Abstract

I extend the full vector pseudospectral-based eigenvalue scheme, based on the transverse magnetic field components, to analyze the mode behaviors of dielectric optical waveguides with transverse, nondiagonal anisotropy. One of the principal axes of the anisotropic materials is thus constrained to point in the longitudinal direction of the waveguide. I expand the guided mode fields in the interior subdomains with finite extent by using Chebyshev polynomials and those in the exterior subdomains with semi-infinite extent by using Laguerre–Gaussian functions with an accurately determined scaling factor. This study analyzes two examples: (1) the circularly-polarized modes of a magneto-optical raised strip waveguide and (2) the guided mode patterns of a nematic liquid-crystal channel waveguide under different orientations of the liquid-crystal molecule. The comparison of the numerical results with those from the vector finite difference approach demonstrates that my numerical approach has a higher computational efficiency and requires less computer memory.

©2010 Optical Society of America

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  5. S. Selleri, L. Vincetti, and M. Zoboli, “Full-vector finite-element beam propagation method for anisotropic optical device analysis,” IEEE J. Quantum Electron. 36(12), 1392–1401 (2000).
    [Crossref]
  6. K. Saitoh and M. Koshiba, “Approximate scalar finite element beam-propagation method with perfectly matched layers for anisotropic optical waveguides,” J. Lightwave Technol. 19(5), 786–792 (2001).
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  7. J. P. da Silva, H. E. Hernandez-Figueroa, and A. M. F. Frasson, “Improved vectorial finite-element BPM analysis for transverse anisotropic media,” J. Lightwave Technol. 21(2), 567–576 (2003).
    [Crossref]
  8. A. B. Fallahkhair, K. S. Li, and T. E. Murphy, “Vector finite difference modesolver for anisotropic dielectric waveguides,” J. Lightwave Technol. 26(11), 1423–1431 (2008).
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    [Crossref] [PubMed]
  11. V. Schulz, “Adjoint high-order vectorial finite elements for nonsymmetric transversally anisotropic waveguides,” IEEE Trans. Microw. Theory Tech. 51(4), 1086–1095 (2003).
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  12. B. G. Ward, “Finite element analysis of photonic crystal rods with inhomogeneous anisotropic refractive index tensor,” IEEE J. Quantum Electron. 44(2), 150–156 (2008).
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    [Crossref]
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    [Crossref]
  15. T. Ando, H. Nakayama, S. Numata, J. Yamauchi, and H. Nakano, “Eigenmode analysis of optical waveguides by a Yee-mesh-based imaginary-distance propagation method for an arbitrary dielectric interface,” J. Lightwave Technol. 20(8), 1627–1634 (2002).
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    [Crossref]
  18. C. C. Huang and C. C. Huang, “An efficient and accurate semivectorial spectral collocation method for analyzing polarized modes of rib waveguides,” J. Lightwave Technol. 23(7), 2309–2317 (2005).
    [Crossref]
  19. 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]
  20. 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(1), 56–66 (2008).
    [Crossref]
  21. C. C. Huang, “Numerical calculations of ARROW structures by pseudospectral approach with Mur’s absorbing boundary conditions,” Opt. Express 14(24), 11631–11652 (2006).
    [Crossref] [PubMed]
  22. P. J. Chiang and Y. C. Chiang, “Pseudospectral frequency-domain formulae based on modified perfectly matched layers for calculating both guided and leaky modes,” IEEE Photon. Technol. Lett. 22(12), 908–910 (2010).
    [Crossref]
  23. 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]
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  26. T. Tang, “The Hermite spectral method for Gauss-type functions,” SIAM J. Sci. Comput. 14(3), 594–605 (1993).
    [Crossref]
  27. T. Tamir, Guides-Wave Optoelectronics (Springer-Verlag, 1988).
  28. M. Loymeyer, N. Bahlmann, O. Zhuromskyy, H. Dotsch, and P. Hertel, “Phase-matched rectangular magnetooptic waveguides for applications in integrated optics isolators: numerical assessment,” Opt. Commun. 158(1-6), 189–200 (1998).
    [Crossref]
  29. B. Bellini and R. Beccherelli, “Modeling, design and analysis of liquid crystal waveguides in preferentially etched silicon grooves,” J. Phys. D Appl. Phys. 42(4), 045111 (2009).
    [Crossref]
  30. A. D’Álessandro, B. Bellini, D. Donisi, R. Beccherelli, and R. Asquini, “Nematic liquid crystal optical channel waveguides on silicon,” IEEE J. Quantum Electron. 42, 1084–1090 (2006).
    [Crossref]
  31. J. Beeckman, R. James, F. A. Í. Fernandez, W. De Cort, P. J. M. Vanbrabant, and K. Neyts, “Calculation of fully anisotropic liquid crystal waveguide modes,” J. Lightwave Technol. 27(17), 3812–3819 (2009).
    [Crossref]
  32. P. J. Vanbrabant, J. Beeckman, K. Neyts, R. James, and F. A. Fernandez, “A finite element beam propagation method for simulation of liquid crystal devices,” Opt. Express 17(13), 10895–10909 (2009).
    [Crossref] [PubMed]
  33. P. Yeh, and C. Gu, Optics of Liquid Crystal Displays (John Wiley and Sons. Inc., New York, 1999).

