Abstract

This work presents a novel full-vectorial imaginary-distance beam propagation method based on the multidomain pseudospectral scheme, for the first time, to study the modal characteristics of dielectric optical waveguides. The proposed method divides the transverse plane into several subdomains with uniform refractive indices, and expands the optical field in each subdomain in terms of a set of suitable basis functions. Accordingly, the complicated cross-coupling terms, which are required by the finite difference or finite element schemes, can be removed from the full-vectorial formulations. However, the coupling effect can be restored by matching the physical interface conditions. Moreover, to identify the higherorder modes, the residual traces of the preceding lower-order modes are subtracted from the calculated optical fields in each propagation step to suppress the rapid growth of the lower-order modes. Numerical examples of the two-dimensional slab waveguides demonstrate that the present approach yields a highly accurate propagation constant. When applied to three-dimensional rib waveguides the present scheme yields results that remarkably agree with the reliable values obtained from the modal transverse resonance method and finite element scheme.

© 2008 Optical Society of America

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  8. J. C. Chen and S. Jüngling, "Computation of high-order waveguide modes by imaginary-distance beam propagation method," Opt. Quantum Electron. 26, S199-S205 (1994).
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  16. J. H. Lee and Q. H. Liu, "A 3-D Spectral-Element Time-Domain Method for Electromagnetic Simulation," IEEE Trans. Microw. Theory Tech. 55, 983-991 (2007).
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  17. 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, 2309-2317 (2005).
    [CrossRef]
  18. C. C. Huang, C. C. Huang, and J. Y. Yang, "A full-vectorial pseudospectral modal analysis of dielectric optical waveguides with stepped refractive index profiles," IEEE J. Sel. Top. Quantum Electron. 11, 457-465 (2005).
    [CrossRef]
  19. P. J. Chiang, C. L. Wu, C. H. Teng, C. S. Yang, and H. C. Chang, "Full-vectorial optical waveguide mode solvers using multidomain pseudospectral frequency-domain (PSFD) formulations," IEEE J. Quantum Electron. 44, 56-66 (2008).
    [CrossRef]
  20. I. Deshmukh and Q. H. Liu, "Pseudospectral beam-propagation method for optical waveguides," IEEE Photon. Technol. Lett. 15, 60-62 (2003).
    [CrossRef]
  21. C. C. Huang and C. C. Huang, "A novel wide-angle beam propagation method based on the spectral collocation scheme for computing tilted waveguides," IEEE Photon. Technol. Lett. 17, 1872-1874 (2005).
    [CrossRef]
  22. W. P. Huang, C. L. Xu, and S. K. Chaudhuri, "A vector beam propagation method based on H fields," IEEE Photo. Technol. Lett. 3, 1117-1120 (1991).
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    [CrossRef]
  25. S. Jungling and J. C. Chen, "A study and optimization of eigenmode calculations using the imaginary-distance beam propgation method," IEEE J. Quantum Electron. 30, 2098-2105 (1994).
    [CrossRef]
  26. M. M. Spuhler, D. Wiesmann, P. Freuler, and M. Diergardt, "Direst computation of higher-order propagation modes using the imaginary-distance beam propagation method," Opt. Quantum Electron. 31, 751-761 (1999).
    [CrossRef]
  27. J. Shibayama, M. Sekiguchi, J. Yamauchi, and H. Nakano, "Eigenmode analysis of optical waveguides by an improved finite-difference imaginary-distance beam propagation method," IEICE Trans. 81, 9-16 (1998).
  28. C. C. Huang, "Numerical calculations of ARROW structures by pseudospectral approach with Mur’s absorbing boundary conditions," Opt. Express 14, 11631-11652 (2006).
    [CrossRef] [PubMed]
  29. G. R. Hadley, "High-accuracy finite-difference equations for dielectric waveguide analysis II: Dielectric corners," J. Lightwave Technol. 20, 1219-1232 (2002).
    [CrossRef]
  30. C. Vassallo, "1993-1995 optical mode solvers," Opt. Quantum Electron. 29, 95-114 (1997).
    [CrossRef]
  31. S. Selleri and J. Petracek, "Modal analysis of rib waveguide through finite element and mode matching methods," Opt. Quantum Electron. 33, 373-386 (2001).
    [CrossRef]
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2008 (1)

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

2007 (2)

