Abstract

A coupled-mode formulation is described in which the radiation fields are represented in terms of discrete complex modes. The complex modes are obtained from a waveguide model facilitated by the combination of perfectly matched boundary (PML) and perfectly reflecting boundary (PRB) condition. By proper choice of the PML parameters, the guided modes of the structure remain unchanged, whereas the continuous radiation modes are discretized into orthogonal and normalizable complex quasi-leaky and PML modes. The complex coupled-mode formulation is identical to that for waveguides with loss and/or gain and can be solved by similar analytical and numerical techniques. By identifying the phase-matching conditions between the complex modes, the coupled mode formulation may be further simplified to yield analytical solutions. The complex coupled-mode theory is applied to Bragg grating in slab waveguides and validated by rigorous mode-matching method. It is for the first time that we can treat guided and radiation field in a unified and straightforward fashion without having to resort to cumbersome radiation modes. Highly accurate and insightful results are obtained with consideration of only the nearly phase-matched modes.

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References

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  1. D. Marcuse, Theory of Dielectric Optical Waveguides. (Academic, New York, 1991).
  2. C. Vassallo, Optical Waveguide Concepts. (Elsevier, Amsterdam,1991).
  3. A. W. Synder, and J. D. Love, Optical Waveguide Theory. (Chapman and Hall, New York, 1983).
  4. H. A. Haus, Waves and Fields in Optoelectronics. (Prentice Hall, Englewood Cliffs, N.J1984).
  5. T. Tamir, Ed., Guided-Wave Optoelectronics. (Springer-Verlag, New York, 1990).
  6. S. E. Miller, “Coupled-wave theory and waveguide applications,” Bell Syst. Tech. J. 33, 661–719 (1954).
  7. H. A. Haus, “Electron beam waves in microwave tubes,” in Proceedings of the Symposium on Electronic Waveguides (Polytechnic Institute of Brooklyn, Brooklyn, N.Y., 1958).
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    [CrossRef]
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    [CrossRef]
  11. H. Kogelnik, “Theory of dielectric waveguides,” in Integrated Optics,T. Tamir, Ed. (Springer-Verlag, New York, 1975), Chap.2.
  12. A. Hardy and W. Streifer, “Coupled mode theory of parallel waveguides,” J. Lightwave Technol. 3(5), 1135–1146 (1985).
    [CrossRef]
  13. H. A. Haus, W. P. Huang, S. Kawakami, and N. A. Whitaker, “Coupled- mode theory of optical waveguides,” J. Lightwave Technol. 5(1), 16–23 (1987).
    [CrossRef]
  14. S. L. Chuang, “A coupled mode formulation by reciprocity and a variational principle,” J. Lightwave Technol. LT-6, 294–303 (1988).
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    [CrossRef] [PubMed]
  16. T. Erdogan, “Cladding-mode resonances in short and long period fiber grating filters,” J. Opt. Soc. Am. A 14(8), 1760–1773 (1997).
    [CrossRef]
  17. T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15(8), 1277–1294 (1997).
    [CrossRef]
  18. C. Lu and Y. Cui, “Fiber Bragg grating spectra in multimode optical fibers,” J. Lightwave Technol. 24(1), 598–604 (2006).
    [CrossRef]
  19. J. A. Besley and J. D. Love, “Supermode analysis of fibre transmission,” IEE Proc., Optoelectron. 144(6), 411–419 (1997).
    [CrossRef]
  20. R. Sammut and A. W. Snyder, “Leaky modes on a dielectric waveguide: Orthogonality and excitation,” Appl. Opt. 15(4), 1040–1044 (1976).
    [CrossRef] [PubMed]
  21. S. L. Lee, Y. Chung, L. A. Coldren, and N. Dagli, “On leaky mode approximations for modal expansion in multilayer open waveguides,” IEEE J. Quantum Electron. 31(10), 1790–1802 (1995).
    [CrossRef]
  22. H. Derudder, F. Olyslager, D. De Zutter, and S. Van den Berghe, “Efficient mode-matching analysis of discontinuities in finite planar outer claddings using perfectly matched layers,” IEEE Trans. Antenn. Propag. 49(2), 185–195 (2001).
    [CrossRef]
  23. K. Jiang and W. P. Huang, “Finite-difference-based mode-matching method for 3-D waveguide structures under semivectorial approximation,” J. Lightwave Technol. 23(12), 4239–4248 (2005).
    [CrossRef]
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    [CrossRef] [PubMed]
  25. W. P. Huang, C. L. Xu, W. Lui, and K. Yokoyama, “The perfectly matched layer (PML) boundary conditions for modal analysis of optical waveguides: leaky mode calculations,” IEEE Photon. Technol. Lett. 8(5), 652–654 (1996).
    [CrossRef]
  26. W. C. Chew, J. M. Jin, and E. Michielssen, “Complex coordinate stretching as a generalized absorbing boundary condition,” Microw. Opt. Technol. Lett. 15(6), 363–369 (1997).
    [CrossRef]
  27. J.-P. Bérenger, “Application of the CFS PML to the absorption of evanescent waves in waveguides,” IEEE Microw. Wireless Compon. Lett. 12(6), 218–220 (2002).
    [CrossRef]

