1. Introduction

Freespace optical (FSO) communications have always been the focus of data transfer applications [1]. [Abbreviations used in this paper and their corresponding full names are shown in Table 1 of the Appendix.] The orbital angular momentum (OAM) is a special light field with phase structure $\textrm{exp} (il\theta )$, where $\theta $ is the azimuth angle, l is the topological charge number, which can be arbitrary in theory, and the OAM modes with different l are orthogonal to each other [2]. OAM beams can be generated by means of spiral phase plates [3], diffractive optical elements [4], mode transducers [5] amongst other approaches. Based on the special spatial structure of OAM beams, they have been widely used in super-resolution microscopy [6], optical tweezers [7], and quantum science [8]. Also, due to the orthogonality of OAM beams [9] as well as their high degree of directionality and narrow divergence angles, multiplexing OAM beams in FSO communication links could significantly improve communication capacity and spectral efficiency, and has been heavily investigated for such purposes during the last decade [10–12]. However, because of atmospheric turbulence (AT), multiplexed OAM beams may experience crosstalk, which results in broadening of the OAM channels and limits the application of OAM modes to FSO communications [13–15]. In order to evaluate the negative impact of AT on OAM beams, modeling the OAM propagation link is necessary, an analysis of which may assist to systematically reduce the distortions [16,17]. Typically, the angular-spectrum propagation method has been adopted to theoretically analyze the multiplexing/demultiplexing of OAM beams in modeling the channel [18,19]. Based on the results of spatial modeling and angular-spectrum analysis, phase retrieval algorithms [20,21], adaptive-optics (AO) systems [22,23], and convolutional neural-network (CNN) models [24–26] have been proposed to improve OAM mode purity and reduce crosstalk. For example, Ref. [26] shows that crosstalk has been shown to be reduced from −23.15 dB to −29.46 dB under the influence of AT.

Nevertheless, AT is characterized not only by spatial-phase chaos properties, but also temporal-evolution chaos. In fact, time-varying (TV) AT leads to different turbulence distributions at different moments, which will cause non-negligible effects on OAM beams. Therefore, TV characteristics of AT in OAM-based FSO communications have attracted increasing attention in recent years [27–30]. In laboratories, TV AT is mimicked by rotating thin phase screen plates in multiplexed OAM-based FSO communication links [27,28]. In outdoor AT environment, time-domain propagation characteristics have also been investigated [29], as well as characterization of seven OAM modes after propagation through a 340-m outdoor FSO link is researched [30] and performance of OAM quantum key distribution with real-time AO correction has also obtained [29]. It has found that real-time AO correction is the most effective method for moderate AT [29], which is different from a previous spatial-turbulence investigation that concluded that the AO system is best suitable for weak AT [22,23]. The aforementioned TV AT studies have revealed that [27–30] power, crosstalk, and BER all have time-domain volatility in OAM-based FSO communication links. Thus, the time-domain propagation characteristics of multiplexed OAM beams in AT should be treated carefully.

Unfortunately, although the temporal evolution properties of AT have been considered in OAM-based FSO communication links [27–30], there remain several unsolved key issues associated with the temporal properties. First, using rotating phase screen plates in a laboratory setting, it is difficult to mimic the continuous variation of either AT strength $C_n^2$ or transverse wind speed v, two key parameters describing TV AT, and the appearance of repeated AT phase screens does not agree with the real AT phenomenon [27,28]. Second, for outdoor experiments, equipment and systems are both expensive and complex, thus limiting most current research to few specific locations at discrete fixed altitudes up to a few hundred meters [29,30]. Third, a real-time AO correction system requires expensive devices and needs excessive computation, thus results in high cost and difficulties in exploring a wide range of parameters [29]. Moreover, in practical, performance of correction is limited by the hardware utilized in the system, such as the refresh frequency of SLM, resolution, the beam diameter, size of receiving screen and so on. In summary, the issues just raised are consequences of the ability to perform only a finite number of experiments under a discrete set of conditions [27–30]. Thus, establishing a TV multiplexed OAM propagation model that conforms to realistic temporal evolution properties of AT could substantially advance our ability to mitigate the effects of TV AT on multiplexed OAM beams for FSO communications.

