Optical Communication Performance of Cylindrical Vector Partially Coherent Laguerre–Gaussian Beams in Atmospheric Turbulence

Abstract

The optical communication performance of cylindrical vector partially coherent Laguerre–Gaussian (PCLG) beams in different atmospheric turbulence models are investigated. Based on the unified theory of coherence and polarization and turbulence theory, analytical formulas for the signal-to-noise ratio (SNR), crosstalk equivalent intensity and bit error rate (BER) of cylindrical vector PCLG beams are derived in Kolmogorov turbulence, non-Kolmogorov turbulence and strong turbulence, respectively. Numerical analyses indicate that selecting a smaller azimuthal index l₀ or a larger radial index p₀ of beams can effectively enhance the SNR. In addition, selecting appropriate beam width, coherence length, wavelength of the beam, propagation distance and receiving aperture diameter enables the acquisition of the optimal signal detection position. Our results are conducive to the application of cylindrical vector PCLG beams in FSO communication.

1. Introduction

The application of structured beams in free-space optical (FSO) communication systems is currently one of the most prevalent research fields in the wireless communication domain, as structured beams can mitigate the adverse effects of turbulence on FSO communication performance [1,2,3,4,5]. Structured beams refer to beams with special spatial structures, such as specific polarization states, phase distributions or intensity distributions. Their unique physical properties endow them with greater advantages in light–matter interaction and information transmission than traditional Gaussian beams. Recent advances in structured beams, such as a programmable spintronic emitter capable of flexibly generating broadband terahertz beams with diverse structured polarization states [6] and optical control of unidirectional unpolarized luminescence using orbital angular momentum (OAM) of photons [7], have further highlighted the potential of structured beams in optical communication. The Laguerre–Gaussian (LG) beam carrying OAM is a typical phase structured beam, which has a distinctive spiral phase distribution [8,9]. Their different OAM modes are mutually orthogonal, and multiplexing LG beams with distinct topological charge numbers can significantly improve the spectral efficiency and capacity of communication systems [10,11]. In addition, the radial indices of LG beams can also be applied to multiplexing, and the encoding system based on high-order radial indices of LG beams are investigated [12]. A 200-Gbit/s free space optical link is experimentally demonstrated by multiplexing two LG beams with different radial indices [13]. However, the OAM modes are highly vulnerable to turbulence interference, which causes phase distortion and mode crosstalk [14,15,16,17].

Light beams exhibit not only phase characteristics but also coherence and polarization properties. Partially coherent beams with low spatial coherence have many advantages, including strong anti-turbulence ability [18], speckle noise suppression [19], enhanced self-healing ability [20] and the ability to break through diffraction limits [21], making them widely used in optical communication, optical imaging and optical capture. Compared with fully coherence beams, partially coherent beams have evidently mitigated intensity scintillation and phase distortion when propagating through turbulence, resulting in a notably lower bit error rate (BER) [22]. For instance, twisted Gaussian Schell-model beams show greatly reduced intensity scintillation and beam wander during propagation in atmospheric turbulence [23], and non-uniformly correlated flat-topped beams have an advantage in improving the mean signal-to-noise ratio (SNR) and reducing the mean BER [24]. Moreover, partially polarized beams and non-uniform polarized beams outperform fully polarized beams in overcoming the negative impact caused by turbulence [25,26,27]. Therefore, partially coherent polarized beams have a positive effect when transmitted in turbulence and have great potential for application in FSO communication.

Cylindrical vector partially coherent Laguerre–Gaussian (PCLG) beams combine cylindrical symmetric vector polarization distribution, spiral phase OAM, partial coherence with statistical fluctuations, robust transmission behavior and a unique field distribution under tight focusing [28,29], which make them promising for application in laser communication, particle manipulation, high-precision processing and biological imaging [30,31]. The statistical properties of cylindrical vector PCLG beams in atmospheric turbulence have been extensively studied [32,33,34]. However, the optical communication performance of cylindrical vector PCLG beams requires further exploration.

To promote the future application of cylindrical vector PCLG beams in optical communication, we conduct theoretical analysis and numerical simulations to investigate their optical communication performance in Kolmogorov turbulence, non-Kolmogorov turbulence and strong turbulence medium. The results indicate that the radial and azimuthal indices, beam width, wavelength of cylindrical vector PCLG beam, receiving aperture diameter, propagation distance and turbulence intensity have an impact on communication performance, such as SNR or crosstalk equivalent intensity and BER.

