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高峰值功率、高光束质量激光器在空间探测[1-2]、强场物理[3-4]、材料加工[5-6]、等离子体物理[7-8]等领域具有广泛应用。作为获得高功率激光途径之一的受激布里渊散射（Stimulated Brillouin Scattering, SBS）是一种高能激光场与入射介质中的声波场相互作用引起的三阶非线性声子散射效应。基于该效应的受激布里渊散射相位共轭镜（SBS-PCM）能够在一定程度上补偿高功率激光的波前畸变问题，实现高光束质量的空间强度分布输出；此外，其饱和增益特性以及弛豫耦合特性使之能够实现光束在空间、时域和频域的整形，因此在工业和科学研究中得到了广泛的应用[9-11]。在常用的SBS-PCM介质中，固体介质材料低损伤阈值特性易产生不可逆损伤；气体介质实验结构复杂、成本高等缺点，因此常采用液体材料作为SBS-PCM介质[12-13]。
虽然液体SBS-PCM具有能量转换率高、可以有效补偿高功率激光束波前畸变提高光束质量等优良特性，但随着激光功率的提升，仍面临着光束畸变和光学击穿等问题[14]，其原因是液体介质中存在不可避免的热量积累。具体来说，当高功率泵浦光注入介质池进行增益过程时，部分能量转化为热量被SBS介质吸收，在介质池内诱发形成自然对流，导致光斑畸变，输出光束质量严重下降；当热积累效应大于扩散效应时，温度达到临界阈值则发生光学击穿。这对SBS-PCM能量反射效率和空间传输稳定性等造成严峻挑战[15]。
为应对液体介质热效应引发的光束畸变和光学击穿等问题，已有部分学者提出了一些改进措施。Yoshida等[16]采用净化液体介质，提高介质纯度的方法使光学击穿阈值得到有效提升。Hasi等[17]采用限制液体介质内杂质体积的方法，减少了介质光学击穿的发生。Kang等[18]则在使用超净池的同时，将杂质颗粒尺寸限制在40 nm以下，结果表明，在500 Hz脉冲重复频率下，SBS-PCM的反射率达92%，且无光击穿现象和明显热效应。目前，杂质微粒的去除在实验中已成为必不可少的环节，此外通过改变系统结构也能在一定程度上缓解热效应。Wang等[19]采用旋转楔形板减小了高功率条件下的光学击穿和热效应的影响，实现了反射率的稳定。Kiriyama等[20]也在实验中采用旋转楔形板来减小热效应的影响。Yoshida等[21]利用循环介质池来抑制热量积累，但结构复杂，且液体介质流动易导致光束不稳定。
虽然采用净化介质、旋转楔形板、循环介质池等方法可以一定程度缓解液体池中的热积累，但介质中的热积累本质上是不可避免的，尤其是在更高功率泵浦下，上述缓解热效应方法已不足以应对，需要对介质池内热效应进行定量分析，为提出高功率泵浦条件下热效应的缓解方法提供理论指导。虽然目前已有部分研究工作针对介质热效应进行了研究 [15,18]，但仅局限于对焦点位置介质温度的定性分析。而Wang等[22]定量分析了焦点处介质的对流换热与导热的主导关系并通过理论计算得到热对流的流动状态为层流，但并没有展现及分析焦点处热对流的强度和流场分布，特别是热对流与泵浦光参数的关系还鲜有报道。因此，文中建立了SBS-PCM介质池内热对流数值模型，对高功率泵浦下的热对流特性进行了定量分析，研究了泵浦光重复频率对介质热对流参数关系，能够为SBS-PCM光束质量优化提供理论依据。
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SBS激光增益系统如图1(a)所示。由于重氟碳系列的电子氟化液具有较好的热稳定性和化学稳定性，其在常见的1 064 nm波段具有较低的吸收系数和较高的布里渊增益系数，因此采用目前广泛应用的FC-770作为SBS介质，如表1所示，其被封装于长为1 000 mm、内径为16 mm的介质池内。当高功率泵浦脉冲进入SBS-PCM介质池后，在声场作用下与Stokes脉冲发生耦合，峰值功率迅速提升，实现激光增益。由于介质内声波场的热耗散，光路附近的介质吸收热量，温度升高并形成不均匀的温度场，进而造成不同区域的介质出现密度差。不同密度的介质在重力的作用下形成浮升力，在浮升力的驱动下介质形成自然对流。由于焦点处激光能量强度最高，此处热效应最为明显，因此文中仅选取焦点所在截面作为研究对象，如图1(b)所示。此外，泵浦光源单脉冲能量采用25 mJ，重复频率采用10、50、100、250 Hz，脉冲宽度为7 ns。
图 1 (a) SBS-PCM激光增益系统；(b) 焦点截面示意图
Figure 1. (a) SBS-PCM laser gain system; (b) Schematic of focus section
表 1 介质物理性质
Table 1. Physical properties of medium
Medium Density/
kg·m−3Thermal conductivity/
W·m−1·K−1Specific heat capacity at constant pressure/
J·kg−1·K−1Dynamic viscosity/
Pa·sAbsorption coefficient/
cm−1FC-770 1789 0.063 1 038 0.001 3 0.001 1 -
描述焦点截面处介质自然对流换热过程的控制方程包括连续性方程、动量方程和能量方程。
连续性方程：
$$ \frac{{\partial (\rho u)}}{{\partial x}} + \frac{{\partial (\rho v)}}{{\partial y}} = 0 $$ (1) 式中：u、v分别为x、y两个方向上的速度；ρ为流体密度。
