Modeling of PCF with multiple reciprocity boundary element method

Xiaoyan Wang; Junjun Lou; Chao Lu; Chun-Liu Zhao; W.T. Ang

doi:10.1364/OPEX.12.000961

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).

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References

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Article Order
Year
Author
Publication

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. Himeno, and A. Wada, “Characteristics of field confined holey fiber analyzed by boundary element method,” in OFC 2002.
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]
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)
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).
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)

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]

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.

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]

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.

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]

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.

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]

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.

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]

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.

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]

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.

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]

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)

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]

Opt. Lett. (1)

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]

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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Fig. 1. 3-ring holey fiber with radius=0.4µm and Λ=2.3µm

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Fig. 2. Root searching figure: (a) direct BEM (b) MRBEM

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Fig. 3. Birefringent PCF with Λ=2.3µm

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Fig. 4. Vector plots of the fields as calculated using MRBEM: (a)

H_{x}^{11}

(b)

H_{y}^{11}

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Fig. 5. Comparison of computation time

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Tables (2)

Table 1. Root searching results with different values of m

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Table 2. Comparison between the MRBEM and direct BEM

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Equations (15)

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(\nabla_{t}^{2} + k^{2}) u = 0,

k = {\begin{matrix} \sqrt{k_{0}^{2} n^{2} - β^{2}} & k_{0}^{2} n^{2} > β^{2} \\ - j \sqrt{β^{2} - k_{0}^{2} n^{2}} & β^{2} > k_{0}^{2} n^{2} \end{matrix}

c (r') u (r', r) = \int_{Γ} u_{0}^{*} (r', r) \frac{\partial u}{\partial n} (r) d Γ - \int_{Γ} \frac{\partial u_{0}^{*}}{\partial n} (r', r) u (r) d Γ,

u_{0}^{*} (r', r) = (- j ⁄ 4) H_{0}^{[2]} (k ∣ r - r' ∣) .

\nabla_{t}^{2} u (x) + b_{0} (x) = 0

c (r') u (r') + \int_{Γ} \frac{\partial u_{0}^{*}}{\partial n} (r', r) u (r) d Γ - \int_{Γ} u_{0}^{*} (r', r) \frac{\partial u}{\partial n} (r) d Γ = \int_{Ω} b_{0} (r) u_{0}^{*} (r', r) d Ω

b_{j + 1} (x) = \nabla^{2} b_{j} (x); \nabla^{2} u_{j + 1}^{*} = u_{j}^{*},

\int_{Ω} b_{0} (r) u_{0}^{*} (r', r) d Ω = \sum_{j = 0}^{m} \int_{Γ} \frac{\partial u_{j + 1}^{*}}{\partial n} (r', t) b_{0} (r) d Γ - \sum_{j = 0}^{m} \int_{Γ} \frac{\partial b_{j + 1}^{*}}{\partial n} (r) u_{j + 1} (r', r) d Γ + \int_{Ω} b_{m + 1} (r) u_{m + 1}^{*} (r', r) d Ω

c (r') u (r') + \sum_{j = 0}^{m} {(- k^{2})}^{j} \int_{Γ} \frac{\partial u_{j}^{*}}{\partial n} (r', r) u (r) d Γ - \sum_{j = 0}^{m} {(- k^{2})}^{j} \int_{Γ} u_{j}^{*} (r', r) \frac{\partial u}{\partial n} (r) d Γ = 0

u_{j}^{*} = - \frac{1}{2 π} r^{2 j} \frac{1}{{4^{j} (j!)}^{2}} (\ln r - s_{j})

\sum_{j = 0}^{m} {(- k^{2})}^{j} [H_{j}] {u} - \sum_{j = 0}^{m} {(- k^{2})}^{j} [G_{j}] {\frac{\partial u}{\partial n}} = {0}

\begin{array}{l} [H] = [H_{0}] - k^{2} [H_{1}] + \dots + {(- k^{2})}^{m} [H_{m}] \\ [G] = [G_{0}] - k^{2} [G_{1}] + \dots + {(- k^{2})}^{m} [G_{m}] \end{array}

[H] {u} - [G] {\frac{\partial u}{\partial n}} = {0}

[A] {x} = {0}

A_{eff} = \frac{{[\int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} {∣ H (x, y) ∣}^{2} dxdy]}^{2}}{{\int_{- \infty}^{+ \infty} \int}_{- \infty}^{+ \infty} {∣ H (x, y) ∣}^{4} dxdy} .

	m for silica region
		15	20	25
m for air region	5	1.429782	1.42978	1.42978
	6	1.429764	1.429762	1.429762
	7	1.429762	1.42976	1.42976
	8	1.429762	1.42976	1.42976

	n₁=16, n₂=16, n₃=8	n₁=32, n₂=16, n₃=8	n₁=32, n₂=20, n₃=8
MRBEM	1.42993	1.42976	1.42975
BEM	1.42994+5.744e-6i	1.429786+5.4378e-6i	1.42976+5.2580e-6i

	m for silica region
		15	20	25
m for air region	5	1.429782	1.42978	1.42978
	6	1.429764	1.429762	1.429762
	7	1.429762	1.42976	1.42976
	8	1.429762	1.42976	1.42976

	n₁=16, n₂=16, n₃=8	n₁=32, n₂=16, n₃=8	n₁=32, n₂=20, n₃=8
MRBEM	1.42993	1.42976	1.42975
BEM	1.42994+5.744e-6i	1.429786+5.4378e-6i	1.42976+5.2580e-6i

Modeling of PCF with multiple reciprocity boundary element method

Abstract

References

2003 (2)

2002 (1)

2001 (1)

1985 (1)

Andoh, E.

Botten, L. C.

Brebbia, C.A.

de Sterke, C. M.

Guan, N.

Habu, S.

Himeno, K.

Kamiya, N.

Lu, Tao

McPhedran, R. C.

Nogae, K.

Nowak, A.J.

Steel, M. J.

Su, C-C

Takenaga, K.

Wada, A.

White, T. P.

Yevick, David

IEEE J. Lightwave Technology (1)

IEEE Trans. Microwave Theory Tech. (1)

J. Lightwave Technol. (1)

Opt. Lett. (1)

Other (3)

Cited By

Figures (5)

Tables (2)

Equations (15)

Metrics

Optics Express

Abstract

References

2003 (2)

2002 (1)

2001 (1)

1985 (1)

Andoh, E.

Botten, L. C.

Brebbia, C.A.

de Sterke, C. M.

Guan, N.

Habu, S.

Himeno, K.

Kamiya, N.

Lu, Tao

McPhedran, R. C.

Nogae, K.

Nowak, A.J.

Steel, M. J.

Su, C-C

Takenaga, K.

Wada, A.

White, T. P.

Yevick, David

IEEE J. Lightwave Technology (1)

IEEE Trans. Microwave Theory Tech. (1)

J. Lightwave Technol. (1)

Opt. Lett. (1)

Other (3)

Cited By

Figures (5)

Tables (2)

Equations (15)

Metrics

Optics Express

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