Abstract

We developed an asymptotic formalism that fully characterizes the propagation and loss properties of a Bragg fiber with finite cladding layers. The formalism is subsequently applied to miniature air-core Bragg fibers with Silicon-based cladding mirrors. The fiber performance is analyzed as a function of the Bragg cladding geometries, the core radius and the material absorption. The problems of fiber core deformation and other defects in Bragg fibers are also addressed using a finite-difference time-domain analysis and a Gaussian beam approximation, respectively.

© 2003 Optical Society of America

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References

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  10. A. Taflove, S. C. Hagness, Computational electrodynamics: the finite-difference time-domain method, (Artech house, Boston, 2000).
  11. See for example, A. Yariv, Optical electronics in modern communications, (Oxford university press, New York, 1997) Chapter 2.

J. Lightwave Technol

Y. Fink, D. J. Ripin, S. Fan, C. Chen, J. D. Joannopoulos, and E. L. Thomas, �??Guiding optical light in air using an all-dielectric structure,�?? J. Lightwave Technol. 17, 2039-2041 (1999).
[CrossRef]

J. Lightwave Technol.

J. Opt. A

J. Marcou, F. Brechet, and P. Roy, �??Design of weakly guiding Bragg fibres for chromatic dispersion shifting towards short wavelengths,�?? J. Opt. A, 3, S144-S153 (2001).
[CrossRef]

J. Opt. Soc. Am.

Opt. Express

Opt. Lett.

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]

Other

A. Taflove, S. C. Hagness, Computational electrodynamics: the finite-difference time-domain method, (Artech house, Boston, 2000).

See for example, A. Yariv, Optical electronics in modern communications, (Oxford university press, New York, 1997) Chapter 2.

J. G. Fleming, S. Y. Lin, and R. Hadley, in Proceeding of the solid-state sensor actuator and microsystems workshop, p.p. 173, (Hilton Head, S.C., 2002).

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Figures (7)

Fig. 1.
Fig. 1.

Schematics of a Bragg fiber. In this paper, we assume the low index core is air (nco =1) and has radius of rco . The refractive index and the thickness of the cladding layers are respectively n 1, L 1, and n 2, L 2. The dashed line represents the interface between the “exact solution” region and the “asymptotic solution” region.

Fig. 2.
Fig. 2.

The asymptotic results of an air-core Bragg fiber with rco =7.5µm, n 1=3.5 and L1=0.11µm, n 2=2.0 and L2=0.21µm. (a) The E θ component of the TE01 mode at λ=1.65µm. (b) The effective indices of the TE01, TE02, TM01, and HE11 modes. (c) The loss of the TE01, TE02, TM01, and HE11 modes. In (a), (b) and (c), we set the absorption loss in the Si layer to be 10dB/cm. (d) The loss of the TE01 mode at λ=1.55µm with Si layer loss varying from 0dB/cm to 10dB/cm. The dashed line is a linear fit of the asymptotic results.

Fig. 3.
Fig. 3.

The loss of the TE01, TE02, TM01, and HE11 modes at λ=1.55µm. The refractive index and thickness of the cladding layers are the same as in Fig. 2. In (a) we choose the air core radius rco =8µm and vary the Bragg cladding pair number. The dashed lines are the fitting of the asymptotic results using Eq. (9). In (b), we use 4 pairs of Bragg cladding layers and vary the air core radius. The dashed lines are the fitting of the asymptotic results for the TM01 and HE11 modes using Eq. (11). The solid lines are the estimation given by Eq. (12).

Fig. 4.
Fig. 4.

Estimation of the modal loss from the picture of photons zigzagging within the Bragg fiber.

Fig. 5.
Fig. 5.

Asymptotic results for the TE01, TE02, TM01, and HE11 modal loss. The solid lines are the estimation of the TE mode loss given by Eq. (12). In (a), the fiber parameters are the same as used in Fig. 2(c). In (b), we change the Si3N4 layer thickness to 0.19µm, while the rest of the parameters remain the same as in (a).

Fig. 6.
Fig. 6.

(a) Dispersion of a Bragg fiber, with air core radius 5.3µm, Si layer thickness 0.11µm and Si3N4 layer thickness 0.21µm. The diamonds and the circles respectively represent the dispersion of the cylindrically symmetric fibers and the deformed fibers, calculated from FDTD simulations. The solid line is given by Eq. (8). In (b) and (c), we show the Bz field of the TE01 mode at λ=1.55µm in the circularly symmetric Bragg fiber and the deformed Bragg fiber.

Fig. 7.
Fig. 7.

Schematics of a Bragg fiber inter-connect.

Equations (15)

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[ E z H θ H z E θ ] = M n ( r ) [ A n B n C n D n ] ,
M n ( r ) = [ J m ( k n r ) Y m ( k n r ) 0 0 i ω ε 0 ε n k n J m ( k n r ) i ω ε 0 ε n k n Y m ( k n r ) m β k n 2 r J m ( k n r ) m β k n 2 r Y m ( k n r ) 0 0 J m ( k n r ) Y m ( k n r ) m β k n 2 r J m ( k n r ) m β k n 2 r Y m ( k n r ) i ω μ 0 k n J m ( k n r ) i ω μ 0 k n Y m ( k n r ) ] ,
k n = ε n ω 2 c 2 β 2 .
M n ( r ) = 1 r [ e i k n r e i k n r 0 0 ω ε 0 ε n k n e i k n r ω ε 0 ε n k n e i k n r 0 0 0 0 e i k n r e i k n r 0 0 ω μ 0 k n e i k n r ω μ 0 k n e i k n r ] .
[ A n + 1 B n + 1 C n + 1 D n + 1 ] = [ M n + 1 ( ρ n ) ] 1 M n ( ρ n ) [ A n B n C n D n ] ,
[ A out B out C out D out ] = [ 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 core B core C core D core ] .
T [ A core C core ] = 0 , T = [ t 21 t 23 t 41 t 43 ] .
n eff T E 0 i = 1 ( x 1 i λ 2 π r co ) 2 ,
Modal Loss Δ N ,
Δ TE = ε l 1 ε h 1 , Δ non TE = ε l 2 ( ε h 1 ) ε h 2 ( ε l 1 ) ,
Modal Loss ( 1 r co ) α .
Modal Loss = 3.46 × 10 3 × x 1 i λ r co 2 ( 1 2 ) ( dB cm ) ,
[ w ( z ) ] 2 = w 0 2 ( 1 + z 2 z 0 2 ) , z 0 = π w 0 2 λ .
[ w 0 w ( z ) ] 2 = 1 1 + z 2 λ 2 x 11 4 16 π 2 r 1 4 .
Throughtput Loss = 23.7 r 2 2 λ 2 r 1 4 ( dB ) .

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