Abstract

Solid core photonic bandgap fibers (SC-PBGFs) consisting of an array of high index cylinders in a low index background and a low index defect core have been treated as a cylindrical analog of the planar anti-resonant reflecting optical waveguide (ARROW). We consider a limiting case of this model in which the cylinders in the SC-PBGF cladding are widely spaced apart, so that the SC-PBGF modal loss characteristics should resemble the antiresonant scattering properties of a single cylinder. We find that for glancing incidence, the single cylinder scattering resonances are Fano resonances, and these Fano resonances do in fact appear in the loss spectra of SC-PBGFs. We apply our analysis to enhance the core design of SC-PBGFs, specifically with an eye towards improving the mode confinement in the fundamental bandgap.

© 2006 Optical Society of America

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References

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  1. R. T. Bise, R. S. Windeler, K. S. Kranz, C. Kerbage, B. J. Eggleton, and D. J. Trevor, "Tunable photonic bandgap fiber," in Optical Fiber Communications Conference, Post Conference Ed., Vol. 70 of OSA Trends in Optics and Photonics Series Technical Digest (Optical Society of America, Washington, D. C., 2002), 466-468.
  2. T. T. Larsen, A. Bjarklev, D. S. Hermann, and J. Broeng, "Optical devices based on liquid crystal photonic bandgap fibres," Opt. Express 11, 2589-2596 (2003).
    [CrossRef] [PubMed]
  3. F. Luan, A. K. George, T. D. Hedley, G. J. Pearce, D. M. Bird, J. C. Knight, and P. St. J. Russell, "All-solid photonic bandgap fiber," Opt. Lett. 29, 2369-2371 (2004).
    [CrossRef] [PubMed]
  4. A. Argyros, T. A. Birks, S. G. Leon-Saval, C. M. B. Cordeiro, F. Luan, and P. St. J. Russell, "Photonic bandgap with an index step of one percent," Opt. Express 13, 309-314 (2005).
    [CrossRef] [PubMed]
  5. G. Bouwmans, L. Bigot, Y. Quiquempois, F. Lopez, L. Provino, and M. Douay, "Fabrication and characterization of an all-solid 2D photonic bandgap fiber with a low-loss region (< 20 dB/km) around 1550 nm," Opt. Express 13, 8452-8459 (2005).
    [CrossRef] [PubMed]
  6. J. M. Stone, G. J. Pearce, F. Luan, T. A. Birks, J. C. Knight, A. K. George, and D. M. Bird, "An improved photonic bandgap fiber based on an array of rings," Opt. Express 14, 6291-6295 (2006).
    [CrossRef] [PubMed]
  7. F. Brechet, P. Roy, J. Marcou, and D. Pagnoux, "Singlemode propagation into depressed-core-index photonic bandgap fibre designed for zero-dispersion propagation at short wavelengths," Electron. Lett. 36, 514-515 (2000).
    [CrossRef]
  8. J. Lægsgaard, "Gap formation and guided modes in photonic bandgap fibres with high-index rods," J. Opt. A, Pure Appl. Opt. 6, 798-804 (2004).
    [CrossRef]
  9. M. A. Duguay, Y. Kokubun, T. L. Koch, and L. Pfeiffer, "Antiresonant reflecting optical waveguides in SiO2-Si multilayer structures," Appl. Phys. Lett. 49, 13-15 (1986).
    [CrossRef]
  10. N. M. Litchinitser, A. K. Abeeluck, C. Headley, and B. J. Eggleton, "Antiresonant reflecting photonic crystal optical waveguides," Opt. Lett. 27, 1 592-1594 (2002).
    [CrossRef]
  11. A. K. Abeeluck, N. M. Litchinitser, C. Headley, and B. J. Eggleton, "Analysis of spectral characteristics of photonic bandgap waveguides," Opt. Express 10, 1320-1333 (2002).
    [PubMed]
  12. N. M. Litchinitser, S. C. Dunn, B. Usner, B. J. Eggleton, T. P. White, R. C. McPhedran, and C. M. de Sterke, "Resonances in microstructured optical waveguides," Opt. Express 11, 1243-1251 (2003).
    [CrossRef] [PubMed]
