Ultrabright light sources have always been a major pursuit because of their applications in various research areas. At the moment, femtosecond lasers based on the chirped pulse amplification (CPA) technique [1] are regarded as the most reliable approach to realize the highest peak power. After being amplified, compressed, and focused, the peak laser intensity can reach up to 1022−1023W/cm2 [2–5]. The 10 PW-class laser facilities, such as ELI [4] (ELI-NP [6] and ELI-BL [7]), Apollo [8], Vulcan [9], and SULF [10], aim at boosting the focused intensity by another tenfold. Ambitious plans of 100 PW-class have been proposed [11–13] worldwide, where the peak intensities of 1025W/cm2 are anticipated. Furthermore, efforts have also been paid in exploring new mechanisms to generate exawatt–zettawatt lasers [14–16]. At such extreme light intensities, particle acceleration towards 10–100 GeV for leptons [17] and 0.1–10 GeV/nucleon for ions [18–20] is to be expected. Nuclear physics [21–23] as well as lab astrophysics [24–26] will also benefit from these extreme laser sources. Laser–plasma interaction at such intensities enters a new regime where photon emission and radiation reaction become significant [27–34] and strong-field quantum electrodynamics (SF-QED) is necessary to account for the quantum effects [35–38]. It is further predicted that copious electron–positron pairs can be generated [37–55].

While high-power lasers are under fast development, a central question regarding the ultimate laser intensities researchers can build arises [56]. Basically, the upper limitation for laser intensity in an ideal vacuum condition is considered as the Schwinger field Es=2πme2c3/eh∼1.32×1018V/m [57]. The QED theory predicts that laser pulses of 1029W/cm2 can provide such field strength in several ways (tight focusing or coherent combining or others), such that they can transfer a large number of virtual particle pairs to real particles [58,59]. Meanwhile, the generated electron–positron pairs further lose their energies by radiating gamma photons. The laser energy is thus rapidly drained in vacuum [39]. Previous studies have shown that even a single pair produced in vacuum by a laser field can lead to rapid depletion of laser energy [44], i.e., the maximum light intensity is much smaller than 1029W/cm2 in vacuum. It points out that full depletion appears when the energy of generated pairs and photons is equivalent to the energy stored in the pulse, at E∼6.6αEs∼0.05Es (corresponding to 5×1026W/cm2 for laser wavelength λ=800nm).

In reality, it is impossible to build a perfect vacuum environment for experiments. Typically, the vacuum electron density in a chamber suitable for PW-class lasers is about 1011cm−3, provided by ordinary pumping technique (e.g., 10−3Pa for SULF [10]). For laser power above 100 PW, the chamber volume is enlarged by more than tenfold, posing a great challenge to the pump. Another potential drawback is the existence of electrons extracted from optical components (focusing mirror, plasma mirror, etc.) by the passing laser fields. These residual electrons could serve as seeds to trigger the QED processes when the laser field surpasses a certain threshold. Specifically, during the laser–electron interaction, nonlinear Compton scattering [27] following e+nω→e+γ will occur, where electrons absorb multiple laser photons and emit high-energy γ photons. The radiated γ photons further interact with the strong laser field, generating electron–positron pairs via the nonlinear Breit–Wheeler process (γ+nω→e++e−) [41]. These two reaction channels build up positive feedback, i.e., the amount of the pairs and γ photons will be avalanche-like amplified and deplete the laser significantly, known as the QED cascade [42–44]. It can be triggered for a single pulse with intensity above 1025W/cm2 [44] or two colliding pulses with intensity above 1023W/cm2 [45–51]. Therefore, finding out the specific restriction on the attainable laser intensity in these conditions is a key question that needs to be answered for developing lasers beyond 100 PW peak power.

For more realistic consideration, the depletion is a dynamic process where the laser intensity gradually decreases during the development of a QED cascade, which changes the rate of photon emission and pair production. The latter would again deplete the laser energy. A self-consistent dynamic description of the process is therefore required. To this end, we developed a set of dynamic equations that take into account the above-mentioned effects self-consistently. We carried out particle-in-cell (PIC) simulations by including the QED models responsible for the two major reaction channels. Both the simulation and our theoretical model show that the attainable peak intensity depends on the vacuity. At electron density about 109cm−3, notable energy drain emerges from 1025W/cm2 and the upper limit of the laser intensity is modified to ∼1026W/cm2.

