Surface plasmon resonance (SPR) technology features high sensitivity, fast response speed, and stable measurement results, and is widely applied in the fields such as biomedical and nanomaterials detection. According to the signal modulation methods, SPR sensors are classified into four types of intensity modulation, wavelength modulation, angle modulation, and phase modulation. Compared with other modulation methods, phase modulation SPR has higher sensitivity. When SPR occurs, the phase of the p-polarized light changes greatly near the resonance angle, and its detection sensitivity is usually two-three orders of magnitude higher than that of other types of SPR sensors. However, previous literature reported phase jumps in the calculated phase sensitivity of prism-structure SPR, and the maximum sensitivity obtained from this calculation is not achievable in practical measurements. Further research and analysis prove that the appearance of jumps is due to the utilization of the arctangent function in calculating the phase angle. When the phase value of an angle infinitely approaches -π, the subsequent angle's phase result jumps to π, which causes an increase in the phase differences between the two angles and the occurrence of phase jumps. We solve the phase jumps by introducing the phase unwrapping function during calculating the phase angle and correct the calculation results of the phase difference of three gold films with different thicknesses. At the same time, the SPR phase demodulation method based on the Mach-Zehnder interferometer is optimized, and the correctness of the phase unwrapping method is experimentally verified. Our proposed method is expected to provide an optimized solution for the design of phase-type SPR sensors.

We propose a phase unwrapping function to solve phase jumps. The basic idea is to employ phase continuity to determine the location of the phase wrap by comparing the values of two adjacent phases in the calculated phase function results, and then adopt the phase unwrapping function to unwrap the phase at the wrap point. After all the phase values are processed in this way, the phase unwrapping process is completed, and continuous phase results can be obtained. Next, a Mach-Zehnder interferometric SPR phase measurement system is designed, in which the interference light paths are divided into a signal path and a reference path. The signal path includes an SPR prism which is rotated by a stepper motor to change the incident angle. In the reference path, modulation is introduced through a piezoelectric ceramic to allow interference between the signal light and reference light at the beam splitter prism. The p and s polarized lights are then separated by a polarizing beam splitter prism, and the signal is processed on a computer to obtain the measured phase differences. In signal processing, the wavelet filtering method is introduced to filter non-standard sinusoidal interference signals, and then the phase difference is calculated by the arcsine function phase extraction method.

The phase unwrapping method is leveraged to obtain continuous SPR phase results without 360° phase jumps as the incident angle increases (Fig. 3b). With the increasing thickness, the maximum value of the phase difference shows a trend of increasing first and then decreasing (Fig. 4b), which indicates the existence of an optimal gold film thickness for maximizing the sensitivity of the phase-type SPR. Considering the dispersion of the gold film, the optimal gold film thickness is inferred to decrease gradually as the incident light wavelength increases (Fig. 5). The arcsine phase extraction method is adopted to demodulate the SPR phase, with stable demodulation results (Table 1). Finally, after the interferometric SPR phase measurement system is employed, the SPR phase differences of three gold film thicknesses are measured (Fig. 8). It is found that the maximum values of the phase differences for gold films with thicknesses of 36.58 nm, 45.43 nm, and 62.86 nm are 117.38°, 258.29°, and 42.72° respectively near the SPR resonance angle. Additionally, the experimental results of the phase differences for different thicknesses of the gold films are consistent with the trend of the theoretical calculation curve. The experimental results also verify the theoretical calculation accuracy of the SPR phase differences through the phase unwrapping method.

We address the phase discontinuity in phase modulation-based SPR sensing systems during phase demodulation by introducing a phase unwrapping function to calculate the phase differences and eliminate phase jumps. Based on this optimized demodulation algorithm, the phase differences of three different thicknesses of gold films are measured with a Mach-Zehnder interferometric SPR phase measurement system. Both theoretical calculations and experimental results show that the phase differences during SPR change continuously. Near the SPR angle, the maximum change in phase difference presents a trend of first increasing and then decreasing with rising gold film thicknesses, which indicates an optimal thickness corresponding to the maximum sensitivity change. The proposed SPR phase difference demodulation method using phase unwrapping correction provides an accurate solution for phase-based SPR sensing.