Abstract

We numerically and theoretically investigate the core modes of two-dimensional solid-core photonic bandgap (PBG) fibers based on hexagonal arrays of high-index circular rods. Such fibers guide light in discrete bandgaps, and the number of core-guided modes depends on the order of the bandgap as well as the position within the bandgap. We first classify the different core-guided modes in such fibers and we discuss the links among band structure, losses, and number and type of modes. We demonstrate that, similar to the case of bandgapless Kagome and ring-based fibers, solid-core bandgap fibers can have core-guided modes that are within photonic bands of the cladding. We discuss the classification of core modes in such fibers, and highlight analogies and differences with that of index-guiding fibers. Through an asymptotic expansion of an analytic model of a fiber’s photonic bands, we show that, in the limit of higher-order gaps (i.e., short wavelengths), the number of modes in the middle of gaps tends to a constant that is independent of refractive index contrast, as is the case for index-guiding photonic crystal fibers. We also discuss the evolution of the effectively single-mode propagation regime with geometrical parameters of structures having constant or variable band diagrams. For small- and large-core PBG fibers, we compute the exact number of core-guided modes within the center of the transmission band. We discuss their evolution with gap orders and coupling strength between high-index inclusions in the cladding. We find good agreement of the core-guided mode number in the center of the gaps computed with our theoretical model and with a numerical method for short wavelengths.

