Abstract

In this paper we consider the problem of computing the eigen-modes for the varying refractive-index profile in an open waveguide. We first approximate the refractive-index by a piecewise polynomial of degree two, and the corresponding Sturm-Liouville problem (eigenvalue problem) of the Helmholtz operator in each layer can be solved analytically by the Kummer functions. Then, analytical approximate dispersion equations are established for both TE and TM cases. Furthermore, the approximate dispersion equations converge fast to the exact ones for the continuous refractive-index function as the maximum value of the subinterval sizes tends to zero. Suitable numerical methods, such as Müller’s method or the chord secant method, may be applied to the dispersion relations to compute the eigenmodes. Numerical simulations show that our method is very practical and efficient for computing eigenmodes.

© 2015 Optical Society of America

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References

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  1. J. S. Bagby, D. P. Nyquist, and B. C. Drachman, “Integral formulation for analysis of integrated dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-33, 906–915 (1985).
    [Crossref]
  2. K. S. Chiang, “Review of numerical and approximate methods for the modal analysis of general optical dielectric waveguides,” Opt. Quantum Electron. 26, S113–S134 (1994).
    [Crossref]
  3. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic Press, 1974).
  4. E. F. Keuster and R. C. Pate, “Fundamental mode propagation on dielectric fibres of arbitrary cross-section,” Proc. IEE 127(1), 41–51 (1980).
  5. A. Snyder and J. Love, Optical Waveguide Theory (Springer, 1983).
  6. J. A. DeSanto, Scalar Wave Theory (Springer-Verlag, 1992).
    [Crossref]
  7. R. Ye and D. Yevick, “Noniterative calculation of complex propagation constants in planar waveguides,” J. Opt. Soc. Am. A 18, 2819–2822 (2001).
    [Crossref]
  8. D. Stowell and J. Tausch, “Variational formulation for guided and leaky modes in multilayer dielectric waveguides,” Commun. Comput. Phys. 7, 564–579 (2010).
  9. M. Koshiba, K. Hayata, and M. Suzuki, “Vectorial finite-element method without spurious solutions for dielectric waveguide problems,” Electron. Lett. 20, 409–410 (1984).
    [Crossref]
  10. Z. E. Abid, K. L. Johnson, and A. Gopinath, “Analysis of dielectric guides by vector transverse magnetic field finite elements,” J. Lightwave Technol. 11, 1545–1549 (1993).
    [Crossref]
  11. 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]
  12. 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]
  13. P. J. Ciang, 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]
  14. S. F. Chiang, B. Y. Lin, C. H. Teng, C. Y. Wang, and S. Y. Chung, “A multidomain pseudospectral mode solver for optical waveguide analysis,” J. Lightwave Technol. 30, 2077–2087 (2012).
    [Crossref]
  15. M. Walz, T. Zebrowski, J. Kuchenmeister, and K. Büsch, “B-spline modal method: A polynomial approach compared to the Fourier modal method,” Opt. Express 21, 14683–14697 (2013).
    [Crossref] [PubMed]
  16. A. Ghatak, “Leaky modes in optical waveguides,” Opt. Quantum Electron. 17, 311–321 (1985).
    [Crossref]
  17. J. Hu and C. R. Menyuk, “Understanding leaky modes: slab waveguide revisited,” Adv. Opt. Photon. 1, 58–106 (2009).
    [Crossref]
  18. J. Zhu and Y. Lu, “Leaky modes of slab waveguides-asymptotic solutions,” J. Lightwave Technol. 24, 1619–1623 (2006).
    [Crossref]
  19. J. Zhu and J. Zheng, “Exact dispersion equation of transverse electric leaky modes for nonhomogeneous optical waveguides,” J. Opt. Soc. Am. B 32, 92–100 (2015).
    [Crossref]
  20. L. Knockaert, H. Rogier, and D. De Zutter, “An FFT-based signal identification approach for obtaining the propagation constants of the leaky modes in layered media,” AEU Int. J. Electron. Commun. 59, 230–238 (2005).
    [Crossref]
  21. 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]
  22. D. Song and Y. Lu, “Pesudospectral modal method for computing optical waveguide modes,” J. Lightwave Technol. 32, 1624–1630 (2014).
    [Crossref]
  23. J. Zhu, Z. Chen, and S. Tang, “Leaky modes of optical waveguides with varied refractive index for mirochip optical interconnect applications – Asymptotic solutions,” Microelecton. Reliab. 48, 555–562 (2008).
    [Crossref]
  24. S. Khorasani and K. Mehrany, “Differential transfer-matrix method for solution of one-dimensional linear non-homogeneous optical structures,” J. Opt. Soc. Am. B 20, 91–96 (2003).
    [Crossref]
  25. M. Eghlidi, K. Mehrany, and B. Rashidian, “Modified differential transfer-matrix method for solution of one-dimensional linear inhomogeneous optical structures,” J. Opt. Soc. Am. B 22, 1521–1528 (2005).
    [Crossref]
  26. J. Zhu and Z. Shen, “Dispersion relation of leaky modes in nonhomogeneous waveguides and its applications,” J. Lightwave Technol. 29, 3230–3236 (2011).
    [Crossref]
  27. J. Zhu and J. Zheng, “Nonlinear equation of the modes in circular slab waveguides and its application,” Appl. Opt. 52, 8013–8023 (2013).
    [Crossref]
  28. S. Pruess, “Estimating the eigenvalues of Sturm-Liouville problems by approximating the differential equation,” SIAM J. Numer. Anal. 10, 55–68 (1973).
    [Crossref]
  29. R. L. Burden and J. D. Faires, Numerical Analysis (Brooks/Cole, 2010).
  30. P. J. Frey and P. George, Mesh Generation (Wiley, 2010).
  31. F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, NIST Handbook of Mathematical Functions (Cambridge University, 2010).

