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目前,评估准分子激光的脉冲能量稳定性主要用σ参数来进行判断。σ参数通常定义为在固定时间窗内脉冲能量偏离脉冲能量均值的程度[10-11]。脉冲能量稳定性是衡量准分子激光光源应用特性的重要指标。然而,目前国内外对激光合束后的脉冲能量稳定性的评估与优化方法仍然缺乏系统的理论研究。因此,通过对激光脉冲能量稳定性进行理论分析,研究激光合束后准分子激光的σ参数,进一步探索对于紫外准分子激光脉冲能量稳定性的评估与优化方法。
准分子激光脉冲能量稳定性是指在一定时间段内,激光器输出脉冲能量偏离均值的情况。根据Pflanz T的研究[12],理想情况下,准分子激光脉冲输出能量应该符合正态分布,即Ej~N (Ei, σi2)。
$$ {E}_{i}=\frac{1}{n}\sum _{j=1}^{n}{E}_{j} $$ (1) $$ {\sigma }_{i}^{2}=\frac{1}{n-1}\sum _{j=1}^{n}({{E}_{j}-{E}_{i})}^{2} $$ (2) 式中:j为脉冲序数;Ei为能量平均值;σi2为能量方差。
目前,评估准分子激光的脉冲能量稳定性主要用相对标准差σ%参数来进行判断。
由于脉冲能量σ%参数为相对值,随激光器输出的平均能量值变化,因此表达式为:
$$ {\sigma }_{\text{%}}=\frac{{\sigma }_{p}}{{E}_{p}}\times 100 {\text{%}} $$ (3) $$ {E}_{p}=\frac{1}{n}\sum _{j=1}^{n}{E}_{j}={E}_{i} $$ (4) $$ {\sigma }_{p}=\sqrt{\dfrac{1}{n-1}\sum _{j=1}^{n}({{E}_{j}-{E}_{p})}^{2}}={\sigma }_{i} $$ (5) 式中:n表示激光脉冲个数;Ep表示n个脉冲能量的平均能量;σp表示n个脉冲能量的标准差。
根据上述表达式可知,当σ%参数的值越小时,说明脉冲能量更加稳定[13]。
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若有两台脉冲激光器合束,理想情况下,由于准分子激光脉冲输出能量波动性符合正态分布,则两台激光器发出的激光脉冲分别服从E1~N (E1, σ12),E2~N (E2, σ22)的正态分布,由于两台激光束是进行非相干合束,可以视为相互独立。由正态分布的可加性可知[14-15],其合束激光脉冲服从Ec~N (E1+E2, σ12+σ22),脉冲激光相对标准差σ%c的表达式为:
$$ {\sigma }_{{\text{%}}{c}}=\frac{\sqrt{{\sigma }_{1}^{2}+{\sigma }_{2}^{2}}}{{E}_{1}+{E}_{2}}\times 100{\text{%}} $$ (6) 同理,n台脉冲激光器进行非相干合束时,脉冲激光相对标准差σ%c的表达式为:
$$ {\sigma }_{{\text{%}}{c}}=\frac{\sqrt{{\sigma }_{1}^{2}+{\sigma }_{2}^{2}+\dots +{\sigma }_{n}^{2}}}{{E}_{1}+{E}_{2}+\dots +{E}_{n}}\times 100{\text{%}} $$ (7) -
实际应用中最常见的合束情况就是把多台能量和稳定性相当的脉冲激光器进行合束。理想情况下,两台激光器发出的脉冲激光符合相同的正态分布,即E1~N (E, σ2),E2~N (E, σ2),由于两台激光束是进行非相干合束,可以视为相互独立,由正态分布的可加性可知,其合束激光脉冲服从Ec~N (2E, 2σ2),其脉冲激光相对标准差σ%c的表达式为:
$$ {\sigma }_{{\text{%}}{c}}=\frac{\sqrt{2{\sigma }^{2}}}{2{E}}\times 100{\text{%}} $$ (8) 相对于合束前的E1~N (E, σ2),脉冲激光相对标准差:
$$ {\sigma }_{{\text{%}}}=\frac{\sigma }{{E}}\times 100{\text{%}} $$ (9) 显然σ%c<σ%,相对标准差参数的值越小,脉冲能量越稳定,即通过合束提高了脉冲激光的输出稳定性。
同理,如果把n台符合相同正态分布的脉冲激光束进行合束,其脉冲激光相对标准差σ%c的表达式为:
$$ {\sigma }_{{\text{%}}{c}}=\frac{\sqrt{n{\sigma }^{2}}}{{n}{E}}\times 100{\text{%}} $$ (10) -
对三台PLD20型准分子激光器进行脉冲能量测试,分别进行10000脉冲能量分布记录,采用Origin软件对数据进行统计分析,结果如表1所示。
表 1 激光器输出脉冲参数统计
Table 1. Statistics of laser output pulse parameters
Data number Energy average/mJ Standard deviation Relative standard deviation 1 152.2 1.62 1.06% 2 152.9 1.34 0.88% 3 153.3 1.46 0.95% 三组脉冲能量分布数据经过曲线拟合,见图1。
图 1 (a) 1号数据正态分布曲线拟合图;(b) 2号数据正态分布曲线拟合图;(c) 3号数据正态分布曲线拟合图
Figure 1. (a) Fitting diagram of normal distribution curve of No.1 data; (b) Fitting diagram of normal distribution curve of No.2 data; (c) Fitting diagram of normal distribution curve of No.3 data
