Research Summary

Group Member: Michail Tzoufras


Intense short laser pulses can produce conditions of extreme energy density that present a tremendous theoretical and computational challenge. Understanding and controlling the physical mechanisms in these regimes can open the way for novel applications including inertial fusion energy and compact particle accelerators. I develop and utilize computer simulation codes—Particle-In-Cell, reduced/hybrid Particle-In-Cell and Vlasov-Fokker-Planck—and analytical theory to explore the complex nonlinear physics of High Energy Density Laboratory Plasmas (HEDLP). My research is in three interrelated areas of HEDLP:

A summary of my main contributions in each area is presented below.



Vlasov-Fokker-Planck Code Development

To investigate electron transport in the context of the interaction of intense (I = 1015 – 1018 W/cm2) short pulse (t = 100fs – 100ps) lasers with plasmas, whose density ranges from less than critical to more than solid, I developed in collaboration with professor Tony Bell (Univeristy of Oxford) the simulation code OSHUN [1]. OSHUN is a parallel object-oriented (C++) relativistic 2D3P Vlasov-Fokker-Planck code that employs the expansion of the distribution function to spherical harmonics and allows one to keep an arbitrary number of terms in the expansion. This makes it possible to treat distributions that substantially deviate from isotropy, such as those generated when short intense laser pulses shine on over-dense plasmas. Moreover, because the code is relativistic, it can capture the behavior of hot tails in the energy spectrum, which often develop in these scenarios. A rigorous Fokker-Planck operator that recovers transport coefficients with excellent accuracy, has been implemented to take collisional physics into account.

The inclusion of an arbitrary number of terms to the spherical harmonic expansion means that OSHUN may be used for similar problems as conventional Cartesian Vlasov codes. For collisional—even weakly collisional—plasmas our code can be very powerful, because collisions tend to isotropize the distribution function by damping the high-order harmonics. (The more directional/anisotropic a spherical harmonic is, the faster it decays.) To demonstrate that OSHUN can model highly anisotropic relativistic distributions we present the evolution of the phase-space px-x from a simulation of the relativistic two-stream instability (see Movie 1).

To extend the simulation capabilities of OSHUN we plan to:

MOVIE 1 - The evolution of the relativistic two-stream instability in the frame of the unstable wave. The axis are X1 = px/(mc) and X2 = kp x.

Non-local Transport in Laser-Solid Interactions

When an intense laser pulse irradiates an over-dense target it generates hot electrons, which travel through and around the target driving strong electromagnetic fields and instabilities. This process of electron transport is determined by the plasma heat conduction, but in regimes in which the laser intensity is high, classical heat conduction does not apply and a fully kinetic description of the electron distribution must be considered. The problem that served as motivation for developing OSHUN was understanding the effect of electron transport on shock ignition. Shock ignition for Inertial Confinement Fusion (ICF) is a method proposed to increase the energy gain of ICF targets without major modifications to conventional ICF designs. In shock ignition, the hot electrons generated near the critical surface due to a spike in the intensity of the drive laser pulse must distribute the energy around the compressed target so as to launch a spherically convergent shock. To evaluate and optimize shock ignition designs electron transport must be well-understood, because non-local transport can effect shock formation as well as the stability and symmetry of the target. Furthermore, reliable modeling of non-local effects in multiple dimensions can help assess the feasibility of polar drive.

One-dimensional Vlasov-Fokker-Planck modeling has already indicated that non-local transport of laser-heated electrons can be beneficial to shock ignition by depositing the energy at higher density and by inhibiting losses to the plasma corona [2]. I am currently using OSHUN to scan the parameter space of laser-solid interactions for simplified geometries in two dimensions and gain insight into the transport physics. Subsequently I plan to model realistic shock ignition configurations. Further studies will involve hand-in-hand progress in understanding the physics and expansion of the simulation capabilities, i.e. 3D, addition of WDM and phenomenological LPI (two-plasmon decay, scattering instabilities etc.)



Short Pulse

In Plasma-Based Acceleration a laser or a particle beam excites a plasma wake, thereby generating a structure that can support accelerating fields more than three orders of magnitude above those of conventional accelerators. A (second) particle beam properly loaded into the wake, can be accelerated to very high energy in a short distance. This mechanism makes it possible to develop compact accelerators for high-energy physics, for studying new materials and for medical applications.

