Particle-in-Cell Simulations of Electron Beam Production from Infrared Ultra-intense Laser Interactions
Gregory K. Ngirmang, Chris Orban, Scott Feister, John T. Morrison, Enam A. Chowdhury, W. M. Roquemore
PParticle-in-Cell Simulations of Electron Beam Production from Infrared Ultra-intenseLaser Interactions
Gregory K. Ngirmang ∗ and Chris Orban Department of Physics, The Ohio State University, Columbus, OH, 43210, USA andInnovative Scientific Solutions, Inc., Dayton, OH, 45459, USA
Scott Feister
Department of Astronomy and Astrophysics, University of Chicago, Chicago, IL, 60637, USA andDepartment of Physics, The Ohio State University, Columbus, OH, 43210, USA
John T. Morrison
Innovative Scientific Solutions, Inc., Dayton, OH, 45459, USA
Enam A. Chowdhury
Department of Physics, The Ohio State University, Columbus, OH, 43210, USA andIntense Energy Solutions, LLC., Plain City, OH, 43064, USA
W. M. Roquemore
Air Force Research Laboratory, Dayton, OH, 45433, USA (Dated: September 4, 2018)Advances in ultra-intense laser technology are enabling, for the first time, relativistic intensities atmid-infrared (mid-IR) wavelengths. Anticipating further experimental research in this domain, wepresent high-resolution two dimensional Particle-in-Cell (PIC) simulation results using the Large-Scale Plasma (LSP) code that explore intense mid-IR laser interactions with dense targets. Wepresent the results of thirty PIC simulations over a wide range of intensities (0 . < a <
39) andwavelengths ( λ =780 nm, 3 μ m, and 10 μ m). Earlier studies, limited to λ =780 nm and a ∼ a (cid:46)
1) and that the pre-plasma scalelength is similar to or longer than the laser wavelength. Under these conditions, ejected electronangular and energy distributions are similar to expectations from an analytic model used in [2].We also find that, for a ∼
1, the mid-IR simulations exhibit a classic ponderomotive steepeningpattern with multiple peaks in the ion and electron density distribution. Experimental validationof this basic laser-plasma interaction process will be possible in the near future using mid-IR lasertechnology and interferometry.
I. INTRODUCTION
While advances in laser technology have allowed ultra-intense laser interactions at near-IR wavelengths to bethoroughly explored, and it is only more recently thatultra-intense laser interactions at mid-IR wavelengthshave become experimentally possible [3]. A variety ofgroups are beginning to examine what may be learnedfrom experiments at these wavelengths and how phenom-ena observed in the near-IR may scale to longer wave-lengths [4]. Some of this interest stems from the existenceof atmospheric “windows” in the mid-IR [5], while othergroups consider how the longer length scale of mid-IRinteractions allows subtle phenomena to be more easilyprobed [6]. Another interesting value of intense mid-IRinteractions is in examining the physics of laser damage[7, 8]. ∗ [email protected] To the best of our knowledge, despite recent inter-est in mid-IR ultra-intense laser interactions, the liter-ature has not focused much attention on intense mid-IRlaser interactions with dense (i.e. solid or liquid den-sity) targets. These interactions are interesting for avariety of reasons, among them the favorable scaling ofthe ponderomotive electron energy with laser wavelength( a ∼ √ Iλ ∼ λ ). However, given the complexity ofultra-intense interactions with dense targets, these scal-ing arguments can only offer an order-of-magnitude ex-pectation for the results of detailed simulations and ex-periments in this regime. With experimental capabili-ties still maturing in the mid-IR, the present work offersa simulation survey of energetic electron ejection frommid-IR laser irradiated dense targets.The work presented here is motivated in part by ear-lier investigations of normal-incidence ultra-intense laserinteractions with liquid targets at the Air Force Re-search Lab which found much larger than expected con-version efficiencies from laser energy to ejected electron a r X i v : . [ phy s i c s . p l a s m - ph ] A p r energy [9]. These experimental results prompted simula-tion work by [1, 2]. [1] presented 2D(3 v ) Particle-in-Cell(PIC) simulations showing significant electron ejection atsuperponderomotive energies and emphasized that ultra-intense laser interactions at the ∼ W cm − ( a ∼ λ ≈
