Abstract

A polynomial expansion approach to the extraction of guided and leaky modes in layered structures including dielectric waveguides and periodic stratified media is proposed. To verify the method we compared the results of analysis of a typical test case with those reported in the literature and found good agreement. Polynomial expansion is a nonharmonic expansion and does not involve harmonic functions or intrinsic modes of homogenous layers. This approach has the benefit of leading to algebraic dispersion equations rather than to a transcendental dispersion equation; therefore, it will be easier to use than other methods such as the argument principle method, the reflection pole method, and the wave-vector density method, which solve the transcendental dispersion equation by means of integrals. Besides, an algebraic dispersion equation can be obtained without any problem of numerical instability, whereas an ordinary transcendental dispersion equation, which is usually derived by the transfer matrix method, is difficult to obtain because of instability in multiplying transfer matrices. A demonstration of the utility of the proposed method when the other methods mentioned are inferior or fail are also given.

© 2003 Optical Society of America

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References

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  1. K. Mehrany and B. Rashidian, “Polynomial expansions of fields for extraction of eigenmodes in layered waveguides,” Proc. SPIE 4833, 769–775 (2003).
    [CrossRef]
  2. K. H. Schlereth and M. Tacke, “The complex propagation constant of multilayer waveguides: An algorithm for a personal computer,” IEEE J. Quantum Electron. 26, 627–630 (1990).
    [CrossRef]
  3. E. Anemogiannis and E. N. Glytsis, “Multilayer waveguides: efficient numerical analysis of general structures,” J. Lightwave Technol. 10, 1344–1351 (1992).
    [CrossRef]
  4. M. Koshiba and H. Kumagami, “Theoretical study of silicon-clad planar diffused optical waveguides,” Proc. Inst. Electr. Eng. J. 134, 333–338 (1987).
  5. A. K. Ghatak, K. Thyagarajan, and M. R. Shenoy, “Numerical analysis of planar optical waveguides using matrix approach,” J. Lightwave Technol. LT-5, 660–667 (1987).
    [CrossRef]
  6. L. M. Delves and J. N. Lyness, “A numerical method for locating the zeros of an analytic function,” Math. Comput. 21, 543–560 (1967).
    [CrossRef]
  7. E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, “Efficient solution of eigenvalue equations of optical waveguiding structures,” J. Lightwave Technol. 12, 2080–2084 (1994).
    [CrossRef]
  8. L. F. Abd-ellal, L. M. Delves, and J. K. Reid, “A numerical method for locating the zeros and poles of a mermomorphic function,” in Numerical Methods for Nonlinear Algebraic Equations, P. Rabinowitz, ed. (Gordon & Breach, London, 1970), pp. 47–59.
  9. E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, “Determination of guided and leaky modes in lossless and lossy planar multilayer optical waveguides: reflection pole method and wavevector density method,” J. Lightwave Technol. 17, 929–941 (1999).
    [CrossRef]
  10. K. Mehrany, S. Khorasani, and B. Rashidian, “Variational approach for extraction of eigenmodes in layered waveguides,” J. Opt. Soc. Am. B 19, 1978–1981 (2002).
    [CrossRef]
  11. S. Khorasani and B. Rashidian, “Modified transfer matrix method for conducting interfaces,” J. Opt. A 4, 251–256 (2002).
    [CrossRef]
  12. T. D. Visser, H. Blok, and D. Lenstra, “Modal analysis of a planar waveguide with gain and losses,” IEEE J. Quantum Electron. 31, 1803–1810 (1995).
    [CrossRef]
  13. J. G. Dil and H. Blok, “Propagation of electromagnetic surface waves in a radially inhomogeneous optical waveguide,” Opt. Electron. 5, 415–428 (1973).
    [CrossRef]
  14. M. J. Adams, “The cladded parabolic-index profile waveguide: analysis and application to stripe geometry lasers,” Opt. Quantum Electron. 10, 17–29 (1978).
    [CrossRef]
  15. R. A. Sammut, “Nonlinear planar waveguides with graded index core: power series solution,” Opt. Quantum Electron. 26, S301–S310 (1994).
    [CrossRef]
  16. C. Vassallo and J. M. van der Keur, “Comparison of a few transparent boundary conditions for finite difference optical mode solvers,” J. Lightwave Technol. 15, 397–402 (1997).
    [CrossRef]
  17. P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).
  18. W. M. Robertson, “Experimental measurement of the effect of termination on surface electromagnetic waves in one-dimensional photonic band gap arrays,” J. Lightwave Technol. 17, 2013–2017 (1999).
    [CrossRef]
  19. W. M. Robertson and M. S. May, “Surface electromagnetic waves on one-dimensional photonic band gap arrays,” Appl. Phys. Lett. 74, 1800–1802 (1999).
    [CrossRef]
  20. D. N. Chigrin, A. V. Lavrinenko, D. A. Yarotsky, and S. V. Gaponenko, “All-dielectric one-dimensional periodic structures for total omnidirectional reflection and partial spontaneous emission control,” J. Lightwave Technol. 17, 2018–2024 (1999).
    [CrossRef]
  21. K. Mehrany, S. Khorasani, and B. Rashidian, “Novel optical devices based on surface wave excitation at conducting interfaces,” Semicond. Sci. Technol. 18, 582–588 (2003).
    [CrossRef]
  22. J. Chiwell and I. Hodgkinson, “Thin films field transfer matrix theory of planar multiplayer waveguides and reflection from prism-loaded waveguides,” J. Opt. Soc. Am. A 1, 742–753 (1984).
    [CrossRef]

