Abstract

In this paper, we compare three analysis methods for Bragg fibers, viz. the transfer matrix method, the asymptotic method and the Galerkin method. We also show that with minor modifications, the transfer matrix method is able to calculate exactly the leakage loss of Bragg fibers due to a finite number of H/L layers. This approach is more straightforward than the commonly used Chew’s method. It is shown that the asymptotic approximation condition should be satisfied in order to get accurate results. The TE and TM modes, and the band gap structures are analyzed using Galerkin method.

© 2004 Optical Society of America

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  1. P. Yeh, A. Yariv, and E. Marom, “Theory of Bragg fiber,” J. Opt. Soc. Am. 68, 1196 (1978).
    [CrossRef]
  2. N. Croitoru, J. Dror, and I. Gannot, “Characterization of hollow fibers for the transmission of infrared radiation,” Appl. Opt. 29, 1805 (1990).
    [CrossRef] [PubMed]
  3. T. Erdogan and D. G. Hall, “Circularly symmetric distribution feedback semiconductor lasers,” IEEE J. Quantum Electron. 26, 473 (1990).
  4. C. M. de Sterke, I. M. Bassett, and A. G. Street, “Differential losses in Bragg fibers,” J. Appl. Phys. 76, 680 (1993).
    [CrossRef]
  5. S. G. Johnsonet al., “low-loss asymptotically single-mode propagation in large core omniguide fibers,” Opt. Express 9, 748 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-748.
    [CrossRef] [PubMed]
  6. Y. Finket al., “A dielectric omnidirectional reflector,” Science 282, 1679 (1998).
    [CrossRef] [PubMed]
  7. J. N. Winnet al., “Omnidirectional reflection from a one-dimensional photonic crystals,” Opt. Lett. 23, 1573 (1998).
    [CrossRef]
  8. Y. Finket al., “Guiding optical light in air using an all dielectric structure,” J. Lightwave Technol. 17, 2039 (1999).
    [CrossRef]
  9. M. Ibanescuet al., “An all-dielectric coaxial waveguide,” Science 289, 415 (2000).
    [CrossRef] [PubMed]
  10. B. Temelkuranet al., “Wavelength-scalable hollow optical fibres with large photonic bandgaps for CO2 laser transmission,” Nature 420, 650 (2002).
    [CrossRef] [PubMed]
  11. T. Kawanishi and M. Izutsu, “Coaxial periodic optical waveguide,” Opt. Express 7, 10 (2000), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-7-1-10.
    [CrossRef] [PubMed]
  12. Y. Xu, R.K. Lee, and A. Yariv, “Asymptotic analysis of Bragg fibers,” Opt. Lett. 25, 1756 (2000).
    [CrossRef]
  13. 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 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-733.
    [CrossRef] [PubMed]
  14. Y. Xuet al., “Asymptotic matrix theory of Bragg fibers,” J. Lightwave Technol. 20, 428 (2002).
    [CrossRef]
  15. G. Ouyang, Y. Xu, and A. Yariv, “Theoretical study on dispersion compensation in air-core Bragg fibers,” Opt. Express 10, 899 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-17-899.
    [CrossRef] [PubMed]
  16. S. Guoet al., “Analysis of circular fibers with arbitrary index profiles by Galerkin method,” To appear in Opt. Lett. 29, 32 (Jan. 2004).
  17. W. C. Chew, Waves and fields in inhomogeneous media (New York: Van Nostrand Reinhold, 1990).
  18. C. Themistoset al., “Loss/gain characterization of optical waveguides,” J. lightwave. Technol. 13, 1760(1995).
    [CrossRef]
  19. A. Argyros, “Guided modes and loss in Bragg fibers,” Opt. Express 10, 1411 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-24-1411.
    [CrossRef] [PubMed]
  20. I. M. Bassett and A. Argyros, “Elimination of polarization degeneracy in round waveguides,” Opt. Express 10, 1342 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-23-1342.
    [CrossRef] [PubMed]
  21. N. A. Mortensen, “Effective area of photonic crystal fibers,” Opt. Express 10, 341 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-7-341.
    [CrossRef] [PubMed]
  22. N. A. Issaet al., “Identifying hollow waveguide guidance in air-cored microstructured optical fibres,” Opt. Express 11, 996 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-9-996.
    [CrossRef] [PubMed]
  23. Y. Xu, R.K. Lee, and A. Yariv, “Asymptotic analysis of dielectric coaxial fibers,” Opt. Lett. 27, 1019 (2002).
    [CrossRef]
  24. Y. Xuet al., “Asymptotic analysis of silicon based Bragg fibers,” Opt. Express 11, 1039 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-9-1039.
    [CrossRef] [PubMed]
  25. J. P. Meunier, J. Pigeon, and J. N. Massot, “A general approach to the numerical determination of modal propagation constants and field distribution of optical fibers,” Opt. Quantum Electron. 13, 71 (1981).
    [CrossRef]
  26. P. K. Mishraet al., “Matrix method for determining propagation characteristics of optical waveguides,” Optica Acta 31, 1041 (1984).
    [CrossRef]
  27. H. Etzkom and T. Heun, “Highly accurate numerical method for determination of propagation dispersion-flattened fibers,” Opt. Quantum Electron. 18, 1 (1986).
    [CrossRef]
  28. A. Sharma and S. Banarjee, “Chromatic dispersion in single mode fibers with arbitrary index profiles: a simple method for exact numerical evaluation,” J. Lightwave Technol. 7, 1919 (1989).
    [CrossRef]
  29. D. Marcuse, “Solution of the vector wave equation for general dielectric waveguides by the Galerkin method,” IEEE J. Quantum Electron. 27, 459 (1992).
    [CrossRef]
  30. A. Weisshaaret al., “Vector and quasi-vector solutions for optical waveguide modes using efficient Galerkin’s method with Hermite-Gauss basis functions,” J. Lightwave Technol. 13, 1795 (1995).
    [CrossRef]
  31. W. Zhiet al., “Compact supercell method based on opposite parity for Bragg fibers,” Opt. Express 11, 3542–3549 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-26-3542.
    [CrossRef] [PubMed]
  32. A. W. Snyder and J. D. Love, Optical Waveguide theory (New York: Chapman Hall, 1983).
  33. G. Arfken, Mathematical methods for physicists (Orlando, FL: Academic Press, 1985).

