1. Introduction

Orbital-angular-momentum (OAM) modes with various indices form an infinite-dimensional orthogonal basis spanning the two-dimensional spatial domain. It is known that optical waves with a helical phase front carry OAM [1–3], which is a spatial degree of freedom of the optical waves that can be utilized in both classical and quantum communications [4–12]. For instance, recently great enthusiasm has been aroused in use of orthogonal OAM modes to implement the OAM mode division multiplexing (MDM) and OAM shift keying in classical optical communications [4–8]. In addition, photonic OAM modes are also used in experimental demonstrations of quantum key distribution (QKD) [9–11]; in comparison with the traditional polarization encoding, encoding with use of photonic OAM modes permits more information to be packed into a photon, and thus can enhance the QKD’s robustness and key generation rate [9,10]. Although OAM modes have the above mentioned advantages, they are susceptible to the turbulence-induced distortion when related application systems are operated in the earth’s atmosphere. Hence, thorough understanding of various effects of turbulence on optical OAM mode propagation is of practical importance to designing and optimizing both the OAM-based classical and quantum communication systems.

Up to now, researchers over the world have studied different aspects of optical OAM mode propagation in atmospheric turbulence from different perspectives, e.g., intermodal crosstalk [5,6,13–16], detection probability [2,17,18], entanglement decay [19–21], etc. Besides these aspects, turbulence makes the transmission coefficient, i.e., the transmittance, of optical OAM channels fluctuate randomly, and thus the transmitted signal through the channels will undergo stochastic fading. Conceptually, this resembles somewhat to the irradiance fluctuations in optical waves propagating through atmospheric turbulence. The statistical behavior of optical irradiance fluctuations is a topical issue for traditional free-space optical communications using a power-in-the-bucket (PIB) receiver [22]. The lognormal and Gamma-Gamma distributions [23] have been widely used to model the optical irradiance fluctuations in weak and weak-to-strong atmospheric turbulence, respectively. It should be pointed out that the mathematical model for the signal power detected by a PIB receiver is completely different from the one for the detected signal power of an optical OAM channel. This will become apparent later. For that reason, currently it is indeed not sure whether the existing statistical distribution models for optical irradiance fluctuations are applicable to description of transmission-coefficient fluctuations of turbulent optical OAM channels.

Fluctuations in the transmission coefficient of turbulent optical OAM channels, on the one hand, may play a role in evaluating the performance of OAM-based classical optical communication systems, and, on the other hand, are also important for analyzing the security and key generation rate of a QKD system with use of OAM photons. To deeply understand the statistical behavior of the transmission-coefficient fluctuations, it is imperative to mathematically formulate the statistical distribution of the transmission coefficient. Funes et al. [24] have fitted the linear-scale transformed Johnson $S_B$ distribution to measured randomly fluctuating signal samples of two turbulent optical OAM channels with propagation distances of 84 m and 400 m, respectively; although the authors have found that the linear-scale transformed Johnson $S_B$ distribution fits well with their specific experimental data, it is unclear whether the linear-scale transformed Johnson $S_B$ distribution is appropriate for statistical description of the transmission coefficient of optical OAM channels under turbulence conditions different from those in [24]. Consequently, knowledge of the statistical distribution of the transmission coefficient for a turbulent optical OAM channel is currently still quite incomplete. The main purposes of this work are first to mathematically characterize the statistical distribution of the transmission coefficient of turbulent optical OAM channels in detail, then to provide an insight into fundamental quantities that play decisive roles in determining the control parameters of the statistical distribution, and finally to explore how these quantities affect the statistical distribution.

2. Approach to modeling the statistical distribution of transmission coefficient of turbulent optical OAM channels

Generally speaking, it is very difficult to directly develop a statistical distribution model appropriate for the fluctuating transmission coefficient of a turbulent optical OAM channel by analytically mathematical deduction. For this reason, we intend to first use Monte Carlo simulations to generate random samples of the transmission coefficient and then explore the histogram of these samples, thereby proposing an appropriate distribution model for the fluctuating transmission coefficient. In what follows, firstly, we theoretically formulate the instantaneous transmission coefficient; subsequently, based on the developed formulations, we describe the Monte-Carlo-based method for generating random transmission-coefficient samples; lastly, we present the identification of the statistical distribution model for the transmission coefficient.