2. Theory

According to the unified theory of coherence and polarization, the cross-spectral density (CSD) matrix in the spatial frequency domain can be used to characterize the statistical properties of a vector partially coherent beam propagating along the z-axis in atmospheric turbulence, and its CSD matrix in the source plane can be expressed as follows [35]:

$$\widehat{W}({\mathit{r}}_1,{\mathit{r}}_2,z)=\left(\begin{array}{cc}{W}_{xx}({\mathit{r}}_1,{\mathit{r}}_2,z)& W_{xy}({\mathit{r}}_1,{\mathit{r}}_2,z)\\ W_{yx}({\mathit{r}}_1,{\mathit{r}}_2,z)& W_{yy}({\mathit{r}}_1,{\mathit{r}}_2,z)\end{array}\right),$$

(1)

where

$$W_{\alpha \beta }({\mathit{r}}_1,{\mathit{r}}_2,z)=〈E_{\alpha }\left({\mathit{r}}_1,{\mathit{r}}_2,z\right)E_{\beta }^{*}\left({\mathit{r}}_1,{\mathit{r}}_2,z\right)〉, \left(\alpha ,\beta =x,y\right),$$

(2)

where r₁ = (r₁, φ₁) and r₂ = (r₂, φ₂) are the polar coordinates of two different positions on the cross-section of the z-plane, E_α(r₁, r₂, z) and E_β(r₁, r₂, z) are the electric field vectors in the α and β directions, ⟨ ⟩ represents the ensemble average and * denotes the complex conjugate. Assuming that the cylindrical vector PCLG beam derives from a Schell-model source, the corresponding CSD matrix elements have the following form [28,36]:

$$W_{\alpha \beta }({\mathit{r}}_1,{\mathit{r}}_2,z)=〈{E_{\alpha }}^{*}({\mathit{r}}_1,z)E_{\beta }({\mathit{r}}_2,z)〉g_{\alpha \beta }\left({\mathit{r}}_1-{\mathit{r}}_2\right), \left(\alpha ,\beta =x,y\right),$$

(3)

where

$$g_{\alpha \beta }\left({\mathit{r}}_1-{\mathit{r}}_2\right)=B_{\alpha \beta }\exp \left[-{\displaystyle \frac{{\left|{\mathit{r}}_1-{\mathit{r}}_2\right|}^2}{2{\sigma }_{\alpha \beta }^2}}\right], (\alpha ,\beta =x,y),$$

(4)

where g_αβ is the spectral degree and satisfies Gaussian distribution, which characterizes the coherence of the light field, σ_αβ denotes the coherence length with σ_xy = σ_xx = σ_yy = σ₀, B_αβ is the complex correlation coefficient between E_α and E_β field components and B_αβ = B^*_βα.

Then, we consider the electric field of a cylindrical vector LG beam propagating through atmospheric turbulence. The influence of turbulence on the beam can be regarded as a pure phase perturbation [37,38]. Thus, the electric field of a cylindrical vector LG beam at the z plane in the turbulence can be given as follows:

$$E(\mathit{r},\phi ,z)=E_{l_0,p_0}(\mathit{r},\phi ,z)\exp \left[{\psi }_1\left(\mathit{r},\phi ,z\right)\right],$$

(5)

where ψ₁(r, φ, z) is the random complex phase perturbation induced by the turbulence, p₀ and l₀ are radial and azimuthal indices of the cylindrical vector LG beam at the source plane, respectively, and E_l0,p0(r, φ, z) is the electric field of the cylindrical vector LG beam at the z plane without turbulence, which can be represented by the Jones matrix method as follows [39,40]:

$$E_{l_0,p_0}(\mathit{r},\phi ,z)=R\left(\mathit{r},z\right)\left(\begin{array}{c}\cos \left(l_0\phi \right)\\ \sin \left(l_0\phi \right)\end{array}\right),$$

