Abstract

We present an overview and generalized analysis through a number of general cases of coupled-wave equations for nonlinear optical guided-wave coupled systems. They are applicable for both symmetric and asymmetric guided-wave coupling structures. An overview of the nonlinear symmetric guided-wave optical coupling systems is given that introduces a generalization of the analysis for nonlinear asymmetric coupling. General cases including those for nonlinear guided modes and power nonorthogonality are considered, and full, power-conserved and nonorthogonal coupled-mode equations (in terms of parameters directly related to the light-wave power contained in the waveguides) are proposed and analytically described. A numerical case study of the nonlinear asymmetric coupling system, which consists of a slab-planar optical waveguide and an optical fiber is described.

© 2001 Optical Society of America

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References

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  1. S. M. Jensen, “The nonlinear coherent coupler,” IEEE J. Quantum Electron. QE-18, 1580–1583 (1982).
    [CrossRef]
  2. L. Thylen, E. M. Wright, G. I. Stegeman, C. T. Seaton, and J. V. Moloney, “Beam propagation method analysis of a nonlinear directional coupler exhibiting a Kerr nonlinearity,” Opt. Lett. 11, 739–741 (1986).
    [CrossRef] [PubMed]
  3. Y. Chen, “Solution to full coupled mode equations for nonlinear coupled systems,” IEEE J. Quantum Electron. 25, 2149–2153 (1989).
    [CrossRef]
  4. M. Cada and J. D. Begin, “An analysis of a planar directional coupler with a lossless Kerr-like coupling medium,” IEEE J. Quantum Electron. 26, 361–371 (1990).
    [CrossRef]
  5. See S. L. Chuang, “A coupled-mode formulation by reciprocity and a variational principle,” J. Lightwave Technol. LT-5, 5–15 (1987).
    [CrossRef]
  6. See S. L. Chuang, “Application of the strongly coupled-mode theory to integrated optical devices,” IEEE J. Quantum Electron. QE-23, 499–509 (1987).
    [CrossRef]
  7. X. J. Meng and N. Okamoto, “Improved coupled-modetheory for nonlinear directional couplers,” IEEE J. Quantum Electron. 27, 1175–1181 (1991).
    [CrossRef]
  8. B. Daino, G. Gregori, and S. Wabnitz, “Stability analysis of nonlinear coherent coupler,” J. Appl. Phys. 58, 4513–4514 (1985).
    [CrossRef]
  9. A. T. Pham and L. N. Binh, “Nonlinear optical directional coupler used as optical amplifier with twin biasing beams,” Int. J. Optoelectron. 5, 381–386 (1990).
  10. A. T. Pham and L. N. Binh, “All-optical modulation and switching using a nonlinear-optical directional coupler,” J. Opt. Soc. Am. B 8, 1914–1931 (1991).
    [CrossRef]
  11. A. W. Snyder, D. J. Mitchell, L. Poladian, D. R. Rowland, and Y. Chen, “Physics of nonlinear fiber couplers,” J. Opt. Soc. Am. B 8, 2102–2118 (1991).
    [CrossRef]
  12. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984).
  13. A. Ankiewicz, “Novel effects in non-linear coupling,” Opt. Quantum Electron. 20, 329–337 (1988).
    [CrossRef]
  14. A. Snyder and Y. Chen, “Nonlinear fiber couplers: switches and polarization splitters,” Opt. Lett. 14, 517–519 (1989).
    [CrossRef] [PubMed]
  15. M. Abramowitz and I. A. Stegam, Handbook of Mathematical Functions (Dover, New York, 1965).
  16. R. A. Sammut and C. Pask, “Gaussian and equivalent-step-index approximations for nonlinear waveguides,” J. Opt. Soc. Am. B 8, 395–402 (1991).
    [CrossRef]
  17. D. Marcuse, “Investigation of coupling between a fiber and an infinite slab,” J. Lightwave Technol. 7, 122–130 (1989).
    [CrossRef]
  18. S. Zheng, G. P. Simon, and L. N. Binh, “Light coupling and propagation in composite optical-fiber slab waveguides,” J. Lightwave Technol. 13, 244–251 (1995).
    [CrossRef]
  19. D. Marcuse, “Directional couplers made of non-identical asymmetric slab: Part I: synchronous couplers,” J. Lightwave Technol. LT-5, 113–118 (1987).
    [CrossRef]
  20. D. Marcuse, AT&T Research, Crawford Hill Laboratory, Holmdel, N.J., 07733 (personal communication, 1996).

