1. Introduction

Optical whispering gallery mode (WGM) microcavities [1–3] are widely used in lasers, sensors, nonlinear optics, and optomechanics [4–10] due to their ultrahigh quality factor, very small mode volume and high optical energy density. Various WGM microcavity morphologies, including microrings, microtoroids, microdisks, microcylinders, microcapillaries, microspheres, microbottles and microbubbles, have been studied. Among them, microcapillaries, microbottles and microbubbles have attracted attention due to their natural optofluidic channels [11–18]. Microbubble resonator (MBR)-based sensors have been developed in physical, chemical and biological sensing, such as for temperature, stress, pressure, flow rate, gas concentration, virus and biomolecules detection [19–24].

Because the WGM fields extend into the surrounding analytes, almost all the resonant modes vary with any changes occurring around the evanescent field. Its sensing mechanism is always to track and record the changes in a single mode, including resonance wavelength shift, mode broadening, and mode splitting [8,23–25]. The single mode tracking method has achieved excellent performance, including high sensitivity, good selectivity, and fast response time [21]. However, there are two main challenges that need to be addressed: one is relative measurement uncertainty, and the other is limited measurement range. First, the single mode tracking method is performed by monitoring the relative resonance shift generated by a target change with respect to a baseline, which must be calibrated previously and regularly due to its lack of references and fluctuations. Second, the mode tracking method often does not work when the mode removes out of the laser scanning range. Therefore, the mode tracking method cannot obtain the absolute measurement without calibration and always greatly limits the dynamic detection range. When only increasing the scanning range in the method, the dynamic range is improved at the expense of sensitivity and resolution. In fact, there is a fundamental contradiction between high sensitivity and large dynamic range for this method so that they cannot be achieved simultaneously.

Fortunately, multiple resonances in the WGM microcavity have been exploited to address the above challenges. An abundance of WGMs in microcavities, such as microbubbles and microspheres, often show different responses to target analytes. Therefore, a large number of WGMs can provide rich sensing information to increase the accuracy and measurement range simultaneously, which has been demonstrated in Ref. [26]. In the reference, Yang et al. have proposed the sensing mechanism, which transforms multiple resonant modes in MBR’s spectra into optical barcode. Prior to the actual measurement, a lookup table should have been built for the later pre-calibration by converting the spectra corresponding to all known temperatures as standard barcodes. Subsequently, by comparing this barcode generated by a particular measurement with the standard barcode in the lookup table, the modes with the best overlap are searched for and the actual temperature is determined. However, the scheme using a lookup table may consume excessive data resources to yield high accuracy. It is clear that the detection accuracy scales seriously with the amount of collected data. In practice, it is always a time- and resource-consuming process to collect a large amount of data. Additionally, the lookup-table approach always overlooks the differences in detection conditions between data in the standard database and the latest measurement data. For example, it is not guaranteed that the precalibration measurements to create a standard database and actual measurements have identical measurement circumstances, including the incident energy, coupling conditions and signal noise ratio in the detection system. Once there is a small difference present, the latest barcode often does not match the data of the lookup table very well so that the measurement error could become large. In addition, this method cannot achieve automated mass data processing in future measurements. Therefore, it is a critical issue to provide another alternative approach to effectively extract sensing information from multimode resonances.

As a powerful tool for information fusion and pattern recognition, machine learning algorithms have been widely used in many fields of spectral analysis [27–29]. The artificial neural network (ANN) automatically extracts features and builds a model from known large spectral datasets and helps the predictive model generalize beyond the training data [30–32]. Aided by a ANN, a relationship between the sensing information and target analytes can be established directly. The sensing information from multiple modes can be effectively fused to improve the detection characteristics, which have been referred to as multimode sensing in the literature [33–37]. However, this multimode sensing is based on modified microring resonators rather than WGM optical microcavities in a general sense. Moreover, the factors affecting the performance of multimode sensing have also not been investigated, and there is rarely an accurate and exhaustive comparison between single mode tracking and multimode sensing [31–41].

In this paper, the multimode sensing method combined with machine learning is studied based on a high-Q MBR. First, we theoretically confirm that multimode sensing contains more Fisher information than the single-mode tracking method. Second, an automated experimental system based on an MBR sensor is built for a great deal of spectral data acquisition. When the temperature is changed from 25.00°C to 40.00°C with a step size of 0.01°C, we collect multiple groups of transmission spectra, which constitutes the original dataset used in the paper. Finally, a GRNN is employed in multimode sensing to exploit their sensing qualities. The scheme achieves a high prediction accuracy with a root-mean-square error (RMSE) of 3.8 × 10⁻³°C in the large temperature range of 25.00°C to 40.00°C. Furthermore, in the following discussions, the conclusion is also drawn that the proposed multimode sensing method can more accurately predict the temperature on testing data from outside the temperature range of the training data.

2. Multimode sensing theory

In mode shift sensing, the uncertainty of the wavelength shift determines its limit of detection (LOD). For a continuously sampled spectral signal, in the presence of photonic noise with Poisson statistics, the variance of the estimated spectral shift is given by [44–45],

Its corresponding Fisher information $J(\Delta \lambda )$ equals $1/\textrm{CRB}$. This means that the variance of any such estimator is at least as high as the inverse of the Fisher information. Fisher information provides a way to measure the amount of information that a random variable contains about some parameter of the random variable’s assumed probability distribution. Here, a larger amount of Fisher information indicates a smaller CRB, which represents a higher achievable estimation accuracy of the spectral shift.

