Abstract

A theory of light scattering by closely spaced parallel radially stratified cylinders embedded in a semi-infinite dielectric matrix is presented. No restriction is placed on the polarization and direction of propagation of the incident plane wave. The diameter of the cylinders, the spacing between the cylinders, and the incident wavelength are comparable to each other. Each cylinder can have an arbitrary number of concentric layers of stratification, and the complex index of refraction in each layer can be different. A rigorous solution of Maxwell’s equations that accounts for depolarization at oblique incidence, diffraction of waves at the dielectric interface, and coherent scattering between the cylinders is developed. The light-scattering characteristics of closely spaced cylinders in a dielectric medium are examined by numerical data.

© 1998 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 cylinders 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. G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge University, Cambridge, 1952).
  15. S. C. Lee, J. A. Grzesik, “Optimization of the thermal performance of high-density fibrous composites,” (National Science Foundation, Washington, D.C., 1994).
  16. 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]
  17. W. G. Driscoll, ed., Handbook of Optics (McGraw-Hill, New York, 1987).
  18. I. H. Malitson, “Interpsecimen comparison of refractive index of fused silica,” J. Opt. Soc. Am. 55, 1206–1210 (1965).
    [CrossRef]

1997

1996

1994

1993

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

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

1990

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

1987

1970

1965

I. H. Malitson, “Interpsecimen comparison of refractive index of fused silica,” J. Opt. Soc. Am. 55, 1206–1210 (1965).
[CrossRef]

1952

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.

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, “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, “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).

Malitson, I. H.

I. H. Malitson, “Interpsecimen comparison of refractive index of fused silica,” J. Opt. Soc. Am. 55, 1206–1210 (1965).
[CrossRef]

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.

Watson, G. N.

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge University, Cambridge, 1952).

J. Acoust. Soc. Am.

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.

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.

I. H. Malitson, “Interpsecimen comparison of refractive index of fused silica,” J. Opt. Soc. Am. 55, 1206–1210 (1965).
[CrossRef]

G. O. Olaofe, “Scattering by an arbitrary configuration of parallel circular cylinders,” J. Opt. Soc. Am. 60, 1233–1236 (1970).
[CrossRef]

J. Opt. Soc. Am. A

J. Quant. Spectrosc. Radiat. Transf.

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

Opt. Commun.

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

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).

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge University, Cambridge, 1952).

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 (9)

Fig. 1
Fig. 1

Polarized light incident on parallel cylinders in a dielectric medium.

Fig. 2
Fig. 2

Geometric configurations for numerical illustration.

Fig. 3
Fig. 3

Convergence of the calculated backscatter cross section with truncation order.

Fig. 4
Fig. 4

Backscatter cross section at λ0=10.6 µm for three cylinders aligned normal to the interface (c/d=0.1, D=15 µm).

Fig. 5
Fig. 5

Backscatter cross section at λ0=10.6 µm for three cylinders aligned normal to the interface (c/d=1, D=15 µm).

Fig. 6
Fig. 6

Backscatter cross section at λ0=10.6 µm for three cylinders aligned normal to the interface (c/d=0.1, D=25 µm).

Fig. 7
Fig. 7

Backscatter cross section at λ0=10.6 µm for three cylinders aligned normal to the interface (c/d=1, D=25 µm).

Fig. 8
Fig. 8

Backscatter cross section at λ0=10.6 µm for three cylinders aligned parallel to the interface (c/d=0.1, D=25 µm).

Fig. 9
Fig. 9

Backscatter cross section at λ0=10.6 µm for three cylinders aligned parallel to the interface (c/d=1, D=25 µm).

