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Abstract
A novel statistical model connected with turbulence strength is proposed to describe the attenuation and crosstalk in a vortex-based multi-channel free space optical (FSO) communication system. In this model, self-channel fading and interference between different orbital angular momentum (OAM) modes are characterized by the mixture exponential-generalized-gamma (EGG) distribution, and the analytical relations between turbulence strength and the distribution function’s parameters are expressed by piecewise functions. The problems of obtaining parameters of this model are converted into optimization problems, and the algorithms based on the trust trigon algorithm are proposed to achieve more optimized parameters. This model is confirmed to have a good fit with the emulated data of OAM attenuation and crosstalk calculated by the square of the scalar product between the fields of two OAM modes. Furthermore, the application of the statistical model to the OAM-multiplexing FSO system with quadrature-phase-shift-keying modulation is presented, in which the theoretical average bit-error rate results match well with Monte Carlo simulation. This model can be used for FSO system design with OAM for continuous weak turbulence condition.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Orbital angular momentum (OAM), as a promising freedom degree, has aroused a widespread interest in many fields, such as optical fiber communications, radio communications and optical wireless communications (OWC) [1–6]. Especially in OWC, or free space optical (FSO) systems, vortex beams with different OAM modes that carry independent data stream can achieve more efficient information transmission spectrum than the classical OWC system [7]. Furthermore, the number of OAM modes is theoretically infinite, and the beams carrying different OAMs are mutually orthogonal with one another. Thus, OAM is considered as a promising technique that allows for a potentially significant increase in the transmission capacity of OWC system in the future network [8].
So far, because of its great potential, numerous reports have been made about FSO systems using OAM [9–20], where the two main applications are OAM shift keying (OAM-SK) and OAM mode division multiplexing (OAM-MDM). For example, in 2012, terabit free-space data transmission was achieved by 16-quadrature amplitude modulation (QAM) encoding based on OAM-MDM, when combined with polarization multiplexing [14]. In 2015, a quadrature-phase-shift-keying (QPSK) OAM-MDM system of 2ns channel hopping with 20 Gbit/s data encoding was demonstrated [15]. In 2017, an OAM-SK FSO communication system was presented to transfer images [16]. Nevertheless, some studies have shown that the performance of the FSO communication system could be dramatically degraded by atmospheric turbulence, which would cause a random change in the wavefront phase of vortex beams [21–25]. In [22], the crosstalk induced by atmospheric turbulence was found to be more serious when the number of OAM states is larger. In [23], the performance of low-density parity-check precoded OAM modulation was studied over a 1- km FSO system subject to OAM modal crosstalk induced by atmospheric turbulence. In classical FSO systems, to describe the impact of turbulence, several statistical distributions have been proposed to describe the random irradiance, including log-normal distribution [26], gamma-gamma distribution [27], m distribution [28], and double-Weibull distribution [29]. On the basis of these distributions, statistical channel information and the closed-form performance expressions of the classical FSO communication system can be derived, such as bit- error rate and outage probability [29]. In particular, the details of various FSO systems and the ways to enhance their performances can be obtained through theoretical analysis. However, for FSO communication system with OAM, the classical statistical models mentioned above cannot effectively describe the impact of turbulence, which will cause eigenstate channel self-fading and interference between different OAM modes, or the attenuation and crosstalk, respectively [30]. Fortunately, some recent works have been carried out about the statistical model of the attenuation and crosstalk for the FSO communication system with OAM [30–36]. In [30], the effect of atmospheric turbulence on the energy among OAM modes was studied by calculating a two-dimensional scalar product of the distorted and undistorted received fields on the basis of the simulating the Laguerre–Gauss (LG) beam propagation. Johnson ${S_B}$ probability density function (PDF) and the exponential distribution were then used to express the attenuation and crosstalk, respectively. This was the first time that PDF was proposed to describe the characteristics of OAM attenuation and crosstalk through the Johnson ${S_B}$ PDF and exponential distributions. In [33], the measurement of self-channel fading was accomplished by photodiode sensor detection and power normalization after single-mode fiber coupling on the basis of the experiment platforms of atmospheric transmission. The result proved that Johnson ${S_B}$ PDF could provide a perfect fit for the experimental data on the attenuation in both an 84 m indoor path and a 400 m outdoor path for partial turbulence conditions. However, the statistical properties of Johnson ${S_B}$ PDF are analytically intractable. Thus, it is impossible to extract closed-form expressions for performance metrics, and the lack of a unified turbulence model makes it difficult to assess the effect of the interference on the performance of OAM modes [36]. In [36], the unified statistical characteristics of the attenuation and crosstalk were described by the generalized gamma distribution (GGD) in which the parameters were calculated by the maximum-likelihood (ML) estimation. Nevertheless, the analytical relations of the parameters of the GGD in OAM FSO systems as a function of the turbulence strength were not given although they are really required for FSO communication systems.
Motivated by the above analysis, the analytically relational models between distribution function parameters of the attenuation or crosstalk and turbulence strength are first built in this work. Specifically, a statistical model connected with turbulence strength is proposed, which describes power attenuation in certain OAM modes and the crosstalk between OAM modes caused by atmospheric turbulence in FSO systems. The effective algorithms to obtain the model’s parameters are also given. The result shows that the mixture exponential generalized-gamma (EGG) distribution can better characterize channel fading for whole weak turbulence regime, and the effective range of this model is finally discussed.
