Abstract

We present a modal method for the computation of eigenmodes of cylindrical structures with arbitrary cross sections. These modes are found as eigenvectors of a matrix eigenvalue equation that is obtained by introducing a new coordinate system that takes into account the profile of the cross section. We show that the use of Hertz potentials is suitable for the derivation of this eigenvalue equation and that the modal method based on Gegenbauer expansion (MMGE) is an efficient tool for the numerical solution of this equation. Results are successfully compared for both perfectly conducting and dielectric structures. A complex coordinate version of the MMGE is introduced to solve the dielectric case.

© 2014 Optical Society of America

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  1. K. Edee, “Modal method based on subsectional Gegenbauer polynomial expansion for lamellar grating,” J. Opt. Soc. Am. A 28, 2006–2013 (2011).
    [Crossref]
  2. K. Edee, I. Fenniche, G. Granet, and B. Guizal, “Modal method based on subsectional Gegenbauer polynomial expansion for lamellar gratings: Weithing function, convergence and stability,” Prog. Electromagn. Res. 133, 17–35 (2013).
    [Crossref]
  3. K. Edee and B. Guizal, “Modal method based on subsectional Gegenbauer polynomial expansion for nonperiodic structures: complex coordinates implementation,” J. Opt. Soc. Am. A 30, 631–639 (2013).
    [Crossref]
  4. M. Nevière and E. Popov, “New theoretical method for electromagnetic wave diffraction by a metallic or dielectric cylinder, bare coated with a thin dielectric layer,” J. Electromagn. Waves. Appl. 12, 1265–1296 (1998).
    [Crossref]
  5. B. Sieger, “Die beugung einer ebenen elektrischen welle an einem schirm von elliptischem querscnitt,” Ann. Phys. 332, 626–664 (1908).
    [Crossref]
  6. P. M. Morse and P. J. Rubenstein, “The diffraction of waves by ribbons and by slits,” Phys. Rev. 54, 895–898 (1938).
    [Crossref]
  7. C. Yeh, “The diffraction of waves by a penetrable ribbon,” J. Math. Phys. 4, 65–71 (1963).
    [Crossref]
  8. C. Yeh, “Scattering of obliquely incident light waves by elliptical fibers,” J. Opt. Soc. Am. 54, 1227–1281 (1964).
    [Crossref]
  9. J.-T. Zhange and Y.-L. Li, “A new research approach of electromagnetic theory and its applications,” Indian J. Radio Space Phys. 35, 249–252 (2008).
  10. Y.-L. Li, J.-Y. Huang, M.-J. Wang, and J. T. Zhang, “Scattering field for the ellipsoidal targets irradiated by an electromagnetic wave with arbitrary polarizing and propagating direction,” Prog. Electromagn. Res. Lett. 1, 221–235 (2008).
    [Crossref]
  11. Y.-L. Li, M.-J. Wang, and G.-F. Tang, “The scattering from an elliptic cylinder irradiated by an electromagnetic wave with arbitrary direction and polarization,” Prog. Electromagn. Res. Lett. 5, 137–149 (2008).
    [Crossref]
  12. L. A. Ferrari, “Scale transformation of Maxwell’s equations and scattering by an elliptic cylinder,” J. Opt. Soc. Am. A 28, 1285–1290 (2011).
    [Crossref]
  13. G. Granet, K. Edee, and D. Felbacq, “Scattering of a plane wave by rough surfaces: a new curvilinear coordinate system based approach,” Prog. Electromagn. Res. 41, 235–250 (2003).
  14. K. Edee, B. Guizal, G. Cranet, and A. Moreau, “Beam implementation in a nonorthogonal coordinate system: application to the scattering from random rough surfaces,” J. Opt. Soc. Am. A 25, 796–804 (2008).
    [Crossref]
  15. K. Edee, J.-P. Plumey, and J. Chandezon, “On the Rayleigh–Fourier method and the Chandezon method: comparative study,” Opt. Commun. 286, 34–41 (2013).
    [Crossref]
  16. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1970), pp. 771–792.