动量方程：
$$ \rho \left(\frac{{\partial u}}{{\partial \tau }} + u\frac{{\partial u}}{{\partial x}} + v\frac{{\partial u}}{{\partial y}}\right) = \eta \left(\frac{{{\partial ^2}u}}{{\partial {x^2}}} + \frac{{{\partial ^2}u}}{{\partial {y^2}}}\right) - \frac{{\partial p}}{{\partial x}} $$ (2) $$ \rho \left(\frac{{\partial v}}{{\partial \tau }} + u\frac{{\partial v}}{{\partial x}} + v\frac{{\partial v}}{{\partial y}}\right) = \eta \left(\frac{{{\partial ^2}v}}{{\partial {x^2}}} + \frac{{{\partial ^2}v}}{{\partial {y^2}}}\right) - \frac{{\partial p}}{{\partial y}} - \rho g $$ (3) 式中：τ为时间；η为动力粘度。
能量方程：
$$ \rho {c_p}\left(\frac{{\partial T}}{{\partial \tau }} + u\frac{{\partial T}}{{\partial x}} + v\frac{{\partial T}}{y}\right) = \lambda \left(\frac{{{\partial ^2}T}}{{\partial {x^2}}} + \frac{{{\partial ^2}T}}{{\partial {y^2}}}\right) + {{\dot \varPhi }}(x,y) $$ (4) 式中：cp为流体定压比热容；T为热力学温度；λ为导热系数；$ {{\dot \varPhi }}(x,y) $为内热源强度。
文中用内热源模拟空间强度分布为高斯分布的泵浦光，实验中采集的焦点处光斑为明显的椭圆形高斯光斑，如图2(a)所示。因此，数值模拟中的内热源强度分布为接近泵浦光的高斯分布，如图2(b)所示，且内热源的加载频率与泵浦光频率保持一致。内热源强度分布$ {{\dot \varPhi }}(x,y) $的表达式如下：
$$ {{\dot \varPhi }}(x,y) = \iint {\frac{P}{{\pi {r^2}(1 - {{\rm{{e}}}^{ - 1}})}} \cdot \exp \left[ { - \left(\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}}\right)} \right]{\rm{d}}x{\rm{d}}y} $$ (5) 式中：P为泵浦脉冲平均功率；a=0.020 8 mm与b=0.029 4 mm分别为椭圆形高斯光斑短轴与长轴；r=0.029 4 mm为椭圆形高斯光斑平均直径。
设定计算域初始条件为300 K，边界条件为绝热壁面。由于介质池外壁面裸露在空气中，理论上与外界空气存在对流换热与辐射换热。随着泵浦光的载入，虽然焦点处温度上升，但内壁面周围介质温升较小，且玻璃的导热性较差，导致介质池外壁面温度基本接近外界环境温度，此时对流换热与辐射换热量很小，故在此忽略外壁面换热，将外壁面简化为绝热壁面，此时计算域与外界没有热量交换。
图 2 (a) 实验采集的焦点处高斯光斑分布图；(b) 数值模拟的焦点处高斯内热源分布示意图
Figure 2. (a) The distribution of Gaussian spot at the focal point collected by the experiment; (b) The distribution diagram of Gaussian internal heat source at the focal point of numerical simulation
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引入无量纲数Se数计算涡通量来量化介质池内热对流强度。涡通量的强弱反映了介质池中热对流的强度，其表达式如下：
$$ S e = \frac{{\rho {D^2}_{{\text{in}}}{J^n}_{{\text{ABS}}}}}{\eta } $$ (6) $$ {J^n}_{{\rm{ABS}}} = \frac{1}{A}\iint\limits_A {\left| {{\omega _n}} \right|}{\rm{d}}A $$ (7) $$ {\omega _{{n}}} = \frac{{\partial u}}{{\partial z}} - \frac{{\partial w}}{{\partial x}} $$ (8) 式中：${D_{{\text{in}}}}$为介质池内径；${J^n}_{{\text{ABS}}}$为绝对涡通量；$A$为介质池截面积；${\omega _{{n}}}$为沿主流方向涡通量的分量。
