1. NUMERICAL SIMULATION OF NONLINEAR EFFECTS IN VOLUME FREE ELECTRON LASER (VFEL) K. Batrakov, S. Sytova Research Institute for Nuclear Problems, Belarusian State University

2. Free Electron Laser Theoretical investigations show that it is one of the effective schemes with n-wave volume distributed feedback (VDFB) FEL lasing is aroused by different types of spontaneous radiation: undulator radiation, Smith-Purcell or Cherenkov radiation and so on. Using positive feedback in FELs reduces the working length and provides oscillation regime of generation. This feedback is usually one-dimensional and can be formed either by two parallel mirrors or by one-dimensional diffraction grating, in which incident and diffracted waves move along the electron beam.

3. What is volume distributed feedback ? Volume (non-one-dimensional) multi-wave distributed feedback is the distinctive feature of Volume Free Electron Laser (VFEL) one-dimensional distributed feedback two-dimensional distributed feedback

4. Benefits provided by the volume distributed feedback This new law provides for noticeable reduction of electron beam current density (10 8 A/cm 2 for LiH crystal against 10 13 A/cm 2 required in G.Kurizki, M.Strauss, I.Oreg, N.Rostoker, Phys.Rev. A35 (1987) 3427 ) necessary for achievement the generation threshold. In X-ray range this generation threshold can be reached for the induced parametric X-ray radiation in crystals, i.e. to create X-ray laser ( V.G.Baryshevsky, K.G.Batrakov, I.Ya. Dubovskaya, J.Phys D24 (1991) 1250) The new law of instability for an electron beam passing through a spatially- periodic medium provides the increment of instability in degeneration points proportional to  1/(3+s), here s is the number of surplus waves appearing due to diffraction. This increment differs from the conventional increment for single-wave system (TWTA and FEL), which is proportional to  1/3. ( V.G.Baryshevsky, I.D.Feranchuk, Phys.Lett. 102A (1984) 141) This law is universal and valid for all wavelength ranges regardless the spontaneous radiation mechanism

5. volume diffraction grating What is Volume Free Electron Laser ? * * Eurasian Patent no. 004665 volume “grid” resonator X-ray VFEL

6. frequency tuning at fixed energy of electron beam in significantly wider range than conventional systems can provide more effective interaction of electron beam and electromagnetic wave allows significant reduction of threshold current of electron beam and, as a result, miniaturization of generator reduction of limits for available output power by the use of wide electron beams and diffraction gratings of large volumes simultaneous generation at several frequencies Use of volume distributed feedback makes available:

7. VFEL experimental history 1996 Experimental modeling of electrodynamic processes in the volume diffraction grating made from dielectric threads 2001 First lasing of volume free electron laser in mm-wavelength range. Demonstration of validity of VFEL principles. Demonstration of possibility for frequency tuning at constant electron energy 2004 New VFEL prototype with volume “grid” resonator. V.G.Baryshevsky, K.G.Batrakov,A.A.Gurinovich, I.I.Ilienko, A.S.Lobko, V.I.Moroz, P.F.Sofronov, V.I.Stolyarsky, NIM 483 A (2002) 21 V.G.Baryshevsky, K.G.Batrakov, I.Ya. Dubovskaya, V.A.Karpovich, V.M.Rodionova, NIM 393A (1997) 71

8. Resonator (2001) side view top view diffraction gratings The interaction of the exciting diffraction grating with the electron beam arouses Smith-Purcell radiation. The second resonant grating provides distributed feedback of generated radiation with electron beam by Bragg dynamical diffraction.

9. New VFEL generator (2004) Main features:  volume gratings (volume “grid”)  electron beam of large cross-section  electron beam energy 180-250 keV  possibility of gratings rotation  operation frequency 10 GHz  tungsten threads with diameter 100  m First observation of generation in the backward wave oscillator with a "grid" diffraction grating and lasing of the volume FEL with a "grid" volume resonator* ( V. Baryshevsky et al., LANL e-print archive: physics/0409125)

