A novel 1D Bragg resonator based on coupling propagated and locked (quasi cut-off) modes should be tested in a JINR- IAP FEM-oscillator to improve selectivity over the transverse mode index. In this scheme the electron beam interacts with only propagating wave, and the latter is coupled with a quasi cut-off mode. This coupling can be realized by either helical or azimuthally-symmetric corrugation. The quasi cut-off mode provides the feedback in the system leading to the absolute instability and the self-excitation of the whole system while efficiency in the steady-state regime of generation is almost completely determined by the propagating mode, synchronous to the beam. Analytical consideration and numerical simulation show that the efficiency of such an FEM can be rather high. The main advantage of this scheme is provision of higher selectivity over the transverse mode index than traditional scheme of Bragg FEL that encourage increasing operating frequency for fixed transverse size of the interaction space.