2003 (2)

K. Mehrany and B. Rashidian, “Polynomial expansions of fields for extraction of eigenmodes in layered waveguides,” Proc. SPIE 4833, 769–775 (2003).
[CrossRef]

K. Mehrany, S. Khorasani, and B. Rashidian, “Novel optical devices based on surface wave excitation at conducting interfaces,” Semicond. Sci. Technol. 18, 582–588 (2003).
[CrossRef]

2002 (2)

K. Mehrany, S. Khorasani, and B. Rashidian, “Variational approach for extraction of eigenmodes in layered waveguides,” J. Opt. Soc. Am. B 19, 1978–1981 (2002).
[CrossRef]

S. Khorasani and B. Rashidian, “Modified transfer matrix method for conducting interfaces,” J. Opt. A 4, 251–256 (2002).
[CrossRef]

1999 (4)

1997 (1)

C. Vassallo and J. M. van der Keur, “Comparison of a few transparent boundary conditions for finite difference optical mode solvers,” J. Lightwave Technol. 15, 397–402 (1997).
[CrossRef]

1995 (1)

T. D. Visser, H. Blok, and D. Lenstra, “Modal analysis of a planar waveguide with gain and losses,” IEEE J. Quantum Electron. 31, 1803–1810 (1995).
[CrossRef]

1994 (2)

R. A. Sammut, “Nonlinear planar waveguides with graded index core: power series solution,” Opt. Quantum Electron. 26, S301–S310 (1994).
[CrossRef]

E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, “Efficient solution of eigenvalue equations of optical waveguiding structures,” J. Lightwave Technol. 12, 2080–2084 (1994).
[CrossRef]

1992 (1)

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

1990 (1)

K. H. Schlereth and M. Tacke, “The complex propagation constant of multilayer waveguides: An algorithm for a personal computer,” IEEE J. Quantum Electron. 26, 627–630 (1990).
[CrossRef]

1987 (2)

M. Koshiba and H. Kumagami, “Theoretical study of silicon-clad planar diffused optical waveguides,” Proc. Inst. Electr. Eng. J. 134, 333–338 (1987).

A. K. Ghatak, K. Thyagarajan, and M. R. Shenoy, “Numerical analysis of planar optical waveguides using matrix approach,” J. Lightwave Technol. LT-5, 660–667 (1987).
[CrossRef]

1984 (1)

1978 (1)

M. J. Adams, “The cladded parabolic-index profile waveguide: analysis and application to stripe geometry lasers,” Opt. Quantum Electron. 10, 17–29 (1978).
[CrossRef]

1973 (1)

J. G. Dil and H. Blok, “Propagation of electromagnetic surface waves in a radially inhomogeneous optical waveguide,” Opt. Electron. 5, 415–428 (1973).
[CrossRef]

1967 (1)

L. M. Delves and J. N. Lyness, “A numerical method for locating the zeros of an analytic function,” Math. Comput. 21, 543–560 (1967).
[CrossRef]

Adams, M. J.

M. J. Adams, “The cladded parabolic-index profile waveguide: analysis and application to stripe geometry lasers,” Opt. Quantum Electron. 10, 17–29 (1978).
[CrossRef]

Anemogiannis, E.

