Speckle Pattern Errors in Self-Mixing Interferometry

We analyze the random errors occurring in interferometric measurements because of the speckle pattern regime, when the remote target is a diffusing surface. First, we review the statistical properties of speckle and discuss amplitude fading that is affecting the self-mixing interferometer (SMI) signal and methods to alleviate it. Second, we derive intra-speckle phase errors using the bivariate conditional probability, and find that the noise-equivalent-displacement for small displacement Δ is proportional to the ratio of Δ to speckle longitudinal size s_l. Last, we extend the analysis to inter-speckle displacements (Δ > s₁) and, after deriving speckle systematic and random errors, show that operation up to meters on a diffusing surface target is possible with a small (≈ λ) error. Results are mainly focussed on SMI, yet they have general validity for any configuration of interferometry.