# Nanometric Plasmonic Rulers Based on Orthogonal Plasmonic Gap Modes in Metal Nanoblocks
## 2 Two-Dimensional Positioning of a Nanoblock  
The proposed 3D plasmonic ruler is composed of three silver nanoblocks. In order to readily illustrate the plasmonic interactions between the three blocks, we first investigated the opticalproperties of simpler structures that consist of two silver nanoblocks, called a 2D plasmonic ruler. After the determination of the lateral movement of two blocks, we extend the concept of the ruler to three silver nanoblocks. Figure 1 shows a schematic diagram of a 2D plasmonic ruler that consists of a large silver nanoblock and a small nanoblock, where the blocks are spaced with an air gap of 10 nm. The upper small nanoblock and the lower large nanoblock have dimensions of 200 nm \(\times\) 150 nm \(\times\) 100 nm and 260 nm \(\times\) 210 nm \(\times\) 100 nm, respectively. In this structure, the lateral movement along the x-y plane of the upper block is estimated by observing the absorption spectra, and the lower block with a fixed position is declared as the reference block. Silver is an optimal material because of its lower absorption losses in noble metals for the spectral range of interest. The center block size was chosen to locate the spectral positions of plasmon resonant mode 1 (M1) and mode 2 (M2) near 1000 nm. The peak wavelengths are insensitive to the upper and lower block sizes. The given differences in \(x\) and \(y\) dimensions of the center block were determined for two peaks, M1 and M2, which were placed with enough spectral spacing to be larger than linewidths of the peaks. In this paper, we calculate higher-order plasmonic resonant modes and their absorption cross-sections using a 3D finite-difference time-domain (FDTD) method. Silver was modeled by the Drude model, which is represented as follows:  
\[\varepsilon(\omega)=\varepsilon_{\infty}-\frac{\omega_{\mathrm{p}}{}^{2}}{ \omega^{2}+\mathrm{i}\gamma\omega}. \tag{1}\]  
The background dielectric constant (\(\varepsilon_{\infty}\)), plasma frequency (\(\omega_{p}\)), and collision frequency (\(\gamma\)) were set to 3.14, 9.13, and 0.021 eV, respectively, by fitting experimentally-determined dielectric functions of silver [14]. The spatial grid along the \(x,y\) and \(z\) axes were 2.5, 2.5, and 1 nm, respectively. When an incident light, linearly polarized along the \(z\)-axis (\(E_{z}\)), is injected into the proposed structure, a strongly confined surface plasmon cavity mode can be excited in the air gaps between nanoblocks. In this case, there are two orthogonal, higher-order plasmon resonant modes with one  
Figure 1: A schematic diagram of a 2D plasmonic ruler. The proposed ruler is made of two silver nanoblocks, a small upper block (200 nm \(\times\) 150 nm \(\times\) 100 nm) and a larger lower block (260 nm \(\times\) 210 nm \(\times\) 100 nm). Incident light with \(E_{z}\) linear polarization is on the \(x\)–\(y\) plane and has an incident angle of 40\({}^{\circ}\) from the \(y\)-axis. intensity node along the \(x\)- or \(y\)-axis. Moreover, compared with a fundamental gap cavity mode, the proposed cavity modes have one intensity node that suppresses radiation loss along the node direction [15; 16]. Suppression of radiation loss provides a narrow linewidth, which can enhance the sensitivity of the gap size sensors [15; 16]. Depending on the direction of the incident light, one of two modes can be selected by mode matching with an incident plane wave. When the linearly polarized light is injected into the proposed structure along the \(y\)-axis or \(x\)-axis, a plasmon resonant mode 1 (M1) with an intensity node along the \(x\)-axis or a mode 2 (M2) with a node along the \(y\)-axis can be excited in the air gap between the two nanoblocks, as shown in Figure 2a,b. To estimate the lateral movement (\(\Delta x\) and \(\Delta y\)) of the upper block, we need to exploit the optical properties of the two plasmonic modes, M1 and M2, simultaneously. In order to excite M1 and M2 at the same time, the incident angle of light was set to 40\({}^{\circ}\) from the \(y\)-axis (see inset of Figure 1).