Abstract

We present a formalism for the wave characteristics in gratings and periodic dielectrics based on the linear superposition of retarded fields. The idea is based on the physical picture that an incident field affects the charges in the material forming the gratings and hence leads to oscillating current and charge densities, which in turn generate more fields via the retarded potential. A set of self-consistent equations for the electric field and current and charge densities is derived. Expressions for the electric field everywhere, including the reflected and transmitted fields, are derived. The formalism is then applied to the calculation of diffraction efficiency so as to illustrate its application and to establish its validity by comparing results with the rigorous coupled-wave method. We further generalize the formalism to include possible anisotropy and nonlinearity in the response of the material forming the grating.

© 2006 Optical Society of America

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  1. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through subwavelength holes arrays," Nature 391, 667-669 (1998).
    [CrossRef]
  2. J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, "Transmission resonances on metallic gratings with very narrow slits," Phys. Rev. Lett. 83, 2845-2848 (1999).
    [CrossRef]
  3. E. Popov, M. Nevière, S. Enoch, and R. Reinisch, "Theory of light transmission through subwavelength periodic hole arrays," Phys. Rev. B 62, 16100-16108 (2000).
    [CrossRef]
  4. L. Martin-Moreno, F. J. Garcia Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, "Theory of extraordinary optical transmission through subwavelength hole arrays," Phys. Rev. Lett. 86, 1114-1117 (2001).
    [CrossRef] [PubMed]
  5. W. L. Barnes, A. Dereux, and T. W. Ebbesen, "Surface plasmon subwavelength optics," Nature 424, 824-830 (2003).
    [CrossRef] [PubMed]
  6. J. Dintinger, A. Degiron, and T. W. Ebbesen, "Enhanced light transmission through subwavelength holes," MRS Bull. 30, 381-384 (2005).
    [CrossRef]
  7. R.Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, 1980).
    [CrossRef]
  8. D. Maystre, "A new general integral theory for dielectric coated gratings," J. Opt. Soc. Am. 68, 490-495 (1978).
    [CrossRef]
  9. D. Maystre, "A new theory for multiprofile, buried gratings," Opt. Commun. 26, 127-132 (1978).
    [CrossRef]
  10. M. Nevière and E. Popov, Light Propagation in Periodic Media: Differential Theory and Design (Marcel Dekker, 2003).
  11. P. Sheng, R. S. Stepleman, and P. N. Sanda, "Exact eigenfunctions for square-wave gratings: application to diffraction and surface-plasmon calculations," Phys. Rev. B 26, 2907-2916 (1982).
    [CrossRef]
  12. J. Chandezon, M. T. Dupuis, G. Cornet, and E. Maystre, "Multicoated gratings: a differential formalism applicable in the entire optical region," J. Opt. Soc. Am. 72, 839-846 (1982).
    [CrossRef]
  13. M. G. Moharam and T. K. Gaylord, "Rigorous coupled-wave analysis of planar-grating diffraction," J. Opt. Soc. Am. 71, 811-818 (1981).
    [CrossRef]
  14. G. Granet and B. Guizal, "Efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization," J. Opt. Soc. Am. A 13, 1019-1023 (1996).
    [CrossRef]
  15. E. G. Loewen and E. Popov, Diffraction Gratings and Applications (Marcel Dekker, 1997), Chap. 10.
  16. M. Nevière, E. Popov, R. Reinisch, and G. Vitrant, Electromagnetic Resonances in Nonlinear Optics (Gordon & Breach, 2000), Chap. II.
  17. D. Y. K. Ko and J. R. Sambles, "Scattering matrix method for propagation of radiation in stratified media: attenuated total reflection studies of liquid crystals," J. Opt. Soc. Am. A 5, 1863-1866 (1988).
    [CrossRef]
  18. L. Li, "Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings," J. Opt. Soc. Am. A 13, 1024-1035 (1996).
    [CrossRef]
  19. L. Li, "Use of Fourier series in the analysis of discontinuous periodic structures," J. Opt. Soc. Am. A 13, 1870-1876 (1996).
    [CrossRef]
  20. E. Popov and M. Nevière, "Grating theory: new equations in Fourier space leading to fast converging results for TM polarization," J. Opt. Soc. Am. A 17, 1773-1784 (2000).
    [CrossRef]
  21. E. Popov and M. Nevière, "Maxwell equations in Fourier space: fast-converging formulation for diffraction by arbitrary shaped, periodic, anisotropic media," J. Opt. Soc. Am. A 18, 2886-2894 (2001).
    [CrossRef]
  22. E. Popov, M. Nevière, B. Gralak, and G. Tayeb, "Staircase approximation validity for arbitrary-shaped gratings," J. Opt. Soc. Am. A 19, 33-42 (2002).
    [CrossRef]
  23. L. Li, "Reformulation of the Fourier modal method for surface-relief gratings made with anisotropic materials," J. Mod. Opt. 45, 1313-1334 (1998).
    [CrossRef]
  24. B. Bai and L. Li, "Reduction of computation time for crossed-grating problems: a group-theoretic approach," J. Opt. Soc. Am. A 21, 1886-1894 (2004).
    [CrossRef]
  25. R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics (Addison-Wesley, 1963), Vol. 1, Secs. 31 and 30-7.
  26. M. Schwartz, Principles of Electrodynamics (Dover, 1987), Chap. 7.
  27. H. M. Lai, Y. P. Lau, and W. H. Wong, "Understanding wave characteristics via linear superposition of retarded fields," Am. J. Phys. 70, 173-179 (2002).
    [CrossRef]
  28. P. M. Hui and N. F. Johnson, "Photonic band-gap materials," in Solid State Physics: Advances in Research and Applications, H.Ehrenreich and D.Turnbull, eds. (Academic, 1995), Vol. 49, pp. 151-203.
  29. M. Cadilhac, "Some mathematical aspects of the grating theory," in Electromagnetic Theory of Gratings, R.Petit, ed. (Springer-Verlag, 1980), pp. 53-62.
    [CrossRef]
  30. E.D.Palik, ed., Handbook of Optical Constants of Solids (Academic, 1985).
  31. R. L. Sutherland, Handbook of Nonlinear Optics (Marcel Dekker, 2003).
    [CrossRef]
  32. H. Yang, P. Xie, S. K. Chan, W. X. Lu, Z. Q. Zhang, I. K. Sou, G. K. L. Wong, and K. S. Wong, "Simultaneous enhancement of the second- and third-harmonic generations in one-dimensional semiconductor photonic crystals," Quantum Electron. 42, 447-452 (2006).
    [CrossRef]
  33. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed. (Academic, 2000), pp. 714-715, Eqs. (6.677-1) and (6.677-2).

