Abstract

We present a theoretical and numerical method for solving problems of inverse scattering in optics: given data on the far field, find the scattering object. This method is applied to perfectly conducting diffraction gratings. From the efficiency curve in a Littrow mounting and in the TE case, we derive the shape of the grating surface. Two different cases must be distinguished. The first problem, which we call "reconstruction," is to compute the profile when the efficiency is experimentally known. In the second one, called "synthesis," we give <i>a priori</i> an efficiency curve and look for the corresponding grating(s), if it actually exists. We show several theoretical reconstructions for various gratings, and present our first results in the very difficult field of synthesis. The relevance of this method in the domain of electromagnetic optics is then outlined by its application to two other problems.

© 1980 Optical Society of America

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  1. Inverse Source Problems in Optics, edited by H. P. Baltes (Springer-Verlag, Heidelberg, 1978).
  2. I. J. Wilson, Ph.D. Thesis, University of Tasmania, Hobart, 1977 (unpublished).
  3. A. Roger and D. Maystre, "The perfectly conducting grating from the point of view of inverse diffraction," Opt. Acta 26, 447–460 (1979).
  4. In this grating formula, the incidence angle θ and the diffracted angles θn are defined using, respectively, the counterclockwise sense and the clockwise sense from 0y axis.
  5. R. Petit, "Electromagnetic grating theories: limitations and successes," Nouv. Rev. Opt. 6, 129–135 (1975).
  6. D. Maystre, "A new general integral theory for dielectric coated gratings," J. Opt. Soc. Am. 68, 490–495 (1978).
  7. D. Maystre, "Sur la diffraction d’une onde plane électromagnetique par un réseau métallique," Opt. Commun. 8, 216–219 (1973).
  8. R. Petit and D. Maystre, "Application des lois de l'électromagnétisme à l'étude des réseaux," Rev. Phys. Ap. 7, 427–441 (1972).
  9. D. Maystre, "Sur la diffraction et l'absorption par les réseaux utilisés dans l'infrarouge, le visible et l'ultraviolet," Thesis, Université Aix-Marseille III, CNRS A.O. 9545, 1974 (unpublished).
  10. Electromagnetic Theory of Gratings, edited by R. Petit (Springer-Verlag, Heidelberg, 1980).
  11. C. Miranda, Partial Differential Equations of Elliptic Type (Springer-Verlag, Heidelberg, 1970).
  12. J. C. Guillot and C. H. Wilcox, "Théorie spectrale du laplacien dans des ouverts coniques et cylindriques non bornés," C.R. Acad. Sci. Paris A 282, 1171–1174 (1976).
  13. L. Schwarz, Théorie des Distributions (Hermann, Paris, 1966).
  14. P. M. Van den Berg, "Diffraction theory of a reflection grating," Appl. Sci. Res. 24, 261–293 (1971), App. A.
  15. L. Y. Kantorovitch, "On Newton's method for functional equations," DAN SSSR 59 (7), 1237–1240 (1948).
  16. Dunford-Schwartz, Linear Operators (Wiley-Interscience, New York, 1966), Chap. II, p. 92.
  17. E. Hille and J. D. Tamarkin, "On the characteristic values of linear integral equations," Ata Math. 57, 1–75 (1931).
  18. A. Roger, D. Maystre, and M. Cadilhac, "On a problem of inverse scattering in Optics: the dielectric inhomogeneous medium," J.Opt. 9 (2), 83–90 (1978).
  19. M. Bertero, C. de Molb and G. A. Viano, "Restoration of optical objects using regularization," Opt. Lett. 3, 51–53 (1978).
  20. D. Maystre and P. Vincent, "Diffraction d'une onde électromagnétique plane par un objet cylindrique non infiniment conducteur de section arbitraire," Opt. Commun. 5, 327–330 (1972).
  21. A. Roger and D. Maystre, "Determination of the index profile of a dielectric plate by optical methods," SPIE 136, 26–28 (1977).
  22. A. Roger, "Determination of the index profile of a dielectric plate from scattering data," Lecture notes in Physics n° 85, Applied Inverse Problems (Springer-Verlag, Heidelberg, 1978).
  23. D. Maystre and M. Cadilhac, "A phenomenological theory for gratings, perfect blazing for polarized light in non-zero deviation mounting," Proceedings of the U.R.S.I. Symposium, Munich, 1980 (unpublished).
  24. A. Roger, "Grating profile optimizations by inverse scattering methods," Opt. Commun. 32, 11–13 (1980).
  25. M. Reed and B. Simon, Methods of Modern Mathematical Physics (Academic, New York, 1972).
  26. A. Tikhonov and V. Arsénine, Méthodes de Résolution de Problémes Mal Posés (Editions de Moscou, Moscow, 1976).
  27. K. Miller, "Least squares method for ill posed problems with a prescribed bound," Siam J. Math. Anal. 1, 52–74 (1970).
  28. R. Petit, "Quelques propriétés des réseaux métalliques," Opt. Acta 14, 301–310 (1967).
  29. M. Breidne and D. Maystre, "Equivalence of ruled, holographic and lamellar gratings in constant deviation mountings," Appl. Opt. (to be published).

