## Modeling of PCF with multiple reciprocity boundary element method

Optics Express, Vol. 12, Issue 5, pp. 961-966 (2004)

http://dx.doi.org/10.1364/OPEX.12.000961

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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

## 1. Introduction

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]

*et al*. [2,3

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]

*k*. During the direct search process, the integrals have to be re-evaluated. Consequently, the scheme is very time-consuming.

*k*from the integration by using a series of fundamental solutions of the Laplace equation instead of the usual fundamental solution of the Helmholtz equation. Consequently, the efficiency in root searching is significantly improved.

## 2. MRBEM formulation

*k*

_{0}and

*β*are free space wave number and propagation constant respectively.

*r*′ can be expressed by the boundary integral as:

*r*is the field point,

*n*is the normal outward direction to the boundary and

*u**

_{0}is the zero order fundamental solution of the two dimensional Helmholtz equation given by,

*k*and |

*r*′-

*r*| (distance between source point and field point) as variables. Here

*k*appears in the argument of the Hankel function.

*c*(

*r*′) is 1 when the field point

*r*is inside the domain

*V*, 0 when

*r*is outside the domain and 1/2 when

*r*is on the boundary, which is smooth in the PCF problem.

*k*are substituted into the Green’s function and the boundary integrals involved are evaluated again and again, making the search very time consuming.

*j*is the order of fundamental solutions which has the value of 0, 1 and so on [4]. With this, the domain integral on the right hand side of Eq. (4) becomes,

*m*increase, the last term in Eq. (5) converges to zero. Thus when a sufficiently large

*m*is used to truncate the number of terms, Eq. (4) becomes, [5]

*u**

_{j}is

*j*th order fundamental solution given by,

*H*

_{j}] and [

*G*

_{j}] (j=0,1,…

*m*) are evaluated once only because the fundamental solution (Eq. (7)) is independent of

*k*.

*H*and

*G*are both real matrix instead of complex matrix in the case of BEM. This is because the fundamental solutions of Laplace equation are of real form instead of the complex fundamental solution for Helmholtz equation. By comparing the final matrix of BEM and MRBEM, one can find that MRBEM is equivalent to BEM but without the imaginary part. By doing so, the time taken to evaluate the integrations is greatly reduced, however, the imaginary part of the root will not be found.

*H*

_{x}and

*H*

_{y}) and longitudinal field components (

*E*

_{z}and

*H*

_{z}), an eigenvalue equation of the following form can be obtained which is the same as in direct BEM [2].

*x*is the unknown field values at the boundaries, and A is the coefficient matrix with unknown propagation constant.

## 3. Simulation results

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]

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]

*m*correctly for different regions. Results obtained with different values of

*m*at wavelength

*λ*=1.55µm are tabulated in Table 1.

*m*for silica region is higher than the one for air region. This is because the maximum distance

*r*between elements in silica region is much larger than that in air region, hence more terms are needed in order to make

*u**

_{j}in Eq. (7) converges to very small values. From Table 1, it is found the results converge with the increase of

*m*.

*λ*=1.55µm are compared in table 2. The results by BEM are in complex form and its accuracy has been verified with multipole method [7

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]

^{st}, 2

^{nd}and 3

^{rd}ring are represented by

*n*

_{1},

*n*

_{2}and

*n*

_{3}respectively. From Table 2, the results of the MRBEM agree well with the ones of the direct BEM. The number of terms

*m*used for silica and air region are 20 and 7 respectively in the MRBEM.

*n*

_{1},

*n*

_{2}and

*n*

_{3}are 32, 16, and 8 respectively. In the MRBEM,

*m*is 20 and 7 for silica and air region respectively. The root is present at the local minimum of the determinant of matrix

*A*. The insets in Fig. 2 are the surface plots of x component of the magnetic field by the corresponding methods.

*A*

_{eff}) can be calculated using

*A*

_{eff}calculated based on the mode profiles obtained with the MRBEM and direct BEM in Fig. 2 are 11.2115 µm

^{2}and 11.1686 µm

^{2}respectively.

*λ*=1.55µm obtained by BEM and MRBEM are 7.4×10

^{-4}and 7.3×10

^{-4}respectively. Figure 4 shows the vector field plots of the

*x*and

*y*polarized modes.

*m*in Eq. (5) used to express the integral in silica and air region are 15 and 5 respectively. It is obvious that the MRBEM outperforms the direct BEM in terms of simulation speed. It is also noteworthy that the speed of MRBEM can be further increased by putting a remainder term when the series starts converging instead of using very large number of terms [5].

## 4. Conclusion

*k*is separated from the integrand. The integrations are evaluated once only, which significantly improves the efficiency of the algorithm. Although only a real propagation constant is considered and MRBEM cannot be used to predicate the leakage loss due to a finite number of rings, it is effective in studying other PCF properties such as birefringence and dispersion as well as effective mode area. In conclusion, when confinement loss is not a design concern, MRBEM is a much faster approach to analyze the properties (e.g., dispersion, birefringence and nonlinearity) of PCFs.

## References and links

1. | Tao Lu and David Yevick, “A vectorial boundary element method analysis of integrated optical waveguides,” IEEE J. Lightwave Technology , |

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. |

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. , |

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. |

**OCIS Codes**

(060.2270) Fiber optics and optical communications : Fiber characterization

(230.3990) Optical devices : Micro-optical devices

(230.7370) Optical devices : Waveguides

**ToC Category:**

Research Papers

**History**

Original Manuscript: October 7, 2003

Revised Manuscript: March 2, 2004

Published: March 8, 2004

**Citation**

Xiaoyan Wang, Junjun Lou, Chao Lu, Chun-Liu Zhao, and W. Ang, "Modeling of PCF with multiple reciprocity boundary element method," Opt. Express **12**, 961-966 (2004)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-5-961

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### References

- Tao Lu and David Yevick, ???A vectorial boundary element method analysis of integrated optical waveguides,??? IEEE J. Lightwave Technol. 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, 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, 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 Technol. MTT-33, 1114-1119 (1985).
- T. P.White, R. C. McPhedran, C. M. de Sterke, L. C. Botten, andM. J. Steel, ???Confinement losses in microstructured optical fibers,??? Opt. Lett. 26, 1660-1662 (2001). [CrossRef]

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