Abstract

We describe a source-model technique for the analysis of the strictly bound modes propagating in photonic crystal fibers that have a finite photonic bandgap crystal cladding and are surrounded by an air jacket. In this model the field is simulated by a superposition of fields of fictitious electric and magnetic current filaments, suitably placed near the media interfaces of the fiber. A simple point-matching procedure is subsequently used to enforce the continuity conditions across the interfaces, leading to a homogeneous matrix equation. Nontrivial solutions to this equation yield the mode field patterns and propagation constants. As an example, we analyze a hollow-core photonic crystal fiber. Symmetry characteristics of the modes are discussed and exploited to reduce the computational burden.

© 2004 Optical Society of America

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  1. T. A. Birks, P. J. Roberts, P. S. J. Russell, D. M. Atkin, T. J. Shepherd, “Full 2-D photonic bandgaps in silica/air structures,” Electron. Lett. 31, 1941–1943 (1995).
    [CrossRef]
  2. P. S. J. Russell, “Photonic crystal fibers,” Science 299, 358–362 (2003).
    [CrossRef] [PubMed]
  3. B. J. Eggleton, P. S. Westbrook, C. A. White, C. Kerbage, R. S. Windeler, G. L. Burdge, “Cladding-mode-resonances in air–silica microstructure optical fibers,” J. Lightwave Technol. 18, 1084–1100 (2000).
    [CrossRef]
  4. N. A. Mortensen, “Effective area of photonic crystal fibers,” Opt. Express 10, 341–348 (2002), http://www.opticsexpress.org .
    [CrossRef] [PubMed]
  5. T. P. White, B. T. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. M. de Sterke, L. C. Botten, “Multipole method for microstructured optical fibers. I. Formulation,” J. Opt. Soc. Am. B 19, 2322–2330 (2002).
    [CrossRef]
  6. B. T. Kuhlmey, T. P. White, G. Renversez, D. Maystre, L. C. Botten, C. M. de Sterke, R. C. McPhedran, “Multipole method for microstructured optical fibers. II. Implementation and results,” J. Opt. Soc. Am. B 19, 2331–2340 (2002).
    [CrossRef]
  7. D. Felbacq, G. Tayeb, D. Maystre, “Scattering by a random set of parallel cylinders,” J. Opt. Soc. Am. A 11, 2526–2538 (1994).
    [CrossRef]
  8. K. Saitoh, M. Koshiba, “Confinement losses in air-guiding photonic bandgap fibers,” IEEE Photon. Technol. Lett. 15, 236–238 (2003).
    [CrossRef]
  9. Z. Altman, H. Cory, Y. Leviatan, “Cutoff frequencies of dielectric waveguides using the multifilament current model,” Microwave Opt. Technol. Lett. 3, 294–295 (1990).
    [CrossRef]
  10. X. E. Lin, “Photonic band gap fiber accelerator,” Phys. Rev. ST Accel. Beams 4, 051301 (2001).
    [CrossRef]
  11. Y. Leviatan, A. Boag, A. Boag, “Generalized formulations for electromagnetic scattering from perfectly conducting and homogeneous material bodies—theory and numerical solution,” IEEE Trans. Antennas Propag. 36, 1722–1734 (1988).
    [CrossRef]
  12. Y. Leviatan, “Analytic continuation considerations when using generalized formulations for scattering problems,” IEEE Trans. Antennas Propag. 38, 1259–1263 (1990).
    [CrossRef]
  13. C. Hafner, The Generalized Multipole Technique for Computational Electromagnetics (Artech House, Norwood, Mass., 1990).
  14. R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, New York, 1961).
  15. Yu. A. Eremin, N. V. Orlov, A. G. Sveshnikov, “Models of electromagnetic scattering problems based on discrete sources method,” in Generalized Multipole Techniques for Electromagnetic and Light Scattering, T. Wriedt, ed. (Elsevier, Amsterdam, 1999), Chap. 4.
  16. B. N. Datta, Numerical Linear Algebra and Applications (Brooks-Cole, Pacific Grove, Calif., 1994).
  17. P. R. McIsaac, “Symmetry-induced modal characteristics of uniform waveguides—I: Summary of results,” IEEE Trans. Microwave Theory Tech. 23, 421–429 (1975).
    [CrossRef]
  18. H. Shigesawa, “The equivalent source model,” in Analysis Methods for Electromagnetic Wave Problems, E. Yamashita, ed. (Artech House, Norwood, Mass., 1990), Chap. 6.
  19. A. A. Maradudin, A. R. McGurn, “Out of plane propagation of electromagnetic waves in a two-dimensional periodic dielectric medium,” J. Mod. Opt. 41, 275–284 (1994).
    [CrossRef]
  20. R. Lehoucq, D. Sorensen, C. Yang, ARPACK Users’ Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1998).

