Fresnel Limits Discussion

An inspection of the limits and figure 2 should clarify when the large limits Fresnel approximations are valid.

• In an estimation context, $N$ and $K$ are in practice fixed and only certain small magnitude values of $k$ will be of interest.
• In our estimator, we will implicitly use $\vert k\vert\leq\ensuremath{\frac{KN\hat{\alpha}}{4\pi}}$ where $\hat{\alpha}$ is at the same time estimated.
• We see then that as $\alpha$ becomes very small, no range of $k$ will be valid. Requiring $\{-l_1,l_2\}\geq 3$ and solving for $\alpha$ we find a constraint of $\alpha N^2 \geq 72\pi$. (For $\{-l_1,l_2\}\geq 2.5$, we require $\alpha N^2 \geq 50\pi$.)
• Figure 2 should help clarify the relationship of the limits to $\alpha$ and $k$. In the top subplot, we see $l_1$ and $l_2$ plotted versus $\alpha$, for each of $k = -1, 0,$ and $1$, with $N$ fixed at 641 and $K$ fixed at 2048. The middle subplot is similar, but considers values of $k$ from -10 to 10. We see that the larger magnitude values of $k$ cause one of the limits to be too small for a given value of $\alpha$. We also see that in general, the model is valid for smaller $\alpha$ when smaller $k$ values are used. The bottom subplot is similar to the top two, but uses $k$ values ``dynamically'' selected by $\vert k\vert\leq\ensuremath{\frac{KN\hat{\alpha}}{4\pi}}$ (see Fresnel estimator discussion below).

Figure 2: The choice of $k$ and the value of $\alpha$ affect the limits $l_1$ and $l_2$, which in turn dictate Fresnel model validity. The values of $l_1$ and $l_2$ giving 1% model error bounds ($\pm 3$) are shown with dash-dot lines.