  Most adaptive optics (AO) systems using pyramid wavefront sensors (PyWFS) to
estimate the phase of the pupil field use mechanical modulation of the beam in
order to increase the dynamic range in low-order modes so the PyWFS can
usefully operate at low Strehl ratio. The tradeoff for this approach is reduced
sensitivity, which, in turn, makes it difficult to attain a high Strehl ratio
once the loop has been closed. We propose an algorithm that increases the
dynamic range of the PyWFS without modulation. The proposed algorithm achieves
this in two ways: 1) it allows the PyWFS to be treated with any desired optical
modeling algorithms, and 2) it employs Newton's method for nonlinear
optimization to create an estimator that is more accurate than the
corresponding linear estimator. Numerical simulations show that nonlinear
optimization can make more accurate estimates of the phase of the wavefront
than the corresponding linear estimator for Strehl ratios of the input beam
that are greater than about 0.2. As the input Strehl ratio increases, so does
the advantage of the nonlinear estimator over the linear one. For example, when
the input beam had a Strehl ratio of 0.4 (corresponding to a standard deviation
of the phase of about 0.96 radians), the linear estimator error had a standard
deviation of about 0.65 radians, while the nonlinear estimator error had a
standard deviation of about 0.27 radians (this depends only weakly on the noise
level, assuming there is enough signal for the PyWFS to work in the linear
regime). The new algorithm can be implemented in massively parallel modes,
since the required calculations have almost no inter-dependencies. It is
suggested that the required computations can be performed quickly enough for
the purposes of AO on modern computer systems.