1980 (2)

D. Maystre and M. Cadilhac, "A phenomenological theory for gratings, perfect blazing for polarized light in non-zero deviation mounting," Proceedings of the U.R.S.I. Symposium, Munich, 1980 (unpublished).

A. Roger, "Grating profile optimizations by inverse scattering methods," Opt. Commun. 32, 11–13 (1980).

1979 (1)

A. Roger and D. Maystre, "The perfectly conducting grating from the point of view of inverse diffraction," Opt. Acta 26, 447–460 (1979).

1978 (4)

A. Roger, "Determination of the index profile of a dielectric plate from scattering data," Lecture notes in Physics n° 85, Applied Inverse Problems (Springer-Verlag, Heidelberg, 1978).

A. Roger, D. Maystre, and M. Cadilhac, "On a problem of inverse scattering in Optics: the dielectric inhomogeneous medium," J.Opt. 9 (2), 83–90 (1978).

M. Bertero, C. de Molb and G. A. Viano, "Restoration of optical objects using regularization," Opt. Lett. 3, 51–53 (1978).

D. Maystre, "A new general integral theory for dielectric coated gratings," J. Opt. Soc. Am. 68, 490–495 (1978).

1977 (2)

I. J. Wilson, Ph.D. Thesis, University of Tasmania, Hobart, 1977 (unpublished).

A. Roger and D. Maystre, "Determination of the index profile of a dielectric plate by optical methods," SPIE 136, 26–28 (1977).

1975 (1)

R. Petit, "Electromagnetic grating theories: limitations and successes," Nouv. Rev. Opt. 6, 129–135 (1975).

1974 (1)

D. Maystre, "Sur la diffraction et l'absorption par les réseaux utilisés dans l'infrarouge, le visible et l'ultraviolet," Thesis, Université Aix-Marseille III, CNRS A.O. 9545, 1974 (unpublished).

1973 (1)

D. Maystre, "Sur la diffraction d’une onde plane électromagnetique par un réseau métallique," Opt. Commun. 8, 216–219 (1973).

1972 (2)

R. Petit and D. Maystre, "Application des lois de l'électromagnétisme à l'étude des réseaux," Rev. Phys. Ap. 7, 427–441 (1972).

D. Maystre and P. Vincent, "Diffraction d'une onde électromagnétique plane par un objet cylindrique non infiniment conducteur de section arbitraire," Opt. Commun. 5, 327–330 (1972).

1971 (1)

P. M. Van den Berg, "Diffraction theory of a reflection grating," Appl. Sci. Res. 24, 261–293 (1971), App. A.

1970 (1)

K. Miller, "Least squares method for ill posed problems with a prescribed bound," Siam J. Math. Anal. 1, 52–74 (1970).

1967 (1)

R. Petit, "Quelques propriétés des réseaux métalliques," Opt. Acta 14, 301–310 (1967).

1948 (1)

L. Y. Kantorovitch, "On Newton's method for functional equations," DAN SSSR 59 (7), 1237–1240 (1948).

1931 (1)

E. Hille and J. D. Tamarkin, "On the characteristic values of linear integral equations," Ata Math. 57, 1–75 (1931).

Arsénine, V.

A. Tikhonov and V. Arsénine, Méthodes de Résolution de Problémes Mal Posés (Editions de Moscou, Moscow, 1976).

Bertero, M.

Breidne, M.

M. Breidne and D. Maystre, "Equivalence of ruled, holographic and lamellar gratings in constant deviation mountings," Appl. Opt. (to be published).

Cadilhac, M.

D. Maystre and M. Cadilhac, "A phenomenological theory for gratings, perfect blazing for polarized light in non-zero deviation mounting," Proceedings of the U.R.S.I. Symposium, Munich, 1980 (unpublished).

A. Roger, D. Maystre, and M. Cadilhac, "On a problem of inverse scattering in Optics: the dielectric inhomogeneous medium," J.Opt. 9 (2), 83–90 (1978).

de Molb, C.

Guillot, J. C.

J. C. Guillot and C. H. Wilcox, "Théorie spectrale du laplacien dans des ouverts coniques et cylindriques non bornés," C.R. Acad. Sci. Paris A 282, 1171–1174 (1976).

Hille, E.

