⇦Chinese Physics Letters, 2020, Vol. 37, No. 9, Article code 094201 Novel Polarization Control Approach to Long-Term Fiber-Optic Frequency Transfer Dong-Jie Wang (王东杰)1,2,3, Xiang Zhang (张翔)1,2,3, Jie Liu (刘杰)1,3, Dong-Dong Jiao (焦东东)1,2,3, Xue Deng (邓雪)1,2,3, Jing Gao (高静)1,2,3, Qi Zang (臧琦)1,2,3, Dan Wang (王丹)1,3, Tao Liu (刘涛)1,2,3*, Rui-Fang Dong (董瑞芳)1,2,3, and Shou-Gang Zhang (张首刚)1,2,3 Affiliations 1National Time Service Center, Chinese Academy of Sciences, Xi'an 710600, China 2University of Chinese Academy of Sciences, Beijing 100049, China 3Key Laboratory of Time and Frequency Standards, Chinese Academy of Sciences, Xi'an 710600, China Received 12 June 2020; accepted 24 July 2020; published online 1 September 2020 Supported by the National Key Research and Development Program of China (Grant No. 2016YFF0200200), the National Natural Science Foundation of China (Grant Nos. 91636101, 91836301 and 11803041), the West Light Foundation of the Chinese Academy of Sciences (Grant No. XAB2016B47), and the Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No. XDB21000000).
*Corresponding author. Email: taoliu@ntsc.ac.cn Citation Text: Wang D J, Zhang X, Liu J, Jiao D D and Deng X et al. 2020 Chin. Phys. Lett. 37 094201    Abstract We demonstrate a novel polarization control system based on a gradient descent algorithm, applied to a 450-km optical frequency transfer link. The power of the out-loop beat note is retrieved by controlling the polarization state of the transferred signal, with a recovery time of 24 ms, thereby ensuring the long-term evaluation of the fiber link. As a result, data utilization is enhanced from 70% to 99% over a continuous measurement period of $\sim$12 h. A fractional transfer instability of $7.2 \times 10^{-20}$ is achieved at an integration time of 10000 s. This work lays the foundation for the comparison of a remote optical clock system via a long-haul optical fiber link. DOI:10.1088/0256-307X/37/9/094201 PACS:42.62.Eh, 42.79.Sz, 06.30.Ft © 2020 Chinese Physics Society Article Text III VI IV Within the last decade, optical clocks have surpassed the best fountain clocks in terms of accuracy and stability by approximately two orders of magnitude.[1–4] An optical clock network is formed using optical fiber links without degradation in terms of stability and accuracy, benefiting various applications such as the dissemination of time,[5,6] geodesy,[7] astronomy,[8] and fundamental and applied research.[9,10] However, due to the non-polarization-maintaining characteristics of the conventional single-mode fiber, the state of polarization (SOP) of the transmitted laser may drift dramatically, and even degrade,[11,12] due to, e.g., environmental temperature fluctuations and stress along the fiber link.[13] In a typical optical frequency transfer system, two Faraday mirrors are respectively applied at a local site and a remote site to ensure that the reflected laser and the reference laser maintain a fixed polarization state.[14,15] Nonetheless, the transmitted laser at the remote site still experiences a deterioration in the SOP, which can interrupt the out-loop frequency measurement. Several active polarization methods have been proposed to reduce this deterioration, but detailed descriptions have so far been lacking.[14,16–18] In this Letter, we report a new approach to cost-efficient polarization control for the long-term frequency measurement of long-haul fiber optical frequency transfer systems. Based on the Gradient Descent (GD) algorithm, the power of the out-loop beat-note signal is utilized as the feedback for controlling the polarization state of the transferred laser. When the power falls below the threshold, the control program is launched to electrically drive the polarization controller to adjust the polarization state of the transmitted laser. By means of this control, the beat-note power thus achieves its maximum value, and the long-term stability of the optical frequency transfer system can be implemented. This polarization control approach has been applied to a 450-km optical frequency transfer experiment. The maximization of the out-loop beat-note power was retrieved by changing the SOP of the transmitted laser. Consequently, the stability of the frequency transfer via a 450-km spooled fiber link is improved from $1.4 \times 10^{-19}$ at 2000 s to $7.2 \times 10^{-20}$ at 10000 s with the polarization controller (PC) implemented at the remote site. A schematic of the experimental setup is shown in Fig. 1; the laser source is a home-made 1550 nm ultra-stable laser with a linewidth of 3 Hz.