2003 (2)

P. S. J. Russell, “Photonic crystal fibers,” Science 299, 358–362 (2003).
[CrossRef] [PubMed]

K. Saitoh, M. Koshiba, “Confinement losses in air-guiding photonic bandgap fibers,” IEEE Photon. Technol. Lett. 15, 236–238 (2003).
[CrossRef]

2002 (3)

2001 (1)

X. E. Lin, “Photonic band gap fiber accelerator,” Phys. Rev. ST Accel. Beams 4, 051301 (2001).
[CrossRef]

2000 (1)

1995 (1)

T. A. Birks, P. J. Roberts, P. S. J. Russell, D. M. Atkin, T. J. Shepherd, “Full 2-D photonic bandgaps in silica/air structures,” Electron. Lett. 31, 1941–1943 (1995).
[CrossRef]

1994 (2)

A. A. Maradudin, A. R. McGurn, “Out of plane propagation of electromagnetic waves in a two-dimensional periodic dielectric medium,” J. Mod. Opt. 41, 275–284 (1994).
[CrossRef]

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

1990 (2)

Y. Leviatan, “Analytic continuation considerations when using generalized formulations for scattering problems,” IEEE Trans. Antennas Propag. 38, 1259–1263 (1990).
[CrossRef]

Z. Altman, H. Cory, Y. Leviatan, “Cutoff frequencies of dielectric waveguides using the multifilament current model,” Microwave Opt. Technol. Lett. 3, 294–295 (1990).
[CrossRef]

1988 (1)

Y. Leviatan, A. Boag, A. Boag, “Generalized formulations for electromagnetic scattering from perfectly conducting and homogeneous material bodies—theory and numerical solution,” IEEE Trans. Antennas Propag. 36, 1722–1734 (1988).
[CrossRef]

1975 (1)

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

Altman, Z.

Z. Altman, H. Cory, Y. Leviatan, “Cutoff frequencies of dielectric waveguides using the multifilament current model,” Microwave Opt. Technol. Lett. 3, 294–295 (1990).
[CrossRef]

Atkin, D. M.

T. A. Birks, P. J. Roberts, P. S. J. Russell, D. M. Atkin, T. J. Shepherd, “Full 2-D photonic bandgaps in silica/air structures,” Electron. Lett. 31, 1941–1943 (1995).
[CrossRef]

Birks, T. A.

T. A. Birks, P. J. Roberts, P. S. J. Russell, D. M. Atkin, T. J. Shepherd, “Full 2-D photonic bandgaps in silica/air structures,” Electron. Lett. 31, 1941–1943 (1995).
[CrossRef]

Boag, A.

Y. Leviatan, A. Boag, A. Boag, “Generalized formulations for electromagnetic scattering from perfectly conducting and homogeneous material bodies—theory and numerical solution,” IEEE Trans. Antennas Propag. 36, 1722–1734 (1988).
[CrossRef]

Y. Leviatan, A. Boag, A. Boag, “Generalized formulations for electromagnetic scattering from perfectly conducting and homogeneous material bodies—theory and numerical solution,” IEEE Trans. Antennas Propag. 36, 1722–1734 (1988).
[CrossRef]

Botten, L. C.

Burdge, G. L.

Cory, H.

