Abstract

The coupling phenomena between two slab waveguides in the presence of ring resonators are investigated through a rigorous integral equation analysis. A Green’s-function-theory approach is utilized to develop the integral equation formulation. The solution is obtained by applying an entire-domain Galerkin technique, using the orthogonality properties of involved wave functions in the ring resonators. The field’s transmission and reflection coefficients are accurately computed and a practical directional-coupler approach is discussed. The presented results reveal strong resonant coupling phenomena between the two waveguides and the resonators.

© 2004 Optical Society of America

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References

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  1. R. Orta, P. Savi, R. Tascone, D. Trinchero, “Synthesis of multiple-ring resonator filters for optical systems,” IEEE Photon. Technol. Lett. 7, 1447–1449 (1995).
    [CrossRef]
  2. K. Oda, N. Tokato, H. Toba, “A wide-FSR waveguide double-ring resonator for optical FDM transmission systems,” J. Lightwave Technol. 9, 728–736 (1991).
    [CrossRef]
  3. B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15, 998–1005 (1997).
    [CrossRef]
  4. C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35, 1322–1331 (1999).
    [CrossRef]
  5. M. K. Chin, S. T. Ho, “Design and modeling of waveguide-coupled single-mode microring resonators,” J. Lightwave Technol. 16, 1433–1446 (1998).
    [CrossRef]
  6. C. Manolatou, S. G. Johnson, S. Fan, P. R. Villeneuve, H. A. Haus, J. D. Joannopoulos, “High-density integrated optics,” J. Lightwave Technol. 17, 1682–1692 (1999).
    [CrossRef]
  7. D. Rafizadeh, J. P. Zhang, S. C. Hagness, A. Taflove, K. A. Stair, S. T. Ho, R. C. Tiberio, “Waveguide-coupled AlGaAs/GaAs microcavity ring and disk resonators with high finesse and 21.6 nm free spectral range,” Opt. Lett. 22, 1244–1246 (1997).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  13. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972).
  14. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).
  15. N. K. Uzunoglu, “Scattering from inhomogeneities inside a fiber waveguide,” J. Opt. Soc. Am. 71, 259–273 (1981).
    [CrossRef]
  16. S. V. Boriskina, A. I. Nosich, “Radiation and absorption losses of the WGM dielectric resonators excited by a dielectric waveguide,” IEEE Trans. Microwave Theory Tech. 47, 224–231 (1999).
    [CrossRef]
  17. M. Cai, O. Painter, K. J. Vahala, “Observation of critical coupling in a fiber taper to a silica-microsphere whispering gallery mode system,” Phys. Rev. Lett. 85, 74–77 (2000).
    [CrossRef] [PubMed]

2000 (1)

M. Cai, O. Painter, K. J. Vahala, “Observation of critical coupling in a fiber taper to a silica-microsphere whispering gallery mode system,” Phys. Rev. Lett. 85, 74–77 (2000).
[CrossRef] [PubMed]

1999 (4)

S. V. Boriskina, A. I. Nosich, “Radiation and absorption losses of the WGM dielectric resonators excited by a dielectric waveguide,” IEEE Trans. Microwave Theory Tech. 47, 224–231 (1999).
[CrossRef]

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

C. Manolatou, S. G. Johnson, S. Fan, P. R. Villeneuve, H. A. Haus, J. D. Joannopoulos, “High-density integrated optics,” J. Lightwave Technol. 17, 1682–1692 (1999).
[CrossRef]

J. E. Heebner, R. W. Boyd, “Enhanced all-optical switching by use of a nonlinear fiber ring resonator,” Opt. Lett. 24, 847–849 (1999).
[CrossRef]

1998 (1)

1997 (2)

1995 (1)

R. Orta, P. Savi, R. Tascone, D. Trinchero, “Synthesis of multiple-ring resonator filters for optical systems,” IEEE Photon. Technol. Lett. 7, 1447–1449 (1995).
[CrossRef]

1991 (1)

K. Oda, N. Tokato, H. Toba, “A wide-FSR waveguide double-ring resonator for optical FDM transmission systems,” J. Lightwave Technol. 9, 728–736 (1991).
[CrossRef]

1982 (1)

1981 (1)

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972).

