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OPLL具有与传统电学锁相环(PLL)相类似的结构,如图1中(a)和(b)所示,传统电学锁相环主要包含鉴相器(Phase detector)、环路滤波器(Loop filter)与压控振荡器(VCO)三部分。而在光学锁相环中,鉴相器之前还需增加光电探测器,将两束激光的拍频信号转换为电信号。此外,传统电学锁相环中的压控振荡器在光学锁相环中为待锁定的可调谐激光器,经锁相环路实现其相对于参考激光的偏频锁定。
图 1 光学锁相环与电学锁相环结构对比。(a) 电学锁相环结构示意图;(b) 光学锁相环结构示意图
Figure 1. Structural comparison of OPLL and PLL. (a) Diagram of PLL structure; (b) Diagram of OPLL structure
光学锁相环的基本工作原理为:待锁定激光器的激光频率与参考激光器的激光频率之间存在一定的频差时,两束激光相干混合后可由光电探测器探测到其拍频信号,然后由鉴相器对该拍频信号与已设定为预期偏频频率值的参考频率信号进行鉴频鉴相处理,得到与相位差成比例的鉴相误差电压信号,经环路滤波器处理后反馈至待锁定激光器进行调谐,从而使待锁定激光器的频率始终跟随参考激光器频率变化,实现两激光器之间的偏频锁定。
假设参考激光和待锁定激光是两个沿
$ {\textit{z}} $ 轴传播的的单色波,频率分别为$ {\omega }_{1} $ 和$ {\omega }_{2} $ ,则它们的波函数可写为:$$ {E_1} = a\;{\rm{cos}}\left( {{k_1}{\textit{z}} - {\omega _1}t} \right)$$ (1) $$ {E_2} = a\;{\rm{cos}}\left( {{k_2}{\textit{z}} - {\omega _2}t} \right) $$ (2) 两激光束合成后的光波强度为:
$$ \begin{split} I = {A^2} =& 4{a^2}{\rm{co}}{{\rm{s}}^2}\left( {{k_m}{\textit{z}} - {\omega _m}t} \right)=\\ & 2{a^2}\left[ {1 + {\rm{cos}}\;2\left( {{k_m}{\textit{z}} - {\omega _m}t} \right)} \right] \end{split} $$ (3) 式中:
${\omega _m} = \dfrac{1}{2}\left( {{\omega _1} - {\omega _2}} \right)$ ;${k_m} = \dfrac{1}{2}\left( {{k_1} - {k_2}} \right)$ 。由此可见,合成波的光强随时间和传播距离在0~
$ 4{a}^{2} $ 之间变化,这种光强交替变化的现象就是“拍频”。由上式可知,拍频频率为$ {2\omega }_{m} $ ,由于$ {\omega }_{m}= $ $ \dfrac{1}{2}\left({\omega }_{1}-{\omega }_{2}\right) $ ,因此拍频频率即为参考激光频率与待锁定激光频率之差。为了便于分析,我们假设参考激光的频率是不变的,且不考虑其他干扰因素,此时拍频信号的频率和相位仅受待锁定激光器频率和相位变化的影响。因此,完成拍频信号与本振频率参考信号之间频率和相位的锁定即可实现待锁定激光相对于参考激光的频率和相位锁定。光学锁相环的反馈部分则与传统电学锁相环类似,因此可以直接采用电学锁相环的理论对光学锁相环的电路部分进行分析。光学锁相环复频域的基本框图如图2所示。
在拉普拉斯变换域中,设输入到鉴相器的拍频信号为
${\theta }_{\rm beat}\left(s\right)$ ,频率参考源信号为${\theta }_{\rm ref}\left(s\right)$ ,则鉴相器输出的鉴相误差信号为:$$ {V}_{e}\left(s\right)={K}_{\rm PD}({\theta }_{\rm ref}-{\theta }_{\rm beat}) $$ (4) 鉴相误差信号经传输特性函数为
$ F\left(s\right) $ 的环路滤波电路以及环路传播延迟${\rm e}^{-s\tau }$ 后,反馈给待锁激光器,从而实现拍频信号的频率和相位锁定至频率参考源。整个系统的开环传递函数为:
$$ G\left(s\right)=\frac{{K}_{\rm PD}{K}_{\rm SL}F\left(s\right){\rm e}^{-s\tau }}{s} $$ (5) 则对应的闭环传递函数为:
$$ H\left(s\right)=\frac{G\left(s\right)}{1+G\left(s\right)}=\frac{{K}_{\rm PD}{K}_{\rm SL}F\left(s\right){\rm e}^{-s\tau }}{s+{K}_{\rm PD}{K}_{\rm SL}F\left(s\right){\rm e}^{-s\tau }} $$ (6) 以及误差传递函数为:
