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Optical microcavity transmission spectrum fitting algorithm based on the implicit function model
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摘要:
光学微腔品质因子高、灵敏度高，在精密生物传感方面有广阔的应用前景。针对洛伦兹拟合算法不能很好地拟合光学微腔输出端非对称波形和劈裂模式波形的问题，提出了隐函数模型算法。该算法首先建立模板波形，然后经平移、放缩理论实现模板波形操作，利用Levenberg-Marquardt (LM)算法优化参数值，能够实现对称波形、非对称波形和劈裂模式波形数据拟合。通过搭建光学微腔数据采集系统，采用高斯、洛伦兹和隐函数模型算法对不同折射率溶液的实验数据进行拟合。结果表明：隐函数模型算法比前两种算法的MSE低1个数量级，且拟合优度(R2)达到了0.99，拟合效果较好；隐函数模型算法谐振频率误差最小，谐振频率偏移量最大，对应的灵敏度最高，有利于提高光学微腔灵敏度。
Abstract:The optical microcavity has high Q factors and high sensitivity, and has a good application prospect in high-precision biosensing. In order to deal with the problem that the Lorentz fitting algorithm cannot fit the asymmetric waveform and the splitting mode waveform of the optical microcavity, the implicit function model algorithm is proposed. Firstly, according to the method, the template waveform was established and operated by panning and zooming.Then the parameter values were optimized by the Levenberg-Marquardt (LM) algorithm. Finally, data fitting of symmetrical waveform, asymmetric waveform and splitting mode waveform could be achieved. Through constructing the data acquisition system of optical microcavity, the Gauss, the Lorentz and the implicit function model algorithm were used to fit the experimental data of different refractive index of solutions. The results show that MSE of the implicit function model algorithm is one order of magnitude lower than other two algorithms, and has a coefficient of determination (R2) of 0.99. The resonant frequency error of implicit function model algorithm is the smallest, the resonant frequency of implicit function model algorithm is the largest, and the sensitivity of implicit function model algorithm is the highest. Therefore, the fitting effect of the implicit function model algorithm is better and it can efficiently improve the sensitivity of the optical microcavity.
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Key words:
- optical microcavity /
- implicit function model /
- LM algorithm /
- data fitting /
- transmission spectrum
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Abstract: Due to its high quality factor and high sensitivity, the optical microcavity has well promising applications in optical sensing, biomedical, nonlinear optics, environmental monitoring and quantum physics. The principle is that when analyses enter the optical microcavity, the effective refractive index of the solution will change, and the resonant wavelength will be shifted. Therefore, it is very important to find out the variation of resonant wavelength to improve the sensing accuracy of the optical microcavity. A traditional method to do this is using the Lorentz algorithm to fit the transmission spectrum of the optical microcavity. However, the Lorentz fitting algorithm cannot well fit the spectrum when it is an asymmetric waveform or there is a splitting mode waveform within the optical microcavity. In order to deal with the problem, the implicit function model algorithm is proposed in this study. The process of our method can be described as follows. The template waveform was selected and established first, followed by the panning and zooming operations. Then, a traditional method was used to set the initial value of the parameter of objective function, and the parameter values were optimized by the Levenberg-Marquardt (LM) algorithm, which could achieve data fitting results of symmetrical waveform, asymmetric waveform and splitting mode waveform. Note that there was no definite mathematical expression according to the implicit function model algorithm, so different methods were used to obtain the partial derivative of the factor in the Jacobian matrix by means of the template data. In this study, experimental platform, including the optical microcavity, tunable laser source and controller, data acquisition and control system, was established. Different concentrations of solutions of dimethyl sulfoxide, glucose and glycerol were tested as the analyte, and the Gauss, the Lorentz and the implicit function model algorithm were used to fit the experimental data of different transmission spectrums. The results show that MSE of the implicit function model algorithm is one order of magnitude lower than other two algorithms, and the coefficient of determination (R2) is 0.99. The resonant frequency error of implicit function model algorithm is the smallest, the resonant frequency of implicit function model algorithm is the largest, and the sensitivity of implicit function model algorithm is the highest. Therefore, the fitting effect of the implicit function model algorithm is better and it can efficiently improve the sensitivity of the optical microcavity and has a reliable basis on the follow-up to find the spectral resonance center to detect the biological components. The digital implicit function model algorithm will have a wide application prospect in any shape waveform data fitting.
