# Introduction

• Kurt E. Oughstun
Chapter
Part of the Springer Series in Optical Sciences book series (SSOS, volume 224)

## Abstract

The dynamical evolution of an electromagnetic pulse as it propagates through a linear, temporally dispersive medium (such as water) or system (such as a dielectric waveguide) is a classical problem in both electromagnetics and optics. With Maxwell’s unifying theory of electromagnetism and optics, Lorentz’s classical model of dielectric dispersion, and Einstein’s special theory of relativity, the stage was set for a long-standing problem of some controversy in classical physics, engineering, and applied mathematics. If the system was nondispersive, an arbitrary plane wave pulse would propagate unaltered in shape at the phase velocity of the wave field in the medium. For example, for distortionless wave propagation along the z-direction of a cartesian coordinate system, a one-dimensional wave is described by a single-valued function of the form f(z ± vt), where the argument φ = z ± vt is called the phase of the wave function. Any fixed value of this phase (for example, the value at the temporal center of the pulse) then propagates undistorted in shape along the ± z-direction with the velocity dzdt = ±v. The first partial derivatives of the wave function f(z ± vt) with respect to the independent variables z and t are then given by ∂f∂z = f′ and ∂f∂t = ±vf′, where f′ = df(ζ)∕.

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