## Abstract

A numerical approach based on the scalar finite element method is applied to analyse the modal properties of photonic crystal fibers having a solid core and a cladding region with either circular or non-circular microstructured holes. A correction which accounts for the polarization effects due to the large refractive index difference between silica materials and air holes is included in the analysis. Numerical results show that the proposed technique is an efficient and accurate alternative to vector ones.

©2006 Optical Society of America

## 1. Introduction

Photonic crystal fibers (PCFs) consisting of a central defect region surrounded by multiple air holes that run along the length of the fiber are attracting much attention in recent years because of unique properties which are not realized in conventional optical fibers. They can be designed to obtain: i) wide range of single mode operation [1, 2]; ii) small mode area, allowing nonlinear effects [3]; iii) large mode area for generation of high-power optical beams [4]; iv) great controllability in chromatic dispersion by varying the hole diameter and hole pitch [5–7].

Due to the relatively complex dielectric cross section of a PCF, considerable attention has been devoted to the development of numerical methods for its analysis. In general, a vector solution is needed to take into account the polarization effects due to the discontinuity in the refractive index. Several full-vector methods have been reported in the literature such as plane-wave expansion method (PWE) [8], localized-function method [9], beam propagation method (BPM) [10, 11], multipole method (MM) [12, 13], finite-difference method (FDM) [14,15], finite difference time-domain method (FDTD) [16], finite-element method (VFEM) [for a actualized review of this method in PCFs see, for example, Ref. 17]. However, recent approximate-scalar models [2, 18–23] have been used to PCFs. Although the scalar treatment can provide good qualitative information about PCFs, this approach is unable to accurately predict modal properties such as birefringence. So it is important to explore the usefulness and limitations of each method.

In this work, first, the scalar approximation to PCFs is revised by using the finite element method (FEM) that reduces the scalar Helmholtz equation into eigenvalue problem which is solved to find the guided modes and their corresponding propagation constants. Next, we propose to improve the scalar approximation to extend its accuracy at both long wavelength and large air filling fraction conditions, by including in the analysis the polarization effects due to the refractive index discontinuity. Evidently this approach is more efficient than the vector ones since it takes advantage from the simplicity of the scalar solution. In order to show how good this new approximation is, the effective index of fundamental mode for different PCFs were computed and compared with full-vector results reported in literature.

## 2. Analysis method

#### 2.1 Scalar solution

The electric field of a mode of an arbitrary waveguide is expressible in the form

where *β* is the propagation constant, *ẑ* is the unit vector parallel to the waveguide axis and * E_{t}*=

*E*

_{x}

*x̂*+

*E*

_{y}

*ŷ*and

*E*

_{z}are transversal and longitudinal components of the electric field, respectively. If we work with the fields of Eq. (1) in the full vectorial wave equation, it is easy to demonstrate that the transverse modal electric field satisfies the vector wave equation

where ${\nabla}_{t}^{2}$
is the Laplacian operator in transverse plane, *n*=*n*(*x*, *y*) is the refractive-index profile, and *k*
_{0}=2π/λ the wave number in the vacuum, λ being the wavelength.

The scalar approximation results when one neglects the so-called vector term on the right hand side of Eq. (2):

Here, * Ẽ_{t}* and

*$\tilde{\beta}$*are the scalar field and its corresponding propagating constant, respectively. This approximation is valid when coupling between orthogonal field components become negligible.

In order to solve Eq. (3) efficiently for PCFs, we use a nodal based FEM with first order triangular element. By applying the variational finite element procedure, the scalar wave Eq. (3) yields the matrix equation:

where the eigenvector {*e*} contains the values of the electric field at the vertices of the triangular elements used for discretization.

By using scalar formulation to solve Eq. (3), it is neglected the fact the electric fields are discontinuous at the interfaces to air holes. With first-order triangular elements, the continuity of the electric field is fulfilled automatically. This fact is also found in recent applications of the scalar treatment to PCFs in the short-wavelength regimen [2, 20–22], where the fraction of the electric field in the air hole is vanishing, while vectorial effects of Eq. (2) become negligible. An approach very different to this scalar treatment was recently proposed by Mortensen [23], suggesting that one can approximate the problem by imposing the boundary condition that the electric field is zero at the interface to the air holes, achieving an excellent agreement with PWE results up to a normalized wavelength λ/Λ=0.4, where Λ is the hole pitch.

#### 2.2. Polarization correction

In order to correct the scalar propagation constant, we applied the relationship between *$\tilde{\beta}$* and *β* as described in Ref. [24]

where *A* is the fiber cross section.

To determine *δβ*
^{2} exactly we would have to solve the vector wave Eq. (2). However, using simple perturbations methods [24], we have that the polarization correction to *x*- or *y*-polarized mode, to first order, reduces to

where *i* and *x*_{i}
are equal to *x* or *y* for *x* and *y* polarized mode, respectively. This term is always negative and therefore tends to shift down the propagation constant obtained from the scalar analysis.

## 3. Results and discussion

As a direct way of testing the accuracy of the proposed calculations, we compare the computed effective index *n*_{eff}
=*β*/*k*
_{0} of the fundamental mode for different PCFs with vector results reported in literature. A single-polarization single mode (SPSM) fiber is also included in our analysis, as a case to predict sensitive vector quantities such as birefringence. All results were obtained by using large computational domains with Dirichlet boundary conditions.

