## 3 Theoretical Background of a New Phase-Sensitive Technique  
THz imaging techniques, utilizing time-domain spectroscopy (TDS), have been used since 2008 by some of the authors [10; 29; 30; 31; 32]. One of the limitations of this powerful technique is related to the time needed to acquire images. For every pixel of the image, in fact, a mechanical scanning of the optical delay line of the TDS system is needed, thus making the raster scanning of large areas time-consuming. This results to be a limit when the object to be imaged is unknown: in the art conservation field every sample is unique and a huge amount of data have to be collected and analyzed to get the required information [33]. On the other hand, each pixel of the image contains information about the stratigraphy of the sample, so that the image is in fact a tomography of the sample. The imaging system developed at the ENEA center of Frascati [26; 27] makes use of a different physical principle for obtaining simultaneously imaging information and information about the optical properties of the sample at the irradiation frequencies. This capability made it possible to apply the said system to the study of works of art [30]. Since the system works well in the low frequency part of the THz region (0.1-0.3 THz), it can be considered complementary, in terms of frequencies, to the standard THz-TDS systems, which usually operate at higher frequencies. The ENEA system works in reflection mode and exploits the fact that reflection from the sample produces a phase shift of the impinging radiation that depends on the optical properties of the investigated material. The imaging system has been described in detail in the previous paragraph, and its operating physical principle is explained in detail in [34]. For a better understanding, a brief description of the main equations that are used to describe the phenomenon is provided below. In considering the radiation propagating along the \(z\)-axis inside a truncated waveguide (TE\({}_{01}\) mode), it should be remembered that, when the radiation reaches the end of the waveguide, it is partly reflected back onto the waveguide and partly launched into free space. If \(E_{i}=E_{0}exp(ik_{z}z)\) is the impinging radiation, at the truncation the expression for the reflected radiation is \(E_{r}=\alpha\ E_{0}exp(-i(k_{z}z-\varphi_{r}))\), where \(\alpha\) is the reflection coefficient of the truncated waveguide and \(\varphi_{r}\) the phase of the reflected radiation. The rest of the radiation is launched into free space and coupled in a free space propagation mode, mainly in a Gaussian mode, then undergoing natural diffraction. The electric field along the \(z\)-axis will then be:  
\[E_{t}(z)=\eta\sqrt[2]{1-\alpha^{2}}E_{0}\frac{w_{0}}{w(z)}exp[i(kz-\varPhi(z))]\]  
where \(\eta\sqrt[2]{1-\alpha^{2}}\) is the fraction of the field coupled in the free space, w\({}_{0}\) is the beam waist at the waveguide exit, w(z) is the beam radius at a distance \(z\) from the waveguide end, and \(\varPhi(z)\) is the Gaussian beam phase term. When this beam impinges on a sample, it is partly reflected back, and the reflected wave is phase-shifted by an amount that depends on the sample optical properties. A fraction of this radiation is injected back into the waveguide, where it interferes constructively or destructively with the radiation reflected back by the truncated waveguide. By taking the zero for the \(z\)-axis at the waveguide end, the mathematical expression that describes the phenomenon is:  
\[E_{\rm s}=\alpha E_{0}exp[i\varphi_{r}]+\beta e^{i\varphi_{\rm s}}\eta\sqrt[2] {1-\alpha^{2}}E_{0}\frac{w_{0}}{w_{2d}}exp[i(2 kd-\varPhi_{2d})]\]  
where the term \(\beta e^{i\varphi_{\rm s}}\) is the complex reflectivity of the sample and contains information on the optical properties of the material.