Abstract

A new rigorous electromagnetic theory is developed for studying the second-harmonic generation process that occurs when a high-power laser beam falls on a periodically corrugated waveguide. The waveguide can consist of several layers with corrugated or plane boundaries, with at least one of the layers being filled with a nonlinear lossless or lossy dielectric. The theory uses a nonorthogonal coordinate transformation, which maps the corrugated interfaces onto planes. The Maxwell equations written in covariant form lead to a set of first-order partial differential equations with nonconstant coefficients. Taking into account the periodicity of the system, this set of equations could be transformed into a set of ordinary differential equations with constant coefficients, which can be resolved by an eigenvalue–eigenvector technique, avoiding numerical integration. The theory is developed for both TE and TM polarizations and for any groove geometry and incidence; it can be used, in particular, when surface waves (guided modes or surface plasmon–polaritons) are excited. It is a powerful tool for studying the enhancement of the second-harmonic generation resulting from local field enhancement by electromagnetic resonances. In addition an approach based on the Rayleigh hypothesis is developed. Both theoretical approaches are compared with previously developed differential theory for gratings in nonlinear optics. A spectacular agreement is obtained between the three theories for both TE and TM polarization. The new method is free of limitations concerning the waveguide thickness, the groove depth, and the conductivity of the grating material. It is demonstrated that the limitations of the Rayleigh hypothesis are stronger at the second-harmonic frequency.

© 1994 Optical Society of America

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References

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  1. J. E. Sipe and G. I. Stegeman, “Nonlinear optical response of metal surfaces,” in Surface Polaritons, Electromagnetic Waves at Surfaces and Interfaces, V. M. Agranovich and D. L. Mills, eds. (North-Holland, Amsterdam, 1982), pp. 661–701 and references therein.
    [CrossRef]
  2. R. Reinisch and M. Nevière, “Gratings as electromagnetic field amplifiers for second-harmonic generation,” in Scattering in Volumes and Surfaces, M. Nieto-Vesperinas and J. C. Dainty, eds. (North-Holland, Amsterdam, 1990), pp. 269–285, and references therein.
  3. R. Reinisch, G. Vitrant, and M. Nevière, “Electromagnetic resonance induced nonlinear phenomena,” in Nonlinear Waves in Solid State Physics, A. D. Boardman, M. Betolotty, and T. Twardowsky, eds. (Plenum, New York, 1990), pp. 435–461, and references therein.
    [CrossRef]
  4. V. M. Agranovich and D. L. Mills, eds., Surface Polaritons, Electromagnetic Waves at Surfaces and Interfaces (North-Holland, Amsterdam, 1982).
  5. T. Suzuki and T. F. Heinz, “Second-harmonic diffraction from a monolayer gratin,” Opt. Lett. 14, 1201–1203 (1989).
    [CrossRef] [PubMed]
  6. H. J. Simon, C. Huang, J. C. Quail, and Z. Chen, “Second-harmonic generation with surface plasmons from a silvered quartz grating,” Phys. Rev. B 38, 7408–7414 (1988).
    [CrossRef]
  7. Z. Chen and H. J. Simon, “Optical second-harmonic generation with coupled surface plasmons from a multilayer silver-quartz grating,” Opt. Lett. 13, 1008–1010 (1988).
    [CrossRef] [PubMed]
  8. H. J. Simon and Z. Chen, “Optical second-harmonic generation with grating-coupled surface plasmons from a quartz-silver-quartz grating structure,” Phys. Rev. B 39, 3077–3085 (1989).
    [CrossRef]
  9. R. Reinisch and M. Nevière, “Electromagnetic theory of diffraction in nonlinear optics and surface enhanced nonlinear optical effects,” Phys. Rev. 28, 1870–1885 (1983).
    [CrossRef]
  10. M. Nevière, P. Vincent, D. Maystre, R. Reinisch, and J. L. Coutaz, “Differential theory for metallic gratings in nonlinear optics: second-harmonic generation,” J. Opt. Soc. Am. B 5, 330–337 (1988).
    [CrossRef]
  11. D. Maystre, M. Nevière, R. Reinisch, and J. L. Coutaz, “Integral theory for metallic gratings in nonlinear optics and comparison with experimental results on second-harmonic generation,” J. Opt. Soc. Am. B 5, 338–346 (1988).
    [CrossRef]
  12. D. Maystre, M. Nevière, and R. Reinisch, “Nonlinear polarization inside metals: a mathematical study of the free electron model,” Appl. Phys. A 39, 115–121 (1986).
    [CrossRef]
  13. H. Akhouayri, M. Nevière, P. Vincent, and R. Reinisch, “Second harmonic generation in a corrugated waveguide,” Mol. Cryst. Liq. Cryst. Sci. Technol. Sec. B (to be published).
  14. J. Chandezon, D. Maystre, and G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. (Paris) 11, 235–241 (1980).
    [CrossRef]
  15. J. Chandezon, M. T. Dupuis, G. Cornet, and D. Maystre, “Multicoated gratings: a differential formalism applicable in the entire optical region,” J. Opt. Soc. Am. 72, 839–847 (1982).
    [CrossRef]
  16. E. Popov and L. Mashev, “Rigorous electromagnetic treatment of planar corrugated waveguides,” J. Opt. Commun. 7, 127–131 (1986).
  17. E. Popov and L. Mashev, “Convergence of Rayleigh–Fourier method and rigorous differential method for relief diffraction gratings,” Opt. Acta 33, 593–605 (1986).
    [CrossRef]
  18. E. Popov and L. Mashev, “Convergence of Rayleigh–Fourier method and rigorous differential method for relief diffraction gratings—non-sinusoidal profile,” Opt. Acta 34, 155–158 (1987).
  19. M. Cadilhac, “Some mathematical aspects of the grating theory,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 52–62, and references cited therein.
  20. B. Spain, Tensor Calculus (Oliver and Boyd, Edinburgh, and Interscience, New York, 1953), p. 60.
  21. D. J. Zvijac and J. C. Light, “R-matrix theory for collinear chemical reactions,” Chem. Phys. 12, 237–251 (1976).
    [CrossRef]
  22. J. C. Light and R. B. Walker, “An R-matrix approach to the solution of coupled equations for atom–molecule reactive scattering,” J. Chem. Phys. 65, 4272–4282 (1976).
    [CrossRef]
  23. P. M. Van den Berg, “Reflection by a grating: Rayleigh methods,” J. Opt. Soc. Am. 71, 1224–1229 (1981).
    [CrossRef]

