Abstract

The propagation properties of light in anisotropic optical planar waveguides with different index distributions are investigated with the analytical transfer-matrix method. Dispersion equations are analytically deduced by the method in terms of different index profiles. It is shown by examples that this method exhibits good accuracy compared with numerical methods while still holding physical insight.

© 2004 Optical Society of America

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References

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  1. D. P. Gia Russo, J. H. Harris, “Wave propagation in anisotropic thin film optical waveguides,” J. Opt. Soc. Am. 63, 138–145 (1973).
    [CrossRef]
  2. J. Ctyroky, 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]
  3. P. Yeh, “Optics of anisotropic layered media: a new 4×4 matrix algebra,” Surf. Sci. 96, 41–53 (1980).
    [CrossRef]
  4. S. Visnovsky, “Magnetic-optic effects in ultrathin structures at longitudinal and polar magnetizations,” Czech. J. Phys. 48, 1083–1104 (1998).
    [CrossRef]
  5. E. A. Kolosovsky, D. V. Petrov, A. V. Tsarev, I. B. Yakovkin, “An exact method for analyzing light propagation in anisotropic inhomogeneous optical waveguides,” Opt. Commun. 43, 21–25 (1982).
    [CrossRef]
  6. L. M. Walpita, “Solutions for planar optical waveguide equations by selecting zero elements in a characteristic matrix,” J. Opt. Soc. Am. A 2, 595–602 (1985).
    [CrossRef]
  7. L. Tsang, S. L. Chuang, “Improved coupled-mode theory for reciprocal anisotropic waveguide,” J. Lightwave Technol. 6, 304–311 (1988).
    [CrossRef]
  8. G. Tartarini, P. Bassi, S. F. Chen, M. P. De Micheli, D. B. Ostrowsky, “Calculation of hybrid modes in uniaxial planar optical waveguides: application to proton exchanged lithium niobate waveguides,” Opt. Commun. 101, 424–431 (1993).
    [CrossRef]
  9. L. Nuno, J. V. Balbastre, H. Castane, “Analysis of general lossy inhomogeneous and anisotropic waveguides by the finite-element method (FEM) using edge elements,” IEEE Trans. Microwave Theory Tech. 45, 446–449 (1997).
    [CrossRef]
  10. S. Selleri, L. Vincetti, A. Cucinotta, M. Zoboli, “Complex FEM modal solver of optical waveguides with PML boundary conditions,” Opt. Quantum Electron. 33, 359–371 (2001).
    [CrossRef]
  11. I. Bardi, O. Biro, “An efficient finite-element formulation without spurious modes for anisotropic waveguides,” IEEE Trans. Microwave Theory Tech. 39, 1133–1139 (1991).
    [CrossRef]
  12. M. Koshiba, K. Hayata, M. Suzuki, “Finite-element formulation in terms of the electric-field vector for electromagnetic waveguide problems,” IEEE Trans. Microwave Theory Tech. MTT-33, 900–905 (1985).
    [CrossRef]
  13. V. Schulz, “Adjoint high-order vectorial finite elements for nonsymmetric transversally anisotropic waveguides,” IEEE Trans. Microwave Theory Tech. 51, 1086–1095 (2003).
    [CrossRef]
  14. S. G. Garcia, T. M. Hung-Bao, R. G. Martin, B. G. Olmedo, “On the application of finite methods in time domain to anisotropic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 44, 2195–2206 (1996).
    [CrossRef]
  15. P. A. Koukoutsaki, I. G. Tigelis, A. B. Manenkov, “Guided-mode analysis by the Lanczos–Fourier expansion,” J. Opt. Soc. Am. A 19, 2293–2300 (2002).
    [CrossRef]
  16. H. P. Uranus, H. J. W. M. Hoekstra, E. Vangroesen, “Finite difference scheme for planar waveguides with arbitrary index profiles and its implementation for anisotropic waveguides with a diagonal permittivity tensor,” Opt. Quantum Electron. 35, 407–427 (2003).
    [CrossRef]
  17. S. Gaal, “Surface integral method to determine guided modes in uniaxially anisotropic embedded waveguides,” Opt. Quantum Electron. 31, 763–780 (1999).
    [CrossRef]
  18. L. Zhan, Z. Cao, “Exact dispersion equation of a graded refractive-index optical waveguide based on the equivalent attenuated vector,” J. Opt. Soc. Am. A 15, 713–716 (1998).
    [CrossRef]
  19. Z. Cao, Y. Jiang, Q. S. Shen, X. M. Dou, Y. L. Chen, “Exact analytical method for planar optical waveguides with arbitrary index profile,” J. Opt. Soc. Am. A 16, 2209–2212 (1999).
    [CrossRef]
  20. Z. Cao, Q. Liu, Q. S. Shen, X. M. Dou, Y. L. Chen, “Quantization scheme for arbitrary one-dimensional potential wells,” Phys. Rev. A 63, 054103 (2001).
    [CrossRef]

