Abstract

For numerical modeling of optical wave-guiding structures, perfectly matched layers (PMLs) are widely used to terminate the transverse variables of the waveguide. The PML modes are the eigenmodes of a waveguide terminated by PMLs, and they have found important applications in the mode matching method, the coupled mode theory, and so on. In this paper, we consider PML modes for two-dimensional slab waveguides. It is shown that the PML modes consist of perturbed propagating modes, perturbed leaky modes, and two infinite sequences of Berenger modes. High-order asymptotic solutions for the Berenger modes are derived using a systematic approach.

© 2013 Optical Society of America

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  1. J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic-waves,” J. Comput. Phys. 114, 185–200 (1994).
    [CrossRef]
  2. W. C. Chew and W. H. Weedon, “A 3D perfectly matched medium from modified Maxwells equations with stretched coordinates,” Microw. Opt. Technol. Lett. 7, 599–604 (1994).
    [CrossRef]
  3. W. P. Huang, C. L. Xu, W. Lui, and K. Yokoyama, “The perfectly matched layer (PML) boundary condition for the beam propagation method,” IEEE Photon. Technol. Lett. 8, 649–651 (1996).
    [CrossRef]
  4. W. P. Huang, C. L. Xu, W. Lui, and K. Yokoyama, “The perfectly matched layer boundary condition for modal analysis of optical waveguides: leaky mode calculations,” IEEE Photon. Technol. Lett. 8, 652–654 (1996).
    [CrossRef]
  5. H. Derudder, D. De Zutter, and F. Olyslager, “Analysis of waveguide discontinuities using perfectly matched layers,” Electron. Lett. 34, 2138–2140 (1998).
    [CrossRef]
  6. P. Bienstman and R. Baets, “Optical modelling of photonic crystals and VCSELs using eigenmode expansion and perfectly matched layers,” Opt. Quant. Electron. 33, 327–341 (2001).
    [CrossRef]
  7. P. Bienstman, H. Derudder, R. Baets, F. Olyslager, and D. De Zutter, “Analysis of cylindrical waveguide discontinuities using vectorial eigenmodes and perfectly matched layers,” IEEE Trans. Microw. Theory Tech. 49, 349–354 (2001).
  8. J. Mu and W. P. Huang, “Simulation of three-dimensional waveguide discontinuities by a full-vector mode-matching method based on finite-difference schemes,” Opt. Express 16, 18152–18163 (2008).
    [CrossRef]
  9. W. P. Huang and J. Mu, “Complex coupled-mode theory for optical waveguides,” Opt. Express 17, 19134–19152 (2009).
    [CrossRef]
  10. W. P. Huang, L. Han, and J. Mu, “A rigorous circuit model for simulation of large-scale photonic integrated circuits,” IEEE Photon. J. 4, 1622–1638 (2012).
    [CrossRef]
  11. T. E. Rozzi, “Rigorous analysis of step discontinuity in a planar dielectric waveguide,” IEEE Trans. Microw. Theory Tech. 26, 738–746 (1978).
  12. R. Mittra, Y. L. Hou, and V. Jamnejad, “Analysis of open dielectric wave-guides using mode-matching technique and variational-methods,” IEEE Trans. Microw. Theory Tech. 28, 36–43 (1980).
  13. G. Sztefka and H. P. Nolting, “Bidirectional eigenmode propagation for large refractive-index steps,” IEEE Photon. Technol. Lett. 5, 554–557 (1993).
    [CrossRef]
  14. J. Willems, J. Haes, and R. Baets, “The bidirectional mode expansion method for 2-dimensional wave-guides: the TM case,” Opt. Quant. Electron. 27, 995–1007 (1995).
    [CrossRef]
  15. F. Olyslager, “Discretization of continuous spectra based on perfectly matched layers,” SIAM J. Appl. Math. 64, 1408–1433 (2004).
    [CrossRef]
  16. L. F. Knockaert and D. De Zutter, “On the completeness of eigenmodes in a parallel plate waveguide with a perfectly matched layer termination,” IEEE Trans. Antennas Propag. 50, 1650–1653 (2002).
    [CrossRef]
  17. H. Derudder, F. Olyslager, and D. De Zutter, “An efficient series expansion for the 2-D Green’s function of a microstrip substrate using perfectly matched layers,” IEEE Microw. Guided Wave Lett. 9, 505–507 (1999).
    [CrossRef]
  18. H. Rogier and D. De Zutter, “Berenger and leaky modes in optical fibers terminated with a perfectly matched layer,” J. Lightwave Technol. 20, 1141–1148 (2002).
    [CrossRef]
  19. H. Rogier, L. Knockaert, and D. De Zutter, “Fast calculation of the propagation constants of leaky and Berenger modes of planar and circular dielectric waveguides terminated by a perfectly matched layer,” Microw. Opt. Technol. Lett. 37, 167–171 (2003).
    [CrossRef]
  20. H. Rogier and D. De Zutter, “Berenger and leaky modes in microstrip substrates terminated by a perfectly matched layer,” IEEE Trans. Microw. Theory Tech. 49, 712–715 (2001).
  21. Y. Y. Lu and J. Zhu, “Propagating modes in optical waveguides terminated by perfectly matched layers,” IEEE Photon. Technol. Lett. 17, 2601–2603 (2005).
    [CrossRef]
  22. J. Zhu and Y. Y. Lu, “Leaky modes of slab waveguides—asymptotic solutions,” J. Lightwave Technol. 24, 1619–1623 (2006).
    [CrossRef]
  23. R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert W function,” Adv. Comput. Math. 5, 329–359 (1996).
    [CrossRef]
  24. R. Z. L. Ye and D. Yevick, “Noniterative calculation of complex propagation constants in planar waveguides,” J. Opt. Soc. Am. A 18, 2819–2822 (2001).
    [CrossRef]
  25. S. B. Gaal, H. J. W. M. Hoekstra, and P. V. Lambeck, “Determining PML modes in 2-D stratified media,” J. Lightwave Technol. 21, 293–298 (2003).
    [CrossRef]
  26. 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,” Int. J. Electron. Commun. 59, 230–238 (2005).
    [CrossRef]

