Abstract

We present an extension of the rigorous coupled-wave analysis technique to analyze second-harmonic generation (SHG) in periodic optical nanostructures in the undepleted-pump approximation. We apply this method to analyze SHG in two example nanostructures for which we predict enhanced nonlinearity due to transverse near-field localization of the fundamental optical field in the nonlinear material. First, we examine a periodic nanostructure that yields up to twice the transmitted SHG intensity output compared with the bulk nonlinear material but only for small nanostructure depths because of mismatch of the fundamental and second-harmonic mode phase velocities. Second, we develop and analyze a modified nanostructure and find that this nanostructure concurrently achieves transverse localization and phase matching for SHG. In principle, this permits an arbitrary coherent interaction length, and for several specific nanostructure depths we predict a transmitted SHG intensity output more than two orders of magnitude greater than that of the bulk material.

© 2002 Optical Society of America

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2002 (1)

2001 (3)

2000 (2)

1999 (1)

1998 (6)

P. Lalanne, J.-P. Hugonin, “High-order effective-medium theory of subwavelength gratings in classical mounting: application to volume holograms,” J. Opt. Soc. Am. A 15, 1843–1851 (1998).
[CrossRef]

K. P. Yuen, M. F. Law, K. W. Yu, P. Sheng, “Enhancement of optical nonlinearity through anisotropic microstructures,” Opt. Commun. 148, 197–207 (1998).
[CrossRef]

H. Ma, R. Xiao, P. Sheng, “Third-order optical nonlinearity enhancement through composite microstructures,” J. Opt. Soc. Am. B 15, 1022–1029 (1998).
[CrossRef]

A. Fiore, S. Janz, L. Delobel, P. van der Meer, P. Bravetti, V. Berger, E. Rosencher, J. Nagle, “Second-harmonic generation at λ=1.6 µm in AlGaAs/Al2O3 waveguides using birefringence phase matching,” Appl. Phys. Lett. 72, 2942–2944 (1998).
[CrossRef]

A. Fiore, V. Berger, E. Rosencher, P. Bravetti, J. Nagle, “Phase matching using an isotropic nonlinear optical material,” Nature (London) 391, 463–466 (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]

1997 (1)

1996 (2)

1995 (2)

G. L. Fischer, R. W. Boyd, R. J. Gehr, S. A. Jenekhe, J. A. Osaheni, J. E. Sipe, L. A. Weller-Brophy, “Enhanced nonlinear optical response of composite materials,” Phys. Rev. Lett. 74, 1871–1874 (1995).
[CrossRef] [PubMed]

M. G. Moharam, E. B. Grann, D. A. Pommet, T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12, 1068–1076 (1995).
[CrossRef]

1994 (1)

1993 (1)

1992 (1)

J. E. Sipe, R. W. Boyd, “Nonlinear susceptibility of composite optical materials in the Maxwell Garnett model,” Phys. Rev. A 46, 1614–1629 (1992).
[CrossRef] [PubMed]

1987 (1)

1982 (1)

Aitchison, J. S.

Arnold, J. M.

Becouarn, L.

L. A. Eyres, P. J. Tourreau, T. J. Pinguet, C. B. Ebert, J. S. Harris, M. M. Fejer, L. Becouarn, B. Gerard, E. Lallier, “All-epitaxial fabrication of thick, orientation-patterned GaAs films for nonlinear optical frequency conversion,” Appl. Phys. Lett. 79, 904–906 (2001).
[CrossRef]

Berger, V.

K. Moutzouris, S. Venugopal Rao, M. Ebrahimzadeh, A. De Rossi, V. Berger, M. Calligaro, V. Ortiz, “Efficient second-harmonic generation in birefringently phase-matched GaAs/Al2O3 waveguides,” Opt. Lett. 26, 1785–1787 (2001).
[CrossRef]

A. Fiore, S. Janz, L. Delobel, P. van der Meer, P. Bravetti, V. Berger, E. Rosencher, J. Nagle, “Second-harmonic generation at λ=1.6 µm in AlGaAs/Al2O3 waveguides using birefringence phase matching,” Appl. Phys. Lett. 72, 2942–2944 (1998).
[CrossRef]

A. Fiore, V. Berger, E. Rosencher, P. Bravetti, J. Nagle, “Phase matching using an isotropic nonlinear optical material,” Nature (London) 391, 463–466 (1998).
[CrossRef]

Boyd, R. W.