2013 (3)

K. Edee, I. Fenniche, G. Granet, and B. Guizal, “Modal method based on subsectional Gegenbauer polynomial expansion for lamellar gratings: Weithing function, convergence and stability,” Prog. Electromagn. Res. 133, 17–35 (2013).
[Crossref]

K. Edee and B. Guizal, “Modal method based on subsectional Gegenbauer polynomial expansion for nonperiodic structures: complex coordinates implementation,” J. Opt. Soc. Am. A 30, 631–639 (2013).
[Crossref]

K. Edee, J.-P. Plumey, and J. Chandezon, “On the Rayleigh–Fourier method and the Chandezon method: comparative study,” Opt. Commun. 286, 34–41 (2013).
[Crossref]

2011 (2)

2008 (4)

J.-T. Zhange and Y.-L. Li, “A new research approach of electromagnetic theory and its applications,” Indian J. Radio Space Phys. 35, 249–252 (2008).

Y.-L. Li, J.-Y. Huang, M.-J. Wang, and J. T. Zhang, “Scattering field for the ellipsoidal targets irradiated by an electromagnetic wave with arbitrary polarizing and propagating direction,” Prog. Electromagn. Res. Lett. 1, 221–235 (2008).
[Crossref]

Y.-L. Li, M.-J. Wang, and G.-F. Tang, “The scattering from an elliptic cylinder irradiated by an electromagnetic wave with arbitrary direction and polarization,” Prog. Electromagn. Res. Lett. 5, 137–149 (2008).
[Crossref]

K. Edee, B. Guizal, G. Cranet, and A. Moreau, “Beam implementation in a nonorthogonal coordinate system: application to the scattering from random rough surfaces,” J. Opt. Soc. Am. A 25, 796–804 (2008).
[Crossref]

2003 (1)

G. Granet, K. Edee, and D. Felbacq, “Scattering of a plane wave by rough surfaces: a new curvilinear coordinate system based approach,” Prog. Electromagn. Res. 41, 235–250 (2003).

1998 (1)

M. Nevière and E. Popov, “New theoretical method for electromagnetic wave diffraction by a metallic or dielectric cylinder, bare coated with a thin dielectric layer,” J. Electromagn. Waves. Appl. 12, 1265–1296 (1998).
[Crossref]

1964 (1)

1963 (1)

C. Yeh, “The diffraction of waves by a penetrable ribbon,” J. Math. Phys. 4, 65–71 (1963).
[Crossref]

1938 (1)

P. M. Morse and P. J. Rubenstein, “The diffraction of waves by ribbons and by slits,” Phys. Rev. 54, 895–898 (1938).
[Crossref]

1908 (1)

B. Sieger, “Die beugung einer ebenen elektrischen welle an einem schirm von elliptischem querscnitt,” Ann. Phys. 332, 626–664 (1908).
[Crossref]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1970), pp. 771–792.

Chandezon, J.

K. Edee, J.-P. Plumey, and J. Chandezon, “On the Rayleigh–Fourier method and the Chandezon method: comparative study,” Opt. Commun. 286, 34–41 (2013).
[Crossref]

Cranet, G.

Edee, K.

K. Edee and B. Guizal, “Modal method based on subsectional Gegenbauer polynomial expansion for nonperiodic structures: complex coordinates implementation,” J. Opt. Soc. Am. A 30, 631–639 (2013).
[Crossref]

K. Edee, J.-P. Plumey, and J. Chandezon, “On the Rayleigh–Fourier method and the Chandezon method: comparative study,” Opt. Commun. 286, 34–41 (2013).
[Crossref]

K. Edee, I. Fenniche, G. Granet, and B. Guizal, “Modal method based on subsectional Gegenbauer polynomial expansion for lamellar gratings: Weithing function, convergence and stability,” Prog. Electromagn. Res. 133, 17–35 (2013).
[Crossref]

K. Edee, “Modal method based on subsectional Gegenbauer polynomial expansion for lamellar grating,” J. Opt. Soc. Am. A 28, 2006–2013 (2011).
[Crossref]

K. Edee, B. Guizal, G. Cranet, and A. Moreau, “Beam implementation in a nonorthogonal coordinate system: application to the scattering from random rough surfaces,” J. Opt. Soc. Am. A 25, 796–804 (2008).
[Crossref]

G. Granet, K. Edee, and D. Felbacq, “Scattering of a plane wave by rough surfaces: a new curvilinear coordinate system based approach,” Prog. Electromagn. Res. 41, 235–250 (2003).