2003 (3)

2002 (7)

2001 (2)

2000 (3)

1999 (1)

1998 (2)

1995 (2)

C. Themistoset al., “Loss/gain characterization of optical waveguides,” J. lightwave. Technol. 13, 1760(1995).
[CrossRef]

A. Weisshaaret al., “Vector and quasi-vector solutions for optical waveguide modes using efficient Galerkin’s method with Hermite-Gauss basis functions,” J. Lightwave Technol. 13, 1795 (1995).
[CrossRef]

1993 (1)

C. M. de Sterke, I. M. Bassett, and A. G. Street, “Differential losses in Bragg fibers,” J. Appl. Phys. 76, 680 (1993).
[CrossRef]

1992 (1)

D. Marcuse, “Solution of the vector wave equation for general dielectric waveguides by the Galerkin method,” IEEE J. Quantum Electron. 27, 459 (1992).
[CrossRef]

1990 (2)

N. Croitoru, J. Dror, and I. Gannot, “Characterization of hollow fibers for the transmission of infrared radiation,” Appl. Opt. 29, 1805 (1990).
[CrossRef] [PubMed]

T. Erdogan and D. G. Hall, “Circularly symmetric distribution feedback semiconductor lasers,” IEEE J. Quantum Electron. 26, 473 (1990).

1989 (1)

A. Sharma and S. Banarjee, “Chromatic dispersion in single mode fibers with arbitrary index profiles: a simple method for exact numerical evaluation,” J. Lightwave Technol. 7, 1919 (1989).
[CrossRef]

1986 (1)

H. Etzkom and T. Heun, “Highly accurate numerical method for determination of propagation dispersion-flattened fibers,” Opt. Quantum Electron. 18, 1 (1986).
[CrossRef]

1984 (1)

P. K. Mishraet al., “Matrix method for determining propagation characteristics of optical waveguides,” Optica Acta 31, 1041 (1984).
[CrossRef]

1981 (1)

J. P. Meunier, J. Pigeon, and J. N. Massot, “A general approach to the numerical determination of modal propagation constants and field distribution of optical fibers,” Opt. Quantum Electron. 13, 71 (1981).
[CrossRef]

1978 (1)

Arfken, G.

G. Arfken, Mathematical methods for physicists (Orlando, FL: Academic Press, 1985).

Argyros, A.

Banarjee, S.

A. Sharma and S. Banarjee, “Chromatic dispersion in single mode fibers with arbitrary index profiles: a simple method for exact numerical evaluation,” J. Lightwave Technol. 7, 1919 (1989).
[CrossRef]

Bassett, I. M.