2012 (1)

W. P. Huang, L. Han, and J. Mu, “A rigorous circuit model for simulation of large-scale photonic integrated circuits,” IEEE Photon. J. 4, 1622–1638 (2012).
[CrossRef]

2009 (1)

2008 (1)

2006 (1)

2005 (2)

Y. Y. Lu and J. Zhu, “Propagating modes in optical waveguides terminated by perfectly matched layers,” IEEE Photon. Technol. Lett. 17, 2601–2603 (2005).
[CrossRef]

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,” Int. J. Electron. Commun. 59, 230–238 (2005).
[CrossRef]

2004 (1)

F. Olyslager, “Discretization of continuous spectra based on perfectly matched layers,” SIAM J. Appl. Math. 64, 1408–1433 (2004).
[CrossRef]

2003 (2)

H. Rogier, L. Knockaert, and D. De Zutter, “Fast calculation of the propagation constants of leaky and Berenger modes of planar and circular dielectric waveguides terminated by a perfectly matched layer,” Microw. Opt. Technol. Lett. 37, 167–171 (2003).
[CrossRef]

S. B. Gaal, H. J. W. M. Hoekstra, and P. V. Lambeck, “Determining PML modes in 2-D stratified media,” J. Lightwave Technol. 21, 293–298 (2003).
[CrossRef]

2002 (2)

H. Rogier and D. De Zutter, “Berenger and leaky modes in optical fibers terminated with a perfectly matched layer,” J. Lightwave Technol. 20, 1141–1148 (2002).
[CrossRef]

L. F. Knockaert and D. De Zutter, “On the completeness of eigenmodes in a parallel plate waveguide with a perfectly matched layer termination,” IEEE Trans. Antennas Propag. 50, 1650–1653 (2002).
[CrossRef]

2001 (4)

H. Rogier and D. De Zutter, “Berenger and leaky modes in microstrip substrates terminated by a perfectly matched layer,” IEEE Trans. Microw. Theory Tech. 49, 712–715 (2001).

