激光自混合干涉技术中颗粒粒度反演算法研究
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摘要
激光自混合干涉测量纳米颗粒粒径属于动态光散射技术中的外差法探测,其功率谱密度反演颗粒粒径分布属于病态问题研究的范畴。颗粒粒度的反演是颗粒测量技术中的难点和重点,对反演算法的研究是亟待解决的问题。本文主要分析和研究了功率谱密度反演的正则化方法,并对算法做了相应的改进和优化。其中完成的主要工作包括：
1、研究和分析了零差法和外差法的测量原理,并从速率方程与三镜法布里一珀罗腔模型结构方面,分析了激光自混合技术的机理和特性。
2、比较了线性形式和对数形式的功率谱核函数矩阵的反演颗粒粒度分布的问题,对数形式的功率谱核函数矩阵得出的分布比线性形式更合理。为了保证结果的非负性,我们采用Chahine迭代算法。在对单峰颗粒分布反演时,本文采用Morrison平滑算法处理Chahine迭代算法的初始分布,处理后的初始分布经迭代后,平滑性要好于未经处理的迭代结果。对于双峰的颗粒分布,Chahine算法很难得出合理的结果,得出的分布表现出过平滑性的特征。
3、为了解决双峰颗粒分布问题采用截断奇异值分解的方法。本文通过Chahine算法形成的参考点改进了L曲线准则选取合理正则化参数的方法。此参考点减少了选取正则化参数时的计算量,并增强了L准则选择正则化参数的正确性。另外,本文将截断奇异分解法得出的颗粒粒度分布,作为Chahine算法的初始分布,使之迭代后得到了明显的优化。
4、将功率谱反问题Tikhonov正则化后,使用和比较了奇异值分解法和总体最小二乘法。通过仿真颗粒粒度的结果比较,总体最小二乘法的稳健性要优于奇异值分解法。为了给这两种方法选择正确的正则化化参数,本文提出了解最小变化法,并用实验验证了该方法的实用性和优点。使用这种新的方法可以增加最优正则化参数附近分辨率,从而选出更好的最优正则化参数。
5、通过SR曲线改进了Chahine算法迭代的停止条件,SR曲线的X轴为均方根误差,Y轴为解的平滑性限制。通过SR曲线,可以选出Chahine算法的最优迭代分布。平滑性限制我们采用离散零阶导数算子和离散一阶导数算子两种解范数形式。离散一阶导数算子的SR曲线选出的Chahine算法的迭代解的相对误差小于由离散零阶导数算子的SR曲线选出的迭代解。
迄今为止,反演复杂的颗粒粒度分布,依然无法得出满意的反演结果,颗粒粒度反演算法极大程度上影响了测量结果的准确性,本文所进行的工作有助于颗粒粒度分布反演算法研究的发展。
Self-mixing interference (SMI) belongs to heterodyne detection of dynamic light scattering (DLS) for measuring the particle-sizing distributions (PSD) of submicron particles and nanometer particles suspended in suspension. And its power spectral density inversion is ill condition problem. In the corresponding fields, the inversion of PSD is an difficult and important factor, and the inversion result is directly related to the accuracy of the measurement results. The research on this aspect meets the needs of the related fields.This work mainly analysed regularization algorithms of power spectrum inversion, and made some modifications on the corresponding inversion algorithms. The main contents are as following:
1、Studied the mechanisms of self-beating and heterodyne detection, and analysed SMI theory and character based on rate equation and Fabry-Perot cavity modal.
2、Studied the PSD problem by comparing linear kernel matrix with logarithmic form kernel matrix, the latter retrieved more reasonable distribution than the former.To guarantee the non-negative constraint, Chahine iteration method served as the optimization algorithm.In the unimodal PSD, Morrison smoothing method was adopted, which made smoothing process on the initial distribution of Chahine algorithm. The processed result was better than the unprocessed. For the bimodal PSD, Chahine algorithm hardly performed well, the derived solution showed over-smoothing appearance.
3、To solve the bimodal PSD problem, we applied truncated singular value decomposition (TSVD) method. The referenced dot formed by Chahine algorithm reduced the calculation and enhance the validity for determining the optimal regularization parameter in the L-curve. In addition, The TSVD retrieved result was obviously optimized after it was made as the Chahine algorithm initial distribution through the iterated process.
4、The power spectrum inversion problem was transferred through Tikhonov regularization, then SVD and total least squares (TLS) behaved and were compared. TLS was verified to have more stability than SVD in simulation.In order to choose the optimal regularization parameter, we proposed the minimum variation of solution method, which is proved to be valid and its advantage for determining regularization parameter in experiment. A better regularization parameter could be gotten through increasing the discrimination between the neighboring ancipital parameters by means of the new method.
