# Flexible Construction of a Partially Coherent Optical Array
## 1 Introduction  
Since [PERSON] et al. proposed sufficient conditions for constructing partially coherent beams [1, 2], various partially coherent beams have been proposed, with different amplitude distributions [3, 4] and spatial coherent structures [5, 6] and carrying different phases [7, 8]. The partially coherent beams can be classified through response functions; when the response functions of these two systems are different, they can be divided into uniformly correlated partially coherent beams and non-uniformly correlated partially coherent beams [9, 10]. In addition, there is a special type of partially coherent beam that carries a twist phase which can be observed through the reasonable adjustment of the system response function [11]. Among them, the complex degree of coherence of uniformly correlated partially coherent beams depends on the absolute position of two points in space, which can also be referred to as partially coherent Schell model beams. After determining the system response function, multiple special correlated partially coherent beams can be generated by designing weight functions. The Gaussian Schell model beam constructed by [PERSON] and [PERSON] in 1978, as a classical correlated partially coherent beam, has been extensively studied for its generation, transmission, and application [12, 13]. As research progresses, various partially coherent beams, including scalar and vector partially coherent beams, have been discovered and widely applied. These beams exhibit unique properties during propagation, including self-healing [14], self-focusing [15], self-splitting [16], and self-shifting [17] properties, and they show extraordinary advantages in overcoming random perturbation when propagating through random media, such as turbulent atmosphere [18], turbulent ocean [19], and weak scattering [20]. Partially coherent beams can be used to improve image quality in classical or correlation images [21] or to reduce bit error rates [22] and have found potential use in laser processes [23], robust imaging through random media [24], and free-space optical communication [25, 26, 27]. Recently, research on optical arrays has received increasing attention; various optical arrays have been proposed to cater to practical applications, and currently there are manymethods for generating optical array. Usually, a spatial light modulator is used to modulate the beam to construct various optical arrays, as this method is not only simple but also efficient [28; 29]. Optical arrays have demonstrated their usefulness in many aspects, such as beam shaping [30], lattice light microscopy [31], and as a universal tool for performing quantum simulations [32]. However, all these discussions are based on completely coherent beam modulation. As typical partially coherent sources, optical coherent lattices have attracted great attention in recent years. Since [PERSON] et al. introduced the optical coherent lattices mode [33], vector optical coherent lattices [34] and the carrying vortex phase [35], as well as experimental generation [36] and paraxial propagation [37] through random media [19; 38], have been studied extensively. The periodicity reciprocity arises between the source coherence and far-zone intensity of optical coherence lattices during propagation. The periodicity reciprocity of optical coherent lattices mean a Gaussian intensity profile of an optical coherence lattices evolves into a lattice-like far-field profile, while the periodic spectral degree of coherence at the source loses its periodicity during propagation [37]. The periodic reciprocity of optical coherent lattices has shown great potential use for the aspects of orientation-selective sub-Rayleigh imaging [39], free space optical communication [37], and optical encryption [36]. Recent research has shown that one can generate perfect optical coherent lattices [40] and tunable polarization optical arrays [41], which indicates that flexible and programmable optical coherent arrays extend the scope of application and are now becoming a future trend.