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 consolidated 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, whereas 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 data set (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 theoretical graphical techniques from the first-order reflectance and transmittance. Our approach is then taken further by considering all the Fabry–Perot reflections with the frequency- and space-domain spectroscopy. With the 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 the solution space that gives multiple mathematical solutions. By knowing threshold conditions, it is possible to gain a much deeper insight, in terms of the 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. In theory, the result from this paper allows for the exact extraction of complex dielectric properties using simpler and lower cost scalar reflection–transmission mode spectroscopy.
Highlights
Spectroscopy represents the study of interaction between matter and electromagnetic radiation
It is worth noting that normal incidence reflection mode measurements are more cumbersome to implement, especially foroptical systems, even though they are ideal for opaque samples under test that cannot be measured with the simpler transmission mode configuration
VECTOR MEASUREMENTS With numerical methods, effective complex dielectric properties can be extracted from the combination of magnitude and phase measurements from the overall reflection and/or transmission transfer functions using (33) & (34) or full 2-port S-parameters using (35) & (36) with ELECTRIC-FIELD PROPAGATION MODEL (EFPM) and VOLTAGE-WAVE PROPAGATION MODEL (VWPM) or (46) & (47) using TRANSMISSION LINE MODEL (TLM)
Summary
Spectroscopy represents the study of interaction between matter and electromagnetic radiation. From the outset and throughout, this paper considers the most common reflection-transmission mode spectroscopic scenario of normal incidence of the electromagnetic wave (or guided-wave mode) onto the sample under test, using well-established assumptions (outlined ). Within this context, our modeling techniques for predicting normalized power spectra and extracting exact values for the effective complex dielectric properties can be adopted for a range of reflection-transmission mode spectroscopic implementations and across the full electromagnetic spectrum; from dielectric-filled transmission line/waveguide sections at microwave [5] and millimeter-wave frequencies to quasi-optical approaches at millimeter-wave and infrared frequencies [6] to optical methods at visible light and ultraviolet wavelengths. It is worth noting that normal incidence reflection mode measurements are more cumbersome to implement, especially for (quasi-)optical systems, even though they are ideal for opaque samples under test that cannot be measured with the simpler transmission mode configuration
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