Abstract

Based on complex-variable analysis of a Fabry–Perot resonator as a multimode nonsymmetric two-port waveguide device, two versions of equivalent-circuit configurations are presented: Starting from a renewed study on single-mode two-pole circuits, we develop two respective multimode equivalent circuits of an almost identical configuration: one for the reflection coefficient and the other for the pass-through transmission coefficient. In the mathematics language of complex-variable analysis, the two models successfully “approximate” the two scattering coefficients through two “uniformly converging” partial-fraction series expansions.

© 2014 Optical Society of America

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  1. C. Fabry and A. Pérot, “Théorie et applications d’une nouvelle méthodes de spectroscopie interférentielle,” Annal. Chim. Phys. 16, 115–144 (1899).
  2. H. A. Haus and Y. Lai, “Narrow-band optical channel-dropping filter,” J. Lightwave Technol. 10, 57–62 (1992).
    [CrossRef]
  3. R. E. Collin, Foundations for Microwave Engineering, 2nd ed. (IEEE, 1992), Fig. 7.30.
  4. E. A. J. Marcatilli, “Bends in optical dielectric waveguides,” Bell Syst. Tech. J. 48, 2103–2132 (1969).
    [CrossRef]
  5. I. Ohtomo and S. Shimada, “A channel channel-dropping filter using ring resonators for the millimeter wave communication system,” Trans. Inst. Electron. Commun. Eng. Jpn. 52-B, 265–272 (1969).
  6. K. Hwang and G. H. Song, “Design of a high-Q channel add-drop multiplexer based on the two-dimensional photonic-crystal membrane structure,” Opt. Express 13, 1948–1957 (2005).
    [CrossRef]
  7. A. Melloni, M. Floridi, F. Morichetti, and M. Martinelli, “Equivalent circuit of Bragg gratings and its applications to Fabry–Pérot cavities,” J. Opt. Soc. Am. A 20, 273–281 (2003).
    [CrossRef]
  8. C. G. Montgomery, R. H. Dicke, and E. M. Purcell, eds., Principles of Microwave Circuits (McGraw-Hill, 1948).
  9. W. Culshaw, “Resonators for millimeter and submillimeter wavelengths,” IRE Trans. Microwave Theory Technol. MTT-9, 135–144 (1961).
    [CrossRef]
  10. R. E. Collin, Foundations for Microwave Engineering, 1st ed. (McGraw-Hill, 1966).
  11. A. E. Siegman, Lasers (University Science, 1986), pp. 934–939.
  12. H. A. Haus and Y. Lai, “Theory of cascaded quarter wave shifted distributed feedback resonators,” IEEE J. Quantum Electron. 28, 205–213 (1992).
    [CrossRef]
  13. H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984), Fig. 9.9.
  14. T. Quarles, D. Pederson, R. Newton, A. Sangiovanni-Vincentelli, and C. Wayne, SPICE3 Version 3f3 User’s Manual (University of California, 1993), The SPICE Page http://bwrcs.eecs.berkeley.edu/Classes/IcBook/SPICE/ .
  15. G. H. Song, “Mathematical modeling of Fabry–Perot resonators: I. Complex-variable analysis by uniformly convergent partial-fraction expansion,” J. Opt. Soc. Am. A31, 404–410 (2014)
  16. C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35, 1322–1331 (1999), Eq. (14).
    [CrossRef]
  17. A. Karp, H. J. Shaw, and D. K. Winslow, “Circuit properties of microwave dielectric resonators,” IEEE Trans. Microwave Theor. Technol. MTT-16, 818–828 (1968).
    [CrossRef]
  18. R. Beringer, “Resonant cavities as microwave circuit elements,” in Principles of Microwave Circuits, C. G. Montgomery, R. H. Dicke, and E. M. Purcell, eds. (McGraw-Hill, 1948), Chap. 7, Sect. 7.3.

