Mathematical and Numerical Explanation of the Nonlinear Acoustic Wave Interaction in Acousto-Optical Cells

In addition to a recently acousto-optical deflector that has been the subject of both theoretical and experimental analysis, this work presents the successful processing of two acousto-optical deflectors that have been orthogonally positioned using two frequency-modulated ultrasonic waves. A comprehensive theoretical analysis is conducted, based on the Collins integral and the ABCD matrix formalism, to explain how the positions of the diffracted orders oscillate in two dimensions as a function of time. The numerical simulation of the derived formula demonstrates the potential for steering a laser beam along Lissajous trajectories. The trajectories in question are observed to exhibit a variety of shapes and velocities. They are sometimes linear with sinusoidal velocities, sometimes circular with constant velocities, and often elliptical with variable velocities. The noteworthy aspect is that all these diffracted orders traverse the spatial domain with an identical sweep frequency, despite the heterogeneity of their trajectories and velocities. Furthermore, these trajectories can be shaped by controlling the phase shift value. This technique can be employed in metrology for rotation measurements based on the Doppler effect. Additionally, it can be used to develop a spatial display that enables tracing Lissajous trajectories, rather than relying on an oscilloscope.