A Scalable, High-Efficiency, Low-Energy-Spread Laser Wakefield Accelerator Using a Tri-Plateau Plasma Channel

The emergence of multi-petawatt laser facilities is expected to push forward the maximum energy gain that can be achieved in a single stage of a laser wakefield acceleration (LWFA) to tens of giga-electron volts, which begs the question—is it likely to impact particle physics by providing a truly compact particle collider? Colliders have very stringent requirements on beam energy, acceleration efficiency, and beam quality. In this article, we propose an LWFA scheme that can for the first time simultaneously achieve hitherto unrealized acceleration efficiency from the laser to the electron beam of >20% and a sub-1% energy spread using a stepwise plasma structure and a nonlinearly chirped laser pulse. Three-dimensional high-fidelity simulations show that the nonlinear chirp can effectively mitigate the laser waveform distortion and lengthen the acceleration distance. This, combined with an interstage rephasing process in the stepwise plasma, can triple the beam energy gain compared to that in a uniform plasma for a fixed laser energy, thereby dramatically increasing the efficiency. A dynamic beam loading effect can almost perfectly cancel the energy chirp that arises during the acceleration, leading to the sub-percent energy spread. This scheme is highly scalable and can be applied to petawatt LWFA scenarios. Scaling laws are obtained, which suggest that electron beams with parameters relevant for a Higgs factory could be reached with the proposed high-efficiency, low-energy-spread scheme.

but not sufficient requirement to build a future collider based on this paradigm changing technology.An accelerator based on such limited energy gain in a single stage necessitates multi-stage operation to reach energies of interest to particle physics and energy recovery of the unspent laser beam.There are of course other challenges such as generating spin polarized beams and an overall rep-rate needed to achieve the necessary luminosity in the collision (interaction) volume.Furthermore, similar considerations need to be addressed for the positron arm of a future e -e+ collider.While a comprehensive design of a LWFA-based e-e+ collider is beyond the scope of any single article, it is possible to address issues for the electron arm.In this article -using particle-in-cell (PIC) code simulations and scaling laws -we show that it is possible to design a single 100 GeV high-efficiency LWFA stage that can provide collider-like beam quality using PW-class lasers that are rapidly coming online around the world.The resulting electron beam would already meet the requirements for the electron arm of a Higgs factory in terms of beam quality, average energy and charge 11,12 .
In the beam-driven plasma-based acceleration concept, often referred to as plasma wakefield acceleration (PWFA), energy transfer efficiencies from the drive beam to the trailing beam of 60% have been demonstrated in simulations 13,14 and maximum efficiencies of 30% have been inferred in experiment 8 .However, for a LWFA stage, simulations and experiments have not demonstrated an energy transfer efficiency from the laser pulse to the accelerated electron beam beyond a few percent.Such low acceleration efficiencies of LWFAs primarily stem from the laser pulse shape distortion that occurs because of group velocity dispersion that downshifts frequency of the photons (photon deceleration) and dephasing between the accelerating beam and the continuously temporally evolving laser pulse.Specifically, the non-uniform density and relativistic mass of background electrons in a plasma wake make the photon deceleration rate vary along the laser, leading to longitudinal and transverse pulse distortion before significant pump depletion can occur.Second, the group velocity of the evolving laser and the resulting wake's phase velocity are lower than the speed of relativistic electron bunches.Therefore, there is significant phase slippage between accelerated particles and the wake well before pump depletion of the driver occurs.Moreover, the accelerated beam often has a large correlated energy spread (energy chirp) due to the non-negligible bunch length relative to the (mm-scale) accelerating field structure.Methods have been proposed to reduce the energy spread [15][16][17][18][19][20] , mitigate dephasing [21][22][23][24][25][26][27][28][29] and pulse depletion [30][31][32] , but a comprehensive proposal is still lacking to simultaneously achieve high overall efficiency, low energy spread and emittance preservation of a LWFA stage.
