Abstract

A theory of light scattering by closely spaced parallel radially stratified cylinders embedded in a finite dielectric slab is presented. The refractive indices of the slab and the half-spaces on both sides of the slab are assumed to be real but arbitrary. No restriction is placed on the polarization and the propagation direction of the incident wave, the diameter of the cylinders, the intercylinder spacing, and the wavelength of the incident radiation. Each cylinder can have any number of concentric layers of stratification, and the complex refractive index of each layer can be different. A rigorous solution of Maxwell’s equations is developed by accounting for depolarization effects that are due to oblique incidence on the cylinders, refraction of scattered waves at the slab boundaries, and coherent scattering between the cylinders. The scattering characteristics of a slab containing a specific configuration of cylinders are examined by numerical data for several combinations of refractive indices of the slab and the half-spaces. In addition, the present scattering theory is applied to obtain the solutions for the cases of cylinders embedded in a semi-infinite medium, cylinders located in front of a reflecting–transmitting plane, and cylinders in an infinite homogeneous medium.

© 1999 Optical Society of America

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References

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  1. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).
  2. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).
  3. M. Barabas, “Scattering of a plane wave by a radially stratified tilted cylinder,” J. Opt. Soc. Am. A 4, 2240–2248 (1987).
    [CrossRef]
  4. V. Twersky, “Multiple scattering of radiation by an arbitrary configuration of parallel cylinders,” J. Acoust. Soc. Am. 24, 42–46 (1952).
    [CrossRef]
  5. G. O. Olaofe, “Scattering by an arbitrary configuration of parallel circular cylinders,” J. Opt. Soc. Am. 60, 1233–1236 (1970).
    [CrossRef]
  6. S. C. Lee, “Dependent scattering of an obliquely incident plane wave by a collection of parallel cylinders,” J. Appl. Phys. 68, 4952–4957 (1990).
    [CrossRef]
  7. S. C. Lee, “Scattering by closely-spaced radially-stratified parallel cylinders,” J. Quant. Spectrosc. Radiat. Transf. 48, 119–130 (1992).
    [CrossRef]
  8. S. C. Lee, “Scattering of polarized radiation by an arbitrary collection of closely spaced parallel nonhomogeneous tilted cylinders,” J. Opt. Soc. Am. A 13, 2256–2265 (1996).
    [CrossRef]
  9. D. Felbacq, G. Tayeb, D. Maystre, “Scattering by a random set of parallel cylinders,” J. Opt. Soc. Am. A 11, 2526–2538 (1994).
    [CrossRef]
  10. R. Borghi, F. Gori, M. Santarsiero, F. Frezza, F. Furno, G. Schettini, “Plane-wave scattering by a perfectly conducting circular cylinder near a plane surface: cylindrical-wave approach,” J. Opt. Soc. Am. A 13, 483–493 (1996).
    [CrossRef]
  11. R. Borghi, F. Gori, M. Santarsiero, F. Frezza, F. Furno, G. Schettini, “Plane-wave scattering by a set of perfectly conducting circular cylinders in the presence of a plane surface,” J. Opt. Soc. Am. A 13, 2441–2452 (1996).
    [CrossRef]
  12. G. Videen, D. Ngo, “Light scattering from a cylinder near a plane interface: theory and comparison with experimental data,” J. Opt. Soc. Am. A 14, 70–78 (1997).
    [CrossRef]
  13. M. K. Moaveni, A. A. Rizvi, B. A. Kamran, “Plane-wave scattering by gratings of conducting cylinders in an inhomogeneous and lossy dielectric,” J. Opt. Soc. Am. A 5, 834–842 (1988).
    [CrossRef]
  14. S. C. Lee, J. A. Grzesik, “Light scattering by closely spaced parallel cylinders embedded in a semi-infinite dielectric medium,” J. Opt. Soc. Am. A 15, 163–173 (1998).
    [CrossRef]
  15. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1993).
  16. S. C. Lee, J. A. Grzesik, “Optimization of the thermal performance of high-density fibrous composites,” (National Science Foundation, Washington, D.C., 1994).
  17. G. Cincotti, F. Gori, M. Santarsiero, F. Frezza, F. Furno, G. Schettini, “Plane wave expansion of cylindrical functions,” Opt. Commun. 95, 192–198 (1993).
    [CrossRef]
  18. W. G. Driscoll, ed., Handbook of Optics (McGraw-Hill, New York, 1987).

