Optical vortices with a vortex core size that is at least two orders of magnitude smaller than the laser beam waist is presented. The optical vortex is generated by a spiral phase plate (SPP) and counter-rotating optical vortex pairs are created in a modified Mach-Zehnder interferometer surrounded by a 4 f lens arrangement. The azimuthal variation of the counter-rotating optical vortex forms a sinusoidal intensity modulation for which the winding number of the optical vortex is deduced accurately and precisely by fitting it to a sinusoidal function. These results are of interest in designing novel optical vortex gratings for manipulating matter waves (e.g. in Kapitza-Dirac scattering). A theory of atomic angular Kapitza-Dirac scattering with optical vortices is presented. The large beam waist combined with a small optical vortex core size would also be of interest when using an optical vortex to perform spectroscopy in a wide range of matter systems including solid, liquid, atomic, and molecular systems, as well as in short range optical communication.