2003 (2)

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

H. P. Uranus, H. J. W. M. Hoekstra, E. Vangroesen, “Finite difference scheme for planar waveguides with arbitrary index profiles and its implementation for anisotropic waveguides with a diagonal permittivity tensor,” Opt. Quantum Electron. 35, 407–427 (2003).
[CrossRef]

2002 (1)

2001 (2)

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

Z. Cao, Q. Liu, Q. S. Shen, X. M. Dou, Y. L. Chen, “Quantization scheme for arbitrary one-dimensional potential wells,” Phys. Rev. A 63, 054103 (2001).
[CrossRef]

1999 (2)

Z. Cao, Y. Jiang, Q. S. Shen, X. M. Dou, Y. L. Chen, “Exact analytical method for planar optical waveguides with arbitrary index profile,” J. Opt. Soc. Am. A 16, 2209–2212 (1999).
[CrossRef]

S. Gaal, “Surface integral method to determine guided modes in uniaxially anisotropic embedded waveguides,” Opt. Quantum Electron. 31, 763–780 (1999).
[CrossRef]

1998 (2)

L. Zhan, Z. Cao, “Exact dispersion equation of a graded refractive-index optical waveguide based on the equivalent attenuated vector,” J. Opt. Soc. Am. A 15, 713–716 (1998).
[CrossRef]

S. Visnovsky, “Magnetic-optic effects in ultrathin structures at longitudinal and polar magnetizations,” Czech. J. Phys. 48, 1083–1104 (1998).
[CrossRef]

1997 (1)

L. Nuno, J. V. Balbastre, H. Castane, “Analysis of general lossy inhomogeneous and anisotropic waveguides by the finite-element method (FEM) using edge elements,” IEEE Trans. Microwave Theory Tech. 45, 446–449 (1997).
[CrossRef]

1996 (1)

S. G. Garcia, T. M. Hung-Bao, R. G. Martin, B. G. Olmedo, “On the application of finite methods in time domain to anisotropic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 44, 2195–2206 (1996).
[CrossRef]

1993 (1)

G. Tartarini, P. Bassi, S. F. Chen, M. P. De Micheli, D. B. Ostrowsky, “Calculation of hybrid modes in uniaxial planar optical waveguides: application to proton exchanged lithium niobate waveguides,” Opt. Commun. 101, 424–431 (1993).
[CrossRef]

1991 (1)

I. Bardi, O. Biro, “An efficient finite-element formulation without spurious modes for anisotropic waveguides,” IEEE Trans. Microwave Theory Tech. 39, 1133–1139 (1991).
[CrossRef]

1988 (1)

L. Tsang, S. L. Chuang, “Improved coupled-mode theory for reciprocal anisotropic waveguide,” J. Lightwave Technol. 6, 304–311 (1988).
[CrossRef]

1985 (2)

L. M. Walpita, “Solutions for planar optical waveguide equations by selecting zero elements in a characteristic matrix,” J. Opt. Soc. Am. A 2, 595–602 (1985).
[CrossRef]

M. Koshiba, K. Hayata, M. Suzuki, “Finite-element formulation in terms of the electric-field vector for electromagnetic waveguide problems,” IEEE Trans. Microwave Theory Tech. MTT-33, 900–905 (1985).
[CrossRef]

1982 (1)

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

1981 (1)

J. Ctyroky, 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]

1980 (1)

P. Yeh, “Optics of anisotropic layered media: a new 4×4 matrix algebra,” Surf. Sci. 96, 41–53 (1980).
[CrossRef]

1973 (1)

Balbastre, J. V.

L. Nuno, J. V. Balbastre, H. Castane, “Analysis of general lossy inhomogeneous and anisotropic waveguides by the finite-element method (FEM) using edge elements,” IEEE Trans. Microwave Theory Tech. 45, 446–449 (1997).
[CrossRef]

Bardi, I.