P. Bienstman and R. Baets, “Optical modelling of photonic crystals and VCSELs using eigenmode expansion and perfectly matched layers,” Opt. Quant. Electron. 33, 327–341 (2001).
[CrossRef]

P. Bienstman, H. Derudder, R. Baets, F. Olyslager, and D. De Zutter, “Analysis of cylindrical waveguide discontinuities using vectorial eigenmodes and perfectly matched layers,” IEEE Trans. Microw. Theory Tech. 49, 349–354 (2001).

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

1999 (1)

H. Derudder, F. Olyslager, and D. De Zutter, “An efficient series expansion for the 2-D Green’s function of a microstrip substrate using perfectly matched layers,” IEEE Microw. Guided Wave Lett. 9, 505–507 (1999).
[CrossRef]

1998 (1)

H. Derudder, D. De Zutter, and F. Olyslager, “Analysis of waveguide discontinuities using perfectly matched layers,” Electron. Lett. 34, 2138–2140 (1998).
[CrossRef]

1996 (3)

W. P. Huang, C. L. Xu, W. Lui, and K. Yokoyama, “The perfectly matched layer (PML) boundary condition for the beam propagation method,” IEEE Photon. Technol. Lett. 8, 649–651 (1996).
[CrossRef]

W. P. Huang, C. L. Xu, W. Lui, and K. Yokoyama, “The perfectly matched layer boundary condition for modal analysis of optical waveguides: leaky mode calculations,” IEEE Photon. Technol. Lett. 8, 652–654 (1996).
[CrossRef]

R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert W function,” Adv. Comput. Math. 5, 329–359 (1996).
[CrossRef]

1995 (1)

J. Willems, J. Haes, and R. Baets, “The bidirectional mode expansion method for 2-dimensional wave-guides: the TM case,” Opt. Quant. Electron. 27, 995–1007 (1995).
[CrossRef]

1994 (2)

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

W. C. Chew and W. H. Weedon, “A 3D perfectly matched medium from modified Maxwells equations with stretched coordinates,” Microw. Opt. Technol. Lett. 7, 599–604 (1994).
[CrossRef]

1993 (1)

G. Sztefka and H. P. Nolting, “Bidirectional eigenmode propagation for large refractive-index steps,” IEEE Photon. Technol. Lett. 5, 554–557 (1993).
[CrossRef]

1980 (1)

R. Mittra, Y. L. Hou, and V. Jamnejad, “Analysis of open dielectric wave-guides using mode-matching technique and variational-methods,” IEEE Trans. Microw. Theory Tech. 28, 36–43 (1980).

1978 (1)

T. E. Rozzi, “Rigorous analysis of step discontinuity in a planar dielectric waveguide,” IEEE Trans. Microw. Theory Tech. 26, 738–746 (1978).

Baets, R.

P. Bienstman and R. Baets, “Optical modelling of photonic crystals and VCSELs using eigenmode expansion and perfectly matched layers,” Opt. Quant. Electron. 33, 327–341 (2001).
[CrossRef]

P. Bienstman, H. Derudder, R. Baets, F. Olyslager, and D. De Zutter, “Analysis of cylindrical waveguide discontinuities using vectorial eigenmodes and perfectly matched layers,” IEEE Trans. Microw. Theory Tech. 49, 349–354 (2001).

J. Willems, J. Haes, and R. Baets, “The bidirectional mode expansion method for 2-dimensional wave-guides: the TM case,” Opt. Quant. Electron. 27, 995–1007 (1995).
[CrossRef]

Berenger, J. P.

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

Bienstman, P.