G. L. Fischer, R. W. Boyd, R. J. Gehr, S. A. Jenekhe, J. A. Osaheni, J. E. Sipe, L. A. Weller-Brophy, “Enhanced nonlinear optical response of composite materials,” Phys. Rev. Lett. 74, 1871–1874 (1995).
[CrossRef] [PubMed]

R. W. Boyd, J. E. Sipe, “Nonlinear optical susceptibilities of layered composite materials,” J. Opt. Soc. Am. B 11, 297–303 (1994).
[CrossRef]

J. E. Sipe, R. W. Boyd, “Nonlinear susceptibility of composite optical materials in the Maxwell Garnett model,” Phys. Rev. A 46, 1614–1629 (1992).
[CrossRef] [PubMed]

Bravetti, P.

A. Fiore, V. Berger, E. Rosencher, P. Bravetti, J. Nagle, “Phase matching using an isotropic nonlinear optical material,” Nature (London) 391, 463–466 (1998).
[CrossRef]

A. Fiore, S. Janz, L. Delobel, P. van der Meer, P. Bravetti, V. Berger, E. Rosencher, J. Nagle, “Second-harmonic generation at λ=1.6 µm in AlGaAs/Al2O3 waveguides using birefringence phase matching,” Appl. Phys. Lett. 72, 2942–2944 (1998).
[CrossRef]

Brown, C. T. A.

Bryce, A. C.

Calligaro, M.

Chen, C.-H.

Cheng, C.-C.

R.-C. Tyan, A. A. Salvekar, H.-P. Chou, C.-C. Cheng, A. Scherer, P.-C. Sun, F. Xu, Y. Fainman, “Design, fabrication and characterization of form-birefringent multilayer polarizing beam splitter,” J. Opt. Soc. Am. A 14, 1627–1636 (1997).
[CrossRef]

F. Xu, R.-C. Tyan, P.-C. Sun, Y. Fainman, C.-C. Cheng, A. Scherer, “Form-birefringent computer-generated holograms,” Opt. Lett. 21, 1513–1515 (1996).
[CrossRef] [PubMed]

R.-C. Tyan, P.-C. Sun, A. A. Salvekar, H.-P. Chou, C.-C. Cheng, F. Xu, A. Scherer, Y. Fainman, “Subwavelength multilayer binary grating design for implementing photonic crystals,” in Quantum Optoelectronics, Vol. 9 of 1997 OSA Technical Digest Series (Optical Society of America, Washington D.C.,1997), pp. 35–37.

Chou, H.-P.

R.-C. Tyan, A. A. Salvekar, H.-P. Chou, C.-C. Cheng, A. Scherer, P.-C. Sun, F. Xu, Y. Fainman, “Design, fabrication and characterization of form-birefringent multilayer polarizing beam splitter,” J. Opt. Soc. Am. A 14, 1627–1636 (1997).
[CrossRef]

R.-C. Tyan, P.-C. Sun, A. A. Salvekar, H.-P. Chou, C.-C. Cheng, F. Xu, A. Scherer, Y. Fainman, “Subwavelength multilayer binary grating design for implementing photonic crystals,” in Quantum Optoelectronics, Vol. 9 of 1997 OSA Technical Digest Series (Optical Society of America, Washington D.C.,1997), pp. 35–37.

De La Rue, R. M.

De Rossi, A.

Delobel, L.