2008 (1)

2006 (1)

2005 (1)

2002 (1)

J.-P. Bérenger, “Application of the CFS PML to the absorption of evanescent waves in waveguides,” IEEE Microw. Wireless Compon. Lett. 12(6), 218–220 (2002).
[CrossRef]

2001 (1)

H. Derudder, F. Olyslager, D. De Zutter, and S. Van den Berghe, “Efficient mode-matching analysis of discontinuities in finite planar outer claddings using perfectly matched layers,” IEEE Trans. Antenn. Propag. 49(2), 185–195 (2001).
[CrossRef]

1997 (4)

J. A. Besley and J. D. Love, “Supermode analysis of fibre transmission,” IEE Proc., Optoelectron. 144(6), 411–419 (1997).
[CrossRef]

T. Erdogan, “Cladding-mode resonances in short and long period fiber grating filters,” J. Opt. Soc. Am. A 14(8), 1760–1773 (1997).
[CrossRef]

T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15(8), 1277–1294 (1997).
[CrossRef]

W. C. Chew, J. M. Jin, and E. Michielssen, “Complex coordinate stretching as a generalized absorbing boundary condition,” Microw. Opt. Technol. Lett. 15(6), 363–369 (1997).
[CrossRef]

1996 (1)

W. P. Huang, C. L. Xu, W. Lui, and K. Yokoyama, “The perfectly matched layer (PML) boundary conditions for modal analysis of optical waveguides: leaky mode calculations,” IEEE Photon. Technol. Lett. 8(5), 652–654 (1996).
[CrossRef]

1995 (1)

S. L. Lee, Y. Chung, L. A. Coldren, and N. Dagli, “On leaky mode approximations for modal expansion in multilayer open waveguides,” IEEE J. Quantum Electron. 31(10), 1790–1802 (1995).
[CrossRef]

1989 (1)

1988 (1)

S. L. Chuang, “A coupled mode formulation by reciprocity and a variational principle,” J. Lightwave Technol. LT-6, 294–303 (1988).

1987 (1)

H. A. Haus, W. P. Huang, S. Kawakami, and N. A. Whitaker, “Coupled- mode theory of optical waveguides,” J. Lightwave Technol. 5(1), 16–23 (1987).
[CrossRef]

1985 (1)

A. Hardy and W. Streifer, “Coupled mode theory of parallel waveguides,” J. Lightwave Technol. 3(5), 1135–1146 (1985).
[CrossRef]

1976 (1)

1973 (2)

D. Marcuse, “Coupled mode theory of round optical fibers,” J Bell Syst Tech 52, 817–843 (1973).

A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. 9(9), 919–933 (1973).
[CrossRef]

1972 (1)

1954 (1)

S. E. Miller, “Coupled-wave theory and waveguide applications,” Bell Syst. Tech. J. 33, 661–719 (1954).

Bérenger, J.-P.

J.-P. Bérenger, “Application of the CFS PML to the absorption of evanescent waves in waveguides,” IEEE Microw. Wireless Compon. Lett. 12(6), 218–220 (2002).
[CrossRef]

Besley, J. A.

J. A. Besley and J. D. Love, “Supermode analysis of fibre transmission,” IEE Proc., Optoelectron. 144(6), 411–419 (1997).
[CrossRef]

Chew, W. C.

W. C. Chew, J. M. Jin, and E. Michielssen, “Complex coordinate stretching as a generalized absorbing boundary condition,” Microw. Opt. Technol. Lett. 15(6), 363–369 (1997).
[CrossRef]

Chuang, S. L.

S. L. Chuang, “A coupled mode formulation by reciprocity and a variational principle,” J. Lightwave Technol. LT-6, 294–303 (1988).

Chung, Y.