10. Electrodynamical properties of a volume resonator that is formed by a periodic structure built from the metallic threads inside a rectangular waveguide depend on diffraction conditions. Electrodynamical properties of a "grid" volume resonator * * V. Baryshevsky, A. Gurinovich, LANL e-print archive: physics/0409107 Then in this system the effect of anomalous transmission for electromagnetic waves could appear similarly to the Bormann effect well-known in the dynamical diffraction theory of X-rays. Resonant grating provides VDFB of generated radiation with electron beam

11. “Grid” volume resonator* V.G.Baryshevsky, N.A.Belous, A.Gurinovich et al., Abstracts of FEL06, (2006), TUPPH012

12. VFEL in Bragg geometry k k 0 u L k 

13. Laue geometry u 

14. Three-wave VFEL, Bragg-Bragg geometry k2 k2 k1 k1 0 u L k

15. Equations for electron beam

16. Main equations

17. System for two-wave VFEL: - detuning from exact Cherenkov condition γ j are direction cosines, χ ±1 are Fourier components of the dielectric susceptibility of the target System parameters The integral form of current is obtained by averaging over two initial phases: entrance time of electron in interaction zone and transverse coordinate of entrance point in interaction zone. In the mean field approximation double integration over two initial phases can be reduced to single integration.

18. Code VOLC - VFEL simulation

19. Some cases of bifurcation points Results of numerical simulation (2002- 2006): Bragg Laue SASE Laue- Laue Bragg- Laue Synchronism of several modes: 1, 2, 3 VFEL Two-wave Three-wave Amplification regime j δ βiβi lili Bragg- Bragg SASE Generation regime with external mirrors Amplification regime Amplification regime Generation regime with external mirrors Generation regime j L β χ Generation threshold Generation threshold Generation regime

20. Dependence of the threshold current on assymetry factor of VDFB  j, A/cm 2 with absorption without absorption -20-18-16-14-12-10-8-6-4-2 0 30 60 90 120 150 180 210 240 270 300 This relation confirms that VDFB allows to control threshold current.

21. Current threshold for two- and three-wave geometry in dependence on L L, c m 10152025303540455055606570 0.01 0.1 1 10 1E+2 1E+3 1E+4 1E+5 |E| two-wave geometry, one mode in synchronism 2-root degeneration point in three-wave geometry 3-root degeneration point For n modes in synchronism*: *V.Baryshevsky, K.Batrakov, I.Dubovskaya, NIM 358A (1995) 493 So, threshold current can be significantly decreased when modes are degenerated in multiwave diffraction geometry

22. References (VFEL theory and experiment)  V.G.Baryshevsky, I.D.Feranchuk, Phys.Lett. 102A (1984) 141-144  V.G.Baryshevsky, Doklady Akademii Nauk USSR 299 (1988) 19  V.G.Baryshevsky, K.G.Batrakov, I.Ya. Dubovskaya, Journ.Phys D24 (1991) 1250-1257  V.G.Baryshevsky, NIM 445A (2000) 281-283  Belarus Patent no. 20010235 (09.10.2001)  Russian Patent no. 2001126186/20 (04.10.2001)  Eurasian Patent no. 004665 (25.09.2001)  V.G.Baryshevsky et al., NIM A 483 (2002) 21-24  V.G.Baryshevsky et al., NIM A 507 (2003) 137-140  V. Baryshevsky et al., LANL e-print archive: physics/ 0409107, physics/0409125, physics/0605122, physics/0608068  V.G.Baryshevsky et al., Abstracts of FEL06, (2006), TUPPH012, TUPPH013

23. References (VFEL simulation)  Batrakov K., Sytova S. Mathematical Modelling and Analysis, 9 (2004) 1-8.  Batrakov K., Sytova S. Computational Mathematics and Mathematical Physics 45: 4 (2005) 666–676  Batrakov K., Sytova S.Mathematical Modelling and Analysis. 10: 1 (2005) 1–8  Batrakov K., Sytova S. Nonlinear Phenomena in Complex Systems, 8 : 4(2005) 359-365  Batrakov K., Sytova S. Nonlinear Phenomena in Complex Systems, 8 : 1(2005) 42–48  Batrakov K., Sytova S. Mathematical Modelling and Analysis. 11: 1 (2006) 13–22  Batrakov K., Sytova S. Abstracts of FEL06, (2006), MOPPH005