Optimally direction-independent numerical dispersion

Although the choice of the free parameter $a$ which gives a maximally direction-independent numerical dispersion profile has been made, in the past, through computerized optimization procedures [157], we note here that it is possible to make a theoretical choice as well, based on a Taylor series expansion of the spectrum.

The spectral amplification factors for the interpolated scheme can be written in terms of the function $B_{\mbox{{\scriptsize\boldmath $\beta$}}}$, or, equivalently, in terms of the function $F_{\mbox{{\scriptsize\boldmath $\beta$}}}$. It should be clear, then, that if $F_{\mbox{{\scriptsize\boldmath $\beta$}}}$ is directionally-independent, then so are the amplification factors, and thus the numerical phase velocity (see §A.1.4) as well. Ideally, we would like $F_{\mbox{{\scriptsize\boldmath $\beta$}}}$ to be a function of the spectral radius $\Vert$$\beta$$\Vert _{2} = (\beta_{x}^{2}+\beta_{y}^{2})^{1/2}$ alone. Now examine the Taylor expansion of $F_{\mbox{{\scriptsize\boldmath $\beta$}}}$ about $\beta$$={\bf0}$:

$$F_{\mbox{{\scriptsize\boldmath$\beta$}}} = -\Delta^{2}\Vert\mbox{... ...eta_{y}^{4}\right)+\frac{1-a}{4}\beta_{x}^{2}\beta_{y}^{2}\right)+O(\Delta^{6})$$

The directionally-independent $O(\Delta^{2})$ term reflects the fact that the scheme is consistent with the wave equation; higher order terms in general show directional dependence. The choice of $a=2/3$, however, gives

$$F_{\mbox{{\scriptsize\boldmath$\beta$}}} = -\Delta^{2}\Vert\mbox{... ...Vert _{2}^{4}+O(\Delta^{6}) \hspace{1.0in} \mbox{{\rm for}} \hspace{0.2in}a=2/3$$

and the directional dependence is confined to higher-order powers of $\Delta$. Thus for this choice of $a$, the numerical scheme is maximally direction independent about spatial DC. Note that this value of $a$ does fall within the required bounds for a passive waveguide mesh implementation. The value of $0.62$ (for which the numerical dispersion profile is plotted in Figure A.2), which is very close to $2/3$, was chosen by visual inspection of dispersion profiles for various values of $a$.