Phase Demodulation Algorithm Based on Multi-Harmonic Mixing and Nonlinear Curve Fitting
Objective Interferometric fiber optic sensors are widely used in underwater target detection,oil and natural gas prospecting,and earthquake monitoring due to their high sensitivity,large dynamic range,immunity to electromagnetic interference,and ease of large-scale array implementation. The phase-generated carrier (PGC) demodulation technique is a crucial signal processing scheme for these systems,offering a simple optical structure,large dynamic range,high resolution,and good linearity. However,traditional PGC arctangent (Arctan) and PGC differential cross multiplication (DCM) algorithms are affected by light intensity disturbance (LID) phase modulation depth (C value) drift and carrier phase delay (θ),leading to nonlinear distortions in the demodulation results. Additionally,if the θ is a singularity (kπ/4),the phase signal may not be recovered.Methods To simultaneously eliminate the effects of these factors,we propose an improved scheme with high stability and low harmonic distortion,which combines a multi-harmonic mixing technique,a nonlinear curve fitting algorithm,and a sign recovery method. First,we mix the in-phase and quadrature components of three pairs of reference carriers with the interference signal,followed by low-pass filtering to eliminate high-frequency carrier components. The signals without the carrier phase delay term are obtained by squaring and summing the filtered signals. The Levenberg-Marquardt (LM) nonlinear curve fitting algorithm is then applied to derive an error compensation equation that relates J3(C)/J1(C) to J2(C)/J1(C). This allows us to obtain J2(C)/J1(C) in real time,compensating for nonlinear errors caused by C value drift. As the multi-harmonic mixing involves square root operation,we use the sign of the filtered signal to recover the demodulated signal. For phase singularities (kπ/4),the filtered signal becomes noise,requiring an efficient sign recovery method. Our method identifies phase singularities,determines the phase delay range,and selects the appropriate sign recovery function to prevent phase signal inversion.Results and Discussions Simulations in MATLAB demonstrate that our algorithm performs stably across C values from 1.5 rad to 3.0 rad and phase delays from 0 to π (Fig. 8). At phase delays of π/4 or π/2,traditional algorithms like PGC-Arctan,PGC-SDD-DSM,and PGC-DSVV algorithms fail (Fig. 9). Additionally,PGC-Arctan and PGC-DSVV algorithms exhibit phase inversion at a delay of 3π/8,while our improved algorithm remains unaffected by simultaneous variations in C value and phase delay,performing well even at singularities. We then construct a PGC demodulation system using a Michelson interferometer with unequal arms and conduct comparison experiments to validate our approach. The C value fluctuated around 2.63 rad due to the unstable power of the electro-optical modulator (EOM),while the initial phase delay resulted from the transmission and conversion time delay of the optical signal. Additionally,the initial phase delay between the interference signal and the reference carrier depends on the transmission and conversion time delay of the optical signal. With the combined effect of C value drift and phase delay,the demodulated waveforms of the PGC-Arctan and the other algorithms become distorted. In contrast,the improved algorithm remains insensitive to nonlinear factors,achieving a signal-to-noise and distortion ratio (SINAD) of 54.52 dB and a total harmonic distortion (THD) of 63.04 dB (Fig. 11). Compared to the PGC-Arctan scheme,the SINAD of the proposed algorithm increases by 6.38 dB,while the THD decreases by 14.30 dB (Table 2). The demodulated waveform of the improved algorithm shows no inversion across the phase delay range from 0 to π,with stable performance at phase singularities (Fig. 12). To test the linearity of the improved algorithm,the amplitude of the phase signal is gradually increased from 100 mV to 1000 mV,and the correlation coefficient between the input and output linearity of the demodulation system exceeds 99.99％ (Fig. 13).Conclusions We propose an improved PGC demodulation algorithm combining the multi-harmonic mixing technique and the LM nonlinear curve fitting method. This algorithm effectively eliminates the influence of modulation depth drift and carrier phase delay on the demodulated signal. Simulation and experimental results align with theoretical predictions,confirming the algorithm's advantages in stability,low harmonic distortion,low computational complexity,and hardware implementation. The proposed method holds great promise for signal processing in interferometric fiber optic sensors.
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