  The working principle of ordinary refractive lenses can be explained in terms
of the space-variant optical phase retardations they introduce, which reshape
the optical wavefront curvature and hence affect the subsequent light
propagation. These phases, in turn, are due to the varying optical path length
seen by light at different transverse positions relative to the lens centre. A
similar lensing behavior can however be obtained when the optical phases are
introduced by an entirely different mechanism. Here, we consider the "geometric
phases" that arise from the polarization transformations occurring in
anisotropic optical media, named after Pancharatnam and Berry. The medium
anisotropy axis is taken to be space-variant in the transverse plane and the
resulting varying geometric phases give rise to the wavefront reshaping and
lensing effect, which however depends also on the input polarization. We
describe the realization and characterization of a cylindrical geometric-phase
lens that is converging for a given input circular polarization state and
diverging for the orthogonal one, which provides one of the simplest possible
examples of optical element based on geometric phases. The demonstrated lens is
flat and only few microns thick (not including the supporting substrates);
moreover, its working wavelength can be tuned and the lensing can be switched
on and off by the action of an external control electric field. Other kinds of
lenses or more general phase elements inducing different wavefront distortions
can be obtained by a similar approach. Besides their potential for
optoelectronic technology, these devices offer good opportunities for
introducing college-level students to an advanced topic of modern physics, such
as the Berry phase, with the help of interesting optical demonstrations.