E. Hille and J. D. Tamarkin, "On the characteristic values of linear integral equations," Ata Math. 57, 1–75 (1931).

Kantorovitch, L. Y.

L. Y. Kantorovitch, "On Newton's method for functional equations," DAN SSSR 59 (7), 1237–1240 (1948).

Maystre, D.

D. Maystre and M. Cadilhac, "A phenomenological theory for gratings, perfect blazing for polarized light in non-zero deviation mounting," Proceedings of the U.R.S.I. Symposium, Munich, 1980 (unpublished).

A. Roger and D. Maystre, "The perfectly conducting grating from the point of view of inverse diffraction," Opt. Acta 26, 447–460 (1979).

D. Maystre, "A new general integral theory for dielectric coated gratings," J. Opt. Soc. Am. 68, 490–495 (1978).

A. Roger, D. Maystre, and M. Cadilhac, "On a problem of inverse scattering in Optics: the dielectric inhomogeneous medium," J.Opt. 9 (2), 83–90 (1978).

A. Roger and D. Maystre, "Determination of the index profile of a dielectric plate by optical methods," SPIE 136, 26–28 (1977).

D. Maystre, "Sur la diffraction et l'absorption par les réseaux utilisés dans l'infrarouge, le visible et l'ultraviolet," Thesis, Université Aix-Marseille III, CNRS A.O. 9545, 1974 (unpublished).

D. Maystre, "Sur la diffraction d’une onde plane électromagnetique par un réseau métallique," Opt. Commun. 8, 216–219 (1973).

D. Maystre and P. Vincent, "Diffraction d'une onde électromagnétique plane par un objet cylindrique non infiniment conducteur de section arbitraire," Opt. Commun. 5, 327–330 (1972).

R. Petit and D. Maystre, "Application des lois de l'électromagnétisme à l'étude des réseaux," Rev. Phys. Ap. 7, 427–441 (1972).

M. Breidne and D. Maystre, "Equivalence of ruled, holographic and lamellar gratings in constant deviation mountings," Appl. Opt. (to be published).

Miller, K.

K. Miller, "Least squares method for ill posed problems with a prescribed bound," Siam J. Math. Anal. 1, 52–74 (1970).

Miranda, C.

C. Miranda, Partial Differential Equations of Elliptic Type (Springer-Verlag, Heidelberg, 1970).

Petit, R.

R. Petit, "Electromagnetic grating theories: limitations and successes," Nouv. Rev. Opt. 6, 129–135 (1975).

R. Petit and D. Maystre, "Application des lois de l'électromagnétisme à l'étude des réseaux," Rev. Phys. Ap. 7, 427–441 (1972).

R. Petit, "Quelques propriétés des réseaux métalliques," Opt. Acta 14, 301–310 (1967).

Electromagnetic Theory of Gratings, edited by R. Petit (Springer-Verlag, Heidelberg, 1980).

Reed, M.

M. Reed and B. Simon, Methods of Modern Mathematical Physics (Academic, New York, 1972).

Roger, A.

A. Roger, "Grating profile optimizations by inverse scattering methods," Opt. Commun. 32, 11–13 (1980).

A. Roger and D. Maystre, "The perfectly conducting grating from the point of view of inverse diffraction," Opt. Acta 26, 447–460 (1979).

A. Roger, D. Maystre, and M. Cadilhac, "On a problem of inverse scattering in Optics: the dielectric inhomogeneous medium," J.Opt. 9 (2), 83–90 (1978).

A. Roger, "Determination of the index profile of a dielectric plate from scattering data," Lecture notes in Physics n° 85, Applied Inverse Problems (Springer-Verlag, Heidelberg, 1978).

A. Roger and D. Maystre, "Determination of the index profile of a dielectric plate by optical methods," SPIE 136, 26–28 (1977).

Schwarz, L.

L. Schwarz, Théorie des Distributions (Hermann, Paris, 1966).

Simon, B.

M. Reed and B. Simon, Methods of Modern Mathematical Physics (Academic, New York, 1972).

Tamarkin, J. D.

E. Hille and J. D. Tamarkin, "On the characteristic values of linear integral equations," Ata Math. 57, 1–75 (1931).

Tikhonov, A.

A. Tikhonov and V. Arsénine, Méthodes de Résolution de Problémes Mal Posés (Editions de Moscou, Moscow, 1976).

Van den Berg, P. M.

P. M. Van den Berg, "Diffraction theory of a reflection grating," Appl. Sci. Res. 24, 261–293 (1971), App. A.

Viano, G. A.

Vincent, P.

D. Maystre and P. Vincent, "Diffraction d'une onde électromagnétique plane par un objet cylindrique non infiniment conducteur de section arbitraire," Opt. Commun. 5, 327–330 (1972).

Wilcox, C. H.

