Abstract

A planar optical waveguide mode solver is established based on a finite-element (FE) formulation for determining the guided and leaky modes that exist on waveguides made of anisotropic materials with an arbitrary permittivity tensor, for example, with arbitrary optic-axis orientation in the uniaxially anisotropic material case. Correct numerical determination of the complex effective index, especially its imaginary part that gives the modal leakage, is particularly emphasized referring to available analytical solutions. For the situation when the optic axis changes its direction only in the plane parallel to the waveguide interface planes, analytical characteristic equations for solving purely guided and leaky modes are separately derived in a more systematic manner compared with prior analytical formulae reported more than three decades ago, with the obtained complex effective indices agreeing excellently with FE solutions. It is found that in the FE analysis of leaky modes, the thickness of the perfectly matched layer (PML) and the PML theoretical reflection coefficient should be properly chosen. The FE formulation is based on either the three electric-field components or the three magnetic-field components using quadratic nodal bases, resulting in a quadratic eigenvalue equation that is then solved by the shift-and-invert Arnoldi method.

© 2014 Optical Society of America

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  1. R. V. Schmidt and H. Kogelnik, “Electro-optically switched coupler with stepped Delta β reversal using Ti-diffused LiNbO3 waveguides,” Appl. Phys. Lett. 28, 503–506 (1976).
    [CrossRef]
  2. A. Knoesen, T. K. Gaylord, and M. G. Moharam, “Hybrid guided modes in uniaxial dielectric planar wave-guides,” J. Lightwave Technol. 6, 1083–1104 (1988).
    [CrossRef]
  3. K. Yamanouchi, T. Kamiya, and K. Shibayama, “New leaky surface waves in anisotropic metal-diffused optical waveguides,” IEEE Trans. Microwave Theory Tech. 26, 298–305 (1978).
    [CrossRef]
  4. S. K. Sheem, W. K. Burns, and A. F. Milton, “Leaky mode propagation in Ti-diffused LiNbO3, and LiNbO3, waveguides,” Opt. Lett. 3, 76–78 (1978).
    [CrossRef]
  5. W. K. Burns, S. K. Sheem, and A. F. Milton, “Approximate calculation of leaky-mode loss coefficients for Ti-diffused LiNbO3 waveguides,” IEEE J. Quantum Electron. QE-15, 1282–1289 (1979).
    [CrossRef]
  6. J. Ctyroky and M. Cada, “Guided and semileaky modes in anisotropic optical waveguides of the LiNbO3 type,” Opt. Commun. 27, 353–357 (1978).
    [CrossRef]
  7. S. Yamamoto, Y. Koyamada, and T. Makimoto, “Normal-mode analysis of anisotropic and gyrotropic thin-film waveguides for integrated optics,” J. Appl. Phys. 43, 5090–5097 (1972).
    [CrossRef]
  8. W. K. Burns and J. Warnert, “Mode dispersion in uniaxial optical waveguides,” J. Opt. Soc. Am. 64, 441–446 (1974).
    [CrossRef]
  9. Y. Satomura, M. Matsuhara, and N. Kumagai, “Analysis of electromagnetic-wave modes in anisotropic slab waveguide,” IEEE Trans. Microwave Theory Tech. 22, 86–92 (1974).
    [CrossRef]
  10. E. A. Kolosovskii, D. V. Petrov, A. V. Tsarev, and I. B. Iakovin, “An exact method for analysing light propagation in anisotropic inhomogeneous optical waveguides,” Opt. Commun. 43, 21–25 (1982).
    [CrossRef]
  11. D. Marcuse, “Electrooptic coupling between TE and TM modes in anisotropic slabs,” IEEE J. Quantum Electron. QE-11, 759–767 (1975).
    [CrossRef]
  12. D. Marcuse, “Modes of a symmetric slab optical waveguide in birefringent media–Part 1: optical axis not in plane of slab,” IEEE J. Quantum Electron. QE-14, 736–741 (1978).
    [CrossRef]
  13. S. Yamamoto and Y. Okamoto, “Guided-radiation mode interaction in off-axis propagation in anisotropic optical waveguides with application to direct-intensity modulators,” J. Appl. Phys. 50, 2555–2564 (1979).
    [CrossRef]
  14. D. Marcuse and I. P. Kaminow, “Modes of a symmetric slab optical waveguide in birefringent media, Part II: slab with coplanar optical axis,” IEEE J. Quantum Electron. QE-15, 92–101 (1979).
    [CrossRef]
  15. D. P. G. Russo and J. H. Harris, “Wave propagation in anisotropic thin-film optical waveguides,” J. Opt. Soc. Am. 63, 138–145 (1973).
    [CrossRef]
  16. M. S. Kharusi, “Uniaxial and biaxial anisotropy in thin-film optical waveguides,” J. Opt. Soc. Am. 64, 27–35 (1974).
    [CrossRef]
  17. J. Ctyroky and M. Cada, “Generalized WKB method for the analysis of light propagation in inhomogeneous anisotropic optical waveguides,” IEEE J. Quantum Electron. QE-17, 1064–1070 (1981).
    [CrossRef]
  18. L. Torner, F. Canal, and J. Hernandez-Marco, “Leaky modes in multilayer uniaxial optical waveguides,” Appl. Opt. 29, 2805–2814 (1990).
    [CrossRef]
  19. M. Lu and M. M. Fejer, “Anisotropic dielectric waveguides,” J. Opt. Soc. Am. A 10, 246–261 (1993).
    [CrossRef]
  20. A. B. Yakovle, G. W. Hanson, and R. L. Byer, “Fundamental modal phenomena on isotropic and anisotropic planar slab dielectric waveguides,” IEEE Trans. Antennas Propag. 51, 888–897 (2003).
    [CrossRef]
  21. M. A. Boroujeni and M. Shahabadi, “Modal analysis of multilayer planar lossy anisotropic optical waveguides,” J. Opt. A 8, 856–863 (2006).
    [CrossRef]
  22. T. A. Maldonado and T. K. Gaylord, “Hybrid guided modes in biaxial planar waveguides,” J. Lightwave Technol. 14, 486–499 (1996).
    [CrossRef]
  23. L. Torner, J. Recolons, and J. P. Torres, “Guided-to-leaky mode transition in uniaxial optical slab waveguides,” J. Lightwave Technol. 11, 1592–1600 (1993).
    [CrossRef]
  24. F. L. Teixeira and W. C. Chew, “General closed-form PML constitutive tensors to match arbitrary bianisotropic and dispersive linear media,” IEEE Microwave Guided Wave Lett. 8, 223–225 (1998).
    [CrossRef]
  25. Y. Tsuji and M. Koshiba, “Guided-mode and leaky-mode analysis by imaginary distance beam propagation method based on finite element scheme,” J. Lightwave Technol. 18, 618–623 (2000).
    [CrossRef]
  26. 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]
  27. M. Koshiba and M. Suzuki, “Numerical-analysis of planar arbitrarily anisotropic diffused optical waveguides using finite-element method,” Electron. Lett. 18, 579–581 (1982).
    [CrossRef]
  28. M. Koshiba, H. Kumagami, and M. Suzuki, “Finite-element solution of planar arbitrarily anisotropic diffused optical waveguides,” J. Lightwave Technol. 3, 773–778 (1985).
    [CrossRef]
  29. M. Koshiba, H. Kumagami, and M. Suzuki, “Correction to finite-element solution of planar arbitrarily anisotropic diffused optical waveguides,” J. Lightwave Technol. 4, 100 (1986).
    [CrossRef]
  30. A. P. Zhao and S. R. Cvetkovic, “Finite-element analysis of hybrid modes in uniaxial planar waveguides by a simple iterative method,” Opt. Lett. 20, 139–141 (1995).
    [CrossRef]
  31. J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic-waves,” J. Comput. Phys. 114, 185–200 (1994).
    [CrossRef]
  32. K. Saitoh and M. Koshiba, “Full-vectorial finite element beam propagation method with perfectly matched layers for anisotropic optical waveguides,” J. Lightwave Technol. 19, 405–413 (2001).
    [CrossRef]
  33. C. H. Lai and H. C. Chang, “Effect of perfectly matched layer reflection coefficient on modal analysis of leaky waveguide modes,” Opt. Express 19, 562–569 (2011).
    [CrossRef]
  34. O. C. Zienkiewitz, The Finite Element Method (McGraw-Hill, 1977).
  35. J. F. Lee, D. K. Sun, and Z. J. Cendes, “Full-wave analysis of dielectric waveguides using tangential vector finite elements,” IEEE Trans. Microwave Theory Tech. 39, 1262–1271 (1991).
    [CrossRef]
  36. K. Kawano and T. Kitoh, Introduction to Optical Waveguide Analysis: Solving Maxwell’s Equation and the Schrödinger Equation (Wiley, 2001).
  37. C. M. Krowne, “Theoretical considerations for finding anisotropic permittivity in layered ferroelectric/ferromagnetic structures from full-wave electromagnetic simulations,” Microw. Opt. Technol. Lett. 28, 63–69 (2001).
    [CrossRef]
  38. R. B. Lehoucq and D. C. Sorensen, “Deflation techniques for an implicitly restarted Arnoldi iteration,” SIAM J. Matrix Anal. Appl. 17, 789–821 (1996).
    [CrossRef]
  39. R. B. Lehoucq, D. C. Sorensen, and C. Yang, ARPACK Users’ Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods (SIAM, 1998).
  40. M. Koshiba and Y. Tsuji, “Curvilinear hybrid edge/nodal elements with triangular shape for guided-wave problems,” J. Lightwave Technol. 18, 737–743 (2000).
    [CrossRef]
  41. J. Hu and C. R. Menyuk, “Understanding leaky modes: slab waveguide revisited,” Adv. Opt. Photon. 1, 58–106 (2009).
    [CrossRef]
  42. R. D. Kekatpure, A. C. Hryciw, E. S. Barnard, and M. L. Brongersma, “Solving dielectric and plasmonic waveguide dispersion relations on a pocket calculator,” Opt. Express 17, 24112–24129 (2009).
    [CrossRef]