(6)

where

$$\begin{array}{ll}R\left(\mathit{r},z\right)=& {\displaystyle \frac{1}{w\left(z\right)}}\sqrt{{\displaystyle \frac{2p_0!}{\pi \left(p_0+\left|l_0\right|\right)!}}}{\left({\displaystyle \frac{\sqrt{2}\mathit{r}}{w\left(z\right)}}\right)}^{l_0}{L}_{p_0}^{l_0}\left[{\displaystyle \frac{2{\mathit{r}}^2}{w^2\left(z\right)}}\right]\\ & \times \exp \left\{{\displaystyle \frac{-{\mathit{r}}^2}{w^2\left(z\right)}}-{\displaystyle \frac{ik{\mathit{r}}^{2}z}{2\left(z^2+z_R^2\right)}}+i\left(2p_0+\left|l_0\right|+1\right){\tan }^{-1}\left({\displaystyle \frac{z}{z_R}}\right)\right\},\end{array}$$

(7)

$$w\left(z\right)=w_0\sqrt{1+{\left(z/z_R\right)}^2},$$

(8)

$$Z_R=k{w}_0^2/2,$$

(9)

where w(z) is the beam radius at distance z and Z_R is the Ray range. k is the wave number and k = 2π/λ, λ is the wave length. $L_{p_0}^{l_0}\left[\right]$ is the Laguerre-polynomial, where l₀ and p₀ represent the azimuthal index and the radial index. Since cylindrical polarized beams can be decomposed into left-handed and right-handed polarized beams with opposite spiral phases, the electric fields of Equation (6) in the x and y direction are written as follows [41]:

$$\begin{array}{l}{E}_{x(l_0,p_0)}(\mathit{r},\phi ,z)={\displaystyle \frac{1}{2\sqrt{2\pi }}}R\left(\mathit{r},z\right)\left[\exp \left(i{l}_0\phi \right)+\exp \left(-i{l}_0\phi \right)\right],\\ E_{y(l_0,p_0)}(\mathit{r},\phi ,z)={\displaystyle \frac{i}{2\sqrt{2\pi }}}R\left(\mathit{r},z\right)\left[-\exp \left(i{l}_0\phi \right)+\exp \left(-i{l}_0\phi \right)\right].\end{array}$$

(10)

Substituting Equations (5) and (10) into Equation (3), the CSD matrix elements of the cylindrical vector PCLG beam in turbulent media are obtained:

$$\begin{array}{ll}{W}_{\alpha \beta }({\mathit{r}}_1,& {\mathit{r}}_2,{\phi }_1,{\phi }_2,z)={E_{\alpha \left(l_0,p_0\right)}}^{*}({\mathit{r}}_1,{\phi }_1,z)E_{\beta \left(l_0,p_0\right)}({\mathit{r}}_2,{\phi }_2,z)\\ & \times 〈\exp \left[{\psi }_1\left({\mathit{r}}_1,{\phi }_1,z\right)+{{\psi }_1}^{*}\left({\mathit{r}}_2,{\phi }_2,z\right)\right]〉\exp \left[-{\displaystyle \frac{{\mathit{r}}_1^2+{\mathit{r}}_2^2-2{\mathit{r}}_1{\mathit{r}}_2\cos \left({\phi }_1-{\phi }_2\right)}{2{\sigma }_0^2}}\right].\end{array}$$

(11)

Based on the quadratic approximation of the wave structure function, the ensemble mean term on the right side of Equation (11) is phase disturbance caused by turbulence, and it can be obtained via the following form [42]:

$$〈\exp \left[{\psi }_1\left({\mathit{r}}_1,{\phi }_1,z\right)+{{\psi }_1}^{*}\left({\mathit{r}}_2,{\phi }_2,z\right)\right]〉=\exp \left[-{\displaystyle \frac{\left({{\mathit{r}}_1}^2-2{\mathit{r}}_1{\mathit{r}}_2\cos \left({\phi }_1-{\phi }_2\right)+{{\mathit{r}}_2}^2\right)}{{\rho }_0^2}}\right],$$

(12)

where ρ₀ represents the coherent length of a spherical wave in turbulent medium and ρ₀ has distinct expressions in different types of turbulent medium. In Kolmogorov turbulence, ρ₀ is written as follows [37]:

$${\rho }_0={\left(0.545C_n^2{k}^{2}z\right)}^{-3/5},$$

(13)