I. Bardi, O. Biro, “An efficient finite-element formulation without spurious modes for anisotropic waveguides,” IEEE Trans. Microwave Theory Tech. 39, 1133–1139 (1991).
[CrossRef]

Bassi, P.

G. Tartarini, P. Bassi, S. F. Chen, M. P. De Micheli, D. B. Ostrowsky, “Calculation of hybrid modes in uniaxial planar optical waveguides: application to proton exchanged lithium niobate waveguides,” Opt. Commun. 101, 424–431 (1993).
[CrossRef]

Biro, O.

I. Bardi, O. Biro, “An efficient finite-element formulation without spurious modes for anisotropic waveguides,” IEEE Trans. Microwave Theory Tech. 39, 1133–1139 (1991).
[CrossRef]

Cada, M.

J. Ctyroky, 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]

Cao, Z.

Castane, H.

L. Nuno, J. V. Balbastre, H. Castane, “Analysis of general lossy inhomogeneous and anisotropic waveguides by the finite-element method (FEM) using edge elements,” IEEE Trans. Microwave Theory Tech. 45, 446–449 (1997).
[CrossRef]

Chen, S. F.

G. Tartarini, P. Bassi, S. F. Chen, M. P. De Micheli, D. B. Ostrowsky, “Calculation of hybrid modes in uniaxial planar optical waveguides: application to proton exchanged lithium niobate waveguides,” Opt. Commun. 101, 424–431 (1993).
[CrossRef]

Chen, Y. L.

Z. Cao, Q. Liu, Q. S. Shen, X. M. Dou, Y. L. Chen, “Quantization scheme for arbitrary one-dimensional potential wells,” Phys. Rev. A 63, 054103 (2001).
[CrossRef]

Z. Cao, Y. Jiang, Q. S. Shen, X. M. Dou, Y. L. Chen, “Exact analytical method for planar optical waveguides with arbitrary index profile,” J. Opt. Soc. Am. A 16, 2209–2212 (1999).
[CrossRef]

Chuang, S. L.

L. Tsang, S. L. Chuang, “Improved coupled-mode theory for reciprocal anisotropic waveguide,” J. Lightwave Technol. 6, 304–311 (1988).
[CrossRef]

Ctyroky, J.

J. Ctyroky, 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]

Cucinotta, A.

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

De Micheli, M. P.

G. Tartarini, P. Bassi, S. F. Chen, M. P. De Micheli, D. B. Ostrowsky, “Calculation of hybrid modes in uniaxial planar optical waveguides: application to proton exchanged lithium niobate waveguides,” Opt. Commun. 101, 424–431 (1993).
[CrossRef]

Dou, X. M.

Z. Cao, Q. Liu, Q. S. Shen, X. M. Dou, Y. L. Chen, “Quantization scheme for arbitrary one-dimensional potential wells,” Phys. Rev. A 63, 054103 (2001).
[CrossRef]

Z. Cao, Y. Jiang, Q. S. Shen, X. M. Dou, Y. L. Chen, “Exact analytical method for planar optical waveguides with arbitrary index profile,” J. Opt. Soc. Am. A 16, 2209–2212 (1999).
[CrossRef]

Gaal, S.

S. Gaal, “Surface integral method to determine guided modes in uniaxially anisotropic embedded waveguides,” Opt. Quantum Electron. 31, 763–780 (1999).
[CrossRef]

Garcia, S. G.

S. G. Garcia, T. M. Hung-Bao, R. G. Martin, B. G. Olmedo, “On the application of finite methods in time domain to anisotropic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 44, 2195–2206 (1996).
[CrossRef]

Gia Russo, D. P.

Harris, J. H.

Hayata, K.

M. Koshiba, K. Hayata, M. Suzuki, “Finite-element formulation in terms of the electric-field vector for electromagnetic waveguide problems,” IEEE Trans. Microwave Theory Tech. MTT-33, 900–905 (1985).
[CrossRef]

Hoekstra, H. J. W. M.

H. P. Uranus, H. J. W. M. Hoekstra, E. Vangroesen, “Finite difference scheme for planar waveguides with arbitrary index profiles and its implementation for anisotropic waveguides with a diagonal permittivity tensor,” Opt. Quantum Electron. 35, 407–427 (2003).
[CrossRef]

Hung-Bao, T. M.