Balanis, C. A.

C. A. Balanis, Advanced Engineering Electromagnetics (Wiley, New York, 1989).

Boriskina, S. V.

S. V. Boriskina, A. I. Nosich, “Radiation and absorption losses of the WGM dielectric resonators excited by a dielectric waveguide,” IEEE Trans. Microwave Theory Tech. 47, 224–231 (1999).
[CrossRef]

Boyd, R. W.

Cai, M.

M. Cai, O. Painter, K. J. Vahala, “Observation of critical coupling in a fiber taper to a silica-microsphere whispering gallery mode system,” Phys. Rev. Lett. 85, 74–77 (2000).
[CrossRef] [PubMed]

Chin, M. K.

Chu, S. T.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15, 998–1005 (1997).
[CrossRef]

Collin, R. E.

R. E. Collin, Field Theory of Guided Waves (McGraw-Hill, New York, 1960).

Fan, S.

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

C. Manolatou, S. G. Johnson, S. Fan, P. R. Villeneuve, H. A. Haus, J. D. Joannopoulos, “High-density integrated optics,” J. Lightwave Technol. 17, 1682–1692 (1999).
[CrossRef]

Fikioris, J. G.

Foresi, J.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15, 998–1005 (1997).
[CrossRef]

Hagness, S. C.

Haus, H. A.

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

C. Manolatou, S. G. Johnson, S. Fan, P. R. Villeneuve, H. A. Haus, J. D. Joannopoulos, “High-density integrated optics,” J. Lightwave Technol. 17, 1682–1692 (1999).
[CrossRef]

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15, 998–1005 (1997).
[CrossRef]

Heebner, J. E.

Ho, S. T.

Joannopoulos, J. D.

C. Manolatou, S. G. Johnson, S. Fan, P. R. Villeneuve, H. A. Haus, J. D. Joannopoulos, “High-density integrated optics,” J. Lightwave Technol. 17, 1682–1692 (1999).
[CrossRef]

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

Johnson, S. G.

Khan, M. J.

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

Laine, J.-P.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15, 998–1005 (1997).
[CrossRef]

Little, B. E.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15, 998–1005 (1997).
[CrossRef]

Manolatou, C.

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

C. Manolatou, S. G. Johnson, S. Fan, P. R. Villeneuve, H. A. Haus, J. D. Joannopoulos, “High-density integrated optics,” J. Lightwave Technol. 17, 1682–1692 (1999).
[CrossRef]

Nosich, A. I.

S. V. Boriskina, A. I. Nosich, “Radiation and absorption losses of the WGM dielectric resonators excited by a dielectric waveguide,” IEEE Trans. Microwave Theory Tech. 47, 224–231 (1999).
[CrossRef]

Oda, K.

K. Oda, N. Tokato, H. Toba, “A wide-FSR waveguide double-ring resonator for optical FDM transmission systems,” J. Lightwave Technol. 9, 728–736 (1991).
[CrossRef]

Orta, R.

R. Orta, P. Savi, R. Tascone, D. Trinchero, “Synthesis of multiple-ring resonator filters for optical systems,” IEEE Photon. Technol. Lett. 7, 1447–1449 (1995).
[CrossRef]

Painter, O.

M. Cai, O. Painter, K. J. Vahala, “Observation of critical coupling in a fiber taper to a silica-microsphere whispering gallery mode system,” Phys. Rev. Lett. 85, 74–77 (2000).
[CrossRef] [PubMed]

Rafizadeh, D.

Savi, P.

R. Orta, P. Savi, R. Tascone, D. Trinchero, “Synthesis of multiple-ring resonator filters for optical systems,” IEEE Photon. Technol. Lett. 7, 1447–1449 (1995).
[CrossRef]

Sommerfeld, A. R.

A. R. Sommerfeld, Partial Differential Equations in Physics (Academic, New York, 1949).

Stair, K. A.

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972).

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

Taflove, A.

Tascone, R.

R. Orta, P. Savi, R. Tascone, D. Trinchero, “Synthesis of multiple-ring resonator filters for optical systems,” IEEE Photon. Technol. Lett. 7, 1447–1449 (1995).
[CrossRef]

Tiberio, R. C.

