Theory of Optical Precursors

In this chapter, we discuss the theoretical framework of optical precursors based on the incident electromagnetic waves interacting with dielectric media. The simplest way to interpret the light-matter interaction is the medium optical response to the incident light characterized by dielectric constant \( \varepsilon (\omega ) \) as a function of incident light frequency \( \omega \). To build a theoretical model of optical precursors, first we derive macroscopic dielectric constant \( \varepsilon (\omega ) \) starting from the microscopic dipole moment \( p(\omega ) \). Based on Maxwell’s equation and transfer function, the transmitted step-modulated electromagnetic field through dielectric media will be derived in general form of inverse Fourier transform of transmitted spectrum. The general expression of transmitted field can be solved numerically or analytically depending on the specific parameter regimes, such as Brillouin regime or resonant regime. The discussion extends from single Lorentz medium to electromagnetically-induced transparency medium, where the main signal transmits without loss.