E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, “Determination of guided and leaky modes in lossless and lossy planar multilayer optical waveguides: reflection pole method and wavevector density method,” J. Lightwave Technol. 17, 929–941 (1999).
[CrossRef]

E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, “Efficient solution of eigenvalue equations of optical waveguiding structures,” J. Lightwave Technol. 12, 2080–2084 (1994).
[CrossRef]

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

Blok, H.

T. D. Visser, H. Blok, and D. Lenstra, “Modal analysis of a planar waveguide with gain and losses,” IEEE J. Quantum Electron. 31, 1803–1810 (1995).
[CrossRef]

J. G. Dil and H. Blok, “Propagation of electromagnetic surface waves in a radially inhomogeneous optical waveguide,” Opt. Electron. 5, 415–428 (1973).
[CrossRef]

Chigrin, D. N.

Chiwell, J.

Delves, L. M.

L. M. Delves and J. N. Lyness, “A numerical method for locating the zeros of an analytic function,” Math. Comput. 21, 543–560 (1967).
[CrossRef]

Dil, J. G.

J. G. Dil and H. Blok, “Propagation of electromagnetic surface waves in a radially inhomogeneous optical waveguide,” Opt. Electron. 5, 415–428 (1973).
[CrossRef]

Gaponenko, S. V.

Gaylord, T. K.

Ghatak, A. K.

A. K. Ghatak, K. Thyagarajan, and M. R. Shenoy, “Numerical analysis of planar optical waveguides using matrix approach,” J. Lightwave Technol. LT-5, 660–667 (1987).
[CrossRef]

Glytsis, E. N.

E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, “Determination of guided and leaky modes in lossless and lossy planar multilayer optical waveguides: reflection pole method and wavevector density method,” J. Lightwave Technol. 17, 929–941 (1999).
[CrossRef]

E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, “Efficient solution of eigenvalue equations of optical waveguiding structures,” J. Lightwave Technol. 12, 2080–2084 (1994).
[CrossRef]

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

Hodgkinson, I.

Khorasani, S.

K. Mehrany, S. Khorasani, and B. Rashidian, “Novel optical devices based on surface wave excitation at conducting interfaces,” Semicond. Sci. Technol. 18, 582–588 (2003).
[CrossRef]

K. Mehrany, S. Khorasani, and B. Rashidian, “Variational approach for extraction of eigenmodes in layered waveguides,” J. Opt. Soc. Am. B 19, 1978–1981 (2002).
[CrossRef]

S. Khorasani and B. Rashidian, “Modified transfer matrix method for conducting interfaces,” J. Opt. A 4, 251–256 (2002).
[CrossRef]

Koshiba, M.

M. Koshiba and H. Kumagami, “Theoretical study of silicon-clad planar diffused optical waveguides,” Proc. Inst. Electr. Eng. J. 134, 333–338 (1987).

Kumagami, H.

M. Koshiba and H. Kumagami, “Theoretical study of silicon-clad planar diffused optical waveguides,” Proc. Inst. Electr. Eng. J. 134, 333–338 (1987).

Lavrinenko, A. V.

Lenstra, D.

T. D. Visser, H. Blok, and D. Lenstra, “Modal analysis of a planar waveguide with gain and losses,” IEEE J. Quantum Electron. 31, 1803–1810 (1995).
[CrossRef]

Lyness, J. N.

L. M. Delves and J. N. Lyness, “A numerical method for locating the zeros of an analytic function,” Math. Comput. 21, 543–560 (1967).
[CrossRef]

May, M. S.

W. M. Robertson and M. S. May, “Surface electromagnetic waves on one-dimensional photonic band gap arrays,” Appl. Phys. Lett. 74, 1800–1802 (1999).
[CrossRef]

Mehrany, K.

K. Mehrany, S. Khorasani, and B. Rashidian, “Novel optical devices based on surface wave excitation at conducting interfaces,” Semicond. Sci. Technol. 18, 582–588 (2003).
[CrossRef]

K. Mehrany and B. Rashidian, “Polynomial expansions of fields for extraction of eigenmodes in layered waveguides,” Proc. SPIE 4833, 769–775 (2003).
[CrossRef]

K. Mehrany, S. Khorasani, and B. Rashidian, “Variational approach for extraction of eigenmodes in layered waveguides,” J. Opt. Soc. Am. B 19, 1978–1981 (2002).
[CrossRef]

Rashidian, B.