A. Fiore, S. Janz, L. Delobel, P. van der Meer, P. Bravetti, V. Berger, E. Rosencher, J. Nagle, “Second-harmonic generation at λ=1.6 µm in AlGaAs/Al2O3 waveguides using birefringence phase matching,” Appl. Phys. Lett. 72, 2942–2944 (1998).
[CrossRef]

Ebert, C. B.

L. A. Eyres, P. J. Tourreau, T. J. Pinguet, C. B. Ebert, J. S. Harris, M. M. Fejer, L. Becouarn, B. Gerard, E. Lallier, “All-epitaxial fabrication of thick, orientation-patterned GaAs films for nonlinear optical frequency conversion,” Appl. Phys. Lett. 79, 904–906 (2001).
[CrossRef]

Ebrahimzadeh, M.

Eyres, L. A.

L. A. Eyres, P. J. Tourreau, T. J. Pinguet, C. B. Ebert, J. S. Harris, M. M. Fejer, L. Becouarn, B. Gerard, E. Lallier, “All-epitaxial fabrication of thick, orientation-patterned GaAs films for nonlinear optical frequency conversion,” Appl. Phys. Lett. 79, 904–906 (2001).
[CrossRef]

Fainman, Y.

Fejer, M. M.

L. A. Eyres, P. J. Tourreau, T. J. Pinguet, C. B. Ebert, J. S. Harris, M. M. Fejer, L. Becouarn, B. Gerard, E. Lallier, “All-epitaxial fabrication of thick, orientation-patterned GaAs films for nonlinear optical frequency conversion,” Appl. Phys. Lett. 79, 904–906 (2001).
[CrossRef]

Fiore, A.

A. Fiore, S. Janz, L. Delobel, P. van der Meer, P. Bravetti, V. Berger, E. Rosencher, J. Nagle, “Second-harmonic generation at λ=1.6 µm in AlGaAs/Al2O3 waveguides using birefringence phase matching,” Appl. Phys. Lett. 72, 2942–2944 (1998).
[CrossRef]

A. Fiore, V. Berger, E. Rosencher, P. Bravetti, J. Nagle, “Phase matching using an isotropic nonlinear optical material,” Nature (London) 391, 463–466 (1998).
[CrossRef]

Fischer, G. L.

G. L. Fischer, R. W. Boyd, R. J. Gehr, S. A. Jenekhe, J. A. Osaheni, J. E. Sipe, L. A. Weller-Brophy, “Enhanced nonlinear optical response of composite materials,” Phys. Rev. Lett. 74, 1871–1874 (1995).
[CrossRef] [PubMed]

Gaylord, T. K.

Gehr, R. J.

G. L. Fischer, R. W. Boyd, R. J. Gehr, S. A. Jenekhe, J. A. Osaheni, J. E. Sipe, L. A. Weller-Brophy, “Enhanced nonlinear optical response of composite materials,” Phys. Rev. Lett. 74, 1871–1874 (1995).
[CrossRef] [PubMed]

Gerard, B.

L. A. Eyres, P. J. Tourreau, T. J. Pinguet, C. B. Ebert, J. S. Harris, M. M. Fejer, L. Becouarn, B. Gerard, E. Lallier, “All-epitaxial fabrication of thick, orientation-patterned GaAs films for nonlinear optical frequency conversion,” Appl. Phys. Lett. 79, 904–906 (2001).
[CrossRef]

Glytsis, E. N.

Grann, E. B.

Harris, J. S.

L. A. Eyres, P. J. Tourreau, T. J. Pinguet, C. B. Ebert, J. S. Harris, M. M. Fejer, L. Becouarn, B. Gerard, E. Lallier, “All-epitaxial fabrication of thick, orientation-patterned GaAs films for nonlinear optical frequency conversion,” Appl. Phys. Lett. 79, 904–906 (2001).
[CrossRef]

Houston, P. A.

Hugonin, J.-P.

Hutchings, D. C.

Janz, S.