2.1 Instantaneous transmission coefficient of turbulent optical OAM channels

Below, we mathematically formulate the instantaneous transmission coefficient of a turbulent optical OAM channel. To this end, here we consider that a given OAM-mode wave field is propagated through an atmospheric free-space channel from a transmitter to a receiver, and the receiver can detect only the power contained in the sent OAM mode; moreover, we further assume that the OAM mode is initially a Laguerre-Gaussian (LG) mode [1,2]

It is noted that many existing theoretical and experimental researches relevant to optical OAM mode propagation have used a random-phase-screen model to approximately represent the effects of turbulence on OAM mode detection (see, e.g., [2,5,6,11,14–21,26,27]). Although this approximation is not strictly valid under strong-scintillation conditions, it is generally an accepted practice when weak-scintillation cases are considered. In this paper, we will follow the said treatment, that is, we model the turbulence-induced distortion via a random phase screen. With this in mind, the wave field of a propagated LG mode with a fixed radial index of zero in the presence of atmospheric turbulence can be expressed by

It is noted that the total power contained in the sent OAM mode is

2.2 Generation of random transmission-coefficient samples based on Monte Carlo simulations

It should be emphasized that $\varphi (r,\theta )$ in Eq. (11) actually denotes a random phase screen. Consequently, $W(r)$ is a random quantity, which further causes that ${{T}_{l}}$ takes on a random value. In this subsection, we mainly elucidate how to generate random samples of ${{T}_{l}}$ by use of Monte Carlo simulations. It is apparent that the key to generating a sample of ${{T}_{l}}$ is the creation of a random realization of the phase screen $\varphi (r,\theta )$. Various types of methods are currently available to us for generating random realizations of phase screens relevant to turbulence-induced phase perturbations, e.g., the Fourier-series-based method, covariance-based method, sparse-spectrum-based method, and so forth (see the literature review in [28]). Notice that, use of uniform spatial sampling grids in polar coordinates instead of Cartesian coordinates in generating $\varphi (r,\theta )$ may benefit the numerical evaluation of the integration over $\theta$ in Eq. (11). For this reason, here we use the sparse-spectrum-based method [29] to generate random realizations of the phase screen, because it can easily accommodate to the said uniform spatial sampling scheme in polar coordinates.

The sparse-spectrum-based method given by [29] expresses the phase screen as follows:

To facilitate the execution of Monte Carlo simulations, here we further reformulate Eqs. (9)–(11) to find nondimensional parameters that play decisive roles in determining ${{T}_{l}}$. In [17], the average detection probability of OAM photons in atmospheric turbulence has been plotted in terms of the ratio of the LG beam width to the Fried’s atmospheric coherence width ${{r}_{0}}$. Inspired by this fact, by letting $\hat {r}=r/{{r}_{0}}$, we find

