Analysis of surface plasmon resonance of arbitrary shape nano-particles on a substrateis important for many engineering applications. The strong optical absorption ofnoble metal nano-particles is due to the localized surface plasmon, which enablesthe development of novel applications such as surface enhanced Raman spectroscopybased biological sensing, optical transparency based sensors, and uni-directional nanoantennas.Size, shape, and distance between nano-particles on a surface are thekey factors in the design of these structure. Usually, these systems utilize noblematerial such as gold which is deposited on a dielectric surface. Tuning the structuresof the noble metals increases the local absorption eciency of light within specicfrequency ranges, depending rely on the nal design of the system. Understandingthe physics of plasmon phenomena and its relation with these parameters should bestudied in depth. This thesis provides a theoretical study and a numerical validationof the coupled phenomena that occurs between gold nano-particles on a dielectricsubstrate where the nano-particles have dierent shapes and separation distances.Numerical study of nano-particles on a substrate is carried out using Discrete DipoleApproximation with Surface Interaction (DDA-SI) approach. The DDA-SI is basedon the discretization of the nano-particles to dipoles for solving volume integral ofMaxwell s equations and Green s dyadic tensor of electric eld within the dipoles tocalculate the optical properties of arbitrarily shaped, non-homogeneous, anisotropicobjects. There are some renowned open sources packages for DDAs. These packagesare mostly used for calculating the interaction of particles in a free space with directwave propagating. However, in order to calculate the light scattering from objectsplaced on a substrate, the substrate itself also is required to be approximated by thedipoles which increase the computational time remarkably. In this study, we usedthe Discrete Dipole Approximation with Surface Interaction (DDA-SI) which is anopen source MATLAB based software package for calculation of optical propertiesof nano-particles (extinction, absorption and scattering) on a substrate. It can beused to investigate both the near- and far-eld eects and accounts for the couplingbetween dierent particles on the surface.This study focuses on specic geometries such as cube, spheroid and triangulargeometries, each of varying sizes and separation distances from each other. Apartfrom studying the plasmon resonance of individual nano-particle on a dielectric substrate,a system with more nano-particles is scrutinized. It is found out that as thedistance between particles decreases, the plasmon resonance frequency is pushed intothe infrared region due to the inter-particle coupling, and the redshift becomes dominant.Furthermore, we found out that the coupling eect becomes negligible if theratio of their distance between nano-particles to the radius, c = d=a, is greater thanthree.During this study, the main DDA-SI toolbox is further developed, vecrorized andoptimized numerically which named DDA-SI-3 in order to calculate the optical propertiesof noble metals. The imaginary component of refractive index for noble metalsfor larger wavelengths is large and make the calculation challenging. Therefore, weapplied the new numerical method to calculate the linear system of DDA-SI in orderto achieve more precise, faster and stable calculations. The structure of interactionmatrix is studied and for this, specic preconditioning matrices are extracted. It isalso found out that the use of the least square method, with the proper preconditioningmatrices for iteratively solving the linear system, yields results to achieve moreaccurate and relatively faster calculation.