Application of Post’s formula to optical pulse propagation in dispersive media

In this paper we consider the propagation of optical pulses in dielectric media with nontrivial dispersion relations. In particular, we implement Post’s Laplace transform formula to invert in time the Fourier–Laplace space coefficients which arise from the joint space solution of the optical dispersive wave equation. Due to the inefficiency of a direct application of this formula, we have considered and present here two more efficient implementations. In the first, the Gaver–Post method, we utilize the well known Gaver functionals but store intermediate calculations to improve efficiency. The second, the Bell–Post method, involves an analytic reformulation of Post’s formula such that knowledge of the dispersion relation and its derivatives are sufficient to invert the coefficients from Laplace space to time. Unlike other approaches to the dispersive wave equation which utilize a Debye–Lorentzian assumption for the dispersion relation, our algorithm is applicable to general Maxwell-Hopkinson dielectrics. Moreover, we formulate the approach in terms of the Fourier–Laplace coefficients which are characteristic of a dispersive medium but are independent of the initial pulse profile. They thus can be precomputed and utilized when necessary in a real-time system. Finally, we present an illustration of the method applied to optical pulse propagation in a free space and in two materials with Cole-type dispersion relations, room temperature ionic liquid (RTIL) hexafluorophosphate and brain white matter. From an analysis of these examples, we find that both methods perform better than a standard Post-formula implementation. The Bell–Post method is the more robust of the two, while the Gaver–Post is more efficient at high precision and Post formula approximation orders.