800 nm) intensities present in the experimentshould create strong standing-wave fields near the tar-get. [2] performed 3D PIC simulations of these targetsand provided an analytic model to explain both the ener-gies and angular distribution of ejected electrons. Morerecently, [10] have reported direct experimental measure-ments of the ejected electron energies, confirming theexistence of multi-MeV electrons in the experiment andfrom this reinforcing conclusions that the conversion effi-ciency in the experiment is large compared to other ultra-intense laser experiments. An interesting question, then,is whether superponderomotive electron ejection occurseven with intense mid-IR laser interactions.The other motivator for this project is the plan to pur-chase and upgrade an intense 3 μ m wavelength laser sys-tem at the Air Force Research Laboratory. The upgradedlaser system will be able to produce ∼ W/cm . We explorea much wider range of laser energies and intensities in aneffort to examine the physics of mid-IR laser interactionswith dense targets.Sec. II describes our simulation setup. Sec. III de-scribes our results. Finally, Sec. V provides a summaryand conclusions. II. PARTICLE-IN-CELL SIMULATIONS
We performed 30 different high-resolution 2D(3 v ) PICsimulations with the LSP code [11]. For all simulationsthe initial conditions included a liquid-density water slabtarget with some assumed pre-plasma scale length sim-ilar to earlier studies [1, 2]. In all simulations, a laseris normally incident onto the water slab. We use thefollowing Cartesian coordinate system for these simula-tions: the positive x -axis is the direction of the laser, the y -axis is the polarization direction, and z -axis is the axisof the water column, which is assumed to be the axis ofsymmetry in the 2D(3 v ) PIC simulations.The simulations involved a normally incident, spa-tially Gaussian, sine-squared envelope pulse with 780 nm,3 μ m, and 10 μ m wavelengths (denoted λ ). These sim-ulations extend the results of earlier investigations with780 nm laser pulses [1, 2] by examining the same phe-nomena with long infrared(IR) wavelengths. For conve-nience we will often refer to the set of all simulationsperformed with a particular laser wavelength incident bysaying, for example, “the 3 μ m simulations”, and like-wise for the other wavelengths. All the 780 nm simula-tions had a laser pulse with a 2.15 μ m Gaussian radiusand a 40 fs temporal full-width-half-maximum (FWHM)pulse duration (similar to the laser system described in[1, 2, 9]). The 3 μ m simulations have laser pulses with 8.25 μ m Gaussian radius and 158 fs FWHM pulse du-ration, and the 10 μ m simulations had laser pulses with27.5 μ m Gaussian radii and 510 fs. These Gaussian radiiand pulse durations were chosen so that the ratio of thewavelength to Gaussian radius and the number of opti-cal periods in a pulse were fixed across all simulationsregardless of laser wavelength. For each wavelength wesimulated a range of pulse energies from 10 − J, 1 mJ ,10 mJ, 1 J, to 10 J. Since the Gaussian radius and pulseduration were fixed for each wavelength this was done bychanging the intensity. Simulation parameters for eachwavelength simulation are summarized in Table I.The target in all simulations consisted of free electrons,protons, and O + ions, with number densities set in rela-tive proportion to make the target match water’s chem-ical composition and to ensure charge neutrality (O + top + to e − ratio of 1:2:3). In all simulations, the plasmadensity only varied along the x direction. In the “target”region the density is constant and in the “pre-plasma”region the density profile is exponentially decreasing in x away from the target region.For every intensity and wavelength considered we per-form a simulation with a 1.5 μ m scale-length, for directcomparison with previous studies [1, 2] which employeda such a scale length pre-plasma. For 3 μ m and 10 μ mwavelengths, we also perform simulations where the pre-plasma scale length is a constant multiple of the wave-length ( L = 1 . λ ), so that we performed 3 μ m sim-ulations with a 5.77 μ m scale length and 10 μ m simu-lations with a 19.2 μ m scale-length. We refer to thesescale lengths are the “scaled” scale lengths. The 780 nmand 3 μ m simulations had a target region electron den-sity of 10 cm − , which with the O + to p + to e − ratiomentioned earlier correspondent to the mass density of1 g/cm as one would expect for liquid water. The 10 μ msimulations with a 1.5 μ m scale length also had a targetregion with this same 10 cm − electron density into thetarget. However, the 10 μ m simulations with a 19.2 μ mscale-length had a target region electron density of 10 cm − in order to reduce the size of the target in order toreduce computational