Abstract

The supercell- based orthonormal basis method is proposed to investigate the modal properties of the Bragg fibers. A square lattice is constructed by the whole Bragg fiber which is considered as a supercell, and the periodical dielectric structure of the square lattice is decomposed using periodic functions (cosine). The modal electric field is expanded as the sum of the orthonormal set of Hermite-Gaussian basis functions based on the opposite parity of the transverse electric field. The propagation characteristics of Bragg fibers can be obtained after recasting the wave equation into an eigenvalue system. This method is implemented with very high efficiency and accuracy.

© 2003 Optical Society of America

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References

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    [CrossRef]

J. Lightwave Technol.

J. Opt. Soc. Am.

Opt. Express

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

G. Ourang, Yong Xu, A. Yariv, �??Theoretical study on dispersion compensation in air-core Bragg fibers,�?? Opt. Express 10, 899-908 (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]

I.M. Bassett, A. Argyros, �??Elimination of polarization degeneracy in round waveguides,�?? Opt. Express 10, 1342-1346 (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]

Shangping Guo, Sacharia Albin, �??Simple plane wave implementation for photonic crystal calculations,�?? Opt. Express 11, 167-175 (2003), <a href="http://www. opticsexpress.org/ abstract. cfm? URI=OPEX-11-2-167">http://www. opticsexpress.org/ abstract. cfm? URI=OPEX-11-2-167</a>.
[CrossRef] [PubMed]

W. Zhi, R.G. Bin, L.S. Qin, and J.S.Sheng, �??Supercell lattice method for photonic crystal fibers,�?? Opt. Express 11, 980-991 (2003), <a href="http:// www.opticsexpress.org/ abstract. cfm? URI=OPEX-11-9-980">http:// www.opticsexpress.org/ abstract. cfm? URI=OPEX-11-9-980</a>.
[CrossRef] [PubMed]

T.D. Engeness, M. Ibanescu, S.G. Johnson, O.Weisberg, M.Skorobogatiy, S.Jacobs, Y.Fink, �??Dispersion tailoring and compensation by modal interactions in omniguide fibers,�?? Opt. Express 11, 1175-1196 (2003), <a href="http://www.opticsexpress.org/ abstract. cfm? URI =OPEX-11�??10-1175">http://www.opticsexpress.org/ abstract. cfm? URI =OPEX-11�??10-1175</a>.
[CrossRef] [PubMed]

R.G. Bin, W. Zhi, L.S. Qin, J.S. Sheng, �??Mode Classification and Degeneracy in Photonic Crystal Fiber,�?? Opt. Express 11, 1310-1321 (2003), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11�??11- 1310">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11�??11- 1310</a>.
[CrossRef]

J.A.Mosoriu, E. Silvestre, A. Ferrando, P.Andres, J.J. Miret, �??High-index-core Bragg fibers: dispersion properties,�?? Opt. Express 11, 1400-1405 (2003), <a href="http://www.opticsexpress.org/ abstract. cfm? URI =OPEX- 11�??12-1400">http://www.opticsexpress.org/ abstract. cfm? URI =OPEX- 11�??12-1400</a>.
[CrossRef]

Opt. Lett.

Phys. Rev. B

R. D. Meade, A.M. Rappe, K.D. Brommer, J.D. Joannopoulos, and O.L. Alerhand, �??Accurate theoretical analysis of photonic band-gap materials,�?? Phys. Rev. B 48, 8434-8437 (1993).
[CrossRef]

Phys. Rev. E

M. Ibanescu, S.G. Johnson, M. Soljacic, J.D. Joannopoulos, Y.Fink, �??Analysis of mode structure in hollow dielectric waveguide fibers,�?? Phys. Rev. E 67, 046608-1-8 (2003).
[CrossRef]

Science

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

S.D. Hart, G.R.Maskaly, B.Temelkuran, P.H. Prideaux, J.D. Joannopoulos, Y.Fink, �??External reflection from omnidirectional dielectric mirror fibers,�?? Science 296, 510-513 (2002).
[CrossRef] [PubMed]

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

J.C. Knight, P. St. Russell, �??New ways to guide light,�?? Science 296, 276-277 (2002).
[CrossRef] [PubMed]

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

Other

J.D. Joannopoulos, R.D. Meade, J.N. Winn, �??Photonic crystals: molding the flow of light,�?? (New York, Princeton University Press, 1995).

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

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Figures (4)

Fig. 1.
Fig. 1.

scheme of the construction of the supercell square lattice of the Bragg fiber, (a) is the radial distribution of the dielectric constant and (b) is the supercell square lattice.

