Dispersion of electromagnetic waves in linear, homogeneous, and isotropic media

TL;DR

This paper systematically describes how electromagnetic wave-packets propagate in linear, homogeneous, isotropic media, focusing on effects of diffraction and dispersion, and provides analytical insights into their evolving shape and trajectory.

Contribution

It offers a detailed analysis of the physical assumptions and mathematical framework for understanding narrowband electromagnetic wave-packets in dispersive media.

Findings

Analytical expressions for wave-packet trajectories under certain conditions.

Clarification of the roles of diffraction and dispersion in wave evolution.

Systematic description of physical and mathematical principles involved.

Abstract

An electromagnetic wave-packet propagating in a linear, homogeneous, and isotropic medium changes shape while its envelope travels with different velocities at different points in spacetime. In general, a wave-packet can be described as a superposition of plane-waves having different frequencies {\omega} and different propagation vectors k. While the angular spread of the k-vectors gives rise to diffractive effects, it is the frequency-dependence of the refractive index of the host medium that is commonly associated with optical dispersion. When the spectral distribution of the wave-packet is confined to a narrow band of frequencies, and also when the spread of the k-vectors is not too broad, it is possible, under certain circumstances, to obtain analytical expressions for the local and/or global trajectory of the packet's envelope as it evolves in time. This paper is an attempt at a systematic description of the underlying physical assumptions and mathematical arguments leading to certain well-known properties of narrowband electromagnetic wave-packets in the presence of diffractive as well as (temporally) dispersive effects.