Scalar Wave Functions Research Articles

The electromagnetic theory of the spherical luneberg lens

A solution has been obtained for the spherical or the three-dimensional Luneberg lens. The paper starts with a general discussion of the vector wave functions which can be used to represent the electromagnetic field in a radially stratified medium. It is then applied specifically to the Luneberg lens. Four essential vector wave functions are defined in terms of two scalar wave functions. One of them contains the confluent hypergeometric function, and the other involves a “generalized” confluent hypergeometric function. The latter satisfies a differential equation which can be obtained by a proper confluence of three regular singularities of an equation with five regular singularities. The confluence differs from the confluences employed to obtain the differential equations commonly encountered in certain branches of mathematical physics. It is, therefore, concluded that the “generalized” confluent hypergeometric function is an entirely new function which is not simply related to other well-known functions. The paper finally presents a number of important formulae pertinent to the problem. The solution to another class of stratified lenses is briefly discussed.

A Scalar Representation of Electromagnetic Fields

It is shown that in a region which is free of currents and charges, any electromagnetic field may be rigorously derived from a single, generally complex, scalar wave function V(x, t). In terms of this function the momentum density g(x, t) and the energy density w(x, t) of the field may be defined in such a way that they are represented by expressions analogous to the formulae for the probability current and the probability density in quantum mechanics; in a homogeneous isotropic medium

On the foundation of the scalar diffraction theory of optical imaging

In this paper the relationship between the rigorous vector theory and the approximate scalar theory is investigated and the usual procedure of calculating the intensity in optical images from a single scalar wave-function is justified. Unlike the earlier discussion of Picht (1925) the present analysis takes fully into account the polarization and amplitude properties which an actual light source and the optical instrument impose upon the radiation field. In the calculations of diffraction images the effect of the non-monochromatic nature of natural light is usually disregarded. We give an estimate for the maximum wave-length spread of the source which will give rise to a diffraction pattern that may be calculated from the single monochromatic wave-function of the scalar theory. We find that one of the main reasons for the validity of the scalar theory for wide class of optical instruments is the fact that each monochromatic component of the spectrum gives rise to E and H fields with the property that at any particular instant of time they are nearly constant over each of the geometrical wave-fronts which are not too close to the source or the image region.

On the stability and magnitude of electronic charges. Part II, scalar wave functions

The question of how electronic particles can exist in space and time under the laws of relativity and quantum theory splits up in two separate problems. First, if the particle is supposed to be characterized by a fundamental momentum (mc) and a fundamental length a reciprocal to m, then how large is the product amc (which is independent of m)? The answer has been found 1 in Part I.

VIII.—Reciprocity. Part II : Scalar Wave Functions

This paper contains an attempt to formulate more rigorously the postulate of reciprocity (Born, 1939) and to solve the problem of finding reciprocal wave functions. We have not yet completely succeeded in this task, though considerable progress has been made; but it turns out that the original discussion has to be modified in several important respects.