# Permittivity of Undoped Silicon in the Millimeter Wave Range
## 3 Generating Quasi-Plan Wave  
The QO technique was originally developed for radio astronomy for low-loss and wideband receivers [19]. Transferring the QO technique to dielectric measurement is a good attempt. The QO theory is based on Gaussian beam description of an electromagnetic wave. The propagation and refocusing of a Gaussian beam are illustrated in Figure 2. A Gaussian beam can be described using [20]  
\[E(r,z)=\frac{w}{w_{0}}\exp\left(\frac{-r^{2}}{w^{2}}-\mathrm{j}\kappa z-\frac{ \mathrm{j}\pi r^{2}}{\lambda R}+\mathrm{j}\phi_{0}\right), \tag{3}\]  
where \(w_{0}\) is the beam waist located at \(z=0\), \(w\) is the beam radius, \(R\) is the radius of curvature and \(\phi_{0}\) is the phase shift. These parameters can be written as  
Figure 1: An illustration of frequency dependent dielectric effect. \[R=z+\frac{1}{z}\bigg{(}\frac{\pi w_{0}^{2}}{\lambda}\bigg{)} \tag{4}\]  
\[w=w_{0}\bigg{[}1+\bigg{(}\frac{\lambda z}{\pi w_{0}^{2}}\bigg{)}^{2}\bigg{]}^{0.5}, \tag{5}\]  
\[\tan\phi_{0}=\frac{\lambda z}{\pi w_{0}^{2}}. \tag{6}\]  
It is seen from Equation (4) that the radius of curvature of the wave front is infinite at the beam waist. This is a property of plane wave. However, as the beam propagates, the size of the beam increases with the propagation distance due to the diffraction effect of the electromagnetic wave, gradually forming a spherical wave front. Therefore, a focusing unit has to be used to refocus the beam to a beam waist. The beam transformation by a focusing unit can be analyzed using the ABCD matrix. In a general case, the transform by a focusing unit of focal length \(f\) can be calculated by  
\[\left\{ \begin{array}{l}\frac{d_{\text{out}}}{f}=1+\frac{d_{\text{in}}/f-1}{(d _{\text{in}}/f-1)^{2}+z_{\text{c}}^{2}/f^{2}}\\ w_{0}\ \text{out}=\frac{w_{0}}{\left(d_{\text{in}}/f-1\right)^{2}+z_{\text{c}}^{2}/ f^{2}\right]^{0.5}}\end{array} \right., \tag{7}\]  
where \(d_{\text{in}}\), \(d_{\text{out}}\), \(w_{0\text{ in}}\), and \(w_{0\text{ out}}\) are the input distance, output distance, input beam waist, and output beam waist, respectively. Additionally, \(z_{\text{c}}\) is the confocal distance  
\[z_{\text{c}}=\frac{\pi w_{0\text{ in}}^{2}}{\lambda}. \tag{8}\]  
It is seen that given \(d_{\text{in}}=f\), the output distance \(d_{\text{out}}\) is always \(f\), independent of the working frequency. Such a feature enables broadband operation. For a special case, if two identical focusing units are used and placed \(2f\) apart, i.e., \(f_{1}=f_{2}=f\) and \(d_{1}+d_{2}=2f\) (see Figure 2), the output beam waist will be the same as the input beam waist \(w_{0\text{ out}}=w_{0\text{ in}}\). Therefore, the transmitting and receiving horns can be fabricated to be the same. Such a special case gives one a symmetrical structure and is referred to as Gaussian telescope. The system in this paper is in the form of Gaussian telescope. Figure 2: The propagation of a Gaussian beam through free space and the refocusing effect by focusing units. For a free space system, low cross polarization, low transmission loss, low reflection, and good bandwidth are preferred. To meet these requirements, metallic reflectors can be used as focusing units. Although a dielectric lens can also be used for focusing, the inherent reflection loss and dielectric loss as well as its narrower bandwidth make it less preferred. To obtain low cross polarization, a high-quality horn antenna is always plausible. Corrugated horns are good candidates to the transmitting/receiving units since good bandwidth and polarization purity can be achieved [21]. In addition, the system has to be capable of creating a local plane wave with acceptable transverse size. Additionally, the field should be close enough to a plane wave over a longitudinal distance to ensure that thick samples can also be measured. Such requirements may demand the reflector to be a paraboloidal surface. However, the ellipsoidal surface is more common in a QO system [20]. Considering these requirements, the following parameters were used for the system design (see Table 1).