## Fourier decomposition algorithm for leaky modes of fibres with arbitrary geometry

Optics Express, Vol. 10, Issue 10, pp. 449-454 (2002)

http://dx.doi.org/10.1364/OE.10.000449

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### Abstract

A new algorithm for calculating the confinement loss of leaky modes in arbitrary fibre structures is presented within the scalar wave approximation. The algorithm uses a polar-coordinate Fourier decomposition method with adjustable boundary conditions (ABC-FDM) to model the outward radiating fields. Leaky modes are calculated for different examples of microstructured fibres with various shaped holes.

© 2002 Optical Society of America

## 1 Introduction

## 2 Structure of algorithm

*D*of radius

*R*is used to encapsulate the waveguide structure, and the index is uniform and equal to

*n*

_{cl}in the infinite region outside the domain. The scalar wave equation in dimensionless coordinates

*x*=

*r*/

*R*is

*V*

^{2}(

*x*,

*θ*) =

*k*

^{2}

*R*

^{2}[

*n*

^{2}(

*x*,

*θ*) -

*n*(

*x*,

*θ*) of the fibre and

*W*

^{2}=

*k*

^{2}

*R*

^{2}[

*n*

_{eff}. Note that

*V*(

*x*,

*θ*) ≡ 0 outside

*D*.

*D*can be expressed exactly in the form

*K*

_{m}(

*Wx*) are modified Bessel functions. For bound modes

*W*is real and these fields are evanescent. For radiating and leaky modes

*W*is complex and the fields express an outward propagating field. In either case, the expansion outside

*D*has the correct physical behaviour at infinity. The field inside

*D*is expanded in some complete set of basis functions

*ϕ*

_{mn}(

*x*,

*θ*) is chosen [8

8. C. A. J. Fletcher, *Computational Galerkin Methods*, (Springer Verlag,1984). [CrossRef]

*A*

_{mn}and the eigenvalue

*W*

^{2}is obtained:

*n*

_{eff}is used to determine the parameter

*W*appearing in the external field expansion. The adjustable parameters in the basis functions (Eq. (7)) are chosen to match the external field. The matrix problem is solved for the eigenvalue and eigenvector of interest. Inverse iteration methods can be used to narrow in on the eigenvalue closest to some chosen value, and additional eigenvalues and eigenvectors can subsequently be found by working in the subspace orthogonal to the modes already found. The eigenvalue then gives an improved estimate of

*n*

_{eff}for the mode of interest. If required, this new estimate can be used to re-adjust the external field and the adjustable boundary conditions again to obtain an improved estimate and continued iteration can be used to increase the accuracy of the answers. Such re-iteration will converge quickly for any mode with the majority of its guided power within the computational domain. Since the imaginary part is often orders of magnitude smaller than the real part our experience was that, in most cases of interest, a single reiteration was sufficient to accurately determine the imaginary part of the index unless the loss is extremely large.

*W*remains real for all bound modes. However, if one is only interested in bound modes, formulations that exploit self-adjointness of bound mode problems will in general be more efficient.

## 3 Numerical results

*μ*m and the inner and outer radii of the annular sector shaped holes are 1 and 2

*μ*m respectively with an angular width of 108 degrees. Throughout we assume the material to be undoped silica, although this is not a restriction of the method. The contours are in decibels, and the fields are also displayed out beyond the domain radius. The phase of the fields is denoted by the colour. These modes were found using 1050 basis functions (50 radial terms, 21 azimuthal terms) which took of the order of 6 mins on a desktop Pentium III.

*i*yielding a confinement loss of 1.7 dB/mm. The higher order mode (b) shows the expected angular variation in phase of the form exp(

*iϕ*). Its effective index and loss are 1.255 + 0.00075

*i*and 27 dB/mm respectively.

*μ*m. The first ring of elliptical holes have radii 0.3 and 0.9

*μ*m at distance 1.65

*μ*m from the origin. The second ring of circular holes have radii 0.3

*μ*m and are 2.55

*μ*m from the centre. These modes were calculated using 100 radial and 41 azimuthal terms taking a few hours. Two different modes are shown: one which is guided in the material around the inner hole Fig. 2(c), and one which is guided within the hole Fig. 2(d). The mode in (c) has effective index and loss of 1.243+0.0075

*i*and 270 dB/mm respectively. The mode in (d) has effective index and loss of 0.848+0.00095

*i*and 35 dB/mm respectively. All of these modes are extremely lossy and would be unlikely to be observed in practice without the addition of many more holes to improve the confinement. Despite this, the algorithm is still able to find the leaky modes, and can describe modes that are guided either by average index or band-gap effects. We next show how loss varies with wavelength, the number of holes and hole dimensions.

*μ*m, as in structure (c) substantially reduces the confinement loss as expected. Note these values are comparable (accounting for material and scattering loss and the scalar approximation) with the measurement [1] of 4 dB/m for a fibre with the same geometry as Fig. 1(c) with

*f*= 0.9 but a higher glass index of 1.8 at 1.55

*μ*m. Fig. 3(c) shows the real part of the effective index of the fundamental mode. Note that it depends weakly on

*f*suggesting such properties can be approximated without including the bridges.