The rest of this paper is arranged as follows. Section 2 illustrates the way to emulate the attenuation and crosstalk by LG propagation in FSO systems. The impact of turbulence, OAM mode interval, and OAM modes on the statistical properties of the attenuation and crosstalk is also studied. Section 3 introduces the unified PDF, namely, the EGG distribution, and the conditions of reduction to GGD and exponential distribution function. The analytically relational models, describing the connection between the distribution function and $C_n^2$, are then given in ${10^{ - 15}}\sim {10^{ - 14}}{\textrm{m}^{{{\textrm{ - 2}} / \textrm{3}}}}$ as an example for different OAM modes interval, respectively. To acquire a quantitative model, the complex problem of solving parameters is divided into several steps to get a better result of model parameters. The PDF calculated from the proposed model is likewise verified to have a good fit with the simulated data. Section 4 represents the application of the proposed model and the numerical simulations. Finally, some conclusions are summarized in Section 5.
2. OAM attenuation and crosstalk
The attenuation and crosstalk ${\eta _{m \to n}}$ describes the fraction of power received on OAM mode $n$, when the transmitted power is carried by the vortex light with OAM mode m [36]. OAM attenuation ${\eta _{m \to m}}$ is regarded as self-channel fading, and the interference fading between different OAM channels is described as ${\eta _{m \to n}}({m \ne n} )$ [30]. Their concrete form is [30–38]
with where $({{u^\textrm{1}},{u^\textrm{2}}} )= \int {\int {{u^{\textrm{1}\ast }}({\rho ,\varphi ,z} ){u^\textrm{2}}({\rho ,\varphi ,z} )\rho d\rho d\varphi } }$, with ${\ast} $ indicating the complex conjugate operation. $u_m^T$ is the optical field with OAM modes m through atmospheric turbulence at distance z, and $u_n^V$ is the optical field with OAM mode n in vacuum. $|{m - n} |$ is the OAM modes interval. There are many studies on the simulation propagation of FSO systems [39–41]. Fresnel integral and multiple-phase screen methods are usually used to simulate respectively the diffraction transmission and random fluctuation induced by turbulence [39–43]. In this work, angular spectrum method and phase screen are adopted as in [44]. In addition to these, the LG beam is set as the light source.For the convenience of computation, the power of light is set as unity: $({u,u} )= \int {\int {{{|u |}^2}\rho d\rho d\varphi } } = 1$. Therefore, the optical fields with different modes are orthogonal mutually in vacuum [29,45]:
The attenuation equals $1$, and the crosstalk equals $0$.
As is known, the atmospheric turbulence induced by atmospheric motion would results in random fluctuations of the refractive index [46], causing random fluctuation on the intensity distribution. The distribution of the light field is more complex with larger OAM modes, and thus the impacts of OAM modes and turbulence strength on the FSO communication system would be more significant. These will be investigated on the empirical PDF of the attenuation and crosstalk ${\eta _{m \to n}}$. First, in order to get the empirical PDF, OAM attenuation and crosstalk with different turbulence strengths are emulated as Eq. (1).
The main parameters are as follows: the whole path is divided into 10 parts, the number of screen points is 256; the inner and outer scale sizes are 0.01 and 20 m, respectively; the light’s wavelength is $\textrm{1550 nm}$; the waist of the LG beam is $\textrm{0}\textrm{.016 m}$; and the propagation distance is $1000\textrm{ m}$. The number of emulations is set to 20,000. Moreover, the empirical PDF is estimated using the histograms.
Figure 1 shows the empirical PDFs of the attenuation, when $C_n^2 = 1.0 \times {10^{ - 15}}\textrm{ }{\textrm{m}^{{{\textrm{ - 2}} / \textrm{3}}}}$, $5.5 \times {10^{ - 15}}\textrm{ }{\textrm{m}^{{{\textrm{ - 2}} / \textrm{3}}}}$ and $1.0 \times {10^{ - 14}}\textrm{ }{\textrm{m}^{{{\textrm{ - 2}} / \textrm{3}}}}$. As can be seen, for different turbulence strengths, the shapes of the empirical PDF are completely different in Figs. 1(a)–1(c). For the same turbulence strength, the shapes of the empirical PDF are similar for the different OAM modes. Therefore, the impact of OAM mode on the empirical PDF is less than that of the atmospheric turbulence strength.
Figure 2 compares the distributions of the attenuation and crosstalk for different OAM mode intervals and turbulence strengths. As is shown, the PDFs of ${\eta _{m \to m - \Delta }}$ and ${\eta _{m \to m + \Delta }}$ ($\Delta = 1,2$) are very alike with the increasing $m$, which is caused by the partial power of light with a certain OAM mode evenly spreading to adjacent OAM modes. In addition, a comparison of Fig. 2(a), 2(b), and 2(c) indicates that the shape of the PDF of ${\eta _{m \to n}}({|{m - n} |= 1} )$ changes gradually from one to another gradually with increasing turbulence strength, whereas the PDF of ${\eta _{m \to n}}$ ($|{m - n} |\ge 2$) is similar to the exponential distribution and remains almost the same. Stronger turbulence and larger $|{m - n} |$ may also have similar impact on the empirical PDFs’ shapes of OAM attenuation and crosstalk but with different degrees. That is, the probability of a small value of ${\eta _{m \to n}}$ will be larger, or the power of beam with the OAM mode m will be more likely to shift to beam with another OAM mode. Therefore, the attenuation and the crosstalk would become OAM-dependent random variables from the constants in vacuum due to the turbulence. Specifically, the empirical PDFs of the attenuation and crosstalk depend mainly on the turbulence strength and OAM mode. In the following content, this phenomenon could be quantified approximately by mathematical formulas.