A. Fiore, S. Janz, L. Delobel, P. van der Meer, P. Bravetti, V. Berger, E. Rosencher, J. Nagle, “Second-harmonic generation at λ=1.6 µm in AlGaAs/Al2O3 waveguides using birefringence phase matching,” Appl. Phys. Lett. 72, 2942–2944 (1998).
[CrossRef]

Jenekhe, S. A.

G. L. Fischer, R. W. Boyd, R. J. Gehr, S. A. Jenekhe, J. A. Osaheni, J. E. Sipe, L. A. Weller-Brophy, “Enhanced nonlinear optical response of composite materials,” Phys. Rev. Lett. 74, 1871–1874 (1995).
[CrossRef] [PubMed]

Kleckner, T. C.

Lalanne, P.

Lallier, E.

L. A. Eyres, P. J. Tourreau, T. J. Pinguet, C. B. Ebert, J. S. Harris, M. M. Fejer, L. Becouarn, B. Gerard, E. Lallier, “All-epitaxial fabrication of thick, orientation-patterned GaAs films for nonlinear optical frequency conversion,” Appl. Phys. Lett. 79, 904–906 (2001).
[CrossRef]

Law, M. F.

K. P. Yuen, M. F. Law, K. W. Yu, P. Sheng, “Enhancement of optical nonlinearity through anisotropic microstructures,” Opt. Commun. 148, 197–207 (1998).
[CrossRef]

Li, L.

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

L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996).
[CrossRef]

Loza-Alvarez, P.

Ma, H.

Mait, J. N.

Marsh, J. H.

Millar, P.

Mirotznik, M. S.

Moharam, M. G.

Moutzouris, K.

Nagle, J.

A. Fiore, V. Berger, E. Rosencher, P. Bravetti, J. Nagle, “Phase matching using an isotropic nonlinear optical material,” Nature (London) 391, 463–466 (1998).
[CrossRef]

A. Fiore, S. Janz, L. Delobel, P. van der Meer, P. Bravetti, V. Berger, E. Rosencher, J. Nagle, “Second-harmonic generation at λ=1.6 µm in AlGaAs/Al2O3 waveguides using birefringence phase matching,” Appl. Phys. Lett. 72, 2942–2944 (1998).
[CrossRef]

Nakagawa, W.

Ortiz, V.

Osaheni, J. A.

G. L. Fischer, R. W. Boyd, R. J. Gehr, S. A. Jenekhe, J. A. Osaheni, J. E. Sipe, L. A. Weller-Brophy, “Enhanced nonlinear optical response of composite materials,” Phys. Rev. Lett. 74, 1871–1874 (1995).
[CrossRef] [PubMed]

Pinguet, T. J.

L. A. Eyres, P. J. Tourreau, T. J. Pinguet, C. B. Ebert, J. S. Harris, M. M. Fejer, L. Becouarn, B. Gerard, E. Lallier, “All-epitaxial fabrication of thick, orientation-patterned GaAs films for nonlinear optical frequency conversion,” Appl. Phys. Lett. 79, 904–906 (2001).
[CrossRef]

Pommet, D. A.

Prather, D. W.

Rafailov, E. U.

Rao, S. Venugopal

Roberts, J. S.

Rosencher, E.

A. Fiore, S. Janz, L. Delobel, P. van der Meer, P. Bravetti, V. Berger, E. Rosencher, J. Nagle, “Second-harmonic generation at λ=1.6 µm in AlGaAs/Al2O3 waveguides using birefringence phase matching,” Appl. Phys. Lett. 72, 2942–2944 (1998).
[CrossRef]

A. Fiore, V. Berger, E. Rosencher, P. Bravetti, J. Nagle, “Phase matching using an isotropic nonlinear optical material,” Nature (London) 391, 463–466 (1998).
[CrossRef]

Saher Helmy, A.

Salvekar, A. A.

