## Methods  
### Fiber-Optic Interferometry and Signal Processing  
Light interference in optical fibers was selected as an underlying phenomenon because it is highly sensitive and allows frequency changes from several Hz to several kHz to be monitored. Figure 1 shows a simplified scheme of the interferometric sensor used based on the two-armed Mach-Zehnder interferometer. A passing railway vehicle near a sensor (Figure 1) causes a response (\(\omega\)) with low frequencies in the range of several Hz up to hundreds of Hz [35; 36; 37]. This vibration and acoustic response excite the measurement fiber of the interferometric sensor and cause a change in its optical length, or more precisely, the product of its refractive index nand geometric length \(L\). The resulting phase change \(\Delta\phi\) is given by the phase changes of the light source (\(\Delta\phi_{1}\)) and the changes in the length of the arm and its refractive index (\(\Delta\phi_{2}\)). The phase changes of the source (\(\Delta\phi_{1}\)) occur in both arms of the interferometric sensor (reference \(L_{2}\) and measurement \(L_{1}\) arms), and one is subtracted from the other, while the phase changes due to a vibration source (\(\Delta\phi_{2}\)) that contain information about measured frequencies (\(\omega\)) caused by a passing railway vehicle only occur in the measurement arm of the interferometric sensor. In the presented solution, the evaluation unit consists of several parts: a laser (MCLS1 Thorlabs, wavelength 1550 nm) with a reference power of 1 mW; a photodetector (Thorlabs PDA10 CS-EC, InGaAs 700-1800 nm); an analogue-digital (A/D) converter, used as a measuring card (NI USB 6210 National Instruments, Austin, TX, USA); an analogue first-order high-pass (HP) filter with a corner frequency of 2 Hz; and a computer (PC). The presented solution (sensor and evaluation unit) was based on conventional single-mode G.652 optical fibers. A simplified block diagram of the sensor system's detection comp  
Figure 1: A simplified scheme of the interferometric sensor based on the Mach–Zehnder interferometer. Optical signals from the reference (\(L_{2}\)) and the sensing (\(L_{1}\)) arms of the interferometric sensor merge with a coupling ratio of 2:1 in the optical coupler (SMF-28E, 9/125 \(\upmu\)m fiber, 3.0-mm jacket protection, length of individual arms 1.5 m), whose output is connected to a photodetector (PD). The resulting intensity of optical radiation falling on the photodetector is sufficient for the phase change between the light beams scattered through the interferometer arms. The output of the photodetector is a voltage signal (with sufficient overall power for collected light, i.e., the phase), which can be described as the sum of two signals:  
\[u(t)=p(t)+e(t), \tag{1}\]  
where \(p(t)\) is the desired signal (vibration and acoustic response) detected by the interferometric sensor and \(e(t)\) is the error signal caused by the instability of light polarization and external influences (temperature). The error signal can be described as a slow change in phase with a frequency that is one order of magnitude less than the lowest frequency of the desired signal \(p(t)\). Because of this difference, the first-order high-pass (HP) filter mentioned above with a corner frequency of 2 Hz can be used to separate \(e(t)\) from \(p(t)\). The use of an analogue filter before the A/D converter with the stated parameters seems to be optimal, as it improves the dynamic range of measurement and practically leads to removal of the DC component and low frequency components (signal \(e(t)\) from the measured signal \(p(t)\).) The output intensity of the interferometer is expressed by the following equation:  
\[I=2I_{0}\left\{1+\cos\left[\frac{2\pi}{\lambda}n\left(L_{1}-L_{2}\right) \right]\right\}, \tag{2}\]  
where \(L_{1}\) and \(L_{2}\) indicate the length of the measurement and reference arms of the interferometer, \(I_{0}\) is the medium signal value, and \(\lambda\) is the wavelength of the radiation source.