Particle Displacement and Velocity Measurement Using Holography

A method of measuring particle displacements and velocities in the transform plane of a double-exposure hologram is examined both theoretically and experimentally. The analysis begins with a study of the transform patterns obtained from single-exposure holograms (both in-line holograms and off-axis holograms are considered). Although the transform of an in-line hologram and an off-axis hologram are similar, the on-axis images of an in-line hologram are shown to produce a zone lens which is not observed in the transform plane of an off-axis hologram. The Fourier transform patterns of double-exposure holograms are then examined, and it is shown that each pair of particle images related by translation contributes a set of cosine fringes to the total diffraction pattern. The frequency and orientation of the fringes are related to the magnitude and direction, respectively, of a given particle's displacement or velocity. The transform patterns corresponding to 1- and 2-dimensional distributions of displacement are examined in detail. For the 1-dimensional case, a theoretical analysis indicates that the probability density function that characterizes the distribution of displacements may be determined by properly processing intensity data measured in the transform plane of a doubly exposed hologram. For the 2-dimensional case it is shown that the marginal density functions may be obtained if the orthogonal components of particle displacement are statistically independent. Experimental results are presented, and good agreement between theory and experiment is observed.