1998 (1)

1997 (1)

1996 (3)

1994 (1)

1993 (1)

G. Cincotti, F. Gori, M. Santarsiero, F. Frezza, F. Furno, G. Schettini, “Plane wave expansion of cylindrical functions,” Opt. Commun. 95, 192–198 (1993).
[CrossRef]

1992 (1)

S. C. Lee, “Scattering by closely-spaced radially-stratified parallel cylinders,” J. Quant. Spectrosc. Radiat. Transf. 48, 119–130 (1992).
[CrossRef]

1990 (1)

S. C. Lee, “Dependent scattering of an obliquely incident plane wave by a collection of parallel cylinders,” J. Appl. Phys. 68, 4952–4957 (1990).
[CrossRef]

1988 (1)

1987 (1)

1970 (1)

1952 (1)

V. Twersky, “Multiple scattering of radiation by an arbitrary configuration of parallel cylinders,” J. Acoust. Soc. Am. 24, 42–46 (1952).
[CrossRef]

Barabas, M.

Borghi, R.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1993).

Cincotti, G.

G. Cincotti, F. Gori, M. Santarsiero, F. Frezza, F. Furno, G. Schettini, “Plane wave expansion of cylindrical functions,” Opt. Commun. 95, 192–198 (1993).
[CrossRef]

Felbacq, D.

Frezza, F.

Furno, F.

Gori, F.

Grzesik, J. A.

S. C. Lee, J. A. Grzesik, “Light scattering by closely spaced parallel cylinders embedded in a semi-infinite dielectric medium,” J. Opt. Soc. Am. A 15, 163–173 (1998).
[CrossRef]

S. C. Lee, J. A. Grzesik, “Optimization of the thermal performance of high-density fibrous composites,” (National Science Foundation, Washington, D.C., 1994).

Kamran, B. A.

Kerker, M.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).

Lee, S. C.

S. C. Lee, J. A. Grzesik, “Light scattering by closely spaced parallel cylinders embedded in a semi-infinite dielectric medium,” J. Opt. Soc. Am. A 15, 163–173 (1998).
[CrossRef]

S. C. Lee, “Scattering of polarized radiation by an arbitrary collection of closely spaced parallel nonhomogeneous tilted cylinders,” J. Opt. Soc. Am. A 13, 2256–2265 (1996).
[CrossRef]

S. C. Lee, “Scattering by closely-spaced radially-stratified parallel cylinders,” J. Quant. Spectrosc. Radiat. Transf. 48, 119–130 (1992).
[CrossRef]

S. C. Lee, “Dependent scattering of an obliquely incident plane wave by a collection of parallel cylinders,” J. Appl. Phys. 68, 4952–4957 (1990).
[CrossRef]

S. C. Lee, J. A. Grzesik, “Optimization of the thermal performance of high-density fibrous composites,” (National Science Foundation, Washington, D.C., 1994).

Maystre, D.

Moaveni, M. K.

Ngo, D.

Olaofe, G. O.

Rizvi, A. A.

Santarsiero, M.

Schettini, G.

Tayeb, G.

Twersky, V.

V. Twersky, “Multiple scattering of radiation by an arbitrary configuration of parallel cylinders,” J. Acoust. Soc. Am. 24, 42–46 (1952).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

Videen, G.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1993).

J. Acoust. Soc. Am. (1)

V. Twersky, “Multiple scattering of radiation by an arbitrary configuration of parallel cylinders,” J. Acoust. Soc. Am. 24, 42–46 (1952).
[CrossRef]

J. Appl. Phys. (1)

S. C. Lee, “Dependent scattering of an obliquely incident plane wave by a collection of parallel cylinders,” J. Appl. Phys. 68, 4952–4957 (1990).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (8)

M. Barabas, “Scattering of a plane wave by a radially stratified tilted cylinder,” J. Opt. Soc. Am. A 4, 2240–2248 (1987).
[CrossRef]

