Matter-Wave Diffraction from a Periodic Array of Half Planes

We report on reflection and diffraction of beams of He and ${\mathrm{D}}_2$ from square-wave gratings of a $400\text{−}\mu \mathrm{m}$ period and strip widths ranging from 10 to $200\mu \mathrm{m}$ at grazing-incidence conditions. In each case we observe fully resolved matter-wave diffraction patterns including the specular reflection and diffracted beams up to the second diffraction order. With decreasing strip width, the observed diffraction efficiencies exhibit a transformation from the known regime of quantum reflection from the grating strips to the regime of edge diffraction from a half-plane array. The latter is described by a single-parameter model developed previously to describe phenomena as diverse as quantum billiards, scattering of radio waves in urban areas, and reflection of matter waves from microstructures. Our data provide experimental confirmation of the widespread model. Moreover, our results demonstrate that neither classical reflection nor quantum reflection are essential for reflective diffraction of matter waves from a structured solid, but it can result exclusively from half-plane edge diffraction.

Figure 1

(a) Schematic of wave scattering from a periodic array of parallel (zero-width) half planes at grazing incidence angle ${\theta }_{\mathrm{in}}$. The grating plane (dashed horizontal line) is defined by the half-plane edges. (b) Diffraction efficiencies $e_n(u)$ for the specular ($n=0$) and for diffracted beams up to the second order calculated by the Bogomolny-Schmit solution given by Eqs. (2) and (3) [3].

Figure 2

(a) Schematic of the experimental setup. The grating normal is chosen as the $z$ axis of our coordinate system. Incidence ${\theta }_{\mathrm{in}}$ and detection angle $\theta$ are measured with respect to the grating surface in the $xz$ plane of incidence. (b) The reflection gratings are 50-mm-long microstructured arrays of 4-mm-long parallel strips made out of $1\text{−}\mu \mathrm{m}$-thick photoresist patterned on a commercial gold mirror. The center-to-center distance of the strips defines the period $d=400\mu \mathrm{m}$ identical for all gratings used. Four gratings with strip width $a=10$, 30, 100, or $200\mu \mathrm{m}$ have been used. For all four gratings, the gold surface between the strips is completely shadowed by the strips for all incidence angles used in this Letter. A sketch of the $a=10\mu \mathrm{m}$ grating is shown in (b). (c) Representative diffraction pattern for $a=10\mu \mathrm{m}$ and ${\theta }_{\mathrm{in}}=0.984\mathrm{mrad}$ measured by rotating the detector around the $y$ axis and integrating the signal for 8 s at each angular position.

Figure 3

Observed diffraction efficiencies (open symbols) for He atoms scattering from a square-wave grating compared with theory (red solid lines) calculated for an array of half planes. The efficiencies are plotted on a logarithmic scale as a function of incidence angle ${\theta }_{\mathrm{in}}$ for diffraction orders $n=2$, 1, 0, $-1$, and $-2$ for different grating strip widths of 10, 30, 100, and $200\mu \mathrm{m}$. The third graph also includes the reflection probability of He atoms from the blank photoresist (full black circles, labeled “mirror”) as well as the calculated quantum reflection (QR) probability (red dashed line [39]). The dotted vertical lines indicate the Rayleigh incident conditions ${\theta }_{R,m}$ and $u_{R,m}$ for $m=-1$, $-2$, and $-3$ sequentially, where the $m$th-order diffraction beam emerges from the grating plane. The corresponding $u_{R,m}$ values according to Eq. (1) are labeled atop the graph.

Figure 4

Specular beam efficiencies observed for He and ${\mathrm{D}}_2$ scattering from the gratings of different strip widths $a$ as indicated. The efficiencies are plotted on a logarithmic scale as a function of incidence angle ${\theta }_{\mathrm{in}}$. As in Fig. 3, red lines represent the prediction by Eqs. (2) and (3), and the dotted vertical lines indicate Rayleigh conditions ${\theta }_{R,m}$ and $u_{R,m}$ of $m$th-order beam emergence for $m=-1$, $-2$, and $-3$.