P. J. Chiang, C. P. Yu, and H. C. Chang, "Analysis of two-dimensional photonic crystals using a multidomain pseudospectral method," Phys. Rev. E 75, 026703 (2007).
[CrossRef]

J. H. Lee and Q. H. Liu, "A 3-D Spectral-Element Time-Domain Method for Electromagnetic Simulation," IEEE Trans. Microw. Theory Tech. 55, 983-991 (2007).
[CrossRef]

2006 (1)

2005 (5)

2003 (1)

I. Deshmukh and Q. H. Liu, "Pseudospectral beam-propagation method for optical waveguides," IEEE Photon. Technol. Lett. 15, 60-62 (2003).
[CrossRef]

2002 (4)

G. R. Hadley, "High-accuracy finite-difference equations for dielectric waveguide analysis II: Dielectric corners," J. Lightwave Technol. 20, 1219-1232 (2002).
[CrossRef]

K. Saitoh and M. Koshiba, "Full-vectorial imaginary-distance beam propagation method based on a finite element scheme: application to photonic crystal fibers," IEEE J. Quantum Electron. 38, 927-933 (2002).
[CrossRef]

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, 1627-1634 (2002).
[CrossRef]

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

2001 (1)

S. Selleri and J. Petracek, "Modal analysis of rib waveguide through finite element and mode matching methods," Opt. Quantum Electron. 33, 373-386 (2001).
[CrossRef]

2000 (1)

1999 (2)

B. Yang and J. S. Hesthaven, "A pseudospectral method for time-domain computation of electromagnetic scattering by bodies of revolution," IEEE Trans. Antennas Propagat. 47, 132-141 (1999).
[CrossRef]

M. M. Spuhler, D. Wiesmann, P. Freuler, and M. Diergardt, "Direst computation of higher-order propagation modes using the imaginary-distance beam propagation method," Opt. Quantum Electron. 31, 751-761 (1999).
[CrossRef]

1998 (1)

J. Shibayama, M. Sekiguchi, J. Yamauchi, and H. Nakano, "Eigenmode analysis of optical waveguides by an improved finite-difference imaginary-distance beam propagation method," IEICE Trans. 81, 9-16 (1998).

1997 (2)

C. Vassallo, "1993-1995 optical mode solvers," Opt. Quantum Electron. 29, 95-114 (1997).
[CrossRef]

Y. Tsuji., M. Koshiba, and T. Shiraishi, "Finite element beam propagation method for three-dimensional optical waveguide structures," J. Lightwave Technol. 15, 1728-1734 (1997).
[CrossRef]

1996 (1)

Y. Tsuji., M. Koshiba, and T. Shiraishi, "A finite element beam propagation method for strongly guiding and longitudinally varying optical waveguides," J. Lightwave Technol. 14, 217-222 (1996).
[CrossRef]

1994 (2)

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

S. Jungling and J. C. Chen, "A study and optimization of eigenmode calculations using the imaginary-distance beam propgation method," IEEE J. Quantum Electron. 30, 2098-2105 (1994).
[CrossRef]

1993 (2)

C. L. Xu, W. P. Huang, and S. K. Chaudhuri, "Efficient and accurate vector mode calculations by beam propagation method," J. Lightwave Technol. 11, 1209-1215 (1993).
[CrossRef]

W. P. Huang and C. L. Xu, "Simulation of three-dimensional optical waveguides by a full-vector beam propagation method," IEEE J. Quantum Electron. 29, 2639-2649 (1993).
[CrossRef]

1992 (1)

W. P. Huang, C. L. Xu, S. T. Chu, and S. K. Chaudhuri, "The finite difference beam propagation method: analysis and assessment," J. Lightwave Technol. 10, 295-304 (1992).
[CrossRef]

1991 (1)

W. P. Huang, C. L. Xu, and S. K. Chaudhuri, "A vector beam propagation method based on H fields," IEEE Photo. Technol. Lett. 3, 1117-1120 (1991).
[CrossRef]

1989 (1)

D. Yevick and B. Hermansson, "New formulation of the matrix beam propagation method: Application to rib waveguides," IEEE J. Quantum Electron. 25, 221-229 (1989).
[CrossRef]

1978 (1)

Ando, T.

Chang, H. C.

