Propagation of laser vortex beams with an arbitrary topological charge in a craded-index parabolic fiber

In this work, we discuss the propagation of the laser vortex beams exp{iμφ} in a parabolic fiber. The relationship between the complex amplitude and the transverse coordinates and the distance on the optical axis is described as the integral operator of propagation in a parabolic medium acting on the input beam amplitude distribution. This integral is analogous to the Fresnel integral that describes the propagation of paraxial laser beams in a uniform medium. The result of action of the integral operator onto the vortex beam can be analytically represented in two ways. In the first case, the kernel of the analytical expression for the amplitude is given by a degenerate hyper-geometric function. In the second case, the amplitude is represented as a composition of an infinite number of Gauss-Laguerre modes, which are eigenmodes of the parabolic fiber. The analytical relations derived are verified by numerical modeling with use of the integral operator and the method for propagating the light beams through thin lenses, which relies on the fast Fourier transform. The results of the numerical modeling and the analytical calculations agree within the method’s accuracy.