**2.1 Implementation of FEM on PCFs**

The full – vectorial wave equation can be derived from Maxwell's equations for an optical waveguide with an arbitrary cross section as (Koshiba 1992):

$$\nabla \times \left( \left[ p \right] \nabla \times \boldsymbol{\phi} \right) - k\_0^2 \left[ q \right] \boldsymbol{\phi} = 0 \tag{1}$$

where represents the electric **E** or magnetic **H** field. The relative permittivity and permeability tensors [p] and [q] can be written as (Koshiba 1992);

$$\begin{aligned} \begin{bmatrix} p \end{bmatrix} &= \begin{bmatrix} p\_x & 0 & 0 \\ 0 & p\_y & 0 \\ 0 & 0 & p\_z \end{bmatrix} \\\\ \begin{bmatrix} q \end{bmatrix} &= \begin{bmatrix} q\_x & 0 & 0 \\ 0 & q\_y & 0 \\ 0 & 0 & q\_z \end{bmatrix} \end{aligned} \tag{3}$$

where 1 *xyz ppp* , <sup>2</sup> *x x q n* , <sup>2</sup> *<sup>y</sup> <sup>y</sup> <sup>q</sup> <sup>n</sup>* , <sup>2</sup> *z z q n* for electric field ( = E ) and <sup>1</sup> *xyz qqq* , 2 1 *x x p n* , 2 1 *y y p n* , 2 1 *z z p n* for magnetic field ( H ). In the above

expressions, *nx* , *ny* and *nx* represent the refractive indices in the x, y and z directions respectively.
