Abstract

We present numerical results on dispersion relation and electromagnetic fields in a Bragg fiber. Discussions are focused on the hybrid modes. Dispersion relations for TE, TM, and hybrid modes are investigated in four different ways. It is shown that the HE11 mode is the lowest-order mode and the TM01 mode is the first higher-order mode. A photonic bandgap is observed in the dispersion relation if the parameters are appropriately changed. We study the influence of a deviation from the quarter-wave stack condition on the dispersion curve and electromagnetic fields. The deviation has little influence on dispersion characteristics. Electromagnetic fields of hybrid modes vanish in the cladding interfaces under the quarter-wave stack condition. Characteristics of the Bragg fiber are related with those in the circular metallic waveguide.

© 2006 Optical Society of America

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  1. P. Russell, "Photonic crystal fibers," Science 299, 358-362 (2003).
    [CrossRef] [PubMed]
  2. T. A. Birks, P. J. Roberts, P. St. J. Russell, D. M. Atkins, and T. J. Sheperd, "Full 2-D photonic bandgaps in silica/air structures," Electron. Lett. 31, 1941-1943 (1995).
    [CrossRef]
  3. P. R. Villeneuve and M. Piché, "Photonic band gaps in two-dimensional square lattices: square and circular rods," Phys. Rev. B 46, 4969-4972 (1992).
    [CrossRef]
  4. R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, "Single-mode photonic band gap guidance of light in air," Science 285, 1537-1539 (1999).
    [CrossRef] [PubMed]
  5. P. Yeh, A. Yariv, and E. Marom, "Theory of Bragg fiber," J. Opt. Soc. Am. 68, 1196-1201 (1978).
    [CrossRef]
  6. M. Ibanescu, Y. Fink, S. Fan, E. L. Thomas, and J. D. Joannopoulos, "An all-dielectric coaxial waveguide," Science 289, 415-419 (2000).
    [CrossRef] [PubMed]
  7. R. Guobin, W. Zhi, L. Shuqin, and J. Shuisheng, "Mode classification and degeneracy in photonic crystal fibers," Opt. Express 11, 1310-1321 (2003).
    [CrossRef] [PubMed]
  8. H. P. Uranus and H. J. W. M. Hoekstra, "Modelling of microstructured waveguides using a finite-element-based vectorial mode solver with transparent boundary conditions," Opt. Express 12, 2795-2809 (2004).
    [CrossRef] [PubMed]
  9. S. G. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, T. D. Engeness, M. Soljacic, S. A. Jacobs, J. D. Joannopoulos, and Y. Fink, "Low-loss asymptotically single-mode propagation in large-core omniguide fibers," Opt. Express 9, 748-779 (2001).
    [CrossRef] [PubMed]
  10. Y. Xu, A. Yariv, J. G. Fleming, and S. Lin, "Asymptotic analysis of silicon based Bragg fiber," Opt. Express 11, 1039-1049 (2003).
    [CrossRef] [PubMed]
  11. W. Zhi, R. Guobin, L. Shuqin, L. Weijun, and S. Guo, "Compact supercell method based on opposite parity for Bragg fibers," Opt. Express 11, 3542-3549 (2003).
    [CrossRef] [PubMed]
  12. G. Ouyang, Y. Xu, and A. Yariv, "Comparative study of air-core and coaxial Bragg fibers: single-mode transmission and dispersion characteristics," Opt. Express 9, 733-747 (2001).
    [CrossRef] [PubMed]
  13. Y. Xu, G. X. Ouyang, R. K. Lee, and A. Yariv, "Asymptotic matrix theory of Bragg fibers," J. Lightwave Technol. 20, 428-440 (2002).
    [CrossRef]
  14. J. Sakai and P. Nouchi, "Propagation properties of Bragg fiber analyzed by a Hankel function formation," Opt. Commun. 249, 153-163 (2005).
    [CrossRef]
  15. Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. D. Thomas, "A dielectric omnidirectional reflector," Science 282, 1679-1682 (1998).
    [CrossRef] [PubMed]
  16. B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, and Y. Fink, "Wavelength-scalable hollow optical fibers with large photonic bandgaps for CO2 laser transmission," Nature 420, 650-653 (2002).
    [CrossRef] [PubMed]
  17. G. Vienne, Y. Xu, C. Jakobsen, H. Deyerl, J. B. Jensen, T. Sørensen, T. P. Hansen, Y. Huang, M. Terrel, R. K. Lee, N. A. Mortensen, J. Broeng, H. Simonsen, A. Bjarklev, and A. Yariv, "Ultra-large bandwidth hollow-core guiding in all-silica Bragg fibers with nano-supports," Opt. Express 12, 3500-3508 (2004).
    [CrossRef] [PubMed]
  18. S. Guo, S. Albin, and R. S. Rogowski, "Comparative analysis of Bragg fibers," Opt. Express 12, 198-207 (2004).
    [CrossRef] [PubMed]
  19. J. Sakai, "Hybrid modes in a Bragg fiber: general properties and formulas under the quarter-wave stack condition," J. Opt. Soc. Am. B 22, 2319-2330 (2005).
    [CrossRef]
  20. P. A. Rizzi, Microwave Engineering: Passive Circuits (Prentice-Hall, 1988), Chap. 5.
  21. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (Dover, 1972), Chap. 9.
  22. P. Yeh, Optical Waves in Layered Media (Wiley, 1988), Chap. 6.
  23. E. Snitzer, "Cylindrical dielectric waveguide modes," J. Opt. Soc. Am. 51, 491-498 (1961).
    [CrossRef]
  24. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, 1983), Sect. 12-9.
  25. T. D. Engeness, M. Ibanescu, S. G. Johnson, O. Weisberg, M. Skorobogatiy, S. Jacobs, and Y. Fink, "Dispersion tailoring and compensation by modal interactions in omniguide fibers," Opt. Express 11, 1175-1196 (2003).
    [CrossRef] [PubMed]

