Differential operator formalism for axial optical vortex beam and the double-phase-ramp converter

A systematic study of the properties of the output dark rays or singular skeleton for the Laguerre-Gaussian beam LG01 passed through double-phase-ramp converter is presented. When the DOE is discontinuous at the origin, as is the case here, the transfer function is not analytical, so that a special theoretical approach is needed. The previously reported formalism of scattering modes, which permitted the analytical calculation of arbitrary multisingular Gaussian beams, requires analyticity everywhere. We present here an adaption of this formalism that overcomes this limitation. The procedure is based on the differential operator algebra used in the previous construction. We give an example of diffraction of LG beams by a discontinuous DOE that can be solved completely in an analytical way using this adapted method. The experimental setup for the investigation of the considered problem with spatial light modulator is discussed. The theoretical and experimental skeletons are similar but not the same because initial conditions are slightly different. In the near future, we plan to obtain analytical solutions that reproduce exactly the initial field parameters used in the experiment.