Lattice Lasing Modes of Finite Grating of Silver Nanowires with Active Shells

We solve a source-free problem of the electromagnetic field in the presence of a grating from finite number of silver nanowires coated with active shells. We use co-called Lasing Eigenvalue Problem (LEP) statement for the analysis of lasing modes. We look for the frequency of the lasing mode, which has real values in the LEP statement, and for the correspondent value of material gain threshold in the active shell. We find such two-component eigenvalues from characteristic equation, built using the each wire field expansions in local coordinates and addition theorems for cylindrical functions. This is Fredholm second kind determinant equation, what provides a convergence of its roots. We look for a lattice lasing modes and study dependences of their threshold values and wavelengths on the variant parameters.