## 1 Introduction  
Recently, optical vortex (OV) beams, owing to their unique properties, have attracted more and more interest and have been utilized in a wide range of fields--from scientific research to advanced technology applications, such as optical communications [1; 2; 3; 4], optical metrology [5; 6; 7], and optical trapping and manipulation [8; 9; 10]. Many specialties of OV beams are due to their phase singularity in the wavefront function, where the intensity drops to zero and the phase is undefined [11; 12]. Moreover, the phase along a closed path enclosing the singularity point varies from 0 to \(2n\pi\), where \(n\) is an integer known as the topological charge (TC) or the orbital angular momentum (OAM). OV beams with different TC values perform diverse characteristics and consequently are used as information carriers in state-of-art optical communication systems, which are used to generate sufficient force to manipulate the molecules and so on. To meet the requirements of these applications, the determination of how to precisely measure the TC value of an OV beam is an important issue, and therefore many methods, such as interferometry-based methods [13], diffraction-based methods [14; 15; 16], and model decomposition-based methods [17; 18] have been proposed and comprehensively studied. As one of the key devices used in adaptive optics systems [19], Shack-Hartmann wavefront sensors (SH-WFS) have also been utilized to determine the TC value of OV beams [12; 20; 21; 22; 23]. An SH-WFS consists of a lenslet array and an image sensor, and directly measures the phase slope of the incident wavefront at each lenslet position. TC determination with an SH-WFS is simple and direct, and the contour-sum method (CSM) has been proposed based on the principle that the net TC value in an area is proportional to the line integral of the phase slope along the closed path circumscribing the area [12]. The basic form of the contour-sum method uses the pre-assigned closed paths, e.g., the closed path associated with the central \(2\times 2\) or \(3\times 3\) lenslet area to calculate the discrete line integral, which is employed to determine the TC value of an OV beam [20; 22]. This approach is successful in measuring OV beams with nearly uniform or quasi-uniform intensity distribution. However, it becomes deficient under some conditions, especially when the OV beams to be measured have donut-shaped intensity profiles, where phase singularities are embedded in the low-intensity region (dark region), and the phase slope measurement could be invalidated. This condition commonly exists in many OV applications such as OV-based optical communication and OV-based optical metrology. In view of this, in a previous study, we generalized the contour-sum method to be workable for a closed path with an arbitrary shape, and proposed a maximum average-intensity circle (MAIC) method to extract a closed path with only valid phase slope data from the annular intensity profile [24]. We experimentally investigated the MAIC method with both aberration-free OV beams as well as the OV beams distorted by simulated atmospheric turbulence, and we concluded that the proposed MAIC method has good robustness against aberrations. Moreover, we found that the closed path used for circulation calculation has a notable influence on the determined TC value when the OV beams are severely distorted by aberrations. Considering that the closed path is vital for the CSM, we also briefly demonstrated the superiority of the MAIC method against the use of perfectly round circles (PRC) as the closed path [24].