Abstract

We extend the differential theory to anisotropic cylindrical structures with an arbitrary cross section. Two cases have to be distinguished. When the anisotropic cylinders do not contain the origin, the scattering matrix of the device is calculated from the extended differential theory with the help of the scattering matrix propagation algorithm. The fields outside the cylinders are described by Fourier–Bessel expansions. When the origin is located in one cylinder, the fields inside the cylinder are expressed from a semi-analytical theory related to a homogeneous anisotropic medium. In this second case, the formalism of the scattering matrix propagation algorithm is not exactly the same and requires suitable change. The numerical results are in good agreement with the ones obtained for the diffraction by one circular cylinder. The theory is then applied on the diffraction by an elliptical cylinder.

© 2013 Optical Society of America

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  1. G. Cerutti-Maori, R. Petit, and M. Cadilhac, “Etude numérique du champ diffracté par un réseau,” C. R. Acad. Sci. Paris 268, 1060–1063 (1969).
  2. M. Nevière and E. Popov, Light Propagation In Periodic Media: Differential Theory and Design (Marcel Dekker, 2003).
  3. E. Popov and M. Nevière, “Grating theory: new equations in Fourier space leading to fast converging result in TM polarization,” J. Opt. Soc. Am. A 17, 1773–1784 (2000).
    [CrossRef]
  4. E. Popov and 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]
  5. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996).
    [CrossRef]
  6. 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]
  7. S. Enoch, E. Popov, and M. Nevière, “3-D photonic crystals dispersion relation: improved convergence using fast Fourier factorization (FFF) method,” Proc. SPIE 4438, 183–190(2001).
    [CrossRef]
  8. S. Enoch, E. Popov, and N. Bonod, “Analysis of the physical origin of surface modes on finite size photonic crystals,” Phys. Rev. B 72, 155101 (2005).
    [CrossRef]
  9. N. Bonod, E. Popov, and M. Nevière, “Factorization of nonlinear Maxwell equations in periodic media: application to the optical Kerr effect,” Opt. Commun. 244, 389–398 (2005).
    [CrossRef]
  10. P. Boyer, E. Popov, M. Nevière, and G. Tayeb, “Diffraction theory in TM polarization: application of the fast Fourier factorization method to cylindrical devices with arbitrary cross section,” J. Opt. Soc. Am. A 21, 2146–2153 (2004).
    [CrossRef]
  11. P. Boyer, E. Popov, M. Nevière, and G. Renversez, “Diffraction theory: application of the fast Fourier factorization to cylindrical devices with arbitrary cross lighted in conical mounting,” J. Opt. Soc. Am. A 23, 1146–1158 (2006).
    [CrossRef]
  12. P. Boyer, G. Renversez, E. Popov, and M. Nevière, “Improved differential method for microstructured optical fibres,” J. Opt. A 9, 728–740 (2007).
    [CrossRef]
  13. B. Stout, M. Nevière, and E. Popov, “Light diffraction by a three-dimensional object: differential theory,” J. Opt. Soc. Am. A 22, 2385–2404 (2005).
    [CrossRef]
  14. B. Stout, M. Nevière, and E. Popov, “Mie scattering by an anisotropic object. part I: homogeneous sphere,” J. Opt. Soc. Am. A 23, 1111–1123 (2006).
    [CrossRef]
  15. B. Stout, M. Nevière, and E. Popov, “Mie scattering by an anisotropic object. part II: arbitrary-shaped object—differential theory,” J. Opt. Soc. Am. A 23, 1124–1134 (2006).
    [CrossRef]
  16. M. Nevière, E. Popov, and P. Boyer, “Diffraction theory of an anisotropic circular cylinder,” J. Opt. Soc. Am. A 23, 1731–1740 (2006).
    [CrossRef]
  17. J. C. Monzon and N. J. Damaskos, “Two-dimensional scattering by homogeneous anisotropic rod,” IEEE Trans. Antennas Propag. 34, 1243–1249 (1986).
    [CrossRef]
  18. J. C. Monzon, “Three-dimensional scattering by an infinite homogeneous anisotropic circular cylinder: a spectral approach,” IEEE Trans. Antennas Propag. 35, 670–682 (1987).
    [CrossRef]

