# Propagation Characteristics of a Twisted Cosine-Gaussian Correlated Radially Polarized Beam
## 1 Introduction  
It is well-known that the coherence properties play an important role in determining the spatial behaviors of light beams under propagation [1; 2; 3; 4; 5; 6; 7; 8; 9; 10]. For instance, there is a Fourier reciprocal relationship between the initial coherence and the far field intensity distribution [1; 2]. Also, it has been proven that the source coherence can apparently influence the evolution behaviors of the degree of polarization and state of polarization of light beams after propagation, even in free space [7; 8]. These results contradict the common belief that polarization is propagation invariant. However, most of the studies concerning the spatial coherence sources are restricted to the so-called Schell-model (SM) correlations, where the spectral degree of coherences obey Gaussian distributions [1; 2; 3]. In recent years, based on the nonnegative definiteness criteria of the cross-spectral density (CSD), [PERSON] and [PERSON] proposed a sufficient condition for devising genuine correlation functions and quickly extended to the vector cases [11; 12]. Manipulation of nontrivial correlation structures to produceprescribed structure beams with tailored intensity, polarization, and phase, and to explore novel spatial behaviors has seen a rapid growth of interest due to their important applications in optical imaging, optical tweezers, free-space optical communications, laser radar, and remote sensing [13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33]. In previous decades, much prominence has been given to radially polarized beams because of their nontrivial properties and potential applications such as lithography, display technologies, optical trapping, optical data storage, and confocal microscopy [34; 35; 36; 37; 38; 39; 40; 41; 42]. A partially coherent radially polarized beam was introduced as a natural extension of a coherent cylindrical vector beam. Paraxial and non-paraxial properties of partially coherent radially polarized beams have been investigated in detail [43; 44; 45; 46; 47; 48; 49]. Statistical properties of partially coherent radially polarized beams in random media such as a turbulent atmosphere and ocean turbulence have been studied [50; 51]. It is demonstrated that partially coherent radially polarized beams are more effective at resisting signal distortion caused by turbulence than general linearly polarized partially coherent beams. Moreover, by manipulating the correlation structures of source beams, the ability to generate tunable partially coherent radially polarized array beams was demonstrated [22; 23; 24; 25; 26]. In 1993, [PERSON] and [PERSON] proposed that a partially coherent beam has the ability to carry a new type of phase-twist phase, and this was later generated experimentally by [PERSON] et al. [52; 53]. Recently, on the basis of the nonnegative constraint of the CSD, [PERSON] et al. introduced a new method to design twisted sources endowed with circular or rectangular symmetry [54; 55]. Due to the existence of the twist phase, twisted partially coherent beams not only have the ability to carry orbital angular momentum, but also have better anti-turbulence self-repairing capabilities, which can find applications in optical tweezers, optical imaging, and optical communications [56; 57; 58; 59; 60; 61; 62; 63; 64]. Besides, owing to their nontrivial angular momentum mechanical properties, twisted partially coherent beams are receiving increasing attention [65; 66]. In this paper, our aim is to study the focusing properties of a new class of partially coherent radially polarized beams with a nontrivial cosine-Gaussian correlation function and a twisted phase. The effects of the twist phase on the spectral intensity, the spectral degree of coherence, the degree of polarization, the state of polarization, and the spectral change are investigated in detail.