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采用有限体积法对上述控制方程等进行计算求解，计算至物理时间40 s。为排除网格数量引起的误差，将模型分为6组网格，使用FC-770介质在25 mJ，10 Hz工况下计算介质池内的Se数。得到Se数随网格数的变化，如图3所示。当网格数为7万时，Se数基本不随网格数明显变化，因此选择网格数量为7万进行后续的数值模拟研究。
图 3 Se数随网格数量变化图
Figure 3. Se number changes with the number of grids
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图4所示为不同重复频率下Se数随相互作用时间的变化曲线。不同重复频率下，自然对流强度都呈现出先增大后减小，最后趋于稳定的趋势。这是由于随着泵浦光的注入，焦点处温度不断上升，介质池内密度梯度不断增大导致浮升力增大，使得自然对流强度增加。与此同时，随着自然对流的不断增强，介质池内的流动使冷热介质不断掺混，导致介质池内密度梯度下降，浮升力下降，致使自然对流强度下降，最后冷热流体充分掺混，介质密度梯度达到平衡，自然对流强度趋于稳定。
图 4 Se数随相互作用时间变化
Figure 4. Se number varies with the interaction time
对比不同重复频率的热对流强度时间演化规律，热对流强度极值和稳定值与重复频率呈正相关，达到极值和稳定值的时间与重复频率呈负相关。具体表现为当重复频率由10 Hz增大至250 Hz时，热对流强度极值由10增至49，稳定值由6增至31，其原因是随着重复频率的提升，相同时间内注入的泵浦能量越多，使介质所能达到的温度更高，进而有更大的密度梯度，形成的热对流强度也更大。当重复频率由10 Hz增大至250 Hz时，热对流强度到达极值所用时间由9 s降至3 s，是由于重复频率的提升使介质升温速率更快，介质达到更高温度所需时间更短，热对流强度能更快到达极值；到达稳定值所用时间由37 s降至19 s，则是因为更大的热对流强度使得冷热介质掺混越剧烈，介质池内密度梯度更快达到平衡，热对流趋于稳定所需时间越短。
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在相互作用时间为40 s时，不同重复频率下热对流速度、密度分布云图与对应的实验观测光斑如图5~图8所示。对比热对流速度、密度分布与光斑的纵向变形不难发现，在低重复频率下，介质池中心速度较小，热流体向上流动较为缓慢，介质池上部低密度区域分布较小，对光的折射影响较小，对应于低重复频率下，光斑仅有向上部的小部分变形偏移。随着重复频率的提升，中心流体流速逐渐升高，介质池上部低密度区域分布变大且密度更低，导致光斑的纵向变形偏移程度更大。对于光斑的横向变形，低重复频率时流向上方的热流体产生堆积，并由于结构限制沿介质池壁面向两侧流动，形成反向流，此时这部分流体温度下降，密度得到回升，所以光斑的横向变形并不明显。而随着重复频率的提升，反向流密度相较于低重复频率时更小，所以光斑在横向上有略微延展变形，同时光斑外侧轮廓部分也逐渐发生折射而扩散。所以，正是由于介质池内温度不均匀所产生的热对流导致了焦点处光斑的变形偏移，且随着重复频率的提升，热对流强度增强，光斑偏移程度越大。
图 5 (a) 速度云图；(b) 密度云图；(c) 光斑图(10 Hz)
Figure 5. (a) Velocity contour; (b) Density contour; (c) Spatial profiles (10 Hz)
图 8 (a) 速度云图；(b) 密度云图；(c) 光斑图 (250 Hz)
Figure 8. (a) Velocity contour; (b) Density contour; (c) Spatial profiles (250 Hz)
图 6 (a) 速度云图；(b) 密度云图；(c) 光斑图 (50 Hz)
Figure 6. (a) Velocity contour; (b) Density contour; (c) Spatial profiles (50 Hz)
图 7 (a) 速度云图；(b) 密度云图；(c) 光斑图 (100 Hz)
Figure 7. (a) Velocity contour; (b) Density contour; (c) Spatial profiles (100 Hz)
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针对高能量高重复频率注入时SBS-PCM中存在的光斑畸变问题，文中对焦点截面处的热对流进行了数值模拟，定量分析了热对流强度随相互作用时间、重复频率的变化规律，解释并分析了热对流强度对光斑偏移程度的影响。数值模拟结果表明：1)热对流强度随相互作用时间的发展趋势呈现出先增大后减小，最后趋于稳定的规律，且热对流强度与重复频率呈正相关；2)热对流是造成光斑发生变形偏移的重要原因，随着热对流强度的增加光斑变形偏移程度越大。综上所述，相较于以往对热对流的定性说明，文中补充了对热对流强度的定量分析，解释了热对流强度对光斑偏移程度的影响，对通过缓解热对流强度等方式进一步提高SBS-PCM输出质量具有重要意义。后续工作将从泵浦能量对热对流强度分布的影响以及热对流强度引发光斑偏移的具体阈值展开研究，建立更为完整的光热效应模型。此外，目前介质池为水平放置，为缓解热效应，后续也将建立三维模型，模拟介质池竖直倾斜一定角度时的热对流情况，此时热对流仅沿竖直方向运动，与光斑偏移方向不重合，热对流对光斑偏移影响可能较小。
