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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 10, Iss. 10 — May. 20, 2002
  • pp: 449–454
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Fourier decomposition algorithm for leaky modes of fibres with arbitrary geometry

Leon Poladian, Nader A. Issa, and Tanya M. Monro  »View Author Affiliations


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

Microstructured or holey optical fibres have attracted much interest recently because they can exhibit properties (dispersion, mode size) very different from conventional fibres. These structures can consist exclusively of air holes in a uniform host material. The central region can confine light if it is surrounded by air holes, or alternatively, it can be effectively air-suspended by fine supporting filaments (Fig. 1). Although the majority of microstructured fibres have been fabricated by stacking capillaries, and so the first approach has been predominantly used to date, the second approach has recently been demonstrated to be practical using alternative fabrication techniques such as extrusion [1

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 Optical Fiber Communications Conference 2002 PD-FA1.

]. Regardless of the fabrication technique, single-material fibres do not have a raised index core, and so cannot support true bound modes. Instead, leaky modes describe the propagation and account for the gradual leakage of light between and across the holes.

Fig. 1. Four different fibre structures investigated with the ABC-FDM. The structures are all to scale and the core radius in structure (a) is 1 micron.

2 Structure of algorithm

A circular computational domain D of radius R is used to encapsulate the waveguide structure, and the index is uniform and equal to ncl in the infinite region outside the domain. The scalar wave equation in dimensionless coordinates x = r/R is

2ψ(x,θ)+V2(x,θ)ψ(x,θ)=W2ψ(x,θ),
(1)

where V 2(x,θ) = k 2 R 2[n 2(x,θ) - ncl2] depends on the index profile n(x,θ) of the fibre and W 2 = k 2 R 2[neff2 -ncl2] is a constant that depends on the effective index of the mode n eff. Note that V(x,θ) ≡ 0 outside D.

The field outside the computational domain D can be expressed exactly in the form

ψ(x,θ)=mBmeimθKm(Wx).
(2)

where Km (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

ψ(x,θ)=m,nAmnψmn(x,θ).
(3)

A set of weighting functions ϕmn (x,θ) is chosen [8

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

], and the following generalised eigenvalue problem for the unknown coefficients Amn and the eigenvalue W 2 is obtained:

MμνmnAmn=W2NμνmnAmn.
(4)

where the matrix elements are given by

Mμνmn=ϕμν2ψmn+ϕμνV2(x,θ)ψmn;Nμνmn=ϕμνψmn
(5)

The choice of inner product and basis functions is discussed in Appendix A. The complex eigenvalues of leaky modes arise from the non-self-adjoint boundary conditions applied to the system, which forces the matrix operators of Eq. (4) to be non-Hermitian. Thus, techniques which exploit Hermitian operator properties and variational theorems are unavailable to us, and as discussed below, we use inverse iteration to find the eigenvalues of interest. The essential properties of the expansion functions ψmn are that they: have the correct behaviour at the origin (such that the coordinate singularity at the origin is not problematic); contain adjustable parameters that allow continuity with the expansion outside the domain (allowing adjustable boundary conditions); and allow computationally efficient evaluation of the matrix elements. Our choice is a polar-coordinate harmonic Fourier decomposition(Eq. (6)). The basis functions can be chosen to exploit rotational symmetries of the structure for computational advantage.

The adjustable boundary condition Fourier decomposition method (ABC-FDM) proceeds as follows. To start the procedure, an initial guess for the value of 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.

Although we apply our algorithm to structures which only have leaky modes, it can be used directly with structures that have both bound and leaky modes. The algorithm proceeds exactly as above except that the adjustable parameter 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

As our first example we calculate the first two modes of the three fold symmetric structure shown in Fig. 1(a). The modes are calculated at λ = 1.55μm and the inner and outer radii of the annular sector shaped holes are 1 and 2 μm 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.

Note that in Fig. 2(a) the central confined region of the field of the first mode has essentially a constant phase (i.e. a planar phase front similar to a bound mode), but that beyond the confining holes the phase front becomes an outward travelling wave. The effective index is 1.374+0.000048i 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(). Its effective index and loss are 1.255 + 0.00075i and 27 dB/mm respectively.

Fig. 2. (a)Fundamental and (b) first higher order mode of 3 hole structure shown in Fig 1(a). (c)Ring mode and (d) hollow core mode of the structure shown in Fig 1(d). The contours are equally spaced at 3 dB intervals. The color indicates phase. The radial phase variation reveals outward propagating waves associated with confinement loss.

