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Abstract
We establish an analytical model for the stable dissemination of radio-frequency (RF) signals via fiber-optic links. Based on the phase-locked loop theory, the contributions from the photonic RF source, transmission-path, and additional system noise have been taken into account, leading to the quantitative analysis of the phase noise evolution in the transmission link. Furthermore, the theoretical analysis reveals the relation between the system instability and the frequency of the transmitted signal, which is further verified. Assisted with the proposed model, the optimization for stabilized dissemination of RF signals with a certain length of transmission link or any specified noise floors can be achieved with minimized timing jitter performance, testifying the potential high stability obtained thanks to the higher transmitted signal frequencies. This quantitative model, enabling precise prediction of the frequency instability and timing jitter from the residual phase noise, can be a useful guide in designing a fiber-optic distribution system and evaluating its fundamental limits.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
With the recent development of state-of-the-art optical and atomic clocks, high-stability, high-accuracy, ultralow-noise time and frequency references [1–3] have been widely applied in scientific researches and engineering systems. However, it is quite costly to build such high-quality clocks for each terminal user due to the complicated system set-ups. The distribution of a stable reference via long-distance fiber-optic links has become an essential and enabling tool in a variety of applications, e.g., comparisons of remote optical clocks [4,5], synchronization of X-ray pulses [6], radio astronomy in deep space network [7], and distributed coherent aperture radars [8]. A fundamental challenge in these fiber-optic distribution systems is how to transfer a stable optical or microwave signal to the remote sites with high phase stability and time synchronization accuracy since any mechanical stress and temperature variations on the transmission fiber path will deteriorate the frequency stability and timing jitter, thus the phase noise performance.
Over the past decades, various compensation techniques have been proposed and applied for the fiber-based distribution of optical [9–15] and radio-frequency (RF) references [16–26]. Among these outstanding works, an analytical theory for predicting the performance of coherent transport of optical frequencies has been proposed [9]. High-stability optical frequency dissemination has been experimentally demonstrated with the limitations discussed in detail [10]. Despite the advantages of the direct optical frequency transmission, it could be relatively sophisticated to achieve frequency division or conversion directly from optical frequencies into an RF signal. Meanwhile, in most practical systems, highly-stable and synchronized RF signals are directly used and thus urgently required. These have made the RF transmission a popular, straight, and sometimes more effective way. However, the phase noise contributions and evolution could be more complicated in the RF transmission systems. The source noise, polarization-mode dispersion, thermal noise, shot noise, the excess amplifier noise and amplitude-to-phase conversion may have a significant impact on the RF transmission performance. To this end, a full investigation on these effects in RF transmission is highly demanded. In connection with the basic principle, techniques, noise processing, the advantage of using a high RF frequency in RF dissemination systems have been introduced and discussed [16]. So far, an analytical model that allows for facilitating the quantitative analysis concerning different noises, limits and phase noise evolution is necessary and important to further evaluate and optimize the RF transmission system.
In this paper, we report on a theoretical model for the dissemination of an optically-carried RF signal via fiber-optic links based on the phase-locked loop (PLL) theory. The contributions to the residual phase noise of the transfer references are analyzed and quantified, including different noise sources such as the delay-interferometry nature, fiber-optic transmission links, and overall system phase noise. The relation between frequency and phase noise is discussed in detail. The noises within a wide integral bandwidth from 0.01 Hz to 1 MHz Fourier frequencies have been accounted for in the proposed model, allowing for a more precise description of the time-domain performance improvement. It also reveals the fact that the relative frequency stability can be enhanced by increasing the ratio of the transmitted signal frequency to the system noise floor. Namely, in a fiber-optic RF distribution system, disseminating higher frequency references or reducing the noise floor can achieve higher stability. The prediction for the RMS timing jitter based on the proposed model has successfully unveiled the impact of the parameters such as transmission length, signal frequency and system noise floor. This model allows to predict the instability and timing jitter from the phase noise and can be regarded as significant guidance to design and optimize the transmission frequency for a given fiber-optic distribution system and as well as an evaluation of its fundamental limits.