1989 (2)

T. Suzuki and T. F. Heinz, “Second-harmonic diffraction from a monolayer gratin,” Opt. Lett. 14, 1201–1203 (1989).
[CrossRef] [PubMed]

H. J. Simon and Z. Chen, “Optical second-harmonic generation with grating-coupled surface plasmons from a quartz-silver-quartz grating structure,” Phys. Rev. B 39, 3077–3085 (1989).
[CrossRef]

1988 (4)

1987 (1)

E. Popov and L. Mashev, “Convergence of Rayleigh–Fourier method and rigorous differential method for relief diffraction gratings—non-sinusoidal profile,” Opt. Acta 34, 155–158 (1987).

1986 (3)

E. Popov and L. Mashev, “Rigorous electromagnetic treatment of planar corrugated waveguides,” J. Opt. Commun. 7, 127–131 (1986).

E. Popov and L. Mashev, “Convergence of Rayleigh–Fourier method and rigorous differential method for relief diffraction gratings,” Opt. Acta 33, 593–605 (1986).
[CrossRef]

D. Maystre, M. Nevière, and R. Reinisch, “Nonlinear polarization inside metals: a mathematical study of the free electron model,” Appl. Phys. A 39, 115–121 (1986).
[CrossRef]

1983 (1)

R. Reinisch and M. Nevière, “Electromagnetic theory of diffraction in nonlinear optics and surface enhanced nonlinear optical effects,” Phys. Rev. 28, 1870–1885 (1983).
[CrossRef]

1982 (1)

1981 (1)

1980 (1)

J. Chandezon, D. Maystre, and G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. (Paris) 11, 235–241 (1980).
[CrossRef]

1976 (2)

D. J. Zvijac and J. C. Light, “R-matrix theory for collinear chemical reactions,” Chem. Phys. 12, 237–251 (1976).
[CrossRef]

J. C. Light and R. B. Walker, “An R-matrix approach to the solution of coupled equations for atom–molecule reactive scattering,” J. Chem. Phys. 65, 4272–4282 (1976).
[CrossRef]

Akhouayri, H.

H. Akhouayri, M. Nevière, P. Vincent, and R. Reinisch, “Second harmonic generation in a corrugated waveguide,” Mol. Cryst. Liq. Cryst. Sci. Technol. Sec. B (to be published).

Cadilhac, M.

M. Cadilhac, “Some mathematical aspects of the grating theory,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 52–62, and references cited therein.

Chandezon, J.

J. Chandezon, M. T. Dupuis, G. Cornet, and D. Maystre, “Multicoated gratings: a differential formalism applicable in the entire optical region,” J. Opt. Soc. Am. 72, 839–847 (1982).
[CrossRef]

J. Chandezon, D. Maystre, and G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. (Paris) 11, 235–241 (1980).
[CrossRef]

Chen, Z.

