A Scalar Representation of Electromagnetic Fields : IIt

It is shown that the scalar representation of electromagnetic fields introduced in an earlier paper leads to a new model for energy transport. The energy may be considered to be carried by two mutually incoherent scalar waves, each of which arises from contributions of circularly polarized components of the same helicity. I n a monochromatic field the energy density and the energy flow of each of these two partial waves are time-independent and the energy flow is at every point of the field in the direction of the normal to the surface of constant phase of the wave. Mathematically, the two partial waves are represented by ' analytic signals ' (containing spectral components of only positive or negative frequencies) into which the complex potential of the field may be decomposed. As an immediate consequence of these results, a new representation of an unpolarized quasi-monochromatic electromagnetic field is obtained and it is shown that, under usual conditions, the ' complex disturbance ' of the classical scalar diffraction theory of optics may be identified with the complex potential of this representation. N an earlier paper (Green and Wolf 1953, to be referred to as I) a new repre-T h e field I was represented by a (generally complex) scalar potential, in terms of which the energy density and the energy flow were defined by expressions similar to the quantum mechanical expressions for the probability density and the probability current. Such a representation was obtained by utilizing,in an appropriate manner, the subsidiary (Lorentz) condition of electromagnetic theory. T h e analysis was extended to regions which include charges and currents by Roman (1955), and the method was used by him and also by Nagy (1955) in quantum mechanical investigations. Focke (1957) employed the theory in the analysis of an optical diffraction problem for which the usual optical scalar theory is inadequate. Recently Roman (1959) has investigated the transformation properties of the complex potential and derived the associate energy-momentum tensor. In the present paper some further aspects of this new representation are discussed. In particular it is shown that the representation leads to an interesting new model for energy transport : The energy may be considered to be carried by two mutually incoherent scalar waves, each of which contains contributions