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当一束频率为$ \nu $的单色光穿过长度为L的充满某种介质的吸收池时,介质分子吸收一部分光能,透射光强$ I $和入射光强$ {I}_{0} $之间满足Lambert-Beer定律[12]:
$$ I={I}_{0}\exp\left[-\alpha \left(\nu \right)CL\right]={I}_{0}\exp\left[-PS\left(T\right)\varphi \left(\nu \right)CL\right] $$ (1) 式中:$ \alpha \left(\nu \right) $为气体的吸收系数,cm−1,该系数与气体的种类以及穿过该气体的光频率有关;C为探测气体的浓度,mol·cm−3;L为吸收光程长度,cm;P为待测气体的压强,atm;S(T)为谱线的吸收强度,cm−2·atm−1,只与温度有关;$ \varphi \left(\nu \right) $为线型函数,表示待测气体吸收谱线的形状,与待测气体温度、压强、种类及各成分含量有关[13]。
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不同展宽机制对应不同的谱线线型,主要分为三种基本线型[14]:Lorentz线型、Gauss线型、Voigt线型。线型函数表示被测气体吸收谱线的形状,主要存在两种谱线展宽机制[15]:碰撞展宽和多普勒展宽。
碰撞展宽是由于辐射原子或分子周围存在大量的干扰粒子,它们之间相互碰撞引起谱线加宽。随着压力的增大,碰撞加宽作用加大导致谱线线宽增加,因此这种展宽又称为压力展宽。碰撞展宽的线型函数是Lorentz函数,其线型函数表达式为:
$$ {g}_{L}\left(\nu ,{\nu }_{0}\right)=\frac{1}{2\pi }\frac{\mathrm{\Delta }{\nu }_{L}}{{\left(\nu -{\nu }_{0}\right)}^{2}+{\left(\dfrac{\mathrm{\Delta }{\nu }_{L}}{2}\right)}^{2}} $$ (2) 式中:$ {\nu }_{0} $为谱线中心吸收频率;$ \mathrm{\Delta }{\nu }_{L} $为Lorentz线型的半高宽。
多普勒展宽是由于分子的无规则热运动,分子吸收激光后引发多普勒频移,分子吸收和发射单频光的机率不再相同引起谱线加宽。气体多普勒展宽的线型函数一般为Gauss函数,谱线的多普勒展宽的线型函数表达式为:
$$ {g}_{D}\left(\nu ,{\nu }_{0}\right)=\frac{2}{\mathrm{\Delta }{\nu }_{D}}\sqrt{\frac{{\rm{ln}} 2}{\pi }}\exp \left[-4 {\rm{ln}} 2{\left(\dfrac{\nu -{\nu }_{0}}{\mathrm{\Delta }{\nu }_{D}}\right)}^{2}\right] $$ (3) 式中:$ {\nu }_{0} $为谱线中心吸收频率;$ \mathrm{\Delta }{\nu }_{D} $为Gauss线型的半高宽。
Gauss线型与压强变化无关;Lorentz线型随压强的增大而减小;Voigt线型随着压强的增大由Gauss线型向Lorentz线型过渡,由Gauss线型和Lorentz线型的卷积得到[16]。其近似表达式为[17]:
$$ {g}_{\nu }\left(\nu ,{\nu }_{0}\right)=\frac{2}{\mathrm{\Delta }{\nu }_{D}}\sqrt{\frac{{\rm{ln}}2}{\pi }}\frac{a}{\pi }{\int }_{-\infty }^{+\infty }\frac{\exp\left(-{y}^{2}\right)}{{a}^{2}+{\left(w-y\right)}^{2}}{\rm{d}}y $$ (4) 式中:$a=\dfrac{\sqrt{{\rm{ln}}2}\mathrm{\Delta }{\nu }_{L}}{\mathrm{\Delta }{\nu }_{G}}$;$w=\dfrac{\sqrt{{\rm{ln}}2}\left(\nu -{\nu }_{0}\right)}{\mathrm{\Delta }{\nu }_{G}}$;$y=\dfrac{\sqrt{{\rm{ln}}2}\left({\nu }_{i}-\nu \right)}{\mathrm{\Delta }{\nu }_{G}}$;$ \mathrm{\Delta }{\mathrm{\nu }}_{G} $、$ \mathrm{\Delta }{\nu }_{L} $分别为多普勒展宽和碰撞展宽的半高宽;$ {\nu }_{0} $为线型中心频率;$ {\nu }_{i} $为激光频率[18]。
基于谐波检测原理得到Voigt线型函数的二次谐波信号,其峰宽用两谷值之间的波长宽度表示,无论Gauss线型还是Lorentz线型,原函数的半高宽与二次谐波的峰宽之比均为常数,因此气体吸收峰的半高宽可由二次谐波的峰宽描述。随着压强的增大,碰撞展宽在谱线展宽中逐渐占据主导因素,而Lorentz线型的半高宽与压强成正比,谱线的宽度会随之变大。也就是说压强对二次谐波信号产生影响,进而反映在吸收线线型上。因此文中基于压强对吸收谱线的影响,通过调节两种线型半高宽占比来模拟压强变化,以Gauss线型和Lorentz线型的表达式分别拟合,通过拟合度之比的测量对气体压强进行实时检测。
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文中对通过卷积方式得到的二阶导数信号进行仿真分析,固定两原函数半高宽和为1,Lorentz原函数半高宽占总峰宽比例为5%~95%,间隔5%,令Gauss原函数峰高与Lorentz二阶导数信号峰高都为1保持不变。
