Proca Metamaterials, Massive Electromagnetism, and Spatial Dispersion

Abstract

A class of electromagnetic metamaterials (MTMs) is investigated, which the author dubs Proca MTMs, constituting a medium behaving like a “relativistic material” for potential use in electromagnetic applications. This work is motivated by numerous previous results on particular structures such as plasma, waveguides, photonic crystals, magnetic materials, where it has been observed that photons may acquire mass in some dispersive domains. In this approach, it is rigorously proved using a field‐theoretic approach that Maxwell theory inside certain classes of nonlocal (spatially‐dispersive) metamaterials is equivalent to Proca theory in vacuum, where in the latter photons acquire a nonzero mass (massive electromagnetism.) An explicit closed‐form general expression for the Proca MTM dielectric function is given. It turns out that the key to the operation of Proca MTM is spatial dispersion, and hence Proca MTMs represent an important example of the more general family of nonlocal MTMs. The author's 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 (spatially‐dispersive) MTMs. As a concrete application, the main ingredients of Proca antennas are developed as an example of the emerging technology of nonlocal antennas, where the author establishes that a single Proca dipole possesses a perfect isotropic radiation pattern, a noteworthy departure from conventional local antennas (radiators in vacuum and temporally dispersive media) where such radiation characteristics is impossible.

Directive Properties of Radiating Source Systems in Massive Electromagnetism

“…Proof sketch. The proof is a direct consequence of applying Definition 2 and Theorem 3, and Appendix A.4 in [8]. The result is obtained after utilizing the radiation formulas (A.…”

“…After all, the fields radiated by a physically realizable antenna system must satisfy Maxwell's equations and the material constitutive relations. However, it was discovered recently that a complementary systems may exist in Proca theory, i.e., the theory of massive electromagnetism [8]. Most noteworthy is the fact that the total radiated field energy emitted by an infinitesimal (point) source are perfectly isotropic radiators (PIRs) regardless to the orientation of the infinitesimal dipole: Theorem 3.…”

“…In such theory, the electromagnetic fields possess a non-Gauge invariant structure where photons have a non-vanishing mass [1], [2]. However, it was recently pointed out that a special type of nonlocal metamaterials (MTMs), hereafter Proca MTM, can lead to exact equivalence between Maxwell's and Proca's field theories, where the electromagnetic fields exactly solves Proca's equation (massive electromagnetism) in vacuum [8]. In other words, massive Proca fields in vacuum are formally equivalent to Maxwell's theory in a nonlocal (spatially dispersive) medium, the Proca MTM.…”

“…The main conditions blocking the existence of PIR patterns in this class of radiators are formally identified and then exploited in Sec. III by introducing the concept of the Proca antenna, a special example of a nonlocal antenna discovered recently [8]. It is carefully shown that for this class of problems a theorem asserting the existence of PIR patterns can be proved by using the concept of complementary radiation patterns introduced in Sec.…”

Fundamental Spacetime Representations of Quantum Antenna Systems

“…In order to expand the quantum field into a proper continuum of plane-wave modes of the form exp ±ip µ x µ , we will need to integrate over all four-vector momenta p µ , i.e., perform four-dimensional integrals over p and E p (or equivalently k and ω). However, for massive particles in general, and Proca waves in particular [94], there is a definite relation between momentum/wavevector and energy/frequency, so the integral over d 3 p might appear at first sight to be essentially reducible to d 3 p, while E p is computed from…”