In this paper, in contrast to previous OAM-based FSO propagation models [16,17], in which only the spatial-distribution properties of AT are considered, the Taylor hypothesis (discussed below) is introduced into the proposed OAM propagation model to include the influence of the temporal evolution of AT. The model simulates time-varying atmospheric turbulence under the influence of wind speed by intercepting subscreens on the parent screen. Using the model, the spiral power spectrum [29] of multiplexed OAM beams in various TV AT is systematically analyzed. Moreover, the time-domain effectiveness of conventional Gaussian-probe wavefront correction in TV AT is evaluated. It was found that each phase correction is effective for a specific time that is subject to both $C_n^2$ and v, and such behavior is used to optimize OAM communication links in TV AT. So this paper presents a period compensation (PDC) method combining with bit error rate (BER) and optimized dynamic correction periods, which will save many computational resources and improve computational efficiency. An optimized result shows that, with a correction period of 0.18 s and a AT changing frequency of 100 Hz, both system BER within the forward-error-correction (FEC) limit and a low iterative computation volume with only 6% of the real-time correction (RTC) could be achieved under the moderate TV AT.

2. TV OAM multiplexed propagation model

In conventional OAM propagation models [16–26], AT is usually assumed to be stationary and only the spatial variation of the turbulence is considered. Here, in our proposed TV propagation model, in addition to considering the spatial properties of AT, the Taylor frozen-turbulence hypothesis is introduced to consider the time evolution of AT. The Taylor hypothesis has been widely used in fields such as meteorology [31] and aerospace [32], which has well explained the temporal evolution properties of the phase distortion through AT [33,34].

The proposed model is shown schematically in Fig. 1. First, several single-mode OAM beams (l₁, l₂, …, l_i, …, l_s) are multiplexed, where l_i is the topological charge of the OAM beam and s indicates the total number of multiplexed OAM modes. Then, the multiplexed OAM beams are transmitted through freespace with AT. According to the Taylor frozen-turbulence hypothesis, multiple continuous AT phase screens influenced by transverse wind speed v are obtained to simulate TV AT. These AT phase screens are arranged in chronological order and introduced into a conventional propagation model, thus multiplexed OAM modes dynamically influenced by TV AT can be treated. Finally, after passing through the AT, the distorted multiplexed OAM beams are demultiplexed back into single-mode OAM beams. To be compatible with the conventional model [16–26], in the proposed TV model, except for the new treatment of TV AT, all other aspects of our treatment conform to classical angular-spectrum theory.

 

Fig. 1. Schematic diagram of the propagation model of multiplexed OAM beams in TV AT. Mux = multiplexed; De-mux = demultiplexed. (a) The process of obtaining subscreens using the Taylor hypothesis. (b) Subscreens arranged in chronological order. (c) Conventional multiplexed OAM propagation link. See main text for further explanation.

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The key to model TV AT is how to introduce the temporal properties of AT into the conventional TV multiplexed OAM communication link. According to the Taylor hypothesis, for a very short period of time, the spatial structure of the AT is by default in a frozen state since the timescales for variation of TV AT are much slower than the relevant optical timescale as discussed below. Thus, during a finite duration, for a specific spatial location, the AT variation can be seen as a spatial atmospheric turbulence translation process caused by the blowing wind. The duration is defined as the atmospheric coherence time ${\tau _0}$ [35],

Assuming the Taylor hypothesis over the timescale set by ${\tau _0}$, the temporal properties of AT for a specific location are considered by translating different parts of the frozen spatial AT, as shown in Fig. 1(b). Distinct from the conventional OAM propagation model [16–26], the frozen spatial AT requires an AT distribution screen over a larger spatial scale (no less than $v \cdot {\tau _0}$) to have enough scale to be split into different parts for representing the AT at different locations, which are then arranged sequentially to model the TV AT. Apart from that, the frozen spatial AT should also satisfy the Kolmogorov turbulence theory as well as the conventional model [37,38]. The Kolmogorov theory describes spatial statistical properties of AT by defining an atmospheric coherence length ${r_0}$ [39,40],

From Eq. (2), ${r_0}$ depends mainly on $C_n^2(z)$. Also, if the Hufnagel-Valley model [41] is introduced at this point, the influence of AT at various altitudes could be considered in our model. Here, only horizontal propagation-link conditions are considered. Thus, the default refractive-index structure function and wind speed are both constant, that is, $C_n^2(z) = C_n^2$, $v(z) = v$. Consequently, Eqs. (1) and (2) can be simplified as

From Eqs. (3) and (4), the Taylor hypothesis relates the temporal properties with the spatial properties of AT depending on a small set of important parameters.