In experiments, transmission spectra can be collected by the photodiode (PD), and each transmission dip represents one resonance mode. The lineshape of a transmission dip can be fitted by a Lorentzian function as

To compare the performance between the traditional wavelength scanning method and the proposed multimode sensing scheme, we theoretically calculated the CRB of spectral shift estimation for three groups of spectra with different numbers of WGMs within the wavelength range. According to Eq. (3), the transmission spectra containing several modes are simulated numerically. Figure 1 (a) shows transmission spectra containing a single mode, two modes, and three modes, respectively. To make a fair comparison, we draw the spectrum with two modes (red curve) by adding the second mode into the spectrum with a single mode (black curve). Similarly, the spectrum with three modes (blue curve) is drawn by introducing the third mode in the spectrum with two modes (red curve). The FWHMs of three modes from left to right are 0.35 pm, 0.32 pm, and 0.36 pm, respectively. When three groups of spectra in Fig. 1 (a) are shifted from right to left by 0-120 pm, the CRBs of wavelength shift are calculated, where the wavelength range w is fixed as [-0.1 nm, 0.1 nm].

 

Fig. 1. Multi-mode information metrics and the flowchart of the proposed multimode sensing scheme. (a) Three groups of transmission spectra with different numbers of WGMs within the wavelength range: single mode, two modes and three modes present in the transmissions, (b) When three groups of spectra are shifted from right to left by 0-120 pm, dependence of the CRB on wavelength shifts. (c) A general overview of the proposed multimode sensing method.

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Figure 1 (b) shows the CRB versus spectral shift for three groups of spectra with different numbers of WGMs. The results show that CRB increases with a decrease in the number of modes when these modes gradually move out of the fixed wavelength range (-0.1∼0.1 nm). As the spectrum with three modes (blue curve in Fig. 1 (a)) is moved out to the right by 30-50 pm, the third mode gradually starts to move out of the wavelength range, and there are continuous increments of the CRB values. When the spectral shift is 60pm, the third mode completely disappears and only two complete modes are left within the fixed wavelength range. If two spectra contain the same number of complete modes, their CRB values are equal. Then, its CRB is equal to that of the spectrum with two complete modes (red curve) in Fig. 1 (a) for some values of spectral shift (0-20pm). Overall, the CRB of the spectrum including three complete modes is the smallest in all cases. Therefore, if there are more modes present within the wavelength range, we can achieve a lower CRB. A lower CRB implies the higher Fisher information and the target parameter can be estimated with a higher accuracy. These theoretical results confirm that the multimode sensing method provides higher Fisher information and holds the potential to achieve a higher-level quality characteristic than the single-mode tracking method.

Subsequently, a machine learning algorithm is introduced to fuse the data from multiple WGMs. Figure 1 (c) shows a general overview of the proposed multimode sensing method. Here, there are two phases to be developed: data preprocessing and model training. The data preprocessing includes data denoising and data generation. In the detecting system, there is frequency noise present in the obtained spectra due to fluctuations in the sweep frequency speed of the tunable laser. Hence, this noise should be removed from the original data as much as possible before use. In addition, it is generally accepted that an accurate prediction model by machine learning algorithms needs plenty of high-quality data. However, it is often difficult to obtain adequate data because of the limitations of many realistic conditions. One way that works well is to generate synthetic data by using machine learning algorithms (for example, generative adversarial networks (GANs)) to supplement training data [46]. In this paper, synthetic spectral data are not generated because there are enough high-quality data produced by the experimental systems. Subsequently, the dataset is divided into two sub-datasets: one part is used for the training dataset to train machine learning algorithms and fit the prediction model in the multimode sensing method; the other part is used as the testing dataset to evaluate the accuracy of the model. When the training error of the model converges to the goal error, the training process is stopped, and the machine learning model is obtained. Eventually, the training model is tested on the testing dataset. It is noted that the classification and regression problems in sensing may be done in the same manner.

3. Fabrication and characterization

The fabrication process of the MBR is as follows: Firstly, one end of a silica capillary is fused and sealed during the discharging operation with a fiber fusion splicer. Secondly, the sealed capillary is exactly repositioned in the middle between the two discharge electrodes of the splicer, and the discharge time and current are adjusted to suitable values. Finally, a syringe is used to concurrently inject air into the capillary tube from the other end while conducting the discharge. The portion of the capillary heated by discharge gradually expands to form a bubble under the pressure of the filled gas [47–48]. We can create a microbubble of the desired size by heating and inflating it repeatedly. The MBR is then placed in a homemade copper box that has the dimensions 3 cm long, 1.5 cm wide, and 1 cm high. As shown in Fig. 1(a), the temperature of the copper box is controlled to modify the MBR's temperature.