P. Bienstman, H. Derudder, R. Baets, F. Olyslager, and D. De Zutter, “Analysis of cylindrical waveguide discontinuities using vectorial eigenmodes and perfectly matched layers,” IEEE Trans. Microw. Theory Tech. 49, 349–354 (2001).

P. Bienstman and R. Baets, “Optical modelling of photonic crystals and VCSELs using eigenmode expansion and perfectly matched layers,” Opt. Quant. Electron. 33, 327–341 (2001).
[CrossRef]

Chew, W. C.

W. C. Chew and W. H. Weedon, “A 3D perfectly matched medium from modified Maxwells equations with stretched coordinates,” Microw. Opt. Technol. Lett. 7, 599–604 (1994).
[CrossRef]

Corless, R. M.

R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert W function,” Adv. Comput. Math. 5, 329–359 (1996).
[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,” Int. J. Electron. Commun. 59, 230–238 (2005).
[CrossRef]

H. Rogier, L. Knockaert, and D. De Zutter, “Fast calculation of the propagation constants of leaky and Berenger modes of planar and circular dielectric waveguides terminated by a perfectly matched layer,” Microw. Opt. Technol. Lett. 37, 167–171 (2003).
[CrossRef]

L. F. Knockaert and D. De Zutter, “On the completeness of eigenmodes in a parallel plate waveguide with a perfectly matched layer termination,” IEEE Trans. Antennas Propag. 50, 1650–1653 (2002).
[CrossRef]

H. Rogier and D. De Zutter, “Berenger and leaky modes in optical fibers terminated with a perfectly matched layer,” J. Lightwave Technol. 20, 1141–1148 (2002).
[CrossRef]

H. Rogier and D. De Zutter, “Berenger and leaky modes in microstrip substrates terminated by a perfectly matched layer,” IEEE Trans. Microw. Theory Tech. 49, 712–715 (2001).

P. Bienstman, H. Derudder, R. Baets, F. Olyslager, and D. De Zutter, “Analysis of cylindrical waveguide discontinuities using vectorial eigenmodes and perfectly matched layers,” IEEE Trans. Microw. Theory Tech. 49, 349–354 (2001).

H. Derudder, F. Olyslager, and D. De Zutter, “An efficient series expansion for the 2-D Green’s function of a microstrip substrate using perfectly matched layers,” IEEE Microw. Guided Wave Lett. 9, 505–507 (1999).
[CrossRef]

H. Derudder, D. De Zutter, and F. Olyslager, “Analysis of waveguide discontinuities using perfectly matched layers,” Electron. Lett. 34, 2138–2140 (1998).
[CrossRef]

Derudder, H.

P. Bienstman, H. Derudder, R. Baets, F. Olyslager, and D. De Zutter, “Analysis of cylindrical waveguide discontinuities using vectorial eigenmodes and perfectly matched layers,” IEEE Trans. Microw. Theory Tech. 49, 349–354 (2001).

H. Derudder, F. Olyslager, and D. De Zutter, “An efficient series expansion for the 2-D Green’s function of a microstrip substrate using perfectly matched layers,” IEEE Microw. Guided Wave Lett. 9, 505–507 (1999).
[CrossRef]

H. Derudder, D. De Zutter, and F. Olyslager, “Analysis of waveguide discontinuities using perfectly matched layers,” Electron. Lett. 34, 2138–2140 (1998).
[CrossRef]

Gaal, S. B.

Gonnet, G. H.

R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert W function,” Adv. Comput. Math. 5, 329–359 (1996).
[CrossRef]

Haes, J.

J. Willems, J. Haes, and R. Baets, “The bidirectional mode expansion method for 2-dimensional wave-guides: the TM case,” Opt. Quant. Electron. 27, 995–1007 (1995).
[CrossRef]

Han, L.

W. P. Huang, L. Han, and J. Mu, “A rigorous circuit model for simulation of large-scale photonic integrated circuits,” IEEE Photon. J. 4, 1622–1638 (2012).
[CrossRef]

Hare, D. E. G.

