Systematic and random errors in self-mixing measurements: effect of the developing speckle statistics

We consider the errors introduced by speckle pattern statistics of a diffusing target in the measurement of large displacements made with a self-mixing interferometer (SMI), with sub-λ resolution and a range up to meters. As the source on the target side, we assume a diffuser with randomly distributed roughness. Two cases are considered: (i) a developing randomness in z-height profile, with standard deviation σz, increasing from ≪λ to ≫λ and uncorrelated spatially (x,y), and (ii) a fully developed z-height randomness (σz≫λ) but spatially correlated with various correlation sizes ρx,y. We find that systematic and random errors of all types of diffusers converge to that of a uniformly illuminated diffuser, independent of the actual profile of radiant emittance and phase distribution, when the standard deviation σz is increased or the scale of correlation ρx,y is decreased. This convergence is a sign of speckle statistics development, as all distributions end up with the same errors of the fully developed diffuser. Convergence is earlier for a Gaussian-distributed amplitude than for other spot distributions. As an application of simulation results, we plot systematic and random errors of SMI measurements of displacement versus distance, for different source distributions standard deviations and correlations, both for intra- and inter-speckle displacements.