Gyrator transform of four-petal Gaussian beam and generation of rectangular hollow beam

Four-petal Gaussian beam is a special type of Gaussian beam, and its propagation properties are widely used in micro optics, optical communication and splitting technology. Recently, the generations and the properties of different types of hollow beams have become a hot research topic, such as research on hollow optical vortex beams. The Gyrator transform can be used to fulfill the mode conversion of laser beam. In this paper, based on the Gyrator transform, the analytical expression of four-petal Gaussian beam passing through such a transform system is derived, and the intensity distribution and the corresponding phase distribution associated with the transforming four-petal Gaussian beam are analyzed by numerical simulations. It is found that the four-petal Gaussian beam can be transformed into rectangular hollow beam by Gyrator transform, under the appropriate conditions of the beam order, the beam parameter, the transform angle of Gyrator transform, and the waist width. For the beam order n=m=3, the transform angle of Gyrator transform = 0.4133, the beam parameter K=30, and the waist width = 0.9, the rectangular hollow optical vortex beams can be obtained. Under such conditions, the maximum intensities appear in the four corners, and they are almost uniform on the four sides. The effects of the beam parameters, the transform angle, and the beam order on the distributions of intensity and phase of the rectangular hollow beam are analyzed in detail. The numerical results show that for the beam parameter K10, the rectangular hollow beam always is obtained, and for a lager beam parameter, the intensity distribution of the rectangular hollow beam is more uniform. Different beam order generates different type of hollow beam. For example, for n=m = 2, = 1.2, K = 30, and = 0.5409, a new strange circular hollow beam with solid circular nucleus can be obtained. The transform angle of Gyrator transform has a significant effect on the energy distribution of the hollow beam. When the transform angle changes in a small range, the uniformity of the intensity distribution of the rectangular hollow beam is lost. The bigger the transform angle change, the more serious the loss of uniformity of the hollow beam intensity is. The size of the hollow beam bright ring is determined by the waist width of the four-petal Gaussian beam: the larger the waist width, the smaller the bright ring is. The results further enriches the applications of Gyrator transform system and the four-petal Gaussian beam in the beam shaping.