# Recirculated Wave Undulators for Compact FELs
## 9 FEL Radiation: Comparison with the Analytic Model  
In this section, we finally review a fully analytical approach to the FEL power evolution [27], in order to make a comparison with the numerical approach used above while highlighting the main differences. The FEL intensity evolution is ruled by the so-called Pierce parameter, which, for the purposes of this paper, can be cast in the following form:  
\[\varrho=\frac{8.36\times 10^{-3}}{\gamma_{e}}\Big{(}j_{e}[A/m^{2}]\lambda_{0}^{2 }[m]K^{2}\Big{)}^{1/3} \tag{31}\]  
Figure 5: Simulation results with FEL1D code in Python, at the entrance of the second interaction point.  
Figure 6: Simulation results with FEL1D code in Python, after the second interaction point.  
where \(j_{e}\) is the electron beam density current. The saturation length can be roughly expressed in terms of a few gain lengths. The \"full saturation\" condition on the emitted power is usually achieved after several times the gain length, the latter given by:  
\[L_{g}=\frac{\lambda_{0}}{4\pi\sqrt{3}\varrho} \tag{32}\]  
The saturation power is:  
\[P_{F}=\sqrt{2}\varrho P_{e} \tag{33}\]  
where \(P_{e}=\gamma_{r}mc^{2}j_{e}\Sigma_{e}/e\). As a consequence of the previous equations, the FEL saturated power reads:  
\[P_{F}(MW)=1.7\times 10^{16}\frac{(\gamma_{r}\varrho)^{4}}{I_{p}[W/m^{2}] \lambda_{0}[m]^{4}}\Sigma_{e}[m^{2}] \tag{34}\]  
Beside the scaling formulas above, a more complete and analytic description of the SASE intensity growth can be specified by the following logistic-like function:  
\[P(z)=\frac{P_{0}}{9}\frac{B(z)}{1+\frac{P_{0}}{9P_{F}^{2}}B(z)} \tag{35}\]  
\[B(z)=2\left[\cosh\left(\frac{z}{L_{g}}\right)-e^{z/2L_{g}}\cos\left(\frac{ \pi}{3}+\frac{\sqrt{3}z}{L_{g}}\right)-e^{-z/2L_{g}}\cos\left(\frac{\pi}{3}- \frac{\sqrt{3}z}{L_{g}}\right)\right] \tag{36}\]  
The radiated FEL power analytically calculated by Formulas (35) and (36) under the same operational conditions used for the numerical simulations in Section 8 is shown in Figure 7. For the sake of rigor, Figure 7 should be compared to the bottom-left plot of Figure 4, since it is related to the assumption of no initial correlation (or bunching) in the phase-space. Therefore, the power level and the power trend after the first interaction point analytically calculated is in good agreement with the one numerically simulated. The same cannot be said at the second interaction point, since the pre-bunching preserved after the magnetic chicane makes the analytic model not longer comparable to the bottom-left plot of Figure 6. In conclusion, the above presented analytic model can be used for evaluating the power emitted at the first interaction point, but to be reliable as well at the second interaction point should be generalized to a situation of pre-bunching, which is not in the focus of this paper. In Ref. [27] the FEL emission was studied only analytically and the free parameter \(P_{0}\) was set to an arbitrarily small value to obtain reasonable output power. On the other hand, in the current paper we have managed to fix this free parameter to a value (\(P_{0}=21\) kW) such that the analytic model coincides in terms of final output power with the more rigorous numerical approach. Therefore, here we demonstrate that actually the performance of the proposed design can be higher than the one already shown in [27]. At the same time we show the limitation of the analytic approach, which cannot be used for more than one interaction point (unless for the case that the pre-bunching is completely destroyed by the chicane), since it does not consider the effects related to the pre-bunching.