智能锁模光纤激光器的原理与研究进展 
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The invention of laser technology has had a transformative impact on society. Mode-locked fiber lasers have been widely used in research and industry, and they play an important role in basic science as a convenient nonlinear system. A mode-locked fiber laser is a complex nonlinear dissipative system with a large number of internal nonlinear dynamical phenomena that, in addition to outputting stable femtosecond pulses, exhibits a series of complex mode-locked states, including the breather locked mode, strange waves, noise-like locked modes, soliton explosions, and self-organized modes arising from soliton interactions, such as soliton molecules, soliton crystals, soliton complexes, and supramolecular structures. Even chaotic states have recently been discovered in mode-locked lasers. The study of these mode-locked states helps to understand the nonlinear dynamical properties of femtosecond fiber lasers. Additionally, because the femtosecond fiber laser is a universal nonlinear dissipative system, studying its dynamics can clarify the complex dynamics in related fields, such as Bose-Einstein condensation, microcavities, and oceanography. The intrinsic dynamics of these systems and the mode-locked laser are described uniformly by the nonlinear Schr?dinger equation and thus have similarities.
Owing to the presence of numerous mode-locked regions in mode-locked lasers, it has long been a challenging problem to control the parameters of the laser and thus access specific mode-locked states. For example, the most commonly used femtosecond fiber laser based on the nonlinear polarization rotational mode-locking technique is mathematically a multidimensional parametric space and experimentally requires tuning of at least seven parameters (pump, loss, dispersion, nonlinearity, and angles of the three waveplates) to traverse the entire parametric space. Because of the lack of a definite functional relationship between the mode-locking state and these parameters, a long trial-and-error process is needed to obtain the desired mode-locking state. In addition, even if the target locked mode is obtained, its repeatability is a problem.
Recently, a major breakthrough was made in intelligent mode-locked lasers, which can resolve the difficulty of precise control of mode-locked states. In 2015, Prof. Grelu’s group in France applied a genetic algorithm to the intelligent control of mode-locked lasers for the first time and realized the intelligent control of soliton pulses and noise-like pulses. Subsequently, the development of intelligent mode-locked lasers has accelerated. Hence, it is necessary to summarize the existing studies to rationally guide subsequent research in this area.
The principle of the commonly used smart locking algorithms and recent scientific research results are summarized. First, the principles of the genetic algorithm, human-like algorithm, and artificial neural network are explained, and a schematic (Fig.1) and architecture diagram (Fig.2) are presented. Then, recent scientific achievements in smart mode-locked lasers are described, including the first implementation of a soliton-locked mode in smart lasers by Andral et al. at the Université de Bourgogne, France (Fig.3); the development of genetic algorithms for soliton-locked mode recovery by Winters et al. at Kapteyn-Murnane Laboratories, USA (Fig.4); the development of the first smart programmable mode-locked laser by Pu et al. at Shanghai Jiao Tong University (Fig.5); and the development of the first smart programmable mode-locked laser using deep learning for intelligent mode-locking recovery by Yan et al. at the National University of Defense Technology (Fig.6). Subsequently, the realization of programmable control of the spectral width and spectral shape by Pu et al. of Shanghai Jiao Tong University (Fig.7) and the intelligent control of spatiotemporal mode-locking by Wei et al. of South China University of Technology (Fig.8) are elaborated. The intelligent regulation of the breather ultrafast laser is summarized, starting with the design of an adaptation function based on the radiofrequency signal of the breather locked mode (Fig.9), in which the relaxation oscillation dynamics and noise-like pulse dynamics in the laser are excluded (Fig.10). Then, experimental results of the genetic algorithm (Fig.11) are discussed, along with the control of the breather breathing ratio, the breathing period, and the number of pulses (Figs.12-14). Finally, the work related to the intelligent control of fractal respiratory subsets is briefly described. The differences in the spectra and stability of frequency-locked and non-frequency-locked breathers are examined (Figs.15 and 16), the evolutionary dynamics of fractal breathers are specified (Fig.17), and the intelligent search for fractal breathers is implemented using a smart laser based on a liquid-crystal phase delay (Figs.18 and 19).
