Abstract

The perfectly matched layer (PML) is a widely used tool to truncate the infinite domain in modal analysis for optical waveguides. Since the PML mimics the unbounded domain, propagation modes and leaky modes of the original unbounded waveguide can be derived. However, the presence of PML will introduce a series of new modes, which depend on the parameters of PML, and they are named as Berenger modes. For two-dimensional step-index waveguides, the eigenmode problem is usually transformed into an algebraic equation by the transfer matrix method (TMM). When the waveguide is nonhomogeneous, in which the refractive index in the core is varied, TMM is not available. In this paper, we use the differential TMM to derive the dispersion relation. We also deduced the asymptotic formulas for leaky modes and Berenger modes separately, which are accurate for large modes.

© 2012 Optical Society of America

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References

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  1. J. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
    [CrossRef]
  2. J. Bérenger, “Perfectly matched layer (pml) for computational electromagnetics,” Synthesis Lect. Comput. Electromagnetics 2, 1–117 (2007).
  3. Z. Sacks, D. Kingsland, R. Lee, and J. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antennas Propag. 43, 1460–1463 (1995).
    [CrossRef]
  4. W. Chew and W. Weedon, “A 3d perfectly matched medium from modified maxwell’s equations with stretched coordinates,” Microw. Opt. Technol. Lett. 7, 599–604 (1994).
    [CrossRef]
  5. F. Olyslager, “Discretization of continuous spectra based on perfectly matched layers,” SIAM J. Appl. Math. 64, 1408–1433 (2004).
    [CrossRef]
  6. P. Bienstman and R. Baets, “Advanced boundary conditions for eigenmode expansion models,” Opt. Quantum Electron. 34, 523–540 (2002).
  7. E. Anemogiannis and E. Glytsis, “Multilayer waveguides: efficient numerical analysis of general structures,” J. Lightwave Technol. 10, 1344–1351 (1992).
    [CrossRef]
  8. J. Zhu and Y. Lu, “Leaky modes of slab waveguides—asymptotic solutions,” J. Lightwave Technol. 24, 1619–1623 (2006).
    [CrossRef]
  9. E. Golant and K. Golant, “New method for calculating the spectra and radiation losses of leaky waves in multilayer optical waveguides,” Tech. Phys. 51, 1060–1068 (2006).
    [CrossRef]
  10. R. Smith and S. Houde-Walter, “Leaky guiding in nontransparent waveguides,” J. Opt. Soc. Am. A 12, 715–724 (1995).
    [CrossRef]
  11. W. Huang, C. 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]
  12. L. Yang, L. Xue, Y. Lu, and W. Huang, “New insight into quasi leaky mode approximations for unified coupled-mode analysis,” Opt. Express 18, 20595–20609 (2010).
    [CrossRef]
  13. H. Rogier and D. De Zutter, “Berenger and leaky modes in microstrip substrates terminated by a perfectly matched layer,” IEEE Trans. Microwave Theory Tech. 49, 712–715 (2001).
    [CrossRef]
  14. D. Yevick, J. Yu, and F. Schmidt, “Analytic studies of absorbing and impedance-matched boundary layers,” IEEE Photon. Technol. Lett. 9, 73–75 (1997).
    [CrossRef]
  15. Y. Lu and J. Zhu, “Propagating modes in optical waveguides terminated by perfectly matched layers,” IEEE Photon. Technol. Lett. 17, 2601–2603 (2005).
    [CrossRef]
  16. S. Khorasani and K. Mehrany, “Differential transfer-matrix method for solution of one-dimensional linear nonhomogeneous optical structures,” J. Opt. Soc. Am. B 20, 91–96 (2003).
    [CrossRef]
  17. J. Zhu and Z. Shen, “Dispersion relation of leaky modes for nonhomogeneous waveguides and its applications,” J. Lightwave Technol. 29, 3230–3236 (2011).
    [CrossRef]
  18. J. Chilwell and I. Hodgkinson, “Thin-films field-transfer matrix theory of planar multilayer waveguides and reflection from prism-loaded waveguides,” J. Opt. Soc. Am. A 1, 742–753(1984).
    [CrossRef]
  19. D. Vande Ginste, H. Rogier, and D. De Zutter, “Efficient computation of TM-and TE-polarized leaky modes in multilayered circular waveguides,” J. Lightwave Technol. 28, 1661–1669 (2010).
    [CrossRef]
  20. H. Rogier and D. Vande Ginste, “A fast procedure to accurately determine leaky modes in multilayered planar dielectric substrates,” IEEE Trans. Microwave Theory Tech. 56, 1413–1422 (2008).
    [CrossRef]
  21. 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 Inter. J. Electron. Commun. 59, 230–238 (2005).
  22. A. L. Fructos, R. R. Boix, R. Rodriguez-Berral, and F. Mesa, “Efficient determination of the poles and residues of spectral domain multilayered Green’s functions that are relevant in far-field calculations,” IEEE Trans. Antennas Propag. 58, 218–222 (2010).
    [CrossRef]
  23. X. Wang, Z. Wang, and Z. Huang, “Propagation constant of a planar dielectric waveguide with arbitrary refractive-index variation,” Opt. Lett. 18, 805–807 (1993).
    [CrossRef]
  24. A. Nissen and G. Kreiss, “An optimized perfectly matched layer for the Schrödinger equation,” Commun. Comput. Phys. 9, 147–179 (2011).