Based on the above method, as shown in Fig. 1(a), an initial static phase screen (the parent screen) is first generated according to ${r_0}$ [35] (the phase screen is generated by the power-spectrum-inversion method [20] and the subharmonic compensation theory [39], and ${r_0}$ is numerical foundation of these methods.), which is much larger than the beam aperture (transverse width of the beam) and represents the AT variations within ${\tau _0}$. Then, assuming a fixed nonzero v, subregions of the parent screen (subscreens), which represent the AT corresponding to different moments within ${\tau _0}$, are successively intercepted and then shifted along the optical axis of the propagation link. During the ${\tau _0}$, the total number of frames of the subscreens intercepted on one parent screen is recorded as N, which is inversely proportional to v.

Once the observation time exceeds ${\tau _0}$, the AT variation will no longer satisfy the Taylor hypothesis. Thus, the initial parent screen cannot continue to be regarded as frozen; it is necessary to replace the previous parent screen with another new parent screen that also satisfies the Kolmogorov statistical property. For this next time interval, a new AT strength and new transverse wind speed corresponding to the new interval should also be employed. In this vein, the total observation time t can be considered as a sequence of M atmospheric coherence periods ${\tau _{{0_1}}}$…, ${\tau _{{0_i}}}$, …, ${\tau _{{0_M}}}$. In the sequence, each ${\tau _{{0_i}}}$ has N_i frames of the subscreens, and each corresponds to different wind speed v_i. Thus, t satisfies

If v and $C_n^2$ are both constant, ${\tau _0}$ would be equal to each other, so Eq. (5) can be simplified as

Therefore, as shown in Fig. 1(b), $\sum\nolimits_{i = 1}^{i = M} {{N_i}} $ frames of subscreens that represent the spatio-temporal evolution of AT over observation time t are arranged in chronological order. Then as shown in Fig. 1(c), TV AT is introduced into the conventional OAM propagation model to constitute the TV OAM multiplexed propagation model. That is to say, the OAM multiplexed propagation model in TV AT is established. In the model, by changing $C_n^2$ and v, various TV AT scenarios are simulated, laying a foundation for the study of propagation characteristics of multiplexed OAM beams in various conditions of TV AT. However, for simplification, only the effects of transverse wind instead of the wind speed in the other direction are studied in the present paper. Therefore, in the sequel, v means the transverse wind speed.

3. Time-domain propagation of multiplexed OAM beams in TV AT

Having established the TV model, further exploration of various propagation characteristics of multiplexed OAM beams in TV AT is possible. The time-domain effects of $C_n^2$ and v on the light intensity, phase, and power of the multiplexed OAM beams after propagation through the TV AT are subsequently analyzed. It is found that the characteristics of the light intensity and phase are similar to those in the static AT [39,40]; consequently, this paper focuses on power characteristics. In all statistical analysis that follows, we assume for the propagation distance L, beam-waist radius w₀, length of one side of the square subscreens D, and wavelength $\lambda $ are 1 km, 5 cm, 1.024 m, and 1550 nm, respectively. The size D of subscreens represents the spatial extent of the AT we consider. While we assume a specific value of L, it can be changed to any value in our model, meaning that the characteristics of OAM beams after any propagating distance in TV AT can be studied. Using the power-spectrum-inversion method [20] and the subharmonic compensation theory [41], which are common approaches to generating phase screen (including the power-spectrum-inversion method [20] is used to generate phase screen, and subharmonic compensation theory [41] is added because the phase screens generated by power-spectrum-inversion method lack low-frequency components), the parent screens are generated with a pixel size of 2048 × 2048, and the pixel size of the intercepted subscreens are set to 512 × 512 pixels. The sampling frequency of the subscreens f, the fluctuation rate of the atmosphere, is uniformly set to 100 Hz, which is convenient for recording and analyzing. When the sampling interval 1/f is larger than ${\tau _0}$, there remains little correlation between two adjacent frames of the subscreens. In this case, TV AT tends to a state of random variation.

