  We have investigated bound modes in finite linear chains of dielectric
particles of various lengths, interparticle spacing and particle materials.
Through a unique application of the multisphere Mie scattering formalism, we
have developed numerical methods to calculate eigen-optical modes for various
arrays of particles. These numerical methods involve the use of the multisphere
scattering formalism as the entries in NxN matrices where N represents the
number of particles in the chain. Eigenmodes of these matrices correspond to
the eigen-optical modes of interest. We identified the eigenmodes with the
highest quality factor by the application of a modified version of the
Newton-Raphson algorithm. We found that convergence is strong using this
algorithm for linear chains of up to several hundreds of particles. By
comparing the dipolar approach with the more complex approach which utilizes a
combination of both dipolar and quadrupolar approaches, we demonstrated that
the dipolar approach has an accuracy of approximately 99%. We found that the
quality factor Q of the mode increases with the cubed value of the number of
particles in chain in agreement with the previously developed theory, the
effects of disordering of particle sizes and inter-particle distances will be
discussed.
