Abstract

Near- and far-field solutions are derived for the diffraction of a Guassian beam by a layer whose permittivity is periodically modulated along the longitudinal direction. We examine a first-order coupled-mode expression of the plane-wave solution by Chu and Tamir and compare it with a highly accurate second-order formulation developed by Kong, who verified its validity and accuracy by means of a rigorous method. Our results show that, while the first-order approach is accurate for Gaussian beams scattered by layers with small periodic modulation, the second-order formulation must be used for strongly modulated media. The latter formulation can easily handle also asymmetric configurations involving a periodic layer which is bounded on its two sides by media having different dielectric constants. Computed results agree qualitatively with experiments carried out by Forshaw, and they account for the observed ripples in the diffracted beam profiles. We also confirm that, in contrast to previous theoretical studies which assumed an incident plane wave of infinite extent, complete conversion of energy into the Bragg field cannot generally occur in the case of bounded beams.

© 1977 Optical Society of America

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  1. M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1965), Chap. 12, p. 579.
  2. P. Phariseau, "On the Diffraction of Light by Progressive Supersonic Waves," Proc. Ind. Acad. Sci. A 44, 165–169 (1956).
  3. R. S. Chu and T. Tamir, "Guided-Wave Theory of Light Diffraction by Acoustic Microwaves," IEEE Trans. MTT-18, 486–504 (1970).
  4. R. S. Chu and J. A. Kong, "Modal Theory of Spatially Periodic Media," IEEE Trans. MTT-25, 18–24 (1977).
  5. J. A. Kong, "Second-Order Coupled-Mode Equations for Spatially Periodic Media," J. Opt. Soc. Am. 67, 825–829 (1977).
  6. R. S. Chu, The Diffraction of Bounded Electromagnetic Beams by Periodically Modulated Media, Polytechnic Institute of Brooklyn, Ph. D. dissertation, 1971 (University Microfilms No. 71–29042, Ann Arbor, Michigan).
  7. R. S. Chu and T. Tamir, "Bragg Diffraction of Gaussian Beams by Periodically Modulated Media," J. Opt. Soc. Am. 66, 220–226 (1976).
  8. R. S. Chu and T. Tamir, "Diffraction of Gaussian Beams by Periodically Modulated Media for Incidence Close to a Bragg Angle," J. Opt. Soc. Am. 66, 1438–1440 (1976).
  9. M. R. B. Forshaw, "Diffraction of a Narrow Laser Beam by a Thick Hologram: Experimental Results," Opt. Commun. 12, 279–281 (1974).
  10. J. A. Kong, Theory of Electromagnetic Waves (Wiley, New York, 1975), p. 259.

1977 (2)

R. S. Chu and J. A. Kong, "Modal Theory of Spatially Periodic Media," IEEE Trans. MTT-25, 18–24 (1977).

J. A. Kong, "Second-Order Coupled-Mode Equations for Spatially Periodic Media," J. Opt. Soc. Am. 67, 825–829 (1977).

1976 (2)

1974 (1)

M. R. B. Forshaw, "Diffraction of a Narrow Laser Beam by a Thick Hologram: Experimental Results," Opt. Commun. 12, 279–281 (1974).

1970 (1)

R. S. Chu and T. Tamir, "Guided-Wave Theory of Light Diffraction by Acoustic Microwaves," IEEE Trans. MTT-18, 486–504 (1970).

1956 (1)

P. Phariseau, "On the Diffraction of Light by Progressive Supersonic Waves," Proc. Ind. Acad. Sci. A 44, 165–169 (1956).

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1965), Chap. 12, p. 579.

Chu, R. S.

R. S. Chu and J. A. Kong, "Modal Theory of Spatially Periodic Media," IEEE Trans. MTT-25, 18–24 (1977).

R. S. Chu and T. Tamir, "Diffraction of Gaussian Beams by Periodically Modulated Media for Incidence Close to a Bragg Angle," J. Opt. Soc. Am. 66, 1438–1440 (1976).

R. S. Chu and T. Tamir, "Bragg Diffraction of Gaussian Beams by Periodically Modulated Media," J. Opt. Soc. Am. 66, 220–226 (1976).

R. S. Chu and T. Tamir, "Guided-Wave Theory of Light Diffraction by Acoustic Microwaves," IEEE Trans. MTT-18, 486–504 (1970).

R. S. Chu, The Diffraction of Bounded Electromagnetic Beams by Periodically Modulated Media, Polytechnic Institute of Brooklyn, Ph. D. dissertation, 1971 (University Microfilms No. 71–29042, Ann Arbor, Michigan).

Forshaw, M. R. B.

M. R. B. Forshaw, "Diffraction of a Narrow Laser Beam by a Thick Hologram: Experimental Results," Opt. Commun. 12, 279–281 (1974).

Kong, J. A.

J. A. Kong, "Second-Order Coupled-Mode Equations for Spatially Periodic Media," J. Opt. Soc. Am. 67, 825–829 (1977).

R. S. Chu and J. A. Kong, "Modal Theory of Spatially Periodic Media," IEEE Trans. MTT-25, 18–24 (1977).

J. A. Kong, Theory of Electromagnetic Waves (Wiley, New York, 1975), p. 259.

Phariseau, P.

P. Phariseau, "On the Diffraction of Light by Progressive Supersonic Waves," Proc. Ind. Acad. Sci. A 44, 165–169 (1956).

Tamir, T.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1965), Chap. 12, p. 579.

IEEE Trans. (2)

R. S. Chu and T. Tamir, "Guided-Wave Theory of Light Diffraction by Acoustic Microwaves," IEEE Trans. MTT-18, 486–504 (1970).

R. S. Chu and J. A. Kong, "Modal Theory of Spatially Periodic Media," IEEE Trans. MTT-25, 18–24 (1977).

J. Opt. Soc. Am. (3)

Opt. Commun. (1)

M. R. B. Forshaw, "Diffraction of a Narrow Laser Beam by a Thick Hologram: Experimental Results," Opt. Commun. 12, 279–281 (1974).

Proc. Ind. Acad. Sci. A (1)

P. Phariseau, "On the Diffraction of Light by Progressive Supersonic Waves," Proc. Ind. Acad. Sci. A 44, 165–169 (1956).

Other (3)

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1965), Chap. 12, p. 579.

R. S. Chu, The Diffraction of Bounded Electromagnetic Beams by Periodically Modulated Media, Polytechnic Institute of Brooklyn, Ph. D. dissertation, 1971 (University Microfilms No. 71–29042, Ann Arbor, Michigan).

J. A. Kong, Theory of Electromagnetic Waves (Wiley, New York, 1975), p. 259.

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