Abstract

A full-vectorial pseudospectral method is reported for solving the mode characteristics of nonlinear dielectric and plasmonic waveguides. The coupled equations are formulated in terms of transverse magnetic-field components, and self-consistent solutions are obtained through an iterative procedure. The proposed scheme applies in a saturable medium with biaxial anisotropy of practical interest. The accuracy and efficiency of this scheme are demonstrated by solving the mode bistability of a nonlinear dielectric optical waveguide, analyzed by the well-known finite-element-method-based imaginary-distance beam propagation method. Furthermore, the relationship between geometry and input power is studied by analyzing the power dispersion curve of the long-range surface plasmon polariton modes of a nonlinear plasmonic waveguide.

© 2012 OSA

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    [CrossRef] [PubMed]
  7. I. D. Rukhlenko, A. Pannipitiya, M. Premaratne, and G. Agrawal, “Exact dispersion relation for nonlinear plasmonic waveguides,” Phys. Rev. B84(11), 113409 (2011).
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  8. 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).
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  9. J. R. Salgueiro and Y. S. Kivshar, “Nonlinear plasmonic directional couplers,” Appl. Phys. Lett.97(8), 081106 (2010).
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    [CrossRef]
  18. P. J. Chiang, C. P. Yu, and H. C. Chang, “Full-vectorial optical waveguide mode solvers using multidomain pseudospectral frequency-domain (PSFD) formulations,” IEEE J. Quantum Electron.44(1), 56–66 (2008).
    [CrossRef]
  19. C. C. Huang, “Simulation of optical waveguides by novel full-vectorial pseudospectral-based imaginary-distance beam propagation method,” Opt. Express16(22), 17915–17934 (2008).
    [CrossRef] [PubMed]
  20. 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]
  21. C. C. Huang, “Numerical investigation of mode characteristics of nanoscale surface plasmon-polaritons using a pseudospectral scheme,” Opt. Express18(23), 23711–23726 (2010).
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  28. S. J. Al-Bader, “Optical transmission on metallic wires-fundamental modes,” IEEE J. Quantum Electron.40(3), 325–329 (2004).
    [CrossRef]
  29. A. Degiron and D. R. Smith, “Numerical simulations of long-range plasmons,” Opt. Express14(4), 1611–1625 (2006).
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    [CrossRef]
  31. G. I. Stegeman and C. T. Seaton, “Nonlinear surface plasmons guided by thin metal films,” Opt. Lett.9(6), 235–237 (1984).
    [CrossRef] [PubMed]
  32. 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]
  33. R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation,” Nat. Photonics2(8), 496–500 (2008).
    [CrossRef]

2012

2011

2010

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]

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

A. Degiron and D. R. Smith, “Nonlinear long-range plasmonic waveguides,” Phys. Rev. A82(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]

C. C. Huang, “Numerical investigation of mode characteristics of nanoscale surface plasmon-polaritons using a pseudospectral scheme,” Opt. Express18(23), 23711–23726 (2010).
[CrossRef] [PubMed]

2009

2008

C. C. Huang, “Simulation of optical waveguides by novel full-vectorial pseudospectral-based imaginary-distance beam propagation method,” Opt. Express16(22), 17915–17934 (2008).
[CrossRef] [PubMed]

A. R. Davoyan, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear plasmonic slot waveguides,” Opt. Express16(26), 21209–21214 (2008).
[CrossRef] [PubMed]

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

R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation,” Nat. Photonics2(8), 496–500 (2008).
[CrossRef]

2006

2005

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

S. J. Al-Bader, “Optical transmission on metallic wires-fundamental modes,” IEEE J. Quantum Electron.40(3), 325–329 (2004).
[CrossRef]

Y. D. Wu, “New all-optical wavelength auto-router based on spatial solitons,” Opt. Express12(18), 4172–4177 (2004).
[CrossRef] [PubMed]

2002

K. Hayata and M. Koshiba, “Full vectorial analysis of nonlinear-optical waveguides,” J. Lightwave Technol.20, 1876–1884 (2002).

