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Abstract
The high-precision time-frequency transfer of the optical atomic clock signals over ground-to-satellite and terrestrial free-space laser paths is of great significance in the fields of fundamental and applied sciences. However, the phase noises caused by atmospheric turbulence severely degrade the measurement precision. In this paper, a new method to simulate the transmission of time-frequency signal propagating through atmospheric turbulence is investigated. An analog transmission system comparable to the practical out-field link has been demonstrated, which can provide a deep insight into the phase distortion of time-frequency signal of free-space optical communication links.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Optical atomic clock technology has the potential to revolutionize various scientific studies in fundamental and applied sciences [1–3], e.g., general relativity testing [4], searching for dark matter [5], fundamental constants measurement [6,7], radio astronomy [8], and global navigation systems [9,10]. Meanwhile, the ability to realize remote time synchronization in long-distance free-space optical (FSO) links, and particularly satellite-ground systems, would be of importance for inter-continental scale and space-borne applications [11]. Efforts have been put on optical time-frequency transfer systems that can be deployed on the open-air ground [12,13], the in-orbit spacecraft [14], and international space station [15].
Similarly, synchronization systems between mobile terrestrial optical clocks, where one or more mobile clocks can be deployed or moved over a restricted area, have been established for applications in resource exploration, environmental monitoring, surveying, and differential gravitational redshift measurements [16,17].
Just like frequency transfer over optical fiber links, free-space time-frequency frequency transfer exhibits satisfactory residual instability. However, the phase noises caused by atmospheric turbulence are much greater than that of optical fibers with comparable length [18,19]. Impacts of atmospheric turbulence on wireless optical links can be characterized by the Fried parameter, which takes into account the atmospheric conditions, altitude, and path length [20]. When the waist size of light beam is smaller than Fried parameter, the centroid of the beam will deviate from the detection area, while in the case where the waist size of the light beam is larger than Fried parameter, destructive interference will lead to deformation of the beam spot and fluctuations of optical power, resulting in the loss of timescale synchronization [21]. Deep fades would occur by 10 to 100 times per second in the case of vertical connection between ground and space stations [18]. Usually, for severe atmospheric turbulence or long-haul optical path, the time-frequency transfer link not only requires a pointing, acquisition, and tracking (PAT) system [22] but also adaptive optics (AO) instruments to dynamically correct the turbulence-induced aberrations [23].
Since the AO system is quite complex, extensive numerical simulation work is required to design the system architecture. Simulation of atmospherically distorted wavefronts is the starting point of all these models [24,25]. The simplest method to simulate atmospheric turbulence is to use the random phase screen. Several methods have been proposed for this purpose: FFT-based method [26–28], Zernike polynomial expansion [29] and low-spatial frequency compensation [30,31], all of which have been widely employed in numerical analysis of wavefront distortion [32,33], ground-based interferometry synthetic aperture radar [34,35] and turbulence correction [36]. Nonetheless, the random phase screen is established on the truth that the laser pulse width is significantly narrower than timescale of refractive index fluctuations [23]. Therefore, the dynamical process of time-frequency signal propagation through atmospheric turbulence cannot be investigated by using the above methods. Moreover, the static assumption would lose its efficacy because of the simple fact that the instantaneous frequency is the derivative of phase versus time. The infinite-time phase screen (ITPS) method presented in this paper gives a feasible scheme to overcome these difficulties.
In this work, we propose an analog transmission system for time-frequency signal propagation through atmospheric turbulence based on the ITPS with a high frame rate. This system can simulate a phase jitter in the vast majority of turbulence states where the waist size of the light beam is smaller than Fried parameter, in cases of the vertical fluctuation component of the arrival angle vector can be ignored. The Monte Carlo spectral method is also introduced to generate random, correlated phase values. In order to verify the reliability of the proposed ITPS, a $124\textrm{ m}$ round-trip free-space link was built in the outfield. The experimental results show that the modified Allan deviation (MDEV) curves are in good agreement, and the root mean square (RMS) relative errors tend to be 10.07% for the power-law scaling of MDEV.
