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辐照激光能量标定光路如图 2所示。激光束经分光镜A,一部分反射辐照在被测样品表面,另一部分透射到分光镜B,分光镜B将一部分激光能量反射到能量探测器供能量测量,另一部分透射供辐照光斑面积测量。
为了计算辐照在被测样品表面激光能量与探测器所测量激光能量之间的关系,设分光镜A和分光镜B的分光比(透射部分与反射部分能量之比,即P=T:R)分别为PA和PB,其中PT, A和PR, A为分光镜A透射部分和反射部分能量,PT, B和PR, B为分光镜B透射部分和反射部分能量,则:
$ {P_{\rm{A}}} = {P_{T, {\rm{A}}}}/{P_{R, {\rm{A}}}} $
(1) $ {P_{\rm{B}}} = {P_{T, {\rm{B}}}}/{P_{R, {\rm{B}}}} $
(2) $ {P_{T, {\rm{A}}}} = {P_{T, {\rm{B}}}}/{P_{R, {\rm{B}}}} $
(3) 根据(1)式、(2)式和(3)式可得:
$ {P_{R, {\rm{A}}}} = {P_{R, {\rm{B}}}}\left( {1 + {P_{\rm{B}}}} \right)/{P_{\rm{A}}} $
(4) 因此,已知两分光镜的分光比PA和PB,以及能量探测器示值PR, B,即可计算得到辐照在被测薄膜表面的激光能量PR, A。
以上能量探测器示值与被测样品表面能量值比值的获得是理论计算得到的,并没有考虑分光镜的分光误差和能量探测器的测量误差。因此,为了消除这部分误差需要对比例关系进行试验标定。具体标定方法为:采用两个能量探测器C和D,它们的示值分别用EC和ED表示,将其分别放置在能量探测器位置和被测样品位置附近,对激光器发出的同一个激光脉冲能量进行测量,则E1=EC, 1∶ED, 1;为了消除能量探测器本身的测量误差,将C和D的位置互换重新测量,则E2=ED, 2∶EC, 2。取两次测量的平均值作为比例关系的标定值,即:
$ E = \left( {{E_1} + {E_2}} \right)/2 $
(5) 通过以上方法,对能量探测器示值与被测样品表面辐照能量之间的实际比例关系进行标定,校正了分光镜的分光误差和能量探测器的测量误差。根据这一比例关系,通过能量探测器的示值PR, B,标定出每个激光脉冲作用时对应的被测样品表面的实际辐照能量。
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辐照光斑面积的标定包括光斑面积的准确测量和测量位置的标定两部分。激光光斑面积的准确测量对薄膜激光损伤阈值的测量精度具有重要意义,常用的测量方法中,相纸削波法和刀口扫描法的测量分辨率较低,线阵旋转扫描法只适用于连续激光的光斑测量,光纤探针扫描法不适用于大尺寸光斑的测量,因此电荷耦合器件(charge-coupled devices,CCD)测量法应用更广泛。
根据ISO11254-1.2的测试要求,当激光光束的空间分布稳定时,有:
$ {A_{{\rm{eff}}}} = \frac{Q}{{{H_{{\rm{max}}}}}} = \frac{{\int {\mathop {}\limits_{ - \infty }^\infty \int {\mathop {}\limits_{ - \infty }^\infty H\left( {{\rm{\backslash user}}1x,y} \right){\rm{d}}{x}{\rm{d}}{y}} } }}{{{H_{{\rm{max}}}}}} $
(6) 式中,Q为激光的总能量,Hmax为最大峰值能量密度,Aeff为光斑有效面积, H(x, y)为能量密度。
以上计算的前提是将激光能量等效为以最高峰值为顶点的均匀分布,该均匀分布的范围即为有效光斑面积[14]。CCD成像法是将被测激光光斑的全部能量直接辐射到CCD的光敏面上测量总能量Q;激光光斑的峰值能量密度Hmax是通过采集每个像元的最大峰值能量除以像元的面积得到的。根据(6)式可计算得到激光光斑的有效面积Aeff。
CCD的像素分辨率决定了光斑面积的精度,为避免高能量激光对CCD探测面的损伤,需要对光能量进行衰减。该方法对于空间能量非高斯型分布的激光光斑测量具有很高的准确性。本系统测试用激光能量服从平顶分布,因此将CCD探测器表面放置在被测样品表面等效位置,光敏面上获得光斑面积即为辐照在被测薄膜表面激光光斑面积。
CCD探测器表面非常敏感,对1064nm波长激光的典型饱和密度大约为9nJ/cm2,当用于面积测量的激光束直接辐照在其表面时,容易使CCD饱和,甚至超过其损伤阈值而导致CCD器件损坏,因此,需要对入射到CCD探测面上的光束进行衰减,直到CCD能够在线性区域范围内工作。测量时,环境的背景光波段较宽,而CCD对可见光的光谱响应灵敏度要比1064nm高得多,因而背景光容易使CCD处于饱和状态,影响测量的精度。为了解决以上问题,需要采用针对1064nm波长的窄带滤光片,主要透过1064nm波长的脉冲激光,滤除掉其它波长的光。测量原理如图 3所示。系统由衰减器、窄带滤光片、CCD探测器和上位机等组成,从分光镜B透射的激光束经衰减后再经窄带滤光片在光敏面上形成光斑,CCD输出的图像视频信号输入计算机供显示、存储、测量和分析。