2011 (1)

2009 (2)

2006 (1)

M. A. Boroujeni and M. Shahabadi, “Modal analysis of multilayer planar lossy anisotropic optical waveguides,” J. Opt. A 8, 856–863 (2006).
[CrossRef]

2003 (1)

A. B. Yakovle, G. W. Hanson, and R. L. Byer, “Fundamental modal phenomena on isotropic and anisotropic planar slab dielectric waveguides,” IEEE Trans. Antennas Propag. 51, 888–897 (2003).
[CrossRef]

2001 (3)

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. M. Krowne, “Theoretical considerations for finding anisotropic permittivity in layered ferroelectric/ferromagnetic structures from full-wave electromagnetic simulations,” Microw. Opt. Technol. Lett. 28, 63–69 (2001).
[CrossRef]

K. Saitoh and M. Koshiba, “Full-vectorial finite element beam propagation method with perfectly matched layers for anisotropic optical waveguides,” J. Lightwave Technol. 19, 405–413 (2001).
[CrossRef]

2000 (2)

1998 (1)

F. L. Teixeira and W. C. Chew, “General closed-form PML constitutive tensors to match arbitrary bianisotropic and dispersive linear media,” IEEE Microwave Guided Wave Lett. 8, 223–225 (1998).
[CrossRef]

1996 (2)

T. A. Maldonado and T. K. Gaylord, “Hybrid guided modes in biaxial planar waveguides,” J. Lightwave Technol. 14, 486–499 (1996).
[CrossRef]

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

1995 (1)

1994 (1)

J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic-waves,” J. Comput. Phys. 114, 185–200 (1994).
[CrossRef]

1993 (2)

L. Torner, J. Recolons, and J. P. Torres, “Guided-to-leaky mode transition in uniaxial optical slab waveguides,” J. Lightwave Technol. 11, 1592–1600 (1993).
[CrossRef]

M. Lu and M. M. Fejer, “Anisotropic dielectric waveguides,” J. Opt. Soc. Am. A 10, 246–261 (1993).
[CrossRef]

1991 (1)

J. F. Lee, D. K. Sun, and Z. J. Cendes, “Full-wave analysis of dielectric waveguides using tangential vector finite elements,” IEEE Trans. Microwave Theory Tech. 39, 1262–1271 (1991).
[CrossRef]

1990 (1)

1988 (1)

A. Knoesen, T. K. Gaylord, and M. G. Moharam, “Hybrid guided modes in uniaxial dielectric planar wave-guides,” J. Lightwave Technol. 6, 1083–1104 (1988).
[CrossRef]

1986 (1)

M. Koshiba, H. Kumagami, and M. Suzuki, “Correction to finite-element solution of planar arbitrarily anisotropic diffused optical waveguides,” J. Lightwave Technol. 4, 100 (1986).
[CrossRef]

1985 (1)

M. Koshiba, H. Kumagami, and M. Suzuki, “Finite-element solution of planar arbitrarily anisotropic diffused optical waveguides,” J. Lightwave Technol. 3, 773–778 (1985).
[CrossRef]

1982 (2)

M. Koshiba and M. Suzuki, “Numerical-analysis of planar arbitrarily anisotropic diffused optical waveguides using finite-element method,” Electron. Lett. 18, 579–581 (1982).
[CrossRef]

E. A. Kolosovskii, D. V. Petrov, A. V. Tsarev, and I. B. Iakovin, “An exact method for analysing light propagation in anisotropic inhomogeneous optical waveguides,” Opt. Commun. 43, 21–25 (1982).
[CrossRef]

1981 (1)

J. Ctyroky and M. Cada, “Generalized WKB method for the analysis of light propagation in inhomogeneous anisotropic optical waveguides,” IEEE J. Quantum Electron. QE-17, 1064–1070 (1981).
[CrossRef]

1979 (3)

S. Yamamoto and Y. Okamoto, “Guided-radiation mode interaction in off-axis propagation in anisotropic optical waveguides with application to direct-intensity modulators,” J. Appl. Phys. 50, 2555–2564 (1979).
[CrossRef]

D. Marcuse and I. P. Kaminow, “Modes of a symmetric slab optical waveguide in birefringent media, Part II: slab with coplanar optical axis,” IEEE J. Quantum Electron. QE-15, 92–101 (1979).
[CrossRef]

