Abstract

Numerical mode solver using a pseudospectral scheme is developed for solving various nonlinear dielectric and plasmonic waveguides with arbitrary nonlinear media. Two nonlinear iterative approaches that use this scheme are implemented; these approaches assign the mode power and effective index as extracted eigenvalues. However, to obtain the complete power dispersion curve including the stable and unstable modal solutions, assigning the mode power as an eigenvalue for a given effective index is required. Moreover, the biaxial feature of the nonlinear refractive index is considered for solving the transverse magnetic (TM) modes in materials of practical interest. Furthermore, the proposed scheme solves the problem of nonlinear surface plasmons guided by a thin metal film with nonlinear cladding, and the mode characteristics of long- and short-range surface plasmon polaritons are analyzed. We also apply the proposed scheme to a 2D strip waveguide with a nonlinear saturation substrate.

© 2012 OSA

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]

2011 (3)

2010 (5)

C. C. Huang, “Numerical investigation of mode characteristics of nanoscale surface plasmon-polaritons using a pseudospectral scheme,” Opt. Express 18(23), 23711–23726 (2010).
[CrossRef] [PubMed]

J. R. Salgueiro and Y. S. Kivshar, “Nonlinear plasmonic directional couplers,” Appl. Phys. Lett. 97(8), 081106 (2010).
[CrossRef]

A. R. Davoyan, I. V. Shadrivov, A. A. Zharov, D. K. Gramotnev, and Y. S. Kivshar, “Nonlinear nanofocusing in tapered plasmonic waveguides,” Phys. Rev. Lett. 105(11), 116804 (2010).
[CrossRef] [PubMed]

A. Degiron and D. R. Smith, “Nonlinear long-range plasmonic waveguides,” Phys. Rev. A 82(3), 033812 (2010).
[CrossRef]

J. B. Xiao and X. H. Sun, “Full-vectorial mode solver for anisotropic optical waveguides using multidomain spectral collocation method,” Opt. Commun. 283(14), 2835–2840 (2010).
[CrossRef]

2009 (2)

S. Roy and P. Roy Chaudhuri, “Analysis of nonlinear multilayered waveguides and MQW structures: a field evolution approach using finite–difference formulation,” IEEE J. Quantum Electron. 45(4), 345–350 (2009).
[CrossRef]

A. R. Davoyan, I. V. Shadrivov, and Y. S. Kivshar, “Quadratic phase matching in nonlinear plasmonic nanoscale waveguides,” Opt. Express 17(22), 20063–20068 (2009).
[CrossRef] [PubMed]

2008 (3)

2007 (1)

2006 (1)

2005 (1)

C.-C. Huang, C.-C. Huang, and J.-Y. Yang, “A full-vectorial pseudospectral modal analysis of dielectric optical waveguides with stepped refractive index profiles,” IEEE J. Sel. Top. Quantum Electron. 11(2), 457–465 (2005).
[CrossRef]

2004 (1)

2002 (1)

S. S. A. Obayya, B. M. A. Rahman, K. T. V. Grattan, and H. A. El-Mikati, “Full vectorial finite–element solution of nonlinear bistable optical waveguides,’,” IEEE J. Quantum Electron. 38(8), 1120–1125 (2002).
[CrossRef]

2001 (1)

P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: bound modes of asymmetric structures,” Phys. Rev. B 63(12), 125417 (2001).
[CrossRef]

1998 (1)

1997 (1)

J. G. Ma and Z. Chen, “Numerically determining the dispersion relations of nonlinear TE slab-guided waves in non-Kerr-like media,” IEEE Trans. Microw. Theory Tech. 45(7), 1113–1117 (1997).
[CrossRef]

1996 (1)

T. Rozzi and L. Zappelli, “Modal analysis of nonlinear propagation in dielectric slab waveguide,” J. Lightwave Technol. 14(2), 229–235 (1996).
[CrossRef]

1994 (1)

1993 (1)

1992 (2)

Q. Y. Li, R. A. Sammut, and C. Pask, “Variational and finite element analyses of nonlinear strip optical waveguides,” Opt. Commun. 94(1-3), 37–43 (1992).
[CrossRef]

A. P. Zhao and S. R. Cvetkovic, “Finite-element solution of nonlinear TM waves in multiple-quantum-well waveguides,” IEEE Photon. Technol. Lett. 4(11), 1231–1234 (1992).
[CrossRef]

1991 (1)

R. D. Ettinger, F. A. Fernandez, B. M. A. Rahman, and J. D. Davies, “Vector finite element solution of saturable nonlinear strip-loaded optical waveguides,” IEEE Photon. Technol. Lett. 3(2), 147–149 (1991).
[CrossRef]

1990 (1)

B. M. A. Rahman, J. R. Souza, and J. B. Davies, “Numerical analysis of nonlinear bistable optical waveguides,” IEEE Photon. Technol. Lett. 2(4), 265–267 (1990).
[CrossRef]

1989 (2)

K. Ogusu, “TM waves guided by nonlinear planar waveguides,” IEEE Trans. Microw. Theory Tech. 37(6), 941–946 (1989).
[CrossRef]

M. R. Ramadas, R. K. Varshney, K. Thyagarajan, and A. K. Ghatak, “A matrix approach to study the propagation characteristics of a general nonlinear planar waveguide,” J. Lightwave Technol. 7(12), 1901–1905 (1989).
[CrossRef]

1988 (3)

K. Hayata, M. Nagai, and M. Koshiba, “Finite-element formalism for nonlinear slab-guided waves,” IEEE Trans. Microw. Theory Tech. 36(7), 1207–1215 (1988).
[CrossRef]

K. Hayata and M. Koshiba, “Full vectorial analysis of nonlinear-optical waveguides,” J. Opt. Soc. Am. B 5(12), 2494–2501 (1988).
[CrossRef]

T. H. Wood, “Multiple quantum well (MQW) waveguide modulators,” J. Lightwave Technol. 6(6), 743–757 (1988).
[CrossRef]

1987 (2)

A. D. Boardman, A. A. Maradudin, G. I. Stegeman, T. Twardowski, and E. M. Wright, “Exact theory of nonlinear p-polarized optical waves,” Phys. Rev. A 35(3), 1159–1164 (1987).
[CrossRef] [PubMed]

R. I. Joseph and D. N. Christodoulides, “Exact field decomposition for TM waves in nonlinear media,” Opt. Lett. 12(10), 826–828 (1987).
[CrossRef] [PubMed]

1986 (1)

