The method for deducing expressions of arbitrary geometrical structures is studied in detail by using the Fourier series expansion. The dispersion curves of the slow-wave structures (SWSs) with the cosinoidal，rapezoidal and rectangular corrugations are obtained by numerical calculation. Moreover，the longitudinal resonance properties of the finite-length coaxial SWS are investigated with the S-parameter method. It is proposed that the introduction of a well designed coaxial extractor to slow-wave devices can help to reduce the period-number of the SWS，which not only can make the devices more compact，but also can avoid the destructive competition between various longitudinal modes. Furthermore，a compact L-band coaxial relativistic backward wave oscillator (RBWO) is investigated and optimized in detail with particle-in-cell (PIC) methods (KARAT code). In the preliminary experiments，the measured microwave frequency is 161 GHz，with a peak power level of above 102 GW，when the diode voltage is 670 kV and the current is 107 kA. The pulse duration (full-width at half-maximum) of the radiated microwave is 22 ns.