Effect of pump light repetition rate on thermal convection characteristics in liquid SBS-PCM (invited)
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摘要: 受激布里渊散射相位共轭镜（SBS-PCM）因能实时补偿静态和动态波前畸变、提高光束质量，在激光领域受到广泛关注，但仍存在高功率泵浦下引发损伤和输出光束质量下降的问题。液体增益介质具有高增益、高抗损伤阈值和尺寸拓展性强的特点，目前是高能高功率激光领域最广泛应用的SBS介质，但随着注入功率的提升，热效应引发的液体介质热对流会导致反射Stokes光中出现波前畸变，降低了其光束质量补偿效果。文中发展了高功率泵浦下介质池内热对流的数值模型，定量分析了热对流强度随相互作用时间的变化规律，着重探讨了泵浦光重复频率对热对流强度分布的影响，并结合热对流强度解释了光斑畸变程度。研究结果表明：泵浦光注入初期，热对流强度在达到极值后小幅下降最后趋于稳定；泵浦光重复频率是影响热对流强度的重要因素，热对流强度与重复频率呈正相关；随着热对流强度的增强，光斑偏移程度逐渐增大。文中从液体介质流动性角度分析了泵浦光重复频率与介质热对流的关系，对完善光热效应模型提供了新的研究方向。Abstract:
Objective The stimulated Brillouin scattering phase conjugated mirror (SBS-PCM) has garnered significant attention in the laser field due to its ability to compensate for both static and dynamic wavefront distortion in real time and enhance beam quality. However, there remain concerns regarding optical breakdown and degradation of output beam quality under high power pumping. Liquid gain medium is currently the most widely used SBS medium due to its characteristics of high gain, high damage threshold resistance and strong size expansion. However, with the increase of injection power, thermal convection caused by absorption of liquid medium will cause wavefront distortion in reflected Stokes light, resulting in reductions of beam quality. Methods The finite element method was involved, and the 2-dimentional thermal convection at the focus section was solved by coupling the continuity equation, momentum equation, energy equation and the internal heat source equation. The boundary condition was adiabatic, and the numerical model of thermal convection in the medium cell under high power pump was developed. The dimensionless Se number is introduced to calculate the eddy flux to quantify the thermal convection intensity in the medium cell. Results and Discussions The variation of the Se number with the interaction time is quantitatively analyzed, and the influence of the pump light repetition rate on the thermal convection intensity distribution is emphatically discussed. The results show that, starting from the pump light injection medium, the Se number firstly increases and then decreases, and finally tends to be stable. In addition, when the repetition rate increases from 10 Hz to 250 Hz,the maximum Se number increases from 10 to 49, and the stable Se number increases from 6 to 