2005 (1)

2003 (1)

1999 (1)

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35, 1322–1331 (1999), Eq. (14).
[CrossRef]

1992 (2)

H. A. Haus and Y. Lai, “Theory of cascaded quarter wave shifted distributed feedback resonators,” IEEE J. Quantum Electron. 28, 205–213 (1992).
[CrossRef]

H. A. Haus and Y. Lai, “Narrow-band optical channel-dropping filter,” J. Lightwave Technol. 10, 57–62 (1992).
[CrossRef]

1969 (2)

E. A. J. Marcatilli, “Bends in optical dielectric waveguides,” Bell Syst. Tech. J. 48, 2103–2132 (1969).
[CrossRef]

I. Ohtomo and S. Shimada, “A channel channel-dropping filter using ring resonators for the millimeter wave communication system,” Trans. Inst. Electron. Commun. Eng. Jpn. 52-B, 265–272 (1969).

1968 (1)

A. Karp, H. J. Shaw, and D. K. Winslow, “Circuit properties of microwave dielectric resonators,” IEEE Trans. Microwave Theor. Technol. MTT-16, 818–828 (1968).
[CrossRef]

1961 (1)

W. Culshaw, “Resonators for millimeter and submillimeter wavelengths,” IRE Trans. Microwave Theory Technol. MTT-9, 135–144 (1961).
[CrossRef]

1899 (1)

C. Fabry and A. Pérot, “Théorie et applications d’une nouvelle méthodes de spectroscopie interférentielle,” Annal. Chim. Phys. 16, 115–144 (1899).

Beringer, R.

R. Beringer, “Resonant cavities as microwave circuit elements,” in Principles of Microwave Circuits, C. G. Montgomery, R. H. Dicke, and E. M. Purcell, eds. (McGraw-Hill, 1948), Chap. 7, Sect. 7.3.

Collin, R. E.

R. E. Collin, Foundations for Microwave Engineering, 2nd ed. (IEEE, 1992), Fig. 7.30.

R. E. Collin, Foundations for Microwave Engineering, 1st ed. (McGraw-Hill, 1966).

Culshaw, W.

W. Culshaw, “Resonators for millimeter and submillimeter wavelengths,” IRE Trans. Microwave Theory Technol. MTT-9, 135–144 (1961).
[CrossRef]

Fabry, C.

C. Fabry and A. Pérot, “Théorie et applications d’une nouvelle méthodes de spectroscopie interférentielle,” Annal. Chim. Phys. 16, 115–144 (1899).

Fan, S.

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35, 1322–1331 (1999), Eq. (14).
[CrossRef]

Floridi, M.

Haus, H. A.

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35, 1322–1331 (1999), Eq. (14).
[CrossRef]

H. A. Haus and Y. Lai, “Narrow-band optical channel-dropping filter,” J. Lightwave Technol. 10, 57–62 (1992).
[CrossRef]

H. A. Haus and Y. Lai, “Theory of cascaded quarter wave shifted distributed feedback resonators,” IEEE J. Quantum Electron. 28, 205–213 (1992).
[CrossRef]

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984), Fig. 9.9.

Hwang, K.

Joannopoulos, J. D.

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35, 1322–1331 (1999), Eq. (14).
[CrossRef]

Karp, A.

A. Karp, H. J. Shaw, and D. K. Winslow, “Circuit properties of microwave dielectric resonators,” IEEE Trans. Microwave Theor. Technol. MTT-16, 818–828 (1968).
[CrossRef]

Khan, M. J.

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35, 1322–1331 (1999), Eq. (14).
[CrossRef]

Lai, Y.

H. A. Haus and Y. Lai, “Narrow-band optical channel-dropping filter,” J. Lightwave Technol. 10, 57–62 (1992).
[CrossRef]

H. A. Haus and Y. Lai, “Theory of cascaded quarter wave shifted distributed feedback resonators,” IEEE J. Quantum Electron. 28, 205–213 (1992).
[CrossRef]

Manolatou, C.

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35, 1322–1331 (1999), Eq. (14).
[CrossRef]

Marcatilli, E. A. J.

E. A. J. Marcatilli, “Bends in optical dielectric waveguides,” Bell Syst. Tech. J. 48, 2103–2132 (1969).
[CrossRef]

Martinelli, M.

Melloni, A.

Morichetti, F.

Newton, R.

