## Modelling of microstructured waveguides using a finite-element-based vectorial mode solver with transparent boundary conditions

Optics Express, Vol. 12, Issue 12, pp. 2795-2809 (2004)

http://dx.doi.org/10.1364/OPEX.12.002795

Acrobat PDF (2632 KB)

### Abstract

A finite-element-based vectorial optical mode solver is used to analyze microstructured optical waveguides. By employing 1^{st}-order Bayliss-Gunzburger-Turkel-like transparent boundary
conditions, both the real and imaginary part of the modal indices can be calculated in a relatively small computational domain. Results for waveguides with either circular or non-circular
microstructured holes, solid- or air-core will be presented, including the silica-air Bragg fiber recently demonstrated by Vienne *et al*. (Post-deadline Paper PDP25,
*OFC* 2004). The results of solid-core structures are in good agreement with the results of other methods while the results of air-core structure agree to the experimental results.

© 2004 Optical Society of America

## 1. Introduction

1. J.C. Knight, T.A. Birks, P.S.J. Russell, and D.M. Atkin, “All-silica single-mode optical fiber with photonic crystal cladding,” Opt. Lett. **21**, 1547–1549 (1996). [CrossRef] [PubMed]

2. P. Russell, “Photonic Crystal Fibers,” Science **299**, 358–362 (2003). [CrossRef] [PubMed]

3. Y. Fink, D.J. Ripin, S. Fan, C. Chen, J.D. Joannopoulos, and E.L. Thomas, “Guiding optical light in air using an all-dielectric structure,” J. Lightwave Technol. **17**, 2039–2041 (1999). [CrossRef]

4. A. Ferrando, E. Silvestre, J.J. Miret, and P. Andres, “Nearly zero ultraflattened dispersion in photonic crystal fibers,” Opt. Lett. **25**, 790–792 (2000). [CrossRef]

5. J.C. Knight, J. Arriaga, T.A. Birks, A. Ortigosa-Blanch, W.J. Wadsworth, and P.S.J. Russell, “Anomalous dispersion in photonic crystal fiber,” Photonics Technol. Lett. **12**, 807–809 (2000). [CrossRef]

6. 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]

7. T.A. Birks, J.C. Knight, and P.S.J. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. **22**, 961–963 (1997). [CrossRef] [PubMed]

8. S.G. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, T.D. Engeness, M. Soljacic, S.A. Jacobs, J.D. Joannopoulos, and Y. Fink, “Low-loss asymptotically single-mode propagation in large-core OmniGuide fibers,” Opt. Express **9**, 748–779 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-748. [CrossRef] [PubMed]

10. A. Argyros, N. Issa, I. Basset, and M. van Eijkelenborg, “Microstructured optical fiber for single-polarization air guidance,” Opt. Lett. **29**, 20–22 (2004). [CrossRef] [PubMed]

11. A. Ferrando, E. Solvestre, 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]

12. W. Zhi, R. Guobin, L. Shuqin, and J. Shuisheng, “Supercell lattice method for photonic crystal fibers,” Opt. Express **11**, 980–991 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-9-980. [CrossRef] [PubMed]

13. T.P. White, B.T. Kuhlmey, R.C. McPhedran, D. Maystre, R. Renversez, C.M. de Sterke, and L.C. Botten, “Multipole method for microstructured optical fibers. I. Formulation,” J. Opt. Soc. Am B **19**, 2322–2330 (2002). [CrossRef]

14. N.A. Issa and L. Poladian, “Vector wave expansion method for leaky modes of microstructured optical fibers,” J. Lightwave Technol. **21**, 1005–1012 (2003). [CrossRef]

15. F. Fogli, L. Saccomandi, P. Bassi, G. Bellanca, and S. Trillo, “Full vectorial BPM modeling of index-guiding photonic crystal fibers and couplers,” Opt. Express **10**, 54–59 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-1-54. [CrossRef] [PubMed]

16. K. Saitoh and M. Koshiba, “Full-vectorial imaginary-distance beam propagation method based on a finite element scheme: application to photonic crystal fibers,” J. Quantum
Electron. **38**, 927–933 (2002). [CrossRef]

17. C.P. Yu and H.C. Chang, “Applications of the finite difference mode solution method to photonic crystal structures,” Opt. Quantum Electron. **36**, 145–163 (2004). [CrossRef]

