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Abstract
Self-mixing interferometry (SMI) is a well-known non-destructive sensing technique that has been widely applied in both laboratory and engineering applications. In a laser SMI sensing system, there are two vital parameters, i.e., optical feedback factor C and line-width enhancement factor α, which influence the operation characteristics of the laser as well as the sensing performance. Therefore, many efforts have been made to determine them. Most of the existing methods of estimating these two parameters can often be operated in a certain feedback regime, e.g., weak or moderate feedback regime. In this paper, we propose a new method to estimate C and α based on back-propagation neural network for all feedback regimes. A parameter predicting model was trained and built. The performance of the proposed predicting model was tested using simulation and experiment data. The results show that the proposed method can estimate C and α with an average error of 2.76% and 2.99%, respectively. Additionally, the proposed method is noise-proof. The method and results are useful for extending the utilization of SMI technology in practical engineering fields.
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1. Introduction
Self-mixing interferometry (SMI) is a promising non-destructive sensing technology with advantages of compactness in system structure, low cost in implementation, and ease in optical alignment. The laser intensity and spectrum are modulated when the emitted light from the laser is back-reflected by an external target (reflector) and partly reenters the laser cavity, which is the well-known self-mixing effect [1]. Since King first reported the use of self-mixing effect for metrology [2], self-mixing theory has been extensively explored and researched in terms of the intrinsic physical nature and its applications. To date, SMI technology has been widely used in experimental and practical engineering environments for the measurement and sensing applications in the fields of physical quality measurements [3–7], mechanical and material metrology [8–11], and biomedical sensing [12–15], etc.
There are two important parameters in an SMI sensing system, i.e., the line-width enhancement factor α and the optical feedback factor C. The line-width enhancement factor α directly affects the spectral line-width of the semiconductor laser, the chirp under current modulation, the mode stability of the laser, the injection locking range, the optical amplification factor, and the behavior of an SMI system [16]. The optical feedback factor C is used to characterize the noise property in a laser [17]. It also characterizes the level of external optical feedback and output intensity waveform in a laser SMI system [1]. A laser SMI system can operate in different feedback regimes according to the value of C [18]. When C ≤ 1, the SMI system is in weak feedback regime, and the SMI signal almost appears in the shape of a sinusoidal wave. When 1 < C < 4.6, the SMI system is in moderate feedback regime, and the SMI signal fringes show asymmetric hysteresis and produces sawtooth-like fringe structure. When C > 4.6, the SMI system is in strong feedback regime, SMI fringes may lose and even totally disappear with a certain high value of C [19,20]. SMI technology has the same measurement resolution with the conventional two-beam interferometries, i.e., one interferometric fringe corresponds to a displacement of half laser wavelength for the target to be measured. While, for higher-resolution SMI based applications, C and α should be estimated in advance for some common algorithms, e.g., phase unwrapping method [21]. Therefore, it is of great importance to determine C and α in a laser SMI system no matter from the aspect of investigating the behavior of a laser or from that of the measurement and sensing applications.
Many researchers have made many efforts to estimate C and α in recent two decades. The widely-accepted methods are often based on the SMI signals although there are some other ones, e.g., the method that are based on relaxation oscillation frequencies in semiconductor lasers [22,23]. However, most of these SMI-based approaches are limited for certain feedback regimes. Also, measurement accuracy is sensitive to noise, hindering the applications in practical engineering. The milestone work [24] of estimating C and α proposed by Yu et.al in 2004 is based on the hysteresis of SMI signals in moderate feedback regime, which is only valid for 1 < C < 3.5. Then a data-fitting algorithm and its improved version were developed in [25,26] for weak feedback regime to ensure the accuracy of measurement results. The data-fitting method may also be used in moderate or strong feedback regime, but they are only applicable when the external target has a pure harmonic vibration. In 2011, a frequency-domain-based method was reported [27], but it is only for measurement of C. The method in [28] is for C > 1.5, and ones in [29,30] are for C > 1. More recently, the methods in [31,32] are developed for weak to strong feedback regime, but the noise performance were not discussed.
