This paper reviews some of the recent techniques investigated by the authors for numerical, asymptotic and approximate evaluations of the Somraerfeld integrals. Only the summaries of the final formulations are given and their numerical characteristics are illustrated using representative numerical data. Due to space limitations, the discussion in this paper is limited to the cases where both the source and the observation points are above the lossy half-space.

Evaluation of sommerfeld integrals for lossy half-space problems

Several approximation techniques for the Hertz potential of an infinitesimal dipole in the presence of a conducting half space are presented. The formal solutions obtained by rigorous mathematical procedures are transformed, with the aid of some approximation, to fast convergent forms. The approximation techniques are different depending on the media where the observer and source are located. All results calculated are compared to those of the brute-force integrations of the formal solutions. Restrictions of the validity of these approximations are discussed.

Numerical approximations of the Sommerfeld integral for fast convergence

The magnetism of iron and other 3-d transition metals

An efficient method for modeling wire antennas of arbitrary shapes and orientations above a lossy ground is presented by using Galerkin method. As opposed to the traditional point matching method, careful treatment is required for the electric-field integral equation (EFIE) and its numerical implementation. Comparing with the point matching method, only three simple Sommerfeld integrals are needed in the new method, and the convergence is much faster. To test the validity of this method, numerous numerical examples are given. Numerical results show that, for equivalent-accuracy solutions, the cell number used in this method is much fewer than that in the point matching method.

Modeling of arbitrary wire antennas above ground

This paper presents the design of a K band frequency scanned array in an RG 96/U waveguide. The radiating elements consist of slots machined in the narrow wall of the guide. experiment-resluts of the measured slot conductances for the antenna elements and the feed system are given for slot spacings of 0.25 inches and various slot orientations and depths. Theoretical optimum efficiencies for uniform, Chebyshev, and Taylor distributions are also presented. Finally, the experimental results for a 400-element planar array are given.

K band frequency-scanned waveguide array

By the application of the method of operators asymptotic expressions of the Hertz vector for the radiation of a dipole in a lossy half-space are obtained under the exclusive condition k 2 R 0 ≫1, where k 2 is the wave number for free space, and R 0 is the distance from the median between the dipole and its image to the observing point. A method to obtain the asymptotic expansion for the Hertz vector in the general case is given, by means of which formulas applicable even to the near field are obtained in a relatively in a relatively simple from under the condition practically admitted, in particular without restrictions on the position of the receiving point in air. The result is compared with others previously appeared, and when k 2 R 0 ≪1 assures its agreement with the one derived by the quasi-static approach.

Radiation of a Dipole in a Lossy Half-Space

Method of operators is utilized effectively to simplify the rigorous integral expression for the Hertz vector, as well as the relation between Hermite polynomials and their integral expressions under the condition that k R ≫1, where R is the distance from the dipole to the point of observation, and k is the wave number for the air. The obtained formulas for the Hertz vector of the radiation field hold without restrictions on the earth's refractive index and heights of the dipole or the point of observation, except the assumption that k R ≫1, for all its simplicity.

Dipole Radiation Effected by the Plane Earth

The method of operators utilized effectively for calculating the field of a dipole above the plane earth is applied to the problem of a line source parallel to a homogeneous plasma layer being modified in order to be adapted for the two-dimensional case. It is pointed out that the generation of the pole waves such as surface waves or leaky waves is denied even in the case of an anisotropic or compressible plasma so far as the continuity of the energy flow is assured. The evidence is based upon the cancellation of the residue term at the pole by the exact calculation of the contribution from the pole in the contour integral in the same manner as in the case of the surface wave called by Sommerfeld.

On the Surface Wave Guided by a Plasma Slab

Addenda to "On the Surface Wave Guided by a Plasma Slab"

By the application of methods of operators and Green's theorem, solutions having the reciprocity property for the radiation field of a dipole in a lossy half-space are obtained. They have no difficulties in the region close to the angle of total reflection and agree with those obtained under the more restricted conditions.

Akira Yokoyama

Papers

Radiation of a Dipole in a Lossy Half-Space

Dipole Radiation Effected by the Plane Earth

On the Surface Wave Guided by a Plasma Slab

Addenda to "On the Surface Wave Guided by a Plasma Slab"

Expressions with Reciprocity of the Radiation Field of a Dipole in a Lossy Half-Space