# Generation and Propagation of a Hermite-Gaussian Correlated Schell-Model LG\({}_{0l}\) Beam
## 1 Introduction  
The light beam with a spiral structure of the phase (i.e., vortex phase), named the vortex beam has continuously been under the spotlight in optical research area, due to its unique properties, e.g., orbital angular momentum (OAM), and phase singularity [1; 2; 3]. The OAM of a vortex beam can be used for particle trapping, detection of spinning object, free-space optical communications, and quantum information processing [4; 5; 6; 7]. At the beginning, research attention was concentrated on the fully coherent vortex beams [8; 9; 10; 11]. The quantitative measurement methods of the topological charge (TC) or the OAM state have been developed [12; 13; 14; 15; 16; 17]. Until recently, a partially coherent (PC) beam, obtained by reducing the spatial coherence width, has demonstrated its superiority to a complete coherent counterpart in various applications [18; 19; 20; 21; 22; 23; 24; 25; 26]. In other words, controlling the spatial coherence width provides an effective way of manipulating beam properties. It is naturally to extend the coherent vortex beams to partially coherent vortex light (PCV). The PCV light, whose correlation function satisfies the Gaussian distribution, is a conventional PCV beam (e.g., Gaussian Schell-model (GSM) vortex light). Compared with PC light without vortex phase, and its complete coherent counterpart, the GSM vortex beam has advantages in many aspects. It can be shaped into prescribed beam profiles by modulating the coherence width of the incident beam to capture particles no matter which index of refraction is larger or smaller than that of the ambient [26; 27]. PCV light has advantages of attenuatingthe beam-width spreading, beam wander, and scintillation induced by atmospheric turbulence [28; 29; 30]. PCV light also displays self-reconstruction abilities in the distribution of the DOC on propagation, which can be used for information encryption and decryption [31].  
Now, more and more attention has been paid to PC light with non-Gaussian distribution correlation function, which is named the special correlated (or nonconventional) PC light. A variety of these type of beams have been constructed and generated experimentally [20; 32; 33; 34; 35; 36; 37; 38; 39], ever since the conditions which guarantee the real existence of cross-spectral density function or matrix, were discussed in [40; 41]. These non-Gaussian distribution correlation functions bring about the new changes and characteristics of the beam on propagation, which can be applied to reduce scintillation in turbulence and realize optical super resolution [42; 43; 44]. In this article, we will discuss the combined action of special correlation structure and the vortex phase on the beam's propagation properties, both theoretically and experimentally. An LG\({}_{0l}\) beam with a Hermite-Gaussian correlated function named HGCSMLG\({}_{0l}\) beam is introduced and explored in detail. We find that the correlation function and TC together determine the focused spot and the DOC distribution on the propagation displays' rotating properties, which are induced by the vortex phase. By measuring the distribution of DOC, the handedness and the TC of the vortex phase can be obtained. The above theory results are verified well by the experiment results.