H. J. Simon and Z. Chen, “Optical second-harmonic generation with grating-coupled surface plasmons from a quartz-silver-quartz grating structure,” Phys. Rev. B 39, 3077–3085 (1989).
[CrossRef]

H. J. Simon, C. Huang, J. C. Quail, and Z. Chen, “Second-harmonic generation with surface plasmons from a silvered quartz grating,” Phys. Rev. B 38, 7408–7414 (1988).
[CrossRef]

Z. Chen and H. J. Simon, “Optical second-harmonic generation with coupled surface plasmons from a multilayer silver-quartz grating,” Opt. Lett. 13, 1008–1010 (1988).
[CrossRef] [PubMed]

Cornet, G.

Coutaz, J. L.

Dupuis, M. T.

Heinz, T. F.

Huang, C.

H. J. Simon, C. Huang, J. C. Quail, and Z. Chen, “Second-harmonic generation with surface plasmons from a silvered quartz grating,” Phys. Rev. B 38, 7408–7414 (1988).
[CrossRef]

Light, J. C.

D. J. Zvijac and J. C. Light, “R-matrix theory for collinear chemical reactions,” Chem. Phys. 12, 237–251 (1976).
[CrossRef]

J. C. Light and R. B. Walker, “An R-matrix approach to the solution of coupled equations for atom–molecule reactive scattering,” J. Chem. Phys. 65, 4272–4282 (1976).
[CrossRef]

Mashev, L.

E. Popov and L. Mashev, “Convergence of Rayleigh–Fourier method and rigorous differential method for relief diffraction gratings—non-sinusoidal profile,” Opt. Acta 34, 155–158 (1987).

E. Popov and L. Mashev, “Convergence of Rayleigh–Fourier method and rigorous differential method for relief diffraction gratings,” Opt. Acta 33, 593–605 (1986).
[CrossRef]

E. Popov and L. Mashev, “Rigorous electromagnetic treatment of planar corrugated waveguides,” J. Opt. Commun. 7, 127–131 (1986).

Maystre, D.

Nevière, M.

M. Nevière, P. Vincent, D. Maystre, R. Reinisch, and J. L. Coutaz, “Differential theory for metallic gratings in nonlinear optics: second-harmonic generation,” J. Opt. Soc. Am. B 5, 330–337 (1988).
[CrossRef]

D. Maystre, M. Nevière, R. Reinisch, and J. L. Coutaz, “Integral theory for metallic gratings in nonlinear optics and comparison with experimental results on second-harmonic generation,” J. Opt. Soc. Am. B 5, 338–346 (1988).
[CrossRef]

D. Maystre, M. Nevière, and R. Reinisch, “Nonlinear polarization inside metals: a mathematical study of the free electron model,” Appl. Phys. A 39, 115–121 (1986).
[CrossRef]

R. Reinisch and M. Nevière, “Electromagnetic theory of diffraction in nonlinear optics and surface enhanced nonlinear optical effects,” Phys. Rev. 28, 1870–1885 (1983).
[CrossRef]

H. Akhouayri, M. Nevière, P. Vincent, and R. Reinisch, “Second harmonic generation in a corrugated waveguide,” Mol. Cryst. Liq. Cryst. Sci. Technol. Sec. B (to be published).

R. Reinisch, G. Vitrant, and M. Nevière, “Electromagnetic resonance induced nonlinear phenomena,” in Nonlinear Waves in Solid State Physics, A. D. Boardman, M. Betolotty, and T. Twardowsky, eds. (Plenum, New York, 1990), pp. 435–461, and references therein.
[CrossRef]

R. Reinisch and M. Nevière, “Gratings as electromagnetic field amplifiers for second-harmonic generation,” in Scattering in Volumes and Surfaces, M. Nieto-Vesperinas and J. C. Dainty, eds. (North-Holland, Amsterdam, 1990), pp. 269–285, and references therein.

Popov, E.

E. Popov and L. Mashev, “Convergence of Rayleigh–Fourier method and rigorous differential method for relief diffraction gratings—non-sinusoidal profile,” Opt. Acta 34, 155–158 (1987).

E. Popov and L. Mashev, “Rigorous electromagnetic treatment of planar corrugated waveguides,” J. Opt. Commun. 7, 127–131 (1986).

E. Popov and L. Mashev, “Convergence of Rayleigh–Fourier method and rigorous differential method for relief diffraction gratings,” Opt. Acta 33, 593–605 (1986).
[CrossRef]

Quail, J. C.

H. J. Simon, C. Huang, J. C. Quail, and Z. Chen, “Second-harmonic generation with surface plasmons from a silvered quartz grating,” Phys. Rev. B 38, 7408–7414 (1988).
[CrossRef]

Raoult, G.

J. Chandezon, D. Maystre, and G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. (Paris) 11, 235–241 (1980).
[CrossRef]

Reinisch, R.

