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Abstract
Benefiting from the coupling between the Surface Nanoscale Axial Photonics (SNAP) microcavity and the waveguide, i.e., influenced by their abrupt field overlap, multiple axial modes in the transmission spectrum form a functional relationship with the coupling position, thus enabling displacement sensing. However, this functional relationship is complex and nonlinear, which is difficult to be fitted using analytical methods. We introduce a back-propagation neural network (BPNN) to model this functional relationship. The numerical results show that the multimode sensing scheme has great potential for practical large-range, high-precision displacement sensing platforms compared with the single-mode sensing based on the whispering gallery mode (WGM) resonators.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Whispering gallery mode (WGM) optical microcavities [1] have been widely studied and applied in the field of sensing due to their extremely high quality factor (Q factor) [2–4], small mode volume, and anti-electromagnetic interference, including biochemical detection [5], refractive index sensing [6], pressure monitoring [7], temperature sensing [8], and displacement sensing [9,10]. In displacement sensing, existing schemes are mainly implemented by detecting the wavelength shift [10], the change of the transmission intensity, or the extinction ratio of a single resonant mode [9]. The sensing strategy based on wavelength shift has a theoretical resolution better than 0.1 nm, but is sensitive to temperature fluctuations, making it difficult to obtain high accuracy. Due to the insensitivity of transmittance to temperature changes, displacement sensing based on transmittance changes has better temperature stability. However, high-sensitivity displacement sensing with a large range cannot be achieved using only a single mode.
Due to the challenges faced by existing sensing strategies, we previously proposed a scheme using multiple axial modes of the Surface Nanoscale Axial Photonics (SNAP) microcavity to realize displacement sensing [11]. Multimode sensing stems from the fact that the SNAP microcavity has multiple axial modes which have wide and different field distributions in the axial direction [12]. The transmission depth of each excited axial mode is determined by the relative position of the coupling waveguide and the SNAP microcavity. By comprehensively using multiple axial modes to realize displacement sensing, we theoretically demonstrate that the resolution can reach 20 nm. This sensing strategy is less affected by temperature and has the advantages of wide sensing range and high measurement accuracy. Although this new sensing scheme is theoretically feasible, it still faces huge challenges to demodulate the displacement from the spectrum composed of multiple resonant modes. Because of the complex and highly nonlinear mapping relationship between displacement and transmission depths of multiple axial modes, it is difficult for traditional nonlinear fitting methods (such as cubic spline interpolation, least square method, etc.) to give an accurate analytical formula.
A solution to overcome the aforementioned limitations is to introduce an artificial neural network (ANN), which is an efficient method for modeling non-linear characteristics of physical parameters. ANNs can be designed and trained to model input/output mapping from measured data. The resulting neural models can process highly nonlinear data with good stability and low sensitivity noise. The ANN has been widely used in many fields to solve difficult and diverse problems [13–15]. Particularly, in the field of microcavity applications, Hu et al. demonstrated a multi-parameter sensing in a multimode self-interference micro-ring resonator by an ANN [16]. Also, Li et al. realized a neural network model to predict the composition for multicomponent analysis using a single label-free ring resonator, with relatively high accuracy of the prediction results [15].
The main objective of this paper is to decode the spectral data to determine the displacement of SNAP microcavity relative to the waveguide. Therefore, we propose a SNAP microcavity multimode displacement sensing technology based on an ANN. The transmission spectrum is obtained by theoretically calculating the coupling between a SNAP microcavity and a tapered fiber, and the functional relationship between the transmission depths of multiple axial modes in the transmission spectrum and displacement is fitted by an ANN to achieve high-precision displacement measurement. This demodulation method can reduce the interference degree of noise, vibration and temperature on the demodulation result to a greater extent, and improve the accuracy of displacement sensing. Through these work, we verify the feasibility and effectiveness of the proposed sensing strategy, and make a positive contribution to the practical application of microcavity displacement sensing.