Z. Altman, H. Cory, Y. Leviatan, “Cutoff frequencies of dielectric waveguides using the multifilament current model,” Microwave Opt. Technol. Lett. 3, 294–295 (1990).
[CrossRef]

Datta, B. N.

B. N. Datta, Numerical Linear Algebra and Applications (Brooks-Cole, Pacific Grove, Calif., 1994).

de Sterke, C. M.

Eggleton, B. J.

Eremin, Yu. A.

Yu. A. Eremin, N. V. Orlov, A. G. Sveshnikov, “Models of electromagnetic scattering problems based on discrete sources method,” in Generalized Multipole Techniques for Electromagnetic and Light Scattering, T. Wriedt, ed. (Elsevier, Amsterdam, 1999), Chap. 4.

Felbacq, D.

Hafner, C.

C. Hafner, The Generalized Multipole Technique for Computational Electromagnetics (Artech House, Norwood, Mass., 1990).

Harrington, R. F.

R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, New York, 1961).

Kerbage, C.

Koshiba, M.

K. Saitoh, M. Koshiba, “Confinement losses in air-guiding photonic bandgap fibers,” IEEE Photon. Technol. Lett. 15, 236–238 (2003).
[CrossRef]

Kuhlmey, B. T.

Lehoucq, R.

R. Lehoucq, D. Sorensen, C. Yang, ARPACK Users’ Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1998).

Leviatan, Y.

Z. Altman, H. Cory, Y. Leviatan, “Cutoff frequencies of dielectric waveguides using the multifilament current model,” Microwave Opt. Technol. Lett. 3, 294–295 (1990).
[CrossRef]

Y. Leviatan, “Analytic continuation considerations when using generalized formulations for scattering problems,” IEEE Trans. Antennas Propag. 38, 1259–1263 (1990).
[CrossRef]

Y. Leviatan, A. Boag, A. Boag, “Generalized formulations for electromagnetic scattering from perfectly conducting and homogeneous material bodies—theory and numerical solution,” IEEE Trans. Antennas Propag. 36, 1722–1734 (1988).
[CrossRef]

Lin, X. E.

X. E. Lin, “Photonic band gap fiber accelerator,” Phys. Rev. ST Accel. Beams 4, 051301 (2001).
[CrossRef]

Maradudin, A. A.

A. A. Maradudin, A. R. McGurn, “Out of plane propagation of electromagnetic waves in a two-dimensional periodic dielectric medium,” J. Mod. Opt. 41, 275–284 (1994).
[CrossRef]

Maystre, D.

McGurn, A. R.

A. A. Maradudin, A. R. McGurn, “Out of plane propagation of electromagnetic waves in a two-dimensional periodic dielectric medium,” J. Mod. Opt. 41, 275–284 (1994).
[CrossRef]

McIsaac, P. R.

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

McPhedran, R. C.

Mortensen, N. A.

Orlov, N. V.

Yu. A. Eremin, N. V. Orlov, A. G. Sveshnikov, “Models of electromagnetic scattering problems based on discrete sources method,” in Generalized Multipole Techniques for Electromagnetic and Light Scattering, T. Wriedt, ed. (Elsevier, Amsterdam, 1999), Chap. 4.

Renversez, G.

Roberts, P. J.

T. A. Birks, P. J. Roberts, P. S. J. Russell, D. M. Atkin, T. J. Shepherd, “Full 2-D photonic bandgaps in silica/air structures,” Electron. Lett. 31, 1941–1943 (1995).
[CrossRef]

Russell, P. S. J.

P. S. J. Russell, “Photonic crystal fibers,” Science 299, 358–362 (2003).
[CrossRef] [PubMed]

T. A. Birks, P. J. Roberts, P. S. J. Russell, D. M. Atkin, T. J. Shepherd, “Full 2-D photonic bandgaps in silica/air structures,” Electron. Lett. 31, 1941–1943 (1995).
[CrossRef]

Saitoh, K.

K. Saitoh, M. Koshiba, “Confinement losses in air-guiding photonic bandgap fibers,” IEEE Photon. Technol. Lett. 15, 236–238 (2003).
[CrossRef]

Shepherd, T. J.