© 2012 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. Knight, T. Birks, P. Russel, and D. Atkin, “All-silica single-mode optical fiber with photonic crystal cladding,” Opt. Lett. 21, 1547–1549 (1996).
    [CrossRef]
  2. T. A. Birks, J. C. Knight, and P. St. J. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. 22, 961–963 (1997).
    [CrossRef]
  3. S. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, T. Engeness, M. Soljacic, S. Jacobs, J. Joannopoulos, and Y. Fink, “Low-loss asymptotically single-mode propagation in large-core omniguide fibers,” Opt. Express 9, 748–779 (2001).
    [CrossRef]
  4. R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allen, “Single-mode photonic band gap guidance of light in air,” Science 285, 1537–1539 (1999).
    [CrossRef]
  5. 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]
  6. N. M. Litchinitser, A. K. Abeeluck, C. Headley, and B. J. Eggleton, “Antiresonant reflecting photonic crystal optical waveguides,” Opt. Lett. 27, 1592–1594 (2002).
    [CrossRef]
  7. 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]
  8. B. T. Kuhlmey, B. J. Eggleton, and D. K. C. Wu, “Fluid-filled solid-core photonic bandgap fibers,” J. Lightwave Technol. 27, 1617–1630 (2009).
    [CrossRef]
  9. V. Pureur and J. M. Dudley, “Nonlinear spectral broadening of femtosecond pulses in solid-core photonic bandgap fibers,” Opt. Lett. 35, 2813–2815 (2010).
    [CrossRef]
  10. T. Taru and J. C. Knight, “Raman gain suppression in all-solid photonic bandgap fiber,” in 33rd European Conference and Exhibition of Optical Communication (IEEE, 2007), pp. 1–2.
  11. V. Pureur, L. Bigot, G. Bouwmans, Y. Quiquempois, M. Douay, and Y. Jaouen, “Ytterbium-doped solid core photonic bandgap fiber for laser operation around 980 nm,” Appl. Phys. Lett. 92, 061113 (2008).
    [CrossRef]
  12. C. B. Olausson, A. Shirakawa, M. Chen, J. K. Lyngs, J. Broeng, K. P. Hansen, A. Bjarklev, and K. Ueda, “167 W, power scalable ytterbium-doped photonic bandgap fiber amplifier at 1178 nm,” Opt. Express 18, 16345–16352 (2010).
    [CrossRef]
  13. A. Isomäki and O. G. Okhotnikov, “Femtosecond soliton mode-locked laser based on ytterbium-doped photonic bandgap fiber,” Opt. Express 14, 9238–9243 (2006).
    [CrossRef]
  14. B. T. Kuhlmey, R. C. McPhedran, and C. M. de Sterke, “Modal cutoff in microstructured optical fibers,” Opt. Lett. 27, 1684–1686 (2002).
    [CrossRef]
  15. K. Saitoh and M. Koshiba, “Empirical relations for simple design of photonic crystal fibers,” Opt. Express 13, 267–274 (2005).
    [CrossRef]
  16. B. T. Kuhlmey, “Theoretical and numerical investigation of the physics of microstructured optical fibres,” Ph.D. dissertation (University of Sydney and Université Aix-Marseille III, 2003), http://hdl.handle.net/2123/560 .
  17. G. Renversez, F. Bordas, and B. T. Kuhlmey, “Second mode transition in microstructured optical fibers: determination of the critical geometrical parameter and study of the matrix refractive index and effects of cladding size,” Opt. Lett. 30, 1264–1266 (2005).
    [CrossRef]
  18. N. A. Mortensen, “Semianalytical approach to short-wavelength dispersion and modal properties of photonic crystal fibers,” Opt. Lett. 30, 1455–1457 (2005).
    [CrossRef]
  19. M. J. F. Digonnet, H. K. Kim, G. S. Kino, and S. Fan, “Understanding air-core photonic-bandgap fibers: analogy to conventional fibers,” J. Lightwave Technol. 23, 4169–4177 (2005).
    [CrossRef]
  20. V. Pureur, J. C. Knight, and B. T. Kuhlmey, “Higher order guided mode propagation in solid-core photonic bandgap fibers,” Opt. Express 18, 8906–8915 (2010).
    [CrossRef]
  21. R. Guobin, W. Zhi, L. Shuqin, and J. Shuisheng, “Mode classification and degeneracy in photonic crystal fibers,” Opt. Express 11, 1310–1321 (2003).
    [CrossRef]
  22. T. P. White, B. T. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. Martijn de Sterke, and L. C. Botten, “Multipole method for microstructured optical fibers. I. Formulation,” J. Opt. Soc. Am. B 19, 2322–2330 (2002).
    [CrossRef]
  23. T. A. Birks, F. Luan, G. J. Pearce, A. Wang, J. C. Knight, and D. M. Bird, “Bend loss in all-solid bandgap fibers,” Opt. Express 14, 5688–5698 (2006).
    [CrossRef]
  24. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals Molding the Flow of Light, 2nd ed. (Princeton University, 2008).
  25. T. A. Birks, G. J. Pearce, and D. M. Bird, “Approximate band structure calculation for photonic bandgap fibers,” Opt. Express 14, 9483–9490 (2006).
    [CrossRef]
  26. P. Steinvurzel, C. Martijn de Sterke, M. J. Steel, B. T. Kuhlmey, and B. J. Eggleton, “Single scatterer Fano resonances in solid core photonic band gap fibers,” Opt. Express 14, 8797–8811 (2006).
    [CrossRef]
  27. A. Snyder and J. Love, Optical Waveguide Theory (Kluwer Academic, 1983).
  28. 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]
  29. F. Couny, F. Benabid, P. J. Roberts, P. S. Light, and M. G. Raymer, “Generation and photonic guidance of multioctave optical-frequency combs,” Science 318, 1118–1121 (2007).
    [CrossRef]
  30. A. Argyros and J. Pla, “Hollow-core polymer fibres with a kagome lattice: potential for transmission in the infrared,” Opt. Express 15, 7713–7719 (2007).
    [CrossRef]
  31. A. Argyros, S. G. Leon-Saval, J. Pla, and A. Docherty, “Antiresonant reflection and inhibited coupling in hollow-core square lattice optical fibers,” Opt. Express 16, 5642–5648 (2008).
    [CrossRef]
  32. T. Grujic, B. T. Kuhlmey, A. Argyros, S. Coen, and C. M. de Sterke, “Solid-core fiber with ultra-wide bandwidth transmission window due to inhibited coupling,” Opt. Express 18, 25556–25566 (2010).
    [CrossRef]
  33. T. White, R. McPhedran, L. Botten, G. Smith, and C. M. de Sterke, “Calculations of air-guided modes in photonic crystal fibers using the multipole method,” Opt. Express 9, 721–732 (2001).
    [CrossRef]
  34. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, 1991).
  35. M. Koshiba and K. Saitoh, “Applicability of classical optical fiber theories to holey fibers,” Opt. Lett. 29, 1739–1741 (2004).
    [CrossRef]
  36. M. Kashiwagi, K. Saitoh, K. Takenaga, S. Tanigawa, S. Matsuo, and M. Fujimaki, “Low bending loss and effectively single-mode all-solid photonic bandgap fiber with an effective area of 650  μm2,” Opt. Lett. 37, 1292–1294 (2012).
    [CrossRef]
  37. B. Ward, “Solid-core photonic bandgap fibers for cladding-pumped Raman amplification,” Opt. Express 19, 11852–11866 (2011).
    [CrossRef]
  38. P. D. Rasmussen, J. Lægsgaard, and O. Bang, “Degenerate four wave mixing in solid core photonic bandgap fibers,” Opt. Express 16, 4059–4068 (2008).
    [CrossRef]
  39. V. Pureur, L. Bigot, G. Bouwmans, Y. Quiquempois, M. Douay, and Y. Jaouen, “Ytterbium-doped solid core photonic bandgap fiber for laser operation around 980 nm,” Appl. Phys. Lett. 92, 061113 (2008).
    [CrossRef]
  40. V. Pureur, A. Bétourné, G. Bouwmans, L. Bigot, A. Kudlinski, K. Delplace, A. Le Rouge, Y. Quiquempois, and M. Douay, “Overview on solid core photonic bandgap fibers,” Fiber Integr. Opt. 28, 27–50 (2009).
    [CrossRef]
  41. Y. Ould-Agha, A. Bétourné, O. Vanvincq, G. Bouwmans, and Y. Quiquempois, “Broadband bandgap guidance and mode filtering in radially hybrid photonic crystal fiber,” Opt. Express 20, 6746–6760 (2012).
    [CrossRef]
  42. V. Pureur and J. M. Dudley, “Design of solid core photonic bandgap fibers for visible supercontinuum generation,” Opt. Commun. 284, 1661–1668 (2011).
    [CrossRef]
  43. O. N. Egorova, S. L. Semjonov, A. F. Kosolapov, A. N. Denisov, A. D. Pryamikov, D. A. Gaponov, A. S. Biriukov, E. M. Dianov, M. Y. Salganskii, V. F. Khopin, M. V. Yashkov, A. N. Gurianov, and D. V. Kuksenkov, “Single-mode all-silica photonic bandgap fiber with 20 m mode-field diameter,” Opt. Express 16, 11735–11740 (2008).
    [CrossRef]
  44. BandSOLVE 4.1 (Rsoft Design Group, Inc., 2008).
  45. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1965).

