Modeling an Electron Beam/Terahertz Wave Interaction in a Waveguide Undulator Caleb Zerger University of Michigan Department of Physics and Astronomy, UCLA August 30, 2013 Abstract This paper presents a look at a preliminary 1- D Matlab model of an electron beam interaction with a THz radiation pulse in a waveguide undulator. Unlike previous models, this model does not use the Slowly Varying Envelope Approximation, which allows the model to account for the changes in the shape of the radiation wave packet at the wavelength scale. The model shows the expected characteristics of an electron beam radiation interaction, including bunching of the electron beam at the radiation wavelength, phase space rotation of the electron beam, and increasing energy spread. 1 Introduction 1.1 Basic FEL principles The interaction described in this paper shares many characteristics with the often studied Free Electron Laser (FEL). In an FEL, an electron beam and laser co- propogate along the axis of an undulator consisting of opposing magnets. The magnetic field causes the electron beam to undulate transverse to the axis, and this acceleration causes the electrons to emit synchrotron radiation and exchange energy with the laser as seen below. From Courant, Pellegrini, Zakowicz 1985 The interaction between the electron beam and laser is most apparent when the resonance condition is satisfied. The resonance condition requires that the laser radiation overtakes the electron beam by one wavelength over the course of an undulator wavelength. The condition is given by the equation (1) λ = λ 2γ (1 + K ) Where λ is the radiation wavelength, λ u is the undulator wavelength, γ is the Lorentz factor, and K is the dimensionless undulator factor K = " ". The radiation field will cause some electrons to accelerate faster than others. The fast electrons will overtake the slower electrons, causing periodic bunching at the radiation wavelength. When the electrons are bunched, the synchrotron radiation released will be coherent, and thus will amplify the radiation from the laser. This in turn causes exponential
bunching/radiation amplification. This is the working principle of an FEL 1. 1.2 Previous Work Most work in simulating FEL type interactions has been done using the Slowly Varying Envelope Approximation (SVEA). This approximation assumes that changes in the field envelope occur slowly relative to the radiation wavelength, and allow the equations of motion of the electron beam to be averaged over a radiation period. These simulations have been successful for many purposes, but the approximation has its limitations. One of the main reasons we will avoid using SVEA is that averaging over a period makes resolution of processes occurring at lengths at the scale of the radiation wavelength impossible. For many experiments this is irrelevant, but in our case the electron bunch length is on the order of a millimeter, as is the radiation wavelength, making the use of SVEA impossible 2,3. 2 The model The model we are attempting to simulate is the interaction between an electron beam and a single Terahertz radiation pulse as they propagate through a waveguide undulator. This was done while ignoring the effect of the electron bunching on the radiation. 2.1 Radiation Profile The first step to building this model is modeling the radiation itself. The THz pulse we are modeling is emitted from a source with a spread of wavelengths given by A(ω), a Gaussian distribution about a peak wavelength. Integrating this over all frequencies we find the vector potential of the THz wave propagating through the waveguide to be (2) A z, t = 1 2π A(ω)e "() e "# e () dω Here α(ω) is a waveguide factor and k(ω)is a frequency dependent wavenumber. These are related to the dispersive properties of the waveguide. The speed of the radiation in the waveguide to a group velocity of v ω = c 1 ( ) where ω is the cutoff frequency of the waveguide. This is less than the speed of light, and keeps the radiation from overtaking the electron beam which moves at less than the speed of light.the normalized vector potential kkl z, t = A z, t profile at time t=0.2/c is shown below: Note that while the time is 0.2/c, the wave packet is not yet to 0.2 m, showing the group velocity principal. The initial value of β, the normalized speed of the electron beam, is.989, which means the electron beam would have travelled about 0.198 m at t=0.2/c, which is about the same as the wave packet, so the velocities are matched as expected. This enables the use of a single, relatively short radiation pulse, because the pulse will travel with the beam. It is also notable that each period of the radiation is different in amplitude from those before and after. This variation on the scale of wavelength is allowed only because a SVEA approximation, which would average over a radiation period, is not applied. 2.2 Modelling the electron beam The electron beam can be modeled as a group of macroparticles each travelling at a
