Abstract

A theoretical investigation of resonant scattering from two-dimensional gratings is presented. Abrupt changes of diffraction efficiency over a small parameter range have been observed by rigorous coupled-wave analysis. The peak reflection or transmission efficiencies can approach unity. This phenomenon is explained in terms of the coupling between the incident plane wave and guided modes that can be supported by the two-dimensional-grating waveguide structure. Because of the double periodicity, the incident field can be coupled into any direction in the grating plane. The guided modes supported by two-dimensional gratings are found by rigorous solution of the homogeneous problem associated with the scattering (inhomogeneous) problem. The complex propagation constants for the guided modes provide estimates of both the resonance angle and width. In addition, to illustrate the implication of the radical change in the phase and amplitude of the propagating waves, we report a study of finite-beam diffraction in the resonant scattering region. Applications for the structures include polarization-independent narrow-band filters and bandwidth-tunable filters. It is shown that, because of the double resonance, the polarization-independent narrow-band filters have a large angular tolerance.

© 1996 Optical Society of America

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References

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  1. R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Philos. Mag. 4, 396–402 (1902).
  2. L. Rayleigh, “On the dynamical theory of gratings,” Proc. R. Soc. London Ser. A 79, 399–416 (1907).
    [CrossRef]
  3. A. Hessel, A. A. Oliner, “A new theory of Wood’s anomalies on optical gratings,” Appl. Opt. 4, 1275–1297 (1965).
    [CrossRef]
  4. N. Nevière, R. Petit, M. Cadilhac, “About the theory of optical grating coupler-waveguide systems,” Opt. Commun. 8, 113–117 (1973).
    [CrossRef]
  5. M. Nevière, R. Petit, M. Cadilhac, “Systematic study of resonances of holographic thin film couplers,” Opt. Commun. 9, 48–53 (1973).
    [CrossRef]
  6. E. Popov, L. Mashev, D. Maystre, “Theoretical study of the anomalies of coated dielectric gratings,” Opt. Acta 33, 607–619 (1986).
    [CrossRef]
  7. H. L. Bertoni, L.-H. S. Cheo, T. Tamir, “Frequency-selective reflection and transmission by a periodic dielectric layer,” IEEE Trans. Antennas Propag. 37, 78–83 (1989).
    [CrossRef]
  8. L. Mashev, E. Popov, “Zero order anomaly of dielectric coated gratings,” Opt. Commun. 55, 377–380 (1985).
    [CrossRef]
  9. M. T. Gale, K. Knop, R. Morf, “Zero-order diffractive microstructures for security applications,” in Optical Security and Anticounterfeiting Systems, W. Fagan, ed., Proc. SPIE1210, 83–89 (1990).
    [CrossRef]
  10. S. S. Wang, R. Magnusson, J. S. Bagby, “Guided-mode resonances in planar dielectric-layer diffraction gratings,” J. Opt. Soc. Am. A 7, 1470–1474 (1990).
    [CrossRef]
  11. S. S. Wang, R. Magnusson, “Design of waveguide-grating filters with symmetrical line shapes and low sidebands,” Opt. Lett. 19, 919–921 (1994).
    [CrossRef] [PubMed]
  12. R. Ulrich, “Modes of propagation on an open periodic waveguide for the far infrared,” in Proceedings of the Symposium on Optical and Acoustical Micro-Electronics (Polytechnic Press, New York, 1974), pp. 359–376.
  13. R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980), pp. 123–157.
  14. P. Vincent, “A finite-different method for dielectric and conducting crossed gratings,” Opt. Commun. 26, 293–296 (1978).
    [CrossRef]
  15. M. G. Moharam, T. K. Gaylord, “Coupled-wave analysis of two-dimensional gratings,” in Holographic Optics: Design and Applications, I. Cindrich, ed., Proc. SPIE883, 8–11 (1986).
    [CrossRef]
  16. G. H. Derrick, R. C. McPhedran, D. Maystrey, M. Nevière, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
    [CrossRef]
  17. O. P. 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]
  18. P. St. J. Russell, “Power conservation and field structures in uniform dielectric gratings,” J. Opt. Soc. Am. A 1, 293–299 (1984).
    [CrossRef]
  19. S. Peng, G. M. Morris, “Efficient implementation of rigorous coupled-wave analysis for surface-relief gratings,” J. Opt. Soc. Am. A 12, 1087–1096 (1995).
    [CrossRef]
  20. N. Chateau, J. Hugonin, “Algorithm for the rigorous coupled-wave analysis of grating diffraction,” J. Opt. Soc. Am. A 11, 1321–1331 (1994).
    [CrossRef]
  21. L. Li, “Multilayer modal method for diffraction gratings of arbitrary profile, depth, and permittivity,” J. Opt. Soc. Am. A 10, 2581–2591 (1993).
    [CrossRef]
  22. J. Chilwell, I. Hodgkinson, “Thin-films field-transfer matrix theory of planar multilayer waveguides and reflection from prism-loaded waveguides,” J. Opt. Soc. Am. A 1, 742–753 (1984).
    [CrossRef]
  23. P. Vincent, M. Nevière, “Corrugated dielectric waveguides: a numerical study of the second-order stop bands,” Appl. Phys. 20, 345–351 (1979).
    [CrossRef]
  24. M. Nevière, E. Popov, R. Reinisch, “Electromagnetic resonances in linear and nonlinear optics: phenomenological study of grating behavior through the poles and zeros of the scattering operator,” J. Opt. Soc. Am. A 12, 513–523 (1995).
    [CrossRef]
  25. S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
    [CrossRef]
  26. K. C. Chang, V. Shah, T. Tamir, “Scattering and guiding of waves by dielectric grating with arbitrary profiles,” J. Opt. Soc. Am. 70, 804–813 (1980).
    [CrossRef]
  27. S. T. Peng, “Rigorous formulation of scattering and guidance by dielectric grating waveguides: general case of oblique incidence,” J. Opt. Soc. Am. A 6, 1869–1883 (1989).
    [CrossRef]
  28. S. Zhang, T. Tamir, “Spatial modifications of Gaussian beams diffracted by reflection gratings,” J. Opt. Soc. Am. A 6, 1368–1381 (1989).
    [CrossRef]
  29. C. W. Hsue, T. Tamir, “Lateral displacement and distortion of beams incident upon a transmitting-layer configuration,” J. Opt. Soc. Am. A 2, 978–988 (1985).
    [CrossRef]
  30. V. Shah, T. Tamir, “Absorption and lateral shift of beams incident upon lossy multilayered media,” J. Opt. Soc. Am. 73, 37–44 (1983).
    [CrossRef]
  31. T. Tamir, H. L. Bertoni, “Lateral displacement of optical beams at multilayered and periodic structures,” J. Opt. Soc. Am. 61, 1397–1413 (1971).
    [CrossRef]
  32. G. A. Evans, N. W. Carlson, J. M. Hammer, S. L. Palfrey, R. Amantea, L. A. Carr, F. Z. Hawrylo, E. A. James, C. J. Kaiser, J. B. Kirk, W. F. Reichert, “Two-dimensional coherent laser arrays using grating surface emission,” IEEE J. Quantum Electron. 25, 1525–1538 (1989).
    [CrossRef]
  33. S. S. Wang, R. Magnusson, “Theory and applications of guided-mode resonance filters,” Appl. Opt. 32, 2606–2613 (1993).
    [CrossRef] [PubMed]

