III. FISKE RESONANCES IN INHOMOGENEOUS JOSEPHSON TUNNEL JUNCTIONS: THEORY Our theoretical analysis is based on the extension of the Kulik approach developed for small low-dissipative Josephson junctions with γ << ω$_{0}$ , [1, 27]. Thus, we present ϕ$_{1}$ ( ⃗ ρ, t ) in the following form where the wave vectors of electromagnetic waves that can be excited in the junction, i.e. k$_{n}$ = πn/W , are determined by the junction’s geometry. We note that in the particular case when the parameter γ is not so small, Fiske resonances overlap each other, and a single Eck peak is formed in the I − V characteristic of a Josephson junction [1]. Such feature has been theoretically analyzed also for Josephson junctions with randomly distributed inhomogeneities [28]. As the parameter γ << ω$_{0}$ the Fiske resonances are well separated, and for the analysis of n -th Fiske step we can choose a single wave vector k$_{n}$ in the Eq. (3). Substituting Eq.(3) in Eq.(2) and using the condition ℏ ω$_{0}$ = c$_{0}$k$_{n}$ , i.e. (1), we obtain the nonlinear equation for the amplitude of Swihart electromagnetic waves as where J$_{0}$ and J$_{1}$ are the Bessel functions, S is the junction area. Similarly we obtain the expression for the amplitude of n -th Fiske step where j$_{c}$ is the Josephson critical current density in the absence of vortices. Since the contribution of Abrikosov vortices ϕ$_{V}$ ( ⃗ ρ ) to the Josephson phase difference is a random function of the coordinate ⃗ ρ , we obtain the averaged quantities only. First we define 6 IV. CRITICAL CURRENT AND FISKE RESONANCES: THE DEPENDENCE ON THE ABRIKOSOV VORTICES DENSITY a$_{n}$ = c 2 0 λ 2 $_{J}$γω$_{0}$ [ J 2 $_{0}$( a$_{n}$ 2 ) + J 2 $_{1}$( a$_{n}$ 2 )] Re [ ∫ d $^{2}$⃗ ρ S e iϕ$_{V}$ ( ⃗ ρ )+ ik 0 x − ik$_{n}$$^{x}$] , (4) I ( n ) F = j$_{c}$J$_{0}$ ( a$_{n}$ 2 ) J$_{1}$ ( a$_{n}$ 2 ) Re [ ∫ d $^{2}$⃗ ρ e iϕ$_{V}$ ( ⃗ ρ )+ ik 0 x − ik$_{n}$$^{x}$] . (5) ϕ$_{1}$ ( ⃗ ρ, t ) = ∑ n ℜ e { a$_{n}$e − ik$_{n}$x − iω 0 $^{t}$} , (3)