# A Metal-Insulator-Metal Deep Subwavelength Cavity Based on Cutoff Frequency Modulation
## 3 Results
### Loss Analysis for the Three Cavities  
To understand the loss mechanisms of the MIM cavity modes, we investigated each type of loss (i.e., radiation loss, mirror loss, and metallic absorption loss) separately in terms of the quality (\(Q\)) factors. The \(Q\) factor is defined by \(2\pi f\times\) (stored energy in the cavity/power loss) [13]. The radiation loss corresponds to radiation from the cavity's sides to free space due to the imperfect horizontal modal confinement of index-guiding. Mirror loss originates from energy loss tunneling through the cutoff frequency mirrors of the narrow waveguides, which have a finite length. In addition, metallic absorption loss results from intrinsic ohmic loss in the silver strips. We were able to obtain \(Q_{\text{tot}}\), \(Q_{\text{rad}}\), \(Q_{\text{m}}\), and \(Q_{\text{abs}}\) by directly calculating the respective losses because the \(Q\) factors were inversely proportional to each loss, the total cavity loss, radiation loss, mirror loss, and absorption loss. We calculated each \(Q\) factor as follows. \(Q_{\text{total}}\) was obtained from the time decay of the total energy in the cavity. \(Q_{\text{abs}}\) could be obtained by directly calculating the absorbed energy in the metal. Next, we calculated \(Q_{\text{optical}}\) with the equation: \(1/Q_{\text{total}}=1/Q_{\text{optical}}+1/Q_{\text{abs}}\). Radiation loss and mirror loss, which consists of the total optical loss, were estimated by calculating the sums of Poynting vectors into free space and through two mirror waveguides, respectively. \(Q_{\text{rad}}\) and \(Q_{\text{m}}\) were obtained by the ratio of the sums of Poynting vectors and the following equation, \(1/Q_{\text{optical}}=1/Q_{\text{rad}}+1/Q_{\text{m}}\). Table 1 shows the \(Q\) factors of the three cavities. The \(Q\) factors for the three different cutoff frequency-modulated cavities (i.e., width-/index-/gap-modulated cavities) were similar and showed only slight differences. Therefore, by analyzing one of the three cavities, the loss properties of all three cavities could be understood. For example, in the case of a width-modulated cavity, the total \(Q\) factors (\(Q_{\text{total}}\)) of the fundamental and the second-order cavity modes were 45.5 and 63.2, respectively. \(Q_{\text{m}}\), which is inversely proportional to the mirror loss, was three orders of magnitude larger than the other \(Q\) factors when \(L_{m}=1000\) nm; therefore, tunneling loss through the cutoff frequency mirror was negligible. \(Q_{\text{abs}}\), which is related to the absorption loss (i.e., the dominant loss of the cavities), was between 80 and 90, regardless of the fundamental/second-order cavity modes of the cavities. Additionally, the radiation loss was comparable with the absorption loss. For example, \(Q_{\text{rad}}\) of the width-modulated cavity was 101 for the fundamental cavity mode and 260 for the second-order cavity mode. Since radiation into free space is difficult to collect (because of its large divergence from 10 nanometer-scale light confinement), it is desirable to minimize this type of loss for practical application of the proposed MIM cavity. Because radiation from the two oppositely-phased \(E_{z}\) intensity antinodes of the second-order mode along the intensity node direction into free space was strongly  
Figure 4: (**a**) Schematic of a cavity consisting of a cavity waveguide (\(t=10\) nm) and two mirror waveguides (\(t=30\) nm). The gap size between the two silver strips is modulated. In the gap, a low-index dielectric layer (\(n=1.5\)) is assumed. Here, \(w=240\) nm, \(L_{m}=1000\) nm, and the heights of the cavity and mirror waveguide silver strips are 100 nm and 80 nm, respectively. The cavity waveguide length is \(L_{c}\); (**b**) Dispersion curves of the first-order waveguide modes for \(t=10\) nm (blue) and \(t=30\) nm (red). The cutoff frequencies (wavelengths) are \(2\pi f=1059\) THz (1779 nm) and 1945 THz (1414 nm).