The Kolmogorov turbulence model is the earliest proposed and most widely applied atmospheric turbulence model. It assumes turbulence is locally homogeneous and isotropic while neglecting the effects of inner and outer scales. In contrast, the non-Kolmogorov model is applicable to beam propagation in vertical directions, along inhomogeneous paths and in high-altitude regions. For non-Kolmogorov turbulence, ρ₀ is written as follows [43]:

$${\rho }_0={\left\{{\displaystyle \frac{2\left(n-1\right)\mathsf{\Gamma }\left(\frac{3-n}{2}\right){\left[\frac{8}{n-2}\mathsf{\Gamma }\left(\frac{2}{n-2}\right)\right]}^{ \left(n-2\right)/2 }}{\sqrt{\pi }{k}^2\mathsf{\Gamma }\left(\frac{2-n}{2}\right)C_n^{2}z}}\right\}}^{1/\left(n-2\right)}(3<n<4),$$

(14)

where n is the power law exponent, Γ() denotes the gamma function and $C_n^2$ is the refraction index structure constant, which represents the strength of turbulence. Kolmogorov and non-Kolmogorov turbulence commonly refer to weak turbulence situations, while for strong turbulence, ρ₀ is written as follows [44]:

$${\rho }_0=1.36C_n^{-1}{k}^{-1}{z}^{-1/2}{l}_n^{1/6},$$

(15)

where l_n represents the inner scale of turbulence.

The random fluctuations in atmospheric refractive index can eventuate distortion of the helical phase of vortex beams, resulting in random shifts in the OAM mode l. In addition, turbulence causes modal energy to diffuse towards other modes, reducing the purity of the signal OAM state and inducing crosstalk. To investigate the influence of turbulence on OAM states, the electric field in Equation (5) is regarded as a series of modes with spiral phase superposition, expressed as follows [38]:

$$E(\mathit{r},\phi ,z)={\displaystyle \frac{1}{\sqrt{2\pi }}}{\displaystyle \sum _l{a}_l\left(\mathit{r},z\right)}\exp \left(-il\phi \right),$$

(16)

where a_l(r, z) is the weight factor and l is the quantum number detected at the receiving plane. Thus, the CSD matrix elements have the following form:

$$W_{\alpha \beta }({\mathit{r}}_1,{\mathit{r}}_2,{\phi }_1,{\phi }_2,z)={\displaystyle \frac{1}{2\pi }}{\displaystyle \sum _l〈{\left|a_{\alpha \beta l}\left(\mathit{r},z\right)\right|}^2〉}\exp \left[-il\left({\phi }_1-{\phi }_2\right)\right],$$

(17)

⟨|a(r,z)|²⟩ reflects the probability distribution of the signal OAM modes with the new OAM quantum number l, expressed as follows:

$$〈{\left|a_{\alpha \beta l}\left(\mathit{r},z\right)\right|}^2〉={\displaystyle \frac{1}{2\pi }}{\displaystyle \underset{0}{\overset{2\pi }{\int }}{\displaystyle \underset{0}{\overset{2\pi }{\int }}}}{W}_{\alpha \beta }({\mathit{r}}_1,{\mathit{r}}_2,{\phi }_1,{\phi }_2,z)\exp \left[-il\left({\phi }_1-{\phi }_2\right)\right]d{\phi }_{1}d{\phi }_2.$$

(18)

When l = l₀, ⟨|a_αβl(r,z)|²⟩ denotes the mode probability density, and when l ≠ l₀, it denotes crosstalk probability density. The intensity of the cylindrical vector PCLG beams with an OAM mode at the receiving plane can be given as follows:

$$I_l=I_{xxl}+I_{yyl}=2I_{xxl}=2{\displaystyle \underset{0}{\overset{D/2}{\int }}}〈{\left|a_{xxl}\left(\mathit{r},z\right)\right|}^2〉\mathit{r}d\mathit{r},$$