longitudinal position (position along the undulator axis) z and a normalized energy γ.these are related by a system of ODEs. The differential equation describing energy is given by (3) dγ dt = Kcos(k z) kkl(z, t) γ Here the term Kcos(k z) comes from the velocity given to the macroparticle in the transverse direction by the undulator and hence depends on cos(k z) where k is the wavenumber of the undulator. kkl(z, t) is simply the aforementioned normalized vector potential. The position is decribed by (4) The following graph shows the energy of each particle vs its initial position for a pulse of length ~.44 mm, the wavelength of the radiation. The red line represents the particles energies after they have travelled for time 0.5/c, and the rest at times evenly distributed down to ~0.02/c. β = 1 1 γ The velocity of the electron makes an angle of K/γ with the z axis, implying (5) cos K γ = β 1 1 γ Substituting cos = 1 for β we obtain (6) β 1 1 γ K γ and solving after eliminating terms dependent on. Here β = ", so (3) and (6) will be the system of "# ODEs used in the simulation. 2.3 The simulation The simulation itself was created in Matlab by creating an array of macroparticles. The peak frequency used was 0.674 THz, with σ =0.15 THz. The undulator wavelength corresponding to the spacing between magnets was λ u = 2 cm. The resonant energy calculated from (1) was γ 0 = 7.828. The undulator parameter K = 0.59 and an electron pulse of randomly distributed macroparticles over 2 mm was used. This graph exemplifies some basic but important principles of the electron/thz radiation interaction. The first is that the electron properties are periodic at the radiation wavelength. The other is that the energy spread about the resonant energy increases with time, but increases more slowly as the energy increases. This is expected from equation (3), as the change in energy is inversely proportional to the energy itself. 3.2 Electron Beam Phase Space The following graphs will show the electron beam phase space, plotted energy vs position. This is for a 2 mm bunch, which is approximately 5 radiation wavelengths. Each graph is for a different time, given as a distance over c, to show the approximate location of the beam along the undulator axis. 3 Results 3.1 Dependence on Initial Position
At t = 0.1/c, the electron phase space looks very similar to the vector potential and does not yet exhibit any noticeable rotation or bunching. By t = 0.3/c the beam profile exhibits very minimal bunching, which can be seen by noting that each peak appears more sparse on the left side than the right. At 0.5/c there is a noticeable rotation in the phase space: that is, the peaks are no longer straight up and down. At t = 0.7/c, there is an even more pronounced phase space rotation and bunching at the radiation wavelength is very obvious, as the electrons are now very concentrated at certain distances and sparse at others. At 0.9/c even more phase space rotation is apparent, and there is defined bunching, as we would expect to see.
radiation field. As mentioned before, the electron beam bunching causes coherent emission which amplifies the field. This aspect has been ignored in the model so far but plays an important role in the interaction. This has been modeled before for electrons in an undulator, but never before in a waveguide. Future work will take into account this coherent emission within the waveguide. 5 Acknowledgements Another interesting feature easiest to see at 0.9/c is that each wavelength long bunch, or bucket looks different from the bucket next to it. This feature is unique to simulations that do not use the SVEA approximation, because changes at the order of a radiation wavelength are allowed. 4 Conclusion The expected phase space rotation and bunching implies that we have a reasonable model of the interaction between the electron beam and the single pulse Terahertz wave. This is an important first step in understanding this interaction, as most previous models have used the slowly varying envelope approximation which is not an option for this particular interaction. Understanding this interaction will enable the use of the THz source to manipulate the electron beam phase space for future experiments, such as pump- probe electron diffraction. I would first like to thank my advisor Professor Musumeci for his support and guidance this summer and for giving me the opportunity to learn about the world of accelerator physics. I would also like to thank Renkai Li and the other members of the Pegasus lab for all of their help this summer, and Françoise Queval for making the REU possible and all of her help along the way. Lastly, I would like to thank UCLA and the NSF for their generosity. 1 Reiche, Sven, 1999, Numerical Studies for a Single Pass High Gain Free- Electron Laser 2 L. T. Campbell and B. W. J. McNeil, 2012, Puffin: A three dimensional, unaveraged free electron laser simulation code, Physics of Plasmas Vol. 19 3 C. Maroli, V. Petrillo, and M. Ferrario, 2011, One- dimensional free- electron laser equations without the slowly varying envelope approximation, Physical Review Special Topics- Accelerators and Beams Vol. 14 The next step in future work is to examine how the electron beam bunching affects the