Abstract

The diffraction of an electromagnetic wave by a cylindrical object with arbitrary cross section is studied by taking advantage of recent progress in grating theories. The fast Fourier factorization method previously developed in Cartesian coordinates is extended to cylindrical coordinates thanks to the periodicity of both the diffracting object and the incident wave with respect to the polar angle θ. Thus Maxwell equations in a truncated Fourier space are derived and separated in TE and TM polarization cases. The new set of equations for TM polarization is resolved numerically with the S-matrix propagation algorithm. Examples of elliptic cross sections and cross sections including couples of nonconcentric circles show fast convergence of the results, for both dielectric and metallic materials, as well as good agreement with previous published results. Thus the method is suitable for an extension to conical (out-of-plane) diffraction, which will allow studying mode propagation along microstructured fibers.

© 2004 Optical Society of America

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References

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  1. R. Petit, Electromagnetic Theory of Grating (Springer-Verlag, Berlin, 1980).
  2. F. Montiel, M. Nevière, “Differential theory of gratings: extension to deep gratings of arbitrary profile and permittivity through the R-matrix propagation algorithm,” J. Opt. Soc. Am. A 11, 3241–3250 (1994).
    [CrossRef]
  3. L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A 13, 1024–1035 (1996).
    [CrossRef]
  4. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996).
    [CrossRef]
  5. E. Popov, M. Nevière, “Maxwell equations in Fourier space: fast converging formulation for diffraction by arbitrary shaped, periodic, anisotropic media,” J. Opt. Soc. Am. A 18, 2886–2894 (2001).
    [CrossRef]
  6. M. Nevière, E. Popov, Light Propagation in Periodic Media: Differential Theory and Design (Marcel Dekker, New York, 2003).
  7. G. Tayeb, “The method of fictitious sources applied to diffraction gratings,” special issue on Generalized Multipole Techniques (GMT), Appl. Comput. Electromagn. Soc. J. 9, 90–100 (1994).
  8. F. Zolla, R. Petit, M. Cadilhac, “Electromagnetic theory of diffraction by a system of parallel rods: the method of fictitious sources,” J. Opt. Soc. Am. A 11, 1087–1096 (1994).
    [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]

2001 (1)

1996 (2)

1994 (4)

Cadilhac, M.

Felbacq, D.

Li, L.

Maystre, D.

Montiel, F.

Nevière, M.

Petit, R.

Popov, E.

Tayeb, G.

G. Tayeb, “The method of fictitious sources applied to diffraction gratings,” special issue on Generalized Multipole Techniques (GMT), Appl. Comput. Electromagn. Soc. J. 9, 90–100 (1994).

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

Zolla, F.

Appl. Comput. Electromagn. Soc. J. (1)

G. Tayeb, “The method of fictitious sources applied to diffraction gratings,” special issue on Generalized Multipole Techniques (GMT), Appl. Comput. Electromagn. Soc. J. 9, 90–100 (1994).

J. Opt. Soc. Am. A (6)

Other (2)

M. Nevière, E. Popov, Light Propagation in Periodic Media: Differential Theory and Design (Marcel Dekker, New York, 2003).

R. Petit, Electromagnetic Theory of Grating (Springer-Verlag, Berlin, 1980).

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Figures (16)

Fig. 1
Fig. 1

Arbitrary shaped cylindrical object.

Fig. 2
Fig. 2

Elliptically shaped cylindrical object.

Fig. 3
Fig. 3

Convergence of a differential cross section evaluated to 0° according to truncation order.

Fig. 4
Fig. 4

Differential cross section according to the angle coordinate D(θ).

Fig. 5
Fig. 5

Cross-section field map.

Fig. 6
Fig. 6

Convergence of a differential cross section evaluated to 0° according to truncation order.

Fig. 7
Fig. 7

Differential cross section according to the angle coordinate D(θ).

Fig. 8
Fig. 8

Device made of two identical circular cylinders.

Fig. 9
Fig. 9

First extension of Nθ2.

Fig. 10
Fig. 10

Convergence of a differential cross section evaluated to 315° according to truncation order.

Fig. 11
Fig. 11

Differential cross section according to angle coordinate D(θ).

Fig. 12
Fig. 12

Second extension of Nθ2.

Fig. 13
Fig. 13

Convergence of a differential cross section evaluated to 315° according to truncation order.

Fig. 14
Fig. 14

Device made of one circular cylinder.

Fig. 15
Fig. 15

Convergence of a differential cross section evaluated to 275° according to truncation order.

Fig. 16
Fig. 16

Differential cross section according to angle coordinate D(θ).