T. A. Birks, P. J. Roberts, P. S. J. Russell, D. M. Atkin, T. J. Shepherd, “Full 2-D photonic bandgaps in silica/air structures,” Electron. Lett. 31, 1941–1943 (1995).
[CrossRef]

Shigesawa, H.

H. Shigesawa, “The equivalent source model,” in Analysis Methods for Electromagnetic Wave Problems, E. Yamashita, ed. (Artech House, Norwood, Mass., 1990), Chap. 6.

Sorensen, D.

R. Lehoucq, D. Sorensen, C. Yang, ARPACK Users’ Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1998).

Sveshnikov, A. G.

Yu. A. Eremin, N. V. Orlov, A. G. Sveshnikov, “Models of electromagnetic scattering problems based on discrete sources method,” in Generalized Multipole Techniques for Electromagnetic and Light Scattering, T. Wriedt, ed. (Elsevier, Amsterdam, 1999), Chap. 4.

Tayeb, G.

Westbrook, P. S.

White, C. A.

White, T. P.

Windeler, R. S.

Yang, C.

R. Lehoucq, D. Sorensen, C. Yang, ARPACK Users’ Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1998).

Electron. Lett. (1)

T. A. Birks, P. J. Roberts, P. S. J. Russell, D. M. Atkin, T. J. Shepherd, “Full 2-D photonic bandgaps in silica/air structures,” Electron. Lett. 31, 1941–1943 (1995).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

K. Saitoh, M. Koshiba, “Confinement losses in air-guiding photonic bandgap fibers,” IEEE Photon. Technol. Lett. 15, 236–238 (2003).
[CrossRef]

IEEE Trans. Antennas Propag. (2)

Y. Leviatan, A. Boag, A. Boag, “Generalized formulations for electromagnetic scattering from perfectly conducting and homogeneous material bodies—theory and numerical solution,” IEEE Trans. Antennas Propag. 36, 1722–1734 (1988).
[CrossRef]

Y. Leviatan, “Analytic continuation considerations when using generalized formulations for scattering problems,” IEEE Trans. Antennas Propag. 38, 1259–1263 (1990).
[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. 23, 421–429 (1975).
[CrossRef]

J. Lightwave Technol. (1)

J. Mod. Opt. (1)

A. A. Maradudin, A. R. McGurn, “Out of plane propagation of electromagnetic waves in a two-dimensional periodic dielectric medium,” J. Mod. Opt. 41, 275–284 (1994).
[CrossRef]

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

J. Opt. Soc. Am. B (2)

Microwave Opt. Technol. Lett. (1)

Z. Altman, H. Cory, Y. Leviatan, “Cutoff frequencies of dielectric waveguides using the multifilament current model,” Microwave Opt. Technol. Lett. 3, 294–295 (1990).
[CrossRef]

Opt. Express (1)

Phys. Rev. ST Accel. Beams (1)

X. E. Lin, “Photonic band gap fiber accelerator,” Phys. Rev. ST Accel. Beams 4, 051301 (2001).
[CrossRef]

Science (1)

P. S. J. Russell, “Photonic crystal fibers,” Science 299, 358–362 (2003).
[CrossRef] [PubMed]

Other (6)

R. Lehoucq, D. Sorensen, C. Yang, ARPACK Users’ Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1998).

H. Shigesawa, “The equivalent source model,” in Analysis Methods for Electromagnetic Wave Problems, E. Yamashita, ed. (Artech House, Norwood, Mass., 1990), Chap. 6.

C. Hafner, The Generalized Multipole Technique for Computational Electromagnetics (Artech House, Norwood, Mass., 1990).

R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, New York, 1961).

Yu. A. Eremin, N. V. Orlov, A. G. Sveshnikov, “Models of electromagnetic scattering problems based on discrete sources method,” in Generalized Multipole Techniques for Electromagnetic and Light Scattering, T. Wriedt, ed. (Elsevier, Amsterdam, 1999), Chap. 4.

B. N. Datta, Numerical Linear Algebra and Applications (Brooks-Cole, Pacific Grove, Calif., 1994).

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Figures (13)

Fig. 1
Fig. 1

Hollow-core PCF. Gray areas, fused silica; white areas, air. Λ is the lattice constant.