1995 (1)

S. Zheng, G. P. Simon, and L. N. Binh, “Light coupling and propagation in composite optical-fiber slab waveguides,” J. Lightwave Technol. 13, 244–251 (1995).
[CrossRef]

1991 (4)

1990 (2)

A. T. Pham and L. N. Binh, “Nonlinear optical directional coupler used as optical amplifier with twin biasing beams,” Int. J. Optoelectron. 5, 381–386 (1990).

M. Cada and J. D. Begin, “An analysis of a planar directional coupler with a lossless Kerr-like coupling medium,” IEEE J. Quantum Electron. 26, 361–371 (1990).
[CrossRef]

1989 (3)

Y. Chen, “Solution to full coupled mode equations for nonlinear coupled systems,” IEEE J. Quantum Electron. 25, 2149–2153 (1989).
[CrossRef]

D. Marcuse, “Investigation of coupling between a fiber and an infinite slab,” J. Lightwave Technol. 7, 122–130 (1989).
[CrossRef]

A. Snyder and Y. Chen, “Nonlinear fiber couplers: switches and polarization splitters,” Opt. Lett. 14, 517–519 (1989).
[CrossRef] [PubMed]

1988 (1)

A. Ankiewicz, “Novel effects in non-linear coupling,” Opt. Quantum Electron. 20, 329–337 (1988).
[CrossRef]

1987 (3)

D. Marcuse, “Directional couplers made of non-identical asymmetric slab: Part I: synchronous couplers,” J. Lightwave Technol. LT-5, 113–118 (1987).
[CrossRef]

See S. L. Chuang, “A coupled-mode formulation by reciprocity and a variational principle,” J. Lightwave Technol. LT-5, 5–15 (1987).
[CrossRef]

See S. L. Chuang, “Application of the strongly coupled-mode theory to integrated optical devices,” IEEE J. Quantum Electron. QE-23, 499–509 (1987).
[CrossRef]

1986 (1)

1985 (1)

B. Daino, G. Gregori, and S. Wabnitz, “Stability analysis of nonlinear coherent coupler,” J. Appl. Phys. 58, 4513–4514 (1985).
[CrossRef]

1982 (1)

S. M. Jensen, “The nonlinear coherent coupler,” IEEE J. Quantum Electron. QE-18, 1580–1583 (1982).
[CrossRef]

Ankiewicz, A.

A. Ankiewicz, “Novel effects in non-linear coupling,” Opt. Quantum Electron. 20, 329–337 (1988).
[CrossRef]

Begin, J. D.

M. Cada and J. D. Begin, “An analysis of a planar directional coupler with a lossless Kerr-like coupling medium,” IEEE J. Quantum Electron. 26, 361–371 (1990).
[CrossRef]

Binh, L. N.

S. Zheng, G. P. Simon, and L. N. Binh, “Light coupling and propagation in composite optical-fiber slab waveguides,” J. Lightwave Technol. 13, 244–251 (1995).
[CrossRef]

A. T. Pham and L. N. Binh, “All-optical modulation and switching using a nonlinear-optical directional coupler,” J. Opt. Soc. Am. B 8, 1914–1931 (1991).
[CrossRef]

A. T. Pham and L. N. Binh, “Nonlinear optical directional coupler used as optical amplifier with twin biasing beams,” Int. J. Optoelectron. 5, 381–386 (1990).

Cada, M.

M. Cada and J. D. Begin, “An analysis of a planar directional coupler with a lossless Kerr-like coupling medium,” IEEE J. Quantum Electron. 26, 361–371 (1990).
[CrossRef]

Chen, Y.

Chuang, S. L.

See S. L. Chuang, “A coupled-mode formulation by reciprocity and a variational principle,” J. Lightwave Technol. LT-5, 5–15 (1987).
[CrossRef]

See S. L. Chuang, “Application of the strongly coupled-mode theory to integrated optical devices,” IEEE J. Quantum Electron. QE-23, 499–509 (1987).
[CrossRef]

Daino, B.

B. Daino, G. Gregori, and S. Wabnitz, “Stability analysis of nonlinear coherent coupler,” J. Appl. Phys. 58, 4513–4514 (1985).
[CrossRef]

Gregori, G.