$$ \begin{array}{l} E\left( s \right) = \dfrac{1}{{1 + G\left( s \right)}} = 1 - H\left( s \right) \dfrac{s}{{s + {K_{\rm PD}}{K_{\rm SL}}F\left( s \right){{\rm e}^{ - s\tau }}}} \end{array} $$ (7) -
光学锁相环可以实现待锁定激光器相对于高精度频率参考源的锁定,在对激光频率稳定度要求很高的应用和研究中有重要作用。1964年,Enloe等人首次利用激光锁相环装置成功实现了两路单频He-Ne激光器的锁定[19],此后随着激光技术的发展,基于光学锁相环的激光偏频锁定方法也在不断进步。下面按照光学锁相环实现方法的不同对其采用的关键技术进行介绍。
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模拟鉴相是早期光学锁相环广泛采用的一种鉴相技术方式[20-22],其基本结构如图3所示。
该鉴相技术主要是通过采用以双平衡混频器(DBM)为代表的模拟鉴相器进行鉴相,并在偏频锁定频率较大时,在鉴相器之前增加另一DBM的方式对高频信号进行频率下转换,将信号的频率降到模拟鉴相器的可处理范围内。值得注意的是,DBM要实现鉴相功能需要其中频为直流耦合,且只有当混频器两输入端口的信号频率相同时才可进行鉴相。鉴相器的中频端可输出与两输入信号的相位差成比例的鉴相误差电压,经环路滤波单元进行滤波和放大等处理后反馈至待锁定激光器进行调谐。1989年,Williams等人采用八次谐波混频器对频率进行下转换,采用DBM作为鉴相器,实现了6~34 GHz大偏频范围的OPLL,将两Nd:YAG(掺钕钇铝石榴石)激光器的拍频线宽保持在mHz量级[20]。1992年,Gliese等人介绍了一种基于模拟鉴相技术的OPLL,偏频频率范围为3~18 GHz,残余相位误差方差为0.04 rad2,并将其用于131 Mbit/s的QPSK/DQBSK(正交相移键控/差分四相相移键控)微波通信系统[22]。由于模拟鉴相技术采用无源器件,所以受自身噪声影响较小,具有很高的鉴相灵敏度,但是抗干扰能力相对较差。此外该方法所用器件体积较大不便于移动,且需要两个高稳定度的本振频率参考源,成本较高。
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随着数字鉴相器和锁相环芯片等集成电路的出现,光学锁相环越来越多地采用基于鉴频鉴相器(PFD)的数字鉴相技术[23-26]。与模拟鉴相技术不同,在数字鉴相中,PFD要求输入的信号通常为方波形式,利用高低电平的触发和跳变进行工作。采用数字鉴相方式的OPLL基本结构如图4所示。
其主要包括分频器、鉴频鉴相器和环路滤波器等部分。其中分频器的作用是将高频拍频信号进行分频,取代了模拟鉴相技术中利用混频器进行频率下转换的方法,大大降低了成本。2008年,Marino等人在原子相干实验中采用了基于数字鉴相方法的OPLL,通过差分电路对PFD输出的反映两信号相位超前滞后情况的UP/DOWN脉冲序列进行运算后得到误差信号[24]。2009年,Höckel等人在EIT实验中同样采用了基于数字鉴相方法的OPLL,所选用的数字鉴相器在偏频10 kHz时具有−153 dBc/Hz的极低相位噪声,实现了残余相位噪声小于0.02 rad2的OPLL系统[25]。2014年,Sternkopf等人采用集成了PFD的商用OPLL芯片搭建了光学锁相环系统,将外腔半导体激光器偏频锁定至633 nm He-Ne激光器,实现了3.2×10−9的相对频率稳定度[26]。采用数字鉴相技术的OPLL相较于模拟鉴相OPLL具有体积小、成本低、抗干扰能力强等优点,是目前光学锁相环中应用最广泛的方法。
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随着FPGA技术的不断发展,其凭借处理速度快、延时低以及功能丰富等优点,被逐渐应用到光学锁相环中。Numata等人率先在对分布式反馈(DFB)激光器与数字-超模分布布拉格反射式(DS-DBR)激光器进行偏频锁定研究时引入FPGA对光学锁相环各部分进行控制[27]。如图5所示,该研究中FPGA并不直接参与锁相环路的鉴相及PID控制等工作,而是主要用来生成前馈信号以及动态控制OPLL的环路参数,如PFD的锁定逻辑、电荷泵(CP)电流的增益和符号、直接数字频率合成器(DDS)的频率以及分频器的分频因数等。这种光学锁相环可在40 GHz的范围内实现精确快速的调谐。
此后Xu等人设计了一种基于半导体激光器的OPLL用于原子干涉测量[28-29],如图6所示。其中,虚线框a中PFD、Loop Filter和PID三部分均通过FPGA实现。虚线框b部分可视为等效的数控振荡器(DCO),它与FPGA的数字控制电路相结合构成全数字锁相环。该锁相环可以实现小于1 Hz的拍频线宽,残余相位误差方差为0.14 rad2。
