Minimum theoretical requirements for three-dimensional scanning-laser Doppler anemometry

An investigation of the minimum number of intersecting beams that is requir ed for laser Doppler anemometry (LDA) incorporating only a single detector is presented. We aim to provide decisive arguments for using four beams as the minimum requirement for complete three-dimensional velocity reconstruct ion even though three beams supply three velocity components. We derive exp ressions for the detected signals of the most general LDA system. From a ma trix analysis of these expressions, we conclude that there is no physically realizable arrangement of three beams that results in complete three-dimen sional velocity reconstruction and that four beams is the minimum number of beams required. We also determine the optimal arrangement of the four inci dent beams for unambiguous LDA and far best signal separation and immunity to minor optical alignment errors. To ascertain the velocity components, we scan the specimen in a precise manner relative to the point of focus of th e beams, whereas some other researchers alter the frequency of the incident beams. The results obtained with these two methods are equivalent. However , scanning is mechanically simpler than frequency shifting and also allows for the formation of velocity images-images of the flow velocity over a reg ion in two- or three-dimensional space. In particular, we examine systems t hat are limited by the common practice of using only a single high-numerica l-aperture objective for both focusing and detection. We show that using hi gh-numerical-aperture objectives results in the best signal differentiation and immunity to minor alignment errors. (C) 2000 Optical Society of Americ a. OCIS codes: 170.3340, 280.3340, 220.1140, 280.2490.