Phase-Shifted Quadrature-Phase demodulation based on a Multi-Longitudinal mode laser Self-Mixing sensor for displacement measurement

•Phase difference is generated by a multi-longitudinal mode laser with a wave plate.•Quadrature phase difference is constructed by a phase-shifted quadrature-phase demodulation algorithm.•The proposed quadrature-phase demodulation is unaffected by the inherent phase shift error produced by other optical components in the optical path.•Simplified displacement demodulation process and might be applied in online displacement measurement.•The max error is 37 nm with a relative error of about 0.37 % for a target displacement of 10 μm.