T. Quarles, D. Pederson, R. Newton, A. Sangiovanni-Vincentelli, and C. Wayne, SPICE3 Version 3f3 User’s Manual (University of California, 1993), The SPICE Page http://bwrcs.eecs.berkeley.edu/Classes/IcBook/SPICE/ .

Ohtomo, I.

I. Ohtomo and S. Shimada, “A channel channel-dropping filter using ring resonators for the millimeter wave communication system,” Trans. Inst. Electron. Commun. Eng. Jpn. 52-B, 265–272 (1969).

Pederson, D.

T. Quarles, D. Pederson, R. Newton, A. Sangiovanni-Vincentelli, and C. Wayne, SPICE3 Version 3f3 User’s Manual (University of California, 1993), The SPICE Page http://bwrcs.eecs.berkeley.edu/Classes/IcBook/SPICE/ .

Pérot, A.

C. Fabry and A. Pérot, “Théorie et applications d’une nouvelle méthodes de spectroscopie interférentielle,” Annal. Chim. Phys. 16, 115–144 (1899).

Quarles, T.

T. Quarles, D. Pederson, R. Newton, A. Sangiovanni-Vincentelli, and C. Wayne, SPICE3 Version 3f3 User’s Manual (University of California, 1993), The SPICE Page http://bwrcs.eecs.berkeley.edu/Classes/IcBook/SPICE/ .

Sangiovanni-Vincentelli, A.

T. Quarles, D. Pederson, R. Newton, A. Sangiovanni-Vincentelli, and C. Wayne, SPICE3 Version 3f3 User’s Manual (University of California, 1993), The SPICE Page http://bwrcs.eecs.berkeley.edu/Classes/IcBook/SPICE/ .

Shaw, H. J.

A. Karp, H. J. Shaw, and D. K. Winslow, “Circuit properties of microwave dielectric resonators,” IEEE Trans. Microwave Theor. Technol. MTT-16, 818–828 (1968).
[CrossRef]

Shimada, S.

I. Ohtomo and S. Shimada, “A channel channel-dropping filter using ring resonators for the millimeter wave communication system,” Trans. Inst. Electron. Commun. Eng. Jpn. 52-B, 265–272 (1969).

Siegman, A. E.

A. E. Siegman, Lasers (University Science, 1986), pp. 934–939.

Song, G. H.

K. Hwang and G. H. Song, “Design of a high-Q channel add-drop multiplexer based on the two-dimensional photonic-crystal membrane structure,” Opt. Express 13, 1948–1957 (2005).
[CrossRef]

G. H. Song, “Mathematical modeling of Fabry–Perot resonators: I. Complex-variable analysis by uniformly convergent partial-fraction expansion,” J. Opt. Soc. Am. A31, 404–410 (2014)

Villeneuve, P. R.

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35, 1322–1331 (1999), Eq. (14).
[CrossRef]

Wayne, C.

T. Quarles, D. Pederson, R. Newton, A. Sangiovanni-Vincentelli, and C. Wayne, SPICE3 Version 3f3 User’s Manual (University of California, 1993), The SPICE Page http://bwrcs.eecs.berkeley.edu/Classes/IcBook/SPICE/ .

Winslow, D. K.

A. Karp, H. J. Shaw, and D. K. Winslow, “Circuit properties of microwave dielectric resonators,” IEEE Trans. Microwave Theor. Technol. MTT-16, 818–828 (1968).
[CrossRef]

Annal. Chim. Phys. (1)

C. Fabry and A. Pérot, “Théorie et applications d’une nouvelle méthodes de spectroscopie interférentielle,” Annal. Chim. Phys. 16, 115–144 (1899).