Fig. 2. Comparison of PDFs of the attenuation and crosstalk between different OAM modes intervals and turbulence strengths.
3. Statistical model
In this section, the statistical models of OAM attenuation and crosstalk are introduced for weak fluctuation. For arbitrary refractive-index spectral model with a Gaussian-beam wave, the weak fluctuations are described by $q < 1$ and $q\Lambda < 1$, where the parameters are $q = {z / {({{k_0}\rho_{\textrm{pl}}^2} )}}$ and $\Lambda = {{2z} / {({{k_0}w{{(z )}^2}} )}}$ with the plane wave spatial coherence radius ${\rho _{\textrm{pl}}} = {({1.46C_n^2k_0^2z} )^{{{ - 3} / 5}}}$ [47], so the propagation distance and turbulence strength $C_n^2$ are limited by $q < 1$ and $q\Lambda < 1$ for the same source light. For the fixed transmission distance $z = 1000\textrm{m}$, when $C_n^2 = {10^{ - 14}}{\textrm{m}^{{{\textrm{ - 2}} / \textrm{3}}}}$, $q = 0.1771$ and $\Lambda = 0.4088$, and when $C_n^2 = {10^{ - 17}}{\textrm{m}^{{{\textrm{ - 2}} / \textrm{3}}}}$, $q = 4.4 \times {10^{ - 5}}$ and $\Lambda = 0.4088$. Thus, the range of weak fluctuation is set as ${10^{ - 17}}{\textrm{m}^{{{\textrm{ - 2}} / \textrm{3}}}} \le C_n^2 \le {10^{ - 14}}{\textrm{m}^{{{\textrm{ - 2}} / \textrm{3}}}}$. In this range, the unified PDF of OAM attenuation and crosstalk could be approximated as the mixture EGG distribution:
On the basis of multiple fitting tests, the form of PDF initially becomes complex and then simple with increasing turbulence strength. In many cases, if the turbulence strength is relatively weak, most of the power would be concentrated on the eigenmode with less leakage. If the turbulence is relatively strong, most of the power would shift to other modes and the value of the attenuation would be close to 0. In a certain range, the unified PDF of ${\eta _{m \to n}}$ could be approximated as GGD or exponential distribution to get a simpler statistical model. When $|{m - n} |$ becomes larger, the form of PDF is simpler. The turbulence affects the vortex light field and causes energy leakage of the OAM eigenmode, thus partial energy is shifted to other OAM modes. Specifically, in this scenario, the difference of the optical field with OAM modes between n and m is so bigger that the light field distortion induced by atmospheric turbulence would be difficult to reflected in the OAM mode with a larger interval. Here, the attenuation and crosstalk represent power fading and transferring, respectively. Thus, the statistical models are built with $m = n$ and $|{m - n} |\ge 1$ in the following content.
For instance, when $C_n^2 \le {10^{ - 15}}{\textrm{m}^{{{\textrm{ - 2}} / \textrm{3}}}}$ or $|{m - n} |\ge 1$ with the parameters given in the Section 2, the unified PDF of ${\eta _{m \to n}}$ could be approximated by GGD, which is the special case of the mixture EGG distribution:
When $|{m - n} |= 1,C_n^2 \le {10^{ - 15}}{\textrm{m}^{{{\textrm{ - 2}} / \textrm{3}}}}$ or $|{m - n} |\ge 2$ with the parameters given in Section 2, the unified PDF of ${\eta _{m \to n}}({|{m - n} |\ge 1} )$ could be approximated as the exponential distribution, which is the special case of GGD:
Here, ${10^{ - 17}} \le C_n^2 \le {10^{ - 14}}{\textrm{m}^{{{\textrm{ - 2}} / \textrm{3}}}}$ is divided into three regions by orders of magnitude for simplicity:${10^{ - 17}} - {10^{ - 16}}{\textrm{m}^{{{\textrm{ - 2}} / \textrm{3}}}}$, ${10^{ - 16}} - {10^{ - 15}}{\textrm{m}^{{{\textrm{ - 2}} / \textrm{3}}}}$, and ${10^{ - 15}} - {10^{ - 14}}{\textrm{m}^{{{\textrm{ - 2}} / \textrm{3}}}}$. In these three regions, the formulas of PDF are not exactly the same; and in fact, they are piecewise functions for the whole weak turbulence regime. In this work, ${10^{ - 15}} - {10^{ - 14}}{m^{{{\textrm{ - 2}} / \textrm{3}}}}$, as the strongest fluctuation range, is selected as the example to be investigated, and the analytically relational model for this range is discussed in the following section.