R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert W function,” Adv. Comput. Math. 5, 329–359 (1996).
[CrossRef]

Hoekstra, H. J. W. M.

Hou, Y. L.

R. Mittra, Y. L. Hou, and V. Jamnejad, “Analysis of open dielectric wave-guides using mode-matching technique and variational-methods,” IEEE Trans. Microw. Theory Tech. 28, 36–43 (1980).

Huang, W. P.

W. P. Huang, L. Han, and J. Mu, “A rigorous circuit model for simulation of large-scale photonic integrated circuits,” IEEE Photon. J. 4, 1622–1638 (2012).
[CrossRef]

W. P. Huang and J. Mu, “Complex coupled-mode theory for optical waveguides,” Opt. Express 17, 19134–19152 (2009).
[CrossRef]

J. Mu and W. P. Huang, “Simulation of three-dimensional waveguide discontinuities by a full-vector mode-matching method based on finite-difference schemes,” Opt. Express 16, 18152–18163 (2008).
[CrossRef]

W. P. Huang, C. L. Xu, W. Lui, and K. Yokoyama, “The perfectly matched layer boundary condition for modal analysis of optical waveguides: leaky mode calculations,” IEEE Photon. Technol. Lett. 8, 652–654 (1996).
[CrossRef]

W. P. Huang, C. L. Xu, W. Lui, and K. Yokoyama, “The perfectly matched layer (PML) boundary condition for the beam propagation method,” IEEE Photon. Technol. Lett. 8, 649–651 (1996).
[CrossRef]

Jamnejad, V.

R. Mittra, Y. L. Hou, and V. Jamnejad, “Analysis of open dielectric wave-guides using mode-matching technique and variational-methods,” IEEE Trans. Microw. Theory Tech. 28, 36–43 (1980).

Jeffrey, D. J.

R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert W function,” Adv. Comput. Math. 5, 329–359 (1996).
[CrossRef]

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,” Int. J. Electron. Commun. 59, 230–238 (2005).
[CrossRef]

H. Rogier, L. Knockaert, and D. De Zutter, “Fast calculation of the propagation constants of leaky and Berenger modes of planar and circular dielectric waveguides terminated by a perfectly matched layer,” Microw. Opt. Technol. Lett. 37, 167–171 (2003).
[CrossRef]

Knockaert, L. F.

L. F. Knockaert and D. De Zutter, “On the completeness of eigenmodes in a parallel plate waveguide with a perfectly matched layer termination,” IEEE Trans. Antennas Propag. 50, 1650–1653 (2002).
[CrossRef]

Knuth, D. E.

R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert W function,” Adv. Comput. Math. 5, 329–359 (1996).
[CrossRef]

Lambeck, P. V.

Lu, Y. Y.

J. Zhu and Y. Y. Lu, “Leaky modes of slab waveguides—asymptotic solutions,” J. Lightwave Technol. 24, 1619–1623 (2006).
[CrossRef]

Y. Y. Lu and J. Zhu, “Propagating modes in optical waveguides terminated by perfectly matched layers,” IEEE Photon. Technol. Lett. 17, 2601–2603 (2005).
[CrossRef]

Lui, W.

W. P. Huang, C. L. Xu, W. Lui, and K. Yokoyama, “The perfectly matched layer (PML) boundary condition for the beam propagation method,” IEEE Photon. Technol. Lett. 8, 649–651 (1996).
[CrossRef]

W. P. Huang, C. L. Xu, W. Lui, and K. Yokoyama, “The perfectly matched layer boundary condition for modal analysis of optical waveguides: leaky mode calculations,” IEEE Photon. Technol. Lett. 8, 652–654 (1996).
[CrossRef]

Mittra, R.

R. Mittra, Y. L. Hou, and V. Jamnejad, “Analysis of open dielectric wave-guides using mode-matching technique and variational-methods,” IEEE Trans. Microw. Theory Tech. 28, 36–43 (1980).

Mu, J.

Nolting, H. P.