It is found from Eqs. (17)–(19) that the nondimensional parameters $\Omega$ and $l$ are two crucial parameters in evaluation of ${{T}_{l}}$. We note that $\hat {W}(\hat {r})$ is visually independent of both $\Omega$ and $l$. However, it seems that $\hat {W}(\hat {r})$ depends on ${{r}_{0}}$. As pointed out in [29], the Fried’s atmospheric coherence width ${{r}_{0}}$ essentially plays a role of a phase “magnitude” scaling factor. If ${{\tilde {\varphi }}^{({{r}_{0}})}}(r,\theta )$ denotes a random realization of a phase screen associated with the Fried’s atmospheric coherence width ${{r}_{0}}$, ${{\varepsilon }^{-5/6}}{{\tilde {\varphi }}^{({{r}_{0}})}}(r,\theta )$ is a random realization of a phase screen associated with the Fried’s atmospheric coherence width ${{{r}'}_{0}}=\varepsilon {{r}_{0}}$. Figure 1 illustrates the random realizations of the weighting factor $\hat {W}(\hat {r})$ in terms of $\hat {r}$ with various ${{r}_{0}}$. To create each subplot of Fig. 1, we first generate random realizations for ${{a}_{n}}$ and ${{\mathbf {K}}_{n}}$, respectively, with ${{r}_{0}}$ specified as 1 cm ($n=1,2,\ldots ,500$); subsequently, we use Eq. (14) to generate phase-screen realizations for ${{r}_{0}}$ = 5 cm and 1 m, respectively, together with the said scaling operation on the phase “amplitude”. Hence, the phase screen realizations for various ${{r}_{0}}$ in the same subplot of Fig. 1 have a similar spatial structure feature but different phase “amplitude”. It is found from Fig. 1 that the curves therein for various ${{r}_{0}}$ are obviously different. This implies that ${{r}_{0}}$ may play a distinctive role in determining the statistical distribution of the transmission coefficient. In the next subsection, we will reveal how ${{r}_{0}}$ affects the statistical distribution of the transmission coefficient by comparing the empirical cumulative distribution functions (CDFs) [30] related to transmission-coefficient samples generated by using different ${{r}_{0}}$. According to Eqs. (17)–(19), it is apparent that we can reuse the generated random realizations for $\hat {W}(\hat {r})$ in calculation of the transmission-coefficient samples for identical ${{r}_{0}}$ but different combinations of $\Omega$ and $l$. In performing our later Monte Carlo simulations, we first generate different random realization series for $\hat {W}(\hat {r})$ with different ${{r}_{0}}$, and then calculate the sample values of ${{T}_{l}}$ for different combinations of $\Omega$, $l$ and ${{r}_{0}}$ according to Eq. (17).

 

Fig. 1. Random realizations of the weighting factor $\hat {W}(\hat {r})$ in terms of $\hat {r}$ with different Fried’s atmospheric coherence width ${{r}_{0}}$, where the number of the spectral components, $N$, of the phase screen is 500, and the sampling grid of the phase screen over the two-dimensional space $(r,\theta )$ contains $1000\times 1000$ points.

Download Full Size | PDF

Before proceeding any further, we need to discuss a subtle issue related to the Monte Carlo simulations described above. We note that, for not heavily burdening the understanding of the essential physics of optical OAM mode propagation in atmospheric turbulence, the canonical pure Kolmogorov model for the turbulence with an inner scale of zero and an outer scale of infinity has been widely considered in the relevant published literature (see, e.g., [2,5,6,11,14,15,17,18,20,21,26,27] where the inner and outer scales were not mentioned at all). We will follow the same practice in this work. However, finite low- and high-frequency cutoff wavenumbers of the turbulence spectrum have to be specified in the said sparse-spectrum-based method for generating random realizations of the phase screen. This in turn means that finite inner and outer scales should be considered in generation of random phase screen realizations. We point out that, due to the finite and discrete sampling nature, it is generally not possible for any phase screen generation method to create random phase screens that can exactly reproduce the pure Kolmogorov spectrum with an inner scale of zero and an outer scale of infinity. In other words, practical random phase screens even generated by other methods, with the pure Kolmogorov spectrum as the intended spectrum, should also be regarded as ones generated with suitable finite inner and outer scales. In our Monte Carlo simulations, we let the inner and outer scales of turbulence be 1 mm and 100 m, respectively. This results in the ratio of the outer scale to inner scale equal to ${{10}^{5}}$, thus providing us with a wide enough spectral interval that is consistent with the canonical pure Kolmogorov spectrum.

