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由质量因子定义可知,激光光束质量因子M2是实际光束束腰宽度和光束远场发散角的乘积比上理想高斯光束的束腰宽度和光束远场发散角的乘积来确定的。对于基模高斯光束,M2的值为1, 但是在实际测量中,无法得到由其定义得到的值,目前常用的方法是首先对光束宽度进行测量, 然后根据得到的数据进行双曲线拟合, 最后由双曲线拟合曲线来获取光束质量因子。但由于飞秒激光光束具有独特的性质,不同于普通激光光束质量的测量,所以用光束能量密度与光斑宽度的关系来检测飞秒激光质量因子是目前比较可信的测量方法之一。由照射到屏幕上的光束能量密度和光束半径的关系, 可以获得沿光束轴的光束半径。
根据Siegman定义, 飞秒激光光束质量因子可表示为:
$ {M^2} = \frac{{{w_{\rm{r}}}(z)\mathit{\Theta }}}{{{w_{\rm{i}}}(z)\theta }} = \frac{{\rm{ \mathsf{ π} }}}{{4\lambda }} \cdot {w_{\rm{r}}}(z)\mathit{\Theta } $
(1) 式中, wr(z), wi(z), Θ, θ分别为实际高斯光束半径、基模高斯光束半径、实际光束发散角和理想光束发散角。经过凸透镜的高斯光束沿光轴的光束模示意图如图 1所示。
沿光轴的高斯光束表达式为:
$ \begin{array}{l} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;E\left( {r, z} \right) = \frac{{{w_0}}}{{w(z)}} \times {\rm{ }}\\ {\rm{exp}}\left\{ { - {\rm{i}}kz + {\rm{i}}\mathit{\Phi }\left( z \right) - {r^2}\left[ {\frac{1}{{{w^2}(z)}} + \frac{{{\rm{i}}k}}{{2R(z)}}} \right]} \right\} \end{array} $
(2) 式中, r, z, w0, k, Φ(z), w(z), R(z)分别表示为径向半径、光束轴、光束腰半径、波数、相位变化、光束轴z处的光束腰半径和波前曲率半径。其中理想光束腰半径与实际光束腰半径分别表示为:
$ {w_{\rm{i}}}(z) = {w_0}{\left[ {1 + {{\left( {\frac{{\lambda z}}{{{\rm{ \mathsf{ π} }}{w_0}^2}}} \right)}^2}} \right]^{\frac{1}{2}}} $
(3) $ {w_{\rm{r}}}(z) = {w_{{\rm{0, r}}}}{\left[ {1 + {{\left( {\frac{{{M^2}\lambda z}}{{{\rm{ \mathsf{ π} }}{w_{{\rm{0, r}}}}^2}}} \right)}^2}} \right]^{\frac{1}{2}}} $
(4) 沿光束轴的能量密度分布为:
$ \begin{array}{l} \;\;\;\;F\left( r \right) = \frac{{2{E_\rm{p}}{k^2}}}{{\rm{ \mathsf{ π} }}} \times {\rm{ }}\\ \int_0^{a{w_0}} {\mathit{\rho }{\rm{exp}}} ( - {\mathit{\rho }^2}){{\rm{J}}_0}(\mathit{kr\rho }){\rm{d}}\mathit{\rho } \end{array} $
(5) 式中, Ep是总脉冲能量,波数k=2πw0/(fλ),a是光束最小位置的截断孔径,J0是零阶贝塞尔函数, ρ是脉冲中心的径向距离,f是脉冲重复频率。高斯光束在光束腰半径为w(z)位置处的脉冲能量密度为:
$ F\left( r \right) = {F_{\rm{p}}}{\rm{exp}}\left[ { - \frac{{{r^2}}}{{{w^2}(z)}}} \right] $
(6) 式中,w(z)为沿光束轴z处的光束腰半径。
$ {F_{\rm{p}}} = \frac{{2{E_{\rm{p}}}}}{{{\rm{ \mathsf{ π} }}{w^2}(z)}} $
(7) 式中, Fp为峰值能量密度,r为径向半径,且代用r=D/2,由(6)式和(7)式可得:
$ {D^2} = 2{w^2}\left( z \right){\rm{ln}}\left( {\frac{{{F_{\rm{p}}}}}{{{F_{{\rm{th}}}}}}} \right) = 2{w^2}\left( z \right){\rm{ln}}\left( {\frac{{{E_{\rm{p}}}}}{{{E_{{\rm{th}}}}}}} \right) $
(8) 式中,D为光斑直径,Eth为阈值能量,由(8)式得:
$ w\left( z \right) = D{\left[ {2{\rm{ln}}\left( {\frac{{{E_{\rm{p}}}}}{{{E_{{\rm{th}}}}}}} \right)} \right]^{ - \frac{1}{2}}} $
