## Abstract

In order to better evaluate the relationship between reciprocity and time delay of the fiber receiving system in the atmospheric turbulence channel, a time-domain signal generation mathematical model is proposed for the first time. A numerical solution of Johnson *SB* probability density distribution (PDF) in time-domain is creatively given for evaluating the reciprocity of both communication ends, which relates to the normalized fluctuation variance of the light intensity and the Greenwood frequency. An experiment is then carried out for verifying the time-domain signal generation model and measuring reciprocity. It shows that the excellent fitting accuracy of Johnson *SB* PDF signal generation model is first experimentally verified. It also indicates that the system reciprocity is improved by 10% after eliminating the system time delay. Meanwhile, the relationship between time delay and reciprocity under different atmospheric environments are analyzed and the relationship between time delay and system reciprocity at different Greenwood frequencies are discussed. This work provides a time parameter reference for the design of adaptive system and free-space optical (FSO) communication system.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

The intensity scintillation and the arrival of angular fluctuations induced by atmospheric turbulence have been a major obstacle in the development of free-space optical (FSO) communications with high speed and long distance for a long time. At present, adaptive techniques are proposed to reduce the effects of atmospheric turbulence, such as adaptive wavefront aberration correction, adaptive power conditioning, and pilot techniques in orthogonal frequency-division multiplexing (OFDM). The application of these technologies are based on the channel reciprocity [1–4]. In recent years, many teams have studied the reciprocity of FSO systems in theoretical analysis and numerical simulation. J. H. Shapiro and A. L. Puryear *et*. *al*. have theoretically proven the reciprocity of the atmospheric turbulence channel based on the Green’s function. Soon afterward, they presented the proofs of reciprocity principles with and without phase compensation conditions. Meanwhile, they discussed the communication architectures and the performance of the system when the reciprocity principles were used in atmospheric turbulence [5–7]. In [8, 9], when atmospheric turbulence exists over the propagation path, the reciprocity of point-source point-receiver (PSPR) have been studied by numerical simulations. In addition, J. M. Conan *et*. *al*. have shown that the reciprocity principle can quantify turbulence effects on ground-space adaptive optics assisted optical links by Monte-Carlo simulations [10]. In 2015 and 2018, our team has also carried out related research on turbulent channel reciprocity. We have given an expression for the correlation coefficient between light-flux fluctuations of two waves counter-propagating along a common path in weak turbulence, and theoretically formulated the correlation coefficient between the angles of arrival in the *x* or *y* direction of two counter-propagating spherical waves [11,12]. However, these studies did not give a model for generating a real-time optical signal to analyze the reciprocity of the system. Because, in practical engineering, reciprocity is time-varying, we need to obtain a real-time signal to characterize this property by comparing and analyzing time-domain signals, which can further uncover the performance of reciprocity.

Moreover, some experiments on the reciprocity of atmospheric turbulence channel have been reported in recent years. In 2010, R. R. Parenti *et*. *al*. have experimentally observed the channel reciprocity in free-space link (25 *km* and 80 *km*) by using single-mode fiber (SMF) receiver system, the reciprocity was above 0.9 [13]. In 2011, D. Giggenbach have experimentally investigated the correlation of both received optical powers with aperture sizes between those two extremes, and have concluded that the reciprocity of light intensity is limited to an axial intensity area comparable to the Fresnel-size [14]. In 2018, S. Parthasarathy *et*. *al*. have measured the channel reciprocity in long-range (63km) turbulent link and the measured results shown that there is strong reciprocity in the turbulent channel [15], but they did not give the mathematical model to analyze the effect of measurement system time delay on reciprocity, since the measurement system response takes time in actual engineering, in particular, adaptive algorithm processing takes a long time when the principle of reciprocity is employed for the bit error rate (BER) reduction in FSO communication . Hence the time delay is a non-negligible factor that reduces the reciprocity of system. In practical engineering, we need to ensure that the adaptive algorithm can be completed within a reasonable reciprocity range, that is, in a certain time range, reciprocity can be used for adaptive optical correction, correction of optical signal distortion, adaptive power adjustment, etc. It is very worthwhile for us to explore the relationships between system time delay and reciprocity, which can provide a time parameter reference for the design of adaptive systems. The objective of this work is to establish a channel reciprocity mathematical model of time-domain signal generation and derive a physical model about the influence of the time delay factor on the reciprocity.