2.3 Statistical distribution model for the transmission coefficient

It has been pointed out in [31] that, under weak-scintillation conditions, the roles of all the adjustable dimension parameters in formulating the output density matrix of OAM photonic quantum states propagating through turbulence can be completely incorporated into a nondimensional parameter like the quantity $\Omega$, meaning that the role of ${{r}_{0}}$ therein can actually be completely played by $\Omega$. In this subsection, we begin by examining whether this still remains valid for the statistical distribution of the transmission coefficient. Figure 2 shows the empirical CDFs of the transmission coefficient with different combinations of $\Omega$, $l$ and ${{r}_{0}}$, which are calculated according to transmission-coefficient samples generated by Monte Carlo simulations. It can be found from Fig. 2 that there exists a discernible difference between the two curves in each subplot associated with given $\Omega$ and $l$. Hence, ${{r}_{0}}$ does have a distinctive impact on the statistical distribution of the transmission coefficient; more specifically, the role of ${{r}_{0}}$ cannot be completely incorporated into $\Omega$. In addition, one can see from Fig. 2 that the ${{r}_{0}}\textrm {-dependence}$ behavior of the statistical distribution of the transmission coefficient is not changed sharply with varying $l$; however, unlike $l$, changing $\Omega$ will obviously vary the ${{r}_{0}}\textrm {-dependence}$ behavior of the statistical distribution of the transmission coefficient.

 

Fig. 2. Empirical CDFs of the transmission coefficient ${{T}_{l}}$ with various combinations of $\Omega$, $l$ and ${{r}_{0}}$, which are calculated according to transmission-coefficient samples generated by Monte Carlo simulations. The number of transmission-coefficient samples used to create each curve is 20000. (a) $\Omega$ = 0.1, $l$ = 0; (b) $\Omega$ = 1, $l$ = 0; (c) $\Omega$ = 10, $l$ = 0; (d) $\Omega$ = 0.1, $l$ = 5; (e) $\Omega$ = 1, $l$ = 5; (f) $\Omega$ = 10, $l$ = 5; (g) $\Omega$ = 0.1, $l$ = 10; (h) $\Omega$ = 1, $l$ = 10; (i) $\Omega$ = 10, $l$ = 10.

Download Full Size | PDF

Now we turn to the mathematical characterization of the statistical distribution of ${{T}_{l}}$. It is evident that ${{T}_{l}}$ = 1 when turbulence vanishes. Further, it is easy to prove that $0\le W(r)\le 1$. By recognizing that $\int _{0}^{\infty }{\rho (r)\textrm {d}r}\equiv 1$, one finds that $0\le {{T}_{l}}={{P}_{l}}\le 1$. With this in mind, one can infer that the statistical distribution of ${{T}_{l}}$ should have a support from 0 to 1. Hence, the common probability distributions with a support from 0 to $\infty$, such as the lognormal and Gamma-Gamma distributions, are not very appropriate for describing the statistical behavior of the transmission coefficient. In [24], the linear-scale transformed Johnson ${{S}_{B}}$ distribution with four independent control parameters has been fitted to the measured random signal fluctuations of turbulent optical OAM channels. We note that the standard Johnson ${{S}_{B}}$ distribution, whose probability density function (PDF) is given by [32]

Figure 3 shows the histogram of transmission-coefficient samples generated by Monte Carlo simulations. One finds from Fig. 3 that the peak of the histogram shifts towards the right side when $\Omega$ is much smaller than 1, and, however, towards the left side when $\Omega$ is close to or larger than 1. It is seen from Fig. 3 that, when $\Omega \ll 1$, ${{T}_{l}}$ occurs with high probability in a subrange near to 1; on the contrary, when $\Omega \gg 1$, ${{T}_{l}}$ has high likelihood to take on a value in a subrange near to 0. When $\Omega$ is within the subrange adjacent to 1, the shape of the histogram of the simulated samples becomes more complicated than those related to $\Omega \ll 1$ and $\Omega \gg 1$. We have tried to fit the standard Johnson ${{S}_{B}}$ distribution based on the maximum likelihood estimate (MLE) method to the simulated samples. It is seen from Figs. 3(b)–3(d) that the histograms therein have a heavy left- or right-side tail besides the main peak when $\Omega$ is somewhat close to 1. In these cases, it is indeed difficult to find an acceptable fit of the standard Johnson ${{S}_{B}}$ distribution to the simulated samples. To overcome this difficulty, by keeping in mind the said heavy left- or right-side tail of the histograms in Figs. 3(b)–3(d), we propose the new distribution

