Spherical shock waveform reconstruction by heterodyne interferometry

The indirect measurement of shock waveforms by acousto-optic sensing requires a method to reconstruct the field from the projected data. Under the assumption of spherical symmetry, one approach is to reconstruct the field by the Abel inversion integral transform. When the acousto-optic sensing modality measures the change in optical phase difference time derivative, as for a heterodyne Mach-Zehnder interferometer, e.g., a laser Doppler vibrometer, the reconstructed field is the fluctuating refractive index time derivative. A technique is derived that reconstructs the fluctuating index directly by assuming plane wave propagation local to a probe beam. With synthetic data, this approach is compared to the Abel inversion integral transform and then applied to experimental data of laser-induced shockwaves. Time waveforms are reconstructed with greater accuracy except for the tail of the waveform that maps spatially to positions near a virtual origin. Furthermore, direct reconstruction of the fluctuating index field eliminates the required time integration and results in more accurate shock waveform peak values, rise times, and positive phase duration.