A 30 GHz “Flying” Electromagnetic Undulator

1. A 30 GHz “Flying” Electromagnetic Undulator • 1. Static and radiofrequency undulators • A concept of “flying” undulator • Calculations of the undulator based on helical corrugated waveguide • Eigen mode excitation • Simulation of pulse propagation in a long waveguide • Effect of transversal focusing due to the pondermotive force • 4. Cold tests • Gigawatt power level, nanosecond BWO Eduard B. Abubakirov1, Ilya V. Bandurkin1, Yury Yu. Danilov1, Andrey N. Denisenko1, Sergey V. Kuzikov1,2,3, Andrey V. Savilov1,2, Alexander A. Vikharev1 1Institute of Applied Physics, Nizhny Novgorod, 603950, Russian Federation2Lobachevsky Sate University of Nizhny Novgorod, 603950, Russian Federation 3Euclid Techlabs LLC, Bolingbrook, IL

2. Static and Radiofrequency Undulators Permanent Magnet Undulator •Small aperture and period (~cm); •Polarization not adjustable; •Undulator period and other parameters cannot be changed; •Permanent magnet damage by radiation; • High cost. • RF Undulator • • less energy of electron beam • •Smaller undulator periods possible • •Larger apertures possible • •Fast dynamic control possible • Polarization • Undulator parameter K • •No magnets damaged by radiation

3. Problems for RF Undulators • High RF fields should be (~1 Т at ~ 1 cm wavelength) to obtainK~1. High-power sources are necessary. • Breakdown and pulse heating must be avoided. • In order to provide the desirable saturation regime in frames of the existing electron beams, one need tens to hundreds of meters long XFEL.Provision of synchronization for a lot of RF sources is necessary. • More complicated field structure and beam dynamics, because usually particles interact with co-propagating waves as well. Transversal defocusing effect can also be important.

4. «Classical» Conceptof RF Undulator High-power RF source RF undulator = storage cavity (high-Q cavity) HE11 Length1 m High-power klystron (SLAC): f=11.4 GHz, P~100 MW, ~10 s, 100 Hz RF undulator in SLAC (S. Tantawi, J. Neilson et al., 2012) XFEL consists of tens of sections. Each is fed by own stand alone high-power RF amplifier.

5. Two Concepts of Undulators Based on Counter- and Co-Propagating, High-Power, Short RF Pulses Short, high-power, travelling RF pulse (t=10-20ns) allows avoid breakdown and pulse heating effects as well at gigawatt power level (K~1). t - Effective interaction length A. Kanareykin et al. HBEB Workshop, 2013. Co-moving RF pulse with high enough group velocity has long interaction length with particles, particles wiggle in field of the -1st harmonic. This is a “flying” undulator: t - effective interaction length - effective period of undulator - group velocity

6. Helical Undulator on ТМ01and ТM11Mode Superposition z R0 = 6.1 mm – average radius a = 0.3 mm – corrugation amplitudeD = 6 mm – period of helics Light cone 1 – operating normal wave 2 – partialTM11 mode 3 – partial TM01 mode 4 – TE11 h D

7. Operating Eigen Mode E/Ez=0.31 E/Es=0.32 x z y

8. Optimization of Parameters Main undulator parameters: 1. - Undulator parameter In the rotating on azimuth wave Power of optical wave in linear regime: - Gain length and Pierce parameters - Gain parameter 2. - Effective period 3. Surface electric and magnetic fields, Esи Hs 4.

9. Optimization Results R0 = 6.1 mm a = 0.3 mm D = 6 mm 1 - K 2 –  3 – vgr 4 - u Pin= 1 GW

10. Comparison of Optimized “Flying” Undulator with Undulators on Travelling Waves The optimized undulator (R0 = 6.1 mm, a = 0.3 mm, D = 6 mm(vgr/c=0.7)forP=1 GW hasK=0.12, gain parameterFandEs=100 MV/m. Let us fix frequency f, power P, pulse length and optical wavelengths: • The undulator on pure counter-propagating TM11 wave in smooth waveguide • (R0=8.2 mm): F/TM11=3.15 EsTM11=140 MV/m 2. The undulator on pure counter-propagating TE11 wave in smooth waveguide (R0=4 mm): F/TE11 =2.2 Es TE11=80 MV/m 3. In flying undulator (R0 = 6.1 mm, a = 0.1 mm, D = 4 mm) with higher group velocity vgr/c=0.8): F/TM11=5 F/TE11 =3.5, But in this case undulator parameter decreses down to K=0.05 (energy spread can be influent).

11. Excitation of Operating Mode in Flying Undulator TM01 Sections with linear variation of corrugation amplitude Length=10periods Length=5 periods

12. Рarticle Dynamics and Irradiation Spectra in Flying Undulator with Sharp Pulse Fronts We = 600 MeV, P= 1 GW, L = 10 m Ez=0 Ez=0 Ez=0 Ez=0 Ez0 Radiation spectra for different injection phases Spectrum:

13. Particle Dynamics in Flying Undulator with Smooth Pulse Fronts Length of pulse front is 0.15 ns. Radiation spectra for different injection phases

14. Pulse Propagation in Flying Undulator z=0 z=10 m z=0 z=10 m

15. Transversal Self-Focusing of Electron Bunch Electrons see the focusing 0-th harmonic (TM01)and the defocusing -1-stharmonic (TM11): - Pondermotive Miller’s force TM11: TM01: defocusing focusing Simulation of particle dynamics for the optimized flying undulator

16. Project of SASE XFEL on a Base of Flying Undulator

17. Low-Power Tests Helical waveguide section for flying undulator Mode filters Helical waveguide TE10-TM11 mode converters Cutoff TM01 reflector Transmission measurement scheme Reflection measurement scheme

18. Transmission of the TM01 wave through helical waveguide section: calculation (black), measurement (red). Phase of transmission: calculation (black), measurement (red).

19. RF Source for the Flying Undulator As a prototype of the RF source for the flying undulator, a relativistic Cherenkov BWO is being developed in IAP. BWO parameters: Voltage - 550 kV Current - 4 kA Guiding magnetic field - 6 T Operating mode - TM01 Results: Operating frequency - 35 GHz Output power ~ 450 MW Efficiency - 20% The use of a sectioned tube can enhance the output power up to 700-800 MW. E.B. Abubakirov, I.V. Bandurkin, A.A. Vikharev, S.V. Kuzikov, et al. Radiophysics and Quantum Electronics, vol. 58, 2016.

20. “Flying”Undulator, Excited by Drive Bunch Drive bunch excited synchronous wake field, witness bunch wigglesin this field and radiates optical radiation. 1 – drive bunch 2 – witness bunch Longitudinal field Transverse field

21. Conclusion • Flying RF undulator concept is proposed for SASE XFEL. In accordance with this concept electron bunch flies in high-power, short, co-propagating RF pulse. Helical corrugation allows to have a space harmonic with negative propagation constant which provides large Doppler’s up-shift. • For electron beam with energy ~1 ГэВ the flying undulator at 1 сm wavelength, pu;lse duration~10 ns, power~1 GWallows to obtain optical radiation at 1 nm wavelength. In this case 10 m of flying undulator can substitute 25 m of typical static magnet undulator. • The optimized flying undulator has high efficiency. • There are RF sources at 30 GHz (gigawatt level BWO) which can feed the proposed flying undulator.