# A Metal-Insulator-Metal Deep Subwavelength Cavity Based on Cutoff Frequency Modulation
## 2 Dispersion Properties and Cutoff Frequency  
Figure 1a shows a schematic diagram of a MIM waveguide consisting of two silver strips with a sandwiched low-index dielectric layer. Each strip has a thickness (_h_) of 100 nm and a width (_w_) of 240 nm. The dielectric layer has a thickness (_t_) of 10 nm. The refractive index of the dielectric layer is set to 1.5. In this MIM waveguide, the waveguide modes can be classified into fundamental and first-order waveguide modes according to the mirror symmetry of the dominant electric field (\(E_{z}\)) profiles for the plane with \(y\) = 0 (dotted lines of Figure 1b,c). Figure 1b,c show the top and side views for the \(E_{z}\) profiles of the fundamental and first-order waveguide modes, respectively, with a wavelength of 1550 nm (\(\lambda_{0}\)). In both modes, the electric fields of the waveguide modes are strongly localized in the deep subwavelength cross-section of the dielectric layer, 240 (_w_) \(\times\) 10 (_t_) (nm\({}^{2}\)) = 0.00099 (\(\lambda_{0}\)\({}^{2}\)), by index guiding in the _y_-direction and plasmonic coupling between the silver strips.  
Fundamental and first-order MIM waveguide modes have distinct dispersion properties, as shown in Figure 1d. The dispersion curve of the fundamental waveguide mode (black) shows a linear dependence between the frequency and the wavevector. The frequency linearly increases from the zero frequency as the wavevector increases. Alternatively, the dispersion curve of the first-order waveguide mode (red) has a nonzero lowest frequency at the zero wavevector; this is referred to as the cutoff frequency. Below the cutoff frequency, the first-order waveguide mode cannot exist in this MIM waveguide. Thus, the waveguide operates as a mirror for the first-order waveguide mode at frequencies below the cutoff frequency. The frequency region can be considered to be the mode gap of the first-order waveguide mode, which is indicated by the yellow box in Figure 1d. Therefore, the cutoff frequency mechanism can be used to make an effective mirror in the MIM-based cavity.  
Figure 1: (**a**) Schematic of a metal-insulator-metal (MIM) waveguide consisting of two silver strips and a low-index dielectric layer. The variables \(w\), \(h\), and \(t\) represent the waveguide width, the thickness of the silver strip, and the thickness of the dielectric layer, respectively; (**b**,**c**) Top and side views of the \(E_{z}\) electric field profiles of the fundamental and first-order waveguide modes with a wavelength of 1550 nm, \(w\) = 240 nm, \(h\) = 100 nm, and \(t\) = 10 nm. The top view (left) is obtained in the center of the dielectric layer and the side view (right) is obtained along the solid line of the top view; (**d**) Dispersion curves of fundamental (black) and first-order (red) waveguide modes, respectively. The yellow box indicates the mode gap region for the odd waveguide mode. Light line is indicated by blue line.  
The cutoff frequency strongly depends on the waveguide width (\(w\)) as well as the thickness (\(t\)) and refractive index (\(n\)) of the dielectric material. This is the case because the effective index of the first-order waveguide mode is sensitive to changes in these structural parameters. By spatially modulating the mode gap with these structural parameters, a photonic well can be formed for the first-order waveguide mode, localizing photons inside of the well. In this paper, we propose three types of deep subwavelength-sized cavities using the difference in cutoff frequency in the first-order waveguide mode.