## 5 Tailoring the Sensitivity of LSPR-Based Sensors Using Simulation Models
### Optical Properties Modelling  
The most widely used methods for modelling the optical properties of nanoparticles with sizes above 5 nm (where quantum mechanical calculations are not required) rely on solutions, numeric or via approximations, of the Maxwell equations, such as Mie theory-based codes, finite-difference time-domain (FDTD) approach, finite element method (FEM), and the discrete dipole approximation (DDA), among others [37, 203, 204, 205, 206]. Some of these methods use computationally intensive algorithms that are currently available to everyone thanks to the processing power of modern computers and clusters that run parallel computations in open access. NanoHUB [207] is an example of a user-friendly and web-based system, controlled by the Purdue University's Network for Computational Nanotechnology and funded by the American National Science Foundation. It hosts a collection of simulation tools, which allow one to study light-matter interactions in different contexts. In particular, there are (i) \"_Nanosphere Optics Lab_\" [208] for an exact solution of Maxwell equations using the Mie theory for single spheres; (ii) \"_Extinction, Scattering and Absorption Efficiencies of Single and Multilayer Nanoparticles_\" [209] (Figure 19a), also using the Mie theory for single spheres, core-shell, and \"nanomatryoshka\" structures; (iii) \"_Molecular Foundry Photonics Toolkit_\" [210] (Figure 19b) for FDTD simulations; (iv) \"_MOOSE_\" [211, 212] (Figure 19c), for advanced FEM simulations; and (v) \"_nanoDDSCAT_+\" [213] (Figure 19d) for DDA simulations with nanoparticles of arbitrary shapes. Commercially available simulation tools include MATLAB (with incorporated Mie theory codes), Lumerical software (FDTD), and COMSOL Multiphysics (FEM), among others. Detailed comparisons of these and other methods may be found elsewhere [37, 214, 215, 216, 217, 218], but the main decisions that one has to make in order to choose the best method are related to the following factors:  
1. Nanoparticle shape (for instance, Mie codes and their extensions work only with high-symmetry shapes and the DDA uses a large amount of memory if the simulations use both large and very small shapes at the same time);
2. Number of simulations to run (for instance, spectral calculations are time consuming for FDTD, FEM, and DDA);
3. Type of output one is interested in, namely, only the local field intensity in the vicinity of a plasmonic structure, to find the most intense hot spots, or angular distribution of the scattered light intensity and polarization in the far field zone (for instance, COMSOL is ideal for the former but not so well-suited for the latter). Figure 18: (**a**) Shapes of nanoparticles synthesized and investigated by [PERSON] et al. [173], including nanodisks, nanopprisms, nanorods, and nanospheres, with different aspect ratio (AR = L/h). (**b**) Measured sensitivities (S, or RIS) of nanoparticles of various sizes, shapes, and compositions. The sensitivity follows a linear relationship with the aspect ratio, regardless of nanoparticle shape, size, and composition (adapted from [PERSON] et al. [173], copyright 2016 American Chemical Society). It is also necessary to consider that none of these tools can be used to directly calculate the response of a statistical ensemble of interacting nanoparticles, considering each of them explicitly (for instance, a composite layer containing arbitrarily shaped particles). For such a task one needs to apply either approximate methods known in the theory of disordered systems, which use either a self-consistent approach or a kind of perturbation theory [219; 220; 221; 222] or solve numerically a system of equations that describes a relatively small, yet representative, set of interacting polarizable particles distributed in space according to some rules mimicking the real system, such as the coupled dipole equations (CDE), where each particle is approximated by a dipole [223].