W. K. Burns, S. K. Sheem, and A. F. Milton, “Approximate calculation of leaky-mode loss coefficients for Ti-diffused LiNbO3 waveguides,” IEEE J. Quantum Electron. QE-15, 1282–1289 (1979).
[CrossRef]

1978 (4)

J. Ctyroky and M. Cada, “Guided and semileaky modes in anisotropic optical waveguides of the LiNbO3 type,” Opt. Commun. 27, 353–357 (1978).
[CrossRef]

K. Yamanouchi, T. Kamiya, and K. Shibayama, “New leaky surface waves in anisotropic metal-diffused optical waveguides,” IEEE Trans. Microwave Theory Tech. 26, 298–305 (1978).
[CrossRef]

S. K. Sheem, W. K. Burns, and A. F. Milton, “Leaky mode propagation in Ti-diffused LiNbO3, and LiNbO3, waveguides,” Opt. Lett. 3, 76–78 (1978).
[CrossRef]

D. Marcuse, “Modes of a symmetric slab optical waveguide in birefringent media–Part 1: optical axis not in plane of slab,” IEEE J. Quantum Electron. QE-14, 736–741 (1978).
[CrossRef]

1976 (1)

R. V. Schmidt and H. Kogelnik, “Electro-optically switched coupler with stepped Delta β reversal using Ti-diffused LiNbO3 waveguides,” Appl. Phys. Lett. 28, 503–506 (1976).
[CrossRef]

1975 (1)

D. Marcuse, “Electrooptic coupling between TE and TM modes in anisotropic slabs,” IEEE J. Quantum Electron. QE-11, 759–767 (1975).
[CrossRef]

1974 (3)

1973 (1)

1972 (1)

S. Yamamoto, Y. Koyamada, and T. Makimoto, “Normal-mode analysis of anisotropic and gyrotropic thin-film waveguides for integrated optics,” J. Appl. Phys. 43, 5090–5097 (1972).
[CrossRef]

Barnard, E. S.

Berenger, J. P.

J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic-waves,” J. Comput. Phys. 114, 185–200 (1994).
[CrossRef]

Boroujeni, M. A.

M. A. Boroujeni and M. Shahabadi, “Modal analysis of multilayer planar lossy anisotropic optical waveguides,” J. Opt. A 8, 856–863 (2006).
[CrossRef]

Brongersma, M. L.

Burns, W. K.

Byer, R. L.

A. B. Yakovle, G. W. Hanson, and R. L. Byer, “Fundamental modal phenomena on isotropic and anisotropic planar slab dielectric waveguides,” IEEE Trans. Antennas Propag. 51, 888–897 (2003).
[CrossRef]

Cada, M.

J. Ctyroky and M. Cada, “Generalized WKB method for the analysis of light propagation in inhomogeneous anisotropic optical waveguides,” IEEE J. Quantum Electron. QE-17, 1064–1070 (1981).
[CrossRef]

J. Ctyroky and M. Cada, “Guided and semileaky modes in anisotropic optical waveguides of the LiNbO3 type,” Opt. Commun. 27, 353–357 (1978).
[CrossRef]

Canal, F.

Cendes, Z. J.

J. F. Lee, D. K. Sun, and Z. J. Cendes, “Full-wave analysis of dielectric waveguides using tangential vector finite elements,” IEEE Trans. Microwave Theory Tech. 39, 1262–1271 (1991).
[CrossRef]

Chang, H. C.

Chew, W. C.

F. L. Teixeira and W. C. Chew, “General closed-form PML constitutive tensors to match arbitrary bianisotropic and dispersive linear media,” IEEE Microwave Guided Wave Lett. 8, 223–225 (1998).
[CrossRef]

Ctyroky, J.

J. Ctyroky and M. Cada, “Generalized WKB method for the analysis of light propagation in inhomogeneous anisotropic optical waveguides,” IEEE J. Quantum Electron. QE-17, 1064–1070 (1981).
[CrossRef]

J. Ctyroky and M. Cada, “Guided and semileaky modes in anisotropic optical waveguides of the LiNbO3 type,” Opt. Commun. 27, 353–357 (1978).
[CrossRef]

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]

Cvetkovic, S. R.

Fejer, M. M.

Gaylord, T. K.

T. A. Maldonado and T. K. Gaylord, “Hybrid guided modes in biaxial planar waveguides,” J. Lightwave Technol. 14, 486–499 (1996).
[CrossRef]

A. Knoesen, T. K. Gaylord, and M. G. Moharam, “Hybrid guided modes in uniaxial dielectric planar wave-guides,” J. Lightwave Technol. 6, 1083–1104 (1988).
[CrossRef]

Hanson, G. W.

A. B. Yakovle, G. W. Hanson, and R. L. Byer, “Fundamental modal phenomena on isotropic and anisotropic planar slab dielectric waveguides,” IEEE Trans. Antennas Propag. 51, 888–897 (2003).
[CrossRef]

Harris, J. H.

Hernandez-Marco, J.

Hryciw, A. C.

Hu, J.

Iakovin, I. B.

E. A. Kolosovskii, D. V. Petrov, A. V. Tsarev, and I. B. Iakovin, “An exact method for analysing light propagation in anisotropic inhomogeneous optical waveguides,” Opt. Commun. 43, 21–25 (1982).
[CrossRef]

Kaminow, I. P.

D. Marcuse and I. P. Kaminow, “Modes of a symmetric slab optical waveguide in birefringent media, Part II: slab with coplanar optical axis,” IEEE J. Quantum Electron. QE-15, 92–101 (1979).
[CrossRef]

Kamiya, T.

K. Yamanouchi, T. Kamiya, and K. Shibayama, “New leaky surface waves in anisotropic metal-diffused optical waveguides,” IEEE Trans. Microwave Theory Tech. 26, 298–305 (1978).
[CrossRef]

Kawano, K.

K. Kawano and T. Kitoh, Introduction to Optical Waveguide Analysis: Solving Maxwell’s Equation and the Schrödinger Equation (Wiley, 2001).

Kekatpure, R. D.

Kharusi, M. S.

Kitoh, T.

K. Kawano and T. Kitoh, Introduction to Optical Waveguide Analysis: Solving Maxwell’s Equation and the Schrödinger Equation (Wiley, 2001).

Knoesen, A.

A. Knoesen, T. K. Gaylord, and M. G. Moharam, “Hybrid guided modes in uniaxial dielectric planar wave-guides,” J. Lightwave Technol. 6, 1083–1104 (1988).
[CrossRef]

Kogelnik, H.

R. V. Schmidt and H. Kogelnik, “Electro-optically switched coupler with stepped Delta β reversal using Ti-diffused LiNbO3 waveguides,” Appl. Phys. Lett. 28, 503–506 (1976).
[CrossRef]

Kolosovskii, E. A.

E. A. Kolosovskii, D. V. Petrov, A. V. Tsarev, and I. B. Iakovin, “An exact method for analysing light propagation in anisotropic inhomogeneous optical waveguides,” Opt. Commun. 43, 21–25 (1982).
[CrossRef]

Koshiba, M.

