Abstract

Using an asymptotic formalism we developed in an earlier paper, we compare the dispersion properties of the air-core Bragg fiber with those of the coaxial Bragg fiber. In particular we are interested in the way the inner core of the coaxial fiber influence the dispersion relation. It is shown that, given appropriate structural parameters, large single-mode frequency windows with a zero-dispersion point can be achieved for the TM mode in coaxial fibers. We provide an intuitive interpretation based on perturbation analysis and the results of our asymptotic calculations are confirmed by Finite Difference Time Domain (FDTD) simulations.

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References

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  1. P. Yeh, A. Yariv, and E. Marom, "Theory of Bragg fiber," J. Opt. Soc. Am. 68, 1196-1201 (1978).
    [CrossRef]
  2. 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]
  3. M. Miyagi, A. Hongo, Y. Aizawa, and S. Kawakami, "Fabrication of germanium-coated nickel hollow waveguides for infrared transmission," Appl. Phys. Lett. 43, 430-432 (1983).
    [CrossRef]
  4. N. Croitoru, J. Dror, and I. Gannot, "Characterization of hollow fibers for the transmission of infrared radiation," Appl. Opt. 29, 1805-1809 (1990).
    [CrossRef] [PubMed]
  5. 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]
  6. M. Ibanescu, Y. Fink, S. Fan, E. L. Thomas, J. D. Joannopoulos, "An all-dielectric coaxial waveguide," Science 289, 415-419 (2000).
    [CrossRef] [PubMed]
  7. Y. Xu, G. Ouyang, R. Lee, and A. Yariv, "Asymptotic matrix theory of Bragg fibers," (submitted to J. Lightwave Technol.).
  8. Roy J. Glauber and M. Lewenstein, "Quantum optics of dielectric media," Phy. Rev. A 43, 467-491 (1991).
    [CrossRef]
  9. F. Zepparelli, P. Mezzanotte, F. Alimenti, L. Roselli, R. Sorrentino, G. Tartarini, and P. Bassi, "Rigorous analysis of 3D optical and optoelectronic devices by the compact-2D-FDTD method." Opt. and Quantum Electron. 31, 827-841 (1999).
    [CrossRef]
  10. K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media," IEEE Trans. Antennas Propag. AP-14, 302-307 (1966).
  11. J. P. Berenger, "A perfectly matched layer for the absorption of electromagnetic waves," J. Computat. Phys. 114, 185-200 (1994).
    [CrossRef]
  12. S. D. Gedney, "An anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD lattices," IEEE Trans. Antennas Propag. 44, 1630-1639 (1996).
    [CrossRef]

Other (12)

P. Yeh, A. Yariv, and E. Marom, "Theory of Bragg fiber," J. Opt. Soc. Am. 68, 1196-1201 (1978).
[CrossRef]

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]

M. Miyagi, A. Hongo, Y. Aizawa, and S. Kawakami, "Fabrication of germanium-coated nickel hollow waveguides for infrared transmission," Appl. Phys. Lett. 43, 430-432 (1983).
[CrossRef]

N. Croitoru, J. Dror, and I. Gannot, "Characterization of hollow fibers for the transmission of infrared radiation," Appl. Opt. 29, 1805-1809 (1990).
[CrossRef] [PubMed]

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, J. D. Joannopoulos, "An all-dielectric coaxial waveguide," Science 289, 415-419 (2000).
[CrossRef] [PubMed]

Y. Xu, G. Ouyang, R. Lee, and A. Yariv, "Asymptotic matrix theory of Bragg fibers," (submitted to J. Lightwave Technol.).

Roy J. Glauber and M. Lewenstein, "Quantum optics of dielectric media," Phy. Rev. A 43, 467-491 (1991).
[CrossRef]

F. Zepparelli, P. Mezzanotte, F. Alimenti, L. Roselli, R. Sorrentino, G. Tartarini, and P. Bassi, "Rigorous analysis of 3D optical and optoelectronic devices by the compact-2D-FDTD method." Opt. and Quantum Electron. 31, 827-841 (1999).
[CrossRef]

K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media," IEEE Trans. Antennas Propag. AP-14, 302-307 (1966).