Felbacq, D.

G. Granet, K. Edee, and D. Felbacq, “Scattering of a plane wave by rough surfaces: a new curvilinear coordinate system based approach,” Prog. Electromagn. Res. 41, 235–250 (2003).

Fenniche, I.

K. Edee, I. Fenniche, G. Granet, and B. Guizal, “Modal method based on subsectional Gegenbauer polynomial expansion for lamellar gratings: Weithing function, convergence and stability,” Prog. Electromagn. Res. 133, 17–35 (2013).
[Crossref]

Ferrari, L. A.

Granet, G.

K. Edee, I. Fenniche, G. Granet, and B. Guizal, “Modal method based on subsectional Gegenbauer polynomial expansion for lamellar gratings: Weithing function, convergence and stability,” Prog. Electromagn. Res. 133, 17–35 (2013).
[Crossref]

G. Granet, K. Edee, and D. Felbacq, “Scattering of a plane wave by rough surfaces: a new curvilinear coordinate system based approach,” Prog. Electromagn. Res. 41, 235–250 (2003).

Guizal, B.

Huang, J.-Y.

Y.-L. Li, J.-Y. Huang, M.-J. Wang, and J. T. Zhang, “Scattering field for the ellipsoidal targets irradiated by an electromagnetic wave with arbitrary polarizing and propagating direction,” Prog. Electromagn. Res. Lett. 1, 221–235 (2008).
[Crossref]

Li, Y.-L.

J.-T. Zhange and Y.-L. Li, “A new research approach of electromagnetic theory and its applications,” Indian J. Radio Space Phys. 35, 249–252 (2008).

Y.-L. Li, J.-Y. Huang, M.-J. Wang, and J. T. Zhang, “Scattering field for the ellipsoidal targets irradiated by an electromagnetic wave with arbitrary polarizing and propagating direction,” Prog. Electromagn. Res. Lett. 1, 221–235 (2008).
[Crossref]

Y.-L. Li, M.-J. Wang, and G.-F. Tang, “The scattering from an elliptic cylinder irradiated by an electromagnetic wave with arbitrary direction and polarization,” Prog. Electromagn. Res. Lett. 5, 137–149 (2008).
[Crossref]

Moreau, A.

Morse, P. M.

P. M. Morse and P. J. Rubenstein, “The diffraction of waves by ribbons and by slits,” Phys. Rev. 54, 895–898 (1938).
[Crossref]

Nevière, M.

M. Nevière and E. Popov, “New theoretical method for electromagnetic wave diffraction by a metallic or dielectric cylinder, bare coated with a thin dielectric layer,” J. Electromagn. Waves. Appl. 12, 1265–1296 (1998).
[Crossref]

Plumey, J.-P.

K. Edee, J.-P. Plumey, and J. Chandezon, “On the Rayleigh–Fourier method and the Chandezon method: comparative study,” Opt. Commun. 286, 34–41 (2013).
[Crossref]

Popov, E.

M. Nevière and E. Popov, “New theoretical method for electromagnetic wave diffraction by a metallic or dielectric cylinder, bare coated with a thin dielectric layer,” J. Electromagn. Waves. Appl. 12, 1265–1296 (1998).
[Crossref]

Rubenstein, P. J.

P. M. Morse and P. J. Rubenstein, “The diffraction of waves by ribbons and by slits,” Phys. Rev. 54, 895–898 (1938).
[Crossref]

Sieger, B.

B. Sieger, “Die beugung einer ebenen elektrischen welle an einem schirm von elliptischem querscnitt,” Ann. Phys. 332, 626–664 (1908).
[Crossref]

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1970), pp. 771–792.