To characterize the MBR-based temperature sensing, a tunable laser at ∼1550 nm is used to excite the resonant modes of the MBR through a tapered fiber, as shown in Fig. 2(a). The wavelength of the laser was finely swept at a speed of 15 MHz/s. A fiber polarization controller (FPC) was used to adjust the polarization state of the input light. Figure 2(b) shows a picture of the MBR with a diameter of 340 µm and a wall thickness of 18 µm. The thermoelectric cooler (TEC) connected to the controller is placed on the surface of the copper box to change its temperature, where the thermistor sensor offers the input signals of the controller by measuring the difference between the real and set temperatures. With the aid of the closed-loop control system, the total device can accurately regulate the MBR temperature with a resolution of 0.01°C. The transmission spectra are recorded by a photodiode (PD). Then, the electrical spectral signals are divided into two channels: one channel is connected to an oscilloscope (OSC) for real-time observation; the other channel is related to a data acquisition (DAQ) card and a computer to collect these data automatically. Figure 2(c) shows the typically calculated distribution of the optical modes at the cross-section of the MBR. Figure 2(d) shows typical transmission in the microbubble with a lower input power of 9.6 $\mathrm{\mu}\textrm{W}$ at 26.00°C. The red line is Lorentz fitting with a linewidth of 0.14 pm, corresponding to the loaded Q factor of $1.07 \times {10^7}$. To avoid thermal and nonlinear effects in the MBR, a laser power of 9.6 $\mathrm{\mu}\textrm{W}$ is coupled into the MBR sensor.

 

Fig. 2. Schematic of the temperature sensor in a high-Q MBR. (a) Experimental setup. (b) The MBR with a diameter of 340 µm and a wall thickness of 18 µm. (c) The calculated distribution of the optical modes at the cross-section of the MBR. (d) Typical transmission spectra around the resonant wavelength of 1541.9328 nm at $T = 26.00^\circ \textrm{C}$.

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The LabVIEW experimental platform realizes the automation of laser parameter control, TEC control, and data storage, enabling us to automatically collect spectral signals when the temperature was varied from 25.00°C to 40.00°C with a change step of 0.01°C. When the MBR temperature is increased, its multiple WGMs in its transmission spectra exhibit different responses due to thermo-optic effects and thermal expansion effects. This implies that the temperature determines a group of unique transmission spectra. The laser wavelength was scanned precisely from 1541.88 nm to 1542.12 nm. A temporal interval corresponding to a minimum temperature change (0.01°C) is set to 1000 s to ensure that the steady-state spectra are obtained for each temperature value (see Appendix A for details). The dataset consists of 1501 groups of collected spectral signals at the corresponding temperature values. In addition, an experiment was conducted to verify the stability and reproducibility of the acquired spectral signals under different temperatures (see Appendix B for more details).

When the frequency of the tunable laser is scanned, the starting point in every sweeping cycle is not absolutely the same due to the control accuracy limit of the tunable laser, which would induce tens of MHz frequency shift between different sweeping cycle. Therefore, to improve the sensing performance, we built an experimental setup to reduce the random noise from the frequency dislocations in the transmission spectra. The wavelength is corrected by referencing an acetylene-stabilized fiber laser at approximately 1542 nm (stabiλlaser 1542, 194.369489384(5) THz) [49]. Figure 3 (a) shows the typical calibrated transmission from 25.00°C to 30.00°C with the step size of 0.5°C. For comparison purposes, the sensitivity and LOD were first investigated with regard to the single-mode tracking approach. The change in temperature causes a change in the size and refractive index of the MBR, both of which result in a change in the resonant wavelength. The resonance wavelength gradually blueshifts with increasing temperature. As shown by the dashed arrow in Fig. 3 (a), a resonance is selected for tracking to perform the temperature measurement. Figure 3 (b) is an enlarged diagram of the spectra with the selected resonance at several temperature values in Fig. 3(a). When the temperature is close to 30.00°C, the selected resonance lineshape has a large distortion due to the influence of another nearest neighbor resonance on it, and the line-width of the selected resonance becomes wide. Figure 3 (c) displays the selected resonance wavelength shift concerning temperature within the temperature range of [25.00∼29.50]°C. The single-mode tracking method obtains good linearity with a sensitivity of 9.535 pm/°C and a LOD as low as $1.5 \times {10^{ - 3}}$°C, where the uncertainty of the wavelength shift is considered to be one-tenth of its linewidth (FWHM = 0.143pm at 25.00°C). At 30.00°C, the FWHM of the selected resonance is 0.459 pm, we get a LOD of . To maintain uniformity of measurement, another resonance needs to be calibrated and tracked for detection, which adds additional measurement burden. Moreover, for single-mode tracking, the laser scanning range is always limited in sensing applications. In essence, there is a fundamental contradiction between high sensitivity and a large dynamic range for the single-mode tracking method. When keeping a limited wavelength-tuning range, it is easier for the resonance to run out of the tuning range with high sensitivity; On the contrary, in order to capture a large dynamic range, it is necessary to reduce the resolution of the spectral signal, which in turn inevitably leads to a reduction in sensitivity. Therefore, a small measurement range often limits its application [26,50–53].

 

Fig. 3. Temperature sensing using the single mode tracking method. (a) When the temperature was increased from 25.00°C to 30.00°C, the resonant mode shown by the arrow was used for tracking and detection; (b) enlarged diagram of the spectra with the selected resonance at several temperature values in Fig3 (a); (c) dependence of the wavelength shift on temperature variations.

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4. Multimode sensing assisted by GRNN

For the proposed multimode sensing method, different types of machine learning algorithms may have different capabilities and effects to fuse multimode sensing information. Hence, it is a critical issue to select a suitable algorithm to achieve good performance. The backpropagation neural network (BPNN) is the most commonly used machine learning algorithm for training feedforward neural networks. The goal of backpropagation is to adjust the weights minimizing the error, thus allowing the neural network to learn how to correctly map arbitrary inputs to outputs. Nevertheless, as BPNN is used in the multimode sensing method, every time BPNN is operated, its results fluctuate greatly, and a large number of experimental results show that BPNN cannot achieve the desired result. GRNN is a variation of the radial basis function (RBF) neural network. The GRNN has a stronger nonlinear mapping ability and learning speed than the RBF neural network. The network ultimately converges to the optimization regression with more data, and the prediction effect is often good as the sample data are small. As shown in Fig. 4 (a), GRNN generally consists of four layers: input layer, pattern layer, summation layer, and output layer. The spread parameter can regulate the number of neurons in the pattern and summation layers. GRNN represents an improved technique in neural networks based on nonparametric regression, and its performance depends on only the spread parameter. Figure 4(a) shows the workflow schematic employing GRNN for temperature prediction in the multimode sensing method. The input variables are these intensity values, and the output variables are the temperature labels in the training process or the prediction values in the testing process.

