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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 12, Iss. 12 — Jun. 14, 2004
  • pp: 2795–2809
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Modelling of microstructured waveguides using a finite-element-based vectorial mode solver with transparent boundary conditions

H.P. Uranus and H.J.W.M. Hoekstra  »View Author Affiliations


Optics Express, Vol. 12, Issue 12, pp. 2795-2809 (2004)
http://dx.doi.org/10.1364/OPEX.12.002795


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Abstract

A finite-element-based vectorial optical mode solver is used to analyze microstructured optical waveguides. By employing 1st-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

2. Description of the method

The details of the scheme, used for mode solving of ordinary waveguides supporting guided and leaky modes has been given elsewhere [23

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).

]. For convenience, it will be briefly explained here. Starting from the H-field-based vectorial wave-equation, for longitudinally-invariant structures composed of non-magnetic anisotropic materials with diagonal permittivity tensors and exp(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:

[y[1nzz2(xHyyHx)]x[1nzz2(xHyyHx)]][1nyy2x(xHx+yHy)1nxx2y(xHx+yHy)]+k02neff2[1nyy2Hx1nxx2Hy]=k02[HxHy]
(1)

BoundaryElemente{Γe1nzz2wy(xHyyHx)dyΓe1nzz2wx(xHyyHx)dx
Γe1nyy2wx(xHx+yHy)dy+Γe1nxx2wy(xHx+yHy)dx}
+InterfaceElemente{Γint,e1nyy2wx(xHx+yHy)dy+Γint,e1nxx2wy(xHx+yHy)dx}
+TriangularElementeΩe{1nzz2(xwyywx)(xHyyHx)+[x(1nyy2wx)+y(1nxx2wy)](xHx+yHy)
+k02neff2(1nyy2wxHx+1nxx2wyHy)k02(wxHx+wyHy)}dxdy=0
(2)

with wx and wy denote the weight functions, Ωe denote the area in each triangular element, Γint,e the line element at the interface between different materials, and Γe the line element at the computational boundaries.

The derivatives of the fields occurring in the boundary term in Eq. (2) will be handled through the 1st-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]

] TBC to reflect the properties of the fields in the exterior domain properly. We use a vector radiation function

H(r,θ)Γ=[HxHy]Γ=p=01rp+12[Hx,p(θ)exp(jkr,xr)Hy,p(θ)exp(jkr,yr)]
(3)

along the computational boundary Γ, which leads to a 1st-order operator on the boundary fields as follows:

B1([HxHy])Γ={(r+12r)[HxHy]+j[kr,xHxkr,yHy]}Γ=O(r52)
(4)

In Eqs. (3) and (4) 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

kr,xΓ=k0nxx2neff2Γ
(5)

and

kr,yΓ=k0nyy2neff2Γ
(6)

with Re(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

xHxΓ=r̂x̂(jkr,x+12r)HxΓ+O(r52)
(7a)
yHxΓ=r̂ŷ(jkr,x+12r)HxΓ+O(r52)
(7b)
xHyΓ=r̂x̂(jkr,y+12r)HyΓ+O(r52)
(7c)
yHyΓ=r̂ŷ(jkr,y+12r)HyΓ+O(r52)
(7d)

where the caret (^) notation denotes the unit vector. Note that the approximation has one extra term (the 1/(2r) 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]

]. For this reason, it is one order more accurate than the latter one in order to satisfy the radiation function (3).

3. Results and discussions

In order to illustrate the ability of the mode solver to model the microstructured waveguides, we choose several types of structures as our samples, including those that employ solid-material and air as the core, with either circular or non-circular shape of microstructured holes. When available, we will compare our results with the results of other methods. We will also show computational results of the air-silica Bragg fiber recently demonstrated experimentally by Vienne 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

As the first sample, we choose a photonic crystal fiber with 6 circular holes arranged in a hexagonal setting as shown in Fig. 1. In order to enable comparison of our results with the results of other methods, we choose the same structure parameters as in the references [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

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]

]. The diameter of the holes is 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 rb =10µm as the computational domain with the curved boundary located just slightly after the hole. The 1st-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.

Fig. 1. The structure with 6 circular holes, its computational domain and triangulation.