Yao等人在基于FPGA的激光偏频锁定方案中采用了上升沿和下降沿同时计数的判别方法,以及高概率均值滤波算法和变步长与分段逼近算法,最终实现拍频频率波动范围在±10 Hz以内[30]。
浙江理工大学的谢建东等人基于锁相放大原理,实现了利用FPGA进行鉴频鉴相及相位锁定的OPLL,使外腔半导体激光器(ECDL)在10 nm波长范围均可偏频锁定至光频梳,锁定后的拍频信号频率波动被控制在±3 Hz以内,激光器频率波动在±6 kHz左右[31]。
目前,基于FPGA的OPLL在激光偏频锁定中已有越来越多的研究和应用,FPGA在系统中主要实现各器件的低延时控制,或对拍频进行计数等工作,而超高频信号的鉴频鉴相工作仍需由专门的鉴频鉴相器件来完成。随着OPLL偏频锁定系统功能的不断丰富,对逻辑控制的要求不断提高,将FPGA应用于OPLL系统是未来的发展趋势。
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光学锁相环的集成化是通过在磷化铟(InP)基底上刻蚀光通道,加工集成激光器和光电探测器来实现光子集成电路(PIC)[17, 32-34],如图7所示。在该PIC中,待锁激光二极管、光电探测器和两个Y形结构的光波导集成在长度为3.1 mm的InP基底上,参考激光和待锁定激光分别通过特制光纤引入PIC中进行拍频。此外,也可将参考激光与待锁激光同时集成在InP上,实现全集成化的OPLL方案[35-37]。
另一种完整的集成OPLL结构如图8所示,参考激光与待锁激光在PIC中完成拍频,集成的光电探测器将拍频信号输出至电子集成电路(EIC)中进行处理,最后反馈至集成的待锁激光器对其进行调谐。这样,整个OPLL系统的体积得到了很大程度的减小,完成了对OPLL系统整体的集成化。
Steed等人报道的单片集成OPLL可以实现0.6~6.1 GHz的偏频可调谐范围,集成的半导体激光器拥有1.1 MHz的线宽,在大于20 kHz偏频锁定时的相位噪声小于−90 dBc/Hz,10 GHz带宽下的残余相位误差方差为0.04 rad2[38]。Lu等人采用取样光栅分布布拉格反射式(SG-DBR)激光器实现了宽带宽调谐范围的集成OPLL,其线宽为10 MHz,可调谐范围高达16.5 GHz,具有0.12 ns的环路传播延时,环路带宽1.1 GHz。利用该集成OPLL,可以将激光器的10 MHz线宽压缩至100 kHz[40]。Balakier等人介绍了一种采用铸造方法和现成的电子元器件制作的光子集成OPLL,在4~12 GHz之间的偏移频率内实现了稳定的锁定,在偏频10 kHz时,外差相位噪声低于−100 dBc/Hz[41],达到了目前光子集成OPLL领域中非常高的水平。
由于集成在芯片上的激光器以及光电探测器等在性能上较传统器件还有一定差距,目前仍处于实验阶段,并未得到广泛的实际应用,但光子集成OPLL是光子芯片前沿研究的热点方向之一,对于未来实现大规模光子集成具有重要意义。
表1总结了上述几种不同类型的OPLL的结构、性能参数以及实验结果。可以看到,光学锁相环的光源可以为多种不同类型的激光器,但基本要求是作为激光参考源的主激光器应具有较高的频率稳定度,而从激光器应具有可调谐功能。此外,随着时间的发展和技术的进步,在光学锁相环的鉴相方式和光路结构上,也表现出了明显的数字化、集成化趋势。
表 1 不同OPLL偏频锁定系统的结构和关键参数
Table 1. Structure and key parameters of different OPLL offset locking systems
Time Laser Offset range Type of phase
discriminatorLoop
bandwidthLinewidth Phase noise
levelPhase error
varianceOptical structure Master Slave 1989[20] Nd:YAG Nd:YAG 6-34 GHz DBM - <1 mHz - - Space light 1992[42] DFB DFB 3-18 GHz DBM 180 MHz - <102 dBc/Hz 0.04 rad2
(1 GHz bandwidth)Space light 1994[43] ECDL ECDL <10 GHz DBM 3.7 MHz 50 kHz - <0.004 rad2 Space light 1999[44] DFB DFB 7-14 GHz DBM 70 MHz - −95 dBc/Hz
(50 MHz offset)0.05 rad2
(1 GHz bandwidth)Space light 2008[24] ECDL ECDL 250 kHz-20 GHz DBM+PFD - <10 Hz - <0.04 rad2 Space light 2008[25] ECDL ECDL 0.01-1.3 GHz PFD - 500 kHz −153 dBc/Hz
(10 kHz offset)<0.02 rad2 Space light 2012[39] - SG-DBR −9-7.5 GHz PFD 400 MHz - - - Integrated optical circuit 2012[27] DFB DS-DBR <40 GHz PFD 100 kHz