2. Structure and principle of sensing
As shown in Fig. 1(a), the displacement sensing system based on a SNAP microcavity is composed of a tunable laser, a coupled waveguide, a SNAP microcavity and a photodetector. The tunable laser operates near 1550.7 nm wavelength. The SNAP microcavity has a parabolic profile with a length of 400 µm and an effective radius variation (ERV) of about 25 nm, which can be formed by processing the surface of a single-mode fiber (SMF-28) by electrode discharge heating [17], CO2 femtosecond laser etching [18], or focused infrared light (CO2 laser) irradiation [19]. In addition, it can also be obtained by irradiating the photosensitive fibers using an ultraviolet laser (248 nm excimer laser) through an amplitude mask [19]. The coupling waveguide is a tapered fiber with a tapered waist diameter of about 1 µm, which can be fabricated by heating and stretching the single-mode fiber using a hydroxide flame or by etching the glass fiber using hydrofluoric acid [20]. As the core component of the sensing system, the SNAP microcavity and the tapered fiber are kept perpendicular to each other and in contact for coupling. Furthermore, the SNAP microcavity can move along its axis, while the tapered fiber remains stationary. During the operation of the system, the tunable laser is coupled into the SNAP microcavity through the tapered fiber. The light waves near the resonant wavelength are bound to the surface of the microcavity, while the non-resonant light waves are output from the tapered fiber and converted into electrical signals by a photodetector to obtain a transmission spectrum containing multiple resonant modes.
Fig. 1. (a) Structure of the displacement sensing system. (b) Transmission spectra at z1 = −180 µm and z2 = −100 µm. (c) Characteristic relationship between the transmission depths of q = 0 - 3 axial modes and displacement.
According to the coupling theory, a variation of the relative coupling position between the SNAP microcavity and tapered fiber will affect the coupling intensity, leading to changes in transmission spectrum. For this reason, we calculated the simulated transmission spectra at z1 = −180 µm and z2 = −100 µm using MATLAB software. The equation for the theoretical transmission spectrum of the SNAP-taper coupling system is given by [21].
Set each parameter according to Ref. [11]: |C|2 = 0.015 µm−1, S(0) = 0.95−0.01i, nf 0 = 1.452, γres = 0.05 pm, λres = 1550.7 nm. As we know, the real part of D will affect the shift of the resonance wavelength. Considering that the shift of the resonance wavelength has no effect on the sensing accuracy, for the convenience of data processing, we set the real part of D to 0 (D = 0.016i µm−1). The wavelength range of the transmission spectrum is λ = 1550 - 1550.7 nm, and the movable range of the SNAP microcavity is z = −200 - 200 µm. As shown in Fig. 1(b), the green curve and the red curve represent the simulated transmission spectra at z1 = −180 µm and z2 = −100 µm, respectively. It can be seen that when the SNAP microcavity is moved along the z-axis, different positions produce transmission spectra with different characteristics.
Due to the symmetry of the SNAP structure with a parabolic profile, only half the length of the SNAP structure can be applied to achieve sensing. By moving the SNAP microcavity at equal spacing along the z-axis direction to obtain a set of transmission spectra and extracting the mode features in the transmission spectra, the characteristic relationship between the transmission depth of each order of axial mode and displacement can be obtained. Figure 1(c) shows the transmission depths of the first four order axial modes as a function of the SNAP displacement. It can be seen that the transmission depth curve of the axial mode of order q = 0 - 3 decreases 1, 2, 3, 4 times and rises 1, 2, 3, 4 times respectively. The variation rule of the transmission depth of the axial mode of order q = 4 - 20 can be deduced in the same way. Because the relationship between transmission depth and displacement is symmetrical, only half of the length of SNAP microcavity (z = 0 - 200 µm) can be used to realize displacement sensing.
