  Gravitational lensing of the cosmic microwave background (CMB) polarization
field has been recognized as a potentially valuable probe of the cosmological
density field. We apply likelihood-based techniques to the problem of lensing
of CMB polarization and show that if the B-mode polarization is mapped, then
likelihood-based techniques allow significantly better lensing reconstruction
than is possible using the previous quadratic estimator approach. With this
method the ultimate limit to lensing reconstruction is not set by the lensed
CMB power spectrum. Second-order corrections are known to produce a curl
component of the lensing deflection field that cannot be described by a
potential; we show that this does not significantly affect the reconstruction
at noise levels greater than 0.25 microK arcmin. The reduction of the mean
squared error in the lensing reconstruction relative to the quadratic method
can be as much as a factor of two at noise levels of 1.4 microK arcmin to a
factor of ten at 0.25 microK arcmin, depending on the angular scale of
interest.
