Abstract

We propose an accurate and computationally efficient rational Chebyshev multi-domain pseudo-spectral method (RC-MDPSM) for modal analysis of optical waveguides. For the first time, we introduce rational Chebyshev basis functions to efficiently handle semi-infinite computational subdomains. In addition, the efficiency of these basis functions is enhanced by employing an optimized algebraic map; thus, eliminating the use of PML-like absorbing boundary conditions. For leaky modes, we derived a leaky modes boundary condition at the guide-substrate interface providing an efficient technique to accurately model leaky modes with very small refractive index imaginary part. The efficiency and numerical precision of our technique are demonstrated through the analysis of high-index contrast dielectric and plasmonic waveguides, and the highly-leaky ARROW structure; where finding ARROW leaky modes using our technique clearly reflects its robustness.

© 2016 Optical Society of America

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  1. Y. C. Shih, “The mode-matching method,” in Numerical Techniques for Microwave and Millimeter-Wave Passive Structures, T. Itoh, ed. (Wiley, 1989), pp. 592–621.
  2. A. S. Sudbo, “Film mode matching: a versatile numerical method for vector mode field calculations in dielectric waveguides,” Pure Appl. Opt. 2(3), 211–233 (1993).
    [Crossref]
  3. P. Bienstman and R. Baets, “Optical modelling of photonic crystals and VCSELs using eigenmode expansion and perfectly matched layers,” Opt. Quantum Electron. 33, 327–341 (2001).
    [Crossref]
  4. Y. P. Chiou, Y. C. Chiang, C. H. Lai, C. H. Du, and H. C. Chang, “Finite difference modeling of dielectric waveguides with corners and slanted facets,” J. Lightwave Technol. 27, 2077–2086 (2009).
    [Crossref]
  5. N. Thomas, P. Sewell, and T. M. Benson, “A new full-vectorial higher order finite-difference scheme for the modal analysis of rectangular dielectric waveguides,” J. Lightwave Technol. 25, 2563–2570 (2007).
    [Crossref]
  6. S. Selleri, L. Vincetti, A. Cucinotta, and M. Zoboli, “Complex FEM modal solver of optical waveguides with PML boundary conditions,” Opt. Quantum Electron. 33, 359–371 (2001).
    [Crossref]
  7. Z. E. Abid, K. L. Johnson, and A. Gopinath, “Analysis of dielectric guides by vector transverse magnetic field finite elements,” J. Lightwave Technol. 11, 1545–1549 (1993).
    [Crossref]
  8. C. C. Huang, “Solving the full anisotropic liquid crystal waveguides by using an iterative pseudospectral-based eigenvalue method,” Opt. Express 14, 3363–3378 (2011).
    [Crossref]
  9. P. J. Chiang, C. L. Wu, C. H. Teng, C. S. Yang, and H. C. Chang, “Full-vectorial optical waveguide mode solvers using multidomain pseudospectral frequency-domain (PSFD) formulations,” IEEE J. Quantum Electron. 44, 56–66 (2008).
    [Crossref]
  10. 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, 457–465 (2005).
    [Crossref]
  11. C. C. Huang, C. C. Huang, and J. Y. Yang, “An efficient method for computing optical waveguides with discontinuous refractive index profile using spectral collocation method with domain decomposition,” J. Lightwave Technol. 20, 2284–2296 (2003).
    [Crossref]