G. I. Stegeman, E. M. Wright, C. T. Seaton, J. V. Moloney, T. P. Shen, A. A. Maradudin, and R. F. Wallis, “Nonlinear slab-guided waves in non-Kerr-like media,” IEEE J. Quantum Electron. 22(6), 977–983 (1986).
[CrossRef]

1985 (4)

C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. Chilwell, and S. D. Smith, “Calculations of nonlinear TE waves guided by thin dielectric films bounded by nonlinear media,” IEEE J. Quantum Electron. 21(7), 774–783 (1985).
[CrossRef]

I. Bennion, M. J. Goodwin, and W. J. Stewart, “Experimental nonlinear optical waveguide device,” Electron. Lett. 21(1), 41–42 (1985).
[CrossRef]

J. Ariyasu, C. T. Seaton, G. I. Stegeman, A. A. Maradudin, and R. F. Wallis, “Nonlinear surface polaritons guided by meal films,” J. Appl. Phys. 58(7), 2460–2466 (1985).
[CrossRef]

C. T. Seaton, J. D. Valera, B. Svenson, and G. I. Stegeman, “Comparison of solutions for TM-polarized nonlinear guided waves,” Opt. Lett. 10(3), 149–150 (1985).
[CrossRef] [PubMed]

1984 (3)

1983 (3)

F. Lederer, U. Langbein, and H. E. Ponath, “Nonlinear waves guided by a dielectric slab: I. TE polarization,” Appl. Phys. B 31(2), 69–73 (1983).
[CrossRef]

F. Lederer, U. Langbein, and H. E. Ponath, “Nonlinear waves guided by a dielectric slab: II. TM–polarization,” Appl. Phys. B 31(3), 187–190 (1983).
[CrossRef]

D. J. Robbins, “TE modes in a slab waveguide bounded by nonlinear media,” Opt. Commun. 47(5), 309–312 (1983).
[CrossRef]

Agrawal, G.

I. D. Rukhlenko, A. Pannipitiya, M. Premaratne, and G. Agrawal, “Exact dispersion relation for nonlinear plasmonic waveguides,” Phys. Rev. B 84(11), 113409 (2011).
[CrossRef]

Agrawal, G. P.

Ariyasu, J.

J. Ariyasu, C. T. Seaton, G. I. Stegeman, A. A. Maradudin, and R. F. Wallis, “Nonlinear surface polaritons guided by meal films,” J. Appl. Phys. 58(7), 2460–2466 (1985).
[CrossRef]

Bennion, I.

I. Bennion, M. J. Goodwin, and W. J. Stewart, “Experimental nonlinear optical waveguide device,” Electron. Lett. 21(1), 41–42 (1985).
[CrossRef]

Berini, P.

P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: bound modes of asymmetric structures,” Phys. Rev. B 63(12), 125417 (2001).
[CrossRef]

Boardman, A. D.

A. D. Boardman, A. A. Maradudin, G. I. Stegeman, T. Twardowski, and E. M. Wright, “Exact theory of nonlinear p-polarized optical waves,” Phys. Rev. A 35(3), 1159–1164 (1987).
[CrossRef] [PubMed]

Cambrell, G. K.

Chang, C. F.

Chang, H.-

P.-J. Chiang, C.-L. Wu, C.-H. Teng, C.-S. Yang, and H.- Chang, “Full-vectorial optical waveguide mode solvers using multidomain pseudospectral frequency-domain (PSFD) formulations,” IEEE J. Quantum Electron. 44(1), 56–66 (2008).
[CrossRef]

Chen, M. H.

Chen, S. Y.

Chen, Z.

J. G. Ma and Z. Chen, “Numerically determining the dispersion relations of nonlinear TE slab-guided waves in non-Kerr-like media,” IEEE Trans. Microw. Theory Tech. 45(7), 1113–1117 (1997).
[CrossRef]

Chiang, P.-J.

P.-J. Chiang, C.-L. Wu, C.-H. Teng, C.-S. Yang, and H.- Chang, “Full-vectorial optical waveguide mode solvers using multidomain pseudospectral frequency-domain (PSFD) formulations,” IEEE J. Quantum Electron. 44(1), 56–66 (2008).
[CrossRef]

Chilwell, J.

C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. Chilwell, and S. D. Smith, “Calculations of nonlinear TE waves guided by thin dielectric films bounded by nonlinear media,” IEEE J. Quantum Electron. 21(7), 774–783 (1985).
[CrossRef]

G. I. Stegeman, C. T. Seaton, J. Chilwell, and S. D. Smith, “Nonlinear waves guided by thin films,” Appl. Phys. Lett. 44(9), 830–832 (1984).
[CrossRef]

Christodoulides, D. N.

Cvetkovic, S. R.

A. P. Zhao and S. R. Cvetkovic, “Finite-element solution of nonlinear TM waves in multiple-quantum-well waveguides,” IEEE Photon. Technol. Lett. 4(11), 1231–1234 (1992).
[CrossRef]

Davies, J. B.

B. M. A. Rahman, J. R. Souza, and J. B. Davies, “Numerical analysis of nonlinear bistable optical waveguides,” IEEE Photon. Technol. Lett. 2(4), 265–267 (1990).
[CrossRef]

Davies, J. D.

R. D. Ettinger, F. A. Fernandez, B. M. A. Rahman, and J. D. Davies, “Vector finite element solution of saturable nonlinear strip-loaded optical waveguides,” IEEE Photon. Technol. Lett. 3(2), 147–149 (1991).
[CrossRef]

Davoyan, A. R.

Degiron, A.

A. Degiron and D. R. Smith, “Nonlinear long-range plasmonic waveguides,” Phys. Rev. A 82(3), 033812 (2010).
[CrossRef]

El-Mikati, H. A.

S. S. A. Obayya, B. M. A. Rahman, K. T. V. Grattan, and H. A. El-Mikati, “Full vectorial finite–element solution of nonlinear bistable optical waveguides,’,” IEEE J. Quantum Electron. 38(8), 1120–1125 (2002).
[CrossRef]

Ettinger, R. D.

R. D. Ettinger, F. A. Fernandez, B. M. A. Rahman, and J. D. Davies, “Vector finite element solution of saturable nonlinear strip-loaded optical waveguides,” IEEE Photon. Technol. Lett. 3(2), 147–149 (1991).
[CrossRef]

Fernandez, F. A.

