Abstract

Rigorous numerical techniques for treating light-scattering problems with one-dimensional rough reentrant surfaces have been developed. We present a modification of one of these techniques to calculate modes for a waveguide with an arbitrarily cross-sectional structure that represents, to our knowledge, a unique work of this kind. We have applied this technique to a regular polygonal structure and also to a triadic Koch fractal structure. The results obtained reproduce the well-known situations of square and circular cross.

© 2001 Optical Society of America

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References

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  1. E. R. Méndez, K. A. O’Donnell, “Observation of depolarization and backscattering enhancement in light scattering from Gaussian random surfaces,” Opt. Commun. 61, 91–95 (1987).
    [CrossRef]
  2. K. A. O’Donnell, E. R. Méndez, “Experimental study of scattering from characterized random surfaces,” J. Opt. Soc. Am. A 4, 1194–1205 (1987).
    [CrossRef]
  3. R. E. Luna, E. R. Méndez, J. Q. Lu, Z.-H. Gu, “Enhanced backscattering due to total internal-reflection at a dielectric–air interface,” J. Mod. Opt. 42, 257–269 (1995).
    [CrossRef]
  4. R. E. Luna, “Scattering by one-dimensional random rough metallic surfaces in a conical configuration: several polarizations,” Opt. Lett. 21, 1418–1420 (1996).
    [CrossRef] [PubMed]
  5. R. E. Luna, S. E. Acosta-Ortiz, L.-F. Zou, “Mueller matrix for characterization of one-dimensional rough perfectly reflecting surfaces in a conical configuration,” Opt. Lett. 23, 1075–1077 (1998).
    [CrossRef]
  6. M. Nieto-Vesperinas, J. M. Soto-Crespo, “Monte Carlo simulations for scattering of electromagnetic waves from perfectly conductive random rough surface,” Opt. Lett. 12, 979–981 (1987).
    [CrossRef] [PubMed]
  7. A. A. Maradudin, E. R. Méndez, T. Michel, “Backscattering effects in the elastic scattering of p-polarized light from a large-amplitude random metallic grating,” Opt. Lett. 14, 151–153 (1989).
    [CrossRef] [PubMed]
  8. S. Savaidis, P. Frangos, D. L. Jaggard, K. Hizanidis, “Scattering from fractally corrugated surfaces: an exact approach,” Opt. Lett. 20, 2357–2359 (1995).
    [CrossRef] [PubMed]
  9. C. J. R. Sheppard, “Scattering by fractal surfaces with an outer scale,” Opt. Commun. 122, 178–188 (1996).
    [CrossRef]
  10. A. Mendoza-Suárez, E. R. Méndez, “Derivation of an impedance boundary condition for a one-dimensional, curved, reentrant surfaces,” Opt. Commun. 134, 241–250 (1997).
    [CrossRef]
  11. A. Mendoza-Suárez, E. R. Méndez, “Light scattering by a reentrant fractal surface,” Appl. Opt. 36, 3521–3531 (1997).
    [CrossRef] [PubMed]
  12. G. Arfken, Mathematical Methods for Physicists (Academic, New York, 1970), pp. 387–390.
  13. A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. 203, 255–307 (1990).
    [CrossRef]
  14. D. J. DiGiovanni, M. H. Muendel, “High power fiber lasers,” Opt. Photon. News 10(1), 26–30 (1999).
    [CrossRef]
  15. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in FORTRAN, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992).

1999 (1)

D. J. DiGiovanni, M. H. Muendel, “High power fiber lasers,” Opt. Photon. News 10(1), 26–30 (1999).
[CrossRef]

1998 (1)

1997 (2)

A. Mendoza-Suárez, E. R. Méndez, “Derivation of an impedance boundary condition for a one-dimensional, curved, reentrant surfaces,” Opt. Commun. 134, 241–250 (1997).
[CrossRef]

A. Mendoza-Suárez, E. R. Méndez, “Light scattering by a reentrant fractal surface,” Appl. Opt. 36, 3521–3531 (1997).
[CrossRef] [PubMed]

1996 (2)

1995 (2)

R. E. Luna, E. R. Méndez, J. Q. Lu, Z.-H. Gu, “Enhanced backscattering due to total internal-reflection at a dielectric–air interface,” J. Mod. Opt. 42, 257–269 (1995).
[CrossRef]

S. Savaidis, P. Frangos, D. L. Jaggard, K. Hizanidis, “Scattering from fractally corrugated surfaces: an exact approach,” Opt. Lett. 20, 2357–2359 (1995).
[CrossRef] [PubMed]

1990 (1)

A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. 203, 255–307 (1990).
[CrossRef]

1989 (1)

1987 (3)

Acosta-Ortiz, S. E.