Computer neural networks are emerging tools for machine learning tasks. These neural networks have been applied in daily life and research fields and their validity and effectiveness have been proved. Recently, different neural networks have been adapted in the SMI fields. They have been used for identifying SMI fringe [33], de-noising SMI signals [34], and retrieving displacement [35] and blood pressure [36] from SMI signals. Back propagation neural network (BPNN) is one of the computer neural networks. The core characteristic of a BPNN is that the signal is transmitted forward and the error is transmitted backward, which has strong learning ability and good robustness, can create a highly nonlinear mapping based on gradient-decrease algorithm and is able to extract the complex relationship between variables [37]. Therefore, it fits well with the estimation of parameters in SMI systems. As a result, in this work, we use the BPNN to estimate C and α in a laser SMI system. The proposed method is able to be applied to all feedback regimes. Moreover, it is noise-proof, and does not require the external target has harmonic vibrations, making it suitable in practical engineering fields.
The remaining part of this paper is organized as follows. Firstly, we introduce the fundamental theory of SMI technology and BPNN. Then, the detailed procedure of estimating C and α using BPNN is presented. Afterward, we verify the proposed method by using simulation and experimental SMI signals. Finally, the measurement performance is discussed.
2. Fundamentals of SMI and BPNN
2.1 Fundamentals of SMI
The widely-accepted mathematic model for a laser SMI system can be derived from the well-known Lang-Kobayashi Equations [38] or three-mirror model [1] which is shown as below:
where ${\phi _F}(t)$ and ${\phi _0}(t)$ are the external light phases at the location of the target for the laser with and without feedback respectively, $P(t)$ and ${P_0}$ is the power emitted by the laser with and without optical feedback respectively, m is the modulation index (with typical values ∼10−3). The directly detected signal is usually the emitting power $P(t)$, whereas the measurands are often contained in ${\phi _0}(t)$. The typical displacement resolution of the SMI technique by fringe-counting is half laser wavelength (${\lambda _\textrm{0}}/2$). To achieve higher resolution than ${\lambda _\textrm{0}}/2$, further signal processing algorithms, e.g., the phase unwrapping method [39], are required. For these methods, C and α are needed to be determined in advance. The optical feedback factor C is defined as:2.2 Fundamentals of BPNN
Back propagation neural network is a multilayer feed-forward neural network, which has strong learning ability and can create a highly nonlinear model system and extract the complex relationship between variables [40]. BPNN consists of many neurons and is generally divided into three layers. i.e., input, hidden and output layer. The input layer obtains information from the outside world, and transfers the information into the middle hidden layer. The middle layer can be composed of a single hidden layer or multiple hidden layers. The signal then passes to the output layer after the calculation of each neuron in the middle layer. The characteristic of a BPNN is that the signal is transmitted forward and the error is transmitted backward. A certain relationship between input and output is found through training and learning of samples, realizing a complex nonlinear mapping from n-dimension to m-dimension. Therefore, it fits well with the estimation of parameters in SMI systems. When using simple neural network with only one middle layer, there may be problems such as over-saturation of neurons, large amount of data calculation and inability to calculate quickly. Compared with single hidden layer neural network, deep layer neural network has better network generalization ability and higher data processing accuracy [37,40]. In order to improve the accuracy of the model, it is necessary to improve the simple neural network to meet the needs of multiple-data processing. In this work, we adapted the deep BPNN. Figure 1 shows the basic structure of a deep BPNN that contains several hidden layers.
3. Proposed method
3.1 Signal preprocessing
We use MATLAB and python to carry out the algorithm for proposed method. Figure 2 is the procedure chart. Firstly, the characteristic parameters are extracted from the SMI signals as the training data. Then, the data are normalized to limit the data within a certain range. After that, the BPNN is trained, and C. Finally, C and α are predicted by using the trained BPNN model.
In order to improve the simulation efficiency, we firstly do preprocessing for the original SMI signals, i.e., finding some special points in the SMI signals as the eigenvalue points. The detailed procedure of obtaining eigenvalue points is as below:
- 1. Down-sample the SMI signals with a down-sampling ratio and apply a five-point average smoothing filter to the original signal. Note that, the down-sampling ratio is adapted according to the number of sampling points in a piece of SMI signal to facilitate the determination of eigenvalue points in step 3. For different vibration amplitudes and frequencies, the down-sampling ratio can be adjusted accordingly. In the example of Fig. 3, it was taken with 1/5.
- 2. Divide the SMI signals after step 1 into ten pieces based on the time length of the signal equally.
- 3. Find the maximum points of each piece of data by using the SciPy function in Python, and the minimum points by data flipping. Then, also use the Python script to find all the zero-crossing points in the whole SMI signals after step 1.
- 4. Down-sample the original SMI signals with a ratio of 1/100 to get an over-down-sampled SMI signal. Note that, the down-sampling ratio here influences the training time and the measurement performance of the BPNN model. It can be adjustable based on the practical measurement situation, e.g., the vibration amplitude and frequency. In the example of Fig. 3, it was taken with 1/100.