2012 (2)

2011 (2)

V. Pureur and J. M. Dudley, “Design of solid core photonic bandgap fibers for visible supercontinuum generation,” Opt. Commun. 284, 1661–1668 (2011).
[CrossRef]

B. Ward, “Solid-core photonic bandgap fibers for cladding-pumped Raman amplification,” Opt. Express 19, 11852–11866 (2011).
[CrossRef]

2010 (4)

2009 (2)

B. T. Kuhlmey, B. J. Eggleton, and D. K. C. Wu, “Fluid-filled solid-core photonic bandgap fibers,” J. Lightwave Technol. 27, 1617–1630 (2009).
[CrossRef]

V. Pureur, A. Bétourné, G. Bouwmans, L. Bigot, A. Kudlinski, K. Delplace, A. Le Rouge, Y. Quiquempois, and M. Douay, “Overview on solid core photonic bandgap fibers,” Fiber Integr. Opt. 28, 27–50 (2009).
[CrossRef]

2008 (5)

2007 (2)

A. Argyros and J. Pla, “Hollow-core polymer fibres with a kagome lattice: potential for transmission in the infrared,” Opt. Express 15, 7713–7719 (2007).
[CrossRef]

F. Couny, F. Benabid, P. J. Roberts, P. S. Light, and M. G. Raymer, “Generation and photonic guidance of multioctave optical-frequency combs,” Science 318, 1118–1121 (2007).
[CrossRef]

2006 (4)

2005 (5)

2004 (2)

2003 (1)

2002 (3)

2001 (2)

1999 (1)

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allen, “Single-mode photonic band gap guidance of light in air,” Science 285, 1537–1539 (1999).
[CrossRef]

1997 (1)

1996 (1)

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]

Abeeluck, A. K.

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1965).

Allen, D. C.

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allen, “Single-mode photonic band gap guidance of light in air,” Science 285, 1537–1539 (1999).
[CrossRef]

Argyros, A.

Atkin, D.

Bang, O.

Benabid, F.

F. Couny, F. Benabid, P. J. Roberts, P. S. Light, and M. G. Raymer, “Generation and photonic guidance of multioctave optical-frequency combs,” Science 318, 1118–1121 (2007).
[CrossRef]

Bétourné, A.

Y. Ould-Agha, A. Bétourné, O. Vanvincq, G. Bouwmans, and Y. Quiquempois, “Broadband bandgap guidance and mode filtering in radially hybrid photonic crystal fiber,” Opt. Express 20, 6746–6760 (2012).
[CrossRef]

V. Pureur, A. Bétourné, G. Bouwmans, L. Bigot, A. Kudlinski, K. Delplace, A. Le Rouge, Y. Quiquempois, and M. Douay, “Overview on solid core photonic bandgap fibers,” Fiber Integr. Opt. 28, 27–50 (2009).
[CrossRef]

Bigot, L.

V. Pureur, A. Bétourné, G. Bouwmans, L. Bigot, A. Kudlinski, K. Delplace, A. Le Rouge, Y. Quiquempois, and M. Douay, “Overview on solid core photonic bandgap fibers,” Fiber Integr. Opt. 28, 27–50 (2009).
[CrossRef]

V. Pureur, L. Bigot, G. Bouwmans, Y. Quiquempois, M. Douay, and Y. Jaouen, “Ytterbium-doped solid core photonic bandgap fiber for laser operation around 980 nm,” Appl. Phys. Lett. 92, 061113 (2008).
[CrossRef]

V. Pureur, L. Bigot, G. Bouwmans, Y. Quiquempois, M. Douay, and Y. Jaouen, “Ytterbium-doped solid core photonic bandgap fiber for laser operation around 980 nm,” Appl. Phys. Lett. 92, 061113 (2008).
[CrossRef]

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]

Bird, D. M.

Biriukov, A. S.

Birks, T.

Birks, T. A.

Bjarklev, A.

Bordas, F.

Botten, L.

Botten, L. C.

Bouwmans, G.

Y. Ould-Agha, A. Bétourné, O. Vanvincq, G. Bouwmans, and Y. Quiquempois, “Broadband bandgap guidance and mode filtering in radially hybrid photonic crystal fiber,” Opt. Express 20, 6746–6760 (2012).
[CrossRef]

V. Pureur, A. Bétourné, G. Bouwmans, L. Bigot, A. Kudlinski, K. Delplace, A. Le Rouge, Y. Quiquempois, and M. Douay, “Overview on solid core photonic bandgap fibers,” Fiber Integr. Opt. 28, 27–50 (2009).
[CrossRef]

V. Pureur, L. Bigot, G. Bouwmans, Y. Quiquempois, M. Douay, and Y. Jaouen, “Ytterbium-doped solid core photonic bandgap fiber for laser operation around 980 nm,” Appl. Phys. Lett. 92, 061113 (2008).
[CrossRef]

V. Pureur, L. Bigot, G. Bouwmans, Y. Quiquempois, M. Douay, and Y. Jaouen, “Ytterbium-doped solid core photonic bandgap fiber for laser operation around 980 nm,” Appl. Phys. Lett. 92, 061113 (2008).
[CrossRef]

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]

Broeng, J.

Chen, M.

Coen, S.

Couny, F.

F. Couny, F. Benabid, P. J. Roberts, P. S. Light, and M. G. Raymer, “Generation and photonic guidance of multioctave optical-frequency combs,” Science 318, 1118–1121 (2007).
[CrossRef]

Cregan, R. F.

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allen, “Single-mode photonic band gap guidance of light in air,” Science 285, 1537–1539 (1999).
[CrossRef]

de Sterke, C. M.

de Sterke, C. Martijn

Delplace, K.

V. Pureur, A. Bétourné, G. Bouwmans, L. Bigot, A. Kudlinski, K. Delplace, A. Le Rouge, Y. Quiquempois, and M. Douay, “Overview on solid core photonic bandgap fibers,” Fiber Integr. Opt. 28, 27–50 (2009).
[CrossRef]

Denisov, A. N.