3.1 Effect of wind speed

We first explore the time-domain propagation characteristics of the OAM multiplexed beams in TV AT with various v. As shown in Fig. 2, for various v (0.5 m/s, 5 m/s, and 50 m/s), instantaneous power [42] variation curves within 1s observation time of the triple-mode multiplexed OAM beams (l = 1, 4, −7) are plotted at a fixed $C_n^2 = 1 \times {10^{ - 16}}{\textrm{m}^{\textrm{ - 2/3}}}$. We have explored many topological charges within [−20,20], which are usually researched by other papers [39,40], and l = 1, 4, −7 are chosen as a typical example to illustrate our approach. In this paper, each transient power of the OAM modes is normalized to the total power of all the modes at that time. From Fig. 2, it is clearly seen that the instantaneous power for each OAM mode fluctuates over time, and the power fluctuation rate increases sharply with increasing v, demonstrating the TV characteristics of our model. We have also studied the instantaneous power variation over longer times, and similar time-domain power-fluctuation characteristics are observed.

 

Fig. 2. Normalized instantaneous power for each mode of the triple-mode multiplexed OAM beams (l = 1, 4, −7) varies with time in various v with $C_n^2 = 1 \times {10^{ - 16}}{\textrm{m}^{\textrm{ - 2/3}}}$ [(a) v = 0.5 m/s; (b) 5 m/s; (c) 50 m/s]. The lower-left corner of each subplot is marked with the corresponding average power P_l.

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Moreover, as shown in Fig. 2(a), when v is low (0.5 m/s), the instantaneous power of the lower-order mode is higher than that of the higher-order mode at most times, which was found in static AT research [39,40]. There are, however, power anomalies in the power curves at a few times, e.g., at t = 0.65 s and 0.75 s, the instantaneous power magnitude of the l₁ mode is smaller than that of l₄. This indicates that, due to the introduction of TV AT, the instantaneous power-fluctuation intensities of different OAM modes are different. When v increases, as shown in Fig. 2(b), not only the power-fluctuation magnitude tends to be more dramatic, but also the occurrence probability of the power anomaly increases significantly. There are more anomalies when v further increases, as shown in Fig. 2(c). This phenomenon can be explained by Fig. 1(a); when the same sampling frequency is fixed, ${\tau _0}$ becomes shorter as v increases, which leads to a gradual weakening of the correlation between adjacent subscreens. As a result, the difference between the instantaneous OAM power through adjacent subscreens also becomes larger, and the power fluctuation magnitudes become larger.

In the following, the statistical time-domain characteristics of the OAM power fluctuations are investigated by calculating the normalized time-average powers (P_l) of the three OAM modes over the same observation time of 1 second. P_l for various wind speeds is indicated in the lower left of Fig. 2(a-c). For all investigated wind speeds v, P_l for higher-order modes is lower than that for lower-order modes, and this behavior of P_l in TV AT is similar to that of the static power under static-AT conditions. What is more noteworthy is that, for the same $C_n^2$, P_l for the same OAM mode only slightly changes even for high v. It can be understood by consideration of a very special case, i.e., if TV AT has a zero-wind speed, the TV AT tends to static AT, and P_l in TV AT is the static power under static-AT conditions.

3.2 Effect of $C_n^2$

In this subsection, the effect of $C_n^2$ on the time-domain characteristics of OAM beam propagation is discussed. After transmission through TV AT with three values of $C_n^2$ ($1 \times {10^{ - 17}}{\textrm{m}^{\textrm{ - 2/3}}}$, $1 \times {10^{ - 15}}{\textrm{m}^{\textrm{ - 2/3}}}$, and $1 \times {10^{ - 13}}{\textrm{m}^{\textrm{ - 2/3}}}$ corresponding to weak, moderate and strong AT), the normalized instantaneous power for each mode of multiplexed OAM beams (l = 1, 4, −7) is recorded and analyzed. Also, to focus on the effect of $C_n^2$, in Fig. 3 the wind speeds for all TV ATs are maintained constant at v = 0.5 m/s. We also investigated the effect of $C_n^2$ at other wind speeds, and similar properties are observed.