  12. J. Zhu, X. Zhang, and R. Song, “A unified mode solver for optical waveguides based on mapped barycentric rational chebyshev differentiation matrix,” J. Lightwave Technol. 28, 1802–1810 (2010).
    [Crossref]
  13. A. F. Oskooi, L. Zhang, Y. Avniel, and S. G. Johnson, “The failure of perfectly matched layers, and towards their redemption by adiabatic absorbers,” Opt. Express 16, 11376–11392 (2008).
    [Crossref] [PubMed]
  14. H. Rogier and D. D. Zutter, “Berenger and leaky modes in optical fibers terminated with a perfectly matched layer,” J. Lightwave Technol. 20, 1141–1148 (2002).
    [Crossref]
  15. D. B. Haidvogel and T. Zang, “The accurate solution of Poisson equation by expansion in Chebyshev polynomials,” J. Comput. Phys. 30, 167–180 (1979).
    [Crossref]
  16. J. P. Boyd, “The optimization of convergence for Chebyshev polynomial methods in an unbounded domain,” J. Comput. Phys. 45, 42–79 (1982).
    [Crossref]
  17. C. C. Huang, “Numerical calculations of ARROW structures by pseudo-spectral approach with Murs absorbing boundary conditions,” Opt. Express 14, 11631–11652 (2006).
    [Crossref] [PubMed]
  18. F. Tisseur and Karl Meerbergen, “The quadratic eigenvalue problem,” SIAM Rev. 43, 235–286 (2001).
    [Crossref]
  19. J. -P. Dedieu and F. Tisseur, “Perturbation theory for homogeneous polynomial eigenvalue problems,” Linear Algebra Appl. 358, 71–94 (2003).
    [Crossref]
  20. G.-X. Fan, Q. H. Liu, and J. S. Hesthaven, “Multidomain pseudospectral time-domain simulations of scattering by objects buried in lossy media,” IEEE Trans. Geosci. Remote Sens. 40, 1366–1373 (2002).
    [Crossref]
  21. L. Wang and C. S. Hsiao, “A matrix method for studying TM modes of optical planar waveguides with arbitrary index profiles,” IEEE J. Quantum Electron. 37, 1654–1660 (2001).
    [Crossref]
  22. K. Kawano and T. Kitoh, “Analytical Methods,” in Introduction to Optical Waveguide Analysis: Solving Maxwell’s Equations and the Schrödinger Equation (John Wiley & Sons, 2004), pp. 15–18.
  23. C. Vassallo, “1993–1995 Optical mode solvers,” Opt. Quantum Electron. 29, 95–114 (1997).
    [Crossref]
  24. S. Banerjee and A. Sharma, “Propagation characteristics of optical waveguiding structure by direct solution of the helmholtz equation for total fields,” J. Opt. Soc. Amer. A 6, 1884–1894 (1989).
    [Crossref]
  25. S. I. Bozhevolnyi and T. Sondergaard, “General properties of slow-plasmon resonant nanostructures: nano-antennas and resonators,” Opt. Express 15, 10869–10877 (2007).
    [Crossref] [PubMed]
  26. E. N. Economou, “Surface plasmons in thin films,” Phys. Rev. 182, 539–554 (1969).
    [Crossref]
  27. R. Zai, M. D. Selker, P. B. Catrysee, and M. L. Brongersma, “Geometries and materials for subwavelength surface plasmon modes,” J. Opt. Soc. Am. A (21), 2442–2446 (2004).
  28. E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, “Determination of guided and leaky modes in lossless and lossy planar multilayer optical waveguides: reflection pole method and wavevector density method,” J. Lightwave Technol. 17, 929–941 (1999).
    [Crossref]
  29. E. I. Golant and K. M. Golant, “New method for calculating the spectra and radiation losses of leaky waves in multilayer optical waveguides,” Tech. Phys. 51, 1060–1068 (2006).
    [Crossref]

