Simulation of Josephson antenna array in two dimensional electrodynamic waveguide

We present the mathematical model and algorithm of simulation of active Josephson antenna, which consists of a few lumped Josephson junctions, in two-dimensional electrodynamic system. First we discuss some problems, connected with the formulation of mathematical models of the considered system. We suggest a system of equations which contains the discrete model of Maxwell equations, known as Yee scheme [1, 2], the equations for the dynamics of lumped Josephson junctions biased by d.c. voltage, and conditions which connect the current and voltage on the junctions with electromagnetic field in the waveguide. We use so-called Perfectly Matched Layer boundary conditions [1, 3] to avoid the reflection of electromagnetic waves at the artificial boundary of calculating domain. To simulate the dynamic of electromagnetic field we use the known FDTD explicit method [1] and semi-implicit scheme for the Josephson equations. We also discuss the implementation of suggested algorithm. By using near-to-far field transformation [1], the antenna diagram at the Josephson frequency is calculated.