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We show a general method to estimate with optimum precision, i.e., the bestprecision determined by the light-matter interaction process, a set ofparameters that characterize a phase object. The method derives from ideaspresented by Pezze et al., [Phys. Rev. Lett. 119, 130504 (2017)]. Our goal isto illuminate the main characteristics of this method as well as itsapplications to the physics community, probably not familiar with the formalquantum language usually employed in works related to quantum estimationtheory. First, we derive precision bounds for the estimation of the set ofparameters characterizing the phase object. We compute the Cr\`amer-Rao lowerbound for two experimentally relevant types of illumination: a multimodecoherent state with mean photon number N, and N copies of a multimodesingle-photon quantum state. We show under which conditions these two modelsare equivalent. Second, we show that the optimum precision can be achieved byprojecting the light reflected/transmitted from the object onto a set of modeswith engineered spatial shape. We describe how to construct these modes, anddemonstrate explicitly that the precision of the estimation using thesemeasurements is optimum. As example, we apply these results to the estimationof the height and sidewall angle of a cliff-like nanostructure, an objectrelevant in semiconductor industry for the evaluation of nanofabricationtechniques.

Optimum parameter estimation of shaped phase objects