2011 (1)

C. C. Huang, “Solving the full anisotropic liquid crystal waveguides by using an iterative pseudospectral-based eigenvalue method,” Opt. Express 14, 3363–3378 (2011).
[Crossref]

2010 (1)

2009 (1)

2008 (2)

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

A. F. Oskooi, L. Zhang, Y. Avniel, and S. G. Johnson, “The failure of perfectly matched layers, and towards their redemption by adiabatic absorbers,” Opt. Express 16, 11376–11392 (2008).
[Crossref] [PubMed]

2007 (2)

2006 (2)

E. I. Golant and K. M. Golant, “New method for calculating the spectra and radiation losses of leaky waves in multilayer optical waveguides,” Tech. Phys. 51, 1060–1068 (2006).
[Crossref]

C. C. Huang, “Numerical calculations of ARROW structures by pseudo-spectral approach with Murs absorbing boundary conditions,” Opt. Express 14, 11631–11652 (2006).
[Crossref] [PubMed]

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, 457–465 (2005).
[Crossref]

2003 (2)

C. C. Huang, C. C. Huang, and J. Y. Yang, “An efficient method for computing optical waveguides with discontinuous refractive index profile using spectral collocation method with domain decomposition,” J. Lightwave Technol. 20, 2284–2296 (2003).
[Crossref]

J. -P. Dedieu and F. Tisseur, “Perturbation theory for homogeneous polynomial eigenvalue problems,” Linear Algebra Appl. 358, 71–94 (2003).
[Crossref]

2002 (2)

G.-X. Fan, Q. H. Liu, and J. S. Hesthaven, “Multidomain pseudospectral time-domain simulations of scattering by objects buried in lossy media,” IEEE Trans. Geosci. Remote Sens. 40, 1366–1373 (2002).
[Crossref]

H. Rogier and D. D. Zutter, “Berenger and leaky modes in optical fibers terminated with a perfectly matched layer,” J. Lightwave Technol. 20, 1141–1148 (2002).
[Crossref]

2001 (4)

F. Tisseur and Karl Meerbergen, “The quadratic eigenvalue problem,” SIAM Rev. 43, 235–286 (2001).
[Crossref]

P. Bienstman and R. Baets, “Optical modelling of photonic crystals and VCSELs using eigenmode expansion and perfectly matched layers,” Opt. Quantum Electron. 33, 327–341 (2001).
[Crossref]

S. Selleri, L. Vincetti, A. Cucinotta, and M. Zoboli, “Complex FEM modal solver of optical waveguides with PML boundary conditions,” Opt. Quantum Electron. 33, 359–371 (2001).
[Crossref]

L. Wang and C. S. Hsiao, “A matrix method for studying TM modes of optical planar waveguides with arbitrary index profiles,” IEEE J. Quantum Electron. 37, 1654–1660 (2001).
[Crossref]

1999 (1)

1997 (1)

C. Vassallo, “1993–1995 Optical mode solvers,” Opt. Quantum Electron. 29, 95–114 (1997).
[Crossref]

1993 (2)

Z. E. Abid, K. L. Johnson, and A. Gopinath, “Analysis of dielectric guides by vector transverse magnetic field finite elements,” J. Lightwave Technol. 11, 1545–1549 (1993).
[Crossref]

A. S. Sudbo, “Film mode matching: a versatile numerical method for vector mode field calculations in dielectric waveguides,” Pure Appl. Opt. 2(3), 211–233 (1993).
[Crossref]

1989 (1)

S. Banerjee and A. Sharma, “Propagation characteristics of optical waveguiding structure by direct solution of the helmholtz equation for total fields,” J. Opt. Soc. Amer. A 6, 1884–1894 (1989).
[Crossref]

1982 (1)

J. P. Boyd, “The optimization of convergence for Chebyshev polynomial methods in an unbounded domain,” J. Comput. Phys. 45, 42–79 (1982).
[Crossref]

1979 (1)

D. B. Haidvogel and T. Zang, “The accurate solution of Poisson equation by expansion in Chebyshev polynomials,” J. Comput. Phys. 30, 167–180 (1979).
[Crossref]

1969 (1)

E. N. Economou, “Surface plasmons in thin films,” Phys. Rev. 182, 539–554 (1969).
[Crossref]

Abid, Z. E.

Z. E. Abid, K. L. Johnson, and A. Gopinath, “Analysis of dielectric guides by vector transverse magnetic field finite elements,” J. Lightwave Technol. 11, 1545–1549 (1993).
[Crossref]

Anemogiannis, E.

Avniel, Y.

Baets, R.

P. Bienstman and R. Baets, “Optical modelling of photonic crystals and VCSELs using eigenmode expansion and perfectly matched layers,” Opt. Quantum Electron. 33, 327–341 (2001).
[Crossref]

Banerjee, S.

S. Banerjee and A. Sharma, “Propagation characteristics of optical waveguiding structure by direct solution of the helmholtz equation for total fields,” J. Opt. Soc. Amer. A 6, 1884–1894 (1989).
[Crossref]

Benson, T. M.

Bienstman, P.