R.-C. Tyan, A. A. Salvekar, H.-P. Chou, C.-C. Cheng, A. Scherer, P.-C. Sun, F. Xu, Y. Fainman, “Design, fabrication and characterization of form-birefringent multilayer polarizing beam splitter,” J. Opt. Soc. Am. A 14, 1627–1636 (1997).
[CrossRef]

R.-C. Tyan, P.-C. Sun, A. A. Salvekar, H.-P. Chou, C.-C. Cheng, F. Xu, A. Scherer, Y. Fainman, “Subwavelength multilayer binary grating design for implementing photonic crystals,” in Quantum Optoelectronics, Vol. 9 of 1997 OSA Technical Digest Series (Optical Society of America, Washington D.C.,1997), pp. 35–37.

Scherer, A.

R.-C. Tyan, A. A. Salvekar, H.-P. Chou, C.-C. Cheng, A. Scherer, P.-C. Sun, F. Xu, Y. Fainman, “Design, fabrication and characterization of form-birefringent multilayer polarizing beam splitter,” J. Opt. Soc. Am. A 14, 1627–1636 (1997).
[CrossRef]

F. Xu, R.-C. Tyan, P.-C. Sun, Y. Fainman, C.-C. Cheng, A. Scherer, “Form-birefringent computer-generated holograms,” Opt. Lett. 21, 1513–1515 (1996).
[CrossRef] [PubMed]

R.-C. Tyan, P.-C. Sun, A. A. Salvekar, H.-P. Chou, C.-C. Cheng, F. Xu, A. Scherer, Y. Fainman, “Subwavelength multilayer binary grating design for implementing photonic crystals,” in Quantum Optoelectronics, Vol. 9 of 1997 OSA Technical Digest Series (Optical Society of America, Washington D.C.,1997), pp. 35–37.

Sheng, P.

K. P. Yuen, M. F. Law, K. W. Yu, P. Sheng, “Enhancement of optical nonlinearity through anisotropic microstructures,” Opt. Commun. 148, 197–207 (1998).
[CrossRef]

H. Ma, R. Xiao, P. Sheng, “Third-order optical nonlinearity enhancement through composite microstructures,” J. Opt. Soc. Am. B 15, 1022–1029 (1998).
[CrossRef]

Sibbett, W.

Sipe, J. E.

G. L. Fischer, R. W. Boyd, R. J. Gehr, S. A. Jenekhe, J. A. Osaheni, J. E. Sipe, L. A. Weller-Brophy, “Enhanced nonlinear optical response of composite materials,” Phys. Rev. Lett. 74, 1871–1874 (1995).
[CrossRef] [PubMed]

R. W. Boyd, J. E. Sipe, “Nonlinear optical susceptibilities of layered composite materials,” J. Opt. Soc. Am. B 11, 297–303 (1994).
[CrossRef]

J. E. Sipe, R. W. Boyd, “Nonlinear susceptibility of composite optical materials in the Maxwell Garnett model,” Phys. Rev. A 46, 1614–1629 (1992).
[CrossRef] [PubMed]

Stanley, C. R.

Sun, P.-C.

Tourreau, P. J.

L. A. Eyres, P. J. Tourreau, T. J. Pinguet, C. B. Ebert, J. S. Harris, M. M. Fejer, L. Becouarn, B. Gerard, E. Lallier, “All-epitaxial fabrication of thick, orientation-patterned GaAs films for nonlinear optical frequency conversion,” Appl. Phys. Lett. 79, 904–906 (2001).
[CrossRef]

Tyan, R.-C.

van der Meer, P.

A. Fiore, S. Janz, L. Delobel, P. van der Meer, P. Bravetti, V. Berger, E. Rosencher, J. Nagle, “Second-harmonic generation at λ=1.6 µm in AlGaAs/Al2O3 waveguides using birefringence phase matching,” Appl. Phys. Lett. 72, 2942–2944 (1998).
[CrossRef]

Weller-Brophy, L. A.

G. L. Fischer, R. W. Boyd, R. J. Gehr, S. A. Jenekhe, J. A. Osaheni, J. E. Sipe, L. A. Weller-Brophy, “Enhanced nonlinear optical response of composite materials,” Phys. Rev. Lett. 74, 1871–1874 (1995).
[CrossRef] [PubMed]

Xiao, R.

