# Proca Metamaterials, Massive Electromagnetism, and Nonlocality

We investigate a new type of electromagnetic meta-materials (MTMs), which we dub Proca MTMs, constituting an interesting medium behaving like a “relativistic material” for potential use in electromagnetic applications. It is rigorously proved using a field-theoretic approach that Maxwell theory inside certain classes of nonlocal metamaterials is equivalent to Maxwell-Proca theory in vacuum, where in the latter photons acquire a nonzero mass (massive electromagnetism.) It turns out that the key to the operation of Proca MTM is nonlocality (here spatial dispersion since the Proca MTM is homogeneous), and hence Proca MTMs represent an important example of the more general family of nonlocal MTMs. Our analysis involves multiphysics aspects, utilizing concepts and methods taken from classical electromagnetism, special relativity, quantum theory, electromagnetic materials, and antenna theory. Extensive discussion of the physics, computational methods, and design parameters of Proca MTMs is provided to further understand the nature of massive electromagnetism in nonlocal MTMs. Proca waves carry an additional polarization degree of freedom and each wave appears to behave like a single mode with two transverse components and one longitudinal. This opens the door for applications in wireless communications and other fields where information could be encoded in polarization. As a concrete application, we develop the main ingredients of Proca antennas as an example of the emerging technology of nonlocal antennas, where we establish that a single Proca dipole possesses a perfect isotropic radiation pattern, a radical departure from conventional local antennas (radiators in vacuum and temporally dispersive media) where such radiation characteristics is impossible. Moreover, the new connection between electromagnetic theory in some nonlocal MTMs and Maxwell-Proca theory allows the use of relativistic techniques developed in the latter in order to efficiently perform calculations like field quantization in nonlocal domains which would be very difficult to perform otherwise.