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Modeling of PCF with multiple reciprocity boundary element method

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Abstract

The multiple reciprocity boundary element method (MRBEM) is applied to the modeling of Photonic Crystal Fiber (PCF). With the MRBEM, the Helmholtz equation is converted into an integral equation using a series of higher order fundamental solutions of the Laplace equation. It is a much more efficient method to analyze the dispersion, birefringence and nonlinearity properties of PCFs compared with the conventional direct boundary element method (BEM).

©2004 Optical Society of America

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Figures (5)

Fig. 1.
Fig. 1. 3-ring holey fiber with radius=0.4µm and Λ=2.3µm
Fig. 2.
Fig. 2. Root searching figure: (a) direct BEM (b) MRBEM
Fig. 3.
Fig. 3. Birefringent PCF with Λ=2.3µm
Fig. 4.
Fig. 4. Vector plots of the fields as calculated using MRBEM: (a) Hx11 (b) Hy11
Fig. 5.
Fig. 5. Comparison of computation time

Tables (2)

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Table 1. Root searching results with different values of m

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Table 2. Comparison between the MRBEM and direct BEM

Equations (15)

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( t 2 + k 2 ) u = 0 ,
k = { k 0 2 n 2 β 2 k 0 2 n 2 > β 2 j β 2 k 0 2 n 2 β 2 > k 0 2 n 2
c ( r ) u ( r , r ) = Γ u 0 * ( r , r ) u n ( r ) d Γ Γ u 0 * n ( r , r ) u ( r ) d Γ ,
u 0 * ( r , r ) = ( j 4 ) H 0 [ 2 ] ( k r r ) .
t 2 u ( x ) + b 0 ( x ) = 0
c ( r ) u ( r ) + Γ u 0 * n ( r , r ) u ( r ) d Γ Γ u 0 * ( r , r ) u n ( r ) d Γ = Ω b 0 ( r ) u 0 * ( r , r ) d Ω
b j + 1 ( x ) = 2 b j ( x ) ; 2 u j + 1 * = u j * ,
Ω b 0 ( r ) u 0 * ( r , r ) d Ω = j = 0 m Γ u j + 1 * n ( r , t ) b 0 ( r ) d Γ j = 0 m Γ b j + 1 * n ( r ) u j + 1 ( r , r ) d Γ + Ω b m + 1 ( r ) u m + 1 * ( r , r ) d Ω
c ( r ) u ( r ) + j = 0 m ( k 2 ) j Γ u j * n ( r , r ) u ( r ) d Γ j = 0 m ( k 2 ) j Γ u j * ( r , r ) u n ( r ) d Γ = 0
u j * = 1 2 π r 2 j 1 4 j ( j ! ) 2 ( ln r s j )
j = 0 m ( k 2 ) j [ H j ] { u } j = 0 m ( k 2 ) j [ G j ] { u n } = { 0 }
[ H ] = [ H 0 ] k 2 [ H 1 ] + + ( k 2 ) m [ H m ] [ G ] = [ G 0 ] k 2 [ G 1 ] + + ( k 2 ) m [ G m ]
[ H ] { u } [ G ] { u n } = { 0 }
[ A ] { x } = { 0 }
A eff = [ + + H ( x , y ) 2 dxdy ] 2 + + H ( x , y ) 4 dxdy .
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