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 theoryB. FEL theoryB.1 OverviewB.2 Low-gain FEL theoryB.3 High-gain FEL theoryThe basic model
Top viewIt will be enough to describe the longitudinal phase space of the particle: Energy W, or equivalently .Longitudinal position in the bunch .
The finally chosen coordinates are first a bit unusual, but will turn out to be very useful.
undulator coordinatebeam 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.
r and defined laterAssumptions and limitations of our 1D modelLaser 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, px, y, py not considered.The case if x, y and energy spread is small enough.
Particle motion (1D model)Particle energyGiven 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 l. Coordinate given as angle (z) (ponderomotive phase).
ExtensionsRelation of low- and high-gain FEL theory1.Low-gain theoryPendulum equations: position and energy of particles.Ex-field stays constant for one passage.
2.High-gain theoryChange 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. Low-gain theoryHigh-gain theoryEx constantEx changing ExtensionsTo handle relaxed assumptions:Energy spreadSASE, Survey of forces acting on the electronsUndulator B-fieldVertical undulator field causes horizontal deflection of electrons (right hand rule). Wiggle motion that is constant along the undulator.
X-ray fieldStraight particle motion No energy transfer since fields orthogonal to beam motion
Wiggle motionFE and FB are in slightly different directions.But cancellation is still good enough for trajectory.Remaining forces will matter for energy transfer.
GeneralField 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 codesThe FEL theory makes many simplifications:
1D effects (longitudinal parameters)Infinitely long uniform bunches
FEL codes avoid this limitations
Full transfers phase space is consideredLongitudinal bunch profileHigher modes and diffraction of laser lightRealistic undulator sections.Including quadrupole focusing and gaps. Include SASE as well as realistic seeding scenarios. Several codes available.
Different domains and limitations.GINGERGENESISPUFFIN
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 theoryParticle motion in undulator 1/2Field in undulator is given by
Close to the centre of the undulator (electron beam is centred)
To evaluate motion use Newtons second law
Particle motion in undulator 2/2First 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.
10Particle and light wave interaction 1/4Energy 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 2/4To 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 l with respect to the electron in one undulator period u.Particle and light wave interaction 3/4For small gain theory Ex is constant for one beam passage (0 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 (t) is a fast oscillating term doesnt cause any net W. On the other hand (t) (called ponderomotive phase) can cause a net W under certain conditions.
Particle and light wave interaction 4/4The ponderomotive phase will in general change linear with t and create no net W
The only possibility for (t) = const. is if
This condition is fulfilled in good approximation if
This is an important result: the light wavelength 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 (t) is always linear in t and never contributes to a net W. Hence
The ponderomotive phase 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 is 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 = [-/2, 3/2]. 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 =0. Off-resonance the particle can change its longitudinal position 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 .
Pendulum equationsThe 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 (=-/2) there is no change of .
On the right side of the FEL bucket, particles loss energy and light gains energy and vice versa. Mandys theorem 1/2 The forming of bunching can be studies by tracking many particle Pi, with different initial conditions (i,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 Mandys theorem. It is an expression for the relative energy gain G(b) of the light wave for one pass of the undulator. Here b is the initial energy deviation of all particles in the beam.
Mandys theorem 2/2
For b = 0 no energy gain.
For 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 (b > 0) to get an light amplification. This is not the case in the high gain theory, where there is a high gain for b = 0 (see later).B. FEL theoryB.1 OverviewB.2 Low-gain FEL theoryB.3 High-gain FEL theoryEvolution of the electric field Ex 1/3In 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 Maxwells eq.)
where j is the current density and 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 2/3Here Ex(z) defines the amplitude of the field that is varying slowly compared to the phase factor. Ex(z) is chosen to be a com