S. L. Lee, Y. Chung, L. A. Coldren, and N. Dagli, “On leaky mode approximations for modal expansion in multilayer open waveguides,” IEEE J. Quantum Electron. 31(10), 1790–1802 (1995).
[CrossRef]

Coldren, L. A.

S. L. Lee, Y. Chung, L. A. Coldren, and N. Dagli, “On leaky mode approximations for modal expansion in multilayer open waveguides,” IEEE J. Quantum Electron. 31(10), 1790–1802 (1995).
[CrossRef]

Cui, Y.

Dagli, N.

S. L. Lee, Y. Chung, L. A. Coldren, and N. Dagli, “On leaky mode approximations for modal expansion in multilayer open waveguides,” IEEE J. Quantum Electron. 31(10), 1790–1802 (1995).
[CrossRef]

De Zutter, D.

H. Derudder, F. Olyslager, D. De Zutter, and S. Van den Berghe, “Efficient mode-matching analysis of discontinuities in finite planar outer claddings using perfectly matched layers,” IEEE Trans. Antenn. Propag. 49(2), 185–195 (2001).
[CrossRef]

Derudder, H.

H. Derudder, F. Olyslager, D. De Zutter, and S. Van den Berghe, “Efficient mode-matching analysis of discontinuities in finite planar outer claddings using perfectly matched layers,” IEEE Trans. Antenn. Propag. 49(2), 185–195 (2001).
[CrossRef]

Erdogan, T.

Hardy, A.

A. Hardy and W. Streifer, “Coupled mode theory of parallel waveguides,” J. Lightwave Technol. 3(5), 1135–1146 (1985).
[CrossRef]

Haus, H. A.

H. A. Haus, W. P. Huang, and A. W. Snyder, “Coupled-mode formulations,” Opt. Lett. 14(21), 1222–1224 (1989).
[CrossRef] [PubMed]

H. A. Haus, W. P. Huang, S. Kawakami, and N. A. Whitaker, “Coupled- mode theory of optical waveguides,” J. Lightwave Technol. 5(1), 16–23 (1987).
[CrossRef]

Huang, W. P.

Jiang, K.

Jin, J. M.

W. C. Chew, J. M. Jin, and E. Michielssen, “Complex coordinate stretching as a generalized absorbing boundary condition,” Microw. Opt. Technol. Lett. 15(6), 363–369 (1997).
[CrossRef]

Kawakami, S.

H. A. Haus, W. P. Huang, S. Kawakami, and N. A. Whitaker, “Coupled- mode theory of optical waveguides,” J. Lightwave Technol. 5(1), 16–23 (1987).
[CrossRef]

Lee, S. L.

S. L. Lee, Y. Chung, L. A. Coldren, and N. Dagli, “On leaky mode approximations for modal expansion in multilayer open waveguides,” IEEE J. Quantum Electron. 31(10), 1790–1802 (1995).
[CrossRef]

Love, J. D.

J. A. Besley and J. D. Love, “Supermode analysis of fibre transmission,” IEE Proc., Optoelectron. 144(6), 411–419 (1997).
[CrossRef]

Lu, C.

Lui, W.

W. P. Huang, C. L. Xu, W. Lui, and K. Yokoyama, “The perfectly matched layer (PML) boundary conditions for modal analysis of optical waveguides: leaky mode calculations,” IEEE Photon. Technol. Lett. 8(5), 652–654 (1996).
[CrossRef]

Marcuse, D.

D. Marcuse, “Coupled mode theory of round optical fibers,” J Bell Syst Tech 52, 817–843 (1973).

Michielssen, E.

W. C. Chew, J. M. Jin, and E. Michielssen, “Complex coordinate stretching as a generalized absorbing boundary condition,” Microw. Opt. Technol. Lett. 15(6), 363–369 (1997).
[CrossRef]

Miller, S. E.

S. E. Miller, “Coupled-wave theory and waveguide applications,” Bell Syst. Tech. J. 33, 661–719 (1954).

Mu, J.

Olyslager, F.

H. Derudder, F. Olyslager, D. De Zutter, and S. Van den Berghe, “Efficient mode-matching analysis of discontinuities in finite planar outer claddings using perfectly matched layers,” IEEE Trans. Antenn. Propag. 49(2), 185–195 (2001).
[CrossRef]

Sammut, R.

Snyder, A. W.

Snyder, W.

Streifer, W.