 

Fig. 4. Temperature predicted by a GRNN. (a) Workflow schematic employing GRNN for temperature prediction in the multimode sensing method. (b) Dependence of the mean squared error (MSE) on the spread parameter. (c) Distribution of squared error between actual values and estimated values by GRNN. (d) Dependence of MSE on wavelength range. (e)The MSE of temperature prediction by GRNN versus the temperature label interval corresponding to different datasets, where the inset shows an enlarged diagram of the curve indicated by the dashed box in (e). (f) Temperature prediction by GRNN for testing data from outside the temperature range of the training data, where the inset shows a comparison between the actual and predicted temperatures when the temperature range of the testing data is 0.20°C.

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The temperature measurement was performed using GRNN with the K-fold cross-validation technique. In the method, the dataset was divided into K folds, where one fold was picked out in turn as the testing data, and the remaining K-1 folds were used as the training data. As previously mentioned, the original dataset contains 1501 groups of spectral signals paired with the corresponding temperature. Here, K was set to 1501, i.e., only one group of spectral signal was singled out in turn from the dataset as the testing data, and the rest of 1500 groups of spectra were the training data. The mean squared error (MSE) is defined as the mean of the square of the difference between the actual and predicted values:

Using the 1501-fold cross-validation technique, the prediction temperature results show a distribution. As shown in Fig. 4(c), the square of the difference between the actual and predicted values is drawn in the box-line plots under different spread parameters. Most values of squared errors fall into the box. When the spread parameter is 2.5, the prediction results are optimal. Its box-line plot is explained in detail. The height of the box is $1.60 \times {10^{ - 5}}$, and it represents the range of these values. A horizontal line drawn in the middle denotes the median, and it is equal to $5 \times {10^{ - 6}}$, which means the “middle” number of these values. The median is much smaller than its MSE ($1.44 \times {10^{ - 5}}$). The reason is that a few “outliers” is located outside the box of the box plot, and they are indicated by some dots outside the box. The outlier’s value is located within the range of $4.53 \times {10^{ - 5}}$ to $2.50 \times {10^{ - 4}}$, and they are far worse than the results within the box. The training model's inaccuracy leads to the occurrence of these outliers. The problem can be handled by enhancing the accuracy of machine learning models in the future. In addition, we investigated the influence of the wavelength range of the spectral data on the measurement results. It is necessary to decide how much wavelength range is suitable to collect the spectral signals for the proposed multimode sensing method. In the article, the center wavelength of the spectrum was always set at 1542.00 nm, and the wavelength range was increased or decreased to evaluate the sensing performance. The larger the wavelength range is, the greater the WGM’s number, which provides more sensing information. Figure 4(d) shows the dependence of the MSE on the wavelength range of the spectral data. MSE achieves the lowest value when the wavelength range is greater than 0.24 nm. The reason is that the negative effect of the noises on the measurement results neutralizes the positive impact of the wavelength range greater than 0.24 nm. The wavelength range of 0.24 nm is sufficient to accurately measure the temperature in the multimode sensing method. Therefore, the wavelength range is selected as 0.24 nm (1541.88 nm∼1542.12 nm) in the remainder of the paper.

Furthermore, we have also investigated the effects of the amount of data used on the measurement results. The used data are constructed by selecting a part of the data from the collected original dataset. In particular, the data are chosen when the corresponding temperature label varies with the step size. Then, different data are obtained by adjusting the step size (temperature label interval), and a specific temperature label interval represents a newly constructed dataset. Subsequently, the temperature is predicted by a GRNN with K-fold cross-validation technique, and K equals the amount of data in each newly constructed dataset. Figure 4 (e) displays the MSE of temperature prediction by GRNN versus the temperature label interval corresponding to different datasets. When the temperature label interval is 0.01°C, the obtained dataset has the largest amount of training data and contains all the collected original data (1501 groups of spectral signals). In this case, the most accurate prediction model is obtained, and the minimum MSE is achieved. The larger the temperature label interval is, the smaller the amount of data and the worse the prediction accuracy. Therefore, the amount of data can be chosen to obtain the expected measurement accuracy. Subsequently, we have also tried to perform temperature prediction for testing data from outside the temperature range of the training data. Under these circumstances, the collected original dataset is repartitioned into two independent sections: the data within a temperature range used as the training dataset and the data outside the temperature range used as the test dataset. Here, the test dataset can also be adjusted by increasing or decreasing its temperature range, and the center temperature of the range is always set at 32.50°C. For example, for the collected original dataset (25.00°C∼40.00°C), when the temperature test range is set to 0.10°C, the testing dataset is chosen as the data whose temperature is between 32.45°C and 32.54°C, and the rest belongs to the training dataset. Figure 4(f) displays the MSE of the temperature prediction by the GRNN versus the temperature test range. As the test range is 0.10°C, we obtain a minimum MSE of $4.14 \times {10^{ - 5}}$. As the test temperature range increased, the MSE worsened. When the temperature test range is 0.20°C (32.40°C∼32.59°C), the comparison between the predicted and actual values is shown in the inset of Fig. 4(f). Although there are considerable differences between the prediction and actual values, its measurement results are far better than the results obtained by the scheme using a lookup table [26]. With this approach, the lookup table contains data that are used as the training data. If the testing data are not in the lookup table, and its predicted temperature value is obtained only by performing an approximate match for the lookup value. It implies that the proposed multimode sensing method has a considerable ability to promote generalization on the testing data from outside the temperature range of the training data. Compared with the lookup table-based scheme, it alleviates the demand for large amounts of data without sacrificing too much accuracy. The GRNN MATLAB code was run on a dedicated server with an Intel i7-8700 k processor and 64 GB of memory, and it took only 7 ms to run each test.