Table 1. Computational results of the structure with 6 circular holes and their comparison with other methods.

table-icon
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Table 1 shows our computed results for the first-ten modes and their comparison to the results of other methods. The mode labeling in the table makes use of the similarity of the mode profiles to those of ordinary step-index fiber with additional superscripts 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]

] and the vector FDM-ABC [14

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]

]. The agreement in the real part of the effective indices is better than that of the imaginary part. By varying the mesh size and the computational boundary position, we observe larger absolute changes in Re(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.

The transverse fields of modes labeled with the same subscript but different superscript (and also 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

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]

,26

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]

], we can see that HE11a - and HE11b -like, HE21a - and HE21b -like, EH11a - and EH11b -like modes are degenerate pairs, while HE31a - and HE31b -like, and also 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]

].

We also take a similar structure and perform a spectral-scan characterization. We assume that pure silica has been used as the host material and take the more realistic value of the refractive index from its 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]

] for each wavelength. In this way, the material dispersion effect is rigorously taken into account. For the holes, we take a constant value of refractive index of 1. Figure 2 shows the real part of the effective indices and the corresponding dispersion parameter as function of wavelength. The dispersion parameter is calculated using [4

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]

]

Dispersion=λc2λ2[Re(neff)]
(8)

Fig. 2. (a) The real part of the mode effective indices and (b) the dispersion parameter of the structure with 6 circular holes.
Fig. 3. The confinement loss of the structure with circular holes. (a) Confinement loss of the first-ten modes of the 1-ring (6-hole) structure. (b) The effect of adding more rings of holes in the cladding.

Fig. 4. Structure with 3 annular-shaped holes.

Table 2. Calculated mode effective indices of the structure with 3 annular-shaped holes.

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Fig. 5. The transverse magnetic fields of the first-six modes of the structure with 3 annular-shaped holes. (a) HE11a -, (b) HE11b -, (c)) TE 01 -, (d) TM 01 -, (e) HE21a -, and (f) HE21b -like modes.
Fig. 6. (a) The real part of the effective indices and (b) the dispersion parameter of the modes of structure with 3 annular-shaped holes.
Fig. 7. (a) The imaginary part of the effective indices and (b) the confinement loss of the modes of structure with 3 annular-shaped holes.

3.2. Air-core structure

Fig. 8. The model of the air-core structure with 3 rings of annular-shaped holes in the cladding.

Table 3. The computational results of the first-six modes of the air-core structure.

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Fig. 9. The transverse magnetic fields of the modes of the air-core structure: (a) HE11a -, (b) HE11b -, (c)) TE 01 -, (d) TM 01 -, (e) HE21a -, and (f) HE21b -like modes.
Fig. 10. The real part of the field profiles of (a) & (b) the HE11a -like and (c) & (d) the TE 01 -like modes of the air-core structure.
Fig. 11. The longitudinal component of the time averaged Poynting vector of (a) the HE11a - and (b) the TE 01 -like modes of the air-core structure. The color-coded scale is in arbitrary unit.

4. Concluding remarks

Acknowledgments

The authors acknowledge the help of N.A. Issa from Univ. Sydney by providing the results of the vector FDM-ABC for structure with 3 annular-shaped holes for comparison. Discussions with E. van Groesen, M. Hammer, and R. Stoffer are also acknowledged. This work is supported by STW Technology Foundation.

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. 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]

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]

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]

19.

Cucinotta, S. Selleri, L. Vincetti, and M. Zoboli, “Holey fiber analysis through the finite element method,” Photonics Tech. Lett. 14, 1530–1532 (2002). [CrossRef]

20.

M. Koshiba, “Full-vector analysis of photonic crystal fibers using the finite element method,” IEICE Trans. Electron. E85-C, 881–888 (2002).

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]

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. 42, 430–451 (1982). [CrossRef]

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]

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]

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]

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]

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. 18, 50–56 (2000). [CrossRef]

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

  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), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-748">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-748</a>. [CrossRef] [PubMed]
  9. I.M. Basset and A. Argyros, �??Elimination of polarization degeneracy in round waveguides,�?? Opt. Express 10, 1342-1346 (2002), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-23-1342">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-23-1342</a>. [CrossRef]
  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, 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), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-9-980">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-9-980</a>. [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), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-1-54">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-1-54</a>. [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]
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