content_image/341129.jpg content_image/341127.jpg content_image/341128.jpg
## 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 

<img src='content_image/341129.jpg'>

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 

<img src='content_image/341127.jpg'>

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 

<img src='content_image/341128.jpg'>

where j$_{c}$ is the Josephson critical current density in the absence of vortices. 

## IV. CRITICAL CURRENT AND FISKE RESONANCES: THE DEPENDENCE ON THE ABRIKOSOV VORTICES DENSITY 

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 

