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Compact supercell method based on opposite parity for Bragg fibers

Wang Zhi, Ren Guobin, Lou Shuqin, Liang Weijun, and Shangping Guo

Open Access Open Access

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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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.

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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.

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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.

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Fig. 4. (a)
Fig. 4. (a) The propagation constant of TE01 mode, and (b) the difference between two different approaches

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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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