  In this work, we generalize the theory of localized surface plasmons to the
case of high-Tc cuprate superconductors, spatially confined in the form of
small spherical particles. At variance from ordinary metals, cuprate
superconductors are characterized by a low-energy bulk excitation known as the
Josephson plasma wave (JPW), arising from interlayer tunneling of the
condensate along the c-axis. The effect of the JPW is revealed in a
characteristic spectrum of surface excitations, which we call Josephson surface
plasmons. Our results, which apply to any material with a strongly anisotropic
electromagnetic response, are worked out in detail for the case of multilayered
superconductors supporting both low-frequency (acoustic) and transverse-optical
JPW. Spatial confinement of the Josephson plasma waves may represent a new
degree of freedom to engineer their frequencies and to explore the link between
interlayer tunnelling and high-Tc superconductivity.
