1. B. FEL theory

2. B. FEL theory B.1 Overview B.2 Low-gain FEL theory B.3 High-gain FEL theory

3. The basic model Top view It will be enough to describe the longitudinal phase space of the particle: -Energy W, or equivalently γ. -Longitudinal position in the bunch ζ. The finally chosen coordinates are first a bit unusual, but will turn out to be very useful. undulator coordinate beam coordinate External seed laser is overlapped with an wiggling bunch. The laser field along undulator is described, considering the interaction with e-beam. We are interested in finding a solution where the light wave is amplified. We first consider only one electron and the use this solution to fill the whole phase space. Comment: The light frequency will not change, but only the power of the light can change. γ r and ψ defined later

4. Assumptions and limitations of our 1D model Laser Light: – FEL process is started with seeding light. – No self-seeding from spontaneous undulator radiation (SASE, see later). – Laser is linear polarized TEM wave. – Light is modeled as perfect plane wave. – Unavoidable diffraction is not considered. Transverse particle motion: – Only wiggle motion is considered, but constant along the undulator (independent solution). – Betatron motion of particles is assumed to be small compared to wiggle motion. – x, p x, y, p y not considered. – The case if ε x, ε y and energy spread is small enough. Particle motion (1D model) Particle energy – Given as relative energy deviation Longitudinal position – Beam assumed to be infinitely long. – Periodic solution in z. – Only one micro-bunch has to be studies. – Distance between micro-bunches λ l. Coordinate given as angle Ψ(z) (ponderomotive phase).

5. Extensions Relation of low- and high-gain FEL theory 1.Low-gain theory – Pendulum equations: position and energy of particles. – E x -field stays constant for one passage. 2.High-gain theory – Change of E x -field is considered. – Coupled 1 st order equations: Pendulum equations. E x -field evolution. Solved numerically. – 3 rd order equation for E x -field: Simplification of coupled first order equation. Can be solved analytically. Low-gain theory High-gain theory E x constant E x changing 3. Extensions -To handle relaxed assumptions: Energy spread SASE, …

6. Survey of forces acting on the electrons Undulator B-field Vertical undulator field causes horizontal deflection of electrons (right hand rule). Wiggle motion that is constant along the undulator. X-ray field Straight particle motion – No energy transfer since fields orthogonal to beam motion Wiggle motion – F E and F B are in slightly different directions. – But cancellation is still good enough for trajectory. – Remaining forces will matter for energy transfer. Space charge General – Field created by charge and current of beam itself. – Electrons are highly relativistic. Transversal: – Defocusing of E-field and focusing of B-field are compensating nearly perfectly. – Can be neglected. Longitudinal: – Due to bunching in electron beam: charge modulation. – This charge modulation causes longitudinal electric fields. – This changes energy of electrons and has to be taken into account. – Effect is small, however, in hard XFELs.

7. FEL simulation codes The FEL theory makes many simplifications: – 1D effects (longitudinal parameters) – Infinitely long uniform bunches – … FEL codes avoid this limitations – Full transfers phase space is considered – Longitudinal bunch profile – Higher modes and diffraction of laser light – Realistic undulator sections. – Including quadrupole focusing and gaps. – Include SASE as well as realistic seeding scenarios. Several codes available. – Different domains and limitations. – GINGER – GENESIS – PUFFIN Most popular is the 3D code GENESIS 1.3 – Full 3D code. – Well benchmarked with measurement results. – Freely available: http://genesis.web.psi.ch/.http://genesis.web.psi.ch/ – Easy to use.

8. B. FEL theory B.1 Overview B.2 Low-gain FEL theory B.3 High-gain FEL theory

9. Particle motion in undulator 1/2 Field in undulator is given by Close to the centre of the undulator (electron beam is centred) To evaluate motion use Newton’s second law

10. Particle motion in undulator 2/2 First order approximation: Solve Eq. (1) with Ansatz This results in Second order approximation (see exercise): – Use solution for x(t) to calculate v x (t). – Calculate v x (t) to calculate v z (t) as – Then use v z (t) to get an improved estimate for z(t) Normalised motion in particles rest frame.

11. Particle and light wave interaction 1/4 Energy exchange is given by Without wiggle motion, fields and forces are perpendicular (no interaction of particle and light) The wiggle motion is necessary to create interaction of particle an light Only energy transfer with E x component of EM wave.

12. Particle and light wave interaction 2/4 To get a light amplification, we search for a solution where energy is transferred from the electron to the light wave. Electron has to loose energy. Possible if v x and E x are in the same directions throughout the undulator. It is not obvious that this can be achieved since light moves faster than particles. The light wave slips with respect to particle beam. But if the electrons fall back by the right amount behind the light continuous energy is possible: Condition: Intuitively, one sees already that the light has to slip by λ l with respect to the electron in one undulator period λ u.