2005

J. Sakai and P. Nouchi, "Propagation properties of Bragg fiber analyzed by a Hankel function formation," Opt. Commun. 249, 153-163 (2005).
[CrossRef]

J. Sakai, "Hybrid modes in a Bragg fiber: general properties and formulas under the quarter-wave stack condition," J. Opt. Soc. Am. B 22, 2319-2330 (2005).
[CrossRef]

2004

2003

2002

Y. Xu, G. X. Ouyang, R. K. Lee, and A. Yariv, "Asymptotic matrix theory of Bragg fibers," J. Lightwave Technol. 20, 428-440 (2002).
[CrossRef]

B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, and Y. Fink, "Wavelength-scalable hollow optical fibers with large photonic bandgaps for CO2 laser transmission," Nature 420, 650-653 (2002).
[CrossRef] [PubMed]

2001

2000

M. Ibanescu, Y. Fink, S. Fan, E. L. Thomas, and J. D. Joannopoulos, "An all-dielectric coaxial waveguide," Science 289, 415-419 (2000).
[CrossRef] [PubMed]

1999

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

1998

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. D. Thomas, "A dielectric omnidirectional reflector," Science 282, 1679-1682 (1998).
[CrossRef] [PubMed]

1995

T. A. Birks, P. J. Roberts, P. St. J. Russell, D. M. Atkins, and T. J. Sheperd, "Full 2-D photonic bandgaps in silica/air structures," Electron. Lett. 31, 1941-1943 (1995).
[CrossRef]

1992

P. R. Villeneuve and M. Piché, "Photonic band gaps in two-dimensional square lattices: square and circular rods," Phys. Rev. B 46, 4969-4972 (1992).
[CrossRef]

1978

1961

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (Dover, 1972), Chap. 9.

Albin, S.

Allan, D. C.

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

Atkins, D. M.

T. A. Birks, P. J. Roberts, P. St. J. Russell, D. M. Atkins, and T. J. Sheperd, "Full 2-D photonic bandgaps in silica/air structures," Electron. Lett. 31, 1941-1943 (1995).
[CrossRef]

Benoit, G.

B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, and Y. Fink, "Wavelength-scalable hollow optical fibers with large photonic bandgaps for CO2 laser transmission," Nature 420, 650-653 (2002).
[CrossRef] [PubMed]

Birks, T. A.