2010 (2)

P. J. Chiang and Y. C. Chiang, “Pseudospectral frequency-domain formulae based on modified perfectly matched layers for calculating both guided and leaky modes,” IEEE Photon. Technol. Lett. 22(12), 908–910 (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 (5)

2008 (3)

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(1), 56–66 (2008).
[Crossref]

A. B. Fallahkhair, K. S. Li, and T. E. Murphy, “Vector finite difference modesolver for anisotropic dielectric waveguides,” J. Lightwave Technol. 26(11), 1423–1431 (2008).
[Crossref]

B. G. Ward, “Finite element analysis of photonic crystal rods with inhomogeneous anisotropic refractive index tensor,” IEEE J. Quantum Electron. 44(2), 150–156 (2008).
[Crossref]

2007 (1)

R. Pashaie, “Fourier decomposition analysis of anisotropic inhomogeneous dielectric waveguide structures,” IEEE Trans. Microw. Theory Tech. 55(8), 1689–1696 (2007).
[Crossref]

2006 (3)

2005 (2)

C. C. Huang and C. C. Huang, “An efficient and accurate semivectorial spectral collocation method for analyzing polarized modes of rib waveguides,” J. Lightwave Technol. 23(7), 2309–2317 (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]

2003 (2)

V. Schulz, “Adjoint high-order vectorial finite elements for nonsymmetric transversally anisotropic waveguides,” IEEE Trans. Microw. Theory Tech. 51(4), 1086–1095 (2003).
[Crossref]

J. P. da Silva, H. E. Hernandez-Figueroa, and A. M. F. Frasson, “Improved vectorial finite-element BPM analysis for transverse anisotropic media,” J. Lightwave Technol. 21(2), 567–576 (2003).
[Crossref]

2002 (2)

2001 (1)

2000 (1)

S. Selleri, L. Vincetti, and M. Zoboli, “Full-vector finite-element beam propagation method for anisotropic optical device analysis,” IEEE J. Quantum Electron. 36(12), 1392–1401 (2000).
[Crossref]

1999 (1)

1998 (1)

M. Loymeyer, N. Bahlmann, O. Zhuromskyy, H. Dotsch, and P. Hertel, “Phase-matched rectangular magnetooptic waveguides for applications in integrated optics isolators: numerical assessment,” Opt. Commun. 158(1-6), 189–200 (1998).
[Crossref]

1996 (2)

R. B. Lehoucq and D. C. Sorensen, “Deflation techniques for an implicitly re-started Arnoldi iteration,” SIAM J. Matrix Anal. Appl. 17(4), 789–821 (1996).
[Crossref]

P. Lüsse, K. Ramm, and H. G. Unger, “Vectorial eigenmode calculation for anisotropic planar optical waveguides,” Electron. Lett. 32(1), 38–39 (1996).
[Crossref]

1994 (2)

C. L. Xu, W. P. Huang, J. Chrostowski, and S. K. Chaudhuri, “A full-vectorial beam propagation method for anisotropic waveguides,” J. Lightwave Technol. 12(11), 1926–1931 (1994).
[Crossref]

J. C. Chen and S. Jüngling, “Computation of high-order waveguide modes by imaginary-distance beam propagation method,” Opt. Quantum Electron. 26(3), S199–S205 (1994).
[Crossref]

1993 (1)

T. Tang, “The Hermite spectral method for Gauss-type functions,” SIAM J. Sci. Comput. 14(3), 594–605 (1993).
[Crossref]

Abo el Maaty, M. I.