D. Felbacq, G. Tayeb, D. Maystre, “Scattering by a random set of parallel cylinders,” J. Opt. Soc. Am. A 11, 2526–2538 (1994).
[CrossRef]

S. C. Lee, J. A. Grzesik, “Light scattering by closely spaced parallel cylinders embedded in a semi-infinite dielectric medium,” J. Opt. Soc. Am. A 15, 163–173 (1998).
[CrossRef]

G. Videen, D. Ngo, “Light scattering from a cylinder near a plane interface: theory and comparison with experimental data,” J. Opt. Soc. Am. A 14, 70–78 (1997).
[CrossRef]

M. K. Moaveni, A. A. Rizvi, B. A. Kamran, “Plane-wave scattering by gratings of conducting cylinders in an inhomogeneous and lossy dielectric,” J. Opt. Soc. Am. A 5, 834–842 (1988).
[CrossRef]

R. Borghi, F. Gori, M. Santarsiero, F. Frezza, F. Furno, G. Schettini, “Plane-wave scattering by a perfectly conducting circular cylinder near a plane surface: cylindrical-wave approach,” J. Opt. Soc. Am. A 13, 483–493 (1996).
[CrossRef]

S. C. Lee, “Scattering of polarized radiation by an arbitrary collection of closely spaced parallel nonhomogeneous tilted cylinders,” J. Opt. Soc. Am. A 13, 2256–2265 (1996).
[CrossRef]

R. Borghi, F. Gori, M. Santarsiero, F. Frezza, F. Furno, G. Schettini, “Plane-wave scattering by a set of perfectly conducting circular cylinders in the presence of a plane surface,” J. Opt. Soc. Am. A 13, 2441–2452 (1996).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transf. (1)

S. C. Lee, “Scattering by closely-spaced radially-stratified parallel cylinders,” J. Quant. Spectrosc. Radiat. Transf. 48, 119–130 (1992).
[CrossRef]

Opt. Commun. (1)

G. Cincotti, F. Gori, M. Santarsiero, F. Frezza, F. Furno, G. Schettini, “Plane wave expansion of cylindrical functions,” Opt. Commun. 95, 192–198 (1993).
[CrossRef]

Other (5)

W. G. Driscoll, ed., Handbook of Optics (McGraw-Hill, New York, 1987).

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1993).

S. C. Lee, J. A. Grzesik, “Optimization of the thermal performance of high-density fibrous composites,” (National Science Foundation, Washington, D.C., 1994).

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Figures (7)

Fig. 1
Fig. 1

Scattering by cylinders inside a finite dielectric slab.

Fig. 2
Fig. 2

Hertz potentials of the external and internal waves inside the slab.

Fig. 3
Fig. 3

Configuration of cylinders for numerical illustration.

Fig. 4
Fig. 4

Scattered intensity distribution for n0=1, n1=1.333, n2=1, ϕi=30°, and θi=0.

Fig. 5
Fig. 5

Scattered intensity distribution for n0=1, n1=1.333, n2=1.334, ϕi=30°, and θi=0.

Fig. 6
Fig. 6

Scattered intensity distribution for n0=1, n1=1.333, n2=1.5, ϕi=30°, and θi=0.

Fig. 7
Fig. 7

Numerical convergence of scattering cross section as a function of truncation order.

Tables (1)

Tables Icon

Table 1 Variation of the Scattering Cross Section for Three Cylinders (n0=1, n1=1.333)

Equations (115)

