Investigation of the Propagation Characteristics of Double-Ring Perfect Vortex Beams in Atmospheric Turbulence

Abstract

Double-ring perfect vortex beams (DR–PVBs) have attracted increasing attention due to their unique characteristics of carrying independent information channels and exhibiting higher security and stability during propagation. In this study, we theoretically simulated and experimentally generated DR–PVBs with various topological charges. We investigated the propagation characteristics of these beams under von Karman spectrum turbulence conditions through numerical simulations based on multiple-phase screen methods. The effects of different inner and outer ring topological charges and varying turbulence intensities on the intensity distribution, beam spreading, and beam wander of DR–PVBs over different propagation distances were examined and compared with double-ring Gaussian vortex beams (DR–GVBs). The simulation results indicate that within a propagation range of 0–500 m, the effective radius of DR–PVBs with different topological charges remains essentially unchanged and stable. For propagation distances exceeding 1000 m, DR–PVBs exhibit superior beam wander characteristics compared to DR–GVBs. Additionally, two occurrences of self-focusing effects were observed during propagation, each enhancing beam stability and reducing the beam spreading and beam wander of the DR–PVBs. This study provides valuable insights for applications of DR–PVBs in optical communication, optical manipulation, and optical measurement.

1. Introduction

In 1992, Allen et al. discovered that Laguerre–Gaussian beams carry a helical phase exp(ilθ) [1], and each photon possesses an orbital angular momentum (OAM) [2] of magnitude $l\hslash$. Since then, these vortex beams with helical wavefront structures have attracted increasing attention from researchers due to their unique properties [3,4].

Common types of vortex beams include Bessel beams, Laguerre–Gaussian beams, Hermite–Laguerre–Gaussian beams, and Mathieu beams [5], among others. In 2013, Ostrovsky et al. [6] introduced the concept of the perfect vortex beam (PVB) and generated it using a spatial light modulator (SLM). PVBs are a novel type of vortex beam where the radius of the bright ring and the ring width are independent of the topological charge, meaning that PVBs with different topological charges have the same radius. PVBs have extensive applications in fields such as microparticle trapping and manipulation, quantum information encoding, biomedicine, and optical information transmission and communication [7,8,9,10].

However, current research on PVBs primarily focuses on the optimization and improvement of generation methods, with few reports on their propagation characteristics in free space. For vortex beams, the propagation characteristics in turbulent atmospheres are crucial for assessing their application value. Historically, the study of beam propagation characteristics in atmospheric turbulence is based on the Kolmogorov power spectrum of refractive index fluctuations, a widely accepted and used model [11,12,13]. However, due to the singularity at the origin in the Kolmogorov spectrum, which is non-integrable, the von Karman spectrum was introduced to facilitate research by avoiding this defect. Moreover, based on the Tatarskii spectrum, a modified von Karman spectrum [14] was developed, which is suitable for the entire turbulence regime.

Therefore, this study starts with a theoretical analysis, employing multiple numerical methods based on phase screens to simulate the propagation of double–ring perfect vortex beams (DR–PVBs) in atmospheric turbulence characterized by the von Karman spectrum. We investigated the propagation characteristics of DR–PVBs with various topological charges, including cases where the inner and outer ring topological charges are equal and unequal, under three different turbulence intensities. The study examines and discusses the intensity distribution, beam spreading, and beam wander characteristics of DR–PVBs in these conditions.

2. Fundamental Theory

2.1. Theoretical Model of Double–Ring Perfect Vortex Beams

The double–ring perfect vortex beam (DR–PVB) can be considered as the linear superposition of two Gaussian-approximated single-ring PVBs. Its expression is given a follows [15]:

$$E\left(r,\theta \right)=A_1\exp \left[-{\displaystyle \frac{{\left(r-w_1\right)}^2}{{\mathsf{\Delta }}_1^2}}\right]\exp \left(i{l}_1\theta \right)-A_2\exp \left[-{\displaystyle \frac{{\left(r-w_2\right)}^2}{{\mathsf{\Delta }}_2^2}}\right]\exp \left(i{l}_2\theta \right)$$

(1)

Here, $A_1$ and $A_2$ are the peak amplitudes of the two rings, $w_1$ and $w_2$ are the radii of the two rings, ${\mathsf{\Delta }}_1$ and ${\mathsf{\Delta }}_2$ are the half-widths of the two rings, and $l_1$ and $l_2$ are the topological charges of the outer and inner rings, respectively. The phase factors $\exp \left(i{l}_1\theta \right)$ and $\exp \left(i{l}_2\theta \right)$ indicate that the DR–PVB possesses a helical phase. The use of the minus sign in the superposition is intended to maintain the clarity of the dark ring.

