Highly directive current distributions: General theory

A theoretical scheme for studying the properties of localized, monochromatic, and highly directive classical current distributions in two and three dimensions is formulated and analyzed. For continuous current distributions, it is shown that maximizing the directivity D in the far field while constraining $C=N/T,$ where N is the integral of the square of the magnitude of the current density and T is proportional to the total radiated power, leads to a Fredholm integral equation of the second kind for the optimum current. This equation is a useful analytical tool for studying currents that produce optimum directivities above the directivity of a uniform distribution. Various consequences of the present formulation are examined analytically for essentially arbitrary geometries of the current-carrying region. In particular, certain properties of the optimum directivity are derived and differences between the continuous and discrete cases are pointed out. When →C∞, the directivity tends to infinity monotonically, in accord with Oseen's ``Einstein needle radiation.''