B. Daino, G. Gregori, and S. Wabnitz, “Stability analysis of nonlinear coherent coupler,” J. Appl. Phys. 58, 4513–4514 (1985).
[CrossRef]

Jensen, S. M.

S. M. Jensen, “The nonlinear coherent coupler,” IEEE J. Quantum Electron. QE-18, 1580–1583 (1982).
[CrossRef]

Marcuse, D.

D. Marcuse, “Investigation of coupling between a fiber and an infinite slab,” J. Lightwave Technol. 7, 122–130 (1989).
[CrossRef]

D. Marcuse, “Directional couplers made of non-identical asymmetric slab: Part I: synchronous couplers,” J. Lightwave Technol. LT-5, 113–118 (1987).
[CrossRef]

Meng, X. J.

X. J. Meng and N. Okamoto, “Improved coupled-modetheory for nonlinear directional couplers,” IEEE J. Quantum Electron. 27, 1175–1181 (1991).
[CrossRef]

Mitchell, D. J.

Moloney, J. V.

Okamoto, N.

X. J. Meng and N. Okamoto, “Improved coupled-modetheory for nonlinear directional couplers,” IEEE J. Quantum Electron. 27, 1175–1181 (1991).
[CrossRef]

Pask, C.

Pham, A. T.

A. T. Pham and L. N. Binh, “All-optical modulation and switching using a nonlinear-optical directional coupler,” J. Opt. Soc. Am. B 8, 1914–1931 (1991).
[CrossRef]

A. T. Pham and L. N. Binh, “Nonlinear optical directional coupler used as optical amplifier with twin biasing beams,” Int. J. Optoelectron. 5, 381–386 (1990).

Poladian, L.

Rowland, D. R.

Sammut, R. A.

Seaton, C. T.

Simon, G. P.

S. Zheng, G. P. Simon, and L. N. Binh, “Light coupling and propagation in composite optical-fiber slab waveguides,” J. Lightwave Technol. 13, 244–251 (1995).
[CrossRef]

Snyder, A.

Snyder, A. W.

Stegeman, G. I.

Thylen, L.

Wabnitz, S.

B. Daino, G. Gregori, and S. Wabnitz, “Stability analysis of nonlinear coherent coupler,” J. Appl. Phys. 58, 4513–4514 (1985).
[CrossRef]

Wright, E. M.

Zheng, S.

S. Zheng, G. P. Simon, and L. N. Binh, “Light coupling and propagation in composite optical-fiber slab waveguides,” J. Lightwave Technol. 13, 244–251 (1995).
[CrossRef]

IEEE J. Quantum Electron. (5)

S. M. Jensen, “The nonlinear coherent coupler,” IEEE J. Quantum Electron. QE-18, 1580–1583 (1982).
[CrossRef]

Y. Chen, “Solution to full coupled mode equations for nonlinear coupled systems,” IEEE J. Quantum Electron. 25, 2149–2153 (1989).
[CrossRef]

M. Cada and J. D. Begin, “An analysis of a planar directional coupler with a lossless Kerr-like coupling medium,” IEEE J. Quantum Electron. 26, 361–371 (1990).
[CrossRef]

See S. L. Chuang, “Application of the strongly coupled-mode theory to integrated optical devices,” IEEE J. Quantum Electron. QE-23, 499–509 (1987).
[CrossRef]

X. J. Meng and N. Okamoto, “Improved coupled-modetheory for nonlinear directional couplers,” IEEE J. Quantum Electron. 27, 1175–1181 (1991).
[CrossRef]

Int. J. Optoelectron. (1)

A. T. Pham and L. N. Binh, “Nonlinear optical directional coupler used as optical amplifier with twin biasing beams,” Int. J. Optoelectron. 5, 381–386 (1990).

J. Appl. Phys. (1)

B. Daino, G. Gregori, and S. Wabnitz, “Stability analysis of nonlinear coherent coupler,” J. Appl. Phys. 58, 4513–4514 (1985).
[CrossRef]

J. Lightwave Technol. (4)

D. Marcuse, “Investigation of coupling between a fiber and an infinite slab,” J. Lightwave Technol. 7, 122–130 (1989).
[CrossRef]

S. Zheng, G. P. Simon, and L. N. Binh, “Light coupling and propagation in composite optical-fiber slab waveguides,” J. Lightwave Technol. 13, 244–251 (1995).
[CrossRef]