2006

H. Yang, P. Xie, S. K. Chan, W. X. Lu, Z. Q. Zhang, I. K. Sou, G. K. L. Wong, and K. S. Wong, "Simultaneous enhancement of the second- and third-harmonic generations in one-dimensional semiconductor photonic crystals," Quantum Electron. 42, 447-452 (2006).
[CrossRef]

2005

J. Dintinger, A. Degiron, and T. W. Ebbesen, "Enhanced light transmission through subwavelength holes," MRS Bull. 30, 381-384 (2005).
[CrossRef]

2004

2003

W. L. Barnes, A. Dereux, and T. W. Ebbesen, "Surface plasmon subwavelength optics," Nature 424, 824-830 (2003).
[CrossRef] [PubMed]

2002

E. Popov, M. Nevière, B. Gralak, and G. Tayeb, "Staircase approximation validity for arbitrary-shaped gratings," J. Opt. Soc. Am. A 19, 33-42 (2002).
[CrossRef]

H. M. Lai, Y. P. Lau, and W. H. Wong, "Understanding wave characteristics via linear superposition of retarded fields," Am. J. Phys. 70, 173-179 (2002).
[CrossRef]

2001

E. Popov and M. Nevière, "Maxwell equations in Fourier space: fast-converging formulation for diffraction by arbitrary shaped, periodic, anisotropic media," J. Opt. Soc. Am. A 18, 2886-2894 (2001).
[CrossRef]

L. Martin-Moreno, F. J. Garcia Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, "Theory of extraordinary optical transmission through subwavelength hole arrays," Phys. Rev. Lett. 86, 1114-1117 (2001).
[CrossRef] [PubMed]

2000

E. Popov, M. Nevière, S. Enoch, and R. Reinisch, "Theory of light transmission through subwavelength periodic hole arrays," Phys. Rev. B 62, 16100-16108 (2000).
[CrossRef]

E. Popov and M. Nevière, "Grating theory: new equations in Fourier space leading to fast converging results for TM polarization," J. Opt. Soc. Am. A 17, 1773-1784 (2000).
[CrossRef]

1999

J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, "Transmission resonances on metallic gratings with very narrow slits," Phys. Rev. Lett. 83, 2845-2848 (1999).
[CrossRef]

1998

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through subwavelength holes arrays," Nature 391, 667-669 (1998).
[CrossRef]

L. Li, "Reformulation of the Fourier modal method for surface-relief gratings made with anisotropic materials," J. Mod. Opt. 45, 1313-1334 (1998).
[CrossRef]

1996

1988

1982

P. Sheng, R. S. Stepleman, and P. N. Sanda, "Exact eigenfunctions for square-wave gratings: application to diffraction and surface-plasmon calculations," Phys. Rev. B 26, 2907-2916 (1982).
[CrossRef]

J. Chandezon, M. T. Dupuis, G. Cornet, and E. Maystre, "Multicoated gratings: a differential formalism applicable in the entire optical region," J. Opt. Soc. Am. 72, 839-846 (1982).
[CrossRef]

1981

1978

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

D. Maystre, "A new theory for multiprofile, buried gratings," Opt. Commun. 26, 127-132 (1978).
[CrossRef]

Bai, B.

