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求解高功率密度连续激光作用下光学元件的温度场是研究热效应的关键。通过温度场,便可求出由温度变化引起的折射率变化;通过基于温度场求解出的热弹场,便可求出由元件变形引起的波前畸变和由热应力引起的应力双折射[10]。
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高功率连续激光作用下,光学元件所反映出的光学、热学和力学特征将会存在非线性温度效应,因此传统基于线性模型的光吸收、热传导、热弹耦合场的偏微分方程都将不再适用。对于基于线性的温度场和热弹场问题,已有不少解析解推导和有限元计算的研究[11-15],且相对较为成熟。但基于非线性的问题,温度场除极少数的情形外[16],几乎求不出解析解,求出热弹场的解析解更是难上加难。所以本文中也通过有限元法来求解。
一般来说,温度场和热弹场的非线性效应主要来源于3个方面:物性参数、热辐射边界条件和光学吸收系数随温度的变化。
以熔石英玻璃为例, 其热导率、密度、比热容、杨氏模量、泊松比和线性热膨胀系数随温度的变化曲线如图 1所示[17-18]。以该材料的热导率为例,它随温度会呈现先缓慢增长后迅速增长的变化趋势。简单来说,是因为总的热导率是传导热导率和辐射热导率之和,温度越高,辐射热导率的贡献越来越大,所以总热导率随温度的增长会越来越快[18]。毫无疑问,各参数复杂的变化趋势会导致温度场和热弹场的求解结果难以预测。
对于热辐射边界条件,即边界热流与温度的四次方成正比,是一种高度非线性的边界条件。显然,温度越高,该非线性因素越不可忽略。
除物性参数外,已有的研究表明[17, 19],光学吸收率[20-21]也会随着光学元件温度下的变化而变化。现假设吸收率α随温度T的变化关系为:
$ α=n+mT $
(1) 式中,n为室温下的吸收率,m为吸收率随温度变化的系数。n和m与波长、光学元件材料等相关,具体取值由实测得到。
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对于温度变化不大的热传导问题,温度场可用一般的温度控制方程(含热源,常物性的均匀各向同性物体区域的热传导微分方程)及其相应条件求解得到,求解的是线性问题。线性模型是忽略了温度及其各类导数的一次方以上项的结果。当温度变化较大时,将导致相应的数学模型为非线性的,或者泛定方程为非线性,或若干定解条件为非线性。基于非线性的定解问题的泛定方程为[16]:
$ \nabla \cdot[\kappa(T) \nabla T]+g(r, t)=\rho(T) c_p(T) \frac{\partial T}{\partial t} $
(2) 式中,$\nabla$为算子, κ(T), ρ(T), cp(T)分别为材料热导率、密度和定压比热容,它们都是温度T的函数,g(r, t)为热源项, r为半径, t为时间。
边界条件考虑对流换热和热辐射:
$ \pm \kappa(T) \frac{\partial T}{\partial s}=h\left(T^{\prime}-T\right)+\varepsilon \sigma\left(T^{\prime 4}-T^4\right) $
(3) 式中,h为换热系数,ε为史蒂芬-玻尔兹曼常数,σ为辐射系数,T′为环境温度, ∂T/∂s为温度沿界面外法线方向的导数。
对于高平均功率连续激光辐照光学元件的热弹场问题,依旧可近似认为具有变形较小、温度变化缓慢的特征。变形较小,也就意味着在推导热弹性运动方程的过程中可以略去非线性项(应变表达式中位移的非线性项和本构方程中的非线性项);温度变化缓慢,也就意味着可以略去热弹性运动方程中的动力项(即位移随时间的2阶导数项)和热弹性材料的热传导方程中的耦合项,这样温度场就可以通过求解前面讨论过的热传导方程来得到,位移场可以通过求解如下拟静态的热弹性运动方程来得到[22-24]。
热弹性运动方程的矢量形式为:
$ \begin{gathered} {[1-2 \nu(T)] \nabla^2 \boldsymbol{u}+\nabla(\nabla \cdot \boldsymbol{u})=} \\ 2[1+\nu(T)] \alpha_{\mathrm{th}}(T) \nabla T \end{gathered} $
(4) 或写成柱坐标系下的标量形式:
$ \left\{\begin{array}{l} \nabla^2 u_r-\frac{u_r}{r^2}-\frac{2}{r^2} \frac{\partial u_\theta}{\partial \theta}+\frac{1}{1-2 \nu} \frac{\partial e}{\partial r}-\frac{2[1+\nu(T)]}{1-2 \nu(T)} \alpha_{\mathrm{th}}(T) \frac{\partial T}{\partial r}=0 \\ \nabla^2 u_\theta-\frac{u_\theta}{r^2}+\frac{2}{r^2} \frac{\partial u_r}{\partial \theta}+\frac{1}{1-2 \nu(T)} \frac{1}{r} \frac{\partial e}{\partial \theta}-\frac{2[1+\nu(T)]}{1-2 \nu(T)} \alpha_{\mathrm{th}}(T) \frac{1}{r} \frac{\partial T}{\partial \theta}=0 \\ \nabla^2 u_z+\frac{1}{1-2 \nu(T)} \frac{\partial e}{\partial z}-\frac{2[1+\nu(T)]}{1-2 \nu(T)} \alpha_{\mathrm{th}}(T) \frac{\partial T}{\partial z}=0 \end{array}\right. $