In this paper, we first propose a time-domain signal generation model for evaluating the performance between time delay and reciprocity of the system. In Section 2, the time-domain signal generation model is deduced and analyzed. In Section 3, the experiment of Johnson *S _{B}* is demonstrated and discussed. Additionally, the numerical simulation of the time-domain signal generation is also implemented and verified. In Section 4, the conclusion is elaborated.

## 2. Theoretical analysis

We begin to consider the measurement principles of the reciprocity in turbulent
channel, as shown in Fig. 1. The
two ends are Alice and Bob, respectively. Taking into account the
time-varying characteristics of the channel, we shall assume that when
light propagates from Alice to Bob (recorded as $\overrightarrow{ab}$), the atmospheric turbulence channel impulse
response function is $H\underset{ab}{\to}\left({\rho}_{a},{\rho}_{b}:t\right)$, then we can know that, in the single mode
fiber (SMF) of Bob, the received light field
*U _{br}* (

*ρ*;

_{b}*t*)comes from the common

*z*axis Alice, which can be written as

*U*(

_{a}*ρ*) is the complex light field of the laser emitted at the Alice’s end, the radius of aperture is

_{a}*D*.

_{b}*P*(

*ρ*) and

_{b}*U*(

_{mb}*ρ*) represent the aperture function of the optical antenna and SMF coupling function at Bob’s end, respectively. They are expressed as follows

_{b}*W*,

_{mb}*f*and

_{b}*λ*denote the SMF mode field radius at Bob’s SMF end face, the focal length of the optical antenna, and the optical wavelength, respectively. Similarly, we can get the complex light field in Alice’s SMF when light propagates from Bob to Alice (recorded as $\overrightarrow{ba}$)

_{b}Assume that Alice and Bob emitted the same light field and their optical system are the same, i.e.,

According to the ABCD optical path and the Green’s formula [16,17], we can know that the channel is reciprocal at time *t*

Equation (6) is a time-varying function of the spatial-domain, that is, the filtering effect of the spatial-domain changes with time. By combining Eqs. (1), (4), (5) and (6), we can prove that the light fields in the SMF of Alice and Bob are the same at this time, as shown below

By observing Eq. (7), we discover that when Alice and Bob have the same emission light field and the optical system, the channel reciprocity can be determined by measuring light field intensity at receiving end. Thus we can use the correlation coefficient (CCF) *η* to express the degree of reciprocity.

*ζ*〉 is the

^{n}*n*moment of the measured variable. However, in actual operation, reciprocity is degraded due to system response delay (e.g. it takes time for the system to process data). Based on such considerations, we shall introduce a time constant

^{th}*τ*to describe this phenomenon, which is given by

_{d}$H\underset{ab}{\to}\left({\rho}_{a},{\rho}_{b};t+{\tau}_{d}\right)$ has a gradual time characteristic, *τ _{d}* is the time delay factor of the measurement system. When

*τ*= 0, Eq. 9 is absolutely equal, i.e.

_{d}*η*= 1. We can evaluate the reciprocity and time-varying characteristics of the channel by measuring the time delay factor

*τ*and the CCF

_{d}*η*.

An important consideration is that the time-domain signal which obeys the distribution of light intensity is employed to depict the time-varying characteristics of the signal at both ends, so as to analyze the performances between reciprocity and time delay [18]. According to our previous work [19], the Johnson *S _{B}* probability density function (PDF) can be utilized to describe the distribution of the received light intensity in SMF, as follows

*< ζ <*1,

*δ*.

*γ*are the two parameters of the function, which can be calculated by the average light intensity

*E*(

*ζ*)and the variance of the light intensity fluctuation ${\sigma}_{\zeta}^{2}$

*n*order moment of

^{th}*ζ*, $\overline{{\sigma}_{\zeta}^{2}}$ is normalized fluctuation variance. According to [20, Chap. 2, Eqs. (1) and (2), Chap. 3, Eq. (26)], we can use $\overline{{\sigma}_{\zeta}^{2}}$

^{to}derive

*Rytov variance*${\sigma}_{Rytov}^{2}$, hence it can be written as

*L*is atmospheric turbulence channel length, ${C}_{n}^{2}$ represents the atmospheric structure constant,

*k*= 2

*π*/

*λ*denotes the optical wave number. In case of $\overline{{\sigma}_{\zeta}^{2}}\le 2$, we can employ the normalized fluctuation variance to approximate

*Rytov variance*, i.e., $\overline{{\sigma}_{\zeta}^{2}}\approx {\sigma}_{Rytov}^{2}$.

