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) [CrossRef https://opg.optica.org/ao/fulltext.cfm?uri=ao-54-28-8459&id=327392] ]. 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) [CrossRef https://opg.optica.org/ao/fulltext.cfm?uri=ao-62-12-3054&id=529080] ], results from the present model are compared with previous exact solutions temporally averaged over a wave cycle and with an approximate method.

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