Toba, H.

K. Oda, N. Tokato, H. Toba, “A wide-FSR waveguide double-ring resonator for optical FDM transmission systems,” J. Lightwave Technol. 9, 728–736 (1991).
[CrossRef]

Tokato, N.

K. Oda, N. Tokato, H. Toba, “A wide-FSR waveguide double-ring resonator for optical FDM transmission systems,” J. Lightwave Technol. 9, 728–736 (1991).
[CrossRef]

Trinchero, D.

R. Orta, P. Savi, R. Tascone, D. Trinchero, “Synthesis of multiple-ring resonator filters for optical systems,” IEEE Photon. Technol. Lett. 7, 1447–1449 (1995).
[CrossRef]

Uzunoglu, N. K.

Vahala, K. J.

M. Cai, O. Painter, K. J. Vahala, “Observation of critical coupling in a fiber taper to a silica-microsphere whispering gallery mode system,” Phys. Rev. Lett. 85, 74–77 (2000).
[CrossRef] [PubMed]

Villeneuve, P. R.

C. Manolatou, S. G. Johnson, S. Fan, P. R. Villeneuve, H. A. Haus, J. D. Joannopoulos, “High-density integrated optics,” J. Lightwave Technol. 17, 1682–1692 (1999).
[CrossRef]

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

Zhang, J. P.

IEEE J. Quantum Electron. (1)

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

IEEE Photon. Technol. Lett. (1)

R. Orta, P. Savi, R. Tascone, D. Trinchero, “Synthesis of multiple-ring resonator filters for optical systems,” IEEE Photon. Technol. Lett. 7, 1447–1449 (1995).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

S. V. Boriskina, A. I. Nosich, “Radiation and absorption losses of the WGM dielectric resonators excited by a dielectric waveguide,” IEEE Trans. Microwave Theory Tech. 47, 224–231 (1999).
[CrossRef]

J. Lightwave Technol. (4)

K. Oda, N. Tokato, H. Toba, “A wide-FSR waveguide double-ring resonator for optical FDM transmission systems,” J. Lightwave Technol. 9, 728–736 (1991).
[CrossRef]

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15, 998–1005 (1997).
[CrossRef]

C. Manolatou, S. G. Johnson, S. Fan, P. R. Villeneuve, H. A. Haus, J. D. Joannopoulos, “High-density integrated optics,” J. Lightwave Technol. 17, 1682–1692 (1999).
[CrossRef]

M. K. Chin, S. T. Ho, “Design and modeling of waveguide-coupled single-mode microring resonators,” J. Lightwave Technol. 16, 1433–1446 (1998).
[CrossRef]

J. Opt. Soc. Am. (2)

Opt. Lett. (2)

Phys. Rev. Lett. (1)

M. Cai, O. Painter, K. J. Vahala, “Observation of critical coupling in a fiber taper to a silica-microsphere whispering gallery mode system,” Phys. Rev. Lett. 85, 74–77 (2000).
[CrossRef] [PubMed]

Other (5)

A. R. Sommerfeld, Partial Differential Equations in Physics (Academic, New York, 1949).

R. E. Collin, Field Theory of Guided Waves (McGraw-Hill, New York, 1960).

C. A. Balanis, Advanced Engineering Electromagnetics (Wiley, New York, 1989).

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972).

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

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Figures (10)

Fig. 1
Fig. 1

Coupled-slab-waveguides geometry in the presence of dielectric ring resonators. The structure is assumed to be infinite along the y direction.

Fig. 2
Fig. 2

Primary TE excitation of the coupled-slab geometry through a current line source of infinite length.

Fig. 3
Fig. 3

(a) Computation of the coupling factors Mn,kij through integration on the complex λ plane and mapping of the integration contour C and the branch cut for φ0(λ) on the (b) w1 and (c) w2 complex plane through the transformation of Eq. (29).

Fig. 4
Fig. 4

Application of the residue theorem for computation of the integral Zkj(x, z) in the case z>zj.

Fig. 5
Fig. 5

(a) Application of the residue theorem for computation of the integral Zkj(x, z) in the case z<zj and mapping of the alternative branch-cut selection for φ0(λ) on the (b) w1 and (c) w2 complex plane through the transformation of Eq. (29).