 

Fig. 3. Normalized instantaneous power for each mode of multiplexed OAM beams (l = 1, 4, −7) as a function of time in three typical TV AT strengths [(a)$C_n^2 = 1 \times {10^{ - 17}}{\textrm{m}^{\textrm{ - 2/3}}}$; (b) $C_n^2 = 1 \times {10^{ - 15}}{\textrm{m}^{\textrm{ - 2/3}}}$; (c)$C_n^2 = 1 \times {10^{ - 13}}{\textrm{m}^{\textrm{ - 2/3}}}$] with v = 0.5 m/s.

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For the case of weak turbulence, as shown in Fig. 3(a), the instantaneous power of each OAM mode has a small fluctuation amplitude. Increase $C_n^2$, first, we see that the instantaneous power amplitude fluctuates more heavily from Fig. 3(b). But when $C_n^2$ increases to $1 \times {10^{ - 13}}{\textrm{m}^{\textrm{ - 2/3}}}$, in Fig. 3(c), the fluctuation amplitudes of the instantaneous powers become small again. The phenomenon demonstrates that the largest instantaneous variability happened during OAM beam transmission in moderate TV AT, instead of in strong or weak TV AT. And the phenomenon that the largest instantaneous variability in moderate TV AT is consistent with the conclusion that the AO correction is the most effective method for moderate AT [29]. The largest instantaneous variability occurred in moderate TV AT may be due to the fact that, while strong AT has an excessive proportion of high spatial-frequency components, weak turbulence has an excessive proportion of low spatial-frequency components, whereas moderate turbulence has more balance between high and low spatial-frequency components.

In most cases of TV AT in Fig. 3, the instantaneous power of the lower-order mode is higher than that of the higher-order mode, which is consistent with previous observations in static AT research [39,40]. In a few cases, however, instantaneous power anomalies are observed, e.g., at t = 0.3 s and 0.5 s in Fig. 3(b), at which time the lower-order mode power is less than the higher-order mode. With the increasing $C_n^2$, the anomalies increase. In distinction from the instantaneous power variation with $C_n^2$, there is no anomaly for the normalized time-averaged powers, P_l. As shown in Fig. 3, for various $C_n^2$, P_l of the lower-order mode is always higher than that of the higher-order OAM mode. Moreover, with increasing $C_n^2$, each P_l of the three OAM modes all decays significantly. The changing behavior of P_l due to $C_n^2$ is similar to the phenomenon of static power found in conventional OAM propagation model [39,40], which verifies the reliability of our model.

From Figs. 2 and 3, we find that though variation of wind speed v has only little effect on P_l, variation of $C_n^2$ has a pronounced effect. In view of this, further study of the influence of $C_n^2$ on power fluctuation for each mode is performed. This time, the standard deviation (${\sigma _l}$) of the instantaneous power for each mode is also introduced as a quantitative index to measure the degree of power-spectrum dispersion under conditions of various $C_n^2$. Figure 4 shows P_l and ${\sigma _l}$ of the respective OAM modes at various $C_n^2$ with constant v = 0.5 m/s. In order to obtain more accurate data, multiplexed OAM beams with topological charge l of +1, + 4, −7, 10, −15 are adopted and a longer observation time of t = 5 s is used. Figure 4(a) shows P_l of the five OAM modes for various $C_n^2$, which shows that with increasing $C_n^2$, P_l of each OAM mode all decreases monotonically. Also, under the same TV AT, P_l of lower-order mode is always higher than that of higher-order mode. These phenomena related to P_l are consistent with the results shown in Fig. 3.

 

Fig. 4. (a) Normalized time-averaged powers (P_l) of the five OAM modes for various $C_n^2$; (b) The instantaneous power standard deviation (${\sigma _l}$) of the five OAM modes for various $C_n^2$.