requirements. The 780 nm simula-tions used a target that was 20 μ m wide in the y direction,the 3 μ m simulations used a 100 μ m wide target, and the10 μ m simulations used a 220 μ m wide target. All tar-gets had a initial temperature of 1 eV. All simulationshad absorbing boundaries with 10 μ m between the initialtarget and the simulation boundaries.The classical formula for the critical electron den-sity is n c = 4 πm e ω /e = m e /πλ e . For the780 nm, 3 μ m, and 10 μ m wavelengths, this corre-sponds to n c =1.74 × cm − , 1.24 × cm − , and1.11 × cm − respectively. In all simulations, the laserfocus was chosen to coincide with the critical density inthe pre-plasma layer. For the 780 nm simulations, a spa-tial resolution with spacing of 33 nm (roughly 23 cellsper wavelength) was used and timesteps of 0.1 fs wasused. For the 3 μ m simulations a spacing of 100 nmand temporal resolution of 0.15 fs was used. Finally, the Wavelength( μ m) Intensity Range(W/cm ) GaussianRadius ( μ m) PulseFWHM(fs) ScaledPre-PlasmaScale ( μ m) SimulationTimestep(fs) SimulationSpatialResolution( μ m)10 1.64 · − · · − · · − · μ m were performed for all wavelengths. μ m simulations utilized a spatial resolution of 250 nmand temporal resolution of 0.5 fs. Although these sim-ulations do not resolve the Debye length in every cell(since there are cells with near-solid densities with sub-nanometer Debye lengths), the phenomena of interest iselectron acceleration in the underdense pre-plasma ex-tending from the target where the Debye length is muchlarger and more easily resolved. The implicit algorithmin LSP avoids grid-heating issues associated with the De-beye instability so that the behavior of near-solid densityregions in the simulation does not ruin the overall energyconservation of the simulation. All simulations had 27macro-particles per cell per species (free electrons, pro-tons, and O + ions). As in earlier work Orban et al. [1],the O + ions in this simulation can be further ionized bystrong electric fields according to the Ammisov-Delone-Krainov rate [12]. In the simulation electron macroparti-cles scatter by a Monte-Carlo algorithm as in Kemp et al.[13] with a scattering rate determined by a Spitzer model[14] except at low temperatures where the scattering rateis limited by the timestep. The 1.5 μ m scale-length simu-lations were run for three times the duration of the simu-lated laser pulse (i.e. three times the full-width full maxduration of the pulse). The scaled simulations were runfor 3.5 times the duration of the simulated laser pulsebecause these were larger targets with a more extendedpre-plasma.The number of pulse energies investigated, the variousscale lengths assumed with the three laser wavelengthsadd up to a total of 30 2D(3 v ) simulations. The param-eters of all simulations are listed in the appendix, andparameters common across given wavelength simulationsare summarized in Table I. III. RESULTSA. Ejected Electron Energies
Fig. 1 shows the energy spectra of back-acceleratedelectrons for all the simulations performed. The leftpanel of Fig. 1 shows the results of the simulations witha 1.5 μ m scale-length pre-plasma, while the right panelplots the results of the scaled scale-length simulations where the scale-length is proportional to the wavelength( L = 1 . λ ). These figures plot the mean and maximumenergy of escaping electrons on the y -axis as a function ofthe normalized vector potential a of the incident laser.Here, a = eE /m e ωc , with the peak electric field valueof the incident pulse is denoted E , c is the speed oflight, m e is the mass of an electron, and ω = 2 πc/λ isthe angular frequency of the laser beam. The mean elec-tron energy is determined by taking the average energyof electrons that reach the boundary of the simulation.Since we are concerned here with back directed electronsonly those electrons with a momentum angle within ± ◦ of the incident laser are counted.These results are found to scale with the Wilks scalingestimate from [15], E wilks = (cid:34)(cid:114) a − (cid:35) m e c . (1)While there are a number of other formulae that describethe typical energy of electrons interacting with an in-tense laser field, we choose to compare with Wilks scal-ing because it is an analytically motivated formula thatis reasonably representative of the various scaling mod-els in this regime [16]. The Wilks model also reducesto the classical ponderomotive energy of an electron in amonochromatic plane wave in the low a limit. A bino-mial approximation yields E wilks ≈ m e c a for small