## 2 Demodulation Principles  
### Modulation Format  
PolSK can generally be categorized into dual-polarization-state polarization shift keying (2-PolSK) and multi-polarization-state polarization shift keying (M-PolSK) [18]. The fundamental form is the 2-PolSK format, utilizing the SOPs at the two ends of the Bloch sphere to represent binary 0 and 1, since they are the easiest to distinguish. After transmission through an optical fiber, the SOP of the received optical signal is analyzed using a polarization beam splitter and demodulation algorithm. M-PolSK uses two or more SOPs, allowing multiple bits of information to be transmitted simultaneously.  
The transmitter module of a typical 2-PolSK system is illustrated in Figure 1a. The laser source outputs a fixed SOP, which is then decomposed into two orthogonally polarized beams via a polarizing beam splitter (PBS). These two optical paths are modulated in parallel through an amplitude modulator (AM), which uses two intensity levels to modulate the input light, in order to represent the binary states \"0\" and \"1\". After modulation, a polarization beam combiner (PBC) merges the two optical signals into a single optical signal with orthogonal polarizations, which now carries the encoded information. This approach enhances the capacity of the communication system.  
The integrated communication-sensing system is shown in Figure 1b, containing both the transmitter and receiver modules. This integration enables distributed sensing to be performed using communication hardware, without adding, removing, or changing any part. The communication system transmits and receives data, which contains vibration information encoded into the optical signal via the optical fiber. The signal demodulation process involves first obtaining the detected parameter (e.g., differential voltage); then using the fast Fourier transform and bandpass filters to separate the low-frequency vibration signals and the medium-high-frequency communication signals; and then performing decision making using predetermined thresholds or calibration curves.  
The receiver module of the 2-PolSK system is illustrated in Figure 1c. The combined signal passes through a PBS, where it is again decomposed into two orthogonally polarized components. Each optical path is received by two photodetectors (PDs). These detectors convert the optical signals into electrical signals, with the voltage being proportional to the optical intensity, followed by a subtractor circuit. Alternatively, a balanced photodetector (BPD) can be used, and the output differential voltage signal is proportional to the change in the polarization azimuth. Frequency-domain filtering can separate the sensing and communication signals. Through threshold detection, binary values are assigned to reconstruct the original communication data.  
The deep integration method in this work should be able to extend to stacked modulation methods [19]. There should be no significant overlap even when the modulation type is the same between communication and sensing, since the sensing aspect occupies low frequencies while the communication aspect uses much higher frequencies (albeit with some lower frequency components due to the abrupt edges of the waveform).  
In terms of multiplexing technology, the deep integration method can still be employed. For example, wavelength-division multiplexing (WDM) and space-division multiplexing (SDM) can be deployed in the same system [20] with sensing. The sensing aspect only needs to use one channel of a multiplexing scheme, such as a single wavelength of a single optical path, and sensing can co-exist with communication without occupying additional bandwidth due to the nature of deep integration that uses a digital copy for separate data analysis.