1995 (2)

1994 (2)

1993 (3)

1990 (1)

1989 (4)

S. Zhang, T. Tamir, “Spatial modifications of Gaussian beams diffracted by reflection gratings,” J. Opt. Soc. Am. A 6, 1368–1381 (1989).
[CrossRef]

S. T. Peng, “Rigorous formulation of scattering and guidance by dielectric grating waveguides: general case of oblique incidence,” J. Opt. Soc. Am. A 6, 1869–1883 (1989).
[CrossRef]

G. A. Evans, N. W. Carlson, J. M. Hammer, S. L. Palfrey, R. Amantea, L. A. Carr, F. Z. Hawrylo, E. A. James, C. J. Kaiser, J. B. Kirk, W. F. Reichert, “Two-dimensional coherent laser arrays using grating surface emission,” IEEE J. Quantum Electron. 25, 1525–1538 (1989).
[CrossRef]

H. L. Bertoni, L.-H. S. Cheo, T. Tamir, “Frequency-selective reflection and transmission by a periodic dielectric layer,” IEEE Trans. Antennas Propag. 37, 78–83 (1989).
[CrossRef]

1986 (1)

E. Popov, L. Mashev, D. Maystre, “Theoretical study of the anomalies of coated dielectric gratings,” Opt. Acta 33, 607–619 (1986).
[CrossRef]

1985 (2)

1984 (2)

1983 (1)

1980 (1)

1979 (2)