R. D. Ettinger, F. A. Fernandez, B. M. A. Rahman, and J. D. Davies, “Vector finite element solution of saturable nonlinear strip-loaded optical waveguides,” IEEE Photon. Technol. Lett. 3(2), 147–149 (1991).
[CrossRef]

Fujisawa, T.

George, N.

Ghatak, A. K.

R. K. Varshney, I. C. Goyal, and A. K. Ghatak, “A simple and efficient numerical method to study propagation characteristics of nonlinear optical waveguides,” J. Lightwave Technol. 16(4), 697–702 (1998).
[CrossRef]

M. R. Ramadas, R. K. Varshney, K. Thyagarajan, and A. K. Ghatak, “A matrix approach to study the propagation characteristics of a general nonlinear planar waveguide,” J. Lightwave Technol. 7(12), 1901–1905 (1989).
[CrossRef]

Goodwin, M. J.

I. Bennion, M. J. Goodwin, and W. J. Stewart, “Experimental nonlinear optical waveguide device,” Electron. Lett. 21(1), 41–42 (1985).
[CrossRef]

Goyal, I. C.

Gramotnev, D. K.

A. R. Davoyan, I. V. Shadrivov, A. A. Zharov, D. K. Gramotnev, and Y. S. Kivshar, “Nonlinear nanofocusing in tapered plasmonic waveguides,” Phys. Rev. Lett. 105(11), 116804 (2010).
[CrossRef] [PubMed]

Grattan, K. T. V.

S. S. A. Obayya, B. M. A. Rahman, K. T. V. Grattan, and H. A. El-Mikati, “Full vectorial finite–element solution of nonlinear bistable optical waveguides,’,” IEEE J. Quantum Electron. 38(8), 1120–1125 (2002).
[CrossRef]

Hayata, K.

K. Hayata and M. Koshiba, “Full vectorial analysis of nonlinear-optical waveguides,” J. Opt. Soc. Am. B 5(12), 2494–2501 (1988).
[CrossRef]

K. Hayata, M. Nagai, and M. Koshiba, “Finite-element formalism for nonlinear slab-guided waves,” IEEE Trans. Microw. Theory Tech. 36(7), 1207–1215 (1988).
[CrossRef]

Huang, C. C.

Huang, C.-C.

C.-C. Huang, C.-C. Huang, and J.-Y. Yang, “A full-vectorial pseudospectral modal analysis of dielectric optical waveguides with stepped refractive index profiles,” IEEE J. Sel. Top. Quantum Electron. 11(2), 457–465 (2005).
[CrossRef]

C.-C. Huang, C.-C. Huang, and J.-Y. Yang, “A full-vectorial pseudospectral modal analysis of dielectric optical waveguides with stepped refractive index profiles,” IEEE J. Sel. Top. Quantum Electron. 11(2), 457–465 (2005).
[CrossRef]

Joseph, R. I.

Khoo, I. C.

Kivshar, Y. S.

J. R. Salgueiro and Y. S. Kivshar, “Nonlinear plasmonic directional couplers,” Appl. Phys. Lett. 97(8), 081106 (2010).
[CrossRef]

A. R. Davoyan, I. V. Shadrivov, A. A. Zharov, D. K. Gramotnev, and Y. S. Kivshar, “Nonlinear nanofocusing in tapered plasmonic waveguides,” Phys. Rev. Lett. 105(11), 116804 (2010).
[CrossRef] [PubMed]

A. R. Davoyan, I. V. Shadrivov, and Y. S. Kivshar, “Quadratic phase matching in nonlinear plasmonic nanoscale waveguides,” Opt. Express 17(22), 20063–20068 (2009).
[CrossRef] [PubMed]

A. R. Davoyan, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear plasmonic slot waveguides,” Opt. Express 16(26), 21209–21214 (2008).
[CrossRef] [PubMed]

Koshiba, M.

Kuo, C. W.

Langbein, U.

F. Lederer, U. Langbein, and H. E. Ponath, “Nonlinear waves guided by a dielectric slab: I. TE polarization,” Appl. Phys. B 31(2), 69–73 (1983).
[CrossRef]

F. Lederer, U. Langbein, and H. E. Ponath, “Nonlinear waves guided by a dielectric slab: II. TM–polarization,” Appl. Phys. B 31(3), 187–190 (1983).
[CrossRef]

Lederer, F.

F. Lederer, U. Langbein, and H. E. Ponath, “Nonlinear waves guided by a dielectric slab: I. TE polarization,” Appl. Phys. B 31(2), 69–73 (1983).
[CrossRef]

F. Lederer, U. Langbein, and H. E. Ponath, “Nonlinear waves guided by a dielectric slab: II. TM–polarization,” Appl. Phys. B 31(3), 187–190 (1983).
[CrossRef]

Li, Q. Y.

Q. Y. Li, R. A. Sammut, and C. Pask, “Variational and finite element analyses of nonlinear strip optical waveguides,” Opt. Commun. 94(1-3), 37–43 (1992).
[CrossRef]

Ma, J. G.

J. G. Ma and Z. Chen, “Numerically determining the dispersion relations of nonlinear TE slab-guided waves in non-Kerr-like media,” IEEE Trans. Microw. Theory Tech. 45(7), 1113–1117 (1997).
[CrossRef]

Maradudin, A. A.

A. D. Boardman, A. A. Maradudin, G. I. Stegeman, T. Twardowski, and E. M. Wright, “Exact theory of nonlinear p-polarized optical waves,” Phys. Rev. A 35(3), 1159–1164 (1987).
[CrossRef] [PubMed]

G. I. Stegeman, E. M. Wright, C. T. Seaton, J. V. Moloney, T. P. Shen, A. A. Maradudin, and R. F. Wallis, “Nonlinear slab-guided waves in non-Kerr-like media,” IEEE J. Quantum Electron. 22(6), 977–983 (1986).
[CrossRef]

J. Ariyasu, C. T. Seaton, G. I. Stegeman, A. A. Maradudin, and R. F. Wallis, “Nonlinear surface polaritons guided by meal films,” J. Appl. Phys. 58(7), 2460–2466 (1985).
[CrossRef]

Moloney, J. V.

G. I. Stegeman, E. M. Wright, C. T. Seaton, J. V. Moloney, T. P. Shen, A. A. Maradudin, and R. F. Wallis, “Nonlinear slab-guided waves in non-Kerr-like media,” IEEE J. Quantum Electron. 22(6), 977–983 (1986).
[CrossRef]

Nagai, M.