2015 (1)

2014 (1)

2013 (2)

2012 (1)

2011 (1)

2010 (1)

D. Stowell and J. Tausch, “Variational formulation for guided and leaky modes in multilayer dielectric waveguides,” Commun. Comput. Phys. 7, 564–579 (2010).

2009 (1)

2008 (2)

P. J. Ciang, 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]

J. Zhu, Z. Chen, and S. Tang, “Leaky modes of optical waveguides with varied refractive index for mirochip optical interconnect applications – Asymptotic solutions,” Microelecton. Reliab. 48, 555–562 (2008).
[Crossref]

2006 (2)

2005 (3)

L. Knockaert, H. Rogier, and D. De Zutter, “An FFT-based signal identification approach for obtaining the propagation constants of the leaky modes in layered media,” AEU Int. J. Electron. Commun. 59, 230–238 (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]

M. Eghlidi, K. Mehrany, and B. Rashidian, “Modified differential transfer-matrix method for solution of one-dimensional linear inhomogeneous optical structures,” J. Opt. Soc. Am. B 22, 1521–1528 (2005).
[Crossref]

2003 (1)

2001 (2)

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]

R. Ye and D. Yevick, “Noniterative calculation of complex propagation constants in planar waveguides,” J. Opt. Soc. Am. A 18, 2819–2822 (2001).
[Crossref]

1994 (1)

K. S. Chiang, “Review of numerical and approximate methods for the modal analysis of general optical dielectric waveguides,” Opt. Quantum Electron. 26, S113–S134 (1994).
[Crossref]

1993 (1)

Z. E. Abid, K. L. Johnson, and A. Gopinath, “Analysis of dielectric guides by vector transverse magnetic field finite elements,” J. Lightwave Technol. 11, 1545–1549 (1993).
[Crossref]

1985 (2)

J. S. Bagby, D. P. Nyquist, and B. C. Drachman, “Integral formulation for analysis of integrated dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-33, 906–915 (1985).
[Crossref]

A. Ghatak, “Leaky modes in optical waveguides,” Opt. Quantum Electron. 17, 311–321 (1985).
[Crossref]

1984 (1)

M. Koshiba, K. Hayata, and M. Suzuki, “Vectorial finite-element method without spurious solutions for dielectric waveguide problems,” Electron. Lett. 20, 409–410 (1984).
[Crossref]

1980 (1)

E. F. Keuster and R. C. Pate, “Fundamental mode propagation on dielectric fibres of arbitrary cross-section,” Proc. IEE 127(1), 41–51 (1980).

1973 (1)

S. Pruess, “Estimating the eigenvalues of Sturm-Liouville problems by approximating the differential equation,” SIAM J. Numer. Anal. 10, 55–68 (1973).
[Crossref]

Abid, Z. E.

