# Principle and Applications of the Coupling of Surface Plasmons and Excitons
## 2 Basic Principle of Surface Plasmons  
The free electron gas [31] in a solid, when moving against a positive background of ions, may be regarded as plasma. Just as a wave of light can be quantified as a photon and a lattice vibration can be quantified as a phonon, the collective oscillation of the free electron gas in a solid's positive ion background can be regarded as an elementary excitation that can also be quantized as a type of quasiparticle, i.e., the plasmon. As we know, the momentum matching condition can be tuned for different wavelengths by changing the incident angle of the plane wave.  
Depending on their different boundary conditions, plasmons in metals can be divided into bulk plasmons and surface plasmons (SPs) [32; 33; 34]. Furthermore, surface plasmons can be subdivided into two categories: the first is conductive surface plasmon polaritons (SPPs), which occur at the metal-dielectric interface in the form of longitudinal waves, and the second is localized surface plasmons (LSPs), which are bound near metal structures. These three different forms are illustrated in Figure 1. As collective photon-electron oscillations makes them attractive candidates for energy applications, when light is incident on metal, it can lead to plasma oscillation inside the metal, and when the metal has boundaries, such as a metal-dielectric interface or a boundary of the metal nanoparticles' geometry, the field will be coupled with the plasmon oscillation within the metal and will form a new oscillation mode, i.e., a conductive surface plasmon oscillation or a localized surface plasmon oscillation.  
Figure 2 depicts the nanostructures of the surface plasmon polaritons (SPPs) and the localized surface plasmons (LSPs). In SPPs, the free electrons interact with the electromagnetic fields to generate dense electron waves that propagate along the metal surface [35], thus requiring one-dimensional metal structure which is close to the excited wavelength. In general, the wave vector of SPP dispersion curve is larger than that of the light while the frequency remains the same, the SPP dispersion relation  
can be calculated for short-pitch metal gratings for various depths, the dispersion of electromagnetic surface mode can be obtained and described by  
\[\text{k}_{\text{SPP}}=\text{k}_{0}\bigg{(}\frac{\varepsilon_{1}\varepsilon_{2}}{ \varepsilon_{1}+\varepsilon_{2}}\bigg{)}^{1/2} \tag{1}\]  
where \(\text{k}_{\text{SPP}}\) is the wave vector of the SPP, \(\text{k}_{0}\) is the wave vector of the incident radiation, and \(\varepsilon_{1}\) and \(\varepsilon_{2}\) are the frequency-dependent dielectric functions of the two media on either side of the interface [36].  
So it is impossible to excite plasmons with plane light waves, and some special structures are required to stimulate the SPPs, and the loss of SPPs is due to propagation [37], resulting from electron collision. Common excitation modes include prism coupling [38; 39], grating coupling [40], waveguide mode coupling [41], strong-focused beams [42], and near-field excitation [43].  
Moreover, the general strong coupling constant is given by [44]  
\[\text{g}=\frac{1}{4\pi\varepsilon_{0}\varepsilon_{r}}\frac{\pi e^{2}f}{mV}, \tag{2}\]  
Figure 1: Three different oscillations. Adapted from Ref. [30].  
Figure 2: Two types of nanostructure. Adapted from Ref. [35].  
where \(f\) is the exciton oscillator strength, and \(V\) is the effective modal volume, m represents the free electron mass, \(e\) is the electron charge and \(\varepsilon_{0}\varepsilon_{r}\) is the dielectric constant. In addition, the momentum matching condition can be tuned for different wavelengths by changing the incident angle of the plane wave [45].  
The theory here is based on the characteristics of surface plasmons when coupled with some excitons, which are confined to the metal surface on a nanometer scale and thus greatly compressed the spatial distribution of the electromagnetic field. This forms a theoretical basis for exploration of the coupling effect between surface plasmons and excitons.