G. H. Derrick, R. C. McPhedran, D. Maystrey, M. Nevière, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
[CrossRef]

P. Vincent, M. Nevière, “Corrugated dielectric waveguides: a numerical study of the second-order stop bands,” Appl. Phys. 20, 345–351 (1979).
[CrossRef]

1978 (1)

P. Vincent, “A finite-different method for dielectric and conducting crossed gratings,” Opt. Commun. 26, 293–296 (1978).
[CrossRef]

1975 (1)

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

1973 (2)

N. Nevière, R. Petit, M. Cadilhac, “About the theory of optical grating coupler-waveguide systems,” Opt. Commun. 8, 113–117 (1973).
[CrossRef]

M. Nevière, R. Petit, M. Cadilhac, “Systematic study of resonances of holographic thin film couplers,” Opt. Commun. 9, 48–53 (1973).
[CrossRef]

1971 (1)

1965 (1)

1907 (1)

L. Rayleigh, “On the dynamical theory of gratings,” Proc. R. Soc. London Ser. A 79, 399–416 (1907).
[CrossRef]

1902 (1)

R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Philos. Mag. 4, 396–402 (1902).

Amantea, R.

G. A. Evans, N. W. Carlson, J. M. Hammer, S. L. Palfrey, R. Amantea, L. A. Carr, F. Z. Hawrylo, E. A. James, C. J. Kaiser, J. B. Kirk, W. F. Reichert, “Two-dimensional coherent laser arrays using grating surface emission,” IEEE J. Quantum Electron. 25, 1525–1538 (1989).
[CrossRef]

Bagby, J. S.

Bertoni, H. L.

H. L. Bertoni, L.-H. S. Cheo, T. Tamir, “Frequency-selective reflection and transmission by a periodic dielectric layer,” IEEE Trans. Antennas Propag. 37, 78–83 (1989).
[CrossRef]

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

T. Tamir, H. L. Bertoni, “Lateral displacement of optical beams at multilayered and periodic structures,” J. Opt. Soc. Am. 61, 1397–1413 (1971).
[CrossRef]

Bruno, O. P.

Cadilhac, M.

N. Nevière, R. Petit, M. Cadilhac, “About the theory of optical grating coupler-waveguide systems,” Opt. Commun. 8, 113–117 (1973).
[CrossRef]

M. Nevière, R. Petit, M. Cadilhac, “Systematic study of resonances of holographic thin film couplers,” Opt. Commun. 9, 48–53 (1973).
[CrossRef]

Carlson, N. W.

G. A. Evans, N. W. Carlson, J. M. Hammer, S. L. Palfrey, R. Amantea, L. A. Carr, F. Z. Hawrylo, E. A. James, C. J. Kaiser, J. B. Kirk, W. F. Reichert, “Two-dimensional coherent laser arrays using grating surface emission,” IEEE J. Quantum Electron. 25, 1525–1538 (1989).
[CrossRef]

Carr, L. A.

G. A. Evans, N. W. Carlson, J. M. Hammer, S. L. Palfrey, R. Amantea, L. A. Carr, F. Z. Hawrylo, E. A. James, C. J. Kaiser, J. B. Kirk, W. F. Reichert, “Two-dimensional coherent laser arrays using grating surface emission,” IEEE J. Quantum Electron. 25, 1525–1538 (1989).
[CrossRef]

Chang, K. C.

Chateau, N.

Cheo, L.-H. S.

H. L. Bertoni, L.-H. S. Cheo, T. Tamir, “Frequency-selective reflection and transmission by a periodic dielectric layer,” IEEE Trans. Antennas Propag. 37, 78–83 (1989).
[CrossRef]

Chilwell, J.

Derrick, G. H.

G. H. Derrick, R. C. McPhedran, D. Maystrey, M. Nevière, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
[CrossRef]

Evans, G. A.

G. A. Evans, N. W. Carlson, J. M. Hammer, S. L. Palfrey, R. Amantea, L. A. Carr, F. Z. Hawrylo, E. A. James, C. J. Kaiser, J. B. Kirk, W. F. Reichert, “Two-dimensional coherent laser arrays using grating surface emission,” IEEE J. Quantum Electron. 25, 1525–1538 (1989).
[CrossRef]

Gale, M. T.

M. T. Gale, K. Knop, R. Morf, “Zero-order diffractive microstructures for security applications,” in Optical Security and Anticounterfeiting Systems, W. Fagan, ed., Proc. SPIE1210, 83–89 (1990).
[CrossRef]

Gaylord, T. K.