This paper reviews the application of intelligent-control technology in passively mode-locked fiber lasers. Using intelligent-control technology, the automatic generation and control of the mode-locked state can be realized without manual tuning, which reduces the tuning time of the laser and improves the tuning accuracy as well as the repeatability of the mode-locked state. This self-optimizing ultrashort pulse laser has promising applications in certain environments. Although the passive mode-locked fiber laser is a complex dynamical system, the successful achievement of accurate tuning of multiple mode-locked states by genetic algorithms indicates the universality of these algorithms. A series of intelligent algorithms, including genetic algorithms, are expected to be applied to the intelligent control of more complex mode-locked states. The current intelligent-control technology focuses on controlling lasers and achieving automatic laser tuning. Whether intelligent-control techniques can have an impact on laser physics remains an open question.
1 引言
1960年，西奥多·梅曼发明了激光，激光给人类社会的发展带来了变革性影响。梅曼发明激光短短几年后，研究人员就报道了激光领域的第一个标志性成果——锁模激光。锁模就是将激光器内成千上万的纵模同时锁定，从而产生飞秒量级的超短脉冲。锁模激光器已在光学频率梳［1-5］、精密制造［6］、光纤通信［7-8］和激光雷达［9］等领域获得了广泛应用。
除了被广泛地用作超短脉冲光源之外，锁模光纤激光器还被作为一个便捷的桌面化非线性系统在基础科学领域发挥着重要作用。例如，它为复杂非线性波的动力学研究提供了理想的平台［10］。事实上，锁模光纤激光器是一个复杂的非线性耗散系统，其内部存在大量的非线性动力学现象：除了输出稳定的飞秒脉冲外，还会呈现出一系列的复杂锁模态，包括呼吸子锁模［11］、怪波［12］、类噪声锁模［13］、孤子爆炸［14-17］以及孤子相互作用产生的自组织模式（如孤子分子［18-21］、孤子晶体［22-23］、孤子复合物［24］和超分子结构［25］等），甚至是混沌态［26］。研究这些锁模态有助于理解飞秒光纤激光器中的非线性动力学特性，同时，由于飞秒光纤激光器是一个普适的非线性耗散系统，对其动力学进行研究也有助于理解其他相关领域（如玻色爱因斯坦凝聚、微腔、海洋学等）的复杂动力学。这些系统和锁模激光器的内在动力学均是由非线性薛定谔方程统一描述的，因此它们之间的动力学具有相似性。
由于锁模激光器中存在众多的锁模区，长期以来，如何控制激光器的参数进而访问特定的锁模态是一个颇具挑战性的难题。以最常用的基于非线性偏振旋转锁模技术的飞秒光纤激光器为例，其在数学上是一个多维参量空间，实验上需要调谐至少7个参量（泵浦、损耗、色散、非线性和三个波片的角度）才能遍历整个参量空间。锁模态与这些参量之间缺乏一个确定的函数关系，因此需要一个漫长的试错过程来获得想要的锁模态。此外，即使获得了目标锁模态，其重复性也是一个问题。而且，激光锁模区的参量空间通常极小，实验上很难精确控制。
最近，智能化锁模激光器取得了重大突破，可以解决锁模态的精确控制难题。2015年，法国Grelu教授课题组［27］首次将遗传算法应用于锁模激光器的智能控制中，实现了孤子脉冲和类噪声脉冲的智能化控制。随后，智能锁模激光器获得了高速发展［27-41］。本文将回顾智能锁模光纤激光器的原理及其主要研究进展，主要介绍遗传算法、类人算法及人工神经网络算法，以及这些算法在孤子、呼吸子和时空锁模智能化控制中的应用。
2 智能锁模算法
2.1 遗传算法
智能锁模激光器可以实现参数的自优化，进而获得目标锁模态，不再需要手动调节激光器参数。其核心就是运用相关算法，有效地调节激光器的参数，直至激光器输出目标锁模态。目前应用最广泛的是遗传算法，该算法是在达尔文进化论“适者生存”理念启发下提出的。遗传算法基于生物进化的概念，为设定的控制目标（目标锁模态）搜寻最合适的“个体”，此处“个体”对应的是一组激光参数。一组激光参数对应单个个体的基因序列。每个个体的优劣由用户定义的适应度函数来评估，进而获得一个适应度数值，适应度高的基因将进行多次繁衍直至系统获得最大适应度值。
在智能锁模激光器中，激光器的控制参数（例如施加到电动偏振控制器［34］或者液晶波片［27］上的一组电压）即是一个基因序列。首先，计算机随机产生一个初始种群，该种群包含多个个体，即多组电压值，将其输入到液晶波片或者偏振控制器后，数据采集模块从激光器输出端探测数据并输入计算机（或者其他在线数据处理模块），然后根据适应度函数计算其适应度数值。系统对个体评价完成后，模仿“适者生存”进行选择、交叉、变异三个步骤产生下一代新种群。整个遗传算法流程图如
式中：
图 1. 遗传算法原理说明。（a）遗传算法流程图；（b）“轮盘赌”示意图