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For the same OAM mode as shown in Fig. 4(b) with increasing $C_n^2$, ${\sigma _l}$ of each OAM mode all increases initially and then decreases, and their maxima are found near moderate TV AT. ($C_n^2$ is between $0.3 \times {10^{ - 15}}{\textrm{m}^{\textrm{ - 2/3}}}$ and $0.5 \times {10^{ - 15}}{\textrm{m}^{\textrm{ - 2/3}}}$.) The phenomenon is in accordance with previous observations shown in Fig. 3(b), which indicates that the largest power-fluctuation magnitudes are found for moderate TV AT. That is, moderate AT has contributions of both high and low spatial-frequency components.

In addition, for the same $C_n^2$, different OAM modes have different degrees of dispersion, as shown in Fig. 4(b), and the larger ${\sigma _l}$, the more dispersive the power spectrum of the OAM mode is. When $C_n^2 < 0.2 \times {10^{ - 15}}{\textrm{m}^{\textrm{ - 2/3}}}$, ${\sigma _l}$ of the lower-order modes is usually smaller than that of higher-order modes, which indicates that the power spectra of higher-order modes are more dispersive in weak TV AT; however, ${\sigma _l}$ of lower-order modes is generally larger than that of higher-order modes when $C_n^2 > 0.6 \times {10^{ - 15}}{\textrm{m}^{\textrm{ - 2/3}}}$. Thus, the power spectra of lower-order modes are more dispersive in strong TV AT. The phenomena shown in Fig. 4(b) may be due to the larger z-dependent width of the beam of the higher-order modes beam in freespace, which will induce an ensemble smoothing effect [15]. Thus the ensemble smoothing effect can effectively weaken the TV AT induced distortion. The result suggests that in FSO communications, multiplexed topological charges for the OAM beams should be carefully chosen according to the TV AT with different $C_n^2$.

4. OAM communication in TV AT and optimization

We have thus find that the distortion of OAM beams after propagating through TV AT is different from that propagating through static AT. It remains to address the distortion caused by TV AT. To suppress OAM distortion caused by static AT, a phase-correction method using a Gaussian probe to detect the AT has been favored in recent years [20,24]. We therefore adopt that phase-correction method [20,24] and optimize the correction mechanism in our TV multiplexed OAM communication link.

4.1 TV multiplexed OAM communication link

A schematic diagram of the TV multiplexed OAM communication link with a dynamic-feedback-phase correction mechanism is shown in Fig. 5. In the link, we consider two information-bearing signals [m₁(t), m₂(t)] first, which are then transformed into OAM beams. In order to better understand the distortion caused by AT, the TV AT influenced by v is introduced. Moreover, a dynamic-feedback-phase correction mechanism is adopted to correct the distortion caused by TV AT. In this correction mechanism, a Gaussian probe is used to obtain the distortion effect of TV AT, so that the correction phase screen (CPS) could be created. At last, the distorted OAM beams will be corrected and then the signals will be recovered.

 

Fig. 5. Schematic diagram of TV multiplexed OAM communication link with a dynamic-feedback-phase correction mechanism. PBS, polarization beam splitter; BS, beam splitter; CPS, correction phase screens; BER, bit error rate.

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First, two intensity-modulated signals [m₁(t), m₂(t)] (messages) are used to modulate the Gaussian beams, which are transformed into OAM beams after passing the spiral phase hologram (l₁, l₂), and are combined with the Gaussian probe by the polarization beam splitter 1 (PBS 1). Then, the multiplexed OAM beams pass through TV AT represented by N(t) on the ATs of Fig. 5. After TV AT, the distorted multiplexed OAM beams are divided into two beams by PBS 2, and then the light intensity and phase information of the distorted Gaussian probe beam are collected to create the CPS, which will be used to correct the aberrated OAM beams. The corrected beams are demultiplexed by the beam splitter (BS) and loaded with opposite spiral phase holograms (-l₁, -l₂) to become Gaussian-like beams. Finally, the Gaussian-like beams undergo coherent detection to recover the signal [m₁(t)´, m₂(t)´].

