# Quantitative Modeling of Near-Field Interactions in Terahertz Near-Field Microscopy
## 1 Introduction  
It has been demonstrated that scattering-scanning near-field optical microscopy (s-SNOM) is a powerful instrument for determining nanoscale dielectric characteristics of the material [1]. The s-SNOM provides a new method for nanoimaging or nanomaterial identification by combining with an infrared spectral source or terahertz source technology, while the traditional imaging system has limited spatial resolution which is roughly half of the incident wavelength. In the past few years, many researchers have investigated some nanomaterial properties, such as electromagnetic field localization [2; 3; 4; 5], nanoscale free-carrier profiling of semiconductors [6; 7], and biomolecular identification [8; 9; 10]. In s-SNOM, the incident light can create a \"hotspot\" around the atomic force microscope (AFM) tip by exciting a modulated and enhanced electric field. The tip-scattered light intensity is determined by the tip radius, the tip length, the sample characteristics, and the tip-to-sample distance. Thus, it is proven that the tip vibration with the incident light would introduce an amplitude modulation signal by considering the tip-sample model as a dipole system [11]. The background scattered signal can be eliminated by acquiring the signal demodulated at higher harmonics of the tip-vibration frequency, and leaving the pure near-field information. However, the dipole model does not account for the influence of the tip geometry or surface properties of the sample, and that cannot accurately represent the experimental findings [12]. Thus, it is necessary to develop an appropriate model considering the tip geometry to explain the experimental phenomenon. [PERSON] developed the finite dipole model (FDM) based on the dipole model [12]. In the finite dipole model, the tip is represented as an ellipse, which may more accurately reflect the form of an expanded tip. In the terahertz band, the finite dipole model cannot accurately predict the experimental result while the AFM tip length is much greater than the tip radius [13]. [PERSON] presented a lightning-rod model for predicting and interpreting the experimental observables of near-field spectroscopy and the probelength influence was given in the article [14]. [PERSON] et al. set different distances between the tip and the sample in their model to simulate the tapping process in an s-SNOM experiment and to obtain the higher harmonics tip-sample scattered signal [15]. As an alternate approach for describing the coupled effect between the tip and the sample, the finite element model (FEM) is not constrained by the complicated geometry of the tip or the heterogeneity of the sample. Althoughthe FEM consumes more computer power than the FDM, simplifying the real model could solve the problem. [PERSON] et al. developed a model which provided a scattering process into the far field at a given demodulation order, with the near-field evolution as the tip-sample distance [16]. [PERSON] et al., in their model, demonstrated that FEM simulations were an appropriate choice for the strong tip-sample coupling and the model could be tailored to specific nanostructured systems and geometries of interest [17]. Moreover, [PERSON] et al. proposed an FEM model to simulate the tip-sample interaction and tip scattering in s-SNOM experiments with the tip geometry on demand [18]. All of these models concentrated on the strong coupling between the tip and sample in the infrared band, but did not account for the antenna effect in the terahertz band. If the length of the tip was consistent, matching with the incident wavelength, there would be a significant resonance effect. It has been shown that a 200 \(\upmu\)m-long tip described in [PERSON]'s terahertz s-SNOM (THz s-SNOM) is superior to a commercial tip [13]. In order to match the experimental parameters of the terahertz s-SNOM system, a new model should be proposed. In this study, a simplified two-dimensional (2D) model is proposed to simulate the interactions between the tip and the sample in THz s-SNOM.