Ultra-broadband material spectroscopy from scattering parameters obtained from time domain measurements

The relaxation phenomena (e.g. free and bond water relaxations) and the frequencies at which they occur (from kHz to GHz) can convey valuable information in dielectric spectroscopy for material process monitoring and biology research. Finding the ultra-broadband frequency domain reflection and transmission coefficients (scattering parameters) from time domain measurements can be challenging. The current approach, often employed by material scientists, involves numerical integration of Laplace transform of the time domain data. On the other hand, the more computationally efficient fast Fourier transform (FFT) techniques have been well-developed and widely used in the engineering community. In this study, we propose a novel method, based on FFT of non-uniformly sampled time domain reflectometry (TDR) data, to obtain the frequency domain information from kHz to GHz. We perform FFT operations on multiple time windows of fixed number of points. The time duration of each window is increased, and the sample rate is decreased progressively. We correct for the truncation errors using Nicolson ramp prior to applying the FFT. We combine the FFT׳s of these multiple windows to obtain a single spectrum for each sample. We test this method to known lossy materials and hydrating cement samples. The results are in close agreement with those obtained using direct numerical integration.