 

Fig. 3. Histogram of transmission-coefficient samples generated by Monte Carlo simulations, where $l$ = 1 and ${{r}_{0}}$ = 10 cm. The number of transmission-coefficient samples used to create each histogram is 20000. The green dashed curve in each subplot is a fit of a standard Johnson ${{S}_{B}}$ distribution to the samples. The red solid curve in each subplot is a fit of a dual Johnson ${{S}_{B}}$ distribution to the samples. (a) $\Omega$ = 0.1; (b) $\Omega$ = 0.4; (c) $\Omega$ = 0.8; (d) $\Omega$ = 1; (e) $\Omega$ = 4; (f) $\Omega$ = 8.

Download Full Size | PDF

At this point, we emphasize that, like the dual Johnson ${{S}_{B}}$ distribution, the said linear-scale transformed Johnson ${{S}_{B}}$ distribution used in [24] also has four independent control parameters. As stated previously, when the transformed lower and upper bounds of the linear-scale transformed Johnson ${{S}_{B}}$ distribution are not equal to 0 and 1, respectively, the distribution will have a support that is not exactly between 0 and 1. Due to the fact $0\le {{T}_{l}}\le 1$, a distribution suitable for statistical description of the fluctuating transmission coefficient should have a support from 0 to 1. For this reason, if we use the linear-scale transformed Johnson ${{S}_{B}}$ distribution to model the statistical behavior of the fluctuating transmission coefficient, strictly speaking, its transformed lower and upper bounds should be specified as fixed values of 0 and 1, respectively. Notice that, the standard Johnson ${{S}_{B}}$ distribution is indeed the linear-scale transformed Johnson ${{S}_{B}}$ distribution with its transformed lower and upper bounds equal to 0 and 1, respectively. In contrast to the linear-scale transformed Johnson ${{S}_{B}}$ distribution, the dual Johnson ${{S}_{B}}$ distribution always has a support from 0 to 1, without the need to let two of its four control parameters take on fixed values in the probability distribution fitting.

