  Presented is a novel way to combine snapshot compressive imaging and lateral
shearing interferometry in order to capture the spatio-spectral phase of an
ultrashort laser pulse in a single shot. A deep unrolling algorithm is utilised
for the snapshot compressive imaging reconstruction due to its parameter
efficiency and superior speed relative to other methods, potentially allowing
for online reconstruction. The algorithm's regularisation term is represented
using neural network with 3D convolutional layers, to exploit the
spatio-spectral correlations that exist in laser wavefronts. Compressed sensing
is not typically applied to modulated signals, but we demonstrate its success
here. Furthermore, we train a neural network to predict the wavefronts from a
lateral shearing interferogram in terms of Zernike polynomials, which again
increases the speed of our technique without sacrificing fidelity. This method
is supported with simulation-based results. While applied to the example of
lateral shearing interferometry, the methods presented here are generally
applicable to a wide range of signals, including Shack-Hartmann-type sensors.
The results may be of interest beyond the context of laser wavefront
characterization, including within quantitative phase imaging.