S. G. Garcia, T. M. Hung-Bao, R. G. Martin, B. G. Olmedo, “On the application of finite methods in time domain to anisotropic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 44, 2195–2206 (1996).
[CrossRef]

Jiang, Y.

Kolosovsky, E. A.

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

Koshiba, M.

M. Koshiba, K. Hayata, M. Suzuki, “Finite-element formulation in terms of the electric-field vector for electromagnetic waveguide problems,” IEEE Trans. Microwave Theory Tech. MTT-33, 900–905 (1985).
[CrossRef]

Koukoutsaki, P. A.

Liu, Q.

Z. Cao, Q. Liu, Q. S. Shen, X. M. Dou, Y. L. Chen, “Quantization scheme for arbitrary one-dimensional potential wells,” Phys. Rev. A 63, 054103 (2001).
[CrossRef]

Manenkov, A. B.

Martin, R. G.

S. G. Garcia, T. M. Hung-Bao, R. G. Martin, B. G. Olmedo, “On the application of finite methods in time domain to anisotropic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 44, 2195–2206 (1996).
[CrossRef]

Nuno, L.

L. Nuno, J. V. Balbastre, H. Castane, “Analysis of general lossy inhomogeneous and anisotropic waveguides by the finite-element method (FEM) using edge elements,” IEEE Trans. Microwave Theory Tech. 45, 446–449 (1997).
[CrossRef]

Olmedo, B. G.

S. G. Garcia, T. M. Hung-Bao, R. G. Martin, B. G. Olmedo, “On the application of finite methods in time domain to anisotropic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 44, 2195–2206 (1996).
[CrossRef]

Ostrowsky, D. B.

G. Tartarini, P. Bassi, S. F. Chen, M. P. De Micheli, D. B. Ostrowsky, “Calculation of hybrid modes in uniaxial planar optical waveguides: application to proton exchanged lithium niobate waveguides,” Opt. Commun. 101, 424–431 (1993).
[CrossRef]

Petrov, D. V.

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

Schulz, V.

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

Selleri, S.

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

Shen, Q. S.

Z. Cao, Q. Liu, Q. S. Shen, X. M. Dou, Y. L. Chen, “Quantization scheme for arbitrary one-dimensional potential wells,” Phys. Rev. A 63, 054103 (2001).
[CrossRef]

Z. Cao, Y. Jiang, Q. S. Shen, X. M. Dou, Y. L. Chen, “Exact analytical method for planar optical waveguides with arbitrary index profile,” J. Opt. Soc. Am. A 16, 2209–2212 (1999).
[CrossRef]

Suzuki, M.

M. Koshiba, K. Hayata, M. Suzuki, “Finite-element formulation in terms of the electric-field vector for electromagnetic waveguide problems,” IEEE Trans. Microwave Theory Tech. MTT-33, 900–905 (1985).
[CrossRef]

Tartarini, G.

G. Tartarini, P. Bassi, S. F. Chen, M. P. De Micheli, D. B. Ostrowsky, “Calculation of hybrid modes in uniaxial planar optical waveguides: application to proton exchanged lithium niobate waveguides,” Opt. Commun. 101, 424–431 (1993).
[CrossRef]

Tigelis, I. G.

Tsang, L.

L. Tsang, S. L. Chuang, “Improved coupled-mode theory for reciprocal anisotropic waveguide,” J. Lightwave Technol. 6, 304–311 (1988).
[CrossRef]

Tsarev, A. V.

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

Uranus, H. P.

H. P. Uranus, H. J. W. M. Hoekstra, E. Vangroesen, “Finite difference scheme for planar waveguides with arbitrary index profiles and its implementation for anisotropic waveguides with a diagonal permittivity tensor,” Opt. Quantum Electron. 35, 407–427 (2003).
[CrossRef]

Vangroesen, E.

H. P. Uranus, H. J. W. M. Hoekstra, E. Vangroesen, “Finite difference scheme for planar waveguides with arbitrary index profiles and its implementation for anisotropic waveguides with a diagonal permittivity tensor,” Opt. Quantum Electron. 35, 407–427 (2003).
[CrossRef]

Vincetti, L.

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

Visnovsky, S.

S. Visnovsky, “Magnetic-optic effects in ultrathin structures at longitudinal and polar magnetizations,” Czech. J. Phys. 48, 1083–1104 (1998).
[CrossRef]

Walpita, L. M.

Yakovkin, I. B.

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

Yeh, P.

