# Full-Field Super-Resolution Terahertz Imaging Based on Rotating Coherent Scattering Microscopy
## 2 Principle of THz ROCS  
The key point of ROCS microscopy is to generate rotatable, highly transverse wave vector illumination. However, unlike visible light, the THz direct intensity imaging method is limited by the sensitivity or scarcity of the optical elements and the detector, making it difficult to modulate the illumination direction by ring illumination or the SLM. Furthermore, in the THz band, there lacks a high RI matching solution and commercial instruments similar to those used in TIRF microscopy. Thus, new illumination schemes need to be developed. TIR is the most common method to generate evanescence waves. Assuming that the wave vector in air is equal to the free-space wave vector \(k_{0}\) (\(k_{0}=2\pi/\lambda\)), a spherical wave with a small divergence angle is generated outside the hemispherical lens made by HRZF-Si, as shown in Figure 1. Under the paraxial approximation, it could be transformed by the spherical surface into a plane beam in the dense medium, whose propagation direction makes an angle of \(\theta\) with the normal to the planar surface of the hemispherical lens. This plane wave is reflected at the planar surface, being the interface between HRZF-Si and the air. The RI of the medium of the hemispherical lens and the air are \(n\) and 1.0, respectively. Then, the transverse wave vector at the reflecting surface is \(k_{\mathrm{x}}=n\mathrm{sin}\theta k_{0}\), according to Maxwell's boundary conditions, and this transverse wave vector should be the same in both the hemispherical lens medium and in the air medium. The wave vector \(k_{0}\) and the component of the wave vector \(k_{\mathrm{z}}\), \(k_{\mathrm{x}}\) satisfy the Pythagorean theorem \({k_{\mathrm{x}}}^{2}+{k_{\mathrm{z}}}^{2}={k_{0}}^{2}\). In the case when \(k_{\mathrm{x}}>k_{0}\), which also means \(\theta\) is greater than the critical angle \(\theta_{\mathrm{c}}\) of the total internal reflection (\(\theta_{\mathrm{c}}=\arcsin(1/n)\)). Then, \({k_{\mathrm{z}}}^{2}<0\), which means that it is in the imaginary field. There exists an evanescent wave in the air medium close to the planar surface and cannot be propagated to the far-field. The transverse wave vector of the evanescent wave becomes bigger when continuously increasing the incident angle \(\theta\). At the same time, when illuminating the sample by the evanescent wave, the interaction with the sample generates a scattered light field that will shift the spatial spectrum of the sample and propagate the part with higher frequencies to the far field. The scattered waves are then collected by an imaging objective, located above, whose optical axis is parallel to the normal of the planar surface of the hemispherical lens. Here, we analyzed the position relationship between the hemispherical lens and the point source of the divergence beam. The radius of the hemispherical lens with refractive index \(n\) is \(r\). Then, the distance along the optical axis from the point source to the planar surface of the hemispherical lens is \(D\), expressed as:  
\[D=r+r/(n-1). \tag{1}\]  
The spatial frequency that can pass through the passband of the system to participate in imaging is shown in Figure 2. In the case of the normal incidence illumination, the NA of the imaging system is determined by the aperture angle \(\alpha\) (NA = sin\(\alpha\)), and is less than 1. The system can only transfer the spatial frequency information inside the gray circular region, and the highest wave vector is NA\(k_{0}\), as shown in Figure 2 denoted by the blue dashed line. When \(\alpha\) = 90\({}^{\circ}\), the cutoff spatial frequency is denoted by the black dashed line, corresponding to the free space wave vector \(k_{0}\). In the case of illumination with the inclined plane wave or the evanescent wave, the transverse wave vector of the illumination beam is no longer equal to zero. The passband will follow the illumination wave vector to shift with the distance \(k_{\mathrm{x}}=n\mathrm{sin}\theta k_{0}\), denoted by the red dashed line.