Bubble acceleration of electrons with few-cycle laser pulses

We present three-dimensional particle-in-cell (3D-PIC) simulations of laser wake field acceleration (LWFA) of electrons in the bubble regime, using the code ILLUMINATION}. The study is motivated by the new Light Wave Synthesizer (LWS), currently under construction at Max-Planck-Institut für Quantenoptik (MPQ), which is expected to produce short (sub-10 fs), high power (multi-terawatt) laser pulses. They will be short enough for optimal drive of bubbles in dense gas jet targets at laser energies as low as 20 mJ and to generate efficiently very bright, quasi-monoenergetic, low-emittance bunches of 10–100 MeV electrons. It is shown how the results change when plasma density, laser amplitude and focusing are varied. The density and the spectral evolution of bubble evolution are presented for a reference case in much detail in terms of a movie. In particular, the formation of peaked spectra with an energy spread in the range of 10% are discussed qualitatively. Also plasma dynamics at the laser front and the rear vertex of the bubble as well as the periodic structures in the stem of accelerated electrons are addressed.


Introduction
Laser wake field acceleration (LWFA) is the most important concept of particle acceleration by laser pulse in dense plasma [1]. For ultra-short pulses interacting with gas jets, the laser pulse ionizes the gas and drives a plasma wave. This is illustrated in figure 1. At sufficiently high intensities, self-trapping of plasma electrons occurs and peaked spectra of accelerated electrons are found. This is the so-called bubble regime [2]. Here, we investigate threshold conditions for bubble formation, in particular in the context of 5 fs, multi-TW pulses. Since laser energy is the dominant parameter (which defines the laser cost) few cycle pulses are best adjusted to this acceleration regime. Corresponding lasers are now becoming available [3].
Bubble acceleration differs from traditional wake field acceleration in that one drives the plasma far beyond the wave-breaking limit, such that a single wake rather than a regular plasma wave train is formed. In this regime, large amounts of plasma electrons (in the order of nano-Coulombs) self-trap in the wake and are accelerated without interacting directly with the laser field. The bubble length is approximately one plasma wavelength λ p , and an important condition to generate electron spectra peaked in energy is that the laser pulse extends over λ p /2 or less. For longer pulses, the accelerating electrons interact directly with the laser pulse, and one obtains broad thermal-like spectral shapes, see e.g. [4].
Peaked electron spectra were detected in recent experiments [5]- [11]. For example, the electron beams obtained by Faure et al [7] with 1 J, 30 fs laser pulses, consisted of low-emittance, 0.5 nC, 170 MeV electron bunches, accelerated over distances of just 3 mm with 15% laser-toelectron conversion efficiency. Still these results came as a surprise, because in all the experiments performed so far the laser pulses were longer than λ p /2. The reason why the bubble regime could be accessed nevertheless is that nonlinear laser plasma interaction tends to shape the incident laser pulses until they fit into the first wake and can produce mono-energetic pulses. Apparently, the bubble regime has the properties of an attractor. This is concluded from simulations presented in [7,10]. The theory of bubble formation, its stability, and its scaling properties has been developed by Gordienko and Pukhov [12]. It applies to the ultra-relativistic regime, a 0 1. The present study primarily considers lower a 0 values, relevant in near-term experiments.
The advent of high-power few-cycle laser pulses makes it possible to drive bubble wake fields optimally adjusted to dense gas jet plasmas and to achieve stable acceleration. This is discussed in the present paper. After defining laser and plasma parameters in section 2, the threshold for bubble formation is explored in section 3. Then a reference case is discussed in much detail in sections 4-6, based on the accompanying movie. Laser focus and intensity of the reference case are chosen in view of the upcoming experiments. Variations to higher intensities and smaller focal radii are addressed in section 8.

