The angular spectrum representation and the Sherman expansion of pulsed electromagnetic beam fields in dispersive, attenuative media

The angular spectrum of plane-waves representation of a pulsed electromagnetic beam field propagating in a dispersive, attenuative medium occupying the half-space expresses that wave field as a superposition of both homogeneous and inhomogeneous plane waves. A generalization of the Sherman expansion for source-free wave fields in lossless media to the lossy, dispersive medium case is used to derive a spatial series representation of a pulsed, source-free electromagnetic beam field from its angular spectrum representation. This spatial series representation displays the temporal evolution of the pulsed beam field explicitly through a single-contour integral that is of the same form as that obtained in the Fourier-Laplace description of a pulsed plane-wave field that is propagating in the positive z-direction in the lossy, dispersive medium. The temporal pulse evolution is found to depend upon the transverse spatial position in the beam field through the derivatives of the field boundary values.