Z. E. Abid, K. L. Johnson, and A. Gopinath, “Analysis of dielectric guides by vector transverse magnetic field finite elements,” J. Lightwave Technol. 11, 1545–1549 (1993).
[Crossref]

Bagby, J. S.

J. S. Bagby, D. P. Nyquist, and B. C. Drachman, “Integral formulation for analysis of integrated dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-33, 906–915 (1985).
[Crossref]

Boisvert, R. F.

F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, NIST Handbook of Mathematical Functions (Cambridge University, 2010).

Burden, R. L.

R. L. Burden and J. D. Faires, Numerical Analysis (Brooks/Cole, 2010).

Büsch, K.

Chang, H. C.

P. J. Ciang, 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]

Chen, Z.

J. Zhu, Z. Chen, and S. Tang, “Leaky modes of optical waveguides with varied refractive index for mirochip optical interconnect applications – Asymptotic solutions,” Microelecton. Reliab. 48, 555–562 (2008).
[Crossref]

Chiang, K. S.

K. S. Chiang, “Review of numerical and approximate methods for the modal analysis of general optical dielectric waveguides,” Opt. Quantum Electron. 26, S113–S134 (1994).
[Crossref]

Chiang, S. F.

Chung, S. Y.

Ciang, P. J.

P. J. Ciang, 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]

Clark, C. W.

F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, NIST Handbook of Mathematical Functions (Cambridge University, 2010).

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]

De Zutter, D.

L. Knockaert, H. Rogier, and D. De Zutter, “An FFT-based signal identification approach for obtaining the propagation constants of the leaky modes in layered media,” AEU Int. J. Electron. Commun. 59, 230–238 (2005).
[Crossref]

DeSanto, J. A.

J. A. DeSanto, Scalar Wave Theory (Springer-Verlag, 1992).
[Crossref]

Drachman, B. C.

J. S. Bagby, D. P. Nyquist, and B. C. Drachman, “Integral formulation for analysis of integrated dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-33, 906–915 (1985).
[Crossref]

Eghlidi, M.

Faires, J. D.

R. L. Burden and J. D. Faires, Numerical Analysis (Brooks/Cole, 2010).

Frey, P. J.

P. J. Frey and P. George, Mesh Generation (Wiley, 2010).

George, P.

P. J. Frey and P. George, Mesh Generation (Wiley, 2010).

Ghatak, A.

A. Ghatak, “Leaky modes in optical waveguides,” Opt. Quantum Electron. 17, 311–321 (1985).
[Crossref]

Gopinath, A.

Z. E. Abid, K. L. Johnson, and A. Gopinath, “Analysis of dielectric guides by vector transverse magnetic field finite elements,” J. Lightwave Technol. 11, 1545–1549 (1993).
[Crossref]

Hayata, K.

M. Koshiba, K. Hayata, and M. Suzuki, “Vectorial finite-element method without spurious solutions for dielectric waveguide problems,” Electron. Lett. 20, 409–410 (1984).
[Crossref]

Hu, J.

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, 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]

Johnson, K. L.

Z. E. Abid, K. L. Johnson, and A. Gopinath, “Analysis of dielectric guides by vector transverse magnetic field finite elements,” J. Lightwave Technol. 11, 1545–1549 (1993).
[Crossref]

Keuster, E. F.

E. F. Keuster and R. C. Pate, “Fundamental mode propagation on dielectric fibres of arbitrary cross-section,” Proc. IEE 127(1), 41–51 (1980).

Khorasani, S.

Knockaert, L.

L. Knockaert, H. Rogier, and D. De Zutter, “An FFT-based signal identification approach for obtaining the propagation constants of the leaky modes in layered media,” AEU Int. J. Electron. Commun. 59, 230–238 (2005).
[Crossref]

Koshiba, M.

M. Koshiba, K. Hayata, and M. Suzuki, “Vectorial finite-element method without spurious solutions for dielectric waveguide problems,” Electron. Lett. 20, 409–410 (1984).
[Crossref]

Kuchenmeister, J.

Lin, B. Y.

Love, J.

A. Snyder and J. Love, Optical Waveguide Theory (Springer, 1983).

Lozier, D. W.

F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, NIST Handbook of Mathematical Functions (Cambridge University, 2010).

Lu, Y.

Marcuse, D.

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic Press, 1974).

Mehrany, K.

Menyuk, C. R.

Nyquist, D. P.