Fig. 1. Principle of genetic algorithm. (a) Genetic algorithm flowchart; (b) schematic of the “roulette wheel” selection
2.2 类人算法
遗传算法虽然应用广泛，但其显著的缺点是运算时间较长。2019年，一种可以实现快速锁模的算法——类人算法被提出［34］。该算法模仿人的逻辑来调节偏振控制器从而实现锁模。具体流程如下：系统随机生成电动偏振控制器的控制参数（一组电压值），然后通过优化后的Rosenbrock搜索算法，模拟人的控制步骤，先将其中一个参数向固定方向进行调整，如果脉冲幅度有所增加，则继续向该方向调节参数，否则向反方向调节参数，直到找到该参数在所有变化方向中使脉冲幅度最高的那个值；然后改变另一个控制参数，重复操作，直到锁模或超出预设的调整次数，进入监测阶段或退出。在监测阶段，通过脉冲计数进行锁模判断，如果出现失锁情况，则采用随机碰撞算法模仿人对激光器进行微调，即以小于之前搜索算法的步进量进行随机尝试，直到恢复锁模或超出尝试次数，回到监测状态或重新进入Rosenbrock搜索算法。该算法不仅可以实现快速锁模，还可以在激光器失锁后迅速恢复到锁模态。
2.3 人工神经网络算法
近年来，各种深度神经网络作为一种新兴的数据分析和处理技术，为处理复杂且对干扰敏感的光纤激光器系统提供了良好的工具。人工神经网络是一种模仿生物神经网络结构和功能的数学模型，通常用于估计或逼近函数［42］。人工神经网络通常由输入层、隐藏层和输出层组成，每层由许多处理元素组成，称为神经元或节点。
3 智能锁模激光器的研究进展
目前，智能锁模激光器中应用的算法大都是遗传算法。遗传算法的最大优势在于，只要目标锁模态可以被适应度函数量化，那么该算法通过模仿生物进化过程一般都可以获得目标锁模态。该算法已经被应用于孤子锁模［27］、时空锁模［36］和呼吸子锁模［37］。下面将介绍相关进展。
3.1 孤子锁模
孤子是传输时脉宽等参数保持不变的一种波包。利用孤子效应可以在激光器中产生稳定的超短脉冲激光。通信波段光纤中天然地包含了产生孤子所需的两个物理效应——反常色散和克尔效应，因此，光纤激光器是产生孤子、研究孤子物理的一个非常便捷的平台。激光器产生孤子的一个标志是脉冲光谱上出现凯利边带。现有的遗传算法尽管可以实现锁模，但是需要精确的适应度函数来区分孤子锁模区和其他锁模区。如果适应度函数特异性不明显，遗传算法可能会产生多种锁模态。例如，如果适应度函数由强度自相关信号定义，而孤子锁模和调Q锁模均会产生强度自相关信号，系统就无法分辨这两种激光状态，因此激光器会输出孤子或者调Q锁模脉冲，如
图 3. 进化算法实现智能锁模［27］。（a）通过进化算法实现非线性偏振偏转（NPR）锁模激光器的实验装置图；（b）（c）利用信号强度设计进化算法适应度发现的锁模（b）和调Q锁模（c）状态；（d）利用射频谱分量强度定义的适应度函数平均值和最佳值的演化；（e）（f）通过进化算法发现的锁模状态的时间脉冲序列和光谱
Fig. 3. Evolutionary algorithm to implement smart mode-locking[27]. (a) Experimental setup for locking an NPR-based MLFL through an evolutionary algorithm; (b)(c) mode-locked (b) and unstable Q-switched mode-locked regime (c) found during the run of the evolutionary algorithm fitness designed with signal intensity; (d) evolution of the average and best values of fitness function defined with radio frequency (RF) component intensity; (e)(f) temporal pulse train (e) and the optical spectrum (f) of the fundamental mode-locking regime found by the evolutionary algorithm
2015年，Andral等［27］利用进化算法首次在被动锁模光纤激光器中实现了智能锁模。实验装置图如
式中：A为射频谱分量的幅度；C为本底噪声阶数的常数，用于限制噪声扰动。这些研究对于智能激光器的推广和应用具有重要意义，但受限于进化算法以及激光器时域和光谱信息的缺失，优化时间较长（约0.5 h），并且无法控制脉冲的参数（如光谱形状和带宽等）。
2017年，Winters等［31］实现了对全正色散光纤激光器的自动锁模。实验装置如
图 4. 全正色散光纤激光器的智能控制［31］。（a）基于液晶的全正色散光纤激光器示意图；（b）液晶相位延迟器的两种配置图，显示前向（左图）和反向（右图）传播方向；（c）通过扫描所有可用电压找到的锁模状态，展示了几个操作点的光谱（4个控制值中的3个表示空间位置，另一个表示标记颜色）；（d）（e）标准谐振腔（d）和液晶控制的谐振腔（e）中作为环境温度函数的脉冲持续时间；（f）通过遗传算法从随机起点使用目标光谱（虚线）恢复的实际锁模光谱（实线）；（g）在温度循环运行期间测量的频率分辨光学开关（FROG）轨迹，数字i~iv对应（e）中标识的时间和温度，光谱以白色插入