G. Sztefka and H. P. Nolting, “Bidirectional eigenmode propagation for large refractive-index steps,” IEEE Photon. Technol. Lett. 5, 554–557 (1993).
[CrossRef]

Olyslager, F.

F. Olyslager, “Discretization of continuous spectra based on perfectly matched layers,” SIAM J. Appl. Math. 64, 1408–1433 (2004).
[CrossRef]

P. Bienstman, H. Derudder, R. Baets, F. Olyslager, and D. De Zutter, “Analysis of cylindrical waveguide discontinuities using vectorial eigenmodes and perfectly matched layers,” IEEE Trans. Microw. Theory Tech. 49, 349–354 (2001).

H. Derudder, F. Olyslager, and D. De Zutter, “An efficient series expansion for the 2-D Green’s function of a microstrip substrate using perfectly matched layers,” IEEE Microw. Guided Wave Lett. 9, 505–507 (1999).
[CrossRef]

H. Derudder, D. De Zutter, and F. Olyslager, “Analysis of waveguide discontinuities using perfectly matched layers,” Electron. Lett. 34, 2138–2140 (1998).
[CrossRef]

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,” Int. J. Electron. Commun. 59, 230–238 (2005).
[CrossRef]

H. Rogier, L. Knockaert, and D. De Zutter, “Fast calculation of the propagation constants of leaky and Berenger modes of planar and circular dielectric waveguides terminated by a perfectly matched layer,” Microw. Opt. Technol. Lett. 37, 167–171 (2003).
[CrossRef]

H. Rogier and D. De Zutter, “Berenger and leaky modes in optical fibers terminated with a perfectly matched layer,” J. Lightwave Technol. 20, 1141–1148 (2002).
[CrossRef]

H. Rogier and D. De Zutter, “Berenger and leaky modes in microstrip substrates terminated by a perfectly matched layer,” IEEE Trans. Microw. Theory Tech. 49, 712–715 (2001).

Rozzi, T. E.

T. E. Rozzi, “Rigorous analysis of step discontinuity in a planar dielectric waveguide,” IEEE Trans. Microw. Theory Tech. 26, 738–746 (1978).

Sztefka, G.

G. Sztefka and H. P. Nolting, “Bidirectional eigenmode propagation for large refractive-index steps,” IEEE Photon. Technol. Lett. 5, 554–557 (1993).
[CrossRef]

Weedon, W. H.

W. C. Chew and W. H. Weedon, “A 3D perfectly matched medium from modified Maxwells equations with stretched coordinates,” Microw. Opt. Technol. Lett. 7, 599–604 (1994).
[CrossRef]

Willems, J.

J. Willems, J. Haes, and R. Baets, “The bidirectional mode expansion method for 2-dimensional wave-guides: the TM case,” Opt. Quant. Electron. 27, 995–1007 (1995).
[CrossRef]

Xu, C. L.

W. P. Huang, C. L. Xu, W. Lui, and K. Yokoyama, “The perfectly matched layer (PML) boundary condition for the beam propagation method,” IEEE Photon. Technol. Lett. 8, 649–651 (1996).
[CrossRef]

W. P. Huang, C. L. Xu, W. Lui, and K. Yokoyama, “The perfectly matched layer boundary condition for modal analysis of optical waveguides: leaky mode calculations,” IEEE Photon. Technol. Lett. 8, 652–654 (1996).
[CrossRef]

Ye, R. Z. L.

Yevick, D.

Yokoyama, K.

W. P. Huang, C. L. Xu, W. Lui, and K. Yokoyama, “The perfectly matched layer (PML) boundary condition for the beam propagation method,” IEEE Photon. Technol. Lett. 8, 649–651 (1996).
[CrossRef]

W. P. Huang, C. L. Xu, W. Lui, and K. Yokoyama, “The perfectly matched layer boundary condition for modal analysis of optical waveguides: leaky mode calculations,” IEEE Photon. Technol. Lett. 8, 652–654 (1996).
[CrossRef]

Zhu, J.

