Small Signal Gain in Tapered Free-Electron Lasers of Oscillator Configuration

A quadrature for small signal gain (SSG) in tapered free electron laser (FEL) is given. In contrast to the uniform case, the optimum of gain is not at v = 1.303 of energy mismatch, and it depend on the taper. Using the antenna array concept, the spontaneous emission is derived and it shows that Madey's theorem is still valid. The generalization horn one to three dimensions is given and degradation of SSG due to the collimation excursion of the axes of the optical, electron beam and undulator can be included as well. The comparision with Boeing's FEL experiment is carried out also.