K. Hayata, M. Nagai, and M. Koshiba, “Finite-element formalism for nonlinear slab-guided waves,” IEEE Trans. Microw. Theory Tech. 36(7), 1207–1215 (1988).
[CrossRef]

Obayya, S. S. A.

S. S. A. Obayya, B. M. A. Rahman, K. T. V. Grattan, and H. A. El-Mikati, “Full vectorial finite–element solution of nonlinear bistable optical waveguides,’,” IEEE J. Quantum Electron. 38(8), 1120–1125 (2002).
[CrossRef]

Ogusu, K.

K. Ogusu, “TM waves guided by nonlinear planar waveguides,” IEEE Trans. Microw. Theory Tech. 37(6), 941–946 (1989).
[CrossRef]

Pannipitiya, A.

I. D. Rukhlenko, A. Pannipitiya, and M. Premaratne, “Dispersion relation for surface plasmon polaritons in metal/nonlinear-dielectric/metal slot waveguides,” Opt. Lett. 36(17), 3374–3376 (2011).
[CrossRef] [PubMed]

I. D. Rukhlenko, A. Pannipitiya, M. Premaratne, and G. Agrawal, “Exact dispersion relation for nonlinear plasmonic waveguides,” Phys. Rev. B 84(11), 113409 (2011).
[CrossRef]

Pask, C.

Q. Y. Li, R. A. Sammut, and C. Pask, “Variational and finite element analyses of nonlinear strip optical waveguides,” Opt. Commun. 94(1-3), 37–43 (1992).
[CrossRef]

Ponath, H. E.

F. Lederer, U. Langbein, and H. E. Ponath, “Nonlinear waves guided by a dielectric slab: I. TE polarization,” Appl. Phys. B 31(2), 69–73 (1983).
[CrossRef]

F. Lederer, U. Langbein, and H. E. Ponath, “Nonlinear waves guided by a dielectric slab: II. TM–polarization,” Appl. Phys. B 31(3), 187–190 (1983).
[CrossRef]

Premaratne, M.

I. D. Rukhlenko, A. Pannipitiya, M. Premaratne, and G. Agrawal, “Exact dispersion relation for nonlinear plasmonic waveguides,” Phys. Rev. B 84(11), 113409 (2011).
[CrossRef]

I. D. Rukhlenko, A. Pannipitiya, and M. Premaratne, “Dispersion relation for surface plasmon polaritons in metal/nonlinear-dielectric/metal slot waveguides,” Opt. Lett. 36(17), 3374–3376 (2011).
[CrossRef] [PubMed]

Radic, S.

Rahman, B. M. A.

S. S. A. Obayya, B. M. A. Rahman, K. T. V. Grattan, and H. A. El-Mikati, “Full vectorial finite–element solution of nonlinear bistable optical waveguides,’,” IEEE J. Quantum Electron. 38(8), 1120–1125 (2002).
[CrossRef]

R. D. Ettinger, F. A. Fernandez, B. M. A. Rahman, and J. D. Davies, “Vector finite element solution of saturable nonlinear strip-loaded optical waveguides,” IEEE Photon. Technol. Lett. 3(2), 147–149 (1991).
[CrossRef]

B. M. A. Rahman, J. R. Souza, and J. B. Davies, “Numerical analysis of nonlinear bistable optical waveguides,” IEEE Photon. Technol. Lett. 2(4), 265–267 (1990).
[CrossRef]

Ramadas, M. R.

M. R. Ramadas, R. K. Varshney, K. Thyagarajan, and A. K. Ghatak, “A matrix approach to study the propagation characteristics of a general nonlinear planar waveguide,” J. Lightwave Technol. 7(12), 1901–1905 (1989).
[CrossRef]

Robbins, D. J.

D. J. Robbins, “TE modes in a slab waveguide bounded by nonlinear media,” Opt. Commun. 47(5), 309–312 (1983).
[CrossRef]

Roy, S.

S. Roy and P. Roy Chaudhuri, “Analysis of nonlinear multilayered waveguides and MQW structures: a field evolution approach using finite–difference formulation,” IEEE J. Quantum Electron. 45(4), 345–350 (2009).
[CrossRef]

Roy Chaudhuri, P.

S. Roy and P. Roy Chaudhuri, “Analysis of nonlinear multilayered waveguides and MQW structures: a field evolution approach using finite–difference formulation,” IEEE J. Quantum Electron. 45(4), 345–350 (2009).
[CrossRef]

Rozzi, T.

T. Rozzi and L. Zappelli, “Modal analysis of nonlinear propagation in dielectric slab waveguide,” J. Lightwave Technol. 14(2), 229–235 (1996).
[CrossRef]

Rukhlenko, I. D.

I. D. Rukhlenko, A. Pannipitiya, M. Premaratne, and G. Agrawal, “Exact dispersion relation for nonlinear plasmonic waveguides,” Phys. Rev. B 84(11), 113409 (2011).
[CrossRef]

I. D. Rukhlenko, A. Pannipitiya, and M. Premaratne, “Dispersion relation for surface plasmon polaritons in metal/nonlinear-dielectric/metal slot waveguides,” Opt. Lett. 36(17), 3374–3376 (2011).
[CrossRef] [PubMed]

Salgueiro, J. R.

J. R. Salgueiro and Y. S. Kivshar, “Nonlinear plasmonic directional couplers,” Appl. Phys. Lett. 97(8), 081106 (2010).
[CrossRef]

Sammut, R. A.

Q. Y. Li, R. A. Sammut, and C. Pask, “Variational and finite element analyses of nonlinear strip optical waveguides,” Opt. Commun. 94(1-3), 37–43 (1992).
[CrossRef]

Seaton, C. T.

G. I. Stegeman, E. M. Wright, C. T. Seaton, J. V. Moloney, T. P. Shen, A. A. Maradudin, and R. F. Wallis, “Nonlinear slab-guided waves in non-Kerr-like media,” IEEE J. Quantum Electron. 22(6), 977–983 (1986).
[CrossRef]

C. T. Seaton, J. D. Valera, B. Svenson, and G. I. Stegeman, “Comparison of solutions for TM-polarized nonlinear guided waves,” Opt. Lett. 10(3), 149–150 (1985).
[CrossRef] [PubMed]

C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. Chilwell, and S. D. Smith, “Calculations of nonlinear TE waves guided by thin dielectric films bounded by nonlinear media,” IEEE J. Quantum Electron. 21(7), 774–783 (1985).
[CrossRef]

J. Ariyasu, C. T. Seaton, G. I. Stegeman, A. A. Maradudin, and R. F. Wallis, “Nonlinear surface polaritons guided by meal films,” J. Appl. Phys. 58(7), 2460–2466 (1985).
[CrossRef]

H. Vach, C. T. Seaton, G. I. Stegeman, and I. C. Khoo, “Observation of intensity-dependent guided waves,” Opt. Lett. 9, 238–240 (1984).
[CrossRef] [PubMed]

G. I. Stegeman, C. T. Seaton, J. Chilwell, and S. D. Smith, “Nonlinear waves guided by thin films,” Appl. Phys. Lett. 44(9), 830–832 (1984).
[CrossRef]

G. I. Stegeman and C. T. Seaton, “Nonlinear surface plasmons guided by thin metal films,” Opt. Lett. 9(6), 235–237 (1984).
[CrossRef] [PubMed]

Shadrivov, I. V.