Three-dimensional particle-in-cell (3D-PIC) simulations with the code ILLUMINATION
The simulations are performed with the 3D-PIC code ILLUMINATION. The typical simulation box is 200 × 200 × 1000 cells and has a volume of 38 × 38 × 50 µm 3 . Each cell contains one macro-particle corresponding to 40 million particles. The simulated box co-moves with the laser at the velocity of light to simulate only the laser plasma interaction region. The plasma is assumed to be fully ionized with immobile ions as a neutralizing background. We have checked that allowing for ion motion gives almost the same results.
Throughout the paper the laser pulse is linear polarized in e x , propagate in e z and has a centre wavelength of λ 0 = 800 nm. The normalized vector potential of the incident laser pulse, a = eA/mc 2 , depends on radius and time in the form a(r, t) = a 0 exp(−(r/w 0 ) 2 ) exp(−2 ln 2 (t/τ 0 ) 2 ). (1) Here, e and m are charge and rest mass of electrons, and c is the velocity of light. For the reference pulse, we choose a focal radius of w 0 = 5 µm and a full-width-at-half-maximum duration of the intensity of τ 0 = 5 fs. This implies a Rayleigh length of z R = πw 2 0 /λ 0 = 98 µm. Further the intensity, the peak power, and the total pulse energy can be calculated as I = 2.2 × 10 18 W cm −2 × a 2 0 , P = πw 2 0 I/2 and W = 1.67Iτ 0 w 2 0 , respectively. The plasma electron density is chosen as n 0 = 1.745 × 10 19 cm −3 , corresponding to 1% of the critical density for λ 0 = 800 nm light, n cr /n 0 = (ω 0 /ω p ) 2 = 100. This determines the group velocity of the laser pulse inside the plasma, which is equal to the phase velocity of the wake field, β ph = 1 − (ω p /ω 0 ) 2 ≈ 0.995, and leads to the wake field γ-factor of Here, ω 0 = 2πc/λ 0 and ω p = 4πe 2 n 0 /m are laser and plasma frequency, respectively, and the plasma wavelength is λ p = 2πc/ω p = 8 µm. At high laser intensities, these parameters will change due to relativistic increase of electron mass. Approximate expressions are obtained from ω rel p ≈ ω p / √ γ av , where the γ-factor averaged over the electron distribution is defined by 1/γ av = 1/γ . For linear polarization, it depends on the laser amplitude approximately in the form γ av ≈ 1 + a 2 0 /2. Figure 1 shows simulations for two different laser amplitudes. The upper frames correspond to a 0 = 3 (≈41 mJ) and the lower ones to a 0 = 5 (≈115 mJ). Both pulses are shorter than the length of the wake field and therefore fulfil the first condition for bubble acceleration. However, the second one of achieving self-trapping of ambient plasma electrons in the first wake is satisfied for a 0 = 5 only. This is evident from the density plots on the left side of figure 1.

Bubble formation
In the first case, a highly nonlinear, but still regular plasma wave is seen in the density plot, trailing the short laser pulse (not shown explicitly). The high-density crests of the wave are curved due to relativistic and other nonlinear effects, but still only a relatively small number of electrons (in the order of 10 7 MeV −1 ) is trapped in the low-density wave buckets and accelerated to energies of 40-80 MeV. This changes drastically when raising a 0 from 3 to 5. Then the pattern of the regular wave train disappears, and a high-density stem of accelerated electrons shows up in the first wake bucket behind the laser pulse. One should notice that the regular wave crests in figure 1(a) are separated by the plasma wavelength of about 8 µm. This changes due to charge loading in figure 1(b). The length of the first wake, the bubble, increases, while the downstream wave pattern is increasingly broken and washed out.
After a certain propagation distance, a peak appears in the energy spectrum of the trapped electrons. In figure 1(b), it is seen at 70 ± 5 MeV after 200 µm of propagation. The total number and energy of accelerated electrons contained in this peak (>60 MeV) is plotted in figure 2 as a function of time together with the laser energy. One observes that the peak starts to build up after 300 fs (z = 100 µm) and to saturate after 600 fs (z = 200 µm), when the number of electrons above 60 MeV approaches 2 × 10 9 . At this time, the total energy in the electron peak is roughly 30 mJ, which amounts to 25% of the incident laser energy. In the subsequent evolution, number and energy of peak electrons still increase somewhat, until the peak decays after 1200 fs (z = 400 µm).