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

2001

P. Berini, “Plasmon-polariton modes guided thin lossy metal films of finite width: bound modes of asymmetric structures,” Phys. Rev. B63(12), 125417 (2001).
[CrossRef]

2000

P. Berini, “Plasmon-polariton modes guided by a metal film of finite width bounded by different dielectrics,” Opt. Express7(10), 329–335 (2000).
[CrossRef] [PubMed]

P. Berini, “Plasmon-polariton modes guided thin lossy metal films of finite width: bound modes of symmetric structures,” Phys. Rev. B61(15), 10484–10503 (2000).
[CrossRef]

1999

1994

1993

1991

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

1988

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

K. Hayata and M. Koshiba, “Full vectorial analysis of nonlinear-optical waveguides,” J. Opt. Soc. Am. B5(12), 2494–2501 (1988).
[CrossRef]

1985

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]

1984

Agrawal, G.

I. D. Rukhlenko, A. Pannipitiya, M. Premaratne, and G. Agrawal, “Exact dispersion relation for nonlinear plasmonic waveguides,” Phys. Rev. B84(11), 113409 (2011).
[CrossRef]

Agrawal, G. P.

Al-Bader, S. J.

S. J. Al-Bader, “Optical transmission on metallic wires-fundamental modes,” IEEE J. Quantum Electron.40(3), 325–329 (2004).
[CrossRef]

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]

Berini, P.

P. Berini, “Long-range surface plasmon polaritons,” Adv. Opt. Photon.1(3), 484–588 (2009).
[CrossRef]

P. Berini, “Plasmon-polariton modes guided thin lossy metal films of finite width: bound modes of asymmetric structures,” Phys. Rev. B63(12), 125417 (2001).
[CrossRef]

P. Berini, “Plasmon-polariton modes guided by a metal film of finite width bounded by different dielectrics,” Opt. Express7(10), 329–335 (2000).
[CrossRef] [PubMed]

P. Berini, “Plasmon-polariton modes guided thin lossy metal films of finite width: bound modes of symmetric structures,” Phys. Rev. B61(15), 10484–10503 (2000).
[CrossRef]

P. Berini, “Plasmon polariton modes guided by a metal film of finite width,” Opt. Lett.24(15), 1011–1013 (1999).
[CrossRef] [PubMed]

Chang, H. C.

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

Chiang, P. J.

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

Davies, J. B.

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

Davoyan, A. R.

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, “Nonlinear plasmonic slot waveguides,” Opt. Express16(26), 21209–21214 (2008).
[CrossRef] [PubMed]

Degiron, A.

A. Degiron and D. R. Smith, “Nonlinear long-range plasmonic waveguides,” Phys. Rev. A82(3), 033812 (2010).
[CrossRef]

A. Degiron and D. R. Smith, “Numerical simulations of long-range plasmons,” Opt. Express14(4), 1611–1625 (2006).
[CrossRef] [PubMed]

Ettinger, R. D.

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

Fujisawa, T.

Gambrell, G. K.

Genov, D. A.

R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation,” Nat. Photonics2(8), 496–500 (2008).
[CrossRef]

George, N.

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. E. Mikati, “Full vectorial finite–element solution of nonlinear bistable optical waveguides,’,” IEEE J. Quantum Electron.38(8), 1120–1125 (2002).
[CrossRef]

Hayata, K.

Huang, C. 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, “Nonlinear plasmonic slot waveguides,” Opt. Express16(26), 21209–21214 (2008).
[CrossRef] [PubMed]

Koshiba, M.

Maradudin, A. A.

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]

Mikati, H. A. E.

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

Obayya, S. S. A.

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

Oulton, R. F.

R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation,” Nat. Photonics2(8), 496–500 (2008).
[CrossRef]

Pannipitiya, A.

I. D. Rukhlenko, A. Pannipitiya, M. Premaratne, and G. Agrawal, “Exact dispersion relation for nonlinear plasmonic waveguides,” Phys. Rev. B84(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]

Pile, D. F. P.

R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation,” Nat. Photonics2(8), 496–500 (2008).
[CrossRef]

Premaratne, M.

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. B84(11), 113409 (2011).
[CrossRef]

Radic, S.

Rahman, B. M. A.

S. S. A. Obayya, B. M. A. Rahman, K. T. V. Grattan, and H. A. E. 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. B. Davies, “Vector finite element solutions of saturable nonlinear strip-loaded optical waveguides,” IEEE Photon. Technol. Lett.3(2), 147–149 (1991).
[CrossRef]

Rukhlenko, I. D.

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. B84(11), 113409 (2011).
[CrossRef]

Salgueiro, J. R.

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

Seaton, C. T.

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]

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.