13. Particle and light wave interaction 3/4 For small gain theory E x is constant for one beam passage (ψ 0 is initial phase of light to beam) Derive the solution for x(t) to get v x (t) and substitute into Develop product of cosine functions in sum of cosine functions It will turn out on the next page that χ(t) is a fast oscillating term doesn’t cause any net ΔW. On the other hand ψ(t) (called ponderomotive phase) can cause a net ΔW under certain conditions. with

14. Particle and light wave interaction 4/4 The ponderomotive phase will in general change linear with t and create no net ΔW The only possibility for ψ(t) = const. is if This condition is fulfilled in good approximation if This is an important result: the light wavelength λ l that allows continuous energy transfer is also the same as the spontaneous undulator radiation. Therefore, spontaneous undulator radiation can be used to seed FEL process. It can be shown that χ(t) is always linear in t and never contributes to a net ΔW. Hence

15. Ponderomotive phase Interpretation: The ponderomotive phase ψ is the relative phase between v x and E x. If the resonance condition is fulfilled it corresponds to a wave along the bunch that moves with the speed of the bunch v z. The the wavelength of ψ is λ l, which is the difference between two micro-bunches. This naturally defines a coordinate system. Due to the made assumptions the longitudinal bunching is periodic and it is enough to study ψ = [-π/2, 3π/2]. That is the range in which one micro-bunch will sit. Particle slippage to different ponderomotive phase: The ponderomotive phase of particle stays constant if on-resonance η=0. Off-resonance the particle can change its longitudinal position ζ and therefore its ponderomotive phase. The expression above can be simplified to The long. position of a particle is dependent on v z and therefore on its energy deviation η.

16. Pendulum equations The longitudinal phase space of one particle can be described by the two pendulum equations The form of these two coupled equations is the one of a pendulum and motion is, e.g. similar to the longitudinal motion in a synchrotron. The trajectories of stable motion are confined in phase space by separatrix, which forms a FEL bucket. Some particle trajectories are computed from the pendulum equation as plotted (red lines). In the centre of the FEL bucket (ψ=-π/2) there is no change of η. On the right side of the FEL bucket, particles loss energy and light gains energy and vice versa.

17. Mandy’s theorem 1/2 The forming of bunching can be studies by tracking many particle P i, with different initial conditions (ψ i,η i ) are tracked. The change of the energy of the light wave is equal to the energy the beam have lost Evaluating this expression in an analytically way (complicated) in an approximated way leads to Mandy’s theorem. It is an expression for the relative energy gain G(η b ) of the light wave for one pass of the undulator. Here η b is the initial energy deviation of all particles in the beam.

18. Mandy’s theorem 2/2 For η b = 0 no energy gain. For η b > 0, more particles loss on average more energy than they gain. From the gain curve it is clear that particles have to be injected with an average energy higher than for the resonance condition (η b > 0) to get an light amplification. This is not the case in the high gain theory, where there is a high gain for η b = 0 (see later).

19. B. FEL theory B.1 Overview B.2 Low-gain FEL theory B.3 High-gain FEL theory

20. Evolution of the electric field E x 1/3 In the high gain theory the E x can not be considered constant anymore for one beam passage but A freely propagating wave EM wave is described by the inhomogeneous wave equation (from Maxwell’s eq.) where j is the current density and ρ is the charge density. Considering the made assumptions: E y = E z = 0 and E x = E x (z,t), this eq. simplify to Note that since we assume a transversally large beam with uniform charge distribution (1D model, no x dependence). According to eq. (3), the following solution is assumed

21. Evolution of the electric field E x 2/3 Here E x (z) defines the amplitude of the field that is varying slowly compared to the phase factor. E x (z) is chosen to be a complex number. This allows to describe small phase variations compared to the plane wave propagation. Inserting the Ansatz Eq. (5) into Eq. (4) gives No the approximation is made that the the variation of the field is small over one undulator period called slowly varying amplitude (SVA) approximation. In this case the the second spatial derivative is even much smaller then the first derivative and can be neglected : Since it will be more practical to use the longitudinal current density (bunching) j x is written as f(j z ) Now an Ansatz for the general expression of j z is made. We assume the form

22. Evolution of the electric field E x 3/3 Equation (7) corresponds to a DC current j 0 and a current modulation j 1 (z) with an wavelength of λ l. This corresponds to a Fourier decomposition if j z where higher harmonics are neglected. Inserting Eq. (7) into Eq. (6) and performing the time derivation gives The second term in the brackets averages out already over half an undulator period and it remains The simplification of exchanging γ with γ r is valid in this case, since high-gain FELs are always operated close to resonance. The amplitude of the first harmonic of the charge density j 1 still has to be determined from the position ψ n of the N particles.