In this article, we propose a novel scheme that combines a nonlinearly chirped driving laser pulse with a tri-plateau density structure.By using a nonlinearly chirped laser pulse, the axial laser distortion can be significantly mitigated over much longer propagation distance 33,34 , leading to a more stable LWFA stage with high laser to wake conversion efficiency.The tri-plateau density structure has three axially uniform sections of progressively higher density connected by upramps, combining with radially parabolic plasma channels (see Fig. 1).The use of such a properly designed plasma structure can significantly mitigate the dephasing effect and thereby increase the energy gain and overall efficiency.These two ideas together lead to energy gains three times larger compared with a single uniform density LWFA.At the same time, we discovered that a dynamic beam loading (DBL) effect 35 , where the loaded wake initially induces a correlated energy spread that can be then naturally removed as the beam loading effect changes during the wake propagation, leads to accelerated beams with extremely low energy spreads (<1%).The cumulative DBL effect is fully controllable and the optimal output energy spread can be achieved by tuning the density and length of each density plateau.Furthermore, simulations have confirmed that the proposed scheme is highly scalable.For tens to hundreds of TW drive laser scenarios, the scheme can output 0.6~10 GeV electron beams with equally high energy efficiency and low energy spread.Scaling the scheme to the PW-laser-driven regime is expected to result in a remarkable 100-GeV energy gain of a 2nC electron beam.
We rely on high fidelity simulations using OSIRIS to illustrate this high laser coupling efficiency, low energy spread concept.In [Fig .1(a)], a 125pC witness electron beam is accelerated through a tri-plateau density profile to ~600MeV in a wake driven by a laser pulse with only 340 mJ energy.The resulting energy transfer efficiency from the laser to the electron beam is >21%.
High efficiency is achieved for relatively low laser normalized vector potential  0 ≡  0    2 in the blowout regime 36,37 .According to the scaling law 38 , the beam energy gain is   ∝  0 5/2 and the laser pulse energy is   ∝  0 7/2 (assuming a round laser pulse  ∼  0 ), and the energy transfer efficiency Here, we chose  0 = 1.67 to ensure the excitation of plasma wake in the blowout regime and a relatively high energy transfer efficiency.The resulting laser power is not sufficient for self-guiding 39 , so a tri-plateau plasma structure with a parabolic transverse density of the form   () =  0 (1 + 2 ) is used where Δ = 0.5 such that the laser with a spot size  0 is free of diffraction throughout the 6-mm-long plasma.Because the density of each plateau increases in discrete steps [Fig.1(b)], the size of the wake cavity shrinks as it transits from one stage to the next stage [Fig.1(a)].The on-axis densities,  0 , of the three plateaus are 2 × 10 18 cm −3 , 3.13 × 10 18 cm −3 and 8.3 × 10 18 cm −3 , and the corresponding lengths of plateau are 1.73 mm, 2.67 mm, 0.15 mm, respectively.Adjacent plateaus are connected by a linear up-ramp of 0.5 mm, and the slopes at both the entrance and the exit for the first and last plateau are 0.25 mm long.Most of the energy gain occurs in the uniform density plateau regions -in this sense the exact lengths of the density upramps between the stages is not critical.The laser pulse has a Gaussian transverse profile, focused to a beam waist  0 = 10.7 μm at the midpoint of the density upramp of the first plateau, and a sin 2 -shaped temporal envelope of the field with a 21.5 fs FWHM pulse length.The laser pulse is nonlinearly chirped to partially compensate the nonlinear dispersion (see Methods).
For ease in interpretation a bi-flattop electron beam with 3.76 μm beam length is initialized behind the drive laser with an initial energy of 51.1 MeV ( = 100) and 0.1% energy spread.The resulting beam energy is insensitive to the specific longitudinal profile for a Gaussian-like laser pulse, and >21% energy efficiency with <1% energy spread can be achieved.The initial emittance is 0.1 mm•mrad and the beam is focused to the midpoint of the first upramp with a spot size of 0.5 μm.