Finally, we comprehensively compare the sensing characteristics between the proposed multimode sensing method and the single mode tracking scheme based on the MBR-based temperature sensor. The single mode tracking scheme is always limited by the laser scanning range and various problems encountered in mode tracking. As we have seen, due to the impact of the adjacent resonance, the single mode tracking scheme achieves a sensitivity of 9.535 pm/°C and a low LOD of $1.5 \times {10^{ - 3}}$°C based on the selected resonance within a small measurement range of 25.00∼29.50°C. Even if the selected mode can remain unchanged in the tracking, the maximum temperature range is about 25.0°C for the tunable laser with a fine sweep range of 0.24 nm (Toptica Pro). The multimode sensing method simultaneously obtains a high accuracy of $3.8 \times {10^{ - 3}}$°C and a large measurement range of 25.00∼40.00°C. The absolute temperature measurement is performed by the multimode sensing method, while only the relative temperature values are measured by the single mode tracking scheme. Then, it is difficult to directly compare the LOD in the single mode tracking method and the accuracy in the multimode sensing method. In this case, we believe that multimode sensing can share the same low LOD with the single-mode tracking scheme. For the multimode sensing method, the measurement range is theoretically independent of the measurement accuracy, and an arbitrarily large measurement range can be obtained. Nevertheless, the multimode sensing method needs to collect more data than the single mode tracking scheme. It is well known that collecting large amounts of data is always difficult and wastes considerable resources and time. Consequently, in specific applications, it is necessary to balance the sensing performance and the required data. In addition, other machine learning algorithms need to be further exploited to enhance their ability to extract more multimode information, and synthetic data can be artificially generated information that can be used in place of part of real data to train models, as actual datasets are lacking in quality and volume. Overall, the proposed multimode method contains more Fisher information than the single-mode approach and has great potential to achieve better sensing characteristics.

5. Conclusion

We systematically investigated the multimode sensing method based on the MBR. An experimental system was established to automatically collect multimode signals of the sensor under various temperatures. The original dataset consists of a large number of spectral signals paired with their corresponding temperatures. The data in the original dataset provide high-capacity multimode information channels, and the proposed multimode sensing scheme contains more Fisher information than the traditional wavelength scanning method. Machine learning algorithms are employed to extract multimode sensing information. In this paper, GRNN is adopted to make temperature predictions. The proposed multimode sensing method has achieved a high accuracy of $3.8 \times {10^{ - 3}}$°C within the large temperature range from 25.00°C to 40.00°C. This study lays a good foundation for the development of intelligent sensing based on WGM-based resonators.

Appendix A: determination of the time interval for the change in temperature

When the temperature is changed from 25.00°C to 40.00°C with a step size of 0.01°C, the spectral signals are collected by the experimental setup in Fig. 5. The DAQ acquisition mode is set to continuous samples, and then the spectral signals are recorded continuously by scanning the laser frequency within the time interval for the change in temperature. During each time interval, we can obtain multiple groups of spectral signals, and the spectral signal at the last moment is regarded as the data corresponding to the current steady-state temperature fields.

 

Fig. 5. Frequency shift of a resonance versus time under different temperature change steps.

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To optimize the time interval for the change in temperature, we studied spectral change versus time under various temperature change steps ($\Delta \textrm{T}$). When the MBR is heated from 26.0°C to 26.1°C in the experiment, a resonance frequency shift versus time curve is displayed in Fig. 5. A simple analytical function can be found by performing experimental curve fitting. According to the obtained analytic expression, we can theoretically calculate the resonance frequency shift curves under several other temperature change steps. Figure 5 also shows the resonance frequency shift versus time curves for the cases of several temperature change steps. The larger the temperature interval is, the longer the stabilization takes. To obtain a large number of spectral signals within the temperature range of 25.00-40.00°C, the temperature change step $\Delta \textrm{T}$ is set to 0.01°C. According to the theoretical curve at $\Delta \textrm{T = }0.01^\circ \textrm{C}$ in Fig. Fig. 5, the resonance frequency shift has basically reached a steady state after 1000 s. Therefore, the time interval for the temperature change step is set to 1000 s in this paper.

