Generation of an approximate TE electromagnetic X-wave in free space and in a planar layered medium using a finite array of electric dipoles

The problem of generating an approximate transverse electric (TE) electromagnetic X-wave using a finite circular array of electric dipoles is investigated as an extension of Huygens´ aperture solution. A comparison is first made between the dominant term of the time domain far field radiated from an electric dipole and from the elementary point source term in Huygens´ formula. The result of this comparison aids in constructing a circular array of electric dipoles for generating an X-wave in free space. Next, the time domain far field radiated due to an electric dipole in free space, and penetrating an air-dielectric interface is evaluated asymptotically for large distances using the saddle point method. Such a far field is used as a basic unit to investigate the behavior of an X-wave generated by an array of dipoles and transmitted inside the dielectric medium through an air-dielectric interface