#### 3.1. Fiber with triangular lattice cladding

The PCF first considered consists of a triangular lattice of air holes. An example of its cross section is shown in Fig. 1(a), where *d* is the hole diameter. The PCF symmetry allows just one quarter of the structure to be considered for the numerical simulation. The scalar and the corrected scalar approaches were tested by comparing the computed curves of the effective index for the H${\mathrm{E}}_{11}^{x}$ mode of this fiber with results reported in Ref. [11], where a full-vector BPM was used. Figure 1(b) shows results where *d*/Λ is taken as a parameter, over a wide range of wavelength from λ=0.4 µm to λ=2.0 µm. As it was expected, the scalar solution is valid in the shorter wavelength region (λ<0.5 µm), and begins to break down with increasing air filling fraction. On the other hand, the results that include the polarization correction show an overall good agreement with vectorial ones, hence justifying the proposed calculation scheme. For large hole size *d*/Λ=0.7, where the polarization effects are important, the maximum relative error at longer wavelength of the studied range is 0.12%, which underlines the relevance of our approximation.

#### 3.2. Annular-shaped holes fiber

In order to verify our methodology in a structure completely different from the typical one with circular holes, we take a structure recently analyzed in Ref. [25] which is a solid-core structure with three annular-shaped holes around it. A model of its cross section is shown in Fig. 2(a). The annular-shaped holes have an inner radius *r*
_{1}=1 µm and outer radius *r*
_{2}=2 µm and angular width of 108°. In this case, we use a half-circle with radius *r*=3.0 µm as computational window. The computed effective-index curve of the HE_{11}-like mode of this structure is shown in Fig. 2(b) with the curve reported in Ref. [25], over a range of wavelength from λ=0.4 µm to λ=1.6 µm. Since this structure has larger air filling fraction than the previous case, the scalar solution is not practically appropriated in the range of analyzed wavelengths, as can be confirmed in Fig. 2(b). In the case of the corrected scalar solutions, the analysis shows that the field of this mode extends further into the air holes thereby strongly increasing the polarizations effects. It is clear that in this condition the validity of the proposed scheme is reduced; nevertheless, the obtained results do not significantly different from the vector solution. In this case, we found that the minimum and maximum of relative error are 0.1% and 0.8%, respectively.

#### 3.3. Cobweb fiber

An extreme example of a PCF with large air filling fraction is the cobweb fiber. Figure 3(a) shows the cross section of the analyzed fiber. The core diameter is about 1 µm and the silica bridges, which separate the holes, have a thickness of 0.12 µm. This fiber with such geometrical characteristics was experimentally evaluated in Ref. [5] and theoretically analyzed through a vector FEM in Ref. [26]. Here, it is necessary to carry out a very accurate fiber cross section representation, so the FEM is a powerful tool able to cope with this kind of geometry. The structure is symmetric, thus, one half of its cross section was considered. The numerical effective-index curves are compared in Fig. 3(b). As this fiber has a comparable air filling fraction as the preceding case, it is expected that the general trend of the obtained results is the same, i.e., non appropriated solution of the scalar approach and that perturbative correction to the scalar solution can be an important tool for observing the properties of PCF even on extreme conditions of air filling fraction. In this case, the relative error of our approximation goes from 0.05% to 0.8% over the range of wavelength studied.

#### 3.4. SPSM fiber

One of the more interesting vector characteristics of PCFs is their birefringence and, as was previously discussed, the scalar treatment is unable to predict it appropriately; thus, for this approximated model is mandatory to include polarization corrections. Here, we considerer a highly birefringent SPSM-PCF as shown in Fig. 4(a), with Λ=2.2 µm, d/Λ=0.5 and d′/Λ=0.95. This fiber is designed such that one polarization of the fundamental mode is unguided [27]. Figure 4(b) shows the modal dispersion curves for the slow-axis and fast-axis modes as a function of wavelength for a PCF with ten rings of array of air holes. The fast-axis mode is unguided in the wavelength range over 1.48 µm. We can see from this example, that perturbative corrections to the scalar solution can give account for fiber birefringence. The birefringence obtained by using our approach is 2.86×10^{-3} at λ=1.45 µm and is in good agreement with the reported value of 3.27×10^{-3} obtained with a full-vector FEM [27].

## 4. Conclusion

In this work, the approximated-scalar model for different PCFs was revised by using the finite element method. For improving accuracy at long wavelength and large air filling fraction, a correction to the scalar solution which accounts for the polarization effects due to the refractive index discontinuity was included in the analysis. We have demonstrated that the proposed calculation scheme can be used to evaluate modal properties in fibers having a solid core and a cladding region with either circular or non-circular microstructured holes. The results at long wavelength where the validity of the method is put on approval, since the polarization effects become extremely important, confirming the utility of the approach. Also, that perturbative correction to the scalar solution can be used to analyze highly birefringent PCFs.

## Acknowledgments

Financial support by Colciencias (1118-05-13632) and DIME (National University of Colombia, Medellín Campus) is gratefully acknowledged.

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