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E=imk0××(ezu)+×(ezv),
H=-m×(ezu)+ik0××(ezv),
u0σv0σ=αIσαIIσexp(-ik0σ·ρ),
E0σH0σ=αIσαIIσmσαIIσ-mσαIσP0NσP0Mσil0σ exp(-ik0σ·ρ),
P0Mσ=sin θσ ex±cos θσ ey,
P0Nσ=-τ sin ϕσ cos θσ ex+sin ϕσ sin θσ ey+cos ϕσ ez,
e0σ=τ cos ϕσ cos θσ ex-cos ϕσ sin θσ ey+sin ϕσ ez,
n0 sin ϕi=n1 sin ϕ=n2 sin ϕt,
n0 cos ϕi sin θi=n1 cos ϕ sin θ=n2 cos ϕt sin θt,
E0σ=(EσP0σ+EσP0σ)exp(-ik0σ·ρ),
H0σ=-mσ(EσP0σ-EσP0σ)exp(-ik0σ·ρ),
P0σ=-sin ξσ ex-τ cos ξσ sin ωσ ey+τ cos ξσ cos ωσ ez,
P0σ=-cos ωσ ey-sin ωσ ez,
e0σ=τ cos ξσ ex-sin ξσ sin ωσ ey+sin ξσ cos ωσ ez
Ei=il0i(αIi sin ϕi cos θi-αIIi sin θi)/[1-(cos ϕi cos θi)2]1/2,
Ei=-il0i(αIi sin θi+αIIi sin ϕi cos θi)/[1-(cos ϕi cos θi)2]1/2.
il0σαIσαIIσ=P0σ·P0NσP0σ·P0NσP0σ·P0MσP0σ·P0MσEσEσ.
S0σ=c08π(l0σ)2(|αIσ|2+|αIIσ|2)e0σ,
uj(Rp)vj(Rp)=τ=±u0τ(Rp)v0τ(Rp)+ujs(Rjp)vjs(Rjp)+kjNukjs(Rjp)vkjs(Rjp)+τ=± k=1Nukjrτ(Rjp)vkjrτ(Rjp),
u0τ(Rp)v0τ(Rp)=jταIταIIτn=-(-τi)nJn(lRjp)×exp[in(θ+τγjp)],
jτ=exp(-ik0τ·Rj)
ujs(Rjp)vjs(Rjp)=-n=-(-i)n exp(inγjp)Hn(lRjp)bjnajn,
ukjs(Rjp)vkjs(Rjp)=-n=- s=-(-i)n exp(inγjp)×Jn(lRjp)Gksjnbksaks,
Gksjn=(-i)s-n exp[i(s-n)γkj]Hs-n(lRjk).
exp(inγ)Hn(lx2+y2)=12π- exp(-ilηy)×Fn(x, η)dη,
Fn(x, η)=- exp(ilηy+inγ)Hn(lx2+y2)dy,
Fn(x, η)=2i(η2-1-η)-τnlη2-1exp(-l|x|η2-1),
uksτ(Rkp)vksτ(Rkp)=-- exp(-ikτ·Rkp)XkτYkτdη,
kτ=τlβex+lηey+hez
XkτYkτ=1πs=-τs(β+τiη)sβbksaks
ukrτ(Rkp)vkrτ(Rkp)=-- exp(ik-τ·Rk-ikτ·Rp)XkrτYkrτdη,
uktτ(Rkp)vktτ(Rkp)=-- exp(ikτ·Rk-iktτ·Rp)XktτYktτdη,
kt-=-γiex+lηey+hez,
kt+=γtex+lηey+hez,
γi=[li2-(lη)2]1/2forli2>(lη)2-i[(lη)2-li2]1/2forli2(lη)2,
γt=[lt2-(lη)2]1/2forlt2>(lη)2-i[(lη)2-lt2]1/2forlt2(lη)2,
hkηhkηβγilhkηE(xk)0βE(xk)0lkli2lki00lkE(xk)000n1βn0γiln1hkηn0hkiηn1βE(xk)0n1hkηE(xk)000n1lkn0li2lki00n1lkE(xk)0hkηE(xk)0βE(xk)0hkηE(L)hkηEˆ(L)βE(L)γtlEˆ(L)lkE(xk)000lkE(L)lt2lktEˆ(L)00n1βE(xk)0n1hkηE(xk)0n1βE(L)n2γtlEˆ(L)n1hkηE(L)n2hktηEˆ(L)00nlkE(xk)000n1lkE(L)n2lt2lktEˆ(L)Xkr+XktYkr+YktXkrXkt+YkrYkt+=hkηlkn1β00000Xk+β0n1hkηn1lk0000Yk+0000hkηlkn1β0Xk++0000β0n1hkηn1lkYk+,