values of a . This is why, on Fig.1, one seesan a dependence for Eq. 1 at low a that transitions tolinear dependence (i.e. ∼ a ) for a (cid:38) μ mwavelength simulations fall closest to the Wilks scalingmodel prediction. The 3 μ m and 780 nm wavelength sim-ulations lie significantly above the prediction, especiallyfor low a values. The 780 nm wavelength simulationshave the most energetic electrons, exceeding Eq. 1 by1-2 orders of magnitude. Thus we say that the ejectedelectrons in the 780 nm simulations are highly ”super-ponderomotive”. -2 -1 a K i n e t i c E n e r g y ( e V ) L =1.5 µ m wilks780 nm3 µ m10 µ mmaxaverage -2 -1 a K i n e t i c E n e r g y ( e V ) L =1.92 · λ wilks780 nm3 µ m10 µ mmaxaverage FIG. 1. Left panel: Ejected electron energies observed in PIC simulations with laser wavelengths 780 nm (red), 3 μ m (dark red),and 10 μ m (black) and with a 1.5 μ m pre-plasma scale length. Right panel: Ejected electron measurements from simulationswhere the pre-plasma scale length is ”scaled” to the wavelength of the laser such that L = 1 . · λ . In both panels results areshown as a function of the a of the incident laser. Expectations from Wilks scaling (Eq. 1) are shown as a solid blue line.The lowest a case for L = 1 . · λ is not shown as accelerated electrons were not detected in that case. Open circles representthe mean kinetic energy of accelerated electron that reach the edge of the simulation space in a 150 degree apex cone in thebackwards direction. The error bars represent 68% of the electrons around the mean energy bin. The triangles represent themaximum ejected electron energy observed. Dashed lines connect the triangles to the error bars so that it is easy to see thatthese measurements come from the same simulation. As identified earlier, the right panel of Fig. 1 shows theresults of the ”scaled” simulations. The 780 nm simula-tions with 1.5 μ m scale-length appear in both panels ofFig. 1, but the longer wavelength simulations shown inthe right panel of Fig. 1 all assume a longer scale lengththan in the left panel. Remarkably, in the right panel theresults from all three wavelengths seem to follow roughlythe same trend and exceed the Wilks prediction by 1-2orders of magnitude.Fig. 2 presents detailed information on the energies andejection angles of electrons that leave the simulation vol-ume. Fig. 2 shows results from the three different wave-lengths, highlighting intensities with a ∼ L = 1 . λ . We also overplot withsolid lines the results from an analytic model describedin [2]. This model considers that the back-directed elec-trons are ejected at high speed into a pulsed plane wavethat approximates the reflected laser pulse. Because ofthe similarity to earlier work in [2], it is unsurprisingthat the model compares favorably to the 780 nm resultsshown in Fig. 2. What is more remarkable is that themodel predictions compare similarly well to the mid-IRsimulations. B. Electron density profiles
Fig. 3 provides snapshots of the electron density in thesimulation after the reflection of the laser (not more thantwo pulse-lengths in time later) but while electrons arestill moving away from the target. The figure demon-strates that the onset of plasma wave phenomena is de-pendent on the a value of the incident laser pulse, andnot only the laser wavelength or laser intensity indepen-dently. For a < . a values between 0.2 and 0.4, the backwards ac-celerated electrons break apart the pre-plasma layer asthey escape. At a values near ∼ . a becomes larger. As discussed in [1],this arises because electrons are only deflected away fromthe target during two specific moments during the lasercycle. In these simulations, it has been observed thatthe onset of these bunches is preceded in the process ofreflection by a significant electron density hole createdby the ponderomotive force, which manifest only as a nears and exceeds unity. As a exceeds 2.0, as shown in pC / cm·rad.·MeV E n e r g y Ejection Angle(Backwards) o (Forwards) - o o o o o o o L =1.5 μ m L =1.92 λ FIG. 2. Summary of the kinetic energies and ejection angles of electrons that escape the simulation space of selected PICsimulations in which the incident laser has a ∼
1. In all panels, the distance from the center represents the kinetic energy ofescaping electrons while the angle to the origin represents the final ejection angle, and the color is proportional to the amountof charge in a particular energy and angular bin. The left panel shows a 780 nm simulation with a = 1 . μ m simulations with a = 0 . μ m simulations with a = 1 .