Al-Begain, K.

Ando, T.

Asquini, R.

A. D’Álessandro, B. Bellini, D. Donisi, R. Beccherelli, and R. Asquini, “Nematic liquid crystal optical channel waveguides on silicon,” IEEE J. Quantum Electron. 42, 1084–1090 (2006).
[Crossref]

Bahlmann, N.

M. Loymeyer, N. Bahlmann, O. Zhuromskyy, H. Dotsch, and P. Hertel, “Phase-matched rectangular magnetooptic waveguides for applications in integrated optics isolators: numerical assessment,” Opt. Commun. 158(1-6), 189–200 (1998).
[Crossref]

Beccherelli, R.

B. Bellini and R. Beccherelli, “Modeling, design and analysis of liquid crystal waveguides in preferentially etched silicon grooves,” J. Phys. D Appl. Phys. 42(4), 045111 (2009).
[Crossref]

A. D’Álessandro, B. Bellini, D. Donisi, R. Beccherelli, and R. Asquini, “Nematic liquid crystal optical channel waveguides on silicon,” IEEE J. Quantum Electron. 42, 1084–1090 (2006).
[Crossref]

Beeckman, J.

Bellini, B.

B. Bellini and R. Beccherelli, “Modeling, design and analysis of liquid crystal waveguides in preferentially etched silicon grooves,” J. Phys. D Appl. Phys. 42(4), 045111 (2009).
[Crossref]

A. D’Álessandro, B. Bellini, D. Donisi, R. Beccherelli, and R. Asquini, “Nematic liquid crystal optical channel waveguides on silicon,” IEEE J. Quantum Electron. 42, 1084–1090 (2006).
[Crossref]

Chang, H. C.

M. Y. Chen, S. M. Hsu, and H. C. Chang, “A finite-difference frequency-domain method for full-vectorial mode solutions of anisotropic optical waveguides with arbitrary permittivity tensor,” Opt. Express 17(8), 5965–5979 (2009).
[Crossref] [PubMed]

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(1), 56–66 (2008).
[Crossref]

Chaudhuri, S. K.

C. L. Xu, W. P. Huang, J. Chrostowski, and S. K. Chaudhuri, “A full-vectorial beam propagation method for anisotropic waveguides,” J. Lightwave Technol. 12(11), 1926–1931 (1994).
[Crossref]

Chen, J. C.

J. C. Chen and S. Jüngling, “Computation of high-order waveguide modes by imaginary-distance beam propagation method,” Opt. Quantum Electron. 26(3), S199–S205 (1994).
[Crossref]

Chen, M. Y.

Chiang, P. J.

P. J. Chiang and Y. C. Chiang, “Pseudospectral frequency-domain formulae based on modified perfectly matched layers for calculating both guided and leaky modes,” IEEE Photon. Technol. Lett. 22(12), 908–910 (2010).
[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(1), 56–66 (2008).
[Crossref]

Chiang, Y. C.

P. J. Chiang and Y. C. Chiang, “Pseudospectral frequency-domain formulae based on modified perfectly matched layers for calculating both guided and leaky modes,” IEEE Photon. Technol. Lett. 22(12), 908–910 (2010).
[Crossref]

Chrostowski, J.

C. L. Xu, W. P. Huang, J. Chrostowski, and S. K. Chaudhuri, “A full-vectorial beam propagation method for anisotropic waveguides,” J. Lightwave Technol. 12(11), 1926–1931 (1994).
[Crossref]

D’Álessandro, A.

A. D’Álessandro, B. Bellini, D. Donisi, R. Beccherelli, and R. Asquini, “Nematic liquid crystal optical channel waveguides on silicon,” IEEE J. Quantum Electron. 42, 1084–1090 (2006).
[Crossref]

da Silva, J. P.

De Cort, W.

Donisi, D.

A. D’Álessandro, B. Bellini, D. Donisi, R. Beccherelli, and R. Asquini, “Nematic liquid crystal optical channel waveguides on silicon,” IEEE J. Quantum Electron. 42, 1084–1090 (2006).
[Crossref]

Dotsch, H.