2. Principle
2.1 RF transmission model based on PLL theory
The operation principle of a typical fiber-based RF dissemination system is exhibited in Fig. 1. An optical fiber link is applied to connect the local and remote end as the transmission medium. At the local site, a high-stability clock is used as the system’s reference source. An optically-carried RF signal, commonly generated through optical modulation or directly extracting two wavelengths from a multi-carriers optical source, is then sent into the fiber link with its phase or transmission delay capable of being instantly and continuously corrected by a double-pass corrector. The light-wave signal travels to the remote site and is output to the terminal user, while a portion of the signal is sent back through the same fiber. After the round-trip transmission, the signal goes through the double-pass corrector once again. The corrector performs phase correction by actively introducing a same amount of phase or time delay on both the transmitted and returned signals. Generally, it can be realized in terms of optical delay lines, such as piezo-electric fiber stretchers or temperature-controlled fiber spools, or acousto-optic modulators that indirectly control the phase delay of the transmitted RF signal. The phase comparison between the returned signal and the initial one is subsequently made in a detection module. The output phase error is filtered into a feedback signal to drive the double-pass corrector. Thus, by locking the returned signal to the reference one or compensating the transmission delay for the forward and backward path, a stabilized and synchronized RF signal can be obtained at the remote end.
For time-domain analysis, the phases of the remote and the returned signals should satisfy the following relation, respectively,
It is assumed that the fiber-optic link is under the hypothesis of reciprocity that the signal passing through the path in both directions experiences the same phase perturbation, referring to ${\phi _{\textrm{Fiber,RT}}}(t )= 2{\phi _{\textrm{Fiber}}}(t )$. Nevertheless, in order to fully satisfy this hypothesis, some preconditions should be clarified. That is, the phase noise are stationary for the forward and backward transmissions through fiber links and the observation time of the phase fluctuation longer than the round-trip time are considered [16]. When the feedback loop is closed, the phase error will satisfy $2[{{\phi_{\textrm{Corrector}}}(t )+ {\phi_{\textrm{Fiber}}}(t )} ]= C$, indicating that the distributed signal is synchronized to the local reference within the effective control bandwidth of the locking system. Furthermore, it should be noted that this locking bandwidth is also limited by the round-trip transmission delay.
For frequency-domain analysis, the open-loop gain of the RF distribution system can be expressed as:
Here, ${H_{\textrm{PD}}}(s)$, ${H_{\textrm{LF}}}(s)$, and ${H_{\textrm{DPC}}}(s)$ are the transfer function of the phase detector, the loop filter, and the double-pass corrector, respectively. $\tau$ is the single-trip fiber propagation delay, and N is the ratio of the frequency divider for the phase-locking in phase discrimination.
For a general discussion here, the gain of the phase detection is chosen as ${H_{\textrm{PD}}}(s) = {3.7\ast }{10^{ - 4}}$, a basic proportional-integrator with ${H_{\textrm{LF}}}(s )\textrm{ = }{K_\textrm{P}} + {{{K_\textrm{I}}} \mathord{\left/ {\vphantom {{{K_\textrm{I}}} s}} \right.} s}$ is used as the loop filter, where ${K_\textrm{P}}\textrm{ = 1000}$ and ${K_\textrm{I}}\textrm{ = 250000}$. The transfer function of the double-pass corrector is defined as ${H_{\textrm{DPC}}}(s) = {{\textrm{3600}} \mathord{\left/ {\vphantom {{\textrm{3600}} s}} \right.} s}$. The transmission delay $\tau = 200\mu s$ and $N = 1$. With above typical parameters, the bode diagram for the open-loop frequency response of the distribution system is shown in Fig. 2 without loss of generality. By letting $s \to j\omega$, a series of zeros can be observed at $f = {{({2n + 1} )} \mathord{\left/ {\vphantom {{({2n + 1} )} {4\tau }}} \right.} {4\tau }}$, where the integer $n \ge 0$. Besides, a $\textrm{2}\pi$ phase shift occurs around those frequencies correspondingly. It is primarily the long transmission delay of the fiber that results in an issue of insufficient phase margin, leading to the deterioration of the stability. In order to avoid this, the cutoff Fourier frequency should be lower than the first zero, which appears at $f = {\textrm{1} \mathord{\left/ {\vphantom {\textrm{1} {4\tau }}} \right.} {4\tau }}$. On the whole, the tradeoff between the loop gain and the phase-margin should be taken into consideration when designing a stable feedback loop.
