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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References

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  1. Tao Lu and David Yevick, “A vectorial boundary element method analysis of integrated optical waveguides,” IEEE J. Lightwave Technology,  21, 1793–1807 (2003).
    [Crossref]
  2. N. Guan, S. Habu, K. Himeno, and A. Wada, “Characteristics of field confined holey fiber analyzed by boundary element method,” in OFC 2002.
  3. N. Guan, S. Habu, K. Takenaga, K. Himeno, and A. Wada, “Boundary element method for analysis of holey optical fibers,” J. Lightwave Technol. 21, 1787–1792 (2003).
    [Crossref]
  4. A.J. Nowak and C.A. Brebbia, “The Multiple Reciprocity Method” in Advanced Formulations in Boundary Element Methods. M.H. Aliabadi and C.A. Brebbia (eds). Chapter 3. Computational Mechanics Publications (1993).
  5. N. Kamiya, E. Andoh, and K. Nogae, “Application of the multiple reciprocity method to eigenvalue analysis of the Helmholtz equation” in The Multiple Reciprocity Boundary Element Method. A.J. Nowak and A.C. Neves (eds). Chapter 5. Computational Mechanics Publications, Southampton (1994)
  6. C-C Su, “A surface integral equations method for homogeneous optical fibers and coupled image lines of arbitrary cross sections,” IEEE Trans. Microwave Theory Tech.,  MTT-33, 1114–1119 (1985).
  7. T. P. White, R. C. McPhedran, C. M. de Sterke, L. C. Botten, and M. J. Steel, “Confinement losses in microstructured optical fibers,” Opt. Lett. 26, 1660–1662 (2001).
    [Crossref]

2003 (2)

Tao Lu and David Yevick, “A vectorial boundary element method analysis of integrated optical waveguides,” IEEE J. Lightwave Technology,  21, 1793–1807 (2003).
[Crossref]

N. Guan, S. Habu, K. Takenaga, K. Himeno, and A. Wada, “Boundary element method for analysis of holey optical fibers,” J. Lightwave Technol. 21, 1787–1792 (2003).
[Crossref]

2002 (1)

N. Guan, S. Habu, K. Himeno, and A. Wada, “Characteristics of field confined holey fiber analyzed by boundary element method,” in OFC 2002.

2001 (1)

1985 (1)

C-C Su, “A surface integral equations method for homogeneous optical fibers and coupled image lines of arbitrary cross sections,” IEEE Trans. Microwave Theory Tech.,  MTT-33, 1114–1119 (1985).

Andoh, E.

N. Kamiya, E. Andoh, and K. Nogae, “Application of the multiple reciprocity method to eigenvalue analysis of the Helmholtz equation” in The Multiple Reciprocity Boundary Element Method. A.J. Nowak and A.C. Neves (eds). Chapter 5. Computational Mechanics Publications, Southampton (1994)

Botten, L. C.

Brebbia, C.A.

A.J. Nowak and C.A. Brebbia, “The Multiple Reciprocity Method” in Advanced Formulations in Boundary Element Methods. M.H. Aliabadi and C.A. Brebbia (eds). Chapter 3. Computational Mechanics Publications (1993).

de Sterke, C. M.

Guan, N.

N. Guan, S. Habu, K. Takenaga, K. Himeno, and A. Wada, “Boundary element method for analysis of holey optical fibers,” J. Lightwave Technol. 21, 1787–1792 (2003).
[Crossref]

N. Guan, S. Habu, K. Himeno, and A. Wada, “Characteristics of field confined holey fiber analyzed by boundary element method,” in OFC 2002.

Habu, S.

N. Guan, S. Habu, K. Takenaga, K. Himeno, and A. Wada, “Boundary element method for analysis of holey optical fibers,” J. Lightwave Technol. 21, 1787–1792 (2003).
[Crossref]

N. Guan, S. Habu, K. Himeno, and A. Wada, “Characteristics of field confined holey fiber analyzed by boundary element method,” in OFC 2002.