[19] The output of the laser is divided into two parts by a $90\!:\!10$ single mode coupler (SMC1). Here, 90% of the output is sent to the receiver module via the 450-km spooled fiber link, while the remaining 10% serves as the reference signal. The phase noise of the 450-km spooled fiber is actively compensated for by an acousto-optic modulator (AOM) in the sender module.[20] The heterodyne beat note between the transferred signal and the reference signal is detected using a fast photo detector (PD), and is then sent to a dead-time free frequency counter (K&K) to calculate the level of transfer instability. In order to achieve long-term transfer stability, the polarization control system is established before the receiver setup. As shown in the dashed frame in Fig. 1, the RF signal output from the PD is power divided into two parts: one part handles the subsequent out-loop frequency measurement, while the other part is converted to a DC voltage signal by a power detector (Mini-Circuits ZX47-60-S$+$). The voltage signal is firstly digitized into 12-bit ADC before being delivered to the STM32 micro-controller. Based on the algorithm demonstrated in our study, the STM32 produces a feedback signal for polarization control, which is subsequently transformed to an analog signal within a range of 0–5 V by four 16-bit DACs and applied to the polarization controller (PC, PCD-M02/MPD-001).
cpl-37-9-094201-fig1.png
Fig. 1. Optical frequency transfer scheme with polarization control system. Sender: the sender module for optical frequency transfer system. Receiver: the receiver module for optical frequency transfer system. PD: photo detector. SMC: single mode coupler. PC: polarization controller (PCD-M02/MPD-001). ADC: analogue-to-digital conversion. DAC: digital-to-analogue conversion. STM32: micro controllers with version of STM32F103RCT6.
Firstly, we address the polarization control program. For a linear retarder with an azimuth angle of $\theta$ and a retardation of $\delta$, its effect on the SOP can be conveniently expressed by Jones Matrix, as given in the following:[21] $$\begin{align} &J=R(-\theta)W(\delta)R(\theta)\\ ={}&\begin{pmatrix} {\cos \theta } & {-\sin \theta } \\ {\sin \theta } & {\cos \theta } \end{pmatrix} \begin{pmatrix} {e^{i\delta /2}} & 0 \\ 0 & {e^{-i\delta /2}} \end{pmatrix} \begin{pmatrix} {\cos \theta } & {\sin \theta } \\ {-\sin \theta } & {\cos \theta } \end{pmatrix} ,~~ \tag {1} \end{align} $$ where $\theta$ is associated with the angle between the fast axis of the retarder and the horizontal $x$-axis of the Poincaré sphere by an index of 2.[22,23] When $\theta$ is 0$^\circ$, the above Jones Matrix is given by $$ J_{0^{\circ}} = \begin{pmatrix} {\exp (i\delta /2)} & 0 \\ 0 & {\exp (-i\delta /2)}\end{pmatrix}.~~ \tag {2} $$ When $\theta$ is 45$^\circ$, Eq. (1) reduces to $$ J_{45^{\circ}} = \begin{pmatrix} {\cos (\delta /2)} & {i\sin (\delta /2)} \\ {i\sin (\delta /2)} & {\cos (\delta /2)} \end{pmatrix}.~~ \tag {3} $$ With regard to the PC, it is composed of four electrically driven wave plates with different extrusion directions, and its Jones Matrix can be given by $$ J=J_{3} J_{2} J_{1} J_{0},~~ \tag {4} $$ where $J_{0}$ and $J_{2}$ represent wave plates with an azimuth angle of 0$^\circ$, and $J_{1}$ and $J_{3}$ denote wave plates with an azimuth angle of 45$^\circ$. When the laser, with a random SOP (noted as ${a \choose b}$) passes through the PC, the SOP of the output laser (noted as ${c \choose d}$) will be controlled by $$ {c \choose d}=J_{3} J_{2} J_{1} J_{0} {a\choose b}.