Fig. 6
Fig. 6

Two examples of coupling geometries studied in present paper. Both structures are symmetrical with respect to the plane x=H/2. The numbers in (a) are used to identify the ports of the structure when the structure is viewed as a directional coupler.

Fig. 7
Fig. 7

(a) Magnitude of the transmission coefficient 1+τ1e and (b) phase of the forward-scattering coefficient τ1e versus t=k0d(n12n02)1/2, for a single ring resonator [Fig. 6(a)] with n2=2.0, b=4.0d and a=3.5d (dashed curves), a=3.0d (dash–dotted curves), a=2.0d (solid curves), and a=1.0d (dotted curves). The parameters of the coupled-slab geometry are H=10d, n1=(2.1)1/2, and n0=1.

Fig. 8
Fig. 8

(a) Magnitude and (b) phase of the reflection coefficient ρ1e versus t for the same parameters and corresponding line styles as Fig. 7.

Fig. 9
Fig. 9

Magnitude of (a) the transmission coefficient 1+τ1e (solid curves) and (b) the reflection coefficient ρ1e (given in logarithmic scale) versus t=k0d(n12-n02)1/2, for a single ring resonator [Fig. 6(a)] with n2=(9.9)1/2, b=4.0d, and a=3.5d. The dashed curve of (a) corresponds to the radiated power’s profile. The parameters of the coupled-slab geometry are H=10d, n1=(2.1)1/2, and n0=1.

Fig. 10
Fig. 10

Magnitude of (a) the transmission and (b) the reflection coefficient of an even (solid curves) and odd (dashed curves) incident mode versus t=k0d(n12-n02)1/2 for two identical, symmetrically placed ring resonators [Fig. 6(b)] with n2=(9.9)1/2, b=2.0d, and a=1.5d. The parameters of the coupled-slab geometry are H=10d, n1=(2.1)1/2, and n0=1.

Tables (2)

Tables Icon

Table 1 Even and Odd Modes’ Normalized Propagation Constants β/k versus t and H, for n1=(2.1)1/2 and n0=1a

Tables Icon

Table 2 Convergence Pattern of the Forward- and Backward-Scattering Coefficients for the Geometry of Fig. 6(a)a

Equations (80)