D. Maystre, M. Nevière, R. Reinisch, and J. L. Coutaz, “Integral theory for metallic gratings in nonlinear optics and comparison with experimental results on second-harmonic generation,” J. Opt. Soc. Am. B 5, 338–346 (1988).
[CrossRef]

M. Nevière, P. Vincent, D. Maystre, R. Reinisch, and J. L. Coutaz, “Differential theory for metallic gratings in nonlinear optics: second-harmonic generation,” J. Opt. Soc. Am. B 5, 330–337 (1988).
[CrossRef]

D. Maystre, M. Nevière, and R. Reinisch, “Nonlinear polarization inside metals: a mathematical study of the free electron model,” Appl. Phys. A 39, 115–121 (1986).
[CrossRef]

R. Reinisch and M. Nevière, “Electromagnetic theory of diffraction in nonlinear optics and surface enhanced nonlinear optical effects,” Phys. Rev. 28, 1870–1885 (1983).
[CrossRef]

R. Reinisch and M. Nevière, “Gratings as electromagnetic field amplifiers for second-harmonic generation,” in Scattering in Volumes and Surfaces, M. Nieto-Vesperinas and J. C. Dainty, eds. (North-Holland, Amsterdam, 1990), pp. 269–285, and references therein.

R. Reinisch, G. Vitrant, and M. Nevière, “Electromagnetic resonance induced nonlinear phenomena,” in Nonlinear Waves in Solid State Physics, A. D. Boardman, M. Betolotty, and T. Twardowsky, eds. (Plenum, New York, 1990), pp. 435–461, and references therein.
[CrossRef]

H. Akhouayri, M. Nevière, P. Vincent, and R. Reinisch, “Second harmonic generation in a corrugated waveguide,” Mol. Cryst. Liq. Cryst. Sci. Technol. Sec. B (to be published).

Simon, H. J.

H. J. Simon and Z. Chen, “Optical second-harmonic generation with grating-coupled surface plasmons from a quartz-silver-quartz grating structure,” Phys. Rev. B 39, 3077–3085 (1989).
[CrossRef]

Z. Chen and H. J. Simon, “Optical second-harmonic generation with coupled surface plasmons from a multilayer silver-quartz grating,” Opt. Lett. 13, 1008–1010 (1988).
[CrossRef] [PubMed]

H. J. Simon, C. Huang, J. C. Quail, and Z. Chen, “Second-harmonic generation with surface plasmons from a silvered quartz grating,” Phys. Rev. B 38, 7408–7414 (1988).
[CrossRef]

Sipe, J. E.

J. E. Sipe and G. I. Stegeman, “Nonlinear optical response of metal surfaces,” in Surface Polaritons, Electromagnetic Waves at Surfaces and Interfaces, V. M. Agranovich and D. L. Mills, eds. (North-Holland, Amsterdam, 1982), pp. 661–701 and references therein.
[CrossRef]

Spain, B.

B. Spain, Tensor Calculus (Oliver and Boyd, Edinburgh, and Interscience, New York, 1953), p. 60.

Stegeman, G. I.

J. E. Sipe and G. I. Stegeman, “Nonlinear optical response of metal surfaces,” in Surface Polaritons, Electromagnetic Waves at Surfaces and Interfaces, V. M. Agranovich and D. L. Mills, eds. (North-Holland, Amsterdam, 1982), pp. 661–701 and references therein.
[CrossRef]

Suzuki, T.

Van den Berg, P. M.

Vincent, P.

M. Nevière, P. Vincent, D. Maystre, R. Reinisch, and J. L. Coutaz, “Differential theory for metallic gratings in nonlinear optics: second-harmonic generation,” J. Opt. Soc. Am. B 5, 330–337 (1988).
[CrossRef]

H. Akhouayri, M. Nevière, P. Vincent, and R. Reinisch, “Second harmonic generation in a corrugated waveguide,” Mol. Cryst. Liq. Cryst. Sci. Technol. Sec. B (to be published).

Vitrant, G.

R. Reinisch, G. Vitrant, and M. Nevière, “Electromagnetic resonance induced nonlinear phenomena,” in Nonlinear Waves in Solid State Physics, A. D. Boardman, M. Betolotty, and T. Twardowsky, eds. (Plenum, New York, 1990), pp. 435–461, and references therein.
[CrossRef]

Walker, R. B.

J. C. Light and R. B. Walker, “An R-matrix approach to the solution of coupled equations for atom–molecule reactive scattering,” J. Chem. Phys. 65, 4272–4282 (1976).
[CrossRef]

Zvijac, D. J.