2007

P. Boyer, G. Renversez, E. Popov, and M. Nevière, “Improved differential method for microstructured optical fibres,” J. Opt. A 9, 728–740 (2007).
[CrossRef]

2006

2005

S. Enoch, E. Popov, and N. Bonod, “Analysis of the physical origin of surface modes on finite size photonic crystals,” Phys. Rev. B 72, 155101 (2005).
[CrossRef]

N. Bonod, E. Popov, and M. Nevière, “Factorization of nonlinear Maxwell equations in periodic media: application to the optical Kerr effect,” Opt. Commun. 244, 389–398 (2005).
[CrossRef]

B. Stout, M. Nevière, and E. Popov, “Light diffraction by a three-dimensional object: differential theory,” J. Opt. Soc. Am. A 22, 2385–2404 (2005).
[CrossRef]

2004

2001

E. Popov and 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]

S. Enoch, E. Popov, and M. Nevière, “3-D photonic crystals dispersion relation: improved convergence using fast Fourier factorization (FFF) method,” Proc. SPIE 4438, 183–190(2001).
[CrossRef]

2000

1996

1987

J. C. Monzon, “Three-dimensional scattering by an infinite homogeneous anisotropic circular cylinder: a spectral approach,” IEEE Trans. Antennas Propag. 35, 670–682 (1987).
[CrossRef]

1986

J. C. Monzon and N. J. Damaskos, “Two-dimensional scattering by homogeneous anisotropic rod,” IEEE Trans. Antennas Propag. 34, 1243–1249 (1986).
[CrossRef]

1969

G. Cerutti-Maori, R. Petit, and M. Cadilhac, “Etude numérique du champ diffracté par un réseau,” C. R. Acad. Sci. Paris 268, 1060–1063 (1969).

Bonod, N.

N. Bonod, E. Popov, and M. Nevière, “Factorization of nonlinear Maxwell equations in periodic media: application to the optical Kerr effect,” Opt. Commun. 244, 389–398 (2005).
[CrossRef]

S. Enoch, E. Popov, and N. Bonod, “Analysis of the physical origin of surface modes on finite size photonic crystals,” Phys. Rev. B 72, 155101 (2005).
[CrossRef]

Boyer, P.

Cadilhac, M.

G. Cerutti-Maori, R. Petit, and M. Cadilhac, “Etude numérique du champ diffracté par un réseau,” C. R. Acad. Sci. Paris 268, 1060–1063 (1969).

Cerutti-Maori, G.

G. Cerutti-Maori, R. Petit, and M. Cadilhac, “Etude numérique du champ diffracté par un réseau,” C. R. Acad. Sci. Paris 268, 1060–1063 (1969).

Damaskos, N. J.

J. C. Monzon and N. J. Damaskos, “Two-dimensional scattering by homogeneous anisotropic rod,” IEEE Trans. Antennas Propag. 34, 1243–1249 (1986).
[CrossRef]

Enoch, S.

S. Enoch, E. Popov, and N. Bonod, “Analysis of the physical origin of surface modes on finite size photonic crystals,” Phys. Rev. B 72, 155101 (2005).
[CrossRef]

S. Enoch, E. Popov, and M. Nevière, “3-D photonic crystals dispersion relation: improved convergence using fast Fourier factorization (FFF) method,” Proc. SPIE 4438, 183–190(2001).
[CrossRef]

Li, L.

Monzon, J. C.

J. C. Monzon, “Three-dimensional scattering by an infinite homogeneous anisotropic circular cylinder: a spectral approach,” IEEE Trans. Antennas Propag. 35, 670–682 (1987).
[CrossRef]

J. C. Monzon and N. J. Damaskos, “Two-dimensional scattering by homogeneous anisotropic rod,” IEEE Trans. Antennas Propag. 34, 1243–1249 (1986).
[CrossRef]

Nevière, M.