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E=yˆΨ(x, z),
J=yˆδ(x-x)δ(z-z),
G0(x, z|x, z)=-j4H0(2)(k0|r-r|)=14π-+dλγ0-1exp[jλ(z-z)-γ0|x-x|],
[2/x2+2/z2+k2n(i)2]Gi(x, z|x, z)=0,
Gi(x, z|x, z)=-+dλ exp[jλ(z-z)]gi(λ, x),
g1(λ, x)=C1(λ)exp[γ0(x+d)],x<-d,g2(λ, x)=A1(λ)cosh(γ1x)+A2(λ)sinh(γ1x),|x|<d,g3(λ, x)=B1(λ)exp[-γ0(x-d)]+B2(λ)exp[γ0(x-H+d)],d<x<H-d,g4(λ, x)=A3(λ)cosh[γ1(x-H)]+A4(λ)sinh[γ1(x-H)],|x-H|<dg5(λ, x)=C2(λ)exp[-γ0(x-H-d)],x<H+d
G(x, z|x, z)=G0(x, z|x, z)+G3(x, z|x, z)=-j4H0(2)(k0|r-r|)+-+dλ×exp[jλ(z-z)]K2πγ0μ(λ, x, x),
μ(λ, x, x)=cosh[γ0(x-H/2)]cosh[γ0(x-H/2)]PePo-K+sinh[γ0(x-H/2)]sinh[γ0(x-H/2)]PePo+K.
Pe=γ0 cosh(γ1d)+γ1 sinh(γ1d),
Po=γ0 sinh(γ1d)+γ1 cosh(γ1d),
K=(k12-k02)sinh(γ1d)cosh(γ1d),
=exp[-γ0(H-2d)].
Ψe/o(x, z)=exp(-jβe/oz)±ja1 exp[-γ0(x-H-d)],x>H+d±Po cos[a1(x-H)]jPe×sin[a1(x-H)],|x-H|<d(2K1/2/γ0)cosh-sinh[γ0(x-H/2)],d<x<H-dPo cos(a1x)+jPe sin(a1x),|x|<dja1 exp[γ0(x+d)],x<-d,
Pe(βe)Po(βe)-K(βe)(βe)=0
(evencoupledTEmode),
Pe(βo)Po(βo)+K(βo)(βo)=0
(oddcoupledTEmode),
Δβ=K(βi)(βi)/[Pe(βi)Po(βi)]ifPe(βi)=0K(βi)(βi)/[Pe(βi)Po(βi)]ifPo(βi)=0,
Ψ(x, z)=Ψinc(x, z)+j[(k2j)2-k02]×RjG(x, z|x, z)Ψ(x, z)dxdz,
Ψi(x, z)=n=-+αni[Jn(k2iρi)+Λi(n)Yn(k2iρi)]exp(jnϕi),
x=xi+ρi sin ϕi,z=zi+ρi cos ϕi,
aiρibi,0ϕi2π,
Ψ(x, z)=n=-+κniJn(k0ρi)exp(jnϕi),
0ρiai,0ϕi2π.
H0(2)(k0|ρj-ρj|)=m=-+Jm(k0ρj<)Hm(2)×(k0ρj>)exp[jm(ϕj-ϕj)],
[(k2j)2-k02]RjG0(ρj, ϕj|ρj, ϕj)Ψ(ρj, ϕj)ρjdϕjdρj=(-πj/2)n=-+αnjLnjHn(2)(k0ρj)exp(jnϕj),jiΨ(ρi, ϕi)+(1/2)n=-+αniKniJn(k0ρi)exp(jnϕi)j=i,
Hn(2)(k0ρj)exp(jnϕj)=λ=-+Hλ-n(2)(k0ρij)exp[-j(λ-n)ϕji]Jλ(k0ρi)exp(jλϕi).
ϕ=02πdϕ exp[-jmϕ+jz cos(ϕ-q)]=2πjmJm(z)exp(-jmq),
[(k2j)2-k02]RjG3(x, z|ρj, ϕj)Ψ(ρj, ϕj)ρjdϕjdρj=(1/2)n=-+k=-+αkjLkjMn,kijJn(k0ρi)exp(jnϕi),
Ψinc(x, z)=n=-+τniJn(k0ρi)exp(jnϕi).
τni(e/o)=Kγ0j-n exp(-jβ(e/o)zi){Z-n exp[-γ0(xi-d)]±Zn exp[-γ0(H-d-xi)]},
Z=β/k0+(β2/k02-1)1/2,
n=-+CniJn(k0ρi)exp(jnϕi)=0,
k=-+αki(LkiMn,kii+δnkKni)+jik=-+αkjLkjFn,kij=-2τni,
Fn,kij=-πjHn-k(2)(k0ρij)exp(-jϕji)+Mn,kij.
αi=[α-Ni, α-N+1,,iαNi],
w1=γ0/λ,w2=(1+w1)/(1-w1),
φ0=-(j/2)ln(w2)
Mn,kij=(-1)n+kM-k,-nji,
Mn,kij=(-1)n+kMn,kji,ifxi+xj=H,
Mn,kij=M-n,-kij,ifzi=zj.