The number of excited axial modes can directly affect the performance of the displacement sensing system. As shown in Fig. 1(c), with the increase of q, the corresponding axial modal field becomes wider, and the sensing region with high sensitivity [the area of green dotted box in Fig. 1(c)] becomes steeper, giving rise to higher sensitivity and a better resolution. For the above reasons, the transmission depths of the first 21 axial modes in the SNAP microcavity are selected as the input of the demodulation model to achieve high-precision displacement detection. However, the functional relationship between displacement and these axial modes is highly nonlinear, which is difficult to be fitted by traditional analytical model methods. Since the back propagation neural network (BPNN) has strong nonlinear generalization ability, we use a BPNN model for multimode sensing to comprehensively utilize these effective sensing information.
3. Multimode displacement sensing based on BPNN
3.1 Training of BPNN
The effective sensing information (transmission depths of multi-order axial modes) resulting from SNAP microcavity displacement sensor system is processed by a BPNN. Theory and practice show that the BPNN with a hidden layer has the ability to approximate a continuous function in any closed interval [22]. As shown in Fig. 2, the BPNN consists of three layers, including an input layer, a hidden layer and an output layer, which are connected by weights. We take the transmission depths of the first 21 order axial modes in the transmission spectrum as the input vector X = (x1, x2, …, x21) of the neural network. According to the composition of the neural network in Fig. 2, the output displacement $\widehat z$ can be expressed as:
Signal processing by BPNN includes two processes: training of neural network and predictive output. The purpose of training is to minimize the loss function and get the optimal weight and threshold of neurons in the hidden layer. According to literature [16], the loss function can be expressed by the mean square error (MSE):
The data set required for neural network training and testing is calculated by Eq. (1). With the displacement z increasing from 0 to 200 µm, we acquired the resonance spectra of the SNAP microcavity at 1001 positions in 0.2 µm intervals (Δz = 0.2 µm). By extracting the transmission depths of the first 21 axial modes in each resonance spectrum, 1001 samples are obtained to be used as the initial training dataset. For the BPNN, the initial parameters are set as follows: (i) the number of neurons in the input layer and output layer is 21 and 1, respectively. (ii) the learning rate is 0.0001 and the training goal error is 1 × 10−9. The number of neurons in the hidden layer and other training parameters need to be adjusted and optimized through multiple trainings.
By training the neural network, the MSE of the training result was obtained as a function of the number of neurons in the hidden layer, the number of samples in the training dataset, and the training goal error, respectively, as shown in Figs. 3(a)–3(c). Since the training of the neural network has the drawback of possibly falling into the local optimal solution due to the randomness, the same training is repeated 1000 times and the minimum MSE value is taken as the training result [16,24]. As can be seen in Fig. 3(a), when the number of neurons in the hidden layer is greater than 20, the MSE is basically in a stable state and no longer decreases with the increase in the number of neurons. Moreover, Fig. 3(a) shows that the optimal number of neurons in the hidden layer is 25. Figure 3(b) reveals the gradual decrease of MSE as the number of samples in the training dataset increases. The training datasets with different sample sizes are obtained by varying the displacement interval (Δz) in calculating the resonant spectrum of the SNAP microcavity. The obtained training datasets are then fed into the neural network for training and it can be found that the MSE no longer decreases significantly with the increase of sample size when the number of samples in the training dataset is increased to 2251. Compared with the number of neurons in the hidden layer and the sample size in the training dataset, the training goal error has the opposite effect on MSE. As shown in Fig. 3(c), the MSE gets larger with the increase of training goal error, especially when the training goal error exceeds 1 × 10−9.
