= Solution
This is \b[<transverse magnetic focusing>]. The depleted gates form reflecting boundaries and two constrictions. A narrow <quantum point contact> collimates carriers into cyclotron trajectories with <cyclotron radius> $r_c=m^*v_F/(e|B|)=\hbar k_F/(e|B|)$. When an integer number of semicircular skipping arcs links the two openings, the collector couples particularly strongly to the injector population. A floating voltage contact then changes its <electrochemical potential> to satisfy zero net current, producing a peak in the measured nonlocal voltage. Sweeping the field changes the arc diameter, moving the focused population into or away from the collector.
The connection condition is $L=2jr_c$ for integer $j\ge1$, with the higher orders requiring specular reflections. The effect needs <ballistic electron transport>, sufficiently narrow openings and a smooth boundary. <Electron phase coherence> can add interference structure, but the basic geometrical focusing condition is semiclassical.<Coherent electron focusing with quantum point contacts in a two-dimensional electron gas by H. van Houten and others>[Physical Review B 39 (1989), pp. 8556–8575, focusing geometry and peak structure]{ref}
There is a polarity qualification in the printed geometry. The <electron> charge is negative: for a field out of the page, an <electron> launched upward through the left opening bends left, not toward the right opening. A direct left-opening-to-right-opening focusing path in the upper region therefore requires the field into the page; the printed outward field supports the reverse connection. Reversing conventional current reverses the occupation perturbation and voltage sign, but does not reverse the curvature of an individual <electron>. Thus the standard injector-to-collector explanation describes the focusing mechanism, while the asserted signed voltage increase requires a consistent field/lead convention. The calculated results below are peak-field magnitudes; taking outward field positive gives negative fields for direct left-to-right electron injection.
\Image[/past-exam-of-the-mphil-in-micro-and-nanotechnology-enterprise-course-of-the-university-of-cambridge/2021/paper-2-electron-focusing.png]
{title=Original semicircular connections for three focusing orders; trajectories are shown without carrier-direction arrows}
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