Shen, T. P.

G. I. Stegeman, E. M. Wright, C. T. Seaton, J. V. Moloney, T. P. Shen, A. A. Maradudin, and R. F. Wallis, “Nonlinear slab-guided waves in non-Kerr-like media,” IEEE J. Quantum Electron. 22(6), 977–983 (1986).
[CrossRef]

Shoemaker, R. L.

C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. Chilwell, and S. D. Smith, “Calculations of nonlinear TE waves guided by thin dielectric films bounded by nonlinear media,” IEEE J. Quantum Electron. 21(7), 774–783 (1985).
[CrossRef]

Smith, D. R.

A. Degiron and D. R. Smith, “Nonlinear long-range plasmonic waveguides,” Phys. Rev. A 82(3), 033812 (2010).
[CrossRef]

Smith, S. D.

C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. Chilwell, and S. D. Smith, “Calculations of nonlinear TE waves guided by thin dielectric films bounded by nonlinear media,” IEEE J. Quantum Electron. 21(7), 774–783 (1985).
[CrossRef]

G. I. Stegeman, C. T. Seaton, J. Chilwell, and S. D. Smith, “Nonlinear waves guided by thin films,” Appl. Phys. Lett. 44(9), 830–832 (1984).
[CrossRef]

Souza, J. R.

B. M. A. Rahman, J. R. Souza, and J. B. Davies, “Numerical analysis of nonlinear bistable optical waveguides,” IEEE Photon. Technol. Lett. 2(4), 265–267 (1990).
[CrossRef]

Stegeman, G. I.

A. D. Boardman, A. A. Maradudin, G. I. Stegeman, T. Twardowski, and E. M. Wright, “Exact theory of nonlinear p-polarized optical waves,” Phys. Rev. A 35(3), 1159–1164 (1987).
[CrossRef] [PubMed]

G. I. Stegeman, E. M. Wright, C. T. Seaton, J. V. Moloney, T. P. Shen, A. A. Maradudin, and R. F. Wallis, “Nonlinear slab-guided waves in non-Kerr-like media,” IEEE J. Quantum Electron. 22(6), 977–983 (1986).
[CrossRef]

C. T. Seaton, J. D. Valera, B. Svenson, and G. I. Stegeman, “Comparison of solutions for TM-polarized nonlinear guided waves,” Opt. Lett. 10(3), 149–150 (1985).
[CrossRef] [PubMed]

C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. Chilwell, and S. D. Smith, “Calculations of nonlinear TE waves guided by thin dielectric films bounded by nonlinear media,” IEEE J. Quantum Electron. 21(7), 774–783 (1985).
[CrossRef]

J. Ariyasu, C. T. Seaton, G. I. Stegeman, A. A. Maradudin, and R. F. Wallis, “Nonlinear surface polaritons guided by meal films,” J. Appl. Phys. 58(7), 2460–2466 (1985).
[CrossRef]

H. Vach, C. T. Seaton, G. I. Stegeman, and I. C. Khoo, “Observation of intensity-dependent guided waves,” Opt. Lett. 9, 238–240 (1984).
[CrossRef] [PubMed]

G. I. Stegeman, C. T. Seaton, J. Chilwell, and S. D. Smith, “Nonlinear waves guided by thin films,” Appl. Phys. Lett. 44(9), 830–832 (1984).
[CrossRef]

G. I. Stegeman and C. T. Seaton, “Nonlinear surface plasmons guided by thin metal films,” Opt. Lett. 9(6), 235–237 (1984).
[CrossRef] [PubMed]

Stewart, W. J.

I. Bennion, M. J. Goodwin, and W. J. Stewart, “Experimental nonlinear optical waveguide device,” Electron. Lett. 21(1), 41–42 (1985).
[CrossRef]

Sun, X. H.

J. B. Xiao and X. H. Sun, “Full-vectorial mode solver for anisotropic optical waveguides using multidomain spectral collocation method,” Opt. Commun. 283(14), 2835–2840 (2010).
[CrossRef]

Svenson, B.

Teng, C.-H.

P.-J. Chiang, C.-L. Wu, C.-H. Teng, C.-S. Yang, and H.- Chang, “Full-vectorial optical waveguide mode solvers using multidomain pseudospectral frequency-domain (PSFD) formulations,” IEEE J. Quantum Electron. 44(1), 56–66 (2008).
[CrossRef]

Thyagarajan, K.

M. R. Ramadas, R. K. Varshney, K. Thyagarajan, and A. K. Ghatak, “A matrix approach to study the propagation characteristics of a general nonlinear planar waveguide,” J. Lightwave Technol. 7(12), 1901–1905 (1989).
[CrossRef]

Twardowski, T.

A. D. Boardman, A. A. Maradudin, G. I. Stegeman, T. Twardowski, and E. M. Wright, “Exact theory of nonlinear p-polarized optical waves,” Phys. Rev. A 35(3), 1159–1164 (1987).
[CrossRef] [PubMed]

Vach, H.

Valera, J. D.

C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. Chilwell, and S. D. Smith, “Calculations of nonlinear TE waves guided by thin dielectric films bounded by nonlinear media,” IEEE J. Quantum Electron. 21(7), 774–783 (1985).
[CrossRef]

C. T. Seaton, J. D. Valera, B. Svenson, and G. I. Stegeman, “Comparison of solutions for TM-polarized nonlinear guided waves,” Opt. Lett. 10(3), 149–150 (1985).
[CrossRef] [PubMed]

Varshney, R. K.