P. Boyer, G. Renversez, E. Popov, and M. Nevière, “Improved differential method for microstructured optical fibres,” J. Opt. A 9, 728–740 (2007).
[CrossRef]

M. Nevière, E. Popov, and P. Boyer, “Diffraction theory of an anisotropic circular cylinder,” J. Opt. Soc. Am. A 23, 1731–1740 (2006).
[CrossRef]

B. Stout, M. Nevière, and E. Popov, “Mie scattering by an anisotropic object. part II: arbitrary-shaped object—differential theory,” J. Opt. Soc. Am. A 23, 1124–1134 (2006).
[CrossRef]

B. Stout, M. Nevière, and E. Popov, “Mie scattering by an anisotropic object. part I: homogeneous sphere,” J. Opt. Soc. Am. A 23, 1111–1123 (2006).
[CrossRef]

P. Boyer, E. Popov, M. Nevière, and G. Renversez, “Diffraction theory: application of the fast Fourier factorization to cylindrical devices with arbitrary cross lighted in conical mounting,” J. Opt. Soc. Am. A 23, 1146–1158 (2006).
[CrossRef]

N. Bonod, E. Popov, and M. Nevière, “Factorization of nonlinear Maxwell equations in periodic media: application to the optical Kerr effect,” Opt. Commun. 244, 389–398 (2005).
[CrossRef]

B. Stout, M. Nevière, and E. Popov, “Light diffraction by a three-dimensional object: differential theory,” J. Opt. Soc. Am. A 22, 2385–2404 (2005).
[CrossRef]

P. Boyer, E. Popov, M. Nevière, and G. Tayeb, “Diffraction theory in TM polarization: application of the fast Fourier factorization method to cylindrical devices with arbitrary cross section,” J. Opt. Soc. Am. A 21, 2146–2153 (2004).
[CrossRef]

S. Enoch, E. Popov, and M. Nevière, “3-D photonic crystals dispersion relation: improved convergence using fast Fourier factorization (FFF) method,” Proc. SPIE 4438, 183–190(2001).
[CrossRef]

E. Popov and 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]

E. Popov and M. Nevière, “Grating theory: new equations in Fourier space leading to fast converging result in TM polarization,” J. Opt. Soc. Am. A 17, 1773–1784 (2000).
[CrossRef]

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

Petit, R.

G. Cerutti-Maori, R. Petit, and M. Cadilhac, “Etude numérique du champ diffracté par un réseau,” C. R. Acad. Sci. Paris 268, 1060–1063 (1969).

Popov, E.

P. Boyer, G. Renversez, E. Popov, and M. Nevière, “Improved differential method for microstructured optical fibres,” J. Opt. A 9, 728–740 (2007).
[CrossRef]

M. Nevière, E. Popov, and P. Boyer, “Diffraction theory of an anisotropic circular cylinder,” J. Opt. Soc. Am. A 23, 1731–1740 (2006).
[CrossRef]

B. Stout, M. Nevière, and E. Popov, “Mie scattering by an anisotropic object. part II: arbitrary-shaped object—differential theory,” J. Opt. Soc. Am. A 23, 1124–1134 (2006).
[CrossRef]

B. Stout, M. Nevière, and E. Popov, “Mie scattering by an anisotropic object. part I: homogeneous sphere,” J. Opt. Soc. Am. A 23, 1111–1123 (2006).
[CrossRef]

P. Boyer, E. Popov, M. Nevière, and G. Renversez, “Diffraction theory: application of the fast Fourier factorization to cylindrical devices with arbitrary cross lighted in conical mounting,” J. Opt. Soc. Am. A 23, 1146–1158 (2006).
[CrossRef]

S. Enoch, E. Popov, and N. Bonod, “Analysis of the physical origin of surface modes on finite size photonic crystals,” Phys. Rev. B 72, 155101 (2005).
[CrossRef]

N. Bonod, E. Popov, and M. Nevière, “Factorization of nonlinear Maxwell equations in periodic media: application to the optical Kerr effect,” Opt. Commun. 244, 389–398 (2005).
[CrossRef]

B. Stout, M. Nevière, and E. Popov, “Light diffraction by a three-dimensional object: differential theory,” J. Opt. Soc. Am. A 22, 2385–2404 (2005).
[CrossRef]

P. Boyer, E. Popov, M. Nevière, and G. Tayeb, “Diffraction theory in TM polarization: application of the fast Fourier factorization method to cylindrical devices with arbitrary cross section,” J. Opt. Soc. Am. A 21, 2146–2153 (2004).
[CrossRef]

S. Enoch, E. Popov, and M. Nevière, “3-D photonic crystals dispersion relation: improved convergence using fast Fourier factorization (FFF) method,” Proc. SPIE 4438, 183–190(2001).
[CrossRef]

E. Popov and 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]

E. Popov and M. Nevière, “Grating theory: new equations in Fourier space leading to fast converging result in TM polarization,” J. Opt. Soc. Am. A 17, 1773–1784 (2000).
[CrossRef]

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

Renversez, G.