当Lorentz原函数半高宽占总峰宽比例为5%、95%时仿真曲线如图1所示。
由图1可知,随着压强的增加,曲线由接近Gauss拟合变为接近Lorentz拟合,与上面论述一致。
依据上述方法计算得到19组仿真结果,其中Gauss拟合度与Lorentz拟合度之比用大写字母Y (Lfit/Gfit)表示。Y随半高宽占比改变而改变,而压强变化用半高宽占比模拟,故可建立压强与Y之间的关系。通过对仿真结果进行拟合,得到压强与Y之间各阶拟合曲线如图2所示。
图 1 仿真曲线:(a) Lorentz原函数占总峰宽比例5%;(b) Lorentz原函数占总峰宽比例95%
Figure 1. Simulation curve: (a) The original Lorentz function accounts for 5% of the total peak width; (b) The original Lorentz function accounts for 95% of the total peak width
由图2可知,压强与Y的三阶拟合有很好的拟合效果,故可运用三阶拟合公式进行模拟压强检测,即可初步确定Y与压强之间近似三阶拟合关系。
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激光器线宽通常远小于吸收峰的峰宽,仿真过程中,设激光器线宽信号模型的幅值为1,峰宽为第一次卷积前Lorentz二阶函数信号的原函数半高宽的1/20、1/10、3/20、1/5。加原函数半高宽的1/20、1/10、3/20、1/5的激光器线宽后,压强与Y的曲线拟合结果对比如图3所示。
由图3可知,加入激光器线宽后Y值变大,在图中表现为曲线纵向伸展,激光器线宽越宽Y值越大,曲线纵向伸展越明显,但整体上压强与Y仍满足三阶拟合关系。
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在实际系统中存在白噪声的影响,向二阶Voigt信号加上不同幅值比例的白噪声信号,通过Y值来分析白噪声对仿真信号的影响。
TLAS系统信噪比为100左右,所以选取白噪声幅值系数分别为信号峰高的1/400、1/200、1/150、1/100、1/80、1/60、1/40、1/30、1/20。对最终信号进行函数拟合,通过Y值分析白噪声对仿真信号的影响。加白噪声后,压强与Y的曲线对比如图4所示。
图 4 加不同幅值系数白噪声后压强与Y的曲线对比图
Figure 4. Curve comparison of pressure and Y after adding white noise with different amplitude coefficients
由于白噪声具有随机性的特点,故考虑白噪声后,图像会有白噪声的无规律波动,但整体上压强与Y仍满足三阶拟合关系。
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在实际情况下会有背景干扰的影响。向二阶Voigt信号加上背景干扰,将结果进行函数拟合,通过Y值分析背景干扰对仿真信号的影响。
用正弦函数仿真背景干扰,分别分析幅值、频率的变化对曲线拟合度的影响。
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保持背景干扰的频率和相位不变,研究幅值变化对拟合度的影响,如图5所示,幅值分别取峰高的5%、10%、15%、20%、30%、40%、60%时,可以看出Y与压强的拟合曲线都有很好的三阶拟合关系,幅值变化对压强与Y的拟合曲线没有影响。
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取背景干扰的幅值为信号峰高的10%,频率分别为信号频率的0.2、0.4、0.6、0.8、1、1.2、1.4、1.6、1.8、2。加不同频率的背景干扰后,压强与Y的拟合曲线如图6所示。
由图6可知,不同频率下压强与Y的拟合曲线都有较高的三阶拟合度,曲线整体有上下的波动。由此说明背景干扰的频率变化对压强与Y的拟合曲线的拟合度无影响。
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在已有的压强测量方法中,主要包括峰宽检测与2f/4f幅值检测。用上述Voigt函数,对压强和谱线峰宽、二次谐波与四次谐波幅值之比的数学模型进行仿真分析。仿真结果与Gauss/Lorentz拟合度之比相近,两种特征值与压强的拟合结果均为三阶,且在固定激光器线宽、噪声与背景的干扰下,三阶拟合度仍能保持在0.998 0以上。
为比较文中提出的Gauss/Lorentz拟合度之比与峰宽、2f/4f幅值面对不同噪声与背景干扰下的抗干扰能力,对生成的同一谱线,加不同噪声与背景,用标准差评价其特征值的起伏程度,结果如表1所示。
由于Gauss/Lorentz拟合度之比的计算过程中,整条谱线的所有点都参与拟合计算,而峰宽和2f/4f幅值的特征值由部分特征点计算得到,易受到各种因素的影响。因而在抗干扰能力方面,Gauss/Lorentz拟合度之比优于峰宽和2f/4f幅值,例如在背景频率变化干扰中,特征值能够保持相对稳定,如图7所示。
表 1 同一谱线三种特征值在不同噪声与背景下的标准差
Table 1. The standard deviation of three eigenvalues of the same spectral line in different noise and background
Influencing factors Noise changes Background amplitude changes Background frequency changes The ratio of the fitting of Gauss and Lorentz 0.000 868 8 0.054 38 0.005 514 Peak width 0.011 28 0.168 9 0.071 97 Amplitude of 2f/4f 0.001 194 0.070 07 0.026 55 -
