Simplified method of numerical simulation of eigenmodes in two-dimensional media

Simplified method of eigenmodes simulation in two-dimensional (2D) media based on numerical solution of the stationary wave equation with a given distribution of dielectric permittivity is suggested. By means of discretization the wave equation was reduced to the system of homogeneous linear equations, for that the eigenvalue problem was worked out. The eigenvalues alpha=(2pib/lambda)², where b is the distance between the nodes of discretization, lambda - the wavelength, were determined and the distribution of relative field amplitudes was calculated. 2D media (ordered and random) were simulated by matrix whose elements take on two different values. One of them corresponds to dielectric permittivity of the material particles, the other to permittivity of the spaces between them. All calculations were made using MATLAB. Different variants of ordered and random media were examined. It was shown that localized modes exist only in disordered systems in a limited range of ratio b/lambda. The dependence of modes character on the value of filling ratio and dielectric permittivity is estimated