M. G. Moharam, T. K. Gaylord, “Coupled-wave analysis of two-dimensional gratings,” in Holographic Optics: Design and Applications, I. Cindrich, ed., Proc. SPIE883, 8–11 (1986).
[CrossRef]

Hammer, J. M.

G. A. Evans, N. W. Carlson, J. M. Hammer, S. L. Palfrey, R. Amantea, L. A. Carr, F. Z. Hawrylo, E. A. James, C. J. Kaiser, J. B. Kirk, W. F. Reichert, “Two-dimensional coherent laser arrays using grating surface emission,” IEEE J. Quantum Electron. 25, 1525–1538 (1989).
[CrossRef]

Hawrylo, F. Z.

G. A. Evans, N. W. Carlson, J. M. Hammer, S. L. Palfrey, R. Amantea, L. A. Carr, F. Z. Hawrylo, E. A. James, C. J. Kaiser, J. B. Kirk, W. F. Reichert, “Two-dimensional coherent laser arrays using grating surface emission,” IEEE J. Quantum Electron. 25, 1525–1538 (1989).
[CrossRef]

Hessel, A.

Hodgkinson, I.

Hsue, C. W.

Hugonin, J.

James, E. A.

G. A. Evans, N. W. Carlson, J. M. Hammer, S. L. Palfrey, R. Amantea, L. A. Carr, F. Z. Hawrylo, E. A. James, C. J. Kaiser, J. B. Kirk, W. F. Reichert, “Two-dimensional coherent laser arrays using grating surface emission,” IEEE J. Quantum Electron. 25, 1525–1538 (1989).
[CrossRef]

Kaiser, C. J.

G. A. Evans, N. W. Carlson, J. M. Hammer, S. L. Palfrey, R. Amantea, L. A. Carr, F. Z. Hawrylo, E. A. James, C. J. Kaiser, J. B. Kirk, W. F. Reichert, “Two-dimensional coherent laser arrays using grating surface emission,” IEEE J. Quantum Electron. 25, 1525–1538 (1989).
[CrossRef]

Kirk, J. B.

G. A. Evans, N. W. Carlson, J. M. Hammer, S. L. Palfrey, R. Amantea, L. A. Carr, F. Z. Hawrylo, E. A. James, C. J. Kaiser, J. B. Kirk, W. F. Reichert, “Two-dimensional coherent laser arrays using grating surface emission,” IEEE J. Quantum Electron. 25, 1525–1538 (1989).
[CrossRef]

Knop, K.

M. T. Gale, K. Knop, R. Morf, “Zero-order diffractive microstructures for security applications,” in Optical Security and Anticounterfeiting Systems, W. Fagan, ed., Proc. SPIE1210, 83–89 (1990).
[CrossRef]

Li, L.

Magnusson, R.

Mashev, L.

E. Popov, L. Mashev, D. Maystre, “Theoretical study of the anomalies of coated dielectric gratings,” Opt. Acta 33, 607–619 (1986).
[CrossRef]

L. Mashev, E. Popov, “Zero order anomaly of dielectric coated gratings,” Opt. Commun. 55, 377–380 (1985).
[CrossRef]

Maystre, D.

E. Popov, L. Mashev, D. Maystre, “Theoretical study of the anomalies of coated dielectric gratings,” Opt. Acta 33, 607–619 (1986).
[CrossRef]

Maystrey, D.

G. H. Derrick, R. C. McPhedran, D. Maystrey, M. Nevière, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
[CrossRef]

McPhedran, R. C.

G. H. Derrick, R. C. McPhedran, D. Maystrey, M. Nevière, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
[CrossRef]

Moharam, M. G.

M. G. Moharam, T. K. Gaylord, “Coupled-wave analysis of two-dimensional gratings,” in Holographic Optics: Design and Applications, I. Cindrich, ed., Proc. SPIE883, 8–11 (1986).
[CrossRef]

Morf, R.

M. T. Gale, K. Knop, R. Morf, “Zero-order diffractive microstructures for security applications,” in Optical Security and Anticounterfeiting Systems, W. Fagan, ed., Proc. SPIE1210, 83–89 (1990).
[CrossRef]

Morris, G. M.

Nevière, M.

M. Nevière, E. Popov, R. Reinisch, “Electromagnetic resonances in linear and nonlinear optics: phenomenological study of grating behavior through the poles and zeros of the scattering operator,” J. Opt. Soc. Am. A 12, 513–523 (1995).
[CrossRef]

G. H. Derrick, R. C. McPhedran, D. Maystrey, M. Nevière, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
[CrossRef]

P. Vincent, M. Nevière, “Corrugated dielectric waveguides: a numerical study of the second-order stop bands,” Appl. Phys. 20, 345–351 (1979).
[CrossRef]

M. Nevière, R. Petit, M. Cadilhac, “Systematic study of resonances of holographic thin film couplers,” Opt. Commun. 9, 48–53 (1973).
[CrossRef]

Nevière, N.

