# Quantitative Modeling of Near-Field Interactions in Terahertz Near-Field Microscopy
## 3 Results and Discussion
### Edge Effect  
In the definition of s-SNOM system resolution, the theoretical resolution of the system is the transition distance between the maximum signal and the minimum signal of the scattering signal between silicon and gold in the silicon-gold grating, which ranges from 10% intensity to 90% intensity, as shown in Figure 6.  
According to the mirror dipole theory that the resolution is just related to the tip radius [25]. However, in our experimentation, we observed that the transition distance decreased when extracting a higher harmonic signal. The phenomenon is explained such that, as the order of the signal extraction increases, the influence of the scattering signal which is unrelated to the amplification of the tip sample diminishes. Thus, a better resolution can be obtained. Our model can simulate the effect of electric field augmentation at any location within the radius of curvature of the tip. This feature enables us to describe the effect of the metal steps on the near-field scattering signals of each order. As shown in Figure 7, the following is a schematic representation of three typical states corresponding to the tip and edge in the scanning process. By simulating the change of electric field intensity of each order's signal under these three conditions, we can further understand the influence of the step effect in the imaging process, so as to find ways to reduce the influence of the edge effect on s-SNOM imaging. The thickness of the gold strip is set to 100 nm. The frequency of the incident wave is 4.2 THz, and the other simulation settings were described in Section 2.2. The scattering signals of each order in three positions are shown in Figure 8.  
Figure 6: Schematic diagram of scattering signal in the scanning process of metal strips on silicon substrate. As shown in Figure 8, there is an electric field enhancement with the tip's side when the tip transitioned from the middle to right. From top to bottom, the retrieved signals (S\({}_{1}\), S\({}_{2}\), and S\({}_{3}\)) are shown in ascending order. The absolute strength of the electric field enhancement between the gold strip and the gold tip is stronger than the electric field enhancement effect of the silicon sample and the gold tip. Even when the needle tip leaves the upper space of the gold sample and is above the silicon sample, the electric field intensity is still strong where the needle tip is close to the gold. For the received signal, an overall signal strength is obtained, which will lead to the conclusion that the needle tip is still in the gold sample during imaging. Due to this reason, the electric field enhancement effect of the interaction between the tip and the sample silicon is hidden. This phenomenon is most pronounced in S1. AS the signal order n increases, the edge effect is steadily reducing. In practical application, with the increase of the order of the demodulated signal, the theoretical resolution that can be achieved in imaging will be further improved as the edge effect of gold is further suppressed. To further comprehend the lowering of the step impact, we extract and process additional image data. The electric field strength values are extracted along the white dashed line, and we extract the electric field strength value and obtain the findings depicted in Figure 9.  
Figure 8: The near-field electric field image for three typical positions of tip corresponding to edge effect in s-SNOM. Figure 7: Three typical positions of tip corresponding to edge effect in s-SNOM. To facilitate the comparison of the influence of the edge effect, we draw the extracted electric field intensities of the edge effect and sample effect together. The blue line represents the edge effect (white vertical line), and the red line represents the sample effect (white horizontal line). As the harmonic signal order n increases, the electric field becomes more confined underneath the tip, while the red line has narrower protrusions.