The propagation properties of various laser beams in a turbulent medium have been widely investigated in past years, and the evolution properties will affect the applications of laser beams in wireless optical communication and sensing. The influences of turbulent medium on propagation properties of coherent laser beams and partially coherent laser beams are both studied. For examples, Cai et al. have reviewed the properties of partially coherent beams in turbulent atmosphere.^[1] Until now, the propagation properties of many partially coherent beams in a turbulent medium have been studied, such as partially coherent electromagnetic beams,^[2,3] off-axis Gaussian–Schell model beam and partially coherent laser array beam,^[4] apertured partially coherent beam,^[5] partially coherent rectangular flat-topped laser array,^[6] partially coherent Lorentz–Gauss beam,^[7] partially coherent elegant Hermite–Gaussian beam,^[8] partially coherent Laguerre–Gaussian beam,^[9] phase-locked partially coherent flat topped laser beam array,^[10] radial phased-locked partially coherent anomalous hollow beam array,^[11] Laguerre–Gaussian correlated Schell-model beam,^[12] partially coherent four-petal Gaussian vortex beam,^[13] radial phased-locked partially coherent flat-topped vortex beam array,^[14] multi-cosine-Laguerre–Gaussian correlated Schell-model beam,^[15] partially coherent Lorentz–Gauss vortex beam,^[16] partially coherent anomalous elliptical hollow Gaussian beam,^[17] partially coherent flat-topped vortex hollow beam,^[18] partially coherent crescent-like optical beam,^[19] partially coherent Lorentz–Gauss beam,^[20] radial phased-locked partially coherent Lorentz–Gauss array beam,^[21] truncated Gaussian–Schell model beam,^[22] partially coherent anomalous hollow vortex beam,^[23] partially coherent elegant Hermite–Cosh–Gaussiands beam,^[24] Hankel–Bessel beams,^[25] Laguerre–Gaussian beams,^[26] multi-Gaussian–Schell model beams carrying an edge dislocation,^[27] and rectangular multi-Gaussian–Schell model array beam.^[28] On the other hand, the studies of off-axis beam have attracted much attention.^[29–31] Recently, a new beam called off-axis hollow vortex Gaussian beam has been introduced in theory.^[32] From previous report,^[1] it can been found that the partially coherent beams have advantage over fully coherent beams in mitigating the effect of atmospheric turbulence. Hence, it is very interesting to investigate what will happen when the spatial coherence length is introduced into the off-axis hollow vortex Gaussian beam. In this paper, a beam called off-axis hollow Gaussian–Schell model vortex beam (HGSMVB) generated by the Gaussian–Schell model source is first introduced. Then, the cross-spectral density of off-axis HGSMVB propagating in turbulent atmosphere is derived. Finally, the influences of beam parameter and turbulent atmosphere on the intensity and coherence properties of off-axis HGSMVB are illustrated and analyzed by using numerical examples.

From previous work, the analytical expression of off-axis hollow vortex beam at the plane z = 0 can be described by^[32]

Consider the method of generating the partially coherent beams, then the cross-spectral density (CSD) of off-axis hollow Gaussian–Schell model vortex beam (HGSMVB) generated by anisotropic Gaussian–Schell model sources will be expressed as^[33]

Under the paraxial approximation, the CSD of off-axis HGSMVB propagating in turbulent atmosphere at the plane z can be written by the extended Huygens–Fresnel integral as follows:^[1–7]

Substituting Eq. (2) into Eq. (3), and considering the following equations:

The average intensity and spectral degree of coherence of off-axis HGSMVB propagating in turbulent atmosphere are illustrated and analyzed in this section. In the numerical analyses, the parameters are set to be w₀ = 1 cm, N = 2, M = 1, R_x = 1 mm, R_y = 0, λ = 800 nm, and = 10⁻¹⁴ m^{−2 / 3}.

To investigate the average intensity properties of off-axis HGSMVB propagating in different media, the intensity of off-axis HGSMVB with σ_x = σ_y = 2 cm propagating in turbulent atmosphere and free space are illustrated in Figs. 1 and 2, respectively. We can see that the dark hollow center of off-axis HGSMVB in turbulent atmosphere gradually disappears and the average intensity in the center increases as the propagation distance z increases (Figs. 1(a) and 1(b)); when the propagation distance increases to a longer distance z = 3000 m, the off-axis HGSMVB in turbulent atmosphere evolves into a Gaussian-like beam (Fig. 1(d)). While the off-axis HGSMVB propagating in free space also gradually loses the dark hollow center as the propagation distance z increases, but average intensity distribution of the off-axis HGSMVB in free space at the propagation distance z = 3000 m does not evolve into that of the Gaussian beam. Thus, the turbulent atmosphere will accelerate the evolution of off-axis HGSMVB into the Gaussian-like beam. The reason why the off-axis HGSMVB in turbulent atmosphere evolves into the Gaussian-like beam at the longer propagation distance is that the general beam propagating in turbulent atmosphere evolves into Gaussian-like beam.^[1,35]

Fig. 1.

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	Fig. 1. Intensity and contour graphs of off-axis HGSMVB propagating in turbulent atmosphere for z = 100 m (a), 300 m (b), 1000 m (c), and 3000 m (d).

Fig. 2.

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	Fig. 2. Intensity and contour graphs of off-axis HGSMVB propagating in free spac for z = 300 m (a) and 3000 m (b).