Next, we employ Johnson *S _{B}* PDF to generate time-domain correlation waveforms to depict the state of the optical signal in the SMF receiving system, so that the influence of time delay on reciprocity can be analyzed based on the time-domain waveform. The stochastic differential equations (SDEs) method shall be employed to solve the Eq. (10), we then propose a time-domain waveform generation model of optical signal about the reciprocal system in FSO communication [18, 21–24]. The SDEs is given by

Ψ(*t*)belongs to white Gaussian noise (WGN). ϒ(*ζ* ) and *f* (*ζ*) can be calculated by [25, 26]

In Eq. (15)*f _{G}* is the Greenwood frequency of the atmospheric channel, which describe the time-varying characteristics of the atmosphere turbulence, its specific form is given by [27–29]

*v*represents the wind speed of the propagation path, the atmospheric structure constant ${\tilde{C}}_{n}^{2}$ is a function of the path,

*θ*is the zenith angle. For the sake of facilitating the simulation, we shall use the equivalent atmospheric structure constant ${C}_{n}^{2}$ to express ${\tilde{C}}_{n}^{2}$ [30, pp. 10]. Based on such analysis, we conclude that Eq. (14) is a time-varying function, which can be modified as

A correlation with Δ*t* time-domain waveform signal can be produced by generating a random noise. Nevertheless, Eq. (18) is an analytical formula and is difficult to express by numerical equations. Therefore, according to the scheme of implicit Milstein, the approximation can be obtained [21, Eq. (40)]

Further simplification, we can obtain an iterative equation about time-factor

By generating a random sequence Ψ* _{k}* , we can perturb the Eq. (20) to simulate the random noise process caused by atmospheric turbulence. Besides, we know that arbitrarily given an initial value

*ζ*

_{0}, the sequence {

*ζ*(

*δ, γ, f*;

_{G}*t*)}can be derived by iterative calculation

*ζ*

_{1},

*ζ*

_{2},⋯,

*ζ*

_{k},

*ζ*

_{k+1}, ⋯. This sequence obeys a Johnson

*S*distribution and is related to the atmospheric Greenwood frequency

_{B}*f*. Without loss of generality, under the condition of absolute reciprocity, we shall assume that the sequence of Alice is {

_{G}*ζ*(

*δ, γ, f*;

_{G}*t*)}and the that of Bob is {

*ζ*(

*δ, γ, f*;

_{G}*t*+

*τ*)}, where

_{d}*τ*is a time delay in the measurement system. According to Eq. (8), we can analyze the degree of reciprocity between them, that is, the sequence can be used to evaluate the relationship between the system time delay and the system reciprocity.

_{d}## 3. Performance analysis

Based on the principle of Section 2, a 6-month measurement was conducted (March 2018September 2018). The experimental site Alice was located at the 13* ^{th}* floor of Block A of Science and Technology Building, South Campus, Changchun University of Science and Technology (CUST), and Bob was located at the 9

*floor of the second teaching building, East Campus. The experimental link length is 864*

^{th}*m*. In order to avoid the influence of background light, rain and fog on reciprocity, we chose a sunny weather time (19:30-22:30, Beijing time). To ensure Eq. (5) is true, we used two lasers with the same wavelength and the same emitted light intensity as the light source at Alice and Bob, as shown in Fig. 2. Two high speed complementary metal oxide semiconductor (CMOS) cameras of the same type (MIKROTRON, CAMMC1360) were employed as detectors with a 1000

*Hz*sampling rate. And two 1064

*nm*circulators (THORLABS, CIR1064-APC) were utilized as Alice and Bob transceiver multiplexers for the optical axis coaxial, where port 1 was connected to the 1064

*nm*laser, port 2 was connected to an optical antenna that receives light from the spatial light coupled to the fiber, port 3 was connected to the CMOS detector. In addition, two transmissive optical systems with the same optical structure and two 100

*mm*diameter were selected as the optical antenna. When the optical antenna of Alice and Bob were on the common optical axis, we adjusted the focusing knob which guaranteed that the optical parameters (caliber size and focal length) of Alice and Bob satisfy the Eq. (5). To ensure the two cameras can simultaneously sample the optical signal from the SMF, we also set up a local area network (LAN) and applied the servers to send synchronous operation instructions to Alice and Bob, as shown in Fig. 3.