K. Mehrany and B. Rashidian, “Polynomial expansions of fields for extraction of eigenmodes in layered waveguides,” Proc. SPIE 4833, 769–775 (2003).
[CrossRef]

K. Mehrany, S. Khorasani, and B. Rashidian, “Novel optical devices based on surface wave excitation at conducting interfaces,” Semicond. Sci. Technol. 18, 582–588 (2003).
[CrossRef]

S. Khorasani and B. Rashidian, “Modified transfer matrix method for conducting interfaces,” J. Opt. A 4, 251–256 (2002).
[CrossRef]

K. Mehrany, S. Khorasani, and B. Rashidian, “Variational approach for extraction of eigenmodes in layered waveguides,” J. Opt. Soc. Am. B 19, 1978–1981 (2002).
[CrossRef]

Robertson, W. M.

W. M. Robertson, “Experimental measurement of the effect of termination on surface electromagnetic waves in one-dimensional photonic band gap arrays,” J. Lightwave Technol. 17, 2013–2017 (1999).
[CrossRef]

W. M. Robertson and M. S. May, “Surface electromagnetic waves on one-dimensional photonic band gap arrays,” Appl. Phys. Lett. 74, 1800–1802 (1999).
[CrossRef]

Sammut, R. A.

R. A. Sammut, “Nonlinear planar waveguides with graded index core: power series solution,” Opt. Quantum Electron. 26, S301–S310 (1994).
[CrossRef]

Schlereth, K. H.

K. H. Schlereth and M. Tacke, “The complex propagation constant of multilayer waveguides: An algorithm for a personal computer,” IEEE J. Quantum Electron. 26, 627–630 (1990).
[CrossRef]

Shenoy, M. R.

A. K. Ghatak, K. Thyagarajan, and M. R. Shenoy, “Numerical analysis of planar optical waveguides using matrix approach,” J. Lightwave Technol. LT-5, 660–667 (1987).
[CrossRef]

Tacke, M.

K. H. Schlereth and M. Tacke, “The complex propagation constant of multilayer waveguides: An algorithm for a personal computer,” IEEE J. Quantum Electron. 26, 627–630 (1990).
[CrossRef]

Thyagarajan, K.

A. K. Ghatak, K. Thyagarajan, and M. R. Shenoy, “Numerical analysis of planar optical waveguides using matrix approach,” J. Lightwave Technol. LT-5, 660–667 (1987).
[CrossRef]

van der Keur, J. M.

C. Vassallo and J. M. van der Keur, “Comparison of a few transparent boundary conditions for finite difference optical mode solvers,” J. Lightwave Technol. 15, 397–402 (1997).
[CrossRef]

Vassallo, C.

C. Vassallo and J. M. van der Keur, “Comparison of a few transparent boundary conditions for finite difference optical mode solvers,” J. Lightwave Technol. 15, 397–402 (1997).
[CrossRef]

Visser, T. D.

T. D. Visser, H. Blok, and D. Lenstra, “Modal analysis of a planar waveguide with gain and losses,” IEEE J. Quantum Electron. 31, 1803–1810 (1995).
[CrossRef]

Yarotsky, D. A.

Appl. Phys. Lett. (1)

W. M. Robertson and M. S. May, “Surface electromagnetic waves on one-dimensional photonic band gap arrays,” Appl. Phys. Lett. 74, 1800–1802 (1999).
[CrossRef]

IEEE J. Quantum Electron. (2)

K. H. Schlereth and M. Tacke, “The complex propagation constant of multilayer waveguides: An algorithm for a personal computer,” IEEE J. Quantum Electron. 26, 627–630 (1990).
[CrossRef]

T. D. Visser, H. Blok, and D. Lenstra, “Modal analysis of a planar waveguide with gain and losses,” IEEE J. Quantum Electron. 31, 1803–1810 (1995).
[CrossRef]

J. Lightwave Technol. (7)

C. Vassallo and J. M. van der Keur, “Comparison of a few transparent boundary conditions for finite difference optical mode solvers,” J. Lightwave Technol. 15, 397–402 (1997).
[CrossRef]

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

A. K. Ghatak, K. Thyagarajan, and M. R. Shenoy, “Numerical analysis of planar optical waveguides using matrix approach,” J. Lightwave Technol. LT-5, 660–667 (1987).
[CrossRef]

E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, “Efficient solution of eigenvalue equations of optical waveguiding structures,” J. Lightwave Technol. 12, 2080–2084 (1994).
[CrossRef]

E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, “Determination of guided and leaky modes in lossless and lossy planar multilayer optical waveguides: reflection pole method and wavevector density method,” J. Lightwave Technol. 17, 929–941 (1999).
[CrossRef]

