Directional pulse propagation in beam, rod, pipe, and disk geometries

TL;DR

This paper develops a unified framework of directional wave equations for pulses in various geometries using cylindrical coordinates, applicable to both ultrashort and long pulses, with methods to derive uni-directional approximations.

Contribution

It introduces a factorization scheme to derive exact bi-directional and first-order wave equations in cylindrical geometries, generalizing paraxial approximations for diverse pulse types.

Findings

Derivation of exact bi-directional wave equations in cylindrical coordinates.

Method to reduce bi-directional equations to uni-directional form with approximations.

Applicability to both ultrashort and long pulses in beam, rod, pipe, and disk geometries.

Abstract

I derive directional wave equations useful for pulses propagating in beam, rod, pipe, and disk geometries by using a cylindrical coordinate system; the scheme works equally well for either long multi-cycle or single-cycle ultrashort pulses. This is achieved by means of a factorization procedure that conveniently generates exact bi-directional and first order wave equations after the selection of propagation direction - either axial, radial, or even angular. I then discuss how to reduce these to a uni-directional form, and discuss the necessary approximation, which is essentially a paraxial approximation as appropriately generalized to the specific geometry.