Equations (87)

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E=imk0xx(ezu)+x(ezv),
H=-mx(ezu)+ik0xx(ezv),
u0iv0i=αIiαIIiexp(-iki·ρ),
E0iH0i=αIiαIIiαIIi-αIiPNiPMiili exp(-iki·ρ),
E0σH0σ=αIσαIIσmσαIIσ-mσαIσPNσPMσilσ exp(-ikσ·ρ)
S0σ=c08πlσ2(|αIσ|2+|αIIσ|2)eσ,
uj(Rp)vj(Rp)=uj0(Rp)vj0(Rp)+ujs(Rjp)vjs(Rjp)+kjNukjs(Rjp)vkjs(Rjp)+k=1Nukjr(Rjp)vkjr(Rjp),
uj0(Rp)vj0(Rp)=αIαIIj exp(-ihz)×n=-(-i)nJn(lRjp)exp[in(θ+γjp)],
j=exp(-ik·Rj)=exp[-ilRj cos(θ+γj)].
ujs(Rjp)vjs(Rjp))=-exp(-ihz)n=-(-i)n exp(inγjp)×Hn(lRjp)bjnajn.
exp(inψk)Hn(lRkp)=s=- exp(isψj)Hn+s(lRjk)Js(lRjp).
ukjs(Rjp)vkjs(Rjp)=-exp(-ihz)n=-s=-(-i)n×exp(inγjp)Jn(lRjp)Gksjnbksaks,
Gksjn=(-i)s-n exp[i(s-n)γkj]Hs-n(lRjk).
Fn(x, η)=- exp(iηy+inγ)Hn(lx2+y2)dy,
exp(inγ)Hn(lx2+y2)=12π- exp(-iηy)Fn(x, η)dη,
Fn(x, η)=2i((η/l)2-1-η/l)-τnη2-l2exp(-|x|η2-l2),
ukbs(Rkp)vkbs(Rkp)=-exp(-ihz)×- exp(-ik-·Rkp)XkYkdη,
XkYk=s=- (-1)sπ(β-iη)sβbksaks.
k-=-lβex+lηey+hez.
ukr(Rkp)vkr(Rkp)=-exp(-ihz)- exp(ik-·Rk)×exp(-ik+·Rp)XkrYkrdη,
ukt(Rkp)vkt(Rkp)=-exp(-ihz)- exp(ik-·Rk)×exp(-ik0-·Rp)XktYktdη,
k+=lβex+lηey+hez,
k0-=-γxex+lηey+hez,
γx2=li2-(lη)2.
-l/kli2/(lk0)00n1βγx/ln1hη/k-hη/k0-hη/khη/k0βγx/l00-n1l/kli2/(lk0)XkrXktYkrYkt=l/kn1βhη/k0Xk+0-n1hη/kβn1l/kYk.
Xkr=RuuXk+RvuYk
Ykr=RuvXk+RvvYk
Xkt=TuuXk+TvuYk
Ykt=TuvXk+TvvYk
ukjr(Rjp)vkjr(Rjp)=-exp(-ihz)n=-s=-(-i)n×exp(inγjp)Jn(lRjp)Ruksujnbks+RvksujnaksRuksvjnbks+Rvksvjnaks
Rσksσjn=(-1)sπ- exp[-ilβ(xj+xk)-ilη(yj-yk)]×(β-iη)s+nβRσσdη.
uj(Rp)vj(Rp)=exp(-ihz)n=-(-i)n×exp(inγjp)Jn(lRjp)UjnVjn-Hn(lRjp)bjnajn
UjnVjn=αIαIIj exp(inθ)-k=1Ns=-IuuRvksujnRuksvjnIvvbksaks,
Iσσ=(l-δjk)Gksjn+Rσksσjn.
uj(m)(r)vj(m)(r)=exp(-ihz)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+imjmrvjn(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)}·(eγ, ez)={Ejn(m+1),Hjn(m+1)}·(eγ, ez)
Ψjn=0ΨjnIUjn+0ΨjnIIVjn,
(δjkδns+0bjnIIuu+0bjnIIRuksvjn)(0bjnIRvksujn+0bjnIIIvv)(0ajnIIuu+0ajnIIRuksvjn)(δjkδns+0ajnIRvksujn+0ajnIIIvv)bjnajn=j exp(inθ)0bjnj exp(inθ)0ajn,
0bjn=αI0bjnI+αII0bjnII,
0ajn=αI0ajnI+αII0ajnII.
Ψjn=αIΨjnI+αIIΨjnII,