R. K. Varshney, I. C. Goyal, and A. K. Ghatak, “A simple and efficient numerical method to study propagation characteristics of nonlinear optical waveguides,” J. Lightwave Technol. 16(4), 697–702 (1998).
[CrossRef]

M. R. Ramadas, R. K. Varshney, K. Thyagarajan, and A. K. Ghatak, “A matrix approach to study the propagation characteristics of a general nonlinear planar waveguide,” J. Lightwave Technol. 7(12), 1901–1905 (1989).
[CrossRef]

Wallis, R. F.

G. I. Stegeman, E. M. Wright, C. T. Seaton, J. V. Moloney, T. P. Shen, A. A. Maradudin, and R. F. Wallis, “Nonlinear slab-guided waves in non-Kerr-like media,” IEEE J. Quantum Electron. 22(6), 977–983 (1986).
[CrossRef]

J. Ariyasu, C. T. Seaton, G. I. Stegeman, A. A. Maradudin, and R. F. Wallis, “Nonlinear surface polaritons guided by meal films,” J. Appl. Phys. 58(7), 2460–2466 (1985).
[CrossRef]

Wang, X. H.

Wood, T. H.

T. H. Wood, “Multiple quantum well (MQW) waveguide modulators,” J. Lightwave Technol. 6(6), 743–757 (1988).
[CrossRef]

Wright, E. M.

A. D. Boardman, A. A. Maradudin, G. I. Stegeman, T. Twardowski, and E. M. Wright, “Exact theory of nonlinear p-polarized optical waves,” Phys. Rev. A 35(3), 1159–1164 (1987).
[CrossRef] [PubMed]

G. I. Stegeman, E. M. Wright, C. T. Seaton, J. V. Moloney, T. P. Shen, A. A. Maradudin, and R. F. Wallis, “Nonlinear slab-guided waves in non-Kerr-like media,” IEEE J. Quantum Electron. 22(6), 977–983 (1986).
[CrossRef]

Wu, C.-L.

P.-J. Chiang, C.-L. Wu, C.-H. Teng, C.-S. Yang, and H.- Chang, “Full-vectorial optical waveguide mode solvers using multidomain pseudospectral frequency-domain (PSFD) formulations,” IEEE J. Quantum Electron. 44(1), 56–66 (2008).
[CrossRef]

Wu, Y. D.

Xiao, J. B.

J. B. Xiao and X. H. Sun, “Full-vectorial mode solver for anisotropic optical waveguides using multidomain spectral collocation method,” Opt. Commun. 283(14), 2835–2840 (2010).
[CrossRef]

Yang, C.-S.

P.-J. Chiang, C.-L. Wu, C.-H. Teng, C.-S. Yang, and H.- Chang, “Full-vectorial optical waveguide mode solvers using multidomain pseudospectral frequency-domain (PSFD) formulations,” IEEE J. Quantum Electron. 44(1), 56–66 (2008).
[CrossRef]

Yang, J.-Y.

C.-C. Huang, C.-C. Huang, and J.-Y. Yang, “A full-vectorial pseudospectral modal analysis of dielectric optical waveguides with stepped refractive index profiles,” IEEE J. Sel. Top. Quantum Electron. 11(2), 457–465 (2005).
[CrossRef]

Zappelli, L.

T. Rozzi and L. Zappelli, “Modal analysis of nonlinear propagation in dielectric slab waveguide,” J. Lightwave Technol. 14(2), 229–235 (1996).
[CrossRef]

Zhao, A. P.

A. P. Zhao and S. R. Cvetkovic, “Finite-element solution of nonlinear TM waves in multiple-quantum-well waveguides,” IEEE Photon. Technol. Lett. 4(11), 1231–1234 (1992).
[CrossRef]

Zharov, A. A.

A. R. Davoyan, I. V. Shadrivov, A. A. Zharov, D. K. Gramotnev, and Y. S. Kivshar, “Nonlinear nanofocusing in tapered plasmonic waveguides,” Phys. Rev. Lett. 105(11), 116804 (2010).
[CrossRef] [PubMed]

Appl. Phys. B (2)

F. Lederer, U. Langbein, and H. E. Ponath, “Nonlinear waves guided by a dielectric slab: I. TE polarization,” Appl. Phys. B 31(2), 69–73 (1983).
[CrossRef]

F. Lederer, U. Langbein, and H. E. Ponath, “Nonlinear waves guided by a dielectric slab: II. TM–polarization,” Appl. Phys. B 31(3), 187–190 (1983).
[CrossRef]

Appl. Phys. Lett. (2)

G. I. Stegeman, C. T. Seaton, J. Chilwell, and S. D. Smith, “Nonlinear waves guided by thin films,” Appl. Phys. Lett. 44(9), 830–832 (1984).
[CrossRef]

J. R. Salgueiro and Y. S. Kivshar, “Nonlinear plasmonic directional couplers,” Appl. Phys. Lett. 97(8), 081106 (2010).
[CrossRef]

Electron. Lett. (1)

I. Bennion, M. J. Goodwin, and W. J. Stewart, “Experimental nonlinear optical waveguide device,” Electron. Lett. 21(1), 41–42 (1985).
[CrossRef]

IEEE J. Quantum Electron. (5)

P.-J. Chiang, C.-L. Wu, C.-H. Teng, C.-S. Yang, and H.- Chang, “Full-vectorial optical waveguide mode solvers using multidomain pseudospectral frequency-domain (PSFD) formulations,” IEEE J. Quantum Electron. 44(1), 56–66 (2008).
[CrossRef]

C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. Chilwell, and S. D. Smith, “Calculations of nonlinear TE waves guided by thin dielectric films bounded by nonlinear media,” IEEE J. Quantum Electron. 21(7), 774–783 (1985).
[CrossRef]

G. I. Stegeman, E. M. Wright, C. T. Seaton, J. V. Moloney, T. P. Shen, A. A. Maradudin, and R. F. Wallis, “Nonlinear slab-guided waves in non-Kerr-like media,” IEEE J. Quantum Electron. 22(6), 977–983 (1986).
[CrossRef]

S. S. A. Obayya, B. M. A. Rahman, K. T. V. Grattan, and H. A. El-Mikati, “Full vectorial finite–element solution of nonlinear bistable optical waveguides,’,” IEEE J. Quantum Electron. 38(8), 1120–1125 (2002).
[CrossRef]

S. Roy and P. Roy Chaudhuri, “Analysis of nonlinear multilayered waveguides and MQW structures: a field evolution approach using finite–difference formulation,” IEEE J. Quantum Electron. 45(4), 345–350 (2009).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