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

P. J. Chiang, C. P. Yu, and H. C. Chang, "Analysis of two-dimensional photonic crystals using a multidomain pseudospectral method," Phys. Rev. E 75, 026703 (2007).
[CrossRef]

Chaudhuri, S. K.

C. L. Xu, W. P. Huang, and S. K. Chaudhuri, "Efficient and accurate vector mode calculations by beam propagation method," J. Lightwave Technol. 11, 1209-1215 (1993).
[CrossRef]

W. P. Huang, C. L. Xu, S. T. Chu, and S. K. Chaudhuri, "The finite difference beam propagation method: analysis and assessment," J. Lightwave Technol. 10, 295-304 (1992).
[CrossRef]

W. P. Huang, C. L. Xu, and S. K. Chaudhuri, "A vector beam propagation method based on H fields," IEEE Photo. Technol. Lett. 3, 1117-1120 (1991).
[CrossRef]

Chen, J. C.

S. Jungling and J. C. Chen, "A study and optimization of eigenmode calculations using the imaginary-distance beam propgation method," IEEE J. Quantum Electron. 30, 2098-2105 (1994).
[CrossRef]

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

Chiang, P. J.

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

P. J. Chiang, C. P. Yu, and H. C. Chang, "Analysis of two-dimensional photonic crystals using a multidomain pseudospectral method," Phys. Rev. E 75, 026703 (2007).
[CrossRef]

Chu, S. T.

W. P. Huang, C. L. Xu, S. T. Chu, and S. K. Chaudhuri, "The finite difference beam propagation method: analysis and assessment," J. Lightwave Technol. 10, 295-304 (1992).
[CrossRef]

Deshmukh, I.

I. Deshmukh and Q. H. Liu, "Pseudospectral beam-propagation method for optical waveguides," IEEE Photon. Technol. Lett. 15, 60-62 (2003).
[CrossRef]

Diergardt, M.

M. M. Spuhler, D. Wiesmann, P. Freuler, and M. Diergardt, "Direst computation of higher-order propagation modes using the imaginary-distance beam propagation method," Opt. Quantum Electron. 31, 751-761 (1999).
[CrossRef]

El-Mikati, H. A.

Feit, M. D.

Fleck, J. A.

Freuler, P.

M. M. Spuhler, D. Wiesmann, P. Freuler, and M. Diergardt, "Direst computation of higher-order propagation modes using the imaginary-distance beam propagation method," Opt. Quantum Electron. 31, 751-761 (1999).
[CrossRef]

Hadley, G. R.

Hermansson, B.

D. Yevick and B. Hermansson, "New formulation of the matrix beam propagation method: Application to rib waveguides," IEEE J. Quantum Electron. 25, 221-229 (1989).
[CrossRef]

Hesthaven, J. S.

B. Yang and J. S. Hesthaven, "A pseudospectral method for time-domain computation of electromagnetic scattering by bodies of revolution," IEEE Trans. Antennas Propagat. 47, 132-141 (1999).
[CrossRef]

Huang, C. C.

C. C. Huang, "Numerical calculations of ARROW structures by pseudospectral approach with Mur’s absorbing boundary conditions," Opt. Express 14, 11631-11652 (2006).
[CrossRef] [PubMed]

C. C. Huang and C. C. Huang, "A novel wide-angle beam propagation method based on the spectral collocation scheme for computing tilted waveguides," IEEE Photon. Technol. Lett. 17, 1872-1874 (2005).
[CrossRef]

C. C. Huang and C. C. Huang, "A novel wide-angle beam propagation method based on the spectral collocation scheme for computing tilted waveguides," IEEE Photon. Technol. Lett. 17, 1872-1874 (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, 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, 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, 457-465 (2005).
[CrossRef]

C. C. Huang, C. C. Huang, and J. Y. Yang, "A full-vectorial pseudospectral modal analysis of dielectric optical waveguides with stepped refractive index profiles," IEEE J. Sel. Top. Quantum Electron. 11, 457-465 (2005).
[CrossRef]

Huang, W. P.