Fig. 3. (a) MSE versus the number of neurons in the hidden layer. (b) MSE versus the number of samples in the training dataset. (c) MSE versus the training goal error. (d) Comparison between the estimated values of the displacement and their corresponding theoretical values
We select the parameter value that minimizes MSE from Figs. 3(a)–3(c) as the optimal parameter of the neural network. As a result, the final optimized parameter values are as follows: The neuron number of 25 in the hidden layer, the sample number of 2251 in the training dataset, and the training goal error of 2.5 × 10−11. The test dataset that consists of many groups of transmission depths are obtained from the spectra calculated by Eq. (1). These corresponding transmission depths are subsequently input into the trained BPNN, and the network output is the predicted value of the displacement. In Fig. 3(d), the theoretical values of displacement are set within the range of 0 - 200 µm with the step of 6.003 µm, 5.333 µm, and 4.333 µm, respectively, and the predicted values of displacement make up of three lines from left to right. Comparing the predicted values with the corresponding theoretical values of displacement, it is clear that the predicted results do agree well with the theoretical value, where their total MSE achieves the value of 2.29 × 10−5. The simulation result implies that the displacement sensing strategy implemented using multiple axial modes can achieve high accuracy localization monitoring of the coupling position between the SNAP microcavity and the tapered fiber.
3.2 Effect of noise
In practice, displacement sensing systems need to withstand the interference of harsh environmental factors on the detection process, which manifest themselves as noise in the transmission spectrum. The noise is mainly divided into two categories: frequency noise and intensity noise [25]. The former leads to uncertainty in the resonant wavelength variation in the transmission spectrum, and the latter leads to variation in the transmission depth [5]. Since the displacement sensing strategy is implemented based on multiple modal transmission depths, the effect of frequency noise can be effectively attenuated by increasing the number of axial modes. Therefore, we only consider the effect of intensity noise on the measurement accuracy.
To simulate the effect of the real environment on the signal processing process, zero-mean Gaussian white noise [26] is added to the noise-free transmission spectrum corresponding to Fig. 3(a). The simulated transmission spectrum for a given signal-to-noise ratio (SNR) is obtained by adjusting the amplitude of the Gaussian white noise. As shown in Fig. 4(a) and 4(b), the SNR of the transmission spectra are 10 dB and 30 dB, respectively. It should be noted that the transmission spectrum with SNR = 30 dB is almost the same as the noiseless transmission spectrum, but the faint noise of the transmission spectrum can still be seen through the local magnification. In order to obtain the transmission depth of each axial mode in the noisy transmission spectrum, the Lorentz function often used in the optical spectrum, was applied here to fit the resonance dips in the transmission spectrum. The Lorentz function is:
Fig. 4. (a) Simulated transmission spectrum with SNR = 10 dB noise. (b) Simulated transmission spectrum with SNR = 10 dB noise. Inset: Magnified view of the resonant mode selected by the rectangle box. (c) Lorentz−fitted curve of the resonance dip (q = 18) in the transmission spectrum containing SNR = 10 dB noise. (d) MSE as a function of SNR.
Figure 4(c) shows the Lorentz fitting curve for a resonant mode in the spectrum containing 10 dB noise, and it can be seen that the fitting curve can significantly remove the interference of noise, thus improving the extraction accuracy of the transmission depth of the resonant mode. To investigate the effect of SNR on the MSE of the training results, we trained the neural network using multiple training datasets through changing the SNR value, and the training results are shown in Fig. 4(d). With the increase of SNR, the MSE value of the training results occurs a significant decrease. Note that, the MSE becomes flat and stabilizes at 6.79 × 10−7 with SNR exceeding 38dB, which is the same as the training result in Fig. 3(a). Additionally, the MSE curve has a small shift up and down centered on the red horizontal line, which is due to the random rise and fall defect in the neural network training.
4. Analysis and discussion
4.1 Error analysis
On the basis of the optimized BPNN, we input the test dataset without noise into the neural network to obtain the predicted displacement. The number of samples in the test dataset is 2001, and the displacement interval between two adjacent samples is 0.1 µm. By subtracting the real displacement from the displacement value predicted by the network, the prediction error of the network can be obtained.
