## 2 Towards Optical Gas Sensing: Plasmonic Nanoparticles
### Background of Surface Plasmons  
Although the aforementioned colour effects have been used for several centuries, their physical interpretation only started to gain relevance by the end of the 19 th century and beginning of the 20 th [37]. Particularly interesting was the theory proposed by [PERSON] in 1908, who worked out a solution of [PERSON]'s equations to calculate the extinction spectrum of a spherical particle illuminated by a time-harmonic electromagnetic field. Figure 5: **(a)** Notre-Dame Cathedral, Paris showing the south facade, and the southern rose window, photographed from **(b-i)** inside and **(c-i)** outside. A detail of the southern rose window photographed from **(b-ii)** inside and **(c-ii)** outside (mirrored) (adapted with permission from [PERSON], copyright 2021). The solution is an infinite series expansion with different terms corresponding to the \(n\)-th order multipoles of the sphere. The coefficients of this expansion are expressed in terms of the Riccati-Bessel functions of the order \(n\) and the argument called the size parameter, \(x=2\pi n_{s}a/\lambda_{0}\) (\(n_{s}=\sqrt{\epsilon_{s}}\) and \(a\) are the refractive index and the radius of the sphere, respectively, and \(\lambda_{0}\) is the wavelength in vacuum). If \(x\ll 1\), i.e., the particle is much smaller than the wavelength of the incident electromagnetic wave, only dipolar response may be considered, that is, the particle acts as an electrical dipole induced by the incident field (Figure 6a), \(\overrightarrow{p}=\epsilon_{\text{m}}\overrightarrow{E_{0}}\), where \(\epsilon_{m}\) is the permittivity of the surrounding medium and \(\alpha\) is particle's polarizability,  
\[\alpha=4\pi a^{3}\frac{\epsilon_{s}-\epsilon_{m}}{\epsilon_{s}+2\epsilon_{m}}.\]  
If the sphere is metallic, the permittivity, \(\epsilon_{s}\), is a complex function of the wavelength, so is the polarizability, and the particle both scatters and absorbs light. The extinction cross-section is a measure of both processes together, although the absorption dominates for small particles (\(x\ll 1\)) and it is given by [38]:  
\[\sigma_{e}=\frac{2\pi}{\lambda_{0}}\text{Im}\;\alpha.\]  
The extinction depends on the dielectric functions of both the metal and host matrix, and the particle volume, \(v=4\pi a^{3}/3\). This expression for the polarizability can be extended to the case of an ellipsoid, where \(\alpha\) is a tensor with the principal components given by similar formulae,  
\[\alpha_{i}=v\frac{\epsilon_{s}-\epsilon_{m}}{\epsilon_{m}+\eta_{i}(\epsilon_{ s}-\epsilon_{m})},\;i=1,\;2,\;3,\]  
where \(\eta_{i}\) are geometrical factors known as the depolarization coefficients [38; 39]. They have the following property: \(\sum_{i}\eta_{i}=1\). These formulae are commonly used to describe elongated nanoparticles, for instance, metallic nanorods. Even though the classical treatment of Mie and its extensions to ellipsoids continue to be used to describe individual nanoparticles, it is not appropriate for more complex systems, where it is necessary to consider electromagnetic interactions between different parts of the system (e.g., different nanoparticles). Such systems, e.g., plasmonic nanostars [40] have more complex absorption and scattering spectra, which, in principle, can be understood by  
Figure 6: (**a**) Schematics of localized surface plasmons of a metal sphere. (**b**) Schematics of a typical LSPR band, measured in transmittance mode, and extinction spectrum. considering a set of interacting dipoles that can support collective electromagnetic modes. Such modes also can be excited by incident light. In the 1950s [PERSON], who studied the characteristic energy losses of accelerated electron beams passing through thin films, triggered new developments on the subject at the experimental level, culminating in the demonstration, by [PERSON] and [PERSON], of the existence of self-sustained collective excitations at metal surfaces, named thereafter as surface plasmons by [PERSON] and [PERSON] [41].