# Tunable Fano Resonances in an Ultra-Small Gap
## 1 Introduction  
When discrete and continuous excited-state energy levels overlap, quantum interference occurs. A zero absorption occurs in a specific optical band, and thereby generates an asymmetric spectrum that is called a Fano resonance [1]. It was first reported by [PERSON] when studying atomic spectra, and he theoretically attributed the asymmetric resonance to a destructive interference that could not be described by the Lorentz formula. The destructive interference suppresses the radiation attenuation of the system, which also greatly increases its sensitivity to the surrounding media and concentrates the system energy on the surface of the structure [2]. Relative to the initial surface plasmon mode [3; 4; 5], the Fano resonance has a stronger near-field enhancement and a finer spectrum, as well as a higher figure of merit (FOM). Therefore, it has potential applications in biosensing and field enhancement [6]. Other potential applications include lasers [7], optical switches [8; 9], nonlinear optics [10; 11], high-quality-factor devices [12], slow-light devices [13], and surface-enhanced Raman scattering [14]. Fano resonances in metallic nanostructures have also been observed in optical responses [15; 16]. These nanostructures have abundant localized surface plasmon resonances (LSPR) that originate from collective electron oscillations on the metal surface. Common LSPR modes include dipoles, quadrupoles, and octupoles. The dipole mode has a large dipole moment, with a large resonance bandwidth in the spectrum, while the quadrupole and the octupole modes have small dipole moments, with small resonance bandwidths in the spectrum. Radiation from the broad resonance mode has high attenuation but can be coupled as a \"bright mode\" into an incident electromagnetic field. Meanwhile, a narrow resonance mode with a small net dipole moment has very low radiation attenuation and cannot be coupled into an incident electromagnetic field; this is the \"dark mode\" [17; 18]. The presence of bright and dark modes gives rise to Fano resonances in metallic nanostructures. In complexstructures, there are abundant plasmon hybridizations among various plasmon modes that lead to Fano resonances, such as a disk-ring nanocavity [19], a metal core-shell nanostructure [20; 21; 22], a self-assembled metal nanostructure [6; 14; 23; 24], a metal nanoparticle (NP) array [25; 26; 27; 28], and a waveguide and cavity resonator coupling structure [29; 30]. These complex structures usually require complicated fabrication methods. For example, self-assembled metallic nanostructured pentamers require electron beam lithography, which is not only costly, but also involves highly controlled precision and considerable time [23]. It would be advantageous to fabricate simple metal nanostructures at low cost to realize Fano resonances. Recently, Fano resonances were observed in silver nanocube dimers [31; 32], where small spacer separations occur via surfactants. However, precise control of these small nanogaps is challenging and the silver nanocube dimers are easily misaligned. Thus, they are poorly tuned and difficult to integrate. A metallic substrate supporting metallic NPs forms a NP-on-mirror (NPoM) structure, which breaks the symmetry [33; 34]. The presence of the substrate creates a non-uniform dielectric environment around the metallic NPs, enabling plasmon mode hybridization that couples broad and narrow plasmon resonances. When the size of the metal NP is large, dipole mode and multipole mode are generated, but these modes will not be coupled with each other. When the dielectric environment around the metal NPs changes, such as placing a substrate near the metal NPs, it creates conditions for interference between the modes. The destructive interference between the dipole mode and the higher-order mode will generate Fano resonance. Due to the presence of the metal film in the NPoM structure, a mirror image of metal NPs is formed in the metal film, which is similar to the dimer structure.