Appendix B: stability and reproducibility of spectral signals

Here, three temperature values (25.000°C, 25.825°C, and 26.155°C) are randomly selected to investigate the stability and reproducibility of spectral signals. For each temperature, we have repeatedly recorded spectral signals five times, where the time interval between each acquisition is 3000 s. Figure 6(a) shows all the collected spectral signals under three temperatures. Principal component analysis (PCA) is implemented to transform high-dimensional spectral signals into two dimensions in Fig. 6(b). In two-dimensional PCA space, 15 groups of spectral signals become 15 points. The three clusters correspond to three temperatures. Within each cluster, 5 points almost overlap. The Davies–Bouldin index (DBI) can be introduced to estimate the cluster separation. For a dataset consisting of m clusters, DBI is defined as:

(5)$$\textrm{DBI} = \frac{1}{m}\sum\limits_i^m {\mathop {\max }\limits_{i \ne j} } \left( {\frac{{{\sigma_i} + {\sigma_j}}}{{d({c_i},{c_j})}}} \right), $$

where ${\; }{c_i}$ denotes the centroid of class i, ${\sigma _i}$ represents the average distance between ${c_i}$ and all elements in class i, and ${\; \; }$ $d({{c_i},{c_j}} )$ is the average distance between the centroids of ${\; }{c_i}$ and ${\; }{c_j}$. Here, the DBI value of 0.277 confirms that the three clusters are well separated from each other. It is proven that the automated experimental acquisition system in Fig. 6(a) can acquire spectral signals with high repeatability and stability.

 

Fig. 6. The evaluation for stability and reproducibility of spectral signals. (a) Five spectral signals obtained at different times for the cases of temperature T = 25.500°C, 25.825°C, and 26.155°C; (b) High-dimensional spectral signals at three temperature values were transformed into two dimensions in principal component analysis (PCA) space.

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Funding

Chinese contract project (KYY-HX-20210893, KYY-HX-20221007); State Key Laboratory of Advanced Optical Communication Systems and Networks (2020GZKF013); Natural Science Foundation of Zhejiang Province (LY16F050009, LY20F050009); National Natural Science Foundation of China (60907032, 61675183, 61675184).

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.

References

1. K. J. Vahala, “Optical microcavities,” Nature 424(6950), 839–846 (2003). [CrossRef]  

2. K. Watanabe, H. Y. Wu, J. Xavier, L. T. Joshi, and F. Vollmer, “Single virus detection on silicon photonic crystal random cavities,” Small 18(15), 2107597 (2022). [CrossRef]  

3. D. Yu, M. Humar, K. Meserve, R. C. Bailey, S. N. Chormaic, and F. Vollmer, “Whispering-gallery-mode sensors for biological and physical sensing,” Nat. Rev. Methods Primers 1(1), 83 (2021). [CrossRef]  

4. L. He, S. K. Ozdemir, and L. Yang, “Whispering gallery microcavity lasers,” Laser Photonics Rev. 7(1), 60–82 (2013). [CrossRef]  

5. T. J. Kippenberg, A. L. Gaeta, M. Lipson, and M. L. Gorodetsky, “Dissipative Kerr solitons in optical microresonators,” Science 361(6402), eaan8083 (2018). [CrossRef]  

6. C.-H. Dong, Y.-D. Wang, and H.-L. Wang, “Optomechanical interfaces for hybrid quantum networks,” Natl. Sci. Rev. 2(4), 510–519 (2015). [CrossRef]  

7. J. Liu, F. Bo, L. Chang, C.-H. Dong, X. Ou, B. Regan, X. Shen, Q. Song, B. Yao, W. Zhang, C. Zou, and Y.-F. Xiao, “Emerging Material Platforms for Integrated Microcavity Photonics,” Sci. China-Phys. Mech. Astron. 65(10), 104201 (2022). [CrossRef]  

8. Y. Zhi, X.-C. Yu, Q. Gong, L. Yang, and Y.-F. Xiao, “Single nanoparticle detection using optical microcavities,” Adv. Mater. 29(12), 1604920 (2017). [CrossRef]  

9. S. Wan, R. Niu, Z.-Y. Wang, J.-L. Peng, M. Li, J. Li, G.-C. Guo, C.-L. Zou, and C.-H. Dong, “Frequency stabilization and tuning of breathing soliton in Si₃N₄ microresonators,” Photonics Res. 8(8), 1342 (2020). [CrossRef]  

10. Y.-F. Xiao, C.-L. Zou, Q. Gong, and L. Yang, Ultra-High-Q Optical Microcavities (WORLD SCIENTIFIC, 2020).

11. Q. Lu, X. Chen, X. Liu, L. Fu, C.-L. Zou, and S. Xie, “Tunable optofluidic liquid metal core microbubble resonator,” Opt. Express 28(2), 2201 (2020). [CrossRef]  

12. Q. Lu, X. Chen, X. Liu, J. Guo, S. Xie, X. Wu, C.-L. Zou, and C.-H. Dong, “Opto-fluidic-plasmonic liquid-metal core microcavity,” Appl. Phys. Lett. 117(16), 161101 (2020). [CrossRef]  

13. Q. Lu, L. Liao, L. Chen, J. Guo, Y. Hu, Y. Zhang, C. Zou, X. Wu, and S. Xie, “Optical Supercontinuum Generation via Rotational and Vibrational Molecular Cavity Optomechanics,” Laser Photonics Rev. 17(2), 2200384 (2023). [CrossRef]  

14. G. Frigenti, L. Cavigli, A. Fernández-Bienes, F. Ratto, S. Centi, T. García-Fernández, G. N. Conti, and S. Soria, “Resonant microbubble as a microfluidic stage for all-optical photoacoustic sensing,” Phys. Rev. Appl. 12(1), 014062 (2019). [CrossRef]  