Compared with the uniform [the yellow line in Fig. 1(b)] and bi-plateau [the red lines in Fig. 1(b)] channel structure, the final energy gain of the tri-plateau structure is almost tripled, significantly increasing the acceleration efficiency.The length of the uniform plasma is 8 mm and the lengths of the two plateaus for the bi-plateau are 1.73 mm and 2.92 mm respectively.Numerical parameters are given in the methods sections.Because ultra-relativistic electrons essentially move at the speed of light, the accelerated ebeam will gradually outrun the accelerating phase of the plasma wake excited by lasers (dephasing).This causes the acceleration gradient felt by the electron beam to decrease or even become negative as the laser propagates in a uniform plasma.Since the dephasing length is typically much shorter than the pump depletion length, the acceleration gradient will be significantly reduced far before the laser energy is depleted, thus greatly limiting the acceleration efficiency.Due to this dephasing process, the spatial gradient of the energy gain in the laboratory frame gradually reduces and appears parabolic as depicted by the yellow dashed in Fig. 1(c).On the other hand, during the acceleration process, the laser loses energy in a nearly linear manner as shown by the yellow dashed in Fig. 1(e) and eventually only 1/3 of the initial energy is transferred to the plasma wake.In our tri-plateau scheme, a rephasing process occurs both in the density upramp and the higher density plateau that follows it.In the rephasing process, the wavelength of the plasma wake shrinks due to the increase of the plasma density and e-beam finds itself once again close to the maximum acceleration phase, significantly improving the acceleration efficiency.Since the acceleration gradient also increases with ~ square root of the density, the slope of the gain curve (blue line) increases sharply from one plateau to the next.As shown in Fig. 1(c).The energy gain of each stage is insensitive to the plasma density albeit the acceleration distance (being proportional to the dephasing length) becomes shorter as the density increases.Using the scaling laws in the blowout regime [38], for the accelerating field,   ∝   1/2  0 1/2 , and the dephasing length,   ∝   −1 , the energy gain scales as Δ ∝   −1/2  0 1/2 .Due to the focusing and steepening of the laser pulse,  0 increases so that Δ does not change much as   increases.As a result, the energy gain is nearly tripled compared with that of the uniform plasma case.
As shown by the blue solid line in Fig. 1(c), the energy gain curve now increases almost linearly in each stage and hence a high beam loading efficiency is maintained continuously for the 125pC beam charge.The beam loading efficiency (wake-to-beam efficiency), defined as the ratio of energy extracted by the particles from the wake to the energy cost of the laser, is calculated by 〈∆〉/(  −   ) in each plateau, where  is the electron number of the beam, 〈∆〉 is the averaged energy gain of the electron beam in this plateau and   /   is the laser energy entering/exiting this plateau.In the first two plateaus, the wake-to-beam efficiency is 49.6% and then drops to 22.3% in the third plateau.More laser energy is utilized in the tri-plateau plasma; 67.5% of the laser energy is consumed in the proposed scheme while only 34.0% is consumed in a single uniform plasma channel.The total efficiency is the product of how much of the laser energy is transferred to the wake (roughly the laser energy utilization percentage) times the beam loading efficiency, which is > 30% in this case.As a result, a total efficiency of > 21% is obtained which is the highest value achieved to date for a fully self-consistent LWFA concept.We note that in this proposed scheme the laser energy is efficiently utilized because the pump laser pulse has a nonlinear chirp with frequency increasing from front to middle and then decreasing from the middle to the back, allowing the pump waveform to self-compensate for photon deceleration due to wake formation and maintain its shape.The nonlinear frequency chirp is of the form 2 , where () and  0 are the local and central wavenumber and  ≡  −  .The chirp coefficients c0, c1, c2 are 1.2, 1.79 × 10 −3  0 and −9.7 × 10 −5  0 