Tang, G.-F.

Y.-L. Li, M.-J. Wang, and G.-F. Tang, “The scattering from an elliptic cylinder irradiated by an electromagnetic wave with arbitrary direction and polarization,” Prog. Electromagn. Res. Lett. 5, 137–149 (2008).
[Crossref]

Wang, M.-J.

Y.-L. Li, M.-J. Wang, and G.-F. Tang, “The scattering from an elliptic cylinder irradiated by an electromagnetic wave with arbitrary direction and polarization,” Prog. Electromagn. Res. Lett. 5, 137–149 (2008).
[Crossref]

Y.-L. Li, J.-Y. Huang, M.-J. Wang, and J. T. Zhang, “Scattering field for the ellipsoidal targets irradiated by an electromagnetic wave with arbitrary polarizing and propagating direction,” Prog. Electromagn. Res. Lett. 1, 221–235 (2008).
[Crossref]

Yeh, C.

C. Yeh, “Scattering of obliquely incident light waves by elliptical fibers,” J. Opt. Soc. Am. 54, 1227–1281 (1964).
[Crossref]

C. Yeh, “The diffraction of waves by a penetrable ribbon,” J. Math. Phys. 4, 65–71 (1963).
[Crossref]

Zhang, J. T.

Y.-L. Li, J.-Y. Huang, M.-J. Wang, and J. T. Zhang, “Scattering field for the ellipsoidal targets irradiated by an electromagnetic wave with arbitrary polarizing and propagating direction,” Prog. Electromagn. Res. Lett. 1, 221–235 (2008).
[Crossref]

Zhange, J.-T.

J.-T. Zhange and Y.-L. Li, “A new research approach of electromagnetic theory and its applications,” Indian J. Radio Space Phys. 35, 249–252 (2008).

Ann. Phys. (1)

B. Sieger, “Die beugung einer ebenen elektrischen welle an einem schirm von elliptischem querscnitt,” Ann. Phys. 332, 626–664 (1908).
[Crossref]

Indian J. Radio Space Phys. (1)

J.-T. Zhange and Y.-L. Li, “A new research approach of electromagnetic theory and its applications,” Indian J. Radio Space Phys. 35, 249–252 (2008).

J. Electromagn. Waves. Appl. (1)

M. Nevière and E. Popov, “New theoretical method for electromagnetic wave diffraction by a metallic or dielectric cylinder, bare coated with a thin dielectric layer,” J. Electromagn. Waves. Appl. 12, 1265–1296 (1998).
[Crossref]

J. Math. Phys. (1)

C. Yeh, “The diffraction of waves by a penetrable ribbon,” J. Math. Phys. 4, 65–71 (1963).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (4)

Opt. Commun. (1)

K. Edee, J.-P. Plumey, and J. Chandezon, “On the Rayleigh–Fourier method and the Chandezon method: comparative study,” Opt. Commun. 286, 34–41 (2013).
[Crossref]

Phys. Rev. (1)

P. M. Morse and P. J. Rubenstein, “The diffraction of waves by ribbons and by slits,” Phys. Rev. 54, 895–898 (1938).
[Crossref]

Prog. Electromagn. Res. (2)

K. Edee, I. Fenniche, G. Granet, and B. Guizal, “Modal method based on subsectional Gegenbauer polynomial expansion for lamellar gratings: Weithing function, convergence and stability,” Prog. Electromagn. Res. 133, 17–35 (2013).
[Crossref]

G. Granet, K. Edee, and D. Felbacq, “Scattering of a plane wave by rough surfaces: a new curvilinear coordinate system based approach,” Prog. Electromagn. Res. 41, 235–250 (2003).