2.2. Atmospheric Turbulence Model

When laser beams propagate through atmospheric turbulence, both the Kolmogorov spectrum and the Tatarskii spectrum functions exhibit non-integrable singularities at the origin, which contradict physical reality. To overcome this limitation, scholars introduced the Von Karman spectrum, which is applicable to the entire turbulence regime [16]:

$${\mathsf{\Phi }}_n\left(\kappa \right)={\displaystyle \frac{0.033C_n^2}{{\left({\kappa }^2+{\kappa }_0^2\right)}^{11/6}}}\exp \left(-{\kappa }^2/{\kappa }_m^2\right), 0\le \kappa \le \infty$$

(2)

Here, ${\kappa }_0=2\pi /L_0$, $L_0$ represents the turbulent outer scale; ${\kappa }_m=5.92/l_0$, $l_0$ represents the turbulent inner scale; $\kappa =\left({\kappa }_x,{\kappa }_y,{\kappa }_z\right)$ is the spatial frequency; and $C_n^2$ is the atmospheric structure constant of the refractive index.

2.3. Beam Spreading and Beam Wander Characteristics

Similar to conventional vortex beams, double–ring perfect vortex beams also experience beam spreading and beam wander when propagating through turbulent atmospheres. The beam spreading characteristics are typically described by the effective radius relative to the beam centroid, as defined by Equation (3) [17]:

$$W_{eff}=\sqrt{2}\sqrt{\iint r^{2}I\left(x,y\right)\mathrm{d}x\mathrm{d}y{\left[\iint I\left(x,y\right)\mathrm{d}x\mathrm{d}y\right]}^{-1}}$$

(3)

In the equation, $r^2$ represents the distance from the beam centroid, and $I\left(x,y\right)$ denotes the intensity at the beam centroid position.

Beam wander characteristics are typically described by the variations in the centroid position of the beam spot. The centroid of the beam spot is defined as follows [17]:

$${\rho }_c={\displaystyle \frac{\iint \rho I\left(\rho \right)\mathrm{d}\rho }{\iint I\left(\rho \right)\mathrm{d}\rho }}$$

(4)

The coordinates of the centroid, $x_c$ and $y_c$, are given as follows [17]:

$$x_c={\displaystyle \frac{\iint xI\left(x,y\right)\mathrm{d}x\mathrm{d}y}{\iint I\left(x,y\right)\mathrm{d}x\mathrm{d}y}}$$

(5)

$$y_c={\displaystyle \frac{\iint yI\left(x,y\right)\mathrm{d}x\mathrm{d}y}{\iint I\left(x,y\right)\mathrm{d}x\mathrm{d}y}}$$

(6)

Therefore, the variance of the beam spot centroid wander can be expressed as follows:

$${\sigma }_{\rho }^2=\langle {\rho }_c^2\rangle =\iint \iint \left({\rho }_1\cdot {\rho }_2\right)I\left({\rho }_1\right)I\left({\rho }_2\right)\mathrm{d}{\rho }_1\mathrm{d}{\rho }_2/{[\iint I\left(\rho \right)\mathrm{d}\rho ]}^2$$

(7)

If the variances of the beam spot centroid wander in the horizontal and vertical directions are ${\sigma }_x$ and ${\sigma }_y$ respectively. Assuming that the wander in the horizontal and vertical directions are statistically independent, the total variance of the beam spot centroid wander can be expressed as follows:

$${\sigma }_{\rho }^2={\sigma }_x^2+{\sigma }_y^2$$

(8)

If the beam propagates over a distance Z, the variance of the wander angle due to turbulence can be expressed as [18]:

$${\sigma }_{\theta \rho }^2={\displaystyle \frac{{\sigma }_{\rho }^2}{Z^2}}$$

(9)

3. Near-Field Experimental Generation

The experimental setup is shown in Figure 1. A spatial light modulator (SLM) is used to generate double–ring perfect vortex beams. A He-Ne laser with a wavelength of 632.8 nm emits a fundamental mode Gaussian beam, which is converted to a horizontally polarized beam after passing through a half-phase plate (HWP). A filter is used to adjust the intensity of the incident Gaussian beam. The fundamental mode Gaussian beam then passes through the SLM, which directly loads a spiral phase pattern. The light field modulated by the SLM forms the desired double–ring perfect vortex beams with different topological charges. Finally, the beams are captured by a CCD camera (CCD).