3.1 Evaluation of the model
To verify model’s correctness, the coefficient of determination ${R^2}$ is used to evaluate the goodness of the fitting with the simulation data [48]. Here, ${R^2}$ is calculated as:
3.2 PDF model of the attenuation
As is known from Eq. (4), OAM attenuation can be expressed by the mixture EGG distribution. In addition, through fitting, the analytical relations between the PDF’s parameters and $C_n^2$ could be approximated as below:
Equations (4) and (8)–(12) constitute the whole model of the attenuation in ${10^{ - 15}} \sim {10^{ - 14}}{m^{{{\textrm{ - 2}} / \textrm{3}}}}$, in which ${a_m},b_m^1,b_m^2,c_m^1,c_m^2,\theta _m^1,\theta _m^2,\theta _m^3,\varpi _m^1,\varpi _m^2$ are the parameters.
In order to calculate the parameters of this proposed model, the attenuation is emulated to calculate the empirical PDF first. Then, there is a fitting problem for obtaining the model’s parameters, which can also be achieved by minimizing the mean square distance from the data points to the fitting functions as the common choice of fitting circles [49]. To get the result conveniently, it is converted into optimization problems on the basis of the least square method.
To get the optimization model, the first step is to take some fixed interval values $\{{{{({C_n^2} )}_1},{{({C_n^2} )}_2}}$, $\ldots ,{{({C_n^2} )}_{{N_c}}} \}$ within ${10^{ - 15}} \sim {10^{ - 14}}{\textrm{m}^{{{\textrm{ - 2}} / \textrm{3}}}}$. The second step is to emulate ${\eta _{m \to m}}$ with ${({C_n^2} )_1}$, ${({C_n^2} )_2},\ldots ,{({C_n^2} )_{{N_c}}}$. The third step is to get the data points $({\bar{x}_{m,{j_c}}^j,\bar{y}_{m,{j_c}}^j} )\,j = 1,2,\ldots ,{N_s}\,{j_c} = 1,2,\ldots ,{N_c}$, which are the points from the empirical PDF of ${\eta _{m \to m}}$, when $C_n^2 = {({C_n^2} )_{{j_c}}}$. Finally, the optimization model of calculating the parameters of the PDF model can be derived:
As can be found, this optimization model in Eq. (13) is too complex to obtain the final result directly and the value of $\bar{y}_{m,{j_c}}^j\,j = 1,2,\ldots ,{N_s}$ and ${j_c} = 1,2,\ldots ,{N_c}$ differ greatly. Thus, this optimization problem is converted into two parts. The first part’s optimizations are as follows:
Owing to the complexity of the optimization process, the suitable parameters of the proposed model can’t be obtained by Eqs. (14)–(19) directly.
Specific steps for getting the better parameters of the statistical model of OAM attenuation are follows
In step 3, ${\theta _{m,{j_c}}}$ and ${({C_n^2} )_{{j_c}}}$, (${j_c} = 1,2,\ldots ,{N_c}$) may not have correspondence with Eq. (10), when $\theta _m^1,\theta _m^2,\theta _m^3$ are being calcul1ated. In detail, the singular values of ${\theta _{m,{j_c}}}\,{j_c} = 1,2,\ldots ,{\bar{N}_c}$ are much larger than the values of .${\theta _{m,{j_c}}}$,${j_c} = {\bar{N}_c} + 1,\ldots ,{N_c}$, as a result of that $\varpi $ is so small that the random of data causes $\theta $ unstable. The best way to avoid this unstable situation is to change the model’s range. Specifically, replace the mixture EGG distribution with GGD in ${({C_n^2} )_1} \sim {({C_n^2} )_{{{\bar{N}}_c}}}$ and set the mixture EGG distribution as the model’s PDF in ${({C_n^2} )_{{{\bar{N}}_c}}} \sim {({C_n^2} )_{{N_c}}}$. In these steps, $C_n^2$ should be divided by ${10^{ - 15}}$ in order to avoid the calculation error induced by $C_n^2$, which is a very small number. In this work, the trust trigon algorithm is adopted to solve the nonlinear optimizations of Eqs. (14) and (17)–(19), and maximum likelihood estimation and simple derivation are used to solve the optimizations of Eqs. (15) and (16). The fit function and polyfit function in MATLAB are used to implement these algorithms.When ${10^{ - 15}} \le C_n^2 \le {10^{ - 14}}{\textrm{m}^{{{\textrm{ - 2}} / \textrm{3}}}}$ and $m = n$, the parameters ${a_m},b_m^1,b_m^2,c_m^1,c_m^2,\theta _m^1,\theta _m^2,$ $\theta _m^3,\varpi _m^1,\varpi _m^2$ can be calculated as those steps and then substituted to Eqs. (8)–(12) to get the attenuation’s statistical model. In some cases, the PDF of the attenuation reduces to GGD, and thus the whole model is composed of Eq. (4), in which $\varpi $ is set to $0$, (8), (11), and (12). For example, when $C_n^2 = {10^{ - 15}}{\textrm{m}^{{{\textrm{ - 2}} / \textrm{3}}}}$, the empirical PDF shape of ${\eta _{3 \to 3}}$ is not the same as when $C_n^2 = 5.5 \times {10^{ - 15}}{\textrm{m}^{{{\textrm{ - 2}} / \textrm{3}}}}$ (Fig. 1). Next, the ${R^2}$ of GGD fitting to ${\eta _{3 \to 3}}$ is calculated when ${10^{ - 15}} \le C_n^2 \le {10^{ - 14}}{\textrm{m}^{{{\textrm{ - 2}} / \textrm{3}}}}$. It is quite close to $1$ when $\textrm{1}\textrm{.0} \times {10^{ - 15}} \le C_n^2 \le \textrm{5}\textrm{.0} \times {10^{ - 15}}{\textrm{m}^{{{\textrm{ - 2}} / \textrm{3}}}}$ and is less than $0.9$ when $7.5 \times {10^{ - 15}} \le C_n^2 \le \textrm{1}\textrm{.0} \times {10^{ - 14}}{\textrm{m}^{{{\textrm{ - 2}} / \textrm{3}}}}$. Therefore, the PDF reduces to GGD when $\textrm{1}\textrm{.0} \times {10^{ - 15}} \le C_n^2 \le \textrm{5}\textrm{.0} \times {10^{ - 15}}{\textrm{m}^{{{\textrm{ - 2}} / \textrm{3}}}}$. At this time, the statistical characteristics of ${\eta _{3 \to 3}}$ could be represented by Eq. (4), in which $\varpi $ is set to $0$, (8), (11), and (12) in $\textrm{1}\textrm{.0} \times {10^{ - 15}} \le C_n^2 \le \textrm{5}\textrm{.0} \times {10^{ - 15}}{\textrm{m}^{{{\textrm{ - 2}} / \textrm{3}}}}$ and expressed by Eqs. (4), (8)–(12) in $5.\textrm{0} \times {10^{ - 15}} \le C_n^2 \le \textrm{1}\textrm{.0} \times {10^{ - 14}}{\textrm{m}^{{{\textrm{ - 2}} / \textrm{3}}}}$. When $C_n^2 \le {10^{ - 15}}{\textrm{m}^{{{\textrm{ - 2}} / \textrm{3}}}}$, the value of $\varpi $ can be determined by the type of PDF, which is the EGG distribution or GGD.
3.3 PDF model of the crosstalk
In this section, two statistical models of the crosstalk are described as the following content. In the first model, the crosstalk can be represented by GGD, and the analytical relations between the PDF’s parameters and $C_n^2$ are approximated as:
The whole statistical model consists of Eq. (5) and Eqs. (20)–(22). In the second model, the distribution of the crosstalk reduces to the exponential distribution, and $C_n^2$ is linear to b, which is approximately:
Similarly, when $|{n - m} |\ge 1$, the first step of getting the parameters of the first model is to achieve several sets of $({\bar{x}_{m \to n,{j_c}}^j,\bar{y}_{m \to n,{j_c}}^j} )$, $j = 1,2,\ldots ,{N_s}$, which are the points from the empirical PDF of ${\eta _{m \to n}}$ with $C_n^2 = {({C_n^2} )_{{j_c}}}$ for ${j_c} = 1,2,\ldots ,{N_c}$. Then, the parameters of the PDF model of OAM crosstalk are calculated by the optimization methods. In the first statistical model, the optimization problem is:
The key steps to get the parameters of the first model of OAM crosstalk are as follows. As for these optimizations of formulas (25)–(28), they are solved in the same way as Section 3.2.
- Step 1. Calculate ${a_{m \to n,{j_c}}},{b_{m \to n,{j_c}}},{c_{m \to n,{j_c}}}$ ${j_c} = 1,2,\ldots,{N_c}$ with the trust trigon algorithm as (25).
- Step 3. Compute $b_{m \to n}^1,b_{m \to n}^\textrm{2},b_{m \to n}^\textrm{3},c_{m \to n}^1,c_{m \to n}^\textrm{2},c_{m \to n}^\textrm{3}$ through (27) and (28).
When ${10^{ - 15}} \le C_n^2 \le {10^{ - 14}}{\textrm{m}^{{{\textrm{ - 2}} / \textrm{3}}}}$ and $|{m - n} |= 1$, OAM crosstalk could be characterized by GGD rather than exponential distribution, and so the first model could be applied to ${\eta _{m \to n}}$ with $|{m - n} |= 1$ and ${10^{ - 15}} \le C_n^2 \le {10^{ - 14}}{\textrm{m}^{{{\textrm{ - 2}} / \textrm{3}}}}$. In the first model, the parameters ${a_{m \to n}},b_{m \to n}^1,b_{m \to n}^\textrm{2}$, $b_{m \to n}^\textrm{3},c_{m \to n}^1,c_{m \to n}^\textrm{2}$, $c_{m \to n}^\textrm{3}$ are calculated by above steps and are subsequently substituted to Eqs. (20)–(22) to get the OAM crosstalk statistical model. When $C_n^2 \le {10^{ - 15}}{\textrm{m}^{{{\textrm{ - 2}} / \textrm{3}}}}$ or $|{m - n} |\ge 2$, the PDF of OAM crosstalk reduces to exponential distribution, and are thus represented by the second model. In the second model, the algorithm for getting the parameters is to get ${b_{m \to n,{j_c}}}\textrm{ }{j_c} = 1,\ldots ,{N_c}$ as Eq. (30), and then $\xi _{m \to n}^1,\xi _{m \to n}^\textrm{2}$ is calculated by Eq. (31). Finally, $\xi _{m \to n}^1,\xi _{m \to n}^\textrm{2}$ are substituted into Eq. (23) to get the second model.
