  Beam displacement measurements are widely used in optical sensing and
communications; however, their performance is affected by numerous intrinsic
and extrinsic factors including beam profile, propagation loss, and receiver
architecture. Here we present a framework for designing a classically optimal
beam displacement transceiver, using quantum estimation theory. We consider the
canonical task of estimating the position of a diffraction-limited laser beam
after passing through an apertured volume characterized by Fresnel-number
product DF. As a rule of thumb, higher-order Gaussian modes provide more
information about beam displacement, but are more sensitive to loss. Applying
quantum Fisher information, we design mode combinations that optimally leverage
this trade-off, and show that a greater than 10-fold improvement in precision
is possible, relative to the fundamental mode, for a practically relevant DF =
100. We also show that this improvement is realizable with a variety of
practical receiver architectures. Our findings extend previous works on
lossless transceivers, may have immediate impact on applications such as atomic
force microscopy and near-field optical communication, and pave the way towards
globally optimal transceivers using non-classical laser fields.
