A Simplified Formulation of the Reflection Coefficient of an Open-Ended Coaxial Probe in Multilayered Media

Permittivity spectroscopy can discern a material's composition based on its response to an electromagnetic field (EMF). The predominant method employed for permittivity spectroscopy relies on measuring the reflection coefficient of an open-ended coaxial probe immersed in the material under test (MUT). While several physics-based models exist to describe the observed reflection coefficient as a function of the material properties, they assume a single, homogeneous layer. However, most applications, particularly in biomedical engineering, often involve heterogeneous media composed of layers of different permittivities. This article addresses a gap in the literature and introduces a new model designed for accurately estimating the reflection coefficient of two-layer media using an open-ended coaxial probe. Adapting a full-wave model initially developed for determining the reflection coefficient of homogeneous media, the proposed model incorporates the influence of the second layer by expanding the spherical waves generated by the probe into planar waves using Bessel-based formulation. The proposed method is validated experimentally in different scenarios within a frequency range of 1-5 GHz. The experimental results indicate that the proposed model can predict the reflection coefficient of the media with an error no higher than 5%, while the average error in the extracted permittivity from the inverse problem is 1.08%.