Метод формирования единственного сфокусированного порядка дифракции при помощи бинарных амплитудных дифракционных элементов без пространственной несущей

The method of synthesis of self-focusing amplitude DOE without the carrier spatial frequency working in divergent beams and forming single focused diffraction order which can occupy whole DOE reconstruction field because there is no necessity for spatial separation of diffraction orders, which is the case with holograms, is presented. Synthesis was carried out in two stages. The first one was carried out by the iteration algorithm similar to Gerchberg-Saxton algorithm, with those differences that synthesizable DOE is an amplitude one and incident wave front is divergent. Then the method of direct search with random trajectory was applied. As a result for binary amplitude DOE synthesis error of 6% and diffraction efficiency of 6% was achieved. Results of experimental DOE implementation using DMD are presented.

Kesten’s theorem for uniformly recurrent subgroups

We prove a lower bound on the difference between the spectral radius of the Cayley graph of a group $G$ and the spectral radius of the Schreier graph $H\backslash G$ for any subgroup $H$. As an application, we extend Kesten’s theorem on spectral radii to uniformly recurrent subgroups and give a short proof that the result of Lyons and Peres on cycle density in Ramanujan graphs [Lyons and Peres. Cycle density in infinite Ramanujan graphs. Ann. Probab.43(6) (2015), 3337–3358, Theorem 1.2] holds on average. More precisely, we show that if ${\mathcal{G}}$ is an infinite deterministic Ramanujan graph then the time spent in short cycles by a random trajectory of length $n$ is $o(n)$.