In 2003 experiments at the Rutherford Appleton Laboratory (UK) using the VULCAN laser demonstrated that when an ultra-intense pico-second laser interacts with a mm-long gas jet, a hot tail in the electron energy spectrum is generated [3]. The maximum energy observed in these experiments was 350MeV, and it was the world record from a plasma-based accelerator. To investigate the mechanism responsible for the acceleration, I incorporated particle-tracking diagnostics in the Particle-In-Cell code OSIRIS. The simulations recovered the experimentally observed spectrum and revealed that the acceleration is not due to the longitudinal wake-field, but comes directly from the electric field of the laser pulse. This is possible due to the extreme intensity of the VULCAN laser pulse, which creates a long plasma channel with strong focusing fields in which some electrons can find themselves oscillating in resonance with the field of the laser. These oscillating electrons can reach very high energy and radiate in a broadband spectrum [4]. Finally, the simulations revealed that the energetic ions, that were also measured in the aforementioned experiments, were due to the collision between ion shocks [5].

FIGURE 1 - Typical electron trajectories for the stochastic resonant acceleration by the laser fields. (See Ref. [3].)

As promising as the VULCAN results are for producing hot electrons and broadband radiation, the stochastic features of the acceleration process do not allow for the generation of the high-quality beams necessary for most of the envisioned applications.


Ultra-short Pulse

Laser-wake-field acceleration of electrons with energy spectra that exhibited beam-like features was first realized in experiments in 2004. All of these experiments involved ultra-short pulses. Our group (F. S. Tsung et al) used OSIRIS to model these experiments in 3D and analyze the physics [6]. At the same time we (W. Lu et al) developed an analytical theory for the excitation of the plasma wake behind an intense driver, be it a particle or laser beam [7]. This allowed us (W. Lu et al) to develop a detailed phenomenological theory for laser wake-field acceleration which includes both the laser evolution and electron acceleration process [8]. This theory allows one to choose the laser and plasma parameters so as to optimize the plasma-based accelerator. Its applicability was demonstrated with the generation of a 1.5GeV electron beam using a 200TW, 30fs laser pulse in a full 3D PIC simulation. The acceleration process is shown in the movie below.

MOVIE 2 - The evolution of the envelope of a 200TW 30fsec laser pulse (orange) in a plasma with electron density (blue) 1.5 × 1018cm-3 is shown on the top. The bottom part of the movie shows the electric field line-out across the center of the plasma bubble, i.e. Ex(x,y=0,z=0). (See Ref. [8].)

Since its publication, this theoretical framework has been proven helpful in designing experiments and accurate in predicting their outcome. However, scaling laws are not sufficient to generate beams with excellent beam quality. To address this issue I started from the description of the wake developed by W. Lu et al [7] and developed an exact analytical theory for the interaction of an electron beam with a 3D nonlinear plasma wake [9][10]. This theory shows how to control the quality of an electron beam produced by a plasma-based accelerator and how to transfer the energy available in the plasma wake to the accelerating electron beam with nearly 100% efficiency. Figure 2 shows how choosing the charge per unit length one can control the shape of the plasma bubble and the accelerating field and Figure 3 the confirmation of the analytical results using PIC simulations. This theory was recently used in experiments at the Laboratoire d' Optique Appliquèe (LOA).

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FIGURE 2 - The beam loading theory allows one to control the shape of the plasma bubble and the accelerating field by choosing the charge per unit length of the accelerating electron bunch appropriately. (See Ref. [10].)
FIGURE 3 - Confirmation of the beam loading theory using Particle-In-Cell simulations. Loading the correct charge per unit length profile into the wake leads to a constant accelerating field within the electron bunch (See Ref. [10].)

Employing the beam loading theory and the phenomenological scaling laws I compiled a list of simulations in order to demonstrate stable acceleration of high-quality beams up to 100GeV. For such simulations using a full PIC code is not practical, and therefore the quasi-static PIC code QuickPIC was used. I performed extensive benchmarks for QuickPIC comparing the results to those from the full-PIC code OSIRIS. This was done to ensure that the laser propagation is modeled accurately using the envelope approximation in QuickPIC, and that the wake-field is reproduced with high fidelity. The movie below shows the acceleration of an electron bunch (green) to 0.4GeV using a 15TW, 35fsec laser pulse in a plasma channel with electron density (blue) which has a minimum at the center of the box and is equal to 2 × 1018cm-3.

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MOVIE 3 - The evolution of the envelope of a 15TW laser pulse (yellow) and an externally injected electron beem (green) in a plasma channel (blue) from a Particle-In-Cell simulation with QuickPIC. (The movie plays twice from different viewing angles.)
FIGURE 4 - Generation of high-quality electron beams with energy up to 100GeV from Particle-In-Cell simulations with QuickPIC. (See Ref. [11].)

This simulation parameters were then scaled up to 25GeV and to 100GeV and the corresponding energy spectrum is shown in Figure 4. These results were included in my dissertation titled “Generation of Multi-Giga-Electron-Volt monoenergetic electron beams via laser wakefield acceleration” (2008) [11]. In collaboration with J. Vieira from IST Portugal I also examined the evolution of the laser pulse to provide an analytic description of the effect of frequency shifts on the early laser dynamics [12].