J. S. Bagby, D. P. Nyquist, and B. C. Drachman, “Integral formulation for analysis of integrated dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-33, 906–915 (1985).
[Crossref]

Olver, F. W. J.

F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, NIST Handbook of Mathematical Functions (Cambridge University, 2010).

Pate, R. C.

E. F. Keuster and R. C. Pate, “Fundamental mode propagation on dielectric fibres of arbitrary cross-section,” Proc. IEE 127(1), 41–51 (1980).

Pruess, S.

S. Pruess, “Estimating the eigenvalues of Sturm-Liouville problems by approximating the differential equation,” SIAM J. Numer. Anal. 10, 55–68 (1973).
[Crossref]

Rashidian, B.

Rogier, H.

L. Knockaert, H. Rogier, and D. De Zutter, “An FFT-based signal identification approach for obtaining the propagation constants of the leaky modes in layered media,” AEU Int. J. Electron. Commun. 59, 230–238 (2005).
[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]

Shen, Z.

Snyder, A.

A. Snyder and J. Love, Optical Waveguide Theory (Springer, 1983).

Song, D.

Stowell, D.

D. Stowell and J. Tausch, “Variational formulation for guided and leaky modes in multilayer dielectric waveguides,” Commun. Comput. Phys. 7, 564–579 (2010).

Suzuki, M.

M. Koshiba, K. Hayata, and M. Suzuki, “Vectorial finite-element method without spurious solutions for dielectric waveguide problems,” Electron. Lett. 20, 409–410 (1984).
[Crossref]

Tang, S.

J. Zhu, Z. Chen, and S. Tang, “Leaky modes of optical waveguides with varied refractive index for mirochip optical interconnect applications – Asymptotic solutions,” Microelecton. Reliab. 48, 555–562 (2008).
[Crossref]

Tausch, J.

D. Stowell and J. Tausch, “Variational formulation for guided and leaky modes in multilayer dielectric waveguides,” Commun. Comput. Phys. 7, 564–579 (2010).

Teng, C. H.

S. F. Chiang, B. Y. Lin, C. H. Teng, C. Y. Wang, and S. Y. Chung, “A multidomain pseudospectral mode solver for optical waveguide analysis,” J. Lightwave Technol. 30, 2077–2087 (2012).
[Crossref]

P. J. Ciang, 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]

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]

Walz, M.

Wang, C. Y.

Wu, C. L.

P. J. Ciang, 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, C. S.

P. J. Ciang, 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]

Ye, R.

Yevick, D.

Zebrowski, T.

Zheng, J.

Zhu, J.

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)

AEU Int. J. Electron. Commun. (1)

L. Knockaert, H. Rogier, and D. De Zutter, “An FFT-based signal identification approach for obtaining the propagation constants of the leaky modes in layered media,” AEU Int. J. Electron. Commun. 59, 230–238 (2005).
[Crossref]

Appl. Opt. (1)

Commun. Comput. Phys. (1)

D. Stowell and J. Tausch, “Variational formulation for guided and leaky modes in multilayer dielectric waveguides,” Commun. Comput. Phys. 7, 564–579 (2010).

Electron. Lett. (1)

M. Koshiba, K. Hayata, and M. Suzuki, “Vectorial finite-element method without spurious solutions for dielectric waveguide problems,” Electron. Lett. 20, 409–410 (1984).
[Crossref]

IEEE J. Quantum Electron. (1)

P. J. Ciang, 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]

IEEE J. Sel. Top. Quantum Electron. (1)

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

IEEE Trans. Microwave Theory Tech. (1)

J. S. Bagby, D. P. Nyquist, and B. C. Drachman, “Integral formulation for analysis of integrated dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-33, 906–915 (1985).
[Crossref]

J. Lightwave Technol. (5)

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

J. Opt. Soc. Am. B (3)

Microelecton. Reliab. (1)

J. Zhu, Z. Chen, and S. Tang, “Leaky modes of optical waveguides with varied refractive index for mirochip optical interconnect applications – Asymptotic solutions,” Microelecton. Reliab. 48, 555–562 (2008).
[Crossref]

Opt. Express (2)

Opt. Quantum Electron. (3)

A. Ghatak, “Leaky modes in optical waveguides,” Opt. Quantum Electron. 17, 311–321 (1985).
[Crossref]

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]

K. S. Chiang, “Review of numerical and approximate methods for the modal analysis of general optical dielectric waveguides,” Opt. Quantum Electron. 26, S113–S134 (1994).
[Crossref]

Proc. IEE (1)

E. F. Keuster and R. C. Pate, “Fundamental mode propagation on dielectric fibres of arbitrary cross-section,” Proc. IEE 127(1), 41–51 (1980).