(open-loop unity-gain bandwidth)- - - Space light 2014[26] He-Ne
LaserECDL 80 MHz-1.4 GHz PFD - 3.2×10−9
(stability)- - Space light 2017[17] OFC SG-DBR <26 GHz XOR phase discriminator - - −80 dBc/Hz
(200 Hz offset)0.08 rad2
(1 kHz-1 GHz
bandwidth)Integrated optical circuit 2018[41] ECDL DBR 4-12 GHz analog phase
discriminator100 MHz - −100 dBc/Hz
>(10 kHz offset)0.012 rad2 Integrated
optical circuit
Research progress of optical phase-locked loop
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摘要: 光学锁相环(OPLL)根据其锁定的两束激光间是否存在频差可分为零差光学锁相环和外差光学锁相环。主要介绍了外差光学锁相环的研究进展,它是一种通过鉴频鉴相方式使激光间的频率差保持相对稳定的偏频锁定方法。相较于其他激光偏频锁定方法,光学锁相环具有结构简单、伺服频率带宽大、频率偏置范围宽、锁定准确度高等优势,在原子相干、冷原子系统、相干功率合成以及外差干涉测量等领域都得到了越来越广泛的应用。首先介绍了激光偏频锁定的主要方法及光学锁相环的特点;其次介绍了光学锁相环的基本模型,分析了光学锁相环的误差反馈过程,并按照光学锁相环实现方法的不同详细介绍了其采用的关键技术和研究进展,对近年来光学锁相环在不同领域的应用进展做了简要介绍;最后对该方法的发展路线进行了总结和展望。Abstract: According to whether there is frequency difference between two laser beams locked by an optical phase-locked loop (OPLL), it can be divided into homodyne optical phase-locked loop and heterodyne optical phase-locked loop. The progress of heterodyne optical phase-locked loop was mainly introduced, which could keep the frequency difference between lasers relatively stable through frequency discrimination and phase discrimination. Compared with other laser offset locking methods, optical phase-locked loop has the advantages of simple structure, large servo bandwidth, wide offset range and high locking accuracy. It plays a very important role in atomic coherence, cold atom system, coherent beam combining, heterodyne interferometry and other fields, and has received more and more extensive attention. The main method of laser offset locking and the characteristics of optical phase-locked loop were introduced firstly, then the basic model of optical phase-locked loop was introduced, and the basic error model of optical phase-locked loop was analyzed. According to the different realization methods of OPLL, the key technologies and research progress of OPLL were introduced in detail. Then the application progress of OPLL in different fields in recent years was introduced. Finally, the development of this method was summarized and prospected.