D. N. Chigrin, A. V. Lavrinenko, D. A. Yarotsky, and S. V. Gaponenko, “All-dielectric one-dimensional periodic structures for total omnidirectional reflection and partial spontaneous emission control,” J. Lightwave Technol. 17, 2018–2024 (1999).
[CrossRef]

W. M. Robertson, “Experimental measurement of the effect of termination on surface electromagnetic waves in one-dimensional photonic band gap arrays,” J. Lightwave Technol. 17, 2013–2017 (1999).
[CrossRef]

J. Opt. A (1)

S. Khorasani and B. Rashidian, “Modified transfer matrix method for conducting interfaces,” J. Opt. A 4, 251–256 (2002).
[CrossRef]

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

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

Math. Comput. (1)

L. M. Delves and J. N. Lyness, “A numerical method for locating the zeros of an analytic function,” Math. Comput. 21, 543–560 (1967).
[CrossRef]

Opt. Electron. (1)

J. G. Dil and H. Blok, “Propagation of electromagnetic surface waves in a radially inhomogeneous optical waveguide,” Opt. Electron. 5, 415–428 (1973).
[CrossRef]

Opt. Quantum Electron. (2)

M. J. Adams, “The cladded parabolic-index profile waveguide: analysis and application to stripe geometry lasers,” Opt. Quantum Electron. 10, 17–29 (1978).
[CrossRef]

R. A. Sammut, “Nonlinear planar waveguides with graded index core: power series solution,” Opt. Quantum Electron. 26, S301–S310 (1994).
[CrossRef]

Proc. Inst. Electr. Eng. J. (1)

M. Koshiba and H. Kumagami, “Theoretical study of silicon-clad planar diffused optical waveguides,” Proc. Inst. Electr. Eng. J. 134, 333–338 (1987).

Proc. SPIE (1)

K. Mehrany and B. Rashidian, “Polynomial expansions of fields for extraction of eigenmodes in layered waveguides,” Proc. SPIE 4833, 769–775 (2003).
[CrossRef]

Semicond. Sci. Technol. (1)

K. Mehrany, S. Khorasani, and B. Rashidian, “Novel optical devices based on surface wave excitation at conducting interfaces,” Semicond. Sci. Technol. 18, 582–588 (2003).
[CrossRef]

Other (2)

P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).

L. F. Abd-ellal, L. M. Delves, and J. K. Reid, “A numerical method for locating the zeros and poles of a mermomorphic function,” in Numerical Methods for Nonlinear Algebraic Equations, P. Rabinowitz, ed. (Gordon & Breach, London, 1970), pp. 47–59.

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Figures (5)

Fig. 1
Fig. 1

Structure of the layered waveguide.

Fig. 2
Fig. 2

Structure of periodic stratified media.

Fig. 3
Fig. 3

Structure of a parallel-plate waveguide loaded with a dielectric film.

Fig. 4
Fig. 4

Propagation constants of a parallel-plate waveguide loaded with a dielectric film: dashed curve, Legendre expansion; solid curve, Chebyshev expansion.

Fig. 5
Fig. 5

Band structure of a one-dimensional photonic crystal, K [Bloch wave number versus wavelength of light (Lambda) in free space].

Tables (4)

Tables Icon

Table 1 Relative Error in Guided Modes of a Parallel-Plate Waveguide

Tables Icon

Table 2 Effective Indices for TE and TM Propagating Eigenmodes for an Open Waveguide

Tables Icon

Table 3 Effective Indices for Leaky Eigenmodes for an Open Waveguide a

Tables Icon

Table 4 Effective Indices for Guided TE Modes in a Three-Layer Structure with Exponentially Growing Details a

Equations (54)