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, “Nonlinear plasmonic slot waveguides,” Opt. Express16(26), 21209–21214 (2008).
[CrossRef] [PubMed]

Smith, D. R.

A. Degiron and D. R. Smith, “Nonlinear long-range plasmonic waveguides,” Phys. Rev. A82(3), 033812 (2010).
[CrossRef]

A. Degiron and D. R. Smith, “Numerical simulations of long-range plasmons,” Opt. Express14(4), 1611–1625 (2006).
[CrossRef] [PubMed]

Sorger, V. J.

R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation,” Nat. Photonics2(8), 496–500 (2008).
[CrossRef]

Stegeman, G. I.

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]

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

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]

Wallis, R. F.

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]

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, 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]

Yu, C. P.

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

Zhang, X.

R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation,” Nat. Photonics2(8), 496–500 (2008).
[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]

Adv. Opt. Photon.

Appl. Phys. Lett.

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

IEEE J. Quantum Electron.

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

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

S. J. Al-Bader, “Optical transmission on metallic wires-fundamental modes,” IEEE J. Quantum Electron.40(3), 325–329 (2004).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

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.

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

J. Appl. Phys.

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]

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Figures (11)

Fig. 1
Fig. 1

(a) Mesh division of an arbitrary interior subdomain and (b) mesh division of a problem with two subdomains (labeled 1 and 2).

Fig. 2
Fig. 2

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

Fig. 3
Fig. 3

Effective index versus input power for fundamental quasi-TE mode H 11 y .

Fig. 4
Fig. 4

(a) Major magnetic field component (Hy) and (b) minor field component (Hx) of H 11 y mode for an input power of P = 80 μW on the lower branch of the power dispersion curve.

Fig. 5
Fig. 5

(a) Major magnetic field component (Hy) and (b) minor field component (Hx) of H 11 y mode for an input power of P = 80 μW on the upper branch of the power dispersion curve.

Fig. 6
Fig. 6

Real (a) and imaginary (b) parts of the effective index of the fundamental LR–SPP mode as a function of power for width w = 4 μm and various thicknesses.

Fig. 7
Fig. 7

Magnetic-field profiles |Hx| of the fundamental LR–SPP mode for the following guided powers: (a) P = 0, (b) P = 50, and (c) P = 100 μW for thickness t = 80nm and width w = 4 μm.

Fig. 8
Fig. 8

Magnetic-field profiles |Hx| of fundamental LR–SPP mode for guided powers (a) P = 0, (b) P = 50, and (c) P = 100 μW for thickness t = 20 nm and width w = 4 μm.

Fig. 9
Fig. 9

Real (a) and imaginary (b) parts of effective index of fundamental LR–SPP mode as a function of power at the thickness t = 50 nm and different widths.

Fig. 10
Fig. 10

Magnetic-field profiles |Hx| of fundamental LR–SPP mode for guided powers (a) P = 0, (b) P = 50, and (c) P = 100 μW for width w = 2 μm and thickness t = 50 nm.

Fig. 11
Fig. 11

Magnetic-field profiles |Hx| of fundamental LR–SPP mode for guided powers (a) P = 0, (b) P = 50, and (c) P = 100 μW for width w = 8 μm and thickness t = 50 nm.

Equations (36)