  13. T. P. White, R. C. McPhedran, C. M. de Sterke, N. M. Litchinitser, and B. J. Eggleton, "Resonance and scattering in microstructured optical fibers," Opt. Lett. 27, 1977-1979 (2002).
    [CrossRef]
  14. J. Kubica, D. Uttamchandani, and B. Culshaw, "Modal propagation within ARROWwaveguides," Opt. Commun. 78, 133-136 (1990).
    [CrossRef]
  15. T. Baba and Y. Kokubun, "Dispersion and radiation loss characteristics of antiresonant reflecting optical waveguides - numerical results and analytical expressions," IEEE J. Quantum Electron. 28,1689-1700 (1992).
    [CrossRef]
  16. A. C. Lind and J. M. Greenberg, "Electromagnetic scattering by obliquely Oriented Cylinders," J. Appl. Phys. 37, 3195-3203 (1966).
    [CrossRef]
  17. A. Argyros, T. A. Birks, S. G. Leon-Saval, C. M. B. Cordeiro, and P. St. J. Russell, "Guidance properties of low-contrast photonic bandgap fibres," Opt. Express 13, 2503-2511 (2005).
    [CrossRef] [PubMed]
  18. P. Steinvurzel, B. T. Kuhlmey, T. P. White, M. J. Steel, C. M. de Sterke, and B. J. Eggleton, "Long wavelength anti-resonant guidance in high index inclusion microstructured fibers," Opt. Express 12, 5424-5433 (2004).
    [CrossRef] [PubMed]
  19. T. A. Birks, F. Luan, G. J. Pearce, A. Wang, J. C. Knight, and D. M. Bird, "Bend loss in all-solid bandgap fibres," Opt. Express 14, 5688-5698 (2006).
    [CrossRef] [PubMed]
  20. G. Renversez, P. Boyer, and A. Sagrini, "Antiresonant reflecting optical waveguide microstructured fibers revisited: a new analysis based on leaky mode coupling," Opt. Express 14, 5682-5687 (2006).
    [CrossRef] [PubMed]
  21. J. R. Wait, "Scattering of a plane wave from a circular dielectric cylinder at oblique incidence," Canadian J. Phys. 33, 189-195 (1955).
    [CrossRef]
  22. C. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1998).
    [CrossRef]
  23. Note that we define ϑ0 such that it approaches zero for near glancing incidence, which we feel is the natural choice in the current context; in most of the literature on scattering by cylinders [16,21,22], however, the conicity angle approaches π=2 for glancing incidence, and our definition is equivalent to π=2¡α in the earlier references.
  24. A. W. Snyder and J.D. Love, Optical Waveguide Theory (Chapman and Hall, London, 1983).
  25. E. Snitzer, "Cylindrical Dielectric Waveguide Modes," J. Opt. Soc. Am. 51, 491-498 (1961).
    [CrossRef]
  26. 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]
  27. B. T. Kuhlmey, T. P. White, D. Maystre, G. Renversez, L. C. Botten, C. M. de Sterke, and R. C. McPhedran, "Multipole method for microstructured optical fibers II. Implementation and results," J. Opt. Soc. Am. B 19, 2331-2340 (2002).
    [CrossRef]
  28. http://www.physics.usyd.edu.au/cudos/mofsoftware
  29. U. Fano, "Effects Of Configuration Interaction On Intensities And Phase Shifts," Phys. Rev. 124, 1866-1878 (1961).
    [CrossRef]
  30. R. V. Andaloro, H. J. Simon, and R. T. Deck, "Temporal pulse reshaping with surface waves," Appl. Opt. 33, 6340-6347 (1994).
    [CrossRef] [PubMed]
  31. S. Fan, and J. D. Joannopoulos, "Analysis of guided resonances in photonic crystal slabs," Phys. Rev. B 65, 235112 (2002).
    [CrossRef]
  32. A. E. Miroshnichenko, S. F. Mingaleev, S. Flach, and Yu. S. Kivshar, "Nonlinear Fano resonance and bistable wave transmission," Phys. Rev. E 71, 036626 (2005).
    [CrossRef]
  33. 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]
  34. T. A. Birks, D. M. Bird, T. D. Hedley, J. M. Pottage, and P. St. J. Russell, "Scaling laws and vector effects in bandgap-guiding fibers," Opt. Express 12, 69-74 (2004).
    [CrossRef] [PubMed]