~~ \tag {5} $$ Numerous algorithms have been developed for the purpose of polarization control, such as simulated annealing (SA),[24] particle swarm optimization (PSO),[25] or the genetic algorithm (GA).[26] The optimization algorithm used in this study is the GD algorithm. The GD algorithm can be briefly described as follows: the control signal for the PC can be represented by the cost function $F=F({v}_{0},{v}_{1},{v}_{2},{v}_{3})$, which is a function of four voltages. By applying the four control voltages to the four electrically driven wave plates, respectively, the SOP of the transmitted laser will be changed on demand. The voltage range for each channel is within 0–5 V and the corresponding polarization variation is 0–4$\pi$. A typical control process based on the GD algorithm is shown in Fig. 2(a). The program mainly comprises an algorithm module and a monitoring module. The monitoring module runs continuously until the beat-note power is below threshold, at which point the algorithm module will be triggered. The algorithm module will operate according to the following iteration cycle: (I) Calculate the value of $\xi =\frac{\partial }{\partial {v}_{j}}F({v}_{0},{v}_{1},{v}_{2},{v}_{3})~(j=0,1,2,3$), which is the gradient of the cost function in the current position for $v_{j}$. (II) Update the control voltage signal with ${v}_{j} ={v}_{j} -\eta \xi$, in which $\eta$ denotes the tuning step and is decided by the iterations. (III) Determine whether the number of iterations reaches the preset value, stop running the algorithm when it does so, otherwise return to step I. The iteration strategy of the GD algorithm is to find the voltage quantities with the smallest gradient. Therefore, the minus sign is used in step II. When the algorithm runs for the first time, it will execute repeatedly, and each subsequent search will continue to lower the gradient, based on the optimal value previously found, to ensure that the optimal solution can be found. As the variation of the control voltages may exceed the allowable range of the hardware during the operation time, a voltage reset program is designed to make the change of the SOP as small as possible during the voltage reset process.[27] The beat-note signal power under the polarization control is also shown in Fig. 2(b). Where the polarization state suffers from a disturbance after point A, this may lead to a signal power degradation. If the power drops to the predefined threshold, e.g., 2 dB from the maximum power in our case (point B), the polarization control based on the GD algorithm will then be launched to tune the four channels of the PC. During the optimization process from B to C, the signal power is maximized by iteratively tuning the polarization rotators in the direction of steepest descent, as defined by the negative of the signal power gradient. As shown in Fig. 2(b), signal power can be recovered after a few cycles, with a typical operation time of 24 ms. A large tuning step is generally helpful to find the global optimum control voltage, and may also induce a relatively larger power change during the optimization process. Furthermore, we note that the power drop may cause a certain offset in the frequency measurement. Therefore, the tuning step should be optimized. In the experiment, the tuning step $\eta$ was adjusted from 0.2$\pi$ (0.25 V) to 0.02$\pi$ (0.025 V) which adheres to a linear decreasing function in the iteration procedure, which considers the capability of the global optimum at the beginning, as well as efficiently avoiding violent fluctuations due to the large step at the end. Therefore, the calculation of the actual tuning step $\Delta v_{j}$ in step two of the GD algorithm is related to the variation of $\eta$ and $\xi$.
cpl-37-9-094201-fig2.png
Fig. 2. (a) Flowchart detailing the polarization control procedure of the program. (b) Variation of the beat-note power under polarization control and recovery time for a given power (inset). (c) The corresponding evolution of the SOP on Poincaré sphere.