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

T. A. Birks, P. J. Roberts, P. St. J. Russell, D. M. Atkins, and T. J. Sheperd, "Full 2-D photonic bandgaps in silica/air structures," Electron. Lett. 31, 1941-1943 (1995).
[CrossRef]

Bjarklev, A.

Broeng, J.

Chen, C.

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. D. Thomas, "A dielectric omnidirectional reflector," Science 282, 1679-1682 (1998).
[CrossRef] [PubMed]

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. Allan, "Single-mode photonic band gap guidance of light in air," Science 285, 1537-1539 (1999).
[CrossRef] [PubMed]

Deyerl, H.

Engeness, T. D.

Fan, S.

M. Ibanescu, Y. Fink, S. Fan, E. L. Thomas, and J. D. Joannopoulos, "An all-dielectric coaxial waveguide," Science 289, 415-419 (2000).
[CrossRef] [PubMed]

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. D. Thomas, "A dielectric omnidirectional reflector," Science 282, 1679-1682 (1998).
[CrossRef] [PubMed]

Fink, Y.

T. D. Engeness, M. Ibanescu, S. G. Johnson, O. Weisberg, M. Skorobogatiy, S. Jacobs, and Y. Fink, "Dispersion tailoring and compensation by modal interactions in omniguide fibers," Opt. Express 11, 1175-1196 (2003).
[CrossRef] [PubMed]

B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, and Y. Fink, "Wavelength-scalable hollow optical fibers with large photonic bandgaps for CO2 laser transmission," Nature 420, 650-653 (2002).
[CrossRef] [PubMed]

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

M. Ibanescu, Y. Fink, S. Fan, E. L. Thomas, and J. D. Joannopoulos, "An all-dielectric coaxial waveguide," Science 289, 415-419 (2000).
[CrossRef] [PubMed]

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. D. Thomas, "A dielectric omnidirectional reflector," Science 282, 1679-1682 (1998).
[CrossRef] [PubMed]

Fleming, J. G.

Guo, S.

Guobin, R.

Hansen, T. P.

Hart, S. D.

B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, and Y. Fink, "Wavelength-scalable hollow optical fibers with large photonic bandgaps for CO2 laser transmission," Nature 420, 650-653 (2002).
[CrossRef] [PubMed]

Hoekstra, H. J. W. M.

Huang, Y.

Ibanescu, M.

Jacobs, S.

Jacobs, S. A.

Jakobsen, C.

Jensen, J. B.

Joannopoulos, J. D.

B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, and Y. Fink, "Wavelength-scalable hollow optical fibers with large photonic bandgaps for CO2 laser transmission," Nature 420, 650-653 (2002).
[CrossRef] [PubMed]

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

M. Ibanescu, Y. Fink, S. Fan, E. L. Thomas, and J. D. Joannopoulos, "An all-dielectric coaxial waveguide," Science 289, 415-419 (2000).
[CrossRef] [PubMed]

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. D. Thomas, "A dielectric omnidirectional reflector," Science 282, 1679-1682 (1998).
[CrossRef] [PubMed]

Johnson, S. G.

Knight, J. C.

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

Lee, R. K.

Lin, S.

Love, J. D.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, 1983), Sect. 12-9.

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. Allan, "Single-mode photonic band gap guidance of light in air," Science 285, 1537-1539 (1999).
[CrossRef] [PubMed]

Marom, E.

Michel, J.

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. D. Thomas, "A dielectric omnidirectional reflector," Science 282, 1679-1682 (1998).
[CrossRef] [PubMed]

Mortensen, N. A.

Nouchi, P.

J. Sakai and P. Nouchi, "Propagation properties of Bragg fiber analyzed by a Hankel function formation," Opt. Commun. 249, 153-163 (2005).
[CrossRef]

Ouyang, G.

Ouyang, G. X.

Piché, M.

P. R. Villeneuve and M. Piché, "Photonic band gaps in two-dimensional square lattices: square and circular rods," Phys. Rev. B 46, 4969-4972 (1992).
[CrossRef]

Rizzi, P. A.