该节通过卷积方式得到的二阶导数信号仿真吸收谱线的二次谐波,通过改变Gauss/Lorentz比例关系仿真压强变化,建立Gauss/Lorentz拟合度之比于压强的数学模型,并对激光器线宽、噪声、背景干扰的影响进行分析。仿真结果表明,Gauss/Lorentz拟合度之比于压强存在三阶拟合关系,不受激光器线宽、噪声、背景干扰的影响,并且在拟合度相近的情况下,相比峰宽和2f/4f幅值表现出更好的抗噪声、背景干扰的能力于稳定性。
Research on harmonic detection pressure inversion based on Gauss/Lorentz line fitting ratio (invited)
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摘要: 调谐激光吸收光谱(TLAS)技术具有非接触、抗干扰、高灵敏度等优势,可对气体进行浓度、温度、压强的测量。目前已有的压强检测模型中多以谱线的有限特征点进行提取与计算,存在易受干扰、测量误差较大的问题,因而有必要建立新的抗干扰、稳定性强的压强检测模型。针对此问题,文中根据吸收线宽的气体压力测量方法,提出了低压与高压范围内压强与谱线线型拟合函数的数学模型。结合谱线展宽原理,对不同压强下的二次谐波吸收线进行仿真研究。通过改变Gauss线型函数和Lorentz线型函数的半高宽比例关系模拟压强变化,分析信号拟合度的变化趋势,仿真结果表明,在理想情况以及激光器线宽、白噪声、背景干扰影响下,Gauss/Lorentz线型拟合度之比与压强之间存在三阶拟合关系,拟合度均保持在0.998 0以上,且与传统模型相比在动态噪声和背景干扰下具有更好的稳定性。最后对CO2气体1 580 nm位置的实测信号进行处理,实验结果表明,实际检测谱线Gauss/Lorentz线型拟合度之比与压强之间的三阶拟合度为0.986 3,略低于仿真的拟合度0.998 7,符合仿真分析结果。文中提出的方法可以根据吸收谱线的拟合比曲线反演气体压强,为气体压强检测提供了解决方案。Abstract:
Objective Tuned laser absorption spectroscopy (TLAS) technology has advantages such as non-contact, anti-interference, and high sensitivity, which can be used for gas concentration, temperature, and pressure measurement. In the existing pressure detection models, limited feature points of spectral lines are mostly extracted and calculated, which can lead to problems such as susceptibility to interference in measurement results and significant measurement errors. Therefore, it is necessary to establish a new anti-interference and stable pressure detection model. To solve this problem, a mathematical model was proposed for fitting the pressure and spectral line shape function within the low and high pressure ranges based on the gas pressure measurement method of absorption line width. Methods Simulation research on the second harmonic absorption lines under different pressures was conducted based on the principle of spectral line broadening. In order to simulate the pressure changes by adjusting the Gauss/Lorentz halfwidth ratio, the second-order derivative signal was obtained by convolution of Gauss and Lorentz functions to simulate the second harmonic of the absorption spectral line. By establishing a mathematical model of the Gauss/Lorentz line fitting ratio and pressure, the fitting relationship between the two was obtained under ideal conditions and the influence of laser line width, white noise, and background interference. The comparative analysis on the stability of the fitting ratio with eigenvalues used to calculate pressure in existing models such as the peak width and 2f/4f amplitude under