# Nanometric Plasmonic Rulers Based on Orthogonal Plasmonic Gap Modes in Metal Nanoblocks
## 2 Two-Dimensional Positioning of a Nanoblock
That is to say, when the upper block is moved in the node direction, the resonant wavelength is slightly red-shifted; when it is moved perpendicular to the node direction, the resonant wavelength is largely blue-shifted. Based on these different dependences of the changes in wavelengths of the two modes, M1 and M2, we expect that the lateral position of the upper block can be estimated by observing the resonant wavelengths of M1 and M2 simultaneously. Figure 3: (**a**) Lateral movement of a small upper nanoblock, indicated by \(\Delta x\) and \(\Delta y\). A transparent grey box represents an original upper block with no movement. The resonant wavelengths of the (**b**) M1 and (**c**) M2 as functions of \(x\)- (black) and \(y\)- (red) displacements of the cavity with a fixed air gap size of 10 nm. Insets show the mode profiles of M1 and M2 in the cavity with no movement. Figure 2: The dominant electric field (\(E_{z}\)) mode profiles of (**a**) mode 1 (M1) and (**b**) mode 2 (M2) in the cavity with an upper block placed at the center. The mode profiles (**c**) M1 and (**d**) M2 in the cavity with a lateral movement (\(\Delta x\) = 30 nm and \(\Delta y\) = 30 nm) of the upper block. Each mode profile is obtained from the center of the air gap. The absorption cross-sections of the cavity structure are strongly enhanced at the resonant wavelength of the plasmonic cavity mode. In order to exploit the absorption cross-section spectra to obtain the resonant wavelength shifts of M1 and M2 for the lateral movement of the upper block, a monochromatic plane wave source with vertical polarization (\(E_{z}\)) was injected into the proposed structure to excite the plasmonic modes M1 and M2 simultaneously. As shown in the inset of Figure 1, the incidence angle of the plane wave was set to 40\({}^{\circ}\) and the absorption cross-section was calculated in the spectral range from 800 to 1250 nm, which includes the resonant wavelengths of M1 and M2. Figure 4a shows the absorption cross-section spectrum for the original position of the upper block, indicated by (0-0) meaning (\(\Delta x\) = 0 nm and \(\Delta y\) = 0 nm). In this absorption cross-section spectrum, each peak position corresponds to the resonant wavelengths of M1 and M2 (882.2 and 1081.3 nm, respectively). Similarly, Figure 4b-d show the absorption cross-section spectra for the lateral movement of the upper block: (30-0), (0-30), and (30-30), respectively. Black curves represent the absorption cross-section spectrum at the reference position, as shown in Figure 4a, and red curves represent the absorption spectra with respect to the corresponding position of the upper block. In the case of the movement of the block only along the \(x\)-axis by 30 nm (denoted (30-0) in Figure 4b), the corresponding absorption peaks in the red curve are 885.1 and 1064.7 nm, respectively, which correspond to M1 and M2. The shorter wavelength mode, M1, has a small red-shift of 2.9 nm and the longer wavelength mode, M2, has a large blue shift of 16.6 nm. In contrast, for the movement only along the \(y\)-axis by 30 nm--(0-30) in Figure 4c--M1 has a large blue-shift of 17.1 nm and M2 has a small red-shift of 2.9 nm. In the case of movement of \(\Delta x\) = 30 nm and \(\Delta y\) = 30 nm, (30-30), the two modes have large blue-shifts of 16.9 and 16.3 nm. Compared to the initial wavelengths, the shifted absorption resonant wavelengths have similar tendencies with the resonant wavelength shifts characteristic of M1 and M2, as depicted in Figure 3b,c. A large wavelength shift of M1 (M2) along the \(y\)- (\(x\)-)axis exploits the resonant wavelength of M1 (M2) as a ruler for the \(y\)- (\(x\)-)directional position of the upper block. Figure 4: (a) Absorption cross-section spectrum of the 2D plasmonic ruler without upper nanoblock movement, (\(\Delta x\) = 0 nm and \(\Delta y\) = 0 nm), denoted by (0-0). Two absorption peaks are observed at 882.2 nm (M1) and 1081.3 nm (M2).