K. Saitoh and M. Koshiba, “Full-vectorial finite element beam propagation method with perfectly matched layers for anisotropic optical waveguides,” J. Lightwave Technol. 19, 405–413 (2001).
[CrossRef]

M. Koshiba and Y. Tsuji, “Curvilinear hybrid edge/nodal elements with triangular shape for guided-wave problems,” J. Lightwave Technol. 18, 737–743 (2000).
[CrossRef]

Y. Tsuji and M. Koshiba, “Guided-mode and leaky-mode analysis by imaginary distance beam propagation method based on finite element scheme,” J. Lightwave Technol. 18, 618–623 (2000).
[CrossRef]

M. Koshiba, H. Kumagami, and M. Suzuki, “Correction to finite-element solution of planar arbitrarily anisotropic diffused optical waveguides,” J. Lightwave Technol. 4, 100 (1986).
[CrossRef]

M. Koshiba, H. Kumagami, and M. Suzuki, “Finite-element solution of planar arbitrarily anisotropic diffused optical waveguides,” J. Lightwave Technol. 3, 773–778 (1985).
[CrossRef]

M. Koshiba and M. Suzuki, “Numerical-analysis of planar arbitrarily anisotropic diffused optical waveguides using finite-element method,” Electron. Lett. 18, 579–581 (1982).
[CrossRef]

Koyamada, Y.

S. Yamamoto, Y. Koyamada, and T. Makimoto, “Normal-mode analysis of anisotropic and gyrotropic thin-film waveguides for integrated optics,” J. Appl. Phys. 43, 5090–5097 (1972).
[CrossRef]

Krowne, C. M.

C. M. Krowne, “Theoretical considerations for finding anisotropic permittivity in layered ferroelectric/ferromagnetic structures from full-wave electromagnetic simulations,” Microw. Opt. Technol. Lett. 28, 63–69 (2001).
[CrossRef]

Kumagai, N.

Y. Satomura, M. Matsuhara, and N. Kumagai, “Analysis of electromagnetic-wave modes in anisotropic slab waveguide,” IEEE Trans. Microwave Theory Tech. 22, 86–92 (1974).
[CrossRef]

Kumagami, H.

M. Koshiba, H. Kumagami, and M. Suzuki, “Correction to finite-element solution of planar arbitrarily anisotropic diffused optical waveguides,” J. Lightwave Technol. 4, 100 (1986).
[CrossRef]

M. Koshiba, H. Kumagami, and M. Suzuki, “Finite-element solution of planar arbitrarily anisotropic diffused optical waveguides,” J. Lightwave Technol. 3, 773–778 (1985).
[CrossRef]

Lai, C. H.

Lee, J. F.

J. F. Lee, D. K. Sun, and Z. J. Cendes, “Full-wave analysis of dielectric waveguides using tangential vector finite elements,” IEEE Trans. Microwave Theory Tech. 39, 1262–1271 (1991).
[CrossRef]

Lehoucq, R. B.

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

R. B. Lehoucq, D. C. Sorensen, and C. Yang, ARPACK Users’ Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods (SIAM, 1998).

Lu, M.

Makimoto, T.

S. Yamamoto, Y. Koyamada, and T. Makimoto, “Normal-mode analysis of anisotropic and gyrotropic thin-film waveguides for integrated optics,” J. Appl. Phys. 43, 5090–5097 (1972).
[CrossRef]

Maldonado, T. A.

T. A. Maldonado and T. K. Gaylord, “Hybrid guided modes in biaxial planar waveguides,” J. Lightwave Technol. 14, 486–499 (1996).
[CrossRef]

Marcuse, D.

D. Marcuse and I. P. Kaminow, “Modes of a symmetric slab optical waveguide in birefringent media, Part II: slab with coplanar optical axis,” IEEE J. Quantum Electron. QE-15, 92–101 (1979).
[CrossRef]

D. Marcuse, “Modes of a symmetric slab optical waveguide in birefringent media–Part 1: optical axis not in plane of slab,” IEEE J. Quantum Electron. QE-14, 736–741 (1978).
[CrossRef]

D. Marcuse, “Electrooptic coupling between TE and TM modes in anisotropic slabs,” IEEE J. Quantum Electron. QE-11, 759–767 (1975).
[CrossRef]

Matsuhara, M.

Y. Satomura, M. Matsuhara, and N. Kumagai, “Analysis of electromagnetic-wave modes in anisotropic slab waveguide,” IEEE Trans. Microwave Theory Tech. 22, 86–92 (1974).
[CrossRef]

Menyuk, C. R.

Milton, A. F.

W. K. Burns, S. K. Sheem, and A. F. Milton, “Approximate calculation of leaky-mode loss coefficients for Ti-diffused LiNbO3 waveguides,” IEEE J. Quantum Electron. QE-15, 1282–1289 (1979).
[CrossRef]

S. K. Sheem, W. K. Burns, and A. F. Milton, “Leaky mode propagation in Ti-diffused LiNbO3, and LiNbO3, waveguides,” Opt. Lett. 3, 76–78 (1978).
[CrossRef]

Moharam, M. G.

A. Knoesen, T. K. Gaylord, and M. G. Moharam, “Hybrid guided modes in uniaxial dielectric planar wave-guides,” J. Lightwave Technol. 6, 1083–1104 (1988).
[CrossRef]

Okamoto, Y.

S. Yamamoto and Y. Okamoto, “Guided-radiation mode interaction in off-axis propagation in anisotropic optical waveguides with application to direct-intensity modulators,” J. Appl. Phys. 50, 2555–2564 (1979).
[CrossRef]

Petrov, D. V.

E. A. Kolosovskii, D. V. Petrov, A. V. Tsarev, and I. B. Iakovin, “An exact method for analysing light propagation in anisotropic inhomogeneous optical waveguides,” Opt. Commun. 43, 21–25 (1982).
[CrossRef]

Recolons, J.

L. Torner, J. Recolons, and J. P. Torres, “Guided-to-leaky mode transition in uniaxial optical slab waveguides,” J. Lightwave Technol. 11, 1592–1600 (1993).
[CrossRef]

Russo, D. P. G.

Saitoh, K.

Satomura, Y.

Y. Satomura, M. Matsuhara, and N. Kumagai, “Analysis of electromagnetic-wave modes in anisotropic slab waveguide,” IEEE Trans. Microwave Theory Tech. 22, 86–92 (1974).
[CrossRef]

Schmidt, R. V.

R. V. Schmidt and H. Kogelnik, “Electro-optically switched coupler with stepped Delta β reversal using Ti-diffused LiNbO3 waveguides,” Appl. Phys. Lett. 28, 503–506 (1976).
[CrossRef]

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]

Shahabadi, M.

M. A. Boroujeni and M. Shahabadi, “Modal analysis of multilayer planar lossy anisotropic optical waveguides,” J. Opt. A 8, 856–863 (2006).
[CrossRef]

Sheem, S. K.

W. K. Burns, S. K. Sheem, and A. F. Milton, “Approximate calculation of leaky-mode loss coefficients for Ti-diffused LiNbO3 waveguides,” IEEE J. Quantum Electron. QE-15, 1282–1289 (1979).
[CrossRef]

S. K. Sheem, W. K. Burns, and A. F. Milton, “Leaky mode propagation in Ti-diffused LiNbO3, and LiNbO3, waveguides,” Opt. Lett. 3, 76–78 (1978).
[CrossRef]

Shibayama, K.