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表 1 不同OPLL偏频锁定系统的结构和关键参数
Table 1. Structure and key parameters of different OPLL offset locking systems
Time Laser Offset range Type of phase
discriminatorLoop
bandwidthLinewidth Phase noise
levelPhase error
varianceOptical structure Master Slave 1989[20] Nd:YAG Nd:YAG 6-34 GHz DBM - <1 mHz - - Space light 1992[42] DFB DFB 3-18 GHz DBM 180 MHz - <102 dBc/Hz 0.04 rad2
(1 GHz bandwidth)Space light 1994[43] ECDL ECDL <10 GHz DBM 3.7 MHz 50 kHz - <0.004 rad2 Space light 1999[44] DFB DFB 7-14 GHz DBM 70 MHz - −95 dBc/Hz
(50 MHz offset)0.05 rad2
(1 GHz bandwidth)Space light 2008[24] ECDL ECDL 250 kHz-20 GHz DBM+PFD - <10 Hz - <0.04 rad2 Space light 2008[25] ECDL ECDL 0.01-1.3 GHz PFD - 500 kHz −153 dBc/Hz
(10 kHz offset)<0.02 rad2 Space light 2012[39] - SG-DBR −9-7.5 GHz PFD 400 MHz - - - Integrated optical circuit 2012[27] DFB DS-DBR <40 GHz PFD 100 kHz
(open-loop unity-gain bandwidth)- - - Space light 2014[26] He-Ne
LaserECDL 80 MHz-1.4 GHz PFD - 3.2×10−9
(stability)- - Space light 2017[17] OFC SG-DBR <26 GHz XOR phase discriminator - - −80 dBc/Hz
(200 Hz offset)0.08 rad2
(1 kHz-1 GHz
bandwidth)Integrated optical circuit 2018[41] ECDL DBR 4-12 GHz analog phase
discriminator100 MHz - −100 dBc/Hz
>(10 kHz offset)0.012 rad2 Integrated
optical circuit -
[1] Li Tianchu, Qian Jin, Zhang Xiaoping, et al. The linear absorption frequency stabilization of 1.5 μm Acetylene (12C2H2) of DFB-LD at NIM [J]. Acta Metrologica Sinica, 2001(3): 161-163. (in Chinese) doi: 10.3321/j.issn:1000-1158.2001.03.001 [2] Zhang Jing, Tao Ye, Wei Dong, et al. A laser diode system stabilized on the saturated absorption lines of rubidium atoms [J]. Acta Optica Sinica, 2003(2): 197-201. (in Chinese) doi: 10.3321/j.issn:0253-2239.2003.02.014 [3] Jin Jie, Zhang Jianwei, Yang Yu, et al. ECDL at 1.5μm with acetylene saturated-absorption frequency stabilization [J]. Laser Technology, 2007, 31(4): 341-343. (in Chinese) doi: 10.3969/j.issn.1001-3806.2007.04.008 [4] Drever R W P, Hall J L, Kowalski F V, et al. Laser Phase and Frequency Stabilization Using an Optical Resonator [J]. Applied Physics B, 1983, 31(2): 97-105. [5] Weiss D S, Young B C, Chu S. Precision measurement of h/mCs based on photon recoil using laser-cooled atoms and atomic interferometry [J]. Applied Physics B, 1994, 59(3): 217-256. doi: https://doi.org/10.1007/BF01081393 [6] Eisaman M D, Andre A, Massou F, et al. Electromagnetically induced transparency with tunable single-photon pulses [J]. Nature, 2005, 438(7069): 837-841. doi: 10.1038/nature04327 [7] Zhao Hui, Pu Zhaobang, Liu Guodong. High precision straightness device based on double-frequency laser interference technique [J]. Chinese Journal of Lasers, 2001, 28(7): 637-640. (in Chinese) doi: 10.3321/j.issn:0258-7025.2001.07.016 [8] Peng Wencui, Zhou Lin, Long Shitong, et al. Locking laser frequency of up to 40GHz offset to a reference with a 10GHz electro-optic modulator [J]. Optics Letters, 2014, 39(10): 2998-3001. doi: 10.1364/OL.39.002998 [9] Li Zhiquan, Chen Xi, Zhu Guofang, et al. Modulation-free frequency locking of a laser based on acousto-optic frequency-shifting method [J]. Opto-Electronic Engineering, 2010, 37(1): 110-114. (in Chinese) doi: 10.3969/j.issn.1003-501X.2010.01.20 [10] Hisai Y, Ikeda K, Sakagami H, et