# Frequency Range Optimization for Continuous Wave Terahertz Imaging
## 2 THz-FDS System for Generating and Detecting THz Waves  
Common THz-FDS is based on beating two optical signals with different wavelengths [22]. Tunable laser sources are used as optical signal sources, as shown in Figure 1. The frequency modulation happens in an optical fiber coupler, where two optical signals are mixed. The resulting optical signal contains THz carrier frequency \(\
u\), estimated as:  
\[\
u=c\cdot n\cdot\frac{\Delta\lambda}{\lambda_{1}\lambda_{2}} \tag{1}\]  
where \(c\) = 299,792,458 m/s is the speed of light in vacuum, \(n\) is the refractive index of the light propagation medium (\(n\approx\) 1.4682 for optical fiber), and \(\Delta\lambda=\lambda_{1}-\lambda_{2}\). Tunable laser sources are usually distributed feedback (DFB) laser diodes. DFB laser diodes can be tuned with temperature--if the DFB laser diode is heated or cooled, the wavelength of the emitted optical signal will shift while the spectral line will not change. In the standard telecom DFB laser diode, wavelength shift is estimated to be around 0.1 nm/\({}^{\circ}\)C [23]. A THz-FDS system can work in two modes: transmission mode, shown in Figure 2, and reflection mode. This paper is focused on transmission mode. In transmission mode, the emitted THz beam from the emitter PCA travels through the sample and is detected using a PCA detector on the other side. There are several ways of focusing the THz beam. One of the most common methods in THz-FDS is the beam's collimation, as shown in Figure 2. A collimation mirror collates the THz beam emitted from the THz emitter. At the detector side, another collimation mirror is used for focusing the THz beam on the PCA's structure. The measured characteristic of the sample in transmission mode is transmittance, or how much of the original field has passed the sample. Transmittance \(T\) can be estimated as:  
\[T=\frac{n_{2}|E_{T}|^{2}}{n_{1}|E_{I}|^{2}} \tag{2}\]  
where \(E_{T}\) is the remaining field after propagation through the medium, and \(E_{I}\) is the initial field. The initial field's value is hard to measure, which is why the reference measurement is performed. The reference field is detected in the same environment as the remaining field after propagation through the sample. Equation (2) can now be expressed as:  
\[T\approx\frac{I_{T}}{I_{0}} \tag{3}\]  
where \(I_{T}\) is the measured intensity with the sample obstructing the THz beam's path, and \(I_{0}\) is the reference intensity with the unobstructed THz beam's path. In the presented THz-FDS system, the emitted THz wave is modulated with a bias signal on the PCA. Because of that, coherent detection can be used using a lock-in amplifier (LIA). The inbound electric field will induce a photocurrent in the PCA receiver, which will be modulated with the bias signal's frequency. The induced photocurrent is then detected using the LIA and can be measured. If the length of both the emitter and receiver arms are the same, the detected intensity can be estimated as:  
\[I_{0}(\
u)=E_{\text{THz}}\cos F(\
u,L) \tag{4}\]  
where \(E_{\text{THz}}\) is the inbound THz field at the receiver PCA, and \(L\) is the distance between the antennas, and \(F\) is the phase fringe frequency. From Equation (4), the phase fringe frequency \(F\) can be estimated as:  
\[F(\
u,L)=2\pi\
u\cdot\frac{L}{c} \tag{5}\]  
Combining Equations (4) and (5), the detected photocurrent depends on the THz carrier frequency \(\
u\) and distance from the PCA emitter and PCA detector \(L\). Sweeping either \(\
u\) or \(L\) will result in phase fringe detection, as shown in Figure 3.  
Figure 1: THz-FDS System. The sample's spectral characteristics will impact the detected intensity and shift the phase. The absorption coefficient \(\alpha\) is introduced, since the classical electromagnetic theory using Maxwell equations can describe the propagation of THz waves on the macro level [1].