18. F. Brechet, J. Marcou, D. Pagnoux, and P. Roy, “Complete analysis of the characteristics of propagation into photonic crystal fibers by finite element method,” Optical Fiber Technol. **6**, 181–191 (2000). [CrossRef]

21. D. Ferrarini, L. Vincetti, M. Zoboli, A. Cucinotta, and S. Selleri, “Leakage properties of photonic crystal fibers,” Opt. Express **10**, 1314–1319 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-23-1314. [CrossRef] [PubMed]

22. K. Saitoh and M. Koshiba, “Leakage loss and group velocity dispersion in air-core photonic bandgap fibers,” Opt. Express **11**, 3100–3109 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-23-3100 [CrossRef] [PubMed]

^{st}-order Bayliss-Gunzburger-Turkel-like (BGT-like) transparent boundary conditions (TBC) [23] to model various kinds of microstructured optical waveguides. Thanks to the boundary conditions, the structure can be analyzed in a relatively small computational domain for its complex-valued modal indices and field profiles. The structures being considered include those with either solid material or air as the core; circular or non-circular microstructured holes arranged around the core. Although not being demonstrated in this paper; in principle, structures with arbitrary cross-section shapes can be handled, including those with non-circularly oriented hole arrangements and quasi-homogeneous exterior domain. This feature might be useful in the future, if the advancement of the fabrication technology would enable the application of the same principle of waveguide engineering in integrated optics.

## 2. Description of the method

*jωt*) time dependent of the field; it is possible to get the vectorial wave-equation expressed only in terms of the transverse components of the magnetic field as follows:

*x*and

*y*denote the transverse Cartesian coordinates associated with the structure cross-section,

*k*

_{0}the vacuum wavenumber,

*n*

_{eff}the complex modal index,

*H*

_{x}and

*H*

_{y}the

*x*and

*y*components of the magnetic field, while

*x*,

*y*, and

*z*components of the electric field. Using the Galerkin procedure and discretizing the computational domain into triangular elements lead to the following discretized weak formulation:

*w*

_{x}and

*w*

_{y}denote the weight functions, Ω

_{e}denote the area in each triangular element, Γ

_{int,e}the line element at the interface between different materials, and Γ

_{e}the line element at the computational boundaries.

^{st}-order BGT-like [24

24. A. Bayliss, M. Gunzburger, and E. Turkel, “Boundary conditions for the numerical solution of elliptic equations in exterior regions,” SIAM J. Appl. Math. **42**, 430–451 (1982). [CrossRef]

^{st}-order operator on the boundary fields as follows:

*r*and

*θ*are the polar coordinates of the cross-section whereby the center of the waveguide has been taken as the origin, and

*k*

_{r,x}and

*k*

_{r,y}are the complex transverse wavenumbers associated with the

*x*and

*y*components of the field. Solving the wave-equation at the elementwise homogeneous anisotropic exterior domain leads to

*k*

_{r})>0 for the leaky-mode case (the one used in this paper) and Im(

*k*

_{r})<0 for the guided-mode case. Using Eqs. (4)–(6) and neglecting the angular dependence of the field, the derivative terms within the boundary term of Eq. (2) can be approximated as

*r*) term) compared to the Sommerfeld-like TBC [25

25. H.E. Hernandez-Figueroa, F.A. Fernandez, Y. Lu, and J.B. Davies, “Vectorial finite element modeling of 2D leaky waveguides,” Trans. Magnetics **31**, 1710–1713 (1995). [CrossRef]

## 3. Results and discussions

*et al*. [31

31. G. Vienne, Y. Xu, C. Jakobsen, H.J. Deyerl, T.P. Hansen, B.H. Larsen, J.B. Jensen, T. Sorensen, M. Terrel, Y. Huang, R. Lee, N.A. Mortensen, J. Broeng, H. Simonsen, A. Bjarklev, and A. Yariv, “First demonstration of air-silica Bragg fiber,” Post Deadline Paper PDP25, Optical Fiber Conference 2004, Los Angeles, 22–27 Feb. 2004.