Chew, W. C.

W. C. Chew, Waves and fields in inhomogeneous media (New York: Van Nostrand Reinhold, 1990).

Croitoru, N.

de Sterke, C. M.

C. M. de Sterke, I. M. Bassett, and A. G. Street, “Differential losses in Bragg fibers,” J. Appl. Phys. 76, 680 (1993).
[CrossRef]

Dror, J.

Erdogan, T.

T. Erdogan and D. G. Hall, “Circularly symmetric distribution feedback semiconductor lasers,” IEEE J. Quantum Electron. 26, 473 (1990).

Etzkom, H.

H. Etzkom and T. Heun, “Highly accurate numerical method for determination of propagation dispersion-flattened fibers,” Opt. Quantum Electron. 18, 1 (1986).
[CrossRef]

Fink, Y.

Gannot, I.

Guo, S.

S. Guoet al., “Analysis of circular fibers with arbitrary index profiles by Galerkin method,” To appear in Opt. Lett. 29, 32 (Jan. 2004).

Hall, D. G.

T. Erdogan and D. G. Hall, “Circularly symmetric distribution feedback semiconductor lasers,” IEEE J. Quantum Electron. 26, 473 (1990).

Heun, T.

H. Etzkom and T. Heun, “Highly accurate numerical method for determination of propagation dispersion-flattened fibers,” Opt. Quantum Electron. 18, 1 (1986).
[CrossRef]

Ibanescu, M.

M. Ibanescuet al., “An all-dielectric coaxial waveguide,” Science 289, 415 (2000).
[CrossRef] [PubMed]

Issa, N. A.

Izutsu, M.

Johnson, S. G.

Kawanishi, T.

Lee, R.K.

Love, J. D.

A. W. Snyder and J. D. Love, Optical Waveguide theory (New York: Chapman Hall, 1983).

Marcuse, D.

D. Marcuse, “Solution of the vector wave equation for general dielectric waveguides by the Galerkin method,” IEEE J. Quantum Electron. 27, 459 (1992).
[CrossRef]

Marom, E.

Massot, J. N.

J. P. Meunier, J. Pigeon, and J. N. Massot, “A general approach to the numerical determination of modal propagation constants and field distribution of optical fibers,” Opt. Quantum Electron. 13, 71 (1981).
[CrossRef]

Meunier, J. P.

J. P. Meunier, J. Pigeon, and J. N. Massot, “A general approach to the numerical determination of modal propagation constants and field distribution of optical fibers,” Opt. Quantum Electron. 13, 71 (1981).
[CrossRef]

Mishra, P. K.

P. K. Mishraet al., “Matrix method for determining propagation characteristics of optical waveguides,” Optica Acta 31, 1041 (1984).
[CrossRef]

Mortensen, N. A.

Ouyang, G.

Pigeon, J.

J. P. Meunier, J. Pigeon, and J. N. Massot, “A general approach to the numerical determination of modal propagation constants and field distribution of optical fibers,” Opt. Quantum Electron. 13, 71 (1981).
[CrossRef]

Sharma, A.

A. Sharma and S. Banarjee, “Chromatic dispersion in single mode fibers with arbitrary index profiles: a simple method for exact numerical evaluation,” J. Lightwave Technol. 7, 1919 (1989).
[CrossRef]

Snyder, A. W.

A. W. Snyder and J. D. Love, Optical Waveguide theory (New York: Chapman Hall, 1983).

Street, A. G.

C. M. de Sterke, I. M. Bassett, and A. G. Street, “Differential losses in Bragg fibers,” J. Appl. Phys. 76, 680 (1993).
[CrossRef]

Temelkuran, B.

B. Temelkuranet al., “Wavelength-scalable hollow optical fibres with large photonic bandgaps for CO2 laser transmission,” Nature 420, 650 (2002).
[CrossRef] [PubMed]

Themistos, C.

C. Themistoset al., “Loss/gain characterization of optical waveguides,” J. lightwave. Technol. 13, 1760(1995).
[CrossRef]

Weisshaar, A.

A. Weisshaaret al., “Vector and quasi-vector solutions for optical waveguide modes using efficient Galerkin’s method with Hermite-Gauss basis functions,” J. Lightwave Technol. 13, 1795 (1995).
[CrossRef]

Winn, J. N.

Xu, Y.

Yariv, A.

Yeh, P.

Zhi, W.

