# Permittivity of Undoped Silicon in the Millimeter Wave Range
## 4 Retrieval Model  
Since we created a quasi-plane wave, the analysis can be conducted using the plane wave method. Such assumption would introduce limited measurement error while significantly reducing the complexity of mathematic manipulation. Reflection on an air-medium interface is an important part in electromagnetism. The reflection model, however, is based on an ideal case where the medium is infinitely long. To apply the ideal model to dielectric measurement, a more precise model has to be built based on the ideal model. This can be simply done by introducing a second interface. The structure can be described by an air-interface-medium-interface-air model, as illustrated in Figure 4.  
The interaction of the electromagnetic wave with matter on interfaces 1 and 2 is actually the same as the ideal model. Differently, the reflected wave inside the medium will bounce forward and back. Additionally, these multi-reflection/transmission effects have to be included in calculating the total reflection and transmission coefficients. Using the same mathematical manipulation of [22], the total reflection and transmission coefficients were found to be  
\[\left\{ \begin{array}{l}R=\frac{\left(1-\mathrm{e}^{-2\gamma L}\right)R_{12} }{1-R_{12}^{2}\mathrm{e}^{-2\gamma L}},\\ T=\frac{\left(1-R_{12}^{2}\right)\mathrm{e}^{-2\gamma L}}{1-R_{12}^{2}\mathrm{e }^{-2\gamma L}},\end{array} \right. \tag{10}\]  
where the reflection of a plane wave on interface 1 is  
\[R_{12}=\frac{1-\sqrt{\varepsilon_{r}}}{1+\sqrt{\varepsilon_{r}}}, \tag{11}\]  
and the wave propagation coefficient is  
\[\gamma=j2\pi\sqrt{\varepsilon_{r}}/\lambda. \tag{12}\]  
Such a structure is identical to a two-port microwave network, where the elements of \(S_{11}\), \(S_{21}\) are the reflection and transmission coefficients of port 1, and \(S_{22}\), \(S_{12}\) are the reflection and transmission coefficients of port 1, respectively. Interface 1 and interface 2 are similar to port 1 and port 2 of a two-port microwave network, so that the scattering parameters \(S_{11}\) and \(S_{21}\) measured using a vector network analyzer (VNA) can be correlated to the reflection and transmission coefficients, respectively. Figure 4: Multi-reflection model. To correlate the transmission coefficient to \(S_{21}\), a measurement on open air has to be conducted prior to sample measurement. In comparison, to correlate the reflection coefficient to \(S_{11}\), a reference measurement of planar metallic plate has to be used. To this end, a link between the wave propagation in a sample and the scattering parameters measured by a vector network analyzer was established. In Figure 5, we present the first few terms in comparison to the total reflection/transmission coefficients to examine the wave interference at different frequencies. It is seen that the first two or three terms make up most of the reflection/transmission. It is also found that at 160 GHz, the first reflection R1 has a phase of \(-180^{\circ}\), while the second term R2 and the third term R3 are \(0^{\circ}\) in phase. Therefore, a destructive interference is formed. At 80 GHz, a constructive interference is formed. Although the third term is out of phase with the first term, the amplitude of R3 is 13 dB smaller than R1, therefore, it would not contribute too much to the interference. The constructive point of the reflection is the destructive point of transmission and vice versa. Retrieve of dielectric permittivity can be fulfilled using Equation (10). For non-magnetic materials, a simple mathematic manipulation can be employed as shown in Reference [23]. The error function can be defined as  
\[E(\varepsilon_{r})=S_{21}-T. \tag{13}\]  
Or using a weighted error function  
\[E(\varepsilon_{r})=S_{21}-T+w(S_{11}-R). \tag{14}\]  
These numerical techniques are available in the literature [22]. In this work, Equation (13) was used as the error function.