(19)

where D is the receiving aperture diameter. Substituting Equations (16)–(18) into Equation (19), we obtain the following:

$$\begin{array}{ll}{I}_l=& {\displaystyle \frac{1}{8{\pi }^2}}{\displaystyle \underset{0}{\overset{D/2}{\int }}{\displaystyle \underset{0}{\overset{2\pi }{\int }}{\displaystyle \underset{0}{\overset{2\pi }{\int }}}}}{R}^{*}\left(\mathit{r},z\right)R\left(\mathit{r},z\right)\\ & \left[\exp \left(i{l}_0{\phi }_0\right)+\exp \left(-i{l}_0{\phi }_0\right)\right]\left[\exp \left(-i{l}_0{\phi }_0\right)+\exp \left(i{l}_0{\phi }_0\right)\right]\exp \left[-il\left({\phi }_1-{\phi }_2\right)\right]\\ & \exp \left[-{\displaystyle \frac{\left({{\mathit{r}}_1}^2-2{\mathit{r}}_1{\mathit{r}}_2\cos \left({\varphi }_1-{\varphi }_2\right)+{{\mathit{r}}_2}^2\right)}{{\rho }_0^2}}-{\displaystyle \frac{{\mathit{r}}_1^2+{\mathit{r}}_2^2-2{\mathit{r}}_1{\mathit{r}}_2\cos \left({\phi }_1-{\phi }_2\right)}{2{\sigma }_0^2}}\right]d{\phi }_{1}d{\phi }_2\mathit{r}d\mathit{r}.\end{array}$$

(20)

Applying the following formulas [45],

$$L_n^{\alpha }\left(x\right)={\displaystyle \sum _{m=0}^n{\left(-1\right)}^m\left(\begin{array}{c}n+\alpha \\ n-m\end{array}\right)}{\displaystyle \frac{x^m}{m!}},$$

(21)

$$I_{l_0}(x)={\displaystyle \sum }_{m=0}^{\infty }{\displaystyle \frac{1}{m!\mathsf{\Gamma }(m+l_0+1)}}{({\displaystyle \frac{x}{2}})}^{l_0+2m},$$

(22)

$$\gamma \left(s,x\right)={\displaystyle \underset{0}{\overset{x}{\int }}{t}^{s-1}{e}^{-t}dt},$$

(23)

$${\displaystyle \underset{0}{\overset{2\pi }{\int }}\exp \left[-il{\theta }_1+n\cos \left({\theta }_1-{\theta }_2\right)\right]}d{\theta }_1=2\pi \exp \left(-il{\theta }_2\right)I_l\left(n\right),$$

(24)

$${\displaystyle \underset{0}{\overset{2\pi }{\int }}\exp \left(im\theta \right)d\theta =\left\{\begin{array}{c}2\pi \left(m=0\right)\\ 0\left(m\ne 0\right)\end{array}\right.}.$$

(25)

We have the analytical expression of I_l:

$$\begin{array}{ll}{I}_l=& {\displaystyle \frac{1}{2w^2\left(z\right)}}{\displaystyle \frac{p_0!}{\pi \left(p_0+\left|l_0\right|\right)!}}{\left({\displaystyle \frac{\sqrt{2}}{w\left(z\right)}}\right)}^{2l_0}\\ & {\displaystyle \sum _{m=0}^{p_0}{\displaystyle \sum _{k=0}^{p_0}{\displaystyle \sum }_{q=0}^{\infty }{\left(-1\right)}^k}{\left(-1\right)}^m\left(\begin{array}{c}p+l_0\\ p-m\end{array}\right)}\left(\begin{array}{c}p+l_0\\ p-k\end{array}\right){\displaystyle \frac{{\left[\frac{2}{w^2\left(z\right)}\right]}^m}{m!}}{\displaystyle \frac{{\left[\frac{2}{w^2\left(z\right)}\right]}^k}{k!}}\\ & \left[\begin{array}{l}{\displaystyle \frac{{(b)}^{l_0+l+2q}}{q!\mathsf{\Gamma }(q+l_0+l+1)}}{\displaystyle \frac{\gamma \left(m+k+2l_0+l+2q+1,c{D}^2/4\right)}{c^{1+m+k+2l_0+l+2q}}}\\ +{\displaystyle \frac{{(b)}^{l-l_0+2q}}{q!\mathsf{\Gamma }(q+l-l_0+1)}}{\displaystyle \frac{\gamma \left(m+k+l+2q+1,c{D}^2/4\right)}{c^{1+m+k+l+2q}}}\end{array}\right],\end{array}$$