P. Yeh, “Optics of anisotropic layered media: a new 4×4 matrix algebra,” Surf. Sci. 96, 41–53 (1980).
[CrossRef]

Zhan, L.

Zoboli, M.

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

Czech. J. Phys. (1)

S. Visnovsky, “Magnetic-optic effects in ultrathin structures at longitudinal and polar magnetizations,” Czech. J. Phys. 48, 1083–1104 (1998).
[CrossRef]

IEEE J. Quantum Electron. (1)

J. Ctyroky, 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 Trans. Microwave Theory Tech. (1)

S. G. Garcia, T. M. Hung-Bao, R. G. Martin, B. G. Olmedo, “On the application of finite methods in time domain to anisotropic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 44, 2195–2206 (1996).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (4)

L. Nuno, J. V. Balbastre, H. Castane, “Analysis of general lossy inhomogeneous and anisotropic waveguides by the finite-element method (FEM) using edge elements,” IEEE Trans. Microwave Theory Tech. 45, 446–449 (1997).
[CrossRef]

I. Bardi, O. Biro, “An efficient finite-element formulation without spurious modes for anisotropic waveguides,” IEEE Trans. Microwave Theory Tech. 39, 1133–1139 (1991).
[CrossRef]

M. Koshiba, K. Hayata, M. Suzuki, “Finite-element formulation in terms of the electric-field vector for electromagnetic waveguide problems,” IEEE Trans. Microwave Theory Tech. MTT-33, 900–905 (1985).
[CrossRef]

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

J. Lightwave Technol. (1)

L. Tsang, S. L. Chuang, “Improved coupled-mode theory for reciprocal anisotropic waveguide,” J. Lightwave Technol. 6, 304–311 (1988).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (4)

Opt. Commun. (2)

G. Tartarini, P. Bassi, S. F. Chen, M. P. De Micheli, D. B. Ostrowsky, “Calculation of hybrid modes in uniaxial planar optical waveguides: application to proton exchanged lithium niobate waveguides,” Opt. Commun. 101, 424–431 (1993).
[CrossRef]

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

Opt. Quantum Electron. (3)

H. P. Uranus, H. J. W. M. Hoekstra, E. Vangroesen, “Finite difference scheme for planar waveguides with arbitrary index profiles and its implementation for anisotropic waveguides with a diagonal permittivity tensor,” Opt. Quantum Electron. 35, 407–427 (2003).
[CrossRef]

S. Gaal, “Surface integral method to determine guided modes in uniaxially anisotropic embedded waveguides,” Opt. Quantum Electron. 31, 763–780 (1999).
[CrossRef]

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

Phys. Rev. A (1)

Z. Cao, Q. Liu, Q. S. Shen, X. M. Dou, Y. L. Chen, “Quantization scheme for arbitrary one-dimensional potential wells,” Phys. Rev. A 63, 054103 (2001).
[CrossRef]

Surf. Sci. (1)

P. Yeh, “Optics of anisotropic layered media: a new 4×4 matrix algebra,” Surf. Sci. 96, 41–53 (1980).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Principal axes of anisotropy in the coordinate system of a waveguide structure.

Fig. 2
Fig. 2

Coordinate system of a three-layer anisotropic waveguide and index distribution in each layer.

Fig. 3
Fig. 3

Index distribution of a four-layer anisotropic waveguide.

Fig. 4
Fig. 4

Graded-index anisotropic waveguide with arbitrary index profile.

Fig. 5
Fig. 5

Graded-index anisotropic waveguide with two turning points.

Fig. 6
Fig. 6

Discontinuous index profile in a graded-index anisotropic waveguide.

Fig. 7
Fig. 7

Normalized mode index bTM=(β2-k02nc2)/(k02nx,max2-k02nc2) with normalized frequency V=k0d(nx,max2-nc2)1/2 of TM modes in a symmetric anisotropic waveguide.