N. Nevière, R. Petit, M. Cadilhac, “About the theory of optical grating coupler-waveguide systems,” Opt. Commun. 8, 113–117 (1973).
[CrossRef]

Oliner, A. A.

Palfrey, S. L.

G. A. Evans, N. W. Carlson, J. M. Hammer, S. L. Palfrey, R. Amantea, L. A. Carr, F. Z. Hawrylo, E. A. James, C. J. Kaiser, J. B. Kirk, W. F. Reichert, “Two-dimensional coherent laser arrays using grating surface emission,” IEEE J. Quantum Electron. 25, 1525–1538 (1989).
[CrossRef]

Peng, S.

Peng, S. T.

S. T. Peng, “Rigorous formulation of scattering and guidance by dielectric grating waveguides: general case of oblique incidence,” J. Opt. Soc. Am. A 6, 1869–1883 (1989).
[CrossRef]

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

Petit, R.

N. Nevière, R. Petit, M. Cadilhac, “About the theory of optical grating coupler-waveguide systems,” Opt. Commun. 8, 113–117 (1973).
[CrossRef]

M. Nevière, R. Petit, M. Cadilhac, “Systematic study of resonances of holographic thin film couplers,” Opt. Commun. 9, 48–53 (1973).
[CrossRef]

Popov, E.

M. Nevière, E. Popov, R. Reinisch, “Electromagnetic resonances in linear and nonlinear optics: phenomenological study of grating behavior through the poles and zeros of the scattering operator,” J. Opt. Soc. Am. A 12, 513–523 (1995).
[CrossRef]

E. Popov, L. Mashev, D. Maystre, “Theoretical study of the anomalies of coated dielectric gratings,” Opt. Acta 33, 607–619 (1986).
[CrossRef]

L. Mashev, E. Popov, “Zero order anomaly of dielectric coated gratings,” Opt. Commun. 55, 377–380 (1985).
[CrossRef]

Rayleigh, L.

L. Rayleigh, “On the dynamical theory of gratings,” Proc. R. Soc. London Ser. A 79, 399–416 (1907).
[CrossRef]

Reichert, W. F.

G. A. Evans, N. W. Carlson, J. M. Hammer, S. L. Palfrey, R. Amantea, L. A. Carr, F. Z. Hawrylo, E. A. James, C. J. Kaiser, J. B. Kirk, W. F. Reichert, “Two-dimensional coherent laser arrays using grating surface emission,” IEEE J. Quantum Electron. 25, 1525–1538 (1989).
[CrossRef]

Reinisch, R.

Reitich, F.

Russell, P. St. J.

Shah, V.

Tamir, T.

Ulrich, R.

R. Ulrich, “Modes of propagation on an open periodic waveguide for the far infrared,” in Proceedings of the Symposium on Optical and Acoustical Micro-Electronics (Polytechnic Press, New York, 1974), pp. 359–376.

Vincent, P.

P. Vincent, M. Nevière, “Corrugated dielectric waveguides: a numerical study of the second-order stop bands,” Appl. Phys. 20, 345–351 (1979).
[CrossRef]

P. Vincent, “A finite-different method for dielectric and conducting crossed gratings,” Opt. Commun. 26, 293–296 (1978).
[CrossRef]

Wang, S. S.

Wood, R. W.

R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Philos. Mag. 4, 396–402 (1902).

Zhang, S.

Appl. Opt. (2)

Appl. Phys. (2)

P. Vincent, M. Nevière, “Corrugated dielectric waveguides: a numerical study of the second-order stop bands,” Appl. Phys. 20, 345–351 (1979).
[CrossRef]

G. H. Derrick, R. C. McPhedran, D. Maystrey, M. Nevière, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
[CrossRef]

IEEE J. Quantum Electron. (1)

G. A. Evans, N. W. Carlson, J. M. Hammer, S. L. Palfrey, R. Amantea, L. A. Carr, F. Z. Hawrylo, E. A. James, C. J. Kaiser, J. B. Kirk, W. F. Reichert, “Two-dimensional coherent laser arrays using grating surface emission,” IEEE J. Quantum Electron. 25, 1525–1538 (1989).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