As shown in Fig. 5(a), the prediction error is between −0.04 µm and 0.04 µm, but mainly distributed between −0.02 µm and 0.02 µm, which verifies the feasibility and effectiveness of SNAP microcavity displacement sensing based on neural network. It can be clearly found that the test error between 0 µm and 100 µm is larger than that between 100 µm and 200 µm, and multiple tests satisfy this rule. Moreover, in the displacement 0 - 100 µm, the prediction error distribution is relatively scattered, while in the displacement 100 - 200 µm, the error distribution is tighter and more uniform. Overall, the error distribution gradually decreases as the displacement increases.
Fig. 5. (a) The error between real displacement and predicted displacement. (b) Transmission depth versus displacement curves for all axial modes (q = 0 - 20). (c) Prediction error of displacement for the neural network trained on the training dataset from 10 µm to 200 µm (d) Expansion of figure (b). (e) Relationship curve between MSE of the training results and the moving endpoints of the training dataset for different displacement ranges.
In the case that the neural network structure as well as the parameters and the number of samples in the dataset are optimal, the key to good training results lies in the selection of the training dataset. Figure 5(b) shows the transmission depth versus displacement curves for the q = 0 - 20 axial modes. In order to see more clearly the distribution pattern of all modes at different displacements, we plotted all the curves overlapping together. It can be easily seen that the training dataset in the range of displacements from 0 µm to 10 µm is significantly different from the other displacement ranges. Among them, the high-sensitivity sensing regions of each axial mode in the 10 - 200 µm range are staggered with each other and cover all the positions in this range, while the slopes of the transmission depth curves of each axial mode in the 0 - 10 µm range largely equal to 0 around the displacement of 0 µm and part of the displacement range is not covered by high-sensitivity regions. The neural network was retrained after removing the training data in the 0 - 10 µm displacement range, and the test results are shown in Fig. 5(c). The prediction error of the neural network for the displacement was smaller than that of the previous model prediction, implying that the training data in the 0 - 10 µm displacement range has a bad effect on the overall fit.
As shown in Fig. 5(c), with the increase of displacement, the prediction error still gradually shrinks. The error near the displacement of 10 µm is larger than that near the displacement of 200 µm. To explain this phenomenon, the unfolding diagram of Fig. 5(b) is plotted as Fig. 5(d). It can be found that the number of high-sensitivity sensing regions [as depicted by the green dashed box region in Fig. 1(c)] increases as the order of the axial mode becomes larger from 0 to 20. At the same time, the slope of the relationship curve between transmission depth and displacement increases, i.e., the displacement sensing sensitivity becomes larger. Consequently, when the displacement increases from 0 µm to 200 µm the number of high-sensitivity sensing regions is decreasing and, in particular, near 200 µm only the high-sensitivity region corresponding to the q = 19 mode can be used for high-resolution displacement sensing. This means that, near the displacement of 200 µm, the weights of the neurons connected to the q = 19 mode are the focus of optimization to achieve the established training goal. Then, as the displacement decreases, although more high-sensitivity regions exist, only those neurons corresponding to the unused high-sensitivity regions are the focus of optimization. Therefore, as the displacement decreases from 200 µm to 10 µm in the neural network training, the order of the axial modes that can be effectively utilized at each displacement gradually decreases, leading to a gradual increase in the prediction error. For example, the displacement range B in Fig. 5(d) mainly relies on the axial modes q = 9 - 20 to achieve neural network fitting, then the displacement range A can no longer use these axial modes to achieve displacement prediction.
To further verify the reasonability of the above explanation, we trained the neural network using training datasets with different lengths of displacement ranges. As shown in Fig. 5(d), A and B are two displacement ranges with variable lengths. The left endpoint of the displacement range A is set at 10 µm and the right endpoint can be shifted, varying from 20 µm to 200 µm. Similarly, the right endpoint of displacement range B is set at 200 µm, and the left endpoint is movable from 10 µm to 190 µm. By varying the length of the displacement range of the training dataset in steps of 10 µm, the curves of MSE variation with the movable endpoint position of the varying displacement range are obtained through the training of the neural network, as shown in Fig. 5(e). The red and green curves indicate the training results for displacement ranges A and B, respectively. It can be seen that the MSE monotonically increases as the right endpoint of the displacement range A is shifted to the right. This is due to the fact that a small displacement range A utilizes the sensing region of higher order modes for displacement sensing, while as the displacement range A increases, parts of A can only utilize the sensing region of lower order modes for displacement sensing, resulting in a decrease in sensitivity and thus an increase in MSE. The same is true for the displacement range B. Therefore, the result suggests that the training and prediction accuracy of the neural network can be improved by reducing the range of the displacement sensing.