C.-C. Huang, C.-C. Huang, and J.-Y. Yang, “A full-vectorial pseudospectral modal analysis of dielectric optical waveguides with stepped refractive index profiles,” IEEE J. Sel. Top. Quantum Electron. 11(2), 457–465 (2005).
[CrossRef]

IEEE Photon. Technol. Lett. (3)

B. M. A. Rahman, J. R. Souza, and J. B. Davies, “Numerical analysis of nonlinear bistable optical waveguides,” IEEE Photon. Technol. Lett. 2(4), 265–267 (1990).
[CrossRef]

R. D. Ettinger, F. A. Fernandez, B. M. A. Rahman, and J. D. Davies, “Vector finite element solution of saturable nonlinear strip-loaded optical waveguides,” IEEE Photon. Technol. Lett. 3(2), 147–149 (1991).
[CrossRef]

A. P. Zhao and S. R. Cvetkovic, “Finite-element solution of nonlinear TM waves in multiple-quantum-well waveguides,” IEEE Photon. Technol. Lett. 4(11), 1231–1234 (1992).
[CrossRef]

IEEE Trans. Microw. Theory Tech. (3)

K. Hayata, M. Nagai, and M. Koshiba, “Finite-element formalism for nonlinear slab-guided waves,” IEEE Trans. Microw. Theory Tech. 36(7), 1207–1215 (1988).
[CrossRef]

K. Ogusu, “TM waves guided by nonlinear planar waveguides,” IEEE Trans. Microw. Theory Tech. 37(6), 941–946 (1989).
[CrossRef]

J. G. Ma and Z. Chen, “Numerically determining the dispersion relations of nonlinear TE slab-guided waves in non-Kerr-like media,” IEEE Trans. Microw. Theory Tech. 45(7), 1113–1117 (1997).
[CrossRef]

J. Appl. Phys. (1)

J. Ariyasu, C. T. Seaton, G. I. Stegeman, A. A. Maradudin, and R. F. Wallis, “Nonlinear surface polaritons guided by meal films,” J. Appl. Phys. 58(7), 2460–2466 (1985).
[CrossRef]

J. Lightwave Technol. (4)

R. K. Varshney, I. C. Goyal, and A. K. Ghatak, “A simple and efficient numerical method to study propagation characteristics of nonlinear optical waveguides,” J. Lightwave Technol. 16(4), 697–702 (1998).
[CrossRef]

M. R. Ramadas, R. K. Varshney, K. Thyagarajan, and A. K. Ghatak, “A matrix approach to study the propagation characteristics of a general nonlinear planar waveguide,” J. Lightwave Technol. 7(12), 1901–1905 (1989).
[CrossRef]

T. Rozzi and L. Zappelli, “Modal analysis of nonlinear propagation in dielectric slab waveguide,” J. Lightwave Technol. 14(2), 229–235 (1996).
[CrossRef]

T. H. Wood, “Multiple quantum well (MQW) waveguide modulators,” J. Lightwave Technol. 6(6), 743–757 (1988).
[CrossRef]

J. Opt. Soc. Am. B (3)

Opt. Commun. (3)

J. B. Xiao and X. H. Sun, “Full-vectorial mode solver for anisotropic optical waveguides using multidomain spectral collocation method,” Opt. Commun. 283(14), 2835–2840 (2010).
[CrossRef]

Q. Y. Li, R. A. Sammut, and C. Pask, “Variational and finite element analyses of nonlinear strip optical waveguides,” Opt. Commun. 94(1-3), 37–43 (1992).
[CrossRef]

D. J. Robbins, “TE modes in a slab waveguide bounded by nonlinear media,” Opt. Commun. 47(5), 309–312 (1983).
[CrossRef]

Opt. Express (7)

Opt. Lett. (6)

Phys. Rev. A (2)

A. Degiron and D. R. Smith, “Nonlinear long-range plasmonic waveguides,” Phys. Rev. A 82(3), 033812 (2010).
[CrossRef]

A. D. Boardman, A. A. Maradudin, G. I. Stegeman, T. Twardowski, and E. M. Wright, “Exact theory of nonlinear p-polarized optical waves,” Phys. Rev. A 35(3), 1159–1164 (1987).
[CrossRef] [PubMed]

Phys. Rev. B (2)

I. D. Rukhlenko, A. Pannipitiya, M. Premaratne, and G. Agrawal, “Exact dispersion relation for nonlinear plasmonic waveguides,” Phys. Rev. B 84(11), 113409 (2011).
[CrossRef]

P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: bound modes of asymmetric structures,” Phys. Rev. B 63(12), 125417 (2001).
[CrossRef]

Phys. Rev. Lett. (1)

A. R. Davoyan, I. V. Shadrivov, A. A. Zharov, D. K. Gramotnev, and Y. S. Kivshar, “Nonlinear nanofocusing in tapered plasmonic waveguides,” Phys. Rev. Lett. 105(11), 116804 (2010).
[CrossRef] [PubMed]

Other (1)

J. P. Boyd, Chebyshev and Fourier Spectral Methods (Springer–Verlag, 2001).

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Figures (13)

Fig. 1
Fig. 1

Schematic diagram of a three-layer dielectric waveguide with a Kerr-like nonlinear cladding.

Fig. 2
Fig. 2

Convergence of effective index versus (a) iteration time and (b) number of terms of the basis function.

Fig. 3
Fig. 3

Effective index versus input power for both the ne- and P-based approaches.

Fig. 4
Fig. 4

Mode profiles for the input powers P = 111 mW/mm < Pth (blue curve labeled A) and P = 112 mW/mm > Pth (green curve labeled B).

Fig. 5
Fig. 5

Guided power versus effective index for the fundamental TE mode of a three-layered dielectric nonlinear waveguide.

Fig. 6
Fig. 6

Guided power versus effective index for the long-range and short-range modes of a thin metal film surrounded by a Kerr-like nonlinear cladding and a linear substrate.

Fig. 7
Fig. 7

Relative magnetic profiles of the LR-SPP mode for the guided powers: (a) P = 0, (b) P = 71.8, and (c) P = 981.2 mW/mm.

Fig. 8
Fig. 8

Relative magnetic profiles of the SR-SPP mode for the input powers: (a) P = 0, (b) P = 19.7, and (c) P = 30.9 mW/mm.

Fig. 9
Fig. 9

Guided power versus effective index for the LR-SPP modes of a thin metal film with different thicknesses surrounded by a Kerr-like nonlinear cladding and a linear substrate. The power scales of the thicknesses, d = 20nm, 30nm, and 50nm, have been magnified by x10.