2006

2005

2004

2003

2002

2000

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]

F. Brechet, P. Roy, J. Marcou, and D. Pagnoux, "Singlemode propagation into depressed-core-index photonic bandgap fibre designed for zero-dispersion propagation at short wavelengths," Electron. Lett. 36, 514-515 (2000).
[CrossRef]

1994

1992

T. Baba and Y. Kokubun, "Dispersion and radiation loss characteristics of antiresonant reflecting optical waveguides - numerical results and analytical expressions," IEEE J. Quantum Electron. 28,1689-1700 (1992).
[CrossRef]

1990

J. Kubica, D. Uttamchandani, and B. Culshaw, "Modal propagation within ARROWwaveguides," Opt. Commun. 78, 133-136 (1990).
[CrossRef]

1986

M. A. Duguay, Y. Kokubun, T. L. Koch, and L. Pfeiffer, "Antiresonant reflecting optical waveguides in SiO2-Si multilayer structures," Appl. Phys. Lett. 49, 13-15 (1986).
[CrossRef]

1966

A. C. Lind and J. M. Greenberg, "Electromagnetic scattering by obliquely Oriented Cylinders," J. Appl. Phys. 37, 3195-3203 (1966).
[CrossRef]

1961

E. Snitzer, "Cylindrical Dielectric Waveguide Modes," J. Opt. Soc. Am. 51, 491-498 (1961).
[CrossRef]

U. Fano, "Effects Of Configuration Interaction On Intensities And Phase Shifts," Phys. Rev. 124, 1866-1878 (1961).
[CrossRef]

1955

J. R. Wait, "Scattering of a plane wave from a circular dielectric cylinder at oblique incidence," Canadian J. Phys. 33, 189-195 (1955).
[CrossRef]

Abeeluck, A. K.

Andaloro, R. V.

Argyros, A.

Baba, T.

T. Baba and Y. Kokubun, "Dispersion and radiation loss characteristics of antiresonant reflecting optical waveguides - numerical results and analytical expressions," IEEE J. Quantum Electron. 28,1689-1700 (1992).
[CrossRef]

Bigot, L.

Bird, D. M.

Birks, T. A.

Bjarklev, A.

Botten, L. C.

Bouwmans, G.

Boyer, P.

Brechet, F.

F. Brechet, P. Roy, J. Marcou, and D. Pagnoux, "Singlemode propagation into depressed-core-index photonic bandgap fibre designed for zero-dispersion propagation at short wavelengths," Electron. Lett. 36, 514-515 (2000).
[CrossRef]

Broeng, J.

Centeno, E.

Cordeiro, C. M. B.

Culshaw, B.

J. Kubica, D. Uttamchandani, and B. Culshaw, "Modal propagation within ARROWwaveguides," Opt. Commun. 78, 133-136 (1990).
[CrossRef]

de Sterke, C. M.

Deck, R. T.

Douay, M.

Duguay, M. A.

M. A. Duguay, Y. Kokubun, T. L. Koch, and L. Pfeiffer, "Antiresonant reflecting optical waveguides in SiO2-Si multilayer structures," Appl. Phys. Lett. 49, 13-15 (1986).
[CrossRef]

Dunn, S. C.

Eggleton, B. J.

Fan, S.

S. Fan, and J. D. Joannopoulos, "Analysis of guided resonances in photonic crystal slabs," Phys. Rev. B 65, 235112 (2002).
[CrossRef]

Fano, U.

