Wave Equation In Materials Research Articles

Wavevector-dependent optical properties from wavevector-independent proper conductivity tensor

We discuss the calculation of the refractive index by means of the ab initio scalar dielectric function and point out its inherent limitations. To overcome these, we start from the recently proposed fundamental, microscopic wave equation in materials in terms of the frequency- and wavevector-dependent dielectric tensor, and investigate under which conditions the standard treatment can be justified. Thereby, we address the question of neglecting the wavelength dependence of microscopic response functions. Furthermore, we analyze in how far the fundamental, microscopic wave equation is equivalent to the standard wave equation used in theoretical optics. In particular, we clarify the relation of the “effective” dielectric tensor used there to the microscopic dielectric tensor defined in ab initio physics.Graphical abstract

Linear electromagnetic wave equations in materials

After a short review of microscopic electrodynamics in materials, we investigate the relation of the microscopic dielectric tensor to the current response tensor and to the full electromagnetic Green function. Subsequently, we give a systematic overview of microscopic electromagnetic wave equations in materials, which can be formulated in terms of the microscopic dielectric tensor.

Microscopic theory of the refractive index

We examine the refractive index from the viewpoint of modern first-principles materials physics. We first argue that the standard formula, n2=ɛrμr, is generally in conflict with fundamental principles on the microscopic level. Instead, it turns out that an allegedly approximate relation, n2=ɛr, which is already being used for most practical purposes, can be justified theoretically at optical wavelengths. More generally, starting from the fundamental, Lorentz-covariant electromagnetic wave equation in materials as used in plasma physics, we rederive a well-known, three-dimensional form of the wave equation in materials and thereby clarify the connection between the covariant fundamental response tensor and the various Cartesian tensors used to describe optical properties. Finally, we prove a general theorem by which the fundamental, covariant wave equation can be reformulated concisely in terms of the microscopic dielectric tensor.

A Global Damage Theory and the Hyperbolicity of the Wave Problem

It is well known that wave equation in materials that suffer damage in the course of deformation loses it hyperbolicity when the damage process is described by a continuum damage theory of the local type. Here we develop a global (nonlocal) damage theory by (a) introducing a damage coordinate which is a spatial functional of the strain field in the material domain and (b) stipulating that the rate of evolution of damage is with respect to the damage coordinate. We then derive the axial wave equation for a thin rod and thereby demonstrate that, while the rod experiences softening, the wave speed is given in terms of the secant modulus and the wave equation retains its hyperbolicity. Various other phenomena, such as the onset of inhomogeneous damage in the presence of homogeneous deformation and the tendency of axial specimens under tension to fracture invariably at the center, are also explained.