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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References

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IEEE J. Selected Topics in Quantum Elect

S.V. Boriskina, T.M. Benson. P. Sewell and A.I. Nosich, "Highly efficient full-vectorial integral equations method solution for the bound, leaky, and complex modes of dielectric waveguides," IEEE J. Selected Topics in Quantum Electron. 8, 1225-1231 (2002).
[CrossRef]

IEEE Photon. Technol. Lett.

A. Cucinotta, S. Selleri, L. Vincent and M. Zoboli, "Holey fiber analysis through the finite element method," IEEE Photon. Technol. Lett. 14, 1530-1532 (2002).
[CrossRef]

J. Comput. Physics

A. Figotin and Y.A. Godin, "The computation of spetra of some 2D photonic crystals," J. Comput. Physics 136, 585-598 (1997).
[CrossRef]

V. Rokhlin, "Rapid solution of integral equations of classical potential theory," J. Comput. Physics 60, 187-207 (1985).
[CrossRef]

J. Lightwave Technol.

J. Opt. Soc. Am. B

Opt. Express

Opt. Lett.

Science

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

SIAM J. Numer. Anal.

S. Kapur and V. Rokhlin, "High-order corrected trapezoidal quadrature rules for singular functions," SIAM J. Numer. Anal. 34, 1331-1356 (1997).
[CrossRef]

SIAM J. Sci. Statist. Comput.

J. Carrier, L. Greengard and V. Rokhlin, "A fast adaptive multipole algorithm for particle simulations," SIAM J. Sci. Statist. Comput. 9, 669 (1988).
[CrossRef]

Other

H. Cheng, W.Y. Crutchfield and L. Greengard, "Sensitivity analysis of photonic crystal fibers," in preparation.

P.M. Morse and H. Feshbach, Methods of Theoretical Physics, Part I, (McGraw-Hill, New York, 1953).

R. B. Guenther and J. W. Lee, Partial Differential Equations of Mathematical Physics and Integral Equations, (Prentice-Hall, Englewood Cliffs, 1988).

J. Stoer and R. Bulirsch, Introduction to numerical analysis, Second Edition, (Springer-Verlag, New York, 1993).

A. Bjarklev, J. Broeng and A.S. Bjarklev, Photonic crystal fibers, (Kluwer Academic Publishers, Boston, 2003).
[CrossRef]

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

H. Cheng, W.Y. Crutch_eld and L. Greengard, "Fast, accurate integral equation methods for the analysis of photonic crystal fibers II: Acceleration techniques," in preparation.

A.W. Snyder and J.D. Love, Optical Waveguide Theory, (Chapman & Hall, London, 1996).

I. Stakgold, Green's Functions and Boundary Value Problems, (John Wiley & Sons, New York, 1979).

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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)

Tables Icon

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

Tables Icon

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

Tables Icon

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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