U. Fano, "Effects Of Configuration Interaction On Intensities And Phase Shifts," Phys. Rev. 124, 1866-1878 (1961).
[CrossRef]

Felbacq, D.

Flach, S.

A. E. Miroshnichenko, S. F. Mingaleev, S. Flach, and Yu. S. Kivshar, "Nonlinear Fano resonance and bistable wave transmission," Phys. Rev. E 71, 036626 (2005).
[CrossRef]

George, A. K.

Greenberg, J. M.

A. C. Lind and J. M. Greenberg, "Electromagnetic scattering by obliquely Oriented Cylinders," J. Appl. Phys. 37, 3195-3203 (1966).
[CrossRef]

Headley, C.

Hedley, T. D.

Hermann, D. S.

Joannopoulos, J. D.

S. Fan, and J. D. Joannopoulos, "Analysis of guided resonances in photonic crystal slabs," Phys. Rev. B 65, 235112 (2002).
[CrossRef]

Kivshar, Yu. S.

A. E. Miroshnichenko, S. F. Mingaleev, S. Flach, and Yu. S. Kivshar, "Nonlinear Fano resonance and bistable wave transmission," Phys. Rev. E 71, 036626 (2005).
[CrossRef]

Knight, J. C.

Koch, T. L.

M. A. Duguay, Y. Kokubun, T. L. Koch, and L. Pfeiffer, "Antiresonant reflecting optical waveguides in SiO2-Si multilayer structures," Appl. Phys. Lett. 49, 13-15 (1986).
[CrossRef]

Kokubun, Y.

T. Baba and Y. Kokubun, "Dispersion and radiation loss characteristics of antiresonant reflecting optical waveguides - numerical results and analytical expressions," IEEE J. Quantum Electron. 28,1689-1700 (1992).
[CrossRef]

M. A. Duguay, Y. Kokubun, T. L. Koch, and L. Pfeiffer, "Antiresonant reflecting optical waveguides in SiO2-Si multilayer structures," Appl. Phys. Lett. 49, 13-15 (1986).
[CrossRef]

Kubica, J.

J. Kubica, D. Uttamchandani, and B. Culshaw, "Modal propagation within ARROWwaveguides," Opt. Commun. 78, 133-136 (1990).
[CrossRef]

Kuhlmey, B. T.

Lægsgaard, J.

J. Lægsgaard, "Gap formation and guided modes in photonic bandgap fibres with high-index rods," J. Opt. A, Pure Appl. Opt. 6, 798-804 (2004).
[CrossRef]

Larsen, T. T.

Leon-Saval, S. G.

Lind, A. C.

A. C. Lind and J. M. Greenberg, "Electromagnetic scattering by obliquely Oriented Cylinders," J. Appl. Phys. 37, 3195-3203 (1966).
[CrossRef]

Litchinitser, N. M.

Lopez, F.

Luan, F.

Marcou, J.

F. Brechet, P. Roy, J. Marcou, and D. Pagnoux, "Singlemode propagation into depressed-core-index photonic bandgap fibre designed for zero-dispersion propagation at short wavelengths," Electron. Lett. 36, 514-515 (2000).
[CrossRef]

Maystre, D.

McPhedran, R. C.

Mingaleev, S. F.

A. E. Miroshnichenko, S. F. Mingaleev, S. Flach, and Yu. S. Kivshar, "Nonlinear Fano resonance and bistable wave transmission," Phys. Rev. E 71, 036626 (2005).
[CrossRef]

Miroshnichenko, A. E.

A. E. Miroshnichenko, S. F. Mingaleev, S. Flach, and Yu. S. Kivshar, "Nonlinear Fano resonance and bistable wave transmission," Phys. Rev. E 71, 036626 (2005).
[CrossRef]

Pagnoux, D.

F. Brechet, P. Roy, J. Marcou, and D. Pagnoux, "Singlemode propagation into depressed-core-index photonic bandgap fibre designed for zero-dispersion propagation at short wavelengths," Electron. Lett. 36, 514-515 (2000).
[CrossRef]

Pearce, G. J.