从图1中可以看出,三组数据均符合正态分布曲线拟合。能量平均值分别为152.2、152.9、153.3 mJ、相对标准差σ%分别为1.06%、0.88%、0.95%。三台激光器脉冲激光输出的能量与稳定性非常相近,可以用于模拟参数相当激光器合束进行下一步实验。
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由于是模拟不相干合束,将上述三组数据任选两组进行叠加模拟,将其叠加后的10000个脉冲的能量分布进行统计记录,将1号与2号数据的合束结果记为4号数据,合束后脉冲能量分布数据经过曲线拟合,如图2所示。
从图2中可以看出,合束后的激光脉冲依旧符合正态分布。采用Origin软件对数据进行统计分析,数据分析结果如表2所示。
表 2 两台激光器合束输出脉冲参数统计
Table 2. Statistics of output pulse parameters of any two laser beams
Parameter Value Data number 4 Energy average/mJ 305.1 Standard deviation 2.14 Relative standard deviation 0.70% Theoretical standard deviation 0.69% Actual theoretical error 1.4% 表2中:相对标准差(Relative standard deviation)σ%是使用能量叠加后的数据实际的标准差以及能量平均值计算得到的结果;理论标准差(Theoretical standard deviation) σ%t是由公式(6)推导出来的公式计算结果;实际理论误差(Actual theoretical error)是相对标准差与理论标准差差值的绝对值除理论标准差得到的,其公式为:
$$ {A}=\frac{\left|{\sigma }_{{\text{%}}{c}}-{\sigma }_{{\text{%}}{t}}\right|}{{\sigma }_{{\text{%}}{t}}}\times 100{\text{%}} $$ (11) 如表2所示,两台激光器合束后激光脉冲的能量平均值分别为305.1 mJ,相对标准差σ%c为0.70%,相对标准差的值都相对于单台激光器的相对标准差的值有所降低,而相对标准差的值降低说明输出的脉冲激光能量更加稳定。同时,可以根据实际理论误差得出,相对标准差与理论标准差的差距很小,在可接受的误差范围内,也证明1.2节中的理论推导是成立的。
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最后,将1号、2号、3号三组数据进行叠加模拟,合束后脉冲能量分布数据经过曲线拟合,如图3所示。
图 3 1号、2号、3号数据模拟合束正态分布曲线拟合图
Figure 3. Fitting diagram of normal distribution curve of No.1, No.2 and No.3 data simulation
从图3可知,合束后的激光脉冲依旧符合正态分布。采用Origin软件对数据进行统计分析,数据分析结果如表3所示。
表3中:相对标准差σ%是使用能量叠加后的数据实际的标准差以及能量平均值计算得到的;理论标准差σ%t是由公式(7)推导出来的公式计算结果;实际理论误差是相对标准差与理论标准差差值的绝对值除理论标准差得到的。
表 3 三台激光器合束输出脉冲参数统计
Table 3. Statistics of combined output pulse parameters of three lasers
Parameter Value Data number 1+2+3 Energy average/mJ 458.5 Standard deviation 2.77 Relative standard deviation 0.60% Theoretical standard deviation 0.56% Actual theoretical error 7.1% 可以看出,将三台激光器合束后的激光脉冲的能量平均值为458.48 mJ,相对标准差σ%c为0.60%。相对于单台激光器以及任意两台激光器合束的激光脉冲,相对标准差的值均有所下降,即输出的激光脉冲更加稳定。同时,可以根据实际理论误差得出相对标准差与理论标准差的差距很小,在可接受的误差范围内,证明1.2节中的理论推导是成立的。
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根据上述实验模拟,为了进一步对推导公式进行验证,对一台PLD20型准分子激光器以及一台PLD30型准分子激光器进行合束脉冲能量测试,按照图4示意搭建非相干合束的光路系统[16],分别进行三次合束实验,每次都统计单台激光器10000个脉冲能量以及合束后的10000个脉冲能量,采用Origin软件对数据进行统计分析,结果如表4所示。
三组合束脉冲能量分布数据经过曲线拟合,如图5所示。
表 4 实际合束激光器输出脉冲参数统计
Table 4. Statistics of output pulse parameters of actual combined beam laser
Data number Equipment
numberEnergy average/mJ Standard deviation Relative standard deviation Theoretical standard deviation Actual theoretical error 1 PLD20 356.6 4.95 1.39% - - PLD30 354.5 4.41 1.24% - - Laser beam combination 687.2 5.95 0.86% 0.93% 7.52% 2 PLD20 368.4 4.19 1.14% - - PLD30 354.5 4.41 1.24% - - Laser beam combination 694.5 5.54 0.79% 0.84% 5.95% 3 PLD20 328.4 4.89 1.49% - - PLD30 331.4 4.50 1.35% - - Laser beam combination 646.8 5.40 0.83% 1.00% 17.00% 图 5 (a)依次为第一次合束时PLD20、PLD30、合束后的正态分布曲线拟合图;(b)依次为第二次合束时PLD20、PLD30、合束后的正态分布曲线拟合图;(c)依次为第三次合束时PLD20、PLD30、合束后的正态分布曲线拟合图