Fig. 4. Intelligent control of all-normal dispersion(ANDi) fiber laser[31]. (a) Schematic of an all-normal dispersion fiber laser with liquid crystal (LC); (b) diagram of the two configurations of the LC phase retarder showing the forward (left) and backward (right) propagation directions; (c) mode-locked states found through a scan over all available voltages, showing spectra for several operating points (three control values represent spatial position and the fourth value represents the color of the marker); (d)(e) pulse duration as a function of environmental temperature in a standard oscillator (d) and an LC stabilized oscillator (e); (f) spectrum of the mode-locked ANDi laser (solid) recovered by genetic algorithm from a random starting point using a target spectrum (dashed); (g) measured FROG traces measured during the temperature cycling run with the numbers i-iv corresponding to the time and temperatures indicated in Fig. (e), where the spectral is inset in white along the appropriate axes
2019年，Pu等［34］首次实现了基于类人算法的锁模激光器，实验装置如
其中，
图 5. 基于类人算法的智能可编程激光器［34］。（a）智能锁模光纤激光器实验装置图；（b）类人算法与其他智能锁模算法初始锁定时间、恢复时间和锁模状态数量的对比；（c）类人算法示意图；（d）锁模运行方式，从左到右依次为基频锁模、二阶谐波锁模、三阶谐波锁模、调Q、调Q锁模
Fig. 5. Intelligent programmable laser based on human-like algorithm[34]. (a) Experiment setup of intelligent mode-locked fiber laser; (b) comparison over initial lock time, recovery time and the number of mode-locking regimes between human-like algorithm and recent automatic mode-locking algorithms; (c) schematic of human-like algorithm; (d) operation regimes, from left to right respectively shows the fundamental mode-locking, the second-order harmonic mode-locking, the third-order harmonic mode-locking, the Q-switch, and the Q-switch mode-locking operation regimes
2021年，Yan等［45］针对自动锁模光纤激光器提出了一种低延迟的深度强化学习算法。该算法结构如
图 6. 基于深度学习算法的智能光纤激光器［45］。（a）深度学习超高速光纤激光器的实验装置图；（b）基于深度确定性梯度策略的低延迟深度强化学习算法结构；（c）搜寻到基频锁模状态时激光器的输出表征；（d）激光器振动失锁后算法恢复效果图，包括最后100轮稳定锁模计算模型训练迭代中奖励值的收敛曲线（左上），振动后激光器重复频率和输出功率的变化（右上），1500次振动实验的恢复时间统计（左下），在每分钟振动1.5 s并一直运行锁模恢复算法条件下系统在10 min内输出功率的变化（右下）
Fig. 6. Intelligent fiber laser based on deep learning algorithm[45]. (a) Experimental setup of deep learning ultrafast fiber laser; (b) structure of the low-latency deep-reinforcement learning algorithm based on deep deterministic gradient strategy in the laser environment; (c) characterization of the output when the laser is in the fundamental mode-locking state; (d) effect diagram of algorithm recovery after the laser loses mode-locked state due to motor vibration, including the convergence curves of the reward values in the last 100 training iterations of the stable mode-locking computational model (top left), the repetition frequency and power change of the laser output during the process of applying vibration to the laser (top right), the recovery time statistics of 1500 vibration tests (down left), the output power change of the system within 10 min under the condition of vibrating for 1.5 s per minute and running the mode-locked recovery algorithm all of the time (down right)