2011

J. Zhu and Z. Shen, “Dispersion relation of leaky modes for nonhomogeneous waveguides and its applications,” J. Lightwave Technol. 29, 3230–3236 (2011).
[CrossRef]

A. Nissen and G. Kreiss, “An optimized perfectly matched layer for the Schrödinger equation,” Commun. Comput. Phys. 9, 147–179 (2011).

2010

A. L. Fructos, R. R. Boix, R. Rodriguez-Berral, and F. Mesa, “Efficient determination of the poles and residues of spectral domain multilayered Green’s functions that are relevant in far-field calculations,” IEEE Trans. Antennas Propag. 58, 218–222 (2010).
[CrossRef]

D. Vande Ginste, H. Rogier, and D. De Zutter, “Efficient computation of TM-and TE-polarized leaky modes in multilayered circular waveguides,” J. Lightwave Technol. 28, 1661–1669 (2010).
[CrossRef]

L. Yang, L. Xue, Y. Lu, and W. Huang, “New insight into quasi leaky mode approximations for unified coupled-mode analysis,” Opt. Express 18, 20595–20609 (2010).
[CrossRef]

2008

H. Rogier and D. Vande Ginste, “A fast procedure to accurately determine leaky modes in multilayered planar dielectric substrates,” IEEE Trans. Microwave Theory Tech. 56, 1413–1422 (2008).
[CrossRef]

2007

J. Bérenger, “Perfectly matched layer (pml) for computational electromagnetics,” Synthesis Lect. Comput. Electromagnetics 2, 1–117 (2007).

2006

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

E. Golant and K. Golant, “New method for calculating the spectra and radiation losses of leaky waves in multilayer optical waveguides,” Tech. Phys. 51, 1060–1068 (2006).
[CrossRef]

2005

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 Inter. J. Electron. Commun. 59, 230–238 (2005).

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

2004

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

2003

2002

P. Bienstman and R. Baets, “Advanced boundary conditions for eigenmode expansion models,” Opt. Quantum Electron. 34, 523–540 (2002).

2001

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

1997

D. Yevick, J. Yu, and F. Schmidt, “Analytic studies of absorbing and impedance-matched boundary layers,” IEEE Photon. Technol. Lett. 9, 73–75 (1997).
[CrossRef]

1996

W. Huang, C. 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]

1995

R. Smith and S. Houde-Walter, “Leaky guiding in nontransparent waveguides,” J. Opt. Soc. Am. A 12, 715–724 (1995).
[CrossRef]

Z. Sacks, D. Kingsland, R. Lee, and J. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antennas Propag. 43, 1460–1463 (1995).
[CrossRef]

1994

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

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

1993

1992

E. Anemogiannis and E. Glytsis, “Multilayer waveguides: efficient numerical analysis of general structures,” J. Lightwave Technol. 10, 1344–1351 (1992).
[CrossRef]

1984

Anemogiannis, E.

E. Anemogiannis and E. Glytsis, “Multilayer waveguides: efficient numerical analysis of general structures,” J. Lightwave Technol. 10, 1344–1351 (1992).
[CrossRef]

Baets, R.

P. Bienstman and R. Baets, “Advanced boundary conditions for eigenmode expansion models,” Opt. Quantum Electron. 34, 523–540 (2002).

Berenger, J.

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

Bérenger, J.