D. Marcuse, “Directional couplers made of non-identical asymmetric slab: Part I: synchronous couplers,” J. Lightwave Technol. LT-5, 113–118 (1987).
[CrossRef]

See S. L. Chuang, “A coupled-mode formulation by reciprocity and a variational principle,” J. Lightwave Technol. LT-5, 5–15 (1987).
[CrossRef]

J. Opt. Soc. Am. B (3)

Opt. Lett. (2)

Opt. Quantum Electron. (1)

A. Ankiewicz, “Novel effects in non-linear coupling,” Opt. Quantum Electron. 20, 329–337 (1988).
[CrossRef]

Other (3)

M. Abramowitz and I. A. Stegam, Handbook of Mathematical Functions (Dover, New York, 1965).

D. Marcuse, AT&T Research, Crawford Hill Laboratory, Holmdel, N.J., 07733 (personal communication, 1996).

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984).

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Figures (6)

Fig. 1
Fig. 1

A cross-sectional schematic of the third-order nonlinear two-mode couplers: (a) the coupler; (b) the constituent waveguides.

Fig. 2
Fig. 2

Cross-sectional view of a fiber–slab waveguide coupler.

Fig. 3
Fig. 3

Propagation of |a0(z)|2: nf=1.4817, ns=1.4745, nc=1.40; (1) solid curve, scalar CMT without nonlinearity; (2) dotted curve, scalar CMT with nonlinearity.

Fig. 4
Fig. 4

Propagation of |a0(z)|2: nf=1.4756, ns=1.4745, nc=1.40; (1) solid curve, scalar CMT without nonlinearity; (2) dotted curve, scalar CMT with nonlinearity.

Fig. 5
Fig. 5

Propagation of |a0(z)|2: nf=1.4745, ns=1.4745, nc=1.40; (1) solid curve, scalar CMT without nonlinearity; (2) dotted curve, scalar CMT with nonlinearity.

Fig. 6
Fig. 6

Propagation of |a0(z)|2: nf=1.4709, ns=1.4745, nc=1.40; (1) solid curve, scalar CMT without nonlinearity; (2) dotted curve, scalar CMT with nonlinearity.

Equations (151)