A. Hardy and W. Streifer, “Coupled mode theory of parallel waveguides,” J. Lightwave Technol. 3(5), 1135–1146 (1985).
[CrossRef]

Van den Berghe, S.

H. Derudder, F. Olyslager, D. De Zutter, and S. Van den Berghe, “Efficient mode-matching analysis of discontinuities in finite planar outer claddings using perfectly matched layers,” IEEE Trans. Antenn. Propag. 49(2), 185–195 (2001).
[CrossRef]

Whitaker, N. A.

H. A. Haus, W. P. Huang, S. Kawakami, and N. A. Whitaker, “Coupled- mode theory of optical waveguides,” J. Lightwave Technol. 5(1), 16–23 (1987).
[CrossRef]

Xu, C. L.

W. P. Huang, C. L. Xu, W. Lui, and K. Yokoyama, “The perfectly matched layer (PML) boundary conditions for modal analysis of optical waveguides: leaky mode calculations,” IEEE Photon. Technol. Lett. 8(5), 652–654 (1996).
[CrossRef]

Yariv, A.

A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. 9(9), 919–933 (1973).
[CrossRef]

Yokoyama, K.

W. P. Huang, C. L. Xu, W. Lui, and K. Yokoyama, “The perfectly matched layer (PML) boundary conditions for modal analysis of optical waveguides: leaky mode calculations,” IEEE Photon. Technol. Lett. 8(5), 652–654 (1996).
[CrossRef]

Appl. Opt. (1)

Bell Syst. Tech. J. (1)

S. E. Miller, “Coupled-wave theory and waveguide applications,” Bell Syst. Tech. J. 33, 661–719 (1954).

IEE Proc., Optoelectron. (1)

J. A. Besley and J. D. Love, “Supermode analysis of fibre transmission,” IEE Proc., Optoelectron. 144(6), 411–419 (1997).
[CrossRef]

IEEE J. Quantum Electron. (2)

S. L. Lee, Y. Chung, L. A. Coldren, and N. Dagli, “On leaky mode approximations for modal expansion in multilayer open waveguides,” IEEE J. Quantum Electron. 31(10), 1790–1802 (1995).
[CrossRef]

A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. 9(9), 919–933 (1973).
[CrossRef]

IEEE Microw. Wireless Compon. Lett. (1)

J.-P. Bérenger, “Application of the CFS PML to the absorption of evanescent waves in waveguides,” IEEE Microw. Wireless Compon. Lett. 12(6), 218–220 (2002).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

W. P. Huang, C. L. Xu, W. Lui, and K. Yokoyama, “The perfectly matched layer (PML) boundary conditions for modal analysis of optical waveguides: leaky mode calculations,” IEEE Photon. Technol. Lett. 8(5), 652–654 (1996).
[CrossRef]

IEEE Trans. Antenn. Propag. (1)

H. Derudder, F. Olyslager, D. De Zutter, and S. Van den Berghe, “Efficient mode-matching analysis of discontinuities in finite planar outer claddings using perfectly matched layers,” IEEE Trans. Antenn. Propag. 49(2), 185–195 (2001).
[CrossRef]

J Bell Syst Tech (1)

D. Marcuse, “Coupled mode theory of round optical fibers,” J Bell Syst Tech 52, 817–843 (1973).

J. Lightwave Technol. (6)

T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15(8), 1277–1294 (1997).
[CrossRef]

C. Lu and Y. Cui, “Fiber Bragg grating spectra in multimode optical fibers,” J. Lightwave Technol. 24(1), 598–604 (2006).
[CrossRef]

A. Hardy and W. Streifer, “Coupled mode theory of parallel waveguides,” J. Lightwave Technol. 3(5), 1135–1146 (1985).
[CrossRef]

H. A. Haus, W. P. Huang, S. Kawakami, and N. A. Whitaker, “Coupled- mode theory of optical waveguides,” J. Lightwave Technol. 5(1), 16–23 (1987).
[CrossRef]

S. L. Chuang, “A coupled mode formulation by reciprocity and a variational principle,” J. Lightwave Technol. LT-6, 294–303 (1988).

K. Jiang and W. P. Huang, “Finite-difference-based mode-matching method for 3-D waveguide structures under semivectorial approximation,” J. Lightwave Technol. 23(12), 4239–4248 (2005).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Microw. Opt. Technol. Lett. (1)

W. C. Chew, J. M. Jin, and E. Michielssen, “Complex coordinate stretching as a generalized absorbing boundary condition,” Microw. Opt. Technol. Lett. 15(6), 363–369 (1997).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Other (7)

H. Kogelnik, “Theory of dielectric waveguides,” in Integrated Optics,T. Tamir, Ed. (Springer-Verlag, New York, 1975), Chap.2.