3.4 Verification of Model Reliability
When $|{m - n} |$ is larger, the statistical characteristics of the attenuation and crosstalk can be expressed by the PDF of the simpler functional form. Next, the shapes of PDF with different $|{m - n} |$ are studied by simulation. When $|{m - n} |= 0$ with $m \in \{{2,3,4,5,6} \}$, the mixture EGG distribution can fit the PDF of ${\eta _{m \to m}}$ in $\bar{C}{_n^2} - \mathord{\buildrel{\hbox{$\scriptscriptstyle\frown$}} \over C}{ _n^2}$, whereas GGD cannot fit it, in which $\bar{C}_n^2$ and $\hat{C}_n^2$ will vary when the OAM mode changes. Figure 3 plots the distribution of the attenuation. For each distribution, the fitting by GGD along with the EGG fitting is also given for comparison. Clearly, the attenuation cannot be represented by GGD, while the mixture EGG distribution performs well. For instance, compared with GGD, the mixture EGG distribution gives a better fit with the attenuation when $C_n^2 = \textrm{5}\textrm{.5} \times {10^{ - 15}}{m^{{{\textrm{ - 2}} / \textrm{3}}}}$. GGD is also the special case of the mixture EGG distribution. Therefore, more situations can be expressed by the EGG distribution, which makes it become more suitable unified PDF. These results on the PDF of the attenuation are consistent with the proposed distributions in Section 3. When $|{m - n} |\ge 1$, GGD can be used to fit ${\eta _{m \to n}}$, when ${10^{ - 15}}\textrm{ } \le C_n^2 \le {10^{ - 14}}\textrm{ }{\textrm{m}^{{{\textrm{ - 2}} / \textrm{3}}}}$ and $m \in [{0,7} ]$, exponential distribution could be used to fit ${\eta _{m \to n}}$ with $|{m - n} |\ge 2$ well. Additionally, when $C_n^2 \le {10^{ - 15}}{\textrm{m}^{{{\textrm{ - 2}} / \textrm{3}}}}$, the PDF of ${\eta _{m \to n}}$ with $|{m - n} |\ge 1$ will reduce to the exponential distribution. Figure 4 plots the ${R^2}$ of GGD fitting and exponential distribution fitting to the simulation data of the crosstalk. ${R^2}$ of GGD to fit ${\eta _{5 \to n}}$ with $|{5 - n} |\ge 1$ and ${R^2}$ of exponential distribution to fit ${\eta _{5 \to n}}$ with $|{5 - n} |\ge 2$ are greater than $0.97$ in ${10^{ - 15}}\textrm{ } \le C_n^2 \le {10^{ - 14}}\textrm{ }{\textrm{m}^{{{\textrm{ - 2}} / \textrm{3}}}}$. In addition, when $|{m - n} |\ge 2$, the ${R^2}$ of exponential distribution to fit ${\eta _{m \to n}}$ decreases as $C_n^2$ increases. Clearly, the PDF shapes of ${\eta _{m \to n}}$,$|{m - n} |= 1$ are fitted terribly by exponential distribution, when ${10^{ - 15}}\textrm{ } \le C_n^2 \le {10^{ - 14}}\textrm{ }{\textrm{m}^{{{\textrm{ - 2}} / \textrm{3}}}}$. While the goodness of exponential distribution fitting has a tendency of decay from $\textrm{C}_n^2\textrm{ = 3}\textrm{.0} \times {10^{ - 15}}\textrm{ }{\textrm{m}^{{{\textrm{ - 2}} / \textrm{3}}}}$ to $C_n^2 = {10^{ - 15}}\textrm{ }{\textrm{m}^{{{\textrm{ - 2}} / \textrm{3}}}}$. Thus, it could be inferred that when $|{m - n} |\ge 1$, ${\eta _{m \to n}}$ with ${10^{ - 17}}\textrm{ } \le C_n^2 \le {10^{ - 15}}\textrm{ }{\textrm{m}^{{{\textrm{ - 2}} / \textrm{3}}}}$ could be described by exponential distribution, and the PDF calculated by the model in Section 3 is consistent with these deductions.
Figure 5 presents the empirical PDF of ${\eta _{m \to m}}$ with $m \in \{{4,6} \}$ for ${10^{ - 15}}\textrm{ } \le C_n^2 \le {10^{ - 14}}\textrm{ }{\textrm{m}^{{{\textrm{ - 2}} / \textrm{3}}}}$ and the corresponding PDF calculated by the proposed model. As is shown, these PDFs are very close to each other, thus confirming the effectiveness of the proposed model and the algorithm of the attenuation in Section 3.2. Figure 6 shows the PDF of ${\eta _{m \to n}}$ with $|{m - n} |= 1$ calculated by the model for different m and $C_n^2$. For each situation, the empirical PDF is also given. As is shown, the model could describe the statistical characteristic of ${\eta _{m \to n}}$ when $|{m - n} |= 1$. Besides, the probability of large value of ${\eta _{m \to n}}$ is higher when the fluctuation is stronger. For example, a comparison of (a) and (c) in Fig. 6 reveals that ${\eta _{2 \to 1}},{\eta _{4 \to 3}}$ are more likely to be closed to $0$ with weaker turbulence.
