## 1 Introduction  
In a groundbreaking theoretical concept in the early 1950s, [PERSON] and [PERSON] [1] proved that excitations in long-range Coulomb interacting systems can be decomposed into two separate sectors, namely the high-energy collective excitations, so-called longitudinal bulk plasmons, and the low-energy single-electron excitations. Well-defined plasmon oscillations exist if the momentum of carriers in the system is much smaller than the Thomas-Fermi wavevector. Moreover, the plasmon branch enters the single-electron excitation region where at this point, the collective energy of the plasmon dissipates into single electron-hole excitations. This process is known as the Landau damping [2]. The [PERSON] result is consistent with the classical plasma picture and was the first demonstration of the idea of the renormalization group theory in physics. Plasmonics is based on interaction processes between electromagnetic radiation and itinerant charges (electrons or holes) at metallic or doped semiconductor interfaces or in small metallic nanostructures. Although it is well known that there are two main ingredients of plasmonics, namely surface plasmon polaritons and localized surface plasmons, it is often far from trivial to appreciate the interlinked nature of many of the phenomena and applications of this field. This is compounded by the fact that throughout the 20 th Century, surface plasmon polaritons have been rediscovered in a variety of different contexts. Accordingly, the science of plasmonics is dealing with generation, manipulation and detection of surface plasmon polaritons. Surface plasmon-polaritons are electromagnetic surface waves coupled to plasmon modes of the itinerant charges (electrons or holes), propagating along the interface between a dielectric and a conductor. Therefore, surface plasmon-polaritons are bound modes whose fields decay exponentially away from the interface, and therefore, plasmonics opened the possibility for manipulating light and controlling light-matter interactions at scales below the diffraction limit. Propagation of electromagnetic waves at the interfaces of plasmas with other dielectrics depends strongly on the interface geometry. A surface plasmon was predicted by [PERSON] in 1957 [3]. The plasmon mode shows dispersions of the various plasmon modes at the metal-vacuum surface. The two dispersion branches of the surface plasmon, the dispersion-less branch of the longitudinal bulk plasmon (\(\omega_{p}\)) and the dispersion of the so-called multipole surface plasmon, can be found along with a linear dispersion for an electromagnetic wave, \(\omega=ck\) where \(c\) is the speed of light in vacuum, parallel to its surface stemming from the coupling of a photon and a plasmon at the interface. The coupling between the light and the longitudinal bulk plasmon leads to a splitting of the \((\omega-k)\) dispersion curves for the excitations, which form a photon dispersion and the bulk plasma mode as the joint of the photon mode and surface plasmon mode. For small wave vectors, dispersion of the surface plasmon mode asymptotically approaches the light-line. For the large wavevector, in the local approximation, this surface plasmon approaches asymptotically a constant frequency, which for metals is \(\omega=\omega_{p}/\sqrt{2}\). Let us focus on the interaction of metals with electromagnetic fields using a classical approach based on [PERSON]'s equations. Small sizes of metallic nanostructures on the order of a few nanometers could be qualitatively described by semiclassical mechanics. The reason for that is the high density of free carriers results in minute spacings of the electron energy levels compared to thermal excitations of energy \(k_{B}T\) at room temperature.