Figure 5. (a) Fitting curves of the normal distribution curves of PLD20, PLD30, and after the first laser beam combination; (b) Fitting curves of the normal distribution curves of PLD20, PLD30, and after the second laser beam combination; (c) Fitting curves of the normal distribution curves of PLD20, PLD30, and after the third laser beam combination
如表4所示,三次合束实验中,PLD20分别输出能量平均值为356.6、368.4、328.4 mJ,相对标准差为1.39%、1.14%、1.49%的激光脉冲;PLD30分别输出能量平均值为354.5、354.5、331.4 mJ,相对标准差为1.24%、1.24%、1.35%的激光脉冲。三次合束后分别得到能量平均值为687.2、694.5、646.8 mJ,相对标准差为0.86%、0.79%、0.83%的激光脉冲。如图5所示,合束前单台激光的激光脉冲与合束后的激光脉冲依旧符合正态分布。可以看出,实际合束后激光脉冲相对标准差的值都相对于单台激光器相对标准差的值有所降低,而相对标准差的值降低说明输出的脉冲激光能量更加稳定。根据前两次合束的实际理论误差得出,相对标准差与理论标准差的差距很小,在可接受的误差范围内。第三次合束可能是由于能量衰减等问题,实际相对标准差相较于理论标准差差距过大,但依旧存在合束后脉冲能量稳定性提高的现象。
Improvement of the stability of excimer laser output pulse energy by beam combination
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摘要: 为了获得稳定性更好、能量更大的准分子激光输出脉冲,对合束方式提高输出脉冲能量稳定性的可行性进行了理论推导、模拟实验以及合束实验研究。从理论推导得知,当激光输出脉冲能量符合正态分布时,多台激光器合束可以降低输出能量相对标准差。对三台输出脉冲能量分布特性符合正态分布的准分子激光器进行合束模拟实验研究,每台输出脉冲能量平均值约为153 mJ,脉冲能量相对标准差约为1%。两台准分子激光器合束时得到输出脉冲能量平均值为305 mJ、能量相对标准差为0.7%的准分子激光脉冲。三台准分子激光器合束时可以得到输出能量平均值为458 mJ、能量相对标准差为0.6%的准分子激光脉冲。使用两台准分子激光器进行三次实际合束实验,其中两台准分子激光器三次合束分别输出能量平均值约为355、350、330 mJ,相对标准差为1.3%、1.2%、1.4%的激光脉冲;三次合束后分别得到能量平均值为687、694、646 mJ,相对标准差为0.86%、0.79%、0.83%的激光脉冲。模拟实验以及合束实验都表明,合束后的能量相对标准差均小于单台能量相对标准差,即合束提高准分子输出脉冲的稳定性。因此,当单台准分子激光输出脉冲能量符合正态分布且平均能量相当时,多台合束可以有效提高激光输出脉冲的能量稳定性。Abstract:
Objective UV excimer lasers have the characteristics of short wavelength, high power and narrow linewidth, and are widely used in semiconductor lithography, new display manufacturing, corneal refractive correction, etc. Whether in the fields of medical treatment, industry, flat panel processing, or scientific research, the pulse energy stability of excimer lasers is a very important parameter, which directly determines the accuracy of surgery, the critical dimensions of processing, and the uniformity of material processing. Limited by the composition and proportion of working gas, gas circulation system, stability of excitation source, laser resonance amplification mechanism, etc., the energy stability of excimer laser pulses output by current excimer lasers generally fluctuates around 1%-2%, and it is difficult to further improve. However, the pulse energy stability directly affects the development of industrial production, medical treatment, and scientific research. For