First stages of bubble formation
Details of bubble evolution are presented in the following by means of the movie. At time t = 0 fs, the centre of the laser pulse is located at z = −10 µm, moving to the right. After 30 fs, the pulse starts to touch the plasma density profile, which rises in a linear ramp (z = 0-10 µm) to the value of 1.745 × 10 19 cm −3 . At 90 fs, the formation of the first wake bucket is almost complete. The imprint of laser action is seen on the right side in the form of vertical high-density layers, separated by half the laser wavelength. These layers correspond to electrons trapped between the peaks of laser intensity. They are expelled mainly in the transverse direction. After expulsion they swing back to the laser axis. In the ramp region, this radially converging flow has already formed a high density spot on the axis. After 100 fs, bubble formation is complete. At this time, the longitudinal bubble extension is λ bub ≈ 11 µm, somewhat longer than the plasma wavelength λ p = c/ω p = 8 µm. This is expected due to the relativistic reduction of ω p . From λ bub /λ p ≈ √ γ av , we obtain γ av ≈ 2; here 1/γ av = 1/γ defines the average taken over the laser-excited electrons involved in the bubble dynamics. At 115 fs, the laser pulse approaches the right border of the simulation box, and now cells are added at this border and dropped at the left border to follow pulse propagation through undisturbed plasma.
Between 100 and 150 fs, a rather regular bubble is seen. It is practically void of electrons, while ions are nearly unperturbed and create a high, mainly longitudinal E-field. The most conspicuous feature is observed at the rear vertex, where electrons accumulate and create a region of high space charge. A portion of the electrons coming in radially is scattered into the bubble and starts to be accelerated in the wake field. During the time period of 150 to 350 fs, the accelerated electrons generate a plateau-like spectrum between 10 and 50 MeV and form a narrow stem along the bubble axis.
The distance between the base of the stem and the peak of the laser pulse is about 11 µm at 100 fs and lengthens to 14 µm at 150 fs. This gradual lengthening is attributed to the effect of beam loading. At this time, the number of trapped electrons has reached about 2 × 10 9 . Electron trapping then drops sharply. Beam loading also changes the shape of the bubble near the vertex from spherical at 100 fs to conical at 150 fs. Details of vertex and stem structure will be discussed further below.