(5) $ e=\frac{\partial u_r}{\partial r}+\frac{1}{r} \frac{\partial u_\theta}{\partial \theta}+\frac{u_r}{r}+\frac{\partial u_z}{\partial z} $
(6) 式中,u为样品变形位移矢量,ur为径向分量,uθ为环向分量,uz为轴向分量,ν(T)为泊松比,αth(T)为样品线性膨胀系数,T(r, θ, z, t)为样品的温度分布。如果是轴对称问题,则:uθ=0, ur=ur(r, z), uz=uz(r, z)。
高功率激光作用下光学元件非线性热效应研究
Nonlinear thermal effects of optical components irradiated by high-power laser beam
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摘要: 为了研究高功率激光作用下光学元件热效应的非线性特性对光束质量的影响, 基于热传导、热弹性力学、物理光学等基础理论, 采用有限元分析方法, 进行了高功率连续激光(功率密度约为500kW/cm2)作用下光学元件的温度场和位移场的计算; 分析对比了各个参数的非线性特性对光学元件热效应的影响; 探讨了不同材料、不同形状光斑辐照下的光学元件非线性热效应。结果表明, 高功率连续激光作用下光学元件所反映出的热学、力学和光学吸收存在着不同程度的非线性效应, 其强弱取决于光学元件材料、光斑形状等因素; 高斯激光辐照熔石英样品且吸收率为100×10-6时, 考虑物性参数和温度边界条件的非线性, 会引起表面最大温升16%的相对误差和表面变形峰谷值10%的相对误差。这一结果为开展与此相关的研究提供了一些新思路。Abstract: In order to investigate the influence of the nonlinear characteristics of thermal effects of optical components on the beam quality irradiated by high-power laser beam, based on the basic theories of heat conduction, thermoelasticity, and physical optics, etc., with the help of finite element analysis, the temperature field and displacement field of optical components irradiated by high-power continuous-wave laser (the power density is approximately 500kW/cm2) were presented. The influences of various parameters on the thermal effects of optical components were analyzed and compared. Moreover, the nonlinear thermal effects under different conditions were discussed. The results show that the thermal, mechanical, and optical absorption of optical components irradiated by high-power continuous-wave laser present nonlinear effects, and the strengths of these nonlinear effects depend on the materials of the optical components, and the spot shape of the laser beam. When fused silica sample is irradiated by Gaussian laser and the absorption rate is 100×10-6, the nonlinearity without the physical parameters and temperature boundary condition will cause 16% and 10% relative errors for the maximum surface temperature rise and the peak-to-valley surface deformation, respectively. The results presented in this paper is expected to provide some new clues for the related research.
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Key words:
- laser optics /
- nonlinear thermal effect /
- finite element method /
- high-power laser /
- beam quality
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