Abstract

We describe the numerical verifications of a multipole formulation for calculating the electromagnetic properties of the modes that propagate in microstructured optical fibers. We illustrate the application of this formulation to calculating both the real and the imaginary parts of the propagation constant. We compare its predictions with the results of recent measurements of a low-loss microstructured fiber and investigate the variations in fiber dispersion with geometrical parameters. We also show that the formulation obeys appropriate symmetry rules and that these rules may be used to improve computational speed.

© 2002 Optical Society of America

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References

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  1. T. P. White, B. T. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. M. de Sterke, and L. C. Botten, “Multipole method for microstructured optical fibers. I. Formulation,” J. Opt. Soc. Am. B 19, 2322–2330 (2002).
    [CrossRef]
  2. M. J. Steel, T. P. White, C. M. de Sterke, R. C. McPhedran, and L. C. Botten, “Symmetry and degeneracy in microstructured optical fibers,” Opt. Lett. 26, 488–490 (2001).
    [CrossRef]
  3. H. Kubota, K. Suzuki, S. Kawanishi, M. Nakazawa, M. Tanaka, and M. Fujita, “Lowloss, 2-km-long photonic crystal fiber with zero GVD in the near IR suitable for picosecond pulse propagation at the 800 nm band,” in Conferenceon Lasers and Electro-Optics (CLEO2001), Vol. 56 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2001), paper C3.
  4. F. Zolla and R. Petit, “Method of fictious sources as applied to the electromagnetic diffraction of a plane wave by a grating in conical mounts,” J. Opt. Soc. Am. A 13, 1087–1096 (1996).
    [CrossRef]
  5. D. Felbacq, G. Tayeb, and D. Maystre, “Scattering by a random set of parallel cylinders,” J. Opt. Soc. Am. A 11, 2526–2538 (1994).
    [CrossRef]
  6. E. Centeno and D. Felbacq, “Rigorous vector diffraction of electromagnetic waves by bidimensional photonic crystals,” J. Opt. Soc. Am. A 17, 320–327 (2000).
    [CrossRef]
  7. P. Kravanja and M. Van Barel, Computing the Zeros of Analytic Functions (Springer-Verlag, Berlin, 2000).
  8. L. C. Botten, M. S. Craig, and R. C. McPhedran, “Complex zeros of analytic functions,” Comput. Phys. Commun. 29, 245–259 (1983).
    [CrossRef]
  9. C. G. Broyden, “A class of methods for solving nonlinear simultaneous equations,” Math. Comput. 19, 577–593 (1965).
    [CrossRef]
  10. G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, Calif., 1995).
  11. E. Yamashita, S. Ozeki, and K. Atsuki, “Modal analysis method for optical fibers with symmetrically distributed multiple cores,” J. Lightwave Technol. 3, 341–346 (1985).
    [CrossRef]
  12. D. Mogilevtsev, T. A. Birks, and P. St. J. Russell, “Group-velocity dispersion in photonic crystal fibers,” Opt. Lett. 23, 1662–1664 (1998).
    [CrossRef]
  13. T. P. White, R. C. McPhedran, C. M. de Sterke, and M. J. Steel, “Confinement losses in microstructured optical fibers,” Opt. Lett. 26, 488–490 (2001).
    [CrossRef]
  14. P. R. McIsaac, “Symmetry-induced modal characteristics of uniform waveguides. I. Summary of results,” IEEE Trans. Microwave Theory Tech. MTT-23, 421–429 (1975).
    [CrossRef]

2002 (1)

2001 (2)

2000 (1)

1998 (1)

1996 (1)

F. Zolla and R. Petit, “Method of fictious sources as applied to the electromagnetic diffraction of a plane wave by a grating in conical mounts,” J. Opt. Soc. Am. A 13, 1087–1096 (1996).
[CrossRef]

1994 (1)

1985 (1)

E. Yamashita, S. Ozeki, and K. Atsuki, “Modal analysis method for optical fibers with symmetrically distributed multiple cores,” J. Lightwave Technol. 3, 341–346 (1985).
[CrossRef]

1983 (1)

L. C. Botten, M. S. Craig, and R. C. McPhedran, “Complex zeros of analytic functions,” Comput. Phys. Commun. 29, 245–259 (1983).
[CrossRef]

1975 (1)

P. R. McIsaac, “Symmetry-induced modal characteristics of uniform waveguides. I. Summary of results,” IEEE Trans. Microwave Theory Tech. MTT-23, 421–429 (1975).
[CrossRef]

1965 (1)

C. G. Broyden, “A class of methods for solving nonlinear simultaneous equations,” Math. Comput. 19, 577–593 (1965).
[CrossRef]

Atsuki, K.