3.2 锁模激光器的光谱智能调控
很多应用需要对超短脉冲的光谱进行调控。虽然可以通过在腔内加入滤波器的方法实现对光谱的控制，但这一方案降低了激光器的集成度，增加了成本，而且滤波器本身的损耗较大，导致激光器的转换效率较低。事实上，锁模激光器通过调节自身参数也可以使光谱在一定范围内调谐。特别地，对于非线性偏振旋转锁模来说，由于腔内天然地存在一个Loyt滤波器（基于双折射效应和起偏器的原理设计），调节偏振控制器可以实现对滤波器特性的改变，进而可以在一定程度上调节锁模脉冲的光谱。然而，由于手动调节偏振控制器存在调谐精度低、重复性差等缺点，该方案在实际应用时不具备吸引力。智能控制技术可以很好地解决手动调节带来的相关问题。
2020年，Pu等［35］利用遗传算法实现了对脉冲光谱和脉冲形状的智能控制。
图 7. 光谱宽度和形状可编程光纤激光器［35］。（a）实验装置示意图；（b）（c）光谱形状编程：双曲正割谱（b），三角形谱（c）；（d）~（f）10~40 nm的光谱宽度控制，包括光谱（d）、自相关轨迹（e）和最大光谱宽度（半峰全宽FWHM）的可重复性测试（f）
Fig. 7. Spectral width and shape programmable fiber lasers[35]. (a) Experiment setup diagram; (b)(c) spectral shape programming: the hyperbolic secant spectrum (b), the triangular spectrum (c); (d)-(f) spectral width programming from 10 to 40 nm, showing the spectra (d), autocorrelation traces (e), and the repeatability test for the maximum spectral FWHM (f)
3.3 时空锁模自调控
锁模通常是指激光器内纵模的锁定，光纤激光器中横模一般只有基模一个模式，不需要锁定。如果激光器内同时存在大量横模和纵模（例如多模光纤构成的谐振腔），由于不同横模的群速度不同，很难产生超短脉冲。2017年，Wright等［47］通过在多模光纤激光器内加上一个空间滤波器，实现了多模激光器中横纵模的同时锁定，因此称之为“时空锁模”。时空锁模不仅可以突破锁模激光器横向模式的限制，同时由于其是多模光纤，芯径较大，因此产生的脉冲能量也远大于单模光纤。时空锁模激光近几年获得了国内外的极大关注［48-53］。然而，多模激光器的参量空间随着模式的增加几乎呈几何式增加，人工调谐已很难满足需求，极大地限制了时空锁模技术的发展［48］。2020年，Wei等［36］利用智能化控制技术很好地解决了这一问题。
图 8. 多模智能激光器［36］。（a）基于波前整形的遗传算法多模激光器示意图；（b）功率随遗传代数增加而增大，插图是在空间光调制器上显示的最佳相位图；（c）光功率增强前后的光谱；（d）~（f）在准连续光状态下利用遗传算法对多模模式进行轮廓清洁，其中（d）是算法优化前的模式轮廓，（e）是算法优化后的模式轮廓，（f）是算法优化前后的模式截面对比
Fig. 8. Multimode intelligent fiber laser[36]. (a) Schematic diagram of a genetic multimode fiber laser using wavefront shaping; (b) power increases with the increase of generation number, where the inset shows the optimal phase map displayed on the spatial light modulator (SLM); (c) optical spectra before and after optical power enhancement; (d)-(f) mode profile cleaning of the genetic multimode fibre laser working in the quasi-CW state, where the Fig.8(d) is mode profile before genetic optimization, Fig.8 (e) is mode profile after genetic optimization, and Fig.8(f) is mode profile comparison before and after genetic optimization
3.4 呼吸子超快激光器的智能调控
3.4.1 呼吸子智能控制
近些年来，呼吸子超快激光取得了快速发展［11，37，41，54-64］。传统锁模激光器输出的是一系列等同的超短脉冲（孤子脉冲），呼吸子激光器输出的脉冲参数如能量、光谱、脉宽等会随时间周期性变化，因此被称为“呼吸子”（或者脉动孤子）。变化周期则被称为“呼吸周期”，对应的频率被称为“呼吸频率”。呼吸频率亦被称为“特征频率”，因为其是呼吸子特有的频率成分。呼吸子激光的射频谱如
式中：
如
式中：
图 10. 弛豫振荡和类噪声脉冲动力学［37］。（a）~（c）弛豫振荡演化动力学：（a）对弛豫振荡时间强度信号进行FFT获得的射频谱；（b）示波器捕捉的弛豫振荡相应的时域轨迹；（c）弛豫振荡在2000个周期内的DFT（dispersive Fourier transformation）光谱演化过程，白色曲线表示能量演变。（d）~（f）类噪声脉冲演化动力学：（d）对弛豫振荡时间强度信号进行快速傅里叶变换获得的射频谱，插图显示了边带的放大；（e）示波器捕捉的类噪声相应的时域轨迹；（f）类噪声脉冲在1000个周期内的DFT光谱演化过程
