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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.

Vortex beam has been widely studied due to its special properties [

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 [

Consider the electric field distribution of an eLG beam at the source plane (z = 0) [

E ( ρ , θ ; 0 ) = ( ρ w 0 ) | l | L p | l | ( ρ 2 w 0 2 ) exp ( − ρ 2 w 0 2 ) exp ( − i l θ ) (1)

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 polynomial with the radial mode index p and topological charge |l|.

Based on the partial coherence theory, the cross spectral density of a partially coherent beam at the source plane can be written as [

W ( ρ 1 , ρ 2 , 0 ) = 〈 E ∗ ( ρ 1 , 0 ) E ( ρ 2 , 0 ) 〉 (2)

where 〈 ⋅ 〉 denotes an ensemble average, ρ i = ( ρ i , θ i ) 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:

C ( ρ 1 , ρ 2 , 0 ) = exp ( − ( ρ 1 − ρ 2 ) 2 δ 2 ) (3)

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:

W ( ρ 1 , ρ 2 , 0 ) = ( ρ 1 ρ 2 w 0 2 ) | l | L p l ( ρ 1 2 w 0 2 ) L p l ( ρ 2 2 w 0 2 ) exp ( − ρ 1 2 + ρ 2 2 w 0 2 ) exp ( i l ( θ 1 − θ 2 ) ) exp ( − ( ρ 1 − ρ 2 ) 2 δ 2 ) (4)

In the framework of the paraxial approximation, the cross spectral density of such beam propagating in free space can be expressed [

W ( ρ 1 , ρ 2 , z ) = ( k 2 π z ) 2 ∫ ∫ ∞ W ( ρ 10 , ρ 20 , 0 ) × exp { − i k 2 z [ ( ρ 1 − ρ 10 ) 2 − ( ρ 1 − ρ 20 ) 2 ] } d 2 ρ 10 d 2 ρ 20 (5)

where k = 2 π / λ is wavenumber; z is the distance in free space.

The spectral degree of coherence is defined as [

μ ( ρ 1 , ρ 2 ) = W ( ρ 1 , ρ 2 ) S ( ρ 1 ) S ( ρ 2 ) (6)

where S ( ρ ) = W ( ρ , ρ ) is the spectral density. Since the spectral density is non-zero for the partially coherent beam, the position of coherence vortices at pairs of points ( x 1 , y 1 ) and ( x 2 , y 2 ) can be expressed as:

Re [ W ( ρ 1 , ρ 2 ) ] = 0 (7)

Im [ W ( ρ 1 , ρ 2 ) ] = 0 (8)

where Re, Im denote the real and imaginary parts, respectively.

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 parameters are set as: λ = 532 nm , w 0 = 1 mm , δ = 1 mm , r 2 = ( 0.5 w 0 , 0 ) , R = π w 0 2 / λ is Rayleigh distance.

represents the value of radial mode index p. With the propagation distance increases, the circular dislocation becomes not clearly. When the beam arrives at the Rayleigh distance, the information of circular dislocation disappears. However, the vortex structure and phase singularity still exist. We can conclude that the radial mode index doesn’t affect the coherent vortices.

a beam with radial index p = 1 in the Rayleigh distance. The other parameters are same as those in

The analytical expression of the cross-spectral density function of partially coherent elegant Laguerre-Gaussian beams propagating in free space has been derived. According to the definition of coherent vortices, the coherent vortices properties of such a beam propagating in free space are investigated. The results show that the distance and the coherence length affect the propagation properties of the coherent vortex. When the propagation distance increases or the coherence length decreases, the coherent vortices of higher order (lth) split to lth first order of coherent vortex, then the corresponding coherent vortex of opposite sign appears. The pairs of coherent vortices form. During propagation, the radial mode index disappears and it doesn’t affect the coherent vortices properties. These finding can help to explore such beams in potential application in optical communication.

The authors declare no conflicts of interest regarding the publication of this paper.

Dong, M. and Yang, Y.J. (2020) Coherent Vortices Properties of Partially Coherent Elegant Laguerre-Gaussian Beams in the Free Space. Optics and Photonics Journal, 10, 159-166. https://doi.org/10.4236/opj.2020.106017