Coherent Vortices Properties of Partially Coherent Elegant Laguerre-Gaussian Beams in the Free Space

The analytical expression of the cross-spectral density of partially coherent elegant Laguerre-Gaussian beams propagating in free space has been derived. The coherence vortex properties of such beams have been investigated. The effect of the beam parameters, including the topological charge, radial mode index and coherence length on the coherent vortex, is analyzed. The results show that the higher order (lth) of coherent vortices split to lth first order of coherent vortex. New coherent vortices of opposite sign appear, and then pairs of coherent vortices form. The propagation distance and coherence length affect the generation of coherent vortices, while the radial mode index doesn’t affect the coherent vortices. These results might be helpful for potential application of such beams in optical communication.

DOI: 10.4236/opj.2020.106017 160 Optics and Photonics Journal typical region of zero intensity, and hence does not possess any obvious phase singularities [7]. This makes it difficult for studying vortex beam with low coherence. In 2003, Schouten et al. [8] examined two-point correlation function of partially coherent beams using the Young's interference experiment. It was shown that there exist phase singularities of the spectral degree of coherence of the field at pairs of points. Such phase singularities are termed as "coherence singularities". Further investigation showed that the phase of spectral degree of coherence possesses a vortex structure around these singular points [9]. The new term "coherence vortices" is used to refer to them. These findings reveal that the vortex structure of partially coherent vortex beam is just "hidden" in the correlation function. Since then, these new physical phenomena and concepts arise a lot of interest [10]- [15]. Many investigations revealed that there is an intimate relationship between optical vortices produced by a coherent field and the corresponding coherent vortices produced by a partially coherent beam [16]. As the coherence length decreases, the optical vortex can evolve into coherent vortex.
Recently, researchers group showed that the phase distribution of the cross spectral density of partially coherent vortex beams possesses rich information [17] [18]. It was shown that the number of coherent vortex is equal to the topological charge. It provides a way to measure the topological charge of partially coherent vortex beams. The results can also apply to other fields, such as information transmission and imaging.
On the other hand, partially coherent elegant Laguerre-Gaussian (eLG) beams, as the natural extension of partially coherent standard Laguerre-Gaussian (SLG) beams, represent a typical kind of partially coherent vortex beams [19]. They have been widely investigated. For example, the propagation properties of partially coherent ELG beams are less affected by the turbulence [20] [21], and the spreading of such beam is slower through the free space and the turbulent atmosphere. The coherence properties of partially coherent eLG beam are quite different from corresponding SLG beam [22]. As far as we know, the coherent vortex properties of partially coherent ELG beams have not been studied. In the paper, we have derived the analytical expression of the cross spectral density function of a partially coherent eLG beam propagating in free space. The effect of topological charge, radial mode index and coherence length on the coherent vortices have been analyzed.

Coherence Vortex of Partially Coherent eLG Beams
Consider the electric field distribution of an eLG beam at the source plane (z = 0) [19]: where ρ and θ are the radial and azimuthal coordinates, respectively, w 0 is the beam waist width of the fundamental Gaussian mode, l p L is the Laguerre poly- Based on the partial coherence theory, the cross spectral density of a partially coherent beam at the source plane can be written as [6]: where ⋅ denotes an ensemble average, is a position vector of a point in the source plane. Here we assume a partially coherent light source in the initial plane with field correlation properties described by a Gaussian-Schell correlator: where δ is the coherence length.
Substituting Equation (1) and Equation (3) into Equation (2), the cross spectral density of a partially coherent eLG beam at the source plane can be expressed as: ( ) In the framework of the paraxial approximation, the cross spectral density of such beam propagating in free space can be expressed [20]: is wavenumber; z is the distance in free space.
The spectral degree of coherence is defined as [9]: , x y can be expressed as: where Re, Im denote the real and imaginary parts, respectively.

Numerical Simulation
Based on equations above, we can investigate the phase distribution of the cross-spectral density of partially coherent elegant LG beams in free space. The

Conclusion
The analytical expression of the cross-spectral density function of partially coherent elegant Laguerre-Gaussian beams propagating in free space has been de-