  The sensitivity of the the pyramid wavefront sensor (PyWFS) has made it a
popular choice for astronomical adaptive optics (AAO) systems, and it is at its
most sensitive when it is used without modulation of the input beam. In
non-modulated mode, the device is highly nonlinear. Hence, all PyWFS
implementations on current AAO systems employ modulation to make the device
more linear. The upcoming era of 30-m class telescopes and the demand for
ultra-precise wavefront control stemming from science objectives that include
direct imaging of exoplanets make using the PyWFS without modulation desirable.
This article argues that nonlinear estimation based on Newton's method for
nonlinear optimization can be useful for mitigating the effects of nonlinearity
in the non-modulated PyWFS. The proposed approach requires all optical modeling
to be pre-computed, which has the advantage of avoiding real-time simulations
of beam propagation. Further, the required real-time calculations are amenable
to massively parallel computation. Numerical experiments simulate a currently
operational PyWFS. A singular value analysis shows that the common practice of
calculating two "slope" images from the four PyWFS pupil images discards
critical information and is unsuitable for the non-modulated PyWFS simulated
here. Instead, this article advocates estimators that use the raw pixel values
not only from the four geometrical images of the pupil, but from surrounding
pixels as well. The simulations indicate that nonlinear estimation can be
effective when the Strehl ratio of the input beam is greater than 0.3, and the
improvement relative to linear estimation tends to increase at larger Strehl
ratios. At Strehl ratios less than about 0.5, the performances of both the
nonlinear and linear estimators are relatively insensitive to noise, since they
are dominated by nonlinearity error.