Tables (3)

Tables Icon

Table 1 Comparison of Calculated Values of b for a Four-Layer Structure

Tables Icon

Table 2 Calculated Results of Effective Index Neff for an Anisotropic Graded-Index Profile

Tables Icon

Table 3 Calculated Values of Effective Index Neff for an Anisotropic TIPE LiNbO3 Waveguide with Discontinuous Profile

Equations (93)

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

=0nx2000ny2000nz2,
Hz(x)=A1exp(pTMx),x0B1cos(κ1TMx)+C1sin(κ1TMx),0<x<hD1exp[qTM(h-x)],xh,
κ1TM=n1zn1x (k02n1x2-β2)1/2,
pTM=n0zn0x (β2-k02n0x2)1/2,
qTM=n2zn2x (β2-k02n2x2)1/2,
Hz(h)1n2z2 Hz(h)=cos(κ1TMh)n1z2κ1TMsin(κ1TMh)-κ1TMn1z2sin(κ1TMh)cos(κ1TMh)×Hz(0)1n0z2 Hz(0).
A11-qTMn2z2=cos(κ1TMh)n1z2κ1TMsin(κ1TMh)-κ1TMn1z2sin(κ1TMh)cos(κ1TMh)×1pTMn0z2D1.
tan(κ1TMh)=qTMn2z2+pTMn0z2κ1TMn1z2-n1z2κ1TMqTMn2z2pTMn0z2;
κ1TMh=mπ+tan-1n1z2n0z2pTMκ1TM+tan-1n1z2n2z2qTMκ1TM(m=0,1,2,).
Ez(x)=A2exp(pTEx),x0B2cos(κ1TEx)+C2sin(κ1TEx),0<x<hD2exp[qTE(h-x)],xh,
A21-qTE=cos(κ1TEh)1κ1TEsin(κ1TEh)-κ1TEsin(κ1TEh)cos(κ1TEh)1pTED2
κ1TEh=mπ+tan-1pTEκ1TE+tan-1qTEκ1TE
(m=0,1,2,).
κ1,p,q=κ1TM,pTM,qTMforTMmodeκ1TE,pTE,qTEforTEmode,
fi=niz2forTMmode1forTEmode(i=0, 1, 2),
κ1h=mπ+tan-1f1f0pκ1+tan-1f1f2qκ1
(m=0,1,2,).
-p0f01M1M21-p3f3=0,
Mi=cos(κihi)-fiκisin(κihi)κifisin(κihi)cos(κihi)(i=1,2),
κi=κiTM=niznix (k02nix2-β2)1/2forTMmodeκiTE=(k02niy2-β2)1/2forTEmode
(i=1,2),
p0=p0TM=n0zn0x (β2-k02n0x2)1/2forTMmodep0TE=(β2-k02n0y2)1/2forTEmode,
p3=p3TM=n3zn3x (β2-k02n3x2)1/2forTMmodep3TE=(β2-k02n3y2)1/2forTEmode,
fi=niz2forTMmode1forTEmode(i=0,1,2,3).
κ1h1=mπ+tan-1f1f0p0κ1+tan-1f1f2p2κ1
(m=0,1,2,),
p2=κ2tantan-1f2f3p3κ2-κ2h2,
κ2h2+Φ2=mπ+tan-1f2f3p3κ2(m=0,1,2,),
κ1h1+κ2h2+Φ(s)
=mπ+tan-1f1f0p0κ1+tan-1f2f3p3κ2
(m=0,1,2,),
M2=cosh(α2h2)-f2α2sinh(α2h2)-α2f2sinh(α2h2)cosh(α2h2).
κ1h=mπ+tan-1f1f0p0κ1+tanh-1f1f2p2κ1
(m=0,1,2,).
nx(x)=nsx+Δnxf1(x/D1),x0ncx,x<0,
ny(x)=nsy+Δnyf2(x/D2),x0ncy,x<0,
nz(x)=nsz+Δnzf3(x/D3),x0ncz,x<0,
-pcfc1i=1lMij=l+1l+mMj1-psfs=0,
Mi=cos(κih)-fiκisin(κih)κifisin(κih)cos(κih)(i=0,1,2,,l),
Mj=cosh(αjh)-fiαjsinh(αjh)-αjfjsinh(αjh)cosh(αjh)
(j=l+1, l+2,,l+m),
κi=κiTM=niznix (k02nix2-β2)1/2forTMmodeκiTE=(k02niy2-β2)1/2forTEmode
(i=0,1,2,,l),
