E α = 0 .
α × [ ( 1 / μ ) α × H α ] α [ ( 1 / μ ) α · H α ] w 2 H α = 0
For all but a sequence of countable frequencies ω j , | ω j | + , the diffraction problem has a unique solution .
If Im 1 > 0 or Im 2 > 0 , then the diffraction problem has a unique solution .
( Δ + k 2 ) u = 0 ,
u S = 0 ,
u f u g H 1 C f g C 1 ,
α ( u α , ϕ ) = ( f , ϕ ) , ϕ H 1 ( Ω ) ,
α ( u α , ϕ ) = Ω u α · ϕ ¯ Ω ( k 2 α 2 ) u α ϕ ¯ 2 i α Ω ( x 1 u α ) ϕ ¯ Γ 1 ( T 1 u α ) ϕ ¯ Γ 2 ( T 2 u α ) ϕ ¯ ,
( f , ϕ ) = Γ 1 2 i β 1 exp ( i β 1 b ) ϕ ¯ .
α ( u α , ϕ ) = Ω 1 k 2 u α · ϕ ¯ + Ω ( ω 2 α 2 k 2 ) u α ϕ ¯ + i α Ω 1 k 2 ( x 1 u α ) ϕ ¯ i α Ω 1 k 2 u α x 1 ϕ ¯ + Γ 1 1 k 1 2 ( T 1 u α ) ϕ ¯ + Γ 1 2 k 2 2 ( T 2 u α ) ϕ ¯ Γ 1 2 i β 1 1 k 1 2 exp ( i β 1 b ) ϕ ¯
( f , ϕ ) = Γ 1 2 i β 1 1 k 1 2 exp ( i β 1 b ) ϕ ¯
a ( u h , υ h ) = ( f , υ h ) .
u h = c 1 ϕ 1 + c 2 ϕ 2 + + c k ϕ k
u α u h L 2 ( Ω ) C h 2 ,
u α u h H 1 ( Ω ) C h 1 ,
a N ( u N h , υ h ) = ( f , υ h ) υ h S h ,
( T j H f ) ( x 1 ) = | n | < N i β j n ( α ) f ( n ) exp ( i α n x 1 ) .
u α u N h L 2 C h ( h + N 1 / 2 ) ,
u α u N h H 1 C h ,
α · [ ( 1 / k 2 ) α u α ] + u α = 0 in R 2 ,
u α u h L 2 ( Ω ) δ u α u h H 1 Ω .
u α u h H 1 ( Ω ) δ f L 2 ( Ω ) .
T j N f ( x 1 ) = | n | < N i β j n f ( n ) exp ( i α n x 1 ) .
a N ( u N h , υ h ) = ( f , υ h ) υ h S h ,
u α u N h L 2 ( Ω ) exp ( γ N ) u α H 1 ( Ω ) + δ u α u N h H 1 ( Ω ) .
u α u N h H 1 ( Ω ) δ f L 2 ( Ω ) + exp [ ( 1 / 2 ) γ N ] u α H 1 ( Ω ) .
u I = exp ( i α x 1 i β 1 x 3 ) ,
( Δ + k 2 ) u = 0 ,
u S = 0 ,
u ν x 3 = b = B ( u x 3 = b ) 2 i β exp ( i β b + i α x 1 ) ,
B ( f ) = n Z i β j n f ( n ) exp [ i ( α n + α ) x 1 ]
k has a nonzero imaginary part ;
k is real , and T satisfies k 2 < 2 ( T 2 + Λ 2 ) .
d ( D 1 , D 2 ) = max { ρ ( D 1 , D 2 ) , ρ ( D 2 , D 1 ) } ,
ρ ( D 1 , D 2 ) = sup x D 1 in f y D 2 | x y | .
C 1 h d ( D , D h ) C 2 h ,
d ( D h , D ) C u h x 3 = b u x 3 = b H 1 / 2 ,
Ω = { ( x 1 , x 3 ) : b < x 3 < b } .
( Δ α + α S ) u = 0 in Ω ,
( T 1 x 3 ) u = 2 i β exp ( i β b ) on { x 3 = b } ,
( T 2 x 3 ) u = 0 on { x 3 = b } ,
( T j f ) ( x 1 ) = n Z i β j n ( α ) f ( n ) exp ( i n x 1 ) .
a s ( x ) = { k 1 2 if x is above S k 2 2 if x is below S .
P r = { n integer : β 1 n ( α ) is real }
P t = { m integer : β 2 m ( α ) is real } .
r n = u n ( b ) exp ( i β 1 b ) for n 0 , n in P r , r 0 = u n ( b ) exp ( i β 1 b ) const . for n = 0 ,
t m = u m ( b ) exp ( i β 2 b ) for m in P t .
r = ( r n ) n P r , t = ( t m ) m P t ,
min S S J ( S ) = F ( S ) g 2 2 ,
A = { a = k 2 2 γ + k 1 2 ( 1 γ ) : γ is bounded and measurable , 0 γ 1 } .
min a A J ( a ) = F ( a ) g 2 2
f ( x 1 ) = j = 1 N f j χ j ,

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