Barnes, W. L.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, "Surface plasmon subwavelength optics," Nature 424, 824-830 (2003).
[CrossRef] [PubMed]

Cadilhac, M.

M. Cadilhac, "Some mathematical aspects of the grating theory," in Electromagnetic Theory of Gratings, R.Petit, ed. (Springer-Verlag, 1980), pp. 53-62.
[CrossRef]

Chan, S. K.

H. Yang, P. Xie, S. K. Chan, W. X. Lu, Z. Q. Zhang, I. K. Sou, G. K. L. Wong, and K. S. Wong, "Simultaneous enhancement of the second- and third-harmonic generations in one-dimensional semiconductor photonic crystals," Quantum Electron. 42, 447-452 (2006).
[CrossRef]

Chandezon, J.

Cornet, G.

Degiron, A.

J. Dintinger, A. Degiron, and T. W. Ebbesen, "Enhanced light transmission through subwavelength holes," MRS Bull. 30, 381-384 (2005).
[CrossRef]

Dereux, A.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, "Surface plasmon subwavelength optics," Nature 424, 824-830 (2003).
[CrossRef] [PubMed]

Dintinger, J.

J. Dintinger, A. Degiron, and T. W. Ebbesen, "Enhanced light transmission through subwavelength holes," MRS Bull. 30, 381-384 (2005).
[CrossRef]

Dupuis, M. T.

Ebbesen, T. W.

J. Dintinger, A. Degiron, and T. W. Ebbesen, "Enhanced light transmission through subwavelength holes," MRS Bull. 30, 381-384 (2005).
[CrossRef]

W. L. Barnes, A. Dereux, and T. W. Ebbesen, "Surface plasmon subwavelength optics," Nature 424, 824-830 (2003).
[CrossRef] [PubMed]

L. Martin-Moreno, F. J. Garcia Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, "Theory of extraordinary optical transmission through subwavelength hole arrays," Phys. Rev. Lett. 86, 1114-1117 (2001).
[CrossRef] [PubMed]

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through subwavelength holes arrays," Nature 391, 667-669 (1998).
[CrossRef]

Enoch, S.

E. Popov, M. Nevière, S. Enoch, and R. Reinisch, "Theory of light transmission through subwavelength periodic hole arrays," Phys. Rev. B 62, 16100-16108 (2000).
[CrossRef]

Feynman, R. P.

R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics (Addison-Wesley, 1963), Vol. 1, Secs. 31 and 30-7.

Garcia Vidal, F. J.

L. Martin-Moreno, F. J. Garcia Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, "Theory of extraordinary optical transmission through subwavelength hole arrays," Phys. Rev. Lett. 86, 1114-1117 (2001).
[CrossRef] [PubMed]

Garcia-Vidal, F. J.

J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, "Transmission resonances on metallic gratings with very narrow slits," Phys. Rev. Lett. 83, 2845-2848 (1999).
[CrossRef]

Gaylord, T. K.

Ghaemi, H. F.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through subwavelength holes arrays," Nature 391, 667-669 (1998).
[CrossRef]

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed. (Academic, 2000), pp. 714-715, Eqs. (6.677-1) and (6.677-2).

Gralak, B.

Granet, G.

Guizal, B.

Hui, P. M.

P. M. Hui and N. F. Johnson, "Photonic band-gap materials," in Solid State Physics: Advances in Research and Applications, H.Ehrenreich and D.Turnbull, eds. (Academic, 1995), Vol. 49, pp. 151-203.

Johnson, N. F.

P. M. Hui and N. F. Johnson, "Photonic band-gap materials," in Solid State Physics: Advances in Research and Applications, H.Ehrenreich and D.Turnbull, eds. (Academic, 1995), Vol. 49, pp. 151-203.

Ko, D. Y. K.

Lai, H. M.

H. M. Lai, Y. P. Lau, and W. H. Wong, "Understanding wave characteristics via linear superposition of retarded fields," Am. J. Phys. 70, 173-179 (2002).
[CrossRef]

Lau, Y. P.