3. Analysis of statistical distribution of the transmission coefficient

The relation between $(\Omega ,l,{{r}_{0}})$ and $({{\gamma }_{1}},{{\delta }_{1}},{{\gamma }_{2}},{{\delta }_{2}})$ can be regarded as a mapping from a three-dimensional space to a four-dimensional space. Although it is difficult to develop analytical expressions for description of this mapping, a graphical display of the dependence of $({{\gamma }_{1}},{{\delta }_{1}},{{\gamma }_{2}},{{\delta }_{2}})$ on $(\Omega ,l)$ with various ${{r}_{0}}$ can be put on by utilization of Monte Carlo simulations and MLE-based probability distribution fitting. Figure 4 illustrates the dependence of $({{\gamma }_{1}},{{\delta }_{1}},{{\gamma }_{2}},{{\delta }_{2}})$ on $(\Omega ,l)$ with various ${{r}_{0}}$. One can find the underlying values for the meshes in Fig. 4 from Code 1 [33] of the supplementary materials used to plot them; the matrices in Code 1 [33] can be regarded as “lookup tables” for ${{\gamma }_{1}}$, ${{\delta }_{1}}$, ${{\gamma }_{2}}$ and ${{\delta }_{2}}$, which may be useful for other relevant researches. We point out that the underlying values of the meshes in Fig. 4 are computed by the MLE-based probability distribution fitting instead of analytical expressions. Although finding a model for the mapping from $(\Omega ,l,{{r}_{0}})$ to $({{\gamma }_{1}},{{\delta }_{1}},{{\gamma }_{2}},{{\delta }_{2}})$ is beyond the scope of the main purposes of this work, it does deserve further study in the future. In plotting Fig. 4, we let ${{\log }_{10}}(\Omega )$ increase from $-2$ to 1.8 in steps of 0.2, and let the OAM index $l$ increase from 0 to 16 in steps of 1. It is evident from Eqs. (17) and (18) that ${{T}_{l}}\equiv {{T}_{-l}}$. For this reason, we do not need to consider a negative OAM index at all. The above sampling ranges of the parameter space $(\Omega ,l)$ actually cover various cases that are often of practical interest to us. Moreover, in Fig. 4, we let the Fried’s atmospheric coherence width ${{r}_{0}}$ achieve the values of 1 cm, 10 cm, 1 m and 10 m, which include typical orders of magnitude of the Fried’s atmospheric coherence width under various practical turbulence conditions. It is found from Fig. 4 that, for a given ${{r}_{0}}$, the mesh related to ${{\gamma }_{1}}$ is not the same as the one corresponding to ${{\gamma }_{2}}$; a similar statement also applies to ${{\delta }_{1}}$ and ${{\delta }_{2}}$. This phenomenon reveals from another perspective that the standard Johnson ${{S}_{B}}$ distribution is not very appropriate for statistical characterization of the transmission coefficient. One can see from Fig. 4 that, although the meshes associated with different ${{r}_{0}}$ in the same column exhibit a degree of resemblance, they do have ${{r}_{0}}\textrm {-dependence}$, i.e., there do exist observable differences between them. This justifies the aforementioned inference from Fig. 2 that the role of ${{r}_{0}}$ in determining the statistical distribution of the transmission coefficient cannot be completely incorporated into the nondimensional parameter $\Omega$. Furthermore, one can find from Fig. 4 that changing $l$ generally does not make the values of ${{\gamma }_{1}}$, ${{\delta }_{1}}$, ${{\gamma }_{2}}$ and ${{\delta }_{2}}$ alter significantly except a few individual cases. Nevertheless, a variation in ${{\log }_{10}}(\Omega )$ will lead to obvious changes in the values of ${{\gamma }_{1}}$, ${{\delta }_{1}}$, ${{\gamma }_{2}}$ and ${{\delta }_{2}}$. Thus, a statement can be made that the meshes in Fig. 4 have relatively weak dependence on $l$ and, however, have relatively strong dependence on ${{\log }_{10}}(\Omega )$.

 

Fig. 4. Dependence of ${{\gamma }_{1}}$, ${{\delta }_{1}}$, ${{\gamma }_{2}}$ and ${{\delta }_{2}}$ on $(\Omega ,l)$ with different ${{r}_{0}}$. ${{r}_{0}}$ = 1 cm for (a)–(d); ${{r}_{0}}$ = 10 cm for (e)–(h); ${{r}_{0}}$ = 1 m for (i)–(l); ${{r}_{0}}$ = 10 m for (m)–(p). For each combination of $(\Omega ,l,{{r}_{0}})$, we first generate 20000 transmission-coefficient samples by employing Monte Carlo simulations, and then calculate the values of ${{\gamma }_{1}}$, ${{\delta }_{1}}$, ${{\gamma }_{2}}$ and ${{\delta }_{2}}$ by fitting the dual Johnson ${{S}_{B}}$ distribution to the histogram of the 20000 transmission-coefficient samples. The underlying values for these meshes can be found in the Matlab codes used to plot them (see Code 1 [33] of the supplementary materials).