E. Yamashita, S. Ozeki, and K. Atsuki, “Modal analysis method for optical fibers with symmetrically distributed multiple cores,” J. Lightwave Technol. 3, 341–346 (1985).
[CrossRef]

Birks, T. A.

Botten, L. C.

Broyden, C. G.

C. G. Broyden, “A class of methods for solving nonlinear simultaneous equations,” Math. Comput. 19, 577–593 (1965).
[CrossRef]

Centeno, E.

Craig, M. S.

L. C. Botten, M. S. Craig, and R. C. McPhedran, “Complex zeros of analytic functions,” Comput. Phys. Commun. 29, 245–259 (1983).
[CrossRef]

de Sterke, C. M.

Felbacq, D.

Kuhlmey, B. T.

Maystre, D.

McIsaac, P. R.

P. R. McIsaac, “Symmetry-induced modal characteristics of uniform waveguides. I. Summary of results,” IEEE Trans. Microwave Theory Tech. MTT-23, 421–429 (1975).
[CrossRef]

McPhedran, R. C.

Mogilevtsev, D.

Ozeki, S.

E. Yamashita, S. Ozeki, and K. Atsuki, “Modal analysis method for optical fibers with symmetrically distributed multiple cores,” J. Lightwave Technol. 3, 341–346 (1985).
[CrossRef]

Petit, R.

F. Zolla and R. Petit, “Method of fictious sources as applied to the electromagnetic diffraction of a plane wave by a grating in conical mounts,” J. Opt. Soc. Am. A 13, 1087–1096 (1996).
[CrossRef]

Renversez, G.

Russell, P. St. J.

Steel, M. J.

Tayeb, G.

White, T. P.

Yamashita, E.

E. Yamashita, S. Ozeki, and K. Atsuki, “Modal analysis method for optical fibers with symmetrically distributed multiple cores,” J. Lightwave Technol. 3, 341–346 (1985).
[CrossRef]

Zolla, F.

F. Zolla and R. Petit, “Method of fictious sources as applied to the electromagnetic diffraction of a plane wave by a grating in conical mounts,” J. Opt. Soc. Am. A 13, 1087–1096 (1996).
[CrossRef]

Comput. Phys. Commun. (1)

L. C. Botten, M. S. Craig, and R. C. McPhedran, “Complex zeros of analytic functions,” Comput. Phys. Commun. 29, 245–259 (1983).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

P. R. McIsaac, “Symmetry-induced modal characteristics of uniform waveguides. I. Summary of results,” IEEE Trans. Microwave Theory Tech. MTT-23, 421–429 (1975).
[CrossRef]

J. Lightwave Technol. (1)

E. Yamashita, S. Ozeki, and K. Atsuki, “Modal analysis method for optical fibers with symmetrically distributed multiple cores,” J. Lightwave Technol. 3, 341–346 (1985).
[CrossRef]

J. Opt. Soc. Am. A (3)

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

Math. Comput. (1)

C. G. Broyden, “A class of methods for solving nonlinear simultaneous equations,” Math. Comput. 19, 577–593 (1965).
[CrossRef]

Opt. Lett. (3)

Other (3)

G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, Calif., 1995).

P. Kravanja and M. Van Barel, Computing the Zeros of Analytic Functions (Springer-Verlag, Berlin, 2000).

H. Kubota, K. Suzuki, S. Kawanishi, M. Nakazawa, M. Tanaka, and M. Fujita, “Lowloss, 2-km-long photonic crystal fiber with zero GVD in the near IR suitable for picosecond pulse propagation at the 800 nm band,” in Conferenceon Lasers and Electro-Optics (CLEO2001), Vol. 56 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2001), paper C3.