31, but the time taken for the Se number to reach the maximum value decreases from 9 s to 3 s. The time taken to reach the stable value is reduced from 37 s to 19 s (Fig.4). The contour of thermal convection velocity and density distribution and corresponding experimental observed spatial profiles at different repetition rates were shown (Fig.5-8). With the increase of repetition rate, the intensity of thermal convection increases, and the distribution of low-density areas in the medium cell expands, leading to the increase of the horizontal and vertical deformation of light spots. Conclusions The relationship between pump light repetition rate and thermal convection in liquid medium is analyzed from liquid medium flow. The pump light repetition rate is an important factor affecting the thermal convection intensity, the thermal convection intensity is positively correlated to the repetition rate, and the time for the thermal convection intensity to reach the extreme value and the stable value is negatively correlated with the repetition frequency. With the increase of thermal convection intensity, the degree of spot migration increases gradually. This study provides a new perspective for perfecting the model of photothermal effect. -
图 1 (a) SBS-PCM激光增益系统；(b) 焦点截面示意图
Figure 1. (a) SBS-PCM laser gain system; (b) Schematic of focus section
图 2 (a) 实验采集的焦点处高斯光斑分布图；(b) 数值模拟的焦点处高斯内热源分布示意图
Figure 2. (a) The distribution of Gaussian spot at the focal point collected by the experiment; (b) The distribution diagram of Gaussian internal heat source at the focal point of numerical simulation
图 5 (a) 速度云图；(b) 密度云图；(c) 光斑图(10 Hz)
Figure 5. (a) Velocity contour; (b) Density contour; (c) Spatial profiles (10 Hz)
图 8 (a) 速度云图；(b) 密度云图；(c) 光斑图 (250 Hz)
Figure 8. (a) Velocity contour; (b) Density contour; (c) Spatial profiles (250 Hz)
图 6 (a) 速度云图；(b) 密度云图；(c) 光斑图 (50 Hz)
Figure 6. (a) Velocity contour; (b) Density contour; (c) Spatial profiles (50 Hz)
图 7 (a) 速度云图；(b) 密度云图；(c) 光斑图 (100 Hz)
Figure 7. (a) Velocity contour; (b) Density contour; (c) Spatial profiles (100 Hz)
表 1 介质物理性质
Table 1. Physical properties of medium
Medium Density/
kg·m−3Thermal conductivity/
W·m−1·K−1Specific heat capacity at constant pressure/
J·kg−1·K−1Dynamic viscosity/
Pa·sAbsorption coefficient/
cm−1FC-770 1789 0.063 1 038 0.001 3 0.001 1 
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