1. Multiple scale analysis of a single-pass free-electron lasers Andrea Antoniazzi (Dipartimento di Energetica, Università di Firenze) High Intensity Beam Dynamics September 12 - 16, 2005 Senigallia (AN), Italy

2. plan 1.Single-pass FEL introduction to the model short overview of the results obtained by our group 2. Multiple scale analysis introduction to this method application to the FEL 3. Conclusions

3. 1.The single-pass FEL

4. Hamiltonian model Bonifacio et al., Riv. del Nuovo Cimento 13, 1-69 (1990) numerics Conjugated to the Hamiltonian that describes the beam-plasma instability I

5. Results Statistical mechanics prediction of the laser intensity (Large deviation techniques and Vlasov statistics) J. Barre ’ et al., Phys. Rev. E, 69 045501 Derivation of a Reduced Hamiltonian (four degrees of freedom) to study the dynamics of the saturated regime Antoniazzi et al., Journal of Physics: Conference series 7 143-153 Multiple-scale approach to characterize the non linear dynamics of the FEL Collaborations: Florence (S.Ruffo, D. Fanelli), Lyon (T. Dauxois), Nice (J. Barre’), Marseille (Y. Elskens)

6. Multiple-scale analysis is a powerful perturbative technique that permits to construct uniformly valid approximation to solutions of nonlinear problems. The idea is to eliminate the secular contributions at all orders by introducing an additional variable  =  t, defining a longer time scale. Multiple scale analysis seeks solution which are function of t and  treated as independent variables When studying perturbed systems with usual perturbation expansion, we can have secular terms in the approximated solution, which diverges in time. 2.Multiple scale analysis

7. Example: approach to limit cycle Consider the Rayleigh oscillator, whose solution approaches a limit cycle in phase-space. Using regular perturbation expansion inserting in (1) and solving order by order This expression is a good approximation of the exact solution only for short time. When t~O(1/ є ) the discrepancy becomes relevant The first order solution contains a term that diverges like εt

8. Multiple-scale analysis permits to avoid the presence of secularities. where  =  t. Assume a perturbation expansion in the form : (1) (2) Inserting the ansatz and equating coefficients of  0 and  1 we obtain:

9. The solution of eq (1) reads : Observe that secular terms will arise unless the coefficients of e  it in the right hand side of eq. (2) vanish. Setting the contribution to zero, after some algebra, one gets: where

10. Coming back to the original time variable, the solution reads : The approximate solution accounts both for the initial growth and for the later saturated regime We have obtain a zero-th order solution that remains valid for time at least of order 1/ є, while usual perturbation method is valid only for t~O( 1 ) Є=0.2

12. Multiple scale analysis of single-pass FEL Vlasov-wave system : where plays the role of the small parameter. Janssen P.et al., Phys. of Fluids, 24 268-273

13. Linear analysis In the linear regime one gets : The dispersion relation reads: Thus motivating the introduction of the slower time scales  2 =  2 t,  4 =  4 t, ….

14. Non-linear regime Following the prescription of the multiple-scale analysis we replace the time derivative by : and develop: where:

15. Avoiding the secularities......at the third order where : And  is the solution of the adjoint problem

16. ...at the fifth order obtaining.. CC Re C Im

17. Coming back to t.... we obtain: where Non linear Landau equation Analytical solution This solution account for both the exponential growth and the limit cycle asymptotic behavior

18. Comparison with numerical results Analytical solution Qualitative agreement with numerical results, both for exponential growth and saturated regime The saturation intensity level increases with δ as observed in numerical simulation the level of the plateau is sensibly higher than the corresponding numerical value: probably some approximations need to be relaxed (quasi-linear approximation)

19. 3.Conclusions and perspectives Developed an analytical approach to study the dynamics and saturated intensity of a single-pass FEL in the steady-state regime. Future direction of investigations: HMF? Next step: improvement of the calculations to have a quantitative matching with numerical results.

20. Project coworkers Duccio Fanelli, Univ. of Florence, Stefano Ruffo, Univ. of Florence. Thierry Dauxois, ENS Lyon. Giovanni De Ninno, Sincrotrone Trieste. Yves Elskens, Univ. de Provence, Marseille. Julien Barré, Univ. Nice.