When the loop is closed, the phase evolutions of the local and the remote outputs are derived as follows based on the PLL theory
Since each part of the phase in Eq. (3) and Eq. (4) is uncorrelated with the others, the phase noise power-spectrum-density (PSD) of the local and the remote signal can be written as
In order to directly evaluate the system performance in terms of the effect on the noise suppression, the residual phase noise PSD can be obtained by extracting the phase of the beat note between the local-generated RF and the remote signal as expressed in the following
From the above equations, it can be figured out that the contributions to the residual phase noise can fall into two categories. One is the frequency-dependent noise, which consists of the phase noise due to the delay self-interferometry of the RF signal and the unsuppressed phase noise of the fiber-link. Both are in proportion to the square of the angular frequency. Thus, with the increase of the signal’s frequency, this part of the phase noise will increase accordingly. For instance, as the frequency of the transferred signal doubles, the noise level of this part will rise by ∼6 dB. The other is relatively insensitive to the frequency of the transmitted signal, including the phase noise originated from optical or electrical devices that accumulates in signal processing such as signal amplification, photo-detection, feedback phase-locking circuits. Different types of noise, such as the white thermal noise, shot noise, and any other extra noise jointly comprise the phase noise of the system that is also referred to as the system noise floor. It is rather because of the different mechanisms on the generation and evolution of these noises that they are barely relevant to the frequency of the transferred RF signal. Nevertheless, the noise floor has a significant impact on the transmission system. Therefore, it can be deduced that with a different frequency of the transmitted signal, the stability and precision of the RF transmission can be dominated by different types of noise sources. By increasing the frequency for the transmitted signal or optimizing the system noise floor, the proportion of frequency-dependent noise will be gradually dominant until the impact of the other part can be negligible.
In addition to the phase noise PSD, the time-domain statistics such as the Allan Deviation (ADEV) and the root-mean-square (RMS) timing jitter of the transmission system are also very important for evaluating the short- and long-term stability. Theoretically, both of them can be derived from the residual phase noise by using a weighting function [29]. The expressions are demonstrated as follows:
From Eq. (7) and Eq. (8), it can be inferred that as the frequency of the transferred signal increases, the performance of the ADEV and RMS timing jitter can be gradually improved as the frequency-dependent noise becomes dominant. Eventually, when the contribution from the system noise floor becomes negligible, further raising the frequency would hardly show any improvements in the system performance. Meanwhile, from the practical point of view, it also results in other problems in signal processing at a high frequency due to the limited bandwidth of the electronics components. Therefore, it is crucial to minimize the system noise floor and subsequently to select the optimum transmitted frequency accordingly to the actual system noise floor in order to fully exploit the potential of the RF transmission system.
2.2 Phase noise of the photonic RF signal
Generally, a low phase noise, highly-stable photonic RF signal is vital for the fiber dissemination system. Not only should it meet the fundamental requirement of the application, but also it should introduce limited excess noise resulting from the delay self-interferometry in the round-trip transmission. Thereby, from the above discussion of Eq. (3), it can be concluded that the noise contribution due to the delay self-interferometry to the local signal should be less than the noise induced from round-trip fiber-link, where we should have ${S_{{\phi _{\textrm{Signal}}}}}(\omega )\cdot {|{1 - {e^{ - j2\omega \tau }}} |^2} < {S_{{\phi _{\textrm{Fiber,RT}}}}}(\omega )$.