2. Theoretical background
When an optical signal passes through a free-space path with link distance L, the variation of propagation time can be expressed as [37]
where $\delta n({z,t} )$ represents the variation in refractive index at time t and position z, L is the total path length, and c is the vacuum speed of light. For optical pulses, $\delta n({z,t} )$ is the change of group refractive index, while for an optical carrier, $\delta n({z,t} )$ refers to the variation in the phase refractive index. The refractive index of air can be roughly estimated by the modified Edlén equation. For given wavelength $\lambda $ (µm), temperature T (℃), atmospheric pressure p (Pa), and partial pressure of water vapor f (Pa), the Edlén equation is described as [38]The parameters in Eq. (2) are can be expressed by the following equations:
Based on Eq. (2), relative difference between phase refractive index and group refractive index $\Delta {n_r}$ is equal to $\lambda /n \cdot \partial n(\lambda )/\partial \lambda $. Considering the following environmental parameters: $T = 27^\circ\textrm{C}$, $p = 101.325\textrm{ kPa}$, $f = 40\%\times 3564.4\textrm{ Pa}$ (air relative humidity is 40%, and the saturated vapor pressure at 27°C is 3564.4 Pa under a standard atmospheric pressure), $\Delta {n_r}$ should be 1.236 × 10−6. By neglecting the dispersion of atmospheric path, the variations in time of flight and envelope of optical pulses propagating through the same transmission link should be almost the same, that is, $\Delta {\tau _g} \approx \Delta {\tau _p}$.
According to turbulence theory, the Kolmogorov spectrum assumes that the outer scale of turbulence ${L_0}$ is infinite, and the inner scale of turbulence ${l_0}$ is negligible [39]. Tatarski gave the Kolmogorov spectrum describing spatial fluctuations of refractive index:
Under Taylor’s hypothesis of frozen turbulence with a constant transverse wind speed V, the three-dimensional spatial spectrum ${\Phi _n}(\kappa )$ is converted into a one-dimensional temporal power spectral density (PSD) ${S_\phi }(f)$. The corresponding ${S_\phi }(f)$ of the Kolmogorov spectrum and the modified von Karman spectrum can be expressed as [41,42]
The conversion relation between timing-jitter PSD and phase PSD is ${S_{\textrm{jitter}}}(f )= {S_\phi }(f )/{({2\pi {v_0}} )^2}$ for an optical carrier frequency ${v_0}$. According to Eqs. (8) and (9), the theoretically predicted values of the timing-jitter PSD are plotted in Fig. 1. It should be noted that there may exist dispersion due to high-order turbulence, which will further increase the timing jitter on optical pulses. However, for the nanosecond pulse lengths and the path length in kilometers, the influences of dispersion can be negligible [43]. In other words, the most effective factor resulting in fluctuations of the pulse-to-pulse arrival time is the path-averaged refractive index variations rather than the temporal broadening of pulses.
Fig. 1. Theoretical predictions for the timing-jitter PSD. The timing-jitter PSDs calculated from the Kolmogorov spectrum (red) and the modified von Karman spectrum (blue), are shown for $\lambda = 1550\textrm{ nm}$, $V = 1\textrm{ m/s}$, $C_n^2 = 1 \times {10^{ - 14}}\textrm{ }{\textrm{m}^{ - 2/3}}$, $L = 2.4\textrm{ km}$ with an inner scale of ${l_0} = 1\textrm{ mm}$ and an outer scale of either ${L_0} = 10\textrm{ m}$ (dashed blue line) or ${L_0} = 100\textrm{ m}$ (solid blue line) for the von Karman spectrum.
