## 2 Towards Optical Gas Sensing: Plasmonic Nanoparticles
### Localized Surface Plasmon Resonance (LSPR)  
Noble metals (e.g., Au, Ag) are excellent electrical conductors due to the typically long mean free path of the electrons in the \(s\)-type conduction band. A quantum of their collective oscillations is called a plasmon. The term is also used to loosely refer to these kinds of elementary excitation, which are oscillations of the electron density accompanied by an electromagnetic field. Plasmons can arise in bulk materials (volume plasmons), or at the interface between two materials (surface plasmons-SPs) [49]. The excitation of SPs occurs when they interact with an incident electromagnetic field of frequency close to their natural oscillation frequency. SPs are divided into two main fundamental excitations: (i) surface plasmon polaritons (SPPs), which are evanescent electromagnetic waves coupled to the electron plasma of a metal and propagating along its surface [41; 49; 50], as already mentioned in the previous section; and (ii) localized surface plasmons (LSPs), Figure 6a, which are nonpropagating excitations of the conduction band electrons in subwavelength metallic nanostructures, coupled to a confined electromagnetic field [49; 51; 52; 53].  
To comply with energy and momentum conservation laws, SPPs cannot be excited directly by a propagating photon in air, since their momenta do not match [54], while LSPs have the advantage of coupling directly to photons. These frequency dependent coupling processes are commonly known as surface plasmon resonance (SPR) and localized surface plasmon resonance (LSPR), whether they occur, respectively, in a metal surface/dielectric interface, or in a metal nanoparticle or nanostructure. The LSPR band can be detected in an optical transmittance spectrum by a depression zone, and the LSPR resonance frequency corresponds to the maximum of the extinction (absorption + scattering) (Figure 6b). Thus, optical transmittance-LSPR (T-LSPR) is a simple technique by which the LSPs can be excited by the traversing light.  
Different-order LSPs are predicted by the Mie theory for a metallic sphere. The dipole resonance is determined by the pole of the particle polarizability, \(\varepsilon_{s}+2\varepsilon_{m}=0\). Noteworthy is the dependence of the LSPR frequencies on the dielectric constant of the surrounding medium, which is the basis for the sensing applications of plasmonic nanoparticles. For a spheroidal nanoparticle, LSPRs can be split into two frequencies, corresponding to the poles of two unequal components of the polarizability tensor, \(\alpha_{i}\)[55]. Taking the noble metal permittivity within the simplest Drude model, the LSPR frequencies are given by:  
\[\omega_{p}^{\prime}=\frac{\omega_{p}}{\sqrt{1+\left(\eta_{||}^{-1}-1\right)}e_{m }};\;\omega_{p}^{\prime\prime}=\frac{\omega_{p}}{\sqrt{1+\left(\eta_{\perp}^{-1 }-1\right)}e_{m}},\]  
where \(\omega_{p}\) is the plasma frequency of the bulk metal and \(\eta_{||}\) and \(\eta_{\perp}\) are the corresponding depolarization coefficients. The two resonances correspond to the electric field directed along or perpendicular to the longer axis of the spheroid, respectively. The separation in frequency between them can be very large for elongated nanoparticles (see Section 5.3).