## 3 Results and Discussions
### Support Interaction  
In general, the support may influence the optical properties in two ways.  
* First, the support changes the dielectric environment and, thus, shifts the plasmon resonance to longer wavelengths. This effect can be taken into account by including the Yamaguchi theory [46] in the quasistatic approximation, using a mixing factor \(m\)[15]. The mixing factor corresponds to the part of the nanoparticle surface in contact with the substrate and has a value between 0 and 1 [15]. For oblate nanoparticles on substrates \(m\) can be determined either from the energetic position of both plasmon modes [47] or due to a modeling of the nanoparticle geometry using the adhesion energy on the substrate and the surface energy of the metal [43,48]. The latter method is necessary for gold nanoparticles, because they exhibit only the (1,1)-mode. The Yamaguchi theory has been proven to be sufficient for most applications, although the dielectric environment is reduced to an average of the dielectric functions of the contributing environments [14,15].  
Figure 2: (**a**) Extinction spectra during growth of gold on sapphire at different coverage \(\Theta\); (**b**) Optical extinction spectrum of gold particles 24 h after preparation (solid line) and calculated absorption cross section of a single particle with the same resonance frequency as the ensemble (dashed line).  
* Second, due to the surface energies, an oblate growth is expected, which cause also a red-shift of the plasmon resonance. However, neither the TEM nor the AFM images yield a clear evidence of an oblate nanoparticle shape. Moreover, since for gold nanoparticles the (1,0)-mode is damped by the interband transition, also from the extinction spectrum the shape cannot be extracted directly.  
In the following, we demonstrate that indeed oblate nanoparticles are grown. As explained in the introduction, the polarizabilities of an oblate nanoparticle along its principal axes \(a\) and \(b\) are determined by \(\alpha_{\rm a}\) and \(\alpha_{\rm b}\). Hence, the absorption cross section depends on the relative orientation of the particles' axes with respect to the angle of incidence and polarization of the incoming light. Consequently, the absorption cross section \(\sigma_{\rm abs}\) of the (1,1)-mode of gold nanoparticles is a function of the angle of incidence \(\vartheta\) and polarization angle \(\varphi\). Thus, \(\sigma_{\rm abs}\) in the range of the (1,1) mode can be written as [13]:  
\[\sigma_{\rm abs}=\frac{2\pi\omega}{c}\mbox{Im}\{\alpha_{\rm b}\cos^{2}\varphi \sin^{2}\vartheta+\alpha_{\rm b}\cos^{2}\vartheta\} \tag{3}\]  
This procedure is valid since \(\alpha_{\rm a}\) and \(\alpha_{\rm b}\) peak at different frequency ranges. Equation (3) reveals, that for normal incidence (\(\vartheta=0^{\circ}\)) the absorption is independent of the polarization angle. This is easily understood due to the symmetry of the system. In contrast, for an angle of incidence of \(\vartheta=45^{\circ}\), the particle absorption follows the dependence \(\sim 1+\cos^{2}\varphi\), due to the oblate shape. In other words, the nanoparticles appear round only under normal incidence, and the symmetry is lost for \(\vartheta\
eq 0^{\circ}\). To demonstrate that the nanoparticles are indeed oblate, the amplitude of the (1,1)-mode has been measured as a function of the polarization angle for \(\vartheta=0^{\circ}\) and \(\vartheta=45^{\circ}\). The experimental results together with the theoretical curves are depicted in Figure 3.  
It is obvious that the experimental data is in accord with the theoretical curve, which demonstrates that the nanoparticles can be approximated as rotational ellipsoids. The small deviation of the experimental data from the theoretical curve is due to experimental uncertainties.