Fig. 2.
Fig. 2.

The simulation result of the dielectric constant of the Bragg fiber, with the parameters ε 1=4.62, ε 2=1.62, ε 3=1.0, Λ=0.434µm, R=30Λ, a=0.78Λ, and m=17, supercell lattice constant D=1.2(2R+18Λ), P=1200.

Fig. 3.
Fig. 3.

The electric field of the modes HE11, TE01, TM01 and HE21 of the Bragg fiber with the structure parameters same as in Fig. 2, the annular dielectric constant is superimposed.

Fig. 4. (a)
Fig. 4. (a)

The propagation constant of TE01 mode, and (b) the difference between two different approaches

Equations (20)

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ε F ( k ) = 1 A A ε ( r ) e i k · r d s ,
ε ( r ) = { ε i , r i 1 < r < r i ε b , r > r m = ε b + { ε i ε b , r i 1 < r < r i 0 , r > r m ,
ε F ( k ) = 1 A A ε b e i k · r d s + 1 A i = 1 m A ( ε i ε b ) e i k · r d s
= ε b δ ( k ) + i = 1 m ( ε i ε b ) [ 2 f i J 1 ( k r i ) k r i 2 f i 1 J 1 ( k r i 1 ) k r i 1 ] ,
ε ( r ) = ε ( x , y ) = a , b = 0 P P ab cos 2 π a x D cos 2 π b y D
ln ε ( r ) = ln ε ( x , y ) = a , b = 0 P P ab ln cos 2 π a x D cos 2 π b y D ,
P ab = ε F ( k a + P , b + P ) + ε F ( k a + P , b + P ) + ε F ( k a + P , b + P ) + ε F ( k a + P , b + P ) ,
for a = 0 or b = 0 , P ab = ε F ( k a + P , b + P ) + ε F ( k a + P , b + P ) ,
for a = 0 and b = 0 , P 00 = ε F ( k P , P ) ,
( t 2 β 2 + k 0 2 ε ) e x = x ( e x ln ε x + e y ln ε y )
( t 2 β 2 + k 0 2 ε ) e y = y ( e x ln ε x + e y ln ε y ) ,
e x ( x , y ) mn = a , b = 0 F 1 ε ab x ψ 2 a + m ( x ) ψ 2 b + n ( y ) , e y ( x , y ) m ¯ n ¯ = a , b = 0 F 1 ε ab y ψ 2 a + m ¯ ( x ) ψ 2 b + n ¯ ( y ) ,
ψ i ( s ) = 2 i 2 π 1 4 i ! ϖ s exp ( s 2 2 ϖ s 2 ) H i ( s ϖ s ) ,
L mn [ ε x ε y ] [ [ I abcd ( 1 ) + k 2 I abcd ( 2 ) + I abcd ( 3 ) x ] mn [ I abcd ( 4 ) x ] mn [ I abcd ( 4 ) y ] mn ¯ [ I abcd ( 1 ) + k 2 I abcd ( 2 ) + I abcd ( 3 ) y ] mn ¯ ] [ ε x ε y ] = β 2 [ ε x ε y ] ,
[ I abcd ( 1 ) ] mn = + ψ 2 a + m ( x ) ψ 2 b + n ( y ) t 2 [ ψ 2 c + m ( x ) ψ 2 d + n ( y ) ] d x d y ,
[ I abcd ( 2 ) ] mn = + ε ψ 2 a + m ( x ) ψ 2 b + n ( y ) ψ 2 c + m ( x ) ψ 2 d + n ( y ) d x d y ,
[ I abcd ( 3 ) x ] mn = + ψ 2 a + m ( x ) ψ 2 b + n ( y ) x [ ψ 2 c + m ( x ) ψ 2 d + n ( y ) ln ε x ] d x d y ,
[ I abcd ( 3 ) y ] mn = + ψ 2 a + m ( x ) ψ 2 b + n ( y ) y [ ψ 2 c + m ( x ) ψ 2 d + n ( y ) ln ε y ] d x d y ,
[ I abcd ( 4 ) x ] mn = + ψ 2 a + m ( x ) ψ 2 b + n ( y ) x [ ψ 2 c + m ¯ ( x ) ψ 2 d + n ¯ ( y ) ln ε y ] d x d y ,
[ I abcd ( 4 ) y ] mn = + ψ 2 a + m ( x ) ψ 2 b + n ( y ) y [ ψ 2 c + m ¯ ( x ) ψ 2 d + n ¯ ( y ) ln ε x ] d x d y .

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