al. Evaluation of laser frequency offset locking using an electrical delay line [J]. Appl Opt, 2018, 57(20): 5628-5634. doi: 10.1364/AO.57.005628 [11] Schünemann U, Engler H, Grimmet R, et al. Simple scheme for tunable frequency offset locking of two lasers [J]. Review of Scientific Instruments, 1999, 70(1): 242-243. doi: 10.1063/1.1149573 [12] Li Yun, Bao Xiaoyi, Ravet F, et al. Distributed Brillouin sensor system based on offset locking of two distributed feedback lasers [J]. Appl Opt, 2008, 47(2): 99-102. doi: 10.1364/AO.47.000099 [13] Rutt H N. A heterodyne frequency offset locking technique for pulsed or CW lasers [J]. Journal of Physics E: Scientific Instruments, 1984, 17(8): 704-709. doi: 10.1088/0022-3735/17/8/017 [14] Wang Hanmu, Cheng Hong, Zhang Shanshan, et al. Laser frequency offset-locking using electromagnetically induced transparency spectroscopy of 85Rb in magnetic field [J]. Chinese Physics B, 2018, 27(9): 1-5. [15] Ying Kang, Niu Yueping, Chen Dijun, et al. Laser frequency offset locking via tripod-type electromagnetically induced transparency [J]. Appl Opt, 2014, 53(12): 2632-2637. doi: 10.1364/AO.53.002632 [16] Seishu Y, Hasegawa T. Robust offset locking of laser frequency with electronically tunable LC circuits for sub-millihertz uncertainty [J]. Applied Physics B, 2019, 125(8): 142. doi: 10.1007/s00340-019-7253-5 [17] Arafin S, Simsek A, Kim S K, et al. Towards chip-scale optical frequency synthesis based on optical heterodyne phase-locked loop [J]. Opt Express, 2017, 25(2): 681-695. doi: 10.1364/OE.25.000681 [18] Balakier K, Shams H, Fice M J, et al. Optical phase lock loop as high-quality tuneable filter for optical frequency comb line selection [J]. Journal of Lightwave Technology, 2019, 36(19): 4646-4654. doi: 10.1109/JLT.2018.2848961 [19] Enloe L H, Rodda J L. Laser phase-locked loop [J]. Proceedings of the IEEE, 1965, 53(2): 165-166. doi: 10.1109/PROC.1965.3585 [20] Williams K J, Goldberg L, Esman R D, et al. 6–34 GHz offset phase-locking of Nd: YAG 1319 nm nonplanar ring lasers [J]. Electronics Letters, 1989, 25(18): 1242-1243. doi: 10.1049/el:19890833 [21] Harrison J, Mooradian A. Linewidth and offset frequency locking of external cavity GaAlAs lasers [J]. IEEE Journal of Quantum Electronics, 1989, 25(6): 1152-1155. doi: 10.1109/3.29240 [22] Gliese U, Nielsen T N, Bruun M, et al. Coherent optical generation of a 6 GHz microwave signal with directly phase locked semiconductor DFB lasers[C]// European Microwave Conference, 1992. [23] Sternkopf C, Diethold C, Gerhardt U, et al. Heterodyne interferometer laser source with a pair of two phase locked loop coupled He–Ne lasers by 632.8 nm [J]. Measurement Science and Technology, 2012, 23(7): 1-6. [24] Marino A M, Stroud C R. Phase-locked laser system for use in atomic coherence experiments [J]. Rev Sci Instrum, 2008, 79(1): 013104. doi: 10.1063/1.2823330 [25] Höckel D, Scholz M, Benson O. A robust phase-locked diode laser system for EIT experiments in cesium [J]. Applied Physics B, 2008, 94(3): 429-435. [26] Sternkopf C, Goellner S, Manske E. Frequency stabilization of an external-cavity diode laser by offset frequency looking to a stabilized He-Ne laser[C]// SPIE Photonics Europe. International Society for Optics and Photonics, 2014, 9134: 91341C . [27] Numata K, Chen J R, Wu S T. Precision and