During the last decade, the concept of the orbital angular momentum (OAM) has been applied for the description of coherent optical vortex beams [1]. In recent publications [2], it has been suggested to decompose the OAM into two parts: the asymmetrical part and the vortex part. The first part describes an astigmatic beam but with a smooth wave front, while the second one is related to the singularity of the wave front. Furthermore, it has recently been reported that some partially coherent light fields also exhibit the vortex behavior. Taking into account that the concept of the OAM can be generalized to the case of partially coherent light beams, the results of the present paper can be applied to both the completely coherent and the partially coherent case.
In this paper we study the evolution of the vortex part of the OAM of partially coherent light beams during their propagation through separable first-order optical systems. After having observed a relationship between this vortex part and the optical twist, we obtain a general expression describing the evolution of the vortex part in first-order optical systems, which expression contains only the parameters a and b of the system's ray transformation matrix. Moreover, we show that in the case of isotropic systems, the evolution of the vortex part is determined only by the ratio b/a. As an example, it is shown that when light propagates through an optical fiber with a quadratic refractive index profile, and is thus undergoing a fractional Fourier transformation, the vortex part of the OAM cannot change its sign more than four times per period.
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