J. Bérenger, “Perfectly matched layer (pml) for computational electromagnetics,” Synthesis Lect. Comput. Electromagnetics 2, 1–117 (2007).

Bienstman, P.

P. Bienstman and R. Baets, “Advanced boundary conditions for eigenmode expansion models,” Opt. Quantum Electron. 34, 523–540 (2002).

Boix, R. R.

A. L. Fructos, R. R. Boix, R. Rodriguez-Berral, and F. Mesa, “Efficient determination of the poles and residues of spectral domain multilayered Green’s functions that are relevant in far-field calculations,” IEEE Trans. Antennas Propag. 58, 218–222 (2010).
[CrossRef]

Chew, W.

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

Chilwell, J.

De Zutter, D.

D. Vande Ginste, H. Rogier, and D. De Zutter, “Efficient computation of TM-and TE-polarized leaky modes in multilayered circular waveguides,” J. Lightwave Technol. 28, 1661–1669 (2010).
[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,” AEU Inter. J. Electron. Commun. 59, 230–238 (2005).

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

Fructos, A. L.

A. L. Fructos, R. R. Boix, R. Rodriguez-Berral, and F. Mesa, “Efficient determination of the poles and residues of spectral domain multilayered Green’s functions that are relevant in far-field calculations,” IEEE Trans. Antennas Propag. 58, 218–222 (2010).
[CrossRef]

Glytsis, E.

E. Anemogiannis and E. Glytsis, “Multilayer waveguides: efficient numerical analysis of general structures,” J. Lightwave Technol. 10, 1344–1351 (1992).
[CrossRef]

Golant, E.

E. Golant and K. Golant, “New method for calculating the spectra and radiation losses of leaky waves in multilayer optical waveguides,” Tech. Phys. 51, 1060–1068 (2006).
[CrossRef]

Golant, K.

E. Golant and K. Golant, “New method for calculating the spectra and radiation losses of leaky waves in multilayer optical waveguides,” Tech. Phys. 51, 1060–1068 (2006).
[CrossRef]

Hodgkinson, I.

Houde-Walter, S.

Huang, W.

L. Yang, L. Xue, Y. Lu, and W. Huang, “New insight into quasi leaky mode approximations for unified coupled-mode analysis,” Opt. Express 18, 20595–20609 (2010).
[CrossRef]

W. Huang, C. 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]

Huang, Z.

Khorasani, S.

Kingsland, D.

Z. Sacks, D. Kingsland, R. Lee, and J. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antennas Propag. 43, 1460–1463 (1995).
[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,” AEU Inter. J. Electron. Commun. 59, 230–238 (2005).

Kreiss, G.

A. Nissen and G. Kreiss, “An optimized perfectly matched layer for the Schrödinger equation,” Commun. Comput. Phys. 9, 147–179 (2011).

Lee, J.

Z. Sacks, D. Kingsland, R. Lee, and J. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antennas Propag. 43, 1460–1463 (1995).
[CrossRef]

Lee, R.

Z. Sacks, D. Kingsland, R. Lee, and J. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antennas Propag. 43, 1460–1463 (1995).
[CrossRef]

Lu, Y.

Lui, W.

W. Huang, C. 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]

Mehrany, K.

Mesa, F.

A. L. Fructos, R. R. Boix, R. Rodriguez-Berral, and F. Mesa, “Efficient determination of the poles and residues of spectral domain multilayered Green’s functions that are relevant in far-field calculations,” IEEE Trans. Antennas Propag. 58, 218–222 (2010).
[CrossRef]

Nissen, A.

A. Nissen and G. Kreiss, “An optimized perfectly matched layer for the Schrödinger equation,” Commun. Comput. Phys. 9, 147–179 (2011).

Olyslager, F.

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

Rodriguez-Berral, R.

A. L. Fructos, R. R. Boix, R. Rodriguez-Berral, and F. Mesa, “Efficient determination of the poles and residues of spectral domain multilayered Green’s functions that are relevant in far-field calculations,” IEEE Trans. Antennas Propag. 58, 218–222 (2010).
[CrossRef]

Rogier, H.

D. Vande Ginste, H. Rogier, and D. De Zutter, “Efficient computation of TM-and TE-polarized leaky modes in multilayered circular waveguides,” J. Lightwave Technol. 28, 1661–1669 (2010).
[CrossRef]

H. Rogier and D. Vande Ginste, “A fast procedure to accurately determine leaky modes in multilayered planar dielectric substrates,” IEEE Trans. Microwave Theory Tech. 56, 1413–1422 (2008).
[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,” AEU Inter. J. Electron. Commun. 59, 230–238 (2005).

