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**Extra resources for Analysis, Manifolds and Physics [Part II] (rev.) [math]**

**Example text**

3. The maximal value of the field growth rate is achieved at exact resonance, at C = 0. At small deviations of the detuning, 0 2 « 1, the field growth rate is approximately equal to ReAc::o A 2J3 [1- 91 c"2] . 38) At large negative values of the detuning parameter (C < 0, ICI » 1) the asymptotic expressions for the field growth rate and for the input coupling factor have the form: The amplification process displays resonance behavior and the power gain depends strongly on the value of the detuning parameter 6.

101) We start with the simplest situation neglecting the influence of the space charge and the energy spread effects on the operation of the FEL amplifier. The plot in Fig. 01. The field stops growing at the saturation point when the beam is overmodulated and a significant fraction of the electrons falls into the accelerating phase of the effective potential. e. when Eext/ Eo « 1. To analyze the dynamics of the particles in the undulator, it is convenient to study their distribution in the phase plane (F, 11'0), where 11'0 ='if+ 'lf 0 .

30) where 24 2. 1 is the jth root of the iJ~~=0. 31) 1- iA~D Cold Electron Beam. In the limit of a small energy spread the distribution function F can be replaced by the delta function, F(P) = o(P). +iC in the entire complex >.. plane. 28), is of the order of 0(>.. i ---+ oo, the conditions of Jordan's lemma are satisfied. ; ' where >.. 32) . 33) Exactly at resonance, 6 = 0, for the case of a negligibly small space charge field, A~ ---+ 0, the solution for E(z) takes the form ~) [ E- ( z = -Eext 3 - exp ( ~) +z i -y'32- + exp ( -J32 + i z') .

### Analysis, Manifolds and Physics [Part II] (rev.) [math] by Y. Choquet-Bruhat, et. al.,

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