Flat polarization-controlled cylindrical lens based on the Pancharatnam-Berry geometric phase

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 followed by light at different transverse positions relative to the lens center. 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 also depends 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 elements based on geometric phases. The demonstrated lens is flat and only a 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.