# Nanometric Plasmonic Rulers Based on Orthogonal Plasmonic Gap Modes in Metal Nanoblocks
## 2 Two-Dimensional Positioning of a Nanoblock
Our modes M1 and M2 have strong electric field in the air gap, not around sharp block edges. Additionally, as shown in the mode profiles of Figure 2, significant enhancements at the edges are not observed. In fact, the resonant wavelengths of M1 and M2 are maintained, except for slight blue-shift due to the decrease of the effective cavity size when the corner of the blocks are rounded. In other words, the proposed plasmonic ruler is robust to the rounding of edges in the realistic experimental nanoparticles. In spite of the similarity in the mode profiles of M1 and M2, the resonant wavelengths of the two modes are different because of the different \(x\)- and \(y\)-directional sizes of the rectangular cavity: 200 and 150 nm. Figure 2a,b present a top-view of the dominant electric field (\(E_{z}\)) mode profiles of M1 and M2 when the upper block is placed just above the center of the lower large block, where the profiles are obtained at the center of the air gap. The resonant wavelengths of M1 and M2 are 882.2 and 1081.3 nm, respectively. M2, with an intensity node along the \(y\)-axis, primarily oscillates along the \(x\)-axis (the longer axis) and has a longer wavelength than that of M1. Next, we investigate changes in the properties of each mode for a lateral movement of the upper block when the air gap between the upper and lower blocks is fixed. As shown in Figure 2c,d, when the upper block moves on the \(x\)- and \(y\)-axes by amounts of \(\Delta x\) = 30 nm and \(\Delta y\) = 30 nm from the original position, the position of the strongly confined field also changes with respect to the upper block position. Therefore, an effective cavity size that determines a resonant wavelength of a cavity mode can be changed because the electric field of the cavity mode partially experiences the placement of the air region out of the lower block. In this regard, M1 and M2 experience distinct resonant wavelength shifts depending on the direction and size of the lateral displacements, \(\Delta x\) or \(\Delta y\), of the upper block. If the changes of the resonant wavelengths of M1 and M2 are measured, the direction and size of the lateral movement of the upper block are also obtained. As a result, the proposed structure can operate as a plasmonic ruler, which enables measurement of the horizontal relative position of the upper block to the lower block. In order to quantify the change in the resonant wavelength of each mode for the lateral movement of the upper block, we calculated the resonant wavelengths of M1 and M2 as functions of the movement of the block along the \(x\)- or \(y\)-direction from 0 to 30 nm at 5 nm increments. Figure 3a shows a schematic diagram of the upper block movement in the \(x\)- and \(y\)-directions. In this case, the air gap thickness between the upper and lower blocks was fixed to 10 nm. First, in the case of M1, as shown in Figure 3b, distinctive resonant wavelength shifts are observed depending on the moving direction, \(x\)- (black) or \(y\)- (red). When the block moves along the \(x\)-direction from 0 to 30 nm, the resonant wavelength of the M1 slightly increases from 882.2 to 885.1 nm. In contrast, when the block moves along the \(y\)-direction from 0 to 30 nm, the resonant wavelength of M1 largely decreases from 882.2 to 865.1 nm. Hence, the wavelength of M1, with an intensity node along the \(x\)-axis, has a larger wavelength shift for the movement of the upper block along the \(y\)-axis. On the other hand, the resonant wavelength of M2 shows opposite dependences on the movement of the block. As demonstrated in Figure 3c, when the upper block moves along the \(x\)-direction from 0 nm to 30 nm (black), the resonant wavelength largely blue-shifts from 1081.3 to 1064.7 nm; when the upper block moves along the \(y\)-direction (red), the wavelength slightly red-shifts from 1081.3 to 1084.2 nm. The wavelength of M2 with a node along the \(y\)-axis has a larger wavelength shift for the movement of the block along the \(x\)-axis.