Spectral bunching
An important feature of bubble acceleration is that it leads to relatively sharp peaks in the electron energy spectra during later phases of spectral evolution. In the reference case presented in the movie, this happens during 350 and 700 fs. At 350 fs, the plateau-like spectrum is divided into a front sector (40-60 MeV), which mainly contains electrons trapped close to the plasma surface in the initial phase of bubble formation, and a second sector (10-40 MeV), from which the spectral peak develops over the next 200 fs. At 700 fs, a sharp peak at 70 ± 5 MeV has formed. It sits on top of a broader spectrum extending to somewhat above 100 MeV. This spectral pattern stays almost invariant until 800 fs. It then broadens and decays, and also the bubble as a whole starts to degrade. The main reason for this is depletion of laser energy and that the laser pulse has passed the focal volume (z = 0-100 µm) and widens again.
In order to better understand the reason for the spectral bunching, we show in figure 3 the trajectories of all electrons that reach energies between 60 and 80 MeV after 700 fs. These are the electrons forming the sharp peak in figure 1. We have plotted energy versus spatial coordinates x (laser-polarization direction) and z (laser-propagation direction). It is seen that all the electrons producing the quasi-mono-energetic peak originate from the first 80 µm of the plasma volume. They were trapped in the time interval of 0-350 fs. They are then accelerated over a distance of 100-120 µm and reach peak energies of 60-100 MeV at times of 700-800 fs (see movie). The acceleration distance is considerably smaller than the dephasing length L d = λ d γ 2 ph ≈ 800 µm; rather than by dephasing it is determined by depletion of laser energy. According to the general analysis of Gordienko and Pukhov [12], this is typical for the bubble regime.
Still we need to understand what determines the formation of the spectral peak at later times. The most immediate explanation is that the electrons in the peak have passed approximately the same electric potential. This is indeed expected for the group of electrons that is trapped after 100 fs, when the bubble has formed and is propagating in a stationary fashion. Trapping stops at 350 fs, and the acceleration for each group of trapped electrons takes about 300-350 fs. Therefore this regular phase of acceleration terminates at 700 fs. The spread of the energy peak may be explained by the fact that the laser energy already decays during this time (to 50% at 700 fs, see figure 2), and the bubble acceleration field decreases correspondingly.
The insert in figure 3 provides some more details about the energy bunching in the present simulation. It shows the trajectories projected into the x = 0 plane and plotted in terms of number density in grey-scale. Careful inspection indicates that electrons trapped late experience  figure 2). The perspective view shows particle energy versus x and z (laboratory) coordinates. The insert shows the main plot projected into the E, z plane; instead of single trajectories, electron density (trajectory number per z interval) is given in grey scale. Also the relative energy spread is given as the solid curve. a somewhat higher acceleration field than those trapped early. This adds to the energy focusing in the region around z = 200 µm. The relative energy spread is also given in the insert. It falls to 10-15% in this region. For reducing the energy spread, one would have to reduce the time interval in which electrons are injected into the bubble. The injection problem has been discussed in more detail by van der Wiel et al [13]. We also refer to Bulanov et al [14] for an approximate analytical description of the spectral evolution.
For making use of the mono-energetic bunch in the present simulation, one would have to extract it from the plasma in time. For this it is best to cut the plasma layer at 200-250 µm and to release the bunch into vacuum at this distance. New methods of how to produce such ultra-thin gas jet layers are currently under investigation [15]. For thicker gas jets, one should shift the laser focus to the rear side of the jet. We have checked that laser pulse propagation through plasma in front of the focus does not significantly change the results seen in the movie. On the other hand, the accelerated electron bunch degrades rapidly when propagating inside the plasma behind the focal region. This is seen in the movie. When the laser pulse loses its function as bubble driver, the electron bunch takes over the role of the driving pulse, and a secondary electron-driven cavity is formed. This is observed in the later stages of the movie between 800 and 1300 fs. A recent discussion on driving wake fields with electron pulses was given by Joshi and Mori [16].

Details of laser electron expulsion and electron trapping
Here, we discuss some finer details observed in the movie. The laser action on the plasma is often viewed as a snowplough, piling up electrons at the front which are then pushed in transverse direction by the ponderomotive force. Here, we observe that the electrons are first trapped in the ponderomotive buckets of the laser wave, forming electron layers separated by half the laser wavelength. Due to the high electron density, the wavelength appears to be about 10-20% longer than λ 0 /2. The electrons are then radially expelled from these layers. Due to the linear polarization in the x, z plane, the distribution of the expelled electrons shows some azimuthal variations, but on the average they form the surface of an axially symmetric bubble, dragged by the laser pulse to the right.
The details of the 3D electron trajectories are quite complex and cannot be followed in the x, z cut of the movie. Nevertheless, one can observe that the electron flow along the bubble surface is not completely uniform, but carries the imprint of the spatially structured expulsion on the laser side. The electrons coming in radially at the vertex and being trapped and accelerated in the stem appear in bunches forming the oscillatory structure in the stem. The period of this structure is 0.45-0.50 µm, close to the period observed on the laser side. Direct laser action on the stem appears to be too weak to explain the structure.