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图 1 隐函数模型算法示意图.
Figure 1. The schematic diagram of implicit function model algorithm diagram.
图 3 光学微腔实验装置. (a)实验装置示意图. (b)显微镜下回音壁模式微腔与光纤耦合图.
Figure 3. The experimental device of optical microcavity. (a) Diagram of the experiment setup. (b) Photograph of the MBR coupling with a fused fiber, taken by microscope.
图 4 隐函数模型算法拟合100组水溶液数据所得调谐频率分布图.
Figure 4. The distribution map of frequency detuning obtained by 100 sets of water solution data fitted by implicit function model algorithm.
图 5 三种不同溶液下高斯、洛伦兹和隐函数模型算法SSE值比较图(IFM：隐函数模型). (a) DMSO RI=1.3526. (b)葡萄糖RI=1.3587. (c)甘油RI=1.3511.
Figure 5. Comparison of SSE values for Gaussian, Lorentz and implicit function model in three different solutions(IFM: Implicit function model). (a) DMSO RI=1.3526. (b) Glucose RI=1.3587. (c) Glycerol RI=1.3511.
图 6 高斯、洛伦兹、隐函数模型算法拟合三组溶液所得对比图(ED：实验数据；IFM：隐函数模型). (a) DMSO RI=1.3526. (b)葡萄糖RI=1.3587. (c)甘油RI=1.3511.
Figure 6. Contrast map of three solutions fitted by Gaussian, Lorentz and implicit function model(ED: Experimental data; IFM: Implicit function model). (a) DMSO RI=1.3526. (b) Glucose RI=1.3587. (c) Glycerol RI=1.3511.
表 1 高斯、洛伦兹、隐函数模型拟合算法调谐频率的比较.
Table 1. The comparison of frequency detuning among Gauss, Lorentz and implicit function model algorithm.
Solution Actual frequency detuning/GHz Frequency detuning/GHz Frequency detuning error/GHz IFM Gauss Lorentz IFM Gauss Lorentz DMSO RI=1.3526 -18.72 -18.7196 -18.7192 -18.7181 0.0004 0.0008 0.0019 Glucose RI=1.3587 -13.82 -13.8226 -13.7944 -13.7999 0.0226 0.0056 0.0001 Glycerol RI=1.3511 -17.76 -17.7703 -17.7931 -17.7949 0.0103 0.0331 0.0349 
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表 2 高斯、洛伦兹和隐函数模型算法拟合程度判断.
Table 2. The fitting degree of Gauss, Lorentz and implicit function model algorithm.
Solution MSE R2 IFM Gauss Lorentz IFM Gauss Lorentz Average 3.37×104 5.3×103 5.38×103 0.998075 0.970083 0.96955
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表 3 高斯、洛伦兹、隐函数模型拟合算法灵敏度的比较.
Table 3. The comparison of sensitivity among Gauss, Lorentz and implicit function model algorithm.
Solution RI Frequency detuning/GHz Frequency detuning offset/GHz Sensitivity /nm·RIU-1 IFM Gauss Lorentz IFM Gauss Lorentz IFM Gauss Lorentz DMSO 1.3430 -19.2211 -19.2335 -19.2395 0.6421 0.6087 0.6373 0.31089 0.2947 0.3085 1.3478 -18.5790 -18.6248 -18.6022 Glycerol 1.3600 -13.8210 -13.7969 -13.7908 1.4545 1.4446 1.4382 0.1888 0.1876 0.1867 1.3690 -12.3665 -12.3523 -12.3526 Glucose 1.3495 -17.7715 -17.7726 -17.7555 1.5317 1.5299 1.5125 0.2094 0.2092 0.2068 1.3665 -16.2398 -16.2427 -16.2430
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