P. Bienstman and R. Baets, “Optical modelling of photonic crystals and VCSELs using eigenmode expansion and perfectly matched layers,” Opt. Quantum Electron. 33, 327–341 (2001).
[Crossref]

Boyd, J. P.

J. P. Boyd, “The optimization of convergence for Chebyshev polynomial methods in an unbounded domain,” J. Comput. Phys. 45, 42–79 (1982).
[Crossref]

Bozhevolnyi, S. I.

Brongersma, M. L.

R. Zai, M. D. Selker, P. B. Catrysee, and M. L. Brongersma, “Geometries and materials for subwavelength surface plasmon modes,” J. Opt. Soc. Am. A (21), 2442–2446 (2004).

Catrysee, P. B.

R. Zai, M. D. Selker, P. B. Catrysee, and M. L. Brongersma, “Geometries and materials for subwavelength surface plasmon modes,” J. Opt. Soc. Am. A (21), 2442–2446 (2004).

Chang, H. C.

Y. P. Chiou, Y. C. Chiang, C. H. Lai, C. H. Du, and H. C. Chang, “Finite difference modeling of dielectric waveguides with corners and slanted facets,” J. Lightwave Technol. 27, 2077–2086 (2009).
[Crossref]

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

Chiang, P. J.

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

Chiang, Y. C.

Chiou, Y. P.

Cucinotta, A.

S. Selleri, L. Vincetti, A. Cucinotta, and M. Zoboli, “Complex FEM modal solver of optical waveguides with PML boundary conditions,” Opt. Quantum Electron. 33, 359–371 (2001).
[Crossref]

Dedieu, J. -P.

J. -P. Dedieu and F. Tisseur, “Perturbation theory for homogeneous polynomial eigenvalue problems,” Linear Algebra Appl. 358, 71–94 (2003).
[Crossref]

Du, C. H.

Economou, E. N.

E. N. Economou, “Surface plasmons in thin films,” Phys. Rev. 182, 539–554 (1969).
[Crossref]

Fan, G.-X.

G.-X. Fan, Q. H. Liu, and J. S. Hesthaven, “Multidomain pseudospectral time-domain simulations of scattering by objects buried in lossy media,” IEEE Trans. Geosci. Remote Sens. 40, 1366–1373 (2002).
[Crossref]

Gaylord, T. K.

Glytsis, E. N.

Golant, E. I.

E. I. Golant and K. M. Golant, “New method for calculating the spectra and radiation losses of leaky waves in multilayer optical waveguides,” Tech. Phys. 51, 1060–1068 (2006).
[Crossref]

Golant, K. M.

E. I. Golant and K. M. Golant, “New method for calculating the spectra and radiation losses of leaky waves in multilayer optical waveguides,” Tech. Phys. 51, 1060–1068 (2006).
[Crossref]

Gopinath, A.

Z. E. Abid, K. L. Johnson, and A. Gopinath, “Analysis of dielectric guides by vector transverse magnetic field finite elements,” J. Lightwave Technol. 11, 1545–1549 (1993).
[Crossref]

Haidvogel, D. B.

D. B. Haidvogel and T. Zang, “The accurate solution of Poisson equation by expansion in Chebyshev polynomials,” J. Comput. Phys. 30, 167–180 (1979).
[Crossref]

Hesthaven, J. S.

G.-X. Fan, Q. H. Liu, and J. S. Hesthaven, “Multidomain pseudospectral time-domain simulations of scattering by objects buried in lossy media,” IEEE Trans. Geosci. Remote Sens. 40, 1366–1373 (2002).
[Crossref]

Hsiao, C. S.

L. Wang and C. S. Hsiao, “A matrix method for studying TM modes of optical planar waveguides with arbitrary index profiles,” IEEE J. Quantum Electron. 37, 1654–1660 (2001).
[Crossref]

Huang, C. C.

