  We present numerical characterizations of the wavefront sensing performance
for few-mode photonic lantern wavefront sensors (PLWFSs). These
characterizations include calculations of throughput, control space, sensor
linearity, and an estimate of maximum linear reconstruction range for standard
and hybrid lanterns with 3 to 19 ports, at a wavelength of 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 ~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, before the sensor response becomes degenerate, to be
around ~1-2 radians RMS. 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. Lastly, 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
recombiners.
