Representations of reflection and transmission functions of canonical electromagnetic waveforms at a conducting half-space

This paper presents analytical expressions for reflected and transmitted electromagnetic fields if a Delta function pulse impinges on a conductive half-space. These impulse response functions were obtained by analytic continuation of the Fresnel coefficients into the whole complex omega-plane and their Fourier transform back to time domain. From a mathematical point of view, the Fresnel coefficients are double-valued with a branch cut, and it is important for all further considerations to remain on the "physical" Riemann surface. Besides the branch cut no further singularities (e. g. poles) appear on this Riemann surface. All response functions for reflection R(tau) and transmission T(tau) can be represented by sums of two terms. The first term stems from the path integral encircling the branch cut. The second term corresponds to the original wave function reduced by a factor identical with the asymptotic value of the Fresnel coefficient. Other representations of the response function are also given (e. g. in terms of modified Bessel functions, and series expansions), and some examples for the convolution with specific canonical waveforms (unit step function, exponential decay, reciprocal sum of exponentials) in the time domain are presented.