New freely available FORTRAN library for evaluating Sommerfeld integrals

Computing the electromagnetic fields produced by an aboveground radiating antenna has been one of the most basic problems in radioengineering for the past century. If the antenna in question is a dipole (i.e., it can be thought of as a pair of electric charges of equal magnitude but opposite sign, separated by some relatively small time-varying distance), the radiated fields are given by the so-called Green’s tensor. Apart from the obvious applications in radiocommunications, the Green’s tensor is also used in geophysical exploration of Earth and in near-field imaging and spectroscopy. Unfortunately, the Green’s tensor is not easy to compute even in the simple half-space geometry. A rather general analytical solution to the problem was given by Sommerfeld about 100 years ago.1 A contemporary exposition of the subject can be found in a paper by Maradudin and Mills.2 However, Sommerfeld’s solution is expressed in terms of oscillatory integrals. (The term ‘oscillatory’ means that the integrated functions change sign or oscillate many times over the domain of integration.) To compute an oscillatory integral numerically, one must sample the integrand at a very large number of points. In some instances, this may become computationally inefficient. Evaluation of Sommerfeld integrals is confounded by the fact that the functions in question are not only oscillatory but also complex, singular (i.e., they diverge at some points in the complex plane), and are otherwise difficult to handle. Still, it may seem that, given the power of modern computers, numerical evaluation of the Sommerfeld integrals should not be a serious problem. However, this is not so. To quote from a paper by Jimenez, Cabrera, and Cuevas del Rio,3 who in 1996 wrote a (now apparently no longer Figure 1. Various tensor components of the reflected (R) part of the Green’s tensor, GR αβ, normalized by the free-space wave number k = ω/c, as functions of the free-space wavelength, λ = 2π/k. (a, b) Imaginary (Im) parts and (c, d) real (Re) parts of GR . The geometry is as follows: The source and the point of observation are located 40nm above a transparent dielectric substrate and 40nm apart. The substrate dielectric permittivity is = 2.5 and assumed to be constant in the spectral range considered. The x-axis of the laboratory frame coincides with the line connecting the two points. The indices α and β label the tensor components and can take the values x, y, or z. In this geometry, the tensor components GR xy and GR yz are identically zero. The results were obtained using the numerical integration capability of the GF package (centered symbols) and the analytical approximation, which is also implemented in the package (lines). The imaginary part of the Green’s tensor is very accurately reproduced by the analytical approximation— a result not obtainable from the electrostatic method of images.