# Nanometric Plasmonic Rulers Based on Orthogonal Plasmonic Gap Modes in Metal Nanoblocks
## 3 Three-Dimensional Positioning of a Nanoblock
In this figure, there are four absorption peaks at 849.0, 949.2, 1032.3, and 1163.2 nm, which are marked P1, P2, P3, and P4, respectively, from shortest to longest wavelength. P1 (849.0 nm) and P3 (1032.2 nm) indicate the  
Figure 7: **(a)** A schematic diagram of a 3D plasmonic ruler. A central larger block moves freely in 3D space with a movement of \(\Delta x\) (nm), \(\Delta y\) (nm), and \(\Delta z\) (nm). Upper and lower blocks have dimensions of 200 nm \(\times\) 150 nm \(\times\) 100 nm, and the central block has dimensions of 260 nm \(\times\) 210 nm \(\times\) 100 nm. Absorption cross-section spectra as functions of wavelength (**b**) without a movement of the central block, denoted by (0-0-0), and with a movement of the central block (**c**) only 2 nm along the \(z\)-direction (0-0-2) and (**d**) 30 and 2 nm along the \(x\)- and \(z\)-directions (30-0-2), respectively. P1 and P3 indicate the absorption peaks of mode 1 and mode 2, respectively, excited at the lower gap of 12 nm and P2 and P4 indicate the peaks of mode 1 and mode 2 at the upper gap of 8 nm. absorption peaks of M1 and M2 excited in the air gap with a thickness of 12 nm, whilst P2 (949.2 nm) and P4 (1163.2 nm) indicate those in the gap with a thickness of 8 nm. The resonant wavelengths can also be confirmed by the curves of the gap dependences of M1 and M2 in Figure 6b. Next, we investigate the absorption spectrum when the lateral displacement of the center block has a vertical movement of 2 nm. Figure 7d shows the absorption spectrum (red) corresponding to the position of the center block, \(\Delta x\) = 30 nm, \(\Delta y\) = 0 nm, and \(\Delta z\) = 2 nm, denoted by (30-0-2). In comparison with the absorption spectrum of \(\Delta x\) = 0 nm, \(\Delta y\) = 0 nm, and \(\Delta z\) = 2 nm, (0-0-2) (black), P1 and P2 corresponding to the M1 peaks of the air gap of 12 and 8 nm, respectively, are slightly red-shifted, and P3 and P4 corresponding to the M2 peaks of the air gap of 12 and 8 nm are largely blue-shifted. Wavelength shifts in M1 and M2 in the \(x\)-directional movement of 30 nm are identical to the wavelength dependences of M1 and M2 in the two-block structure, as shown in Figure 4b. Therefore, we can precisely estimate the position of the center block three-dimensionally using the single absorption spectrum of the three-block structure. On the other hand, four peaks--M1 and M2 modes for upper and lower air gaps--exist independently without any hybridization of each mode. Because M1 and M2 in the same air gap are orthogonal, the two M1 or M2 modes excited in different air gaps cannot couple each other due to large the block height of 100 nm. The method of estimating the position of the center block from the absorption spectrum of the three-block structure is analogous to that of the two-block structure. From the absorption spectra (e.g., those in Figure 7c,d), the wavelengths of the four peaks are determined as P1, P2, P3, and P4. Similar to the 2D color map of the wavelengths of M1 and M2 in Figure 5, the absorption peak wavelengths of M1 and M2 in an air gap can be calculated for all of the 3D space of \(\Delta x\) and \(\Delta y\) and for \(\Delta z\) (the air gap thickness). For 3D mapping of the absorption peak wavelengths, the wavelength surfaces for P1 and P2 are plotted in the M1 mapping. In addition, the wavelength surfaces for P3 and P4 are plotted in the M2 mapping. For example, in the absorption spectrum for the position of the center block, \(\Delta x\) = 30 nm, \(\Delta y\) = 0 nm, and \(\Delta z\) = 2 nm (30-0-2), P1 (852.1 nm), P2 (953.3 nm), P3 (1017.5 nm), and P4 (1148.3 nm) are observed. In Figure 8a, the wavelength surfaces of P1 (red) and P2 (yellow) are plotted on the M1 mapping, and the surfaces of P3 (cyan) and P4 (blue) are plotted on the M2 mapping. Once the surfaces of P1 and P3 are obtained for the same air gap, a blue line--the line of the intersection of P1 and P3 surfaces--can be displayed, which is on the plane of z = 12 nm.