Capabilities and limits of surface roughness measurements with monochromatic speckles

For coherent light illumination, surface roughness leads to speckles in the scattered light image. By evaluating the speckle contrast or image auto-correlation, a measurement of the roughness parameter S is possible. While these measurement principles have been well known for decades, a fundamental understanding of the minimal achievable measurement uncertainty is missing. Therefore, the measurement uncertainty limits for four unavoidable sources of uncertainty are derived by means of theoretical and numerical approaches. The study is focused on the case of monochromatic speckles, which provide the highest sensitivity, as well as on the case of planar surface and isotropic surface roughness with a Gaussian height distribution and Gaussian correlation function. The considered uncertainty sources are the natural randomness of surface roughness itself, speckle noise, quantum shot noise, and camera noise. As a result, for the studied experimental configuration, speckle noise is determined as the largest contribution to measurement uncertainty, which leads to a minimal achievable relative uncertainty of 1%-2% for S =(0.03-0.15)λ. According to theoretical studies, the speckle noise limit of the relative uncertainty is inversely proportional to four times the square root of the independent number of evaluated speckles. In addition, an absolute uncertainty limit is reached for ever-smoother surfaces, which amounts to λ divided by 64 times the square root of the independent number of evaluated speckles. Furthermore, systematic errors due to cross-sensitivity with respect to other parameters of surface roughness (height distribution, correlation length) as well as the surface position and shape (axial offset, tilt, curvature) are quantified and discussed. For the considered small deviations of different influencing quantities, the quantified errors are one order of magnitude smaller than the speckle noise limit.