# Nanometric Plasmonic Rulers Based on Orthogonal Plasmonic Gap Modes in Metal Nanoblocks
## 3 Three-Dimensional Positioning of a Nanoblock  
Until now, we assumed the air gap thickness to be fixed at 10 nm. However, due to the strong plasmonic coupling in close proximity to metal nanoparticles, the resonant wavelength of the proposed structure has strong gap dependence [1; 2; 3; 4; 5; 15; 16]. For the same structure as is in Figure 3, we calculated the resonant wavelengths of M1 and M2 by varying the vertical position of the upper block along the \(z\)-direction, as shown in Figure 6a. The resonant wavelengths of M1 (black) and M2 (red) are plotted as functions of the gap thickness in Figure 6b, with a thickness range from 2 to 20 nm. The wavelengths increase as the gap decreases, and the wavelengths dramatically increase at small gap size. For example, the resonance of M1 red-shifts from 882.2 to 949.2 nm with a change of 66.8 nm; for a decrease of the gap from 10 to 8 nm, there is a change of 2 nm. In contrast, for the same 2 nm change of the gap from 4 to 2 nm, a large red-shift of 359.8 nm is observed from 1164 to 1523.8 nm. This nonlinear dependence of the wavelength on the gap is widely observed in the gap plasmon resonances [1; 2; 3; 4; 5]. The wavelength of the two modes, M1 and M2, have similar dependences on gap thickness. As shown in Figure 5, the lateral displacement of the upper block can be measured precisely using a two-nanoblock structure. However, because of the degrees of freedom in the 3D positioning of a nanoblock relative to the reference block, a combination of lateral and vertical movement cannot be estimated simultaneously in the two-block scheme. Thus, we introduce an extended nanoblock structure that can measure the position of a block three-dimensionally using the same mechanism: the measurement of the absorption peaks of orthogonal plasmonic modes in two silver nanoblocks. Figure 7a shows the extended nanoblock structure, which consists of three silver nanoblocks. In this structure, we added a small silver nanoblock to the larger block of the two-block structure with the same air gap thickness of 10 nm. Top and bottom small blocks have a size of 200 nm \(\times\) 150 nm \(\times\) 100 nm, and the center large block has a size of 260 nm \(\times\) 210 nm \(\times\) 100 nm. Contrary to the two-block  
Figure 6: (**a**) A schematic diagram of the vertical movement of the small upper block with a shift of \(\Delta z\) without a lateral movement. (**b**) Resonant wavelengths of mode 1 (M1) (black) and mode 2 (M2) (red) as functions of gap thickness. structure with one air gap, the extended structure consists of two air gaps: between the top small and the center large block, and between the center large and the bottom small blocks. Here, the top and bottom blocks are reference blocks and are assumed to have fixed positions. A center large block is moved three-dimensionally with values of \(\Delta x\) (nm), \(\Delta y\) (nm), and \(\Delta z\) (nm), which is measured by the absorption peaks due to the plasmonic modes in the two air gaps. First, we calculate the absorption spectrum when the center block is placed at the origin with no movement, \(\Delta x\) = 0 nm, \(\Delta y\) = 0 nm, and \(\Delta z\) = 0 nm, denoted by (0-0-0), as shown in Figure 7b. Because the thicknesses of the two air gaps are the same and both gaps are 10 nm, the absorption cross-section spectrum of the structure shows the same absorption peak wavelengths as those of the two-block structure in Figure 4a: 882.2 and 1081.3 nm, the wavelengths of M1 and M2. However, when the center block moves along the \(z\)-axis to 2 nm from the origin, the upper and lower air gaps become 8 and 12 nm, respectively. Due to the strong gap thickness dependence of the resonant wavelengths in the gap plasmonic cavity, as shown in Figure 6b, there are large resonant wavelength differences between the plasmonic modes in the upper and lower air gaps. Figure 7c shows the absorption spectrum of the center block with \(\Delta x\) = 0 nm, \(\Delta y\) = 0 nm, and \(\Delta z\) = 2 nm (the 0-0-2 position).