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The resonant frequency of a free electron laser may be modulated via the undulator or electron beam parameters. This modulation may generate sidebands which can subsequently undergo amplification, analogous to frequency modulation in a conventional cavity laser. However, due to the relative slippage of the light through the relativistic electron beam, the FM-FEL system has a more complex behavior than its conventional laser counterpart. The system may be described in the linear regime by adoi:10.18429/jacow-fel2017-frb05 fatcat:6cy2mjfblbalfikoyxhbxmrrbu