1. The upper right panel shows two mid-IR simulations with a L = 1 . λ exponential scalelength, while the lower right panel shows two mid-IR simulations with L = 1 . μ m. An analytic model from [2] is overlayed ineach panel and labeled by the normalized momentum similar to figures shown in [2]. the top two plots, the laser begins to penetrate beyondthe (non-relativistic) critical density surface due to rela-tivistic transparency. Hole-boring [17] does not occur inthis case due to the ultrashort pulse in all cases, and thelaser pulse begins to penetrate only when the last few cy-cles of the main pulse are present on the non-relativisticcritical density surface.To comment on another aspect of the plots in Fig. 3,in essentially all of the plots shown (0 . < a <
4) thelaser ionizes the target, moving the critical density (whitecontour) towards the incoming laser, especially along thelaser axis. Because the laser intensity decreases awayfrom the laser axis according to a gaussian spatial profile,this causes the critical density to assume a curved shapeas seen in the figure.
C. Ponderomotive Steepening
While fig. 3 demonstrates that the onset of plasmawave phenomena is determined by the a value of theincident laser, it also demonstrates that at a given a , plasma features observed scale in physical size with theincident laser wavelength. This finding can be used toscale the physical size the laser plasma interaction tofacilitate experimental observation of phenomena whichwould not be observable with short wavelength pulses.One such phenomenon is ponderomotive steepening, awell-known laser-plasma interaction process where theradiation pressure from the laser modifies the electrondensity profile, which, over time, will substantially mod-ify the ion density profile [18] on the scale of half thewavelength. A related process is ”hole boring” which hasbeen studied theoretically and experimentally [e.g. 17,and references therein]. Unlike hole boring, ponderomo-tive steepening, as originally described by [18], involvesa series of peaks in the ion density profile. Ponderomo-tive steepening has been observed in a number of PICsimulations in the literature [e.g. 1] but to the best ofour knowledge it has never been experimentally observedgiven the size of such features would be on the sub micronscale. To be able to see this phenomenon, one needs anintense laser and high spatial resolution interferometry ofthe target that can resolve half-wavelength-scale features. FIG. 3. Electron density snapshots (colormapped rectangles) from twelve unique simulations ranging over wavelengths and a values. The snapshots are arranged on the grid according to wavelength and a value. Left, middle, and center columns classifythe snapshots by the wavelength of the incident laser as 780 nm, 3 μ m, and 10 μ m, respectively. The vertical axis is the a value of the incident laser, with the bottom edge of a snapshot aligned along the vertical axis with the peak a of the incidentlaser in that simulation. All colormaps are logarithmic, and roughly chosen normalized to the critical density. Colorbars (top)and spatial scale bars (within each snapshot) are consistent for each column. The snapshots demonstrate two main trendsobserved across simulations performed, as discussed in Section IV, that the onset of plasma phenomenon is determined by the a value of the incident pulse (as observed scanning up the y-axis) while the size scale of these plasma phenomenon scales withthe wavelength (as observed scanning across the x-axis). FIG. 4. Panel (a) shows a 2D pseudo-color of ion charge density, i.e., Zn I where Z is the mean charge state of the plasma and n I is the ion density, in the sub-critical pre-plasma region for a 3 μ m wavelength simulation with a 6 . · W/cm laser pulse( a = 0 . L = 5.77 μ m). Panel (b) shows a central line out of the ioncharge density (red) and electron density (black), with the lineout taken from the highlighted red line in (a). The bottom row(panels (c) and (d)) are similar plots for a 10 μ m wavelength simulation with a 1 . · W/cm laser pulse. The displayedsnapshots shows the target state well after the laser reflection has left the simulation space. On panels (a) and (c), the chargedensity equivalent to critical electron density is highlighted with white contours and the quarter of critical electron densityis highlighted with an orange contour. In all panels, the spatial scale of λ/ Mid-IR lasers therefore significantly relax the spatial res-olution requirements of such an experiment compared tolasers in the near-IR.With this in mind we looked for ponderomotive steep-ening in our simulations and found that both the 3 μ mwavelength simulation with a = 0 . L = 5 . μ mand the 10 μ m wavelength simulation with a = 1 .