Dianov, E. M.

Digonnet, M. J. F.

Docherty, A.

Douay, M.

V. Pureur, A. Bétourné, G. Bouwmans, L. Bigot, A. Kudlinski, K. Delplace, A. Le Rouge, Y. Quiquempois, and M. Douay, “Overview on solid core photonic bandgap fibers,” Fiber Integr. Opt. 28, 27–50 (2009).
[CrossRef]

V. Pureur, L. Bigot, G. Bouwmans, Y. Quiquempois, M. Douay, and Y. Jaouen, “Ytterbium-doped solid core photonic bandgap fiber for laser operation around 980 nm,” Appl. Phys. Lett. 92, 061113 (2008).
[CrossRef]

V. Pureur, L. Bigot, G. Bouwmans, Y. Quiquempois, M. Douay, and Y. Jaouen, “Ytterbium-doped solid core photonic bandgap fiber for laser operation around 980 nm,” Appl. Phys. Lett. 92, 061113 (2008).
[CrossRef]

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]

Dudley, J. M.

V. Pureur and J. M. Dudley, “Design of solid core photonic bandgap fibers for visible supercontinuum generation,” Opt. Commun. 284, 1661–1668 (2011).
[CrossRef]

V. Pureur and J. M. Dudley, “Nonlinear spectral broadening of femtosecond pulses in solid-core photonic bandgap fibers,” Opt. Lett. 35, 2813–2815 (2010).
[CrossRef]

Eggleton, B. J.

Egorova, O. N.

Engeness, T.

Fan, S.

Fink, Y.

Fujimaki, M.

Gaponov, D. A.

George, A. K.

Grujic, T.

Guobin, R.

Gurianov, A. N.

Hansen, K. P.

Headley, C.

Hedley, T. D.

Ibanescu, M.

Isomäki, A.

Jacobs, S.

Jaouen, Y.

V. Pureur, L. Bigot, G. Bouwmans, Y. Quiquempois, M. Douay, and Y. Jaouen, “Ytterbium-doped solid core photonic bandgap fiber for laser operation around 980 nm,” Appl. Phys. Lett. 92, 061113 (2008).
[CrossRef]

V. Pureur, L. Bigot, G. Bouwmans, Y. Quiquempois, M. Douay, and Y. Jaouen, “Ytterbium-doped solid core photonic bandgap fiber for laser operation around 980 nm,” Appl. Phys. Lett. 92, 061113 (2008).
[CrossRef]

Joannopoulos, J.

Joannopoulos, J. D.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals Molding the Flow of Light, 2nd ed. (Princeton University, 2008).

Johnson, S.

Johnson, S. G.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals Molding the Flow of Light, 2nd ed. (Princeton University, 2008).

Kashiwagi, M.

Khopin, V. F.

Kim, H. K.

Kino, G. S.

Knight, J.

Knight, J. C.

Koshiba, M.

Kosolapov, A. F.

Kudlinski, A.

V. Pureur, A. Bétourné, G. Bouwmans, L. Bigot, A. Kudlinski, K. Delplace, A. Le Rouge, Y. Quiquempois, and M. Douay, “Overview on solid core photonic bandgap fibers,” Fiber Integr. Opt. 28, 27–50 (2009).
[CrossRef]

Kuhlmey, B. T.

T. Grujic, B. T. Kuhlmey, A. Argyros, S. Coen, and C. M. de Sterke, “Solid-core fiber with ultra-wide bandwidth transmission window due to inhibited coupling,” Opt. Express 18, 25556–25566 (2010).
[CrossRef]

V. Pureur, J. C. Knight, and B. T. Kuhlmey, “Higher order guided mode propagation in solid-core photonic bandgap fibers,” Opt. Express 18, 8906–8915 (2010).
[CrossRef]

B. T. Kuhlmey, B. J. Eggleton, and D. K. C. Wu, “Fluid-filled solid-core photonic bandgap fibers,” J. Lightwave Technol. 27, 1617–1630 (2009).
[CrossRef]

P. Steinvurzel, C. Martijn de Sterke, M. J. Steel, B. T. Kuhlmey, and B. J. Eggleton, “Single scatterer Fano resonances in solid core photonic band gap fibers,” Opt. Express 14, 8797–8811 (2006).
[CrossRef]

G. Renversez, F. Bordas, and B. T. Kuhlmey, “Second mode transition in microstructured optical fibers: determination of the critical geometrical parameter and study of the matrix refractive index and effects of cladding size,” Opt. Lett. 30, 1264–1266 (2005).
[CrossRef]

T. P. White, B. T. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. Martijn de Sterke, and L. C. Botten, “Multipole method for microstructured optical fibers. I. Formulation,” J. Opt. Soc. Am. B 19, 2322–2330 (2002).
[CrossRef]

B. T. Kuhlmey, R. C. McPhedran, and C. M. de Sterke, “Modal cutoff in microstructured optical fibers,” Opt. Lett. 27, 1684–1686 (2002).
[CrossRef]

B. T. Kuhlmey, “Theoretical and numerical investigation of the physics of microstructured optical fibres,” Ph.D. dissertation (University of Sydney and Université Aix-Marseille III, 2003), http://hdl.handle.net/2123/560 .

Kuksenkov, D. V.

Lægsgaard, J.

Le Rouge, A.

V. Pureur, A. Bétourné, G. Bouwmans, L. Bigot, A. Kudlinski, K. Delplace, A. Le Rouge, Y. Quiquempois, and M. Douay, “Overview on solid core photonic bandgap fibers,” Fiber Integr. Opt. 28, 27–50 (2009).
[CrossRef]

Leon-Saval, S. G.