5、Chahine algorithm stopping criterion was modified with SR-curve, the SR-curve horizontal axis shown root mean square deviation, and the vertical axis shown the solution smoothing constraint, then the modest iterative distribution was chosen with SR-curve. The zero and the first derivative operators were employed as the smoothing constraint. The estimated particle size distribution (PSD) corresponding to the first derivative operator had less relative size distribution error.
So far, the satisfied result can not be got if deriving complicated PSD, yet the PSD inversion algorithm largely affect the accuracy of the particle size measurement, this work will have a promoted effect on the particle-sizing distributions inversion algorithm.
1、研究和分析了零差法和外差法的测量原理,并从速率方程与三镜法布里一珀罗腔模型结构方面,分析了激光自混合技术的机理和特性。
2、比较了线性形式和对数形式的功率谱核函数矩阵的反演颗粒粒度分布的问题,对数形式的功率谱核函数矩阵得出的分布比线性形式更合理。为了保证结果的非负性,我们采用Chahine迭代算法。在对单峰颗粒分布反演时,本文采用Morrison平滑算法处理Chahine迭代算法的初始分布,处理后的初始分布经迭代后,平滑性要好于未经处理的迭代结果。对于双峰的颗粒分布,Chahine算法很难得出合理的结果,得出的分布表现出过平滑性的特征。
3、为了解决双峰颗粒分布问题采用截断奇异值分解的方法。本文通过Chahine算法形成的参考点改进了L曲线准则选取合理正则化参数的方法。此参考点减少了选取正则化参数时的计算量,并增强了L准则选择正则化参数的正确性。另外,本文将截断奇异分解法得出的颗粒粒度分布,作为Chahine算法的初始分布,使之迭代后得到了明显的优化。
4、将功率谱反问题Tikhonov正则化后,使用和比较了奇异值分解法和总体最小二乘法。通过仿真颗粒粒度的结果比较,总体最小二乘法的稳健性要优于奇异值分解法。为了给这两种方法选择正确的正则化化参数,本文提出了解最小变化法,并用实验验证了该方法的实用性和优点。使用这种新的方法可以增加最优正则化参数附近分辨率,从而选出更好的最优正则化参数。
5、通过SR曲线改进了Chahine算法迭代的停止条件,SR曲线的X轴为均方根误差,Y轴为解的平滑性限制。通过SR曲线,可以选出Chahine算法的最优迭代分布。平滑性限制我们采用离散零阶导数算子和离散一阶导数算子两种解范数形式。离散一阶导数算子的SR曲线选出的Chahine算法的迭代解的相对误差小于由离散零阶导数算子的SR曲线选出的迭代解。
迄今为止,反演复杂的颗粒粒度分布,依然无法得出满意的反演结果,颗粒粒度反演算法极大程度上影响了测量结果的准确性,本文所进行的工作有助于颗粒粒度分布反演算法研究的发展。
Self-mixing interference (SMI) belongs to heterodyne detection of dynamic light scattering (DLS) for measuring the particle-sizing distributions (PSD) of submicron particles and nanometer particles suspended in suspension. And its power spectral density inversion is ill condition problem. In the corresponding fields, the inversion of PSD is an difficult and important factor, and the inversion result is directly related to the accuracy of the measurement results. The research on this aspect meets the needs of the related fields.This work mainly analysed regularization algorithms of power spectrum inversion, and made some modifications on the corresponding inversion algorithms. The main contents are as following:
1、Studied the mechanisms of self-beating and heterodyne detection, and analysed SMI theory and character based on rate equation and Fabry-Perot cavity modal.
2、Studied the PSD problem by comparing linear kernel matrix with logarithmic form kernel matrix, the latter retrieved more reasonable distribution than the former.To guarantee the non-negative constraint, Chahine iteration method served as the optimization algorithm.In the unimodal PSD, Morrison smoothing method was adopted, which made smoothing process on the initial distribution of Chahine algorithm. The processed result was better than the unprocessed. For the bimodal PSD, Chahine algorithm hardly performed well, the derived solution showed over-smoothing appearance.
3、To solve the bimodal PSD problem, we applied truncated singular value decomposition (TSVD) method. The referenced dot formed by Chahine algorithm reduced the calculation and enhance the validity for determining the optimal regularization parameter in the L-curve. In addition, The TSVD retrieved result was obviously optimized after it was made as the Chahine algorithm initial distribution through the iterated process.
4、The power spectrum inversion problem was transferred through Tikhonov regularization, then SVD and total least squares (TLS) behaved and were compared. TLS was verified to have more stability than SVD in simulation.In order to choose the optimal regularization parameter, we proposed the minimum variation of solution method, which is proved to be valid and its advantage for determining regularization parameter in experiment. A better regularization parameter could be gotten through increasing the discrimination between the neighboring ancipital parameters by means of the new method.