P. A. Rizzi, Microwave Engineering: Passive Circuits (Prentice-Hall, 1988), Chap. 5.

Roberts, P. J.

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

T. A. Birks, P. J. Roberts, P. St. J. Russell, D. M. Atkins, and T. J. Sheperd, "Full 2-D photonic bandgaps in silica/air structures," Electron. Lett. 31, 1941-1943 (1995).
[CrossRef]

Rogowski, R. S.

Russell, P.

P. Russell, "Photonic crystal fibers," Science 299, 358-362 (2003).
[CrossRef] [PubMed]

Russell, P. St. J.

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

T. A. Birks, P. J. Roberts, P. St. J. Russell, D. M. Atkins, and T. J. Sheperd, "Full 2-D photonic bandgaps in silica/air structures," Electron. Lett. 31, 1941-1943 (1995).
[CrossRef]

Sakai, J.

J. Sakai and P. Nouchi, "Propagation properties of Bragg fiber analyzed by a Hankel function formation," Opt. Commun. 249, 153-163 (2005).
[CrossRef]

J. Sakai, "Hybrid modes in a Bragg fiber: general properties and formulas under the quarter-wave stack condition," J. Opt. Soc. Am. B 22, 2319-2330 (2005).
[CrossRef]

Sheperd, T. J.

T. A. Birks, P. J. Roberts, P. St. J. Russell, D. M. Atkins, and T. J. Sheperd, "Full 2-D photonic bandgaps in silica/air structures," Electron. Lett. 31, 1941-1943 (1995).
[CrossRef]

Shuisheng, J.

Shuqin, L.

Simonsen, H.

Skorobogatiy, M.

Snitzer, E.

Snyder, A. W.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, 1983), Sect. 12-9.

Soljacic, M.

Sørensen, T.

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (Dover, 1972), Chap. 9.

Temelkuran, B.

B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, and Y. Fink, "Wavelength-scalable hollow optical fibers with large photonic bandgaps for CO2 laser transmission," Nature 420, 650-653 (2002).
[CrossRef] [PubMed]

Terrel, M.

Thomas, E. D.

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. D. Thomas, "A dielectric omnidirectional reflector," Science 282, 1679-1682 (1998).
[CrossRef] [PubMed]

Thomas, E. L.

M. Ibanescu, Y. Fink, S. Fan, E. L. Thomas, and J. D. Joannopoulos, "An all-dielectric coaxial waveguide," Science 289, 415-419 (2000).
[CrossRef] [PubMed]

Uranus, H. P.

Vienne, G.

Villeneuve, P. R.

P. R. Villeneuve and M. Piché, "Photonic band gaps in two-dimensional square lattices: square and circular rods," Phys. Rev. B 46, 4969-4972 (1992).
[CrossRef]

Weijun, L.

Weisberg, O.

Winn, J. N.

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. D. Thomas, "A dielectric omnidirectional reflector," Science 282, 1679-1682 (1998).
[CrossRef] [PubMed]

Xu, Y.

Yariv, A.

Yeh, P.

Zhi, W.

Electron. Lett.

T. A. Birks, P. J. Roberts, P. St. J. Russell, D. M. Atkins, and T. J. Sheperd, "Full 2-D photonic bandgaps in silica/air structures," Electron. Lett. 31, 1941-1943 (1995).
[CrossRef]

J. Lightwave Technol.

J. Opt. Soc. Am.

J. Opt. Soc. Am. B

Nature

B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, and Y. Fink, "Wavelength-scalable hollow optical fibers with large photonic bandgaps for CO2 laser transmission," Nature 420, 650-653 (2002).
[CrossRef] [PubMed]

Opt. Commun.