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

Sacks, Z.

Z. Sacks, D. Kingsland, R. Lee, and J. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antennas Propag. 43, 1460–1463 (1995).
[CrossRef]

Schmidt, F.

D. Yevick, J. Yu, and F. Schmidt, “Analytic studies of absorbing and impedance-matched boundary layers,” IEEE Photon. Technol. Lett. 9, 73–75 (1997).
[CrossRef]

Shen, Z.

Smith, R.

Vande Ginste, D.

D. Vande Ginste, H. Rogier, and D. De Zutter, “Efficient computation of TM-and TE-polarized leaky modes in multilayered circular waveguides,” J. Lightwave Technol. 28, 1661–1669 (2010).
[CrossRef]

H. Rogier and D. Vande Ginste, “A fast procedure to accurately determine leaky modes in multilayered planar dielectric substrates,” IEEE Trans. Microwave Theory Tech. 56, 1413–1422 (2008).
[CrossRef]

Wang, X.

Wang, Z.

Weedon, W.

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

Xu, C.

W. Huang, C. 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]

Xue, L.

Yang, L.

Yevick, D.

D. Yevick, J. Yu, and F. Schmidt, “Analytic studies of absorbing and impedance-matched boundary layers,” IEEE Photon. Technol. Lett. 9, 73–75 (1997).
[CrossRef]

Yokoyama, K.

W. Huang, C. 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]

Yu, J.

D. Yevick, J. Yu, and F. Schmidt, “Analytic studies of absorbing and impedance-matched boundary layers,” IEEE Photon. Technol. Lett. 9, 73–75 (1997).
[CrossRef]

Zhu, J.

AEU Inter. J. Electron. Commun.

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 Inter. J. Electron. Commun. 59, 230–238 (2005).

Commun. Comput. Phys.

A. Nissen and G. Kreiss, “An optimized perfectly matched layer for the Schrödinger equation,” Commun. Comput. Phys. 9, 147–179 (2011).

IEEE Photon. Technol. Lett.

W. Huang, C. 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]

D. Yevick, J. Yu, and F. Schmidt, “Analytic studies of absorbing and impedance-matched boundary layers,” IEEE Photon. Technol. Lett. 9, 73–75 (1997).
[CrossRef]

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

IEEE Trans. Antennas Propag.

Z. Sacks, D. Kingsland, R. Lee, and J. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antennas Propag. 43, 1460–1463 (1995).
[CrossRef]

A. L. Fructos, R. R. Boix, R. Rodriguez-Berral, and F. Mesa, “Efficient determination of the poles and residues of spectral domain multilayered Green’s functions that are relevant in far-field calculations,” IEEE Trans. Antennas Propag. 58, 218–222 (2010).
[CrossRef]

IEEE Trans. Microwave Theory Tech.

H. Rogier and D. Vande Ginste, “A fast procedure to accurately determine leaky modes in multilayered planar dielectric substrates,” IEEE Trans. Microwave Theory Tech. 56, 1413–1422 (2008).
[CrossRef]

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

J. Comput. Phys.

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

J. Lightwave Technol.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Microw. Opt. Technol. Lett.

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

Opt. Express

Opt. Lett.

Opt. Quantum Electron.

P. Bienstman and R. Baets, “Advanced boundary conditions for eigenmode expansion models,” Opt. Quantum Electron. 34, 523–540 (2002).

SIAM J. Appl. Math.

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

Synthesis Lect. Comput. Electromagnetics

J. Bérenger, “Perfectly matched layer (pml) for computational electromagnetics,” Synthesis Lect. Comput. Electromagnetics 2, 1–117 (2007).

Tech. Phys.

E. Golant and K. Golant, “New method for calculating the spectra and radiation losses of leaky waves in multilayer optical waveguides,” Tech. Phys. 51, 1060–1068 (2006).
[CrossRef]

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Figures (5)

Fig. 1.
Fig. 1.

Waveguide with varied RIP, and the boundary conditions are given in x=a and x=a. It could be treated as the black box described by the transfer matrix T.

Fig. 2.
Fig. 2.