M. Loymeyer, N. Bahlmann, O. Zhuromskyy, H. Dotsch, and P. Hertel, “Phase-matched rectangular magnetooptic waveguides for applications in integrated optics isolators: numerical assessment,” Opt. Commun. 158(1-6), 189–200 (1998).
[Crossref]

Fallahkhair, A. B.

Farrell, G.

Fernandez, F. A.

Fernandez, F. A. Í.

Frasson, A. M. F.

Hameed, M. F. O.

Hernandez-Figueroa, H. E.

Hertel, P.

M. Loymeyer, N. Bahlmann, O. Zhuromskyy, H. Dotsch, and P. Hertel, “Phase-matched rectangular magnetooptic waveguides for applications in integrated optics isolators: numerical assessment,” Opt. Commun. 158(1-6), 189–200 (1998).
[Crossref]

Hsu, S. M.

Huang, C. C.

C. C. Huang, “Numerical calculations of ARROW structures by pseudospectral approach with Mur’s absorbing boundary conditions,” Opt. Express 14(24), 11631–11652 (2006).
[Crossref] [PubMed]

C. C. Huang and C. C. Huang, “An efficient and accurate semivectorial spectral collocation method for analyzing polarized modes of rib waveguides,” J. Lightwave Technol. 23(7), 2309–2317 (2005).
[Crossref]

C. C. Huang and C. C. Huang, “An efficient and accurate semivectorial spectral collocation method for analyzing polarized modes of rib waveguides,” J. Lightwave Technol. 23(7), 2309–2317 (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]

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]

Huang, W. P.

C. L. Xu, W. P. Huang, J. Chrostowski, and S. K. Chaudhuri, “A full-vectorial beam propagation method for anisotropic waveguides,” J. Lightwave Technol. 12(11), 1926–1931 (1994).
[Crossref]

James, R.

Jüngling, S.

J. C. Chen and S. Jüngling, “Computation of high-order waveguide modes by imaginary-distance beam propagation method,” Opt. Quantum Electron. 26(3), S199–S205 (1994).
[Crossref]

Koshiba, M.

Lehoucq, R. B.

R. B. Lehoucq and D. C. Sorensen, “Deflation techniques for an implicitly re-started Arnoldi iteration,” SIAM J. Matrix Anal. Appl. 17(4), 789–821 (1996).
[Crossref]

Li, K. S.

Liu, Q. H.

Q. H. Liu, “A pseudospectral frequency-domain (PSFD) method for computational electromagnetics,” IEEE Antennas Wirel. Propag. Lett. 1(6), 131–134 (2002).
[Crossref]

Loymeyer, M.

M. Loymeyer, N. Bahlmann, O. Zhuromskyy, H. Dotsch, and P. Hertel, “Phase-matched rectangular magnetooptic waveguides for applications in integrated optics isolators: numerical assessment,” Opt. Commun. 158(1-6), 189–200 (1998).
[Crossref]

Lüsse, P.

P. Lüsse, K. Ramm, and H. G. Unger, “Vectorial eigenmode calculation for anisotropic planar optical waveguides,” Electron. Lett. 32(1), 38–39 (1996).
[Crossref]

Murphy, T. E.

Nakano, H.

Nakayama, H.

Nasr, A. M.

Neyts, K.

Numata, S.

Obayya, S. S. A.

Pashaie, R.

R. Pashaie, “Fourier decomposition analysis of anisotropic inhomogeneous dielectric waveguide structures,” IEEE Trans. Microw. Theory Tech. 55(8), 1689–1696 (2007).
[Crossref]

Ramm, K.

P. Lüsse, K. Ramm, and H. G. Unger, “Vectorial eigenmode calculation for anisotropic planar optical waveguides,” Electron. Lett. 32(1), 38–39 (1996).
[Crossref]

Saitoh, K.

Schulz, V.

V. Schulz, “Adjoint high-order vectorial finite elements for nonsymmetric transversally anisotropic waveguides,” IEEE Trans. Microw. Theory Tech. 51(4), 1086–1095 (2003).
[Crossref]

Selleri, S.

S. Selleri, L. Vincetti, and M. Zoboli, “Full-vector finite-element beam propagation method for anisotropic optical device analysis,” IEEE J. Quantum Electron. 36(12), 1392–1401 (2000).
[Crossref]

Semenova, Y.

Sorensen, D. C.

