Optimization of astigmatic particle tracking velocimeters

Astigmatic particle tracking velocimetry (APTV) has been developed in the last years to measure the three-dimensional displacement of tracer particles using a single-camera view. The measurement principle relies on an astigmatic optical system that provides aberrated particle images with a characteristic elliptical shape univocally related to the corresponding particle depth position. Because of the precision of this method, this concept is well established for measuring and controlling the distance between a CD/DVD and the reading head. The optical arrangement of an APTV system essentially consists of a primary stigmatic optics (e.g., a microscope, or a camera objective) and an astigmatic optics, typically a cylindrical lens placed in front of the camera sensor. This paper focuses on the uncertainty of APTV in the depth direction. First, an approximated analytical model is derived and experimentally validated. From the model, a set of three non-dimensional parameters that are the most significant in the optimization of the APTV performance are identified. Finally, the effect of different parameter settings and calibration approaches are studied systematically using numerical Monte Carlo simulations. The results allow for the derivation of general criteria to minimize the overall error in APTV measurements and provide the basis for reliable uncertainty estimation for a wide range of applications.