E(±xk)=exp(±i2lβxk),
E(L)=exp(i2lβL),
Eˆ(L)=exp[i(lβ-γt)L].
(Xkr+, Ykr+)=τ=±(Ruτu+Xkτ+Rvτu+Ykτ),τ=±(Ruτv+Xkτ+Rvτv+Ykτ),
(Xkr-, Ykr-)=τ=±(Ruτu-Xkτ+Rvτu-Ykτ),τ=±(Ruτv-Xkτ+Rvτv-Ykτ),
(Xkt+, Ykt+)=τ=±(Tuτu+Xkτ+Tvτu+Ykτ),τ=±(Tuτv+Xkτ+Tvτv+Ykτ),
(Xkt-, Ykt-)=τ=±(Tuτu-Xkτ+Tvτu-Ykτ),τ=±(Tuτv-Xkτ+Tvτv-Ykτ),
ukjrτ(Rjp)vkjrτ(Rjp)=-- exp(-ikτ·Rjp)×exp(ik-τ·Rk-ikτ·Rj)XkrτYkrτdη,
ukjr+(Rjp)vkjr+(Rjp)=-n=- s=-(-i)n exp(inγjp)Jn(lRjp)×Fuksujnbks+FvksujnaksFuksvjnbks+Fvksvjnaks,
ukjr-(Rjp)vkjr-(Rjp)=-n=- s=-(-i)n exp(inγjp)Jn(lRjp)×Buksujnbks+BvksujnaksBuksvjnbks+Bvksvjnaks,
Fνksνjn=1π-1βfjk+{Rν+ν+ exp[i(s-n)δ]+Rν-ν+(-1)s exp[-i(s+n)δ]}dη,
Bνksνjn=(-1)nπ-1βfjk-{Rν+ν- exp[i(s+n)δ]+Rν-ν-(-1)s exp[-i(s-n)δ]}dη,
fjkτ=exp[-τilβ(xk+xj)+ilη(yk-yj)],
exp(iδ)=β+iη,
uj(Rp)vj(Rp)=n=-(-i)n exp(inγjp)Jn(lRjp)UjnVjn-Hn(lRjp)bjnajn,
UjnVjn=τ=±jταIτjταIIττn exp(τinθ)-k=1Ns=-IuuRvksujnRuksvjnIvvbksaks,
Rνksνjn=Fνksνjn+Bνksνjn,
Iνν=(1-δjk)Gksjn+Rνksνjn,
uj(m)(r)vj(m)(r)=n=-(-i)n exp(inγjp)Bjn(m)Ajn(m)Jn(ljmr)-bjn(m)ajn(m)Hn(ljmr),
bjn(1)=ajn(1)=0,
Ejn(m)=erhmjmkujn(m)r+inrvjn(m)+eγinhmjmkrujn(m)-vjn(m)r+eziljm2mjmkujn(m),
Hjn(m)=er-inmjmrujn(m)+hkvjn(m)r+eγmjmujn(m)r+inhkrvjn(m)+eziljm2kvjn(m),
{Ejn(m),Hjn(m)}·et={Ejn(m+1),Hjn(m+1)}·et,
ψjn=0ΨjnIUjn+0ΨjnIIVjn,
δjkδns+0bjnIIuu+0bjnIIRuksvjn0bjnIRvksujn+0bjnIIIvv0ajnIIRuksvjn+0ajnIIuuδjkδns+0ajnIRvksujn+0ajnIIIvvbksaks=τ=±jτ 0bjnττn exp(τinθ)jτ 0ajnττn exp(τinθ),
0bjnτ=αIτ 0bjnI+αIIτ 0bjnII,
0ajnτ=αIτ 0ajnI+αIIτ 0ajnII.
Ψjn=τ=±(αIτΨjnI+αIIτΨjnII),
uktτ(Rkp)vktτ(Rkp)=-1π- exp-iltτRp cos(γ-τΔtτ)×Tukτ(η)Tvkτ(η)dη,
exp(iΔtτ)=γtτ/ltτ+ilη/ltτ
TukτTvkτ=exp[il(τβxk+ηyk)]τ=± s=-(τ)s×(β+iτη)sβTuτuτbks+TvτuτaksTuτvτbks+Tvτvτaks