3. Monte Carlo spectral method
3.1 Generation of random, correlated phases
The theoretical phase PSD of optical pulses can be defined under specific turbulence parameters. To determine the phase jitter along the turbulent path, the discrete phase values were generated according to the corresponding PSD first. A phase generation approach developed by Thorsos [44] was re-invented several times [45], which introduces some randomness into the desired PSD. Due to the need to generate a lot of random numbers, this approach is sometimes referred as the Monte Carlo spectral method. Considering the one-dimensional case, for given sampling number N and sampling interval $\Delta t$, the sampling time T is equal to $N\Delta t$, and the discrete phase value $\phi ({{t_n}} )$ is given by [46]
The mean values of random number ${\eta _1}$ and ${\eta _2}$ are both 0 and independent of each other. It can be easily concluded that ${\mu _\phi } = \left\langle \phi \right\rangle$, $\left\langle \cdot \right\rangle $ means the temporal average. Since changing a fixed phase value in the overall time scale does not affect the timing jitter of pulses, for simplicity, here we take ${\mu _\phi } = 0$.
3.2 Analysis of phase sampling
According to Parseval’s theorem, the variance of a continuous variable $\phi (t )$ is equal to the area under the PSD curve:
Generating discrete values ${\phi _n}$ based on Monte Carlo spectral method is practically one kind of sampling operation. Due to the limitation of calculation resources, the influences of $\Delta t$ and N on the sampling accuracy could be found in that the value of temporal frequency f is limited to the range $[{{f_{\min }},{f_{\max }}} ]$ in the Fourier frequency domain, where ${f_{\min }} = 1/N\Delta t = 1/T$ and ${f_{\max }} = 1/({2\Delta t} )$ corresponds to the Nyquist sampling ${f_N}$. The relative error term caused by sampling interval $\Delta t$ can be estimated by defining the error term ${\varepsilon _\textrm{H}}$:
Fig. 2. Analysis of theoretically calculated PSD error. (a) The PSD error term $|{{\varepsilon_\textrm{L}} + {\varepsilon_\textrm{H}}} |$ caused by the loss of Fourier frequency for different sampling intervals $\Delta t$ and sampling numbers N. (b) PSD error as a function of sampling interval $\Delta t$ in logarithm scale when the sampling time T is constant. (c) The PSD error term ${\varepsilon _\textrm{D}}$ caused by discretization of $\phi (t )$ in linear scale (above) and logarithmic scale (below).
By sampling the continuous function $\phi (t )$, the relative error term ${\varepsilon _\textrm{D}}$ can be described as
Similarly, the overall sampling time T was fixed as a constant and the error term ${\varepsilon _\textrm{D}}$ as a function of sampling interval $\Delta t$ could be obtained, as shown in Fig. 2(c). It can be seen from that when $\Delta t$ reaches a specific value, the error term ${\varepsilon _\textrm{D}}$ starts to oscillate, and the different error terms ${\varepsilon _\textrm{D}}$ plotted in logarithm scale exhibit similar oscillation behavior. Therefore, it can be concluded that the error term ${\varepsilon _\textrm{D}}$ oscillates as the sampling interval $\Delta t$ increases because of insufficient sampling points, and the oscillation intensity is inversely proportional to the order of magnitude of sampling time T.
3.3 Convergence of the phase PSD
Considering the overall error caused by the loss of Fourier frequency and discretization, it can be calculated from the above discussion that the total theoretical error $|{{\varepsilon_\textrm{L}} + {\varepsilon_\textrm{H}} + {\varepsilon_\textrm{D}}} |$ is about 1.17% for a phase sampling with a sampling frequency of 60 Hz and sampling time of 2 h (i.e., the number of sampling points of ${\phi _n}$ is 4.32 × 105), which is far below the endurable error. Standard FFT was performed on the discrete phase values ${\phi _n}$, Fig. 3 shows the typical PSD results based on the modified von Karman spectrum. As shown in Fig. 3(b). The PSD of the data set converges commendably to the input PSD $S_\phi ^{\textrm{mvK}}$ by averaging the results of 100 trials.
Fig. 3. Typical PSD retrieved from the generated phase values ${\phi _n}$ based on the modified von Karman spectrum. (a) one trial and (b) an average of 100 trials. The smooth red curve gives the input PSD. (c) Convergence of the generated phase PSD to the input PSD $S_\phi ^{\textrm{mvK}}$ as a function of the number of PSDs with M being averaged ($\Delta t = 1/60\textrm{ s}$, $T = 7200\textrm{ s}$). Dashed red lines shows the theoretical convergence trend.
