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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Appl. Opt. (1)

IEEE J. Quantum Electron. (2)

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

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

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)

A. Weisshaar et 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]

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

Y. Xu et al., "Asymptotic matrix theory of Bragg fibers," J. Lightwave Technol. 20, 428 (2002).
[CrossRef]

J. Lightwave. Technol. (1)

C. Themistos et al., "Loss/gain characterization of optical waveguides," J. Lightwave. Technol. 13, 1760 (1995).
[CrossRef]

J. Opt. Soc. Am. (1)

Nature (1)

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

Opt. Express (10)

N. A. Mortensen, "Effective area of photonic crystal fibers," Opt. Express 10, 341 (2002), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-7-341">.http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-7-341</a>
[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), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-733">.http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-733</a>
[CrossRef] [PubMed]

S. G. Johnson et al., "Low-loss asymptotically single-mode propagation in large core omniguide fibers," Opt. Express 9, 748 (2001), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-748">.http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-748</a>
[CrossRef] [PubMed]

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

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

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

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

Y. Xu et al., "Asymptotic analysis of silicon based Bragg fibers," Opt. Express 11, 1039 (2003), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-9-1039">.http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-9-1039</a>
[CrossRef] [PubMed]

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

W. Zhi et al., "Compact supercell method based on opposite parity for Bragg fibers," Opt. Express 11, 3542-3549 (2003), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-26-3542">.http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-26-3542</a>
[CrossRef] [PubMed]

Opt. Lett. (4)

Opt. Quantum Electron. (2)

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

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]

Optica Acta (1)

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

Science (2)

Y. Fink et al., "A dielectric omnidirectional reflector," Science 282, 1679 (1998).
[CrossRef] [PubMed]

M. Ibanescu et al., "An all-dielectric coaxial waveguide," Science 289, 415 (2000).
[CrossRef] [PubMed]

Other (3)

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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