### 3.1. Solid-core structure

13. T.P. White, B.T. Kuhlmey, R.C. McPhedran, D. Maystre, R. Renversez, C.M. de Sterke, and L.C. Botten, “Multipole method for microstructured optical fibers. I. Formulation,” J. Opt. Soc. Am B **19**, 2322–2330 (2002). [CrossRef]

14. N.A. Issa and L. Poladian, “Vector wave expansion method for leaky modes of microstructured optical fibers,” J. Lightwave Technol. **21**, 1005–1012 (2003). [CrossRef]

*d*=5µm with pitch length of Λ=6.75µm. The refractive index of the background material is

*n*

_{bg}=1.45, while the refractive index of the holes is

*n*

_{hole}=1. The vacuum wavelength used in the calculation is 1.45µm. Taking advantage of the symmetry of the structure; we only use a quarter circle with a radius of

*r*

_{b}=10µm as the computational domain with the curved boundary located just slightly after the hole. The 1

^{st}-order BGT-like TBC is applied to the curved boundary while symmetry boundary conditions which consist of a perfect electric conductor (PEC) and/or a perfect magnetic conductor (PMC) are applied at the two boundaries coinciding with the structure symmetry planes. The computational domain is discretized into 1648 triangular elements as shown in Fig. 1.

*a*and

*b*denoting the results obtained using PMC and PEC at the horizontal symmetry plane, respectively. The table shows that our results using rather modest mesh size and small computational domain are in good agreement with the results of the multipole method [13

13. T.P. White, B.T. Kuhlmey, R.C. McPhedran, D. Maystre, R. Renversez, C.M. de Sterke, and L.C. Botten, “Multipole method for microstructured optical fibers. I. Formulation,” J. Opt. Soc. Am B **19**, 2322–2330 (2002). [CrossRef]

14. N.A. Issa and L. Poladian, “Vector wave expansion method for leaky modes of microstructured optical fibers,” J. Lightwave Technol. **21**, 1005–1012 (2003). [CrossRef]

*n*

_{eff}) than Im(

*n*

_{eff}), indicating that the absolute error in the latter is a few order of magnitude better than that in the former. This is attributed in the following: the error in Re(

*n*

_{eff}) is due to the contributions from the interior and the boundary of the domain, while the error in the Im(

*n*

_{eff}) is dominated by the contribution from the boundary. The boundary terms themselves occupy only a small fraction of the non-zero entries within the FEM matrices and have relatively small values. However, although their absolute errors are small, due to their very small values, it is very hard to get high relative accuracy for the Im(

*n*

_{eff}). Since only modest mesh size and a relatively small computational window were used in the results shown in Table 1, the convergence test indicates that in general only the order of magnitude (for some modes also the first non-zero digit) of the computed Im(

*n*

_{eff}) is significant, while for the Re(

*n*

_{eff}), at least 4 decimal digits for the high order modes and 5 decimal digits for the low order modes are significant. Of course, better accuracy can be expected by using a larger computational window and finer mesh, but with the price of more expensive computational effort.

*TE*- and

*TM*-like modes) are perpendicular to each other at every point in the structure cross-section (see, e.g., Fig. 5 below), hence they can be regarded as pairs. From the computed results (see Table 1, and also later on Figs. 2 and 3) and group theoretical arguments [13

**19**, 2322–2330 (2002). [CrossRef]

26. R. Guobin, W. Zhi, L. Shuqin, and J. Shuisheng, “Mode classification and degeneracy in photonic crystal fibers,” Opt. Express **11**, 1310–1321 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-11-1310. [CrossRef] [PubMed]

*TE*

_{01}- and

*TM*

_{01}-like modes are non-degenerate pairs. The table also shows that using different symmetry boundary conditions at the two symmetry planes, the computed effective indices of the degenerate modes agree to each other up to 5 decimal digits in Re(

*n*

_{eff}). The small numerical birefringence comes from the fact that the mesh used in the discretization has broken the structure symmetry; the effect of which will show up in the discrepancies of the discretization error of the computed degenerate modes. By reducing the discretization error, e.g. using finer mesh, one can expect smaller numerical birefringence [27

27. M. Koshiba and K. Saitoh, “Numerical verification of degeneracy in hexagonal photonic crystal fibers,” Photonics Technol. Lett. **13**, 1313–1315 (2001). [CrossRef]

28. I.H. Malitson, “Interspecimen comparison of the refractive index of fused silica,” J. Opt. Soc. Am. **55**, 1205–1209 (1965). [CrossRef]

4. A. Ferrando, E. Silvestre, J.J. Miret, and P. Andres, “Nearly zero ultraflattened dispersion in photonic crystal fibers,” Opt. Lett. **25**, 790–792 (2000). [CrossRef]