为了准确测量激光光斑的面积,需要CCD在规定的时间开启快门以记录激光脉冲图像,因此需要解决脉冲同步问题。由于激光器的脉冲宽度只有几到十几纳秒,采用连续拍摄的方法无法捕捉到一个完整的脉冲光斑图像,这将导致光斑面积的测量产生较大的误差。为保证采集到的光斑图像是连续、完整的,CCD的电子快门需要采用外触发同步控制的方式。为保证CCD相机有效捕获激光脉冲信号,利用激光电源向脉冲激光二极管(laser diode,LD)提供抽运脉冲作为外同步触发信号,在激光电源向激光器提供抽运能量的同时也发出一个同步触发信号给CCD相机,从而实现激光脉冲的有效捕获。由于从抽运脉冲开始到输出调Q脉冲,持续时间取决于延时电路的延时时长,根据激光器的抽运脉冲宽度设置CCD相机的积分时间。
光斑面积测量光路中,CCD探测面位置与被测样品表面位置的等效性影响光斑面积的测量精度,进而影响损伤阈值的测量精度,因此需要对等效位置进行标定。具体方法为:将两个CCD相机1和2分别放置在样品测量光路和光斑面积测量光路中;将相机1上下移动,找到CCD探测面上光斑尺寸最小的位置,即聚焦光学系统的焦平面,测量此时光斑尺寸;将相机2左右移动,找到CCD探测面上光斑尺寸与相机1测量结果相同的位置;标定相机1此时的位置为被测样品位置,相机2此时的位置为光斑测量装置中与被测样品的等效位置。标定完成后,此次测量试验中,样品位置与光斑测量光路中CCD探测面位置等效,位置不再移动。
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根据辐照激光能量PR, A、测得的光斑面积S以及光束平顶部分所占能量比例η,可以计算得辐照的激光能量密度为:
$ F = \frac{{{P_{R, {\rm{A}}}}}}{S} \times \eta $
(7) 首先,计算辐照在样品表面的的激光能量,根据实际标定的能量探测器示值与辐照在样品表面的激光脉冲能量的比例关系,标定出此时辐照在样品表面的激光脉冲能量。辐照到样品表面的激光脉冲能量中只有比例为η的平顶部分能量,计算时只用平顶部分包络的激光脉冲能量值。然后,根据平顶激光束光斑面积的测量原理,通过精确同步控制被测样品表面与CCD探测面位置等效,测量激光光斑的直径,计算得到光斑面积。
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零几率损伤阈值的确定需要根据激光能量密度与损伤几率的关系建立坐标,横坐标表示激光能量密度,纵坐标表示每个能量密度所对应的损伤几率,对这些离散点进行拟合,该拟合曲线与横坐标的交点所对应的能量密度就是损伤阈值。常用的拟合方法有拉格朗日插值法、牛顿插值法、埃尔米特插值法和最小二乘法等,其中最小二乘法拟合是通过最小化误差的平方和以寻找数据的最佳函数匹配[15-16]。
给定数据点的坐标(xi, yi),其中i=1, 2, …, m。求解近似曲线y=p(x),并使得近似曲线与实际曲线y=f(x)的偏差最小。近似曲线在点(xi, yi)处的偏差为:
$ {\delta _i} = p\left( {{x_i}} \right) - {y_i}, \left( {i = 1, 2, \cdots , m} \right) $
(8) 当按照偏差平方和最小的原则选取拟合曲线时,有:
$ \mathop {{\rm{min}}}\limits_\mathit{\Phi } \sum\limits_{i = 1}^m {{\delta _i}^2 = {{\sum\limits_{i = 1}^m {\left[ {p\left( {{x_i}} \right) - {y_i}} \right]} }^2}} $
(9) 设Φ为次数不超过n(n≤m)的多项式所构成的函数类,现求$ {p_n}\left( x \right) = \sum\limits_{k = 0}^n {{a_k}{x^k} \in \mathit{\Phi }} $,使得:
$ I = \sum\limits_{i = 0}^m {{{\left[ {{p_n}\left( {{x_i}} \right) - {y_i}} \right]}^2} = \sum\limits_{i = 0}^m {{{\left( {\sum\limits_{k = 0}^n {{a_k}{x_i}^k - {y_i}} } \right)}^2}} } $
(10) 满足(10)式的p(x)为最小二乘拟合多项式,当n=1时,为线性拟合或直线拟合。
(10) 式为(a0, a1, …, an)的多元函数,其求解即为求I=I(a0, a1, …, an)的极值问题。根据多元函数求极值的必要条件,得:
$ \begin{array}{c} \frac{{\partial I}}{{\partial {a_j}}} = 2\sum\limits_{i = 0}^m {\left( {\sum\limits_{k = 0}^n {{a_k}{x_i}^k - {y_i}} } \right){x_i}^j = 0, } \\ \left( {j = 0, 1, \cdots , n} \right) \end{array} $