J. Zhu and Y. Y. Lu, “Leaky modes of slab waveguides—asymptotic solutions,” J. Lightwave Technol. 24, 1619–1623 (2006).
[CrossRef]

Y. Y. Lu and J. Zhu, “Propagating modes in optical waveguides terminated by perfectly matched layers,” IEEE Photon. Technol. Lett. 17, 2601–2603 (2005).
[CrossRef]

Adv. Comput. Math. (1)

R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert W function,” Adv. Comput. Math. 5, 329–359 (1996).
[CrossRef]

Electron. Lett. (1)

H. Derudder, D. De Zutter, and F. Olyslager, “Analysis of waveguide discontinuities using perfectly matched layers,” Electron. Lett. 34, 2138–2140 (1998).
[CrossRef]

IEEE Microw. Guided Wave Lett. (1)

H. Derudder, F. Olyslager, and D. De Zutter, “An efficient series expansion for the 2-D Green’s function of a microstrip substrate using perfectly matched layers,” IEEE Microw. Guided Wave Lett. 9, 505–507 (1999).
[CrossRef]

IEEE Photon. J. (1)

W. P. Huang, L. Han, and J. Mu, “A rigorous circuit model for simulation of large-scale photonic integrated circuits,” IEEE Photon. J. 4, 1622–1638 (2012).
[CrossRef]

IEEE Photon. Technol. Lett. (4)

W. P. Huang, C. L. Xu, W. Lui, and K. Yokoyama, “The perfectly matched layer (PML) boundary condition for the beam propagation method,” IEEE Photon. Technol. Lett. 8, 649–651 (1996).
[CrossRef]

W. P. Huang, C. L. Xu, W. Lui, and K. Yokoyama, “The perfectly matched layer boundary condition for modal analysis of optical waveguides: leaky mode calculations,” IEEE Photon. Technol. Lett. 8, 652–654 (1996).
[CrossRef]

Y. Y. Lu and J. Zhu, “Propagating modes in optical waveguides terminated by perfectly matched layers,” IEEE Photon. Technol. Lett. 17, 2601–2603 (2005).
[CrossRef]

G. Sztefka and H. P. Nolting, “Bidirectional eigenmode propagation for large refractive-index steps,” IEEE Photon. Technol. Lett. 5, 554–557 (1993).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

L. F. Knockaert and D. De Zutter, “On the completeness of eigenmodes in a parallel plate waveguide with a perfectly matched layer termination,” IEEE Trans. Antennas Propag. 50, 1650–1653 (2002).
[CrossRef]

IEEE Trans. Microw. Theory Tech. (4)

P. Bienstman, H. Derudder, R. Baets, F. Olyslager, and D. De Zutter, “Analysis of cylindrical waveguide discontinuities using vectorial eigenmodes and perfectly matched layers,” IEEE Trans. Microw. Theory Tech. 49, 349–354 (2001).

H. Rogier and D. De Zutter, “Berenger and leaky modes in microstrip substrates terminated by a perfectly matched layer,” IEEE Trans. Microw. Theory Tech. 49, 712–715 (2001).

T. E. Rozzi, “Rigorous analysis of step discontinuity in a planar dielectric waveguide,” IEEE Trans. Microw. Theory Tech. 26, 738–746 (1978).

R. Mittra, Y. L. Hou, and V. Jamnejad, “Analysis of open dielectric wave-guides using mode-matching technique and variational-methods,” IEEE Trans. Microw. Theory Tech. 28, 36–43 (1980).

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,” Int. J. Electron. Commun. 59, 230–238 (2005).
[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. (3)

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

Microw. Opt. Technol. Lett. (2)

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[CrossRef]

H. Rogier, L. Knockaert, and D. De Zutter, “Fast calculation of the propagation constants of leaky and Berenger modes of planar and circular dielectric waveguides terminated by a perfectly matched layer,” Microw. Opt. Technol. Lett. 37, 167–171 (2003).
[CrossRef]

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[CrossRef]

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[CrossRef]

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Figures (3)

Fig. 1.
Fig. 1.