H. L. Bertoni, L.-H. S. Cheo, T. Tamir, “Frequency-selective reflection and transmission by a periodic dielectric layer,” IEEE Trans. Antennas Propag. 37, 78–83 (1989).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (11)

M. Nevière, E. Popov, R. Reinisch, “Electromagnetic resonances in linear and nonlinear optics: phenomenological study of grating behavior through the poles and zeros of the scattering operator,” J. Opt. Soc. Am. A 12, 513–523 (1995).
[CrossRef]

S. T. Peng, “Rigorous formulation of scattering and guidance by dielectric grating waveguides: general case of oblique incidence,” J. Opt. Soc. Am. A 6, 1869–1883 (1989).
[CrossRef]

S. Zhang, T. Tamir, “Spatial modifications of Gaussian beams diffracted by reflection gratings,” J. Opt. Soc. Am. A 6, 1368–1381 (1989).
[CrossRef]

C. W. Hsue, T. Tamir, “Lateral displacement and distortion of beams incident upon a transmitting-layer configuration,” J. Opt. Soc. Am. A 2, 978–988 (1985).
[CrossRef]

O. P. 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]

P. St. J. Russell, “Power conservation and field structures in uniform dielectric gratings,” J. Opt. Soc. Am. A 1, 293–299 (1984).
[CrossRef]

S. Peng, G. M. Morris, “Efficient implementation of rigorous coupled-wave analysis for surface-relief gratings,” J. Opt. Soc. Am. A 12, 1087–1096 (1995).
[CrossRef]

N. Chateau, J. Hugonin, “Algorithm for the rigorous coupled-wave analysis of grating diffraction,” J. Opt. Soc. Am. A 11, 1321–1331 (1994).
[CrossRef]

L. Li, “Multilayer modal method for diffraction gratings of arbitrary profile, depth, and permittivity,” J. Opt. Soc. Am. A 10, 2581–2591 (1993).
[CrossRef]

J. Chilwell, I. Hodgkinson, “Thin-films field-transfer matrix theory of planar multilayer waveguides and reflection from prism-loaded waveguides,” J. Opt. Soc. Am. A 1, 742–753 (1984).
[CrossRef]

S. S. Wang, R. Magnusson, J. S. Bagby, “Guided-mode resonances in planar dielectric-layer diffraction gratings,” J. Opt. Soc. Am. A 7, 1470–1474 (1990).
[CrossRef]

Opt. Acta (1)

E. Popov, L. Mashev, D. Maystre, “Theoretical study of the anomalies of coated dielectric gratings,” Opt. Acta 33, 607–619 (1986).
[CrossRef]

Opt. Commun. (4)

N. Nevière, R. Petit, M. Cadilhac, “About the theory of optical grating coupler-waveguide systems,” Opt. Commun. 8, 113–117 (1973).
[CrossRef]

M. Nevière, R. Petit, M. Cadilhac, “Systematic study of resonances of holographic thin film couplers,” Opt. Commun. 9, 48–53 (1973).
[CrossRef]

L. Mashev, E. Popov, “Zero order anomaly of dielectric coated gratings,” Opt. Commun. 55, 377–380 (1985).
[CrossRef]

P. Vincent, “A finite-different method for dielectric and conducting crossed gratings,” Opt. Commun. 26, 293–296 (1978).
[CrossRef]

Opt. Lett. (1)

Philos. Mag. (1)

R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Philos. Mag. 4, 396–402 (1902).

Proc. R. Soc. London Ser. A (1)

L. Rayleigh, “On the dynamical theory of gratings,” Proc. R. Soc. London Ser. A 79, 399–416 (1907).
[CrossRef]

Other (4)

R. Ulrich, “Modes of propagation on an open periodic waveguide for the far infrared,” in Proceedings of the Symposium on Optical and Acoustical Micro-Electronics (Polytechnic Press, New York, 1974), pp. 359–376.

R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980), pp. 123–157.

M. G. Moharam, T. K. Gaylord, “Coupled-wave analysis of two-dimensional gratings,” in Holographic Optics: Design and Applications, I. Cindrich, ed., Proc. SPIE883, 8–11 (1986).
[CrossRef]

M. T. Gale, K. Knop, R. Morf, “Zero-order diffractive microstructures for security applications,” in Optical Security and Anticounterfeiting Systems, W. Fagan, ed., Proc. SPIE1210, 83–89 (1990).
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

Geometry of a 2D grating with an incident plane wave.