Fig. 2
Fig. 2

General dielectric waveguide geometry.

Fig. 3
Fig. 3

Simulated equivalence for the optically dense region.

Fig. 4
Fig. 4

Simulated equivalence for the optically thin region.

Fig. 5
Fig. 5

Accuracy of the solution (ΔE-1) as a function of source location. The contraction and dilation ratios, αcon and αdil, are defined in Section 3. Acceptable values of ΔE-1 are enclosed by the thickened contour. Λ/λ and β/k0 are the same as in the lower row of Fig. 7.

Fig. 6
Fig. 6

Minimal sectors for fibers with C6v symmetry. Dashed lines, a perfect magnetic conductor, or Hz=0 boundary condition. Solid lines, a perfect electric conductor, or Ez=0 boundary condition. The class numbering, p, is after McIsaac.17

Fig. 7
Fig. 7

Fundamental p=1 mode, which is outside the band gap, is shown in the top row (β/k0=1.201, Λ/λ=0.235). A mode inside the bandgap is shown in the lower row (β/k0=1.03, Λ/λ=1.353).

Fig. 8
Fig. 8

Comparison between the SMT solution of the PCF surrounded by an air jacket and the supercell model solution in Ref. 10. A cross-sectional plot of Ez along the y axis is shown.

Fig. 9
Fig. 9

Magnetic field of the mode shown in the lower row of Fig. 7.

Fig. 10
Fig. 10

Absolute value of the longitudinal magnetic field of a mode of class p=5. The point on the dispersion curve is Λ/λ=0.5, β/k0=1.139.

Fig. 11
Fig. 11

Linearly polarized (class p=3) mode. Black arrows, the transverse electric field; white arrows, the transverse magnetic field. The intensity of light is shown in the background. The point on the dispersion curve is Λ/λ=1.305, β/k0=1.011.

Fig. 12
Fig. 12

Dispersion curves for modes belonging to the symmetry class p=1. The leftmost curve corresponds to the mode shown in the upper row of Fig. 7.

Fig. 13
Fig. 13

Dispersion curves for modes belonging to the symmetry class p=1. The bandgap for out-of-plane propagation is marked by the shaded area, and the dashed curve corresponds to the mode shown in the lower row of Fig. 7.

Tables (1)

Tables Icon

Table 1 Convergence of β/k0 with the Number of Sources a

Equations (25)

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

A=I4j H0(2)(kρρ)uz,
k2=β2+kρ2,
H=×A,E=-jωμ0A+1jωε0εr (A).
ETM(I)=βkρI4jωε0εr H1(2)(kρρ)uρ-kρ2I4ωε0εr H0(2)(kρρ)uz,
HTM(I)=kρI4j H1(2)(kρρ)uϕ,
F=K4j H0(2)(kρρ)uz.
E=-×F,H=-jωε0F+1jωμ0 (F).
ETE(K)=jkρK4 H1(2)(kρρ)uϕ,
HTE(K)=βkρK4jωμ0 H1(2)(kρρ)uρ-kρ2K4ωμ0 H0(2)×(kρρ)uz.
Ed=iETM(Iid)+iETE(Kid),
Hd=iHTM(Iid)+iHTE(Kid),
E(j)=iETM(Ii(j))+iETE(Ki(j)),
j=1, 2,,NS orjack,
H(j)=iHTM(Ii(j))+iHTE(Ki(j)),
j=1, 2,, NS orjack,
n^(j)×(Ed-E(j))=0onS(j),
j=1, 2,, NS orjack,
n^(j)×(Hd-H(j))=0onS(j),
j=1, 2,, NS orjack,
[Z]I=0,
[Zsym]=m=1Mimg[Zm],
ri(j)=rc(j)+αdilRvcos(ϕi)ux+αdilRvsin(ϕi)uy,
ϕi=πN(j)+2πN(j) (i-1),
Verr=[Z˜]I,
ΔE=|[Z]I||Verr|,

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