Focal-plane wavefront sensing with photonic lanterns II: numerical characterization and optimization

We present numerical characterizations of the wavefront sensing performance for few-mode photonic lantern wavefront sensors (PLWFSs). These characterizations include calculations of the throughput, control space, sensor linearity, and an estimate of the maximum linear reconstruction range for standard and hybrid lanterns with between 3 and 19 ports, at lambda = 1550 nm. We additionally consider the impact of beam-shaping optics and a charge-1 vortex mask placed in the pupil plane. The former is motivated by the application of PLs to high-resolution spectroscopy, which could enable efficient injection into the spectrometer along with simultaneous focal-plane wavefront sensing; similarly, the latter is motivated by the application of PLs to vortex fiber nulling (VFN), which can simultaneously enable wavefront sensing and the nulling of on-axis starlight. Overall, we find that the PLWFS setups tested in this work exhibit good linearity out to similar to 0.25-0.5 radians of RMS wavefront error (WFE). Meanwhile, we estimate the maximum amount of WFE that can be handled by these sensors to be around similar to 1-2 radians RMS before the sensor response becomes degenerate. In the future, we expect these limits can be pushed further by increasing the number of degrees of freedom, either by adopting higher mode-count lanterns, dispersing lantern outputs, or separating polarizations. Finally, we consider optimization strategies for the design of the PLWFS, which involve both modification of the lantern itself and the use of pre- and post-lantern optics like phase masks and interferometric beam