Fig. 2
Fig. 2

Diffraction efficiency of the zeroth-order reflected wave for a 2D binary grating in the cases of (a) α = 0 deg and ψ = 90 deg, (b) α = 3 deg and ψ = 90 deg, (c) α = 3 deg and ψ = 0 deg, and (d) α = 0 deg and ψ = 45 deg, and Λ x = Λ y = 0.4 2. The values of other parameters are Λx = Λy = 0.4 [except in (d)], d = 0.17, I = 1.0, III = 1.462, II = 4, II′ = 3.24, fx = fy = 0.5, and δ = 0.

Fig. 3
Fig. 3

Dispersion relation of a planar waveguide with indices of refraction of 1.0, 1.46, and 1.8520 for the cover, the substrate, and the film layer, respectively. The intersection points indicate the resonance condition.

Fig. 4
Fig. 4

Normalized complex transverse wave number N + for grating duty cycle fx = fy changing from 0 to 100% in increments of 2.5% for the points from left to right. Other values of parameters are λ = 0.7, Λx = Λy = 0.4, d = 0.17, I = 1.0, III = 1.462, II = 4, and II′ = 3.24.

Fig. 5
Fig. 5

Coordinate systems for diffraction of optical beams from gratings. The z = 0 plane is the interface between the grating and the incident medium.

Fig. 6
Fig. 6

Diffraction of a Gaussian beam in a 2D grating. (a) The angular dependence of the diffraction efficiency and the y-component phase of the zeroth-order reflected wave; ψ = 90 deg, δ = 0. (b) The contour plot of the intensity of the incident Gaussian beam at the z = 0 plane; the peak intensity is unity. (c) The contour plot of the intensity of the reflected beam at the z = 0 plane. Other values of parameters are λ = 0.7, Λx = Λy = 0.4, d = 0.17, I = 1.0, III = 1.462, II = 4, II′ = 3.24, fx = fy = 0.5, α = 9.730 deg, ψ = 90 deg, δ = 0, w0 = 500, and h = 0.

Fig. 7
Fig. 7

Diffraction of a Gaussian beam in the region of TE guided-mode resonance: (a) the contour plot of the intensity of the reflected beam at the z = 0 plane, (b) the diffraction efficiency of each plane wave of the angular spectrum. The values of parameters are λ = 0.640035, Λx = Λy = 0.4, d = 0.17, I = 1, 0, III = 1.462, II = 4, II′ = 3.24, fx = fy = 0.5, α = 0, ψ = 90 deg, δ = 0, w0 = 200, and h = 0.

Fig. 8
Fig. 8

Diffraction of a Gaussian beam in the region of TM guided-mode resonance: (a) the contour plot of the intensity of the reflected beam at the z = 0, (b) the diffraction efficiency of each plane wave of the angular spectrum. The values of parameters are the same as in Fig. 7, except that λ = 0.607269.

Fig. 9
Fig. 9

Response of a two-layer polarization-independent narrow-band grating resonance filter. The uniform layer between the grating layer and the substrate has an index of refraction of 2.0 and a thickness of 0.2084. The values of the parameter of the incident-wave and the grating layers are Λx = Λy = 0.6367, d = 0.1308 (grating layer), I = 1.0, III = 2.25, II = 1, II′ = 2.25, fx = fy = 0.5, α = 0, and δ = 0.

Fig. 10
Fig. 10

Response of a dual-bandwidth narrow-band grating resonance filter. The values of parameters are Λx = 0.6619, Λy = 0.50, d = 0.3025, I = 1.462, III = 1.462, II = 4, II′ = 1.522, fx = fy = 0.5, α = 0, δ = 0; ψ = 90 deg for TE and ψ = 0 for TM.

Equations (38)