fast wavelength tuning of a dynamically phase-locked widely-tunable laser [J]. Opt Express, 2012, 20(13): 14234-14243. doi: 10.1364/OE.20.014234 [28] Xu Z, Zhang X, Huang K, et al. A digital optical phase-locked loop for diode lasers based on field programmable gate array [J]. Rev Sci Instrum, 2012, 83(9): 093104. doi: 10.1063/1.4750143 [29] Xu Z, Huang K, Lu X. A digital optical phase-locked loop based on field programmable gate array and its applications[C]// International Conference on Information Science. IEEE, 2014. [30] Yao R, Li Q, Xue K, et al. Investigation on offset frequency locking system for a short-pulse laser[C]//Proceedings of SPIE, 2009, 7382: 77. [31] Xie jiandong, Yan liping, Chenbenyong, et al. Automatic offset-frequency locking of external cavity diode laser in wide wavelength range [J]. Optics and Precision Engineering, 2021, 29(2): 211-219. (in Chinese) doi: 10.37188/OPE.20212902.0211 [32] Balakier K, Fice M J, Ponnampalam L, et al. Monolithically integrated optical phase lock loop for microwave photonics [J]. Journal of Lightwave Technology, 2014, 32(20): 3893-3900. doi: 10.1109/JLT.2014.2317941 [33] Lu M, Park H C, Parker J S, et al. A heterodyne optical phase-locked loop for multiple applications[C]//Optical Fiber Communication Conference and Exposition and the National Fiber Optic Engineers Conference (OFC/NFOEC), 2013, IEEE, 2013. [34] Coldren L A, Lu M, Park H C, et al. New Opportunities for Optical Phase-Locked Loops in Coherent Photonics[C]// Optical Fiber Communication Conference & Exposition & the National Fiber Optic Engineers Conference. IEEE, 2013. [35] Ristic S, Bhardwaj A, Rodwell M J, et al. An optical phase-locked loop photonic integrated circuit [J]. Journal of Lightwave Technology, 2010, 28(4): 526-538. doi: 10.1109/JLT.2009.2030341 [36] Ristic S, Bhardwaj A, Rodwell M J, et al. Integrated optical phase-locked loop[C]//2009 Conference on Optical Fiber Communication, 2009: 1-3. [37] Arafin S, Simsek A, Lu M, et al. Heterodyne locking of a fully integrated optical phase-locked loop with on-chip modulators [J]. Opt Lett, 2017, 42(19): 3745-3748. doi: 10.1364/OL.42.003745 [38] Steed R J, Pozzi F, Fice M J, et al. Monolithically integrated heterodyne optical phase-lock loop with RF XOR phase detector [J]. Opt Express, 2011, 19(21): 20048-20053. doi: 10.1364/OE.19.020048 [39] Steed R J, Ponnampalam L, Fice M J, et al. Hybrid integrated optical phase-lock loops for photonic terahertz sources [J]. IEEE Journal of Selected Topics in Quantum Electronics, 2011, 17(1): 210-217. doi: 10.1109/JSTQE.2010.2049003 [40] Lu M, Park H, Bloch E, et al. Highly integrated optical heterodyne phase-locked loop with phase/frequency detection [J]. Opt Express, 2012, 20(9): 9736-9741. doi: 10.1364/OE.20.009736 [41] Balakier K, Ponnamapalam L, Fice M J, et al. Integrated Semiconductor Laser Optical Phase Lock Loops [J]. IEEE Journal of Selected Topics in Quantum Electronics, 2017, 24(1): 1-12. [42] Gliese U, Nielsen T N, Bruun M, et al. A wideband heterodyne optical phase-locked loop for generation of 3-18 GHz microwave carriers [J]. IEEE Photonics Technology Letters, 1992, 4(8): 936-938. doi: 10.1109/68.149915 [43] Santarelli G, Clairon A, Lea S N, et al. Heterodyne optical phase-locking of extended-cavity semiconductor lasers at 9 GHz [J]. Optics Communications, 1994, 104(4-6): 339-344. doi: 10.1016/0030-4018(94)90567-3 [44] Langley L N, Elkin M D, Edge C, et al. Packaged semiconductor laser optical phase-locked loop (OPLL) for photonic generation, processing and transmission of microwave signals [J]. IEEE Transactions on Microwave Theory and Techniques, 1999, 47(7): 1257-1264. doi: 10.1109/22.775465 [45] 吴凯悦. 