## 5 Conclusion

## Acknowledgments

## A Choice of Basis Functions

8. C. A. J. Fletcher, *Computational Galerkin Methods*, (Springer Verlag,1984). [CrossRef]

*ψ*

_{mn}and weight functions

*ϕ*

_{mn}. The presence of leakage destroys the self-adjointness of

*M*regardless of the choice of basis functions or inner product. Thus, we have the freedom to choose functions purely on the basis of computational efficiency. We investigated a large variety of different basis function sets and finally chose

*α*and

*β*provide for the correct behaviour at the origin and also satisfy the boundary condition

*ψ*′(

*x*, 0)/

*ψ*(

*x*, 0) =

*WK*′

_{m}(

*Wx*)/

*K*

_{m}(

*W*) at

*x*= 1. Our treatment differs from that in [4

4. Y-H. Wang and C. Vassallo, “Circular fourier analysis of arbitrarily shaped optical fibers,” Opt. Lett. **14**, 1377–1379 (1989). [CrossRef] [PubMed]

4. Y-H. Wang and C. Vassallo, “Circular fourier analysis of arbitrarily shaped optical fibers,” Opt. Lett. **14**, 1377–1379 (1989). [CrossRef] [PubMed]

*m*vary over all integers and the radial indices

*n*vary over all positive integers. Appropriate subsets of the azimuthal indices can be used if the modes have rotational symmetries. The second set of functions is complete over the space of bounded functions within the computational domain using the standard inner product

## References and links

1. | T.M. Monro, K.M. Kiang, J.H. Lee, K. Frampton, Z. Yusoff, R. Moore, J. Tucknott, D.W. Hewak, H.N. Rutt, and D.J. Richardson, “High nonlinearity extruded single-mode holey optical fibers” in |

2. | A. Ferrando, E. Silvestre, J. J. Miret, P. Andrés, and M. V. Andrés, “Full-vector analysis of a realistic photonic crystal fiber,” Opt. Lett. |

3. | T. M. Monro, D. J. Richardson, N. G. R. Broderick, and P.J. Bennet, “Holey optical fibers: An efficient modal method,” J. Lightwave Technol. |

4. | Y-H. Wang and C. Vassallo, “Circular fourier analysis of arbitrarily shaped optical fibers,” Opt. Lett. |

5. | B. J. Eggleton, P. S. Westbrook, C. A. White, C. Kerbage, R. S. Windeler, and G. L. Burdge, “Cladding-mode-resonances in air-silica microstructure optical fibers,” J. Lightwave Technol. |

6. | M. J. Steel and R. M. Osgood Jr., “Polarization and dispersive properties of elliptical-hole photonic crystal fibers,” J. Lightwave Technol. |

7. | T. P. White, R. C. McPhedran, C. M. de Sterke, L. C. Botten, and M. J. Steel,“Confinement losses in microstructured optical fibres,” Opt. Lett. |

8. | C. A. J. Fletcher, |

9. | A. W. Snyder and J. D. Love, |

**OCIS Codes**

(060.2430) Fiber optics and optical communications : Fibers, single-mode

(230.3990) Optical devices : Micro-optical devices

**ToC Category:**

Research Papers

**History**

Original Manuscript: April 24, 2002

Revised Manuscript: May 14, 2002

Published: May 20, 2002

**Citation**

Leon Poladian, Nader Issa, and Tanya Monro, "Fourier decomposition algorithm for leaky modes of fibres with arbitrary geometry," Opt. Express **10**, 449-454 (2002)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-10-10-449

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### References

- T.M. Monro, K.M. Kiang, J.H. Lee, K. Frampton, Z. Yuso., R. Moore, J. Tucknott, D.W. Hewak, H.N. Rutt and D.J. Richardson,�??High nonlinearity extruded single-mode holey optical fibers�??in Optical Fiber Communications Conference 2002 PD-FA1.
- A. Ferrando, E. Silvestre, J. J. Miret, P. Andres and M. V. Andres, �??Full-vector analysis of a realistic photonic crystal fiber,�?? Opt. Lett. 24, 276�??278 (1999). [CrossRef]
- T. M. Monro, D. J. Richardson, N. G. R. Broderick and P.J. Bennet, �??Holey optical fibers: An efficient modal method,�?? J. Lightwave Technol. 17, 1093�??1102 (1999). [CrossRef]
- Y-H. Wang and C. Vassallo, �??Circular fourier analysis of arbitrarily shaped optical fibers,�?? Opt. Lett. 14, 1377�??1379 (1989). [CrossRef] [PubMed]
- B. J. Eggleton, P. S. Westbrook, C. A. White, C. Kerbage, R. S. Windeler and G. L. Burdge, �??Cladding-mode-resonances in air-silica microstructure optical fibers,�?? J. Lightwave Technol. 18, 1084�??1100 (2000). [CrossRef]
- M. J. Steel and R. M. Osgood Jr., �??Polarization and dispersive properties of elliptical-hole photonic crystal fibers,�?? J. Lightwave Technol. 19, 495�??503 (2001). [CrossRef]
- T. P. White, R. C. McPhedran, C. M. de Sterke, L. C. Botten and M. J. Steel,�??Confinement losses in microstructured optical fibres,�?? Opt. Lett. 26, 1660�??1662 (2001). [CrossRef]
- C. A. J. Fletcher, Computational Galerkin Methods, (Springer Verlag, 1984). [CrossRef]
- A. W. Snyder and J. D. Love, Optical Waveguide Theory, (Chapman and Hall, 1983) p. 376.

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