Stout, B.

Tayeb, G.

C. R. Acad. Sci. Paris

G. Cerutti-Maori, R. Petit, and M. Cadilhac, “Etude numérique du champ diffracté par un réseau,” C. R. Acad. Sci. Paris 268, 1060–1063 (1969).

IEEE Trans. Antennas Propag.

J. C. Monzon and N. J. Damaskos, “Two-dimensional scattering by homogeneous anisotropic rod,” IEEE Trans. Antennas Propag. 34, 1243–1249 (1986).
[CrossRef]

J. C. Monzon, “Three-dimensional scattering by an infinite homogeneous anisotropic circular cylinder: a spectral approach,” IEEE Trans. Antennas Propag. 35, 670–682 (1987).
[CrossRef]

J. Opt. A

P. Boyer, G. Renversez, E. Popov, and M. Nevière, “Improved differential method for microstructured optical fibres,” J. Opt. A 9, 728–740 (2007).
[CrossRef]

J. Opt. Soc. Am. A

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]

L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996).
[CrossRef]

E. Popov and M. Nevière, “Grating theory: new equations in Fourier space leading to fast converging result in TM polarization,” J. Opt. Soc. Am. A 17, 1773–1784 (2000).
[CrossRef]

E. Popov and 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]

P. Boyer, E. Popov, M. Nevière, and G. Tayeb, “Diffraction theory in TM polarization: application of the fast Fourier factorization method to cylindrical devices with arbitrary cross section,” J. Opt. Soc. Am. A 21, 2146–2153 (2004).
[CrossRef]

B. Stout, M. Nevière, and E. Popov, “Light diffraction by a three-dimensional object: differential theory,” J. Opt. Soc. Am. A 22, 2385–2404 (2005).
[CrossRef]

B. Stout, M. Nevière, and E. Popov, “Mie scattering by an anisotropic object. part I: homogeneous sphere,” J. Opt. Soc. Am. A 23, 1111–1123 (2006).
[CrossRef]

B. Stout, M. Nevière, and E. Popov, “Mie scattering by an anisotropic object. part II: arbitrary-shaped object—differential theory,” J. Opt. Soc. Am. A 23, 1124–1134 (2006).
[CrossRef]

P. Boyer, E. Popov, M. Nevière, and G. Renversez, “Diffraction theory: application of the fast Fourier factorization to cylindrical devices with arbitrary cross lighted in conical mounting,” J. Opt. Soc. Am. A 23, 1146–1158 (2006).
[CrossRef]

M. Nevière, E. Popov, and P. Boyer, “Diffraction theory of an anisotropic circular cylinder,” J. Opt. Soc. Am. A 23, 1731–1740 (2006).
[CrossRef]

Opt. Commun.

N. Bonod, E. Popov, and M. Nevière, “Factorization of nonlinear Maxwell equations in periodic media: application to the optical Kerr effect,” Opt. Commun. 244, 389–398 (2005).
[CrossRef]

Phys. Rev. B

S. Enoch, E. Popov, and N. Bonod, “Analysis of the physical origin of surface modes on finite size photonic crystals,” Phys. Rev. B 72, 155101 (2005).
[CrossRef]

Proc. SPIE

S. Enoch, E. Popov, and M. Nevière, “3-D photonic crystals dispersion relation: improved convergence using fast Fourier factorization (FFF) method,” Proc. SPIE 4438, 183–190(2001).
[CrossRef]

Other

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

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Figures (9)

Fig. 1.
Fig. 1.

Schematic representation of the two kinds of anisotropic cylinders studied.

Fig. 2.
Fig. 2.

Configuration Cin: anisotropic cylinder containing the origin.

Fig. 3.
Fig. 3.

Configuration Cout: anisotropic cylinder not containing the origin.

Fig. 4.
Fig. 4.

Anisotropic circular cylinder not containing the origin.

Fig. 5.
Fig. 5.