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本文中运用的实验系统由飞秒激光器、凸透镜、滑动平台、介电材料玻璃屏幕所组成的,如图 2所示。实验设计中用焦距为100mm凸透镜对飞秒激光光束进行聚焦变换,聚焦过程中基于光斑宽度的大小选出3个光斑点对其宽度进行测量。测量步骤如下:首先将屏幕放入透镜后面使固定不变,然后在光束沿线轴上,将透镜慢慢向屏幕方向移动, 透镜和屏幕间距恰好等于焦距时, 第1次在屏幕上出现的光斑宽度用光学显微镜放大,并进行尺寸测量。光斑宽度最小时对光斑宽度进行第2次测量,超越此位置时进行第3次测量(与第1次测量方法相同)。实验中透镜焦点和超越光斑宽度最小位置的测量目的就是为了确认光束腰,因为光束腰位置是高斯光束绝对平行传输的地方,光斑宽度比其它两个位置的光斑宽度小、光强最大及明显可以保证光束质量因子的精度。实验中测得的数据如表 1所示。不同测量点处的x,y轴的测量值分别为Dx, Dy。
Table 1. Spot width of sample surface
Dx/mm 0.03645 0.02700 0.03649 Dy/mm 0.02360 0.01550 0.02355
基于高斯光束的飞秒激光光束质量检测技术
Detection technology of femtosecond laser beam quality based on Gaussian beam
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摘要: 为了研究飞秒激光光束质量因子M2基于高斯光束的传播特性和能量密度分布, 在对飞秒激光光束质量因子进行理论研究的基础上, 进行了相应的数据计算, 给出了飞秒激光脉冲照射屏幕表面时所形成的环形光斑宽度的测量方法, 设计了一套由飞秒激光器、透镜、介电材料玻璃屏幕所组成的实验平台; 将该方法与刀口法和CCD法测量值进行了对照, 并用刀口法、CCD法确认了飞秒激光束腰在不同位置时的激光光束质量因子取值范围。结果表明, 光束质量因子M2在x和y方向上的测量值分别为2.04, 1.24。该实验结果与理论分析基本一致, 比刀口法和CCD法结构简单, 所得结论数据可靠、执行方便, 对精密测量有一定的参考价值。Abstract: In order to study the propagation characteristics and energy density distribution of femtosecond laser beam quality factor M2 based on Gaussian beam, on the basis of theoretical study of femtosecond laser beam quality factor, the corresponding data were calculated. The measurement method of annular spot width formed by femtosecond laser pulse irradiating the screen surface was given. An experimental platform consisting of femtosecond laser, lens and dielectric glass screen was established. The method was compared with the knife-edge method and the CCD method.The range of laser beam quality factor at different positions of femtosecond laser beam waist was confirmed by knife-edge method and CCD method. The results show that the measured values of beam quality factor M2 in x and y directions are 2.04 and 1.24, respectively. The experimental results are basically consistent with the theoretical analysis. Compared with knife-edge method and CCD method, the structure is simple. The data obtained are reliable and easy to carry out. It has certain reference value for precise measurement.
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Key words:
- ultrafast optics /
- beam quality /
- quality factor M2 /
- beam waist /
- spot width /
- dielectric material screen
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Table 1. Spot width of sample surface
Dx/mm 0.03645 0.02700 0.03649 Dy/mm 0.02360 0.01550 0.02355 -
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