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s0=|a1|2+|a2|2=P1+P2,
s1=|a1|2-|a2|2=P1-P2,
s2=a1a2*+a1*a2,
s3=j(a1a2*-a1*a2),j=(-1)1/2.
ja1=Qa1+Q2a2+(Q3|a1|2+2Q4|a2|2)a1,
ja2=Q1a2+Q2a1+(Q3|a2|2+2Q|a1|2)a2,
Q1=[ω/(4πP0)] dxdyδ|E1|2,
Q2=[ω/(4πP0)] dxdy(ε+δ)|E1E2*,
Q3=[n0n2ω/(πP0)]dxdy|E1|4,
Q4=[n0n2ω/(πP0)]dxdy|E1|2|E2|2,
a1=A1 exp[j(ϕ1+Q1z)],
a2=A2 exp[j(ϕ2+Q1z)],
Pt=A12+A22
Γ=4A1A2 cos Ψ-2(Q3-2Q4)A12A22/Q2,
Γ=4[P(Pt-P)]1/2 cos Ψ-2(Q3-2Q4)×P(Pt-P)/Q2.
P={Q2[4Q2-Γ(Q3-2Q4)]P(Pt-P)-Γ2 Q22/4-(Q3-2Q4)2P2(Pt-P)2}1/2.
P(Z)=Pc{1/2+γδ (γ2+δ2)-1/2sd[Z(γ2+δ2)1/2+F(ϕ0|m)|m]},
(γPc)2=-4Pt2+2Pc(Pc-Γ)+2Pc(Pc-2Γ Pc)1/2,
(δPc)2=4Pt2-2Pc(Pc-Γ)+2Pc(Pc-2Γ Pc)1/2,
sin2 (ϕ0)=(γ2+δ2)[P(0)-Pt/2]2/{δ2[P(0)-Pt/2]2+(γδ Pc/4)2},
m=δ2/(γ2+δ2),
Pc=4P2/(Q3-2Q4).
P1(Z)=P1(0)[1+cn(2Z|m)]/2,
P1(Z)=P1(0)[1+cos(2Z)]/2,
z=π (2Q2).
j(a1+Na2)=(β+C11+Q1|a1|2+2Q2|a2|2)a1+(C12+Nβ+2Q3|a1|2+Q3|a22|)a2+Q2a1*a22+Q3a12a2*,
j(a2+Na1)=(β+C11+Q1|a2|2+2Q2|a1|2)a2+(C12+Nβ+2Q3|a2|2+Q3|a12|)a1+Q2a2*a12+Q3a22a1*,
Q1=kn2Aψ14dA,Q2=kn2Aψ12ψ22dA,
Q3=kn2Aψ1ψ23dA.
jA+jPabB=Q1A+[Q2+kc(AB*+A*B)]B+[(kc-ks)|B|2+kt(AB*+A*B)]A,
jB+jPabA)=Q1B+[Q2+kc(AB*+A*B)]A+[(kc-ks)|A|2+k1(AB*+A*B)]B,
P=14[Et(a)×Ht(a)*+Et(a)*×Ht(a)*]zdxdy,
Pab=14[Et(a)×Ht(b)*+Et(b)*×Ht(a)]zdxdy,
Q1=ω/(4P)(Δε+ΔεNL)E(a)E(a)*dxdy,
Q2=ω/(4P)(Δε+ΔεNL)E(a)E(b)*dxdy,
ks=ωε0/(4P)α|E(a)|4dxdy,
kc=ωε0/(4P)α|E(a)|2|E(b)|2dxdy,
kt=ωε0/(4P)α|E(a)|2E(a)E(b)*dxdy,
Δε=ε0[n2-(n(a))2],ΔεNL=ε1(α-α(a))|E(a)|2.
s0=0,s1=-2(Q2+kcs2)s3,
s2=(ks-kc)s1s3,
s3=2(Q2+kcs2)s1-(ks-kc)s1s2,
s2=4Q2[η(ζ-η)(s2-α1)s2(s2-α2)(s2-β2)]1/2,
|ζ||η|, |η|1,ζη1, |α1||α2|,
|α1||β2|.
Pa=P2{1+cn[2Q2(4ζη+1)1/4z|m]}m1P2 (1+dn{2Q2[ζ(ζ-η)]1/2z|m-1})m1,
m=α2(β2-α1)/[(β2(α2-α1)]ζ(ζ-η).
Lc=k(m)/[Q2(4ζη+1)1/4]m1k(m-1)/{2Q2[ζ(ζ-η)]1/2}m1.
Pa(z)=a*(z)a(z),Pb(z)=b*(z)b(z),
Pr(z)=Re[a*(z)b(z)],
Pi(z)=Im[a*(z)b(z)].
Pa(s0+s1)/2,Pb(s0-s1)/2,
Prs2/2,Pis3/2.
a=-j(Qa+Qaa|a|2+2Qba|b|2)a-jKabb,
b=-j(Qb+Qbb|b|2+2Qab|a|2)b-jKbaa,
Pa=2KabPi,Pb=-2KbaPi,
Pr=[Qb-Qa+(2Qab-Qaa)Pa-(2Qba-Qbb)Pb]Pi,
