  We show a general method to estimate with optimum precision, i.e., the best
precision determined by the light-matter interaction process, a set of
parameters that characterize a phase object. The method derives from ideas
presented by Pezze et al., [Phys. Rev. Lett. 119, 130504 (2017)]. Our goal is
to illuminate the main characteristics of this method as well as its
applications to the physics community, probably not familiar with the formal
quantum language usually employed in works related to quantum estimation
theory. First, we derive precision bounds for the estimation of the set of
parameters characterizing the phase object. We compute the Cr\`amer-Rao lower
bound for two experimentally relevant types of illumination: a multimode
coherent state with mean photon number N, and N copies of a multimode
single-photon quantum state. We show under which conditions these two models
are equivalent. Second, we show that the optimum precision can be achieved by
projecting the light reflected/transmitted from the object onto a set of modes
with engineered spatial shape. We describe how to construct these modes, and
demonstrate explicitly that the precision of the estimation using these
measurements is optimum. As example, we apply these results to the estimation
of the height and sidewall angle of a cliff-like nanostructure, an object
relevant in semiconductor industry for the evaluation of nanofabrication
techniques.