Figure 4 shows a valid data waveform which was acquired on July 16, 2018. Due to the existence of atmospheric turbulence, the optical signal fluctuates with time when the light passes through the atmospheric channel. Note that the optical signal at each moment (denoted as *ζ _{k}*) is correlated with the optical signal at the next moment (denoted as

*ζ*

_{k}_{+1}). As the intensity of turbulence increases, this correlation diminishes. The measured data was processed by maximum normalization, we then analyzed it according to Eq. (8). The CCF is

*η*= 0.8558, and this shows that there is significant reciprocity, which proves that Eq. (7) is true. However, there is a time delay in the signal waveforms at Alice and Bob, because there is a delayed response at both ends of the system when the network server sends a synchronous operation instruction. By offline data processing, the reciprocity

*η*= 0.9734 which is improved by about 0.1 after the time delay is eliminated. For the sake of generality, 78 valid measured data were depicted in Fig. 5. We can know that the reciprocity of data without time delay (the mean of correlation) is about 10% higher than that with time delay, which further confirms the importance of Eq. (9), that is, the time delay caused by the measurement system has an important influence on the channel reciprocity.

Furthermore, we performed a PDF curve analysis on the measured data, as depicted in Fig. 6, where Fig. 6(c) is the PDF of Fig. 5. The measured data of Alice and Bob have the same distribution at the same time. According to Eqs. (12) and (13), after approximate calculation, the normalized fluctuation variance $\overline{{\sigma}_{\zeta}^{2}}$ are 0.093, 0.2022 and 0.9369, as shown in Figs. 6(a)-6(c), respectively. Besides, the atmospheric structure constant ${C}_{n}^{2}$ of Figs. 6(a)-6(c) are 9.56 ×10^{−15}*m*^{−2/3}, 2.07 ×10^{−14}*m*^{−2/3} and 9.63 ×10^{−14}*m*^{−2/3}, respectively. It shows that, under strong atmospheric turbulence, the peak-to-peak value of the PDF is biased toward the vertical axis and the normalized fluctuation variance increases. These experimental results indicate that when the light passes through various intensities atmospheric turbulence channel, Johnson *S _{B}* PDF can describe the optical signal distribution of the coupled light in SMF, and the fitting efficiency is above 98%. In the case of weak turbulent, the fitting efficiency is 5% higher than the lognormal. It is worth noting that the advantage of Johnson

*S*PDF is particularly noticeable under strong turbulence.

_{B}In order to further investigate the influence of time delay on reciprocity, a numerical simulation of time-domain light intensity signal generation was proposed for a SMF receiving system. By combining Eqs. (10), (15), (16), and (20), the numerical simulations were carried out. At the beginning of numerical simulation, we proved that the time-domain waveform, PDF, and frequency characteristics of the generated sequence are consistent with the laser propagation characteristics in the atmosphere channel, as shown in Fig. 7. Here, the simulation parameters *f _{G}* were set to 50

*Hz*and the sampling frequency is 10

^{4}

*Hz*. Figures 7(a)-7(c) represent

*δ*= 3.0,

*γ*= 4.5;

*δ*= 2.1,

*γ*= 3.0; and

*δ*= 0.8,

*γ*= 2.8. The normalized fluctuation variances are 0.0724, 0.1385 and 1.7186, respectively. According to Eqs. (12), (13) and (17), we shall assume that the channel length

*L*= 5000

*m*, light wavelength

*λ*= 1064

*nm*, we then can derive that the atmospheric structural constants ${C}_{n}^{2}$ are 6.71 ×10

^{−17}

*m*

^{−2/3}, 1.26 ×10

^{−15}

*m*

^{−2/3}, and 1.56 ×10

^{−14}

*m*

^{−2/3}, respectively. Thus the turbulence intensity of Fig. 7(a) is the strongest, and Fig. 7(c) is the weakest. The random fluctuation of the waveform increases when the turbulence intensity deteriorates, which results in a weak correlation between

*ζ*and next moment

_{k}*ζ*

_{k}_{+1}. It is worth noting that the signal random fluctuation trend is similar to Fig. 4, indicating that the simulated time-domain curve and the experimentally measured curve have the same time-domain variation characteristics. For better verification of the statistical distribution characteristics, Figs. 7(d)-7(f) were plotted. The simulated curve can well fit the theoretical value, where the fitting efficiencies $\Re $ are 0.9987, 0.99515 and 0.9901. It shows that the waveform simulated by Eq. (20) conforms to the theoretical PDF. Figures 7(g)-7(i) denote power spectral density (PSD) curves, where the horizontal and vertical coordinates are logarithmic. The slopes of the fitting curves are −2.67, −2.67 and −2.65, the logarithmic frequency obeys −8/3 characteristics, which indicates that the simulated time-domain optical signal conforms to the frequency characteristics [17, chap. 8, Eq. (57)]. Thus, after passing through atmospheric turbulence, the proposed model can be used to describe the real-time light intensity state of light waves coupled to the SMF, which proves the correctness of this model.