4.2 Optimizing phase correction periods in TV AT

Due to the TV nature of AT, OAM phase correction should be dynamic. As a preliminary matter, we consider two simple extreme correction modes of OAM phase. One is the RTC, requiring that CPS needs to follow the conditions of the TV AT. The other is the initial correction (ILC), which means that the initial frame of CPS is always used to correct distorted OAM beams regardless of the change of AT. Therefore, when the changing frequency of TV AT has been set, the correction times of RTC equal to the value of changing frequency of atmospheric turbulence, while ILC only need once correction.

As shown in Fig. 6(a), in TV AT ($C_n^2 = 1 \times {10^{ - 15}}{\textrm{m}^{\textrm{ - 2/3}}}$ with v = 0.5 m/s), the normalized instantaneous powers of these two extreme-case correction modes are compared with the case where no correction is made [34]. The coherence time ${\tau _0}$ in this case is 0.1996s, and the sampling frequency f is 100 Hz. From the figure, although the normalized instantaneous power of RTC is always higher than the other cases, it requires more computation. On the other hand, the normalized power with ILC is higher than that with no correction for early times, but with the continuous change of TV AT, the correction effects gradually disappear. For mode 2, the correction effects last for 0.4 s, in which the normalized power with ILC is higher than that with no correction, and the effects last for 0.26 s for mode −5, as shown in Fig. 6(a). This is because the evolution of TV AT depends on both position and time, and the CPS corresponding to the previous frame of AT phase screen is equally valid for the subsequent AT during a certain period. That is, when the correction method is not effective, we could send the probe beam again and create corresponding CPS by using Gerchberg-Saxton algorithm [43]. Thus, if PDC is performed over a period to be found below, the correction effects might be maintained at an acceptable level. The optimized dynamic correction period (ODCP) is found next.

 

Fig. 6. (a) Comparison of the effects of real-time correction (RTC), initial correction (ILC) mode and no correction for double multiplexed OAM beams (l = 2 at upper panel, −5 at lower panel) in TV AT ($C_n^2 = 1 \times {10^{ - 15}}{\textrm{m}^{\textrm{ - 2/3}}}$, v = 0.5 m/s). (b) BER for multiplexed OAM beams (l = 2, −5) with and without period correction in TV AT ($C_n^2 = 1 \times {10^{ - 15}}{\textrm{m}^{\textrm{ - 2/3}}}$, v = 0.5 m/s). (c) Optimized dynamic correction periods (ODCP) of multiplexed OAM beams (l = 2, −5) in various TV AT.

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To obtain the ODCP, a dynamic BER detection and feedback platform are added in the correction stage of the system, as shown in Fig. 5. When system BER exceeds a set threshold, the probe beam will be launched. When the system BER is below the set threshold, however, the probe beam is stopped to send and the CPS of the previous moment is continued to be used, ensuring that system BER is always below the threshold and at the same time greatly saves computational resources. The BER threshold is chosen according to the FEC limit ($3 \times {10^{ - 3}}$). The effective time (ET) of the CPS is obtained by comparing the BER with the FEC, and the ODPC should be no more than the ET for one frame of CPS.

With the dynamic feedback probe correction mechanism, the correction effect of PDC for TV AT with different $C_n^2$ but the same v = 0.5 m/s is studied. In order to show the effect of PDC, we take the effect of PDC under moderate TV AT ($C_n^2 = 1 \times {10^{ - 15}}{\textrm{m}^{\textrm{ - 2/3}}}$, v = 0.5 m/s) as an example in Fig. 6(b). Figure 6(b) illustrates BER under moderate TV AT with and without PDC. The ODCP in this case is 0.18 s. It is clearly seen that without PDC, BER of both channels m₁(t)´ and m₂(t)´ on OAM beams with l₁ = 2, l₂ = −5 are almost above FEC. With PDC, all BERs drop nearly two orders of magnitudes and can reach 10⁻⁵. Moreover, when the AT changes at a frequency of 100 Hz, for this case, PDC only requires 6 corrections in 1 s while RTC needs 100 corrections, which means that the computational effort required for PDC is reduced to only 6% of that for RTC.

