1. W.Lu, M.Tzoufras, F.S.Tsung, C.Joshi, W.B.Mori
Generation of a GeV, nC monoenergetic beam using laser plasma acceleration
Hi, I am michail tzoufras. I will present a simulation with the purpose of demonstrating a nC GeV monoenergetic beam using laser plasma accelerator. This is actually the second part of the talk presented before by Wei Lu. Our collaboration consists of F.S.Tsung, C.Joshi, W.B.Mori and the two of us from UCLA and L.O.Silva, R.A.Fonseca IST.
W.Lu, M.Tzoufras, F.S.Tsung, C.Joshi, W.B.Mori
UCLA, USA
L.O.Silva, R.A.Fonseca
IST, Portugal

2. Outline
Motivation.
Simulation parameters : Show how to choose the simulation parameters given the laser power (“matched profile”, guiding). Find the expected results based on the theory.
Physical picture : Description of the physical picture of the simulation. Additional effects that could lead to results different than the theoretical predictions.
Simulation results : Phasespace of the self generated electron beam. Evolution of the beam with time and its characteristics at the end of the simulation.
Conclusions.
First I will mention some exciting recent results in this area. Then ,based on the theory, I will show how you can choose all of the simulation parameters given the laser power, and what results ought to be expected. I will present the physical picture of the problem and discuss some effects that do not show up in the theory and may affect the outcome. I will finally show the evolution of the self injected electron beam and its properties and summarize our conclusions.

3. Motivation Recent results
Phys. Rev. Lett. by Tsung et al. (September 2004), where energy up to 0.8 GeV and monoenergetic beam with energy 260 MeV were observed.
3 Nature papers (September 2004), where monoenergetic electron beams with energy exceeding 100 MeV were measured.
Last september there was a simulation paper by frank, where …
At the same time 3 nature papers with both experimental and simulation results where …
It seems that the laser induced ….
The laser induced ultrarelativistic blowout regime is very effective in accelerating particles.

4. c Explicit Particle-In-Cell code OSIRIS Typical simulation parameters:
~109 particles
~105 time steps
The code we used for these simulation is an explicit PIC code osiris. The idea is that you place the simulation particles on a grid. You calculate the fields to advance the particle positions and calculate the fields again. Each loop is one time step. Typical simulations for this problem require 10^5 timesteps for 10^9 particles. In order simulate long distances of plasma we follow the laser pulse with a simulation box moves with the speed of light. c

5. Nature papers, agreement with experiment
Simulations
Nature papers, agreement with experiment
3D Simulations for Nature 431, 541 (S.P.D. Mangles et al)
In experiments, the # of electrons in the spike is
In our 3D simulations, we estimate of electrons in the bunch.
Here you can see the comparison between results from our simulation code and one of the nature experiments. The broken blue line is the experimental energy spectrum that was given to us by Stuart Mangles and the red line the result of the simulation. The peak energy agrees excellent and the total number of particles is similar. The large energy spread that was observed in the simulation can to the biggest part be attributed to numerical cherenkov noise associated to numerical dispersion of the speed of light. We are rerunning this simulation right now after we have greatly reduced the numerical dispersion. For the parameters used for the 200TW run numerical dispersion is negligible (we already knew we had to be careful about that).

6. Laser parameters and plasma density
Simulations
Laser parameters and plasma density
What can we do with a 200 TW laser?
“Matched” laser profile
First: Given a laser power of 200TW for a 0.8 um laser, how do you pick the rest of the laser parameters and plasma density?
We chose the critical power to ensure self focused propagation (we chose P/Pcrit = 10)
Used a “matched” laser profile to get the normalized vector potential and the from this the spotsize. The pulse length can be chosen so that it is smaller than the blowout radius. We chose it to be 30 fsec.
Require self focused
propagation
“Matched” laser profile

7. Simulations Expected results
Local pump depletion at the front of the pulse leads to the etching back velocity. We can calculate the pump depletion and the dephasing length. The central energy is expected to be 1.5 GeV. We can see that the pump depletion length is a little bit smaller than the dephasing length. If we had chosen a laser with longer than 40 fsec then the dephasing length would be smaller than the pump depletion length and still fit in the first half of the sphere.
For a 30fs pulse the depletion length is shorter than the dephasing length (we could have chosen a longer pulse).
Energy about 1.5GeV, however what is the energy spread since there is no dephasing?

