An effective technique to the design of dual band coaxial waveguide feed horn for reflector antenna excitation has been developed and tested. Matching the feed horn with free space in upper frequency range is achieved by dielectric insert disposed inside the circular waveguide fabricated in interior conductor of open ended coaxial waveguide. The presence of dielectric insert causes the high level of reflection in lower frequency range from the boundary between the open end of coaxial waveguide and free space. A novel technical solution of highly complicated matching problem has been proposed. The matching design was performed as the series of finite thickness diaphragm arranged on outer surface of interior conductor of coaxial waveguide. To obtain the rigorous solution of given electromagnetic problem for the complicated structure consisting of a great number of connected coaxial waveguide sections and dielectric insert, a new effective approach has been proposed. The distinctive feature of this approach consists in combination of rigorous procedure based on computation of generalized scattering matrices and another rigorous procedure based on the finite difference time domain method. To combine these essentially different approaches, the metallic discontinuity equivalent to overall investigated structure was employed. An optimization of equivalent matching structure has been performed by the variant of the evolution strategy method. The obtained design with metallic discontinuities was converted into real structure comprising the dielectric insert. The final calculations of real structure were performed by the finite difference time domain methods. The effectiveness of proposed technique was confirmed by the results of testing the developed feed horn in reflector antenna system.