D. J. Zvijac and J. C. Light, “R-matrix theory for collinear chemical reactions,” Chem. Phys. 12, 237–251 (1976).
[CrossRef]

Appl. Phys. A (1)

D. Maystre, M. Nevière, and R. Reinisch, “Nonlinear polarization inside metals: a mathematical study of the free electron model,” Appl. Phys. A 39, 115–121 (1986).
[CrossRef]

Chem. Phys. (1)

D. J. Zvijac and J. C. Light, “R-matrix theory for collinear chemical reactions,” Chem. Phys. 12, 237–251 (1976).
[CrossRef]

J. Chem. Phys. (1)

J. C. Light and R. B. Walker, “An R-matrix approach to the solution of coupled equations for atom–molecule reactive scattering,” J. Chem. Phys. 65, 4272–4282 (1976).
[CrossRef]

J. Opt. (Paris) (1)

J. Chandezon, D. Maystre, and G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. (Paris) 11, 235–241 (1980).
[CrossRef]

J. Opt. Commun. (1)

E. Popov and L. Mashev, “Rigorous electromagnetic treatment of planar corrugated waveguides,” J. Opt. Commun. 7, 127–131 (1986).

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. B (2)

Opt. Acta (2)

E. Popov and L. Mashev, “Convergence of Rayleigh–Fourier method and rigorous differential method for relief diffraction gratings,” Opt. Acta 33, 593–605 (1986).
[CrossRef]

E. Popov and L. Mashev, “Convergence of Rayleigh–Fourier method and rigorous differential method for relief diffraction gratings—non-sinusoidal profile,” Opt. Acta 34, 155–158 (1987).

Opt. Lett. (2)

Phys. Rev. (1)

R. Reinisch and M. Nevière, “Electromagnetic theory of diffraction in nonlinear optics and surface enhanced nonlinear optical effects,” Phys. Rev. 28, 1870–1885 (1983).
[CrossRef]

Phys. Rev. B (2)

H. J. Simon and Z. Chen, “Optical second-harmonic generation with grating-coupled surface plasmons from a quartz-silver-quartz grating structure,” Phys. Rev. B 39, 3077–3085 (1989).
[CrossRef]

H. J. Simon, C. Huang, J. C. Quail, and Z. Chen, “Second-harmonic generation with surface plasmons from a silvered quartz grating,” Phys. Rev. B 38, 7408–7414 (1988).
[CrossRef]

Other (7)

J. E. Sipe and G. I. Stegeman, “Nonlinear optical response of metal surfaces,” in Surface Polaritons, Electromagnetic Waves at Surfaces and Interfaces, V. M. Agranovich and D. L. Mills, eds. (North-Holland, Amsterdam, 1982), pp. 661–701 and references therein.
[CrossRef]

R. Reinisch and M. Nevière, “Gratings as electromagnetic field amplifiers for second-harmonic generation,” in Scattering in Volumes and Surfaces, M. Nieto-Vesperinas and J. C. Dainty, eds. (North-Holland, Amsterdam, 1990), pp. 269–285, and references therein.

R. Reinisch, G. Vitrant, and M. Nevière, “Electromagnetic resonance induced nonlinear phenomena,” in Nonlinear Waves in Solid State Physics, A. D. Boardman, M. Betolotty, and T. Twardowsky, eds. (Plenum, New York, 1990), pp. 435–461, and references therein.
[CrossRef]

V. M. Agranovich and D. L. Mills, eds., Surface Polaritons, Electromagnetic Waves at Surfaces and Interfaces (North-Holland, Amsterdam, 1982).

M. Cadilhac, “Some mathematical aspects of the grating theory,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 52–62, and references cited therein.

B. Spain, Tensor Calculus (Oliver and Boyd, Edinburgh, and Interscience, New York, 1953), p. 60.

H. Akhouayri, M. Nevière, P. Vincent, and R. Reinisch, “Second harmonic generation in a corrugated waveguide,” Mol. Cryst. Liq. Cryst. Sci. Technol. Sec. B (to be published).

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Figures (6)

Fig. 1
Fig. 1

Schematic representation of the corrugated waveguide under consideration.

Fig. 2
Fig. 2

Angular dependence of the modulus of the reflected zeroth-order amplitude at the second-harmonic frequency for a TM-polarized wave with wavelength 1.06 μm incident upon the flat interface, a sinusoidal corrugation at the substrate–layer interface with period of d = 1.5 μm and a groove depth of 0.05 μm, and refractive indices at ω for the cladding of n3 = 1, the layer, n2 = 1.57, and the substrate, n1 = 0.15 + i7.31 and at 2ω for the cladding of n3 = 1, the layer, n2 = 1.6 + i0.0005, and the substrate n1 = 0.05 + i3.16. The thickness of the layer is t2 = 0.5 μm; χxxx ≠ 0, and all other components are equal to 0. Solid curve, rigorous method; dashed curve, Rayleigh method; dotted curve, classical differential theory. The solid and the dashed curves are nearly coincident.