4.2 Experiments and discussion
The aforementioned results demonstrate that high-resolution displacement sensing based on SNAP microcavity multi-order modes can be achieved using a BPNN. However, the training and testing datasets are obtained only through theoretical calculations. To realize the experimental validation of the proposed sensing scheme here, the following three technologies are faced to be solved: (i) The precise control of the SNAP microcavity profile to ensure that a sufficient number of axial modes can be supported. (ii) The construction of a high-precision displacement scanning device and a real-time resonance spectrum data acquisition system to realize the rapid acquisition of a large number of resonance spectra required for neural network training and testing. (iii) The realization of a precision scanning laser with a large dynamic range to ensure that a sufficient number of axial modes are acquired without distortion. The first has been solved by techniques that precisely and repeatedly modify the effective radius of the fiber on the picometer scale [19,27]. The second is a common technique in electromechanics field and can be achieved by code control. The third one is currently more problematic.
It is well known that the maximum range of mode-hopping-free fine frequency scanning that can be achieved is currently about 30 GHz (New Focus TLB series lasers). This severely limits the number of SNAP microcavity modes that can be obtained at a fine scanning resolution. As before [17] [28], we used the TLB6728 tunable laser to test the SNAP microcavity fabricated by arc discharge. The SNAP resonator was moved axially from 0 µm to 429 µm and transmission spectra were collected in 1 µm steps. The ambient temperature was kept within 25 ± 0.5 °C throughout the test so that all monitored modes did not shift out of the scanning range. Figure 6(a) plots the transmission spectra at 4 typical positions from 118 µm to 121 µm, showing the evolution of the transmittance and Q factor for q = 0∼7 axial modes. The change in transmittance of resonant modes caused by unit displacement determines the resolution of displacement sensing. As can be seen from Fig. 6(a), the axial modes have different transmission depths and respond differently to the displacement. This is because they are in different coupling states. Clearly, the q = 4 and q = 5 modes are in the undercoupling state, accompanied by high Q fators and high sensitivities to the displacement. In contrast, the other modes are in a critical or overcoupling state, showing insensitivity to displacement.
Fig. 6. (a) Experimentally measured transmission spectra with SNAP displacement z = 118 µm∼121 µm, respectively. (b) Transmission depth versus displacement curve for q = 4 mode. (c) Simulated transmission depth versus displacement curves for the first 8 modes (q = 0 ∼ 7). (d) Errors between real displacement and predicted displacement using the first 8 modes and the first 21 modes under different noise intensities, respectively.
By extracting the transmission depth at each displacement, the transmission depth variation versus displacement for the q = 4 mode is obtained, as shown in Fig. 6(b). It can be seen that the displacement sensing sensitivity, represented by the slope of the curve, exhibits an approximately periodic variation with displacement. Only in the B region (undercoupling state), high-sensitivity displacement sensing can be achieved. However, it is clear that the high-sensitivity intervals in the B region are discrete, making continuous measurements impossible. To achieve high-sensitivity displacement sensing over a wide range, it is necessary to increase the number of modes used so that the high-sensitivity interval of each mode fills the entire range. Limited by the precise scanning range of the tunable laser, only the first 8 modes are scanned, and the high-sensitivity area of the first 8 modes cannot completely cover the entire range. Figure 6(c) is the curves of the transmission depth of the first 8 modes obtained by simulation as a function of displacement. Obviously, the slope of the curve in the interval selected by the dotted box is small, and high-sensitivity displacement sensing cannot be realized. Therefore, the experimentally obtained resonance spectrum containing only the first 8 modes cannot achieve full-range high-sensitivity displacement sensing by applying ANN.
