In the following, only the pre-ionized scenario will be considered, as it clearly pro-vides the best stability and controllability besides producing the highest quality wit-ness bunches [240].
As has been shown in the case of the injection of electrons into the blowout, the trap-ping condition,∆ψ6 γ1ph −γi 1− vphc2vi
, also dependents on the phase velocity of the wake, vph (andγph= (1−v2ph/c2)−1/2). This phase velocity vphc =
1+ kξ
p
∂kp
∂z
−1
is a function of the density gradient, and density ramps can therefore facilitate trapping.
Given a strong drive beam that can excite a nonlinear wakefield for which the trapping condition (122) is fulfilled within a significant portion of the blowout, naturally trap-ping disappears when the perturbation exerted by the Plasma Torch becomes too weak.
This is the case if the ratio of the Plasma Torch density, and the background density becomes too small, or if the Plasma Torch is too short. Thus, one important figure of merit is the ratio of torch width,LT, to plasma wavelength within the torchλp,T, which should be bigger than one.
LT > λp,T (140)
However, the occurrence of trapping is also dependent on the strength of the plasma wave (and therefore its driver) and its capability to quickly accelerate electrons to a relativistic velocity. The inference might therefore be formulated as
3.3 plasma-torch trapping conditions 67
“Trapping occurs if the blowout fits into the Plasma Torch (LT > λp,T), and its rear is located at a position where electrons would have been trapped if they were released
at this position outside the Plasma Torch (∆ψ <−1)”.
The last part of this condition ensures that the driver is strong enough to excite a wakefield that can trap electrons.
In the following, a simulation series will be shown where a FACET-type drive beam (23GeV energy,2% total energy spread,1 nCof charge and2.25 mm mradnormalized rms emittance) drives a plasma wake in a pre-ionized hydrogen plasma asLITmedium of1×1016cm−3 density, and helium asHITmedium whose density was varied from 0.1×1016cm−3 to 10×1016cm−3. The diameter of the Plasma Torch stays approxi-mately the same at≈400 µm, while the torch density is altered from1/10to10 times the background density. For the helium densities from0.1to0.6×1016cm−3, no trap-ping was observed. For helium densities from 0.6 to 1×1016cm−3, electrons were trapped at a position too far towards the rear of the blowout, where they were radially pushed outside the blowout again by its defocusing fields.
This behavior is illustrated in figure13, where the ratio of helium density to hydrogen density is plotted against the amount of stably trapped charge (left), and the position of the first trapped electron (right). Note that the torch density isnH+nHe, but only the helium density was varied, so for convenience the aforementioned ratio of torch density to background density (nH+nHe)/nH is simplified to nHe/nH. The
simula-Figure13: The amount of stably trapped charge behind the torch as a function of density within the torch (left), and the position of the first trapped electrons (right). The threshold for stable injection for aFACET-type driver wasnH = nHe = 1×1016cm−3 (blue line).
tion withnHe = 6nH was removed due to technical difficulties. After stable trapping sets in atnH =nHe=1×1016cm−3, the trapped charge scales approximately linearly with the helium density until beam-loading forbids further injection (not yet reached in the figure).
The right sub-figure shows that, in addition to increasing the trapped charge, increas-ing the density of the Plasma Torch also shifts the trappincreas-ing position of the foremost electron closer to the blowout center, and longer witness bunches are generated.
It can also be seen in this figure that there is a first and a last possible trapping posi-tion in the blowout, limited at the rear by the point were the transverse fields switch to defocusing, and at the front by the distance that electrons need to become trapped when coming from the blowout center. This can be understood as a consequence of the shrinking of the blowout inside the torch when the helium density is increased. So electrons are situated further towards the front of the blowout—compared with a sce-nario with less helium density—which is their starting point for forward acceleration and trapping when leaving the Plasma Torch. So the higher the helium density within the Plasma Torch, the smaller the plasma blowout and the earlier the trapping position.
Figure14: Stably injected charge against the Plasma Torch diameter in units of the plasma wavelength inside the torch, showing that it is not sufficient that the blowout fits inside the Plasma Torch (LT/λp,T =1) to get injection, but also the driver strength must be taken into account. ForLT/λp,T .1.7no stable injection was found.
In conclusion, the amount of trapped charge increases with increase in density within the Plasma Torch up to severalnC’s, so the wakefield behind the first cavity is mostly compensated by the wakefield of the trapped charge, and only one plasma cavity exists.
