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Fast, accurate integral equation methods for the analysis of photonic crystal fibers I: Theory

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Abstract

We present a new integral equation method for calculating the electromagnetic modes of photonic crystal fiber (PCF) waveguides. Our formulation can easily handle PCFs with arbitrary hole geometries and irregular hole distributions, enabling optical component manufacturers to optimize hole designs as well as assess the effect of manufacturing defects. The method produces accurate results for both the real and imaginary parts of the propagation constants, which we validated through extensive convergence analysis and by comparison with previously published results.

©2004 Optical Society of America

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Figures (10)

Fig. 1.
Fig. 1. Cross-section of a model PCF, whose longitudinal axis runs in the z-direction. V 0 denotes the glass matrix with index n 0. In this case, there are only four “holes” in the fiber V 1,…,V 4 made of materials with refractive indices n 1,…,n 4. At each point on a material interface, ν denotes the unit normal vector and τ denotes the unit tangent vector.
Fig. 2.
Fig. 2. Six circular holes.
Fig. 3.
Fig. 3. Convergence study for the second mode
Fig. 4.
Fig. 4. Six cookie-shaped holes.
Fig. 5.
Fig. 5. Convergence study of the PCF depicted in Fig.4 (h=6%)
Fig. 6.
Fig. 6. A PCF with circular holes.
Fig. 7.
Fig. 7. A PCF with irregular shaped holes.
Fig. 8.
Fig. 8. Convergence study of the PCF depicted in Fig.7
Fig. 9.
Fig. 9. Dispersion curve (real part).
Fig. 10.
Fig. 10. Dispersion curve (imaginary part).

Tables (3)

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Table 1. Effective index of the PCF depicted in Fig. 2

Tables Icon

Table 2. Effective index of the PCF depicted in Fig. 4

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Table 3. Effective index of the fundamental mode of PCFs with one to three layers of irregular holes and irregular locations (Fig.7). The results in last row are for the PCF with three layers of regular holes (Fig.6).

Equations (53)