09 and L = 19 . μ m do exhibit a classic ponderomotive steepen-ing pattern with multiple peaks in both the ion and elec-tron density as shown in Fig. 4. Ponderomotive steepen-ing was noticed earlier in 800 nm wavelength simulationspresented in [1, c.f. Fig.12]. Ponderomotive steepen-ing was also present in simulations that we published in[2], although for brevity we did not highlight this result.Fig. 4 provides essentially the first compelling evidencethat this effect should persist in mid-IR experiments ofthis kind. IV. DISCUSSION
We present 30 high-resolution PIC simulations, whichcomprise a parameter scan over laser wavelengths, inten-sities, and target scale-lengths, designed to explore thephysics of intense, normally-incident near-IR (780 nmwavelength) and mid-IR (3 μ m and 10 μ m wavelength)laser plasma interactions in the creation of superpondero-motive elections. The simulations support three majorfindings. The first major conclusion is that the back-wards acceleration of elections in the sub-relativisiticregime is more efficient when the scale length is longerthan the wavelength, expressed by the condition that theratio of the scale length to the wavelength L/λ (cid:38)
1. Sec-ond, the onset of plasma phenomenon scales with the a value of the incident pulse, which of course takes con-tributions from both the intensity of the incident lightand its wavelength. Finally, the physical scale of plasmaphenomena scales with the wavelength, facilitating theexperimental observation of features such as ponderomo-tive steepening.The importance of the condition L/λ (cid:38) a values and scale lengths determinedby L = 1 . λ , shown on the right panel of Fig. 1 to sim-ulations with similar a values on the left panel with afixed scale length L = 1 . μ m. In the sub-relativisticregime ( a < L = 1 . λ produce ejected electronsthat happen to lie well above expectations from [15] andthat they also exceed that estimate by a greater amountthan the corresponding fixed L = 1 . μ m simulations(with similar a value lasers) exceed that estimate. More-over, from examining just the simulations with a fixed L = 1 . μ m, one finds that longer wavelength simulationsproduce less energetic electrons than shorter wavelengthsimulations. The longer wavelength cases represent val-ues of L/λ = 0 . , .
15 for λ =3 μ m,10 μ m, respectively.This further reinforces the idea that the L/λ ratio needsto be unity or greater for back-directed electron acceler-ation to be effective in this a regime.The wilks scaling estimate is linear in a in the regimewhere a > a in the non-relativisticcase ( a < L/λ = 1 .