I am currently studying the effect that imperfections of the initial beam profile can have on the final beam quality and I am working to produce truly monoenergetic beams and to demonstrate the potential of using such accelerators as radiation sources.




Fast Ignition Simulations

In Inertial Confinement Fusion a capsule of hydrogen isotopes (DT) implodes under ablation pressure and the isotopes fuse, releasing energy. Fast Ignition has been proposed to decouple the compression from the ignition process. It relies on the electrons at the surface of the compressed target to absorb the energy of an ultra-intense laser pulse, travel through the dense plasma, and deposit that energy to the pellet’s core. We (Ren et al) performed 2D Particle-In-Cell simulations with the OSIRIS for the first pico-second of the interaction in a scaled-down fast ignition target isolated from the boundaries, in order to identify the key absorption physics [13] - [14]. I developed an extensive library of diagnostic and visualization tools to post-process the data produced by these large simulations, which helped disentangle the mechanisms at work.

MOVIE 4 - Hole boring in a scaled-down fast ignition simulation. Filamentation of the electron density is seen near the laser-plasma interface. (See Ref. [13] - [14].)

A movie of the electron density near the laser-plasma interaction region is presented above. The laser drills a hole in the plasma and the plasma density increases considerably near the laser-plasma interface. Filaments in the electron density can also be observed in this high density shock-like structure. These correspond one-to-one to filaments in the transverse profile of the laser intensity as shown in Figure 5. The laser filaments thus act as a seed for the Weibel instability. This is corroborated by a simulation (Movie 5) in which the plasma boundary is sharp, and where the plasma density still filaments even though the laser does not. In this simulation the effect observed is exclusively due to the Weibel instability.

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FIGURE 5 - There is a one-to-one correspondence between laser intensity filaments and plasma density filaments. (See Ref. [14])
MOVIE 5 - Filamentation of the electron density is observed in a simulation where the plasma-vacuum boundary is very sharp and the laser intensity does not filament at all. This is a manifestation of the current filamentation (Weibel) instability.

Another unexpected effect observed in the fast ignition simulation was that not only did the electrons filament but the ions did as well. To explain this I developed an analytical theory for electromagnetic instabilities that includes the coupling to electrostatic modes (see below).

I am currently using the hybrid code OSIRIS-H to perform large fast ignition simulations. This code enables integrated simulations of realistic targets and time-scales.


Electromagnetic Instabilities

To explain the apparent fluid-like behavior of the current filamentation instability in fast ignition scenarios I developed analytical theory for the coupling of electromagnetic instabilities to electrostatic modes [15] - [16]. This theory shows that as cold electrons tend to filament at a faster rate than hot ones they pull with them the ions. Because hot electrons are usually in the mionority, the filaments of the bulk of the electron population overlap with those of the ions, which is exactly what one would have expected from a fluid instability (such as Rayleigh-Taylor.) Nevertheless, this is a purely kinetic electromagnetic phenomenon. The predicted growth rate was confirmed using Particle-In-Cell simulations and the physics is illustrated in Figure 6.

FIGURE 6 - Filamentation of counter-streaming electron beams (red/blue density isosurfaces) leads to filamentation of the background ions (green isosurfaces). The colder (blue) electron filaments attract the ions. (See Ref. [15].)

Finally, all of these effects in fast ignition were observed for electron distribution functions that did not exhibit any beam-like features, but analytical theory was only available for beam-like distributions, such as drifting Gaussian and Waterbag distribution. To resolve this I calculated the stability properties (growth/damping rate) of electromagnetic modes for arbitrary electron distribution functions [17].




[1] M. Tzoufras, A. R. Bell, P. A. Norreys, F. S. Tsung, accepted in J. Comp. Phys. (2011) “A Vlasov-Fokker-Planck code for high energy density physics.”

[2] A. R. Bell and M. Tzoufras, Plasma Phys. Control. Fusion 53 (4), 045010 (2011) “Electron transport and shock ignition.”

[3] S. P. D. Mangles, B. R. Walton, M. Tzoufras, Z. Najmudin, R. J. Clarke, A. E. Dangor, R. G. Evans, S. Fritzler, A. Gopal, C. Hernandez-Gomez, W. B. Mori, W. Rozmus, M. Tatarakis, A. G. R. Thomas, F. S. Tsung, M. S. Wei and K. Krushelnick, Phys. Rev. Lett. 94, 245001 (2005) “Electron Acceleration in Cavitated Channels Formed by a Petawatt Laser in low-Density Plasma.”