23. Expression for the current density at the first harmonic The position ψ n of the electrons and hence of the charge is known from the solution of the pendulum equations. The current density j z (ψ) is related to the charge density n e as where A b is the transversal beam area. This expression can be in an Fourier series as Evaluating c k for k = 0 and k = 1 finally gives expressions for j 0 and j 1

24. Longitudinal space charge The current density modulation j z creates a lonitudinal field E z that can be calculated by solving For the made assumptions this leads to Assuming E z (z,t) to have the same form as j 1 (z,t) leads to the Ansatz Using again SVA approximation leads to This SC field leads to an particle energy variation of

25. Modified undulator parameter So far the particle motion in the undulator has been treated only in first order. But the longitudinal oscillations due to the second order solution have an effect on the coupling between light an particles. Therefore, the undulator parameter K has to be exchange with the modified undulator parameter where J 0 and J 1 are Bessel functions of the first kind.

26. The 4 coupled 1 st order equations Collecting the different equations This corresponds to 2N+2 equations. Since many particles have to be simulated these equations can only be solved numerically (1D codes). 3D codes also include the tracking of the transverse particles coordinates and solve the wave equations in 3D. Also many buckets are solved and not just one. To get analytical estimates of the FEL processes, the simplified 3 rd order equation is better suited. Space charge field term Light field term

27. The 3 rd order FEL equation One equation for E x (z) in the high-gain regime with gain parameter Γ, the space charge parameter k p, and the beam energy deviation η b given by There are two possibilities to derive this equation (skipped in this lecture) 1.From the 4 coupled first order equations (assume certain form for ψ n and η n ) 2.Starting from the Vaslov equation (evaluation of a distribution function) Another way of writing Eq. (8) introduced the important FEL parameter ρ FEL as The beam energy deviation η b only appears relative to the constant ρ FEL, which is hence important for the energy acceptance of the FEL.

28. Solution of the 3 rd order FEL equation The general solution to the 3 rd order FEL equation has to form where c 1, c 2 and c 3 are coefficients that depend on the initial conditions. The complex numbers α 1, α 2 and α 3 depend on the values of the parameters Γ, k p and η b. For negligible space charge effect (k p << Γ) and a beam on resonance (η b = 0), the solution is simply given by (exercise) Only the first exponential functions causes exponential growth since. After a certain distance the field therefore grows as Here the important gain length Lg0 has been introduced. The light power is then given by

29. Comparison of results of coupled 1 st order equations and 3 rd order equation Coupled 1 st order eq. give accurate estimates and also predict the saturation of the light power P(z). 3 rd order equation gives in general good estimates, but cannot predict the saturation regime. This is due to the fact that for its derivation a small bunching was assumed, which is not the case in the saturation regime.

30. Examples of applications of the 3 rd order equation 1.By solving the 3 rd order equation for different η b, the gain function can be calculate as 2.From the shape of G(ηb, z), the energy acceptance and equivalent the relative light bandwidth Δω/ω l (FWHM) can be computed analytically. It turns out that in the high gain regime ρ FEL is a good approximation. The more detailed estimate is 3.The FEL process reaches full saturation when the beam is fully bunched. The reached saturation power P SAT is independent of the used seed power P in. A rough estimate can be give by considering that most of the light intensity is created in the last field gain length which is 2L g0 see next page for examples

31. Gain curves for low-gain and high-gain theory Good agreement for z < 2L g0. Then strong deviation. For high gain regime G(η,z) drops quickly if |η b |>ρ FEL. Note that highest gain is for η > 0. This will be used for detuning.

32. Specification of beam parameters Without going into details, the 1D theory can also provide limits for beam parameters that have been assumed negligible, when deriving the theory, e.g.: Space charge: Acts as a counterforce again bunching Detuning: Gain reduction can be seen from gain curves. Energy spread: Particles far from resonance condition will have low gain (see G(η b,z)). Emittance: Betatron motion slows down particles and adds spread in v z, (violation of resonance condition). Light divergence: Light should travel over field gain length 2 Lg0 with beam before it diffracts and to stay diffraction limited

33. Comment on gain length increase If the limits on the last slide cannot be fulfilled, the gain length is increased compared to the idealised conditions. One talks about 3D gain length L g instead of 1D gain length L g0. The 3D gain length cannot be studied with the developed theory but relies on fits to simulations. Two estimates are commonly used, which are named according to the developer: – M. Xie parameterization – Saldin parameterization Both estimates have a quite complex form (not given here), and include laser diffraction, energy spread and emittance.