# Spatial-Temporal Self-Focusing of Partially Coherent Pulsed Beams in Dispersive Medium
## 1 Introduction  
As we all know, the coherence properties of a light source have a marked impact on the characteristics of the propagated field, especially on the intensity profile. Hence, it is a possible way to modulate the beam shape by regulating the coherence of the source [1; 2; 3; 4; 5; 6; 7]. In the past few decades, extensive work have been done to control the coherence distribution of light sources in order to obtain desired far-field intensity profile or extraordinary propagation behaviors [8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32], such as self-splitting, self-focusing, self-steering, and flat-topping, ring-shaped, rectangular, gridded, dark-hollow, or cusped intensity profiles, which have many applications in remote sensing [33], imaging [34], optical trapping and optical communications [35].  
In 2017, a new class of partially coherent light sources with circular coherence was introduced [36] and synthesized through a time multiplexing approach [37]. Then, [PERSON] et al. proposed another alternative approach for synthesizing this class of sources [38]. The self-focusing propagation phenomenon of this class of sources was also founded, whose propagation properties through oceanic turbulence is also being explored [39]. All the above-mentioned references are confined to stationary optical fields or beams.  
On the other hand, random optical pulses with partial temporal or spectral coherence represent more widespread partially coherent light fields, which have many applications in pulse shaping, fiber optics, ghost imaging etc. [40; 41; 42]. In practice, many real sources, such as excimer lasers, free-electron lasers, or random lasers generate partially coherent pulse trains [43]. Usually, the Gaussian Schell-model (GSM) pulses with Gaussian correlation function are used to describe the partially coherent pulsed (PCP) field, which is called partially coherent GSM pulsed beams [44; 45; 46], whose propagation propertiesare investigated in detailed [47; 48; 49; 50; 51; 52; 53; 54; 55]. In 2013, the non-uniformly PCP source with non-conventional correlation functions are introduced by [PERSON] and [PERSON] [56]. The propagation properties of various non-uniformly PCP source exhibit many interesting behaviors, such as pulse self-splitting, self-focusing and adjustable pulse profile [57; 58; 59; 60; 61].  
In this paper, we present a novel class of spatially and temporally PCP source with circular spatial coherence distribution and sinc temporal coherence distribution. The propagation properties of pulsed beams generated by this source in two common dispersive media, i.e., water and air are investigated. It is found that, the pulsed beams exhibit spatial-temporal self-focusing behavior upon propagation in water or air. The condition that the self-focusing takes place at spatial domain and temporal domain simultaneously is acquired. In Section 2, based on the optical coherence theory of non-stationary field, we obtain the spatial-temporal intensity distribution of the proposed PCP beams. In Section 3, we give numerical calculations, and present the spatial-temporal self-focusing behavior of the pulsed beams in water and air, respectively. In addition, a physical interpretation of the spatial-temporal self-focusing phenomenon is presented. In Section 4, we summarize the results.