B. FEL theory. B. FEL theory B.1 Overview B.2 Low-gain FEL theory B.3 High-gain FEL theory.
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B. FEL theory
B. FEL theoryB.1 OverviewB.2 Low-gain FEL theoryB.3 High-gain FEL theory
The basic model
• 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.
• 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.
γr and ψ defined later
Assumptions and limitations of our 1D model
– 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, px, y, py not considered.
– The case if εx, εy 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 λl. Coordinate
given as angle Ψ(z) (ponderomotive phase).
Relation of low- and high-gain FEL theory1. Low-gain theory
– Pendulum equations: position and energy of particles.
– Ex-field stays constant for one passage.
2. High-gain theory– Change of Ex-field is considered.– Coupled 1st order equations:
• Pendulum equations. • Ex-field evolution.• Solved numerically.
– 3rd order equation for Ex-field:• Simplification of coupled first order
equation. • Can be solved analytically.
3. Extensions- To handle relaxed assumptions:
• Energy spread• SASE, …
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
X-ray field• Straight particle motion
– No energy transfer since fields orthogonal to beam motion
• Wiggle motion– FE and FB are in slightly different directions.– But cancellation is still good enough for trajectory.– Remaining forces will matter for energy transfer.
• 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.
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/.– Easy to use.
B. FEL theoryB.1 OverviewB.2 Low-gain FEL theoryB.3 High-gain FEL theory
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
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 vx(t).
– Calculate vx(t) to calculate vz(t) as
– Then use vz(t) to get an improved estimate for z(t)
Normalised motion in particles rest frame.
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 Ex component of EM wave.
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 vx and Ex 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:
Intuitively, one sees already that the light has to slip by λl with respect to the electron in one undulator period λu.
Particle and light wave interaction 3/4
For small gain theory Ex is constant for one beam passage (ψ0 is initial phase of light to beam)
Derive the solution for x(t) to get vx(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.
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 λl 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
The ponderomotive phase ψ is the relative phase between vx and Ex. If the resonance condition is fulfilled it corresponds to a wave along the bunch that moves with the speed of the bunch vz. The the wavelength of ψ is λl, 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 vz and therefore on its energy deviation η.
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.
Mandy’s theorem 1/2
The forming of bunching can be studies by tracking many particle P i, with different initial conditions (ψi,ηi) 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(ηb) of the light wave for one pass of the undulator. Here ηb is the initial energy deviation of all particles in the beam.
Mandy’s theorem 2/2
For ηb = 0 no energy gain. For ηb > 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 (ηb > 0) to get an light amplification.
This is not the case in the high gain theory, where there is a high gain for ηb = 0 (see later).
B. FEL theoryB.1 OverviewB.2 Low-gain FEL theoryB.3 High-gain FEL theory
Evolution of the electric field Ex 1/3
In the high gain theory the Ex 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: Ey = Ez = 0 and Ex = Ex(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
Evolution of the electric field Ex 2/3
Here Ex(z) defines the amplitude of the field that is varying slowly compared to the phase factor. Ex(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) jx is written as f(jz)
Now an Ansatz for the general expression of jz is made. We assume the form
Evolution of the electric field Ex 3/3
Equation (7) corresponds to a DC current j0 and a current modulation j1(z) with an wavelength of λl. This corresponds to a Fourier decomposition if jz 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 γr 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 j1 still has to be determined from the position ψn of the N particles.
Expression for the current density at the first harmonic
The position ψn of the electrons and hence of the charge is known from the solution of the pendulum equations.
The current density jz(ψ) is related to the charge density ne as
where Ab is the transversal beam area. This expression can be in an Fourier series as
Evaluating ck for k = 0 and k = 1 finally gives expressions for j0 and j1
Longitudinal space charge
The current density modulation jz creates a lonitudinal field Ez that can be calculated by solving
For the made assumptions this leads to
Assuming Ez(z,t) to have the same form as j1(z,t) leads to the Ansatz
Using again SVA approximation leads to
This SC field leads to an particle energy variation of
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 J0 and J1 are Bessel functions of the first kind.
The 4 coupled 1st 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 3rd order equation is better suited.
Space charge field term
Light field term
The 3rd order FEL equation
One equation for Ex(z) in the high-gain regime
with gain parameter Γ, the space charge parameter kp, and the beam energy deviation ηb 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 ψn and ηn)
2. Starting from the Vaslov equation (evaluation of a distribution function)Another way of writing Eq. (8) introduced the important FEL parameter ρFEL as
The beam energy deviation ηb only appears relative to the constant ρFEL, which is hence important for the energy acceptance of the FEL.
Solution of the 3rd order FEL equation
The general solution to the 3rd order FEL equation has to form
where c1, c2 and c3 are coefficients that depend on the initial conditions. The complex numbers α1, α2 and α3 depend on the values of the parameters Γ, kp and ηb.
For negligible space charge effect (kp << Γ) and a beam on resonance (ηb = 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
Comparison of results of coupled 1st order equations and 3rd order equation
• Coupled 1st order eq. give accurate estimates and also predict the saturation of the light power P(z).
• 3rd 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.
Examples of applications of the 3rd order equation
1. By solving the 3rd order equation for different ηb, 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/ωl (FWHM) can be computed analytically. It turns out that in the high gain regime ρFEL 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 PSAT is independent of the used seed power Pin. A rough estimate can be give by considering that most of the light intensity is created in the last field gain length which is 2Lg0
see next page for examples
Gain curves for low-gain and high-gain theory
• Good agreement for z < 2Lg0.
• Then strong deviation.
• For high gain regime G(η,z) drops quickly if |ηb|>ρFEL.
• Note that highest gain is for η > 0.
• This will be used for detuning.
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(ηb,z)).
Emittance: Betatron motion slows down particles and adds spread in vz, (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
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 Lg instead of 1D gain length Lg0.
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.