Fundamental flow measurement capabilities of optical Doppler and time-of-flight principles

Abstract

Abstract In order to understand the fundamental measurement capabilities of different flow velocity measurement principles based on Mie scattering, a fundamental equation of how to calculate the shot noise limit for a respective signal model is derived. The derivation is based on the well-known rules of uncertainty propagation and yields the Cramér–Rao bound without the necessity to calculate the Fisher information. The derived equation is next applied to compare the shot noise limit for Doppler and time-of-flight principles including laser Doppler anemometry (LDA), planar Doppler velocimetry (PDV), laser-two-focus velocimetry (L2F), particle tracking velocimetry (PTV) and particle image velocimetry (PIV). The comparison is performed for an identical mean laser power, while two cases are studied in detail: measuring on a single seeding particle as well as measuring on multiple seeding particles and averaging. LDA, L2F and PTV/PIV obey a similar shot noise limit. For the case of a measurement on multiple seeding particles, the minimal achievable measurement uncertainty is directly proportional to the absolute value of the measured velocity component and inversely proportional to the spatial resolution. The respective shot noise limit for PDV is almost independent of the measured flow velocity component and the spatial resolution. Since PDV is sensitive with respect to a different flow velocity component depending on the observation direction, a comparison with the other principles is only reasonable to a certain extent. However, all shot noise limits in case of measuring on multiple seeding particles show the expected inverse proportionality to the square root of the total number of detected photons and thus also to the square root of the measurement time. Considering a comparable spatiotemporal resolution, an identical mean light power and typical measurement configurations, the PDV shot noise limit is the largest. As a final result, it is derived that each measurement principle obeys an uncertainty principle between position and the respective component of the wave vector, which is in agreement with Heisenberg’s uncertainty principle. Therefore, a common basis is provided to assess the fundamental measurement capabilities of Doppler and time-of-flight measurement systems on the basis of what is possible within the quantum mechanical constraints. Graphic abstract