Effective Electrodynamics Theory for the Hyperbolic Metamaterial Consisting of Metal–Dielectric Layers

In this work, we study the dynamical behaviors of the electromagnetic fields and material responses in the hyperbolic metamaterial consisting of periodically arranged metallic and dielectric layers. The thickness of each unit cell is assumed to be much smaller than the wavelength of the electromagnetic waves, so the effective medium concept can be applied. When electromagnetic (EM) fields are present, the responses of the medium in the directions parallel to and perpendicular to the layers are similar to those of Drude and Lorentz media, respectively. We derive the time-dependent energy density of the EM fields and the power loss in the effective medium based on Poynting theorem and the dynamical equations of the polarization field. The time-averaged energy density for harmonic fields was obtained by averaging the energy density in one period, and it reduces to the standard result for the lossless dispersive medium when we turn off the loss. A numerical example is given to reveal the general characteristics of the direction-dependent energy storage capacity of the medium. We also show that the Lagrangian density of the system can be constructed. The Euler–Lagrange equations yield the correct dynamical equations of the electromagnetic fields and the polarization field in the medium. The canonical momentum conjugates to every dynamical field can be derived from the Lagrangian density via differentiation or variation with respect to that field. We apply Legendre transformation to this system and find that the resultant Hamiltonian density is identical to the energy density up to an irrelevant divergence term. This coincidence implies the correctness of the energy density formula we obtained before. We also give a brief discussion about the Hamiltonian dynamics description of the system. The Lagrangian description and Hamiltonian formulation presented in this paper can be further developed for studying the elementary excitations or quasiparticles in other hyperbolic metamaterials.