## 1 Introduction  
The vortex beam is characterized by the helical distribution of isophase lines along the azimuth angle, and the convergence of the isophase lines depends on the coherence singularities [1]. The phase of spiral wavefronts can be described as \(\exp(\mathrm{i}l\varphi)\), where \(l\) and \(\varphi\) represent the topological charge (TC) and azimuth coordinate, respectively. TC \(l\) is typically an integer that can be defined as the number of turns of phase twists at a specified wavelength. The sign of \(l\) depends on the rotation direction of the phase; a larger \(l\) value indicates that light rotates faster along the optical axis. In 1992, [PERSON] et al. [2] hypothesized that each photon of a vortex beam exhibits an orbital angular momentum (OAM) with a value of \(\hbar\) (h is the reduced Planck constant), thus revealing the relationship between macroscopic optics and quantum effects. Hence, optical vortices are widely used in optical micromapulation [3; 4; 5], large-capacity optical communication [6; 7; 8], super-resolution imaging [9; 10], nanostructures [11], artificial spin ice systems [12], spin measurement of M87 black holes [13], and ptychographic imaging of highly periodic structures [14]. However, high spatial coherence causes adverse factors, such as speckles, wandering, and scintillation, in turbulence media [15]. [PERSON] et al. [16] introduced a vortex structure into a partially coherent beam. Compared with fully coherent fields, partially coherent fields are advantageous in complex media propagation [17; 18] and exhibit a higher signal-to-noise ratio. In addition, some partially coherent vortex beams exhibit unique propagation properties, such as self-focusing, self-splitting, and self-remodeling effects [19; 20; 21], which can result in a lower bit error rate in photon communication. For a partially coherent vortex beam with a Shell model, such as a partially coherent Laguerre-Gaussian (PCLG) vortex beam, the vortex structure, which features a dark core and bright rings, vanishes as the coherence decreases and beam propagates. By regulating parameters such as the coherent width and TC, the light intensity can be modulated into hollow, flat top, and Gaussian distributions. These diverse beam-shaping capabilities have promoted the development of particle-manipulation techniques. The accurate and rapid detection of the TC is extremely important owing to the close relationship between the application of the vortex beam and the properties of the TC or OAM. TC measurement schemes under fully coherent cases have been widely investigated in recent years [22; 23; 24], such as the Shack-Hartmann measurement [25], interferometry [26; 27; 28; 29], and ptychography [30]. However, when the spatial coherence decreases, the interference and diffraction of the beam are no longer evident. Therefore, the TC of vortex beams with low coherence must be obtained [31; 32; 33; 34]. Cylindrical lenses have been widely used to determine the TCs of coherent vortex beams [35; 36]. For partially coherent vortex beams, [PERSON] et al. [32] proposed a simple method to determine the magnitude and the sign of TCs by measuring the complex degree of coherence of a partially coherent vortex beam after it propagated through two mutually perpendicular cylindrical lenses. [PERSON] et al. [33; 34] proposed a method using self-reference holography to measure cross-spectral density (CSD) and obtained the magnitude and the sign of TCs from the number of coherence singularities. However, the quantitative relationships between bright fringes and TCs based on the cylindrical lens method are only valid for cases of extremely low coherence, whereas self-reference holography relies on the appropriate choice of reference point. Subsequently, the introduction of double slits allows one to observe coherence singularities with an on-axis reference point, whereas the double-slit scheme has only been validated in cases of medium coherence (coherence greater than 0.3 times the beam width) [34].