5、Chahine algorithm stopping criterion was modified with SR-curve, the SR-curve horizontal axis shown root mean square deviation, and the vertical axis shown the solution smoothing constraint, then the modest iterative distribution was chosen with SR-curve. The zero and the first derivative operators were employed as the smoothing constraint. The estimated particle size distribution (PSD) corresponding to the first derivative operator had less relative size distribution error.
So far, the satisfied result can not be got if deriving complicated PSD, yet the PSD inversion algorithm largely affect the accuracy of the particle size measurement, this work will have a promoted effect on the particle-sizing distributions inversion algorithm.
引文
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[2]王刚.用于纳米颗粒粒度测量的模拟探测动态光散射技术研究[D]：博士学位论文.吉林：吉林大学,2006.
[3]王远.基于动态光散射的超细颗粒测试技术研究[D]：硕士学位论文.重庆：重庆：重庆大学,2005.7-12.
[4]王乃宁.颗粒粒径的光学测量技术及应用[M].北京：原子能出版社,2000(1).
[5]F Scheffold. A Shalkevich, R Vavrin,et al. PCS particle sizing in turbid suspensions: scope and limitations[M].2004:ACS Publications.
[6]李绍新.动态光散射技术在生物大分子测量上的应用[D]：硕士学位论文.广州：华南师范大学,2004.30-39.
[7]K Otsuka, K Abe, N Sano, S Sudo, J-Y Ko. Two-channel self-mixing laser Doppler measurement with carrier-frequency-division multiplexing [J]. Applied Optics,2005.44 (9):1709-1714.
[8]P G R King. Metrology with an optical master [J]. Review of Scientific Instruments,1963.17:180-182.
[9]M J Rudd. A later doppler velocimeter employing the laser as a mixer-oscilator [J].Journal of Physics E,1968.1:723-726.
[10]R Lang,K Kobayashi.External optical feedback effects on semiconductor injection laser properties [J]. IEEE Journal of Quantum Electronics,1980,16(3):347-355.
[11]J H Churnside. Laser doppler velocimetry by modulating a CO2 laser with back scattered light [J].Applied Optics,1984,23(1):61-65.
[12]Edson T. Shimizu.Directional discrimination in the self-mixing type laser doppler velocimeter [J].Applied Optics,1987,26(2):4541-4544.
[13]P J de Groot.Ranging and velocimetry signal generation in a backscatter modulated laser diode[J].Applied Optics,1988,27(21):4475-4480.
[14]P J de Groot.Range-dependent optical feedback effects on the multimode spectrum of laser diodes[J].Journal of Modern Optics,1990,37(7):1199-1214.
[15]W M Wang,W J O Boyle,K T V Grattan.Fiber-optic doppler velocimeterthat incorporates active optical feedback from a diode laser [J].Optical Society of America,1992,17(11):819-821.
[16]W M Wang,K T V Grattan,A W Palmer. Self-mixing interferenceinside a single mode diode laser for optical sensing applications[J].IEEE Journal of Light wave Technology,1994,12(9):1577-1587.
[17]J A Smith.Lasers with optical feedback as displacement sensors[J].Optical Engineering,1995,34(9):2802-2810.
[18]R C Addy,A W Palmer.Effects of external reflector alignment in sensing applications of optical feedback in laser diode[J].IEEE J.Lightwave,1996,14(12):2672-2676.
[19]G Giuliani,M Norgia.Laser diode linewidth measurement by means of self-mixing interferometry[J].IEEE Photonics Technology Letters,2000,12(8):1028-1030.
[20]Kenju Otsuka,Kazutaka Abe,Jing-Yuan Ko.Real-time nanometer-vibration measurement with a self-mixing microchip solid-state laser[J].Optics Letters,2002, 27(15):1339-1341.
[21]G Plantier,C Bes,T Bosch.Behavioral Model of a Self-Mixing Laser Diode Sensor [J].IEEE Journal of Quantum Electronics,2005,41(9):1157-1166.
[22]丁迎春,张书练,李岩He-Ne激光自混合干涉的模式研究[J].激光杂志,2003,24(6):16-19
[23]G Liu,S L Zhang,J Zhu.Theoretical and experimental study of intensity branch phenomena in self-mixing interference in a He-Ne laser[J].Optics Communications,2003,221:387-393.
[24]G Liu,S L Zhang,T Xu.Optical feedback signals involving the misalignment information [J]. Optics and Lasers in Engineering,2006,44(6):567-576.
[25]张勇,姚建铨,王鹏等.LD泵浦单模绿光激光器自混合干涉效应的研究[J].光电子·激光,2003,14(7):679-682.
[26]禹延光,叶会英,姚建铨.激光自混合干涉位移测量系统的稳态解[J].光学学报,2003,23(1):80-84.
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