Fundamental uncertainty limit for speckle displacement measurements.
The basic metrological task in speckle photography is to quantify displacements of speckle patterns, allowing for instance the investigation of the mechanical load and modification of objects with rough surfaces. However, the fundamental limit of the measurement uncertainty due to photon shot noise is unknown. For this reason, the Cramér-Rao bound (CRB) is derived for speckle displacement measurements, representing the squared minimal achievable measurement uncertainty. As result, the CRB for speckle patterns is only two times the CRB for an ideal point light source. Hence, speckle photography is an optimal measurement approach for contactless displacement measurements on rough surfaces. In agreement with a derivation from Heisenberg's uncertainty principle, the CRB depends on the number of detected photons and the diffraction limit of the imaging system described by the speckle size. The theoretical results are verified and validated, demonstrating the capability for displacement measurements with nanometer resolution.