The solid and dashed curves in Fig. 3(a) compare the leakage loss for the lowest order mode as a function of wavelength for the structures shown in Fig. 1(a) and (b) respectively. The loss is shown for three different air fractions: f=0.8, 0.9 and 1.0. Notice that for the complete annulus (f=1.0) the 3 and 6 bridge structures become identical. Observe that the leakage loss produced by the bridges becomes relatively more important for shorter wavelengths, and that 6 thin bridges produces less loss than 3 thick bridges for the same air fraction. Likewise, the solid and dashed curves in Fig. 3(b) compare the loss for the structures shown in Fig. 1(a) and (c). Notice that increasing the outer radius to 3 μm, 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

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 Optical Fiber Communications Conference 2002 PD-FA1.

] 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 μm. 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.

Fig. 3. (a) Comparison of leakage loss for 3 and 6 hole structures shown in Fig. 1(a) and (b) as a function of wavelength and air fraction f. (b) Similar comparison of leakage loss for 3 holes with outer radii r = 2.0 and 3.0 μm. (c) Effective index.

4 Inverse Error Analysis

Fig. 4. (a) Reconstruction of waveguide from computed solution of Fig. 2(a). (b) Reconstruction of hollow core waveguide from solution shown in Fig. 2(d). (c) Contour map of the same reconstruction as in (b). Dimensions are in microns.

5 Conclusion

The ABC-FDM can be used to find leaky modes of a large variety of structures. We intend to explore more efficient algorithms for finding eigenvectors of non-self-adjoint systems in future work. Detailed studies of convergence, efficiency and the generalisation to the vector case will also be forthcoming.

Acknowledgments

We acknowledge the support of the Australian Research Council, Sydney VisLab and our colleagues Ian Bassett, Steven Manos and Whayne Padden.

A Choice of Basis Functions

We us a generalised Galerkin technique [8

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

] wish expansion functions ψmn and weight functions ϕmn . 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

ψmn(x,θ)=eimθ[sin(nπx)+αmn(W)+βmn(W)x]
ϕmn(x,θ)=eimθsin(nπx)
(6)

where

αmn(W)=δm,0[1+(1)n1WK0(W)K0(W)],
βmn(W)=δm,0+(1δm,0)(1)nWKm(W)Km(W)Km(W).
(7)

The adjustable parameters α and β provide for the correct behaviour at the origin and also satisfy the boundary condition ψ′(x, 0)/ψ(x, 0) = WKm(Wx)/Km (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]

] because we expand both the azimuthal and radial dependence in complete Fourier series whereas 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]

] the radial dependence is treated using Runge-Kutta techniques. Both treatments allow matching to arbitrary boundary conditions.

The azimuthal indices 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

fg=02π01f(x,θ)*g(x,θ)dxdθ,
(8)

and the first set can be shown to be complete by using standard tests.

The integrals for 〈ϕj |∇2 ψi 〉 and 〈ϕj |ψi 〉 do not depend on the specific structure and involve simple trigonometric functions and so can be pre-tabulated for computational efficiency. The integrals for 〈ϕj |V 2(x,θ)ψi 〉 depend on the structure and can also be done analytically if the holes are in the shape of annular sectors, otherwise they can be done numerically. Most importantly though, because the trig functions appearing are all harmonically related, the number of distinct sum and difference frequencies that arise when products of trig functions appear in the inner products is only of order 2n max where n max is the largest radial index used in the basis set. This number is much less than the number of elements in the matrices and makes the scheme computationally viable even if the integrals need to be calculated numerically.

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 Optical Fiber Communications Conference 2002 PD-FA1.

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. 24, 276–278 (1999). [CrossRef]

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. 17, 1093–1102 (1999). [CrossRef]

4.

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

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. 18, 1084–1100 (2000). [CrossRef]

6.

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]

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. 26, 1660–1662 (2001). [CrossRef]

8.

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

9.

A. W. Snyder and J. D. Love, Optical Waveguide Theory, (Chapman and Hall,1983) p. 376.

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

  1. 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.
  2. 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]
  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. 17, 1093�??1102 (1999). [CrossRef]
  4. Y-H. Wang and C. Vassallo, �??Circular fourier analysis of arbitrarily shaped optical fibers,�?? Opt. Lett. 14, 1377�??1379 (1989). [CrossRef] [PubMed]
  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. 18, 1084�??1100 (2000). [CrossRef]
  6. 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]
  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. 26, 1660�??1662 (2001). [CrossRef]
  8. C. A. J. Fletcher, Computational Galerkin Methods, (Springer Verlag, 1984). [CrossRef]
  9. A. W. Snyder and J. D. Love, Optical Waveguide Theory, (Chapman and Hall, 1983) p. 376.

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