Bell Syst. Tech. J. (1)

E. A. J. Marcatilli, “Bends in optical dielectric waveguides,” Bell Syst. Tech. J. 48, 2103–2132 (1969).
[CrossRef]

IEEE J. Quantum Electron. (2)

H. A. Haus and Y. Lai, “Theory of cascaded quarter wave shifted distributed feedback resonators,” IEEE J. Quantum Electron. 28, 205–213 (1992).
[CrossRef]

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35, 1322–1331 (1999), Eq. (14).
[CrossRef]

IEEE Trans. Microwave Theor. Technol. (1)

A. Karp, H. J. Shaw, and D. K. Winslow, “Circuit properties of microwave dielectric resonators,” IEEE Trans. Microwave Theor. Technol. MTT-16, 818–828 (1968).
[CrossRef]

IRE Trans. Microwave Theory Technol. (1)

W. Culshaw, “Resonators for millimeter and submillimeter wavelengths,” IRE Trans. Microwave Theory Technol. MTT-9, 135–144 (1961).
[CrossRef]

J. Lightwave Technol. (1)

H. A. Haus and Y. Lai, “Narrow-band optical channel-dropping filter,” J. Lightwave Technol. 10, 57–62 (1992).
[CrossRef]

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

Opt. Express (1)

Trans. Inst. Electron. Commun. Eng. Jpn. (1)

I. Ohtomo and S. Shimada, “A channel channel-dropping filter using ring resonators for the millimeter wave communication system,” Trans. Inst. Electron. Commun. Eng. Jpn. 52-B, 265–272 (1969).

Other (8)

R. E. Collin, Foundations for Microwave Engineering, 2nd ed. (IEEE, 1992), Fig. 7.30.

C. G. Montgomery, R. H. Dicke, and E. M. Purcell, eds., Principles of Microwave Circuits (McGraw-Hill, 1948).

R. E. Collin, Foundations for Microwave Engineering, 1st ed. (McGraw-Hill, 1966).

A. E. Siegman, Lasers (University Science, 1986), pp. 934–939.

R. Beringer, “Resonant cavities as microwave circuit elements,” in Principles of Microwave Circuits, C. G. Montgomery, R. H. Dicke, and E. M. Purcell, eds. (McGraw-Hill, 1948), Chap. 7, Sect. 7.3.

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984), Fig. 9.9.

T. Quarles, D. Pederson, R. Newton, A. Sangiovanni-Vincentelli, and C. Wayne, SPICE3 Version 3f3 User’s Manual (University of California, 1993), The SPICE Page http://bwrcs.eecs.berkeley.edu/Classes/IcBook/SPICE/ .

G. H. Song, “Mathematical modeling of Fabry–Perot resonators: I. Complex-variable analysis by uniformly convergent partial-fraction expansion,” J. Opt. Soc. Am. A31, 404–410 (2014)

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Figures (10)

Fig. 1.
Fig. 1.

Simple lumped-element equivalent circuit approximately representing a Fabry–Perot resonator (FPR) near one resonance frequency ωx1/LxC among infinitely many such resonance frequencies. To match with the external wave admittance Yc, two ideal admittance transformers with winding ratios θA and θB are attached.

Fig. 2.
Fig. 2.

Circuit configuration that is dual to that of Fig. 1. To simulate the two paired zeros of Γ⃗I(s), circuit parameters are not chosen exactly dual to those of Fig. 1

Fig. 3.
Fig. 3.

Multimode reflection-equivalent lumped-circuit model for a FPR supporting a large number of modes by implementing as many unit resonators, in the shape of Fig. 2.

Fig. 4.
Fig. 4.

Spectral responses of the reflection-equivalent circuit for a FPR in thick curves from the circuit model of Fig. 3 in comparison with the actual response in thin dotted curves. (a) Real and imaginary parts of the reflection coefficient Γ⃗I(6)(iω) of Eq. (56). (b) Those of the pass-through coefficient for the circuit-based interpretation ϒ⃗P(6)(iω) at Eq. (64). (c) Reflectivity |Γ⃗I(6)(iω)|2 and the pass-through transmittivity |Γ⃗I(6)(iω)+1|2ZB/ZA. For all plots in this study, we use hA=hB=1, rA=rB=0.7304, giving γA/ω1=γB/ω1=0.05 and γin/ω1=0.01. In comparison, the two thin curves are made for the real and imaginary parts of R⃗(ω) in Eq. (15) for the corresponding FPR.

Fig. 5.
Fig. 5.

Real and imaginary parts of [Γ⃗I(6)(iω)+1] eiωτ/2ZB/ZA. The thin curves represent T(ω) from the original FPR.