[4] S. Kneip, S. R. Nagel, C. Bellei, N. Bourgeois, A. E. Dangor, A. Gopal , R. Heathcote, S. P. D. Mangles, J. R. Marquès, A. Maksimchuk, P. M. Nilson, K. Ta Phuoc, S. Reed, M. Tzoufras, F. S. Tsung, L. Willingale, W. B. Mori, A. Rousse, K. Krushelnick, and Z. Najmudin, Phys. Rev. Lett. 100, 105006 (2008) “Observation of Synchrotron Radiation from Electrons Directly Accelerated in a Petawatt-Laser-Generated Plasma Cavity.”

[5] M. S. Wei, S. P. D. Mangles, Z. Najmudin, B. Walton, A. Gopal, M. Tatarakis, A. E. Dangor, E. L. Clark, R. G. Evans, S. Fritzler, R. J. Clarke, C. Hernandez-Gomez, D. Neely, W. B. Mori, M. Tzoufras and K. Krushelnick, Phys. Rev. Lett. 93, 155003 (2004) “Ion Acceleration by Collisionless Shocks in High-Intensity-Laser-Underdense-Plasma Interaction.”

[6] F. S. Tsung, W. Lu, M. Tzoufras, W. B. Mori, C. Joshi, J. M. Vieira, L. O. Silva and R. A. Fonseca, Phys. Plasmas 13, 056708 (2006) “Simulations of monoenergetic electron generation via laser wakefield acceleration for 5-25 TW lasers.”

[7] W. Lu, C. Huang, M. Zhou, M. Tzoufras, F. S. Tsung, W. B. Mori and T. Katsouleas, Phys. Plasmas 13, 056709 (2006) “A nonlinear theory for multi-dimensional relativistic plasma wave wakefields.”

[8] W. Lu, M. Tzoufras, C. Joshi, F. S. Tsung, W. B. Mori, J. Vieira, R. A. Fonseca and L. O. Silva, Phys. Rev. ST Accel. Beams 10, 061301 (2007) “Generating multi-GeV electron bunches using single stage laser wakefield acceleration in a 3D nonlinear regime.”

[9] M. Tzoufras, W. Lu, F. S. Tsung, C. Huang, W. B. Mori, T. Katsouleas, J. Vieira, R. A. Fonseca and L. O. Silva, Phys. Rev. Lett. 101, 145002 (2008), also included in the Virtual Journal of Biological Physics Research, Vol. 17, Issue 6, (2008) “Beam Loading in the Nonlinear Regime of Plasma-Based Acceleration.”

[10] M. Tzoufras, W. Lu, F. S. Tsung, C. Huang, W. B. Mori, T. Katsouleas, J. Vieira, R. A. Fonseca and L. O. Silva, Phys. Plasmas 16, 056705 (2009) “Beam loading by electrons in nonlinear plasma wakes.”

[11] M. Tzoufras, Ph.D. thesis, UCLA, June 2008, Advisor: W. B. Mori “Generation of Multi-Giga-Electron-Volt monoenergetic electron beams via laser wakefield acceleration.”

[12] J. Vieira, F. Fiúza, L. O. Silva, M. Tzoufras, W. B Mori, New J. Phys. 12, 045025 (2010) “Onset of self-steepening of intense laser pulses in plasmas.”

[13] C. Ren, M. Tzoufras, F. S. Tsung, W. B. Mori, S. Amorini, R. A. Fonseca, L. O. Silva, J. C. Adam and A. Heron, Phys. Rev. Lett. 93, 185004 (2004) “Global Simulation for Laser-Driven MeV Electrons in Fast Ignition.”

[14] C. Ren, M. Tzoufras, J. Tonge, W. B. Mori, F. S. Tsung, M. Fiore, R. A. Fonseca, L. O. Silva, J.-C. Adam and A. Heron, Phys. Plasmas 13, 056308 (2006) “A global simulation for laser driven MeV electrons in 50µm-diameter fast ignition targets.”

[15] M. Tzoufras, C. Ren, F. S. Tsung, J. W. Tonge, W. B. Mori, M. Fiore, R. A. Fonseca and L. O. Silva, Phys. Rev. Lett. 96, 105002 (2006) “Space-Charge Effects in the Current-Filamentation or Weibel Instability.”

[16] M. Fiore, L. O. Silva, M. Tzoufras, C. Ren and W. B. Mori, Monthly Notices of the Royal Astronomical Society Volume 372, Issue 4, Page 1851-1855 (2006) “Baryon loading and the Weibel instability in gamma-ray bursts.”

[17] M. Tzoufras, C. Ren, F. S. Tsung, J. W. Tonge, W. B. Mori, M. Fiore, R. A. Fonseca and L. O. Silva, Phys. Plasmas 14, 062108 (2007) “Stability of arbitrary electron velocity distribution functions to electromagnetic modes.”