  Multiple antennas arrays play a key role in wireless networks for
communications but also for localization and sensing applications. The use of
large antenna arrays at high carrier frequencies (in the mmWave range) pushes
towards a propagation regime in which the wavefront is no longer plane but
spherical. This allows to infer the position and orientation of a transmitting
source from the received signal without the need of using multiple anchor
nodes, located in known positions. To understand the fundamental limits of
large antenna arrays for localization, this paper combines wave propagation
theory with estimation theory, and computes the Cram\'er-Rao Bound (CRB) for
the estimation of the source position on the basis of the three Cartesian
components of the electric field, observed over a rectangular surface area. The
problem is referred to as holographic positioning and is formulated by taking
into account the radiation angular pattern of the transmitting source, which is
typically ignored in standard signal processing models. We assume that the
source is a Hertzian dipole, and address the holographic positioning problem in
both cases, that is, with and without a priori knowledge of its orientation. To
simplify the analysis and gain further insights, we also consider the case in
which the dipole is located on the line perpendicular to the surface center.
Numerical and asymptotic results are given to quantify the CRBs, and to
quantify the effect of various system parameters on the ultimate estimation
accuracy. It turns out that surfaces of practical size may guarantee a
centimeter-level accuracy in the mmWave bands.