R. B. Lehoucq and D. C. Sorensen, “Deflation techniques for an implicitly re-started Arnoldi iteration,” SIAM J. Matrix Anal. Appl. 17(4), 789–821 (1996).
[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]

Takimoto, N.

Tang, T.

T. Tang, “The Hermite spectral method for Gauss-type functions,” SIAM J. Sci. Comput. 14(3), 594–605 (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(1), 56–66 (2008).
[Crossref]

Tsuji, Y.

Unger, H. G.

P. Lüsse, K. Ramm, and H. G. Unger, “Vectorial eigenmode calculation for anisotropic planar optical waveguides,” Electron. Lett. 32(1), 38–39 (1996).
[Crossref]

Vanbrabant, P. J.

Vanbrabant, P. J. M.

Vincetti, L.

S. Selleri, L. Vincetti, and M. Zoboli, “Full-vector finite-element beam propagation method for anisotropic optical device analysis,” IEEE J. Quantum Electron. 36(12), 1392–1401 (2000).
[Crossref]

Wang, Q.

Ward, B. G.

B. G. Ward, “Finite element analysis of photonic crystal rods with inhomogeneous anisotropic refractive index tensor,” IEEE J. Quantum Electron. 44(2), 150–156 (2008).
[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(1), 56–66 (2008).
[Crossref]

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]

Xu, C. L.

C. L. Xu, W. P. Huang, J. Chrostowski, and S. K. Chaudhuri, “A full-vectorial beam propagation method for anisotropic waveguides,” J. Lightwave Technol. 12(11), 1926–1931 (1994).
[Crossref]

Yamauchi, J.

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(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]

Zhuromskyy, O.

M. Loymeyer, N. Bahlmann, O. Zhuromskyy, H. Dotsch, and P. Hertel, “Phase-matched rectangular magnetooptic waveguides for applications in integrated optics isolators: numerical assessment,” Opt. Commun. 158(1-6), 189–200 (1998).
[Crossref]

Zoboli, M.

S. Selleri, L. Vincetti, and M. Zoboli, “Full-vector finite-element beam propagation method for anisotropic optical device analysis,” IEEE J. Quantum Electron. 36(12), 1392–1401 (2000).
[Crossref]

Electron. Lett. (1)

P. Lüsse, K. Ramm, and H. G. Unger, “Vectorial eigenmode calculation for anisotropic planar optical waveguides,” Electron. Lett. 32(1), 38–39 (1996).
[Crossref]

IEEE Antennas Wirel. Propag. Lett. (1)

Q. H. Liu, “A pseudospectral frequency-domain (PSFD) method for computational electromagnetics,” IEEE Antennas Wirel. Propag. Lett. 1(6), 131–134 (2002).
[Crossref]

IEEE J. Quantum Electron. (4)

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(1), 56–66 (2008).
[Crossref]

S. Selleri, L. Vincetti, and M. Zoboli, “Full-vector finite-element beam propagation method for anisotropic optical device analysis,” IEEE J. Quantum Electron. 36(12), 1392–1401 (2000).
[Crossref]

B. G. Ward, “Finite element analysis of photonic crystal rods with inhomogeneous anisotropic refractive index tensor,” IEEE J. Quantum Electron. 44(2), 150–156 (2008).
[Crossref]

A. D’Álessandro, B. Bellini, D. Donisi, R. Beccherelli, and R. Asquini, “Nematic liquid crystal optical channel waveguides on silicon,” IEEE J. Quantum Electron. 42, 1084–1090 (2006).
[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. (1)

P. J. Chiang and Y. C. Chiang, “Pseudospectral frequency-domain formulae based on modified perfectly matched layers for calculating both guided and leaky modes,” IEEE Photon. Technol. Lett. 22(12), 908–910 (2010).
[Crossref]

IEEE Trans. Microw. Theory Tech. (2)

R. Pashaie, “Fourier decomposition analysis of anisotropic inhomogeneous dielectric waveguide structures,” IEEE Trans. Microw. Theory Tech. 55(8), 1689–1696 (2007).
[Crossref]

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Figures (10)

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

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Fig. 2
Fig. 2 (a) Cross section of a rib waveguide. (b) Equivalent slab waveguide to original rib waveguide after using EIM along the y direction.

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Fig. 3
Fig. 3 (a) Schematic showing cross section of magneto-optical raised strip waveguide with permittivity tensor [ε] in the core region and the refractive indices of GGG substrate ns and air na , and (b) division of computational domain for magneto-optical raised strip waveguide.