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E ( x , y , z , t ) = E β ( x , y ) e i ( β z ω t ) = e i ( β z ω t ) ( E 1 β ( x , y ) , E 2 β ( x , y ) , E 3 β ( x , y ) ) , H ( x , y , z , t ) = H β ( x , y ) e i ( β z ω t ) = e i ( β z ω t ) ( H 1 β ( x , y ) , H 2 β ( x , y ) , H 3 β ( x , y ) ) .
[ ν × E β ] = 0 , [ ν × H β ] = 0 ,
2 E + ( k 2 β 2 ) E = 0 ,
2 H + ( k 2 β 2 ) H = 0 ,
[ E ] = 0 ,
[ H ] = 0 ,
[ β k 2 β 2 E τ ] = [ k vac k 2 β 2 H n ] ,
[ β k 2 β 2 H τ ] = [ k 2 k vac k 2 β 2 E n ] .
E j ( P ) = Γ j ( s E ( 2 , j ) G j ( P , Q ) σ j E ( Q ) + d E ( 2 , j ) G j ( P , Q ) n Q μ j E ( Q ) ) d s Q ,
H j ( P ) = Γ j ( s H ( 2 , j ) G j ( P , Q ) σ j H ( Q ) + d H ( 2 , j ) G j ( P , Q ) n Q μ j H ( Q ) ) d s Q .
E 0 ( P ) = j = 1 M Γ j ( s E ( 1 , j ) G 0 ( P , Q ) σ j E ( Q ) + d E ( 1 , j ) G 0 ( P , Q ) n Q μ j E ( Q ) ) d s Q ,
H 0 ( P ) = j = 1 M Γ j ( s H ( 1 , j ) G 0 ( P , Q ) σ j H ( Q ) + d H ( 1 , j ) G 0 ( P , Q ) n Q μ j H ( Q ) ) d s Q .
lim P P 0 P 0 Γ j P glass E 0 ( P ) = 1 2 d E ( 1 , j ) μ j E ( P 0 )
+ j = 1 N Γ j ( s E ( 1 , j ) G 0 ( P 0 , Q ) σ j E ( Q ) + d E ( 1 , j ) G 0 ( P 0 , Q ) n Q μ j E ( Q ) ) d s Q .
lim P P 0 P 0 Γ j P glass E 0 ( P ) τ = 1 2 d E ( 1 , j ) μ j E ( P 0 ) τ
+ j = 1 N Γ j ( s E ( 1 , j ) G 0 ( P 0 , Q ) τ σ j E ( Q ) + d E ( 1 , j ) 2 G 0 ( P 0 , Q ) τ n Q μ j E ( Q ) ) d s Q ,
lim P P 0 P 0 Γ j P glass E 0 ( P ) n P = 1 2 s E ( 1 , j ) σ j E ( P 0 )
+ j = 1 N Γ j ( s E ( 1 , j ) G 0 ( P 0 , Q ) n P σ j E ( Q ) + d E ( 1 , j ) 2 G 0 ( P 0 , Q ) n P n Q μ j E ( Q ) ) d s Q ,
lim P P 0 P 0 Γ j P V j E j ( P ) = 1 2 d E ( 2 , j ) μ j E ( P 0 )
+ Γ j ( s E ( 2 , j ) G j ( P 0 , Q ) σ j E ( Q ) + d E ( 2 , j ) G j ( P 0 , Q ) n Q μ j E ( Q ) ) d s Q .
lim P P 0 P 0 Γ j P V j E j ( P ) τ = 1 2 d E ( 2 , j ) μ j E ( P 0 ) τ
+ Γ j ( s E ( 2 , j ) G j ( P 0 , Q ) τ σ j E ( Q ) + d E ( 2 , j ) 2 G j ( P 0 , Q ) τ n Q μ j E ( Q ) ) d s Q ,
lim P P 0 P 0 Γ j P V j E j ( P ) n P = 1 2 s E ( 2 , j ) σ j E ( P 0 )
+ Γ j ( s E ( 2 , j ) G j ( P 0 , Q ) n P σ j E ( Q ) + d E ( 2 , j ) 2 G j ( P 0 , Q ) n P n Q μ j E ( Q ) ) d s Q ,
0 = 1 2 ( d E ( 1 , j ) + d E ( 2 , j ) ) μ j E ( P 0 )
+ j ' = 1 , j ' j N Γ j ' ( s E ( 1 , j ' ) G 0 ( P 0 , Q ) σ j ' E ( Q ) + d E ( 1 , j ' ) G 0 ( P 0 , Q ) n Q μ j ' E ( Q ) ) d s Q
+ Γ j ( [ s E ( 1 , j ) G 0 ( P 0 , Q ) s E ( 2 , j ) G j ( P 0 , Q ) ] · σ j E ( Q ) )
+ [ d E ( 1 , j ) G 0 ( P 0 , Q ) n Q d E ( 2 , j ) G j ( P 0 , Q ) n Q ] · μ j E ( Q ) ) d s Q ,
0 = 1 2 ( d H ( 1 , j ) + d H ( 2 , j ) ) μ j H ( P 0 )
+ j ' = 1 , j ' j N Γ j ' ( s H ( 1 , j ' ) G 0 ( P 0 , Q ) σ j ' H ( Q ) + d H ( 1 , j ' ) G 0 ( P 0 , Q ) n Q μ j ' H ( Q ) ) d s Q
