Optical vortex beams (OVs) possess an infinite number of orthogonal orbital angular momentum (OAM) states, making them highly beneficial for increasing communication capacity. In this work, a new type of power-exponent-phase-like vortex beam (PLB) with both quadratic and cubic changes in phase is investigated, and the two cases where the phase gradient increases or decreases along the azimuthal angle are analyzed. Furthermore, based on the changes in phase between 0 and 2π for the direction angle, the single-period and multi-period scenarios are discussed. Both simulations and experiments demonstrate that the intensity distributions are noticeably different when the gradient direction increases or decreases along the azimuth angle. Additionally, the OAM density, the orthogonality, and the OAM of these nonlinear beams are assessed. The intensity distribution of the PLB is not uniform along the angle direction, however, it offers a distinct orthogonality, and the total OAM remains constant, which provides new ideas for applications such as particle manipulation, optical communications, OAM encryption, and more.
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