92 appearsto have a linear power law that extends below the non-relativistic limit at a ∼ a ∼ a simulation did not haveaccelerated electrons that reached the edge of the simula-tion space by the end of the simulation, so it is doubtfulthat this power law extends to a values well below 10 − ,but this lower energy regime will be the focus of futurework.Regarding the ejected electron energy and angle spec-tra shown in Fig. 2, the analytic model actually cor-responds more closely to the 1.5 μ m scale length re-sults for the mid-IR simulations (bottom panel) than the L = 1 . λ results (upper panel) which are quite a bitmore energetic than expected from the model. A carefullook at the upper and lower panels of Fig. 2 plot showsthat there is a larger total number of ejected electrons aswell for the L = 1 . λ simulations. As explained in [2],this model is purely electromagnetic, without consider-ing plasma effects which become more important as theratio L/λ increases.The importance of the ratio
L/λ could be due to anumber of factors. It is well known that intense laserinteractions are highly sensitive to the assumed scalelength; from a physics perspective [c.f. 19] one expectsthat if
L/λ (cid:28)
1, then the laser will reflect off a sharpinterface (much like a mirror) and only accelerate elec-trons that reside on the surface of the target. If in- stead
L/λ (cid:38)
1, then the laser will interact with a moreextended region of near-critical plasma that provides amore suitable environment for accelerating large numbersof electrons in the back direction, as we consider here. Itshould also be emphasized that the “scaled” scale lengthtargets are significantly larger than the L = 1 . μ m tar-gets and this extended pre-plasma can decrease the elec-trostatic potential at the edge of the target where theelectrons are ejected. A large pre-plasma layer can alsoprovide a larger return current for escaping elections.These are possible explanation for both the increased en-ergy for the simulations where L = 1 . λ as well as forthe larger number of ejected electrons, as discussed earlierand shown in Fig. 2. Finally, the ratio L/λ is relevantespecially in the sub-relativistic regime ( a < a regimes. Theseeffects, as well as the confluence of the a value with theratio L/λ , will be investigated in future work.Finally, as discussed in the previous section and shownin Fig. 4, we anticipated the scaling of plasma featureswith the indicident laser wavelength, and observed pon-deromotive steepening in the longer wavelength IR sim-ulations with a ∼ μ m wavelength ultra-intense laser sys-tem recently purchased by AFRL would be for detectingthese density modulations when used with the existing100 Hz acquisition rate interferometric system [21] andcoupled to the existing liquid target setup there. V. CONCLUSIONS
In anticipation of future experiments utilizing ultra-intense, mid-infrared laser pulses and their interactionwith dense targets, we used LSP 2D(3v) simulationsto explore these interactions over a range of intensitiesand wavelengths. Similar to earlier investigations withnear-IR light [1, 2], we find that intense longer IR wave-length interactions also produce highly superponderomo-tive electrons. Moreover, the acceleration is much moreeffective when the pre-plasma scale length is in similarscale to the laser wavelength, or longer. In some casesthe typical ejected electron energies exceed ponderomo-tive expectations by orders of magnitude.The longer IR simulations also indicate that pondero-motive steepening should occur in experiments of thiskind when a ∼ λ/ ACKNOWLEDGMENTS
This research was sponsored by the Air Force Office ofScientific Research (AFOSR) through program managersDr. Enrique Parra and Dr. Jean-Luc Cambier. Theauthors acknowledge significant support from the De-partment of Defense High Performance Computing Mod-ernization Program (DOD HPCMP) Internship Programand the AFOSR summer faculty program. Supercom-puter time was used on the DOD HPC Armstrong andGarnet supercomputers. The authors would also like tothank The Ohio State Department of Physics Informa-tion Technology support, specifically, Keith A. Stewart. [1] C. Orban, J. T. Morison, E. D. Chowdhury, J. A. Nees,K. Frische, and W. M. Roquemore, Physics of Plasmas(2015).[2] G. K. Ngirmang, C. Orban, S. Feister, J. T. Morrison,K. D. Frische, E. A. Chowdhury, and W. M. Roquemore,Physics of Plasmas , 043111 (2016), 1510.05000.[3] Mid-ir lasers: Power and pulse capability rampup for mid-ir lasers , accessed: 2017-1-1, URL .[4]
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The following is an exhaustive list of the 2D(3 v ) PIC simulations presented in this paper.0 Wavelength( μ m) Intensity(W/cm ) a GaussianRadius( μ m) PulseFWHM (fs) LaserEnergy(J) Pre-PlasmaScale( μ m) TargetDensity(cm − ) SimulationTimestep(fs) SimulationSpatialResolution( μ m)10 1.64 · − · − · − · − · − · − · − · − · − · − · − · − · − · − · − · − · − · − · − · · · − · · · · · · · · v ) LSP PIC simulation performed sorted according to the value of a for each simulation. The different laser parameters were chosen to range from 0.1 mJ to 10 J in laser energy for each of thethree laser wavelengths, 780 nm, 3 μ m, and 10 μ m. A number of simulations were performed with a L = 1 . μ m scale lengthtarget. Other simulations were performed with L = 1 . λλ