Recovering density disturbance spectra from FLDI. Part 1

Abstract

Focused laser differential interferometry (FLDI) measures the phase shift corresponding to localized fluctuations in the refractive index of a medium. The sensitivity, bandwidth, and spatial filtering properties of FLDI make it particularly suited to applications in high-speed gas flows. Such applications often require the quantitative measurement of density fluctuations, which are related to changes in the refractive index. In a two-part paper, a method is presented for the recovery of a spectral representation of density disturbances from the measured time-dependent phase shift for a particular class of flows able to be modeled using sinusoidal plane waves. The approach is based on the ray-tracing model of FLDI due to Schmidt and Shepherd [Appl. Opt. 54, 8459 (2015)APOPAI0003-693510.1364/AO.54.008459]. In this first part, the analytical results for FLDI response to single- and multiple-frequency plane waves are derived and validated against a numerical implementation of the instrument. A spectral inversion method is then developed and validated, including consideration for the frequency-shifting effects of any underlying convective flows. In the second part [Appl. Opt. 62, 3054 (2023)APOPAI0003-693510.1364/AO.480354], results from the present model are compared with previous exact solutions temporally averaged over a wave cycle and with an approximate method.