  Most current astronomical adaptive optics (AO) systems rely on the
availability of a bright star to measure the distortion of the incoming
wavefront. Replacing the guide star with an artificial laser beacon alleviates
this dependency on bright stars and therefore increases sky coverage, but it
does not eliminate another serious problem for AO observations. This is the
issue of PSF variation with time and field position near the guide star. In
fact, because a natural guide star is still necessary for correction of the
low-order phase error, characterization of laser guide star (LGS) AO PSF
spatial variation is more complicated than for a natural guide star alone. We
discuss six methods for characterizing LGS AO PSF variation that can
potentially improve the determination of the PSF away from the laser spot, that
is, off-axis. Calibration images of dense star fields are used to determine the
change in PSF variation with field position. This is augmented by AO system
telemetry and simple computer simulations to determine a more accurate off-axis
PSF. We report on tests of the methods using the laser AO system on the Lick
Observatory Shane Telescope. [Abstract truncated.]