(26)

with

$$b=\left({\displaystyle \frac{1}{{\rho }_0^2}}+{\displaystyle \frac{1}{2{\sigma }_{\alpha \beta }^2}}\right),$$

(27)

$$c={\displaystyle \frac{2}{w^2\left(z\right)}}+2b.$$

(28)

At the receiver, the equivalent intensity is defined as the ratio of the intensity corresponding to a specific quantum number to the sum of intensities of all other quantum numbers. The equivalent intensity is applied to evaluate the performance of communication systems and can be expressed as follows [46]:

$$P_l={\displaystyle \frac{I_l}{{\displaystyle \sum _{m=-\infty }^{\infty }{I}_m,\mathrm{Re}\left(m\ne l\right)}}}.$$

(29)

If l = l₀, P_l is the SNR and reflects the quality of optical communication. Otherwise, P_l is the crosstalk equivalent intensity of the OAM crosstalk mode, which describes the noise intensity in the channel. The SNR and BER are vital components for evaluating the performance of communication systems, and the SNR is a key factor determining the system BER. Then, the expression of BER is expressed as [47]

$$BER={\displaystyle \frac{1}{2}}erfc\left(\sqrt{P_{l=l_0}/2}\right),$$

(30)

where erfc( ) is the complementary error function.

3. Results and Discussion

In this section, based on the theoretical analysis in Section 2, we have conducted numerical simulations to explore the communication performance of cylindrical vector PCLG beams propagating through turbulence, including mode probability density, crosstalk probability density, equivalent intensity and BER. The default parameter values are set as follows: wavelength λ = 632.8 nm, beam width w₀ = 0.01 m, coherence length σ = 0.01 m, radial index p₀ = 1, azimuthal index l₀ = 1, refraction index structure constant $C_n^2$ = 5 × 10⁻¹⁵ m^−2/3, power law exponent n = 3.67, turbulence inner scale l_n = 0.001, receiving aperture diameter D = 0.01 m and propagation distance z = 1 km. Unless otherwise specified, these parameters remain unchanged. If the parameter values are changed, special instructions will be provided in the figures.

The variation in the mode probability density of signal OAM for a cylindrical vector PCLG beam with different p₀ and l₀ values against the detection position r in Kolmogorov turbulence is presented in Figure 1. The impacts of p₀ and l₀ on mode probability density are obviously distinct. From Figure 1a, with an increase in l₀, the maximum value of mode probability density decreases, the corresponding optimal detection position moves away from the center and the half-width broadens. It suggests that the larger the OAM mode of the cylindrical vector PCLG beam, the more pronounced the degradation of the received signal quality. Meanwhile, the maximum value of mode probability density trend in Figure 1b is opposite to that in Figure 1a. In Figure 1b, the maximum value of mode probability density increases and the optimal detection position shifts toward the center with the increase in p₀. Comparing Figure 1a,b, we found that the mode probability density curve against r that varies with l₀ exhibits a single peak, while the curve varying with p₀ has multiple peaks and the number of peaks satisfies the condition of p₀ + 1. These results indicate that the detection efficiency of signal OAM mode of a cylindrical vector PCLG beam with different p₀ and l₀ values in turbulence can be effectively improved by selecting the optimal detection position.

Figure 2 depicts the crosstalk probability density of signal OAM mode for a cylindrical vector PCLG beam in Kolmogorov turbulence against the detection position r for OAM modes with difference ∆l = 0, 1, 2, 3. In the case of ∆l = 0, the crosstalk probability density is equivalent to the mode probability density. The first peak of crosstalk probability density rapidly decreases with the increase in ∆l, especially when ∆l changes from 0 to 1, and the detection position of the first peak also slightly moves away from the center. However, the second peak of crosstalk probability density exhibits an approximately linear descent. Additionally, a larger value of l leads to a higher crosstalk probability density. An increase in p can effectively reduce the crosstalk probability density. The number of peaks also satisfies the condition of p₀ + 1. For different ∆l values, the detection position of the minimum value between the two peaks remains consistent. It reveals that the signal OAM mode is prone to crosstalk with adjacent OAM modes and better communication quality can be obtained by moving detection locations.