H. M. Lai, Y. P. Lau, and W. H. Wong, "Understanding wave characteristics via linear superposition of retarded fields," Am. J. Phys. 70, 173-179 (2002).
[CrossRef]

Leighton, R. B.

R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics (Addison-Wesley, 1963), Vol. 1, Secs. 31 and 30-7.

Lezec, H. J.

L. Martin-Moreno, F. J. Garcia Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, "Theory of extraordinary optical transmission through subwavelength hole arrays," Phys. Rev. Lett. 86, 1114-1117 (2001).
[CrossRef] [PubMed]

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through subwavelength holes arrays," Nature 391, 667-669 (1998).
[CrossRef]

Li, L.

Loewen, E. G.

E. G. Loewen and E. Popov, Diffraction Gratings and Applications (Marcel Dekker, 1997), Chap. 10.

Lu, W. X.

H. Yang, P. Xie, S. K. Chan, W. X. Lu, Z. Q. Zhang, I. K. Sou, G. K. L. Wong, and K. S. Wong, "Simultaneous enhancement of the second- and third-harmonic generations in one-dimensional semiconductor photonic crystals," Quantum Electron. 42, 447-452 (2006).
[CrossRef]

Martin-Moreno, L.

L. Martin-Moreno, F. J. Garcia Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, "Theory of extraordinary optical transmission through subwavelength hole arrays," Phys. Rev. Lett. 86, 1114-1117 (2001).
[CrossRef] [PubMed]

Maystre, D.

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

D. Maystre, "A new theory for multiprofile, buried gratings," Opt. Commun. 26, 127-132 (1978).
[CrossRef]

Maystre, E.

Moharam, M. G.

Nevière, M.

E. Popov, M. Nevière, B. Gralak, and G. Tayeb, "Staircase approximation validity for arbitrary-shaped gratings," J. Opt. Soc. Am. A 19, 33-42 (2002).
[CrossRef]

E. Popov and M. Nevière, "Maxwell equations in Fourier space: fast-converging formulation for diffraction by arbitrary shaped, periodic, anisotropic media," J. Opt. Soc. Am. A 18, 2886-2894 (2001).
[CrossRef]

E. Popov and M. Nevière, "Grating theory: new equations in Fourier space leading to fast converging results for TM polarization," J. Opt. Soc. Am. A 17, 1773-1784 (2000).
[CrossRef]

E. Popov, M. Nevière, S. Enoch, and R. Reinisch, "Theory of light transmission through subwavelength periodic hole arrays," Phys. Rev. B 62, 16100-16108 (2000).
[CrossRef]

M. Nevière and E. Popov, Light Propagation in Periodic Media: Differential Theory and Design (Marcel Dekker, 2003).

M. Nevière, E. Popov, R. Reinisch, and G. Vitrant, Electromagnetic Resonances in Nonlinear Optics (Gordon & Breach, 2000), Chap. II.

Pellerin, K. M.

L. Martin-Moreno, F. J. Garcia Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, "Theory of extraordinary optical transmission through subwavelength hole arrays," Phys. Rev. Lett. 86, 1114-1117 (2001).
[CrossRef] [PubMed]

Pendry, J. B.

L. Martin-Moreno, F. J. Garcia Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, "Theory of extraordinary optical transmission through subwavelength hole arrays," Phys. Rev. Lett. 86, 1114-1117 (2001).
[CrossRef] [PubMed]

J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, "Transmission resonances on metallic gratings with very narrow slits," Phys. Rev. Lett. 83, 2845-2848 (1999).
[CrossRef]

Popov, E.

E. Popov, M. Nevière, B. Gralak, and G. Tayeb, "Staircase approximation validity for arbitrary-shaped gratings," J. Opt. Soc. Am. A 19, 33-42 (2002).
[CrossRef]

E. Popov and M. Nevière, "Maxwell equations in Fourier space: fast-converging formulation for diffraction by arbitrary shaped, periodic, anisotropic media," J. Opt. Soc. Am. A 18, 2886-2894 (2001).
[CrossRef]

E. Popov and M. Nevière, "Grating theory: new equations in Fourier space leading to fast converging results for TM polarization," J. Opt. Soc. Am. A 17, 1773-1784 (2000).
[CrossRef]

E. Popov, M. Nevière, S. Enoch, and R. Reinisch, "Theory of light transmission through subwavelength periodic hole arrays," Phys. Rev. B 62, 16100-16108 (2000).
[CrossRef]

M. Nevière and E. Popov, Light Propagation in Periodic Media: Differential Theory and Design (Marcel Dekker, 2003).

M. Nevière, E. Popov, R. Reinisch, and G. Vitrant, Electromagnetic Resonances in Nonlinear Optics (Gordon & Breach, 2000), Chap. II.