Continuous frequency measurements over a period of 11.5 hours, with and without the polarization control system, in a 450-km spooled fiber link, are shown in Fig. 3. The beat-note signal between the transmitted optical field and local laser was measured with a dead-time free frequency counter (K&K). As shown in the upper curve of Fig. 3, polarization variation induces drastic interruptions with durations ranging from tens of minutes to several hours, depending on the magnitude of the polarization variation. For comparison, a typical frequency measurement with polarization control is shown as the lower curve of Fig. 3. Here, it is evident that the long-term disturbance due to polarization variation vanishes, leaving only transient disturbance, possibly induced by mechanical vibration. We can also clearly observe in this measurement that the data utilization ratio has increased from 70% to 99%. The long-term measurement of optical frequency transfer via a long-haul noisy fiber link is therefore anticipated to be feasible.
cpl-37-9-094201-fig3.png
Fig. 3. Frequency deviation measurement of optical frequency transfer via a 450-km spooled fiber link without PC (upper curve, purple line) and with PC (lower curve, red line).
cpl-37-9-094201-fig4.png
Fig. 4. Frequency instability of the optical frequency transfer system via a 450-km spooled fiber link. Triangles (pink line): ADEV of the unlocked system; diamonds (blue line): ADEV of portion data for the locked system without PC; circles (red line): ADEV of locked system with PC; squares (black line): system floor; dots: 1/$\tau$ slope guide line.
Figure 4 shows the measured transferred frequency instability over a 450-km spooled fiber. Without PC, the long-term frequency measurement was disrupted by serial blank-outs. The Allan deviation (ADEV) of the partial data during a silent period without PC is indicated by a diamond curve, and only achieves $1.4 \times 10^{-19}$ at 2000 s. Once the PC is applied, the long-term frequency data can be obtained. The frequency transfer instability in the 450-km spooled fiber link is $3 \times 10^{-16}$ at 1 s, and the long-term instability scales down to $7.2 \times 10^{-20}$ over an integration period of 10000 s, with a slope of nearly 1/$\tau$. With an increase in measurement duration, frequency instability is close to the system floor. We note that, within the same time scale, the long-term frequency instability with PC is slightly worse than that without PC. We attribute this result to the phase slip induced by the operation of polarization rotation. In conclusion, we have developed a polarization control system to control the out-loop beat-note power, which is essential for long-term frequency measurement. Such a system has been efficiently applied to optical frequency transfer via a 450-km spooled fiber link. The results show that the beat-note power can be recovered in 24 ms, and the utilization of the measurement data has increased from 70% to 99%. The transfer instability ultimately achieved is $7.2 \times 10^{-20}$ at an average time of 10000 s. This polarization control system may also be expanded to include the optical clock comparison system and the cascaded optical frequency transfer system. References High-Accuracy Optical Clock Based on the Octupole Transition in Yb + 171 High-Accuracy Measurement of Atomic Polarizability in an Optical Lattice ClockAn optical lattice clock with accuracy and stability at the 10−18 levelA Fermi-degenerate three-dimensional optical lattice clockOptical Clocks and RelativityOptical clock networksGeodesy and metrology with a transportable optical clockNew bounds on dark matter coupling from a global network of optical atomic clocksEinstein Gravity Explorer–a medium-class fundamental physics missionImproved Measurement of the Hydrogen 1 S – 2 S Transition FrequencyTransmission of linearly polarized light through a single-mode fiber with random fluctuations of birefringenceAdaptive Polarization Control of Fiber Amplifier Based on SPGD AlgorithmUltra-stable long distance optical frequency distribution using the Internet fiber networkSimultaneous remote transfer of accurate timing and optical frequency over a public fiber networkOptical-Frequency Transfer over a Single-Span 1840 km Fiber LinkOptical frequency transfer via a 660 km underground fiber link using a remote Brillouin amplifierCascaded multiplexed optical link on a telecommunication network for frequency disseminationDevelopment and application of communication band narrow linewidth lasersCoherent Transfer of Optical Frequency over 112 km with Instability at the 10 −20 LevelPolarization State Evolution in Fibre Polarization Transformer Influenced by Phase DifferencePoincaré sphere analysis of a ferroelectric liquid crystal optical modulator: application to optimize the contrast ratioPoincaré sphere representation of birefringent networksParticle swarm optimization used as a control algorithm for adaptive PMD compensationRealization of microcontroller-based polarization control system with genetic algorithmEndless polarization control systems for coherent optics
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