*r*

_{1}=1µm and outer radius

*r*

_{2}=2µm and angular width of 108°. We take a half-circle with radius

*r*

_{b}=2.5µm as our computational domain with the 1

^{st}-order BGT-like TBC at the curved boundary, and either PEC or PMC at the boundary located at the structure symmetry plane. The computational domain is discretized into 1937 triangular elements as shown in Fig. 4. The results of the first-six modes of this structure are given in Table 2 with comparison to the results obtained using the vector FDM-ABC [14]. The vector FDM-ABC used in this calculation is the one that uses Fourier decomposition only in the azimuthal direction, while in the radial direction it uses a finite difference discretization. Due to the vectorial character of the modes, scalar method [29

29. L. Poladian, N.A. Issa, and T.M. Monro, “Fourier decomposition algorithm for leaky modes of fibres with arbitrary geometry,” Opt. Express **10**, 449–454 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-10-449. [CrossRef] [PubMed]

*TE*

_{01}- and

*TM*

_{01}-like modes are non-degenerate pairs. The vector plots of the transverse magnetic fields of these modes are given in Fig. 5. The real part of the effective indices and the dispersion parameter of this structure are depicted in Fig. 6, while the imaginary part of the effective indices and the confinement loss are shown in Fig. 7. Again, to obtain these spectral plots, we used the Sellmeier’s equation [28

28. I.H. Malitson, “Interspecimen comparison of the refractive index of fused silica,” J. Opt. Soc. Am. **55**, 1205–1209 (1965). [CrossRef]

*n*

_{eff}of

*HE*

_{21}- and

*TM*

_{01}-like modes cross over, which indicates their rather dissimilar dispersion properties as shown in Fig. 6(b). Since this structure has much smaller core size and larger (local) air filling fraction than the previous sample, the effect of the air holes is stronger, hence the zero-dispersion wavelength of the

*HE*

_{11}-like mode is located at a shorter wavelength. Besides, the vectorial character is also more pronounced as indicated by more divergent curves of

*TE*

_{01}-,

*TM*

_{01}-, and

*HE*

_{21}-like modes, both in their real and imaginary part of the effective indices.

### 3.2. Air-core structure

*et al*. [31

31. G. Vienne, Y. Xu, C. Jakobsen, H.J. Deyerl, T.P. Hansen, B.H. Larsen, J.B. Jensen, T. Sorensen, M. Terrel, Y. Huang, R. Lee, N.A. Mortensen, J. Broeng, H. Simonsen, A. Bjarklev, and A. Yariv, “First demonstration of air-silica Bragg fiber,” Post Deadline Paper PDP25, Optical Fiber Conference 2004, Los Angeles, 22–27 Feb. 2004.

*r*

_{core}=10µm. We assume that the holes are of annular-shaped with uniform thickness of

*t*

_{annular}=2.3µm. The corners of the holes are rounded with circles tangent to the side of the annular-shaped holes with radius of

*r*

_{corner}=

*t*

_{annular}/4. The thin rings of the host material located in-between two neighboring rings of holes have thickness of

*t*

_{ring}=0.2µm. These thin rings of solid material are supported by thin bridges with a mid position thickness of

*t*

_{bridge}=45nm. For computational efficiency, we assume that the number of holes are 24, 34, and 44 in the first, second, and third rings of holes, respectively, which are slightly different than the fabricated one that has 24, 35, and 46 holes at the corresponding rings. We use a computational window of a quarter circle with a radius of

*r*

_{b}=19µm, with the curved boundary located not too far from the last ring of holes. The 1st-order BGT-like TBC is applied on this curved boundary, while the PEC and/or PMC are used at the boundaries coinciding with the structure symmetry planes. The computational domain is discretized into 17638 triangular elements. This large number of elements is induced by the very thin bridges located between two adjacent holes in the same ring. In the calculation, we used the wavelength of 1.06µm, which corresponds to the laser used in the experiments to measure the modal near field intensity pattern. The refractive index of the silica as the host material is taken from the Sellmeier’s equation [28

28. I.H. Malitson, “Interspecimen comparison of the refractive index of fused silica,” J. Opt. Soc. Am. **55**, 1205–1209 (1965). [CrossRef]

*TE*

_{01}-like mode exhibits lowest confinement loss; a property that used to be found in the Bragg fibers [8

8. S.G. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, T.D. Engeness, M. Soljacic, S.A. Jacobs, J.D. Joannopoulos, and Y. Fink, “Low-loss asymptotically single-mode propagation in large-core OmniGuide fibers,” Opt. Express **9**, 748–779 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-748. [CrossRef] [PubMed]

