Extracting complex dielectric properties from reflection-transmission mode spectroscopy

Abstract

Material characterization of homogeneous dielectric slabs using reflection-transmission mode spectroscopy can be problematic due to the ambiguity from a phasor term. A comprehensive analytical review of methods for calculating the normalized power spectra, to extract the effective complex dielectric properties of a sample, is undertaken. Three generic power response models (zero-order, power propagation and electric field propagation) are derived; these models act as a common mathematical framework for the whole paper. With our unified engineering approach, the voltage-wave propagation, transmission line and telegrapher’s equation transmission line models are then independently derived; the first two giving the same mathematical solutions, while the third generates the same numerical results, as the exact electric field propagation model. Mathematically traceable simulation results from the various models are compared and contrasted using an arbitrarily chosen dataset (window glass) from 1 to 100 THz. We show how to extract the approximate effective complex dielectric properties using time-gated time-domain spectroscopy and also the exact values with our graphical techniques from the first-order reflectance and transmittance. Our approach is then taken further by considering all the Fabry-Perot reflections with frequency- and space-domain spectroscopy. With scalar reflection-transmission mode infrared spectroscopy, we model the threshold conditions between the solution space that gives the single (exact) solution for the complex refractive index and that which gives multiple mathematical solutions. By knowing threshold conditions, it is possible to gain a much deeper insight, in terms of sample constraints and metrology techniques that can be adopted, to determine the single solution. Finally, we propose a simple additional measurement/simulation step to resolve the ambiguity within the multiple solution space. Here, sample thickness is arbitrary and no initial guesses are required. The result from this work allows for the exact extraction of complex dielectric properties using simpler and lower cost scalar reflection-transmission mode spectroscopy.