Decoupling substrate thickness and refractive index measurement in THz time-domain spectroscopy

Terahertz time-domain spectroscopy (THz–TDS) is an attractive choice for characterizing electrical properties of materials owing to its contactless, non-destructive nature. For efficient application, either the thickness or refractive index of the material being studied must be known precisely. The specimen is typically placed on top of a substrate, such as quartz or high-resistivity silicon, after which their characteristics are determined by studying the light-matter interactions at the substrate-sample interface. Since THz–TDS measures the electric field in the time domain, any uncertainty in the value of the substrate thickness or refractive index can cause errors to cascade into sample property calculations related to the detected phase. Therefore, in practice, one of these values has to be assumed, or their product must be numerically optimized to converge on suitable values. Whatsoever the approach, errors are inherent and may mask some real features or properties of the material being measured. This is one of the stumbling blocks for implementing THz spectroscopy in industrial applications.

Farah Vandrevala (PhD student in Electrical Engineering) and Professor Erik Einarsson (jointly appointed by the Department of Electrical Engineering and the Department of Materials Design and Innovation) at the University at Buffalo found a way to mitigate this problem. They utilized THz–TDS in reflection geometry to accurately and independently determine both thickness and refractive index by illuminating the step-edge of a substrate atop a metal stage, thereby minimizing error in both values. Their work has been published in the research journal Optics Express.

In order to achieve their goal, the researchers focused a THz beam with a spot size of approximately 3 mm on the top surface of a substrate. They then positioned the substrate such that the step-edge was illuminated. A ten-degree incidence angle ensured there were three reflected signals: one from the top of the substrate, one from the underlying metal stage, and a third that was delayed due to travel inside the substrate.

The research pair explain that the time delay between the first two reflections corresponds only to the substrate thickness. “We were quite delighted to learn that the math involved in finding the substrate thickness was nothing more than simple trigonometry. The simplicity of our method is what makes our solution so elegant,” said Vandrevala. As a consequence of independently determining the thickness, the researchers were then able to calculate the optical properties of the substrate with high confidence.

The Vandrevala–Einarsson study presents a fractional-reflection measurement, taken using THz–TDS at the step-edge of a substrate, that allows the thickness and refractive index measurements to be decoupled. Their novel approach has been seen to be advantageous in that all the necessary information is obtained in a single time-domain measurement. In addition, this new technique can be easily applied to a wide range of substrates and still yield precise results regardless of how transparent or opaque the substrate is. Altogether, this technique can come in handy when the effects of the supporting substrate must be carefully accounted for, such as applications involving the study of coatings, thin-films, two-dimensional materials, or biological samples.