Diffraction by Circular and Triangular Apertures as a Diagnostic Tool of Twisted Matter Waves

TL;DR

This work demonstrates that diffraction by a triangular aperture can read out the orbital angular momentum $\ell$ of twisted matter waves, providing a sign-sensitive, high-information readout that complements circular-aperture benchmarks. By combining Kirchhoff–Fresnel theory with Fraunhofer mapping, analytic triangular-mask Fourier amplitudes, and cross-checks via split-step Fourier simulations for both Bessel and Laguerre–Gaussian inputs, the authors derive concrete design rules for achieving a clean far-field lattice at MeV-scale electron and ion energies. The key contributions include a symmetry-based explanation of the triangular OAM signatures, practical guidance on aperture size, propagation distance, and detector sampling, and validation of robustness across particle types and input states. The results establish triangular diffraction as a simple, passive, and practical tool for diagnosing the OAM content of structured quantum beams in high-energy regimes, with implications for electron microscopy, accelerator beamlines, and ion diagnostics.

Abstract

We study diffraction of twisted matter waves (electrons and light ions carrying orbital angular momentum $\ell/\hbar=0,\pm1,\pm2,\ldots$ by circular and triangular apertures. Within the scalar Kirchhoff-Fresnel framework, circular apertures preserve cylindrical symmetry and produce ringlike far-field profiles whose radii and widths depend on $|\ell|$ but are insensitive to its sign. In contrast, equilateral triangles break axial symmetry and yield structured patterns that encode both the magnitude and the sign of $\ell$. A transparent Fraunhofer mapping links detector coordinates to the Fourier plane, explaining the $(|\ell|+1)$-lobe rule and the sign-dependent rotation of the pattern. We validate these results for both ideal Bessel beams and localized Laguerre-Gaussian packets, and we cross-check them by split-step Fourier propagation of the time-dependent Schr"odinger equation. From these analyses we extract practical design rules (Fraunhofer distance, lattice pitch, detector sampling) relevant to OAM diagnostics with moderately relativistic electrons with $E_{\rm kin}\sim0.1$ to $5$ MeV and light ions with $E_{\rm kin}\sim0.1$ to $1$ MeV/u. Our results establish triangular diffraction as a simple, passive, and robust method for reading out the OAM content of structured quantum beams.

Figures (13)

• Figure 1: Schematic of the diffraction geometry for twisted matter waves: source $\rightarrow$ equilateral triangular aperture (side $L$) at $z=0$$\rightarrow$ detection screen at distance $z$. The source–aperture distance is $d_{\mathrm{sa}}$, and $\sigma_{\perp}(0)$ denotes the initial transverse beam size. Not to scale.
• Figure 2: Circular-aperture benchmark at $E_{\rm kin}=100$ keV, $\kappa=15$ eV. Expected-count maps for twisted electron Bessel beams with $\ell=0,1,2,5,9,18$ transmitted by a circular aperture of radius $a=400$ nm; aperture–screen distance $z=0.4$ m. Axial symmetry implies insensitivity to $\mathrm{sign}(\ell)$; the radius of the first bright ring increases with $|\ell|$.
• Figure 3: The same setup as Fig. \ref{['fig:bessel_100keV']} but for $E_{\rm kin}=1$ MeV, $\kappa=70$ eV. The shorter de Broglie wavelength yields a tighter radial scale and more closely spaced rings, reducing the overall field of view on the detector.
• Figure 4: Triangular-aperture benchmark at $E_{\rm kin}=100$ keV, $\kappa=15$ eV. Expected-count maps for twisted Bessel electron beams with $\ell=0,1,2,5,-5,10$ transmitted by an equilateral triangular aperture of side $L=400$ nm; aperture–screen distance $z=0.2$ m. The $(|\ell|+1)$-lobe count per side and the orientation flip under $\ell\!\to\!-\ell$ provide a direct readout of both the magnitude and the sign of the OAM.
• Figure 5: The same geometry as Fig. \ref{['fig:triangle_bessel_100keV']} but for $E_{\rm kin}=1$ MeV, $\kappa=70$ eV and $z=2$ m; triangular side length $L=400$ nm. The shorter de Broglie wavelength yields finer fringe spacing, while the longer propagation distance magnifies the triangular pattern on the detector.