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Abstract
The performance of long-distance two-way fiber-optic time and frequency (T/F) transfer is directly affected by link parameters. In this paper, a fast on-line scheme to maximize the signal-to-noise ratio at the T/F transfer link ends (denoted by SNRW and SNRE) is proposed by optimizing the gains of bidirectional Erbium-doped fiber amplifiers (Bi-EDFAs). The dedicatedly defined objective optimization functions, ln(1/SNRW + 1/SNRE) for microwave and time transfer, and ln(1/SNRW) for optical frequency transfer, are proven to be a convex function and enable the computation time to be reduced significantly by employing the fast and mature convex optimization algorithm. The linear equations about the objective optimization function and the fiber parameters required by optimization are constructed, and then the fiber parameters are proposed to be on-line obtained accordingly by measuring the concerned SNR at the link end under different EDFA gain configurations and solving the linear equations. The transmission matrixes for the proposed optimization scheme are further derived, which can simplify the implementation in the long-distance multiple EDFAs links. The simulation shows that the proposed algorithm can obtain the optimal parameters consistent with the reported experimental results, and can reduce the computation time by more than four orders compared with the scanning optimization method for a link with more than 6 Bi-EDFAs.
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1. Introduction
Fiber-optic time and frequency (T/F) transfer has underpinned the fundamental and applied sciences, such as clock-based geodesy [1,2], radio astronomy [3], and very long baseline interferometry [4,5]. Up to now, thousands of kilometers fiber-optic T/F transfer has been demonstrated for establishing the long-distance intercontinental clock networks [6–8].
In the long-distance fiber-optic T/F transfer, bidirectional wavelength division multiplex transmission is typically conducted to remove the single Rayleigh scattering, and bidirectional Erbium-doped fiber amplifiers (Bi-EDFAs) are generally used to compensate the propagating attenuation [3,8–10]. However, the Bi-EDFA will introduce the amplified spontaneous emission (ASE) noise. Moreover, when the symmetry Bi-EDFA is used as presented in [3,8,9], the double Rayleigh scattering (DRS) noise with the same wavelength as the transferred T/F signal will be amplified and degenerate the SNR at the link ends. A degraded SNR will directly increase the time or frequency jitters, and also influence the working states of the employed phase locking servos, causing frequent cycle slips and even lock-losing [11,12]. In 2013, Ł. Śliwczyński et al. modeled the SNR of bidirectional fiber-optic link for T/F transfer [13]. To maximize the SNR, the gain of each Bi-EDFA is optimized by scanning within a certain range in the case of knowing the fiber losses and scattering coefficients of each fiber span. However, the theoretical scanning computation complexity and time will increase exponentially with the number ($N$) of the employed Bi-EDFAs following the relationship of $s^N$, where $s$ is the scanning step number of each Bi-EDFA’s gain. Furthermore, the above optimization process requires the loss and Rayleigh scattering coefficient of each fiber span. The off-line precise measurement of these parameters is time-consuming and laborious in practical applications. Thus, an on-line optimization method, which remotely adjusts the employed Bi-EDFAs one-by-one by tuning their gains to increase the SNR, is presented in [14], and was tested in real telecom fiber links. A 7-dB SNR improvement is obtained in compared with the initial set of gains. However, the obtained final gains may not be the globally optimal results, since the tuning process doesn’t cover all the gain combination of all the Bi-EDFAs and the sub-optimal results may exist.
