# Nanogap Plasmon Resonator: An Analytical Model
## 1 Introduction  
The calculation of the electric field in the gap between two metallic cylinders is a classical problem in electrodynamics. However, the metal permittivity is typically negative in the optical spectral range, and plasmons are excited between the cylinders, which provide a great enhancement of the local electric field in the gap [1]. Later, it was shown that the large optical field fluctuations in random semicontinuous metallic films were due to the local field concentration in between large metallic clusters [2]. The two metallic nanostructures supporting localized plasmons behave as an open resonator when they are separated by a spatial gap. Exact spectral control is achieved by changing the nanogap's width [3; 4]. [PERSON] et al. reported the use of plasmon nanogaps in order to increase the sensitivity of photodetectors [5]. [PERSON] group showed that the frequency transformation due to the non-linearity was determined by the value of the local electric field and its spatial localization within the gap [6]. The physical processes that are crucially important for sensing can be observed in quantum plasmonics for gaps having a subnanometric size between the metallic particles [7]. [PERSON]'s group reported the use of graphene to tune the subnanometric size of the gaps within metallic dimers [8]. Metallic nanoparticle arrays with a 1-5 nm range of the gap size were prepared by using the atomic layer deposition of aluminum oxide, allowing the control of this gap size [9]. Several groups have demonstrated the achievement of strong electric fields thanks to plasmonic nanoparticles/nanostructures deposited on a metallic film serving as a mirror [10; 11; 12; 13; 14; 15; 16]. Furthermore, a great number of research groups have studied the coupling of plasmonic nanoparticles with nanogaps or subnanogaps [9; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33; 34]. These various groups have theoretically and experimentally investigated great enhancements of the electromagnetic field concerning the coupling of these plasmonic nanoparticles with nanogaps (or subnanogaps) [35; 36; 37; 38; 39; 40; 41; 42; 43; 44]. [PERSON] et al. reported the generation of high-order modes of plasmons obtained with gold nanoparticles deposited on a gold disk of a micrometric size with a gap between them, which can be adjusted by the dimensions of the molecule serving as a spacer [30]. [PERSON] et al. presented studies based on the metasurfaces of a gap plasmon composed of nanostructure arrays and a thick film of metal spaced apart by a thin film of dielectric, thus permitting the parameter adjustment of the reflected light (i.e., polarization, amplitude, and phase) [45]. Furthermore, the principle of the gap plasmon can be applied to several domains, such as strong coupling between the plasmon and exciton [46; 47], the spectroscopy of infrared absorption enhanced by the surface [48], the improvement of electroluminescence [49] and fluorescence [50], non-local effects [51], hyperspectral imaging [52], and the detection of hot electrons [53]. A strong motivation for the investigation of gap plasmons is the effect of surface-enhanced Raman scattering (SERS), which relies on plasmonic enhancement to enable the identification of trace molecules captured within gaps [54]. SERS is extremely important for medical diagnostics, for instance for cancer detection, imaging, and therapy, drug delivery, and the quantitative control of biomarkers, including glycated proteins and cardiovascular biomarkers, and viruses [34; 55; 56; 57; 58; 59]. For instance, [PERSON] et al. demonstrated a sensitive detection of less than 1 pg of a biological molecule (the spike glycoprotein of SARS-CoV-2) by employing a nanometric cavity formed by an Ag film and a layer of a receptor-binding domain of this glycoprotein [34]. [PERSON] et al.