Pi=KabPb-KbaPa+[Qa-Qb+(2Qba-Qbb)Pb-(2Qab-Qaa)Pa]Pr.
P=Pa+Pb+Pab=0.
Pab=2(Kba-Kab)Pi
Γ=KbaPa+KabPb.
n=nL+n3NL,I,
nν=nLν+n3NL,Iν(ν=a, b),
nL=n1n3n2,
n3NL,I=0for|x|>s/2+tn3,IIfors/2|x|s/2+t0for|x|s/2,
nLa=n1n2n3,
n3NL,Ia=0forx>s/2+t0forsxs/2+tn3,IaIaforx<s/2,
nLb=n3n2n1,
n3NL,Ib=n3,IbIbforx>-s/20for-s/2-tx-s/20forx<s/2,
n2=nL2+n3NL,E,nν(2)=nLν2+n3NL,Eν,
n3NL,E=0for|x|>s/2+tn3,E|E|2fors/2|x|s/2+t0for|x|s/2,
n3NL,Ea=0forx>s/2+t0fors/2xs/2+tn3,Ea|Ea|2forx<s/2,
n3NL,Eb=n3,Eb|Eb|2forx>-s/20for-s/2-tx-s/20forx<s/2,
n3,E=cε0nL2n3,I,n3,Eν=cε3nLν2n3Iν(ν=a, b).
a(z)+Pabb(z)=-jQaza(z)-jKzb(z),
b(z)+Paba(z)=-jQbzb(z)-jKza(z),
Qaz=Q+(Kc-Ks)|b(z)|2+Kt[a(z)b(z)*+a(z)*b(z)],
Qbz=Q+(Kc-Ks)|a(z)|2+Kt[a(z)b(z)*+a(z)*b(z)],
Kz=K+Kc[a(z)b(z)*+a(z)*a(z)],
Q=ε0ω4P0 A(n2-nν2)|Eν|2dA(ν=aorb),
K=ε0ω4P0 A(n2-nμ2)EμEν*dA(μ, ν=a, bbutμν),
Ks=ε0ω4P0 An3NLE|Eν|4dA(ν=aorb),
Kc=ε0ω4P0 An3NL,E|Eμ|2|Eν|2dA(μ, ν=a, bbutμν),
Kt=ε0ω4P0 An3NL,E|Eμ|2EμEν*dA(μ, ν=a, bbutμν),
Pab=14P0 A(Eμ×Hν*+Eν*×Hμ)·zˆdA(μ, ν=a, bbutμν),
P0=14 A(Eν×Hν*+Eν*×Hν)·zˆdA(ν=aorb),
a(z)=-j(Q̲az-PabK̲z)a(z)-j(K̲z-PabQ̲bz)b(z),
b(z)=-j(Q̲bz-PabK̲z)b(z)-j(K̲z-PabQ̲az)a(z),
A̲A/(1-Pab).
Pa=(Q̲ba+2K̲ctPr+PabK̲scPa)Pi,
Pb=-(Q̲ba+2K̲ctPr+PabK̲scPb)Pi,
Pr=-K̲sc(Pa-Pb)Pi,
Pi=(-Q̲ba+2K̲stPr)(Pa-Pb).
K̲sc=K̲s-K̲c,Q̲ba=Q̲bz-PabQ̲az,
K̲ct=K̲c-PabK̲t,
K̲st=K̲sc-2K̲ct=K̲s-3K̲c+2PabK̲t.
Pa+Pb=-2PabPr,Γ1=Pa+Pb+2PabPr=P,
Γ2=(Pa-Pb)2-[1-4(2Q̲21/K̲sc+Pab)Pr-4(2K̲ct/K̲sc-Pab2)Pr2],
Γ3=Pi2-[2(Q̲ba/K̲sc)Pr-(K̲st/K̲sc)Pr2].
Pa(0)=1,Pb(0)=Pr(0)=Pi(0)=0,
Pa(0)=Pb(0)=Pr(0),Pi(0)=-Qba,
(Pa-Pb)=[-(η̲/ζ̲)(2Pr-α̲1)(2Pr-α̲2)]1/2,
Pi=[γ̲Pr(β̲2-2Pr)]1/2.
ζ=(4Q̲ba/K̲sc+2Pab)-1,
η̲=(2K̲ct/K̲sc-Pab2)(4Q̲ba/K̲sc+2Pab)-1,
α1=-[1+(1+4ζ̲η̲)1/2]/(2η)-1/η̲,
α̲2=-[1-(1+4ζ̲η̲)1/2]/(2η̲)ζ̲,
β̲2=4Q̲ba/K̲st={1-Pab[2(ζ̲-η̲)+Pab]/γ̲}/(ζ̲-η̲)(1-2Pabζ̲)/(ζ̲-η̲),
γ̲=K̲st/K̲sc=(ζ̲-η̲)/ζ̲+Pab2(ζ̲-η̲)/ζ̲.
|ζ̲||η̲|,|η̲|1,ζ̲η̲1,
|α̲1||α̲2|,|α̲1||β̲2|.
2Pr=K̲sc[(η̲γ̲/ζ̲)2Pr(2Pr-α̲1)(2Pr-α̲2)×(2Pr-β̲2)]1/2,
02Pr(z)[2Pr(2Pr-α̲1)(2Pr-α̲2)(2Pr-β̲2)]-1/2d(2Pr)
=K̲sc(η̲γ̲/ζ̲)1/2z.