Fig. 6.
Fig. 6.

Circuit dual to the one in Fig. 3. Rj,Lj,G,C,GΓ,R¯Γ here replace Gj,CjR,L,RΓ,G¯Γ, respectively, in Fig. 3.

Fig. 7.
Fig. 7.

Pass-through-equivalent circuit for a true FPR supporting infinitely many modes.

Fig. 8.
Fig. 8.

Spectral responses of the pass-through-equivalent circuit for a FPR in thick curves from the circuit model with six RLGC resonators and an amplified/attenuated phase-delay line in comparison with the actual response in thin dotted curves. (a) Real and imaginary parts of the pass-through coefficient ϒP(6)(iω) from Eq. (66), (b) the resulting reflection coefficient Γ⃗I(6)(iω), and (c) their power spectra in thick curves from the pass-through-equivalent circuit model.

Fig. 9.
Fig. 9.

Real and imaginary parts of (a) Γ⃗(6)(iω), (b) [Γ⃗(6)(iω)+1]ZB/ZA, and (c) the reflectivity |Γ⃗(6)(iω)|2 and the pass-through transmittivity |Γ⃗(6)(iω)+1|2ZB/ZA all with Γ⃗(6)(iω) in solid thick curves from the circuit of Fig. 3 without the auxiliary conductance, viz., G¯Γ=0. The thin dotted curves represent the original FPR.

Fig. 10.
Fig. 10.

Spectral response of a common equivalent circuit with G¯Γ=0 in Fig. 3, for simulating both the reflection coefficient and the pass-through coefficient in thick solid curves over a range 2<ω/ω1<3 for a FPR with more reflective mirrors with rA=rB=0.9391 than those in the FPR with rA=rB=0.7304 in all other plots in the paper. The curves almost coincide with the thin dotted curved from the analytic responses, which are actually hidden behind the thick curves: (a) Γ⃗(6)(iω). (b) ϒ⃗(6)(iω)eiωτ/2.

Equations (77)

