The Angular Spectrum Representation of the Pulsed Radiation Field in a Temporally Dispersive Medium

Abstract

Attention is now directed to the rigorous solution of the electromagnetic field that is radiated by a general current source in a homogeneous, isotropic, locally linear, temporally dispersive medium occupying all of space that is characterized by the time-dependent dielectric permittivity response function $$\hat \in \left(t \right)$$, magnetic permeability response function $$\hat \mu \left(t \right)$$, and electric conductivity response function $$\hat \sigma \left(t \right)$$. The applied current source J 0(r, t) is assumed here Fig. 6.1. to be a well-behaved known function of both position r and time t that identically vanishes for |z| ≥ Z, where Z is a positive constant, as illustrated in Figure 6.1. Furthermore, it is assumed that the current source is turned on at time t = 0, so that (6.1)$$J_0 \left({r,t} \right) = 0, t \leqslant 0.$$. Because the present analysis is concerned only with the electromagnetic field that is radiated by this current source, it is then required that both field vectors E(r, t) and B(r, t) also vanish for t ≤ 0; viz., (6.2)$$\left. \begin{gathered} E\left({r,t} \right) = 0 \hfill \\ B\left({r,t} \right) = 0 \hfill \\ \end{gathered} \right\} t \leqslant 0.$$ These requirements on the current source are not restrictive in any physical sense because all real radiation problems may be cast so as to satisfy them.