Equations (57)

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A(i)(r, θ, t)=A0exp[i(α0r cos θ+β0r sin θ)]exp(-iωt),
A{Er, Eθ, Ez, Hr, Hθ, Hz}.
u(r, θ, t)=exp(-iωt)n=-N+Nun(r)exp(inθ),
un(r)=12π02πu(r, θ)exp(-inθ)dθ.
D=(r, θ)E.
hn=[f(x)g(x)]n=m=-N+Nfn-mgm.
[fg]=f[g].
[f(x)g(x)]n=m=-N+N1f-1n,mgm.
[fg]=1f-1[g].
D=(ET+EN).
[D]=[ET]+1-1[EN].
[D]=[E]--1-1[(N  E)N].
[D]=Q(r)[E],
Q(r)=Nθ2+1-1Nr2--1-1NrNθ0--1-1NrNθNr2+1-1Nθ2000.
N(r=g(θ), θ)=grad(f)|grad(f)|r=g(θ),
f(r, θ)=r-g(θ)=0.
r[R1, R2],N(r, θ)=grad(f)|grad(f)|.
[A]/θ=iα[A].
1r α[Ez]=ωμ0[Hr],
d[Ez]dr=-iωμ0[Hθ],
1r[Eθ]+r d[Eθ]dr-iα[Er]=iωμ0[Hz],
1r α[Hz]=-ω[Dr],
d[Hz]dr=iω[Dθ],
1r[Hθ]+r d[Hθ]dr-iα[Hr]=-iω[Dz].
Q=Q,rrQ,rθ0Q,θrQ,θθ000Q,zz,
d[Hz]dr=iωQ,θr[Er]+iωQ,θθ[Eθ].
[Er]=-1rω Q,rr-1α[Hz]-Q,rr-1Q,rθ[Eθ].
d[Hz]dr=iω(Q,θθ-Q,θrQ,rr-1Q,rθ)[Eθ]-ir Q,θrQ,rr-1α[Hz].
d[Eθ]dr=-1r (iαQ,rr-1Q,rθ+Id)[Eθ]+iωμ0Id-1r2ω αQ,rr-1α[Hz],
ddr[Eθ][Hz]=M11M12M21M22[Eθ][Hz].
(kjr)2d2Hz,ndz2+kjr dHz,ndz+[(kjr)2-n2]Hz,n=0.
Hz,n=Ah,n(j)Jn(kjr)+Bh,n(j)Hn(j)(kjr),
Er=iωrHzθ,
Eθ=-iωHzr,
Hr=-iμ0ωrEzθ,
Hθ=iμ0ωEzr.
[F(Rj)]=ψ11(j)ψ12(j)ψ21(j)ψ22(j)[V(j)],
(Ψ11)nm=-iωjnRj Jn(kjRj)-kjJn+1(kjRj)δnm,
(Ψ12)nm=-iωjnRj Hn+(kjRj)-kjHn+1+(kjRj)δnm,
(Ψ21)nm=Jn(kjRj)δnm,
(Ψ22)nm=Hn+(kjRj)δnm.
(V^p)i=δpi,i[1, 2(2N+1)].
T=Ψ(2)-1[F^int(R2)].
Hz(i)=n=-+Ah,zexp(-inθi)inJn(k2r)exp(inθ),
Ah,n(1)=T11-1Ah,zinexp(-inθi),
Bh,n(2)=T21T11-1Ah,zinexp(-inθi).
r(θ)=ab[a2+(b2-a2)cos2 θ]1/2,
Nr(θ)=a2+(b2-a2)cos2 θ{[a2+(b2-a2)cos2 θ]2+(a2-b2)2sin2 θ cos2 θ}1/2,
Nθ(θ)=(a2-b2)sin θ cos θ{[a2+(b2-a2)cos2 θ]2+(a2-b2)2sin2 θ cos2 θ}1/2.
Hzd(r, θ, t)=exp(-iωt)g(θ) exp(ik2r)r,
g(θ)=2πk21/2n=-+Bh,n(2)expinθ-π2.
D(θ)=2π|g(θ)|2.
j[2, M],Bh,n(j)Ah,n(1)=S11(j)S12(j)S21(j)S22(j)Bh,n(1)Ah,n(j).
S22(j)=S22(j-1)[T11(j)+T12(j)S12(j-1)]-1,
S12(j)=[T21(j)+T22(j)S12(j-1)][T11(j)+T12(j)S12(j-1)]-1.
Ah,n(1)=S22Ah,zinexp(-inθi),
Bh,n(2)=S12Ah,zinexp(-inθi).

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