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k 1 = k x x ^ + k y y ^ + k 1 z z ^ = k 1 ( sin α cos δ x ^ + sin α sin δ y ^ + cos α z ^ ) ,
u ^ = u x x ^ + u y y ^ + u z z ^ = ( cos ψ cos α cos δ - sin ψ sin δ ) x ^ + ( cos ψ cos α sin δ + sin ψ cos δ ) y ^ - ( cos ψ sin α ) z ^ ,
E 1 = u ^ exp [ - j k 1 · ( r + ½ d z ^ ) ] + m , n R m n exp [ - j k 1 , m n · ( r + ½ d z ^ ) ] ,
E 3 = m , n T m n exp [ - j k 3 , m n · ( r - ½ d z ^ ) ] ,
k l , m n = k x m x ^ + k y n y ^ + k z l , m n z ^ ,             l = 1 , 3 ,
k x m = k x - m K x ,
k y n = k y - n K y ,
k z l , m n 2 = k l 2 - k x m 2 - k y n 2 ,             l = 1 , 3 ,
k x m x ^ + k y n y ^ = β .
E 2 x x ^ + E 2 y y ^ = m , n [ S x m n ( z ) x ^ + S y m n ( z ) y ^ ] exp ( - j σ m n · r ) ,
H 2 x x ^ + H 2 y y ^ = 0 / μ 0 m , n [ U x m n ( z ) x ^ + U y m n ( z ) y ^ ] × exp ( - j σ m n · r ) ,
σ m n = k x m x ^ + k y n y ^
( k x m 2 + k y n 2 ) 1 / 2 β ,
S x m n ( z ) = q C q ω 1 , m n q exp ( λ q z ) ,
S y m n ( z ) = q C q ω 2 , m n q exp ( λ q z ) ,
U x m n ( z ) = q C q ω 3 , m n q exp ( λ q z ) ,
U y m n ( z ) = q C q ω 4 , m n q exp ( λ q z ) ,
q C q [ k x m k y n ω 1 , m n q + ( k z 1 , m n 2 + k y n 2 ) ω 2 , m n q + k 0 k z 1 , m n ω 3 , m n q ] × exp ( - λ q d / 2 ) = δ m 0 δ n 0 [ k z 1 , m n ( k y u z - k 1 z u y ) + k z 1 , m n 2 u y + k x m k y n u x + k y n 2 u y ] ,
q C q [ ( k z 1 , m n 2 + k x m 2 ) ω 1 , m n q + k x m k y n ω 2 , m n q - k k z 1 , m n ω 4 , m n q ] × exp ( - λ q d / 2 ) = δ m 0 δ n 0 [ ( k x u x - k 1 z ) k z 1 , m n + k z 1 , m n 2 u x + k x m 2 u x + k x m k y n u y ] ,
q C q [ k x m k y n ω 1 , m n q + ( k z 3 , m n 2 + k y n 2 ) ω 2 , m n q + k k z 3 , m n ω 3 , m n q ] × exp ( - λ q d / 2 ) = 0 ,
q C q [ ( k z 3 , m n 2 + k x m 2 ) ω 1 , m n q + k x m k y n ω 2 , m n q - k k z 3 , m n ω 4 , m n q ] × exp ( - λ q d / 2 ) = 0 ,
MC = U .
M = 0 ,
N + j γ = k x / k ,
r y = Γ k x - k x = Γ k 1 sin α - N - j γ ,
G i ( x i , y i , z i ) = a ( ξ , η ) u ^ ( ξ , η ) × exp [ - i k ( ξ x i + η y i + w z i ) ] d ξ d η ,
w = 1 - ξ 2 - η 2 , ξ 2 + η 2 1 ; w = - j ξ 2 + η 2 - 1 , ξ 2 + η 2 > 1 ,
sin α cos δ = ξ cos α cos δ - η sin δ + w sin α cos δ ,
sin α sin δ = ξ cos α sin δ + η cos δ + w sin α sin δ ,
cos α = - ξ sin α + w cos α ,
tan ψ = - cos α cos δ e x - cos α sin δ e y + sin α e z - sin δ e x + cos δ e y ,
e x = - sin ψ cos α cos δ - cos ψ sin δ , e y = - sin ψ cos α sin δ + cos ψ cos δ , e z = sin ψ sin α .
O r ( x r , y r , z r ) = a ( ξ , η ) R 00 ( ξ , η ) × exp [ - i k ( ξ x r + η y r + w z r ) ] d ξ d η ,
G ( x , y , z ) = a ( ξ , η ) u ^ ( ξ , η ) × exp { - i k [ ξ ( cos α cos δ x + cos α sin δ y - sin α z ) + η ( - sin δ x + cos δ y ) + w ( sin α cos δ x + sin α sin δ y + cos α z + h / cos α ) ] } × d ξ d η ,
O ( x , y , z ) = a ( ξ , η ) R 00 ( ξ , η ) × exp { - i k [ ξ ( cos α cos δ x + cos α sin δ y + sin α z ) + η ( - sin δ x + cos δ y ) + w ( sin α cos δ x + sin α sin δ y - cos α z + h / cos α ) ] } × d ξ d η ,
a ( ξ , η ) = k 2 w 0 2 4 π exp [ - 1 4 k 2 w 0 2 ( ξ 2 + η 2 ) ] ,
d P d ξ = cos α γ .
Δ x r = cos α k γ ,             Δ x = 1 k γ ,

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