Intuitive Validity of Fresnel Analysis

The above expressions suggest - with their sums of sinusoids whose argument has an $n^2$ term - that a Fresnel analysis is appropriate. To illustrate this visually, we restate equations 20 and 21 as

$$\Re{Y(k)} = \sum_{n=-(N-1)/2}^{(N-1)/2} f_r(n,k)$$ (22)

$$\Im{Y(k)} = \sum_{n=-(N-1)/2}^{(N-1)/2} f_i(n,k).$$ (23)

where $f_r(n,k) = \cos( \alpha n^2 -2\pi k n/ K)$ and $f_i(n,k) = \sin( \alpha n^2 -2\pi k n/ K)$. We can now plot $f_r(n,k)$ and $f_i(n,k)$ for various values of $k$, along with the cumulative sums whose terminal values by definition reflect $\Re {Y(k)}$ and $\Im{Y(k)}$, respectively. Due to the similarity of $f_r(n,k)$ and $f_i(n,k)$, we presently plot only $f_r(n,k)$ for a simple example case in figure 3 below, in which $N=201, K=N, a=20,$ and $F=8000$. For the values of $k$ in this figure, we see that the cumulative sum function appears visually similar to the Fresnel integrals (figure 1). We also see that its cumulative sum's terminal value appears to approach certain deterministic values. This is the intuition that motivates our Fresnel analysis.

Figure 3: $f_r(n,k)$ (solid) and its cumulative sum (dashed) with terminal value $\Re {Y(k)}$ for various values of $k$ (shown to the left of each plot). We notice that for these $k$ values, the cumulative sum appears to be a modified Fresnel integral. This fact will motivate our Fresnel analysis in this section.

To make note of the fact that the Fresnel analysis will not always be appropriate, we show a similar plot where the values of $k$ render Fresnel analysis irrelevant. This is shown in figure 4, where we observe that the cumulative sums do not appear similar to Fresnel integrals in the way necessary to use Fresnel properties.

Figure 4: $f_r(n,k)$ (solid) and its cumulative sum (dashed) with terminal value $\Re {Y(k)}$ for various values of $k$ (shown to the left of each plot). We notice that for these $k$ values, the cumulative sum does not appear to be a modified Fresnel integral. This fact will motivate constraints on our Fresnel analysis in this section.