# Nanometric Plasmonic Rulers Based on Orthogonal Plasmonic Gap Modes in Metal Nanoblocks
## 2 Two-Dimensional Positioning of a Nanoblock
Absorption cross-section spectra with respect to \(\Delta x\) (nm) and \(\Delta y\) (nm) values of (b) 30 nm, 0 nm (30-0), (c) 0 nm, 30 nm (0-30), and (d) 30 nm, 30 nm (30-30). The black line indicates a reference absorption spectrum (0-0) and the red line indicates the absorption spectrum of the cavity with a moved upper block. The intensity of the mode which has a large wavelength shift for a given movement of the block changes considerably. For example, in Figure 4b, the peak intensity of M2 which shows large wavelength shift decreases from 0.012 to 0.08 in contrast to the no significant change of M1 intensity because light coupling into each mode changes depending on the block positions. In order to calculate the absorption spectra, the absorbed energy in the block is calculated by subtracting the flux of the scattered light from the integrated flux of the incident light at each wavelength. When the angle of incidence changes, the peak positions and linewidths of M1 and M2 are maintained, while the peak intensities change. For example, in the case of the incidence angle of 0\({}^{\circ}\), along the \(y\)-axis, only M1 can be observed and M2 disappears. In contrast, for the incidence angle of 90\({}^{\circ}\), along the \(x\)-axis, only M2 is observable. Therefore, in changing the incidence angle, one can excite M1 or M2 separately in the experimental measurement, which allows measurement of the spectral position of each peak more accurately by removing the overlap of peaks, resulting in higher sensitivity. In the experiment, a thin oxide layer could be observed on the surfaces of the silver metal blocks. The layer in the air gap induced a slight red-shift of the wavelength due to an increase of the effective optical path length in the gap; however, the operation mechanism of the 3D plasmonic ruler was maintained, except for the shift. In order to determine the lateral position of the upper block more precisely via the two absorption peaks M1 and M2, we need to calculate the absorption cross-section peak wavelengths of M1 and M2 for 2D movement of the upper block using 2D color mapping of the wavelengths, as shown in Figure 5a,b. In these 2D maps, the absorption peak wavelengths are displayed as different colors for \(\Delta x\) and \(\Delta y\), ranging from 0 to 30 nm. In the wavelength map of M1 (Figure 5a), the color of the corresponding wavelength is almost maintained along the \(x\)-axis, \(\Delta x\). However, the color changes dramatically along the \(y\)-axis, \(y\)-directional movement, \(\Delta y\), which is demonstrated in Figures 3b and 4. In the case of M2 in Figure 5b, the dependencies are opposite, as shown in Figures 3c and 4. Based on two-color mapping of M1 and M2 for the lateral movement of the upper block, it is simple to determine the position of the block if measuring the absorption cross-section spectrum of the incident light with an incidence angle of 40\({}^{\circ}\) and with vertical polarization. The method is as follows:  
1. Determine the wavelengths of two peaks--M1 and M2--of an absorption cross-section spectrum. 2. Draw lines corresponding to the two wavelengths, shorter and longer wavelengths, on the 2D color mapping spaces of M1 and M2 (Figure 5a,b, respectively). 3. Determine the corresponding movement of \(\Delta x\) (nm) and \(\Delta y\) (nm) from a point of intersection of the two lines. Figure 5: 2D color mapping of the absorption cross-section peak wavelengths of (**a**) mode 1 (M1) and (**b**) mode 2 (M2) as functions of \(\Delta x\) (nm) and \(\Delta y\) (nm). Cyan lines in (**a**) M1 and (**b**) M2 indicate resonant wavelengths—882.2 and 1081.3 nm—for the movement of \(\Delta x=0\) nm, \(\Delta y=0\) nm. Black lines indicate 885.1 and 1064.7 nm for the movement of \(\Delta x=30\) nm and \(\Delta y=0\) nm, respectively. The color bar at the right represents the absorption cross-section peak wavelengths. For example, in the case of \(\Delta x\) = 0 nm and \(\Delta y\) = 0 nm, the absorption peak wavelengths of M1 and M2 are 882.2 and 1081.3 nm, respectively.