αj=αiTM=njznjx (β2-k02njx2)1/2forTMmodeαiTE=(β2-k02njy2)1/2forTEmode
(j=l+1, l+2,,l+m),
pc=pcTM=nczncx (β2-k02ncx2)1/2forTMmodepcTE=(β2-k02ncy2)1/2forTEmode,
ps=psTM=nsznsx (β2-k02nsx2)1/2forTMmodepsTE=(β2-k02nsy2)1/2forTEmode,
fk,fc,fs=nkz2,ncz2,nsz2forTMmode1forTEmode
(k=0,1,2,,l+m).
-pcfc1i=1lMi1-ptf(xt)=0.
pt=pl+1,
pj=αjpj+1αjfjfj+1+tanh(αjh)1+pj+1αjfjfj+1tanh(αjh)
pl+m+1=ps,(j=l+1, l+2,,l+m),
i=0lκih+Φs=Nπ+tan-1f0fcpcκ0+tan-1f(xt)fl+1ptκt
(N=0,1,2,).
i=0lκih+Φs=Nπ+tan-1f0fcpcκ0+π2,
Φs=i=0l-1Φi+1-tan-1fifi+1κi+1κitan(Φi+1),
Φi=tan-1piκi,
pi=κitantan-1fifi+1pi+1κi-κih
(i=1,2,,l).
Φs=i=0l-1pi+1fi+1pi+12+κi+12×(κi/fi-κi+1/fi+1),
pi+1/fi+1-pi/fih=pi2+κi2fi2.
limh0x=0x=xt-hp(x+h)f(x+h)p2(x+h)+κ2(x+h)×[κ(x)/f(x)-κ(x+h)/f(x+h)]h h
=0xtp(x)f(x)p2(x)+κ2(x)d[κ(x)/f(x)]dxdx.
limh0p(x+h)/f(x+h)-p(x)/f(x)h=p2(x)+κ2(x)f2(x),
0xtκ(x)dx+0xtp(x)f(x)p2(x)+κ2(x)d[κ(x)/f(x)]dxdx=Nπ+tan-1f0fcpcκ0+π2(N=0,1,2,),
κ(x)
=κTM(x)=nz(x)nx(x) [k02nx2(x)-β2]1/2forTMmode κTE(x)=[k02ny2(x)-β2]1/2forTEmode.
d[p(x)/f(x)]dx=κ2(x)+p2(x)f2(x),p(xt)=pt,
f(x)=nz2(x)forTMmode1forTEmode.
xt1xt2κ(x)dx+xt1xt2p(x)f(x)p2(x)+κ2(x)d[κ(x)/f(x)]dxdx
=(N+1)π(N=0,1,2,).
p(x)=κ(x)tantan-1f(x)f(x+h)p(x+h)κ(x)-κ(x)h,
Φ(x)=tan-1p(x+h)κ(x+h)-tan-1f(x)f(x+h)p(x+h)κ(x).
p(x-0)=f(x-0)p(x+0)f(x+0),
Φ(x)=tan-1p(x+0)κ(x+0)-tan-1f(x-0)f(x+0)p(x+0)κ(x-0).
Φd=tan-1p-(xd)κ-(xd)-tan-1f+(xd)f-(xd)p-(xd)κ+(xd),
κ±(xd)=[κ±(xd)]TM=nz±(xd)nx±(xd) {k02[nx±(xd)]2-β2}1/2forTMmode[κ±(xd)]TE={k02[ny±(xd)]2-β2}1/2forTEmode,
f±(xd)=[nz±(xd)]2forTMmode1forTEmode.
xt1xt2κ(x)dx+xt1xdp(x)f(x)p2(x)+κ2(x)d[κ(x)/f(x)]dxdx
+xdxt2p(x)f(x)p2(x)+κ2(x)d[κ(x)/f(x)]dxdx+Φd=(N+1)π(N=0,1,2,).
d[p(x)/f(x)]dx=κ2(x)+p2(x)f2(x),
p(xt)=pt,
p+(xd)=f+(xd)f-(xd) p-(xd).
xt1xdκ(x)dx+xt1xdp(x)f(x)p2(x)+κ2(x)d[κ(x)/f(x)]dxdx=N+12π+tan-1f+(xd)f-(xd)ptκ+(xd)
(N=0,1,2,).
d[p(x)/f(x)]dx=κ2(x)+p2(x)f2(x),p+(xd)=f+(xd)f-(xd) pt
ne(x)=nse+Δneexp-x2D12,x0nce,x<0,
no(x)=nso+Δnoexp-x2D22,x0nco,x<0.
ne(x)
=nc,x-1nb,-1<x0nes+0.014 exp(-x2/32)+0.11,0<x1nes+0.014 exp(-x2/32),x>1,
no(x)
=nc,x-1nb,-1<x0nos+0.009 exp(-x2/32)-0.04,0<x1nos+0.009 exp(-x2/32),x>1,

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