Fig. 10. Dynamics of relaxation oscillation and noise-like pulse[37]. (a)-(c) Dynamics of relaxation oscillation: (a) RF spectrum extracted by fast Fourier transform (FFT) of the relaxation oscillation temporal intensity signal, featuring many sidebands around the fundamental repetition frequency; (b) corresponding time trace of relaxation oscillation captured by the oscilloscope; (c) DFT optical spectrum evolution of the relaxation oscillation over 2000 cavity periods, where the white curve represents the energy evolution. (d)-(f) Dynamics of noise-like pulse emission: (d) RF spectrum of the output extracted by FFT of the temporal intensity signal, where the inset shows a magnified version of the sidebands; (e) corresponding time trace captured by the oscilloscope; (f) evolution of DFT optical spectrum roundtrip by roundtrip obtained by DFT, revealing the typical noisy spectrum of noise-like pulse mode locking
为了确保激光器处于单脉冲状态，需要对脉冲数量
式中：
综上，对单呼吸子锁模的搜索可通过三部分优化程序实现：1）对每个个体的适应度值
由于锁模激光器中的呼吸子比孤子的参量空间小［65］，可通过实际测试设置初始种群大小为100，后续种群大小保持为50。通常情况下，为了保留尽可能多的优良个体的基因，变异概率不宜设置得过大。此处设置交叉概率
图 11. 单呼吸子智能搜寻实验结果［37］。（a）连续几代的平均（红色圆圈）和最大（蓝色方块）适应度值的演化。（b）~（d）优化状态的特征：（b）通过对光电二极管的信号进行FFT获得的射频谱；（c）多次腔内往返的光谱（DFT方法测量），白色曲线代表能量演化；（d）在连续腔内演化中脉冲时域强度的变化
Fig. 11. Experimental results of the intelligent search for single breathers[37]. (a) Evolution of the average (red circles) and maximum (blue squares) fitness function value over successive generations, for the merit function given in equation (6). (b)-(d) Characteristics of the optimized state: (b) RF spectrum obtained by FFT of the signal from the photodiode; (c) dispersive Fourier transform recording of single-shot spectra over consecutive cavity round trips (RTs), where the white curve represents the energy evolution; (d) temporal evolution of the intensity relative to the average RT time over consecutive RTs
3.4.1.1 呼吸比控制
呼吸比定义为一个周期内脉冲光谱的最大和最小宽度之比（类似于调制深度）。呼吸比越高，脉冲射频谱的呼吸频率强度越大。因此，优化呼吸比的关键在于控制射频谱中重复频率和呼吸频率的峰值强度之比。呼吸比由
式中：
呼吸比控制的实验结果如
图 12. 呼吸比智能控制［37］。（a）~（c）呼吸比为1.076的呼吸子动力学；（d）~（f）呼吸比为1.471的呼吸子动力学；（g）~（i）呼吸比为1.816的呼吸子动力学；（a）（d）（g）多次腔内往返光谱（DFT方法测量）；（b）（e）（h）多次腔内往返时域记录；（c）（f）（i）单个呼吸周期内最宽和最窄谱宽的单次光谱截面
Fig. 12. Intelligent control of breathing ratio[37]. (a)-(c) Dynamics of breathers with small breathing ratio of 1.076; (d)-(f) dynamics of breathers with moderate breathing ratio of 1.471; (g)-(i) dynamics of breathers with large breathing ratio of 1.816; (a)(d)(g) dispersive Fourier transform recording of single-shot spectra over consecutive cavity round trips (RTs); (b)(e)(h) temporal evolution of the intensity relative to the average RT time over consecutive RTs; (c)(f)(i) single-shot spectra at the RT numbers of maximal and minimal spectrum extents within a period