Subsequently, the equivalent intensity of a cylindrical vector PCLG beam against the coherence length σ₀ for different OAM modes difference ∆l under conditions of Kolmogorov turbulence, non-Kolmogorov turbulence and strong turbulence, respectively, are calculated, as shown in Figure 3. To observe the trend of curves more clearly, the local regions in Figure 3a,b are enlarged, as shown in the embedded subgraphs. When Δl = 0, the curves represent the SNR, which changes rapidly with σ₀, and the values of the SNR increase with the increase in σ_0. As the coherence length increases, the equivalent intensity of non-Kolmogorov turbulence has the highest magnitude, followed by Kolmogorov turbulence and the lowest in strong turbulence. The quality of signal OAM mode in non-Kolmogorov turbulence surpasses that in both Kolmogorov turbulence and strong turbulence. When Δl ≠ 0, the curves denote the crosstalk equivalent intensity, whose values decrease with increasing σ₀ and Δl. It also demonstrates that the crosstalk primarily originates from adjacent modes of the signal OAM mode. The outcomes indicate that the equivalent intensity of a cylindrical vector PCLG beam is related to turbulence models and the coherence length of the beam.

Figure 4 plots the influence of receiving aperture diameter D and propagation distance z on the SNR of a cylindrical vector PCLG beam with different coherence lengths under Kolmogorov turbulence, non-Kolmogorov turbulence and strong turbulence conditions, respectively. The SNR increases with increasing σ₀ as D decreases and z shortens, which indicates excellent signal quality and strong anti-interference ability of the communication system. The trends of the SNR curves related to D are consistent with those related to z under identical turbulence conditions.

Figure 5 shows the crosstalk equivalent intensity of a cylindrical vector PCLG beam against w₀ for different values of refraction index structure constant $C_n^2$ in Kolmogorov turbulence, power law parameter n in the non-Kolmogorov spectrum and the inner scale of turbulence l_n in strong turbulence, respectively, and the OAM modes difference ∆l is set as 1. The three parameters C_n, n and l_n represent the turbulence intensity under different turbulence modes, and the larger C_n and the smaller l_n, the stronger the turbulence. For the non-Kolmogorov spectrum, the turbulence strength first decreases, then increases, and subsequently decreases with increasing n. When n ranges from 3.3 to 3.7, the atmospheric coherence length shows an increasing trend, indicating weakened turbulence strength and correspondingly reduced crosstalk intensity. Comparing Figure 5a–c, it can be observed that the stronger the turbulence intensity, the higher the noise intensity, and consequently, the worse the signal quality. As w₀ increases, the crosstalk equivalent intensity decreases to a minimum value and then increase to a constant. This change is most pronounced in the non-Kolmogorov spectrum and least significant in strong turbulent medium, which illustrates that the impact of beam width on crosstalk is minimal in strong turbulent medium. In addition, selecting an appropriate beam width can effectively enhance communication quality.

To analyze the BER of a cylindrical vector PCLG beam against coherence length σ₀, we calculated the BER for different values of azimuthal index l₀, radial index p₀ and wavelength λ under conditions of Kolmogorov turbulence, non-Kolmogorov turbulence and strong turbulence, respectively, as shown in Figure 6. The values of BER decrease with the increase in σ₀, and the larger the l₀, the smaller the p₀, the shorter the wavelength and the higher the BER. Therefore, to ameliorate the performance of communication system, we can prefer to the cylindrical vector PCLG beam with smaller azimuthal mode order, larger radial mode order and longer wavelength.

4. Conclusions

In this paper, we theoretically analyzed and numerically simulated the influence of different parameters, such as wavelength, coherence length, beam width, propagation distance and receiving aperture diameter, on optical communication performance of cylindrical vector PCLG beams under different atmospheric turbulence spectra. We find that both mode probability density and crosstalk probability density are affected by azimuthal index l₀ and radial index p₀. When l = l₀, it reveals that increased propagation distance and enhanced turbulence strength lead to a decrease in the SNR. Notably, larger wavelength λ or radial index p₀ correspond to a higher SNR. Additionally, reducing the receiving aperture diameter D or azimuthal index l₀ can also improve the SNR. The crosstalk intensity is higher in the modes adjacent to the signal mode. Therefore, we can select appropriate parameters of cylindrical vector PCLG beams to improve the SNR to improve the quality of optical communication.