Since the ODCP is derived from the spatiotemporal dependence of the TV AT, it must be related to $C_n^2$ and v. To explore the connection between them, ODCP and ${\tau _0}$ corresponding to various $C_n^2$ and v have been obtained and analyzed, as shown in Fig. 6(c). It is found that ODCP is inversely proportional to $C_n^2$ and v, which is consistent with the trend of ${\tau _0}$. Moreover, at the same v, it is shown that the ODCP is generally much longer than ${\tau _0}$ when the turbulence is weak, while the ODCP gradually approaches ${\tau _0}$ as the TV AT becomes stronger. For example, when $C_n^2 = 1 \times {10^{ - 16}}{\textrm{m}^{\textrm{ - 2/3}}}$ and v = 0.5 m/s, ${\tau _0}$ is more than 1.2 s and ODCP approaches 0.8 s, but when $C_n^2 = 1 \times {10^{ - 15}}{\textrm{m}^{\textrm{ - 2/3}}}$ and v = 0.5 m/s, ${\tau _0}$ and ODCP are almost the same. This is because weak turbulence itself has little effect on OAM beams, which can be seen in Fig. 3, and performing ILC is enough to maintain BER within FEC limit. However, as the increasing of $C_n^2$, the ET of ILC progressively becomes shorter. Therefore, in order to maintain BER within the FEC limit while saving resource in TV AT, PDC with ODCP is needed, especially in real AT which changes continually.

In order to more visually show the effect of PDC and verify the results in Fig. 6(c), we studied the OAM mode purity (e.g., normalized power spectrum) under moderate TV AT ($C_n^2 = 1 \times {10^{ - 15}}{\textrm{m}^{\textrm{ - 2/3}}}$, v = 0.5 m/s) in 1s by using the results of Fig. 6(c). In this case, ODCP = 0.18 s, so we choose the OAM mode purity at t = 0.19 s, 0.37 s, 0.55 s, 0.73 s and 0.91 s, as shown in Fig. 7. From Fig. 7, it is found that OAM mode purity is obviously improved with PDC, which proved that our method works. However, there are also anomalies due to the spatio-temporal characteristics of AT, which is consistent with the phenomena in Fig. 2 and Fig. 3.

 

Fig. 7. (a1-a6) The OAM mode purity of multiplexed OAM beams (l = 2, −5) after TV AT($C_n^2 = 1 \times {10^{ - 15}}{\textrm{m}^{\textrm{ - 2/3}}}$, v = 0.5 m/s); (b1-b6) The OAM mode purity of multiplexed OAM beams (l = 2, −5) after period correction.

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5. Conclusion

In summary, a TV multiplexed OAM propagation model considering AT that is influenced by v is established for the first time, and optimized dynamic correction periods for various TV AT are found to improve the transmission efficiency. Compared with the conventional OAM spatial propagation model, the TV model presented here can be used to discuss the time-domain propagation characteristics of multiplexed OAM beams in TV AT. Through modeling and power analysis, it is found that while ${\sigma _l}$ of lower-order modes is usually smaller than that of higher-order modes in weak TV AT, the phenomenon in strong TV AT follows the opposite trend. The distributions of ${\sigma _l}$ for each OAM mode all increase first then decrease with the increasing $C_n^2$, which opens a new study perspective for OAM propagation in freespace. Besides, by using the ODCP for dynamic-feedback-phase correction, the computational efficiency of the system can be greatly improved, which does not require a large number of iterative calculations. The validity of the PDC has been demonstrated in the numerical simulation of the TV multiplexed OAM communication link. The results show that BER of OAM transmission can reach 10⁻⁵, and also with a low computation volume that is only 6% of the RTC when the AT changing frequency and the ODCP equal to 100 Hz and 0.18 s, respectively. These studies are useful for quantitatively characterizing the time-domain effect of multiplexed OAM beams in TV AT, and may lay the groundwork for the practical application in phase distortion correction due to TV AT.

Appendix: abbreviations and their according full names

Table 1. Acronyms used in this paper

View Table

Funding

Fundamental Research Funds for the Central Universities of South-Central Minzu University; National Natural Science Foundation of China (11504435, 62171478).

Disclosures

The authors declare that there are no conflicts of interest related to this paper.

Data Availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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