Fig. 3
Fig. 3

Angular dependence of the modulus of the reflected zeroth-order amplitude at the second-harmonic frequency for a TE-polarized wave with wavelength 1.060 μm incident upon the corrugated interface, a sinusoidal corrugation at the cladding-layer interface with period of d = 0.4 μm and a groove depth of h = 0.12 μm, and refractive indices at ω for the cladding of n3 = 1, the layer, n2 = 2.01 + i0.0005, and the substrate, n1 = 1.7 and at 2ω for the cladding of n3 = 1, the layer, n2 = 2.01 + i0.0005, and the substrate, n1 = 1.905. The thickness of the layer is t2 = 0.58 μm. Solid curve, rigorous method; dashed curve, Rayleigh method; dotted curve, classical differential theory.

Fig. 4
Fig. 4

Convergence of the reflected zeroth-order amplitude with respect to the truncation parameter N. All the parameters of the system and of the incident wave are the same as in Fig. 3 except for the groove depth, h = 0.06 μm. The angle of incidence is equal to 48.05°: (a) at ω, (b) at 2ω. Solid curves, rigorous method; dashed curves, Rayleigh method.

Fig. 5
Fig. 5

Same as in Fig. 4 except for the groove depth h equal to 0.16 μm: (a) at ω,(b) at 2ω.

Fig. 6
Fig. 6

Modulus of the amplitude of the reflected zeroth-order as a function of the layer thickness. All the parameters are given in Fig. 3 except the groove depth h is equal to 0.16 μm; the angle of incidence is 48.05°: rigorous method, (a) at ω, (b) at 2ω.

Equations (75)

Equations on this page are rendered with MathJax. Learn more.