E. G. Loewen and E. Popov, Diffraction Gratings and Applications (Marcel Dekker, 1997), Chap. 10.

Porto, J. A.

J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, "Transmission resonances on metallic gratings with very narrow slits," Phys. Rev. Lett. 83, 2845-2848 (1999).
[CrossRef]

Reinisch, R.

E. Popov, M. Nevière, S. Enoch, and R. Reinisch, "Theory of light transmission through subwavelength periodic hole arrays," Phys. Rev. B 62, 16100-16108 (2000).
[CrossRef]

M. Nevière, E. Popov, R. Reinisch, and G. Vitrant, Electromagnetic Resonances in Nonlinear Optics (Gordon & Breach, 2000), Chap. II.

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed. (Academic, 2000), pp. 714-715, Eqs. (6.677-1) and (6.677-2).

Sambles, J. R.

Sanda, P. N.

P. Sheng, R. S. Stepleman, and P. N. Sanda, "Exact eigenfunctions for square-wave gratings: application to diffraction and surface-plasmon calculations," Phys. Rev. B 26, 2907-2916 (1982).
[CrossRef]

Sands, M.

R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics (Addison-Wesley, 1963), Vol. 1, Secs. 31 and 30-7.

Schwartz, M.

M. Schwartz, Principles of Electrodynamics (Dover, 1987), Chap. 7.

Sheng, P.

P. Sheng, R. S. Stepleman, and P. N. Sanda, "Exact eigenfunctions for square-wave gratings: application to diffraction and surface-plasmon calculations," Phys. Rev. B 26, 2907-2916 (1982).
[CrossRef]

Sou, I. K.

H. Yang, P. Xie, S. K. Chan, W. X. Lu, Z. Q. Zhang, I. K. Sou, G. K. L. Wong, and K. S. Wong, "Simultaneous enhancement of the second- and third-harmonic generations in one-dimensional semiconductor photonic crystals," Quantum Electron. 42, 447-452 (2006).
[CrossRef]

Stepleman, R. S.

P. Sheng, R. S. Stepleman, and P. N. Sanda, "Exact eigenfunctions for square-wave gratings: application to diffraction and surface-plasmon calculations," Phys. Rev. B 26, 2907-2916 (1982).
[CrossRef]

Sutherland, R. L.

R. L. Sutherland, Handbook of Nonlinear Optics (Marcel Dekker, 2003).
[CrossRef]

Tayeb, G.

Thio, T.

L. Martin-Moreno, F. J. Garcia Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, "Theory of extraordinary optical transmission through subwavelength hole arrays," Phys. Rev. Lett. 86, 1114-1117 (2001).
[CrossRef] [PubMed]

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through subwavelength holes arrays," Nature 391, 667-669 (1998).
[CrossRef]

Vitrant, G.

M. Nevière, E. Popov, R. Reinisch, and G. Vitrant, Electromagnetic Resonances in Nonlinear Optics (Gordon & Breach, 2000), Chap. II.

Wolff, P. A.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through subwavelength holes arrays," Nature 391, 667-669 (1998).
[CrossRef]

Wong, G. K. L.

H. Yang, P. Xie, S. K. Chan, W. X. Lu, Z. Q. Zhang, I. K. Sou, G. K. L. Wong, and K. S. Wong, "Simultaneous enhancement of the second- and third-harmonic generations in one-dimensional semiconductor photonic crystals," Quantum Electron. 42, 447-452 (2006).
[CrossRef]

Wong, K. S.

H. Yang, P. Xie, S. K. Chan, W. X. Lu, Z. Q. Zhang, I. K. Sou, G. K. L. Wong, and K. S. Wong, "Simultaneous enhancement of the second- and third-harmonic generations in one-dimensional semiconductor photonic crystals," Quantum Electron. 42, 447-452 (2006).
[CrossRef]

Wong, W. H.

H. M. Lai, Y. P. Lau, and W. H. Wong, "Understanding wave characteristics via linear superposition of retarded fields," Am. J. Phys. 70, 173-179 (2002).
[CrossRef]

Xie, P.

H. Yang, P. Xie, S. K. Chan, W. X. Lu, Z. Q. Zhang, I. K. Sou, G. K. L. Wong, and K. S. Wong, "Simultaneous enhancement of the second- and third-harmonic generations in one-dimensional semiconductor photonic crystals," Quantum Electron. 42, 447-452 (2006).
[CrossRef]

Yang, H.

