Quadrature signal construction by vector composition for laser self-mixing interferometry

The quadrature phase unwrapping method (QPUM) is pivotal for achieving ultra-precise measurements in laser self-mixing (SM) interferometry systems. However, traditional quadrature construction methods demand a pair of strict quadrature signals; any mismatch creates error. In this paper, a vector composition quadrature construction algorithm, with only simple operations of addition and subtraction, is presented for strictly quadrature signal construction for two SM signals enclosing a certain phase shift. Taking a multi-longitudinal laser SM phase-shifting generation system as an example, we verified the feasibility of this method through experiments and simulations. The results demonstrate that, even under speckle interference, the vector composition algorithm achieves a displacement reconstruction error of 65 nm for the non-stationary motion target with a 3.193 µm peak-to-peak displacement. Moreover, for a pair of 100-second signals, our algorithm consumes only 1/34.2 of the time required by the conventional method-achieving a 97% reduction in computational cost. Combining computational simplicity, rapid signal processing, precise quadrature phase shifting, speckle immunity, and zero calibration, this technique is an ideal candidate for robust, high real-time self-mixing demodulation in online measurements and low-power scenarios.