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Fig. 4
Fig. 4 Mode profiles of (a) |Hx | and (b) |Hy | of the first order mode for ζ = 0.005.

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Fig. 5
Fig. 5 Mode profiles of (a) |Hx | and (b) |Hy | of the second order mode for ζ = 0.005.

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Fig. 6
Fig. 6 Mode profiles of (a) |Hx − jHy |/2and (b) |Hx + jHy |/2of the first order mode for ζ = 0.005.

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Fig. 7
Fig. 7 Mode profiles of (a) |Hx − jHy |/2and (b) |Hx + jHy |/2of the second order mode for ζ = 0.005.

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Fig. 8
Fig. 8 (a) Schematic showing cross section of the nematic LC channel optical waveguide with permittivity tensor of the core region [ε] and the refractive indices of glass substrate ns and air na , (b) Schematic showing the twist angle φ of LC molecule and orientation of director n ^ .

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Fig. 9
Fig. 9 |Hx | and |Hy | mode patterns of the mode 1 for four twist angles φ.

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Fig. 10
Fig. 10 |Hx | and |Hy | mode patterns of the mode 2 to mode 5 for four twist angles φ.

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Tables (3)

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Table 1 Effective refractive indices of the first and the second order modes and the computational time versus the term of the basis functions for expanding the interior subdomains Nint , while ζ = 0.005

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Table 2 The calculated effective indices obtained by this study and the vector FD method [8] for different values of ζ

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Table 3 Calculated effective indices at φ = 0°, 30°, 45°, 60°, and 90° for the first seven modes using Nex = 10 and Nint = 20 in each subdomain

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Equations (36)