Pfeiffer, L.

M. A. Duguay, Y. Kokubun, T. L. Koch, and L. Pfeiffer, "Antiresonant reflecting optical waveguides in SiO2-Si multilayer structures," Appl. Phys. Lett. 49, 13-15 (1986).
[CrossRef]

Pottage, J. M.

Provino, L.

Quiquempois, Y.

Renversez, G.

Roy, P.

F. Brechet, P. Roy, J. Marcou, and D. Pagnoux, "Singlemode propagation into depressed-core-index photonic bandgap fibre designed for zero-dispersion propagation at short wavelengths," Electron. Lett. 36, 514-515 (2000).
[CrossRef]

Russell, P. St. J.

Sagrini, A.

Simon, H. J.

Snitzer, E.

Steel, M. J.

Steinvurzel, P.

Stone, J. M.

Usner, B.

Uttamchandani, D.

J. Kubica, D. Uttamchandani, and B. Culshaw, "Modal propagation within ARROWwaveguides," Opt. Commun. 78, 133-136 (1990).
[CrossRef]

Wait, J. R.

J. R. Wait, "Scattering of a plane wave from a circular dielectric cylinder at oblique incidence," Canadian J. Phys. 33, 189-195 (1955).
[CrossRef]

Wang, A.

White, T. P.

Appl. Opt.

Appl. Phys. Lett.

M. A. Duguay, Y. Kokubun, T. L. Koch, and L. Pfeiffer, "Antiresonant reflecting optical waveguides in SiO2-Si multilayer structures," Appl. Phys. Lett. 49, 13-15 (1986).
[CrossRef]

Canadian J. Phys.

J. R. Wait, "Scattering of a plane wave from a circular dielectric cylinder at oblique incidence," Canadian J. Phys. 33, 189-195 (1955).
[CrossRef]

Electron. Lett.

F. Brechet, P. Roy, J. Marcou, and D. Pagnoux, "Singlemode propagation into depressed-core-index photonic bandgap fibre designed for zero-dispersion propagation at short wavelengths," Electron. Lett. 36, 514-515 (2000).
[CrossRef]

IEEE J. Quantum Electron.

T. Baba and Y. Kokubun, "Dispersion and radiation loss characteristics of antiresonant reflecting optical waveguides - numerical results and analytical expressions," IEEE J. Quantum Electron. 28,1689-1700 (1992).
[CrossRef]

J. Appl. Phys.

A. C. Lind and J. M. Greenberg, "Electromagnetic scattering by obliquely Oriented Cylinders," J. Appl. Phys. 37, 3195-3203 (1966).
[CrossRef]

J. Opt. A, Pure Appl. Opt.

J. Lægsgaard, "Gap formation and guided modes in photonic bandgap fibres with high-index rods," J. Opt. A, Pure Appl. Opt. 6, 798-804 (2004).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Opt. Commun.

J. Kubica, D. Uttamchandani, and B. Culshaw, "Modal propagation within ARROWwaveguides," Opt. Commun. 78, 133-136 (1990).
[CrossRef]

Opt. Express

A. K. Abeeluck, N. M. Litchinitser, C. Headley, and B. J. Eggleton, "Analysis of spectral characteristics of photonic bandgap waveguides," Opt. Express 10, 1320-1333 (2002).
[PubMed]

N. M. Litchinitser, S. C. Dunn, B. Usner, B. J. Eggleton, T. P. White, R. C. McPhedran, and C. M. de Sterke, "Resonances in microstructured optical waveguides," Opt. Express 11, 1243-1251 (2003).
[CrossRef] [PubMed]

A. Argyros, T. A. Birks, S. G. Leon-Saval, C. M. B. Cordeiro, and P. St. J. Russell, "Guidance properties of low-contrast photonic bandgap fibres," Opt. Express 13, 2503-2511 (2005).
[CrossRef] [PubMed]