J. P. Berenger, "A perfectly matched layer for the absorption of electromagnetic waves," J. Computat. Phys. 114, 185-200 (1994).
[CrossRef]

S. D. Gedney, "An anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD lattices," IEEE Trans. Antennas Propag. 44, 1630-1639 (1996).
[CrossRef]

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Figures (8)

Fig. 1.
Fig. 1.

Schematics of air-core and coaxial Bragg waveguide cross sections. (A) Air-core Bragg waveguide (B) Coaxial Bragg waveguide. Cladding layer 1 has an index of refraction n 1=4.6 and a thickness d 1=0.333a (not shown), whereas layer 2 an index of refraction n 2=1.6 and a thickness d 2=0.667a (not shown). Here, a=d 1+d 2 is the unit length of periodicity of the cladding. The white region in the core is assumed to be filled with air, with an index of refraction no =1. Three parameters are subject to change in this paper: ni , the index of the refraction of the inner core in (B); ri , the radius of the inner core in (B); and ro , the outer core radius in both (A) and (B). Note that an air-core Bragg fiber can be thought as a coaxial Bragg fiber with ri =0.

Fig. 2.
Fig. 2.

Band diagrams for ni =4.6 and ro =1.4a. ri is varied as follows: (A) ri =0 (air-core Bragg fiber), (B) ri =0.133a, (C) ri =0.2a, and (D) ri =0.267a. Note in (D) the TM mode exhibits single-mode behaviour in the lower half of the bandgap, which does not include the zero-dispersion point identified by the arrow.

Fig. 3.
Fig. 3.

E field profiles for ni =4.6 and ro =1.4a. The air-core Bragg fiber has a ri =0 whereas for the coaxial Bragg fiber ri =0.133a. The propagation constant β=0.05(2π/a) for all four modes and the eigen frequencies ω are found to be: (A) ω=0.2430(2πc/a), (B) ω=0.2040(2πc/a), (C) ω=0.1943(2πc/a), (D) ω=0.2018(2πc/a). Note in (D) the large jump discontinuity at the core boundary for Er .

Fig. 4.
Fig. 4.

Band diagrams for ni =1.45 and ro =1.4a. ri is varied as follows: (A) ri =0 (air-core Bragg fiber), (B) ri =0.133a, (C) ri =0.267a, and (D) ri =0.4a. Note the downward shifts of the two bands are small compared to those of Fig. 2. Also note that the shifts are of the same order of magnitude for the two bands.

Fig. 5.
Fig. 5.

E field profiles for ni =1.45, ri =0.4a, and ro =1.4a. The propagation constant β=0.05(2π/a) for both plots and the eigen frequencies ω are found to be: (A) ω=0.2267(2πc/a), (B) ω=0.1966(2πc/a). Note that in (B) the jump discontinuity for Er at the core boundary is not as sharp as that of Fig. 3

Fig. 6.
Fig. 6.

Band diagrams for ni =4.6 and ro =0.867a. ri is varied as follows: (A) ri =0 (air-core Bragg fiber), (B) ri =0.133a, (C) ri =0.2a, and (D) ri =0.267a. Note that in (D) the TM mode exhibits single-mode behaviour almost all the way through the bandgap, with the zero-dispersion point (identified by the arrow) falling inside the single-mode frequency range.

Fig. 7.
Fig. 7.

Coaxial fiber band diagram for ni =4.6, ri =0.267a, and ro =0.867a. The asymptotic curves are copied from Fig. 6(D) and the FDTD structure is defined with 5 cladding pairs.

Fig. 8.
Fig. 8.

The Ez field distribution of a TM mode based on FDTD simulation. The parameters of the Bragg fiber are given in the caption of Fig. 1 with ni =4.6, ri =0.267a, and ro =0.867a. The Bragg cladding consists of 5 cladding pairs and the whole fiber is immersed in air. The frequency and propagation constant of the mode are ω=0.2238(2πc/a) and β=0.2(2π/a).