The RMS relative error can be calculated by comparing the PSD retrieved from the data set with the input PSD. Figure 3(c) shows the convergence of the generated phase PSD to the input PSD $S_\phi ^{\textrm{mvK}}$ as a function of the number of PSDs with M being averaged. Linear fitting of data points in this figure indicates that simulation results converge to the curve $1.3{M^{ - 0.6}}$. Nevertheless, the theoretical convergence trend of RMS should be ${M^{ - 0.5}}$ [46]. The distinctions between simulation results and theoretical prediction are mainly due to the fact that the calculated RMS has an uncertainty of $1/\sqrt {NM} $.
4. Experimental setup
4.1 Analog transmission system of time-frequency signal
The analog transmission system utilizes the discrete phases ${\phi _n}$ loaded onto the spatial light modulator (SLM) to simulate the phase fluctuations of time-frequency signals propagating through atmospheric turbulence. The signal source used in our experiments was a 250 MHz mode-locked laser with a central wavelength of 1544.2 nm and a bandwidth of 140 nm, which was synchronized with a 10 MHz microwave signal produced by a rubidium atomic clock by using a fiber loop-based optical-microwave phase detector (FLOM-PD) [47]. A fractional frequency instability of 2.12 × 10−14 could be achieved within 103 s averaging time. The principle of the phase-locking optical pulses based on FLOM-PD will be introduced in the next section.
As shown in Fig. 4, The phase-locked optical pulses were connected to the transmitter terminal through polarization-maintaining fibers (PMFs). Since the SLM can only modulate the horizontally polarized light, a half-wave plate (HWP) with central wavelength at 1550 nm was placed behind the transmitter terminal to adjust the polarization state of the generated optical pulses. In order to verify the applicability of the proposed dynamic ITPS, the discrete phase values ${\phi _n}$ for specific turbulence condition where $C_n^2 = 1 \times {10^{ - 14}}\textrm{ }{\textrm{m}^{ - 2/3}}$ and $L = 2\textrm{ km}$, or $C_n^2 = 2 \times {10^{ - 14}}\textrm{ }{\textrm{m}^{ - 2/3}}$ and $L = 10\textrm{ km}$, was applied onto the SLM. Rytov variances at these two turbulence conditions are 0.7 and 27.1 [48], corresponding to the cases of moderate turbulence and strong turbulence respectively. The SLM used in our experiment has an image resolution of 1920 × 1080, a pixel size of 8 µm, and a frame rate of 60 Hz. The receiver terminal is connected to a NEW FOCUS 1611 low-noise photo-detector (PD) and a KEYSIGHT 53230A high-precision universal frequency counter to measure the repetition frequency of the free-space optical pulses coupled into the optical fibers.
Fig. 4. Analog transmission system for time-frequency signal based on dynamic ITPS with high frame rate. The solid black lines represent single-mode fibers, the solid blue line refer to PMF, the dashed black line illustrates electronic (radio frequency) connection and the dashed gray lines give free-space optical paths. PMF, polarization-maintaining fiber; HWP, half-wave plate; SLM, spatial light modulator; PD, photo-detector; BPF, band-pass filter; AMP, amplifier; LO, local oscillator.
In our experiments, the transmitted optical power was 30.67 mW, and the received light power at the output port of single-mode fiber reached 2.32 mW. The transmission loss was 11.21 dB, which mainly includes polarization-dependent loss of the SLM and spatial coupling loss. Due to the impact of environmental temperature and mechanical vibration, the received light power exhibits slight drifts at different instantaneous observation moment. A variable optical attenuator was employed at the receiver terminal to maintain the light power at 1 mW level. The residual-phase difference between the local oscillator (LO) and the transferred radio frequency (RF) signal was measured by a digital phase difference discriminator based on a modified version of the analog discriminator method. In this way, the measurement error caused by the amplitude-phase conversion can be limited to some degree.
