General diffraction properties of aperiodic slit arrays

TL;DR

This paper analyzes Fraunhofer diffraction from aperiodic slit arrays, showing that quasiperiodic maxima can appear at multiple distance scales in the far-field. By modeling the near-field as a sum of slits and applying a Fourier-transform framework, the authors derive periodicity scales $L'_j = \frac{2π}{ξ |x_{\bar{n}}^{(j)}|} j!$ that govern constructive interference, while the single-slit envelope $\tilde{Π}(x')$ modulates these peaks. A key suppression condition, $s = \frac{1}{j!}|x_{\bar{n}}^{(j)}|$, indicates when a given order's maxima vanish due to envelope zeros, with the $j=1$ case yielding $s \lesssim |x_{\bar{n}}^{(1)}|$. The authors validate these predictions experimentally using a spatial light modulator to create aperiodic slit distributions, measuring scales such as $L'_0 \approx 37\ \mu\mathrm{m}$ and $L'_1 \approx 1.48\ \mathrm{mm}$, and demonstrating the suppression mechanism by tuning slit width and spacing. These results offer design rules for diffractive devices and materials characterization involving quasiperiodic structures, linking geometry, illumination, and Fraunhofer spectra via the Fourier transform.

Abstract

Fraunhofer diffraction plays a vital role in experimental physics not only because it accurately describes the behaviour of light in the usual propagation limit, but also because it links the diffracted light with the scattering object through one of the most important mathematical transformations in physics: the Fourier transform. Acting as a probe in material characterisation as well as used as a tool for particle trapping or sensing, the pattern of interference maxima resulting from the Fraunhofer diffraction through periodic scattering is an ubiquitous routine. In this paper we analyse the Fraunhofer diffraction resulting from the much less studied aperiodic scatter of the light. We provide general conditions for the experimental observation of the peaks of interference maxima featured into patterns that display periodic structures on a number of distance scales. Our theoretical analysis is supported by thorough experimental demonstrations.

Figures (4)

• Figure 1: Fig. (a) illustrates a simplified version of our experimental setup. A laser beam ($\lambda = 660$nm) is sent to a Spatial Light Modulator. The unmodulated light is filtered out, and a CMOS camera records the Fraunhofer diffraction pattern of the modulated light. We also present images of the gratings for a slit distribution described by Eq. \ref{['Lsqrtn']}, with $L=(0.560 \pm 0.008)$mm and slit widths of (b) $s = 0.024$mm (3 pixels of the SLM); (c) $s = 0.040$mm (5 pixels of the SLM); (d) $s = 0.056$mm (7 pixels of the SLM). The Gaussian illumination in the outlined regions is shown on the right side of the figure for $\bar{n} = 25$, and $\sigma = 2\sqrt{2}$. The position of the centre of each slit is represented by a vertical line.
• Figure 2: Experimental results obtained using a grating with slit distribution given by Eq. \ref{['Lsqrtn']} with $L=(0.800 \pm 0.008)$mm and slit width $s=0.024$mm (corresponding to 3 pixels of our SLM). The images show the Fraunhofer diffraction pattern obtained when illuminating the (a) right-hand side ($\bar{n}=20$), (b) left-hand side ($\bar{n}=-20$), and (c) both sides of the grating. A standard deviation of $\sigma = \sqrt{2}$ was used for the Gaussian illumination. In (d) we plot the graphs resulting from the sum of the recorded images over 80 pixels in the vertical direction, each of which normalised to its maximum value. The solid pink, dash-dotted blue, and the thinner solid-red lines correspond to the images (a), (b), and (c) respectively. The black-dashed line corresponds to the plot of the sinc function, Eq. \ref{['Lsqrtn']}, with the experimental value of the slit width used in this experiment.
• Figure 3: Experimental results obtained using a grating with slit distribution given by Eq. \ref{['Lsqrtn']} with $(L=0.400 \pm 0.008)$mm and slit width $s=0.04$mm (corresponding to 5 pixels of our SLM). The images correspond to the Fraunhofer diffraction patterns obtained by adjusting the illumination coefficients $A_n$ as the Gaussian function, Eq. \ref{['A_nGauss']}, with $\sigma = \sqrt{2}$ and (a) $\bar{n}=\pm 10$, for (b) $\bar{n}=\pm 17$, and (c) $\bar{n}=\pm 25$. In (d), (e) and (f), we plot the graphs resulting from the sum of recorded images (a), (b) and (c), respectively, over 80 pixels in the vertical direction. Each graph was normalised so that the area under the curve equals one and then divided by the maximum value among all three graphs. The black-dashed line corresponds to the plot of the sinc function described by Eq. \ref{['Lsqrtn']}.
• Figure 4: Experimental results obtained using a grating with slit distribution given by Eq. \ref{['Lsqrtn']} with $L=(0.560 \pm 0.008)$mm, a Gaussian illumination distribution with $\bar{n}=\pm 25$ and $\sigma = 2\sqrt{2}$ The images correspond to the Fraunhofer diffraction patterns obtained with three different slits widths: (a) $s = 0.024$mm (3 pixels of the SLM); (b) $s = 0.040$mm (5 pixels of the SLM); (c) $s = 0.056$mm (7 pixels of the SLM). In (d), (e) and (f), we plot the graphs resulting from the sum of recorded images (a), (b) and (c), respectively, over 80 pixels in the vertical direction. Each graph was normalised so that the area under the curve equals one and then divided by the maximum value among all three graphs. The dashed black line corresponds to the plot of the sinc function described by Eq. \ref{['Lsqrtn']}.