curl E = - B t , curl H = D t ,
curl E = i ω ˜ μ ˜ H , curl H = - i ω ˜ ˜ E ,
curl E = i ω μ H , curl H = - i ω E - i ω 0 P .
P i = χ i j k E j E k ,
χ x x x , χ y y y , χ z z z 0.
( curl A ) i = i j k A k x j ,
μ ˜ i j = μ i j = μ 0 g i j , ˜ i j = 0 n ˜ 2 g i j , i j = 0 n 2 g i j .
x 1 = x , x 2 = y - f ( x ) , x 3 = z .
E 3 x 2 = f 1 + f 2 E 3 x 1 + i 1 + f 2 ω ˜ μ 0 H 1 , ω ˜ μ 0 E 1 x 2 = i k ˜ 2 n ˜ E 3 + i x 1 ( 1 1 + f 2 E 3 x 1 ) + x 1 ( f 1 + f 2 ω ˜ μ 0 H 1 ) ,
E 3 x 2 = f 1 + f 2 E 3 x 1 + i 1 + f 2 ω μ 0 H 1 , ω μ 0 H 1 x 2 = i k 2 n 2 E 3 + i x 1 ( 1 1 + f 2 E 3 x 1 ) + x 1 ( f 1 + f 2 ω μ 0 H 1 ) + i k 2 χ z E z 2
ω ˜ μ 0 H 3 x 2 = - i k ˜ 2 n ˜ 2 1 + f 2 E 1 + f 1 + f 2 ω ˜ μ 0 H 3 x 1 , - E 1 x 2 = i ω ˜ μ 0 H 3 - x 1 ( f 1 + f 2 E 1 ) + i k ˜ 2 n ˜ 2 x 1 ( 1 1 + f 2 ω ˜ μ 0 H 3 x 1 ) ,
ω μ 0 H 3 x 2 = - i k 2 n 2 1 + f 2 E 1 + f 1 + f 2 ω μ 0 H 3 x 1 - i k 2 1 1 + f 2 ( χ x E x 2 + f χ y E y 2 ) , - E 1 x 2 = i ω μ 0 H 3 - x 1 ( f 1 + f 2 E 1 ) + i k 2 n 2 x 1 ( 1 1 + f 2 ω μ 0 H 3 x 1 ) - 1 n 2 x 1 f χ x E x 2 - χ y E y 2 1 + f 2
H 1 = H x + f ( x ) H y n × H , E 1 = E x + f ( x ) E y n × E .
F = ( E 3 ω ˜ μ 0 H 1 ) ,             TE case , F = ( ω ˜ μ 0 H 3 - E 1 ) ,             TM case .
F ( x 1 , x 2 ) = m exp ( i α m x 1 ) F m ( x 2 ) ,
α m = α 0 + m 2 π d ,             m = 0 , ± 1 , ± 2 , ,
d F d x 2 = i T F ( x 2 ) ,
T m n = [ α n D m - n q C m - n ( k 2 n 2 δ m n - α m α n C m - n ) / q α n D m - n ] ,
q = { 1 TE case k 2 n 2 TM case ,
F = M Φ ˜ ( x 2 ) F 0 ,
Φ ˜ m n = δ m n exp ( i ρ m x 2 ) ,
L m ( β n ) = 1 d 0 d exp [ i β n f ( x 1 ) - i m 2 π d x 1 ] d x 1 , K m ( β n ) = 1 d 0 d f ( x 1 ) exp [ i β n f ( x 1 ) - i m 2 π d x 1 ] d x 1 ,
F 1 , m i = [ L m ( β 0 ) [ β 0 L m ( β 0 ) - α 0 K m ( β 0 ) ] q ] .
F 3 , m i [ E z or H z ω μ 0 H x or - E x ] = [ δ m 0 δ m 0 β 0 q ] .
M = [ U + U - L + L - ] ,
U ^ m n = p L m - p ( - ρ n ) U p n , L ^ m n = - p ρ n L m - p ( - ρ n ) U p n / q .
[ U 1 - U 2 + U 2 - e ( + ) 0 L 1 - L 2 + L 2 - e ( + ) 0 0 U ^ 2 + e ( + ) U ^ 2 - U ^ 3 + 0 L ^ 2 + e ( + ) L ^ 2 - L ^ 3 + ] [ b 1 - b 2 + b 2 - e ( - ) b 3 + ] = ( F 1 i F 3 i ) ,
e m n ( ± ) = δ m n exp ( ± i ρ m + t 2 ) ,
E z E 3 ( x 1 , x 2 ) = m , n exp ( i α m x 1 ) U 2 , m n exp ( i ρ n x 2 ) b 2 , n
E x ( x 1 , x 2 ) = - 1 k ˜ 2 n ˜ 2 m , n exp ( i α m x 1 ) U 2 , m n exp ( i ρ n x 2 ) ρ n b 2 , n , E y ( x 1 , x 2 ) = 1 k ˜ 2 n ˜ 2 m , n exp ( i α m x 1 ) U 2 , m n exp ( i ρ n x 2 ) × [ α m - f ( x 1 ) ρ n ] b 2 , n .
k 2 χ x , y , z E x , y , z 2 ( x 1 , x 2 ) = m , n , n exp ( i ξ m x 1 ) P m n n x , y , z exp [ i ( ρ n + ρ n ) x 2 ] ,
ξ m = 2 α 0 + m 2 π d ,
F ( x 1 , x 2 ) = m exp ( i ξ m x 1 ) F m ( x 2 ) ,
d d x 2 F ( x 2 ) = i T F ( x 2 ) + i P ( x 2 ) ,
P m ( x 2 ) = n n P m n n exp [ i ( ρ n + ρ n ) x 2 ] ,
P m n n = [ 0 P m n n z ] ,             TE case ,
P m n n = p [ C m - p P p n n x + D m - p P p n n y - 1 k 2 n 2 ξ m ( C m - p P p n n y - D m - p P p n n x ] ,             TM case .
F G ( x 2 ) = M Φ ( x 2 ) ,             Φ m n = δ m n exp ( i r m x 2 ) .
F P ( x 2 ) = F G ( x 2 ) Ψ ( x 2 ) .
F G ( x 2 ) Ψ ( x 2 ) = P ( x 2 ) .
F m P ( x 2 ) = n , n Q m n n exp [ i ( ρ n + ρ n ) x 2 ] ,
Q m n n = p , p M m p M p p - 1 P p n n 1 i ( ρ n + ρ n - r p ) ,
F m ( x 2 ) = n M m n exp ( i r n x 2 ) B n + F m P ( x 2 ) .
( F m P ( 0 ) F m P ( t 2 ) ) .
