# A Metal-Insulator-Metal Deep Subwavelength Cavity Based on Cutoff Frequency Modulation
## 3 Results  
### Waveguide Width-Modulated Cavity  
In this paragraph we introduce a MIM first-order mode cavity with width modulation by using the strong width-dependence of the cutoff frequency. Figure 2a shows a schematic of a cavity consisting of a broad waveguide (\(w=240\) nm) and two narrow mirror waveguides (\(w=120\) nm), where the waveguides consist of two silver strips and a low-index (\(n=1.5\)) dielectric layer (\(t=10\) nm). Here, the mirror waveguide length (\(L_{m}\)) is set to \(1000\) nm.  
Figure 2: (**a**) Schematic of a cavity consisting of a broad waveguide (\(w=240\) nm) and two narrow mirror waveguides (\(w=120\) nm). Here, \(h=100\) nm, \(t=10\) nm, \(L_{m}=1000\) nm, and the cavity length is \(L_{c}\): (**b**) Dispersion curves of first-order waveguide modes for waveguide widths of \(w=240\) nm (black) and \(w=120\) nm (red); their cutoff frequencies (cutoff wavelengths) are \(2\pi f=1059\) THz (\(1779\) nm) and \(1945\) THz (\(968\) nm), respectively. The horizontal blue line indicates a frequency (wavelength) of \(1215\) THz (\(1550\) nm). The light line is indicated by the blue line; (**c**) Cutoff frequency of the first-order waveguide mode along the \(x\)-axis for the cavity in (**d**). The allowed and forbidden regions are indicated by white and yellow colors, respectively. The allowed frequency region for the first-order waveguide mode is spatially changed by modulating the waveguide width; (**d,e**) Top and side views of the electric field profiles of the fundamental (\(L_{c}=210\) nm) and second-order (\(L_{c}=770\) nm) cavity modes, respectively. The solid lines indicate the cross-sections of the side views.  
The dispersion curves of the first-order waveguide modes for broad (\(w\) = 240 nm, black) and narrow (\(w\) = 120 nm, red) waveguides are plotted in Figure 2b. Since the first-order mode in the narrower waveguide experienced more air outside of the MIM waveguide, the dispersion curve moves upward due to the smaller effective index. In particular, the cutoff frequencies (cutoff wavelength) of the broad/narrow waveguides are 2\(\pi\)f = 1059 THz (1779 nm) and 1945 THz (968 nm), respectively. Propagation of first-order waveguide modes with a frequency between the two cutoff frequencies is allowed for the broad waveguide (\(w\) = 240 nm); however, this is forbidden for the narrow waveguide (\(w\) = 120 nm). For example, light with a wavelength of 1550 nm (1215 THz), indicated by the horizontal blue line in Figure 2b, is only allowed in the broad waveguide region of Figure 2a. Indeed, the cavity formed a photonic well by modulating the cutoff frequency with the waveguide width, as shown in Figure 2c. The allowed frequency region from 1059 THz (1779 nm) to 1945 THz (968 nm) was localized in the broad waveguide with a length of \(L_{c}\).  
The resonant wavelength of the cavity mode can be controlled by changing the length (\(L_{c}\)) of the cavity region. In addition, the cavity mode is classified as a fundamental and higher-order cavity mode, depending on the number of intensity nodes along the \(y\)-direction. Figure 2d,e show the electric field profiles of the fundamental and second-order cavity modes with the same resonant wavelengths (1552 nm) where the dominant electric field is orthogonal to the two metal surfaces. Because both modes are based on the first-order waveguide mode, there is a common intensity node along the \(x\)-axis. However, the fundamental cavity mode in the short cavity (\(L_{c}\) = 210 nm) has one intensity antinode along the \(y\)-direction. Alternatively, in the long cavity (\(L_{c}\) = 770 nm), the second-order cavity mode has two intensity antinodes along the \(y\)-direction.