# Coupling Efficiency of a Partially Coherent Radially Polarized Vortex Beam into a Single-Mode Fiber
## 1 Introduction  
Optical fiber, as the fundamental optical components of a coupling process, have been widely applied in various fields, such as networking [1; 2], ladar [3; 4], imaging [5], free-space optical communications [6; 7], and biomedical optics [8; 9; 10]. In the above applications, the input beam must be coupled into a single-mode or multi-modes fiber for propagation and detection before being amplified. Therefore, the coupling efficiency of the input light beam into optical fiber plays an important role. One can improve the signal-to-noise ratio and save energy through increasing the coupled light beam. During the past decades, many theoretical methods and experimental techniques were used to improve the coupling efficiency, such as a fiber array [11], cylindrical glass fiber [12], wedge-shaped fiber endface [13], combination lens [14], chemically etched self-centered diffracting element [15], and a microlens [16]. The degree of freedom of the input light beam, such as phase, amplitude, coherence, and polarization, is one of the key factors when analyzing the coupling efficiency. For example, the coupling efficiency of the plane-wave to fiber was studied as the input light in detail, the influences caused by the misalignments of the fibers and mode's field distribution are considered [17]. [PERSON] and [PERSON] discussed the coupling efficiency when the stellar is coupled into the single mode fiber [18]. [PERSON] and [PERSON] developed analytic equations that describe the mean and normalized variance of the coupling efficiency of the Gaussian Schell-model (GSM) beams into single-mode optical fibers [19]. The light sources in the above studies were scalar beams and did not consider the vector beams. In 2009, the polarization state of the beam was first taken into account by [PERSON] and [PERSON]. They studied the problem of coupling an electromagnetic beam of any state of coherence and polarization into a multimode optical fiber. The coherence and polarization properties of the electromagnetic Gaussian Schell-model (EGSM) beams are discussed in detail in Refs. [20; 21]. In 2012, [PERSON] and othersdemonstrated experimentally the procedure of coupling of an EESM beam into a single-mode optical fiber. The results showed that the coupling efficiency depends closely on the coherence and polarization properties of the EESM beam, which is consistent with [PERSON] and [PERSON]'s theoretical prediction [PERSON] and [PERSON] (2016). Furthermore, in the past several years, due to the partially coherent radially polarized (PCRP) beam having an advantage over a linearly polarized partially coherent beam for reducing turbulence-induced scintillation, which is useful in free-space optical communications, the PCRP beam was introduced in theory and generated in experiment ([PERSON] _et al._, 2012; [PERSON] _et al._, 2013; [PERSON] _et al._, 2014). The propagation properties of a PCRP beam are quite different from those of a radially polarized beam. On the other hand, it is well known that any beam with a helical phase factor, called a vortex beam, possesses an optical vortex and carries orbital angular momentum (OAM). Thus, there are growing interests in focusing partially coherent radially polarized vortex beams ([PERSON] _et al._, 2012; [PERSON] _et al._, 2013; [PERSON] _et al._, 2014). However, to the best of our knowledge, there are no papers dealing with the coupling efficiency of the PCRP vortex beam into a single-mode fiber. In this paper, our aim is to study the coupling efficiency of PCRP vortex beams into a single-mode optical fiber. Analytical expressions for the coupling efficiency are derived based on the concept of the cross-spectral density matrix. The effects caused by the coherence, topological charge, numerical aperture (NA), and wavelength on the coupling efficiency are studied in detail. We calculate the peak coupling efficiency versus the NA for different coherence length and topological charge.