Fig. 1. Schematic of ergodic spectra sensing for pressure measurement, including (a) the experimental setup and (b) data analysis process. VOA, variable optical attenuator; FPC, fiber polarization controller; PD, photodetector; OSC, oscilloscope; AFG, arbitrary function generator.

Fig. 2. (a) Microscope images of the hollow silica capillary (left) and the fabricated MBR (right); (b) measured spectra around the resonant wavelength at 778.9532 nm; (c) transmission spectra of the MBR with the wavelength ranging from 778 to 780 nm; (d) long-term stability of MBR (inset, Allan variance of resonant wavelength); (e) dependence of the wavelength shift on pressure variations; (f) real-time wavelength shift with progressively increasing and decreasing pressure.

Fig. 3. (a) The transmission spectra are collected as raw data to form a database. (b) Conversion of transmission spectra to matrix, where the m and n are the number of samples and input neurons, respectively; (c) schematic diagram of the fully connected three-layer perceptron neural network used for ergodic spectra sensing; (d) MSE over different epochs; (e) effects of neurons’ number in hidden layer on prediction accuracy; (f) learning rate versus prediction accuracy.

Fig. 4. Pressure predicted by a fully connected three-layer perceptron neural network. (a) Dependence of predicted pressure on ground truth; (b) histogram of prediction errors of 3150 testing samples in (a); (c) comparison between predictions and ground truth of arbitrary unknown pressure; (d) histogram of prediction errors of 1650 testing samples in (c).

Fig. 5. (a) The prediction accuracy of 98.94% is obtained by extracting global features of a full spectrum with a fully connected three-layer perceptron neural network. (b) As a comparison, the prediction accuracy of only 87.43% is obtained by simply employing the resonant wavelength of multiple modes with a fully connected four-layer perceptron neural network.