When the loop is closed, it can be further written as
2.3 Phase noise induced from fiber-links
The single-trip phase fluctuation for disseminating a stable photonic RF signal over an optical fiber can be described in time-domain as follows:
As known, the ambient temperature variation is regarded as a random process, while the magnitude of the environmental perturbation also changes constantly. This results in the fact that the phase noise of a certain length of fiber-link can hardly be modeled as one single type of Power-law noise. The thermodynamic fiber noise and thermomechanical fiber noise primary behaving as white phase noise and flicker phase noise dominate in large frequency region (from Hz to kHz), while other high-order types of the fiber noises can been observed at lower frequency in practical systems due to the slow temperature fluctuation and drift [28]. In our case, we propose an empirical expression to quantify these fiber noises and subsequently to predict their evolutions in a transmission system. Based on the Power-laws dependence [29], we have that $\left\langle {{{|{{{\tilde{\tau }}_L}(\omega )} |}^2}} \right\rangle \textrm{ = }L\sum\limits_{\alpha ={-} 3}^0 {{h_\alpha }} {\left( {\frac{\omega }{{2\pi }}} \right)^\alpha }$, where ${h_\alpha }$ holds for the coefficient of the corresponding $\alpha$-order of noises. That is, for a predetermined length of the fiber, the phase noise could be quantified by the sum of four dominant Power-laws noise terms with different coefficients. Furthermore, the model has been experimentally verified with different transmission frequencies and fiber lengths at large Fourier frequencies. The PSD can be expressed as:
As shown in Fig. 3(a), the red curve presents the phase noise measured from the transfer of a 10 GHz signal over a 10 km spooled fiber link. It can be observed that the flicker frequency noise and the flicker phase noise dominate from 0.01 Hz to 1 Hz and 10 Hz to 10 kHz, respectively. Some spurs surround about 50 Hz-200 Hz are considered as the coupled noise from the power supply. The black curve represents the simulated residual phase noise spectrum of the 10 GHz RF signal after 10 km fiber link transmission. To obtain this, we first experimentally measured the phase noise of a commercial RF synthesizer (Keysight E8257D) at 10 GHz. It is worth noting that this synthesizer is also applied as the source for the fiber noise measurement. In order to individually evaluate the noise evolution of the RF source, with this measured phase noise, the phase noise of the mixed signal between the original 10 GHz signal and that after a 10-km transmission (corresponding to ∼50us delay) is numerically obtained, referring to as the residual phase noise after 10-km transmission as shown in the Fig. 3(a). As a matter of fact, this can be regarded as a 10-km delay interference of the 10 GHz RF signal. It can be observed that above 10 kHz, the measured phase noise of 10 GHz via 10 km transmission is almost consistent with the simulated 10 km residual phase noise of the RF signal which proves that the phase noise of the delay self-interferometry is dominant. Below 10 kHz, the red curve is higher than black one, indicating that within this frequency range the fiber noise dominated.
Fig. 3. (a) Measured and Fitted fiber noise by transferring a 10 GHz signal over a 10 km spooled fiber, simulated residual phase noise of 10 GHz via 10 km transmission. (b) Predicted and measured fiber-link phase noise by transmitting a 10GHz signal over a 20km spooled fiber and 20GHz signal over 100km spooled fiber, respectively.
Thus, the coefficients h-3∼h0 in Eq. (11) are obtained by fitting the blue curve to the red one, where ${h_{ - \textrm{3}}} = {10^{ - 32}}$, ${h_{ - 2}} = {2\ast }{10^{ - 32}}$, ${h_{ - 1}} = 5\ast {10^{ - 33}}$, ${h_0} = 2.5\ast {10^{ - 38}}$ are obtained. In the frequency region from 0.01 Hz to 10 kHz, the fitted and measured curves show a good agreement in terms of the magnitude and spectral shape. Above 10 kHz, the fiber noise falls off quickly below the phase noise of the delay self-interferometry.
Next, more experiments are carried out to prove that the fiber-link induced phase noise is proportional to the square of the frequency of the transferred signal and is also scaled with the transmission length. The phase noises of the fiber links are measured by transmitting a 10 GHz signal over a 20 km and a 20 GHz signal over 100 km, respectively. The experimental results are shown in Fig. 3(b). In the frequency region from 0.01 Hz to 10 kHz, the predicted results using the proposed model and the coefficients obtained from Fig. 3(a) are in line with the experimental measurement, confirming the validity of the proposed fiber noise model.
2.4 System phase noise floor
The system noise floor is defined as the measured phase noise under the locking operation of the back-to-back transmission link, which is usually connected through a very short fiber patchcord. It comprises different types of noises such as thermal noise, shot noise, amplifier noise, amplitude to phase noise and any other noises which originate from the optical or electronic components and continuously accumulate along the fiber transmission system. These frequency-independent noises behave as different types of Power-Law noise, which are often considered to be the fundamental limits for a stable distribution of an RF reference. Therefore, a detailed analysis of the system noise floor could be of great help to optimize the design of the RF transmission scheme, and it could be beneficial to choose lower phase noise devices and to evaluate the overall performance of the system.
The measurement results of the phase noise floor of a typical RF distribution system are shown in Fig. 4. The optical carrier frequency is around 1550 nm, while the frequency of transmitted photonic RF signal is 25 GHz. At the Fourier frequencies above 10 kHz, the white phase noise is observed to be about -130 dBc/Hz, indicating that the thermal noise and the shot noise are dominant at these frequencies. Around 200 Hz, a bump can be observed as a result of the phase-locking operation. Below 10 Hz, the flicker phase noise and other high orders of Power-law noises can be observed, which mainly come from the signal processing such as the amplification. Based on these results, the residual phase noise and the ADEV can be numerically estimated in closed-loop operation for an RF dissemination system.