3.4.1.2 呼吸周期控制
呼吸子的呼吸周期可以通过对脉冲输出的射频谱进行计算得到（1/| f±1-fr|）。腔重复频率
式中：
图 13. 具有可调振荡周期的呼吸子的基因算法优化结果［37］。（a）（b）振荡周期大的呼吸子的动力学；（c）（d）振荡周期中等的呼吸子的动力学；（e）（f）振荡周期小的呼吸子的动力学；（a）（c）（e）多次腔内往返光谱（DFT方法测量）；（b）（d）（f）在连续腔内演化的脉冲时域强度的变化
Fig. 13. Genetic algorithm optimization results for breathing solitons with a tunable oscillation period[37]. (a)(b) Dynamics of breathers with large oscillation period; (c)(d) dynamics of breathers with moderate oscillation period; (e)(f) dynamics of breathers with small oscillation period; (a)(c)(e) dispersive Fourier transform recording of single-shot spectra over consecutive RTs; (b)(d)(f) temporal evolution of the intensity relative to the average RT time over consecutive RTs
3.4.1.3 呼吸子分子复合物产生
与大多数多孤子结构类似［19-20，24，66］，激光器中的呼吸子也存在相互作用，并在特定的腔参数范围内能够形成多呼吸子束缚态［11，55］，将其称为“呼吸子分子”。孤子分子在光纤激光器中产生的方法是将泵浦功率增大到单孤子锁模状态所需功率以上，分子内的孤子数量和泵浦功率成正比［19，24］。然而，呼吸子分子的激发过程相比于孤子分子更加复杂。此外，因为在正色散激光器中的呼吸子不会产生色散波，所以在正色散光纤激光器中产生多呼吸子复合物更加困难［11］。因此，利用遗传算法产生呼吸子分子是非常有意义的。增加泵浦功率，结合优化程序，利用
图 14. 基于遗传算法的呼吸子分子优化结果［37］。（a）~（d）增加型相位呼吸子分子动力学；（e）~（h）振荡型相位呼吸子分子动力学；（a）（e）多次腔内往返光谱（DFT方法测量），其中（a）中摩尔干涉条纹出现的根源在于光谱调制过于密集；（b）（f）DFT记录的光谱数据的特写；（c）（g）多次腔内往返光谱的一阶自相关轨迹演化；（d）（h）两个呼吸子分子之间的相对相位差（红色曲线）和分子总能量（黑色曲线）随时间的变化
Fig. 14. Typical genetic algorithm optimization results for breather molecules[37]. (a)-(d) Dynamic of “increasing-phase” breather molecule; (e)-(h) dynamic of “oscillating-phase” breather molecule; (a)(e) dispersive Fourier transform (DFT) recording of single-shot spectra over consecutive cavity RTs, over-dense spectral modulation causes a Moiré interference pattern in Fig.14(a); (b)(f) close-up view of the DFT recorded spectral data; (c)(g) evolution of the first-order single-shot autocorrelation trace over consecutive RTs; (d)(h) evolution of the relative phase difference between the two breathers (red curve) and the total energy of the molecule (black curve) as a function of the time
当泵浦功率增大到150 mW时，结合遗传算法可以获得三呼吸子的束缚态（呼吸子分子复合物）。三呼吸子的束缚态包括三种不同类型的三呼吸子，分别是2+1、1+2以及1+1+1结构。特别地，1+1+1型呼吸子分子复合物由三个几乎等间距的呼吸子组成。在所有的呼吸子复合物中，基于DFT的单次光谱测量和时空强度演化动力学都明确地表明光谱的周期性变化伴随着脉冲能量的同步变化。
当泵浦功率设置为170 mW时，利用遗传算法可以实现四呼吸子束缚态的产生。四呼吸子束缚态的类型包括1+3、3+1和2+2结构，其中有两种间距差别很大的1+3结构呼吸子分子复合物，而且间距较小的呼吸子的呼吸比更大，这也说明脉冲距离与呼吸比之间存在着潜在的联系。
3.4.2 分形呼吸子智能控制
上一节所述的呼吸子超快激光是在正常色散条件下产生的，不具备分形特性。最近发现：当激光器色散趋近于零时，将会产生分形呼吸子［54］。本节将介绍分形呼吸子及其智能控制。
1975年，数学家Benoit Mandelbrot首次提出“分形”一词，并由此诞生了一门新的学科——分形几何学。Mandelbrot将分形定义为“局部和整体以某种方式相似的形体”。分形在自然界中的一个典型例子是西兰花，西兰花的每一个分支与整体的形状都是相似的。虽然分形最初只是一个数学概念，但其已在众多系统中被观察到，如材料学、生物学、神经学、电路、网络、地理等。2000年，光孤子和分形之间的联系在理论上被建立起来［68］。由于光孤子较为稳定，将其“分形”为多个分支需要对传输介质（例如光纤）的物理性质进行多次改变，而且需要在多个位置对光孤子进行探测。这些条件在实验上很难实现，相关实验几乎无法开展。最近的相关工作表明：当激光器工作在近零色散区时，产生的呼吸子具有分形特性［54］。研究发现，通过控制激光器的泵浦电流，激光器输出光场可以在频率锁定和未锁定呼吸子这两类呼吸子之间切换，如
