Covariant theory of light in a dispersive medium

The relativistic theory of the time- and position-dependent energy and momentum densities of light in glasses and other low-loss dispersive media, where different wavelengths of light propagate at different phase velocities, has remained a largely unsolved challenge until now. This is astonishing in view of the excellent overall theoretical understanding of Maxwell's equations and the abundant experimental measurements of optical phenomena in dispersive media. The challenge is related to the complexity of the interference patterns of partial waves and to the coupling of the field and medium dynamics by the optical force on the medium atoms. In this work, we use the mass-polariton theory of light [Phys. Rev. A 96, 063834 (2017)] to derive the stress-energy-momentum (SEM) tensors of the field and the dispersive medium. Our starting point, the fundamental local conservation laws of energy and momentum densities in classical field theory, is close to that of a recent theoretical work on light in dispersive media by Philbin [Phys. Rev. A 83, 013823 (2011)], which, however, excludes the power-conversion and force density source terms describing the coupling between the field and the medium. In the general inertial frame, we present the SEM tensors in terms of Lorentz scalars, four-vectors, and field tensors that reflect in a transparent way the Lorentz covariance of the theory. The SEM tensors of the field and the medium are symmetric, form-invariant for all inertial observers, and in full accordance with the covariance principle of the special theory of relativity. When the power-conversion and force density source terms are accounted for, there is no need to introduce asymmetric SEM tensors or heuristic symmetrization procedures even for the field and medium subsystems. Therefore, asymmetric SEM tensors based on strictly classical energy and momentum densities, which have been studied in previous literature, do not account for all aspects in the field-medium coupling of electromagnetic waves. However, being based on classical field theory, the present work prompts for further groundwork on the classical limit of the quantum mechanical spin of light in a medium. The SEM tensor of the coupled field-medium state of light also has zero four-divergence. Therefore, light in a dispersive medium has a well-defined four-momentum and rest frame. The volume integrals of the total energy and momentum densities of light agree with the model of mass-polariton quasiparticles having a nonzero rest mass. The coupled field-medium state of light drives forward an atomic mass density wave, which makes the constant center-of-energy velocity law of an isolated system—a fundamental conservation law of nature—satisfied. This provides strong evidence for the consistency of the theory. The predictions of our work, such as the atomic mass density wave associated with light, are accessible to experiments.

Figure 1

Simulation of the atomic MDW in the L frame of a dispersive dielectric medium, where the dispersion is linear around the central frequency with the phase refractive index $n_{\mathrm{p}}^{\left(\mathrm{L}\right)}=1.2$ and the group refractive index $n_{\mathrm{g}}^{\left(\mathrm{L}\right)}=2$. (a) The mass density of the atomic MDW and its lower envelope, (b) the atomic velocity, and (c) the atomic displacement as a function of the position. All figures correspond to the same instance of time when the center of the Gaussian light pulse is at $z=0\mu \mathrm{m}$. The Gaussian light pulse has a central wavelength of ${\lambda }_0^{\left(\mathrm{L}\right)}=800$ nm, a field energy $E_{\mathrm{field}}^{\left(\mathrm{L}\right)}=500$ nJ per circular cross-sectional area of diameter $d=100\mu \mathrm{m}$, and a temporal FWHM of $\mathrm{\Delta }{t}_{\mathrm{FWHM}}^{\left(\mathrm{L}\right)}=14$ fs.

Figure 2

Local energy velocity of light in the L frame as a function of the position in units of the wavelength of light in the medium for phase refractive indices $n_{\mathrm{p}}^{\left(\mathrm{L}\right)}=1.2$, 1.5, 1.9, 1.99, when the group refractive index is fixed to $n_{\mathrm{g}}^{\left(\mathrm{L}\right)}=2$. The solid curves depict the local velocities for each refractive index, and the dashed lines with the same colors are the corresponding phase velocities. The horizontal black solid line presents the group velocity. Note that the horizontal scaling of each graph depends on the refractive index of the medium through the central wavelength of light in the medium, given by $\lambda ={\lambda }^{\left(\mathrm{L}\right)}={\lambda }_0^{\left(\mathrm{L}\right)}/n_{\mathrm{p}}^{\left(\mathrm{L}\right)}$, where ${\lambda }_0^{\left(\mathrm{L}\right)}$ is the central wavelength in vacuum.

Figure 3

(a) The MDW energy, the field energy, the total MP energy, and the total exploitable energy and (b) the field momentum, the MDW momentum, and the total MP momentum as a function of the velocity of an inertial observer moving in the L frame parallel to the propagation velocity of light. The energies and momenta in this figure are volume integrals of the corresponding densities. The energy is given in units of $E_0=E_{\mathrm{ex}}^{\left(\mathrm{L}\right)}$, which is the exploitable energy of light in the L frame corresponding to observer velocity $v=0$. The momentum is given in units of $p_0=E_0/c$. The phase refractive index is $n_{\mathrm{p}}^{\left(\mathrm{L}\right)}=1.2$ and the group refractive index is $n_{\mathrm{g}}^{\left(\mathrm{L}\right)}=2$ in the L frame, for which $v=0$. The vertical dashed line corresponds to the group velocity and the vertical dash-dotted line corresponds to the phase velocity. From this figure, we conclude that the total MP energy obtains its minimum value and the MP momentum becomes zero at the group velocity. Thus, the inertial frame propagating with the group velocity of light is the rest frame of light in accordance with the definition of the rest frame in the special theory of relativity.