Fig. 3. SMI signal with α = 4, C = 4, SNR = 10 dB. (a) the original simulated SMI signal, (b) maximum, minimum and zero-crossing points obtained after down-sampling and smooth filtering (c) down-sampled SMI signal with a ratio of 1/100.
As a result, for an SMI signal, the eigenvalue points include two parts. The first part is the maximum, minimum and zero-crossing points obtained in step 3. The other part is the down-sampled signal obtained in step 4, which is used to remain more characteristics of the original SMI signals. Figure 3 shows an example of determining the eigenvalue points in an SMI signal with noise level of SNR = 10 dB, where Fig. 3 (b) shows the obtained eigenvalue points in step 3 and Fig. 3(c) is the other part of the eigenvalue points by down-sampling the original SMI signal as done in step 4. All the eigenvalue points in steps 3 and 4 are then normalized to the range of [-1, 1], and then used to train the BPNN model.
3.2 Predicting model training and simulation results
In order to build an effective BPNN predicting model, the BPNN is required to be trained. The procedure of training a BPNN is as follows. Firstly, initiate the BPNN, i.e., determine the initial weights. Then we compare the predicted value with the true value after the training signal is propagated forward. If the expected output is not obtained after the signal is propagated forward. It will be converted to the error back propagation process. Finally, minimize the error to update the weights and thresholds, and retrain the network.
We used MATLAB to build the predicting BPNN model. The eigenvalues of SMI signals are the inputs, C and α are outputs. To train the neural network better, large simulated dataset was prepared for training and testing the BPNN model. Considering the practical situations, some noises were also added with preset SNRs. The dataset was obtained from the simulations on the SMI model. We set different C, α and SNR, and then got the dataset for training the neural network. The parameter C ranges from 0.1 to 10 with a step of 0.1; α ranges from 2 to 7 with a step of 0.1; and four SNRs, i.e., 10, 15, 20 and 25 dB, are adapted. Therefore, we got a dataset with 20400 SMI signals. The training dataset consists of 15000 SMI signals, while testing dataset contains 5400 examples of SMI signals. Note that the training and testing contains different kinds of displacements, including harmonic vibrations with different amplitudes and frequencies, and even arbitrary displacements, which means the proposed method is applicable to not only harmonic vibrations but also arbitrary displacement.
The accuracy of the BPNN is mainly restricted by the number of hidden layer neurons. If the number of hidden layer nodes is too small, the BPNN cannot establish the correct mapping relationship, resulting in large network predicting errors. Conversely, if there are too many hidden layer nodes, the learning time of the network will be greatly increased, and the phenomenon of “over-fitting” may occur, that is, the trained model has high accuracy in the training set, but in the test set, the realization is very different from the training set, which is also called the poor generalization ability of the model. The number of neurons in the hidden layer is usually determined by the number of input and output layers, and it is an empirical constant [40]. Based on our simulation, the best performance is achieved when the number of hidden neurons is set at 25. Table 1 shows the parameters of the neural network taken for the simulations. Since the weight initialization is random, the model needs to be trained many times to get a smaller error. After repeated testing and training, the BPNN has good predicting ability. The execution time of the training procedure is nearly 2 hours by using a common-used laptop with a CPU (AMD R7 5800H). Once the prediction model is trained and built successfully, the model can be saved for the next prediction, and the predicting will be completed within several seconds.
Table 1. BPNN parameter setting
Table 2 shows the measurement results for the simulated SMI signals by using the trained predicting BPNN model. It can be seen that C and α can be estimated with maximum relative error of 4.62% and 3.06% respectively. Figure 4 shows the relative errors of different preset C and α. Figure 5 shows the estimated C and α for all the testing SMI dataset with respect to the preset true values. It can be found that the estimated values are in good coincidence with the true values, with an average error of 2.76% and 2.99% for C and α respectively. After estimating C and α, we use the phase unwrapping method in [21] to retrieve the displacement (D) also as shown in Table 2. It can be seen that the retrieved displacement has a relative error less than 8%. Then Table 3 presents the measurement performance when the simulated SMI signals with different noise levels with a preset α = 5.0, C = 3.0, D = 2λ0. It can be seen that the predicting performance is better when the SMI signal has a higher SNR. Even when the SNR is low as to 10 dB, the relative error of the measurement results is low as to 4.40% and 3.85% for α and C, respectively. The displacement was still retrieved by the same method in [21], and it shows that the measurement error of D is 7.50% even with SNR = 10 dB. Note that at least 3 fringes are needed in a testing sample for predicting C and α. Actually, the SNR of 10 dB is a severe condition in the actual experimental situation. It can be seen that the signal in Fig. 3(c) is quite distorted, but C and α can be still predicted as shown in Table 3. Even when SMI signals suffering speckle noise, a common noise that leads to fluctuations of the amplitude of the SMI signals [41], the proposed method is still applicable because amplitude of SMI signals in a small range may be approximately considered as constant.
