Geometry of the electron beam-dielectric guide interaction in a CFEL as an amplifier. If two mirrors are added at both ends, the CFEL device will work as an oscillator.
Variations of the gain coefficient with the normalized distance for different values of in the small-signal approximation. The interaction regions induced by the stimulated emission can be divided into transition and steady states. In the transition state, the gain coefficient increases almost linearly with the spatial variation. In the steady state, when the passing distance reaches several times of the distance , the gain coefficient saturates at certain values.
Numerical examples of the saturated gains and the relative angular frequency with the normalized distance . (a) Variations of the gain coefficients and . A high input power of is supposed to clarify the saturation effect. increases along the electron beam and reaches a maximum value and after that it will gradually decrease due to the shifting of the averaged velocity of the electron beam from the synchronism condition. The gain coefficient is proportional to the power of the EM wave; thus, it have the same behavior as for the gain coefficient . (b) The attenuation of the wave frequency as seen by traversing electrons vs. the normalized distance .
Variations of the gain coefficient with for different values of in the saturation regime. Due to the attenuation of the averaged velocity of electrons , the gain cannot reach to the steady state region observed in the small-signal analysis and its maximum values occur at larger distances for larger values of .
Comparison between the gain coefficient in the small-signal gain and saturation limits. The small-signal case is represented by dotted line. At saturation, numerical solutions to the gain coefficient given by Eq. (27) is shown by solid line and by dashed line when the term of is neglected in the denominator of Eq. (27).
Spatial variation of the power amplification . The small-signal gain is depicted by the dotted line, whereas the power amplification increases in the form of the exponential function. In the case of the saturation regime, the power amplification increases almost linearly with the normalized interaction length and decreases after reaching a maximum value as shown by the dashed line. In the saturation regime, if the averaged electron velocity is not significantly reduced, the amplification power would simultaneously vary linearly with the interaction length as shown by the solid line.
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