# Surface Plasmon Nanolaser: Principle, Structure, Characteristics and Applications
## 2 Basic Principle of Surface Plasmons  
The surface plasmon is a collective oscillation caused by the free electrons in the metal surface under the action of an external electromagnetic field. According to the form of expression, surface plasmons can be divided into propagation-type surface plasmon polaritons (SPPs) and non-propagating localized surface plasmons (LSPs) [48; 49]. In this paper, the basic principles of surface plasmons are characterized by taking SPPs as an example. LSPs are generated by the interaction between the electrons in the particles and the photons of the incident light when the light is incident on the metal nanoparticles. SPPs are electromagnetic waves propagating along a metal surface generated by the interaction of external photons with free electrons on the metal surface., thus satisfying Maxwell equations.  
in the region of z < 0,  
\[H_{1}=(0,H_{y1},0)\exp[i(k_{x1}x-k_{z1}z-\omega t)], \tag{1}\]\[E_{1}=(E_{x1},0,E_{z1})\exp[i(k_{x1}x-k_{z1}z-\omega t)], \tag{2}\]  
in the region of z \(>\) 0,  
\[H_{2}=(0,H_{y2},0)\exp[i(k_{x2}x+k_{z2}z-\omega t)], \tag{3}\]  
\[E_{2}=(E_{x2},0,E_{z2})\exp[i(k_{x2}x+k_{z2}z-\omega t)]. \tag{4}\]  
According to the continuity of the electromagnetic field at the boundary z = 0, we obtain E\({}_{x1}\) = E\({}_{x2}\), H\({}_{y1}\) = H\({}_{y2}\), k\({}_{x1}\) = k\({}_{x2}\). Let k\({}_{x1}\) = k\({}_{x2}\) = k\({}_{spp}\), and the electric field is determined by the Maxwell equation:  
\[\
abla\times H=-ik_{0}\epsilon E, \tag{5}\]  
thus, at x = 0,  
\[\frac{k_{z1}}{\epsilon_{1}}+\frac{k_{z2}}{\epsilon_{2}}=0. \tag{6}\]  
In the above formula, k\({}_{x1}\) and k\({}_{x2}\) are positive numbers. Thus, only when the dielectric constant symbols of metals and dielectrics are opposite can surface plasmon waves be generated. The relationship between the wave vectors at the interface is \(k_{xi}^{2}+k_{zi}^{2}=\epsilon_{i}k_{0}^{2}\), which gives [50]  
\[k_{spp}=k_{0}\sqrt{\frac{\epsilon_{1}\epsilon_{2}}{\epsilon_{1}+\epsilon_{2}}}. \tag{7}\]  
It is noted that \(\epsilon_{1}\) \(<\) 0, \(\mid\)\(\epsilon_{2}\)+ \(\epsilon_{1}\)\(\mid\) \(\epsilon_{1}\) \(\mid\) ; therefore, k\({}_{spp}\) \(>\) k\({}_{0}\). In the dielectric material, k\({}^{2}\)\({}_{22}\) \(<\) 0, because k\({}_{x2}\) is an imaginary number. In the metal, k\({}^{2}\)\({}_{x1}\) = \(\epsilon_{1}\) k\({}^{2}\)\({}_{0}\) \(-\) k\({}^{2}\)\({}_{spp}\)\(<\) 0, because \(\epsilon_{1}\) \(<\) 0. It can be inferred that the surface plasmons are exponentially weakened in both z and -z directions perpendicular to the metal surfaces, either through the medium (vacuum) or the metal, and can only propagate along the metal surface [51; 52]. Due to the ohmic loss effect in the metal, the electric field strength decreases as the SPPs propagate along the metal surface. If a large propagation distance is desired, precious metals with small imaginary parts and large real parts, such as Ag and Au, should be used.  
Using this characteristic of surface plasmons, some metal nanostructures can be fabricated, which can couple photons into surface plasmons and confine them to the metal surface at the nanometer scale, thus greatly compressing the spatial distribution scale of the electromagnetic field. The above theory provides a theoretical basis for the realization of nanolasers.