Slab waveguide terminated by PMLs.

Fig. 2.
Fig. 2.

Comparison of the exact (marked by plus signs) and approximate (marked by open circles) propagation constants of the TE Berenger modes.

Fig. 3.
Fig. 3.

Comparison of the exact (marked by plus signs) and approximate (marked by open circles) propagation constants of the TM Berenger modes.

Tables (4)

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Table 1. Example 1: Exact Propagation Constants of the TE Berenger Modes and Relative Errors (R.E.) of Formula (20)

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Table 2. Example 2: Exact Propagation Constants of the TE Berenger Modes and Relative Errors (R.E.) of Formula (18)

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Table 3. Example 1: Exact Propagation Constants and Relative Errors of Formula (31) for TM Berenger Modes

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Table 4. Example 2: Exact Propagation Constants and Relative Errors of Formula (31) for TM Berenger Modes

Equations (33)

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n(x)={n1,x<b1;n0,b1<x<b2;n2,x>b2,
x^=0xs(τ)dτ,s(x)=1+iσ(x),
1sddx(1sdϕdx)+k02n2ϕ=β2ϕ,d1<x<d2,
ϕ(d1)=ϕ(d2)=0.
γj=k02nj2β2,j=0,1,2,
d2ϕdx^2+k02nj2ϕ=β2ϕ
γ0iγ1cot(ρ1γ1)γ0+iγ1cot(ρ1γ1)·γ0iγ2cot(ρ2γ2)γ0+iγ2cot(ρ2γ2)=e2i(b1b2)γ0,
d^j=x^(dj)=dj+icjdjσ(τ)dτ,
ρ1=b1d^1,ρ2=d^2b2.
ρ1=|ρ1|eiφ1,ρ2=|ρ2|eiφ2,
γ0iγ1cot(ρ1γ1)γ0+iγ1cot(ρ1γ1)=±ei(b1b2)γ0.
limm|βm|=andlimmθm=θ*.
γj=|βm|ei(θmπ)/21k02nj2/βm2.
ρjγj=|ρjβm|ei(θm/2+φjπ/2)1k02nj2/βm2
γ0γ1γ0+γ1·γ0γ2γ0+γ2e2i(b1b2)γ0.
γ0iγjcot(ρjγj)0.
γ0γjγ0+γje2iρjγj.
γ0iγ2cot(ρ2γ2)γ0+iγ2cot(ρ2γ2)=e2i(b1b2)γ0.
δj=k02(n02nj2),Wj=LambertW(p,±iρj2δj),
γjWjA0A0A2Wj2A02A3Wj3A03A4Wj4,
A0=iρj,A2=δj4,A3=δj4A0,A4=δj4A02δj216.
n2sddx(1sn2dϕdx)+k02n2ϕ=β2ϕ,d1<x<d2,
ϕ(d1)=ϕ(d2)=0.
μ0iμ1cot(ρ1γ1)μ0+iμ1cot((ρ1γ1)·μ0iμ2cot(ρ2γ2)μ0+iμ2cot(ρ2γ2)=e2i(b1b2)γ0,
μ0μ1μ0+μ1·μ0μ2μ0+μ2e2i(b1b2)γ0.
μ0iμjcot(ρjγj)0,j=1or2.
μ0μjμ0+μje2iρjγj,j=1or2.
2iρjγj=B0B1γj2+B2γj4+,
B0=ln(n02nj2n02+nj2)+(2m+1)πi,m0,B1=k02n02nj2n02+nj2,B2=k04n02nj2(nj43n04)4(n02+nj2)2.
γjγj(0)=B02iρj,
γjγj(1)=B02iρj2iρjB1B02,
γjγj(2)=12iρj[B0B1(γj(1))2+B2(γj(1))4].
σ(x)=Cjη31+η2,η=xcjdjcj,j=1,2

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