C. L. Xu, W. P. Huang, and S. K. Chaudhuri, "Efficient and accurate vector mode calculations by beam propagation method," J. Lightwave Technol. 11, 1209-1215 (1993).
[CrossRef]

W. P. Huang and C. L. Xu, "Simulation of three-dimensional optical waveguides by a full-vector beam propagation method," IEEE J. Quantum Electron. 29, 2639-2649 (1993).
[CrossRef]

W. P. Huang, C. L. Xu, S. T. Chu, and S. K. Chaudhuri, "The finite difference beam propagation method: analysis and assessment," J. Lightwave Technol. 10, 295-304 (1992).
[CrossRef]

W. P. Huang, C. L. Xu, and S. K. Chaudhuri, "A vector beam propagation method based on H fields," IEEE Photo. Technol. Lett. 3, 1117-1120 (1991).
[CrossRef]

Jungling, S.

S. Jungling and J. C. Chen, "A study and optimization of eigenmode calculations using the imaginary-distance beam propgation method," IEEE J. Quantum Electron. 30, 2098-2105 (1994).
[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, S199-S205 (1994).
[CrossRef]

Koshiba, M.

K. Saitoh and M. Koshiba, "Full-vectorial imaginary-distance beam propagation method based on a finite element scheme: application to photonic crystal fibers," IEEE J. Quantum Electron. 38, 927-933 (2002).
[CrossRef]

Lee, J. H.

J. H. Lee and Q. H. Liu, "A 3-D Spectral-Element Time-Domain Method for Electromagnetic Simulation," IEEE Trans. Microw. Theory Tech. 55, 983-991 (2007).
[CrossRef]

Liu, Q. H.

J. H. Lee and Q. H. Liu, "A 3-D Spectral-Element Time-Domain Method for Electromagnetic Simulation," IEEE Trans. Microw. Theory Tech. 55, 983-991 (2007).
[CrossRef]

I. Deshmukh and Q. H. Liu, "Pseudospectral beam-propagation method for optical waveguides," IEEE Photon. Technol. Lett. 15, 60-62 (2003).
[CrossRef]

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

Mugita, T.

Nakano, H.

Nakayama, H.

Numata, S.

Obayya, S. S. A.

Petracek, J.

S. Selleri and J. Petracek, "Modal analysis of rib waveguide through finite element and mode matching methods," Opt. Quantum Electron. 33, 373-386 (2001).
[CrossRef]

Rahman, B. M. A.

Saitoh, K.

K. Saitoh and M. Koshiba, "Full-vectorial imaginary-distance beam propagation method based on a finite element scheme: application to photonic crystal fibers," IEEE J. Quantum Electron. 38, 927-933 (2002).
[CrossRef]

Sekiguchi, M.

J. Shibayama, M. Sekiguchi, J. Yamauchi, and H. Nakano, "Eigenmode analysis of optical waveguides by an improved finite-difference imaginary-distance beam propagation method," IEICE Trans. 81, 9-16 (1998).

Selleri, S.

S. Selleri and J. Petracek, "Modal analysis of rib waveguide through finite element and mode matching methods," Opt. Quantum Electron. 33, 373-386 (2001).
[CrossRef]

Shibayama, J.

J. Shibayama, T. Yamazaki, J. Yamauchi, and H. Nakano, "Eigenmode analysis of a light-guiding metal line loaded on a dielectric substrate using the imaginary-distance beam-propagation method," J. Lightwave Technol. 23, 1533-1539 (2005).
[CrossRef]

J. Shibayama, M. Sekiguchi, J. Yamauchi, and H. Nakano, "Eigenmode analysis of optical waveguides by an improved finite-difference imaginary-distance beam propagation method," IEICE Trans. 81, 9-16 (1998).

Spuhler, M. M.

M. M. Spuhler, D. Wiesmann, P. Freuler, and M. Diergardt, "Direst computation of higher-order propagation modes using the imaginary-distance beam propagation method," Opt. Quantum Electron. 31, 751-761 (1999).
[CrossRef]

Teng, C. H.

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

Tsuji, Y.

Y. Tsuji., M. Koshiba, and T. Shiraishi, "Finite element beam propagation method for three-dimensional optical waveguide structures," J. Lightwave Technol. 15, 1728-1734 (1997).
[CrossRef]

Y. Tsuji., M. Koshiba, and T. Shiraishi, "A finite element beam propagation method for strongly guiding and longitudinally varying optical waveguides," J. Lightwave Technol. 14, 217-222 (1996).
[CrossRef]

Vassallo, C.

C. Vassallo, "1993-1995 optical mode solvers," Opt. Quantum Electron. 29, 95-114 (1997).
[CrossRef]

Wiesmann, D.

