## Abstract

We numerically investigate the scintillation index of an elliptical vortex beam in both modest turbulence and strong turbulence. Numerical simulations are realized with random phase screen scheme. It is shown that the on-axis scintillation index can be effectively reduced by an elliptical vortex beam if crucial parameters are properly chosen. The mechanisms of scintillation reduction in turbulence of different strengths are different. We find that the topological charge and the ratio of minor axis to major axis of an elliptical vortex beam are important in determining the on-axis scintillation index. Our simulation results indicate that using an elliptical vortex beam is a promising strategy to alleviate atmospheric influence on free space optical communication link.

©2011 Optical Society of America

## 1. Introduction

Much attention has been paid to propagation of laser beams through turbulent atmosphere because of applications in many areas, such as free space optical communications, remote sensing, imaging systems, laser radar, etc. It has been found that atmospheric turbulence strongly influences beam characteristics, including intensity, degree of coherence, degree of polarization and beam spreading etc [1–3]. Scintillations of a light beams are important since scintillations degrade ratio of signal to noise and may increase bit error rate. Beam scintillations are closely related to beam properties, like beam shape, phase, polarization, coherence etc. It is discovered that the on-axis scintillation index of an elliptical Gaussian beam or a dark hollow beam can be smaller than that of a circular Gaussian beam in weak turbulences under certain conditions [4, 5]. Decrease of the degree of source coherence may cause scintillations to be suppressed significantly, which means that partially coherent beams are less affected by turbulence than coherent beams [6]. An appropriately chosen coherent beam of non-uniform polarization also has smaller scintillation index than a beam of uniform polarization [7]. Another way to reduce scintillation is substituting beam arrays for a single incident beam. Scintillations can be effectively suppressed by adjusting the spatial separation of beams [8].

Beams with spiral phase are known as vortex beams, each photon carrying orbital angular momentum (OAM) [9]. OAM can be used to encode data for transmitting information in free-space optical systems [10, 11]. It has been demonstrated that the OAM of a beam can be well preserved in a long propagation distance in weak turbulence [12]. The scintillations of circular vortex beams in turbulence have also been studied [13, 14]. In this paper, we simulated the propagation of elliptical vortex beams in turbulent atmosphere, and studied how elliptical vortex beams behave in turbulences of different strengths and how beam parameters influence scintillations. Random phase screen scheme is used to simulate random medium [15, 16]. Analyzing the data of numerical experiments, we find that utilizing elliptical vortex beams is a new way to reduce scintillation index.

## 2. Numerical simulation method

We assume that there are many thin random phase screens which change the phase of the field in the beam propagation path and the distance between two adjacent screens is$\Delta z=50m$. A beam passes through a thin phase screen and then propagates for a distance Δz in free space. Then the beam reaches the next phase screen and again propagates for a distance Δz. The same process would be repeated until the beam reaches the receiver plane. The change of field due to the p-th phase screen is described as

where $E(x,y,p\Delta {z}_{-})$ and$E(x,y,p\Delta {z}_{+})$are the field before and after the phase screen, respectively. ${\phi}_{r}(x,y)$ is the random phase induced by thin phase screen of negligible width.The electric field is uniformly sampled at *N × N* points in the x-y plane and is described by an *N × N* matrix. The sampling interval is $\Delta x=\Delta y=d=0.5mm$ and the sampled area is $-Nd/2<x<Nd/2,-Nd/2<y<Nd/2.$ We define the matrix dimension by keeping a large enough distance between the simulation dimension edge and an intensity contour of the beam to relieve the aliasing effect. The contour is chosen to enclose more than 95% of the total energy. Since the matrix dimension will expand as the width of a propagating beam increases, *N* always increases after propagating between two adjacent phase screens. The newly-added elements of the matrix are assigned to zero.

We will elaborate how to generate a random phase screen. Von Karman spectrum is used as the refractive-index spectrum [17]. It is expressed as

*L*is 10m and the inner scale

_{0}*l*is 2mm.The spectrum of random phase can be expressed as [15]

_{0}*i*b is generated, a and b obeying normal distribution

*N*(0,1). Multiplying the gained matrix by${\Delta}_{\kappa}^{-1}\sqrt{{\Phi}_{\phi}({\kappa}_{x},{\kappa}_{y},{\kappa}_{z})}$ and then inversely Fourier transforming the result, we get a matrix of complex numbers ${\theta}_{1}+i{\theta}_{2}$whose real part is used as random phase screen in our simulation. ${\Delta}_{\kappa}^{-1}=2\pi /(N\Delta )$is the spatial frequency increment.

Beam propagation between two phase screens can be modeled with angular spectrum method. Given the field $E(x,y,p\Delta {z}_{+})$ right after the p-th phase screen, we first calculate $A({k}_{x}^{},{k}_{y}^{},p\Delta {z}_{+})$which is the 2D Fourier transform of$E(x,y,p\Delta {z}_{+})$ in the x-y plane. Multiplying $A({k}_{x}^{},{k}_{y}^{},p\Delta {z}_{+})$ by$\mathrm{exp}(-i\sqrt{{k}^{2}-{k}_{x}^{2}-{k}_{y}^{2}}\Delta z),$ we get $A[{k}_{x}^{},{k}_{y}^{},(p+1)\Delta {z}_{-}]$ which is the 2D Fourier transform of $E[x,y,(p+1)\Delta {z}_{-}]$.After inversely transforming the result we obtain $E[x,y,(p+1)\Delta {z}_{-}]$.