H. Yang, P. Xie, S. K. Chan, W. X. Lu, Z. Q. Zhang, I. K. Sou, G. K. L. Wong, and K. S. Wong, "Simultaneous enhancement of the second- and third-harmonic generations in one-dimensional semiconductor photonic crystals," Quantum Electron. 42, 447-452 (2006).
[CrossRef]

Zhang, Z. Q.

H. Yang, P. Xie, S. K. Chan, W. X. Lu, Z. Q. Zhang, I. K. Sou, G. K. L. Wong, and K. S. Wong, "Simultaneous enhancement of the second- and third-harmonic generations in one-dimensional semiconductor photonic crystals," Quantum Electron. 42, 447-452 (2006).
[CrossRef]

Am. J. Phys.

H. M. Lai, Y. P. Lau, and W. H. Wong, "Understanding wave characteristics via linear superposition of retarded fields," Am. J. Phys. 70, 173-179 (2002).
[CrossRef]

J. Mod. Opt.

L. Li, "Reformulation of the Fourier modal method for surface-relief gratings made with anisotropic materials," J. Mod. Opt. 45, 1313-1334 (1998).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

MRS Bull.

J. Dintinger, A. Degiron, and T. W. Ebbesen, "Enhanced light transmission through subwavelength holes," MRS Bull. 30, 381-384 (2005).
[CrossRef]

Nature

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through subwavelength holes arrays," Nature 391, 667-669 (1998).
[CrossRef]

W. L. Barnes, A. Dereux, and T. W. Ebbesen, "Surface plasmon subwavelength optics," Nature 424, 824-830 (2003).
[CrossRef] [PubMed]

Opt. Commun.

D. Maystre, "A new theory for multiprofile, buried gratings," Opt. Commun. 26, 127-132 (1978).
[CrossRef]

Phys. Rev. B

E. Popov, M. Nevière, S. Enoch, and R. Reinisch, "Theory of light transmission through subwavelength periodic hole arrays," Phys. Rev. B 62, 16100-16108 (2000).
[CrossRef]

P. Sheng, R. S. Stepleman, and P. N. Sanda, "Exact eigenfunctions for square-wave gratings: application to diffraction and surface-plasmon calculations," Phys. Rev. B 26, 2907-2916 (1982).
[CrossRef]

Phys. Rev. Lett.

L. Martin-Moreno, F. J. Garcia Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, "Theory of extraordinary optical transmission through subwavelength hole arrays," Phys. Rev. Lett. 86, 1114-1117 (2001).
[CrossRef] [PubMed]

J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, "Transmission resonances on metallic gratings with very narrow slits," Phys. Rev. Lett. 83, 2845-2848 (1999).
[CrossRef]

Quantum Electron.

H. Yang, P. Xie, S. K. Chan, W. X. Lu, Z. Q. Zhang, I. K. Sou, G. K. L. Wong, and K. S. Wong, "Simultaneous enhancement of the second- and third-harmonic generations in one-dimensional semiconductor photonic crystals," Quantum Electron. 42, 447-452 (2006).
[CrossRef]

Other

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed. (Academic, 2000), pp. 714-715, Eqs. (6.677-1) and (6.677-2).

R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics (Addison-Wesley, 1963), Vol. 1, Secs. 31 and 30-7.

M. Schwartz, Principles of Electrodynamics (Dover, 1987), Chap. 7.

P. M. Hui and N. F. Johnson, "Photonic band-gap materials," in Solid State Physics: Advances in Research and Applications, H.Ehrenreich and D.Turnbull, eds. (Academic, 1995), Vol. 49, pp. 151-203.

M. Cadilhac, "Some mathematical aspects of the grating theory," in Electromagnetic Theory of Gratings, R.Petit, ed. (Springer-Verlag, 1980), pp. 53-62.
[CrossRef]

E.D.Palik, ed., Handbook of Optical Constants of Solids (Academic, 1985).

R. L. Sutherland, Handbook of Nonlinear Optics (Marcel Dekker, 2003).
[CrossRef]

M. Nevière and E. Popov, Light Propagation in Periodic Media: Differential Theory and Design (Marcel Dekker, 2003).

R.Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, 1980).
[CrossRef]

E. G. Loewen and E. Popov, Diffraction Gratings and Applications (Marcel Dekker, 1997), Chap. 10.

M. Nevière, E. Popov, R. Reinisch, and G. Vitrant, Electromagnetic Resonances in Nonlinear Optics (Gordon & Breach, 2000), Chap. II.

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Figures (6)

Fig. 1
Fig. 1

System under consideration is shown schematically. The film has a thickness d and a periodic ϵ in the x direction with period L. The z direction is taken to be normal to the film. Also shown is the propagation vector k of the incident electromagnetic waves and the corresponding angles θ and ϕ specifying the direction of k. The medium above and below the film is taken to be vacuum.