uktτ(Rkp)vktτ(Rkp)=-exp(-iltτRp)2iτπltτRp1/2ltτ cos γlTuk0τTvk0τ,
γt=lt cos γ,
lη =lt sin γ,
lβ =[l2-(lt sin γ)2]1/2,
γi=[li2-(lt sin γ)2]1/2,
γi=-li cos γ,
lη=li sin γ,
lβ=[l2-(li sin γ)2]1/2,
γt=[lt2-(li sin γ)2]1/2,
Ektτ=ilσ(PNtτuktτ+PMtτvktτ),
Hktτ=ilσmσ(PNtτvktτ-PMtτuktτ),
PNtτ=-sin ϕσ cos γ ex-sin ϕσ sin γ ey+cos ϕσ ez,
PMtτ=-sin γ ex+cos γ ey.
Sstτ=c08πRej=1NEjtτ×k=1NHktτ*,
Isτ(γ)=2πlσRplσlcos γ2(lσ/li)2|αIi|2+|αIIi|2k=1NTuk0τ2+k=1NTvk0τ2estτ
estτ=cos ϕσ eR+sin ϕσ ez,
eR=cos γ ex+sin γ ey,
Csf=-π/2π/2[Is+(γ)·eR]Rp dγ
=2πktltl2(lt/li)2|αIi|2+|αIIi|2-π/2π/2k=1NTuk0+2+k=1NTvk0+2cos2 γ dγ.
Csb=π/23π/2[Is-(γ)·eR]Rp dγ
=2πki(li/l)2|αIi|2+|αIIi|2π/23π/2k=1NTuk0-2+k=1NTvk0-2cos2 γ dγ.
αI-=αII-=0.
Rν+ν-=Rν+ν+=Rν-ν-=Tν+ν-=0,
Tν+ν+=1,
Tν+ν+=0forνν.
-lkli2lki00n1βn0γiln1hkη-n0hkiη00-n1lkn0li2lkihkη-hkiη-β-γilXkr+Xkt-Ykr+Ykt-
=lkn1β0-hkηXk-+0-n1hkηn1lk-βYk-,
(Xkr+, Ykr+)=((Ru-u+Xk-+Rv-u+Yk-), (Ru-v+Xk-+Rv-v+Yk-)),
(Xkt-, Ykt-)=((Tu-u-Xk-+Tv-u-Yk-), (Tu-v-Xk-+Tv-v-Yk-)).
Bνksνjn=0,
Rνksνjn=(-1)sπ-1βexp[-ilβ(xk+xj)+ilη(yk-yj)]×exp[-i(s+n)δ)]Rν-ν+ dη,
Iνν=(1-δjk)Gksjn+Rνksνjn.
Rντντ=Tντντ=0forνν,
αI+=αIi,αII+=αIIi,
αI-=αIr,αII-=αIIr,
Rν-ν-=Rν-ν+=Rν+ν+=Tν-ν+=0,
Tν-ν-=1,
Tν-ν+=0forνν.
-lkE(L)lt2lktEˆ(L)00-n1βE(L)-n2γtlEˆ(L)n1hkηE(L)-n2hktηEˆ(L)00-n1lkE(L)n2lt2lktEˆ(L)hkηE(L)-hktηEˆ(L)βE(L)γtlEˆ(L)Xkr-Xkt+Ykr-Ykt+=lk-n1β0-hkηXk++0-n1hkηn1lkβYk+,
(Xkr-, Ykr-)=((Ru+u-Xk++Rv+u-Yk+), (Ru+v-Xk++Rv+v-Yk+)),
(Xkt+, Ykt+)=((Tu+u+Xk++Tv+u+Yk+), (Tu+v+Xk++Tv+v+Yk+)).
Fνksνjn=0,
Rνksνjn=(-1)nπ-1βexp[ilβ(xk+xj)+ilη(yk-yj)]×exp[i(s+n)δ)]Rν+ν- dη,
Iνν=(1-δjk)Gksjn+Rνksνjn.
Rντντ=Tντντ=0forνν,
Tντντ=1,
δjkδns+0bjnI(1-δjk)Gksjn0bjnII(1-δjk)Gksjn0ajnI(1-δjk)Gksjnδjkδns+0ajnII(1-δjk)Gksjnbksaks=j+ 0bjn+ exp(inθ)j+ 0ajn+ exp(inθ),

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