Arfken, G.

G. Arfken, Mathematical Methods for Physicists (Academic, New York, 1970), pp. 387–390.

DiGiovanni, D. J.

D. J. DiGiovanni, M. H. Muendel, “High power fiber lasers,” Opt. Photon. News 10(1), 26–30 (1999).
[CrossRef]

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in FORTRAN, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992).

Frangos, P.

Gu, Z.-H.

R. E. Luna, E. R. Méndez, J. Q. Lu, Z.-H. Gu, “Enhanced backscattering due to total internal-reflection at a dielectric–air interface,” J. Mod. Opt. 42, 257–269 (1995).
[CrossRef]

Hizanidis, K.

Jaggard, D. L.

Lu, J. Q.

R. E. Luna, E. R. Méndez, J. Q. Lu, Z.-H. Gu, “Enhanced backscattering due to total internal-reflection at a dielectric–air interface,” J. Mod. Opt. 42, 257–269 (1995).
[CrossRef]

Luna, R. E.

Maradudin, A. A.

A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. 203, 255–307 (1990).
[CrossRef]

A. A. Maradudin, E. R. Méndez, T. Michel, “Backscattering effects in the elastic scattering of p-polarized light from a large-amplitude random metallic grating,” Opt. Lett. 14, 151–153 (1989).
[CrossRef] [PubMed]

McGurn, A. R.

A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. 203, 255–307 (1990).
[CrossRef]

Méndez, E. R.

A. Mendoza-Suárez, E. R. Méndez, “Derivation of an impedance boundary condition for a one-dimensional, curved, reentrant surfaces,” Opt. Commun. 134, 241–250 (1997).
[CrossRef]

A. Mendoza-Suárez, E. R. Méndez, “Light scattering by a reentrant fractal surface,” Appl. Opt. 36, 3521–3531 (1997).
[CrossRef] [PubMed]

R. E. Luna, E. R. Méndez, J. Q. Lu, Z.-H. Gu, “Enhanced backscattering due to total internal-reflection at a dielectric–air interface,” J. Mod. Opt. 42, 257–269 (1995).
[CrossRef]

A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. 203, 255–307 (1990).
[CrossRef]

A. A. Maradudin, E. R. Méndez, T. Michel, “Backscattering effects in the elastic scattering of p-polarized light from a large-amplitude random metallic grating,” Opt. Lett. 14, 151–153 (1989).
[CrossRef] [PubMed]

K. A. O’Donnell, E. R. Méndez, “Experimental study of scattering from characterized random surfaces,” J. Opt. Soc. Am. A 4, 1194–1205 (1987).
[CrossRef]

E. R. Méndez, K. A. O’Donnell, “Observation of depolarization and backscattering enhancement in light scattering from Gaussian random surfaces,” Opt. Commun. 61, 91–95 (1987).
[CrossRef]

Mendoza-Suárez, A.

A. Mendoza-Suárez, E. R. Méndez, “Light scattering by a reentrant fractal surface,” Appl. Opt. 36, 3521–3531 (1997).
[CrossRef] [PubMed]

A. Mendoza-Suárez, E. R. Méndez, “Derivation of an impedance boundary condition for a one-dimensional, curved, reentrant surfaces,” Opt. Commun. 134, 241–250 (1997).
[CrossRef]

Michel, T.

A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. 203, 255–307 (1990).
[CrossRef]

A. A. Maradudin, E. R. Méndez, T. Michel, “Backscattering effects in the elastic scattering of p-polarized light from a large-amplitude random metallic grating,” Opt. Lett. 14, 151–153 (1989).
[CrossRef] [PubMed]

Muendel, M. H.

D. J. DiGiovanni, M. H. Muendel, “High power fiber lasers,” Opt. Photon. News 10(1), 26–30 (1999).
[CrossRef]

Nieto-Vesperinas, M.

O’Donnell, K. A.

K. A. O’Donnell, E. R. Méndez, “Experimental study of scattering from characterized random surfaces,” J. Opt. Soc. Am. A 4, 1194–1205 (1987).
[CrossRef]

E. R. Méndez, K. A. O’Donnell, “Observation of depolarization and backscattering enhancement in light scattering from Gaussian random surfaces,” Opt. Commun. 61, 91–95 (1987).
[CrossRef]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in FORTRAN, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992).

Savaidis, S.

Sheppard, C. J. R.

C. J. R. Sheppard, “Scattering by fractal surfaces with an outer scale,” Opt. Commun. 122, 178–188 (1996).
[CrossRef]

Soto-Crespo, J. M.

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in FORTRAN, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992).

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in FORTRAN, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992).

Zou, L.-F.