The ratio of the density within, and outside the Plasma Torch—or the ratio of both plasma wavelengths—also determines the position of the leading edge, and the length of the witness bunch. Further, it is not sufficient that the blowout fits into the length (and width) of the Plasma Torch (LT/λp,T > 1), as shown in figure 14. Depending on the drive beam, the blowout must be significantly smaller than the diameter of the Plasma Torch, which means a higher density of theHITcomponent. Using aFACET-type drive beam, stable trapping has been observed forLT/λp,T &1.7.
3.3 plasma-torch trapping conditions 69 Smaller Plasma Torch density provides the opportunity to produce very short witness bunches, with less charge (compared to the observednC-level) and trapping in further plasma-wave buckets becomes possible. However, if the trapping position is too far towards the rear of the blowout, the transverse defocusing fields of the blowout can lead to complete destruction of the witness bunch. The witness bunch quality that has been achieved in the simulation series shown above was at the single digit mm mrad emittance level, and the energy spread shortly after the torch was on the order of10%
to several 10%, at approximately 10 MeV energy. The best witness bunch properties that have been measured in the computer simulations so far were from a bunch with 257 pCcharge,25 MeV energy, and6.4% energy spread,2.7 mm mrademittance, and 3.62 kApeak current extracted10 mmafter the Plasma Torch [240]. Note that the work discussed here was focused on showing the injection process and resulting witness-bunch quality, but did not consider the acceleration, during which the total energy spread can be continuously decreased (see equation (172)). Further enhancements of these results are possible, for instance with controlled beam loading by applying spe-cial Plasma Torch profiles. Also, previous research was performed based only on a strongFACET-type drive beam (of1−3 nCcharge and manyGeV’s of energy); varying the driver properties and background plasma density could therefore be another way to enhance the quality of the resulting witness bunch.
4
T H E O R Y O F F R E E E L E C T R O N L A S E R S
4.1 introduction
For a comprehensive list of currentFEL facilities see http://www.
lightsources.
org/fels
One promising application of the high-energy, low-emittanceTH-bunches is the gener-ation of high-power (GW), short-wavelength (Å) radigener-ation in a free-electron laser (FEL) [126]. TheFELcontains an undulator, a long structure of magnets with a periodically switching orientation of the magnetic field (visualized in figure15). When traversing the undulator, the electrons are forced into a sinusoidal trajectory and therefore emit radiation in the forward direction. The wavelength of this radiation is proportional to the undulator period over the square of the energy of the bunchλr ∝ λu/γ2, and therefore very short wavelengths can be generated. For example, for γ = 1000 (corre-sponding to an electron energy of≈511 MeV), an undulator period ofλu =1 cm, and an undulator parameter of au ≈ 1, the emitted radiation has a wavelength as small as 5 nm. One other major advantage over conventional laser systems is that the gen-erated wavelength can be continuously tuned by easily accessible parameters. This is one reason whyFELs are of growing interest today.
Figure15: FEL principle by Horst Frank, XFEL.
However, other than conventional laser systems, theFEL requires a very high-quality electron bunch the production of which, so far, needs a large-scale accelerator. Plasma-based accelerators could thus help to make FELs less expensive, more available and smaller. Because the size of the accelerator cavity is much smaller in a plasma-based accelerator than in a conventional accelerator and the accelerating fields are much stronger, the slope of the accelerating field is much steeper. Therefore, an accelerat-ing bunch must either be accordaccelerat-ingly shorter in a plasma-based accelerator, or it will feel a larger difference in the accelerating fields within the bunch, which significantly changes the energy gain between the front and the back. Bunches that are accelerated
71
in a plasma wakefield have therefore typically a large correlated energy spread (chirp), which can be compensated by letting the bunch propagate through a reversed field slope. Even without the compensation of the correlated energy spread, theFEL-process is possible if the fractional energy spread (which is the energy spread within a slice of the bunch), is low enough [204]. In addition, the bunch parameters must be adjusted behind the plasma to match the requirements of theFEL, in particular the large diver-gence needs to be lowered. Using a plasma accelerator to drive aFELis therefore not as easy as using a conventional accelerator. First experiments have already shown undula-tor radiation from bunches produced by plasma wakefield acceleraundula-tors [72,198], but for theFEL, significant advances in the quality of the accelerated bunches are required [45].