Ultimate error sources in self-mixing interferometry

In high-accuracy interferometric measurements, once the main sources of error (misalignement, refractive index fluctuation) are removed or compensated for, residual errors due to field curvature and to the statistical nature of the scattered field, typical of mirrorless self-mixing interferometry (SMI), persist. Field curvature (systematic) error can be compensated for from the knowledge of the mean phase space derivative dPhi/dz; the (random) error due speckle-pattern of the scattered field can be tamed by exploiting second order statistics of intensity and phase conditioned to intensity. Starting from knowledge of speckle pattern statistics we derive intra-speckle phase errors using the bivariate conditional probability, finding that the noise-equivalent-displacement (NED) for small displacement delta is proportional to the ratio of delta to speckle longitudinal size sl. Than we extend the analysis to inter-speckle displacements (delta>sl) and, after deriving speckle systematic and random errors, show that operation up to meters on a diffusing surface target is possible with a small (≈lambda) error. Results are of general validity for any configuration of interferometry, even if discussed in a SMI context.