Himeno, K.

N. Guan, S. Habu, K. Takenaga, K. Himeno, and A. Wada, “Boundary element method for analysis of holey optical fibers,” J. Lightwave Technol. 21, 1787–1792 (2003).
[Crossref]

N. Guan, S. Habu, K. Himeno, and A. Wada, “Characteristics of field confined holey fiber analyzed by boundary element method,” in OFC 2002.

Kamiya, N.

N. Kamiya, E. Andoh, and K. Nogae, “Application of the multiple reciprocity method to eigenvalue analysis of the Helmholtz equation” in The Multiple Reciprocity Boundary Element Method. A.J. Nowak and A.C. Neves (eds). Chapter 5. Computational Mechanics Publications, Southampton (1994)

Lu, Tao

Tao Lu and David Yevick, “A vectorial boundary element method analysis of integrated optical waveguides,” IEEE J. Lightwave Technology,  21, 1793–1807 (2003).
[Crossref]

McPhedran, R. C.

Nogae, K.

N. Kamiya, E. Andoh, and K. Nogae, “Application of the multiple reciprocity method to eigenvalue analysis of the Helmholtz equation” in The Multiple Reciprocity Boundary Element Method. A.J. Nowak and A.C. Neves (eds). Chapter 5. Computational Mechanics Publications, Southampton (1994)

Nowak, A.J.

A.J. Nowak and C.A. Brebbia, “The Multiple Reciprocity Method” in Advanced Formulations in Boundary Element Methods. M.H. Aliabadi and C.A. Brebbia (eds). Chapter 3. Computational Mechanics Publications (1993).

Steel, M. J.

Su, C-C

C-C Su, “A surface integral equations method for homogeneous optical fibers and coupled image lines of arbitrary cross sections,” IEEE Trans. Microwave Theory Tech.,  MTT-33, 1114–1119 (1985).

Takenaga, K.

Wada, A.

N. Guan, S. Habu, K. Takenaga, K. Himeno, and A. Wada, “Boundary element method for analysis of holey optical fibers,” J. Lightwave Technol. 21, 1787–1792 (2003).
[Crossref]

N. Guan, S. Habu, K. Himeno, and A. Wada, “Characteristics of field confined holey fiber analyzed by boundary element method,” in OFC 2002.

White, T. P.

Yevick, David

Tao Lu and David Yevick, “A vectorial boundary element method analysis of integrated optical waveguides,” IEEE J. Lightwave Technology,  21, 1793–1807 (2003).
[Crossref]

IEEE J. Lightwave Technology (1)

Tao Lu and David Yevick, “A vectorial boundary element method analysis of integrated optical waveguides,” IEEE J. Lightwave Technology,  21, 1793–1807 (2003).
[Crossref]

IEEE Trans. Microwave Theory Tech. (1)

C-C Su, “A surface integral equations method for homogeneous optical fibers and coupled image lines of arbitrary cross sections,” IEEE Trans. Microwave Theory Tech.,  MTT-33, 1114–1119 (1985).

J. Lightwave Technol. (1)

Opt. Lett. (1)

Other (3)

A.J. Nowak and C.A. Brebbia, “The Multiple Reciprocity Method” in Advanced Formulations in Boundary Element Methods. M.H. Aliabadi and C.A. Brebbia (eds). Chapter 3. Computational Mechanics Publications (1993).

N. Kamiya, E. Andoh, and K. Nogae, “Application of the multiple reciprocity method to eigenvalue analysis of the Helmholtz equation” in The Multiple Reciprocity Boundary Element Method. A.J. Nowak and A.C. Neves (eds). Chapter 5. Computational Mechanics Publications, Southampton (1994)

N. Guan, S. Habu, K. Himeno, and A. Wada, “Characteristics of field confined holey fiber analyzed by boundary element method,” in OFC 2002.

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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)

Tables Icon

Table 1. Root searching results with different values of m

Tables Icon

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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