Download Full Size | PDF

To further explore how variations in $\Omega$ and ${{r}_{0}}$ affect the statistical distribution of the transmission coefficient, we plot the PDF of the transmission coefficient with different $\Omega$ and ${{r}_{0}}$ in Fig. 5. It is seen from Fig. 5 that, with the same $\Omega$, the curves corresponding to different ${{r}_{0}}$ basically take on a somewhat similar shape despite the existence of small distinction between them; however, with the same ${{r}_{0}}$, the curves corresponding to various $\Omega$ distinctly show a conspicuous difference. Accordingly, although the role of ${{r}_{0}}$ in determining the statistical distribution of the transmission coefficient cannot be completely incorporated into $\Omega$, the effect of ${{r}_{0}}$ on the PDF of the transmission coefficient is much weaker than that of $\Omega$ on the PDF of the transmission coefficient. We point out that Figs. 5(a) and 5(f) indeed correspond to two opposite extreme cases where the RMS OAM-beam radius is much smaller and greater than ${{r}_{0}}$, respectively. A comparison of Fig. 5(a) with Fig. 5(f) reveals that ${{T}_{l}}$ has high likelihood to achieve a value near to 1 and 0 when $\Omega \ll 1$ and $\Omega \gg 1$, respectively. The signal-power reception scheme described in subsection 2.1 is essentially coherent mode detection. Generally speaking, the higher the probability of ${{T}_{l}}$ achieving a value close to 1 is, the fainter the turbulence-induced channel-transmittance fading actually becomes. In fact, $\Omega \ll 1$ implies that the whole field within the transverse spot of the OAM beam is almost coherent, thus causing a very weak effect of the turbulence-induced distortion on OAM mode detection. On the other hand, $\Omega \gg 1$ means that the field within the transverse spot of the OAM beam contains many independent coherent patches, hence resulting in a very strong effect of the turbulence-induced distortion on OAM mode detection. One finds from Fig. 5 that the spread of the statistical distribution of ${{T}_{l}}$ first widens with increasing $\Omega$ and subsequently begins to narrow after $\Omega$ grows beyond a specific value. Further, when $\Omega$ is of the same order of magnitude as 1 (see, e.g., Figs. 5(c) and 5(d)), the PDF may have an extraordinarily heavy tail, which cannot be well depicted by other existing PDF models. It is observed from Fig. 5 that the PDFs are obviously left skewed (i.e., with their heavier tails towards the left) when $\Omega \lesssim 1$, and are right skewed (viz., with their heavier tails towards the right) when $\Omega \ge 1$. Moreover, although the PDFs associated with $\Omega ={{10}^{-1.2}}$ show obvious asymmetry, those corresponding to $\Omega ={{10}^{1.4}}$ only feature every weak asymmetry (indeed, they look like normal distributions somewhat).

 

Fig. 5. PDFs of the transmission coefficient ${{T}_{l}}$ with $l\equiv 4$ and different $\Omega$ and ${{r}_{0}}$. The four control parameters of the dual Johnson ${{S}_{B}}$ distribution are determined according to the results shown in Fig. 4. (a) $\Omega ={{10}^{-1.2}}$; (b) $\Omega ={{10}^{-0.8}}$; (c) $\Omega ={{10}^{-0.4}}$; (d) $\Omega ={{10}^{0}}$; (e) $\Omega ={{10}^{0.6}}$; (f) $\Omega ={{10}^{1.4}}$.

Download Full Size | PDF

Although the variation in the OAM index $l$ only results in a change in the statistical distribution of the transmission coefficient that is rather small compared with the one caused by the variation in $\Omega$, the OAM index $l$ does play an independent role in determining the statistical distribution of the transmission coefficient. Now we turn to further examining how variations in $l$ affect the statistical distribution of the transmission coefficient with a given $\Omega$. The PDFs of the transmission coefficient with different $l$ and $\Omega$ are plotted in Fig. 6. It is seen from Fig. 6(a) that the curves with various $l$ basically merge together, implying that the variation in $l$ does not change the PDF obviously when ${{r}_{0}}$ is rather large compared to the RMS OAM-beam radius. On the other hand, there are observable differences between the curves in Figs. 6(b) and 6(c), where ${{r}_{0}}$ becomes not large in comparison with the RMS OAM-beam radius. We note that the spatial structure of an OAM wave field is dependent on $l$, and the transmission coefficient ${{T}_{l}}$ is always equal to 1 for various $l$ in vacuum. When $\Omega$ is much smaller than 1, the transverse spot of the OAM beam comprises approximately only one coherent patch, implying that the effect of turbulence-induced distortion on OAM mode detection is extremely small; hence, $l$ only plays a trivial role in determining the statistical distribution of the transmission coefficient with the given $\Omega$. However, when $\Omega \gtrsim 1$, the transverse spot of the OAM beam comprises more than one independent coherent patch and the combined effect of these patches on OAM mode detection depends on the spatial structure of the OAM wave field; consequently, the statistical distribution of the transmission coefficient with the given $\Omega$ has discernible $l\textrm {-dependence}$. Moreover, by comparing the curves in either Fig. 6(b) or Fig. 6(c), one finds that the magnitude of the distinction between any two curves therein is not completely determined by the absolute difference between their corresponding OAM indices; e.g., the distinction between the curves with $l$ = 0 and 1 is greater than that between the curves with $l$ = 1 and 2. Indeed, we can find from Figs. 6(b) and 6(c) that the change in the PDF, caused by an increment in $l$, gets smaller and smaller when $l$ becomes increasingly large. As a consequence, the independent role of $l$ in determining the statistical distribution of the transmission coefficient with a given $\Omega$ turns trivial when $l$ increases beyond a large enough value. This statement can also be justified by the meshes in Fig. 4; i.e., the meshes tend to be unchanged in the $l\textrm {-axis}$ direction when $l\gtrsim 12$.