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

d2Ψidx2+(k02ni2-β2)Ψi=0,Xi-1<x<Xi,
Ψi(x)=m=0Mi qm(i)Pm(ξ),
ξ=2x-Xi-1-XiXi-Xi-1,Xi-1<x<Xi,
m=0Mi-2 rm(i)Pm(ξ)=di22(β2-k02)m=0Mi qm(i)Pm(ξ),
rn(i)=n+12p=n+2,n+4,Mi(p+n+1)(p-n)qp(i),
di=Xi-Xi-1.
r0(i)=di22(β2-ni2k02)q0(i),
r1(i)=di22(β2-ni2k02)qi(i),
rMi-2(i)=di22(β2-ni2k02)qMi-2(i),
Ψi(x)=m=0Mi qm(i)Tm(ξ),
ξ=2x-Xi-1-XiXi-Xi-1,Xi-1<x<Xi,
Ψi-1(Xi-1)=Ψi(Xi-1),Ψi(Xi)=Ψi+1(Xi),
Ψi-1(Xi-1)=Ψi(Xi-1),Ψi(Xi)=Ψi+1(Xi),
Ψi-1(Xi-1)=Ψi(Xi-1),Ψi(Xi)=Ψi+1(Xi),
1ni-12 Ψi-1(Xi-1)=1ni2 Ψi(Xi-1),
1ni2 Ψi(Xi)=1ni+12 Ψi+1(Xi),
(Q0+β2Q1)q¯=0,
Ψ1(X0)=(β2-k02ns2)1/2Ψ1(X0),
ΨN(XN)=(β2-k02nc2)1/2ΨN(XN).
Ψ1(X0)=n12ns2 (β2-k02ns2)1/2Ψ1(X0),
ΨN(XN)=nN2nc2 (β2-k02nc2)1/2ΨN(XN).
Q0+β2Q1+(β2-k02ns2)1/2Q2
+(β2-k02nc2)1/2Q3)q¯=0,
ΨN+1(x)=m=0MN+1 qm(N+1)Pm(ξ),
ξ=2x-XN-XN+1XN+1-XN,XN<x<XN+1,
ΨN+2(x)=m=0MN+2 qm(N+2)Pm(ξ),
ξ=2x-XN+2-XN+1XN+2-XN+1,XN+1<x<XN+2,
ΨN+1(x)=m=0MN+1 qm(N+1)Tm(ξ),
ξ=2x-XN-XN+1XN+1-XN,XN<x<XN+1,
ΨN+2(x)=m=0MN+2 qm(N+2)Tm(ξ),
ξ=2x-XN+2-XN+1XN+2-XN+1,XN+1<x<XN+2.
ΨN+1(XN+1)=ΨN+2(XN+1),
ΨN+1(XN+1)=ΨN+2(XN+1),
ΨN+1(XN+1)=ΨN+2(XN+1),
1n12 ΨN+1(XN+1)=1n22 ΨN+2(XN+1),
Ey(x+Λ)=Ey(x)exp(-jkΛ),
Hz(x+Λ)=Hz(x)exp(-jkΛ)
EZ(x+Λ)=EZ(x)exp(-jkΛ),
Hy(x+Λ)=HY(x)exp(-jkΛ)
ΨN+1(XN)=ΨN+2(XN+2)exp(jkΛ),
ΨN+1(XN)=ΨN+2(XN+2)exp(jkΛ)
ΨN+1(XN)=ΨN+2(XN+2)exp(jkΛ),
1n12 ΨN+1(XN)=1n22 ΨN+2(XN+2)exp(jkΛ)
βn=k02-nπd21/2,
β=k0,k02-10d21/2.
β=k0,k02-10d21/2,k02-42d21/2.
β=k0,k02-323d21/2,k02-48d21/2.
Ψi(x)=[A exp(-γix)+B exp(+γix)]×exp(jωt-jk0Nz),
γi2=k02(N2-ni2).
(1+γ1/γ3)cosh(γ2d1)+(γ1/γ2+γ2/γ3)sinh(γ2d1)(1-γ5/γ3)cosh(γ4d3)+(γ5/γ4-γ4/γ3)sinh(γ4d3)exp(2γ3d2)
=(1-γ1/γ3)cosh(γ2d1)+(γ1/γ2-γ2/γ3)sinh(k2d1)(1+γ5/γ3)cosh(γ4d3)+(γ1/γ4+γ4/γ3)sinh(k4d3)exp(-2γ3d2)
(n32/n12+γ1/γ3)cosh(γ2d1)+[(n22/n32)(γ1/γ2)+(n12/n22)(γ2/γ3)]sinh(γ2d1)(n52/n12-γ5/γ3)cosh(γ4d3)+[(n42/n32)(γ5/γ4)-(n52/n42)(γ4/γ3)]sinh(γ4d3)exp(2γ3d2)
=(n32/n12-γ1/γ3)cosh(γ2d1)+[(n22/n32)(γ1/γ2)-(n12/n22)(γ2/γ3)]sinh(k2d1)(n52/n12+γ5/γ3)cosh(γ4d3)+[(n42/n32)(γ1/γ4)+(n52/n42)(γ4/γ3)]sinh(k4d3)exp(-2γ3d2).

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