# Nanoplasmonic Sensor Based on Surface Plasmon-Coupled Emission: Review
## 3 Plasmonic and Nanoplasmonic  
The collective oscillation of the free electron gas (charges) density is plasmon. On the other hand, plasmon is quasiparticles and a quantum of plasma oscillation. Plasmons can be obtained from the interaction of light and free charges. The topic, which is related to the application of this issue, is called plasmonic. Nanoplasmonic corresponds to charges density oscillation in nanomaterial, metal nanoparticle, metamaterials, and quantum dots. Sometimes, the light beam has an interaction with the charge density that is distributed on the surface of metal nanoparticles, thereby resulting in the absorption of light beam (Figure 2). This phenomenon, called localized surface plasmon resonance (LSPR) [7], is mainly used in explaining the UV-vis spectrum. When the size of particle is smaller than the wavelength of the light, the charge distributed on the surface of metal cluster oscillates collectively with the light beam. The LSPR occurs when frequency of this oscillation is matched with the frequency of photons [8]. In this case, the unique absorption of UV-vis spectrum and Rayleigh scattering of light are observed in relation to plasmonic effect [9; 10]. As mentioned above, the surface plasmon resonance occurs not only in the boundary of two materials with different dielectric sign, such as metal (gold or silver) and dielectric (glass), but also on the surface of metal cluster structure. When light incidents to the metal cluster in nano-size, the electrical field of electromagnetic wave interacts with free electron of metal cluster. The functions of the electric field and distribution of potential can be derived from Laplace equation in polar coordinate. Typically, the potential function is assumed of as follows:  
Figure 2: Metal cluster in electrical field. \[\phi(r,\theta)=\sum_{l=0}^{\infty}\left[A_{l}r^{l}+B_{l}r^{-(l+1)}\right]P_{1}( \cos\theta) \tag{3}\]  
where \(r\) and \(P_{l}(\cos\theta)\) are the radius part and angular parts (Legendre polynomials) of solution of Laplace equation, respectively. According to electromagnetism theory, the coefficient of Equation (3) will be calculated using boundary condition. To simplify the Equation (3), the potential of outside the metal cluster and inner one can be written as follows:  
\[\phi_{1}(r,\theta)=\sum_{l=0}^{\infty}A_{l}r^{l}P_{1}(\cos\theta) \tag{4}\]  
\[\phi_{2}(r,\theta)=\sum_{l=0}^{\infty}\left[B_{l}r^{l}+C_{l}r^{-(l+1)}\right] P_{1}(\cos\theta) \tag{5}\]  
where \(\phi_{1}\) and \(\phi_{2}\) are the potential inside and outside of the metal cluster, respectively. If the boundary conditions are considered, the potential function for inside and outside the metal cluster will be derived as follows [11]:  
\[\phi_{in}=-\frac{3\varepsilon_{d}}{\varepsilon_{m}+2\varepsilon_{d}}E_{0}r\cos\theta \tag{6}\]  
\[\phi_{out}=-E_{0}r\cos\theta+\frac{\varepsilon_{m}-\varepsilon_{d}}{ \varepsilon_{m}+2\varepsilon_{d}}E_{0}d^{3}\frac{\cos\theta}{r^{2}} \tag{7}\]  
The optical properties of metal cluster related to plasmonic effect can be justified in accordance with the potential of inside and outside of the metal cluster. The optical properties of metal nanoparticle and semiconductor depend on the plasmons. When the light beam interacts with materials, the electrical field of electromagnetic wave produces a force on the carrier charges in the material thereby inducing the dipole momentum in the material. If the particle size is smaller than the wavelength, the polarizability of the nanoparticle is [7]  
\[a=4\pi d^{3}\frac{\varepsilon_{m}-\varepsilon_{d}}{\varepsilon_{m}+2 \varepsilon_{d}} \tag{8}\]  
where \(a\), \(d\), \(\varepsilon_{m}(\varepsilon_{mr}+i\varepsilon_{mi})\), \(\varepsilon_{d}\) are polarizability, nanoparticle diameter, relative permittivity of the metal and surrounding medium. The optical response of the nanoparticle depends on polarizability. The strongest mode will be obtained when \(\varepsilon_{m}\) is near \(-2\varepsilon_{d}(\varepsilon_{m}=-2\varepsilon_{d})\) or the denominator of Equation (8) is near to zero.