Ψ(x, z)=Ψinc(x, z)+(1/2)jk=-+αkj j-kLkjZkj(x, z),
Zkj(x, z)=-+dλexp[jλ(z-zj)]Pe2Po2-K22U(λ, x)×PePoKexp[-γ0(xj-d)+jkφ0]+KPePoexp[-γ0(H-xj-d)-jkφ0],
U(λ, x)=Po cosh(γ1x)+Pe sinh(γ1x),|x|<dPo cosh[γ1(x-H)]-Pe sinh[γ1(x-H)]|x-H|<d.
Zkj(x, z)=-IBj(x, z)+2πj residues,
Ψ(x, z)Ψince/o(x, z)+p=1NeτpeoeU(βpe, x)exp(-jβpez)+p=1Noτpeoo(±)U(βpo, x)exp(-jβpoz),z+p=1NeρpeoeU(βpe, x)exp(jβpez)+p=1Noρpeoo(±)U(βpo, x)exp(jβpoz),z-,
Ψ(x, z)U(β1e, x)exp(-jβ1ez)±CU(β1o, x)exp(-jβ1oz)+τ1eU(β1e, x)exp(-jβ1ez)±Cτ1oU(β1o, x)exp(-jβ1oz),z+ρ1eU(β1e, x)exp(jβ1ez)±Cρ1oU(β1o, x)exp(jβ1oz),z-,
S(24, 1)=12exp[-jβ1e(z2-z1)]×{1+τ1e±(1+τ1o)exp[jΔβ(z2-z1)]},
S(13, 1)=12exp(2jβ1ez1)[ρ1e±ρ1oexp(-2jΔβz1)],
|1+τ1e|2+|ρ1e|2<1,
|1+τ1o|2+|ρ1o|2<1,
H/d>2+(cot t-tan t)/t.
α-n=±(-1)nαn,n=0, 1, 2,,N,
α-n1=±(-1)nαn2,n=-N, -1, 0, 1,,N,
Λi(n)=n0Jn+1(k0ai)Jn(k2iai)-n2iJn(k0ai)Jn+1(k2iai)n2iYn+1(k2iai)Jn(k0ai)-n0Yn(k2iai)Jn+1(k0ai).
(kbi)-1Lni=n2iJn(k0bi)[Jn+1(k2ibi)+Λi(n)Yn+1(k2ibi)]-n0Jn+1(k0bi)×[Jn(k2ibi)+Λi(n)Yn(k2ibi)],
(πjkbi)-1Kni=n0Hn+1(2)(k0bi)[Jn(k2ibi)+Λi(n)Yn(k2ibi)]-n2iHn(2)(k0bi)×[Jn+1(k2ibi)+Λi(n)Yn+1(k2ibi)],
jk-nMn,kij=Ieij(n, k)+Ioij(n, k),
Ie/oij(n, k)=-+dλ2K exp[jλ(zi-zj)]γ0(PePoK)fe/oij(λ, n, k),
fe/oij(λ, n, k)=coshsinh[γ0(H/2-xi)+jnφ0]×coshsinh[γ0(H/2-xj)+jkφ0],
λ=k0 cos φ0γ0=jk0 sin φ0  φ0=tan-1(-jγ0/λ),
Ie/oij(n, k)=0+dλ2Kγ0(PePoK)×{exp[jλ(zi-zj)]fe/oij(λ, n, k)+(-1)n+k×exp[-jλ(zi-zj)]fe/oij(λ, -n, -k)}.
-1<Re(w2)<0,Im(w2)=0
λ=-±jk0(1+w2)/(-4w2)1/2,
w2=exp(jϕ),0<ϕ<π
λ=±k0/[1+tan2(ϕ/2)]1/2+j,
1<Re(w2)<+,Im(w2)=0
λ=k0(1+w2)/(4w2)1/2,
0<Re(w2)<1,Im(w2)=0
λ=-k0(1+w2)/(4w2)1/2,
w2=exp(jϕ),-π<ϕ<0
λ=±k0/(1+tan2(ϕ/2))1/2-j,
-<Re(w2)<-1,Im(w2)=0
λ=±jk0(1+w2)/(-4w2)1/2.
γ0(λ)=(λ2-k02)1/2,|λ|>k0j(k02-λ2)1/2,0<|λ|<k0,
φ0(λ)=π-φ0(-λ),-<λ<0tan-1[(k02/λ2-1)1/2],0<λ<k0-j ln[λ/k0+(λ2/k02-1)1/2]k0<λ<+.
τpρp(incidente/o)=±πj2Φ(βpe/o)i=1Rexp(±jβpe/ozi)×k=-+αki j-kLkiΩe/oi(k, βpe/o),
Φ(λ)=λγ0γ1((1+γ0d)(Pe2+Po2)+PePo{γ1(H-2d)-γ0d[coth(γ1d)+tanh(γ1d)]}),
Ωei(k, βpe)=(1)k{Zk exp[-γ0(xi-d)]+Z±k exp[-γ0(H-d-xi)]},
Ωoi(k, βp0)=(1)k{Zk exp[-γ0(xi-d)]-Z±k exp[-γ0(H-d-xi)]},

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