K. Yamanouchi, T. Kamiya, and K. Shibayama, “New leaky surface waves in anisotropic metal-diffused optical waveguides,” IEEE Trans. Microwave Theory Tech. 26, 298–305 (1978).
[CrossRef]

Sorensen, D. C.

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

R. B. Lehoucq, D. C. Sorensen, and C. Yang, ARPACK Users’ Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods (SIAM, 1998).

Sun, D. K.

J. F. Lee, D. K. Sun, and Z. J. Cendes, “Full-wave analysis of dielectric waveguides using tangential vector finite elements,” IEEE Trans. Microwave Theory Tech. 39, 1262–1271 (1991).
[CrossRef]

Suzuki, M.

M. Koshiba, H. Kumagami, and M. Suzuki, “Correction to finite-element solution of planar arbitrarily anisotropic diffused optical waveguides,” J. Lightwave Technol. 4, 100 (1986).
[CrossRef]

M. Koshiba, H. Kumagami, and M. Suzuki, “Finite-element solution of planar arbitrarily anisotropic diffused optical waveguides,” J. Lightwave Technol. 3, 773–778 (1985).
[CrossRef]

M. Koshiba and M. Suzuki, “Numerical-analysis of planar arbitrarily anisotropic diffused optical waveguides using finite-element method,” Electron. Lett. 18, 579–581 (1982).
[CrossRef]

Teixeira, F. L.

F. L. Teixeira and W. C. Chew, “General closed-form PML constitutive tensors to match arbitrary bianisotropic and dispersive linear media,” IEEE Microwave Guided Wave Lett. 8, 223–225 (1998).
[CrossRef]

Torner, L.

L. Torner, J. Recolons, and J. P. Torres, “Guided-to-leaky mode transition in uniaxial optical slab waveguides,” J. Lightwave Technol. 11, 1592–1600 (1993).
[CrossRef]

L. Torner, F. Canal, and J. Hernandez-Marco, “Leaky modes in multilayer uniaxial optical waveguides,” Appl. Opt. 29, 2805–2814 (1990).
[CrossRef]

Torres, J. P.

L. Torner, J. Recolons, and J. P. Torres, “Guided-to-leaky mode transition in uniaxial optical slab waveguides,” J. Lightwave Technol. 11, 1592–1600 (1993).
[CrossRef]

Tsarev, A. V.

E. A. Kolosovskii, D. V. Petrov, A. V. Tsarev, and I. B. Iakovin, “An exact method for analysing light propagation in anisotropic inhomogeneous optical waveguides,” Opt. Commun. 43, 21–25 (1982).
[CrossRef]

Tsuji, Y.

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]

Warnert, J.

Yakovle, A. B.

A. B. Yakovle, G. W. Hanson, and R. L. Byer, “Fundamental modal phenomena on isotropic and anisotropic planar slab dielectric waveguides,” IEEE Trans. Antennas Propag. 51, 888–897 (2003).
[CrossRef]

Yamamoto, S.

S. Yamamoto and Y. Okamoto, “Guided-radiation mode interaction in off-axis propagation in anisotropic optical waveguides with application to direct-intensity modulators,” J. Appl. Phys. 50, 2555–2564 (1979).
[CrossRef]

S. Yamamoto, Y. Koyamada, and T. Makimoto, “Normal-mode analysis of anisotropic and gyrotropic thin-film waveguides for integrated optics,” J. Appl. Phys. 43, 5090–5097 (1972).
[CrossRef]

Yamanouchi, K.

K. Yamanouchi, T. Kamiya, and K. Shibayama, “New leaky surface waves in anisotropic metal-diffused optical waveguides,” IEEE Trans. Microwave Theory Tech. 26, 298–305 (1978).
[CrossRef]

Yang, C.

R. B. Lehoucq, D. C. Sorensen, and C. Yang, ARPACK Users’ Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods (SIAM, 1998).

Zhao, A. P.

Zienkiewitz, O. C.

O. C. Zienkiewitz, The Finite Element Method (McGraw-Hill, 1977).

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]

Adv. Opt. Photon. (1)

Appl. Opt. (1)

Appl. Phys. Lett. (1)

R. V. Schmidt and H. Kogelnik, “Electro-optically switched coupler with stepped Delta β reversal using Ti-diffused LiNbO3 waveguides,” Appl. Phys. Lett. 28, 503–506 (1976).
[CrossRef]

Electron. Lett. (1)

M. Koshiba and M. Suzuki, “Numerical-analysis of planar arbitrarily anisotropic diffused optical waveguides using finite-element method,” Electron. Lett. 18, 579–581 (1982).
[CrossRef]

IEEE J. Quantum Electron. (5)

W. K. Burns, S. K. Sheem, and A. F. Milton, “Approximate calculation of leaky-mode loss coefficients for Ti-diffused LiNbO3 waveguides,” IEEE J. Quantum Electron. QE-15, 1282–1289 (1979).
[CrossRef]

D. Marcuse and I. P. Kaminow, “Modes of a symmetric slab optical waveguide in birefringent media, Part II: slab with coplanar optical axis,” IEEE J. Quantum Electron. QE-15, 92–101 (1979).
[CrossRef]

D. Marcuse, “Electrooptic coupling between TE and TM modes in anisotropic slabs,” IEEE J. Quantum Electron. QE-11, 759–767 (1975).
[CrossRef]

D. Marcuse, “Modes of a symmetric slab optical waveguide in birefringent media–Part 1: optical axis not in plane of slab,” IEEE J. Quantum Electron. QE-14, 736–741 (1978).
[CrossRef]

J. Ctyroky and M. Cada, “Generalized WKB method for the analysis of light propagation in inhomogeneous anisotropic optical waveguides,” IEEE J. Quantum Electron. QE-17, 1064–1070 (1981).
[CrossRef]

IEEE Microwave Guided Wave Lett. (1)

F. L. Teixeira and W. C. Chew, “General closed-form PML constitutive tensors to match arbitrary bianisotropic and dispersive linear media,” IEEE Microwave Guided Wave Lett. 8, 223–225 (1998).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

A. B. Yakovle, G. W. Hanson, and R. L. Byer, “Fundamental modal phenomena on isotropic and anisotropic planar slab dielectric waveguides,” IEEE Trans. Antennas Propag. 51, 888–897 (2003).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (3)

J. F. Lee, D. K. Sun, and Z. J. Cendes, “Full-wave analysis of dielectric waveguides using tangential vector finite elements,” IEEE Trans. Microwave Theory Tech. 39, 1262–1271 (1991).
[CrossRef]

Y. Satomura, M. Matsuhara, and N. Kumagai, “Analysis of electromagnetic-wave modes in anisotropic slab waveguide,” IEEE Trans. Microwave Theory Tech. 22, 86–92 (1974).
[CrossRef]

K. Yamanouchi, T. Kamiya, and K. Shibayama, “New leaky surface waves in anisotropic metal-diffused optical waveguides,” IEEE Trans. Microwave Theory Tech. 26, 298–305 (1978).
[CrossRef]

J. Appl. Phys. (2)

S. Yamamoto, Y. Koyamada, and T. Makimoto, “Normal-mode analysis of anisotropic and gyrotropic thin-film waveguides for integrated optics,” J. Appl. Phys. 43, 5090–5097 (1972).
[CrossRef]