To highlight the advantages of multi-order mode sensing, we used the first 8 and first 21 modes for displacement sensing validation under different noise intensities, respectively. As seen in Fig. 6(d), the error of displacement sensing using the first 21 modes is much smaller than using the first 8 modes. When the SNR is greater than 30 dB, the displacement measurement error tends to be stable. Note that, the above simulation results are all obtained by adding intensity noise to the resonance spectrum, and only considering the intensity noise can not reflect the actual experiment well. In fact, the resonant spectrum obtained by the experiment also includes frequency noise, polarization noise, light intensity fluctuation caused by imperfect tapered fiber, and so on. In order to simplify the calculation model and truly reflect the actual experiment, we further used the method of adding noise to the curve of transmission depth versus displacement to calculate the displacement sensing error. The noise contained in the curve in Fig. 6(b) is added to the simulated transmission depth curve, and the displacement errors obtained after ANN training are 3 µm (using the first 8 modes) and 1 µm (using the first 21 modes), respectively. Obviously, compared with the theoretical value, the displacement error still has a lot of room for optimization. In the future, displacement sensing errors will mainly be reduced by improving the quality of tapered fibers and using polarization-maintaining fibers. Moreover, subsequent studies will focus on the modulation properties of the SNAP microcavity profile on the axial modes to obtain a narrower FSR, which in turn ensures a sufficient number of axial modes in a smaller wavelength range.
Finally, it needs to be emphasized that our proposed displacement sensing scheme is robust to the temperature noise. The temperature insensitivity here is because the scheme is based on the change of transmission depth to realize displacement sensing. As we know, the change in temperature leads to an obvious shift of the resonant wavelength through the thermo-optic effect and thermal expansion effect, but causes little change in the transmittance, which has been experimentally verified many times including our previous studies. The shift of the resonant wavelength will bring huge interference to the commonly used systems for displacement sensing based on the resonant wavelength, and greatly reduce the accuracy of the displacement sensing. In comparison, the wavelength shift does not affect the transmittance extraction accuracy, so the displacement sensing scheme based on the mode transmittance can still maintain high accuracy when the temperature changes.
5. Conclusion
We propose a BPNN-based SNAP microcavity multimode displacement sensing technique. The displacement sensing technique is based on the functional relationship between the displacement and the transmission depth of the multi-order axial modes. A neural network is used to fit this complex functional relationship to achieve high-precision detection of the displacement. By extracting the transmission depths of the first 21 order axial modes in the spectrum as the input of the neural network, the displacement can be demodulated. Through training, the obtained optimal parameters of the neural network model are: the number of neurons in the hidden layer is 25, the number of samples in the training dataset is 2251, and the training goal error is 2.5 × 10−11. In this case, the prediction accuracy reaches ± 0.04µm. The feasibility and effectiveness of this method is verified by numerical simulation tests. In addition, we also investigated the characteristic relationship between the measurement accuracy of the sensing system and the intensity noise of the transmission spectrum using zero-mean Gaussian white noise, and the training results were close to the theoretical results with SNR exceeding 38dB. Through the analysis of test errors, we found the phenomenon that the displacement range covered by the sensing regions corresponding to the lower-order modes could not be predicted using the higher-order modes (e.g., q = 15 - 21), making the sensing accuracy lower. Although this new sensing scheme is theoretically feasible, it still faces a large challenge in experimental implementation, such as how to achieve scanning measurements of a sufficient number of axial modes in the absence of a fine-scanning laser with a large dynamic range, which is a prerequisite for high-resolution displacement sensing. Follow-up work will focus on this issue, and we believe this can be achieved in the future.
Funding
National Natural Science Foundation of China (61801129).
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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