## 5 Tailoring the Sensitivity of LSPR-Based Sensors Using Simulation Models
### Optical Properties Modelling
Let us focus on the DDA approach, which is suitable for describing nanoparticles of arbitrary shape, both in the near-and far-field zone, and calculate the absorption and extinction cross sections. This is attractive in the context of modelling a sensing system because the plasmonic effect slightly perturbed by the presence of an analyte may be detected via a change in the position and shape of the LSPR band, but also through an altered polarization state of the resonantly scattered light. The DDA has been demonstrated to be one of the most powerful and flexible electrodynamic methods for computing the optical spectra of nanoparticles with an arbitrary geometry [37]. Figure 19: Simulation tools available at Nanohub using the computational power of super computers: (**a**) Extinction, Scattering and Absorption Efficiencies of Single and Multilayer Nanoparticles [209]; (**b**) Molecular Foundry Photonics Toolkit [210], (**c**) MOOSE [211; 212]; and (**d**) nanoDDSCAT+ [213]. The idea was introduced by [PERSON], 1964 [224] who applied it to study the optical properties of molecular aggregates. DDA with retardation effects was presented in 1973 by [PERSON] and [PERSON] [225] to study the scattering and absorption of light of interstellar dust grains. They presented it as a method in which the continuum target is approximated by a finite array of polarizable points. These points acquire dipole moments due to the electric field of an incident wave. The dipoles interact with the neighbouring dipoles via their electric fields. These dipoles can be thought of as polarizable particles of simple geometries, such as spheres or spheroids (actually, it is very similar to the CDE approach for composite materials, mentioned above). The polarizability of the dipoles is related to properties of the material via its refractive index \(n_{s}\). The domain discretization is a very important first step. The method is expected to produce an acceptable response (almost a continuum target) on length scales much larger than the lattice spacing, \(d\ll\lambda_{0}\) (more rigorously, this criterion is \(2\pi n_{s}d/\lambda_{0}\leq 1\)). Roughness of a target can be mimicked by randomly removing some elements from the surface using a Monte Carlo sampling. An obvious problem that appears for a target with a large size is that the number of small spheres necessary to describe the target is huge. Therefore, the memory/CPU requirements may be a problem (computational effort scales with the number of small spheres as \(N^{3}\) and with memory needs scale as \(N^{2}\)). LU decomposition, as stated by [PERSON] and [PERSON] (1994) [226], is a solution for relatively small problems. However, for larger problems, iterative methods such as complex conjugate gradient-type methods might be used. This is the case of the method presented by [PERSON] (1988) [227], where the iterative conjugate gradient algorithm (CGA) to find the dipole polarizations is described. When the number of particles \(N\) is large, this algorithm converges rapidly due to symmetries of the matrix. Since \(P\) has \(3N\) unknowns, this method converges in \(3N\) iterations, which is much faster than a direct method that must solve a linear system of equations \([A]P=E\), where \([A]\) is a \(3N\times 3N\) dense matrix. Using fast-Fourier-transform (FFT) methods, as suggested by [PERSON] et al. (1991) [228], accelerates computations of scattering and absorption by particles of arbitrary shape using DDA and diminishes the CPU price that comes from a CGA method (multiplications matrix-vector, which are mostly convolutions). As an illustration, we used the DDA method to estimate the sensitivity (RIS) of gold spheroids, with different AR, embedded in a TiO\({}_{2}\) dielectric matrix. The spheroids were exposed to environments with different refractive indices and two sets of simulations of the optical properties were undertaken.