4.2 Generation of phase-locked optical pulses based on FLOM-PD
The black block diagram in Fig. 5 shows the operation principle of the mode-locked laser based on FLOM-PD, which was first proposed by K. Jung and J. Kim in 2012 [47]. The main part of FLOM-PD consists of a Sagnac fiber loop and a high-speed phase modulator. Frequency of the microwave signal ${f_\textrm{M}}$ applied onto the phase modulator was set to multiple times of the mode-lock laser repetition frequency ${f_\textrm{R}}$, i. e.${f_\textrm{M}} = l \times {f_\textrm{R}}$, where l is a preset integer. In our experiments, ${f_\textrm{M}}$ and ${f_\textrm{R}}$ are 1 GHz and 250 MHz, respectively. The phase-locked loop (PLL) uses the relative phase difference between the microwave signals and optical pulses as the error signal that drives the piezoelectric transducer (PZT) to control the laser repetition frequency. The microwave signal was locked to a rubidium atomic clock, reaching an instability of 1.43 × 10−14 in 103 s averaging period so as to ensure that the high detection accuracy of FLOM-PD. A non-reciprocal $\pi /2$ phase shifter was placed inside the Sagnac loop to balance the output light intensity and generate the error feedback signal in the linear region. In order to suppress the impact of environmental temperature on the cavity length of the mode-locked laser, a temperature control module was employed in conjunction with the PPL and FLOM-PD to maintain the system temperature at 25°C.
Fig. 5. Illustrated operation principle of the 250 MHz mode-locked laser using a fiber loop-based optical-microwave phase detector (FLOM-PD). FM, frequency multiplier; PZT, piezoelectric transducer.
10 MHz microwave signal from the rubidium atomic clock served as the input signal of the two-stage tenfold frequency multiplier (FM) to finally produce 1 GHz signal as the driving microwave of FLOM-PD module. It was also employed as the time base of the frequency counter to eliminate frequency deviation of the internal time-base signal of the frequency counter from the driving microwave signal of FLOM-PD.
4.3 Out-field time-frequency transfer system
In order to verify the reliability of our proposed time-frequency transfer system, an out-field short-distance time-frequency transfer system was built, the schematic diagram of experimental setup is shown in Fig. 6. The optical pluses were amplified by a low-noise erbium-doped fiber amplifier (EDFA) and coupled into the free space from the east building of College of Electronic Information and Optical Engineering, Nankai University. The optical terminal was aimed at a corner-cube reflector with gold coating for efficient reflection at 1550 nm, which was located 62 m away, to set up a free-space path of 124 m in length. The 250 MHz signal was transmitted to a frequency counter to average the signal frequency in 1 s gate time. The output of the counter was recorded to estimate the fractional frequency instability. The phase difference discriminator was also used to measure the PSD of the phase noise.
The large aperture transmitter terminal with adjustable focus has a typical $1/{e^2}$ intensity diameter of 14.5 mm. This beam was coupled into free space, resulting in a beam waist with a divergence of 10.7 µrad. Diffraction-limited divergence over the whole 124 m atmospheric path results in a typical $1/{e^2}$ intensity diameter of 17.2 mm for the beam returning to the receiver aperture. The corner-cube reflector has a clear aperture of 63.5 mm, and the receiver terminal has a nominal aperture of 42.5 mm, which is much smaller than the Fried scale and large enough to fully cover the receiving spots. The optical power launched into the transmitter is 16.6 dBm (approx. 46 mW). Theoretical loss induced by absorption and scattering of the 1550 nm light in the air is ∼0.5 dB for the 124 m path [18]. The corner-cube reflector introduced an optical attenuation of ∼1.5 dB. The coupling loss from free space into optical fibers is ∼11.5 dB. An additional ∼3 dB loss should be considered because of the optical splitter used to respectively measure optical power and fractional frequency instability. Taking into account all of the above-mentioned losses, the 124 m link has total loss of 16.5 dB, and the received optical power should be about 0 dBm (approx. 1 mW).