 

Fig. 6. PDFs of the transmission coefficient ${{T}_{l}}$ with ${{r}_{0}}\equiv 1$ cm and different $l$ and $\Omega$. The four control parameters of the dual Johnson ${{S}_{B}}$ distribution are determined according to the results shown in Fig. 4. (a) $\Omega ={{10}^{-0.8}}$; (b) $\Omega ={{10}^{0}}$; (c) $\Omega ={{10}^{0.8}}$.

Download Full Size | PDF

4. Conclusions

In this paper, by employing Monte Carlo simulations and MLE-based probability distribution fitting, we have mathematically characterized the statistical distribution of the transmission-coefficient fluctuations of turbulent optical OAM channels. In this endeavor, the instantaneous transmission coefficient of a turbulent optical OAM channel has been theoretically formulated in the form of a weighted integration. It has been shown that the fluctuating transmission coefficient should range between 0 and 1, and thus cannot be properly statistically described by the common probability distribution models for optical irradiance fluctuations that possess a support from 0 to $\infty$. The novel dual Johnson ${{S}_{B}}$ distribution with a support from 0 to 1 has been found to be appropriate for description of the statistical behavior of the fluctuating transmission coefficient. Each of the four independent control parameters of the dual Johnson ${{S}_{B}}$ distribution can be regarded as a function of the OAM index $l$, the Fried’s atmospheric coherence width ${{r}_{0}}$, and the ratio $\Omega$ of the RMS OAM-beam radius to ${{r}_{0}}$; wherein, $\Omega$ plays the most prominent role in determining these control parameters and hence the shape of the PDF for the fluctuating transmission coefficient.

It has been revealed that, with given $\Omega$ and $l$, the PDFs with various ${{r}_{0}}$ rather resemble each other in their shapes although small differences between them exist. It has been demonstrated that an increment in $l$ can lead to a discernable change in the PDF only when $\Omega \gtrsim 1$, and the change turns smaller and smaller as $l$ grows increasingly great. It has also been found that the peak of the PDF shifts close to 1 and 0, respectively, when $\Omega \ll 1$ and $\Omega \gg 1$; the PDF with $\Omega \ll 1$ is obviously left skewed, whereas it with $\Omega \gg 1$ tends to take on a shape similar to a symmetric normal distribution; the PDF with either $\Omega \ll 1$ or $\Omega \gg 1$ can achieve an appreciable value only within a very narrow subrange of [0, 1], whereas the PDF with $\Omega \sim 1$ can achieve a considerable value basically over the whole range of [0, 1]; moreover, the PDF may have a heavy left- or right-side tail when $\Omega \sim 1$.

Our results are useful for deeply understanding the randomly fluctuating behavior of the transmittance of a turbulent optical OAM channel involved in both classical and quantum communications.