Light, P. S.

F. Couny, F. Benabid, P. J. Roberts, P. S. Light, and M. G. Raymer, “Generation and photonic guidance of multioctave optical-frequency combs,” Science 318, 1118–1121 (2007).
[CrossRef]

Litchinitser, N. M.

Lopez, F.

Love, J.

A. Snyder and J. Love, Optical Waveguide Theory (Kluwer Academic, 1983).

Luan, F.

Lyngs, J. K.

Mangan, B. J.

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allen, “Single-mode photonic band gap guidance of light in air,” Science 285, 1537–1539 (1999).
[CrossRef]

Marcuse, D.

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, 1991).

Matsuo, S.

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.

McPhedran, R. C.

Meade, R. D.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals Molding the Flow of Light, 2nd ed. (Princeton University, 2008).

Mortensen, N. A.

Okhotnikov, O. G.

Olausson, C. B.

Ould-Agha, Y.

Pearce, G. J.

Pla, J.

Provino, L.

Pryamikov, A. D.

Pureur, V.

V. Pureur and J. M. Dudley, “Design of solid core photonic bandgap fibers for visible supercontinuum generation,” Opt. Commun. 284, 1661–1668 (2011).
[CrossRef]

V. Pureur, J. C. Knight, and B. T. Kuhlmey, “Higher order guided mode propagation in solid-core photonic bandgap fibers,” Opt. Express 18, 8906–8915 (2010).
[CrossRef]

V. Pureur and J. M. Dudley, “Nonlinear spectral broadening of femtosecond pulses in solid-core photonic bandgap fibers,” Opt. Lett. 35, 2813–2815 (2010).
[CrossRef]

V. Pureur, A. Bétourné, G. Bouwmans, L. Bigot, A. Kudlinski, K. Delplace, A. Le Rouge, Y. Quiquempois, and M. Douay, “Overview on solid core photonic bandgap fibers,” Fiber Integr. Opt. 28, 27–50 (2009).
[CrossRef]

V. Pureur, L. Bigot, G. Bouwmans, Y. Quiquempois, M. Douay, and Y. Jaouen, “Ytterbium-doped solid core photonic bandgap fiber for laser operation around 980 nm,” Appl. Phys. Lett. 92, 061113 (2008).
[CrossRef]

V. Pureur, L. Bigot, G. Bouwmans, Y. Quiquempois, M. Douay, and Y. Jaouen, “Ytterbium-doped solid core photonic bandgap fiber for laser operation around 980 nm,” Appl. Phys. Lett. 92, 061113 (2008).
[CrossRef]

Quiquempois, Y.

Y. Ould-Agha, A. Bétourné, O. Vanvincq, G. Bouwmans, and Y. Quiquempois, “Broadband bandgap guidance and mode filtering in radially hybrid photonic crystal fiber,” Opt. Express 20, 6746–6760 (2012).
[CrossRef]

V. Pureur, A. Bétourné, G. Bouwmans, L. Bigot, A. Kudlinski, K. Delplace, A. Le Rouge, Y. Quiquempois, and M. Douay, “Overview on solid core photonic bandgap fibers,” Fiber Integr. Opt. 28, 27–50 (2009).
[CrossRef]

V. Pureur, L. Bigot, G. Bouwmans, Y. Quiquempois, M. Douay, and Y. Jaouen, “Ytterbium-doped solid core photonic bandgap fiber for laser operation around 980 nm,” Appl. Phys. Lett. 92, 061113 (2008).
[CrossRef]

V. Pureur, L. Bigot, G. Bouwmans, Y. Quiquempois, M. Douay, and Y. Jaouen, “Ytterbium-doped solid core photonic bandgap fiber for laser operation around 980 nm,” Appl. Phys. Lett. 92, 061113 (2008).
[CrossRef]

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]

Rasmussen, P. D.

Raymer, M. G.

F. Couny, F. Benabid, P. J. Roberts, P. S. Light, and M. G. Raymer, “Generation and photonic guidance of multioctave optical-frequency combs,” Science 318, 1118–1121 (2007).
[CrossRef]

Renversez, G.

Roberts, P. J.

F. Couny, F. Benabid, P. J. Roberts, P. S. Light, and M. G. Raymer, “Generation and photonic guidance of multioctave optical-frequency combs,” Science 318, 1118–1121 (2007).
[CrossRef]

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allen, “Single-mode photonic band gap guidance of light in air,” Science 285, 1537–1539 (1999).
[CrossRef]

Russel, P.

Russell, P. St. J.

Saitoh, K.

Salganskii, M. Y.

Semjonov, S. L.

Shirakawa, A.

Shuisheng, J.

Shuqin, L.

Skorobogatiy, M.

Smith, G.

Snyder, A.

A. Snyder and J. Love, Optical Waveguide Theory (Kluwer Academic, 1983).

Soljacic, M.

Steel, M. J.

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1965).

Steinvurzel, P.

Takenaga, K.

Tanigawa, S.

Taru, T.

T. Taru and J. C. Knight, “Raman gain suppression in all-solid photonic bandgap fiber,” in 33rd European Conference and Exhibition of Optical Communication (IEEE, 2007), pp. 1–2.

Ueda, K.

Vanvincq, O.

Wang, A.

Ward, B.

Weisberg, O.

White, T.

White, T. P.

Winn, J. N.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals Molding the Flow of Light, 2nd ed. (Princeton University, 2008).

Wu, D. K. C.

Yashkov, M. V.

Zhi, W.

