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 OSA

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  3. Q. Wang, G. Farrell, and Y. Semenova, “Modeling liquid-crystal devices with the three-dimensional full-vector beam propagation method,” J. Opt. Soc. Am. A 23(8), 2014–2019 (2006).
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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).
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  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).
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  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]
  10. 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]
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  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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  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).
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    [CrossRef]
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    [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]
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2010

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

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]

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]

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]

M. F. O. Hameed, S. S. A. Obayya, K. Al-Begain, M. I. Abo el Maaty, and A. M. Nasr, “Modal properties of an index guiding nematic liquid crystal based photonic crystal fiber,” J. Lightwave Technol. 27(21), 4754–4762 (2009).
[CrossRef]

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]

2008

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. 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]

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]

2007

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

2006

Q. Wang, G. Farrell, and Y. Semenova, “Modeling liquid-crystal devices with the three-dimensional full-vector beam propagation method,” J. Opt. Soc. Am. A 23(8), 2014–2019 (2006).
[CrossRef]

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]

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]

2005

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

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

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).
[CrossRef]

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

2001

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).
[CrossRef]

2000

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

Y. Tsuji, M. Koshiba, and N. Takimoto, “Finite element beam propagation method for anisotropic optical waveguides,” J. Lightwave Technol. 17(4), 723–728 (1999).
[CrossRef]

1998

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

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

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

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.

M. F. O. Hameed, S. S. A. Obayya, K. Al-Begain, M. I. Abo el Maaty, and A. M. Nasr, “Modal properties of an index guiding nematic liquid crystal based photonic crystal fiber,” J. Lightwave Technol. 27(21), 4754–4762 (2009).
[CrossRef]

Al-Begain, K.

M. F. O. Hameed, S. S. A. Obayya, K. Al-Begain, M. I. Abo el Maaty, and A. M. Nasr, “Modal properties of an index guiding nematic liquid crystal based photonic crystal fiber,” J. Lightwave Technol. 27(21), 4754–4762 (2009).
[CrossRef]

Ando, T.

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).
[CrossRef]

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.

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]

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]

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.

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]

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.

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]

De Cort, W.

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]

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.

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]

Farrell, G.

Q. Wang, G. Farrell, and Y. Semenova, “Modeling liquid-crystal devices with the three-dimensional full-vector beam propagation method,” J. Opt. Soc. Am. A 23(8), 2014–2019 (2006).
[CrossRef]

Fernandez, F. A.

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]

Fernandez, F. A. Í.

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]

Frasson, A. M. F.

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]

Hameed, M. F. O.

M. F. O. Hameed, S. S. A. Obayya, K. Al-Begain, M. I. Abo el Maaty, and A. M. Nasr, “Modal properties of an index guiding nematic liquid crystal based photonic crystal fiber,” J. Lightwave Technol. 27(21), 4754–4762 (2009).
[CrossRef]

Hernandez-Figueroa, H. E.

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]

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.

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]

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, 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]

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]

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.

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]

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]

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.

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).
[CrossRef]

Y. Tsuji, M. Koshiba, and N. Takimoto, “Finite element beam propagation method for anisotropic optical waveguides,” J. Lightwave Technol. 17(4), 723–728 (1999).
[CrossRef]

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.

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]

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.

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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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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).
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[CrossRef]

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[CrossRef]

J. Lightwave Technol.

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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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]

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[CrossRef]

M. F. O. Hameed, S. S. A. Obayya, K. Al-Begain, M. I. Abo el Maaty, and A. M. Nasr, “Modal properties of an index guiding nematic liquid crystal based photonic crystal fiber,” J. Lightwave Technol. 27(21), 4754–4762 (2009).
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[CrossRef]

Y. Tsuji, M. Koshiba, and N. Takimoto, “Finite element beam propagation method for anisotropic optical waveguides,” J. Lightwave Technol. 17(4), 723–728 (1999).
[CrossRef]

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).
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J. Opt. Soc. Am. A

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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.

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.

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.

Fig. 4
Fig. 4

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

Fig. 5
Fig. 5

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

Fig. 6
Fig. 6

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

Fig. 7
Fig. 7

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

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 ^ .

Fig. 9
Fig. 9

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

Fig. 10
Fig. 10

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

Tables (3)

Tables Icon

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

Tables Icon

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

Tables Icon

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

Equations (32)

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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 ] ,
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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