M>kR0 cos ϕ,
ukt(Rkp)vkt(Rkp)=-exp(-ihz)π- exp[iRp(γx cos γ-lη sin γ)]TukTvkdη,
TukTvk=exp[-il(βxk-ηyk)]×n=-(-1)n (β-iη)nβTuubkn+TvuaknTuvbkn+Tvvakn.
TukTvk=αITukI+αIITukIIαITvkI+αIITvkII,
ukt(Rkp)vkt(Rkp)=-exp(-ihz)×2iπliRp1/2 li cos γlexp(-iliRp)Tuk0Tvk0
γx0=-li cos γ,
lη0=li sin γ,
lβ0=(l2-(li sin γ)2)1/2.
S0t=c08πRej=1NEjtxk=1NHkt*,
S0t=c0li28π2πliRpli2 cos 2γl2×k=1NTuk02+k=1NTvk02ec.
ec=cos ϕieR+sin ϕiez,
eR=cos γex+sin γey.
Ibs(ϕi, γ)=S0t/S0i·ei=2πliRpli2 cos2 γ/l2|αIi|2+|αIIi|2×k=1NTuk02+k=1NTvk02ec.
Cbs(ϕi)=-π/2π/2Ibs(ϕi, γ)·eRRpdγ
=2πk0li2/l2|αIi|2+|αIIi|2π/23π/2k=1NTuk02+k=1NTvk02cos2 γdγ.
{δjkδns+0bjnI[(1-δjk)Gksjn+Ruksujn]}(bjn)=j exp(inθ)0bjn,
{δjkδns+0ajnII[(1-δjk)Gksjn+Rvksvjn]}[ajn]=j exp(inθ)0ajn,
Tuk0Tvk0=n=-(-1)n (β0-iη0)nβ0×exp[-il(β0xk-η0yk)]Tuu(β0, η0)bknTvv(β0, η0)akn
δjkδns+0bjnI(1-δjk)Gksjn0bjnII(1-δjk)Gksjn0ajnI(1-δjk)Gksjnδjkδns+0ajnII(1-δjk)Gksjnbjnajn=j exp(inθ)0bjnj exp(inθ)0ajn.
TukTvk=exp[iliRk cos(γ-γk)]cos γn=- exp(inγ)bknakn,
Is(ϕi, γ)=2πliRp1|αIi|2+|αIIi|2(|Tu|2+|Tv|2)ec
TuTv=k=1Nn=- exp[inγ+iliRk cos(γ-γk)]bknakn.
Cs(ϕ)=4k01(|αIi|2+|αIIi|2)j=1Nk=1Nn=-s=-×exp[i(n-s)(γkj-π/2)]Js-n(liRjk)×(bjnbks*+ajnaks*).
(PMi, PMr)=sin θiex(+, -)cos θiey,
(PNi, PNr)=(-,+)sin ϕi cos θiex+sin ϕi sin θiey+cos ϕiez,
(ei, er)=(+, -)cos ϕi cos θiex-cos ϕi sin θiey+sin ϕiez,
PM=sin θex+cos θey,
PN=-sin ϕ cos θex+sin ϕ sin θey+cos ϕez,
e=cos ϕ cos θex-cos ϕ sin θey+sin ϕez.
sin ϕ=sin ϕi/n1,
sin θ=cos ϕi sin θi/n12-sin 2ϕi.
E0σ=(EσPσ+EσPσ)exp(-ikσ·ρ),
H0σ=-m(EσPσ-EσPσ)exp(-ikσ·ρ),
(Pi, Pr)=-sin αiex(-, +)cos αi sin ωiey(+, -)cos αi cos ωiez,
(Pi, Pr)=-cos ωiey-sin ωiez,
(ei, er)=(+, -)cos αiex-sin αi sin ωiey+sin αi cos ωiez
P=-sin αex-cos α sin ωey+cos α cos ωez,
P=-cos ωey-sin ωez,
e=cos αex-sin α sin ωey+sin α cos ωez,
ilαIαII=P·PNP·PNP·PMP·PMTEiTEi,
iliαIrαIIr=Pr·PNrPr·PNrPr·PMrPr·PNrREiREi,
Ei=ili(αIi sin ϕi cos θi-αIIi sin θi)/[1-(cos ϕi cos θi)2]1/2,
Ei=-ili(αIi sin θi+αIIi sin ϕi cos θi)/[1-(cos ϕi cos θi)2]1/2.

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