(11) 即:
$ \sum\limits_{k = 0}^n {\left( {\sum\limits_{i = 0}^m {{x_i}^{j + k}} } \right){a_k} = \sum\limits_{i = 0}^m {{x_i}^j{y_i}, \left( {j = 0, 1 \cdots , n} \right)} } $
(12) (12) 式为关于(a0,a1,…,an)的线性方程,用矩阵表示为:
$ \left[ {\begin{array}{*{20}{c}} {m + 1}&{\sum\limits_{i = 0}^m {{x_i}} }& \cdots &{\sum\limits_{i = 0}^m {{x_i}^n} }\\ {\sum\limits_{i = 0}^m {{x_i}} }&{\sum\limits_{i = 0}^m {{x_i}^2} }& \cdots &{\sum\limits_{i = 0}^m {{x_i}^{n + 1}} }\\ \vdots & \vdots &{}& \vdots \\ {\sum\limits_{i = 0}^m {{x_i}^n} }&{\sum\limits_{i = 0}^m {{x_i}^{n + 1}} }& \cdots &{\sum\limits_{i = 0}^m {{x_i}^{2n}} } \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{a_0}}\\ {{a_1}}\\ \vdots \\ {{a_n}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {\sum\limits_{i = 1}^m {{y_i}} }\\ {\sum\limits_{i = 0}^m {{x_i}{y_i}} }\\ \vdots \\ {\sum\limits_{i = 0}^m {{x_i}^n{y_i}} } \end{array}} \right] $
(13) 方程组(13)的系数矩阵为对称正定矩阵,故存在唯一解,解出ak(k=0,1,…,n),可得多项式:
$ {p_n}\left( x \right) = \sum\limits_{k = 0}^n {{a_k}{x^k}} $
(14) 在实验测试中,有时采用线性拟合可以合理地得到损伤阈值,但大多数测试采用线性拟合得到的损伤阈值结果与实际阈值偏差较大,因此,采用非线性拟合更接近实际的阈值[17]。
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损伤阈值的计算流程如图 4所示。输入计算得到的激光能量密度值与其对应的损伤几率,并对损伤几率非零非百的有效数据点进行最小二乘法拟合,求取曲线方程为:
$ y = {a_0} + {a_1}x + \cdots + {a_k}{x^k} $
(15) 式中,x为能量密度,y为损伤几率。当y=0时对应的x值即为损伤阈值。
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实验中采用的样品为电子束蒸发技术制备的TiO2/SiO2高反射膜,采用1064nm波长激光辐照,测得光斑直径为755μm,采用上述方法得到光斑面积为4.48×10-3cm2,并计算出每次辐照的激光能量密度。表 1中为辐照激光能量密度与损伤几率之间对应的测量数据。*为拟合可用数据。
Table 1. Measurement data of antireflection film
number energy density/(J · cm-2) damage probability/% 1 2.25 0 2 44.36 100 3 23.3 0 4* 33.87 90 5* 33.08 90 6* 32.16 80 7* 31.54 75 8* 30.49 65 9* 29.02 45 10* 28.11 45 11* 27.15 30 12* 26.38 25 13* 25.47 15 14* 24.66 15 15* 23.42 5 根据以上测量数据,采用最小二乘法对测量结果进行拟合得损伤阈值为23.0164J/cm2,如图 5所示。
若未进行上述标定,则会引入一系列影响测量精度的因素[18-19]。其中,两分光镜的分光误差为±0.5%、能量探测器的测量误差为±5%、光斑非平顶部分占比为7.7%、光斑测量等效位置误差为1%。根据方和根法,对以上误差合成:
$ \sqrt {{{0.5}^2} + {{0.5}^2} + {5^2} + {{7.7}^2} + {1^2}} {\rm{\% = }}9.26{\rm{\% }} $
(16) 因此,薄膜激光损伤阈值标定技术使测量精度提高了9.26%。
薄膜激光损伤阈值标定技术
Calibration technology of laser-induced damage threshold of thin films