M. M. Spuhler, D. Wiesmann, P. Freuler, and M. Diergardt, "Direst computation of higher-order propagation modes using the imaginary-distance beam propagation method," Opt. Quantum Electron. 31, 751-761 (1999).
[CrossRef]

Wu, C. L.

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

Xu, C. L.

W. P. Huang and C. L. Xu, "Simulation of three-dimensional optical waveguides by a full-vector beam propagation method," IEEE J. Quantum Electron. 29, 2639-2649 (1993).
[CrossRef]

C. L. Xu, W. P. Huang, and S. K. Chaudhuri, "Efficient and accurate vector mode calculations by beam propagation method," J. Lightwave Technol. 11, 1209-1215 (1993).
[CrossRef]

W. P. Huang, C. L. Xu, S. T. Chu, and S. K. Chaudhuri, "The finite difference beam propagation method: analysis and assessment," J. Lightwave Technol. 10, 295-304 (1992).
[CrossRef]

W. P. Huang, C. L. Xu, and S. K. Chaudhuri, "A vector beam propagation method based on H fields," IEEE Photo. Technol. Lett. 3, 1117-1120 (1991).
[CrossRef]

Yamauchi, J.

Yamazaki, T.

Yang, B.

B. Yang and J. S. Hesthaven, "A pseudospectral method for time-domain computation of electromagnetic scattering by bodies of revolution," IEEE Trans. Antennas Propagat. 47, 132-141 (1999).
[CrossRef]

Yang, C. S.

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

Yang, J. Y.

C. C. Huang, C. C. Huang, and J. Y. Yang, "A full-vectorial pseudospectral modal analysis of dielectric optical waveguides with stepped refractive index profiles," IEEE J. Sel. Top. Quantum Electron. 11, 457-465 (2005).
[CrossRef]

Yevick, D.

D. Yevick and B. Hermansson, "New formulation of the matrix beam propagation method: Application to rib waveguides," IEEE J. Quantum Electron. 25, 221-229 (1989).
[CrossRef]

Yu, C. P.

P. J. Chiang, C. P. Yu, and H. C. Chang, "Analysis of two-dimensional photonic crystals using a multidomain pseudospectral method," Phys. Rev. E 75, 026703 (2007).
[CrossRef]

Appl. Opt. (1)

IEEE Antennas Wireless Propag. Lett. (1)

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

IEEE J. Quantum Electron. (5)

K. Saitoh and M. Koshiba, "Full-vectorial imaginary-distance beam propagation method based on a finite element scheme: application to photonic crystal fibers," IEEE J. Quantum Electron. 38, 927-933 (2002).
[CrossRef]

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Figures (8)

Fig. 1.
Fig. 1.

Relative error of the effective index of the (a) TE0 and (b) TM0 modes in the three-layer waveguide as a function of the propagation distance for different terms of basis functions N

Fig. 2.
Fig. 2.

Relative error of the effective index of the (a) TE0 and (b) TM0 modes in the three-layer waveguide as a function of the propagation distance under different scheme parameter α.

Fig. 3.
Fig. 3.

Relative error of the effective index of the (a) TE1 and (b) TE2 modes in the three-layer waveguide as a function of the propagation distance for different terms of basis functions N

Fig. 4.
Fig. 4.

The field profiles of the (a) TE0 (b) TE1 (c) TE2 (d) TE3 modes in the directional coupler

Fig. 5.
Fig. 5.

(a) The cross section of the rib waveguide with refractive indices of core nc , substrate ns , and air na . (b) The division of computational domain for the rib waveguide.

Fig. 6.
Fig. 6.

Convergence of the effective indices of the (a) Hy 11 and (b) Hx 11 modes as a function of the propagation distance.

Fig. 7.
Fig. 7.

The field profiles of the (a) minor field Hx and (b) major field Hy of the Hy 31 mode for W=6µm and t=0.5µm

Fig. 8.
Fig. 8.

The field profiles of the (a) major field Hx and (b) minor field Hy of the Hx 31 mode for W=6µm and t=0.5µm.

Tables (5)

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Table 1. Calculated Effective Index Values of the Four Modes of the Directional Coupler

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Table 2. Convergence of the Effective Index Values neff and Normalized Propagation Constants b of the Hy 11 and Hx 11 Modes of the Rib Waveguide Calculated by the Present Scheme with Different Terms of Basis Functions.

