We investigate an X-ray free electron laser which relies on a TEM optical mode confined by a Bragg waveguide. The electric field of the laser induces transverse oscillations of the electron beam, resulting in X-ray emission. As compared with conventional wigglers, based on alternating magnets to induce oscillations, which typically are on the order of tens of meters in length, the laser-based wiggler enables reduction of the device into a cm-scale length. The interaction dynamics between the laser and the electron beam in the wiggler is examined in an amplifier mode, in which a pre-designed X-ray wavelength with low input power is inserted. The latter results in a stimulated coherent X-ray radiation. We show that optimizing the initial parameters facilitates saturation within 3 cm for a 50 MeV electron beam. The generated X-ray radiation wavelength for the optimal parameters is 0.25 nm, the output power reaches the scale of 1 Watt, and the number of photons generated per electron is on the order of unity.