Experimental validation of the use of Kramers-Kronig relations to eliminate the phase sheet ambiguity in broadband phase spectroscopy

The technique of broadband phase spectroscopy proposed in 1978 by Sachse and Pao [J. Appl. Phys. 49, 4320-4327 (1978)] determines the phase velocity as a function of frequency from the Fourier transforms of a received reference and through-sample signal. Although quite successful, this approach can be influenced by an ambiguity in the phase velocity calculation which stems from the boundedness of the inverse tangent operation used to calculate phase. Several empirical approaches to resolve the phase ambiguity have been reported. An alternative approach that has not previously been considered appeals to the causal nature of the measurements. This article experimentally validates a method which uses the causally consistent Kramers-Kronig relations to eliminate the ambiguity in phase spectroscopy-derived phase velocity calculations. Broadband pulse and narrow-band tone burst measurements were performed on three gelatin-based phantoms containing different concentrations of graphite particles (0%, 10%, and 20% by volume). The phantoms were constructed to have attenuation coefficients which vary approximately linear-with-frequency, a dependence exhibited by many soft tissues. The narrow-band phase velocity measurements do not suffer from a phase ambiguity, and thus they serve as a "gold standard" against which the broadband phase velocity measurements are compared. The experimental results illustrate that using the Kramers-Kronig dispersion relations in conjunction with phase spectroscopy-derived phase velocity measurements is an effective means by which to resolve the phase sheet ambiguity in broadband phase spectroscopy.