2 respectively.Such a nonlinear chirp can either be realized by nonlinear cross-phase modulation 40 or nonlinear pulse compression by customized gratings 41,42 .In the blowout regime 36,37 , the driving laser pulse with the optimal pulse length resides at the first half of the ion cavity where the photon deceleration rate  ℎ < 0 (see Methods).The occurrence and effects on pulse distortion of photon deceleration/pump depletion are illustrated in Fig. 2. The left column frames are taken from a simulation with a transform limited pulse while the right column is for a laser with a nonlinear frequency chirp.For these examples a laser with identical parameters as above is sent through a single plateau with a  0 = 3 × 10 18 cm −3 and Δ = 0.5 .For each case, the axial lineout and Such a pulse can propagate much further than the pump depletion limit of a transform-limited pulse of similar pulse width and peak intensity.The importance of this chirp is seen in the grey line of Fig 1(c) where it can be seen that no energy gain arises in the third stage without the chirp (transform limited pulse).The reason for saturation of the energy gain can be seen in Fig. 1(d) to be that the pulse length increases rapidly for the laser pulse that has no chirp in the final stage.In addition to the high efficiency, the beam energy spread of <1% can be obtained owing to the DBL.The mechanism of how the DBL effect maintains a low energy spread will be elaborated on in the later section.The use of stepwise plasma density plateaus connected with density up-ramps not only significantly increases the beam energy gain, but also is capable of accelerating a large amount of beam charge with an extremely small energy spread through a dynamic beam loading (DBL) effect.The beam loading effect can be understood by tracking the accelerating field felt by the beam 13 .In Figs.3(a 3(c) and (d)] such that the energy chirp caused by the first plateau is almost perfectly removed.The transition from an over-to under-loaded wake happens in the second plateau [between Figs.3(b) and (c)].
This mechanism can be qualitatively described by the nonlinear theory of the blowout regime 36 .We find that the gradient of   (see Methods) can be expressed as where is the normalized linear charge density of the beam,   ≡      is the characteristic electric field in the plasma, and   is the blowout radius.The linear charge density  which is independent of plasma density can be assumed to not depend on the propagation distance.However, the value of   at the location of the beam changes (increases) from dephasing within an individual plateau, and   increases when crossing into the next plateau.(3) with respect to the propagation distance, and eventually vanishes at the optimal distance [Fig.3(i)].As seen Figs 3(e)-(i), the beam energy chirp increases first and then decreases, and the residual energy chirp is almost fully compensated, which agrees with the analysis above.The final relative energy spread reaches as low as 0.83% (FWHM) while a considerable beam charge of 125pC is accelerated.
Assuming a fixed  0 and  0 , and that the laser is guided by a channel, the key physics can be viewed approximately self-similar, and scaling laws can be straightforwardly obtained (see Methods).When the plasma density changes from   to ′  , the characteristic lengths of the key physics can be scaled by a factor  = �  /  ′ .If we proportionally scale the focal waist and pulse length of the laser through  0 ′ =  •  0 and ′ =  • , the acceleration gradient, maximum beam energy gain and the number of electrons that can be accelerated scale as,   ′ =  −1   , 〈∆〉′ =  2 • 〈∆〉 and  ′ =  •  respectively according to the scaling law of LWFAs [38] (see the Methods for the derivation).The energy transfer efficiency ′ is an invariant under this scaling since We carried out simulations with  = 2 and  = 4 to validate the scaling law, and the results are summarized in table 1.In these simulations, the initial parameters of laser, plasma and electron beam are scaled from case 1 and the results (energy gain, efficiency) show excellent agreement with what the scaling law predicts as shown in Fig. 4(b).In each case the final energy spread remains below 1%.Case 3 corresponds to a 500pC electron beam being accelerated