Ksc(ηγ/ζ)1/2z4Qbc[η(ζ-η)]1/2/(1-2Pabζ).
n3,E>0,α1<0,α2>0,n3,E>0,β2>0.
Pr=P2[1+cn(κ+z|m)]m1(α2β2)P2[1+dn(κ-z|m-1)]m>1(α2>β2),
κ+=2Q̲ba(1+4ζ̲η̲)1/4/(1-2Pabζ̲),
κ-=2Q̲ba[ζ̲(ζ̲-η̲)]1/2/(1-2Pabζ̲),
m=α̲2(β̲2-α̲1)/[β̲2(α̲2-α̲1)]α̲2/β̲2ζ̲(ζ̲-η̲)/(1-2Pabζ̲).
Lc=(1-2Pabζ̲)k(m)/[Q2(4ζ̲η̲+1)1/4]m1(α2β2)(1-2Pabζ̲)k(m-1)/{2Q̲2[ζ̲(ζ̲-η̲)]1/2}m1(α2>β2).
n2=nL2+n2|E|2(x, yA),
nν2=nLν2+n2ν|Eν|2(x, yAν),
n2{n2o, n2s, n2c, n2 f}(x, yA),
a0=-jβf0+Qf00+If0|a0|2+m,nNIfsmnbmbn*a0-jnNKf0nbn,
bm=-jnδmnβsn+Qsmn+p,lNImnplbpbl*+Isfmn|a0|2bn-jKsm0a0,
If0=k22βf0 An2F04dA,
Imnpl=k22βs An2SmSnSpSldA,
Ifsmn=k22βf0 An2F02SmSndA,
Isfmn=k22βs An2F02SmSndA,
Imnpl=k22n2oAsSmSnSpSldA.
Sn|Ao=Nskst2Vso exp-{-γa[x-(a+s+t)]} cos(σny).
Imnpl=α0Dk2γ0Nskst̲4Vso4{δ(m+n-p-l),0+δ(m+n+p-l+1),0+δ(m+n-p+l+1,0)+δ(m-n+p+l+1),0+δ(m-n-p--l-1),0+δ(m-n+p-l),0+δ(m-n-p+l),0}.
E(x, y, z)=a0(z)F0(x, y)+nbn(z)Sn(x, y),
F0=NfJ0(kfr)J0(kfa)forraK0(γfr)K0(γfa)forr>a,
Nf=γfJ0(kfa)πVfJ1(kfa)
Vf2=a2(kf2+γf2).
Sn(x, y)=Ns cos(σny)Vso exp[γc(x-h)]/Vscforx<a+s{cos[ks(x-h-t)]-(γo/ks)sin[ks(x-h-t)]}fora+sxa+s+texp[-γo(x-h-t)]forx>a+s+t,
Ns=2γcγo(γo+γc+γcγot)D1/2
Vsc2=t2(ks2+γc2)/4,Vso2=t2(ks2+γo2)/4.
kst tan-1(γc/ks)-tan-1(γo/ks)=mπ(m=0, 1, 2, ).
a0=-j(βf00+Qf00)a0-jnKf0nbn,
bm=-jNn(δmnβsn+Qsmn)bn-jKsn0a0,
Qf00=k22βf0 A[n2(x, y)-nf2(x, y)]F0F0dA,
Qsmn=k22βsm A[n2(x, y)-ns2(x, y)]SmSndA,
Kf0n=k22βf0 A[n2(x, y)-ns2(x, y)]F0SndA,
Ksm0=k22βsm A[n2(x, y)-nf2(x, y)]SmF0dA,
Qf00=π2Nf2Vsc22βf0γf2t3K02(γfa) (erf([2γf(a+s+t)])1/2-erf([2γf(a+s)])1/2+As erfc([2γf(a+s+t)])1/2),
Qsmn=πNs2ks2Vf2t2 exp[-2γc(a+s)]8βsmVsc2×I1([4γc2-(σm+σn)2]1/2a)[4γc2-(σm+σn)2]1/2a+I1([4γc2-(σm-σn)2]1/2a)[4γc2-(σm-σn)2]1/2a,
Kf0n=πNfNsVf2kst exp[-γc(a+s)]2βf0aVscJ0(kfa)(kf2+γc2-σn2)×{(γc2-σn2)1/2J0(kfa)I1[(γc2-σn2)1/2a]+kfJ1(kfa)I0[(γc2-σn2)1/2a]},
Ksn0=πNfNsVscks exp(γf2+σn2)1/12(a+s)βsnK0(γfa)t(γf2+σn2)1/2(γf2+σn2)1/2+γc-Bs[(γf2+σn2)1/2-γ0]exp[-(γf2+σf2)1/2]ks2+γf2+σn2+AsBs exp[-(γf2+σn2)1/2]γ0+(γf2+σn2)

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