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Table 3. Effective Index Values neff and Normalized Propagation Constants b of the Hy 11 Mode of the Rib Waveguide for Various t Obtained by this Work, FVFD [27], and MTRM [28].

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Table 4. Effective Index Values neff and Normalized Propagation Constants b of the Hx 11 Mode of the Rib Waveguide for Various t Obtained by this Work, MTRM [28], and FEM [29].

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Table 5. Effective Index Values neff and Normalized Propagation Constants b of the Six Modes of the Rib Waveguide with W=6µm and t=0.5µm Obtained by this Work.

Equations (35)

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j H z = 1 2 k 0 n 0 PH
H = [ H x H y ] , P = [ P xx P xy P yx P yy ] ,
P xx = 2 x 2 + n 2 [ y ( 1 n 2 y ) ] + k 0 2 ( n 2 n 0 2 )
P xy = 2 y x n 2 [ y ( 1 n 2 x ) ]
P yx = 2 x y n 2 [ x ( 1 n 2 y ) ]
P yy = 2 y 2 + n 2 [ x ( 1 n 2 x ) ] + k 0 2 ( n 2 n 0 2 ) .
P = 2 x 2 + 2 y 2 + k 0 2 ( n 2 n 0 2 ) .
j H z = 1 2 k 0 n 0 [ P 0 0 P ] H
H τ = 1 2 k 0 n 0 [ P 0 0 P ] H .
τ [ H 1 H 2 H r ] = [ Q 1 0 0 0 0 Q 2 0 0 0 0 0 0 0 0 Q r ] [ H 1 H 2 H r ] ,
Q k = [ P k 0 0 P k ] , H k = [ H x k H y k ] .
n y + 2 H x y y n y 2 H x y y + = ( n y + 2 n y 2 ) H y x
H y y y + = H y y y
n x + 2 H y x x n x 2 H y x x + = ( n x + 2 n x 2 ) H x y
H x x x + = H x x x
H x k ( x , y ) = p = 0 N x q = 0 N y θ p k ( x ) ψ q k ( y ) H p , q x , k
H y k ( x , y ) = p = 0 N x q = 0 N y θ p k ( x ) ψ q k ( y ) H p , q y , k
P k = i = 1 N x 1 j = 1 N y 1 [ 2 x 2 + 2 y 2 + k 0 2 ( n k 2 n 0 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 { θ p k ( 2 ) ( x ) ψ q k ( y ) + θ p k ( x ) ψ q k ( 2 ) ( y ) +
k 0 2 ( n k 2 ( x , y ) n 0 2 ) θ p k ( x ) ψ q k ( y ) } ] x = x i , y = y j .
Λ τ = S Λ ,
Λ = [ H 1 H 2 H r ]
θ p ( x ) = ( 1 ) p + 1 ( 1 x 2 ) T n ' ( 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 n ( α x ) α ( x x p ) [ x L n ' ( α x ) ] x = x p e α ( x x p ) 2 ,
Λ ( x , y , τ ) = Λ ( x , y , 0 ) e S τ
Λ ( x , y , 0 ) = i = 0 c i Ψ i ( x , y )
S Ψ i = λ i Ψ i
λ i = k 0 2 n 0 [ ( n e i ) 2 n 0 2 ]
Λ ( x , y , τ ) = i = 0 c i Ψ i ( x , y ) e λ i τ .
Λ l + 1 = [ 1 + ( 1 ρ ) Δ τ S ] [ 1 ρ Δ τ S ] Λ l
Λ l + 1 = i = 0 [ 1 + ( 1 ρ ) Δ τ S i ] [ 1 ρ Δ τ S i ] Λ l i
F i = [ 1 + ( 1 ρ ) Δ τ S i ] [ 1 ρ Δ τ S i ] .
n e ( τ + Δ τ ) = n 0 ( τ ) + ln [ Λ ( x , y , τ + Δ τ ) d x d y ] ln [ Λ ( x , y , τ ) d x d y ] k 0 Δ τ .
Λ ̂ ( x , y , τ ) = Λ ( x , y , τ ) i = 1 N c i Ψ i ( x , y ) e λ i τ .
c i e λ i τ = Ψ i ( x , y ) * Λ ( x , y , τ ) d x d y Ψ i ( x , y ) * Ψ i ( x , y ) d x d y

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