The DR–PVB is a vortex beam characterized by a beam radius that is independent of the topological charge and features two concentric bright rings. The simulated intensity distributions at the source plane for double-ring perfect vortex beams, under three conditions—equal topological charges for the inner and outer rings ($l_1=l_2=3$), inner ring topological charge less than that of the outer ring ($l_1=8,l_2=3$), and inner ring topological charge greater than that of the outer ring ($l_1=3,l_2=8$)—are shown in Figure 2(a1–a3). It is evident that the beam radius of the double–ring perfect vortex beams at the source plane is unaffected by changes in topological charge. Figure 2(c1–c3) show the near-field simulated intensity distributions at a propagation distance of Z = 30 cm. When the corresponding phase diagrams in Figure 2(b1–b3) are loaded onto the SLM and propagated over a distance of Z = 30 cm, the near-field experimental intensity distributions in Figure 2(d1–d3) are obtained. It can be observed that, regardless of whether the topological charges of the inner and outer rings are equal or not, the near-field experimental intensity distributions in Figure 2(d1–d3) are consistent with the near-field intensity simulation distributions in Figure 2(c1–c3). However, the experimental intensity patterns in Figure 2(d2,d3) clearly exhibit an interference pattern with spiral petals. This occurs because, when the topological charges of the inner and outer rings are unequal, the experimental optical setup effectively performs a coaxial interference-based topological charge detection for the inner and outer ring perfect vortex beams. The number of petals corresponds to the difference in topological charges, $\Delta l=l_1-l_2$ [19]. When $\Delta l>0$, the petals appear in the inner ring, and when $\Delta l<0$, the petals appear in the outer ring.

However, it is still necessary to validate the numerical simulation results through outfield experiments. Future research should aim to generate the required double-ring perfect vortex beams using the experimental setup shown in Figure 1. The inner and outer scales of atmospheric turbulence and the atmospheric coherence length should be measured using sounding balloons and atmospheric coherence length measuring instruments, respectively. Outfield propagation experiments should then be conducted at distances of 2000 m and 5000 m under conditions consistent with the numerical simulation data presented in this paper. Finally, these results should be compared with our simulation results to validate our theoretical model.

4. Numerical Simulation and Results Analysis

4.1. Selection of Simulation Parameters

Due to the stochastic nature of beam propagation in atmospheric turbulence, the beam spreading and beam wander characteristics of vortex beams require multiple simulations to obtain statistical results. In this study, each simulation was iterated 500 times. The parameters used in the numerical simulations are listed in

Table 1. Parameters used in numerical simulations.

Parameter	Data	Parameter	Data
Wavelength $\lambda /\mathrm{nm}$	632.8	Outer ring topological charge $l_1$	1–9
Number of samples	512 × 512	Inner ring topological charge $l_2$	1–9
Propagation distance $Z/\mathrm{m}$	0–2000	Radius of the outer ring $w_1/\mathrm{mm}$	80
Inner scale $l_0/\mathrm{mm}$	5	Radius of the inner ring $w_2/\mathrm{mm}$	30
Outer scale $L_0/\mathrm{m}$	5	Width of outer ring ${\Delta }_1/\mathrm{mm}$	8
Turbulence intensity $C_n^2/{\mathrm{m}}^{-2}$	$1\times {10}^{-14}$, $1\times {10}^{-15}$ $1\times {10}^{-16}$	Width of inner ring ${\Delta }_2/\mathrm{mm}$	4
Grid size of phase screen $/\mathrm{m}$	0.001	Maximum amplitude of outer ring $A_1$	0.75
		Maximum amplitude of inner ring $A_2$	1

.