5. Results
5.1 Phase noise measurement of dynamic ITPS
The experiment was conducted over a three-week period from April to May of 2022. Data acquisition runs were performed under two conditions: (1) moderate turbulence ($\sigma _R^2 = 0.7$), and (2) strong turbulence ($\sigma _R^2 = 27.1$). Runs were conducted for up to 2 hours and only paused to change the system configuration. Although the phase data were measured on different days, the environmental conditions, such as indoor temperature, atmospheric turbulence, and receiving optical power, were similar.
Figure 7(a) shows the phase noise PSDs ${S_\phi }(f )$ of the analog time-frequency transfer system. The major concern is the frequency jitter caused by the loading phase of SLM, defined as ${S_{{\phi _{\textrm{SLM}}}}} = {\left( {\sqrt {{S_\phi }} - \sqrt {{S_{{\phi_0}}}} } \right)^2}$, where ${S_{{\phi _0}}}$ is the measured noise floor. From this figure, it can be found that ${S_\phi }$ is several orders of magnitude larger than ${S_{{\phi _0}}}$ throughout a wide frequency range, and thus we have ${S_{{\phi _{\textrm{SLM}}}}} \approx {S_\phi }$. The solid lines in Fig. 7(a) give phase noises for different turbulence intensities, which can be fitted using $\log [{S(f )} ]= \log [{A{f^{ - \gamma }}} ]$. A is dependent on the turbulence intensity, and the scaling exponent $\gamma $ should be 2.67 according to the Kolmogorov spectrum. The PSDs were measured between 10 mHz and 100 Hz to avoid the impact of fiber noise, which is 10∼20 dB lower than the measurement level over this frequency range [37]. The noise floor (gray line) mainly results from intrinsic jitter of the optical frequency comb, mechanical vibration of the optical platform and shot noise of the photoelectric detector. The PSDs show an obvious transition from ${f^{ - 2/3}}$ scaling turbulence to Kolmogorov ${f^{ - 8/3}}$ turbulence at corner frequency ${f_c} = 5.4\textrm{ Hz}$. The PSDs also show roll-off in low-frequency outer scale below $f = 1\textrm{ Hz}$ in both cases of moderate and strong turbulence. The integrated timing jitters from 1 mHz to 100 Hz are 1.6 ps and 5.5 ps, respectively. Figure 7(b) gives fluctuations of the imposed phases loaded onto the SLM for 2 hours, respectively.
Fig. 7. Phase noise analysis of the analog time-frequency transfer link. (a) Phase noise PSDs under different turbulence conditions. The gray line represents the measured noise floor. The red and blue lines represent the measured phase noise measured under moderate ($\sigma _R^2 = 0.7$) and strong ($\sigma _R^2 = 0.7$) turbulence conditions, respectively. The transition from ${f^{ - 2/3}}$ scaling turbulence to Kolmogorov ${f^{ - 8/3}}$ scaling turbulence (dashed) occurs at ${f_c} = 5.4\textrm{ Hz}$ and the outer scale roll-off is below $f = 1\textrm{ Hz}$. The integrated timing jitters from 1 mHz to 100 Hz are 1.6 ps and 5.5 ps, respectively. (b) Imposed phases loaded onto the SLM for 2 hours under different turbulence conditions. (c) Evolution of the MDEV for noise floor (gray), moderate turbulent link (red), and strong turbulent link (blue), respectively. The shaded region represents the power-law response caused by the low-frequency roll-off in the PSDs.
Atmospheric turbulence has a significant impact on the stability and accuracy of optical time-frequency transfer. The frequency counting data were used to evaluate the stability of the round-trip time of flight, which is characterized by the MDEV related to the spectral density of the fractional frequency fluctuations [49]
5.2 Comparison with the out-field frequency transfer system
In order to validate our proposed ITPS, a 124 m out-field time-frequency transfer link was set up. The results of timing jitter PSDs and fractional frequency stability are shown in Fig. 8. The orange curves give the measurement results of the dynamic ITPS, while the cyan curves show the results of the 124 m out-field link.