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Figures (12)

Fig. 1
Fig. 1

Top, internal and bottom, Wijngaard expansions compared for R(Ez) and I(Ez), respectively, for an air core MOF, with M=5 for both the central air hole and all other air holes (3 rings; 54 air holes of diameter 4.0271 µm; core hole diameter, 13.0714 µm; jacket diameter, 50 µm; ne=1.39; ni=1.00; n0=1.39+10-8i; Λ=5.78157 µm; λ=3.846 µm).

Fig. 2
Fig. 2

As for Fig. 1 but with M=19 for the core hole and M=5 for all other holes. Note that the Wijngaard and internal expansions now match with graphic accuracy.

Fig. 3
Fig. 3

Dispersion as a function of wavelength for solid-core MOFs in silica with three rings of holes for various values of d/Λ, with Λ=2.3 µm, n0=nsilica, and the Sellmeier model for dispersion of silica.

Fig. 4
Fig. 4

Confinement loss figures for the fundamental mode of a MOF with three rings of air holes of various diameters and pitches. All curves are for λ=1.55 µm.

Fig. 5
Fig. 5

Confinement loss figures for the first three modes of a MOF with three rings of air holes as a function of pitch. Here d/Λ=0.4, λ=1.55 µm.

Fig. 6
Fig. 6

Scanning-electron micrograph of a cleaved end face of the MOF fabricated by Kubota et al.3 used in our comparisons. Figure supplied by H. Kubota.

Fig. 7
Fig. 7

Loss of the first three modes as a function of number of rings for the structure published by Kubota et al.,3 at a wavelength of 0.76 µm. Only confinement losses are included.

Fig. 8
Fig. 8

Zero-dispersion wavelength as a function of pitch for the structure published by Kubota et al.,3 with constant diameter d=1.51 µm.

Fig. 9
Fig. 9

Zero-dispersion wavelength as a function of hole diameter for the structure published by Kubota et al.,3 with constant pitch Λ=2.26 µm.

Fig. 10
Fig. 10

Zero-dispersion wavelength as a function of pitch for the structure published by Kubota et al.,3 with constant diameter/pitch ratio d/Λ=0.67.

Fig. 11
Fig. 11

Primary (bold circles) and secondary cylinders of the nondegenerate p=1 and p=2 mode classes of a two-ring MOF structure.

Fig. 12
Fig. 12

Primary (bold circles) and secondary cylinders of the degenerate p=3 and p=4 mode classes of a two-ring MOF structure.

Tables (1)

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Table 1 Convergence of neff with Ma

Equations (19)

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[I-R(H˜+J˜B0R˜0J˜0B)]BMB=0.
Ez=m=-MM[AmElJm(kerl)+BmElHm(1)(kerl)]exp(imθl),
Ez=l=1Ncm BmElHm(1)(ke|rl|)exp[im arg(r-cl)]+m AmE0Jm(ker)exp(imθ).
W=C1|Ezlocal(θ1)-EzWijngaard(θ1)|dθ1C1|EzWijngaard(θ1)|dθ1.
vg=cneff+ω dneffdω-1
D=-λc d2R(neff)dλ2
θ,ρ<RR[S(ρ, θ, z)]·uˆzρdρdθ
=δz θR[S(R, θ, z)]·uˆrRdθ+θ,ρ<R R[S(ρ, θ, z+δz)]·uˆzρdρdθ,
S(ρ, θ, z+δz)(1-αδz)S(ρ, θ, z),
αρ<R,θ R[S(ρ, θ, z)]·uˆzρdρdθ
=θR[S(R, θ, z)]·uˆrRdθ.
α=θR[Sr(R, θ, z)]Rdθθ,ρ<RR[Sz(ρ, θ, z)]ρdρdθ,
I(neff)=α2k0.
bmE(PS)=bmE(P1) exp[im(S-1)π/3],
bmE(P2)=-b-mE(P1),bmK(P2)=b-mK(P1).
bmE(P2)=(-1)mb-mE(P1),bmK(P2)=(-1)m+1b-mK(P1).
bmE(P2)=-b-mE(P1),bmK(P2)=b-mK(P1),
bmE(P3)=(-1)m+1bmE(P1),bmK(P3)=(-1)m+1bmK(P1),
bmE(P4)=(-1)mb-mE(P1),bmK(P4)=(-1)m+1b-mK(P1).

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