J. Sakai and P. Nouchi, "Propagation properties of Bragg fiber analyzed by a Hankel function formation," Opt. Commun. 249, 153-163 (2005).
[CrossRef]

Opt. Express

G. Vienne, Y. Xu, C. Jakobsen, H. Deyerl, J. B. Jensen, T. Sørensen, T. P. Hansen, Y. Huang, M. Terrel, R. K. Lee, N. A. Mortensen, J. Broeng, H. Simonsen, A. Bjarklev, and A. Yariv, "Ultra-large bandwidth hollow-core guiding in all-silica Bragg fibers with nano-supports," Opt. Express 12, 3500-3508 (2004).
[CrossRef] [PubMed]

S. Guo, S. Albin, and R. S. Rogowski, "Comparative analysis of Bragg fibers," Opt. Express 12, 198-207 (2004).
[CrossRef] [PubMed]

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

H. P. Uranus and H. J. W. M. Hoekstra, "Modelling of microstructured waveguides using a finite-element-based vectorial mode solver with transparent boundary conditions," Opt. Express 12, 2795-2809 (2004).
[CrossRef] [PubMed]

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

Y. Xu, A. Yariv, J. G. Fleming, and S. Lin, "Asymptotic analysis of silicon based Bragg fiber," Opt. Express 11, 1039-1049 (2003).
[CrossRef] [PubMed]

W. Zhi, R. Guobin, L. Shuqin, L. Weijun, and S. Guo, "Compact supercell method based on opposite parity for Bragg fibers," Opt. Express 11, 3542-3549 (2003).
[CrossRef] [PubMed]

G. Ouyang, Y. Xu, and A. Yariv, "Comparative study of air-core and coaxial Bragg fibers: single-mode transmission and dispersion characteristics," Opt. Express 9, 733-747 (2001).
[CrossRef] [PubMed]

T. D. Engeness, M. Ibanescu, S. G. Johnson, O. Weisberg, M. Skorobogatiy, S. Jacobs, and Y. Fink, "Dispersion tailoring and compensation by modal interactions in omniguide fibers," Opt. Express 11, 1175-1196 (2003).
[CrossRef] [PubMed]

Phys. Rev. B

P. R. Villeneuve and M. Piché, "Photonic band gaps in two-dimensional square lattices: square and circular rods," Phys. Rev. B 46, 4969-4972 (1992).
[CrossRef]

Science

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

M. Ibanescu, Y. Fink, S. Fan, E. L. Thomas, and J. D. Joannopoulos, "An all-dielectric coaxial waveguide," Science 289, 415-419 (2000).
[CrossRef] [PubMed]

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. D. Thomas, "A dielectric omnidirectional reflector," Science 282, 1679-1682 (1998).
[CrossRef] [PubMed]

P. Russell, "Photonic crystal fibers," Science 299, 358-362 (2003).
[CrossRef] [PubMed]

Other

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, 1983), Sect. 12-9.

P. A. Rizzi, Microwave Engineering: Passive Circuits (Prentice-Hall, 1988), Chap. 5.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (Dover, 1972), Chap. 9.

P. Yeh, Optical Waves in Layered Media (Wiley, 1988), Chap. 6.

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Figures (14)

Fig. 1
Fig. 1

Schematic structure of the Bragg fiber. r c , core radius; n c , refractive index of core; n a and n b , indices of layer with thickness a and b, respectively; Λ = a + b , period in the cladding.

Fig. 2
Fig. 2

P parameter versus β k 0 for HE 11 and EH 11 modes. The solid curve indicates P for the HE 11 mode and the dashed curve indicates P for the EH 11 mode. n c = 1.0 , n a = 2.5 , n b = 1.5 , λ 0 = 1.55 μ m , n t = 0.8 , a = 0.1636 μ m , and b = 0.3054 μ m .

Fig. 3
Fig. 3

Comparison of normalized propagation constant β k 0 versus the core radius r c for HE 1 μ and EH 1 μ modes. Solid curves, derived from Eq. (11a); dotted curves, n t = 0.8 ; dotted–dashed curves, n t = 0.2 . Parameters common among the three cases are n a = 2.5 , n b = 1.5 , n c = 1.0 , and λ 0 = 1.55 μ m . Thickness of cladding layers: a = 0.1636 μ m for n t = 0.8 and a = 0.155 μ m for n t = 0.2 while b = 0.3054 μ m for n t = 0.8 and 0.2. Thicknesses a and b for the QWS condition change according to Eqs. (51) in Ref. [19]. n b = 1.5 and n c = 1.0 are used throughout the following calculations. Double numbers in the figure denote the azimuthal and radial mode numbers ν and μ.