Fig. 8. Time-frequency signal measurement results of dynamic ITPS (orange) and 124 m out-field path (cyan), respectively. (a) 500 MHz RF phase drift of dynamic ITPS and 124 m out-field path. The thin line gives the raw data. (b) Measured phase noise PSDs. (c) The beam wandering behavior measured within 30 mins. (d) Fractional frequency stability results in terms of MDEV.
The atmospheric refractive index structure parameter $C_n^2$ was obtained according to the intensity-weighted centroid measurement results within $30\textrm{ mins}$, which is used to produce discrete phase values ${\phi _n}$ [50]. To avoid the meteorological factors such as wind speed and direction, temperature, humidity, and rainfall, we chose a clear day to conduct the experiment. The beam wander was recorded by an imaging charge-coupled device (CCD) with a sampling frequency of 50 Hz. The drift variance of intensity weighted centroid $\sigma _\rho ^2$ can be defined as
where $\sigma _x^2$ and $\sigma _x^2$ refer to the drift variances of intensity-weighted centroid in horizontal and vertical directions, respectively. In this case, the atmospheric refractive index parameter $C_n^2$ can be calculated according to the following equationThe beam wandering behavior were measured within $30\textrm{ mins}$, as shown in Fig. 8(c). The calculated $C_n^2$ is equal to 1.45 × 10−16 m−2/3, corresponding to the case of weak turbulence.
The long-term phase drift of the 500 MHz RF signal was also measured for 2 hours, as shown in Fig. 8(a). The thin line gives the raw data. Compared with the laboratory environment, the relative mechanical vibration between the transmitter terminal and receiver terminal in the outfield is more severe. As a result, the set of out-field data measured in 124 m round-trip transfer path becomes more volatile. It is noteworthy that the PSD of free-space transfer link (orange) in Fig. 8(b) shows no strong evidence of a roll-off at low Fourier frequencies, even at very low Fourier frequency of 10 mHz. For a typical wind speed of $V = 1\textrm{ m/s}$ and an outer scale ${L_0} = 10\textrm{ m}$, a roll-off should begin at 100 mHz (equals to $V/{L_0}$). The absence of a low-frequency roll-off in the PSD of a practical out-field path does not preclude the existence of an outer scale, due to several referenced works do find an outer scale in the spatial structure. This simply indicates a breakdown in Taylor’s hypothesis at low Fourier frequency.
Figure 8(d) shows the MDEVs. On the contrary, a roll-off of the MDEV curve of dynamic ITPS (orange) in long-time scale indicates the existence of an outer scale. The measured MDEV curves based on dynamic ITPS and 124 m out-field path agree very well with the RMS relative error of 10.07% for the power-law scaling of MDEV.
6. Discussion and conclusion
According to the pulse counting results, the MDEV follows an ordinary power-law scaling that can be expressed as $\sigma _{_y}^2(\tau )\propto {\tau ^{ - \gamma }}$ from 1 s to 100 s. Since our data have no hidden phase slips, the simple power-law response of the phase noise PSDs is uncontaminated by potential systematics and the measured power-law scaling can be completely attributed to our proposed ITPS. Except for the phase jitter caused by dynamic ITPS, there are a lot of additional phase jitters due to fiber path vibration, such as temperature cycling, amplitude-phase conversion [51], and the Doppler shift between the transmitter and the receiver. We are only concerned about how well the simple power-law scaling $\gamma $ of the PSDs caused by the ITPS matches the phase disturbance of out-field atmospheric turbulence. And as for the suppression of additional phase jitters, that is another subject to be addressed.
