Recently, much attention has been paid to the propagation properties of a laser beam in turbulent atmosphere [1]. It is found that the intensity and spreading of a laser beam are affected by atmospheric turbulence [2-8], and the laser beam with a vortex has been widely studied, due to its potential applications in free-space laser communication. In past years, Wang et al. studied the focusing properties of a Gaussian Schell-model vortex beam in experiments [9]. Zhou et al. studied the partially coherent hollow vortex Gaussian beam through a paraxial ABCD optical system in turbulent atmosphere [10]. Wang and Qian studied the spectral properties of a random electromagnetic partially coherent flat-topped vortex beam in turbulent atmosphere, based on the extended Huygens-Fresnel principle [11]. Gu studied the transverse position of an optical vortex upon propagation through atmospheric turbulence [12]. Zhou and Ru studied the angular momentum density of a linearly polarized Lorentz-Gauss vortex beam [13]. Huang et al. studied the intensity distributions and spectral degree of polarization of partially coherent electromagnetic hyperbolic-sine-Gaussian vortex beams through non-Kolmogorov turbulence using numerical examples [14]. Wu et al. developed an expression for the wandering of random electromagnetic Gaussian-Schell model beams propagating in atmospheric turbulence, and studied the properties of the beams [15]. Recently, a new dark hollow beam called the partially coherent flat-topped vortex hollow beam has been proposed, which has advantages over a flat-topped hollow beam, and which has potential applications in free-space wireless laser communication. However, to the best of our knowledge, the propagation properties of a partially coherent flat-topped vortex hollow beam in turbulent atmosphere have not been reported.

In this work, we first introduce the partially coherent flat-topped vortex hollow beam based on the theory of coherence, and then investigate the its propagation properties in turbulent atmosphere.

In the Cartesian coordinate system with the z-axis set as the axis of propagation, a circular or elliptical flat-topped vortex hollow beam in the source plane can be described as [16]

where N is the order of the elliptical flat-topped vortex hollow beam, M is the topological charge, w_x and w_y are the beam width in the x and y directions respectively, and denotes the binomial coefficient.

Based on the theory of coherence, a fully coherent flat-topped vortex hollow beam can be extended to a partially coherent flat-topped vortex hollow beam. The second-order correlation properties of an electromagnetic beam can be characterized by the cross-spectral density function introduced by Wolf [17],

where g(x₁-x₂, y₁-y₂) is the spectral degree of coherence, assumed to have a Gaussian profile, and

where σ is the transverse coherence length

Substituting Eq. (1) into Eq. (2), the partially coherent flat-topped vortex hollow beam can be written as

where r₁₀ = (x₁₀, y₁₀) and r₂₀ = (x₂₀, y₂₀) are the position vectors at the source plane z = 0.

According to the extended Huygens-Fresnel principle, the spectral density of a laser beam propagating through turbulent atmosphere can be expressed as follows [1-9]:

where k = 2π/λ is the wave number; ψ(x₀, y₀, x, y) is the solution to the Rytov method that represents the random part of the complex phase (the asterisk denoting complex conjugation), and r = (x, y) and r₀ = (x₀, y₀) are respectively the position vectors at the output plane z and the input plane z=0. The ensemble average in Eq. (5) can be expressed as [5]

where ρ₀ is the spherical-wave lateral coherence radius due to the turbulence, and

with is the constant of refraction index structure, which describes the turbulence strength.

Upon substituting Eq. (4) into Eq. (5), and recalling the integral formulas [18]

after tedious integral calculations, we can obtain

with

and

Eqs. (11)~(15) make up the main analytical expression for a partially coherent circular or elliptical flat-topped vortex beam propagating in turbulent atmosphere. Using the derived equations we can investigate the propagation and transformation of a partially coherent circular or elliptical flat-topped vortex hollow beam in turbulent atmosphere.