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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 ],
ε ˜ i = ε i +( c 0 ε 0 ε i n ¯ )f( E x , E y , E z ),(i=x,y,z),
[ P xx P xy P yx P yy ][ H x H y ]= β 2 [ H x H y ],
P xx = 2 x 2 + ε ˜ y ε ˜ z 2 y 2 + ε ˜ y y ( 1 ε ˜ z ) y + k 0 2 ε ˜ y ,
P xy =(1 ε ˜ y ε ˜ z ) 2 xy ε ˜ y y ( 1 ε ˜ z ) x ,
P yx =(1 ε ˜ x ε ˜ z ) 2 xy ε ˜ x x ( 1 ε ˜ z ) y ,
P yy = ε ˜ x ε ˜ z 2 x 2 + 2 y 2 + ε ˜ x x ( 1 ε ˜ z ) x + k 0 2 ε ˜ x .
H z = 1 jβ ( H x x + H y y ),
E z = j ω ε 0 ε ˜ z ( H x y H y x ).
E x = n e Z0 ε ˜ x [ H y 1 β 2 y ( H x x + H y y ) ],
E y = n e Z 0 ε ˜ y [ H x 1 β 2 x ( H x x + H y y ) ],
E z = j n e Z 0 ε ˜ z [ 1 β y ( H x y H y x ) ],
P= 1 2 ( E x H y * E y H x * ) dxdy = n e Z 0 2 [ 1 ε ˜ y | H x | 2 + 1 ε ˜ x | H y | 2 1 ε ˜ y β 2 x ( H x x + H y y ) H x * 1 ε ˜ x β 2 y ( H x x + H y y ) H y * ]dxdy,
H x (x,y)= p=0 n x q=0 n y θ p (x) ψ q (y) H x,pq ,
H y (x,y)= p=0 n x q=0 n y θ p (x) ψ q (y) H y,pq .
P xx = i=0 n x j=0 n y [ p=0 n x q=0 n y { θ p (2) (x) ψ q (y)+ ε ˜ y ε ˜ z θ p (x) ψ q (2) (y)+ ε ˜ y y ( 1 ε ˜ z ) θ p (x) ψ q (1) (y)+ k 0 2 ε ˜ y θ p (x) ψ q (y) } ] x= x i ,y= y j ,
P xy = i=0 n x j=0 n y [ p=0 n x q=0 n y { (1 ε ˜ y ε ˜ z ) θ p (1) (x) ψ q (1) (y) ε ˜ y y ( 1 ε ˜ z ) θ p (1) (x) ψ q (y) } ] x= x i ,y= y j ,
P yx = i=0 n x j=0 n y [ p=0 n x q=0 n y { (1 ε ˜ x ε ˜ z ) θ p (1) (x) ψ q (1) (y) ε ˜ x x ( 1 ε ˜ z ) θ p (x) ψ q (1) (y) } ] x= x i ,y= y j ,
P yy = i=0 n x j=0 n y [ p=0 n x q=0 n y { ε ˜ x ε ˜ z θ p (2) (x) ψ q (y)+ θ p (x) ψ q (2) (y)+ ε ˜ x x ( 1 ε ˜ z ) θ p (1) (x) ψ q (y) + k 0 2 ε ˜ x θ p (x) ψ q (y) } ] x= x i ,y= y j ,
[ Q 1 0 0 0 0 Q 2 0 0 0 0 0 0 0 0 Q m ][ H 1 H 2 H m ]= β 2 [ H 1 H 2 H m ],
Q r =[ P xx r P xy r P yx r P yy r ], Η r =[ H x r H y r ],(r=1,2,3,,m).
[ R x (1) R y (1) ][ H x (1) H y (1) ]=[ R x (2) R y (2) ][ H x (2) H y (2) ],
R s (1) =[ R s (1) ( ψ 0 ( y 0 (1) )) R s (1) ( ψ 1 ( y 0 (1) )) . . . R s (1) ( ψ n y ( y 0 (1) )) R s (1) ( ψ 0 ( y 1 (1) )) R s (1) ( ψ 1 ( y 1 (1) )) . . . R s (1) ( ψ n y ( y 1 (1) )) . . . . . . . . . . . . . . . . . . R s (1) ( ψ 0 ( y n y (1) )) R s (1) ( ψ 1 ( y n y (1) )) . . . R s (1) ( ψ n y ( y n y (1) )) ],s=x,y,
R x (1) ( ψ i ( y 0 (1) ))=[ θ 0 (1) ( x n x (1) ) ψ i ( y 0 (1) ) θ 1 (1) ( x n x (1) ) ψ i ( y 0 (1) ) . . . θ n x (1) ( x n x (1) ) ψ i ( y 0 (1) ) ], R y (1) ( ψ i ( y 0 (1) ))=[ θ 0 ( x n x (1) ) ψ i (1) ( y 0 (1) ) θ 1 ( x n x (1) ) ψ i (1) ( y 0 (1) ) . . . θ n x ( x n x (1) ) ψ i (1) ( y 0 (1) ) ],
