Everything is a precursor: The asymptotic singular value decomposition for transmission through absorbing media

Sommerfeld and Brillouin introduced electromagnetic precursors in 1914. Their result showed that “precursors” were not exponentially absorbed by the medium, but rather were only absorbed at a very slow polynomial rate. We will present an asymptotic singular value decomposition (SVD) of the electromagnetic absorption and dispersion operator, which supports this work. This SVD is relatively independent of the type of absorption model which is assumed. We will show that there are no causal functions which are exponentially absorbed in any medium in which the absorption approaches zero at the origin in frequency. Thus, all causal functions are “precursors” in the sense that they are not exponentially absorbed.