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

Area of Science:

• Optical Metrology
• Interferometry
• Speckle Statistics

Background:

• Self-mixing interferometry (SMI) enables sub-wavelength resolution measurements of large displacements.
• Speckle pattern statistics from diffusing targets can introduce significant errors in SMI measurements.
• Understanding these errors is crucial for accurate displacement sensing over extended ranges.

Purpose of the Study:

• To analyze errors in self-mixing interferometer (SMI) displacement measurements caused by target speckle statistics.
• To investigate how target roughness and spatial correlation affect measurement errors.
• To determine the conditions under which these errors converge to a predictable model.

Main Methods:

• Simulated speckle pattern statistics for diffusing targets with varying z-height profiles and spatial correlations.
• Analyzed two cases: developing randomness (increasing σ(z)) and fully developed randomness (σ(z)≫λ) with spatial correlation (ρ(x,y)).
• Evaluated systematic and random errors in self-mixing interferometer (SMI) displacement measurements.

Main Results:

• Measurement errors converge to those of a uniformly illuminated diffuser as target roughness (σ(z)) increases or spatial correlation (ρ(x,y)) decreases.
• This convergence indicates the development of speckle statistics, leading to predictable error behavior.
• Gaussian-distributed amplitude sources show earlier convergence compared to other distributions.

Conclusions:

• Speckle statistics of diffusing targets significantly impact self-mixing interferometer (SMI) displacement measurements.
• A universal error model based on uniform illumination emerges under conditions of high roughness or low spatial correlation.
• Simulation results provide insights into displacement measurement errors versus distance for various source distributions.