SIAM J. Numer. Anal. (1)

S. Pruess, “Estimating the eigenvalues of Sturm-Liouville problems by approximating the differential equation,” SIAM J. Numer. Anal. 10, 55–68 (1973).
[Crossref]

Other (6)

R. L. Burden and J. D. Faires, Numerical Analysis (Brooks/Cole, 2010).

P. J. Frey and P. George, Mesh Generation (Wiley, 2010).

F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, NIST Handbook of Mathematical Functions (Cambridge University, 2010).

A. Snyder and J. Love, Optical Waveguide Theory (Springer, 1983).

J. A. DeSanto, Scalar Wave Theory (Springer-Verlag, 1992).
[Crossref]

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic Press, 1974).

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Figures (2)

Fig. 1
Fig. 1 Subdivision diagram of a waveguide
Fig. 2
Fig. 2 Results from Example. Left: the propagation constants: ‘*’ stands for β obtained from the four-term WKB approximations; ‘o’ stands for results of β with k = 8; ‘+’ stands for results of β with k = 16. Right: relative errors for different step sizes: Er stands for the absolute value of the relative error; h = 2/k is the step size; ‘o’ represents log2(Er) for β̂1; ‘+’ represents log2(Er) for β̂2; ‘*’ represents log2(Er) for β̂3.

Tables (1)

Tables Icon

Table 1 Propagation constants of Example for TE case, k = 2, 4, 8, 16.

Equations (63)

