Solving the maxwell-bloch equations with rk4 and a centered finite difference scheme

Abstract

The Maxwell-Bloch equations are used to describe the quantum behavior of electrons in anatom in the presence of an electric field. Such systems are of interest to fields that study optically active media, for example lasers. Solving these equations is a non-trivial task; only asymptotic solutions are known explicitly, and even then only for a few special cases. Two of these special cases are examined in chapters 4 and 5. Since exact solutions are generally unknown, numerical methods are used to compute an approximation of the solution. In this thesis, the semi-classical Maxwell-Bloch equations (i.e. the electric field is modeled in the classical way and the electrons are modeled using quantum mechanics) are derived from Schrodinger’s equation and Maxwell’s equations. In quantum mechanics, energy is absorbed and emitted in discrete packets known as photons. When an atom absorbs or emits a photon, its electrons transition up or down discrete energy levels. The frequency of these transitions determine the color of the light. The electric field is modeled as a wave, as in classical electromagnetism. In this thesis, a fourth order accurate in space and time algorithm for modeling the Maxwell-Bloch equations is discussed, using both a fully numerical approach and a hybrid approach that utilizes numerical envelopes and analytic oscillations.