In this paper, a fast optimization algorithm for calculating the optimal gains of Bi-EDFAs is first proposed to maximize the SNR of the link end (denoted by ${SNR_{\textrm{W}}}$ and ${SNR_{\textrm{E}}}$) for the long-distance fiber-optic T/F transfer. The objective optimization functions, $\ln ({1/SNR_{\textrm{W}}+1/SNR_{\textrm{E}}})$ for microwave and time transfer, and $\ln ({1/SNR_{\textrm{W}}})$ for optical frequency transfer, are dedicatedly defined and is proven to be a convex function. Thereby, the optimization of the Bi-EDFAs’ gains can be fast solved by mature and efficient convex optimization method, and then the computation complexity and time can be dramatically decreased. To avoid the off-line laborious measurement of loss and Rayleigh scattering coefficients of each fiber, an on-line coefficient calculation method is proposed. It is realized by constructing the linear equations about the objective optimization function and the fiber parameters required by optimization, measuring the SNR at the link end ($SNR_{\textrm{W}}$ and $SNR_{\textrm{E}}$ for the microwave and time transfer, the local SNR ($SNR_{\textrm{W}}$) for the optical frequency transfer) under different EDFA gain configurations, and solving the linear equations. Combining with the fast optimization algorithm, a fast on-line link optimization can be enabled for adjustable transmission link. The proposed algorithm is further derived by using transmission matrixes for conveniently modeling the long-distance multiple EDFAs links. In detail, the paper is organized as follows: In section 2, the conversion from the maximization of SNR to a convex optimization problem is given, and the detailed procedure of the on-line optimization is presented. In section 3, the matrix-expressed optimization algorithm is derived. Section 4 presents the simulation results, validating the proposed optimization algorithm with the reported experimental results, and comparing the computation time with the scanning optimization method. Section 5 gives conclusion.
2. Fast and on-line optimization algorithm for the T/F transfer link
Figure 1 illustrates the schematic structure of the T/F transfer for time, microwave, and optical frequency transfer. The two-way T/F signals propagate along a common fiber link to guarantee the bidirectional reciprocity. In time and microwave transfer, as shown in Fig. 1(a), the outputted optical signals by the local and remote transmitters could be unrelated with each other, since different lasers at different wavelengths may be employed to remove the influence of single Rayleigh scattering noise. In the optical frequency transfer as shown in Fig. 1(b), the remotely transmitted power is related to the local transmitter, since it is the reflection of the received signal at the remote site. In the three main noises in the fiber link of three schemes, however, the propagation and the generation of DRS noise and ASE noise are the same, and the generation of the noises related to receivers are also almost same except the PM-IM noise [13]. Thus, the time and microwave transfer link will be first modeled in the following, and then the results will be generalized to describe the optical frequency transfer with proper modifications.
Fig. 1. Schematic diagram of the fiber-optic T/F transfer systems equipped with N Bi-EDFAs. (a) The transfer of microwave and time signals. TX: transmitting terminal, RX: receiving terminal, WDM: wavelength division multiplex, DRS: double Rayleigh scattering, S-RS: single Rayleigh scattering, ASE: amplified spontaneous emission. (b) The transfer of optical frequency signal. FRM: Faraday rotating mirror, AOM: acousto-optic modulator.
Fig. 2. Fast optimization algorithm for obtaining the optimal gains of Bi-EDFAs in the T/F transfer link.
Fig. 3. On-line optimization algorithm for obtaining the optimal gains of Bi-EDFAs in the T/F transfer link.
2.1 Modeling the SNR of T/F transfer
In the fiber-optic time and microwave transfer, assuming that the transmitted powers of local and remote sites are $P_{\textrm{TX1}}$ and $P_{\textrm{TX2}}$ respectively, the locally and remotely received signal powers are given by
respectively. In Eqs. (1) and (2), $L_m$ ($m=1, 2, 3, \ldots, N+1$) is the loss of the $m$th fiber span, and $G_n$ ($n=1, 2, 3, \ldots, N$) is the gain of the $n$th Bi-EDFA [13].In the transfer link, the generated DRS and ASE noises propagate together with T/F signals, whose powers at the local receiver are
Therefore, in time and microwave transfer, the photocurrent outputted by the local and remote receivers can be written as
Table 1. Variances of the link-induced noises in time and microwave transfer.