Moreover, we utilized Eq. (8) to analyze the reciprocal relationship between Alice and Bob, as presented in Fig. 8. When the delay time is constant, the reciprocity is weakened with the increase of turbulence intensity. These results once again prove the correctness of the simulation results by comparing the measured data (see, Fig. 5). We also know that as the delay time increases, the reciprocity decreases. The CCF can maintain above 0.6 within 10 *ms*. In this case, the measured normalized fluctuation variance $\overline{{\sigma}_{\zeta}^{2}}$ is 0.137 (*δ*= 2.1,, *γ*= 3.0) and the reciprocity of the system needs to be kept above 0.75, the time required to correct the system must be shortened within 5 *ms*.

Finally, the effects of different atmospheric Greenwood frequency *f _{G}* and system time delay

*τ*on the reciprocity were explored, as depicted in Fig. 9. According to the [27, 30, 31], the Greenwood frequencies were set as 50

_{d}*Hz*, 25

*Hz*, 16.6

*Hz*, 12.5

*Hz*and 10

*Hz*, respectively. the atmospheric turbulence channel parameters are

*δ*= 0.8

*, γ*= 2.8, $\overline{{\sigma}_{\zeta}^{2}}=1.828$. As can be seen from Fig. 9, if the delay time is constant, the degree of reciprocity decreases as Greenwood frequency increases. In particular, under

*f*= 50

_{G}*Hz*, the downward trend of reciprocity is more serious, it indicates that the system at this time must ensure a relatively short time delay. In addition, the reciprocity of the entire system degrades as the time delay increases. In the time delay of 10

*ms*, the reciprocity performance reaches 0.6 or more. When

*f*= 25

_{G}*Hz*, and

*τ*= 10

_{d}*ms*, the reciprocity is maintained above 0.9, that is, within 10

*ms*, we can use the measured information at Alice to accurately estimate the light intensity signal emitted at Bob, which is of great significance for the application of the reciprocal system. Above all, these results show that the proposed model can be used to predict the channel state during this time. In particular, if the time delay is eliminated or the data processing time is further shortened in the experiment, the system reciprocity will be greatly improved, this verified the measured data in the previous real experiment (see, Fig. 5). It can also provide the time parameter for adaptive power emission correction systems design and can supply the time requirement of the system processing for the adaptive optics system.

## 4. Conclusion

In this paper, we have explored the effect of FSO communication system time delay on reciprocity. By theoretical modeling, the mathematical model for system time delay and reciprocity was deduced. The system reciprocity was measured to be at 0.83 while observing the time delay in the measurement system. After eliminating the time delay by offline, the reciprocity was improved by 10%. Subsequently, we employed Johnson *S _{B}* PDF to express the time-domain intensity distribution in a SMF receiving system, and experimentally demonstrated the applicability of Johnson

*S*for the first time. Most importantly, the time-domain light intensity generation model was given by PEDs method. The availability and effectiveness of the Johnson SB scheme was proved by numerical simulation. These results also proved that, in a SMF receiving system, the generated time domain optical signal waveform satisfies the PDF and PSD characteristics when the light passes through the atmospheric turbulence. Finally, the relationship between time delay and reciprocity under different atmospheric environments were analyzed and the relationship between time delay and system reciprocity at different Greenwood frequency were discussed. When the normalized fluctuation variance is 1.828 (

_{B}*δ*= 0.8

*, γ*= 2.8) and the Greenwood frequency is 25

*Hz*, the reciprocity can be maintained above 0.9 within 10

*ms*of the system delay. Our work is extremely useful for adaptive optics and FSO communication.

## Funding

National Natural Science Foundation of China (61775022, 61475025); Development Program of Science and Technology of Jilin Province of China (20180519012JH, 20190201271JC); Postdoctoral Science Foundation of China (2017M621179).

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