Figure 7 plots the ${R^2}$ of the fit test for the PDF of ${\eta _{m \to n}}$ from the model when $|{m - n} |\ge 2$. The value of ${R^2}$ rises rapidly and is above $0.97$ when $C_n^2$ reaches $2 \times {10^{ - 15}}\textrm{ }{\textrm{m}^{{{\textrm{ - 2}} / \textrm{3}}}}$. The model has excellent fit for the measured data when $2 \times {10^{ - 15}} \le C_n^2 \le 1 \times {10^{ - 14}}{\textrm{m}^{{{\textrm{ - 2}} / \textrm{3}}}}$ but does not perform well when $1 \times {10^{ - 15}} \le C_n^2 \le 2 \times {10^{ - 15}}{\textrm{m}^{{{\textrm{ - 2}} / \textrm{3}}}}$, which can be avoided by building another model to cover this range. In all, the proposed model and algorithm can enable the PDF to be calculated from the proposed model of the attenuation and crosstalk to express ${\eta _{m \to n}}$ when $2 \times {10^{ - 15}} \le C_n^2 \le 1 \times {10^{ - 14}}{\textrm{m}^{{{\textrm{ - 2}} / \textrm{3}}}}$. Therefore, the algorithms for getting the parameters of the statistical models in Section 3 are proved to be effective.
Then, the attenuation ${\eta _{6 \to 6}}$, which describes the fraction of power received on OAM mode 6 when the transmitted power is carried by the vortex light with OAM mode 6, is taken as an example, and the statistical models of the attenuation for ${10^{ - 16}} \le C_n^2 \le {10^{ - 15}}{\textrm{m}^{{{\textrm{ - 2}} / \textrm{3}}}}$ and ${10^{ - 17}} \le C_n^2 \le {10^{ - 16}}{\textrm{m}^{{{\textrm{ - 2}} / \textrm{3}}}}$ are then built. Because $C_n^2 \le {10^{ - 15}}{\textrm{m}^{{{\textrm{ - 2}} / \textrm{3}}}}$, the attenuation ${\eta _{6 \to 6}}$ could be expressed by the GGD distribution, and the statistical model with $\varpi $ being set as 0 in Section 3.2 is chosen as the statistical model of the attenuation ${\eta _{6 \to 6}}$. Figure 8 and Fig. 9 plot the empirical PDF and the PDF calculated from statistical model of the attenuation for ${10^{ - 16}} \le C_n^2 \le {10^{ - 15}}{\textrm{m}^{{{\textrm{ - 2}} / \textrm{3}}}}$ and ${10^{ - 17}} \le C_n^2 \le {10^{ - 16}}{\textrm{m}^{{{\textrm{ - 2}} / \textrm{3}}}}$, respectively. As can be seen, the PDF calculated from the statistical model of attenuation has a perfect match with the corresponding empirical PDF for different turbulence strengths in weak turbulence region, thus confirming the effectiveness of the proposed statistical model.
Fig. 8. The partial result of the statistical model for ${10^{ - 16}} \le C_n^2 \le {10^{ - 15}}{\textrm{m}^{{{\textrm{ - 2}} / \textrm{3}}}}$.
Fig. 9. The partial result of the statistical model for ${10^{ - 17}} \le C_n^2 \le {10^{ - 16}}{\textrm{m}^{{{\textrm{ - 2}} / \textrm{3}}}}$.
As for the crosstalk, there are two statistical models in Section 3.3. Specifically, the second model would be adopted when the PDF of the crosstalk could reduce to exponential distribution. For example, when $C_n^2 \le {10^{ - 15}}{\textrm{m}^{{{\textrm{ - 2}} / \textrm{3}}}}$, the crosstalk ${\eta _{6 \to m}}$ with $|{6 - m} |\ge \textrm{1}$ can be represented by the exponential distribution, then the second model is used as the statistical model of ${\eta _{6 \to m}}$. Otherwise, the first statistical model would be adopted. Figure 10 plots the ${R^2}$ of model fitting to the empirical PDF for ${10^{ - 16}} \le C_n^2 \le {10^{ - 15}}{\textrm{m}^{{{\textrm{ - 2}} / \textrm{3}}}}$ and ${10^{ - 17}} \le C_n^2 \le {10^{ - 16}}{\textrm{m}^{{{\textrm{ - 2}} / \textrm{3}}}}$. As can be seen, the value of ${R^2}$ is greater than 0.98 in most cases, confirming the effectiveness of the proposed statistical model for ${10^{ - 17}} \le C_n^2 \le {10^{ - 15}}{\textrm{m}^{{{\textrm{ - 2}} / \textrm{3}}}}$.
Fig. 10. ${R^2}$ of the empirical and model PDFs of the crosstalk for ${10^{ - 17}} \le C_n^2 \le {10^{ - 15}}{\textrm{m}^{{{\textrm{ - 2}} / \textrm{3}}}}$.
4. Simulation result and comparison
Here, the outage probability has been calculated with the SISO transmission scheme, in which the vortex light with OAM mode m is transmitted, and the transmit optical power is P with photodetector responsivity $\zeta $. The variance of Gaussian white noise is ${\sigma ^2}$. The received electrical SNR can be expressed as
The outage probability is the probability when SNR is failing to achieve a prescribed threshold ${\gamma _{th}}$, and it can be represented as
Then, set the $\frac{{\sqrt {\textrm{2}{\sigma ^2}{\gamma _{th}}} }}{{P\zeta }}$ as the normalized threshold $\bar{x}$. In this work, ${\eta _{mm}}$ can be expressed by the mixture EGG distribution. Thus, the outage probability can be represented as:
In this work, the vortex beam with OAM mode 4 is considered. Figure 11 plots the simulation and theoretical outage probability results in the SISO transmission scheme. For each turbulence strength condition, the simulation has a good fit with the corresponding theoretical result, which further confirming the effectiveness of the proposed statistical model and algorithm. Moreover, the variation of the outage probability versus the normalized threshold were also given for different turbulence strengths. As can be found, for fixed normalized threshold, the outage probability increases with the increasing turbulence strength. That is, when the turbulence becomes stronger, SNR is more likewise failing to achieve a prescribed threshold for a fixed normalized threshold.
