# Quantitative Modeling of Near-Field Interactions in Terahertz Near-Field Microscopy
## 3 Results and Discussion  
### Electric Enhanced Image  
To validate the accuracy of the simplified model, we present simulation results for two common materials, silicon and gold. As an exceptional scattering metal, gold is a better scattering material than silicon, which is represented in the simulation results. In many s-SNOM imaging investigations, silicon substrates are etched with metal strips to test the imaging effect [23]. The frequency of the incident wave is 4.2 THz, and the other simulation settings were described in Section 2.2. Figure 3 depicts the electric field enhancement factor as a function of the tip-sample distance with the gold or silicon sample. The electric field value of each point at different tip-sample distances is simulated, and the maximum value of the electric field is selected to represent the electric field enhancement effect. There are 100 values for the tip-sample distances (d from 2-200 nm). For the purpose of better data comparison, we only extracted the data where the d is less than 100 nm in Figure 3, and did not show the data indicating that the maximum electric field intensity does not change significantly after the d exceeds 100 nm. The results show that the enhancement effect of the electric field decreases sharply with the increase of the tip-sample distance. Thus, the stronger electric field enhancement proves that gold is a better scattering material than silicon.  
Through the data processing of the simulation, the signal of each order is extracted, and the near-field electric field intensity image near the tip is reconstructed. Figure 4 depicts the near-field electric field distribution image for the two materials at the extracted harmonic order signal (S\({}_{1}\), S\({}_{2}\), and S\({}_{3}\)), which corresponds to the vertical section electric field division diagram produced by others in the 3D simulation [16]. The incident electric field is confined to a tiny region close to the tip, and a \"hotspot\" is created by the effect. Figure 5 depicts the width of the field distribution with the gold sample. For the increasing order \(n=1-3\), the lateral full width (2 nm above the sample) at half-maximum (fwhm) 2\(\Gamma\) of the field profiles decreases continuously and the value could be smaller than the tip radius (100 nm). All the results of our 2D model are in good agreement with those of others' 3D models [16; 18], which proves that our 2D model can reduce the amount of calculation and ensure its correctness. The results depicted in Figure 3 indicate that the electric field enhancement can penetrate non-metallic materials such as silicon, and that the scattering electric field of gold is stronger than that of such materials, so it can be used to detect metal structures covered by layers of penetrable materials. This is also an application direction for s-SNOM technology, and significant development has been carried out in this field [24]. It can image the nm antenna covered by layers of penetrable materials and detect whether the antenna is damaged without damaging the surface. Thus, with the development of nm  
Figure 3: Electric field enhancement with different tip-sample distance.  
technology, there will be more scenarios that need to use this technology to solve more nm imaging problems.  
Figure 4: The near-field electric field distribution image for gold and silicon at harmonic order signal (S\({}_{1}\),S\({}_{2}\),S\({}_{3}\)).  
Figure 5: Line profiles with full width at half-maximum 2\(\Gamma\) extracted at 2 nm above the sample.  
By obtaining a higher harmonic signal, the action range of the strong scattering electric field at the tip is condensed. Although extracting higher-order signals will reduce the signal intensity, this part of the energy information more concentrated on the tip can be obtained to improve the resolution during imaging. Thus, through this model, we could further explore the influencing factors of the near-field electric field.