Analytical approximations of the dispersion relation of a linear chain of metal nanoparticles

We find some useful analytical approximations of the dispersion relation of a linear chain of metal nanoparticles in the subwavelength limit where the dipolar approximation can be used. We also approximate the group velocity without a direct estimation of the derivative of the dispersion relation, that carries unavoidable error amplifications. In the end we use these results in order to get some simple recipes that evaluate the sensitivity of the dispersion relation and the propagation losses with respect to the main parameters of the chain.