C. C. Huang, “Solving the full anisotropic liquid crystal waveguides by using an iterative pseudospectral-based eigenvalue method,” Opt. Express 14, 3363–3378 (2011).
[Crossref]

C. C. Huang, “Numerical calculations of ARROW structures by pseudo-spectral approach with Murs absorbing boundary conditions,” Opt. Express 14, 11631–11652 (2006).
[Crossref] [PubMed]

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, 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, 457–465 (2005).
[Crossref]

C. C. Huang, C. C. Huang, and J. Y. Yang, “An efficient method for computing optical waveguides with discontinuous refractive index profile using spectral collocation method with domain decomposition,” J. Lightwave Technol. 20, 2284–2296 (2003).
[Crossref]

C. C. Huang, C. C. Huang, and J. Y. Yang, “An efficient method for computing optical waveguides with discontinuous refractive index profile using spectral collocation method with domain decomposition,” J. Lightwave Technol. 20, 2284–2296 (2003).
[Crossref]

Johnson, K. L.

Z. E. Abid, K. L. Johnson, and A. Gopinath, “Analysis of dielectric guides by vector transverse magnetic field finite elements,” J. Lightwave Technol. 11, 1545–1549 (1993).
[Crossref]

Johnson, S. G.

Kawano, K.

K. Kawano and T. Kitoh, “Analytical Methods,” in Introduction to Optical Waveguide Analysis: Solving Maxwell’s Equations and the Schrödinger Equation (John Wiley & Sons, 2004), pp. 15–18.

Kitoh, T.

K. Kawano and T. Kitoh, “Analytical Methods,” in Introduction to Optical Waveguide Analysis: Solving Maxwell’s Equations and the Schrödinger Equation (John Wiley & Sons, 2004), pp. 15–18.

Lai, C. H.

Liu, Q. H.

G.-X. Fan, Q. H. Liu, and J. S. Hesthaven, “Multidomain pseudospectral time-domain simulations of scattering by objects buried in lossy media,” IEEE Trans. Geosci. Remote Sens. 40, 1366–1373 (2002).
[Crossref]

Meerbergen, Karl

F. Tisseur and Karl Meerbergen, “The quadratic eigenvalue problem,” SIAM Rev. 43, 235–286 (2001).
[Crossref]

Oskooi, A. F.

Rogier, H.

Selker, M. D.

R. Zai, M. D. Selker, P. B. Catrysee, and M. L. Brongersma, “Geometries and materials for subwavelength surface plasmon modes,” J. Opt. Soc. Am. A (21), 2442–2446 (2004).

Selleri, S.

S. Selleri, L. Vincetti, A. Cucinotta, and M. Zoboli, “Complex FEM modal solver of optical waveguides with PML boundary conditions,” Opt. Quantum Electron. 33, 359–371 (2001).
[Crossref]

Sewell, P.

Sharma, A.

S. Banerjee and A. Sharma, “Propagation characteristics of optical waveguiding structure by direct solution of the helmholtz equation for total fields,” J. Opt. Soc. Amer. A 6, 1884–1894 (1989).
[Crossref]

Shih, Y. C.

Y. C. Shih, “The mode-matching method,” in Numerical Techniques for Microwave and Millimeter-Wave Passive Structures, T. Itoh, ed. (Wiley, 1989), pp. 592–621.

Sondergaard, T.

Song, R.

Sudbo, A. S.

A. S. Sudbo, “Film mode matching: a versatile numerical method for vector mode field calculations in dielectric waveguides,” Pure Appl. Opt. 2(3), 211–233 (1993).
[Crossref]

Teng, C. H.

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

Thomas, N.

Tisseur, F.

J. -P. Dedieu and F. Tisseur, “Perturbation theory for homogeneous polynomial eigenvalue problems,” Linear Algebra Appl. 358, 71–94 (2003).
[Crossref]

F. Tisseur and Karl Meerbergen, “The quadratic eigenvalue problem,” SIAM Rev. 43, 235–286 (2001).
[Crossref]

Vassallo, C.

C. Vassallo, “1993–1995 Optical mode solvers,” Opt. Quantum Electron. 29, 95–114 (1997).
[Crossref]

Vincetti, L.

S. Selleri, L. Vincetti, A. Cucinotta, and M. Zoboli, “Complex FEM modal solver of optical waveguides with PML boundary conditions,” Opt. Quantum Electron. 33, 359–371 (2001).
[Crossref]

Wang, L.