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

n ( x ) = { n 1 , x < 0 ; n 0 ( x ) , 0 < x < d ; n 2 , x > d .
ρ d d x ( 1 ρ d ϕ d x ) + κ 0 2 n 2 ( x ) ϕ = β 2 ϕ , < x < ,
d 2 ϕ d x 2 + κ 0 2 n 0 2 ( x ) ϕ = β 2 ϕ , 0 < x < d ,
d ϕ d x = i γ 1 ϕ , x = 0 + ,
d ϕ d x = i γ 2 ϕ , x = d ,
d 2 y j d x 2 + ( a j x 2 + b j x + c j ) y j ( x ) = 0 , ( j 1 ) h < x < j h ,
{ a j = 2 κ 0 2 / h 2 [ n 0 2 ( x 0 ) 2 n 0 2 ( x 1 ) + n 0 2 ( x 2 ) ] , b j = κ 0 2 / h [ ( 1 4 j ) n 0 2 ( x 0 ) + ( 8 j 4 ) n 0 2 ( x 1 ) + ( 3 4 j ) n 0 2 ( x 2 ) ] , c j = κ 0 2 [ ( 2 j 2 j ) n 0 2 ( x 0 ) + 4 ( j j 2 ) n 0 2 ( x 1 ) + ( 2 j 2 3 j + 1 ) n 0 2 ( x 2 ) ] β 2 .
y j ( x ) = A j α j ( x ) + B j β j ( x ) ; j = 1 , 2 , , k ,
y j ( j h ) = y j + 1 ( j h ) , d y j d x | x = j h = d y j + 1 d x | x = j h , ( j = 1 , 2 , k 1 ) .
d y 1 d x | x = 0 = i ( k 0 2 n 1 2 β 2 ) 1 / 2 y 1 ( 0 )
d y k d x | x = d = i ( k 0 2 n 2 2 β 2 ) 1 / 2 y k ( d ) .
{ α 1 ( 0 ) A 1 + β 1 ( 0 ) B 1 = i ( k 0 2 n 1 2 β 2 ) 1 / 2 [ α 1 ( 0 ) A 1 + β 1 ( 0 ) B 1 ] , α j ( j h ) A j + β j ( j h ) B j α j + 1 ( j h ) A j + 1 β j + 1 ( j h ) B j + 1 = 0 , α j ( j h ) A j + β j ( j h ) B j α j + 1 ( j h ) A j + 1 β j + 1 ( j h ) B j + 1 = 0 , α k ( d ) A k + β k ( d ) B k = i ( k 0 2 n 2 2 β 2 ) 1 / 2 [ α k ( d ) A k + β k ( d ) B k ] ,
{ T 1 = α 1 ( 0 ) + i α 1 ( 0 ) γ 1 β 1 ( 0 ) + i β 1 ( 0 ) γ 1 , T j + 1 = α j + 1 ( j h ) [ α j ( j h ) + β j ( j h ) T j ] α j + 1 ( j h ) [ α j ( j h ) + β j ( j h ) T j ] β j + 1 ( j h ) [ α j ( j h ) + β j ( j h ) T j ] β j + 1 ( j h ) [ α j ( j h ) + β j ( j h ) T j ] , T k = α k ( d ) i α k ( d ) γ 2 β k ( d ) i β k ( d ) γ 2 ,
α 1 ( d ) i α 1 ( d ) γ 2 β 1 ( d ) i β 1 ( d ) γ 2 = α 1 ( 0 ) + i α 1 ( 0 ) γ 1 β 1 ( 0 ) + i β 1 ( 0 ) γ 1 ,
exp ( 2 i γ 0 d ) γ 0 + γ 2 γ 2 γ 0 = γ 1 γ 0 γ 1 + γ 0 ,
( γ 0 + γ 1 ) ( γ 0 + γ 2 ) ( γ 0 γ 1 ) ( γ 0 γ 2 ) = exp ( 2 i γ 0 d ) .
[ κ 0 2 n 0 2 ( 0 + ) β 2 ] 1 / 2 W a 0 a 0 a 2 W 2 a 0 2 a 3 W 3 a 0 3 a 4 W 4 + ,
a 0 = i d 2 , a 2 = δ 0 + δ 1 + δ 2 8 , a 3 = i 4 d ( δ 0 + δ 1 + δ 2 ) , a 4 = i d b 4 2 a 3 2 a 0 a 2 2 2 , b 4 = 1 128 ( 5 δ 1 2 + 5 δ 2 2 + δ 0 2 + 2 δ 2 δ 1 + 2 δ 2 δ 0 + 2 δ 1 δ 0 ) , δ 0 = κ 0 2 n 0 2 ( 0 + ) κ 0 2 n 0 2 ( d ) , δ 1 = κ 0 2 n 0 2 ( 0 + ) κ 0 2 n 1 2 , δ 2 = κ 0 2 n 0 2 ( 0 + ) κ 0 2 n 2 2 , δ 3 = δ 2 δ 0 ;
n 0 2 ( x ) d d x ( 1 n 0 2 ( x ) d ϕ d x ) + κ 0 2 n 2 ( x ) ϕ = β 2 ϕ , 0 < x < d ,