The degree of coherence of the laser beam is written as [19]

and the position of coherence vortices at the propagation L is expressed as [20]

where Re and Im are respectively the real and imaginary parts of µ(r₁, r₂, z).

In this section we study the propagation properties of a partially coherent flat-topped vortex hollow beam in turbulent atmosphere. In this work, the calculation parameters λ and w_x throughout the text are set to be λ = 1064 nm (Nd:YAG laser) and w_x = 20 mm.

Figures 1 and 2 show the normalized average intensity and corresponding contour graphs of, respectively, partially coherent circular and elliptical flat-topped vortex hollow beams propagating in turbulent atmosphere; the calculation parameters are , N=2, M =1, σ =10 mm and w_y = 40 mm (Fig. 2). As can be seen from Figs. 1 and 2, a partially coherent circular or elliptical flat-topped vortex hollow beam can keep its original intensity pattern over a short propagation distance (Figs. 1(a) and 2(a)), and with increasing propagation distance either beam loses its initial dark, hollow centre, and the flat-topped vortex dark hollow beam evolves into a flat-topped beam (Figs. 1(b), 1(c), 2(b), and 2(c)). The partially coherent circular and elliptical flat-topped vortex hollow beams eventually evolve respectively into circular and elliptical Gaussian beams in the far field, due to the influence of the coherence length.

Figure 3 shows the cross section (y=0) of normalized average intensity for a partially coherent circular flat-topped vortex hollow beam propagating in turbulent atmosphere for various values of the coherence length σ, with , N=2, and M=1. It can be seen from Fig. 3 that a partially coherent circular flat-topped vortex hollow beam spreads more rapidly than a fully coherent beam (σ =inf), with the initial coherence length σ decreasing during propagation;, and that a partially coherent beam with a small coherence length will evolve into a Gaussian beam faster than a beam with a large coherence length in the far field (Fig. 3(d)).

Figure 4 presents the cross section (y = 0) of normalized average intensity for a partially coherent circular flat-topped vortex hollow beam propagating in turbulent atmosphere for various values of , with σ =10 mm, N=2, and M=1. It can be seen that a beam propagating in turbulent atmosphere and free space can almost keep its initial dark hollow profile over a short distance (Figs. 4(a) and (b)), and with increasing propagation distance a partially coherent flat-topped vortex hollow beam loses its initial dark hollow profile faster with increasing structure constant in the far field (Fig. 4(d)).

Figure 5 depicts the cross section (y=0) of the normalized average intensity for a partially coherent circular flat-topped vortex hollow beam propagating in turbulent atmosphere for different values of M and N with σ =10 mm and . As can be seen, abeam with higher order M (Figs. 5(a) and (b)) loses its initial dark hollow center more slowly, while a beam propagating in turbulent atmosphere with different orders N has similar evolution properties with increasing propagation distance.

Figure 6 presents the curves for Re µ=0 and Im µ=0 for a partially coherent flat-topped vortex hollow beam propagating in turbulent atmosphere with , σ =10 mm, N=2, and M=1, and r₂ = (10 mm, 20 mm). From Fig. 6(a) it can be seen that the beam propagation at a distance of z = 10 m has a coherent vortex, and with increasing propagation distance the beam at a distance of z = 500 m has two coherent vortices. Thus a partially coherent flat-topped vortex hollow beam will experience a change in its number of coherent vortices with increasing propagation distance in turbulent atmosphere.

In this paper the partially coherent flat-topped vortex hollow beam is introduced, and then the propagation Eq. for a partially coherent flat-topped vortex hollow beam in turbulent atmosphere is derived. The average intensity of a beam propagating in turbulent atmosphere is examined using numerical examples. It is found that a partially coherent flat-topped vortex hollow beam will evolve into a Gaussian beam in the far field, and that a beam propagation in turbulent atmosphere with small coherence length or large structure constant will evolve into a Gaussian beam more rapidly. We also find that a beam with higher order M loses its initial dark, hollow center more slowly, while beams of different order N have similar evolution properties.