P. Steinvurzel, B. T. Kuhlmey, T. P. White, M. J. Steel, C. M. de Sterke, and B. J. Eggleton, "Long wavelength anti-resonant guidance in high index inclusion microstructured fibers," Opt. Express 12, 5424-5433 (2004).
[CrossRef] [PubMed]

T. A. Birks, F. Luan, G. J. Pearce, A. Wang, J. C. Knight, and D. M. Bird, "Bend loss in all-solid bandgap fibres," Opt. Express 14, 5688-5698 (2006).
[CrossRef] [PubMed]

G. Renversez, P. Boyer, and A. Sagrini, "Antiresonant reflecting optical waveguide microstructured fibers revisited: a new analysis based on leaky mode coupling," Opt. Express 14, 5682-5687 (2006).
[CrossRef] [PubMed]

T. T. Larsen, A. Bjarklev, D. S. Hermann, and J. Broeng, "Optical devices based on liquid crystal photonic bandgap fibres," Opt. Express 11, 2589-2596 (2003).
[CrossRef] [PubMed]

A. Argyros, T. A. Birks, S. G. Leon-Saval, C. M. B. Cordeiro, F. Luan, and P. St. J. Russell, "Photonic bandgap with an index step of one percent," Opt. Express 13, 309-314 (2005).
[CrossRef] [PubMed]

G. Bouwmans, L. Bigot, Y. Quiquempois, F. Lopez, L. Provino, and M. Douay, "Fabrication and characterization of an all-solid 2D photonic bandgap fiber with a low-loss region (< 20 dB/km) around 1550 nm," Opt. Express 13, 8452-8459 (2005).
[CrossRef] [PubMed]

J. M. Stone, G. J. Pearce, F. Luan, T. A. Birks, J. C. Knight, A. K. George, and D. M. Bird, "An improved photonic bandgap fiber based on an array of rings," Opt. Express 14, 6291-6295 (2006).
[CrossRef] [PubMed]

T. A. Birks, D. M. Bird, T. D. Hedley, J. M. Pottage, and P. St. J. Russell, "Scaling laws and vector effects in bandgap-guiding fibers," Opt. Express 12, 69-74 (2004).
[CrossRef] [PubMed]

Opt. Lett.

Phys. Rev.

U. Fano, "Effects Of Configuration Interaction On Intensities And Phase Shifts," Phys. Rev. 124, 1866-1878 (1961).
[CrossRef]

Phys. Rev. B

S. Fan, and J. D. Joannopoulos, "Analysis of guided resonances in photonic crystal slabs," Phys. Rev. B 65, 235112 (2002).
[CrossRef]

Phys. Rev. E

A. E. Miroshnichenko, S. F. Mingaleev, S. Flach, and Yu. S. Kivshar, "Nonlinear Fano resonance and bistable wave transmission," Phys. Rev. E 71, 036626 (2005).
[CrossRef]

Other

http://www.physics.usyd.edu.au/cudos/mofsoftware

C. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1998).
[CrossRef]

Note that we define ϑ0 such that it approaches zero for near glancing incidence, which we feel is the natural choice in the current context; in most of the literature on scattering by cylinders [16,21,22], however, the conicity angle approaches π=2 for glancing incidence, and our definition is equivalent to π=2¡α in the earlier references.

A. W. Snyder and J.D. Love, Optical Waveguide Theory (Chapman and Hall, London, 1983).

R. T. Bise, R. S. Windeler, K. S. Kranz, C. Kerbage, B. J. Eggleton, and D. J. Trevor, "Tunable photonic bandgap fiber," in Optical Fiber Communications Conference, Post Conference Ed., Vol. 70 of OSA Trends in Optics and Photonics Series Technical Digest (Optical Society of America, Washington, D. C., 2002), 466-468.

Supplementary Material (2)

» Media 1: GIF (1156 KB)     
» Media 2: GIF (949 KB)     

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Figures (9)

Fig. 1.
Fig. 1.

