General diffraction properties of aperiodic slit arrays

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.

💡 Research Summary

This paper presents a comprehensive theoretical and experimental investigation of Fraunhofer diffraction from aperiodic slit arrays, extending the well‑established analysis of periodic gratings to structures lacking translational symmetry. Starting from the scalar field representation of a one‑dimensional array of apertures, the authors write the near‑field as a sum of individual slit contributions Ψ(x) ∝ ∑ₙAₙe^{iϕₙ}Π(x−xₙ), where Π denotes a rectangular aperture of width s. In the Fraunhofer limit the far‑field amplitude is obtained via a Fourier transform, yielding ˜Ψ(x′)=˜Π(x′)∑ₙAₙe^{-iκₙx′} with κₙ=ξxₙ and ξ=2π/(fλ). The envelope ˜Π(x′)=s sinc(ξs x′/2) originates from the single‑slit diffraction and imposes a finite bandwidth on observable features.

The novelty lies in treating the slit positions xₙ as a smooth, generally nonlinear function of the integer index n. By expanding xₙ around a central illuminated slit ¯n using a Taylor series, xₙ≈x̄ₙ+ x′(¯n)(n‑¯n)+½x″(¯n)(n‑¯n)²+…, the authors identify a hierarchy of characteristic length scales L′_j = λ f |x^{(j)}(¯n)|/j! (j ≥ 1). When the observation coordinate x′ is an integer multiple of a given L′j, all terms with the same j interfere constructively, producing a set of diffraction peaks at that scale. Distinct peaks are resolvable only if successive scales are well separated, i.e. L′{j+1} ≫ L′_j, which typically requires the central index ¯n to be large (¯n ≫ 1). Thus, even a fully aperiodic array can generate multiple, hierarchically spaced periodicities in its far‑field pattern.

A crucial practical limitation is the sinc envelope of the single slit. Because sinc(ξs x′/2) vanishes at its zeros, any peak whose position lies beyond the first zero will be strongly suppressed. By equating a given L′_j to the first zero of the sinc function, the authors derive a suppression condition s = λ f |x^{(j)}(¯n)|/j!. The most restrictive case (j = 1) yields s ≲ λ f |x′(¯n)|, meaning that to observe the first‑order periodicity the slit width must be sufficiently small; otherwise the peak merges with the central lobe and becomes invisible. This relationship provides a direct design rule linking the geometry of the aperiodic array to the observable diffraction features.

Experimental validation is performed with a spatial light modulator (SLM) illuminated by a 660 nm laser. The authors program a specific aperiodic distribution xₙ = ±L√n (L ≈ 0.560 mm) and apply Gaussian illumination centered on a chosen ¯n. Three slit widths (3, 5, and 7 SLM pixels corresponding to 0.024 mm, 0.040 mm, and 0.056 mm) are tested. The resulting far‑field patterns, recorded by a CMOS camera in a 4f configuration, display clearly the predicted first‑order (L′_1) and second‑order (L′_2) peaks at positions ±L′_1 and ±L′_2, respectively. As the slit width increases, higher‑order peaks are progressively attenuated, confirming the sinc‑based suppression condition.

The paper also addresses the “zero‑order” periodicity L′0, which appears only as a global phase factor in the amplitude and is invisible in intensity measurements. By constructing a mirror‑symmetric array (x{‑n}=‑x_n, A_{‑n}=A_n) and interfering the field with its mirror copy, the authors convert this global phase into a measurable cosine modulation, enabling observation of the zero‑order feature.

In summary, the work demonstrates that aperiodic slit arrays possess a rich set of diffraction periodicities dictated by the higher‑order derivatives of the slit‑position function. The interplay between these intrinsic scales and the sinc envelope of individual slits yields clear criteria for the visibility or suppression of each diffraction order. These insights provide a powerful framework for engineering custom diffraction patterns in applications such as metasurface design, optical trapping, broadband spectroscopy, and the generation of structured light fields.