Inverse Reflection Symmetry Odd Phase Aperture Excitation for Nondiffracting and Asymmetric Sidelobe Beam Forming Arrays

An approach for beam forming arrays with radiated electric fields that are solutions to second order linear differential (SOD) equations of the type ψ(2)+x^2n-1ψ=0 and higher even ordered differential (HEOD) equations ψ(2n)-xψ=0, where n is an integer, has been developed. The fields represented by the solutions of these two types of generalizations propagate without diffraction within a specific range, self-accelerate, and result in asymmetric side lobes. In both the cases, the nondiffracting beams are generated over an appreciable range of a few thousand wavelengths, which translates to hundreds of meters at mm-wave frequencies. Based on the new mathematical approach presented here, near field nondiffracting and far field beam forming arrays are produced using aperture excitations that belong to a family of odd functions characterized by odd (cubic, quantic, heptic, etc.) inverse reflection symmetry phase distribution of the form (x²ⁿ⁺¹+y²ⁿ⁺¹)/2n+1 followed by Fourier lenses. These arrays perform more efficiently than conventional phased arrays with better than 1/r power attenuation in the nondiffracting range. The mainlobe and the dominant sidelobes are contained in one quadrant of the azimuthal plane, whereas the sidelobes in the remaining three quadrants are reduced significantly (by 9 dB at least, for the considered array size).