L. Wang and C. S. Hsiao, “A matrix method for studying TM modes of optical planar waveguides with arbitrary index profiles,” IEEE J. Quantum Electron. 37, 1654–1660 (2001).
[Crossref]

Wu, C. L.

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

Yang, C. S.

P. J. Chiang, C. L. Wu, C. H. Teng, C. S. Yang, and H. C. Chang, “Full-vectorial optical waveguide mode solvers using multidomain pseudospectral frequency-domain (PSFD) formulations,” IEEE J. Quantum Electron. 44, 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, 457–465 (2005).
[Crossref]

C. C. Huang, C. C. Huang, and J. Y. Yang, “An efficient method for computing optical waveguides with discontinuous refractive index profile using spectral collocation method with domain decomposition,” J. Lightwave Technol. 20, 2284–2296 (2003).
[Crossref]

Zai, R.

R. Zai, M. D. Selker, P. B. Catrysee, and M. L. Brongersma, “Geometries and materials for subwavelength surface plasmon modes,” J. Opt. Soc. Am. A (21), 2442–2446 (2004).

Zang, T.

D. B. Haidvogel and T. Zang, “The accurate solution of Poisson equation by expansion in Chebyshev polynomials,” J. Comput. Phys. 30, 167–180 (1979).
[Crossref]

Zhang, L.

Zhang, X.

Zhu, J.

Zoboli, M.

S. Selleri, L. Vincetti, A. Cucinotta, and M. Zoboli, “Complex FEM modal solver of optical waveguides with PML boundary conditions,” Opt. Quantum Electron. 33, 359–371 (2001).
[Crossref]

Zutter, D. D.

IEEE J. Quantum Electron. (2)

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

L. Wang and C. S. Hsiao, “A matrix method for studying TM modes of optical planar waveguides with arbitrary index profiles,” IEEE J. Quantum Electron. 37, 1654–1660 (2001).
[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, 457–465 (2005).
[Crossref]

IEEE Trans. Geosci. Remote Sens. (1)

G.-X. Fan, Q. H. Liu, and J. S. Hesthaven, “Multidomain pseudospectral time-domain simulations of scattering by objects buried in lossy media,” IEEE Trans. Geosci. Remote Sens. 40, 1366–1373 (2002).
[Crossref]

J. Comput. Phys. (2)

D. B. Haidvogel and T. Zang, “The accurate solution of Poisson equation by expansion in Chebyshev polynomials,” J. Comput. Phys. 30, 167–180 (1979).
[Crossref]

J. P. Boyd, “The optimization of convergence for Chebyshev polynomial methods in an unbounded domain,” J. Comput. Phys. 45, 42–79 (1982).
[Crossref]

J. Lightwave Technol. (7)

Z. E. Abid, K. L. Johnson, and A. Gopinath, “Analysis of dielectric guides by vector transverse magnetic field finite elements,” J. Lightwave Technol. 11, 1545–1549 (1993).
[Crossref]

H. Rogier and D. D. Zutter, “Berenger and leaky modes in optical fibers terminated with a perfectly matched layer,” J. Lightwave Technol. 20, 1141–1148 (2002).
[Crossref]

C. C. Huang, C. C. Huang, and J. Y. Yang, “An efficient method for computing optical waveguides with discontinuous refractive index profile using spectral collocation method with domain decomposition,” J. Lightwave Technol. 20, 2284–2296 (2003).
[Crossref]

J. Zhu, X. Zhang, and R. Song, “A unified mode solver for optical waveguides based on mapped barycentric rational chebyshev differentiation matrix,” J. Lightwave Technol. 28, 1802–1810 (2010).
[Crossref]

Y. P. Chiou, Y. C. Chiang, C. H. Lai, C. H. Du, and H. C. Chang, “Finite difference modeling of dielectric waveguides with corners and slanted facets,” J. Lightwave Technol. 27, 2077–2086 (2009).
[Crossref]

N. Thomas, P. Sewell, and T. M. Benson, “A new full-vectorial higher order finite-difference scheme for the modal analysis of rectangular dielectric waveguides,” J. Lightwave Technol. 25, 2563–2570 (2007).
[Crossref]