Xu, F.

R.-C. Tyan, A. A. Salvekar, H.-P. Chou, C.-C. Cheng, A. Scherer, P.-C. Sun, F. Xu, Y. Fainman, “Design, fabrication and characterization of form-birefringent multilayer polarizing beam splitter,” J. Opt. Soc. Am. A 14, 1627–1636 (1997).
[CrossRef]

F. Xu, R.-C. Tyan, P.-C. Sun, Y. Fainman, C.-C. Cheng, A. Scherer, “Form-birefringent computer-generated holograms,” Opt. Lett. 21, 1513–1515 (1996).
[CrossRef] [PubMed]

R.-C. Tyan, P.-C. Sun, A. A. Salvekar, H.-P. Chou, C.-C. Cheng, F. Xu, A. Scherer, Y. Fainman, “Subwavelength multilayer binary grating design for implementing photonic crystals,” in Quantum Optoelectronics, Vol. 9 of 1997 OSA Technical Digest Series (Optical Society of America, Washington D.C.,1997), pp. 35–37.

Yablonovitch, E.

Yanson, D. A.

Yu, K. W.

K. P. Yuen, M. F. Law, K. W. Yu, P. Sheng, “Enhancement of optical nonlinearity through anisotropic microstructures,” Opt. Commun. 148, 197–207 (1998).
[CrossRef]

Yuen, K. P.

K. P. Yuen, M. F. Law, K. W. Yu, P. Sheng, “Enhancement of optical nonlinearity through anisotropic microstructures,” Opt. Commun. 148, 197–207 (1998).
[CrossRef]

Appl. Phys. Lett. (2)

A. Fiore, S. Janz, L. Delobel, P. van der Meer, P. Bravetti, V. Berger, E. Rosencher, J. Nagle, “Second-harmonic generation at λ=1.6 µm in AlGaAs/Al2O3 waveguides using birefringence phase matching,” Appl. Phys. Lett. 72, 2942–2944 (1998).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Schematic diagram of the transverse field localization nanostructure for SHG enhancement. The nanostructure is a subwavelength periodic square grating with period Λ=0.65λ, fill factor F=0.09, index of refraction nω=3.346 at the fundamental frequency and n2ω=3.539 at the second-harmonic frequency, and depth ranging from d=0.005λ to d=5.00λ in units of the fundamental wavelength in vacuum. The structure is assumed to be infinite and periodic in the xˆ direction and infinite and uniform in the yˆ direction. The nanostructure is illuminated by a normally incident plane wave at the fundamental frequency with wave vector kinc and electric field Einc polarized in the yˆ direction.

Fig. 2
Fig. 2

Total transmitted SHG output intensity for the nanostructure shown in Fig. 1 and bulk nonlinear material as a function of depth.

Fig. 3
Fig. 3

Properties of the nonevanescent energy-carrying eigenmodes of the nanostructure shown in Fig. 1 for the (a) fundamental and (b), (c) second-harmonic wavelengths computed by using the modified RCWA tool: For each mode the transverse profile in one period of the nanostructure as well as the attenuation coefficient and propagation constant (real and imaginary parts of the corresponding eigenvalue Z respectively) are shown.

Fig. 4
Fig. 4

Schematic diagram of the modified transverse field localization nanostructure with improved phase matching, consisting of a periodic subwavelength grating having two nonlinear material ridges with fill factors F1=0.063 and F2=0.03 per period Λ=0.65λ. The index of refraction is nω=3.346 at the fundamental frequency and n2ω=3.539 at the second-harmonic frequency, and the depth ranges from d=0.005λ to d=5.00λ in units of the fundamental wavelength in vacuum. The structure is assumed to be infinite and periodic in the xˆ direction and infinite and uniform in the yˆ direction. The nanostructure is illuminated by a normally incident plane wave at the fundamental frequency with wave vector kinc and electric field Einc polarized in the yˆ direction.