+ Γ j ( [ s H ( 1 , j ) G 0 ( P 0 , Q ) s H ( 2 , j ) G j ( P 0 , Q ) ] · σ j H ( Q ) )
+ [ d H ( 1 , j ) G 0 ( P 0 , Q ) n Q d H ( 2 , j ) G j ( P 0 , Q ) n Q ] · μ j H ( Q ) ) d s Q ,
0 = 1 2 ( b t E ( 1 , j ) d E ( 1 , j ) + b t E ( 2 , j ) d E ( 2 , j ) ) μ j E ( P 0 ) τ
+ b t E ( 1 , j ) j ' = 1 , j ' j N Γ j ' ( s E ( 1 , j ' ) G 0 ( P 0 , Q ) τ σ j ' E ( Q ) + d E ( 1 , j ' ) 2 G 0 ( P 0 , Q ) τ n Q μ j ' E ( Q ) ) d s Q
+ Γ j ( [ b t E ( 1 , j ) s E ( 1 , j ) G 0 ( P 0 , Q ) τ b t E ( 2 , j ) s E ( 2 , j ) G j ( P 0 , Q ) τ ] · σ j E ( Q )
+ [ b t E ( 1 , j ) d E ( 1 , j ) 2 G 0 ( P 0 , Q ) τ n Q b t E ( 2 , j ) d E ( 2 , j ) 2 G j ( P 0 , Q ) τ n Q ] · μ j E ( Q ) ) d s Q
1 2 ( b t H ( 1 , j ) s H ( 1 , j ) + b n H ( 2 , j ) s H ( 2 , j ) ) σ j H ( P 0 )
+ b n H ( 1 , j ) j ' = 1 , j ' j N Γ j ' ( s H ( 1 , j ' ) G 0 ( P 0 , Q ) n P σ j ' E ( Q ) + d H ( 1 , j ' ) 2 G 0 ( P 0 , Q ) n P n Q μ j ' H ( Q ) ) d s Q
+ Γ j ( [ b n H ( 1 , j ) s H ( 1 , j ) G 0 ( P 0 , Q ) n P b n H ( 2 , j ) s H ( 2 , j ) G j ( P 0 , Q ) n P ] · σ j H ( Q )
+ [ b n H ( 1 , j ) d H ( 1 , j ) 2 G 0 ( P 0 , Q ) n P n Q b n H ( 2 , j ) d H ( 2 , j ) 2 G j ( P 0 , Q ) n P n Q ] · μ j H ( Q ) ) d s Q ,
0 = 1 2 ( b t H ( 1 , j ) d H ( 1 , j ) + b t H ( 2 , j ) d H ( 2 , j ) ) μ j H ( P 0 ) τ
+ b t H ( 1 , j ) j ' = 1 , j ' j N Γ j ' ( s H ( 1 , j ' ) G 0 ( P 0 , Q ) τ σ j ' H ( Q ) + d H ( 1 , j ' ) 2 G 0 ( P 0 , Q ) τ n Q μ j ' H ( Q ) ) d s Q
+ Γ j ( [ b t H ( 1 , j ) s H ( 1 , j ) G 0 ( P 0 , Q ) τ b t H ( 2 , j ) s H ( 2 , j ) G j ( P 0 , Q ) τ ] · σ j H ( Q )
+ [ b t H ( 1 , j ) d H ( 1 , j ) 2 G 0 ( P 0 , Q ) τ n Q b t H ( 2 , j ) d H ( 2 , j ) 2 G j ( P 0 , Q ) τ n Q ] · μ j H ( Q ) ) d s Q
1 2 ( b t E ( 1 , j ) s E ( 1 , j ) + b n E ( 2 , j ) s E ( 2 , j ) ) σ j E ( P 0 )
+ b n E ( 1 , j ) j ' = 1 , j ' j N Γ j ' ( s E ( 1 , j ' ) G 0 ( P 0 , Q ) n P σ j ' E ( Q ) + d E ( 1 , j ' ) 2 G 0 ( P 0 , Q ) n P n Q μ j ' E ( Q ) ) d s Q
+ Γ j ( [ b n E ( 1 , j ) s E ( 1 , j ) G 0 ( P 0 , Q ) n P b n E ( 2 , j ) s E ( 2 , j ) G j ( P 0 , Q ) n P ] · σ j E ( Q )
+ [ b n E ( 1 , j ) d E ( 1 , j ) 2 G 0 ( P 0 , Q ) n P n Q b n E ( 2 , j ) d E ( 2 , j ) 2 G j ( P 0 , Q ) n P n Q ] · μ j E ( Q ) ) d s Q ,
b n E ( 1 , j ) d E ( 1 , j ) = b n E ( 2 , j ) , d E ( 2 , j ) b n H ( 1 , j ) d H ( 1 , j ) = b n H ( 2 , j ) d H ( 2 , j ) ,
b n E ( 1 , j ) s E ( 1 , j ) + b n E ( 2 , j ) s E ( 2 , j ) 0 , b n H ( 1 , j ) s H ( 1 , j ) + b n H ( 2 , j ) s H ( 2 , j ) 0 .
A ( β ) · x = 0 .
L = 20 ln ( 10 ) · 2 π λ · ( n eff ) · 10 9 ,
r ( θ ) c i r
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