## Appendix A  
This appendix gives formal derivations of the formulas that were used in Section 2.2.2 to estimate the Rayleigh lengths, \(Z_{R}\), of the THz beams delivered to the samples by means of the \(4\!f\) and \(2\!f\) quasi-optical beam delivery setups. In the Gaussian beam approximation, a beam waist, \(\omega\), of a diverging one-dimensional beam with the intensity profile described by the Gaussian function is given by the formula [48]:  
\[\omega(z)=\omega_{0}\sqrt{1+\left(\frac{z}{Z_{R}}\right)^{2}}. \tag{1}\]  
The beam waist parameter describes the so-called Gaussian beam radius for which the intensity of the beam decreased to \(1/e^{2}\) of its axial or peak value. The origin of the coordinate system related to Equation (1), \(z=0\), is chosen at the point where the beam waist parameter is the smallest, corresponding to \(\omega_{0}\). The distance \(z=Z_{R}\), which is symmetrical on both positive and negative sides around \(z=0\), defines another important parameter-the Rayleigh range, within which the beam waist does not increase above \(\omega_{0}\sqrt{2}\). The \(Z_{R}\) parameter is also known as the Rayleigh length. It is related to the beam waist, \(\omega_{0}\), and the wavelength of the radiation, \(\lambda\), by the following formula:  
\[Z_{R}=\frac{\pi{(\omega_{0})}^{2}}{\lambda}. \tag{2}\]  
Inside the Rayleigh range, the beam is considered to be approximately collimated, and the propagating wave is approximated by the plane wave approximation. The half-divergence angle of the beam outside the Rayleigh length, \(\theta\), is given by:  
\[\theta=\frac{\lambda}{\pi\omega_{0}}. \tag{3}\]  
The above equation is valid for slowly diverging beams and uses the approximation \(\tan(\theta)\approx\theta\). Far beyond the Rayleigh range, a divergent Gaussian beam appears to have its origin at \(z=0\). It propagates in the form of a radial wave. Since its source can be approximated by a point, the full divergence angle, \(2\theta\), can also be calculated by means of the geometrical optics as a diameter, \(d\), illuminated on a mirror divided by the mirror's focal length, \(f\):  
\[2\theta\approx\frac{d}{f}. \tag{4}\]Therefore, a beam waist of a divergent THz wave and its diameter on a surface of a mirror shifted by a distance of the focal length from the THz emitter are related by the formula:  
\[\omega_{0}(\lambda)\approx\frac{2\lambda f}{\pi d}. \tag{10}\]  
A THz pulse emitted by a photoconductive antenna of a THz-TDS system is a superposition of a wide range of wavelengths. According to Equation (10), a diameter of each of these wavelength components at the input of an off-axis collimating mirror in a 4\(f\) quasi-optical system (Figure 1a) can be calculated as:  
\[d(\lambda)\approx\frac{4\lambda f_{trans}}{2\omega_{trans}\pi}, \tag{11}\]  
With \(f_{trans}\) denoting the focal length of the mirror and \(2\omega_{trans}\) indicating the output aperture of Tx-transmitter, which is typically approximated by a diameter of a Si lens attached to the antenna to reduce the divergence of the beam. Note that the collimated THz beam propagates in the form of concentric cylinders, with the longer wavelengths being outside the shorter wavelengths. Such a beam is next focused by the second of the off-axis mirrors, and the wavelength-dependent beam waist in the focal spot can be calculated again by means of Equation (10):  
\[\omega_{focus}(\lambda)\approx\frac{2\lambda f_{focus}}{\pi d}. \tag{12}\]  
Here,\(f_{focus}\) corresponds to the focal length of the focusing mirror and \(\omega_{focus}\) corresponds to the focused beam waist. By inserting Equation (11) into Equation (12), \(\omega_{focus}\) can be expressed in terms of the transmitted beam waist as:  
\[\omega_{focus}\approx\frac{f_{focus}}{f_{trans}}\omega_{trans}. \tag{13}\]  
A remarkable conclusion resulting from Equation (13) is that despite substantially different divergences of the emitted THz pulse spectral components, the beam waists at the Fourier plane after focusing are the same.