E z E 3 ,             ω μ 0 H x = i E 3 x 2 ,             TE case ,
H z H 3 ,             - E x = 1 k 2 n 2 ( - i ω μ 0 H 3 x 2 + i χ x E x 2 ) ,             TM case .
F m P ( t 2 ) = p , n , n L m - p ( - ρ n - ρ n ) Q ^ p n n exp [ i ( ρ n + ρ n ) t 2 ] ,
Q ^ m n p = [ Q m n p U ( ρ n + ρ p ) Q m n p U ] ,             TE case ,
Q ^ m n p = [ Q m n p U ( ρ n + ρ p ) k 2 n 2 Q m n p U + i k 2 n 2 P m n p x ] ,             TM case .
η ˜ 3 m ( ω ) = b 3 m 2 β ˜ 3 m β ˜ j 0 ( q ˜ 3 q ˜ j ) 1 / 2 , η 3 m ( ω ) = B 3 m 2 β 3 m β ˜ j 0 ( q 3 q ˜ j ) 1 / 2 ,
β ˜ p m 2 = k 2 n ˜ p 2 - α m 2 , β p m 2 = k 2 n p 2 - ξ m 2 ,             p = 1 , 3 ,
b l m R = n U ^ l m n ( ω ) b l n , B l m R = n U ^ l m n ( 2 ω ) B l n ,
E z y = i ω μ 0 H x , E z x = - i ω μ 0 H y , H y x - H x y = - i ω 0 n 2 E z - i ω 0 χ z E z 2
H z y = - i ω 0 n 2 E x - i ω 0 χ x E x 2 , H z x = i ω 0 n 2 E x + i ω 0 χ y E y 2 , E y x - E x y = - i ω μ 0 H z
F ( x , y ) = m exp ( i ξ m x + i β m y ) B m R ,
F { [ E z ω μ 0 H x ] [ H z - E x ] = ξ ( x ) M Φ ( y ) B 0 ,
ξ ( x ) m n = δ m n exp ( i ξ m x ) , Φ ( y ) m n = δ m n exp ( i β m y ) ,
M = [ I I β q - β q ] .
M ^ = [ U ^ + U ^ - L ^ + L ^ - ] ,
U ^ m n ± = L m - n ( ± β n ) , L ^ m n ± = q [ ± β n L m - n ( ± β n ) - ξ m K m - n ( ± β n ) ] .
[ U ^ 1 - U ^ 2 + U ^ 2 - e ( + ) 0 L ^ 1 - L ^ 2 + L ^ 2 - e ( + ) 0 0 U 2 + e ( + ) U 2 - U 3 + 0 L 2 + e ( + ) L 2 - L 3 + ] [ B 1 - B 2 + B 2 - e ( - ) B 3 + ] = [ F ^ U P F ^ L P F U P F L P ] .
E z = m , ± exp ( i α m x ) exp ( ± i β ˜ 2 , m y ) b 2 , m R ±
E x ( x , y ) = - 1 k ˜ 2 n ˜ 2 m , ± exp ( i α m x ) exp ( ± i β ˜ 2 , m y ) × ( ± β ˜ 2 , m ) b 2 , m R ± , E y ( x , y ) = 1 k ˜ 2 n ˜ 2 m , ± exp ( i α m x ) exp ( ± i β ˜ 2 , m y ) α m b 2 , m R ± ,
F m + n P [ F U P F L P ] = χ z ± , ± C m n ± [ 1 β ˜ m ± + β ˜ n ± ] ,
F ^ m + n + p P [ F ^ U P F ^ L P ] = χ z ± , ± C m n ± [ L p ( β ˜ m ± + β ˜ n ± ) ( β ˜ m ± + β ˜ n ± ) L p ( β ˜ m ± + β ˜ n ± ) - ξ m + n K p ( β ˜ m ± + β ˜ n ± ) ] ;
F m + n P = ± , ± C m n ± [ χ x β ˜ m ± β ˜ n ± ( β ˜ m ± + β ˜ n ± ) - χ y α m α n ξ m + n χ x β ˜ m ± β ˜ n ± β m + n 2 k 2 n 2 - χ y α m α n ( β ˜ m ± + β ˜ n ± ) ξ m + n k 2 n 2 ] ,
F ^ m + n + p P = ± , ± C m n ± ( L p ( β ˜ m ± + β ˜ n ± ) [ χ x β ˜ m ± β ˜ n ± ( β ˜ m ± + β ˜ n ± ) - χ y α m α n ξ m + n χ x β ˜ m ± β ˜ n ± β m + n 2 k 2 n 2 - χ y α m α n ( β ˜ m ± + β ˜ n ± ) ξ m + n k 2 n 2 ] + K p ( β ˜ m ± + β ˜ n ± ) { 0 χ x β ˜ m ± β ˜ n ± ξ m + n ( β ˜ m ± + β ˜ n ± ) k 2 n 2 - χ y α m α n [ 1 - ( β ˜ m ± + β ˜ n ± ) 2 k 2 n 2 ] } ) ,
C m n ± = - k 2 q b m ± b n ± exp [ i ( β ˜ m ± + β ˜ n ± ) t 2 ] ( β ˜ m ± + β ˜ n ± ) 2 - β m + n 2
χ [ x , x , z ] = χ [ x , y , z ] = χ [ y , y , z ] = 0 ,
P x = χ x x x E x 2 + ( χ x x y + χ x y x ) E x E y + χ x y y E y 2 , P y = χ y x x E x 2 + ( χ y x y + χ y y x ) E x E y + χ y y y E y 2 .
χ x β ˜ m ± β ˜ n ± χ x x x β ˜ m ± β ˜ n ± - ( χ x x y + χ x y x ) β ˜ m ± α n + χ x y y α m α n , χ y α m α n χ y x x β ˜ m ± β ˜ n ± - ( χ y x y + χ y y x ) β ˜ m ± α n + χ y y y α m α n .
g i j = x i x x j x + x i y x j y + x i z x j z ,
g i j = [ 1 - f 0 - f 1 + f 2 0 0 0 1 ] .
E 3 x 1 = - i ω μ 0 ( g 21 H 1 + g 22 H 2 ) , E 3 x 2 = i ω μ 0 ( g 11 H 1 + g 12 H 2 ) , H 2 x 1 - H 1 x 2 = - i ω 0 n ˜ 2 g 33 E 3 .
χ i j k = i j k 3 x i x i x j x j x k x k χ i j k .

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