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摘要: 为了满足薄膜激光损伤阈值客观、准确、高精度测量的要求, 提出了损伤阈值标定技术。通过互换两个能量探测器位置的试验标定方法, 消除分光镜的分光误差和能量探测器的测量误差, 获得准确的辐照能量。再通过调整两个CCD位置获得相同光斑尺寸的测量方法标定被测样品表面与光斑面积测量面的等效, 并剔除激光光斑中非平顶部分, 获得准确的辐照光斑面积。最后采用最小二乘法对计算得到的能量密度及其对应的损伤几率进行拟合, 获得损伤阈值。通过对TiO2/SiO2高反射膜1064nm激光辐照测量实验, 得到23.0164J/cm2的薄膜激光损伤阈值。结果表明, 采用标定技术使薄膜激光损伤阈值的测量精度提高了9.26%, 满足高精度的测量要求。此研究有助于薄膜激光损伤阈值的准确标定。Abstract: In order to meet the requirements of objective, accurate, and high-precision measurement of laser damage threshold of thin films, the calibration technology of damage threshold was proposed. Firstly, the experimental calibration method of exchanging the positions of the two energy detectors was used to eliminate the error of the spectroscope and the measurement of the energy detector, so as to obtain the accurate irradiation energy. Then, the measurement method of adjusting the positions of two CCD devices to obtain the same spot size was used to calibrate the equivalence between the measured sample surface and the spot area measurement surface, and the non flat top part of the laser spot was eliminated, so as to obtain the accurate irradiation spot area. Finally, the least square method was used to fit the calculated energy density and the corresponding damage probability to obtain the damage threshold. The 1064nm laser irradiation experiment of TiO2/SiO2 high reflection film was carried out, and the laser damage threshold was 23.0164J/cm2. The results show that the measurement accuracy of laser damage threshold is increased by 9.26% by using calibration technology, which meets the high precision measurement requirements. This study is helpful for accurate calibration of laser damage threshold of thin films.
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Key words:
- thin films /
- damage threshold /
- calibration /
- laser /
- fitting
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Table 1. Measurement data of antireflection film
number energy density/(J · cm-2) damage probability/% 1 2.25 0 2 44.36 100 3 23.3 0 4* 33.87 90 5* 33.08 90 6* 32.16 80 7* 31.54 75 8* 30.49 65 9* 29.02 45 10* 28.11 45 11* 27.15 30 12* 26.38 25 13* 25.47 15 14* 24.66 15 15* 23.42 5 -
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