Based on the noise variances in Table 1, the local and remote SNRs can be calculated by
In optical frequency transfer, SNR mainly affects the robustness of the local phase feedback servo. To avoid the frequent cycle slip caused by low SNR [11], the local SNR is mainly considered in this technique. Based on the classic scheme of the optical frequency transfer shown in Fig. 1(b), the relationship between the remotely transmitted power $P_{\textrm{TX2}}$ and the local power $P_{\textrm{TX1}}$ can be defined as
where $R_{\textrm{r}}$ is the reflectivity of the remote Faraday rotating mirror. Then, the locally received DRS and ASE noises in optical frequency transfer can be obtained by modifying the expressions of Eqs. (3) and (4),Therefore, the photocurrent outputted by the local detector in this technique can be written as
Table 2. Variances of the link-induced noises in optical frequency transfer.
2.2 Optimizing the T/F transfer link by convex optimization algorithm
In order to maximize the SNR and fast obtain the optimal gains of each Bi-EDFA, the function, $\ln (1/SNR_{\textrm{W}}+1/SNR_{\textrm{E}})$, is dedicatedly defined as the objective optimization function for the time and microwave transfer. The minimization of the objective optimization function is equivalent to the maximization of SNR. More importantly, this objective optimization function will be proven to be a convex function in the following, whose globally optimal values can be fast calculated by mature and efficient convex optimization method. Besides, considering that the typical gain is higher than 15 dB in practical applications of long-distance T/F transfer [3,8,9], the ASE noise power can be approximated by $ASE_k\approx 2n_{\textrm{sp}}G_khvB_{\textrm{opt}}=ASE_0G_k$. Then, according to Eqs. (7) and (8), the function $f_{\textrm{opt1}}=1/SNR_{\textrm{W}}+1/SNR_{\textrm{E}}$ for the time and microwave transfer can be expressed as Eq. (14). Likewise, for optical frequency transfer, the function $f_{\textrm{opt2}}=1/SNR'_{\textrm{W}}$ and can be expressed by Eq. (15) based on Eq. (13).
According to Eqs. (14) and (15), the functions, $f_{\textrm{opt1}}$ and $f_{\textrm{opt2}}$ can be can be summarized by a common positive polynomial $f_{\textrm{opt}}$ with
where the constant coefficients $h_i$, $g_{k,i}$, and $l_i$ ($i=1,\ldots,M$, $k=1,\ldots, N$) can be calculated from the corresponding expression in Eqs. (14) and (15), and $M$ is the number of the monomial, respectively.By replacing the variables ($G_1, G_2, \ldots, G_N$) by $y_i=\ln G_i (i=1,2,3,\ldots,N)$, the defined objective optimization function, $\textrm{min}\ \ln (f_{\textrm{opt}})$, becomes a convex log-sum-exp function [17],
In order to avoid stimulated Brillouin scattering (SBS) of the transferred signals, the power threshold of SBS can be chosen as one of the constraint conditions [18]. Thus, we can establish the optimization problem for the T/F transfer link as a convex minimizing problem,
Theoretically, the above inequality-constrained minimization problem can be strictly solved by the interior-point method [17]. Using this method, the inequality-constrained minimization problem expressed in Eq. (18) is converted into an unconstrained minimization problem given by
To solve the optimization problem, the gradient descent algorithm can be employed. First, the derivative of $f''_{\textrm{opt}}\left ( \mathbf {y} \right )$ can be expressed as
Then, the iteration process of the gradient descent algorithm can be formulated as
Thus, the optimization process of Bi-EDFAs can be conducted by the Algorithm 1 shown in Fig. 2, where the circulation stops when the variation of the $y_k^i$ is below certain threshold ($\alpha$).
