Self-mixing interference (SMI) is a novel optical measurement technique that utilizes the light reflected back into the laser cavity. Due to the interaction between the feedback light and the medium in the cavity, SMI has high sensitivity. It can obtain high quality signals for targets with rough surfaces without the need for additional target mirrors. SMI also features high resolution, low cost, and compactness, making it widely used in sensing applications. The effective signal of SMI is related to the phase of the laser, so phase demodulation is particularly important. Recently, researchers have used electro-optic crystals and mixers for orthogonal phase demodulation, which can eliminate baseline noise, simplify the algorithm, and increase the measurement speed. However, this method requires the orthogonal signal of the electro-optic modulation signal. Due to the high-frequency nature of electro-optic signals, it is difficult to construct the orthogonal signal. This method does not consider the influence of the initial phase on the harmonic amplitude, while the variation of the harmonic amplitudes can lead to a decrease in accuracy. To address the challenge of constructing orthogonal signals and to solve the problem of random fluctuations in harmonic amplitudes, we propose a new signal processing method based on the Hilbert transform, which can simultaneously simplify the system and improve measurement accuracy.

First, the basic SMI model and the fundamental principle of electro-optic modulation are introduced. Subsequently, based on the principle of heterodyning, the reason for the inconsistency in the amplitudes of the two harmonics is discussed, and the influence of this inconsistency on the measurement variance is analyzed by numerical simulation. Then, a self-mixing interference electro-optic modulation device is constructed using a 532 nm solid-state laser, a 251 kHz resonant electro-optic phase modulator, and a photodetector. The laser beam is irradiated onto an aluminum reflector mounted on a piezoelectric transducer (PZT), causing the reflector to make sinusoidal vibrations at a frequency of 5 Hz and a peak-to-peak value of 4 μm. Comparisons are made among three scenarios: no normalization, normalization with fixed coefficients, and normalization using the Hilbert transform. The peak-to-peak interval is gradually increased from 1 to 10 μm, and seven datasets are randomly collected to test the improvement of Hilbert normalization on measurement accuracy. Finally, the applicability of this method is tested when the reflector is subjected to triangular and square wave vibrations.

Based on the principle of heterodyning, the random fluctuation of the difference between the initial phase of heterodyning and the initial phase of electro-optic modulation will cause fluctuations in the amplitudes of the two harmonics (Fig. 3), and the inconsistency of the harmonic amplitudes will increase the standard deviation of the measurement (Fig. 4). Measurements are made on a sinusoidal vibration with a frequency of 5 Hz and a peak-to-peak value of 4 μm. A comparison is made between the fixed coefficient normalization and Hilbert transform normalization (Fig. 8). Using the fixed coefficient normalization, the difference in amplitudes between the two harmonics is reduced. However, due to the fluctuations in the harmonic amplitudes caused by laser power fluctuations and feedback light speckle effects with time, it is not possible to find a fixed coefficient that could normalize all harmonic amplitudes over all time periods. With Hilbert normalization, the amplitudes of the harmonics are normalized within each time period, completely eliminating inconsistencies in the harmonics. It also eliminates the random fluctuations in amplitude over time. As a result, without normalization, the standard deviation is 18.2 nm. After normalization with fixed parameters, the standard deviation is reduced to 15.0 nm, and after Hilbert normalization, it is further reduced to 12.5 nm. The measurement accuracy is the highest after Hilbert normalization, indicating that using the Hilbert transform for normalization has certain advantages. The effect of Hilbert normalization under different amplitude conditions obtained in tests shows a significant reduction in the standard deviation after Hilbert normalization (Fig. 10), with a 30% reduction compared to non-normalized data. In addition, this method shows good applicability to both square waves and triangular waves (Fig. 11).

In this study, we analyze the cause of the inconsistency in harmonic amplitudes and propose a normalization method based on the Hilbert transform to solve the problem of harmonic asymmetry and improve the measurement accuracy of the SMI system. The SMI signal modulated by the electro-optic modulator (EOM) is directly heterodyned with the first-order and second-order electro-optic modulation signals to extract the first- and second-order harmonics. Then, the extracted harmonics are subjected to Hilbert normalization to eliminate the asymmetry in harmonic amplitudes caused by the randomness of the initial phase. This method effectively improves the measurement accuracy by reducing the standard deviation by 30%. Measurements of sinusoidal vibrations with amplitudes ranging from 1 to 10 μm peak-to-peak values achieve a measurement accuracy of λ/42 after normalization. In addition, this method can also reconstruct non-sinusoidal waveforms such as square waves and triangular waves, demonstrating strong applicability. The proposed method features high measurement accuracy, simple algorithms, low sampling rate requirements, and strong applicability, providing valuable exploration for high-precision online displacement measurement.