dynamic noise and background interference were conducted. Finally, the measured signal at 1 580 nm of CO2 gas was processed to verify the simulation results. Results and Discussions The simulation results show that under ideal conditions and the influence of laser linewidth, white noise, and background interference, there is a third-order fitting relationship between the Gauss/Lorentz line fitting ratio and pressure, and the fitting degree remains above 0.998 0 (Fig.3-6). Compared with traditional models, it has better stability under dynamic noise and background interference (Tab.1). The experimental results show that the third-order fit between the Gauss/Lorentz line fitting ratio of the actual detection spectral line and the pressure is 0.986 3 (Fig.9), slightly lower than the simulated fit of 0.998 7 (Fig.2), which is consistent with the simulation analysis results. Conclusions In order to establish a more effective pressure detection method, based on the principle of spectral line broadening, the pressure change is simulated using the ratio of Gauss function half width to Lorentz function half width, and the Voigt function is used to describe the absorption spectral line shape. A mathematical model was established for the pressure to fitting ratio under ideal conditions, laser linewidth, white noise, and background interference. Through simulation analysis, the relationship between pressure and fitting ratio satisfies a third-order fitting relationship, which is not only affected by laser linewidth, white noise, and background interference, but also maintain stability under dynamic noise and background interference, which exhibits advantages in pressure detection compared to traditional models. The experimental validation was carried out using CO2 absorption spectra. The curve fitting obtained from analyzing the experimental data was slightly lower, but its trend was consistent, which indicated the effectiveness of the established mathematical model. The proposed method has certain theoretical significance and practical value in pressure measurement, providing new ideas for pressure detection. -
表 1 同一谱线三种特征值在不同噪声与背景下的标准差
Table 1. The standard deviation of three eigenvalues of the same spectral line in different noise and background
Influencing factors Noise changes Background amplitude changes Background frequency changes The ratio of the fitting of Gauss and Lorentz 0.000 868 8 0.054 38 0.005 514 Peak width 0.011 28 0.168 9 0.071 97 Amplitude of 2f/4f 0.001 194 0.070 07 0.026 55 -
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