Some background from classical electrodynamics

The principal intent of this chapter is to arrive at the classical Hilbert transform connections that apply between the real and imaginary components of the generalized (complex) refractive index, and for the complex dielectric constant. Connections of this type are frequently termed dispersion relations in the physics literature. But for the two functions just mentioned, and for many associated results, they are most often referred to as the Kramers–Kronig relations. Historically, these were the first applications of the Hilbert transform concept in the physical sciences, and were discovered by Kronig (1926) and independently by Kramers (1927). These authors were interested in issues connected with the dispersion of light, and from this emerged the term dispersion relation to describe the Hilbert transform relations found by Kramers and Kronig. The reader will recall that dispersion refers to the frequency variation of the refractive index (or some other optical property), and dispersion formulas provide a connection between the refractive index and the frequency. Functions such as the dielectric constant, refractive index, and permeability, which will be defined shortly, are referred to as optical constants. These functions characterize the interaction of electromagnetic radiation with matter. Though in widespread use, this terminology is somewhat of a misnomer, since the optical constants actually depend on the frequency of the incident electromagnetic radiation interacting with the material, and are hence not true constants.