R s (2) =[ R s (2) ( ψ 0 ( y 0 (2) )) R s (2) ( ψ 1 ( y 0 (2) )) . . . R s (2) ( ψ n y ( y 0 (2) )) R s (2) ( ψ 0 ( y 1 (2) )) R s (2) ( ψ 1 ( y 1 (2) )) . . . R s (2) ( ψ n y ( y 1 (2) )) . . . . . . . . . . . . . . . . . . R s (2) ( ψ 0 ( y n y (2) )) R s (2) ( ψ 1 ( y n y (2) )) . . . R s (2) ( ψ n y ( y n y (2) )) ], s=x,y,
R x (2) ( ψ i ( y 0 (2) ))=[ θ 0 (1) ( x 0 (2) ) ψ i ( y 0 (2) ) θ 1 (1) ( x 0 (2) ) ψ i ( y 0 (2) ) . . . θ n x (1) ( x 0 (2) ) ψ i ( y 0 (2) ) ], R y (2) ( ψ i ( y 0 (2) ))=[ θ 0 ( x 0 (2) ) ψ i (1) ( y 0 (2) ) θ 1 ( x 0 (2) ) ψ i (1) ( y 0 (2) ) . . . θ n x ( x 0 (2) ) ψ i (1) ( y 0 (2) ) ],
[ U x (1) U y (1) ][ H x (1) H y (1) ]=[ U x (2) U y (2) ][ H x (2) H y (2) ],
U s (1) =[ U s (1) ( ψ 0 ( y 0 (1) )) U s (1) ( ψ 1 ( y 0 (1) )) . . . U s (1) ( ψ n y ( y 0 (1) )) U s (1) ( ψ 0 ( y 1 (1) )) U s (1) ( ψ 1 ( y 1 (1) )) . . . U s (1) ( ψ n y ( y 1 (1) )) . . . . . . . . . . . . . . . . . . U s (1) ( ψ 0 ( y n y (1) )) U s (1) ( ψ 1 ( y n y (1) )) . . . U s (1) ( ψ n y ( y n y (1) )) ], s=x,y,
U x (1) ( ψ i ( y 0 (1) ))= 1 ε ˜ z (1) [ θ 0 ( x n x (1) ) ψ i (1) ( y 0 (1) ) θ 1 ( x n x (1) ) ψ i (1) ( y 0 (1) ) . . . θ n x ( x n x (1) ) ψ i (1) ( y 0 (1) ) ], U y (1) ( ψ i ( y 0 (1) ))= 1 ε ˜ z (1) [ θ 0 (1) ( x n x (1) ) ψ i ( y 0 (1) ) θ 1 (1) ( x n x (1) ) ψ i ( y 0 (1) ) . . . θ n x (1) ( x n x (1) ) ψ i ( y 0 (1) ) ],
U s (2) =[ U s (2) ( ψ 0 ( y 0 (2) )) U s (2) ( ψ 1 ( y 0 (2) )) . . . U s (2) ( ψ n y ( y 0 (2) )) U s (2) ( ψ 0 ( y 1 (2) )) U s (2) ( ψ 1 ( y 1 (2) )) . . . U s (2) ( ψ n y ( y 1 (2) )) . . . . . . . . . . . . . . . . . . U s (2) ( ψ 0 ( y n y (2) )) U s (2) ( ψ 1 ( y n y (2) )) . . . U s (2) ( ψ n y ( y n y (2) )) ], s=x,y,
U x (2) ( ψ i ( y 0 (2) ))= 1 ε ˜ z (2) [ θ 0 ( x 0 (2) ) ψ i (1) ( y 0 (2) ) θ 1 ( x 0 (2) ) ψ i (1) ( y 0 (2) ) . . . θ n x ( x 0 (2) ) ψ i (1) ( y 0 (2) ) ], U y (2) ( ψ i ( y 0 (2) ))= 1 ε ˜ z (2) [ θ 0 (1) ( x 0 (2) ) ψ i ( y 0 (2) ) θ 1 (1) ( x 0 (2) ) ψ i ( y 0 (2) ) . . . θ n x (1) ( x 0 (2) ) ψ i ( y 0 (2) ) ].
θ p (x)= (1) p+1 ( 1 x 2 ) T v ' (x) c p n 2 (x x p ) , c p ={ 2 if p=0,N 1 if 1pN1,
θ p (αx)= x L v (αx) α(x x p ) [ x L v ' (αx) ] | x= x p e α(x x p )/2 ,
[ 1 ε ˜ y | H ¯ x | 2 + 1 ε ˜ x | H ¯ y | 2 1 ε ˜ y β 2 x ( H ¯ x x + H ¯ y y ) H ¯ x * 1 ε ˜ x β 2 y ( H ¯ x x + H ¯ y y ) H ¯ y * ]dxdy=1,
ε ˜ sub = ε sub +Δ ε sat [ 1exp( γ | E | 2 Δ ε sat ) ],

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