H. A. Haus, “Electron beam waves in microwave tubes,” in Proceedings of the Symposium on Electronic Waveguides (Polytechnic Institute of Brooklyn, Brooklyn, N.Y., 1958).

D. Marcuse, Theory of Dielectric Optical Waveguides. (Academic, New York, 1991).

C. Vassallo, Optical Waveguide Concepts. (Elsevier, Amsterdam,1991).

A. W. Synder, and J. D. Love, Optical Waveguide Theory. (Chapman and Hall, New York, 1983).

H. A. Haus, Waves and Fields in Optoelectronics. (Prentice Hall, Englewood Cliffs, N.J1984).

T. Tamir, Ed., Guided-Wave Optoelectronics. (Springer-Verlag, New York, 1990).

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Figures (12)

Fig. 1
Fig. 1

Ideal waveguide models

Fig. 2
Fig. 2

(a) Real and imaginary parts of the effective indices for the first twenty (20) modes (in descent order based on the values of the real part for the mode effective indices) supported in the waveguide for the different outer cladding indices. (b) A blow-up view for the portion of graph in (a) enclosed by the dashed circle.

Fig. 3
Fig. 3

Mode leakage loss for the quasi-leaky modes.

Fig. 4
Fig. 4

Mode field patterns for the three waveguide structures. (a) the fundamental guided mode; (b) the quasi-leaky cladding modes; (c) the PML modes.

Fig. 5
Fig. 5

Field confinement factor as a function of PML parameters.

Fig. 6
Fig. 6

Field confinement factors in core and cladding region as functions of computation window ( ns = 1.455).

Fig. 7
Fig. 7

Mode field overlap integrals for different modes. (a)Non-conjugate overlapping integral using Eqs. (13); (b) Conjugate overlapping integral using Eqs. (12). ( ns=1.45 .)

Fig. 8
Fig. 8

Volume Bragg grating structure based on slab waveguides

Fig. 9
Fig. 9

The transmission spectrum for Case A with lower outer cladding index ns = 1.0. (a) Phase matching wavelengths, corresponding coupling strengths, and the transmission spectrum predicted by the reduced CMT involving only two phase matching modes; (b) The transmission spectrum calculated by the reduced CMT (dash lines), the full CMT (dotted lines) and the rigorous MMM (solid lines).

Fig. 10
Fig. 10

The transmission spectrum for Case B with equal outer cladding index ns = 1.450. (a) Phase matching wavelengths, corresponding coupling strengths, and the transmission spectrum predicted by the full CMT involving from 2 up to 11 modes; (b) The transmission spectrum calculated by the full CMT (dotted lines) and the rigorous MMM (solid lines).

Fig. 12
Fig. 12

Transmission spectra with ns lager than ncl. (a) ns = 1.60; (b) ns = 1.90.

Fig. 11
Fig. 11

Transmission spectra with index of the outer cladding ns slightly larger than the index of the inner cladding ncl (ns = 1.455).

Equations (81)