d ϕ d x = i γ 1 n 0 2 ( 0 + ) n 1 2 ϕ , at x = 0 ,
d ϕ d x = i γ 2 n 0 2 ( d ) n 2 2 ϕ , at x = d .
p x x + ( κ 0 2 s 2 ( x ) β 2 ) p = 0 , 0 < x < d ,
p x = [ i γ 1 n 0 2 ( 0 + ) n 1 2 n 0 x ( 0 + ) n 0 ( 0 + ) ] p A ^ ( β ) p , at x = 0 ,
p x = [ i γ 2 n 0 2 ( d ) n 2 2 n 0 x ( d ) n 0 ( d ) ] p B ^ ( β ) p , at x = d ,
s 2 ( x ) = n 0 2 + n 0 x x κ 0 2 n 0 2 n 0 x 2 κ 0 2 n 0 2 ,
d 2 p j d x 2 + ( a ^ j x 2 + b ^ j x + c ^ j ) p j ( x ) = 0 , ( j 1 ) h < x < j h ,
{ a ^ j = 2 κ 0 2 h 2 [ s 2 ( x 0 ) 2 s 2 ( x 1 ) + s 2 ( x 2 ) ] , b ^ j = κ 0 2 h [ ( 1 4 j ) s 2 ( x 0 ) + ( 8 j 4 ) s 2 ( x 1 ) + ( 3 4 j ) s 2 ( x 2 ) ] , c ^ j = κ 0 2 [ ( 2 j 2 j ) s 2 ( x 0 ) + 4 ( j j 2 ) s 2 ( x 1 ) + ( 2 j 2 3 j + 1 ) s 2 ( x 2 ) ] β 2 .
p j ( x ) = A ^ j α j ( x ) + B ^ j β j ( x ) ; j = 1 , 2 , , k .
p j ( j h ) = p j + 1 ( j h ) , d p j d x | x = j h = d p j + 1 d x | x = j h , ( j = 1 , 2 , , k 1 ) .
d p 1 d x | x = 0 = A ^ ( β ) p 1 ( 0 ) ,
d p k d x | x = d = B ^ ( β ) p k ( d ) .
{ α 1 ( 0 ) A ^ 1 + β 1 ( 0 ) B ^ 1 = A ^ ( β ) [ α 1 ( 0 ) A ^ 1 + β 1 ( 0 ) B ^ 1 ] , α j ( j h ) A ^ j + β j ( j h ) B ^ j α j + 1 ( j h ) A ^ j + 1 β j + 1 ( j h ) B ^ j + 1 = 0 , α j ( j h ) A ^ j + β j ( j h ) B ^ j α j + 1 ( j h ) A ^ j + 1 β j + 1 ( j h ) B ^ j + 1 = 0 , α k ( d ) A ^ k + β k ( d ) B ^ k = B ^ ( β ) [ α k ( d ) A ^ k + β k ( d ) B ^ k ] ,
{ T ^ 1 = α 1 ( 0 ) A ^ ( β ) α 1 ( 0 ) β 1 ( 0 ) A ^ ( β ) β 1 ( 0 ) , T ^ j + 1 = α j + 1 ( j h ) [ α j ( j h ) + β j ( j h ) T ^ j ] α j + 1 ( j h ) [ α j ( j h ) + β j ( j h ) T ^ j ] β j + 1 ( j h ) [ α j ( j h ) + β j ( j h ) T ^ j ] β j + 1 ( j h ) [ α j ( j h ) + β j ( j h ) T ^ j ] , T ^ k = α k ( d ) B ^ ( β ) α k ( d ) β k ( d ) B ^ ( β ) β k ( d ) ,
α 1 ( d ) B ^ ( β ) α ( d ) β 1 ( d ) B ^ ( β ) β 1 ( d ) = α 1 ( 0 ) A ^ ( β ) α 1 ( 0 ) β 1 ( 0 ) A ^ ( β ) β 0 ,
exp ( 2 i γ 0 d ) n 0 2 n 2 2 γ 2 + γ 0 n 0 2 n 2 2 γ 2 γ 0 = n 0 2 n 1 2 γ 1 γ 0 n 0 2 n 1 2 γ 1 + γ 0 ,
( μ 0 + μ 1 ) ( μ 0 + μ 2 ) ( μ 0 μ 1 ) ( μ 0 γ 2 ) = exp ( 2 i γ 0 d ) ,
[ κ 0 2 n 0 2 ( 0 + ) β 2 ] 1 / 2 π m d i ln A 0 ˜ 2 d b 2 ˜ K m ;
[ κ 0 2 n 0 2 ( 0 + ) β 2 ] 1 / 2 K m + b 2 ˜ A i ˜ 2 d A 0 ˜ K m K m 1 ;
[ κ 0 2 n 0 2 ( 0 + ) β 2 ] 1 / 2 K m 1 i ( 2 A 2 ˜ A 0 ˜ A 1 ˜ 2 ) 2 d A 0 ˜ 2 K m 1 2 ;