4.2. Effects of Atmospheric Turbulence on Beam Intensity Distribution

Figure 3 and Figure 4 show the normalized intensity distributions of DR–PVB in the $x-y$ plane at varying propagation distances under three turbulence intensities, $C_n^2=1\times {10}^{-14}$, $C_n^2=1\times {10}^{-15}$, and $C_n^2=1\times {10}^{-16}$, when the topological charges of the inner and outer rings are equal, $l_1=l_2=5$. As seen in the figures, DR–PVB exhibits a noticeable self-focusing effect under all three turbulence intensities. Figure 3 clearly shows that when $C_n^2=1\times {10}^{-15}$, the DR–PVB experiences a significant reduction in the size of the inner ring due to the self-focusing effect after propagating approximately 500 m, while the size of the outer ring remains largely unchanged. Simultaneously, the intensity of both the inner and outer rings increases rather than decreases. Subsequently, as the propagation distance increases, the intensity of both rings begins to decrease, and the inner ring’s size increases. Even at a propagation distance of 2000 m, the DR–PVB maintains a clear double-ring beam profile.

When the topological charges of the inner and outer rings are unequal, as shown in Figure 5, where $l_1=3, l_2=8$ under turbulence intensity and $C_n^2=1\times {10}^{-15}$, it can be observed that the DR–PVB still exhibits a significant self-focusing effect for the inner ring during propagation through atmospheric turbulence. The self-focusing focal point is located at approximately Z = 500 m. Subsequently, as the propagation distance increases, the inner ring’s size gradually enlarges, and coaxial interference between the inner and outer rings occurs. Around Z = 1500 m, the interference pattern manifests as a petal-like shape, which is consistent with Figure 2(c3).

4.3. Effects of Atmospheric Turbulence on Beam Spreading

When the topological charges of the inner and outer rings are equal, assume that $l_1=l_2=L$. As shown in Figure 6, the horizontal axis represents the propagation distance Z, and the vertical axis represents the effective radius $W_{eff }$ of the beam obtained from numerical simulations. Figure 6a shows the trend of the effective radius of a double–ring Gaussian vortex beam (DR–GVB) under turbulence intensity $C_n^2=1\times {10}^{-14}$. Figure 6b–d depict the trends of the effective radius of a double-ring perfect vortex beam (DR–PVB) under three turbulence intensities, $C_n^2=1\times {10}^{-14}$, $C_n^2=1\times {10}^{-15}$, and $C_n^2=1\times {10}^{-16}$, respectively. The topological charges for Figure 6a–d are $L=3,4,5,6,7$, respectively. It can be observed that for the DR–PVB, the effective radius $W_{eff }$ remains almost unchanged before Z = 500 m and only begins to gradually increase after Z = 500 m. This indicates that the DR–PVB’s beam spreading is stable before Z = 500 m. Compared to the double-ring gaussian vortex beam, the DR–PVB’s beam radius is independent of the topological charge, and due to the self-focusing effect of the inner ring, changing the topological charge does not significantly affect the effective radius $W_{eff }$ before Z = 500 m for different $L$. After Z = 500 m, as the propagation distance increases, the effective radius $W_{eff }$ exhibits an exponential growth trend. At this stage, it is evident that the beam spreading of the DR–PVB increases with the increase in topological charge $L$. The main reason for these phenomena is that, during short-distance propagation in atmospheric turbulence, beam spreading is primarily influenced by free-space diffraction effects; as the propagation distance increases, the cumulative effect of turbulence becomes dominant. Therefore, after Z = 500 m, $W_{eff }$ shows an exponential growth trend with the increase in propagation distance.

When the topological charges of the inner and outer rings are different, as shown in Figure 7, Figure 7a,b depict the trend of the effective radius of the DR–PVB beam spot under turbulence intensities $C_n^2=1\times {10}^{-14}$ and $C_n^2=1\times {10}^{-15}$ with the $l_1$ being greater than the $l_2$. Figure 7c,d illustrate the trend of the effective radius of the DR–PVB beam spot under turbulence intensities $C_n^2=1\times {10}^{-14}$ and $C_n^2=1\times {10}^{-15}$ with the $l_1$ being less than the $l_2$. As can be seen from the figures, the beam spreading characteristics of the DR–PVB with unequal inner and outer ring topological charges under various turbulence conditions are consistent with those of the DR–PVB with equal inner and outer ring topological charges described above.