3. Results and analysis
3.1 Phase noise and frequency instability versus transmitted frequency
In order to verify the optimization enabled by the proposed model, a series of experiments have been carried out. The fiber length is assumed to be 40 km, which corresponds to ∼0.2 ms of the transmission delay. Phase noise of a commercial RF synthesizer (Keysight E8257D) is applied to the model, which has been calibrated at 25 GHz in advance to confirm its validity with respect to the specifications. The fiber noise is defined according to the Eq. (11), while the system noise floor is experimentally measured from a typical RF transmission system. The transferred frequency covers from 6.25, 12.5 25, 50, 100, 200, 400, 800, 1600 to 3200 GHz. For the ADEV, the integral range is set from 0.01 Hz to 1 MHz. The residual phase noise and the ADEV are plotted respectively in Figs. 5(a) and 5(b).
Fig. 5. (a) The simulated residual phase noise of distributing a series frequency of RF signals over a 40 km fiber-link. (b) The simulated Allan deviation of the distribution system.
As is shown in Fig. 5(a), the phase noise exhibits different trends at Fourier frequencies below 10 Hz, from 10 to10 kHz, and over 10 kHz, respectively. Firstly, at high Fourier frequencies regime above 10 kHz, the residual phase noise rises by ∼6 dB when the signal frequency is doubled, inferring that the frequency-dependent noise is in domination. Secondly, around the range of 1 to 10 kHz, when the transmitted frequency is from 6.25 GHz to 400 GHz, the residual phase noise increases by less than 6 dB. While with a further increase of the transmitted frequency to 3.2 THz, the tendency approaches 6 dB. This means that the proportion of frequency-dependent noise arises gradually. At last, the noise performance almost has no signs of change within frequencies below 1 Hz when frequency increases from 3.125 GHz to 800 GHz, indicating that the system noise floor dominates at this Fourier frequency region, while above 800 GHz, the frequency-dependent noise dominates.
Furthermore, the relationship between the ADEV and the signal frequency has been unveiled, as shown in Fig. 5(b). The short-term instability (below 0.1 s) changes almost with the same trend. Around averaging time 0.1s and 10s, the stability is gradually enhanced as the frequency rises. While long-term instability (above 10s) has a noticeable improvement, which promotes from 2×10−18 of 6.25 GHz to 1×10−20 of 3.2 THz at 1000 s averaging time. In this case, we can further conclude that considering a certain level of the system noise floor, the performance of the transmission system can hardly be further boosted, which means that the system noise is almost negligible at this moment. Thus, the optimum frequency can be determined for such an RF transmission system.
Despite the advantage it offers, due to the wide bandwidth and the ultra-high frequencies, the transfer of high frequencies also brings about difficulties in signal processing such as phase and frequency discrimination as well as the control of the high-frequency optically-carried signals. Few works have been reported to solve this critical issue. As far as we know, schemes by utilizing the optical frequency comb are demonstrated to establish a frequency chain between optical and radio-frequency, and high-frequency optical-carried RF signals are generated with low phase noise [30]. Meanwhile, thanks to the dual-heterodyning phase error transfer scheme and the acousto-optic frequency-shifter -based phase correction technique, the phase discrimination and fast compensation regardless of the signal frequency have been achieved, thus allowing for the high frequency dissemination over fiber links [26]. To this end, the idea of disseminating a higher frequency reference to achieve better system stability are theoretically dicussed and further verified.
3.2 RMS timing jitters
In order to further discuss the limit for the RF dissemination system, the prediction for the RMS timing jitter versus the transmitted signal frequency and fiber length are presented in Fig. 6. Based on the proposed model, all the results are numerically calculated by applying the same RF noise and system phase noise floor. The length of optical fiber is set to be 20, 40, and 160 km, respectively, corresponding to short-, medium-, and long-distance transmission. The signal frequency multiplies from 1 to 800 GHz, and the integration range is from 0.01 Hz to 1 MHz.
Fig. 6. The prediction RMS timing jitter (dot-dash curve) and theoretical limit RMS timing jitter (dash line).