Appl. Opt. (1)

IEEE J. Quantum Electron. (2)

T. Erdogan and D. G. Hall, “Circularly symmetric distribution feedback semiconductor lasers,” IEEE J. Quantum Electron. 26, 473 (1990).

D. Marcuse, “Solution of the vector wave equation for general dielectric waveguides by the Galerkin method,” IEEE J. Quantum Electron. 27, 459 (1992).
[CrossRef]

J. Appl. Phys. (1)

C. M. de Sterke, I. M. Bassett, and A. G. Street, “Differential losses in Bragg fibers,” J. Appl. Phys. 76, 680 (1993).
[CrossRef]

J. Lightwave Technol. (4)

Y. Finket al., “Guiding optical light in air using an all dielectric structure,” J. Lightwave Technol. 17, 2039 (1999).
[CrossRef]

Y. Xuet al., “Asymptotic matrix theory of Bragg fibers,” J. Lightwave Technol. 20, 428 (2002).
[CrossRef]

A. Weisshaaret al., “Vector and quasi-vector solutions for optical waveguide modes using efficient Galerkin’s method with Hermite-Gauss basis functions,” J. Lightwave Technol. 13, 1795 (1995).
[CrossRef]

A. Sharma and S. Banarjee, “Chromatic dispersion in single mode fibers with arbitrary index profiles: a simple method for exact numerical evaluation,” J. Lightwave Technol. 7, 1919 (1989).
[CrossRef]

J. lightwave. Technol. (1)

C. Themistoset al., “Loss/gain characterization of optical waveguides,” J. lightwave. Technol. 13, 1760(1995).
[CrossRef]

J. Opt. Soc. Am. (1)

Nature (1)

B. Temelkuranet al., “Wavelength-scalable hollow optical fibres with large photonic bandgaps for CO2 laser transmission,” Nature 420, 650 (2002).
[CrossRef] [PubMed]

Opt. Express (10)

T. Kawanishi and M. Izutsu, “Coaxial periodic optical waveguide,” Opt. Express 7, 10 (2000), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-7-1-10.
[CrossRef] [PubMed]

Y. Xuet al., “Asymptotic analysis of silicon based Bragg fibers,” Opt. Express 11, 1039 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-9-1039.
[CrossRef] [PubMed]

W. Zhiet al., “Compact supercell method based on opposite parity for Bragg fibers,” Opt. Express 11, 3542–3549 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-26-3542.
[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 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-733.
[CrossRef] [PubMed]

S. G. Johnsonet al., “low-loss asymptotically single-mode propagation in large core omniguide fibers,” Opt. Express 9, 748 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-748.
[CrossRef] [PubMed]

A. Argyros, “Guided modes and loss in Bragg fibers,” Opt. Express 10, 1411 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-24-1411.
[CrossRef] [PubMed]

I. M. Bassett and A. Argyros, “Elimination of polarization degeneracy in round waveguides,” Opt. Express 10, 1342 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-23-1342.
[CrossRef] [PubMed]

N. A. Mortensen, “Effective area of photonic crystal fibers,” Opt. Express 10, 341 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-7-341.
[CrossRef] [PubMed]

N. A. Issaet al., “Identifying hollow waveguide guidance in air-cored microstructured optical fibres,” Opt. Express 11, 996 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-9-996.
[CrossRef] [PubMed]

G. Ouyang, Y. Xu, and A. Yariv, “Theoretical study on dispersion compensation in air-core Bragg fibers,” Opt. Express 10, 899 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-17-899.
[CrossRef] [PubMed]

Opt. Lett. (3)

Opt. Quantum Electron. (2)

J. P. Meunier, J. Pigeon, and J. N. Massot, “A general approach to the numerical determination of modal propagation constants and field distribution of optical fibers,” Opt. Quantum Electron. 13, 71 (1981).
[CrossRef]

H. Etzkom and T. Heun, “Highly accurate numerical method for determination of propagation dispersion-flattened fibers,” Opt. Quantum Electron. 18, 1 (1986).
[CrossRef]

Optica Acta (1)

P. K. Mishraet al., “Matrix method for determining propagation characteristics of optical waveguides,” Optica Acta 31, 1041 (1984).
[CrossRef]

Science (2)

M. Ibanescuet al., “An all-dielectric coaxial waveguide,” Science 289, 415 (2000).
[CrossRef] [PubMed]

Y. Finket al., “A dielectric omnidirectional reflector,” Science 282, 1679 (1998).
[CrossRef] [PubMed]

Other (4)

S. Guoet al., “Analysis of circular fibers with arbitrary index profiles by Galerkin method,” To appear in Opt. Lett. 29, 32 (Jan. 2004).