Comparison between DCS calculated with CAM and ADM for a biaxial medium. (a) DCS versus angle for electric and magnetic fields computed with ADM and CAM and (b) convergence tests of σH(270°) and σE(301°) versus truncation order N.

Fig. 6.
Fig. 6.

Diffracted field maps for an anisotropic circular cylinder computed with ADM for N=60. τ is the relative error between CAM and ADM field maps (average relative error computed at each point of field maps). (a) |Ez|·τ=0.17% and (b) |Hz|·τ=0.022%.

Fig. 7.
Fig. 7.

Comparison between DCS calculated with CAM and ADM for a uniaxial medium. (a) DCS versus angle for electric and magnetic fields computed with ADM and CAM and (b) convergence tests of σH(270°) and σE(321°) versus truncation order N.

Fig. 8.
Fig. 8.

Study of DCS for anisotropic elliptical cylinders with different values of the semi-major axis a. The semi-minor axis b is fixed at 1. (a) DCS for anisotropic circular and elliptical cylinders and (b) evolution of σH(270°) and σH(90°) versus a. The small boxes show the value for the circular cylinder.

Fig. 9.
Fig. 9.

σH(270o) versus NADM for different values of NCAM and for a=1.1.

Equations (24)

Equations on this page are rendered with MathJax. Learn more.

ϵ¯¯cyl=(ϵ¯¯xxϵ¯¯xyϵ¯¯xzϵ¯¯yxϵ¯¯yyϵ¯¯yzϵ¯¯zxϵ¯¯zyϵ¯¯zz),
ϵ˜˜cyl=Rϵ¯¯cylRT,
R=(er·exer·eyer·ezeθ·exeθ·eyeθ·ezez·exez·eyez·ez)=(cosθsinθ0sinθcosθ0001).
γ0=kextcosθinc,
[F(r)]=Ψ˜(r)[A˜]
Ψ˜(r)=(Ψ˜11(r)Ψ˜12(r)Ψ˜21(r)Ψ˜22(r)Ψ˜31(r)Ψ˜32(r)Ψ˜41(r)Ψ˜42(r)),
{Ψ˜1j(r)}n,ν=nrkj,ν,ρ(αn,j,νβn,j,νkj,ν,ρ)Jn(kj,ν,ρr)βn,j,νkj,ν,ρJn+1(kj,ν,ρr),
{Ψ˜2j(r)}n,ν=γn,j,νJn(kj,ν,ρr),
{Ψ˜3j(r)}n,ν=1iωμ0[nrkj,ν,ρ(kj,ν,ργn,j,ν+βn,j,νγ0kj,ν,ραn,j,νγ0)Jn(kj,ν,ρr)+(αn,j,νγ0kj,ν,ργn,j,ν)Jn+1(kj,ν,ρr)],
{Ψ˜4j(r)}n,ν=βn,j,νiωμ0Jn(kj,ν,ρr)
[F(r)]=Ψ(ext)(r)[V(ext)(r)],
[V(ext)(R)]=T(ani)[A˜],
T(ani)={Ψ(ext)(R)}1Ψ˜(R),
Cϵ=1δ(Nrϵrθ+NθϵθθNrNr(Nrϵrz+Nθϵθz)Nrϵrr+NθϵθrNθNθ(Nrϵrz+Nθϵθz)00δ),
δ=Nr2ϵrr+Nθ2ϵθθ+NrNθ(ϵθr+ϵrθ),
T(s)={Ψ(ext)(rs+1)}1[Finteg(rs+1)]Ψ(ext)(rs),
T(s)={Ψ(ext)(rs+1)}1Ψ(iso)(rs+1)C(iso)(rs,rs+1){Ψ(iso)(rs)}1Ψ(ext)(rs)
C(iso)(rs,rs+1)=(J0000J0000H0000H),
(J)n,m=Jn(kt,isors+1)Jn(kt,isors)δnm,
(H)n,m=Hn+(kt,isors+1)Hn+(kt,isors)δnm.
([B(ext)(rs)][A˜])=S(s)[A(ext)(rs)],
Z(s)=[T11(s)+T12(s)S1(s)]1,
S2(s+1)=S2(s)Z(s),
S1(s+1)=[T21(s)+T22(s)S1(s)]Z(s).

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