However, the power-law scaling of ${f^{ - \gamma }}$ is contradictory to the standard turbulence theory presented in Sec. 2, where an assumed power-law scaling of the turbulence is mapped to the Fourier frequency domain through Taylor’s hypothesis of frozen turbulence. In Fig. 1, the theoretical timing-jitter PSD $S_{\textrm{jitter}}^{\textrm{mvK}}$ could be classified into two distinct regions: (1) a Kolmogorov power-law scaling of ${f^{ - 8/3}}$ over the frequency domain $V/{L_0} < f < V/{l_0}$, corresponding to the inertial region; (2) a flat power-law scaling of ${f^0}$ at low frequencies, $f < V/{L_0}$, corresponding to the energy-input region. Unlike the aforementioned theory, our measured phase noise PSD follows much simpler power-law scaling for a frequency range of 10 mHz to1 Hz, which covers the energy-input frequency region. In other words, the phase noise PSD consistently follows a power-law scaling of ${f^{ - \gamma }}$ which lies between the Kolmogorov ${f^{ - 8/3}}$ scaling and a random-walk ${f^{ - 2}}$ scaling [52], with no evidence of a distinct energy-input region. This disagreement is caused by the fact that considering the assumptions in Sec. 2, the simple Kolmogorov scaling of ${f^{ - 8/3}}$ in the inertial region cannot well deal with the full complexity of the outdoor wind patterns, thermal drifts, and path obstructions. More importantly, Taylor's hypothesis is no longer invalid for the time scale of more than 0.1 s. The frequency dependence of the measurement results should be caused by the translational effects of wind and the intrinsic time-dependent evolution of the turbulent vortexes.
Besides whether the transmission link is in analog operation mode, the difference between practical out-field and analog system is the whether an EDFA is used before the transmitter terminal to compensate for the power loss. It is generally recognized that the amplification process of EDFA will not bring about additional phase noise [53]. But in practice, the phase noise definitely causes linewidth broadening because of the presence of amplified spontaneous emission (ASE). The mechanism for the linewidth broadening of the output optical pulses from an EDFA is generally regarded as that the spontaneously emitted photons in the EDFA fall into the input laser linewidth after being amplified [54]. The experimental results also show that with the assistance of EDFA, the power equalization mechanism can be established to suppress the phase noise caused by the amplitude-phase conversion. Experimental results indicate that the phase noise PSD curves do not show observable change with the variations in the pump power of EDFA and attenuation. And therefore, the nonlinear effect induced by optical amplification is negligible.
The fractional frequency stability is limited by thermal instability with integration time longer than 10 s [55]. The temperature of the optical assemblies was maintained at 25°C, but experimental results show that temperature control would not help to improve the fractional frequency stability. The possible reason for this is that the residual thermal instability is caused by thermal instabilities of the electronic instruments (e.g., the internal PPL, etc.) and electric cables in used for the phase stabilization and phase noise measurement. However, within the two-hour measurement process, the thermal instability did not exceed the modulation range of PZT driving voltage, and thus the measurement accuracy would not be severely affected.
A few phase-compensation systems have been verified to be compatible with stabilized optical time-frequency transfer systems [19,56,57]. Furthermore, with the introduction of a phase-compensation system, the time-frequency transfer distance is limited due to the phase distortion and optical power degradation would be improved.
In summary, an analog transmission system has been demonstrated for time-frequency signal transfer under atmospheric turbulence environment, which can be used to simulate the phase jitters induced by refractive index fluctuations comparable to the practical phase jitters over a 124 m round-trip link. The repetition frequency of the optical pulses is locked to 250 MHz by using the FLOM-PD along with a rubidium atomic clock with high fractional frequency stability as the frequency reference. Our proposed dynamic ITPS technique is anticipated to be used to analyze the horizontal fluctuation component of the arrival angle vector, which provides a deep insight into the phase distortion of time-frequency signal of free-space optical communication links.
Funding
National Natural Science Foundation of China (11774181, 11904180, 61875091, 62105164); Natural Science Foundation of Tianjin City (19JCYBJC16700, 20JCQNJC01480, 21JCQNJC00210).
Acknowledgments
Z. G. thanks Kang Jin and Bofu Xie for their careful readings and insightful suggestions.
Disclosures
The authors declare no conflicts of interest.
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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