8. Simulations Parameters After 5 Zr / 7.5 mm Total charge = 1.1 nC
We need to resolve (~30 cells/wavelength):
The laser wavelength in the direction of the laser propagation.
The plasma wavelength in the plane perpendicular to it.
After 5 Zr / 7.5 mm
The main restriction when we choose tha parameters for the grid is that we need to resolve the appropriate wavelength in each direction.
This is the laser wavelength in the direction of the laser propagation and the plasma wavelength in the direction perpendicular to it. We use 32 cells/wavelength. Here you can see our simulation parameters. We use about half a billion particles and run the system for timesteps. This took 1 month on 200G5s on our “dawson” cluster. The results after about 5 rayleigh lengths is a beam with mean energy around 1.5 GeV (a bit smaller) and charge 1.1 nC.
(Took about 1 month on 200 G5s)
Total charge = 1.1 nC

9. Physical picture Geometry - fields
An ion channel is generated due to the ponderomotive blowout of the electrons.
Its shape is almost a sphere.
This structure moves with the speed of (laser) light, supporting huge accelerating fields.
Particles at rear of the channel are injected in the blowout region.
The force on these particles is both accelerating and focusing.
This is a cartoon made from the simulation data. We have taken a slice along the center of the simulation box and we plot the density with blue, the laser envelope with orange and the accelerating wakefield with green. You can see the structure generated from the electrons blown out by the laser ponderomotive force. The shape of the ion channel is almost a sphere. The sphere moves with the group velocity of the pulse. The fields in the back of the sphere are accelerating and focusing. Particles that are in this area can get trapped even if they had low initial velocity.

10. Evolution of the nonlinear structure
Physical picture
Evolution of the nonlinear structure
The front of the laser pulse looses energy to the particles and etches back.
Beam loading eventually shuts down the self injection.
The pulse forms its own channel and remains self-focused until its power falls below a certain value.
The laser can be chosen long enough so that the pump depletion length is longer than the dephasing length.
In this movie you can see the evolution of the laser envelope and the electron density. You can see how the pulse shortens as looses energy to the particles locally in the front and etches back. You can also see that self injection eventually shuts down and particles form a bunch completely distinct form the background density. This bunch keeps accelerating until it reaches the end of the accelerating region or if the pulse is chosen too short (as here) until the pump power is not enough for laser to self focus and starts diffracting.

11. Laser Guiding 3D movie - not exactly matched but stable
If the laser profile is not exactly “matched”, the laser size of the envelope oscillates and as a result so does the blowout spheroid. However the propagation is still stable!
Electron density isosurfaces from 3D simulation for a 200 TW 30 fsec pulse
Here you can see a movie of electron density isosurfaces. You can see the particles pushed to the front of the pulse by the ponderomotive force. You can notice that the isosurfaces oscillate during the first two rayleigh lengths (half an oscillation in every rayleigh length). This is because the laser profile is not perfectly “matched”. In the first rayleigh length the pulse focuses and the blowout radius becomes larger (there is an isosurface moving to the front). Particles get injected and accelerated. In the second rayleigh length as the pulse defocuses the blowout radius decreases. The back of the accelerated particles is then “cut” by the spheroid and the self injected beam loses almost half of its initial charge.

12. Simulations
The 200 TW run
These plots are: momentum in the direction of the laser propagation VS perpendicular momentum px-py, px Vs y - you can see the width of the accelerating bunch and px-x the longitudinal profile of the momentum. There is also a lineout of the wakefield taken at the center of the box (where the beam is). You can see the particles injected after a rayleigh length have reached energy about 500MeV. A beam has not been formed yet. The particles gain so much energy so fast because as you can see from the wakefield lineout they are in the region with the highest wakefield.
At early times the accelerating fields are higher.
A beam has not been formed yet.
Beam loading

13. Simulations
The 200 TW run
After 2 rayleigh lengths there is a beam formed. Because of the oscillation of the speroid, that I showed earlier, the back part of the beam experiences very high accelerating wakefields that exist in the rear of the ion channel for the second time and as you can see in the px-x plot they start to catch up with the front.
A beam starts to form as beam loading becomes significant.
Some particles feel the spike of the wakefield again!
Beam loading

14. Simulations
The 200 TW run
At the end of the simulation the particles in the back of the beam have higher energy than the front, this is what causes this “U” shape in the px-x plot. Even though dephasing was not reached the beam that emerges has low longitudinal spread and divergence. The beam loading is also significant. Of course ideally there would be a short beam in a perfectly matched structure with no beam loading that rotates nicely when it reaches dephasing and becomes really monoenergetic. However even with all this complecated phenomena going on we got a very beautiful beam.
Even though dephasing wasn’t reached a beam with low longitudinal spread does emerge.
The divergence is also very low.
Beam loading

15. Conclusions
We have shown how the theory allows us to design laser plasma accelerators operating in the ultrarelativistic blowout regime.
Given the power of a laser we can:
Pick the density for self-focused propagation.
Choose the rest of the laser parameters by assuming a ”matched” profile.
Predict the energy and the charge of the monoenergetic beam.
Our theoretical estimates are very robust, despite of the very complicated interplay of phenomena that occur in this regime.
For these accelerators, since the energy is proportional to the laser power:
we can expect beams, with energy above 10 GeV and 5nC charge using 2 PW lasers.
To summarize: We have shown how the theory allows us to design laser plasma accelerators operating in the ultrarelativistic blowout regime.
Particularly given the laser power we can:
pick the density by requiring self - focused propagation.
choose a “matched” profile
and then predict the energy of the beam.
Our theoretical estimates are very robust for the very complicated interplay of phenomena that occur in this regime.
Using our scaling we hope that soon with the development of ultrashort PW laser systems, beams with tens of GeV energy will be realized.
(Cplicated phenomena that arise during the interaction may somewhat alter the results but overall the )