15. K. H. Kim, G. Bahl, W. Lee, J. Liu, M. Tomes, X. Fan, and T. Carmon, “Cavity optomechanics on a microfluidic resonator with water and viscous liquids,” Light: Sci. Appl. 2(11), e110 (2013). [CrossRef]  

16. Q. Lu, M. Li, J. Liao, S. Liu, X. Wu, L. Liu, and L. Xu, “Strong coupling of hybrid and plasmonic resonances in liquid core plasmonic micro-bubble cavities,” Opt. Lett. 40(24), 5842–5845 (2015). [CrossRef]  

17. L. T. Hogan, E. H. Horak, J. M. Ward, K. A. Knapper, S. N. Chormaic, and R. H. Goldsmith, “Toward real-time monitoring and control of single nanoparticle properties with a microbubble resonator spectrometer,” ACS Nano 13(11), 12743–12757 (2019). [CrossRef]  

18. N. Toropov, G. Cabello, M. P. Serrano, R. R. Gutha, M. Rafti, and F. Vollmer, “Review of biosensing with whispering-gallery mode lasers,” Light: Sci. Appl. 10(1), 42 (2021). [CrossRef]  

19. X. Jiang, A. J. Qavi, S. Huang, and L. Yang, “Whispering-gallery sensors,” Matter 3(2), 371–392 (2020). [CrossRef]  

20. Y. Wang, S. Zeng, G. Humbert, and H. P. Ho, “Microfluidic Whispering Gallery Mode Optical Sensors for Biological Applications,” Laser Photonics Rev. 14(12), 2000135 (2020). [CrossRef]  

21. X. Yu, S. Tang, W. Liu, Y. Xu, Q. Gong, Y. Chen, and Y. Xiao, “Single-molecule optofluidic microsensor with interface whispering gallery modes,” Proc. Natl. Acad. Sci. 119(6), e2108678119 (2022). [CrossRef]  

22. X. Zhao, Z. Guo, Y. Zhou, J. Guo, Z. Liu, Y. Li, M. Luo, and X. Wu, “Optical Whispering-Gallery-Mode Microbubble Sensors,” Micromachines 13(4), 592 (2022). [CrossRef]  

23. L. Labrador-Páez, K. Soler-Carracedo, M. Hernández-Rodríguez, I. R. Martín, T. Carmon, and L. L. Martin, “Liquid whispering-gallery-mode resonator as a humidity sensor,” Opt. Express 25(2), 1165–1172 (2017). [CrossRef]  

24. Z.-D. Peng, C.-Q. Yu, H.-L. Ren, C.-L. Zou, G.-C. Guo, and C.-H. Dong, “Gas identification in the high-Q microbubble resonators,” Opt. Lett. 45(16), 4440 (2020). [CrossRef]  

25. J. Zhu, S. K. Ozdemir, Y. Xiao, L. Li, L. He, D. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics 4(1), 46–49 (2010). [CrossRef]  

26. J. Liao and L. Yang, “Optical whispering-gallery mode barcodes for high-precision and wide-range temperature measurements,” Light: Sci. Appl. 10(1), 32 (2021). [CrossRef]  

27. D. Wang, M. Zhang, Z. Zhang, J. Li, H. Gao, F. Zhang, and X. Chen, “Machine learning-based multifunctional optical spectrum analysis technique,” IEEE Access 7, 19726–19737 (2019). [CrossRef]  

28. P. O. Bojang, T. Yang, Q. B. Pham, and P. Yu, “Linking singular spectrum analysis and machine learning for monthly rainfall forecasting,” Appl. Sci. 10(9), 3224 (2020). [CrossRef]  

29. W. Naku, C. Zhu, A. K. Nambisan, R. E. Gerald, and J. Huang, “Machine learning identifies liquids employing a simple fiber-optic tip sensor,” Opt. Express 29(24), 40000–40014 (2021). [CrossRef]  

30. D. L. Smith, L. V. Nguyen, D. J. Ottaway, T. D. Cabral, E. Fujiwara, C. M. Cordeiro, and S. C. Warren-Smith, “Machine learning for sensing with a multimode exposed core fiber specklegram sensor,” Opt. Express 30(7), 10443–10455 (2022). [CrossRef]  

31. A. V. Saetchnikov, E. A. Tcherniavskaia, A. V. Saetchnikov, and A. Ostendorf, “Deep-learning powered whispering gallery mode sensor based on multiplexed imaging at fixed frequency,” Opto-Electron. Adv. 3(11), 200048 (2020). [CrossRef]  

32. J. Lu, R. Niu, S. Wan, C.-H. Dong, Z. Le, Y. Qin, Y. Hu, W. Hu, C.-L. Zou, and H. Ren, “Experimental demonstration of multimode microresonator sensing by machine learning,” IEEE Sensors J. 21(7), 9046–9053 (2021). [CrossRef]  

33. D. Hu, C.-L. Zou, H. Ren, J. Lu, Z. Le, Y. Qin, S. Guo, C.-H. Dong, and W. Hu, “Multi-parameter sensing in a multimode self-interference microring resonator by machine learning,” Sensors 20(3), 709 (2020). [CrossRef]  

34. Y. Zhang, J. Lu, Z. Le, C.-H. Dong, H. Zheng, Y. Qin, P. Yu, W. Hu, C.-L. Zou, and H. Ren, “Proposal of unsupervised gas classification by multimode microresonator,” IEEE Photonics J. 13(2), 5800111 (2021). [CrossRef]  

