Since the concept of orbital angular momentum (OAM) was brought to the realm of optics in the 1990s [1], it propelled a number of important technologies, the most influential one being the micro-particle manipulation [2], [3], [4], [5]. Later applications of light’s OAM involve super-resolution in imaging [6], [7], coronagraphy [8], rotational object’s sensing [9], laser communications [10] and quantum memory [11].

The complete characterization of the OAM in spatially partially coherent beams [12] and the related applications are still being in development. A rather general approach of decomposing the beam’s spatial correlation function (cross-spectral density) into the spiral basis [13] has recently provided an insight into the possibilities of modeling [14], generation [15], transformation [16], [17] and measurement [18] of various OAM states in random light, in the language of l×l matrices, Coherence-OAM (COAM) matrices, l being the largest(smallest) OAM index present in the optical field. In particular, the COAM matrix clearly illustrated the relations among the already known classes of the OAM-carrying beams as well as exploited the ideas for the design of the new beam classes.

Twisted Gaussian Schell-model (TGSM) beams occupy a very special place among all the partially coherent beam types. Introduced in 1993 [19], [20], a TGSM beam was the first known type of a random beam to carry the OAM, discovered for deterministic beams just a year prior [1]. Unlike other random beams carrying OAM, for instance, those with a separable phase [21], the TGSM beams do not have an analog among the deterministic beams, i.e., they are intrinsically random. In fact, they can be represented as the infinite sum of radially varying contributions possessing coherence vortices of all orders [22]. The TGSM beams were shown to be more resilient to atmospheric and oceanic turbulence perturbations as compared with their twistless counterparts [23], [24]. They were also shown to enhance the visibility of ghost images [25] and to produce a remarkable sub-Rayleigh resolution in telescopic imaging systems [26]. Later studies established that twisted beams do not need to be of Gaussian Schell-model class and that the twist phase can furnish other random beam families, still being a subject to certain constraints inherent in the structure of the coherence matrices, but providing with some new beam structuring and rotation opportunities [27], [28], [29], [30], [31].

The complete characterization of free-space evolution of the OAM content in a TGSM beam can be done by propagating its COAM matrix. In this paper, we aim to derive the analytical expressions for its elements, for any range from the source. In fact, in a recent publication [32] the COAM matrix of the TGSM was examined in detail for source points. We will now extend these results to the field.