Fig. 10
Fig. 10

Relative magnetic profiles |Hx| of the LR-SPP mode at different thicknesses for the input powers: (a) P = 0 mW/mm for all thicknesses and (b) P = 110 (d = 20nm), P = 194.3 (d = 30nm), P = 981.2 (d = 50nm), and P = 15092.5 (d = 80nm) mW/mm.

Fig. 11
Fig. 11

Schematics showing (a) the cross section of a nonlinear strip waveguide with a nonlinear substrate and (b) the division of its computational domain.

Fig. 12
Fig. 12

Power dispersion curve as a function of the effective index of the strip waveguide.

Fig. 13
Fig. 13

Mode contours for input powers (a) P = 82 μW (<Pth) and (b) P = 86 μW (>Pth) for the 2D strip waveguide with a nonlinear substrate.

Equations (43)

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×( [ε] 1 ×H) ω 2 μ 0 H=0,
[ε]= ε 0 [ ε r ]= ε 0 [ ε ˜ x 0 0 0 ε ˜ y 0 0 0 ε ˜ z ],
d 2 H y d y 2 + k 0 2 ( ε ˜ x n e 2 ) H y =0,
ε ˜ x = ε x +af( E x ),
a= c 0 ε 0 ε x n ¯ ,
P= 1 2 E x H y * dy= Z 0 β 2 k 0 1 ε ˜ x | H y | 2 dy,
E x = Z 0 β k 0 ε ˜ x H y .
d dy ( 1 ε ˜ z d H x dy )+( k 0 2 β 2 ε ˜ y ) H x =0,
ε ˜ y = ε y +ag( E y )+bh( E z ),
ε ˜ z = ε z +bg( E y )+ah( E z ),
P= 1 2 E y H x * dy= Z 0 β 2 k 0 1 ε ˜ y | H x | 2 dy.
E y = Z 0 β k 0 ε ˜ y H x ,
E z = j Z 0 ε ˜ z d H x dy .
2 ξ x 2 + 2 ξ y 2 + k 0 2 ( ε ˜ n e 2 )ξ=0,
ε ˜ =ε+cf(ξ),
P= β 2 Z 0 k 0 | ξ(x,y) | 2 dxdy .
ψ(y)= j=0 n θ j (y) ψ j ,
θ j (y)= ρ n+1 (y) ρ n+1 ' (y)(y y j ) ,0in.
θ j (y)= (1) j+1 (1 y 2 ) T n ' (y) c j n 2 (y y j ) ,y y j .
θ j (αy)= e αy/2 e α y j /2 (αy) L n (αy) (αy L n ) ' (α y j )(αyα y j ) ,y y j .
D{ ψ ¯ }= ( k 0 n e ) 2 { ψ ¯ },
| ψ ¯ (y) | 2 dy=1,
D ij = θ j (2) ( y i )+ k 0 2 ε ˜ x ( y i ) δ ij ,
D ij = ε ˜ y ( y i ) ε ˜ z ( y i ) θ j (2) ( y i ) ε ˜ y ( y i ) ε ˜ z 2 ( y i ) ( d ε ˜ z (y) dy | y= y i ) θ j (1) ( y i )+ k 0 2 ε ˜ y ( y i ) δ ij .
[ D 1 0 0 0 0 D 2 0 0 0 0 0 0 0 0 D t ][ ψ ¯ 1 ψ ¯ 2 ψ ¯ t ]= ( k 0 n e ) 2 [ ψ ¯ 1 ψ ¯ 2 ψ ¯ t ].
ψ ¯ j ( y r + )= ψ ¯ j ( y r ), ψ ¯ j (1) ( y r + )= ψ ¯ j (1) ( y r ),
ψ ¯ j ( y r + )= ψ ¯ j ( y r ), ε ˜ z ( y r ) ψ ¯ j (1) ( y r + )= ε ˜ z ( y r + ) ψ ¯ j (1) ( y r ),
d 2 H y d y 2 + k 0 2 ( ε x n e 2 ) H y = k 0 2 af( E x ) H y ,
D ˜ { H ¯ y }=P B ˜ { H ¯ y }.
D ˜ ij = θ j (2) ( y i )+ k 0 2 ( ε x ( y i ) n e 2 ) δ ij ,
B ˜ ij = k 0 2 a P f( E x ( y i )) δ ij .
ξ(x,y)= i=0 n x j=0 n y φ i (x) θ j (y) ξ i,j ,
A{ ξ ¯ }= ( k 0 n e ) 2 { ξ ¯ },
| ξ ¯ (x,y) | 2 dxdy=1.
A= i=0 n x j=0 n y [ 2 x 2 + 2 y 2 + k 0 2 ε ˜ ] x = x i ,y= y j = i=0 n x j=0 n y [ p=0 n x q=0 n y { φ p (2) (x) θ q (y) + φ p (x) θ q (2) (y)+ k 0 2 ε ˜ (x,y) φ p (x) θ q (y)}] | x= x i ,y= y j ,
[ A 1 0 0 0 0 A 2 0 0 0 0 0 0 0 0 A t ][ ξ ¯ 1 ξ ¯ 2 ξ ¯ t ]= ( k 0 n e ) 2 [ ξ ¯ 1 ξ ¯ 2 ξ ¯ t ].
ξ ¯ ( y r + )= ξ ¯ ( y r ), ξ ¯ ( y r + )/y= ξ ¯ ( y r )/y,
ξ ¯ ( x r + )= ξ ¯ ( x r ), ξ ¯ ( x r + )/x= ξ ¯ ( x r )/x,
2 ξ x 2 + 2 ξ y 2 + k 0 2 (ε n e 2 )ξ= k 0 2 cf(ξ)ξ,
A ˜ { ξ ¯ }=P Q ˜ { ξ ¯ },
A ˜ = i=0 n x j=0 n y [ 2 x 2 + 2 y 2 + k 0 2 ε ] x = x i ,y= y j = i=0 n x j=0 n y [ p=0 n x q=0 n y { φ p (2) (x) ϕ q (y) + φ p (x) ϕ q (2) (y)+ k 0 2 ε(x,y) φ p (x) ϕ q (y)}] | x= x i ,y= y j
Q ˜ = i=0 n x j=0 n y [ k 0 2 c P f( ξ ¯ ( x i , y j )) ] δ ij .
ε ˜ = n s 2 +2 n s Δ n sat [ 1exp( a | ψ | 2 2 n s Δ n sat ) ],

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