Prog. Electromagn. Res. Lett. (2)

Y.-L. Li, J.-Y. Huang, M.-J. Wang, and J. T. Zhang, “Scattering field for the ellipsoidal targets irradiated by an electromagnetic wave with arbitrary polarizing and propagating direction,” Prog. Electromagn. Res. Lett. 1, 221–235 (2008).
[Crossref]

Y.-L. Li, M.-J. Wang, and G.-F. Tang, “The scattering from an elliptic cylinder irradiated by an electromagnetic wave with arbitrary direction and polarization,” Prog. Electromagn. Res. Lett. 5, 137–149 (2008).
[Crossref]

Other (1)

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1970), pp. 771–792.

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Figures (6)

Fig. 1.
Fig. 1.

Geometry of the problem. Example of the coordinate transform of the profile described by P(θ) into concentric circles.

Fig. 2.
Fig. 2.

Relative error in unmN and unmN, that is, the number of digits in common with the reference values unmref and unmref in Table 1 (θ=0).

Fig. 3.
Fig. 3.

Relative error in unmN and unmN, that is, the number of digits in common with the reference values unmref and unmref in Table 2 (θ=2i).

Fig. 4.
Fig. 4.

Comparison of the zeroth-order Bessel functions and the eigenfunctions computed from the eigenvalue problem for different numbers of polynomials.

Fig. 5.
Fig. 5.

Spatial distribution of the Ez [Fig. (5a)] and Hz [Fig. (5b)] components for γ1/k0=1.14623 in the case of an elliptical dielectric waveguide with radii of ra=0.5λ and rb=0.45λ.

Fig. 6.
Fig. 6.

Spatial distribution of the Ez [Fig. (6a)] and Hz [Fig. (6b)] for γ2/k0=1.242352 in the case of an elliptical dielectric waveguide with radii of ra=0.5λ and rb=0.45λ.

Tables (5)

Tables Icon

Table 1. First Five Zeros un0ref of the Zeroth-order Bessel Function and Its Numerical Derivative un0ref Computed from the Eigenvalues γ and for N=30a

Tables Icon

Table 2. First Five Zeros un2ref of the Second-order Bessel Function and Its Numerical Derivative un2ref Computed from the Eigenvalues γ and for N=30a

Tables Icon

Table 3. Influence of the PMLs Functiona

Tables Icon

Table 4. Convergence of the Effective Index γ1/k0=1.14623 of an Elliptical Dielectric Waveguide (Radii of ra=0.5λ and rb=0.45λ) with Respect to Both N and Mmax=2M+1a

Tables Icon

Table 5. Convergence of the Effective Index γ1/k0=1.24235 of an Elliptical Dielectric Waveguide (Radii of ra=0.5λ and rb=0.45λ) with Respect to Both N and Mmax=2M+1a

Equations (45)