W. C. Chew, Waves and fields in inhomogeneous media (New York: Van Nostrand Reinhold, 1990).

A. W. Snyder and J. D. Love, Optical Waveguide theory (New York: Chapman Hall, 1983).

G. Arfken, Mathematical methods for physicists (Orlando, FL: Academic Press, 1985).

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Figures (6)

Fig. 1.
Fig. 1.

Bragg modes and band gap structures calculated by transfer matrix method. Left: TE, Right: TM.

Fig. 2.
Fig. 2.

TE01 and TM01 at k=1.2 in the Bragg fiber, calculated by transfer matrix method. Left: TE, Right: TM.

Fig. 3.
Fig. 3.

Mode field of TE01, TM01 at k=1.2 by asymptotic method. Left: TE, Right: TM.

Fig. 4.
Fig. 4.

The index profile of a Bragg fiber in Galerkin method.

Fig. 5.
Fig. 5.

Band gap and Bragg modes obtained by Galerkin method. Left: TE, Right: TM.

Fig. 6.
Fig. 6.

Mode fields of Bragg mode by Galerkin method. Left: TE, Right: TM.

Tables (1)

Tables Icon

Table 1. Comparison of calculated effective indices by three methods at k 0 =1.2

Equations (16)

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[ E z H ϕ H z E ϕ ] = [ J m ( k i r ) Y m ( k i r ) 0 0 i ω ε k i J m ( k i r ) i ω ε k i Y m ( k i r ) m β k i 2 r J m ( k i r ) m β k i 2 r Y m ( k i r ) 0 0 J m ( k i r ) Y m ( k i r ) m β k i 2 r J m ( k i r ) m β k i 2 r Y m ( k i r ) i ω μ k i J m ( k i r ) i ω μ k i Y m ( k i r ) ] [ A B C D ]
[ A 1 B 1 C 1 D 1 ] = [ T 11 T 12 T 13 T 14 T 21 T 22 T 23 T 24 T 31 T 32 T 33 T 34 T 41 T 42 T 43 T 44 ] [ A N B N C N D N ]
[ E z H ϕ H z E ϕ ] = [ H m I ( k i r ) H m II ( k i r ) 0 0 i ω ε k i H m I ( k i r ) i ω ε k i H m II ( k i r ) m β k i 2 r H m I ( k i r ) m β k i 2 r H m II ( k i r ) 0 0 H m I ( k i r ) H m II ( k i r ) m β k i 2 r H m I ( k i r ) m β k i 2 r H m I ( k i r ) i ω μ k i H m i ( k i r ) i ω μ k i H m II ( k i r ) ] [ A N B N C N D N ]
[ A 1 B 1 C 1 D 1 ] = [ T 11 T 12 T 13 T 14 T 21 T 22 T 23 T 24 T 31 T 32 T 33 T 34 T 41 T 42 T 43 T 44 ] [ A N B N C N D N ]
[ T ] = [ T ] × [ 1 1 0 0 i i 0 0 0 0 1 1 0 0 i i ]
[ T 21 T 23 T 41 T 43 ] [ A N C N ] = 0
det [ T 21 T 23 T 41 T 43 ] = 0 .
Loss = 40 π λ ln 10 Im ( n eff )
d 2 f dr 2 + 1 r df dr + ( k 0 2 n 2 β 2 1 r 2 ) f = 0
d 2 g dr 2 + 1 r dg dr + ( k 0 2 n 2 β 2 1 r 2 ) g d ln n 2 dr ( dg dr + 1 r g ) = 0
x = σ r 2 / a 2 , h ( r ) = n 2 ( r ) n cl 2 n co 2 n cl 2 ,
V 2 = k 0 2 a 2 ( n co 2 n cl 2 ) , b = ( β / k 0 ) 2 n cl 2 n co 2 n cl 2
f ( x ) = i = 0 N 1 a i φ i ( x ) , g ( x ) = i = 0 N 1 b i φ i ( x )
φ i ( x ) = i ! ( i + m ) ! e x / 2 x m / 2 L i ( m ) ( x )
L i ( m ) ( x ) = k = 0 i ( i + m ) ! ( i k ) ! ( k + m ) ! k ! ( x ) k
[ M ] [ A ] = b [ A ]

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