2.3 On-line optimization of the Bi-EDFAs for the T/F transfer link
In order to get an accurate optimal result using the above optimization algorithm, fiber parameters, such as the Rayleigh scattering coefficient and fiber loss (i.e. $R_k$ and $L_k$, $k=1,2,\ldots, N+1$) should be precisely known in advance. In practical applications, it is complicated to precisely measure these fiber parameters span by span. Moreover, these parameters may change with the replacement of fiber spans. Thus, an on-line optimization algorithm that obtains the required link parameters through measuring the concerned SNR of $f_{\textrm{opt}}$ at the corresponding link end is proposed here.
According to Eq. (16), the link-related parameters $h_i$ required by the optimization process can be obtained by solving the linear equations when giving the gains (i.e. $G_1, G_2, \ldots, G_N$) and the corresponding values of function ${{f}_{\text {opt}}}$. In detail, the proposed on-line optimization algorithm can be performed by the following Algorithm 2 shown in Fig. 3.
As demonstrated in [9], the Bi-EDFAs can be controlled remotely. Thus, all the procedures in Algorithm 2 can be operated on line, including the gain adjustment of each Bi-EDFA and the measurement of SNR. Besides, it is worth emphasizing that the optimization procedure of the proposed on-line scheme takes the power constrain of the SBS effect into consideration, so the calculated optimal gains will make sure that the signal power is lower than the power threshold of the SBS, which avoids the risk of unstable link operation.
3. Matrix-modeled optimization algorithm for the T/F transfer link
Using the algebraic method presented in the above section, the modeling and optimizing processes will become complicated for the long-distance multiple Bi-EDFAs links or the complex T/F transfer links, such as using the wavelength selective Bi-EDFAs (WS-Bi-EDFAs) [19]. The method of transmission matrix has been used to conveniently estimate the SNR at the link end of the T/F transfer link in Ref. [16] for the above complex cases. In the section, we further develop the method of transmission matrix to describe the proposed optimization algorithm to fast obtain the optimum gains of EDFAs for convenience.
To perform Algorithm 1, the derivative of objective optimization function with respect to each optimization variable should be first calculated. According to the Eqs. (14) and (15), it can be seen that each monomial only contains part of the optimization variables. Thus, when calculating the derivative of $f'_{\textrm{opt}}$ via the $y_k$, the schematic diagrams of the transmission matrixes should be modified to extract the terms of $f'_{\textrm{opt}}$ related to $y_k$, as shown in Fig. 4.
Fig. 4. Diagram of the transmission matrix method for extracting the double Rayleigh scattering in Eq. (14) and (15) related to the $k$th Bi-EDFA.
When calculating the derivative of DRS terms about $y_k$ ($k=1, 2, \ldots,N$) for the forward propagation signals, the odd scatterings are only generated at the right side of the $k$th Bi-EDFA, and the even scattering only takes place at the left side of the $k$th Bi-EDFA. Likewise, for the backward propagation signals, the odd and even scattering can only be generated at the left and right sides of the $k$th Bi-EDFA, respectively. In this case, the left-side and right-side transmission matrixes of the $k$th Bi-EDFA can be rewritten as
Likewise, the derivative of the function $f_{\textrm{opt}}$ related to ASE noise with respect to $G_k$ can be calculated by Eq. (26), where, the vector $\mathbf {ASE}_k$ presents the contributions of the ASE noises from $k$th Bi-EDFA to the local and remote sites and it is expressed as
For the noises related to the variances $\sigma _{\textrm{NL}}^2$ and $\sigma _{\textrm{NR}}^2$, the relevant derivative can be written as
Consequently, the derivative of the function $f_{\textrm{opt}}$ with respect to the optimization variable $G_k$ is
Then, the derivative of the function $f'_{\textrm{opt}}$ with respect to the $y_k$ is given by
Thus, the derivative of the function $f''_{\textrm{opt}}$ via the $y_k$ can be calculated by
Based on the Eq. (32), the Algorithm 1 can be performed to obtain the optimal gains of each Bi-EDFA.