Fig. 11. The relationship between the outage probability and normalized threshold for different turbulence strengths in weak turbulence.
In the SISO transmission scheme, the vortex light with OAM mode m is transmitted. The average SNR $\bar{\gamma }$ is set as ${{{P^2}{\zeta ^2}} / {\textrm{2}{\sigma ^2}}}$ and the received electrical SNR can be given as
In this work, ${\eta _{mm}}$ can be also expressed by mixture EGG distribution and when $\varpi$ is set to 0, EGG distribution would reduce to GGD. Therefore, the PDF of ${\gamma _1}$ is
Thus, the average BER for OOK modulation scheme with IM/DD technique can be expressed as [48]:
Here, the BER characteristic of the OOK modulated SISO FSO communication system with different weak turbulence strengths has been given in Fig. 12, in which the vortex light with OAM mode 6 is transmitted. For each turbulence strength, the approximate BER result for high SNR and the simulation are also given. As can be observed, the BER result from Eq. (39) would have good match with the simulation when the average SNR increases. In addition, the theoretical result from Eq. (38) shows a good match with the simulation, further confirming the effectiveness of the proposed statistical model and algorithm in this work.
Then, the proposed models are briefly applied to the OAM-multiplexing with QPSK modulation in FSO systems. The transmission scheme is described in Fig. 13. First, the LG beam is split into several portions and modulated with information. They are then merged into one beam and transmitted through the atmospheric turbulence. At the receiving terminal, they are de-multiplexed into several beams with different OAM modes and converted into Gaussian beam. The power loss introduced by multiplexing and de-multiplexing processes is not considered in the simulations. Thus, the received signal is [50] as follows:
According to Eq. (40), the average electrical signal-to-noise ratio of channel $\hat{m}$ is defined as [51,52]
Here, an OAM-multiplexing $\{{3,4,5,6} \}$ communication system is considered for example when $C_n^2 = 4.9 \times {10^{ - 15}}\textrm{ }{\textrm{m}^{{{\textrm{ - 2}} / \textrm{3}}}}$. Therefore, the statistical models of ${\eta _{{l_{\hat{m}}} \to {l_{\hat{n}}}}}$ can be chosen according to different values of $|{{l_{\hat{m}}} - {l_{\hat{n}}}} |$. Specifically, when $|{{l_{\hat{m}}} - {l_{\hat{n}}}} |= 0,1\textrm{ and } \ge 2$, ${\eta _{{l_{\hat{m}}} \to {l_{\hat{n}}}}}$ can be expressed by the model given in Section 3.2 and the first and second models in Section 3.3, respectively. For example, ${\eta _{3 \to 3}}$ described by the model in Section 3.2 can express the statistical information of channel ${l_{\hat{m}}} = 3$ self-fading, and ${\eta _{3 \to 4}}$ described by the first model in Section 3.3 can express the statistical information of channel interference from channel ${l_{\hat{m}}} = 3$ to ${l_{\hat{n}}} = 4$. Figure 14 compares the BER calculated by Eq. (42) with the simulated data when $C_n^2 = 4.9 \times {10^{ - 15}}{\textrm{m}^{{{\textrm{ - 2}} / \textrm{3}}}}$. With ${{{P^2}{\zeta ^2}} / {2{\sigma ^2}}}$ increasing, the BER is lower, and the theoretical results and simulations match well, which further confirming the effectiveness of the proposed model to evaluate the performance of communication systems.
Fig. 14. BER vs ${{{P^2}{\zeta ^2}} / {2{\sigma ^2}}}$ for OAM multiplexing with $\{{3,4,5,6} \}$ when $C_n^2 = 4.9 \times {10^{ - 15}}\textrm{ }{\textrm{m}^{{{\textrm{ - 2}} / \textrm{3}}}}$.
5. Conclusions
In this work, a new statistical model was proposed, in which the parameters could be expressed by the analytical functions of $C_n^2$ to characterize the attenuation and crosstalk for the whole weak turbulence regime in FSO communication systems. First, the mixture EGG distribution as the unified PDF was used to describe the attenuation and crosstalk. Then, the conditions of reduction to GGD and the exponential distribution function were given, and the analytical formulas about the relations between $C_n^2$ and the parameters of PDF were provided when $n = m,|{n - m} |= 1,|{n - m} |\ge 2$, respectively. Next, the statistical models and algorithms for getting their parameters were proposed. For the algorithms, the problems of obtaining parameters were converted into optimizations and solved based on the trust region algorithm. The statistical models were also built in ${10^{ - 15}} \le C_n^2 \le {10^{ - 14}}{\textrm{m}^{{{\textrm{ - 2}} / \textrm{3}}}}$ as an example, and the channel statistical information was obtained. Furthermore, this analytical model was applied in an OAM multiplexing FSO system, and its reliability was verified. This work is helpful to study FSO time-varying systems using OAM.
Funding
National Natural Science Foundation of China (62071365).
Disclosures
The authors declare no conflicts of interest.
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