To investigate the effects of strength of turbulent atmosphere on evolution properties, the cross sections of off-axis HGSMVB propagation in the different turbulent atmospheres are illustrated in Fig. 3. One can see that the off-axis HGSMVB propagating in free space ( = 0) evolves into the Gaussian-like beam slower than the off-axis HGSMVB propagating in turbulent atmosphere, and the off-axis HGSMVB propagating in stronger turbulent atmosphere loses the dark hollow center faster, and the off-axis HGSMVB propagating in stronger turbulent atmosphere has a larger beam spot for the same propagation distance (Fig. 3(b)).

Fig. 3.

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	Fig. 3. Cross sections of off-axis HGSMVB propagating in turbulent atmosphere for different values of and z = 400 m (a) and 3000 m (b).

To investigate the effects of different values of beam parameters σ_x = σ_y = σ, N, M, and R = R_x on the spreading properties of off-axis HGSMVB, the cross sections of off-axis HGSMVB propagating in turbulent atmosphere for the different values of parameters σ, N, M, and R are shown in Figs. 4–7, respectively. In Fig. 4, it is shown that the off-axis HGSMVB with smaller coherence length σ will lose the dark hollow center and evolve into a beam with intensity in the center faster (Fig. 4(a)); as the propagation distance z increases to a larger distance z = 2000 m, the off-axis HGSMVBs with different values of coherence length σ can all evolve into the Gaussian-like beams, but the off-axis HGSMVB with larger coherence length σ has a smaller beam spot. The off-axis HGSMVB with smaller order N has a smaller hollow center (Fig. 5(a)), and the influence of order N on the intensity of off-axis HGSMVB will disappear at the longer propagation distance (Fig. 5(b)). The off-axis HGSMVB with smaller topological charge M propagating in turbulent atmosphere evolves into Gaussian-like beam faster, but which will have a smaller beam spot when the off-axis HGSMVBs with the different values of M all evolve into the Gaussian-like beams (Fig. 6(b)). As the R increases, the circular symmetry of off-axis HGSMVB is destroyed, and the influence of R on the far field distribution of average intensity of off-axis HGSMVB disappears as the off-axis HGSMVBs all evolve from dark hollow center beam into the Gaussian-like beam (Fig. 7(b)).

Fig. 4.

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	Fig. 4. Cross sections of off-axis HGSMVB propagating in turbulent atmosphere for different values of σx = σy and z = 400 m (a) and 2000 m (b).

Fig. 5.

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	Fig. 5. Cross sections of off-axis HGSMVB propagating in turbulent atmosphere for different values of N and z = 400 m (a) and 2000 m (b).

Fig. 6.

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	Fig. 6. Cross sections of off-axis HGSMVB propagating in turbulent atmosphere for different values of M and z = 400 m (a) and 4000 m (b).

Fig. 7.

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	Fig. 7. Cross sections of off-axis HGSMVB propagating in turbulent atmosphere for different values of R = Rx and z = 200 m (a) and 3000 m (b).

The spectral degrees of coherence of off-axis HGSMVB propagating in turbulent atmosphere for the different values of parameters M, N, σ, and R are illustrated in Figs. 8 and 9, the two points are set to be (x,0) and (0,0). As can be seen from Fig. 8, the spectral degree of coherence of off-axis HGSMVB propagating in turbulent atmosphere for the different values of M decreases as the distance between the two points (x,0) and (0,0) increases to the longer propagation distance z = 3000 m. The influences of beam parameters M, N, σ, and R on the spectral degrees of coherence of off-axis HGSMVB propagating in turbulent atmosphere at the longer propagation distance z = 3000 m are given in Fig. 9. One can see that the influences of the different values of parameters M, N, σ, and R on the spectral degree of coherence disappear as the distance between two points (x,0) and (0,0) increases to a certain distance. When the distance between two points (x,0) and (0,0) is smaller, the spectral degrees of coherence of off-axis HGSMVB with larger values of M, N or R decrease more rapidly than the off-axis HGSMVB with smaller values of M, N or R. While, the spectral degree of coherence of off-axis HGSMVB is not affected by the coherence length σ at the longer propagation distance z = 3000 m.

Fig. 8.

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	Fig. 8. Spectral degrees of coherence of off-axis HGSMVB propagating in turbulent atmosphere for different values of distance z and M = 1 (a) and 3 (b).

Fig. 9.

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	Fig. 9. Spectral degrees of coherence of off-axis HGSMVB propagating in turbulent atmosphere at propagation distance z = 3000 m for different values of parameters M, N, σ, and R.

In this paper, the off-axis HGSMVB generated by anisotropic Gaussian–Schell model source is defined by using analytical expressions. The CSD of off-axis HGSMVB propagating in turbulent atmosphere is derived. The evolution properties of average intensity and spectral degree of coherence are illustrated and discussed. As the propagation distance z increases, the off-axis HGSMVB propagating in turbulent atmosphere evolves from dark hollow beam into the Gaussian-like beam due to the effects of turbulent atmosphere and coherence length σ. The off-axis HGSMVB with smaller coherence length σ or the off-axis HGSMVB propagating in stronger turbulent atmosphere (larger ) evolves from dark hollow beam into the Gaussian-like beam with a larger beam spot more rapidly. The influences of parameters N and R on the distribution of average intensity gradually disappears as the beam evolves into the Gaussian-like beam. The obtained results also show that the spectral degrees of coherence for the different points (x,0) and (0,0) decrease with the distance between two points at the longer propagation distance, and the spectral degrees of coherence of off-axis HGSMVB with larger M, N or R decrease more rapidly than the off-axis HGSMVB with smaller M, N or R. The research results may be useful in the real optical communication and remote sensing in turbulent atmosphere.