基于偏频锁定的激光合成波长干涉纳米位移测量技术研究[D]. 浙江理工大学, 2018. Wu Kaiyue. Study on nanometer displacement measurement with laser synthetic wavelength interferometer based on offset-frequency locking[D]. Hangzhou: Zhejiang Sci-Tech University, 2018. (in Chinese) [46] Liang W, Yariv A, Kewitsch A, et al. Coherent combining of the output of two semiconductor lasers using optical phase-lock loops [J]. Optics Letters, 2007, 32(4): 370-372. doi: 10.1364/OL.32.000370 [47] Liang W, Satyan N, Yariv A, et al. Coherent power combination of two Master-oscillator-power-amplifier (MOPA) semiconductor lasers using optical phase lock loops [J]. Opt Express, 2007, 15(6): 3201-3205. doi: 10.1364/OE.15.003201 [48] Guionie M, Frein L, Carre A, et al. Beat note stabilization in dual-polarization DFB fiber lasers by an optical phase-locked loop [J]. Opt Express, 2018, 26(3): 3483-3488. doi: 10.1364/OE.26.003483 [49] Wei Chunhua, Yan Shuhua. Simple and robust method for rapid cooling of 87Rb to quantum degeneracy [J]. Chinese Physics B, 2020, 29(6): 244-248. [50] Ashok R, Ananth G S, Gupta S. Optical phase-locked loop based carrier phase recovery and compensation for 8-PSK coherent optical links[C]//2019 Workshop on Recent Advances in Photonics (WRAP). IEEE, 2020. [51] Zhu G, Wang Q, Dong H, et al. 80Gb/s clock recovery with phase locked loop based on LiNbO3 modulators [J]. Opt Express, 2004, 12(15): 3488-3492. doi: 10.1364/OPEX.12.003488 [52] Jhon Y M, Ki H J, Kim S H. Clock recovery from 40 Gbps optical signal with optical phase-locked loop based on a terahertz optical asymmetric demultiplexer [J]. Optics Communications, 2003, 220(4-6): 315-319. doi: 10.1016/S0030-4018(03)01408-1 [53] Kim D H, Kim S H, Jo C H, et al. Ultrahigh-speed clock recovery with optical phase lock loop based on four-wave-mixing in a semiconductor optical amplifier [J]. Optics Communications, 2000, 182(4-6): 329-334. doi: 10.1016/S0030-4018(00)00837-3 [54] Kim D H, Kim S H, Jo C H, et al. 40 GHz optical PLL based on four-wave-mixing in a semiconductor optical amplifier[C]//1999 IEEE LEOS Annual Meeting Conference Proceedings. LEOS'99. 12th Annual Meeting. IEEE Lasers and Electro-Optics Society 1999 Annual Meeting (Cat. No.99CH37009), 1999, 1: 347-348. [55] Okamura Y, Takada A. Simultaneous carrier extraction from wavelength-division-multiplexed phase-conjugated twin waves by using sum-frequency generation with second-order optical nonlinear medium for phase-sensitive optical amplifier repeaters [J]. Japanese Journal of Applied Physics, 2019, 58(SJ): SJJA04. doi: 10.7567/1347-4065/ab27b4 [56] Zhao Yi, Tong Shoufeng, Song Yansong, et al. Research progress of optical phase locked loop in space laser communication [J]. Laser & Optoelectronics Progress, 2015, 52(8): 20-28. (in Chinese) [57] Xu Nan, Liu Liren, Liu Dean, et al. Optical phase locked loop in intersatellite coherent optical communication [J]. Laser & Optoelectronics Progress, 2008, 45(4): 25-33. (in Chinese)