Our investigation is based on two-dimensional (2D) PIC simulations using the code VLPL (Virtual Laser Plasma Lab) [60]. It has implemented a local constant cross-field approximation (LCFA) [35–37,40] QED–Monte Carlo model accounting for nonlinear Compton scattering and Breit–Wheeler processes. Under LCFA, the newly generated particles gain energies from the parent particles rather than directly from the laser photons. The latter transfer their energies when accelerating the leptons. In our simulations, laser propagates from the left side of a moving simulation window along the x direction. The window size is 40μm(x)×80μm(y) resolved by 4000cells×1000cells. We set two macroparticles for electrons and protons in each cell. The laser beam is linearly polarized along the y axis [EL=EGcos(ωt−kx)eyxz, BL=EGcos(ωt−kx)ez], following a Gaussian profile EG=[aw0/w(x)]cos2[π(t−tf)/2τ0]×exp[−r2/w2(x)] focused at xf=240μm with normalized peak amplitude a=eE/mωc (the corresponding peak intensity Ipeak=(a2/λ2)×1.38×1018W/cm2, with wavelength λ in μm, where m is the mass of electron, c is the velocity of light in vacuum, ω is the laser frequency, and k is the laser wave vector. Here r2=y2+z2, the laser wavelength is λ=800nm, beam width w0=3λ=2.4μm, w(x)=w0{[(x−xf)2+xR2]/xR2}1/2, Rayleigh length xR=πw02/λ, focusing time tf=xf/c, and pulse duration τ0=10λ/c=26.7fs, respectively. The peak laser field amplitude a is varied from 1500 to 20,000, while the vacuum electron density ne is tuned between 1011 and 1015cm−3. The simulation time step is Δt=0.008T0=0.008λ/c. In our 2D simulation configuration, the laser pulse is assumed to be uniform in δz=1λ finite depth along the z axis. We set periodic boundary conditions for particles such that the latter do not escape from the simulation area after surpassing the z boundary. The particle number counted in a mesh is calculated as Nr=ne×δxδyδz, where δx=0.01μm and δy=0.08μm are the mesh size.

Two challenges should be addressed while carrying out these simulations: (i) initialization of the low-density electrons and (ii) the memory cost for generated new particles (γ photons and electron–positron pairs). It should be noted that at extremely low electron densities (e.g., 1011cm−3), the average weight of electrons w located in one cell is much less than 1, i.e., it is not physical to start the simulations with simple homogeneous initialization. Therefore, we take the following initialization strategy: first, the particle weight w is calculated once the electron density is given; then, a [0, 1] uniformly distributed random value ra is generated, by which the weight of the macroparticle is set to w=int(w)+rank(w−ra), where rank is the step function with rank(x≥0)=1, while rank(x<0)=0; finally, if w=0, no macroparticles will be placed in the cell. To mitigate the memory issue in simulation, clusters particle merging is turned on when the macroparticle number of one element per cell surpasses 4 [60]. Moreover, modeling the QED cascade processes via the Monte Carlo algorithm and initialization of low-density plasma induce stochastic features. To avoid contingency of the stochastic effects, ten simulation examples with randomly distributed seeds are carried out at each set of parameters.

In the following, we derive the theory that describes the evolution of particle numbers from the QED cascade and give the criterion for laser energy depletion. We consider the γ photon and electron–positron pair generation rates satisfying the expression, dNe+pdt=2ΓeNγ,(1)dNγdt=ΓγNe+p−ΓeNγ,(2)where Γe and Ne+p are the generated rate coefficient of electron–positron pairs and number of their total particles, correspondingly; Γγ and Nγ are the coefficient and number of γ photons, respectively. The generation rate of cascade processes is determined by the QED parameter χi=|(FuvPiv)2|1/2/Esmec(i=±eorγ) [35–37], where Fuv is the EM field tensor [61] and Piv is the particle’s four-momentum. According to previous research, the QED parameter can be approximated by χi∝a3/2 [44], and the generation rate Γ is proportional to χi2/3∝a [44,45,52]. Considering the generation rate deviating from the χi2/3 scaling, especially for small χ values [52], we introduce an exponential modification term exp(−aph/a) into the generation rate, i.e., Γ∝aexp(−aph/a) in our model. The aph is chosen as 0.01as (as is the normalized Schwinger field), corresponding to the threshold where the cascade occurs. From empirical approximation, the Γγ is about 1/T0 (T0=λ/c is the laser period) with aph=0.01as. Combining with the Γγ∼4Γe [52], we obtain the generation rates Γγ∼4Γe∼aexp(1-aph/a)/(aphT0). Considering the energy of γ photons and electron–positron pairs is about amec2/2 [33,62,63], the laser depletion for such processes can be roughly evaluated as dE∼−ad(Ne+p+Nγ)mec2/2. Assuming the Gaussian profile remains the same during focusing a=G(t)ξ=[1+(t−tf)2/tR2]−1/2ξ and taking dE∼2c1Vdξdξ/c with tf=300T0, tR=xR/c=9πT0, Vd is the depletion region volume where a>aRRT and c1=me2c3ω2ε0/2e2=1.38×1018×(1μm/λ)2W/cm2, the evolution of ξ is derived as follows: dξdt=−amec34c1Vdξ(4Ne+p+Nγ)Γe.(3)