9. I.M. Basset and A. Argyros, “Elimination of polarization degeneracy in round waveguides,” Opt. Express **10**, 1342–1346 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-23-1342. [CrossRef]

*TE*

_{01}-like mode agrees very well to the measured results in the experiment [31

31. G. Vienne, Y. Xu, C. Jakobsen, H.J. Deyerl, T.P. Hansen, B.H. Larsen, J.B. Jensen, T. Sorensen, M. Terrel, Y. Huang, R. Lee, N.A. Mortensen, J. Broeng, H. Simonsen, A. Bjarklev, and A. Yariv, “First demonstration of air-silica Bragg fiber,” Post Deadline Paper PDP25, Optical Fiber Conference 2004, Los Angeles, 22–27 Feb. 2004.

*HE*

_{11}- and

*TE*

_{01}-like modes, while their longitudinal component of time averaged Poynting vectors are given in Fig. 11. The profile of the

*HE*

_{11}-like mode shows the existence of bright resonance spots at the cladding layers that were also observed in the nearfield patterns measured in the experiments of Vienne

*et al*. [31

## 4. Concluding remarks

^{st}-order BGT-like TBC has been applied to analyze the microstructured optical waveguides. The boundary conditions allow the calculation of both the real and imaginary part of the modal indices in a relatively small computational domain. Both the solid- and air-core structures with either circular or non-circular holes have been used as the samples. The computed results agree to the results obtained using other methods or experiments.

## Acknowledgments

## References and links

1. | J.C. Knight, T.A. Birks, P.S.J. Russell, and D.M. Atkin, “All-silica single-mode optical fiber with photonic crystal cladding,” Opt. Lett. |

2. | P. Russell, “Photonic Crystal Fibers,” Science |

3. | Y. Fink, D.J. Ripin, S. Fan, C. Chen, J.D. Joannopoulos, and E.L. Thomas, “Guiding optical light in air using an all-dielectric structure,” J. Lightwave Technol. |

4. | A. Ferrando, E. Silvestre, J.J. Miret, and P. Andres, “Nearly zero ultraflattened dispersion in photonic crystal fibers,” Opt. Lett. |

5. | J.C. Knight, J. Arriaga, T.A. Birks, A. Ortigosa-Blanch, W.J. Wadsworth, and P.S.J. Russell, “Anomalous dispersion in photonic crystal fiber,” Photonics Technol. Lett. |

6. | 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. |

7. | T.A. Birks, J.C. Knight, and P.S.J. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. |

8. | S.G. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, T.D. Engeness, M. Soljacic, S.A. Jacobs, J.D. Joannopoulos, and Y. Fink, “Low-loss asymptotically single-mode propagation in large-core OmniGuide fibers,” Opt. Express |

9. | I.M. Basset and A. Argyros, “Elimination of polarization degeneracy in round waveguides,” Opt. Express |

10. | A. Argyros, N. Issa, I. Basset, and M. van Eijkelenborg, “Microstructured optical fiber for single-polarization air guidance,” Opt. Lett. |

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

12. | W. Zhi, R. Guobin, L. Shuqin, and J. Shuisheng, “Supercell lattice method for photonic crystal fibers,” Opt. Express |

13. | T.P. White, B.T. Kuhlmey, R.C. McPhedran, D. Maystre, R. Renversez, C.M. de Sterke, and L.C. Botten, “Multipole method for microstructured optical fibers. I. Formulation,” J. Opt. Soc. Am B |

14. | N.A. Issa and L. Poladian, “Vector wave expansion method for leaky modes of microstructured optical fibers,” J. Lightwave Technol. |

15. | F. Fogli, L. Saccomandi, P. Bassi, G. Bellanca, and S. Trillo, “Full vectorial BPM modeling of index-guiding photonic crystal fibers and couplers,” Opt. Express |

16. | K. Saitoh and M. Koshiba, “Full-vectorial imaginary-distance beam propagation method based on a finite element scheme: application to photonic crystal fibers,” J. Quantum
Electron. |

17. | C.P. Yu and H.C. Chang, “Applications of the finite difference mode solution method to photonic crystal structures,” Opt. Quantum Electron. |

18. | F. Brechet, J. Marcou, D. Pagnoux, and P. Roy, “Complete analysis of the characteristics of propagation into photonic crystal fibers by finite element method,” Optical Fiber Technol. |