Equations (31)

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× × E m = ε ( r , θ ) ( ω m c ) 2 E m .
d r ε ( r ) F n * ( r ) · F m ( r ) = δ m , n .
× × ( E m + δ E m ) = ( ε + Δ ε ) ( ω m + δ ω m c ) 2 ( E m + δ E m ) .
× × ( δ E m ) = ( ω m c ) 2 ε δ E m + ( ω m c ) 2 Δ ε ( E m + δ E m ) + ( 2 ω m c 2 ) δ ω m ( ε + Δ ε ) ( E m + δ E m ) .
δ E m = n a mn E n ,
n a mn [ ( ω n c ) 2 ( ω m c ) 2 ] ε E n = ( ω m c ) 2 Δ ε ( E m + δ E m ) + ( 2 ω m c 2 ) δ ω m ( ε + Δ ε ) ( E m + δ E m ) .
d r ( 2 ω m c 2 ) δ ω m ( ε + Δ ε ) E m * · ( E m + δ E m ) = d r ( ω m c ) 2 Δ ε E m * · ( E m + δ E m ) .
δ ω m = ( ω m 2 ) d r Δ ε E m * · ( E m + δ E m ) d r ( ε + Δ ε ) E m * · ( E m + δ E m ) .
( + i β e z ) × E ( x , y , t ) = μ 0 t H ( x , y , t ) ,
( + i β e z ) × H ( x , y , t ) = ε ( x , y ) t E ( x , y , t ) ,
ψ ( r , θ , z , t ) = ψ ( r , θ ) e i ( β z ω t ) ,
E r = i β ω 2 μ β 2 ( r E z + ω μ β r θ H z ) ,
E θ = i β ω 2 μ β 2 ( r θ E z ω μ β r H z ) .
E z = A J 0 ( k co r ) ,
k co = n 2 ω 2 c 2 β 2 .
E r = i β n 2 ω 2 c 2 β 2 ( r E z )
= i β n 2 ω 2 c 2 β 2 d dr [ A J 0 ( k co r ) ]
= i β k co A n 2 ω 2 c 2 β 2 J 0 ( k co r )
= i β k co A n 2 ω 2 c 2 β 2 J 1 ( k co r ) .
E z = A J 1 ( k co r ) cos θ ,
H z = CJ 1 ( k co r ) sin θ ,
E r = i β n 2 ω 2 c 2 β 2 ( r E z + ω μ β r θ H z )
= i β n 2 ω 2 c 2 β 2 { r [ AJ 1 ( k co r ) cos θ ] + ω μ r β θ [ CJ 1 ( k co r ) sin θ ] }
= i β cos θ n 2 ω 2 c 2 β 2 { Ak co J 1 ( k co r ) + ωC r β J 1 ( k co r ) }
= i β cos θ n 2 ω 2 c 2 β 2 { Ak co J 0 ( k co r ) + ( ωC r β Ak co k co r ) J 1 ( k co r ) }
i β cos θ n 2 ω 2 c 2 β 2 { A + ω C β } k co 2 as r 0 ,
E θ = i β n 2 ω 2 c 2 β 2 ( r θ E z ω μ β r H z )
= i β sin θ n 2 ω 2 c 2 β 2 { A r J 1 ( k co r ) + ωC β r [ J 1 ( k co r ) ] }
= i β sin θ n 2 ω 2 c 2 β 2 { A r J 1 ( k co r ) + ω k co C β [ J 0 ( k co r ) 1 k co r J 1 ( k co r ) ] }
i β sin θ n 2 ω 2 c 2 β 2 { ω k co C β J 0 ( k co r ) [ A r ω k co C β k co r ] J 1 ( k co r ) }
i β sin θ n 2 ω 2 c 2 β 2 { A + ω C β } k co 2 as r 0 .

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