Self image and simulation of a PR-box using high-order paraxial beams

Abstract

From Maxwell equations (for a free of charge and current, isotropic and homoge- neous medium) and the paraxial approximation, which is to suppose the beam of light moves towards a preferred direction (longitudinal propagation), we ar- rive at the paraxial wave equation, which depending of the constraints of the situation, can be solved by different type of beams. We are intersested in higher- order mode paraxial beams. If we solve the equation with cartesian coordinates, we arrive at Hermite-Gauss beams, if we solve with cilindrical coordinates, we obtain Laguerre-Gauss beams. Each of them has specific characteristics which motivated their use in the two phenomenons presented here: Self Image and the Simulation of a PR box. We call self image to the phenomenon where we are capable of replicating an initial image, over free space longitudinal propagation. What we propose here is a self image produced by the collinear and coherent interference of paraxial Laguerre Gauss (LG) beams, which constrasts with the usage of a fundamen- tal Gaussain beam in Talbot’s self image. Gouy phases, which are the key component that make this phenomenon possible, are exclusive of Higherorder paraxial beams. We show, experimentally, the phenomenon of self image using the superposition of 3 LG beams with specific mode orders. Because of the arct- angent dependence of the Gouy phases, in Laguerre- Gaussian beams, self image distances won’t be periodic over propagation and its number will be limited by the mode orders of the LG beams. Additionally, we use this superposition of the 3 LG beams as dots, to write a word, which can be read only in self image. This application of self image can be thought of as concealing information, and then revealing it only for specific distances. The most controversial feature of quantum mechanics non-locality, has gain much attention over the last years, because of the development of quantum information. Nowadays non-locality is widely accepted and used in many other exciting applications like teleportation, swapping, etc. Nevertheless, this opens other questions, like why is nature just as nonlocal as to reach the Tsirelson’s bound, but can’t surpass it. The algebraical maximum of the CHSH inequality is 4, and quantum mechanics can only reach up to 2 2. What happens in this gap that seems empty and without a theory that can describe it? In 1993, Popescu and Rhorlich proved that from non-locality and relativistic causality, quantum mechanics was not the only theory that emerged. Relativistic causality, meaning that no information is transmitted with superluminal velocities. This means that there are super-quantum correlations, that surpass the Tsirelson’s bound, and are still causal. The super-quantum correlations that maximally surpass the Tsirelson’s bound, making the Bell parameter S = 4, are known as PR boxes. Markovitch et al, showed that, in a bipartite quantum system, post-selecting an entangled state will fake the maximal surpass of the Tsirelson’s bound in the Bell inequality. Here, we propose an experimental setup capable of simulating a PR box using polarization and transverse-mode (Hermitian-Gauss beams of first order) of light as vector spaces that are analogue to Hilbert spaces in quantum mechanics.