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d·Reβ(ωj)jπ,j=0,1,2,.
1R⃗(ω)=1rA+hA2/rArAei2β(ω)drA/rBhA2
hArA2+tA2
T(ω)=rA+R⃗(ω)rBtBtAeiβ(ω)d.
ωjjω1=2πj/τ,j=0,1,2,
γAlnrA/τ,γBlnrB/τ,
γin2dImβ(ω)/τ
1R⃗(ω)ηA+[1rArAhA2]j=ω1/2ijω1+γ⃗iω,
ηA12[1rA+rAhA2]
γ⃗γinγA+γBτ1lnhA2
Σ⃗(N)(s)=[1/rArA/hA2]τ1×{1γ⃗+s+j=1N2s+2γ⃗[s+γ⃗]2+j2ω12}
1/R⃗(ω)=ηA+Σ⃗()(iω).
hA=hB1.
T(ω)sinh[γAτ]sinh[γBτ]sinh{[γxiω]τ/2},
R⃗(ω)sinh{[γ⃗iω]τ/2}sinh{[γxiω]τ/2}=γ⃗γxj=1iω/[γ⃗+ijω1]1iω/[γx+ijω1],
γxγin+γA+γB
1/rArA/hA21/rArA2sinh[γAτ]
Γ⃗(s)2γAs+γx+iωx1
ϒ⃗V(s)=2YA[YA+YB+G]+sC+1/[Rx+sLx].
GG+YB+YA
ϒ⃗V(s)=2YAC·Rx/Lx+s[ssx][ssx*]
sx=12[GC+RxLx]+i1CLx14[GCRxLx]2
Γ⃗V(s)ϒ⃗V(s)1=ss⃗ssx·ss⃗*ssx*
s⃗12[G⃗C+RxLx]+i1CLx14[RxLxG⃗C]2,
G⃗G+YBYA=G2YA.
ϒP(s)=ϒ⃗V(s)YB/YA,
LxC1/ωx2,
{YA/CYB/C}2τ{sinh[γAτ]sinh[γBτ]}{2γA2γB}.
Rx/2Lx+G/2Cγin
Imsx=ωx,
Rx/Lx=G/Cγxγin+γB+γA,
G/CγinγBγA,
sxγx+iωxG/C+i/CLx
s⃗=γ⃗+iωx24γA2
ϒ⃗V(s)2sinh[γAτ]τ·2s+2γx[s+γx]2+ωx24γAsxsx*[sxssxsx*ssx*],
Γ⃗V(s)=ϒ⃗V(s)1.
Γ⃗V(iωx)ϒ⃗V(iωx)12γAγx1=γ⃗γx,
YAYcθA2,YBYcθB2,
ϒP(s)2γAγB[1+iγx/ωxs+γxiωx+1iγx/ωxs+γx+iωx].
Ims⃗=ω1,
ϒ⃗I(s)=2ZAZA+ZB+RΓ+R+sL+1G1+sC1
ZA+ZB+RΓ=2ZA
1Γ⃗I(s)=1+2ZAR+sL+1/[G1+sC1].
LC11/ω12,
{ZA/LZB/L}2τ{sinh[γAτ]sinh[γBτ]}{2γA2γB},
R/L=G1/C1γ⃗γinγA+γB
1Γ⃗I(s)1+2γA[1s+γ⃗+iω1+1s+γ⃗iω1],
1Γ⃗I(iω1)1+2γAγ⃗=1+γxγ⃗γ⃗=γxγ⃗,
ZAZc/θA2,ZBZc/θB2
Y⃗Γ(N)(s)G¯Γ+1/Z0(s)++1/ZN(s)
1Z0(s)12R+s2L=1/2LR/L+s,
1Zj(s)=1/LR/L+s+[1/LCj]/[Gj/Cj+s]
ϒ⃗I(N)(s)=2ZAZA+ZB+RΓ+1/Y⃗Γ(N)(s),
Γ⃗I(N)(s)1ϒ⃗I(N)(s)
RΓ=ZAZB
1/Γ⃗I(N)(s)=1+2ZAY⃗Γ(N)(s)ηA+Σ⃗(N)(s)
2ZAG¯ΓηA1,
2ZAj=0N1Zj(s)Σ⃗(N)(s),
2ZAY⃗Γ(N)(s)Σ⃗(N)(s)+ηA1
{ZA/LZB/L}1τ{1/rArA/hA21/rBrB/hB2}
1/LCjωj2j2ω12
Gj/Cj=R/Lγ⃗γin+γBγAτ1lnhA2
2LG¯Γτ[ηA1]1/rArA/hA2=τ21/rA+rA/hA221/rArA/hA2
ϒ⃗P(N)(s)[Γ⃗I(N)(s)+1]ZB/ZA.
T(ω)tBeiβ(ω)dtArB[rA+1ηA+Σ⃗()(iω)]
T(ω)=eγinτ/2iωτ/2tBrA/tArB×{1+11[1ηArA]+rAΣ⃗()(iω)},
1ηArA=1/2rA2/2hA2tA2/2hA2
ϒ⃗I(N)(s)2ZAZA+ZB+Rϒ+1/Y⃗ϒ(N)(s)=2ZAZA+ZB+Rϒ·Y⃗ϒ(N)(s)Y⃗ϒ(N)(s)+1ZA+ZB+Rϒ
Y⃗ϒ(N)(s)G¯ϒ+1/Z0(s)+1/Z1(s)+
ϒP(N)(s)=2ZAZBZA+ZB+Rϒ×{1+11+[ZA+ZB+Rϒ]Y⃗ϒ(N)(s)}
[ZA+ZB+Rϒ][G¯ϒ+1/Z0(s)+1/Z1(s)+]rAΣ⃗()(s)tA2/2hA2
[ZA+ZB+Rϒ]/2ZArA,
RϒZA=2rA11/rBrB/hB21/rArA/hA22rA1γBγA,
2ZAZBZA+ZB+RϒA⃗tBrAtArBeγinτ/2,
A⃗rArA/rB·eγinτ/2hB/hA,
[ZA+ZB+Rϒ]G¯ϒ=2ZA+ZB+Rϒ2ZAZA2L2LG¯ϒ,
2LG¯ϒτ/2.

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