to ~10 GeV by a 22-J laser pulse which is within the capability of modern PW lasers.The calculated energy transfer efficiency greatly exceeds that obtained in previous LWFA experiments and that described in previous simulations.Experimental verification of case 3 can be attempted in the near term.Furthermore, we can extrapolate the proposed scheme to  = 16 in column 4. In this case, an LWFA driven by a 1.4kJ, 340fs laser pulse would generate a 2nC electron beam with ~150GeV energy with an average acceleration gradient of 6 GeV/m.Laser with such energy and beams with such charge are already available.The proposed scheme could thus provide the electron arm of a Higgs factory, with ~20% efficiency and sub-1% energy spread.For comparison, a scaled case (Case 5) using a uniform plasma channel is also listed in Table 2.In this case, only about one third of the energy gain in Case 4 is achieved, and the energy spread grows up to 15 %.Table 2. Parameter designs of Case 1 to Case 4 for  0 = 2 and  0 = 800 nm according to the scaling law, in which case 1~3 are PIC simulation results and case 4 is an extrapolation to obtain the 145.9 GeV electron beam.As a comparison, Case 5 uses the same laser pulse as Case 4 but in a matched uniform plasma channel, which is scaled from the case of the yellow dashed in Fig. 1 2. Calculation of the gradient of   .To derive the expression of   , the nonlinear theory of the bubble regime 36 is followed here.In the ultra-relativistic limit where the maximum blowout radius     ≫ 1, the trajectory of the inner-most particle is given by where   () is the radial position of the inner-most particle in the bubble sheath, and  =   2 ∫   /   +∞ 0 is the normalized linear charge density of the electron bunch.The longitudinal electric field   can be expressed as a function of   38 , where  ≡  −   / is the pseudo-potential of the plasma wake.The gradient of the accelerating field   / is obtained via differential of Eq. ( 5), In equation ( 6), the term is omitted since it is negligible for most part of the bubble.
Scaling of the simulations.The initial parameters for the scaled simulations are guided by LWFA scaling laws 38 , starting from the scaling of the accelerating wakefield structure.For convenience, real and normalized units are used in this section.Through simulations it has been found that a relatively stable propagation of the ion cavity is realized when    0 ≃     ≃ 2� 0 , where  0 is the initial laser beam waist, and   is the blowout radius of the ion cavity 36,37 .For the parameters being considered here, these relationships hold in a parabolic density channel if the density at the bottom of the channel is used.The connection between the plasma density and the size of the ion cavity is established as   ∝  −2 because   ∝ �   , where  is the scaling factor of the radius of the ion cavity.To efficiently drive the wake, the laser pulse length should fill the first half of the ion cavity, thus  FWHM ∝   , where  FWHM is the pulse duration.Hence the laser pulse energy scales with  3 .The length of the plasma structure   is characterized by the dephasing length   = , providing   ∝  3 .Using the scaling law of the accelerating field,   ∝      ∝ , the expected beam energy gain is obtained as 〈∆〉 ∝  2 .Another initial parameter to be determined is the loaded electron number , and based on [13,36,51] it should scale with the electron number that is expelled from the ion cavity, leading to  ∝     3 ∝ .The relevant formulas and scaling factors are summarized in Table 2.
As for the PIC simulations, the simulation window should scale with the ion cavity, e.g. window ∝  and  window ∝  .The transverse cell size ∆ scales with   to resolve the plasma wavelength while the longitudinal cell size ∆ is fixed to resolve the laser wavelength.

Figure 1 .
Figure 1.(a) Schematic of the high-efficiency LWFA scheme in the tri-plateau plasma channel.(b) On-axis density distribution of the tri-plateau structure.(c) Averaged energy gain in different cases.(d) Pulse length (rms) evolution of the nonlinear chirped pulse (blue line) and the transform-limited pulse (red line) in the tri-plateau structure.(e) Energy consumption of the nonlinear chirped laser pulse in a uniform plasma channel and the tri-plateau structure.

