Gaussian beam propagation in a Lorentz-violating vacuum in the presence of a semi-transparent mirror

Abstract

In this paper we study the propagation of structured optical scalar beams in a Lorentz-violating (LV) vacuum parametrized by a constant 4-vector $u^{\mu}$ and in the presence of a semi-transparent mirror. The two bosonic degrees of freedom of the electromagnetic field can be described by a LV extension of the massless scalar field theory, whose LV part is characterized by the term $(u \cdot \partial \phi ) ^{2}$. The mirror at a surface $\Sigma$ is modelled by a delta-type potential in the Lagrange density for the LV scalar field, i.e. $\lambda \delta (\Sigma ) \phi ^{2}$, where the parameter $\lambda$ controls the degree of transparency of the mirror. Using Green's function techniques, we investigate the propagation of a Gaussian beam in the presence of a mirror which is perpendicular to the propagation direction and for two particular choices of the background 4-vector: parallel and perpendicular to the propagation direction. To quantify the Lorentz-violating effects we introduce the fidelity as a measurement of the closeness of the propagated field distribution with respect to that in the conventional vacuum. In the absence of the mirror ($\lambda = 0$) the fidelity is found to be close to one, and hence LV effects are quite small. However in the presence of the mirror, there are regions where the fidelity drops to zero, thus implying that LV effects could be clearly differentiated from the propagation in vacuum. Within the paraxial approximation we determine analytically the LV effects upon the Rayleigh range, the radius of the beam, the Gouy phase and the radius of curvature of the wavefronts. We discuss possible scenarios where our results could apply, by using optically transparent multiferroic materials, which offer unprecedented opportunities to tailor structured beam propagation, as well as to simulate an LV vacuum.