Fig. 5
Fig. 5

Properties of the nonevanescent energy-carrying eigenmodes of the modified nanostructure shown in Fig. 4 for the (a), (b) fundamental and (c), (d) second-harmonic wavelengths computed by using the modified RCWA tool: For each mode the transverse profile in one period of the nanostructure as well as the attenuation coefficient and propagation constant (real and imaginary parts of the corresponding eigenvalue Z) are shown.

Fig. 6
Fig. 6

Phase velocities of the two fundamental (F-1 and F-2) and two second-harmonic (SH-1 and SH-2) nonevanescent energy-carrying eigenmodes of the nanostructure shown in Fig. 4 as a function of fill factor F2 with F1=0.063 constant.

Fig. 7
Fig. 7

Total transmitted SHG output intensity for the modified nanostructure shown in Fig. 4 and bulk nonlinear material as a function of depth.

Equations (42)

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E(r, t)=Eω(r)exp(jωt)+E2ω(r)exp(j2ωt),
H(r, t)=Hω(r)exp(jωt)+H2ω(r)exp(j2ωt),
D(r, t)=ϵE(r, t)+PNL(r, t).
PNL(r, t)=χ(2)2Eω(r)exp(jωt)Eω(r)exp(jωt)
×[Eω(r)exp(jωt)+E2ω(r)exp(j2ωt)]=-μ t[Hω(r)exp(jωt)+H2ω(r)exp(j2ωt)]
×[Hω(r)exp(jωt)+H2ω(r)exp(j2ωt)]=t[ϵωEω(r)exp(jωt)+ϵ2ωE2ω(r)exp(j2ωt)+PNL(r, t)],
×Eω(r)=-jωμHω(r)
×Hω(r)=jωϵωEω(r).
×E2ω(r)=-j2ωμH2ω(r),
×H2ω(r)=j2ωϵ2ωE2ω(r)+jωχ(2)Eω(r)Eω(r).
χ(2)(x)=kχ˜k exp(jkKgx).
PNL(r)ihkχ˜kSω,iSω,h×exp{-j[(σω,i+σω,h)·r-kKgx]}.
(σω,i+σω,h)·r-kKgx=[2kinc,ω,x-(i+h+k)Kg]x+2kinc,ω,yy.
PNL,v(r, t)=iPi,v(z)exp{-j[(2kinc,x-iKg)x+2kinc,yy]},
Pi,v(z)=hk{χ˜vxx,i-h-kSω,h,xSω,k,x+χ˜vxy,i-h-kSω,h,xSω,k,y+χ˜vxz,i-h-kSω,h,xSω,k,z+χ˜vyx,i-h-kSω,h,ySω,k,x+χ˜vyy,i-h-kSω,h,ySω,k,y+χ˜vyz,i-h-kSω,h,ySω,k,z+χ˜vzx,i-h-kSω,h,zSω,k,x+χ˜vzy,i-h-kSω,k,zSω,k,y+χ˜vzz,i-h-kSω,k,zSω,k,z}
E2ω(r)=i{S2ω,i,x(z)xˆ+S2ω,i,y(z)yˆ+S2ω,i,z(z)zˆ}exp(-jσ2ω,i·r)
