# Propagation of Optical Coherence Vortex Lattices in Turbulent Atmosphere
## 1 Introduction  
Optical lattices can be viewed as a typical kind of beam arrays, which display sufficiently small beam spots. Recently, a novel class of optical lattices, which are named optical coherence lattices (OCLs), were introduced by [PERSON] and [PERSON] [1]. The OCLs were described with the help of the complex Gaussian representation and the time-domain OCLs were generated through superposition of Gaussian pulses [2]. The property of periodic reciprocity of the OCLs on propagation through the free-space was found [3]. Periodic reciprocity means the periodic degree of coherence at the source plane transfers the periodicity to the transverse intensity distribution after a long propagation distance. Recently, both scalar and vector OCLs were generated in experiment [4, 5]. Furthermore, other kinds of optical lattices were also proposed [6, 7, 8]. The optical lattices were found to be useful in various applications, e.g., image transmission and encryption [4], trapping and cooling atoms [9], photonic crystals engineering [10], lattice light-sheet microscopy [11], and ultracold trapping of quantum gas [12].  
On the other hand, partially coherent beams have been studied extensively over the last few decades due to their novel physical properties and widely uses in various fields, such as free-space optical communications, optical imaging, particle trapping, particle scattering, and remotedetection. The statistical properties, e.g., degree of coherence [13], beam spreading [14; 15], propagation factor [16; 17; 18], Rayleigh range [19], and scintillation index [20; 21; 22] of different kinds of partially coherent beams in free space and in various turbulent atmosphere have been studied in detail. The propagation properties of the OCLs (i.e., a typical kind of partially coherent beams with periodical degree of coherence) in atmospheric turbulence were explored [23], and the results showed that the OCLs have an advantage of mitigating scintillation index in atmospheric turbulence on propagation.  
A vortex beam with twisted wavefront carries an orbital angular momentum and has various applications ranging from optical tweezers to optical communications [24; 25; 26; 27; 28]. The topological charge (which defines how the wavefront of the vortex beam twists) on propagation in atmospheric turbulence is robust [29]. In 1998, [PERSON] et al. introduced a class of partially coherent beams with helical wavefront, which were named partially coherent vortex beams (PCVB) [30]. Subsequently, efforts have been made to the PCVBs [31; 32; 33; 34; 35; 36; 37; 38; 39; 40; 41; 42; 43; 44]. A review on propagation, generation, and measurement of PCVBs was given [41], and self-healing properties of a PCV was found [42]. More recently, PCV with a fractional topological charge [43] and partially coherent vortex beam with periodical degree of coherence (i.e., optical coherence vortex lattices) [44] were introduced, respectively. In this paper, we investigate the statistical properties (i.e., spectral density distribution, normalized M\({}^{2}\)-factor, and beam wander) of optical coherence vortex lattices (OCVLs) in turbulent atmosphere by using the recently developed convolution approach. We find that the initial Gaussian beam distribution of the OCVLs evolves into hollow array distribution on propagation and finally becomes a Gaussian beam spot again in the far field in turbulent atmosphere. The OCVLs with prescribed beam parameters are less affected by the turbulence, which will be useful for free-space optical communications.