19. | Cucinotta, S. Selleri, L. Vincetti, and M. Zoboli, “Holey fiber analysis through the finite element method,” Photonics Tech. Lett. |

20. | M. Koshiba, “Full-vector analysis of photonic crystal fibers using the finite element method,” IEICE Trans. Electron. |

21. | D. Ferrarini, L. Vincetti, M. Zoboli, A. Cucinotta, and S. Selleri, “Leakage properties of photonic crystal fibers,” Opt. Express |

22. | K. Saitoh and M. Koshiba, “Leakage loss and group velocity dispersion in air-core photonic bandgap fibers,” Opt. Express |

23. | H.P. Uranus, H.J.W.M. Hoekstra, and E. van Groesen, “Galerkin finite element scheme with Bayliss-Gunzburger-Turkel-like boundary conditions for vectorial optical mode solver,” J. Nonlinear Optical Phys. Materials (to be published). |

24. | A. Bayliss, M. Gunzburger, and E. Turkel, “Boundary conditions for the numerical solution of elliptic equations in exterior regions,” SIAM J. Appl. Math. |

25. | H.E. Hernandez-Figueroa, F.A. Fernandez, Y. Lu, and J.B. Davies, “Vectorial finite element modeling of 2D leaky waveguides,” Trans. Magnetics |

26. | R. Guobin, W. Zhi, L. Shuqin, and J. Shuisheng, “Mode classification and degeneracy in photonic crystal fibers,” Opt. Express |

27. | M. Koshiba and K. Saitoh, “Numerical verification of degeneracy in hexagonal photonic crystal fibers,” Photonics Technol. Lett. |

28. | I.H. Malitson, “Interspecimen comparison of the refractive index of fused silica,” J. Opt. Soc. Am. |

29. | L. Poladian, N.A. Issa, and T.M. Monro, “Fourier decomposition algorithm for leaky modes of fibres with arbitrary geometry,” Opt. Express |

30. | T.M. Monro, D.J. Richardson, N.G.R. Broderick, and P.J. Bennett, “Modeling large air fraction holey optical fibers,” J. Lightwave Technol. |

31. | G. Vienne, Y. Xu, C. Jakobsen, H.J. Deyerl, T.P. Hansen, B.H. Larsen, J.B. Jensen, T. Sorensen, M. Terrel, Y. Huang, R. Lee, N.A. Mortensen, J. Broeng, H. Simonsen, A. Bjarklev, and A. Yariv, “First demonstration of air-silica Bragg fiber,” Post Deadline Paper PDP25, Optical Fiber Conference 2004, Los Angeles, 22–27 Feb. 2004. |

**OCIS Codes**

(000.4430) General : Numerical approximation and analysis

(060.2310) Fiber optics and optical communications : Fiber optics

(060.2400) Fiber optics and optical communications : Fiber properties

(230.3990) Optical devices : Micro-optical devices

**ToC Category:**

Research Papers

**History**

Original Manuscript: May 14, 2004

Revised Manuscript: June 8, 2004

Published: June 14, 2004

**Citation**

Henri Uranus and H. Hoekstra, "Modelling of microstructured waveguides using a finite-element-based vectorial mode solver with transparent boundary conditions," Opt. Express **12**, 2795-2809 (2004)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-12-2795

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### References

- J.C. Knight, T.A. Birks, P.S.J. Russell, and D.M. Atkin, �??All-silica single-mode optical fiber with photonic crystal cladding,�?? Opt. Lett. 21, 1547-1549 (1996). [CrossRef] [PubMed]
- P. Russell, �??Photonic Crystal Fibers,�?? Science 299, 358-362 (2003). [CrossRef] [PubMed]
- Y. Fink, D.J. Ripin, S. Fan, C. Chen, J.D. Joannopoulos, and E.L. Thomas, �??Guiding optical light in air using an all-dielectric structure,�?? J. Lightwave Technol. 17, 2039-2041 (1999). [CrossRef]
- A. Ferrando, E. Silvestre, J.J. Miret, and P. Andres, �??Nearly zero ultraflattened dispersion in photonic crystal fibers,�?? Opt. Lett. 25, 790-792 (2000). [CrossRef]
- J.C. Knight, J. Arriaga, T.A. Birks, A. Ortigosa-Blanch, W.J. Wadsworth, and P.S.J. Russell, �??Anomalous dispersion in photonic crystal fiber,�?? Photonics Technol. Lett. 12, 807-809 (2000). [CrossRef]
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