Appl. Phys. Lett. (2)

V. Pureur, L. Bigot, G. Bouwmans, Y. Quiquempois, M. Douay, and Y. Jaouen, “Ytterbium-doped solid core photonic bandgap fiber for laser operation around 980 nm,” Appl. Phys. Lett. 92, 061113 (2008).
[CrossRef]

V. Pureur, L. Bigot, G. Bouwmans, Y. Quiquempois, M. Douay, and Y. Jaouen, “Ytterbium-doped solid core photonic bandgap fiber for laser operation around 980 nm,” Appl. Phys. Lett. 92, 061113 (2008).
[CrossRef]

Fiber Integr. Opt. (1)

V. Pureur, A. Bétourné, G. Bouwmans, L. Bigot, A. Kudlinski, K. Delplace, A. Le Rouge, Y. Quiquempois, and M. Douay, “Overview on solid core photonic bandgap fibers,” Fiber Integr. Opt. 28, 27–50 (2009).
[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. (2)

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

Opt. Commun. (1)

V. Pureur and J. M. Dudley, “Design of solid core photonic bandgap fibers for visible supercontinuum generation,” Opt. Commun. 284, 1661–1668 (2011).
[CrossRef]

Opt. Express (18)

T. White, R. McPhedran, L. Botten, G. Smith, and C. M. de Sterke, “Calculations of air-guided modes in photonic crystal fibers using the multipole method,” Opt. Express 9, 721–732 (2001).
[CrossRef]

S. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, T. Engeness, M. Soljacic, S. Jacobs, J. Joannopoulos, and Y. Fink, “Low-loss asymptotically single-mode propagation in large-core omniguide fibers,” Opt. Express 9, 748–779 (2001).
[CrossRef]

R. Guobin, W. Zhi, L. Shuqin, and J. Shuisheng, “Mode classification and degeneracy in photonic crystal fibers,” Opt. Express 11, 1310–1321 (2003).
[CrossRef]

V. Pureur, J. C. Knight, and B. T. Kuhlmey, “Higher order guided mode propagation in solid-core photonic bandgap fibers,” Opt. Express 18, 8906–8915 (2010).
[CrossRef]

C. B. Olausson, A. Shirakawa, M. Chen, J. K. Lyngs, J. Broeng, K. P. Hansen, A. Bjarklev, and K. Ueda, “167 W, power scalable ytterbium-doped photonic bandgap fiber amplifier at 1178 nm,” Opt. Express 18, 16345–16352 (2010).
[CrossRef]

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

P. Steinvurzel, C. Martijn de Sterke, M. J. Steel, B. T. Kuhlmey, and B. J. Eggleton, “Single scatterer Fano resonances in solid core photonic band gap fibers,” Opt. Express 14, 8797–8811 (2006).
[CrossRef]

A. Isomäki and O. G. Okhotnikov, “Femtosecond soliton mode-locked laser based on ytterbium-doped photonic bandgap fiber,” Opt. Express 14, 9238–9243 (2006).
[CrossRef]

T. A. Birks, G. J. Pearce, and D. M. Bird, “Approximate band structure calculation for photonic bandgap fibers,” Opt. Express 14, 9483–9490 (2006).
[CrossRef]

A. Argyros and J. Pla, “Hollow-core polymer fibres with a kagome lattice: potential for transmission in the infrared,” Opt. Express 15, 7713–7719 (2007).
[CrossRef]

P. D. Rasmussen, J. Lægsgaard, and O. Bang, “Degenerate four wave mixing in solid core photonic bandgap fibers,” Opt. Express 16, 4059–4068 (2008).
[CrossRef]

A. Argyros, S. G. Leon-Saval, J. Pla, and A. Docherty, “Antiresonant reflection and inhibited coupling in hollow-core square lattice optical fibers,” Opt. Express 16, 5642–5648 (2008).
[CrossRef]

O. N. Egorova, S. L. Semjonov, A. F. Kosolapov, A. N. Denisov, A. D. Pryamikov, D. A. Gaponov, A. S. Biriukov, E. M. Dianov, M. Y. Salganskii, V. F. Khopin, M. V. Yashkov, A. N. Gurianov, and D. V. Kuksenkov, “Single-mode all-silica photonic bandgap fiber with 20 m mode-field diameter,” Opt. Express 16, 11735–11740 (2008).
[CrossRef]

T. Grujic, B. T. Kuhlmey, A. Argyros, S. Coen, and C. M. de Sterke, “Solid-core fiber with ultra-wide bandwidth transmission window due to inhibited coupling,” Opt. Express 18, 25556–25566 (2010).
[CrossRef]

B. Ward, “Solid-core photonic bandgap fibers for cladding-pumped Raman amplification,” Opt. Express 19, 11852–11866 (2011).
[CrossRef]

Y. Ould-Agha, A. Bétourné, O. Vanvincq, G. Bouwmans, and Y. Quiquempois, “Broadband bandgap guidance and mode filtering in radially hybrid photonic crystal fiber,” Opt. Express 20, 6746–6760 (2012).
[CrossRef]

K. Saitoh and M. Koshiba, “Empirical relations for simple design of photonic crystal fibers,” Opt. Express 13, 267–274 (2005).
[CrossRef]

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]

Opt. Lett. (10)

G. Renversez, F. Bordas, and B. T. Kuhlmey, “Second mode transition in microstructured optical fibers: determination of the critical geometrical parameter and study of the matrix refractive index and effects of cladding size,” Opt. Lett. 30, 1264–1266 (2005).
[CrossRef]

N. A. Mortensen, “Semianalytical approach to short-wavelength dispersion and modal properties of photonic crystal fibers,” Opt. Lett. 30, 1455–1457 (2005).
[CrossRef]