Odd modes of the waveguide in Fig. 1, in which the asymptotic leaky modes given by Eq. (26) are marked by “○,” the Berenger modes given by Eq. (29) are marked by “Δ,” and the iterated solutions are marked by “×.”

Fig. 3.
Fig. 3.

Iterated solutions with different PML thickness L.

Fig. 4.
Fig. 4.

Iterated solutions with different PML absorbing strength factor σ0 and PML position a.

Fig. 5.
Fig. 5.

Comparing the iterated solutions and the exact solutions, in which the PML parameters are L=1μm, σ0=30, and a=10μm.

Tables (2)

Tables Icon

Table 1. Relative Errors of the Previous Example

Tables Icon

Table 2. Relative Errors of the Solutions of Eqs. (22) and (20)

Equations (43)

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

ρz(1ρuz)+ρx(1ρux)+κ02n2(x)u=0,
n(x)={ns,x<d,n0(x),d<x<d,nc,x>d.
d2ϕdx2+κ02n2(x)ϕ=β2ϕ,<x<,
ϕ(x)=Aseiksx+Bseiksx,x<d,ϕ(x)=Aceikcx+Bceikcx,x>d,
[AcBc]=[T11T12T21T22][AsBs].
rsAsBs=0,x=a.
AcrcBc=0,x=a.
[1rc][T11T12T21T22][1rs]=0,
[10][T11T12T21T22][10]=0.
d2ϕ^dx^2+κ02nc2ϕ^=β2ϕ^,
ϕ^(x)=APMLeikcx^+BPMLeikcx^=APMLeikc[x+iaxσ(t)dt]+BPMLeikc[x+iaxσ(t)dt].
ϕ^(a+L)=APMLeikc(a+L^)+BPMLeikc(a+L^)=0.
r^c=rPML=APML/BPML=e2ikcae2ikcL^.
r^s=e2iksae2iksL^.
[1r^c][T11T12T21T22][1r^s]=0,
ϕ(x)=ϕ(x+),ϕ(x)=ϕ(x+).
T0=12[(1+ks/k0)ei(k0ks)d(1ks/k0)ei(k0+ks)d(1ks/k0)e+i(k0+ks)d(1+ks/k0)e+i(k0ks)d],
T1=12[(1+k1/kc)e+i(kck1)d(1k1/kc)e+i(kc+k1)d(1k1/kc)ei(kc+k1)d(1+k1/kc)ei(kck1)d],
Td=exp[ddU(t)dt],
U(x)=k(x)2k(x)[1+i2k(x)xexp[+i2k(x)x]exp[i2k(x)x]1i2k(x)x].
Us(x)=ik(x)x[1001]
T=T1·Td·T0=T1·exp[ddU(t)dt]·T0
[1r^c][T11T12T21T22][11]=0,
[1r^c][T11T12T21T22][11]=0,
T=[T11T12T21T22]=T1exp0dU(t)dt
e2ikc(ad+L^)=(k1kc)+(k1+kc)e2i0dk(t)dt(k1+kc)+(k1kc)e2i0dk(t)dt.
kck1kc+k1=e2i0dk(t)dt,
ei(k0+k1)d=kc2k12(kc+k1)2=κ02(nc2n02(d))(kc+k1)2.
±d2κ0n0(d)2nc2=id2(kc+k1)eid2(k0+k1).
k0=k12ξ0=k1(1ξ02k12ξ028k14+),kc=k12ξc=k1(1ξc2k12ξc28k14+).
(a0k1+a1k1+a2k12+)e(a0k1+a1k1+a2k12+),
k1Wa0,
k1Wa0a1Wa0a2W2,
a0=id,a1=id4ξ0,a2=ξ0ξc4.
kck1kc+k1=e2ikc(ad+L^).
k1=kc2+ξc=kc(1+ξc2kc2ξc28kc4+),
±iκ0a^2n02(d)nc2=ikca^2(2+ξc2kc2ξc28kc4+)eikca^,
(b0kc+b1kc+b2kc2+b3kc3+)eb0kc+b1kc+b2kc2+b3kc3+.
b0=ia^,b1=0,b2=ξc4,b3=iξc4a^.
kcWbb0b0b2Wb2b02b3Wb3,
e2ikc(ad+L^)=(kck1)+(kc+k1)e2i0dk(t)dt(kc+k1)+(kck1)e2i0dk(t)dt.
n0(x)=1.51(x/10)2
|[1r^c][T11T12T21T22][11]|

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