Experimental Detection of ∆x Variations on the Complex s-Plane Based on a Michelson Interferometer

In this paper, a Michelson interferometer was employed to generate interference patterns, the modulated function was extracted by digital filtering and the Laplace transform was subsequently applied to obtain the complex function F(s)=F(σ+ωi), which changes its zero structure in response to physical perturbations. This study presents a novel method for detecting millimetric displacements based on analyzing the movement of zeros in the complex s-plane. The proposed technique demonstrates that an unperturbed system displays two zeros on the imaginary axis, whereas the introduction of a displacement ∆x generates a third zero with a real component. The position of this third zero in the s-plane directly correlates with the magnitude of the perturbation. Finally, the measurement vector smeas, calculated as the vector difference between reference and perturbed zeros, quantifies the displacement with high precision. Experimentally, displacements ranging from 0.8 mm to 8.8 mm were introduced into one of the interferometer arms, allowing for the systematic observation of the zero migration within the complex plane. The obtained results demonstrate that this methodology transforms the displacement measurement problem into a singularity localization task in the complex domain, offering enhanced robustness against intensity fluctuations compared to conventional techniques based on Fourier analysis or fringe counting.