4.4. Effects of Atmospheric Turbulence on Beam Wander

Figure 8 shows the relationship between the variance of the wander angle ${\sigma }_{\theta \rho }^2$ and the propagation distance when the topological charges of the inner and outer rings are equal, assuming $l_1=l_2=L$. The selected turbulence intensity is $C_n^2=1\times {10}^{-14}$. In Figure 8a, the selected beam is DR–GVB, while in Figure 8b, it is DR–PVB, with both beams having topological charges of $L=3,4,5,6,7$. It can be observed that under a constant turbulence intensity, the variance ${\sigma }_{\theta \rho }^2$ of DR–PVB exponentially increases with propagation distance, consistent with the trend observed for DR–GVB. Figure 8c,d show that when both the turbulence intensity and topological charge $L$ remain constant, the variances ${\sigma }_{\theta \rho }^2$ of the wander angle for DR–PVB and DR–GVB are identical before Z = 1000 m. However, beyond 1000 m, the variance ${\sigma }_{\theta \rho }^2$ for DR–PVB is lower than that for DR–GVB, indicating that when the propagation distance exceeds 1000 m, the wander angle variance of DR–PVB is smaller, and the beam is more stable.

When the turbulence intensities are between $C_n^2=1\times {10}^{-15}$ and $C_n^2=1\times {10}^{-16}$ and the topological charges of the inner and outer rings of the DR–PVB are equal, as shown in Figure 9a,c, the variance of the wander angle ${\sigma }_{\theta \rho }^2$ remains nearly constant before Z = 1000 m, and then exhibits an exponential growth trend beyond Z = 1000 m, and shows a tendency to decrease near 2000 m. Taking the DR–PVB with a topological charge $l_1=l_2=5$ as an example, as shown in Figure 9b,d, ${\sigma }_{\theta \rho }^2$ initially remains stable, then increases exponentially, and finally decreases as the propagation distance increases. This is because, at turbulence intensities $C_n^2=1\times {10}^{-15}$ and $C_n^2=1\times {10}^{-16}$, the DR–PVB undergoes a second self-focusing effect around a propagation distance of 2000 m, as shown in Figure 10. Figure 10a,b illustrate the normalized intensity distributions in the $x-y$ plane at a propagation distance of 5000 m for DR–PVBs with equal topological charges $l_1=l_2=5$ under turbulence intensities $C_n^2=1\times {10}^{-15}$ and $C_n^2=1\times {10}^{-16}$. It is evident that the DR–PVB experiences two self-focusing effects at propagation distances Z = 500 m and Z = 2000 m. Therefore, at turbulence intensities $C_n^2=1\times {10}^{-15}$ and $C_n^2=1\times {10}^{-16}$, the variance of the wander angle ${\sigma }_{\theta \rho }^2$ is reduced by the self-focusing effect of the DR–PVB, which weakens the beam wander and enhances beam stability when the topological charges of the inner and outer rings are equal.

5. Conclusions

In this study, we conducted numerical simulations using multiple phase screen methods to investigate the propagation characteristics of double–ring perfect vortex beams (DR–PVBs) under von Karman turbulence with three different turbulence intensities and various conditions of equal and unequal topological charges for the inner and outer rings. The simulation analysis revealed that DR–PVBs experience two self-focusing effects during propagation. After the first self-focusing effect, DR–PVBs with unequal topological charges for the inner and outer rings exhibit coaxial interference. Before the first self-focusing effect occurs (at Z = 500 m), DR–PVBs exhibit negligible beam spreading and beam wander. Within the 0–500 m propagation distance, their beam spreading is smaller and more stable compared to double–ring gaussian vortex beams (DR–GVBs). Under constant turbulence intensity and topological charge, the variance of the wander angle for DR–PVBs remains smaller than that for DR–GVBs, resulting in a more stable beam. When DR–PVBs propagate to approximately 2000 m, a second self-focusing effect occurs, reducing beam wander at 2000 m. These findings fill a gap in the research on the turbulence propagation characteristics of DR–PVBs and provide valuable insights for analyzing the applications of DR–PVBs in free-space optical communication processing.