An elliptical vortex beam can be defined as [18]

*m*indicates the topological charge of an optical vortex. If $\alpha =1$, the beam is a circular vortex beam; and is an elliptical vortex beam if $\alpha $ is less than one. The scintillation index is defined as [17]where the angular brackets denote averaging over the ensemble of turbulent media. To calculate the scintillation index, 100 propagation realizations are calculated with the method described above to take the average of

*I*and

*I*.

^{2}## 3. Performances of elliptical vortex beams with different topological charges

#### 3.1 Scintillation in modest turbulence

Since source size has an influence on the fluctuation of intensity, investigation concerning topological charge would make sense only if source size is fixed. Note that when altering the parameter $\alpha $ from one to a number smaller than one to obtain an elliptical vortex beam, the source size changes if *w* is not adjusted. This is because one axis keeps the original length while the perpendicular axis is stretched, rendering the area of the source larger. Therefore, the value of *w* should be multiplied by $\sqrt{\alpha}$ in order to keep the source size unchanged.

In the simulation, ${C}_{n}^{2}$ is chosen to be$2.5\times {10}^{-14}{m}^{-2/3}.$ In Fig. 1 , beam 1 is a Gaussian beam and beam 2 is a circular vortex beam. The scintillation indices of beam 2, beam 3 and beam 5 are large because of their hollow cores. The indices of the three beams start to decrease at the distance of around 2000m to 3000m, meaning that auto-compensation effect takes place [19]. However, this effect is not strong enough to significantly suppress intensity fluctuations. As a more direct demonstration, Fig. 2 shows that the on-axis intensities of beam 2, beam 3 and beam 5, which have odd topological charge, are zero in free space. This characteristic leads to high scintillation index when auto-compensation effect is not evident enough in modest turbulence. In contrast, the scintillation indices of beam 4 and beam 6 are the two smallest. Obviously, this is not the result of auto-compensation but because of the nonzero intensity along the z-axis, which is shown in Fig. 2(c) and Fig. 2(e). To make the on-axis intensity nonzero, topological charge must be an even number.

#### 3.2 Scintillation in strong turbulence

Then beam propagation in strong turbulent atmosphere where ${C}_{n}^{2}$ is $8.0\times {10}^{-13}{m}^{-2/3}$ is investigated and there are phenomena different from those in modest turbulence. Figure 3 is intended to show that the on-axis scintillation indices of both the circular vortex beam and the elliptical vortex beams are smaller than that of the Gaussian beam at long distance because of strong auto-compensation in strong turbulence. Compared with modest turbulence situation in Fig. 4(a) , there are evident increases of on-axis intensity of certain beams in Fig. 4(b) due to scattering and beam wandering in strong turbulence. For example, the intensities of beam 2 and beam 3 are smaller than those of beam 4 and beam 6 in Fig. 4(a). But the intensities of beam 2 and beam 3 become larger than those of beam 4 and beam 6 in Fig. 4(b). Because of this effect, whether the core of a vortex beam is originally hollow or not, as well as whether the topological charge is even or odd, does not dominantly influence scintillation index any more. Thus curves of beam 3-6 in Fig. 3 overlap each other, which form a sharp contrast to Fig. 1. Note that the overall intensities of each beam at the source plane are normalized to one unit and the intensities in Fig. 4 are calculated in a 0.5mm$\times $0.5mm square on the z-axis.

## 4. Dependence of scintillation index on the ratio of minor axis to major axis

Then attention is focused on parameter $\alpha $ which also strongly impacts scintillation index. In the simulation, the source size is kept fixed in the way described in section 3.1. Figure 5 shows the scintillation indices of elliptical vortex beams with the same topological charge but different$\alpha $. In both figures of Fig. 5, the scintillation index of beam 2 is even larger than that of the Gaussian beam, when $\alpha $ is not small enough. Only when $\alpha $ is smaller than a certain value, the scintillation index of an elliptical vortex beam is smaller than that of the Gaussian beam. It is easily observed from both Fig. 5(a) and Fig. 5(b) that the smaller $\alpha $ is, the smaller the scintillation index is. Figure 6 illustrates how $\alpha $ changes beam pattern. Smaller $\alpha $ results in larger lobe at the core which reduces on-axis scintillation index. This is because larger lobe is less influenced by intensity fluctuations. However, when $\alpha $ is already very small, scintillation index will not change obviously even if $\alpha $ continues decreasing. There is only slight difference between the scintillation indices of beam 5 and beam 6.

## 5. Conclusion

In this paper, we investigated the properties of elliptical vortex beams in turbulent atmosphere. It is demonstrated that elliptical vortex beams can suppress scintillations. Since the on-axis intensity of an elliptical vortex beam with even topological charge is nonzero, it fluctuates less heavily than that of a Gaussian beam in modest turbulence. However, an elliptical vortex beam with odd topological charge has high on-axis scintillation index due to its hollow core. Since beam wandering is weak in modest turbulence, auto-compensation is not strong enough to significantly reduce the scintillation index of an elliptical vortex beam with odd topological charge. So the scintillation index of an elliptical vortex beam with odd topological charge may be larger than that of a Gaussian beam. However, when turbulence is strong, whether the topological charge of an elliptical vortex beam is even or odd does not cause significant difference in scintillations because strong auto-compensation and scattering fills original hollow core with considerable intensity. The scintillation index of an elliptical vortex beam is smaller than that of a Gaussian beam at long distance. In addition, smaller ratio of minor axis to major axis results in smaller scintillation index. If this ratio is too large, the scintillation index of an elliptical vortex beam is even larger than that of a Gaussian beam. These properties suggest potential application of elliptical vortex beams as a space communication channel to improve laser communication performance.

## Acknowledgments

This research is supported by National Natural Science Foundation of China (Grant Nos. 60977068 and 61178015) and Open Research Fund of State Key Laboratory of Atmospheric Composition and Optical Radiation.

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