图 15. 锁频和未锁频呼吸子对应的射频谱测量图（参考频率为基频的1/5）［54］。（a）（b）在50 kHz和100 Hz跨度上测量到的锁频呼吸子的单模振荡；（c）（d）在50 kHz和10 kHz跨度上测量到的未锁频呼吸子的多模振荡
Fig. 15. RF spectral measurements of frequency-locked and frequency-unlocked breathers (the reference frequency is one-fifth of the fundamental repetition frequency)[54]. (a)(b) Single-mode oscillation of frequency-locked breather measured over spans of 50 kHz and 100 Hz, respectively; (c)(d) multi-mode oscillation of frequency-unlocked breathers measured over 50 kHz and 10 kHz spans, respectively
图 16. 通过频率计数器测量的锁定呼吸子与未锁定呼吸子频率随时间的变化，前者的稳定性是后者的3500倍左右，其中SD表示为标准差［54］
Fig. 16. Change in breathing frequency over time for the frequency-locked and frequency-unlocked breathers, as measured with a cymometer. The former is 3500 times more stable than the latter, and SD means standard deviation[54]
图 17. 呼吸频率/环绕数随泵浦功率的改变出现一系列平台。平台对应的环绕数构成了法里树，如插图所示。环绕数是指呼吸频率（呼吸子演化周期的倒数）与激光器重复频率（由激光器长度决定）的比值；环绕数等于1/5表示呼吸频率是激光器重复频率的1/5［54］
Fig. 17. Breathing frequency/winding number with pump power in a series of plateaus. The plateaus corresponding winding number forms the Farey tree, as shown in the inset. The winding number refers to the ratio of breathing frequency (fb) and repetition frequency fr; the winding number equal to 1/5 means that
人工寻找频率锁定呼吸子需要一定的经验且耗时费力，而利用智能控制系统可以实现频率锁定呼吸子的快速搜寻。如
图 18. 基于液晶相位延迟器偏振控制的智能激光器实验装置［54］
Fig. 18. Experiment setup of the intelligent laser with polarization control based on liquid crystal phase delayers[54]
其中
式中：
图 19. 机器学习的结果［54］。（a）每代种群的平均和最大适应度的演化，以及呼吸频率信噪比SNR的相应演化；（b）（c）呼吸频率不随泵浦功率和偏振（通过改变液晶LC2上的电压）而变化，最优状态（锁频呼吸子）持续稳定存在
Fig. 19. Machine-learning results[54]. (a) Evolution of the mean and best fitness function value of the individuals for each generation, as well as the corresponding evolution of the SNR (signal-to-noise ratio) of the breathing frequency; (b)(c) breathing frequency is constant with the variation of pump power and polarization, showing stable optimal state (locked-frequency breather)
4 结束语
本文回顾了智能控制技术在被动锁模光纤激光器中的应用。基于智能控制技术，可以实现对锁模态的自动化产生和控制，不需要手动调谐，缩短了激光器的调谐时间，提升了调谐的精度以及锁模态的可重复性。此外，这种自优化的超短脉冲激光器在某些特殊环境下也具有一定的应用前景。尽管被动锁模光纤激光器是一个非常复杂的动力学系统，但是通过遗传算法成功实现了对多种锁模态的精准调控，这说明该算法具有普适性。包括遗传算法在内的一系列智能算法也有望应用于其他更为复杂的锁模态的智能化控制。此外，当前智能控制技术的重点在于控制激光器，实现激光器的自动调节。智能控制技术能否对激光物理产生影响是一个开放性问题。
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Article Outline
吴修齐, 彭俊松, 张颖, 曾和平. 智能锁模光纤激光器的原理与研究进展[J]. 中国激光, 2023, 50(11): 1101006. Xiuqi Wu, Junsong Peng, Ying Zhang, Heping Zeng. Principles and Research Advances of Intelligent Mode‐Locked Fiber Lasers[J]. Chinese Journal of Lasers, 2023, 50(11): 1101006.
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