Figure 2 . 2 �
Figure 2. On-axis spectrograms (in color-obtained by Wigner-wavelet transforming the electric field) of the laser pulse at various times for an initially transform-limited laser pulse (a, c and e) and a chirped laser pulse (b, d and f) propagating through a single plateau.(a), (c) and (e) show the spectrograms of the original unchirped or transform limited laser pulse at  = 0,  = 7.39  and  = 11.36 , where the Rayleigh length   of the original laser pulse is 447 μm.The inset in (b) shows the laser field   by the red line and the frequency shift rate by the blue line.As a comparison, (b), (d) and (f) show the spectrograms of the initially chirped laser pulse at the same distances.The spectrograms in all the subplots are obtained by carrying out wavelet transform of   , i.e., |  (  )| = �∫   �′�  ′ ′ +  2 −  2 spectrogram of the laser are shown at propagation distances, 0 mm, 3.81 mm and 5.08 mm respectively.Initially the middle of the pulse experiences a more intense photon deceleration than the head and tail of the pulse [see the inset in Fig.2(a)].This can be seen from Fig.2(a), 2(c) and 2(e), where the on-axis spectrogram for the transform-limited laser evolves into a V-shape, indicating as expected that the middle part suffers from a faster photon deceleration.The pulse is significantly lengthened as a result of the resulting envelope distortion [Fig.2(e)].On the other hand, as shown in Fig 2(b), (d) and (f), the initial nonlinear chirp nearly perfectly balances the redshifting, and the waveform is well preserved throughout the simulation.Off-axis evolution of the spectrogram has a similar pattern shown by Fig. 2, which is supplied in Methods.

Figure 3 .
Figure 3. (a)-(d) Snapshots of the density and the electric field distributions before/after the second [(f) and (g)] and third plateau [(h) and (i)], respectively.(e)-(i) Snapshots of beam longitudinal phase space corresponding to the tagged points in (j).(j) Absolute beam energy spread through the tri-plateau ) to 3(d), the laser field, lineout of the accelerating field (green line), plasma, and beam density are plotted at the start to end points of each transition section (a to d) in Fig.1(b).In Figs. 3 (e)-(i) the beam energy vs. axial position is shown at the same propagation distances.It can be seen that the plasma wake is over-loaded (    < 0) during the first and at the entrance of the second plateau [Figs.3(a) and (b)], leading to a negative energy chirp in the electron beam.On the other hand, in the third plateau, and during the exit of the second and third plateaus the wake is underloaded (    > 0) [Fig.
For the parameters simulated, during the first plateau the dominant term is − 2     2 while it becomes   2 in the third plateau.In the second plateau, the dominant term changes from − 2     2 to   2 as   increases because of the increased plasma density and dephasing.The final energy chirp is obtained by integrating Eq.

Figure 4 .
Figure 4. (a) The energy spectra and (b) the averaged beam energy gain and transfer efficiency of Case 1 to 3 in Table2.

Figure 5 .
Figure 5. Off-axis spectrograms (in color-obtained by Wigner-wavelet transforming the electric field) of the nonlinearly chirped laser pulse at various times propagating through a single plateau with different transverse positions.The first, second and third rows show the spectrograms at  = 0,  = 7.39  and  = 11.36 , where the Rayleigh length   of the original laser pulse is 447 μm.The different columns show the spectrograms for different transverse positions with  2 = 3.7 μm, 7.4μm and 11.1μm.Other parameters of the wavelet transform are same with Fig. 2. The white lines are the corresponding lineouts of laser transverse electric field.Calculation of the gradient of   .To derive the expression of   , the nonlinear theory of the bubble regime 36 is followed here.In the ultra-relativistic limit where the maximum blowout radius     ≫ 1, the trajectory of the inner-most particle is given by (c).

Table 2 .
Scaling of the laser, plasma and beam parameters with scaling factor  for fixed a 0 and λ 0 .