the phases, are proportional to small parameters $(\Gamma q)^{-1}$ and $(\Gamma p)^{-1}$ and therefore provide small corrections to the current. For this reason, in the first approximation we can assume that the phases in superconducting electrodes are constants independent of x. In the further analysis we denote the phases at the s-film by $\chi $ and at the right S-electrode by $\varphi $ (see Fig.\[design\]c).

The details of calculations are summarized in the Appendix \[Appendix\]. These calculations show that the considered SIsFS junction has two modes of operation depending on relation between s layer thickness $d_{s}$ and the critical thickness $d_{{sc}}=(\pi /2)\xi _{S}(T).$ For $d_{s}$ larger than $d_{sc}$, the s-film keeps its intrinsic superconducting properties (*mode (1)*), while for $d_{s}\leq d_{sc}$ superconductivity in the s-film exists only due to proximity effect with the bulk S electrodes (*mode (2)*).

Mode (1): SIs $+$ sFS junction $d_{s}\geq d_{sc}$
-------------------------------------------------

We begin our analysis with the regime when the intermediate s-layer is intrinsically superconducting. In this case it follows from the solution of GL equations that supercurrent flowing across SIs, sF and FS interfaces ($J(-d_{s}),$ $J(0)$ and $J(d_{F}),$ respectively) can be represented in the form (see Appendix \[Appendix\]) $$\frac{J_{S}(-d_{s})}{J_{G}}=\frac{\delta _{s}(-d_{s})}{\Gamma _{BI}\Delta
_{0}}\sin \left( 