Fig. 2
Fig. 2

System considered in the model calculations. A lamellar grating with permittivity ϵ, period L, slit width s, and thickness d is incident by a plane wave with a wave vector perpendicular to the grating slits ( ϕ = 0 ) .

Fig. 3
Fig. 3

(a) Order 0 and order 1 transmission efficiency of a grating made of quartz ( ϵ = 3.8 ) as a function of incident angle. The system has parameters L = 1000 nm , d = 200 nm , s = 300 nm , and is incident by TE-polarized light of wavelength λ = 700 nm . The symbols give the results obtained by the present method and the curves give the results obtained by the RCW method. (b) The fractional difference between the results obtained by the present approach and the RCW method.

Fig. 4
Fig. 4

(a) Zeroth-order transmission efficiency of a silver grating as a function of incident frequency. The system has parameters L = 900 nm , d = 200 nm , s = 100 nm , and is normally incident by TM-polarized light. The symbols give the results obtained by the present method, and the curves give the results obtained by the RCW method. (b) The fractional difference between the results obtained by the present approach and the RCW method.

Fig. 5
Fig. 5

(a) Order 0 and order 1 transmission efficiency of a grating made of an anisotropic K H 4 As O 4 crystal as a function of incident angle. The system has parameters L = 900 nm , d = 300 nm , s = 450 nm , and is incident by TE-polarized light of wavelength λ = 600 nm . The symbols give the results obtained by the present method and the curves give the results obtained by the RCW method generalized to anisotropic gratings.[23] (b) The fractional difference between the results obtained by the present approach and the RCW method.

Fig. 6
Fig. 6

Second-harmonic transmission efficiency of a model system consisting of a grating made of semiconductors that possess second-order nonlinearity. The material is taken to be ZnSe and the slits are filled with ZnTe. The system has parameters L = 8000 nm , d = 2500 nm , s = 4000 nm , and is incident by TM-polarized light of wavelength λ = 10,600 nm with a field strength of 10 9 V m . The results are calculated within the undepleted pump approximation.

Equations (43)