35. B. Duan, H. Zou, J.-H. Chen, C. Ma, X. Zhao, X. Zhao, X. Zheng, C. Wang, L. Liu, and D. Yang, “High-precision whispering gallery microsensors with ergodic spectra empowered by machine learning,” Photonics Res. 10(10), 2343–2348 (2022). [CrossRef]  

36. Z. Li, H. Zhang, B. Nguyen, S. Luo, P. Liu, J. Zou, Y. Shi, H. Cai, Z. Yang, Y. Jin, Y. Hao, Y. Zhang, and A.-Q. Liu, “Smart ring resonator–based sensor for multicomponent chemical analysis via machine learning,” Photonics Res. 9(2), B38–B44 (2021). [CrossRef]  

37. A. Venketeswaran, N. Lalam, J. Wuenschell, P. R. Ohodnicki, M. Badar, K. P. Chen, P. Lu, Y. Duan, B. Chorpening, and M. Buric, “Recent Advances in Machine Learning for Fiber Optic Sensor Applications,” Adv. Intell. Syst. 4(1), 2100067 (2022). [CrossRef]  

38. L. V. Nguyen, C. C. Nguyen, G. Carneiro, H. Ebendorff-Heidepriem, and S. C. Warren-Smith, “Sensing in the presence of strong noise by deep learning of dynamic multimode fiber interference,” Photonics Res. 9(4), B109–B118 (2021). [CrossRef]  

39. Z. Peng, J. Jian, H. Wen, A. Gribok, M. Wang, H. Liu, S. Huang, Z. Mao, and K. P. Chen, “Distributed fiber sensor and machine learning data analytics for pipeline protection against extrinsic intrusions and intrinsic corrosions,” Opt. Express 28(19), 27277–27292 (2020). [CrossRef]  

40. V. V. Kornienko, I. A. Nechepurenko, P. N. Tananaev, and E. D. Chubchev, “Machine learning for optical gas sensing: a leaky-mode humidity sensor as example,” IEEE Sensors J. 20(13), 6954–6963 (2020). [CrossRef]  

41. B. Wang, L. Wang, N. Guo, Z. Zhao, C. Yu, and C. Lu, “Deep neural networks assisted BOTDA for simultaneous temperature and strain measurement with enhanced accuracy,” Opt. Express 27(3), 2530–2543 (2019). [CrossRef]  

42. S. M. Kay, “Fundamentals of Statistical Signal Processing: Estimation Theory,” Upper Saddle River, USA (Prentice Hall PTR, 1993, Chap 3).

43. H. Ren, C. L. Zou, J. Lu, Z. Le, Y. Qin, S. Guo, and W. Hu, “Dissipative sensing with low detection limit in a self-interference microring resonator,” J. Opt. Soc. Am. B 36(4), 942–951 (2019). [CrossRef]  

44. S. Karbasivalashani and A. Mafi, “Optimizing plasmonic grating sensors for limit of detection based on a Cramer–Rao bound,” IEEE Photonics J. 2(4), 543–552 (2010). [CrossRef]  

45. W. C. Karl and H. H. Pien, “High-resolution biosensor spectral peak shift estimation,” IEEE Trans. Signal Process. 53(12), 4631–4639 (2005). [CrossRef]  

46. M. Marouf, P. Machart, V. Bansal, C. Kilian, D. S. Magruder, C. F. Krebs, and S. Bonn, “Realistic in silico generation and augmentation of single-cell RNA-seq data using generative adversarial networks,” Nat. Commun. 11(1), 166 (2020). [CrossRef]  

47. M. Li, X. Wu, L. Liu, and L. Xu, “Kerr parametric oscillations and frequency comb generation from dispersion compensated silica micro-bubble resonators,” Opt. Express 21(14), 16908–16913 (2013). [CrossRef]  

48. J. Jiang, Y. Liu, K. Liu, S. Wang, Z. Ma, Y. Zhang, P. Niu, L. Shen, and T. Liu, “Wall-thickness-controlled microbubble fabrication for WGM-based application,” Appl. Opt. 59(16), 5052–5057 (2020). [CrossRef]  

49. R. Niu, M. Li, S. Wan, Y. R. Sun, S. M. Hu, C. L. Zou, G. C. Guo, and C. H. Dong, “kHz-precision wavemeter based on reconfigurable microsoliton,” Nat. Commun. 14(1), 169 (2023). [CrossRef]  

50. X. Xu, W. Chen, G. Zhao, Y. Li, C. Lu, and L. Yang, “Wireless whispering-gallery-mode sensor for thermal sensing and aerial mapping,” Light: Sci. Appl. 7(1), 62–330 (2018). [CrossRef]  

51. Y. Chen, J. Li, X. Guo, L. Wan, J. Liu, Z. Chen, J. Pan, B. Zhang, Z. Li, and Y. Qin, “On-chip, high-sensitivity photonic temperature sensor based on a GaAs microresonator,” Opt. Lett. 45(18), 5105–5108 (2020). [CrossRef]  

52. Z. Ding, P. Liu, J. Chen, D. Dai, and Y. Shi, “On-chip simultaneous sensing of humidity and temperature with a dual-polarization silicon microring resonator,” Opt. Express 27(20), 28649 (2019). [CrossRef]  

53. Z. Liu, L. Liu, Z. Zhu, Y. Zhang, Y. Wei, X. Zhang, E. Zhao, Y. Zhang, J. Yang, and L. Yuan, “Whispering gallery mode temperature sensor of liquid microresonator,” Opt. Lett. 41(20), 4649–4652 (2016). [CrossRef]