Fig. 4
Fig. 4

Normalized propagation constant β k 0 versus the core radius r c for TE, TM, and hybrid modes. Solid curves, HE mode; dashed curves, EH mode; dotted curves, TE mode; dotted dashed curves, TM mode. n a = 2.5 , n b = 1.5 , and λ 0 = 1.55 μ m . The tentative index is n t = 0.8 .

Fig. 5
Fig. 5

Normalized propagation constant β k 0 versus the core radius r c for TE, TM, and hybrid modes. Line classification and parameters are same as those in Fig. 4 except for n a = 2.0 .

Fig. 6
Fig. 6

Normalized propagation constant β k 0 versus the core radius r c in a case where only the HE 11 mode is operated under the QWS condition. (a) β k 0 versus r c . (b) Cladding thicknesses a and b for each r c . n a = 2.5 and n b = 1.5 .

Fig. 7
Fig. 7

Normalized propagation constant β k 0 versus normalized core radius r c λ 0 for TM and hybrid modes. Parameters are first determined for r c = 0.757 μ m , λ 0 = 1.55 μ m , n a = 2.5 , n b = 1.5 , and n t = 0.8 . Next the abscissa is varied under r c λ 0 = 0.4884 , a r c = 0.2161 , and b r c = 0.4034 . Open circle indicates the point where the QWS condition is satisfied.

Fig. 8
Fig. 8

Normalized propagation constant β k 0 versus normalized core radius r c λ 0 for hybrid modes. Parameters are same as those in Fig. 7 except that r c = 1.55 μ m is at first prescribed, and n a is prescribed for 2.5, 2.0, and 1.51.

Fig. 9
Fig. 9

ω - β curve. Cladding parameters are prescribed in the same way as in Fig. 7. Parameters are fixed at n a = 2.5 , n b = 1.5 , r c Λ = 1.614 , a Λ = 0.3488 , and b Λ = 0.6512 . The open circle indicates the point where the QWS condition holds. The solid line indicates the light line.

Fig. 10
Fig. 10

Comparison of ω - β curve in an air-core Bragg fiber. Parameters are r c = 2 Λ , n a = 4.6 , n b = 1.6 , a = 0.33 Λ , and b = 0.67 Λ . The solid curves indicate results derived by the present method and the dashed curves indicate results obtained by the transfer-matrix method. The solid line indicates the light line.

Fig. 11
Fig. 11

Electromagnetic fields E z , H z , i E θ , and i H θ for the HE 11 mode. (a) β k 0 = 0.7 ( r c = 0.6194 μ m ) , (b) β k 0 = n t = 0.8 (QWS condition, r c = 0.7570 μ m ), (c) β k 0 = 0.9 ( r c = 1.0800 μ m ) . Fiber parameters are prescribed for n c = 1.0 , n a = 2.5 , n b = 1.5 , λ 0 = 1.55 μ m , and n t = 0.8 . Then a = 0.1636 μ m , b = 0.3054 μ m , and Λ = 0.4689 μ m . The solid curves indicate the E z component, dotted curves indicate the H z component, dotted–dashed curves indicate the i E θ component, and dashed curves indicate the i H θ component. Cladding bilayers a and b are indicated by two different types of dot meshing.

Fig. 12
Fig. 12

Electromagnetic fields E z , H z , i E θ , and i H θ for the EH 11 mode. (a) β k 0 = 0.7 ( r c = 1.3015 μ m ) , (b) β k 0 = n t = 0.8 (QWS condition, r c = 1.5754 μ m ), (c) β k 0 = 0.9 ( r c = 2.2500 μ m ) . Parameters are same as those in Fig. 11 except for the core radius r c .

Fig. 13
Fig. 13

Electromagnetic fields of the HE 22 mode under the QWS condition. β k 0 = n t = 0.8 and r c = 2.7572 μ m . Parameters are same as those in Fig. 11 except for the core radius r c .