Schematic of (a) single layer planar ARROW with antiresonant mode in low index core, (b) SC-PGBF index profile, and (c) transmission loss through a SC-PBGF. Vertical lines correspond to cutoff frequencies of the vector modes of the high index cylinders: red=TE/TM0,p,HE2,p, blue=HE1,p+1, and green=EH1,p;HE3,p

Fig. 2.
Fig. 2.

Geometry of plane wave scattering by a dielectric cylinder at conical incidence. The variable φ is defined such that regions of forward or backward scattering are given by cos(φ)>0 or cos(φ)<0, respectively. The polarization is chosen such that the cylinder axis lies in the plane defined by H and k. In the related SC-PBGF geometry, the region below the cylinder corresponds to the SC-PBGF core.

Fig. 3.
Fig. 3.

Im(n eff) versus V for the first 6 bands of a SC-PBGF with d/Λ=0.15 (upper panel), scattering cross section σsc (middle panel) and asymmetry factor g (lower panel) for ϑ0=0.025 rad; the scattering data are calculated using 3 Fourier-Bessel orders. Red and blue vertical lines correspond to TE0,p /HE2,p and HE1,p+1 cutoff frequencies, respectively.

Fig. 4.
Fig. 4.

3-D line plots of B0H (red) and B1H (blue) as a function of V, with projections on the real axis (right panel) and complex plane (bottom panel) shown.

Fig. 5.
Fig. 5.

Sine of the phase-dependent terms which contribute to g using same simulation parameters as in Fig. 3 and 4.

Fig. 6.
Fig. 6.

Im(n eff) versus V for the first 6 bands of a 3 ring SC-PBGF with (a) d/Λ=0.6, (b) d/Λ=0.4, (c) d/Λ=0.2, and (d) d/Λ=0.15.

Fig. 7.
Fig. 7.

Animations of (a) scattering cross section σsc (b) and asymmetry factor g for three cylinders with d/Λ varying from 0.1 to 0.8 and ϑ0=0.025 rad is constant. Vertical lines indicate modal cutoff frequencies of the single cylinder. Figure shown in text corresponds to d/Λ=0.4. [Media 1] [Media 2]

Fig. 8.
Fig. 8.

Im(n eff) versus V for the first 6 bands of a SC-PBGF with d/Λ=0.4 and different types of defect cores. The discontinuity at the center of the 5th band in (b) is due to the fact that the numerical precision of the simulation does not allow one to find values of Im(n eff) below 10-14. The core designs in (c) and (d) show Fano resonance-like corners at the edges of the odd order bands and improved confinement in the 1st order band.

Fig. 9.
Fig. 9.

Im(n eff) versus V for the first order band of a SC-PBGF with d/Λ=0.4 and different types of defect cores.

Equations (9)

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E z ( r , φ , z , t ) = [ ψ i E ( r , φ ) + ψ s E ( r , φ ) ] e i ( β z ω t ) + c . c . ,
ψ i f ( r , φ ) = q = A q f J q ( k r ) e iq φ ,
ψ s f ( r , φ ) = q = B q f H q ( 1 ) ( k r ) e iq φ ,
A q E = E 0 sin ( δ 0 ) sin ( ϑ 0 ) e iq ( π 2 φ 0 ) ,
A q H = n low E 0 Z 0 cos ( δ 0 ) sin ( ϑ 0 ) e iq ( π 2 φ 0 ) ,
σ d ( φ ) = 2 π E 0 2 k sin 2 ϑ 0 ( g E ( φ ) 2 + g H ( φ ) 2 ) ,
g f = q = B q f e i [ q ( φ π 2 ) π 4 ] .
g = 1 σ sc 0 2 π σ d ( φ ) cos φ d φ ;
g = b 0 H b 1 H sin ( ϕ 0 H ϕ 1 H ) + b 1 H b 2 H sin ( ϕ 1 H ϕ 2 H ) + b 1 E b 2 E sin ( ϕ 1 E ϕ 2 E ) 1 2 ( b 0 H ) 2 + ( b 1 H ) 2 + ( b 2 H ) 2 + ( b 1 E ) 2 + ( b 2 E ) 2 ,

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