Two-frequency undulator brightness and gain with the aid of induced betatron oscillations

Summary form only given. In this problem we discuss the theory of emission of the relativistic electrons in a two frequency undulator with betatron motion components included in the analysis. In recent years free electron lasers with a two frequency undulator have been proposed to generate a laser field with a large extraction efficiency and a narrower spectrum. The results presented have indicated that TFU and OFU brightness exhibit significant differences but still share important analogies such as the absence of even on-axis harmonics and a distinct set of odd harmonics. Further the TFU provides an unique but quite distinct and interesting possibility such as operating on a continuous band operation. We employ Maxwell equations to include the dependence of the undulator field on the transverse co-ordinates. The equation of motion derived with the off-axis structure of the undulator. We outline two procedure to enhance the gain of the system via induced betatron oscillations. In the first scheme an auxiliary undulator has been introduced in to the system. The betatron motion with the undulator provides extra axial force thereby increasing the bunching and gain. In the second process, a d.c. magnetic field does induce the forced oscillation of the betatron motion. An essential key and tool to our analysis approach is use of gBF which allows a precise expression for the undulator brightness and gain. We discuss the comparison of the two schemes and relevant physical consequences.