We propose a new, three-parameter family of diffraction-free asymmetric elegant Bessel modes (aB-modes) with an integer and fractional orbital angular momentum (OAM). The aB-modes are described by the th-order Bessel function of the first kind with complex argument. The asymmetry degree of the nonparaxial aB-mode is shown to depend on a real parameter : when , the aB-mode is identical to a conventional radially symmetric Bessel mode; with increasing , the aB-mode starts to acquire a crescent form, getting stretched along the vertical axis and shifted along the horizontal axis for . On the horizontal axis, the aB-modes have a denumerable number of isolated intensity zeros that generate optical vortices with a unit topological charge of opposite sign on opposite sides of 0. At different values of the parameter , the intensity zeros change their location on the horizontal axis, thus changing the beam’s OAM. An isolated intensity zero on the optical axis generates an optical vortex with topological charge . The OAM per photon of an aB-mode depends near-linearly on , being equal to , where is the Planck constant and is a modified Bessel function.
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