Ann. Phys. (1)

A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. 203, 255–307 (1990).
[CrossRef]

Appl. Opt. (1)

J. Mod. Opt. (1)

R. E. Luna, E. R. Méndez, J. Q. Lu, Z.-H. Gu, “Enhanced backscattering due to total internal-reflection at a dielectric–air interface,” J. Mod. Opt. 42, 257–269 (1995).
[CrossRef]

J. Opt. Soc. Am. A (1)

Opt. Commun. (3)

E. R. Méndez, K. A. O’Donnell, “Observation of depolarization and backscattering enhancement in light scattering from Gaussian random surfaces,” Opt. Commun. 61, 91–95 (1987).
[CrossRef]

C. J. R. Sheppard, “Scattering by fractal surfaces with an outer scale,” Opt. Commun. 122, 178–188 (1996).
[CrossRef]

A. Mendoza-Suárez, E. R. Méndez, “Derivation of an impedance boundary condition for a one-dimensional, curved, reentrant surfaces,” Opt. Commun. 134, 241–250 (1997).
[CrossRef]

Opt. Lett. (5)

Opt. Photon. News (1)

D. J. DiGiovanni, M. H. Muendel, “High power fiber lasers,” Opt. Photon. News 10(1), 26–30 (1999).
[CrossRef]

Other (2)

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in FORTRAN, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992).

G. Arfken, Mathematical Methods for Physicists (Academic, New York, 1970), pp. 387–390.

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Figures (4)

Fig. 1
Fig. 1

Hollow cylindrical waveguide of arbitrary cross-sectional shape. The cross-sectional size and shape are assumed constant along the cylinder axis (z direction), and the intersection of the cylinder with the plane xy is closed profile C.

Fig. 2
Fig. 2

Logarithm of the determinant of matrix Lmnγ[D(γ)] versus γ for polygonal profiles. The interval considered for γ eigenvalues is [0.10, 2.50]. The minimum relative points of D(γ) give the position of the eigenvalues. The polygons are inscribed in a circumference of radius R=32 (arbitrary units). Profiles associated with (A) triangle, (B) square, (C) pentagon, (D) hexagon, and (E) circumference are shown.

Fig. 3
Fig. 3

Squared absolute value of the ground-state eigenfunctions of the waveguides with the polygonal cross section in Fig. 2. Eigenfunctions associated with a polygonal waveguide for the case of (A) triangle, (B) square, (C) pentagon, (D) hexagon, and (E) circumference. The ground eigenfunctions are nondegenerate symmetric functions that correspond to the symmetry of the respective geometries considered.

Fig. 4
Fig. 4

Prefractal triadic Koch structure for a waveguide sequence. The structure is assumed to be formed by four triadic Koch prefractals of orders (A) ν=0, (B) ν=1, (C) ν=2, (D) ν=3, and (E) ν=4. The number of line segments that correspond to (E) is 1024.

Tables (2)

Tables Icon

Table 1 Eigenvalues Associated with the Waveguides with Polygonal Cross Section of Fig. 2a

Tables Icon

Table 2 Eigenvalues Associated with the Waveguides with Triadic Koch Structure Shown in Fig. 4a

Equations (27)

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

A(x, y, z, t)=A(x, y)exp(±ikz-iωt),
E=Ez+Et,
Ez=(e3·E)e3,
Et=(e3×E)×e3,
Bz=0,
Et=iktEz,
Bt=ωcke3×Et
Ψγ|S=0
(t2+γ2)Ψγ=0.
t2=2-2/z2,
γ2=ω2/c2-k2.
(t2+γ2)Gγ(r, r)=-4πδ(r-r),
Gγ(r, r)=iπH0(1)(γ|r-r|).
Ψγ(r)=-14πCGγ(r, R)Fγ(R)ds,
Fγ(R)=nˆ (R).[tΨγ(r)]|r=R,
R=[ξ(s),η(s)],
0=limv0+CGγ[R+vnˆ(R),R]Fγ(R)ds.
0=n=1NLmnγFγ(sn),m=1, 2,, N,
Lmnγ=-Δs4iH0(1)(γ{[ξ(sm)-ξ(sn)]2+[η(sm)-η(sn)]2}1/2)(1-δmn)-Δs4iH0(1)γΔs2eδmn,
|Lmnγ|=0.
D(γ)=ln(|Lmnγ|),
Ψγ(x, y)=Δs4in=1NFγ(sn)H0(1)(γ{[x-ξ(sn)]2+[y-η(sn)]2}1/2).
Ez(x, y, z, t)=jAγjΨγj(x, y)exp(ikz-iωt),
k=±(ω2/c2-γj2)1/2.
γmn=πL(m2+n2)1/2,m, n=1, 2, 3,,
Jm(γmnR)=0,
e=j=1N(γja-γj)21/2,

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