S. Yamamoto and Y. Okamoto, “Guided-radiation mode interaction in off-axis propagation in anisotropic optical waveguides with application to direct-intensity modulators,” J. Appl. Phys. 50, 2555–2564 (1979).
[CrossRef]

J. Comput. Phys. (1)

J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic-waves,” J. Comput. Phys. 114, 185–200 (1994).
[CrossRef]

J. Lightwave Technol. (8)

K. Saitoh and M. Koshiba, “Full-vectorial finite element beam propagation method with perfectly matched layers for anisotropic optical waveguides,” J. Lightwave Technol. 19, 405–413 (2001).
[CrossRef]

Y. Tsuji and M. Koshiba, “Guided-mode and leaky-mode analysis by imaginary distance beam propagation method based on finite element scheme,” J. Lightwave Technol. 18, 618–623 (2000).
[CrossRef]

T. A. Maldonado and T. K. Gaylord, “Hybrid guided modes in biaxial planar waveguides,” J. Lightwave Technol. 14, 486–499 (1996).
[CrossRef]

L. Torner, J. Recolons, and J. P. Torres, “Guided-to-leaky mode transition in uniaxial optical slab waveguides,” J. Lightwave Technol. 11, 1592–1600 (1993).
[CrossRef]

M. Koshiba, H. Kumagami, and M. Suzuki, “Finite-element solution of planar arbitrarily anisotropic diffused optical waveguides,” J. Lightwave Technol. 3, 773–778 (1985).
[CrossRef]

M. Koshiba, H. Kumagami, and M. Suzuki, “Correction to finite-element solution of planar arbitrarily anisotropic diffused optical waveguides,” J. Lightwave Technol. 4, 100 (1986).
[CrossRef]

A. Knoesen, T. K. Gaylord, and M. G. Moharam, “Hybrid guided modes in uniaxial dielectric planar wave-guides,” J. Lightwave Technol. 6, 1083–1104 (1988).
[CrossRef]

M. Koshiba and Y. Tsuji, “Curvilinear hybrid edge/nodal elements with triangular shape for guided-wave problems,” J. Lightwave Technol. 18, 737–743 (2000).
[CrossRef]

J. Opt. A (1)

M. A. Boroujeni and M. Shahabadi, “Modal analysis of multilayer planar lossy anisotropic optical waveguides,” J. Opt. A 8, 856–863 (2006).
[CrossRef]

J. Opt. Soc. Am. (3)

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

Microw. Opt. Technol. Lett. (1)

C. M. Krowne, “Theoretical considerations for finding anisotropic permittivity in layered ferroelectric/ferromagnetic structures from full-wave electromagnetic simulations,” Microw. Opt. Technol. Lett. 28, 63–69 (2001).
[CrossRef]

Opt. Commun. (2)

E. A. Kolosovskii, D. V. Petrov, A. V. Tsarev, and I. B. Iakovin, “An exact method for analysing light propagation in anisotropic inhomogeneous optical waveguides,” Opt. Commun. 43, 21–25 (1982).
[CrossRef]

J. Ctyroky and M. Cada, “Guided and semileaky modes in anisotropic optical waveguides of the LiNbO3 type,” Opt. Commun. 27, 353–357 (1978).
[CrossRef]

Opt. Express (2)

Opt. Lett. (2)

Opt. Quantum Electron. (1)

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]

SIAM J. Matrix Anal. Appl. (1)

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

Other (3)

R. B. Lehoucq, D. C. Sorensen, and C. Yang, ARPACK Users’ Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods (SIAM, 1998).

K. Kawano and T. Kitoh, Introduction to Optical Waveguide Analysis: Solving Maxwell’s Equation and the Schrödinger Equation (Wiley, 2001).

O. C. Zienkiewitz, The Finite Element Method (McGraw-Hill, 1977).

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Figures (14)

Fig. 1.
Fig. 1.

Planar optical waveguide structure with artificial PML regions assumed in the numerical analysis shown. The propagation of the guided mode is along the z direction. The optic-axis orientation is described by the spherical-coordinate angles ϕ and θ.

Fig. 2.
Fig. 2.

Range of the allowable Re[neff] in a uniaxially anisotropic planar waveguide for (a) noF(noS)>neF(neS) and (b) noF(noS)<neF(neS).

Fig. 3.
Fig. 3.

(a) neff’s versus θ for the four modes of the uniaxially anisotropic planar waveguide with noC=noS=2.28, neC=neS=2.17, noF=2.28114, neF=2.17434, and film thickness of 6 μm when ϕ=0° (the optic axis in the xz plane). (b) Expanded plot of (a) for θ from 0° to 10°.

Fig. 4.
Fig. 4.

(a) Re[neff]’s versus θ for the four modes of the uniaxially anisotropic planar waveguide with noC=noS=2.28, neC=neS=2.17, noF=2.28114, neF=2.17434, and film thickness of 6 μm when ϕ=90° (the optic axis in the yz plane). (b) Expanded plot of (a) for θ from 0° to 10°.

Fig. 5.
Fig. 5.

Losses in dB/cm versus θ of (a) TE fundamental mode and (b) TE secondary mode in Fig. 4. The PML thickness in the FEM analysis is 30 μm. The analytical results of [14] are shown as dotted curves for comparison.

Fig. 6.
Fig. 6.

Re[Ey] profiles versus the x position of (a) TE fundamental mode when θ=5.5° and (b) TE secondary mode when θ=5° in Fig. 5. The regions, 80μm<x<50μm and 50μm<x<80μm, are PML ones.

Fig. 7.
Fig. 7.

(a) Re[neff]’s versus θ for the two modes of the uniaxially anisotropic planar waveguide with noC=noS=2.28, neC=neS=2.17, noF=2.28114, neF=2.17434, and film thickness of 3 μm when ϕ=90° (the optic axis in the yz plane). (b) Expanded plot of (a) for θ from 0° to 5°.

Fig. 8.
Fig. 8.

Losses in dB/cm versus θ of TE mode in Fig. 7. The PML thickness in the FEM analysis is 30 μm. The analytical result of [14] is shown as dotted curves for comparison.

Fig. 9.
Fig. 9.

Losses in dB/cm versus θ for the same modes of the same waveguide in Fig. 5, but the theoretical reflection coefficient R in the FEM analysis is set to be 108. The analytical results derived in this work are shown as dotted curves.

Fig. 10.
Fig. 10.

Losses in dB/cm versus θ for the same modes of the same waveguide in Figs. 5 and 9. The theoretical reflection coefficient R in the FEM analysis is taken to be 10300, and the PML thickness, dPML, is 10 or 15 μm. The analytical results derived in this work are shown as dotted curves. (a) Fundamental mode using dPML=10μm. (b) Fundamental mode using dPML=15μm. (c) Secondary mode using dPML=10μm. (d) Secondary mode using dPML=15μm.

Fig. 11.
Fig. 11.

Re[neff]’s versus θ for ϕ from 10° to 90° for the hybrid leaky fundamental mode of the same waveguide of Figs. 36.

Fig. 12.
Fig. 12.