M. Kashiwagi, K. Saitoh, K. Takenaga, S. Tanigawa, S. Matsuo, and M. Fujimaki, “Low bending loss and effectively single-mode all-solid photonic bandgap fiber with an effective area of 650  μm2,” Opt. Lett. 37, 1292–1294 (2012).
[CrossRef]

V. Pureur and J. M. Dudley, “Nonlinear spectral broadening of femtosecond pulses in solid-core photonic bandgap fibers,” Opt. Lett. 35, 2813–2815 (2010).
[CrossRef]

M. Koshiba and K. Saitoh, “Applicability of classical optical fiber theories to holey fibers,” Opt. Lett. 29, 1739–1741 (2004).
[CrossRef]

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]

B. T. Kuhlmey, R. C. McPhedran, and C. M. de Sterke, “Modal cutoff in microstructured optical fibers,” Opt. Lett. 27, 1684–1686 (2002).
[CrossRef]

N. M. Litchinitser, A. K. Abeeluck, C. Headley, and B. J. Eggleton, “Antiresonant reflecting photonic crystal optical waveguides,” Opt. Lett. 27, 1592–1594 (2002).
[CrossRef]

T. A. Birks, J. C. Knight, and P. St. J. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. 22, 961–963 (1997).
[CrossRef]

J. Knight, T. Birks, P. Russel, and D. Atkin, “All-silica single-mode optical fiber with photonic crystal cladding,” Opt. Lett. 21, 1547–1549 (1996).
[CrossRef]

Science (2)

F. Couny, F. Benabid, P. J. Roberts, P. S. Light, and M. G. Raymer, “Generation and photonic guidance of multioctave optical-frequency combs,” Science 318, 1118–1121 (2007).
[CrossRef]

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allen, “Single-mode photonic band gap guidance of light in air,” Science 285, 1537–1539 (1999).
[CrossRef]

Other (7)

T. Taru and J. C. Knight, “Raman gain suppression in all-solid photonic bandgap fiber,” in 33rd European Conference and Exhibition of Optical Communication (IEEE, 2007), pp. 1–2.

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, 1991).

B. T. Kuhlmey, “Theoretical and numerical investigation of the physics of microstructured optical fibres,” Ph.D. dissertation (University of Sydney and Université Aix-Marseille III, 2003), http://hdl.handle.net/2123/560 .

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals Molding the Flow of Light, 2nd ed. (Princeton University, 2008).

A. Snyder and J. Love, Optical Waveguide Theory (Kluwer Academic, 1983).

BandSOLVE 4.1 (Rsoft Design Group, Inc., 2008).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1965).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (18)

Fig. 1.
Fig. 1.

Schematic transverse section of a 2D solid-core PBG fiber. The bright gray region represents the low refractive index background (nlow), whereas dark gray circles are high-index inclusions (nhigh) (ncore=nlow). A typical refractive index profile through the cross section of the cladding is shown on the left. The refractive index contrast Δn is defined as nhighnlow.

Fig. 2.
Fig. 2.

Schematic of effective index and photonic bands in the cross section of a solid-core PBG fiber, for a fixed wavelength. Gray regions (silica) and cross-hatched regions (photonic crystal cladding): propagation of light is allowed. White regions: propagation of light is forbidden. Red: effective index of core-guided modes. Dark arrows symbolize the tunneling rate through the cladding. (a) Propagation of only one mode in the fiber core; (b) several guided modes.

Fig. 3.
Fig. 3.

Top: photonic band structure of an infinite triangular lattice consisting of high-index inclusions embedded in a pure silica background (without material dispersion). Here d/Λ=0.5, nhigh=1.48, and nlow=1.45, k=2π/λ. Dark gray regions: propagation of light is allowed in the microstructured cladding. White regions: propagation of light is forbidden. The blue horizontal line represents the core refractive index (nlow). Black solid curves: effective index of the fundamental core-guided mode within the five first bandgaps. Bottom: associated imaginary part of the effective index, for a 1-cell core with four confinement rings.

Fig. 4.
Fig. 4.

Transverse intensity distributions Sz for modes (a,1) EH21 and (b,1) HE41. (a,2) and (b,2) Associated transverse electric field distribution Ex.

Fig. 5.
Fig. 5.

Transverse intensity distributions Sz for modes (a) TE02 (neff=1.4492451), (b) the first cladding mode (neff=1.4491178), and (c) EH311 (neff=1.4490818) found below nU in PBG4.

Fig. 6.
Fig. 6.

Equivalence between a hexagonal unit cell of width Λ consisting of an high-index inclusion with a radius a, and a circular unit cell of radius ρUC.

Fig. 7.
Fig. 7.

Plots of different band structures for a ratio d/Λ of (a.1) and (a.2) 0.2, and (b.1) and (b.2) 0.5, as well as for both refractive index contrasts of (a.1) and (b.1) 5103, and (a.2) and (b.2) 5102 between the rods and pure silica background (without material dispersion). The black curves are the effective index nU˜ calculated using the proposed analytical method (see Appendix A) for both kinds of LP0m (solid curves) and LP1m (dashed curves) rod modes.

Fig. 8.
Fig. 8.

Cross sections of the solid-core PBG fibers with a 1- (left) and 7- (right) cell(s) core. Four confinement rings are used in both cases.

Fig. 9.
Fig. 9.

Band diagrams and Im(neff) evolution of the different core-guided modes (within PBG3) for (a) and (b) a 1-cell and (c) and (d) a 7-cells PBG fiber. Each curve inside the bandgap [(a) and (c)] represents the effective index of each LP1m-like mode. (b) Dotted vertical line shows the minimum loss frequency of the fundamental mode (1-cell structure), whereas the dashed vertical line in (c) corresponds to the effective indices shown in Table 2 (7-cells structure).