E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, “Determination of guided and leaky modes in lossless and lossy planar multilayer optical waveguides: reflection pole method and wavevector density method,” J. Lightwave Technol. 17, 929–941 (1999).
[Crossref]

J. Opt. Soc. Amer. A (1)

S. Banerjee and A. Sharma, “Propagation characteristics of optical waveguiding structure by direct solution of the helmholtz equation for total fields,” J. Opt. Soc. Amer. A 6, 1884–1894 (1989).
[Crossref]

Linear Algebra Appl. (1)

J. -P. Dedieu and F. Tisseur, “Perturbation theory for homogeneous polynomial eigenvalue problems,” Linear Algebra Appl. 358, 71–94 (2003).
[Crossref]

Opt. Express (4)

Opt. Quantum Electron. (3)

P. Bienstman and R. Baets, “Optical modelling of photonic crystals and VCSELs using eigenmode expansion and perfectly matched layers,” Opt. Quantum Electron. 33, 327–341 (2001).
[Crossref]

S. Selleri, L. Vincetti, A. Cucinotta, and M. Zoboli, “Complex FEM modal solver of optical waveguides with PML boundary conditions,” Opt. Quantum Electron. 33, 359–371 (2001).
[Crossref]

C. Vassallo, “1993–1995 Optical mode solvers,” Opt. Quantum Electron. 29, 95–114 (1997).
[Crossref]

Phys. Rev. (1)

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[Crossref]

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[Crossref]

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[Crossref]

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[Crossref]

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Figures (11)

Fig. 1
Fig. 1 Schematic diagram of a multi-layer 2D planar waveguide.
Fig. 2
Fig. 2 Application of zero-boundary conditions for guided modes at y = ±∞.
Fig. 3
Fig. 3 Application of leaky modes condition of Eq. (16) at the guide-substrate interface.
Fig. 4
Fig. 4 (a) Normalized fundamental TM field, Hx; (b) Convergence of the normalized propagation constant.
Fig. 5
Fig. 5 Collocation points distribution for proposed method for M = 1, N1 = NM+1 = 8.
Fig. 6
Fig. 6 (a) Continuous Gaussian refractive index profile.; (b) First four TE mode profiles obtained by the proposed method.
Fig. 7
Fig. 7 (a) Discontinuous refractive index profile.; (b) Normalized fundamental TE field for the discontinuous RIP.
Fig. 8
Fig. 8 Long and short SPP modes in the symmetric IMI nano-structure.
Fig. 9
Fig. 9 G-SPP mode in the symmetric MIM nano-structure.
Fig. 10
Fig. 10 Comparison of (a) the effective refractive index and (b) the propagation length; calculated by the domain truncation method [12] with variable computational window [−W, W] and the proposed method with fixed computational window, W = ∞ for gap of thickness t = 20 nm.
Fig. 11
Fig. 11 (a) Anti-resonant reflecting optical waveguide structure; (b) Normalized TE1 and TE18 modes at λ = 0.6328μm.

Tables (10)

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Table 1 Comparison of Normalized Propagation Constant and Relative Error for Step-index Slab Waveguide at λ = 1.5μm

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Table 2 Convergence of the Effective Refractive Index of the First TE Mode

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Table 3 Comparison of the Effective Indices for the TE Modes at λ = 0.6328 μm

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Table 4 Convergence of the Effective Index for the Discontinuous Planar Waveguide at λ = 0.6328μm

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Table 5 Convergence of nl and Ll Values for t = 40nm

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Table 6 Convergence of ns and Ls Values for Film Thickness t = 40nm

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Table 7 Relative Error for Different Metal Thickness at N = 100

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Table 8 Convergence of ngsp and Lgsp for Gap Thickness t = 20nm

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Table 9 Convergence of ngsp and Lgsp for Gap Thickness t = 1000nm

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Table 10 Normalized Propagation Constant, β/k0 for TE Modes in ARROW

Equations (34)