The peak intensity during focusing processes is measured from PIC simulations and compared to our analytical model. Again, the results in Fig. 3(c) illustrate the consistency between the two. According to the systematic scanning, the reduction of peak intensity emerges from 1025W/cm2, indicating that the depletion effects should be taken into consideration for above a hundred PW class laser facility. The ratio between the simulated peak intensity and the designed intensity decreases sharply when approaching 1026W/cm2 for density from 1011 to 1015cm−3, corresponding to the energy depletion threshold. As seen in Fig. 3(d), when the designed light intensity surpasses the threshold, the attainable one is restricted to 1026W/cm2 for vacuity down to 109cm−3 according to our theoretical model, exhibiting a clear ceiling. The attainable intensity reaches 2×1026W/cm2 for vacuity ∼108cm−3. It should be noted that at even lower electron densities (<107cm−3), the average electron number in the focusing area is less than 1. The cascading effect only occurs when the seeding particle sits in the focal region. In this case, one may not be able to give a definite threshold.

In fact, the rising and falling edges of the laser pulse should be symmetric around t=tf in the time domain if depletion is negligible. Nevertheless, strong depletion breaks down the symmetrical profile such that the maximum intensity observed in simulations is not exactly at the designed focal position. The behavior is even more obvious with higher residual density [Fig. 4(a)] or at intensities beyond the threshold [Fig. 4(b)]. We choose a=6000 as an example and present the peak intensity at different simulation times. As depicted in Fig. 4(a) from both simulations and the theoretical model, the laser peak intensity appears near the focal position for 1011cm−3, while much earlier for the 1013cm−3 (by ∼10T0) and 1015cm−3 (by ∼15T0) cases. Since laser intensity at the pulse rising front does not reach the threshold at 1011cm−3, the distortion caused by depletion is negligible. At higher electron densities, the intensity exceeds the threshold before t=tf and the cascade develops quickly. Significant depletion in the laser front induces intensity peak shifting to an earlier time than designed. The highest intensity found from simulations as a function of the propagation time is presented from our theoretical model for ne0=109cm−3. As one notices in Fig. 4(b), the symmetrical time profile of peak intensity becomes asymmetric when approaching the threshold. In this case, the attainable intensity is restricted to below 1026W/cm2. Moreover, results of a 3D simulation are also presented in Fig. 4(a) for comparison. Here memory overflow occurs in the later stage due to the enormous number of particles created in the cascade; we therefore show the data before the simulation collapses. One sees the 3D results are in reasonable agreement with the 2D simulations.

It should be mentioned that the cascade process is affected by laser polarization to a certain extent [52]. Since circular polarization essentially requires 3D simulations, we restrain our analysis on linear polarization. One may refer to Ref. [44] for further information on circular polarization. The laser polarization [52], electron seeding [53,54], and saturation [55] can also affect the cascading process. Besides, we employ cos2 profiles in the time domain as a close approximation to avoid cutoff for Gaussian distribution in simulations. The results between the two profiles show a negligible difference (not shown here). Moreover, the pre-pulse may affect the local electron density at these laser intensities. We take 1013W/cm2 pre-pulse level with the duration of a nanosecond to estimate the drifting distance due to ponderomotive scattering, which is approximately ∼a2cT/2∼1.5μm. The drifting distance is at the same order for picosecond duration laser foot (typically 3 orders of magnitude stronger than the nanosecond pre-pulse). In this case, electrons are still within the laser focal region.

In summary, we have explored the attainable highest laser intensity under different vacuum conditions for the first time, to the best of our knowledge. It is found that the avalanche-like QED cascade and RRT effect pose a strong limit on the achievable light intensity due to the residual electrons the laser pulses meet. Our study suggests that the observed peak intensity is suppressed starting from ∼1025W/cm2, and an upper limit emerges at 1026W/cm2 for vacuum electron densities above 109cm−3. These laser intensity thresholds can be approached by focusing the optical laser pulses of multiple hundreds of PW peak power. The cases for building lasers beyond hundreds of PW peak power are therefore not well justified, considering the vacuum conditions for a typical PW-class laser experimental environment.

It is worth noting that light intensities at ∼1024W/cm2 can readily support the research on strong-field QED physics (e.g., the radiation-reaction effects, electron–positron pair production, QED cascade), particle acceleration towards the high-energy frontier, laser-driven nuclear physics and high-energy density physics. The featured intensity is already accessible with a 100-PW laser, such as the SEL 100-PW laser under construction in China [64].

Acknowledgment. The authors would like to thank Prof. Alexander Pukhov for the use of the PIC code VLPL.