Correlation functions in heterodyne optical fiber dynamic light scattering: A laser velocimetry method for tracking active matter at the nanoscale

Heterodyne dynamic light scattering is a powerful interferometer allowing determination of velocities and directions of nano-sized objects that are too small to be tracked using conventional optical microscopy. Here, their scattered signal is directly mixed in optical fibers with that of a local oscillator (LO) originating from the same laser source, thus offering easy and precise control of the LO reference signal and high detection sensitivity over the entire range of scattering vectors. In this paper, we detail the expression of the heterodyne intensity correlation functions, which depend both on the modulation produced by the particle scattering processes but also on the phase shift introduced by the difference in optical path taken by the two interfering fields in the fiber. We subsequently study various situations ranging from pure Brownian diffusion to ballistic motion. In particular, we detail the analysis of the motion of photocatalytic active nanoparticles capable of autonomous propulsion when exposed to UV irradiation. In such a case, the heterodyne intensity-intensity correlation function is an oscillating function of the lag time with a velocity-dependent frequency allowing both speed and propulsion direction of active nanoparticles to be accurately determined.