Wave breaking versus self-trapping
A characteristic feature of the bubble regime is the very efficient self-trapping of background electrons. This is often referred to as wave breaking, but it is important to understand that self-trapping occurring in the 3D bubble regime is quite different from the wave breaking of 1D plasma waves [17,18]. In 1D geometry, breaking of laser-driven plasma waves sets in for a 0 > 2(ω 0 /ω p ) 2/3 (see Bulanov [14], here given for linear polarization).
For the present value of ω 0 /ω p = 10, this condition is neither met by a 0 = 3 nor a 0 = 5. Nevertheless, very efficient electron trapping in the wake is observed for a 0 = 5. This is attributed to the fact that electron motion close to the vertex is transverse rather than longitudinal. The curved plasma wave fronts lead to transverse wave breaking [18] occurring for reduced threshold values; it also generates the swallow-tail density structures behind the vertex, clearly seen in figure 1 and the movie.   figure 1, the corresponding propagation distances mark the acceleration length where the peak in the electron spectrum reaches the highest energy.
For fully developed bubbles at even higher laser intensities, these structures also disappear, the downstream electron flow becomes completely turbulent, and the electron distribution function close to the vertex is thermal-like, different from the multi-fluid flow characteristic for wave breaking. Strictly speaking, there is no wave breaking at all during particle trapping. This was clearly pointed out by Gordienko and Pukhov [12]. Wave breaking in the sense of multi-velocity flow only occurs at later times during the decay of the bubble. The scattering of radially inflowing electrons by the vertex potential, either downstream into the breakwaters or upstream to feed the stem, was described by Kostyukov et al [19].

Changing laser amplitude and focus
Here we add results on electron spectra when changing a 0 between 3 and 30, keeping the parameters w 0 = 5 µm, τ = 5 fs and n 0 = 1.745 × 10 19 cm −3 fixed. The spectra obtained for a 0 = 3, 5, 10, 30, are plotted in figure 4. For a 0 = 5, 10, 30, they have similar shapes. The spectra are taken at distances l acc , at which the energy E peak of the peak in the spectrum reaches the highest value. Propagation beyond this acceleration length leads to rapid degradation of the peak.
The key parameters from this set of simulations are summarized in table 1. The acceleration length and the peak electric field scale approximately like l acc ∝ √ a 0 and E z ∝ a 0.25 0 , and the peak energy follows E peak ≈ 0.3E z l acc . We have then also varied the laser spot radius w 0 . For the bubble radius we find R b ≈ w 0 a 0.25 0 and for the acceleration length l acc (µm) ≈ 40 √ a 0 w 0 (µm). Table 1. Key parameters for different a 0 as presented in figure 4. E z is the peak of the acceleration field at the bubble boundary, N e,peak is the number of electrons forming the peak after l acc , and W peak is the spectral energy of the peak. The numbers in parentheses are obtained from self-similarity theory Combining results and making use of E z ∝ n 0 R b , we obtain for the peak energy as a function of the external parameters, plasma density n 0 , laser amplitude a 0 , and focus radius w 0 , This formula is in reasonable agreement with the simulation results. Here, one should keep in mind, however, that w 0 and n 0 cannot be varied independently over a larger range. In order to stay in the stable bubble regime, one has to match the plasma wavelength λ p roughly with bubble diameter: Always assuming that the laser pulse is shorter than λ p , the bubble diameter can be slightly tuned by laser intensity, once the plasma density is chosen, but has to stay in the range of λ p . Expressing it in more experimental terms, for a given laser pulse, there exists a small density window for bubble acceleration. It is interesting to compare the present results with the scaling relations derived by Gordienko and Pukhov [12]. Their central observation is that, in the ultra-relativistic regime (a 0 1), laser plasma dynamics evolves in a self-similar way when keeping S = n 0 /(a 0 n cr ) = (ω p /ω 0 ) 2 /a 0 constant. This means one compares situations of same relativistic plasma frequency. This relativistic similarity is well confirmed by simulations [12]. When changing a 0 at constant n 0 , also S ∝ a −1 0 changes. For this case, the following scaling relations are given in [12]: transverse bubble radius R ∝ S −1/2 ∝ a 1/2 0 , bubble acceleration field E z ∝ n 0 R ∝ a 1/2 0 , acceleration length l acc ∝ S −3/2 (cτ 0 /R) ∝ a 0 , peak electron energy E peak ∝ E z l acc ∝ a 3/2 0 , peak electron number N e,peak ∝ n 0 R 3 ∝ a 3/2 0 . In table 1, numbers obtained from these relations are given in brackets. In view of the asymptotic character of these scalings a 0 1, we have normalized to values at a 0 = 30. While results for E z and N e,peak compare reasonably with the present simulations, acceleration length l acc and peak energy E peak scale much less with a 0 in the present simulations.