H2ω(r)=ϵ0μ0i{U2ω,i,x(z)xˆ+U2ω,i,y(z)yˆ+U2ω,i,z(z)zˆ}exp(-jσ2ω,i·r)
σ2ω,i=(2kinc,x-iKg)xˆ+2kinc,yyˆ.
z S2ω,x(z)S2ω,y(z)U2ω,x(z)U2ω,y(z)=A2ωS2ω,x(z)S2ω,y(z)U2ω,x(z)U2ω,y(z)+k¯2ω,xϵ¯2ω,zz-1Pz(z)k¯2ω,yϵ¯2ω,zz-1Pz(z)Py(z)-ϵ2ω,yzϵ¯2ω,zz-1Pz(z)-Px(z)+ϵ¯2ω,xzϵ¯2ω,zz-1Pz(z),
S2ω,x(z)S2ω,y(z)U2ω,x(z)U2ω,y(z)=W2ωQ(z),
zQ(z)=W2ω-1A2ωW2ωQ(z)+W2ω-1Y(z),
Y(z)=k¯2ω,xϵ¯2ω,zz-1Pz(z)k¯2ω,yϵ¯2ω,zz-1Pz(z)Py(z)-ϵ¯2ω,yzϵ¯2ω,zz-1Pz(z)-Px(z)+ϵ¯2ω,xzϵ¯2ω,zz-1Pz(z).
Q(z)=Q(h)(z)+Q(ih)(z).
Qi(h)(z)=c2ω,i exp[λ2ω,i(z-z0,i)],
z0,i=d,0i<2p0,2pi<4p.
Qi(ih)(z)=exp(λ2ω,iz)dz exp(-λ2ω,iz)×jW2ω,ij-1Yj(z)dz,0i<2p,exp(λ2ω,iz)0z exp(-λ2ω,iz)×jW2ω,ij-1Yj(z)dz,2pi<4p,
Qi(0)=c2ω,i- exp(-λ2ω,id)-0d exp(-λ2ω,iz)jW2ω,ij-1Yj(z)dz,0i<2p,c2ω,i+,2pi<4p,
Qi(d)=c2ω,i-,0i<2p,c2ω,i+ exp(λ2ω,id)+exp(λ2ω,id)×0d exp(-λ2ω,iz)jW2ω,ij-1Yj(z)dz,2pi<4p.
S2ω,x(d+)S2ω,y(d+)U2ω,x(d+)U2ω,y(d+)=0I0Hrt,
S2ω,x(0-)S2ω,y(0-)U2ω,x(0-)U2ω,y(0-)=I0H0rt+I0H0b0,
I0H0rt+IbHb=W2ω00[exp(-λ2ωd)]W2ω01W2ω10[exp(-λ2ωd)]W2ω11c2ω,i-c2ω,i++W2ω00W2ω01W2ω10W2ω11-Q(ih),-(0)0,
W2ω=W2ω00W2ω01W2ω10W2ω11
0I0Hrt=W2ω00W2ω01[exp(λ2ωd)]W2ω00W2ω11[exp(λ2ωd)]c2ω-c2ω++W2ω00W2ω01W2ω10W2ω110Q(ih),+(d),
W2ω00[exp(-λ2ωd)]W2ω01-I0W2ω10[exp(-λ2ωd)]W2ω11-H0W2ω00W2ω01[exp(λ2ωd)]0-IW2ω10W2ω11[exp(λ2ωd)]0-H×c2ω-c2ω+rt=W2ω00Q(ih),-(0)+IbW2ω10Q(ih),-(0)+Hb-W2ω01Q(ih),+(d)-W2ω11Q(ih),+(d).
ϵ¯(r)=ϵ¯(r+Λxˆ)=ϵ¯ω(x),0<z<dϵ0z<0;z>d.
Eω(r)=i{Sω,i,x(z)xˆ+Sω,i,y(z)yˆ+Sω,i,z(z)zˆ}exp(-jσω,i·r),
Hω(r, t)=ϵ0μ0i[Uω,i,x(z)xˆ+Uω,i,y(z)yˆ+Uω,i,z(z)zˆ]exp(-jσω,i·r),
σω,i=(kinc,x-iKg)xˆ+kinc,yyˆ,
ϵω,vw(x)=ϵ0kϵ˜ω,k,vw exp(jkKgx).
z Sω,x(z)Sω,y(z)Uω,x(z)Uω,y(z)=AωSω,x(z)Sω,y(z)Uω,x(z)Uω,y(z),
Sω,x(z)=Sω,-a(z)Sω,-a+1(z)Sω,a(z),
Sω,x(z)Sω,y(z)Uω,x(z)Uω,y(z)=Wω exp(Dz)cω,

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