We present the study of non-diffracting Mathieu beams, also known as Pendulum beams. In previous studies we studied pendulum beams that corresponded to stationary states of the quantum pendulum. In this work we present the results of superpositions of pendulum states with phases correlating to the time evolution of the quantum states of the pendulum. Thus, as a function of time, the quantum probability extracted from the beams mimicked the classical pendulum, liberating or rotating depending on whether the states involved in the superposition were below the potential barrier or above it. We also study the simple pendulum states for large values of the potential barrier.
Parallels between the Helmholtz and Schrodinger equations can be exploited for using light beams to investigate quantum problems. We present the study of a type of non-diffracting beams known as pendulum beams, where the optical modes satisfy a form of the Helmholtz equation that is identical to the Schrodinger equation for the mechanical pendulum. We prepared optical beams in the corresponding eigenmodes and made measurements of their Fourier spectrum. We find remarkable quantitative agreement between the measured angular spectrum and the quantum mechanical probabilities.