Lower limit for bubble acceleration for 5 fs pulses
Finally we address the question of to what extent bubble dimensions and required energy can be even further reduced, keeping the amplitude of the 5 fs laser pulse at the reference value Results obtained with these parameters are shown in figure 5 as a snapshot after ≈50 µm propagation. They should be compared with the reference case in figure 1. Again we observe a bubble with a stem of accelerating electrons. The peak in the spectrum is located at 20 MeV and contains N peak = 3.5 × 10 8 electrons and a total energy of W peak = 1 mJ. There are differences, however, when following this case for longer times. Although the bubble remains stable over ≈100 µm, the spectrum rapidly changes, e.g. the peak observed at 20 MeV in figure 5 splits into two broader peaks at 25 and 40 MeV after 70 µm; the spectral range between 10 and 60 MeV then contains 6 × 10 8 electrons and a total energy of 3 mJ. After ≈100 µm, the bubble starts to decay due to laser depletion. The estimations of acceleration length and peak energy peak (equation (2)) apply to this phase. They give l acc ≈ 125 µm and E peak = 42 MeV which are in good agreement with the simulation.
This parameter set gives more or less the lower limit for bubble acceleration in terms of laser pulse energy. The bubble dimensions are close to the laser wavelength and cannot be further decreased. Decreasing the laser intensity would quickly lead to the breakdown of the bubble regime.

Conclusions and outlook
We have investigated bubble acceleration with few-cycle TW-to-PW laser pulses in view of experiments coming up in the near future. Based on 3D-PIC simulations, we have emphasized issues of immediate interest to experimentalists, such as the minimum requirements in laser intensity and energy for reaching the bubble regime and peaked electron spectra with 5 fs pulses. Presenting a movie of a reference case, we also provide much more detailed information than has been published so far. This includes the late phases of laser depletion and bubble decay. A critical issue is to extract the pulse, when it has an optimal spectrum, and this requires ultra-thin gas jet layers, at least for the few-cycle laser pulses considered here. These pulses are optimally adjusted to drive wake field acceleration in the bubble regime at pulse energies in the milli-joule range which allows for table-top devices and high repetition rate. The fully self-organizing acceleration regime is remarkably stable, energy-efficient, and simple to install. It is therefore very attractive for many applications.
The simulations presented here show that peaked energy spectra can be obtained with an energy spread of about 10%. The energy width is related to the characteristics of electron selftrapping and laser energy depletion in the bubble regime. Of course, for certain applications, one would like to have much sharper energy spectra. For this goal, one has to control the process of feeding electrons into the wake field. One way would be to work at lower wake amplitudes, below the self-trapping limit, and to use either external injection or interaction with a second laser pulse [13,20,21]. It has still to be demonstrated that sufficient amounts of electrons in controlled space-time volumes can be generated in this way to achieve narrow energy spectra.
In principle, the bubble regime can be scaled to high particle energies, far above the GeV level [12]. This would imply, however, larger bubble volumes and, correspondingly, larger laser drive energies. Given the highly nonlinear character of this regime, in which all parameters are tightly interlinked with each other, bubble acceleration appears somewhat inflexible when trying to optimize particular parameters, in particular to reach GeV energies. A way to overcome these limitations may be staged acceleration, taking the bubble regime as an electron source producing nano-Coulombs of 10-1000 MeV electron bunches efficiently, and then to use either preformed laser plasma channels [22] to increase the acceleration length or vacuum laser acceleration schemes [23]. The vision is to create multi-GeV and TeV beams in this way.