Fig. 5. The relationship between the estimated values and true values for all the simulation testing dataset, (a) optical feedback factor C, (b) line-width enhancement factor α.
Table 2. The estimated results with simulation signals
Table 3. The estimated results with different noise levels
4. Experiment
The SMI experimental setup is shown in Fig. 6. The laser diode (LD) is biased with a DC current above the threshold by using a combined LD current and temperature controller (Thorlabs, ITC4001). A lens is used to focus the light emitted by the LD on the piezoelectric transducer (PZT), (Thorlabs, PAS009), which is assemble on a linear stage and used as the target. An optical variable attenuator (VA), (Thorlabs, NDC-50C-2M-B) is used to adjust the feedback strength to get different optical feedback factors. The SMI signals are detected by the monitor photodiode (PD) and the followed detection circuit. The SMI signals are then acquired by a personal computer via a data acquisition card with sampling frequency of 200 kHz. In the experiment, we tested two LDs, i.e., HL8325G from Hitachi and DL4140-001S from Sanyo. The PZT was driven by the PZT driver with a sinusoidal driving signal with a peak-peak value of 3.5 V. The temperature of the LD was maintained at 25°C±0.1 °C by the LD controller. The above experiment conditions were remained fixed throughout our experiments. Figure 7 shows a group of typical experimental SMI signals with different optical feedback factors.
After getting the SMI signals, we then used the trained BPNN model to estimate C and α for the experimental SMI signals. We firstly divided each SMI signals into ten pieces, and then applied the proposed algorithm to each piece to yield estimated parameter values for C and α. Finally, we took the average values of the ten estimated values as the final estimated C and α. Also, we measured the accuracy of the estimation. The final estimation results and their associated accuracy are given in Table 4. It shows that the estimated values of α are almost the same for the same laser diode even though it may be slightly related to the optical feedback factor. We use the ratio of standard deviation (SD) to average value (AVG), i.e., SD/AVG to evaluate the measurement uncertainty and accuracy. It can be seen that the estimated results of the trained predicting BPNN model for the experiments data have good consistence and accuracy. Also, we compared our measurement results with those by method in [24] and [25] as in Table 4, which shows that the results by these three methods are close to each other except some situations that method in [24] is not applicable. During the experiments, the PZT was driving by a sinusoidal voltage with frequency of 200 Hz and peak-peak value of 3.5 V. For the used PZT, each 0.1 V generate a displacement of 53 nm. As a result, the displacement of the PZT was 1855 nm. After we obtained values of C and α, we used the phase unwrapping method in [21] to retrieve the displacement. Taken Group 2 of the LD HL8325G for the example, the maximal error of the retrieved displacement is about 45 nm, 60 nm and 55nm by using the obtained C and α from the proposed method in this work, in [24] and [25] respectively.
Table 4. The estimated results with the experimental SMI signals
5. Conclusion
This paper proposed a new method for estimating optical feedback factor C and line-width enhancement factor α in a laser SMI system based on BPNN. The training and testing dataset were firstly obtained by simulating the SMI mathematic model. After the BPNN predicting model was trained by using the training dataset, the testing dataset from the simulations was then applied to evaluate the performance of the trained model. The results show that the proposed method can estimate C and α with an average error of 2.76% and 2.99% respectively. Then, we built an experimental SMI system with two laser diodes. Different experimental SMI signals were taken as the testing dataset. The estimated results show good consistency and accuracy. The proposed method is able to be applied for all SMI feedback regimes, i.e., weak, moderate and even strong feedback regime, and it is also noise-proof. The results in this work are useful for extending the utilization of SMI technology in practical engineering fields. Note that, fringe loss may happen when a laser SMI system operates in strong feedback regime, in this case, the trained BPNN model may suffer unexpected measurement errors and it sometimes may be totally invalid for the measurement in some extreme conditions. The problem of fringe loss in the process of parameter prediction needs to be carefully addressed in the future.
Funding
National Natural Science Foundation of China (62005234); Scientific Research Foundation of Hunan Provincial Education Department (20C1791).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request
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