Losses in dB/cm versus θ of the hybrid leaky fundamental mode in Fig. 11 when (a) ϕ=5°, (b) ϕ=10°, (c) ϕ=15°, (d) ϕ=20°, (e) ϕ=25°, (f) ϕ=30°, (g) ϕ=35°, (h) ϕ=40°, (i) ϕ=45°, (j) ϕ=50°, (k) ϕ=55°, (l) ϕ=60°, (m) ϕ=65°, (n) ϕ=70°, (o) ϕ=75°, (p) ϕ=80°, (q) ϕ=85°, and (r) ϕ=90°.

Fig. 13.
Fig. 13.

neff’s versus θ for (a) first 10 modes and (b) last four modes of the uniaxially anisotropic planar waveguide with noC=noS=1.0, neC=neS=1.0, noF=2.0, and neF=1.9, and film thickness of 2 μm at λ=1μm when ϕ=90° (the optic axis in the yz plane).

Fig. 14.
Fig. 14.

(a) Re[neff]’s versus θ for the six modes of the uniaxially anisotropic planar waveguide with noC=noS=1.0, neC=neS=3.88163j0.018969, noF=2.28114, neF=2.17434, and film thickness of 0.5 μm at λ=0.6328μm when ϕ=90° (the optic axis in the yz plane). (b) Losses in dB/μm versus θ of (a).

Tables (4)

Tables Icon

Table 1. Shape Functions for Quadratic Line Elements

Tables Icon

Table 2. Comparison of FEM Numerical Data of neff for Fig. 5(a) with Analytical Results from Appendix A

Tables Icon

Table 3. Comparison of FEM Numerical Data of neff for Fig. 5(b) with Analytical Results from Appendix A

Tables Icon

Table 4. Comparison of FEM Numerical Data of neff for Fig. 8 with Analytical Results from Appendix A

Equations (116)

Equations on this page are rendered with MathJax. Learn more.