example, in the field of integrated circuit manufacturing, the lithography process determines the key dimensions of the device, while the exposure light source directly affects the quality of lithography. The instability of the laser light source may cause pattern deformation and poor registration effect. Therefore, it is of great significance to further improve the output energy stability of excimer lasers based on the existing technology. In practical use, it has been found that when two excimer lasers are used in beam combination, the pulse energy stability is also improved to some extent while the output power is increased. Methods In order to obtain more stable and powerful excimer laser output pulses, theoretical derivation, simulation experiments, and beam combination experiments were conducted to investigate the feasibility of improving the stability of output pulse energy through beam combination. Theoretical derivation showed that when the laser output pulse energy follows a normal distribution, beam combination of multiple lasers with similar parameters can reduce the relative standard deviation of output energy. Subsequently, beam combination simulation experiments and actual beam combination experiments were conducted. Results and Discussions The simulation experiment of beam combination for three excimer lasers with output pulse energy distribution characteristics consistent with normal distribution was conducted. The output pulse energies were 152.2 mJ, 152.9 mJ, and 153.3 mJ, respectively, with relative standard deviations of 1.06%, 0.88%, and 0.95%. Among them, two excimer lasers achieved an average output pulse energy of 305 mJ and a relative standard deviation of 0.7% when combined. Three excimer lasers achieved an average output energy of 458 mJ and a relative standard deviation of 0.6% when combined. Three actual beam combination experiments were conducted using one PLD20 excimer laser and one PLD30 excimer laser (Fig.4). The average output energies of PLD20 were 356.6 mJ, 368.4 mJ, and 328.4 mJ, with relative standard deviations of 1.39%, 1.14%, and 1.49%, respectively. The average output energies of PLD30 were 354.5 mJ, 354.5 mJ, and 331.4 mJ, with relative standard deviations of 1.24%, 1.24%, and 1.35%, respectively. After three beam combinations, the average output energies were 687.2 mJ, 694.5 mJ, and 646.8 mJ, with relative standard deviations of 0.86%, 0.79%, and 0.83%, respectively. The relative standard deviation values of the laser pulses after actual beam combination were all lower than those of the single laser (Tab.4). The decrease in the relative standard