Fig. 10.
Fig. 10.

(a) Evolution of the minimum of losses αmin in PBG3 versus the number of confinement rings Nr computed for the LP01-like (red curve) and LP11-like (blue curves) modes; (b) Rloss is computed at λmin(LP01).

Fig. 11.
Fig. 11.

(a) Evolutions of Δnmode (triangles), Rloss (squares), and (b) Λ/λmin ratio (circles) versus PBG order all computed at λmin(LP01) (1-cell structure).

Fig. 12.
Fig. 12.

Triangle (diamond): exact number of core-guided modes versus the PBG order computed at λmin(LP01) for a 1- (7-) cell(s) core structure. Δn and d/Λ are set to 3102 and 0.5, respectively. Solid and dotted curves: estimation of the core-guided mode number using the analytic model proposed in Section 3 for the LP0m and LP1m rod modes, respectively.

Fig. 13.
Fig. 13.

Evolutions of (a) Δnmode (triangles), Rloss (squares), and (b) Δω=ωLP11ωLP01 (circles) versus the refractive index contrast Δn (PBG3). (a) Solid red (dashed) curve represents Δnmode evolution for a 1- (7)-cell(s) core.

Fig. 14.
Fig. 14.

Confinement loss evolution (Im(neff)) versus wavelength for d/Λ of 0.2, 0.3, and 0.4 (d=7.75μm, Δn=15103, four confinements rings, PBG3). Solid (dashed) curve: LP01-like (LP11-like) mode.

Fig. 15.
Fig. 15.

(a) Triangle (diamonds): exact number of core-guided modes versus d/Λ computed at λmin(LP01) for a 1- (7)-cell-core structure. Δn is set to 15103 (PBG3, four confinement rings). (b) Squares: evolution of the quantity Δn=neff(LP01)nU computed at λmin(LP01) versus d/Λ (1-cell-core structure).

Fig. 16.
Fig. 16.

Influence of d/Λ on the PBG edges versus wavelength. (a) d/Λ=0.6, 0.7, and 0.8, and (b) d/Λ=0.9 (d=7.75μm, Δn=15103 without material dispersion). Horizontal lines show the background refractive index.

Fig. 17.
Fig. 17.

Markers: exact number of core-guided modes computed at λmin(LP01) versus frequency for the six first PBGs. Triangles, d/Λ=0.2; circles, d/Λ=0.4; and squares, d/Λ=0.6 (Δn=3.102). Solid and dotted curves: estimation of the core-guided mode number using the analytic model proposed in Section 3 for the LP0m and LP1m rod modes, respectively.

Fig. 18.
Fig. 18.

(a) Δnmode (triangles), Rloss (squares), and (b) Λ/λmin evolutions versus d/Λ ratio (PBG3). (a) Solid (dotted) curves: Δnmode and Rloss are computed both at λmin(LP01) (at different wavelengths). Dashed curves correspond to d/Λ values where at least one of the LP01- and LP11-like modes is out of the bandgap. The structures are constituted of a 1-cell core (Δn=15103).

Tables (3)

Tables Icon

Table 1. Summary of the Trends of Parameters Determining the Rate of Optical Tunneling with Order of the Guided Mode, PBG Order and Core Size Around the Minimum of Loss, and Resulting Trends in Minimum Loss αmin

Tables Icon

Table 2. Real and Imaginary Parts of the Effective Index (neff), Mode Class (p) (McIsaac Notation [21]), Degeneracy, Label [21], and Equivalent LPlm-Like Notation for the 24 Guided Modes (λmin(LP01)=1487nm)a

Tables Icon

Table 3. Effective Indices and Losses of the Two First Core-Guided Modes Computed at λmin(LP01) (PBG3)a

Equations (20)

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

Losses(dB/m)=40πλln(10)Im(neff),
N(λ)12(2πrcoreλ)2[(nLncore)2(nUncore)2],ifnL<ncore,
12(2πrcoreλ)2[1(nUncore)2]otherwise.
nU˜(λ)=nlowν(λ/Λ)2,
Nν4π2rcore2nlowΛ2+O(λΛ)2,
V=2πrcoreλ×ncore2ncl2,
Ψ=lAlJl(kr)eilθforra
=l{Al+Jl(k+r)+Bl+Hl(1)(k+r)}eilθforra
k+=k0(nlow2nU2)1/2,
k=k0(nhigh2nU2)1/2,
AlJl(ka)=Al+Jl(k+a)+Bl+Hl(1)(k+a).
k+Al+Jl(k+ρUC)+k+Bl+Hl(1)(k+ρUC)=0.
k+Jl(ka)kJl(ka)=Jl(k+ρUC)Hl(1)(k+a)Hl(1)(k+ρUC)Jl(k+a)Jl(k+ρUC)Hl(1)(k+a)Hl(1)(k+ρUC)Jl(k+a),
nU˜=nlowν(λΛ)2.
k=2πλδ+O(λΛ)2,
k+=2πΛ(2νnlow)1/2+O(λΛ)2,
k+J0(ka)kJ0(ka)λ2nlowνδΛtan(ka+π4)forLP0m  modes,
k+J1(ka)kJ1(ka)λ2nlowνδΛtan(kaπ4)forLP1mmodes.
H0(1)(qx)J0(qx)=H1(1)(x)J1(x)forLP0mmodes(l=0),
H1(1)(qx)J1(qx)=H0(1)(x)H1(1)(x)/xJ0(x)J1(x)/xforLP1mmodes(l=1).

Metrics