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.