deviation value indicates that the output pulse laser energy is more stable. Conclusions From theoretical derivation, simulated beam combination data, and actual beam combination data, it can be concluded that multiple lasers with similar output laser pulse parameters can improve the output laser pulse energy of the lasers while reducing the relative standard deviation of the output laser pulses, thereby improving the stability of the laser pulses. When two lasers with similar parameters are combined, the energy can reach 1.9 times that of a single laser, and the relative standard deviation can be reduced to 0.6-0.7 times that of a single laser. -
Key words:
- excimer laser /
- pulse energy /
- stability /
- relative standard deviation
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图 5 (a)依次为第一次合束时PLD20、PLD30、合束后的正态分布曲线拟合图;(b)依次为第二次合束时PLD20、PLD30、合束后的正态分布曲线拟合图;(c)依次为第三次合束时PLD20、PLD30、合束后的正态分布曲线拟合图
Figure 5. (a) Fitting curves of the normal distribution curves of PLD20, PLD30, and after the first laser beam combination; (b) Fitting curves of the normal distribution curves of PLD20, PLD30, and after the second laser beam combination; (c) Fitting curves of the normal distribution curves of PLD20, PLD30, and after the third laser beam combination
表 1 激光器输出脉冲参数统计
Table 1. Statistics of laser output pulse parameters
Data number Energy average/mJ Standard deviation Relative standard deviation 1 152.2 1.62 1.06% 2 152.9 1.34 0.88% 3 153.3 1.46 0.95% 表 2 两台激光器合束输出脉冲参数统计
Table 2. Statistics of output pulse parameters of any two laser beams
Parameter Value Data number 4 Energy average/mJ 305.1 Standard deviation 2.14 Relative standard deviation 0.70% Theoretical standard deviation 0.69% Actual theoretical error 1.4% 表 3 三台激光器合束输出脉冲参数统计
Table 3. Statistics of combined output pulse parameters of three lasers
Parameter Value Data number 1+2+3 Energy average/mJ 458.5 Standard deviation 2.77 Relative standard deviation 0.60% Theoretical standard deviation 0.56% Actual theoretical error 7.1% 表 4 实际合束激光器输出脉冲参数统计
Table 4. Statistics of output pulse parameters of actual combined beam laser
Data number Equipment
numberEnergy average/mJ Standard deviation Relative standard deviation Theoretical standard deviation Actual theoretical error 1 PLD20 356.6 4.95 1.39% - - PLD30 354.5 4.41 1.24% - - Laser beam combination 687.2 5.95 0.86% 0.93% 7.52% 2 PLD20 368.4 4.19 1.14% - - PLD30 354.5 4.41 1.24% - - Laser beam combination 694.5 5.54 0.79% 0.84% 5.95% 3 PLD20 328.4 4.89 1.49% - - PLD30 331.4 4.50 1.35% - - Laser beam combination 646.8 5.40 0.83% 1.00% 17.00% -
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