a 2 ˜ = δ 0 γ x x ( d ) + γ x 2 ( d ) , b 2 ˜ = γ x 2 ( 0 + ) γ x x ( 0 + ) , γ ( x ) = ln n 0 ( x ) , c 2 ˜ = n 2 2 + n 0 2 ( d ) n 2 2 n 0 2 ( d ) , c 1 ˜ = 2 i γ x ( d ) n 2 4 ( n 2 2 n 0 2 ( d ) ) 2 , c 2 ˜ = 2 n 2 6 γ x 2 ( d ) ( n 2 2 n 0 2 ( d ) ) 3 + ( a 2 ˜ δ 2 ) n 0 2 ( d ) n 2 2 ( n 2 2 n 0 2 ( d ) ) 2 , e 0 = n 1 2 + n 0 2 ( 0 + ) n 1 2 n 0 2 ( 0 + ) , e 1 = 2 i γ x ( 0 + ) n 1 4 ( n 1 2 n 0 2 ( 0 + ) ) 2 , e 2 = 2 n 1 6 γ x 2 ( 0 + ) ( n 1 2 n 0 2 ( 0 + ) ) 3 + ( b 2 ˜ δ 1 ) n 0 2 ( 0 + ) n 1 2 ( n 1 2 n 0 2 ( 0 + ) ) 2 .
g = T k + α k ( d ) i α k ( d ) γ 2 β k ( d ) i β k ( d ) γ 2 ,
y ( x ) + ( a x 2 + b x + c ) y ( x ) = 0 ,
d y 2 d t 2 + ( a t 2 + C ) y = 0 ,
z = ( a ) 1 / 2 t 2 and w ( z ) exp ( z / 2 ) = y ( t )
z d 2 w d z 2 + ( 1 2 z ) d w d z ( 1 4 C 4 ( a ) 1 / 2 ) w ( z ) = 0 ,
z d 2 w d z 2 + ( B z ) d w d z A w ( z ) = 0
{ α ( x ) = exp [ 1 2 z ( x ) ] M ( A , 1 2 , z ( x ) ) β ( x ) = [ z ( x ) ] 1 2 exp [ 1 2 z ( x ) ] M ( A + 1 2 , 3 2 , z ( x ) ) ,
z ( x ) = ( a ) 1 / 2 ( x + b 2 a ) 2 , A = 1 4 C 4 ( a ) 1 / 2 = 1 4 + b 2 4 a c 16 a ( a ) 1 / 2 ;
d d z M ( a , b , z ) = a b M ( a + 1 , b + 1 , z ) ;
{ α ( x ) = 1 2 z ( x ) α ( x ) + 2 A z ( x ) exp [ z ( x ) 2 ] M ( A + 1 , 3 2 , z ( x ) ) β ( x ) = 1 z ( x ) 2 z ( x ) z ( x ) β ( x ) + 2 A 1 3 [ z ( x ) ] 1 2 z ( x ) exp [ z ( x ) 2 ] M ( A + 3 2 , 5 2 , z ( x ) ) ,
y ( x ) + ( b x + c ) y ( x ) = 0 ,
α ( x ) = Ai ( w ( x ) ) , β ( x ) = Bi ( w ( x ) )
w ( x ) = b x + c b 2 / 3 .
α ( x ) = b 1 / 3 Ai ( w ( x ) ) , β ( x ) = b 1 / 3 Bi ( w ( x ) )
airy ( z ) = Ai ( z ) , airy ( 1 , z ) = Ai ( z ) , airy ( 2 , z ) = Bi ( z ) , airy ( 3 , z ) = Bi ( z ) .
y ( x ) + c y ( x ) = 0 ,
α ( x ) = sin ( c 1 / 2 x ) , β ( x ) = cos ( c 1 / 2 x ) .
n ( x 0 ) = 1 6 h [ n ( x 0 h ) 8 n ( x 0 0.5 h ) + 8 n ( x 0 + 0.5 h ) n ( x 0 + h ) ] + O ( h 4 ) ,
n ( x 0 ) = 1 3 h 2 [ n ( x 0 h ) 16 n ( x 0 0.5 h ) 30 f ( x 0 ) + 16 n ( x 0 + 0.5 h ) n ( x 0 + h ) ] + O ( h 4 ) .
n ( x 0 ) = 1 6 h [ 25 n ( x 0 ) + 48 n ( x 0 + 0.5 h ) 36 n ( x 0 + h ) + 16 n ( x 0 + 1.5 h ) 3 n ( x + 2 h ) ] + O ( h 4 ) ,
n ( x 0 ) = 1 3 h 2 [ 35 n ( x 0 ) 104 n ( x 0 + 0.5 h ) + 114 n ( x 0 + h ) 56 n ( x 0 + 1.5 h ) + 11 n ( x + 2 h ) ] + O ( h 3 ) .
n ( x 0 ) = 1 6 h [ 25 n ( x 0 ) + 48 n ( x 0 0.5 h ) 36 n ( x 0 h ) + 16 n ( x 0 1.5 h ) 3 n ( x 2 h ) ] + O ( h 4 ) ,
n ( x 0 ) = 1 3 h 2 [ 35 n ( x 0 ) 104 n ( x 0 0.5 h ) + 114 n ( x 0 h ) 56 n ( x 0 1.5 h ) + 11 n ( x 2 h ) ] + O ( h 3 ) .

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