With the transmitted frequency increasing from 1 to 10 GHz, significant improvements on the RMS Timing jitter from approximate 800 fs to 80 fs can be directly observed from the predictions, proving the effectiveness of the increase in transmitted signal frequency. However, the length of fiber shows no impact on the results around this frequency range, which indicates that to a certain degree, the system noise floor is still dominant though the proportion of the frequency-dependent noise is enlarged by the extended fiber length. However, with the continuous increase of the signal frequency, all trends of the improvement begin to slow down. Gradually, the estimations approach each limit at about 13, 17, and 32 fs by comparing the solid curve to the dash lines. Besides, it is worth mentioning that the downward trends of the three curves behave differently in speed at this frequency range, and the improvement is much more apparent in the case of short-distance transmission. At last, we discover the relationship between frequency and fiber length when the same timing jitter is necessitated to be achieved. For instance, in the case of 10% error with respect to the theoretical limit, the transmitted frequency shows f1<f2<f3 for 160 km, 40 km, 20 km link, respectively, as marked in Fig. 6. It reveals that a relatively higher frequency is required at shorter length transmission. In general, it can be inferred that in a longer-length case of fiber transmission, a lower frequency can be used to to achieve a low timing jitter performance. Indeed, these results have confirmed that the proposed model is of practical importance and can be instructive for selecting a competent frequency for transmission links with a certain length of the fiber link. We believe these fruitful and detailed conclusions can be used as a guideline for the design of an RF distribution system.
Besides, the analysis of the system noise floor on the total transmission timing jitter is discussed to investigate the limits and jitter performance under the conditions of different system noises are applied in the proposed model. The two different cases, indicated by the blue and red curves, respectively, in Fig. 7 correspond to the cases when two different system noise floors (see inset in Fig. 7) are used in the numerical simulation. Both of noise floors are experimentally measured using different schemes. The main difference between the two systems are the implementation of the phase discrimination and the loop control for the double-pass corrector where different photodetectors and the subsequent signal processing schemes are utilized. As is shown, the noise of the red curve is higher than the other from 0.01 Hz to 1MHz Fourier frequency. The transmission fiber length is determined at 40 km for both cases.
Fig. 7. The prediction for RMS timing jitters (blue and red dot-dash line) and theoretical limit (green dash line).
When the transmitted frequency increases from 1 to 100 GHz, the timing jitter performances in both systems have exhibited significant improvements. On the one hand, thanks to the lower system noise floor, the blue curve has a lower timing jitter when the transmitted frequency is below somewhat 200 GHz. This indicates that the transmission system does have better time-domain stability compared with the red one at the same frequency. On the other hand, to achieve a low timing jitter, a relatively low frequency could satisfy the transmission more efficiently for systems with a lower noise floor. In other words, the noise floor of the transmission system needs to be reduced as much low as possible. Finally, above 200 GHz, both of the systems approach the limits at ∼17 fs, meaning there is no need to further increase the transfer frequency in either case.
To this end, the prediction results indicates that the proposed model is versatile for different kinds of system noises. The estimations for the RMS timing jitter obtained from the proposed model have successfully unveiled the impact of the parameters such as transmission length and signal frequency under different system noises. The model, therefore, can be regarded as a useful tool and can provide a guiding significance to evaluate the fundamental limits and optimizing the fiber-optic distribution system for different applications.
4. Conclusions
We propose an analytical model for the dissemination of a photonic RF signal via a certain-distance fiber-link based on the PLL theory. The phase noise contributions from the photonic RF source, transmission-path and system noise floor are further analyzed and quantified. The theoretical analysis with numerical simulation results reveal the relation between the transmitted frequency and phase noise as well as the system instability, proving the concept that disseminating higher frequency reference and reducing the systems noise floor can achieve better frequency stability. In addition, assisted with the proposed model, the impact of the parameters such as transmission length and signal frequency under different system noises can be directly quantified and discussed, while the optimization for stabilized dissemination of RF signals with a certain length of transmission link or any specified noise floors can be achieved with minimized timing jitter performance. This quantitative model, enabling precise prediction of the phase noise, frequency instability and timing jitter, can be a useful guide in designing a fiber-optic distribution system and evaluating its fundamental limits.
Funding
National Natural Science Foundation of China (61690193, 61805014, 61827807); China Postdoctoral Science Foundation (2018M630082, 2019T120051).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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