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

t×enjγnen=jωμo[Λ]hn
t×hnjγnhn=+jωε[Λ]en
[Λ]=[Sy/Sx000Sx/Sy000SxSy]
Sx=κxjσxωε
Sy=κyjσyωε
σ=σmax(ρTPML)m
RPML=exp{2σmaxnεo/μo0TPML(ρTPML)mdρ}
S=κjλ4πnTPML[(m+1)ln(1RPML)](ρTPML)m
γn=βnjαn
t×enjγnen=jωμo[Λ]hn
t×hnjγnhn=+jωε[Λ]en
γn=γn
etn=+et,n
ezn=ez,n
htn=ht,n
hzn=+hz,n
(etm×htn)z^da=0
12(etn×htn)z^da=1
(etm×htn)z^da0
ε˜(x,y,z)=ε(x,y)+Δε(x,y,z)
×E(x,y,z)=jωμoH(x,y,z)
×H(x,y,z)=+jωε˜(x,y,z)H(x,y,z)
Et(x,y,z)=n=1[an(z)+bn(z)]etn(x,y)
Ht(x,y,z)=n=1[an(z)bn(z)]htn(x,y)
Ez(x,y,z)=n=1[an(z)bn(z)]εε˜ezn(x,y)
Hz(x,y,z)=n=1[an(z)+bn(z)]hzn(x,y)
Nmdamdz+jγmam=jn=1κmnanjn=1χmnbn
Nmdbmdzjγmbm=+jn=1κmnbn+jn=1χmnan
κmn=ωεo4A(n˜2n2)(etnetmn2n˜2eznezm)da
χmn=ωεo4A(n˜2n2)(etnetm+n2n˜2eznezm)da
Nm=12(etm×htm)z^da
κmn=κnm
χmn=χnm
an=Anexp(jγnz)
bn=Bnexp(+jγnz)
NmdAmdz=jn=1κmnAnexp[j(γnγm)z]jn=1χmnBnexp[+j(γn+γm)z]
NmdBmdz=+jn=1κmnBnexp[+j(γnγm)z]+jn=1χmnAnexp[j(γn+γm)z]
κmn=l=+Dmn(l)exp(jl2πΛz)
χmn=l=+Cmn(l)exp(jl2πΛz)
NmdAmdz=jn=1Anl=+Dmn(l)exp[j(γnγml2πΛ)z]jn=1Bnl=+Cmn(l)exp[+j(γn+γm+l2πΛ)z]
NmdBmdz=+jn=1Bnl=+Dmn(l)exp[+j(γnγm+l2πΛ)z]+jn=1Anl=+Cmn(l)exp[j(γn+γml2πΛ)z]
NmdAmdz=jn=1Bnl=+Cmn(l)exp[+j(γn+γm+l2πΛ)z]
NmdBmdz=+jn=1Anl=+Cmn(l)exp[j(γn+γml2πΛ)z]
NmdAmdz=jn=1Anl=+Dmn(l)exp[j(γnγml2πΛ)z]
NmdBmdz=+jn=1Bnl=+Dmn(l)exp[+j(γnγm+l2πΛ)z]
NmdAmdz=jn=1BnCmn(1)exp[+j(γn+γm2πΛ)z]
NmdBmdz=+jn=1AnCmn(+1)exp[j(γn+γm2πΛ)z]
NmdAmdz=jn=1AnDmn(1)exp[j(γnγm+2πΛ)z]
NmdBmdz=+jn=1BnDmn(+1)exp[+j(γnγm+2πΛ)z]
N1dA1dz=jC1n(1)Bnexp[+j(γn+β12πΛ)z]
NndBndz=+jCn1(+1)A1exp[j(γn+β12πΛ)z]
N1dA1dz=jD1n(1)Anexp[j(γnβ1+2πΛ)z]
NndAndz=jDn1(+1)A1exp[+j(γnβ1+2πΛ)z]
Δβn=12(γ1+γn2πΛ)
Δβn=12(γ1γn2πΛ)
(Δβn)=0
Λ=2π(γ1+γn)
Λ=2π(γ1γn)
N1dA1dz=jC1n(1)Bnexp[+j(2Δβn)z]
NndBndz=+jCn1(+1)A1exp[j(2Δβn)z]
N1dA1dz=jD1n(1)Anexp[j(2Δβn)z]
NndAndz=jDn1(+1)A1exp[+j(2Δβn)z]
a1(z)=a1(0)ΔβnsinhS(zL)jScoshS(zL)ΔβnsinhSLjScoshSLej(β1Δβn)z
bn(z)=a1(0)Cn1(+1)NnsinhS(zL)ΔβnsinhSL+jScoshSLej(γnΔβn)z
S=κn2(Δβn)2
κn=C1n(1)Cn1(+1)N1Nn
a1(z)=a1(0)jΔβsin(Qnz)+Qncos(Qnz)Qnej(β1Δβn)z
an(z)=jejΔβnzDn1(+1)Nnsin(Qnz)Qna1(0)ej(γn+Δβn)z
Qn=χn2+(Δβn)2
χn=Dn1(+1)D1n(1)NnN1
P(z)=12A[Et(x,y,z)×Ht(x,y,z)]z^da
P(z)=m=1n=1Mmn[aman*bmbn*]m=1n=1Nmn[ambn*bman*]
Mmn=14A(etm×htn*+etn*×htm)z^da
Nmn=14A(etm×htn*etn*×htm)z^da
P(z)m=1n=1Mmn[aman*bmbn*]
P(z)n=1[|an|2|bn|2]
P(0)=1R(λ)
R(λ)m=1n=1Mmnbm(0)bn*(0)
R(λ)n=1|bm(0)|2
P(L)=|a1(L)|2=T(λ)
T(λ)=1R(λ)

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