J. C. Guillot and C. H. Wilcox, "Théorie spectrale du laplacien dans des ouverts coniques et cylindriques non bornés," C.R. Acad. Sci. Paris A 282, 1171–1174 (1976).

Wilson, I. J.

I. J. Wilson, Ph.D. Thesis, University of Tasmania, Hobart, 1977 (unpublished).

Appl. Sci. Res. (1)

P. M. Van den Berg, "Diffraction theory of a reflection grating," Appl. Sci. Res. 24, 261–293 (1971), App. A.

Ata Math. (1)

E. Hille and J. D. Tamarkin, "On the characteristic values of linear integral equations," Ata Math. 57, 1–75 (1931).

C.R. Acad. Sci. Paris A (1)

J. C. Guillot and C. H. Wilcox, "Théorie spectrale du laplacien dans des ouverts coniques et cylindriques non bornés," C.R. Acad. Sci. Paris A 282, 1171–1174 (1976).

DAN SSSR (1)

L. Y. Kantorovitch, "On Newton's method for functional equations," DAN SSSR 59 (7), 1237–1240 (1948).

J. Opt. Soc. Am. (1)

J.Opt. (1)

A. Roger, D. Maystre, and M. Cadilhac, "On a problem of inverse scattering in Optics: the dielectric inhomogeneous medium," J.Opt. 9 (2), 83–90 (1978).

Nouv. Rev. Opt. (1)

R. Petit, "Electromagnetic grating theories: limitations and successes," Nouv. Rev. Opt. 6, 129–135 (1975).

Opt. Acta (2)

A. Roger and D. Maystre, "The perfectly conducting grating from the point of view of inverse diffraction," Opt. Acta 26, 447–460 (1979).

R. Petit, "Quelques propriétés des réseaux métalliques," Opt. Acta 14, 301–310 (1967).

Opt. Commun. (3)

D. Maystre and P. Vincent, "Diffraction d'une onde électromagnétique plane par un objet cylindrique non infiniment conducteur de section arbitraire," Opt. Commun. 5, 327–330 (1972).

D. Maystre, "Sur la diffraction d’une onde plane électromagnetique par un réseau métallique," Opt. Commun. 8, 216–219 (1973).

A. Roger, "Grating profile optimizations by inverse scattering methods," Opt. Commun. 32, 11–13 (1980).

Opt. Lett. (1)

Rev. Phys. Ap. (1)

R. Petit and D. Maystre, "Application des lois de l'électromagnétisme à l'étude des réseaux," Rev. Phys. Ap. 7, 427–441 (1972).

Siam J. Math. Anal. (1)

K. Miller, "Least squares method for ill posed problems with a prescribed bound," Siam J. Math. Anal. 1, 52–74 (1970).

SPIE (1)

A. Roger and D. Maystre, "Determination of the index profile of a dielectric plate by optical methods," SPIE 136, 26–28 (1977).

Other (13)

A. Roger, "Determination of the index profile of a dielectric plate from scattering data," Lecture notes in Physics n° 85, Applied Inverse Problems (Springer-Verlag, Heidelberg, 1978).

D. Maystre and M. Cadilhac, "A phenomenological theory for gratings, perfect blazing for polarized light in non-zero deviation mounting," Proceedings of the U.R.S.I. Symposium, Munich, 1980 (unpublished).

M. Breidne and D. Maystre, "Equivalence of ruled, holographic and lamellar gratings in constant deviation mountings," Appl. Opt. (to be published).

Inverse Source Problems in Optics, edited by H. P. Baltes (Springer-Verlag, Heidelberg, 1978).

I. J. Wilson, Ph.D. Thesis, University of Tasmania, Hobart, 1977 (unpublished).

D. Maystre, "Sur la diffraction et l'absorption par les réseaux utilisés dans l'infrarouge, le visible et l'ultraviolet," Thesis, Université Aix-Marseille III, CNRS A.O. 9545, 1974 (unpublished).

Electromagnetic Theory of Gratings, edited by R. Petit (Springer-Verlag, Heidelberg, 1980).

C. Miranda, Partial Differential Equations of Elliptic Type (Springer-Verlag, Heidelberg, 1970).

Dunford-Schwartz, Linear Operators (Wiley-Interscience, New York, 1966), Chap. II, p. 92.

L. Schwarz, Théorie des Distributions (Hermann, Paris, 1966).

M. Reed and B. Simon, Methods of Modern Mathematical Physics (Academic, New York, 1972).

A. Tikhonov and V. Arsénine, Méthodes de Résolution de Problémes Mal Posés (Editions de Moscou, Moscow, 1976).

In this grating formula, the incidence angle θ and the diffracted angles θn are defined using, respectively, the counterclockwise sense and the clockwise sense from 0y axis.

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