Phase optimization problems in antenna synthesis theory

A short overview of the results obtained by the authors and their colleagues for a long time in the field of the antenna synthesis theory and the related problems of the field transformation is presented. The main common subject in all problems are the free phase functions ascribed to the given amplitude patterns of the antennas, or appearing as the phase corrections in the quasi-optical field transformers. The systems of specific types such as wave beam transformers and antennas with semitransparent surfaces, in particular, resonant and leaky-wave antennas are described. It is shown that some nonlinear integral equations arisen in the antenna synthesis problems with free phase (and their generalizations) have analytical solutions