Optical measurement of ultrasonic Poynting and velocity vector fields

This report describes a method for estimating several wide bandwidth ultrasonic field parameters from optical measurements of the local, acoustically induced, refractive index perturbation in water. These parameters include Poynting and particle velocity vector fields as well as pressure and density fields at any temporal delay under mild (forward-propagating) assumptions on the angular plane-wave spectrum of the ultrasound field. A sampling theorem is derived stating that two complete measurements of the three-dimensional pressure field separated in time by Δt allow release of the forward-propagating assumption for every acoustic wave number κ satisfying κ ≠ nπ/(cΔt), where c is the acoustic wave speed in the medium and n an integer greater than zero. The approach provides detailed measurements of very general ultrasound fields. Two optical measurement methods that acquire the Radon transform of the three-dimensional refractive index perturbation are briefly reviewed. It is shown that the Radon transform of the field itself satisfies a two-dimensional wave equation and may be propagated independently forward or backward in time under a source-free model. Conversely, the Radon transform of the ultrasound field measurement at several known time delays provides a means of applying a filter to the data based on known ultrasound propagation models. Each two-dimensional distribution may be propagated to a common time point and the ensemble averaged, thus incorporating the propagation model into the measurement. We support the presented theory with several experiments.