# Quantitative Modeling of Near-Field Interactions in Terahertz Near-Field Microscopy
## 3 Results and Discussion
### Antenna Effect  
This model can also be used to investigate the antenna effect, a key contributor to the increase of the electric field in terahertz s-SNOM [26]. In the theoretical model, the interaction effect between the scattering probe and the sample corresponds to a model of a mirror dipole. As part of the transmission of the dipole antenna, there is an impedance  
Figure 9: The near-field electric field image of the edge effect at harmonic signal (S\({}_{1}\), S\({}_{2}\), S\({}_{3}\)) and electric field intensity comparison curves.  
matching condition with the scattering probe in the actual work process. When the tip length aligns with an excited electromagnetic wave, resonance will occur. In engineering applications, the typical antenna length is \(\frac{1}{4}\lambda\) (\(\lambda\) is the incident wavelength) [27]. For the s-SNOM, the tip is not a circuit-matching emission. It is stimulated by the incident electric field to produce a new emission source. Thus, the engineering experience is no longer relevant. Nevertheless, using the FEM model, we can obtain the influence of the tip length on the electric field enhancement in the original signal. In the model, we extract the maximum electric field intensity as the evaluation standard of the electric field enhancement, and simulate the results under different needle tip lengths.  
Shown in Figure 10 is the result of the influence of varying the tip length in the electric field at different frequencies. The electric field enhancement effect shows a positive correlation with the increase of tip length at the beginning, and when it reached a peak, its enhancement effect began to decline. Therefore, in practical applications, we expect to find the tip length corresponding to the point with the strongest electric field enhancement effect as the actual scattering probe to obtain a better imaging effect. With the condition that the tip radius does not change, there is a clear correlation between the maximum electric field intensity corresponding to the tip length and the incident wave frequency. The influence of the electric field enhancement is illustrated by matching the length of the tip to the incident wavelength, as shown in Figure 11.  
This is performed to visualize the effect more clearly. The strongest enhancement effect is found when the length of the tip is equal to 0.3\(\lambda\), where \(\lambda\) is the wavelength of the simulated electromagnetic wave that associated with it. Notably, this finding is inconsistent with the maximum enhancement effect of \(\frac{1}{2}\lambda\) described in the study [14]. A possible reason is that the simulation frequency in our model is lower, resulting in a longer electromagnetic wave. Our model's tip length was therefore increased to match the actual tip model. Concurrently, the probe model is defined as the longitudinal section of the cone, which differs from the ellipsoid probe in the B. [PERSON] model [28]. These factors may cause the different result. The design of the scattering tip has always been a focus of this kind of research. Although our model is simplified and designed to reduce the amount of calculation, we can explore some parameters of the scattering tip. Because it is an FEM model, the shape of the tip can also be different from some traditional shapes. We can change the shape of the tip according to our own needs to explore its impact on near-field signals. Therefore, it is possible to find a new scattering tip structure through this model to achieve a better imaging effect, which is no longer limited to the traditional commercial  
Figure 10: Simulation results of electric field enhancement effect of changing tip length at different frequencies.  
tip shape. In the subsequent experiment, we will validate the result which we discovered in our 2D model, and try to explain the reasons for the different results clearly to verify whether our conclusion is universal in terahertz systems.