Photonic crystal fibres are capable of special light guiding properties that ordinary optical fibres do not possess, and efforts have been made to numerically model these properties. The plane wave expansion method is one of the numerical methods that has been used. Unfortunately, the function that describes the material in the fibre n(x)
is discontinuous, and convergence of the plane wave expansion method is adversely affected by this. For this reason, the plane wave expansion method may not be every
applied mathematician’s first choice method but we will show that it is comparable in implementation and convergence to the standard finite element method. In particular,an optimal preconditioner for the system matrix A can easily be obtained and matrixvector products with A can be computed in O(N logN) operations (where N is the
size of A) using the Fast Fourier Transform. Although we are always interested in the efficiency of the method, the main contribution of this thesis is the development
of convergence analysis for the plane wave expansion method applied to 4 different 2nd-order elliptic eigenvalue problems in R and R2 with discontinuous coefficients.
To obtain the convergence analysis three issues must be confronted: regularity of the eigenfunctions; approximation error with respect to plane waves; and stability of the
plane wave expansion method. We successfully tackle the regularity and approximation error issues but proving stability relies on showing that the plane wave expansion
method is equivalent to a spectral Galerkin method, and not all of our problems allow this. However, stability is observed in all of our numerical computations.
It has been proposed in , ,  and  that replacing the discontinuous coefficients in the problem with smooth coefficients will improve the plane wave expansion method, despite the additional error. Our convergence analysis for the method in and  shows that the overall rate of convergence is no faster than before. To define A we need the Fourier coefficients of n(x), and sometimes these must be approximated, thus adding an additional error. We analyse the errors for a method
where n(x) is sampled on a uniform grid and the Fourier coefficients are computed with the Fast Fourier Transform. We then devise a strategy for setting the grid-spacing that
will recover the convergence rate of the plane wave expansion method with exact Fourier coefficients.
|Date of Award||1 Sep 2008|
|Supervisor||Robert Scheichl (Supervisor)|