4. Model simulation for the T/F transfer link
In this section, the optimization algorithm is verified by comparing the calculated parameters with the published experimental results in [16]. The relevant parameters, including the RS coefficient, electrical bandwidth, and optical bandwidth, are the same with those in [16]. For the scenario of four Bi-EDFAs, the corresponding fiber lengths are set as 100 km, 20 km, 50 km, 50 km, and 100 km, respectively. The gain updating of each Bi-EDFA in the optimization process is shown in Fig. 5(a), whose total computation time is only about 0.024 s. The calculated optimal gains of the four Bi-EDFAs are 13.2 dB, 8.2 dB, 10.3 dB, and 16.5 dB, respectively, which present a good consistent with the experiment results in [16]. Changing the gain of each Bi-EDFA around its calculated value while fixing the others at the calculated optimal values, the calculated variations of the SNR are shown in Fig. 5(b)–(e). It can be concluded that the calculated results converge to the globally optimal solutions rapidly. Moreover, it should be noted that the approximation of the ASE noise ($ASE_k=2n_{\textrm{sp}}(G_k-1)hvB_{\textrm{opt}}\approx 2n_{\textrm{sp}}G_khvB_{\textrm{opt}}$) in Eqs. (14) and (15) presents ignorable influence on the calculated optimal gains. It is because that the ASE noises outputted by the first and the fourth Bi-EDFAs are dominant, and their gains are relatively high.
Fig. 5. Simulation results of investigating the proposed optimization algorithm for the T/F transfer link with four Bi-EDFAs. (a) The variation of the gains via the iteration calculated by Eq. (21). The initial gains of the four Bi-EDFAs are set at 17 dB. (b-e) Variations of the local SNR (blue dashed curves), the remote SNR (blue dot-dashed curves), and the objective optimization function defined in Eq. (14) (red solid curves).
The computation time of the scanning optimization method and the proposed optimization algorithm performed on the same computer are shown in Fig. 6. The gain scanning range of the former method is from 3 dB to 27 dB with the tuning step of 0.25 dB. From the results, the calculation time of the scanning method increases exponentially with the increasing of the number of amplifiers. A slope of $96^N$ corresponding to 96 scanning steps for each Bi-EDFA is obvious in the simulation result. Although a large tuning step can reduce the computation time of the scanning optimization method, the precision of the searched optimal gains will be influenced. By contrast, the computation time of the proposed optimization algorithm presents little increase as the number of amplifiers increases, which is more than 10000 times lower than that of the scanning optimization method and is still below 1 s even for 10 Bi-EDFAs.
5. Conclusion
In conclusion, a fast on-line optimization scheme for the long-distance bidirectional T/F transfer link is proposed to maximize the SNRs. By choosing the function $\ln ({1/SNR_{\textrm{W}}+1/SNR_{\textrm{E}}})$ or $\ln ({1/SNR_{\textrm{W}}})$ as the objective optimization function, the computation complexity and time can be dramatically decreased, which can fast calculate out the optimal gains of Bi-EDFAs. Then, an on-line coefficient calculation method is proposed to avoid the off-line laborious measurement of loss and Rayleigh scattering coefficients of each fiber. It is realized by constructing the linear equations about the objective optimization function and the fiber parameters required by optimization, measuring the concerned SNR at the corresponding link end under different EDFA gain configurations, and solving the linear equations, which facilitates the practical applications. Combining with the fast optimization algorithm, a fast on-line link optimization can be enabled for adjustable transmission link. The proposed algorithm is further derived by using transmission matrixes for conveniently modeling the long-distance multiple EDFAs links. The simulation result shows that the calculated optimal parameters are consistent with the reported experimental results, and the computational time shows a dramatic decrease compared with the conventional scanning optimization method. The proposed fast optimization algorithm can also allow for the optimization of more parameters. Our future work will investigate the joint location and gain optimization, which has the potential of further improving the SNR.
Funding
National Natural Science Foundation of China (61627817, 61905143).
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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