# Sub-Diffraction Focusing Using Metamaterial-Based Terahertz Super-Oscillatory Lens
## 1 Introduction  
In recent years, the number of applications using terahertz (THz) and millimeter waves has continued to increase, particularly for 6 G technology in telecommunications [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15]. The THz frequency band (from 0.1 to 10 THz) is also expected to be highly effective for industrial product inspections, as THz waves have the potential to detect cracks in asphalt walls and foreign metallic objects in products. However, THz applications are currently limited by resolution constraints. Since the THz wavelengths (from 30 \(\upmu\)m to 3000 \(\upmu\)m) are longer than the visible and near-infrared light wavelengths (from 0.360 \(\upmu\)m to 1.4 \(\upmu\)m) conventionally used for inspection, the resolution of THz-based inspection cannot be as small as that of visible light when using ordinary lenses. This is known as the diffraction limit. Terahertz lenses are generally made of polytetrafluoroethylene (PTFE), polyethylene, silicon, etc., and the spot size made by these lenses is restricted by the diffraction limit \(\delta\):  
\[\delta=0.61\lambda/\text{NA} \tag{1}\]  
where \(\lambda\) is the wavelength in THz. NA is the numerical aperture of the lens, which is determined by  
\[\text{NA }=n\ \sin\theta \tag{2}\]  
where \(n\) is the refractive index of air (the medium surrounding the lens) and \(\theta\) is the maximum half-angle of the light cone defined by an apex at the focal point and a base on the lens. Because of the evolving strict inspection resolution requirements, methods to achieve higher resolution are required. Various methods have been studied to achieve subwavelength resolution in the THz frequency band [16; 17; 18; 19; 20], including superlenses and bull's-eye lenses [17; 18]. These methods use evanescent waves to achieve resolution at the expenseof the working distance (WD). Though trivial for microscopy applications, a short WD can limit the applicability of industrial product inspection methods. A few metamaterial-based lenses with negative refractive indices and which also utilize evanescent waves have also been reported [19; 20]. In this work, we focused on other metamaterial approaches to achieve subwavelength resolution and a long WD. Some studies have revealed that the diffraction of light using a binary mask produces a subwavelength spot that allows imaging with a higher resolution than that of conventional lenses [21; 22; 23]. This physical phenomenon is known as super-oscillation, in which band-limited functions can oscillate arbitrarily faster than the highest contained Fourier component. In 2006, [PERSON] and [PERSON] were the first to propose that light fields diffused through subwavelength gratings can generate arbitrarily small spatial energy regions without resorting to evanescent waves, theoretically proving the feasibility of resolution enhancement in imaging systems using super-oscillation methods [24]. Recently, [PERSON] demonstrated this phenomenon for the first time by observing a focal spot below the diffraction limit generated by a quasiperiodic metallic nanohole array mask [25]. Since then, various devices and design methods for generating super-oscillating electric fields have been proposed one after another. A lens that employs this phenomenon was developed to achieve subwavelength resolution in the visible and ultraviolet (UV) light regions, called \"Super-oscillatory lenses (SOLs)\" [26; 27]. One technique in SOL design is to control the interference of incident waves by optimizing the pattern of concentric ring slits. However, this method requires complex design calculations [28]. Moreover, the lens produces only one focal point at the focal length. On the other hand, a binary mask with multiple holes smaller than the wavelength [29] reportedly achieves a similar effect by controlling the transmission and retardation of the incident wave using metamaterials [30], which is simpler to design than the former. [PERSON] et al.