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2 u y 2 θ 1 n 2 n 2 y u y + k 0 2 ( n 2 ( y ) n eff 2 ) u = 0 ,
u ( y ) = { E x ( y ) and θ = 0 for TE modes ; H x ( y ) and θ = 1 for TM modes .
E x ( d i ) = E x ( d i + ) , E x y ( d i ) = E x y ( d i + ) ,
H x ( d i ) = H x ( d i + ) , ( 1 n 2 H x y ) ( d i ) = ( 1 n 2 H x y ) ( d i + ) ,
C j ( y ˜ ) = ( 1 ) N j + 1 ( 1 y ˜ 2 ) T N ( y ˜ ) σ j N 2 ( y ˜ y ˜ j ) , y ˜ y ˜ j ,
ϕ i ( y ) = 2 ( y d i 1 d i d i 1 ) 1 , 2 i M ,
y j ( i ) = ϕ i 1 ( y ˜ j ) = d i 1 + ( d i d i 1 2 ) ( 1 cos ( j π / N i ) ) .
ϕ 1 ( y ; N 1 ) = τ ( N 1 ) + ( y d 1 ) τ ( N 1 ) ( y d 1 ) ,
y j ( 1 ) = d 1 τ ( N 1 ) 1 + cos ( j π / N 1 ) 1 cos ( j π / N 1 ) ,
ϕ M + 1 ( y ; N M + 1 ) = ( y d m ) α ( N M + 1 ) ( y d m ) + α ( N M + 1 ) ,
y j ( M + 1 ) = d m + α ( N M + 1 ) 1 cos ( j π N M + 1 ) 1 + cos ( j π N M + 1 ) .
u ( i ) ( y ) = j = 0 N i u j ( i ) ( C j ϕ i ) ( y ) , y Ω i ,
u ( M + 1 ) ( y ) = u 0 e j γ s ( y y M ) ,
d u ( M + 1 ) d y | y = d M = j k 0 2 n s 2 β 2 u ( M + 1 ) | y = d M
γ s = k 0 2 n s 2 β 2 k 0 n s ( 1 1 2 β 2 k 0 2 n s 2 + )
d u ( M + 1 ) d y + j k 0 n s u ( M + 1 ) | y = d M = j 2 β 2 k 0 2 n s 2 u ( M + 1 ) | y = d M
d u ( 1 ) d y j k 0 n s u ( 1 ) | y = d 1 + = j 2 β 2 k 0 2 n s 2 u ( 1 ) | y = d 1 +
A 0 λ A 1 u + λ 2 A 2 u + + λ p A p u = 0 , λ = β 2 .
D ( 2 ) = D D = D 2 .
Λ i ( f ) = diag ( f ( y 0 ( j ) ) , , f ( y N i ( i ) ) ) .
D i = Λ i ( d ϕ i d y ) D ,
D i ( 2 ) = Λ i ( ( d ϕ i d y ) 2 ) D 2 + Λ i ( d 2 ϕ i d y 2 ) D .
I ^ i = [ I i ] k j , 1 k N i 1 , 0 j N i ,
D ^ i = [ D i ] k j , 1 k N i 1 , 0 j N i ,
D ^ i ( 2 ) = [ D i ( 2 ) ] k j , 1 k N i 1 , 0 j N i ,
[ K ] u ( i ) = β 2 [ I ^ i ] u ( i )
K = D ^ i ( 2 ) θ Λ i ( 1 n 2 n 2 y ) D ^ i + Λ i ( k 0 2 n 2 ( y ) ) I ^ i .
H τ ( x , y ) = i = 0 N j = 0 M h i j τ C i ( ϕ ( x ) ) C j ( ψ ( y ) ) , τ = x or y ,
Δ b = | b b exact | b exact
n 2 ( y ) = n s 2 + 2 n s C e y 2 / t 2 + C 2 e 2 y 2 / t 2
n 2 ( y ) = { n s 2 + 2 n s C e y 2 / t 2 + C 2 e 2 y 2 / t 2 , y 0 ; 1 , y < 0 ,
n l ( s ) = ( k l ( s ) r s p ) / k 0 , L l ( s ) = [ 2 ( k l ( s ) r s p ) ] 1 .
δ n l ( s ) = | n l ( s ) n l ( s ) exact | n l ( s ) exact , δ L l ( s ) = | L l ( s ) L l ( s ) exact | L l ( s ) exact .
tanh ( α d t 2 ) = ε d α m ε m α d

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