×E=jωμ0[μ]H,
×H=jωϵ0[ϵ]E,
·(ϵ0[ϵ]E)=0,
·(μ0[μ]H)=0,
[ϵ]=[ϵxxϵxyϵxzϵyxϵyyϵyzϵzxϵzyϵzz],
[μ]=[μxx000μyy000μzz],
[ϵ]PML=[syszsxϵxxszϵxysyϵxzszϵyxszsxsyϵyysxϵyzsyϵzxsxϵzysxsyszϵzz],
[μ]PML=[syszsxμxxszμxysyμxzszμyxszsxsyμyysxμyzsyμzxsxμzysxsyszμzz]=[1sxμxx000szsxsyμyy000sxsyszμzz],
sx=1jσeωϵ0ϵxx=1j3λ4πnxxdPML(ρdPML)2ln1R
×([p]×Φ)k02[q]Φ=0,
[p]=[pxxpxypxzpyxpyypyzpzxpzypzz]=[syszsxμxx000szsxsyμyy000sxsyszμzz]1=[1sx000sx000sx]1,
[q]=[qxxqxyqxzqyxqyyqyzqzxqzyqzz]=[syszsxϵxxszϵxysyϵxzszϵyxszsxsyϵyysxϵyzsyϵzxsxϵzysxsyszϵzz]=[1sxϵxxϵxyϵxzϵyxsxϵyysxϵyzϵzxsxϵzysxϵzz],
[p]=[pxxpxypxzpyxpyypyzpzxpzypzz]=[syszsxϵxxszϵxysyϵxzszϵyxszsxsyϵyysxϵyzsyϵzxsxϵzysxsyszϵzz]1=[1sxϵxxϵxyϵxzϵyxsxϵyysxϵyzϵzxsxϵzysxϵzz]1,
[q]=[qxxqxyqxzqyxqyyqyzqzxqzyqzz]=[syszsxμxx000szsxsyμyy000sxsyszμzz]=[1sx000sx000sx],
Φ(x)=[Φx(x)x^+Φy(x)y^+Φz(x)z^]exp(jk0neffz)=[m=13{N}m{Φx}mx^+m=13{N}m{Φy}my^+jm=13{N}m{Φz}mz^]exp(jk0neffz)=[{N}T{Φx}x^+{N}T{Φy}y^+j{N}T{Φz}z^]exp(jk0neffz),
F(Φ)=Γ[(×Φ)·([p]×Φ)k02Φ·[q]Φ]dx+{Φ·[n⃗×([p]×Φ)]}|x=(d+dcover+dPML)+{Φ·[n⃗×([p]×Φ)]}|x=(d+dsub+dPML),
[K]{Φ}jβ[L]{Φ}β2[M]{Φ}={0}
{Φ}={{Φt}{Φz}}={{Φx}{Φy}{Φz}},
[K]=[[Ktt][0][0][0]]=[[Kxx][Kxy][0][Kyx][Kyy][0][0][0][0]],
[L]=[[Ltt][Ltz][Lzt][0]]=[[0][Lxy][Lxz][Lyx][Lyy][Lyz][Lzx][Lzy][0]],
[M]=[[Mtt][Mtz][Mzt][Mzz]]=[[Mxx][Mxy][Mxz][Myx][Myy][Myz][Mzx][Mzy][Mzz]],
[Kxx]=e[k02qxx{N}{N}T]dx,
[Kxy]=e[k02qxy{N}{N}T]dx,
[Kyx]=e[k02qyx{N}{N}T]dx,
[Kyy]=e[k02qyy{N}{N}Tpzz{N}x{N}Tx]dx,
[Lxy]=e[pyz{N}{N}Tx]dx,
[Lxz]=e[k02qxz{N}{N}T]dx,
[Lyx]=e[pzy{N}x{N}T]dx,
[Lyy]=e[pxz{N}{N}Tx+pzx{N}x{N}T]dx,
[Lyz]=e[pzy{N}x{N}Txk02qyz{N}{N}T]dx,
[Lzx]=e[k02qzx{N}{N}T]dx,
[Lzy]=e[pyz{N}x{N}Tx+k02qzy{N}{N}T]dx,
[Mxx]=e[pyy{N}{N}T]dx,
[Mxy]=e[pyx{N}{N}T]dx,
[Mxz]=e[pyy{N}{N}Tx]dx,
[Myx]=e[pxy{N}{N}T]dx,
[Myy]=e[pxx{N}{N}T]dx,
[Myz]=e[pxy{N}{N}Tx]dx,
[Mzx]=e[pyy{N}x{N}T]dx,
[Mzy]=e[pyx{N}x{N}T]dx,
[Mzz]=e[pyy{N}x{N}Txk02qzz{N}{N}T]dx.
[A]{X}=β[B]{X}
{X}={{ψ}{Φ}}={β{Φ}{Φ}},
[A]=[j[L][K][I][0]],
[B]=[[M][0][0][I]],
([A]τ[B])1[B]{X}=1βτ{X}
[ϵ]=[ϵ](θ,ϕ)=[ϵxx(θ,ϕ)ϵxy(θ,ϕ)ϵxz(θ,ϕ)ϵyx(θ,ϕ)ϵyy(θ,ϕ)ϵyz(θ,ϕ)ϵzx(θ,ϕ)ϵzy(θ,ϕ)ϵzz(θ,ϕ)]=R(θ,ϕ)[no2000no2000ne2]RT(θ,ϕ),
R(θ,ϕ)[cosϕsinϕ0sinϕcosϕ0001][cosθ0sinθ010sinθ0cosθ]=[cosθcosϕsinϕsinθcosϕcosθsinϕcosϕsinθsinϕsinθ0cosθ].
ϵxx(θ,ϕ)=no2+(ne2no2)sin2θcos2ϕ,
ϵyy(θ,ϕ)=no2+(ne2no2)sin2θsin2ϕ,
ϵzz(θ,ϕ)=no2+(ne2no2)cos2θ,
ϵxy(θ,ϕ)=ϵyx(θ,ϕ)=(ne2no2)sin2θsinϕcosϕ,
ϵxz(θ,ϕ)=ϵzx(θ,ϕ)=(ne2no2)sinθcosθcosϕ,
ϵyz(θ,ϕ)=ϵzy(θ,ϕ)=(ne2no2)sinθcosθsinϕ.
nUB(θ,ϕ)=neF[noF2+(neF2noF2)sin2θcos2ϕneF2(neF2noF2)sin2θsin2ϕ]1/2,
nLB(θ,ϕ)=neS[noS2+(neS2noS2)sin2θcos2ϕneS2(neS2noS2)sin2θsin2ϕ]1/2.
Kx(2d)=mπ+arctanϵzzFPxϵzzCKx+arctanϵzzFQxϵzzSKx,m=0,1,2,,
Kx2(k02noF2β2),
Px2(β2k02noC2),
Qx2(β2k02noS2),
Kx(2d)=mπ+arctanPxKx+arctanQxKx,m=0,1,2,,
Kx2(k02ϵF(θ,ϕ)β2),
Px2(β2k02ϵC(θ,ϕ)),
Qx2(β2k02ϵS(θ,ϕ)),
ϵi(θ,ϕ)=ni2(θ,ϕ),i=F,C,S,
ni(θ,ϕ)=nei[ϵxxinei2+noi2ϵyyi]1/2,i=F,C,S
=nei[noi2+(nei2noi2)sin2θcos2ϕnei2(nei2noi2)sin2θsin2ϕ]1/2,i=F,C,S.
E(x)=[Ex(x)x^+Ey(x)y^+Ez(x)z^]exp(jβz),
H(x)=[Hx(x)x^+Hy(x)y^+Hz(x)z^]exp(jβz).
jβEy(x)=jωμ0Hx(x),
jβEx(x)Ez(x)x=jωμ0Hy(x),
Ey(x)x=jωμ0Hz(x),
jβHy(x)=jωϵ0ϵxxEx(x),
jβHx(x)Hz(x)x=jωϵ0[ϵyyEy(x)+ϵyzEz(x)],
Hy(x)x=jωϵ0[ϵzyEy(x)+ϵzzEz(x)].
Hx(x)=1jωμ0jβEy(x),
Hy(x)=1jωμ0[jβEx(x)Ez(x)x],
Hz(x)=1jωμ0Ey(x)x,
[β2k02ϵxx0jβx0β2k02ϵyy2x2k02ϵyzjβxk02ϵzyk02ϵzz2x2]{Ex(x)Ey(x)Ez(x)}={0}.
Ez(x)=(β2k02ϵyy2x2)Ey(x)k02ϵyz,
Ex(x)=jββ2k02ϵxxxEz(x),
jβxEx(x)k02ϵzyEy(x)k02ϵzzEz(x)2x2Ez(x)=0.
4x4Ey(x)[(β2k02ϵyy)+ϵzzϵxx(β2k02ϵxx)]2x2Ey(x)+ϵzzϵxx(β2k02ϵxx)(β2k02ϵyy+k02ϵzyϵyzϵzz)Ey(x)=0,
[2x2Ey(x)(β2k02ϵxx)Ey(x)][2x2Ey(x)(ϵzzϵxxβ2k02ϵe)Ey(x)]=0.
Ey(x)Aocos(κox)+Aecos(κex)
Ey(x)Aosin(κox)+Aesin(κex),
Ey(x)Boexp[γo(xd)]+Beexp[γe(xd)]
Ey(x)Boexp[jγo(xd)]+Beexp[γe(xd)],
κo2(k02ϵoFβ2)
κe2(k02ϵeFϵzzFϵoFβ2).
γo2±(β2k02ϵoS)
γe2(ϵzzSϵoSβ2k02ϵeS).
Exojβκo2tanθxEyo,
EzotanθEyo,
Exejβk02ϵoFcotθxEye,
Ezeκo2k02ϵoFcotθEye.
Exojβγo2tanθxEyo,
EzotanθEyo,
Exejβk02ϵoScotθxEye,
Ezeγo2k02ϵoScotθEye.
κe(2d)=2mπ+2arctan[γe+γoBoeκosin(κod)Aoeκe(1+Eoe)],m=0,1,2,,
Aoe[k02ϵoSγo2tanθ+cotθ]γoEoe[k02ϵoFκo2tanθ+cotθ]κosin(κod)[k02ϵoSγo2tanθ+cotθ]γocos(κod),
Boe[k02ϵoFκo2tanθ+cotθ]κosin(κod)Eoe[k02ϵoFκo2tanθ+cotθ]κosin(κod)[k02ϵoSγo2tanθ+cotθ]γocos(κod),
Eoeκo2cotθk02ϵoFγo2cotθk02ϵoSκo2cotθk02ϵoF+tanθ.
κe(2d)=(2m1)π+2arctan[γe+γoBoe+κocos(κod)Aoeκe(1+Eoe)],m=1,2,,
Aoe[k02ϵoSγo2tanθ+cotθ]γoEoe[k02ϵoFκo2tanθ+cotθ]κocos(κod)[k02ϵoSγo2tanθ+cotθ]γosin(κod),
Boe[k02ϵoFκo2tanθ+cotθ]κocos(κod)Eoe[k02ϵoFκo2tanθ+cotθ]κocos(κod)[k02ϵoSγo2tanθ+cotθ]γosin(κod),
Eoeκo2cotθk02ϵoFγo2cotθk02ϵoSκo2cotθk02ϵoF+tanθ.
κe(2d)=2mπ+2arctan[γe+jγoBoeκosin(κod)Aoeκe(1+Eoe)],m=0,1,2,,
Aoej[k02ϵoSγo2tanθ+cotθ]γoEoe[k02ϵoFκo2tanθ+cotθ]κosin(κod)j[k02ϵoSγo2tanθ+cotθ]γocos(κod),
Boe[k02ϵoFκo2tanθ+cotθ]κosin(κod)Eoe[k02ϵoFκo2tanθ+cotθ]κosin(κod)j[k02ϵoSγo2tanθ+cotθ]γocos(κod),
Eoeκo2cotθk02ϵoF+γo2cotθk02ϵoSκo2cotθk02ϵoF+tanθ.
κe(2d)=(2m1)π+2arctan[γe+jγoBoe+κocos(κod)Aoeκe(1+Eoe)],m=1,2,,
Aoej[k02ϵoSγo2tanθ+cotθ]γoEoe[k02ϵoFκo2tanθ+cotθ]κocos(κod)j[k02ϵoSγo2tanθ+cotθ]γosin(κod),
Boe[k02ϵoFκo2tanθ+cotθ]κocos(κod)Eoe[k02ϵoFκo2tanθ+cotθ]κocos(κod)j[k02ϵoSγo2tanθ+cotθ]γosin(κod),
Eoeκo2cotθk02ϵoF+γo2cotθk02ϵoSκo2cotθk02ϵoF+tanθ.

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