# Quantitative Modeling of Near-Field Interactions in Terahertz Near-Field Microscopy
## 2 Methods
### Scattering Model  
The interactions between the tip and the sample can be described as two electric dipoles. The relationship between the scattering electric field and the incident electric field can be expressed in Equation (1).  
\[E_{s}=\alpha_{eff}E_{i} \tag{1}\]  
\[\alpha_{eff}=\frac{\alpha(1+\beta)}{1-\frac{\alpha\beta}{16\pi r^{3}}} \tag{2}\]  
The \(\alpha_{eff}\) is the effective polarizability of the coupled tip-sample system, and \(\alpha\) and \(\beta\) are the tip and sample parameters, respectively (further details can be found in [PERSON] 1995, [20] and [PERSON], 2000 [21]). In Equation (2), \(r\) denotes the distance between the dipoles.  
The Equation (2) shows that the scattering signal becomes progressively smaller with the \(r\) increase. As a consequence, the information of the near-field interactions will be included in the scattered signal while the tip is vibrating on the sample. By demodulating this part of the received signal, we can theoretically remove the influence of noise and recover the near-field interactions signal. The \(r\) in the Equation (2) can be expressed by introducing the information of tip vibration amplitude \(A\), frequency \(f\), and curvature radius \(a\).  
\[r=A\cos(2\pi ft)+a \tag{3}\]  
By varying the tip sample distance d, we replicate the vibration of the tip in the model. Some experimental results indicate that satisfactory experimental results can be achieved when the vibration amplitude is nearly equal or over the radius of curvature of the tip [7; 22]. Therefore, in Equations (4) and (5), we choose \(N\) points between 0 and \(2a\) to mimic the electric field intensity at various positions of tip vibration, and recover the electric field intensity \(S(r_{l})\) at the corresponding points. Thus, the harmonic signals demodulating of tip-vibration frequency are obtained by computing the Fourier demodulation transformation [15].  
\[t_{l}=\frac{1}{2\pi f}\arccos(\frac{A-r_{l}}{A}),l\in[1,N] \tag{4}\]  
\[S_{n}\propto\sum_{N}S(r_{l})e^{-in2\pi ft_{l}}(t_{l+1}-t_{l}) \tag{5}\]  
In this model, the electric field of each point in the space range when the tip vibrates is obtained by acquiring values at equal intervals for fast Fourier transform analysis, and the signal demodulated at higher harmonics of tip-vibration frequency of each point can be extracted to obtain \(S_{n}\) signal imaging in the small area around the tip. Through the theoretical demonstration and experiments, the relationship between near-field electric field  
Figure 1: (**a**) Schematic illustration of the probe in s-SNOM; (**b**) near-field scattering simulation model.  
strength and far-field reception strength is demonstrated in 2020 study [16]. By comparing the spatially dispersed electric field of each space point to the incident electric field, we are able to obtain the electric field enhancement data of each point and reconstruct the image of the harmonic signal, and explore some factors which affect the near-field electric field in s-SNOM. Different from the previous methods, this method is more intuitive for showing the effect of near-field signals. By changing some parameters, we can see its impact on near-field signals and explore the near-field interaction. However, the calculation amount of this method is greatly increased, so we also focus on simplifying the model to reduce the calculation amount in the subsequent model building.