Focused laser differential interferometry (FLDI) and its relative two-point focused laser differential interferometry (2pFLDI) are completely non-intrusive (i.e. seedless) optical techniques for measuring density fluctuations and velocity respectively that offer high frequency response (>10MHz). Developed in the 1970s, FLDI is receiving renewed attention today for its potential usefulness in measuring turbulent fluctuations and velocity in hypersonic flows. In the technique, two focused, closely-spaced (~100microns), orthogonally-polarized beams pass through a region of interest and are subsequently combined and focused onto a photodetector. Differences in refractive index between the two focal volumes cause a phase shift, thus interference, between the beams which is measured by the detector. In this way the instrument is sensitive to the gradient in refractive index along a line between the two focal volumes perpendicular to the beams (dn/dx). Since gradients in index of refraction arise from gradients in density (in homogeneous flows), fluctuations in the FLDI signal are proportional to local fluctuations in density. If the fluctuations are due to localized eddies convecting through the FLDI measurement volume, then the cross-correlation of the FLDI signal with a that from a second FLDI instrument located a known distance downstream of the first provides a measure of convection velocity (2pFLDI). The ability to measure density fluctuations and velocity simultaneously and at the same point in the flow is critical because it enables one to relate the temporal scales measured by the instrument to the spatial scales present in the flow.

In spite of the technique's age, a unified theory for the FLDI operation and sensitivity limits which is simple and easy to use does not exist so the first objective of this thesis is to develop such a theory. It does so using transfer functions that enable one to isolate the effects of focusing, beam separation, and disturbance frequency on the performance (i.e. sensitivity and spatial resolution) of the instrument. While the transfer functions have been previously proposed by others, an application of these functions which accounts for velocity variation in space (u_c(z)) and frequency (u_c(f)) is unique to this work.

The theory is validated via comparison to experimental measurements in a canonical turbulent jet where the distributions of velocity and density fluctuations are well known. Measurements made using different FLDI instruments collapse when the differences between them are accounted for, indicating that the unified theory is correctly capturing the effects of instrument parameters like beam separation and beam diameter. FLDI response to the jet is also modeled by substituting the velocity distribution for a dispersion relation, u_c(f), measured by 2pFLDI. The advantage of the latter procedure is it allows for signal interpretation in flows where historical measurements are unavailable. This is demonstrated by comparing modeled FLDI response to experimental measurements in the flow downstream of a ramp in a small (6.4cm square) Mach 3 wind tunnel.

The second objective of this thesis is to demonstrate 2pFLDI in other industrially-relevant flows. To this end, density fluctuations and convection velocities are measured in the near-wall flow in a 61cm square Mach 4 wind tunnel (Ludwieg tube) and in the free-stream flow of a 1.5m diameter Mach 18 tunnel. In each case, a method for estimating the spatiotemporal resolution using transfer functions is demonstrated. The spatiotemporal resolution of the instrument was not well understood prior to this work so quantifying it is an important contribution. Achieving acceptable signal/noise at Mach 18 was difficult because densities were so low. However, convection velocities of ~75-80 of the freestream velocity are measured above 200kHz in two runs. Spatiotemporal analysis suggests these measurements are the result of freestream disturbances; the first measurement of its kind in a Mach 18 flow.