Laguerre-Gaussian beams with complex shift in Cartesian coordinates

We studied a generalization of the family of Laguerre-Gaussian (LG) laser modes with asymmetrical intensity distribution. As the asymmetrical LG beam is propagating in a homogeneous medium, the asymmetry of its main (central) bright ring reduces, while the contrast of the rest rings increases. The number of bright rings coincides with that of a standard (symmetrical) LG mode. Using the expansion of the complex amplitude into the angular spectrum of plane waves, we calculated analytically the power of the asymmetrical LG beams and projection of their orbital angular momentum (OAM) on the optical axis. It is found that the normalized OAM (OAM per photon) is completely determined by the topological charge and the ratios between the shifts and the waist radius. We also found the conditions under which the normalized OAM coincides with the topological charge (as is the case for the optical vortices).