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

{ξijkjHk=iωεggijEj,ξijkjEk=iωμggijHji,j,k=1,2,3,
iggijjF+k2gF=0,
{(32+k2g33)F1=(13+k2g13)F3kg(g211+g222+g233)G3(32+k2g33)F2=(23+k2g23)F3+kg(g111+g122+g133)G3,
{E3=(32+k2g33)ΠeH3=(32+k2g33)Πh.
(iggijj+k2g)Πe,h=0,
{E1=(13+k2g13)ΠeiZkg(g211+g222+g233)ΠhE2=(23+k2g23)Πe+iZkg(g111+g122+g133)ΠhH1=(13+k2g13)Πh+ikZg(g211+g222+g233)ΠeH2=(23+k2g23)ΠhikZg(g111+g122+g133)Πe.
{x1=u=ρ0P(x2)ρx2=x2=θx3=x3=z.
gij=[f2(x2)f˙(x2)f(x2)x10f˙(x2)f(x2)x1(f(x2)x1)2+(f˙(x2)x1)20001],
ggij=[x1(1+a˙2(x2))a˙(x2)0a˙(x2)1x1000f(x2)f(x2)x1],
[(1+a˙a˙)x11x11(2a˙+a˙2)x11+22+k2ffx1x1]Πe,h=ffx1x132Πe,h,
ΔTΠe,h+32Πe,h=0.
{E3=(ΔT+k2)ΠeH3=(ΔT+k2)ΠhE2=2Πe+ikZ(˜1+a˙2˜1a˙2)ΠhH2=2ΠhikZ(˜1+a˙2˜1a˙2)Πe,,
u˜(u)=(χiη)(uu1)+u1,u[u1,u2].
|Θ=m=MMΘm|em,
|Πj(z)=eiγzn=1Nm=MMΠnmj|BnmΛ,
R=[G2][P]2,
L=[DG01D][Iθ+a˙2][D][a˙2Dθ+Dθa˙2]+[G0][Dθ]2+k2ε[G2][P]2,
[Lj0000Lj0000Ij0000Ij][ΠejΠhjΠejΠhj]=γ[00Rj0000RjIj0000Ij00][ΠejΠhjΠejΠhj],
[Lleft100Lleft2][Φ[1:N],[12M+1]1Φ[1:N],[12M+1]2]=γ[Lright100Lright2][Φ[1:N],[12M+1]1Φ[1:N],[12M+1]2],
Φj=[Πej,Πhj,Πej,Πhj]t,
{3E3=(ΔT+k2g33)Πe3H3=(ΔT+k2g33)Πm3E2=2ΔTΠe+ikZ(˜1+a˙2˜1a˙2)Πm3H2=2ΔTΠmikZ(˜1+a˙2˜1a˙2)Πe.
[Φ[N],[12M+1]1Φ[N1,N],[12M+1]2]=[T][Φ[1:N1],[12M+1]1Φ[1:N2],[12M+1]2].
[Φ[1:N],[12M+1]1Φ[1:N],[12M+1]2]=[C][Φ[1:N1],[12M+1]1Φ[1:N2],[12M+1]2].
[Lleft][C][Φ[1:N1],[12M+1]1Φ[1:N2],[12M+1]2]=γ[Lright][C][Φ[1:N1],[12M+1]1Φ[1:N2],[12M+1]2].
Πnm(u,θ,z)=Jm(kρnmu)eimθeiγnmz,
(kρnm)2+γnm2=k2.
unmNorunmN=ρ0k2(γnmN)2.
J0(unmrefρ0u)
ΠnmN(u)=p=1NΠpnmCpΛ(u),
f(θ)=b1e2cos2θ,
e2=1rb2ra2.
ζdCnΛ(ζ)dζ=dCn1Λ(ζ)dζ+nCnΛ(ζ),
|ζdCnΛdζ=|dCn1Λdζ+n|CnΛ.
CmΛ|ζdCnΛdζ=CmΛ|dCn1Λdζ+nCmΛ|CnΛ.
nCnΛ(ζ)=2(n+Λ1)ζCn1Λ(ζ)(n+2Λ2)Cn2Λ(ζ),
|ζCnΛ=(n+1)2(n+Λ)|Cn+1Λ+(n1+2Λ)2(n+Λ)|Cn1Λ.
CmΛ|ζCnΛ=(n+1)2(n+Λ)CmΛ|Cn+1Λ+(n1+2Λ)2(n+Λ)CmΛ|Cn1Λ,
CmΛ|ζ2CnΛ=(n+1)2(n+Λ)CmΛ|ζCn+1Λ+(n1+2Λ)2(n+Λ)CmΛ|ζCn1Λ.
u(ζ)=(u2u12)(ζ+u2+u1u2u1),
u˜(u)=(χiη)(uu1)+u1,
u˜(ζ)=A(ζ+B),
A=(χiη)(u2u12),
B=u2+u1u2u12u1u2u1+2u1(χiη)(u2u1).
u˜ddu˜=[ζ+B]ddζ,
{CmΛ|u˜dCnΛdu˜[u˜(u1),u˜(u2)]=A[CmΛ|ζdCnΛdζ+BCmΛ|dCnΛdζ]CmΛ|u˜CnΛ[u˜(u1),u˜(u2)]=A2[CmΛ|ζCnΛ+BCmΛ|CnΛ]CmΛ|u˜2CnΛ[u˜(u1),u˜(u2)]=A3[CmΛ|ζ2CnΛ+2BCmΛ|ζCnΛ+B2CmΛ|CnΛ].

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