Fig. 14
Fig. 14

Several electromagnetic fields of the HE 11 mode for n a = 4.6 and β k 0 = n t = 0.8 (QWS condition, r c = 0.7570 μ m ). Parameters are same as those in Fig. 11 except for the n a and core radius r c .

Equations (23)

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r a , m = r [ r c + ( m 1 ) Λ ] for layer a ( 0 r a , m a ) ,
r b , m = r [ r c + ( m 1 ) Λ + a ] for layer b ( 0 r b , m b ) .
[ 1 κ c r c J ν ( κ c r c ) J ν ( κ c r c ) + 1 i κ a r c Σ j TE ] [ n c 2 κ c r c J ν ( κ c r c ) J ν ( κ c r c ) + n a 2 i κ a r c Σ j TM ] = ν 2 [ ( 1 κ c r c ) 2 + ( 1 i κ a r c ) 2 ] [ ( n c κ c r c ) 2 + ( n a i κ a r c ) 2 ] ,
Σ j S exp ( i K j S Λ ) X S Y S exp ( i K j S Λ ) X S + Y S ( S = TE or TM ; j = 1 , 2 ) ,
κ i ( n i k 0 ) 2 β 2 ( i = a , b , c ) .
exp ( i K j S Λ ) = Re ( X S ) ± [ Re ( X S ) ] 2 1
for S = TE or TM ; j = 1 , 2 .
X S = [ cos ( κ b b ) i 2 ( ζ b κ b ζ a κ a + ζ a κ a ζ b κ b ) sin ( κ b b ) ] exp ( i κ a a ) ,
Y S = i 2 ( ζ b κ b ζ a κ a ζ a κ a ζ b κ b ) sin ( κ b b ) exp ( i κ a a ) ,
ζ i = { 1 S = TE 1 n i 2 ( i = a , b ) S = TM } .
E z = { J ν ( κ c r ) α M + TM } cos ( ν θ ) ,
H z = β ω μ 0 P { J ν ( κ c r ) α M + TE } sin ( ν θ ) ,
i E θ = β κ { 1 P 2 J ν 1 ( κ c r ) + 1 + P 2 J ν + 1 ( κ c r ) α ( 1 P 2 M 1 + P 2 M + ) } sin ( ν θ ) ,
i H θ = ( n i k 0 ) 2 ω μ 0 κ { 1 P ( β n c k 0 ) 2 2 J ν 1 ( κ c r ) 1 + P ( β n c k 0 ) 2 2 J ν + 1 ( κ c r ) α [ 1 P ( β n i k 0 ) 2 2 N + 1 + P ( β n i k 0 ) 2 2 N + ] } cos ( ν θ ) .
α = π κ a r c 2 J ν ( κ c r c ) exp ( i K j TM Λ ) X TM + Y TM .
P ω μ 0 β H z E z = ν [ 1 + ( κ c i κ a ) 2 ] κ c r c { [ J ν ( κ c r c ) J ν ( κ c r c ) ] + ( κ c i κ a ) Σ j TE } ,
κ i i = 2 π i λ 0 n i 2 ( β k 0 ) 2 = π 2 ( i = a , b ) ,
κ c r c = 2 π r c λ 0 n c 2 ( β k 0 ) 2 = U QWS ,
U QWS = { j 1 , μ : TE 0 μ mode j 0 , μ : TM 0 μ mode j ν , μ ( ν 1 ) : HE ν μ mode j ν , μ ( ν 1 ) : EH ν μ mode } .
M ± S = 2 π κ i r exp [ i K j S ( m 1 ) Λ ] × { Y S exp ( i κ i r i , m ) ± [ exp ( i K j S Λ ) X S ] exp ( i κ i r i , m ) } ,
M ± = M TE ± ν i κ i r M + TM ,
N ± = M TM ± ν i κ i r M + TE .
E θ P [ Y TE exp ( i K j TE Λ ) + X TE ] exp [ i K j TE ( m 1 ) Λ ] + ν i κ a r [ Y TM + exp ( i K j TM Λ ) X TM ] exp [ i K j TM ( m 1 ) Λ ]

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