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Eigenmodes analysis in two-dimensional random media

Simplified method of eigenmodes simulation in random media based on numerical solution of the stationary wave equation for two-dimensional (2D) medium with a random distribution of dielectric permittivity is suggested. By means of discretization the wave equation can be reduced to the system of homogeneous linear equations that includes parameter α=(2πb/λ)2, where b is the spacing between the nodes of discretization, λ - the wavelength. The values of α (and corresponding b/λ) were determined as eigenvalues of this system of linear equations. The relative field amplitudes in all discretization nodes i.e. eigenvectors were calculated with this α. 2D random medium was simulated by matrix whose elements randomly 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. Under the assumption made, elements of such matrix represent material particles and spaces between them, quantity b - particles size. All calculations were made using MATLAB. Different variants of disordered (and ordered) media were examined. It was shown that localized modes exist only in disordered systems and in a limited range of ratio b/λ . The dependence of modes character on the value of filling ratio and dielectric permittivity is estimated. Some results for one- and three-dimensional media are represented.