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J = σ E .
Z 0 J = i k ( 1 ϵ ) E ,
E ( r , t ) = n E n ( z ) exp ( i k x n x + i k y y i ω t ) ,
k x n = k x + n 2 π L .
J ( r , t ) = n J n ( z ) exp ( i k x n x + i k y y i ω t ) ,
Z 0 J n ( z ) = i k m ( δ n m ϵ n m ( z ) ) E m ( z ) .
ρ ϵ 0 = { E , 0 < z < d ( ϵ 1 ) E z ̂ δ ( z ) , 0 < z < 0 + ( ϵ 1 ) E z ̂ δ ( z d ) , d < z < d + } .
ρ ( r , t ) = n ρ n ( z ) exp ( i k x n + i k y y i ω t ) ,
ρ n ( z ) ϵ 0 = { i k x n E x n ( z ) + i k y E y n ( z ) + E z n ( z ) , 0 < z < d m ( ϵ n m δ n m ) E z m ( 0 ) δ ( z ) , 0 < z < 0 + m ( ϵ n m δ n m ) E z m ( d ) δ ( z d ) , d < z < d + } .
A ( r , t ) = μ 0 J ( r , t ) 4 π R d r , Φ ( r , t ) = ρ ( r , t ) ϵ 0 4 π R d r ,
A ( r , t ) = n 0 d d z μ 0 J n ( z ) d x d y exp ( i k R + i k x n x + i k y y i ω t ) 4 π R ,
Φ ( r , t ) = n 0 d d z ρ ( z ) ϵ 0 d x d y exp ( i k R + i k x n x + i k y y i ω t ) 4 π R .
A ( r , t ) = n [ 0 d d z μ 0 J n ( z ) i exp ( i k z n z z ) 2 k z n ] exp ( i k x n x + i k y y i ω t ) ,
Φ ( r , t ) = n [ 0 d d z ρ n ( z ) ϵ 0 i exp ( i k z n z z ) 2 k z n ] exp ( i k x n x + i k y y i ω t ) ,
E ( r , t ) = n [ 0 d d z ( k Z 0 J n + k n ± ρ n ϵ 0 ) exp ( i k z n z z ) 2 k z n ] exp ( i k x n x + i k y y i ω t ) ,
k n ± = k x n x ̂ + k y y ̂ ± k z n z ̂ ,
E n ( z ) = δ n 0 E ( i ) exp ( i k z z ) + 0 d d z ( k Z 0 J n + k n ± ρ n ϵ 0 ) exp ( i k z n z z ) 2 k z n ,
Z 0 J n ( z ) = i k m ( δ n m ϵ n m ( z ) ) E m ( z ) ,
ρ n ( z ) ϵ 0 = { i k x n E x n ( z ) + i k y E y n ( z ) + E z n ( z ) , 0 < z < d m ( ϵ n m δ n m ) E z m ( 0 ) δ ( z ) , 0 < z < 0 + m ( ϵ n m δ n m ) E z m ( d ) δ ( z d ) , d < z < d + } .
E ( t ) ( r , t ) = n E n ( d ) exp ( i k x n x + i k y y + i k z n z i ω t ) ,
E ( r ) ( r , t ) = n [ E n ( 0 ) δ n 0 E ( i ) ] exp ( i k x n x + i k y y + i k z n z i ω t ) .
η n ( t ) = Re ( k z n k z 0 ) [ E x n ( t ) 2 + E y n ( t ) 2 + k x n E x n ( t ) + k y E y n ( t ) k z n 2 ] ,
η n ( r ) = Re ( k z n k z 0 ) [ E x n ( r ) 2 + E y n ( r ) 2 + k x n E x n ( r ) + k y E y n ( r ) k z n 2 ] ,
sin θ n = sin θ i + n λ L ,
Z 0 J x n ( z ) = i k m = N N ( δ n m [ [ 1 ϵ ( z ) ] ] n m 1 ) E x m ( z ) ,
Z 0 J x n = i k m = N N [ ( Q n m 11 δ n m ) E x m + Q n m 12 E y m + Q n m 13 E z m ] ,
Z 0 J y n = i k m = N N [ Q n m 21 E x m + ( Q n m 22 δ n m ) E y m + Q n m 23 E z m ] ,
Z 0 J z n = i k m = N N [ Q n m 31 E x m + Q n m 32 E y m + ( Q n m 33 δ n m ) E z m ] ,
Z 0 J = i k [ ( 1 ϵ ) E χ ( 2 ) E E ] .
E ( r , t ) = n , μ E n μ ( z ) exp ( i k x n μ + i μ k y y i μ ω t ) ,
J ( r , t ) = n , μ J n μ ( z ) exp ( i k x n μ + i μ k y y i μ ω t ) ,
k x n μ = μ k x + n 2 π L .
Z 0 J n μ ( z ) = i μ k [ m ( δ n m ϵ n m ( μ ω ) ) E m μ ( z ) m , p , ν χ n m p ( 2 ) ( μ ω ; ( μ ν ) ω , ν ω ) E m μ ν ( z ) E p ν ( z ) ] ,
ρ n μ ( z ) ϵ 0 = { i k x n μ E x n μ ( z ) + i μ k y E y n μ ( z ) + E z n μ ( z ) , 0 < z < d [ m ( ϵ n m ( μ ω ) δ n m ) E m μ + m , p , ν χ n m p ( 2 ) ( μ ω ; ( μ ν ) ω , ν ω ) E m μ ν E p ν ] z ̂ δ ( z ) , 0 < z < 0 + [ m ( ϵ n m ( μ ω ) δ n m ) E m μ + m , p , ν χ n m p ( 2 ) ( μ ω ; ( μ ν ) ω , ν ω ) E m μ ν E p ν ] z ̂ δ ( z d ) , d < z < d + } ,
E n μ ( z ) = E n μ ( i ) ( z ) + 0 d d z ( μ k Z 0 J n μ + k n μ ± ρ n μ ϵ 0 ) exp ( i k z n μ z z ) 2 k z n μ .
E μ = 0 ( r ) = n 0 d d z ± z ̂ i sign ( n ) x ̂ 2 ρ n μ = 0 ϵ 0 exp ( n 2 π L z z + i n 2 π x L ) ,
sign ( n ) = { + 1 , n > 0 0 , n = 0 1 , n < 0 } .
I = d x d y exp ( i k R + i k x n x + i k y y ) 4 π R ,
x = x + ρ cos θ ,
y = y + ρ sin θ .
I = exp ( i k x n x + i k y y ) 4 π z z d R exp ( i k R ) 0 2 π d θ exp ( i ρ ( k x n cos θ + k y sin θ ) ) .
0 2 π d θ exp ( i ρ k x n 2 + k y 2 cos ( θ θ 0 ) ) = 2 π J 0 ( k x n 2 + k y 2 ρ ) ,
I = exp ( i k x n x + i k y y ) 2 z z d R exp ( i k R ) J 0 ( k x n 2 + k y 2 R 2 ( z z ) 2 ) = i exp ( i k 2 k x n 2 k y 2 z z + i k x n x + i k y y ) 2 k 2 k x n 2 k y 2 ,

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