## 3 Results and Discussion
### Numerical Predictions  
The discrete dipole approximation (DDA) was employed to calculate the optical properties of a single GNR for each particular shape and orientations and also GNR dimers. The optical properties of a single GNR highly depend on the angle (\(\alpha\)) between its optical axis and the propagation direction of the light. Figure 7 presents the calculated absorption efficiency of a single GNR with various ARs at different orientations. The arrow in each figure indicates the longitudinal SPR of GNR. It is clearly shown in each figure that the wavelength LSPR did not vary with the direction of incident light. As can be seen from Figure 7c, the wavelength of the longitudinal SPR of a single GNR was around 800 nm and did not change with the angle. The wavelength of the transverse SPR was around 500 nm, which was same as the value obtained in the experiment. The solid curves in Figure 7 indicate the average absorption efficiency of GNR at different orientations. The highest absorption efficiency was achieved when the axis of GNR was perpendicular to the incident light. However, the absorption efficiency at the wavelength of longitudinal SPR was nearly 0, when the axis of GNR was parallel to the incident light. The magnitude of the transverse SPR was less than 1 so it was insignificant compared to the longitudinal SPR. Although the GNR showed high absorption efficiency at the wavelength of longitudinal SPR, the orientations of the GNRs suspended in water can cause the absorption efficiency vary from 12 to nearly 0. The spectral of GNRs shown in Figure 7a-c experience a clear red shift with the increase of aspect ratio. This red shift evidences that the SPR of a single GNR can be significantly affected by the shape of the nanoparticles. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Sample** & **Absorbance** & **Temperature Increase (K)** & **Error of Temperature Measurements (K)** \\ \hline D & 0.61 & 0.2 & \(+\)0.015 \\ E & 0.80 & 0.4 & \(-\)0.014 \\ F & 0.96 & 0.5 & \(-\)0.013 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Results from optical and thermal experiments obtained at 780 nm wavelength [35]. The absorption efficiency of GNRs (AR = 2, 3 and 4) for five different orientations at each plasmonic resonance wavelength is presented in Figure 8. The predictions indicate that optical properties of GNRs highly dependent on \(\alpha\), which is the angle between its optical axis and the propagation direction of the light source. Figure 8 shows the linear relationship between the absorption efficiency and \(\sin^{2}\alpha\) of the GNRs. More specifically, the highest absorption efficiency of an independent GNR occurred when the GNR was perpendicular to the propagation direction of the incident light. However, this also indicates that the absorption efficiency could be negligible if the optical axis of the GNR is parallel to the propagation direction of the light. Figure 7: Spectral dependency of absorption efficiency of a single GNR with different aspect ratios (AR), (**a**) AR = 2, (**b**) AR = 3, (**c**) AR = 3.8 and (**d**) AR = 4. As can be seen in Figure 9, the wavelength of longitudinal SPR and the extinction efficiency at this wavelength increased linearly with the aspect ratio of the GNR. Results shown in Figure 9 were obtained for the perpendicular orientation of the GNR (\(\alpha=90^{\circ}\)), with the effective radius being fixed at 8.9 nm. The wavelength of the SPR for GNR with aspect ratio AR = 3.8 calculated from the linear equation shown in Figure 9 was 810 nm, which was the same as obtained from the DDA predictions shown in Figure 6. The plasmonic resonance of absorption efficiency also showed a linear relationship with the aspect ratio in Figure 9. It should be noted that the absorption efficiency presented by red marks in Figure 9 was obtained for each aspect ratio at correspondent SPR wavelength shown as blue marks in the same figure. The wavelength of the longitudinal SPR of AR = 3.8 GNR solutions, as measured in the optical experiment, was equal to 780 nm.