# Effects of Atmospheric Turbulence on Lensless Ghost Imaging with Partially Coherent Light
## 1 Introduction  
Ghost imaging (GI) is a novel technique to retrieve the image of an object by measuring the correlation function of light intensities from two distinct paths. The conventional GI geometry is that the light beam is first divided into two parts entering two optical paths; an object is located in one path, and a bucket detector with no spatial resolution collects the light intensity transmitting from the object; in another path, a high spatial resolution detector receives the light intensity, but no object exists. By making a correlating calculation of the optical signals from two detectors, an image of the object can be obtained. This phenomenon was firstly observed using entangled photon pairs in spontaneous parametric down-conversion (SPDC) [1; 2]. It was thought for a long time that GI can only be realized with entangled quantum light source. In 2002, [PERSON] et al. presented their ghost imaging and interference experiments using classical light [3]. Since then, much work has been devoted to GI with classical light both in experiment and theory [4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24] due to classical light being able to be obtained easily compared to entangled quantum light. Various GI schemes such as lensless GI, reflective GI, computational GI and high-order correlation GI have been proposed [21; 22; 23; 24]. Meanwhile, researchers have developed different algorithms, e.g., compressed sensing, differential, normalized, iterative and pseudo-inverse algorithms to improve the image visibility, quality or the speed of the image acquisition [25; 26; 27; 28; 29]. The concept of the GI was extended to the temporal domain using classical non-stationary pulsed light by [PERSON] et al. [30] and was demonstrated in an experiment recently [31]. In 2009, [PERSON] firstly investigated GI through atmospheric turbulence and found that the quality of the image degraded due to turbulence [32]. Since then, considerable attention has been paid to GI through turbulence with classical or quantum light due to its important applications in remote sensing and communications [33; 34; 35; 36; 37; 38; 39; 40]. The effects of turbulence on the visibility and quality of the image have been extensively studied both theoretically and experimentally [33; 34; 35; 36; 37; 38]. The results in these studies are similar to those in [32], i.e., the turbulence has a negative effect on the quality of the image in the GI system. However, [PERSON] et al. found from experiment that the quality of the ghost image seems to be immune to turbulence [39], while the visibility of the image decreases. This result is a contradiction to the previous theoretical analysis [32; 37; 38]. The motivation of this paper is to try to solve the controversy about the effect of atmospheric turbulence on ghost imaging. Several studies in [33; 34; 35; 36; 37; 38] declared that the turbulence has negative effects on the quality of the ghost image. However, the experiment results in [39] showed that the quality of the ghost image seems immune to the atmospheric turbulence. Up to now, there is still a lack of a clear understanding about the different results between in [33; 34; 35; 36; 37; 38] and in [39]. In this manuscript, we present our explanations that the different results in [33; 34; 35; 36; 37; 38] and in [39] are due to different types of atmospheric turbulences, i.e., bistatic turbulence and monostatic turbulence. In bistatic turbulence, the visibility and the quality of the image all degrade, compared to those in free space. Our experimental results for bistatic turbulence are consistent with those reported in [33; 34; 35; 36; 37; 38]. In monostatic turbulence, we establish the theoretical model for the four-order correlation function between two detectors. With the help of this model, we theoretically predict that the quality of the image is nearly immune to the atmospheric turbulence.