  Adaptive optics (AO) restore the angular resolution of ground-based
telescopes, but at the cost of delivering a time- and space-varying point
spread function (PSF) with a complex shape. PSF knowledge is crucial for
breaking existing limits on the measured accuracy of photometry and astrometry
in science observations. In this paper, we concentrate our analyses on
anisoplanatism signature only onto PSF: for large-field observations (20") with
single- conjugated AO, PSFs are strongly elongated due to anisoplanatism that
manifests itself as three different terms for Laser-guide star (LGS) systems:
angular, focal and tilt. We propose a generalized model that relies on a
point-wise decomposition of the phase and encompasses the non-stationarity of
LGS systems. We demonstrate it is more accurate and less computationally
demanding than existing models: it agrees with end-to-end physical-optics
simulations to within 0.1% of PSF measurables, such as Strehl-ratio, FWHM and
fraction of variance unexplained.Secondly, we study off-axis PSF modelling is
with respect to $C_n^2(h)$ profile (heights and fractional weights). For 10m
class telescope, PSF morphology is estimated at 1% level as long as we model
the atmosphere with at least 7 layers whose heights and weights are known
respectively with 200m and 10% precision. As a verification test we used the
Canada's NRC-Herzberg HeNOS testbed data, featuring four lasers. We highlight
capability of retrieving off-axis PSF characteristics within 10% of fraction of
variance unexplained, which complies with the expected range from the
sensitivity analysis. Our new off-axis PSF modelling method lays the
ground-work for testing on-sky in the near future.