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∇ × ( [ ε ] − 1 ∇ × H ) − ω 2 μ 0 H = 0 ,
[ ε ] = ε 0 [ ε r ] = ε 0 [ ε x x ε x y 0 ε y x ε y y 0 0 0 ε z z ] ,
[ P x x P x y P y x P y y ] [ H x H y ] = β 2 [ η y y − η y x − η x y η x x ] [ H x H y ] ,
P x x H x = η y y ∂ 2 H x ∂ x 2 + η z z ∂ 2 H x ∂ y 2 − η y x ∂ 2 H x ∂ y ∂ x + k 0 2 H x ,
P x y H y = ( η y y − η z z ) ∂ 2 H y ∂ x ∂ y − η y x ∂ 2 H y ∂ y 2 .
P y x H x = ( η x x − η z z ) ∂ 2 H x ∂ y ∂ x − η x y ∂ 2 H x ∂ x 2 ,   and
P y y H y = η z z ∂ 2 H y ∂ x 2 + η x x ∂ 2 H y ∂ y 2 + η x y ∂ 2 H y ∂ x ∂ y + k 0 2 H y ,
η x x = ε y y ε z z Δ ,   η x y = η y x = − ε x y ε z z Δ ,   η y y = ε x x ε z z Δ ,   η z z = ε x x ε y y − ε x y ε y x Δ ,   Δ = ε z z ( ε x x ε y y − ε x y ε y x ) .
H z = 1 j β ( ∂ H x ∂ x + ∂ H y ∂ y ) .
E z = j ω ε 0 ε z z ( ∂ H x ∂ y − ∂ H y ∂ x ) .
H x r ( x , y ) = ∑ p = 0 n x ∑ q = 0 n y θ p   r ( x ) ψ q   r ( y ) H x , p q r ,
H y r ( x , y ) = ∑ p = 0 n x ∑ q = 0 n y θ p   r ( x ) ψ q   r ( y ) H y , p q r ,
[ P x x r P x y r P y x r P y y r ] [ H x r H y r ] = β 2 [ η y y r − η y x r − η x y r η x x r ] [ H x r H y r ] ,
P x x r H x r = [ η y y r ∂ 2 H x r ∂ x 2 + η z z r ∂ 2 H x r ∂ y 2 − η y x r ∂ 2 H x r ∂ y ∂ x + k 0 2 H x r ] | x = x i , y = y j , = ∑ i = 1 n x − 1 ∑ j = 1 n y − 1 [ ∑ p = 0 n x ∑ q = 0 n y { η y y r θ p r   ( 2 ) ( x ) ψ q r ( y ) + η z z r θ p r ( x ) ψ q r   ( 2 ) ( y ) − η y x r θ p r   ( 1 ) ( x ) ψ q r   ( 1 ) ( y ) ... + k 0 2 θ p r ( x ) ψ q r ( y ) } ] | x = x i , y = y j ⋅ [ H x , p q r ] ,
P x y r H y r = [ ( η y y r − η z z r ) ∂ 2 H y r ∂ x ∂ y − η y x r ∂ 2 H y r ∂ y 2 ] | x = x i , y = y j , = ∑ i = 1 n x − 1 ∑ j = 1 n y − 1 [ ∑ p = 0 n x ∑ q = 0 n y { ( η y y r − η z z r ) θ p r   ( 1 ) ( x ) ψ q r   ( 1 ) ( y ) − η y x r θ p r ( x ) ψ q r   ( 2 ) ( y ) } ] | x = x i , y = y j ⋅ [ H y , p q r ] ,
P y x r H x r = [ ( η x x r − η z z r ) ∂ 2 H x r ∂ x ∂ y − η x y r ∂ 2 H x r ∂ x 2 ] | x = x i , y = y j , = ∑ i = 1 n x − 1 ∑ j = 1 n y − 1 [ ∑ p = 0 n x ∑ q = 0 n y { ( η x x r − η z z r ) θ p r   ( 1 ) ( x ) ψ q r   ( 1 ) ( y ) − η x y r θ p r   ( 2 ) ( x ) ψ q r ( y ) } ] | x = x i , y = y j ⋅ [ H x , p q r ] ,
P y y r H y r = [ η z z r ∂ 2 H y r ∂ x 2 + η x x r ∂ 2 H y r ∂ y 2 + η x y r ∂ 2 H y r ∂ y ∂ x + k 0 2 H y r ] | x = x i , y = y j , = ∑ i = 1 n x − 1 ∑ j = 1 n y − 1 [ ∑ p = 0 n x ∑ q = 0 n y { η z z r θ p r   ( 2 ) ( x ) ψ q r ( y ) + η x x r θ p r ( x ) ψ q r   ( 2 ) ( y ) + η x y r θ p r   ( 1 ) ( x ) ψ q r   ( 1 ) ( y ) ... + k 0 2 θ p r ( x ) ψ q r ( y ) } ] | x = x i , y = y j ⋅ [ H y , p q r ] ,
[ 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 [ η 1 0 0 0 0 η 2 0 0 0 0 ⋱ 0 0 0 0 η m ] [ H 1 H 2 ⋮ H m ] ,
Q r = [ P x x r P x y r P y x r P y y r ] ,   Η r = [ H x r H y r ] ,     η r = [ η y y r − η y x r − η x y r η x x 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 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 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 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 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 1 ≤ p ≤ N -1
θ p ( α x ) = x L v ( α x ) α ( x − x p ) [ x L v ' ( α x ) ] | x = x p e − α ( x − x p ) / 2 ,
α =   max 0 ≤ i ≤ n { x i } M
γ y s i = k 0 n y i 2 − ε x x , s ,   i =1, 2, 3 ,
γ y a i = k 0 n y i 2 − ε x x , a ,   i =1, 2, 3,
γ x i = k 0 n e q 2 − n y i 2 ,   i =1, 3,
H ( y t + M y s i ) H ( y t ) = exp [ − ∫ y t y t + M y s i γ y s i   d y ] = κ ,   i =1, 2, 3,
H ( y t + M y a i ) H ( y t ) = exp [ − ∫ y t y t + M y a i γ y a i   d y ] = κ ,   i =1, 2, 3,
H ( x t + M x i ) H ( x t ) = exp [ − ∫ x t x t + M x i γ x i   d x ] = κ ,   i =1, 3,
[ ε ] = ε 0 [ n f 2 j ς 0 − j ς n f 2 0 0 0 n f 2 ]
[ ε ] = ε 0 [ ε x x ε x y 0 ε y x ε y y 0 0 0 ε z z ] = ε 0 [ n o 2 + ( n e 2 − n o 2 ) cos 2 φ ( n e 2 − n o 2 ) cos φ sin φ 0 ( n e 2 − n o 2 ) cos φ sin φ n o 2 + ( n e 2 − n o 2 ) sin 2 φ 0 0 0 n o 2 ] ,

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