Abstract
Keywords
1 Introduction
Progress in high-power laser technology in recent decades has made it possible, through the generation of extraordinarily strong electromagnetic (EM) fields, to investigate radiation and particle-production processes in the nonlinear quantum regime[1–10]. In addition, this has opened up new opportunities for the creation of exotic particle and radiation sources[11–21], as well as for studies of electron–positron plasmas[22,23], which may help one to understand various astrophysical processes[24–26].
The nature of laser–matter (or laser–light) interactions is determined by several parameters, including the ratio between the electric field strength
However, whether it is even possible to attain the needed high field strengths in the laboratory frame is an open question[29–34]. This is because such fields would be expected to trigger an electron–positron pair cascade, forming a dense pair plasma that would screen or absorb the laser radiation being focused, preventing the further increase of the field strength[30,31,35]. Avoiding the triggering of such a cascade (ultraintense, tightly focused lasers can ponderomotively eject stray particles that might seed a cascade, but this is dependent on the interaction geometry[32,36]) will be essential for maximising the reachable field strength[32].
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In this paper we investigate the possibility to generate supercritical fields by a combination of three essential ideas: advanced focusing, plasma-based conversion of optical or near-infrared (IR) light to extreme ultraviolet (XUV) frequencies and coherent combination of multiple laser pulses (see Figure 1). The conversion to higher frequencies has been discussed as a means of reducing the focal volume, which increases field strength at a fixed power[33,34,37–41] (a more detailed discussion can be found in Ref. [42]). Moreover, EM processes in high-strength fields demonstrate a strong dependence on the field wavelength[3]. In combination with
Figure 1.The main principle behind maximising field strength starting from laser sources with optical frequencies.
Our goal here is to provide a far-future outlook on the field strengths that could be attained in ‘best-case’ scenarios that combine currently known concepts and approaches. We demonstrate numerically that, given advanced focusing, the physics of laser–plasma interactions itself provides the possibility to reach
2 Setups
In this section we provide an order of magnitude estimate for the field strength hypothetically attainable with future laser systems, with the help of the known theoretical ideas outlined above. We start by considering the concepts of
2.1 Advanced focusing
The maximal attainable field strength for a given power of focused radiation is limited by the so-called dipole wave[43] that can also be extended to a time-limited solution known as the dipole pulse[44]. The dipole wave can be seen as time-reversed emission of a dipole antenna and thus can be approximated by several focused beams or by focusing an intensity-shaped radially polarised beam with a parabolic mirror[44,45]. Let us start from considering the benefits of using tight focusing of the laser radiation, characterised by small values of
where the radial
In our setup, the smoothing angle
The radiation intensity at focus is proportional to the power
where a wavelength-agnostic, dimensionless parameter
A significant improvement can be achieved by splitting the power into six pulses and focusing them with
The maximal field strength is achieved either for the electric or magnetic field component (pointing along the
2.2 Plasma converter
The idea and particular concepts for field intensification through plasma-based high-order harmonic generation and focusing have been being discussed by several research groups since the beginning of the 2000s. One possibility is to use Doppler frequency upshifting during the reflection of laser radiation from so-called relativistic flying mirrors formed either by the cusp of the preceding plasma wave breaking[38,50,51] or by the ejection of electrons from thin plasma layers[52–56]. In both cases a counter-propagating laser pulse is used to produce the flying mirror that can be shaped to focus the reflected radiation. Another possibility is to use the highly nonlinear reflection of laser radiation from dense plasma naturally formed by the ionisation of solid targets[57–59]. The early discussions and models also appealed to the Doppler frequency upshifting, which in this case occurs during the reflection from the oscillating effective boundary[60,61] that can also be shaped for harmonic focusing by tailoring the pulse intensity shape[39,62] . It was later recognised that the conversion can be more generally seen as coherent synchrotron emission (CSE) of electrons from a self-generated peripheral layer of electrons[63], while the layer’s spring-like dynamics and sought-after emission can be described by a set of differential equations forming the so-called relativistic electronic spring (RES) model[41,64,65]. Further studies[66] showed that optimal conversion is achievable with an incidence angle of
Laser | Conversion parameters | Yield after focusing | |||||
---|---|---|---|---|---|---|---|
Publication, | Peak | Incident | Working plasma | Incidence | Duration, | Intensification | Peak intensity, |
geometry | power, PW | intensity, W/cm | density, cm | angle | as | factor | W/cm |
Naumova et al. (2004)[ | – | 200 | 2.5 | ||||
plasma denting | – | ||||||
Gordienko et al. (2005)[ | |||||||
spherical converter | |||||||
Gonoskov et al. (2011)[ | 10 | 4000 | |||||
groove-shaped converter | |||||||
Baumann et al. (2019)[ | 35 | 150 | 16 | ||||
plasma denting | |||||||
Vincenti (2019)[ | 3 | – | 100 | 1000 | |||
plasma denting |
Table 1. Some of the reported numerical results on focusing plasma-generated XUV pulses.
We performed a number of simulations using a 1D version of the ELMIS PIC code[68] with the quantum radiation reaction accounted for via the QED event generator described in Ref. [69]. (The oblique incidence is transformed to normal in a moving reference frame[70].) We assumed a single-cycle laser pulse (
The amplitude increase becomes larger with the increase of incident wave amplitude
Figure 2.(a) The numerical result for the dipole focusing of XUV pulse. (b) The total laser power of 200 PW is split into six beams and each is focused to at from the focus, where the plasma converters provide an amplitude boost by a factor of 15 and frequency upshift by a factor of approximately . The conversion is followed by the MCLP (e-dipole) focusing using six beams at . (c) The dependency of the field strength on the -coordinate (green curve), -coordinate (blue curve) and time (red curve) is shown in panel (c) together with the fit (black solid curves) and the threshold for cascaded pair-generation (dashed black line).
Although our goal here is to assess the capabilities of the conversion physics itself, we do not consider higher incident intensities, even though these could lead to even higher intensification factors and even shorter pulse duration. One reason for this is that, at higher intensities, QED effects can start to play a detrimental role, due to an increasingly large part of the incident energy being converted into gamma photons (see Ref. [73] and Fig. 2 in Ref. [69]). Ion motion is another factor that becomes more prominent with the rise of intensity and makes it difficult to efficiently exploit the conversion mechanism. This motivates further studies on the generation of short pulses[74], exploiting pulse steepening[75–79] and possibilities to use high-
2.3 Focusing of XUV pulses
We now continue our analysis by considering the possibility of focusing the XUV pulses generated at the curved plasma surfaces of the six focusing mirrors with
In order to resolve the singular XUV peak we use a sequence of adaptive sub-grids that are arranged in the following way. First, we surround the XUV peak with a frame and deduce there the field multiplying by a mask function that smoothly goes from 1 to 0 and the ends of the frame. In such a way we cut out the XUV pulse, and the remaining field with narrower spectral content can be sampled with the first grid. We then take the deduced field within the frame and repeat the procedure, introducing another subframe in a closer vicinity of the XUV peak and sampling the remaining field with another thinner sub-grid. We perform this procedure seven times to reach a sufficient resolution, which in our case corresponds to the space step of 0.064 nm. Every deduced field is advanced first analytically (as a spherical wave) to the distance of four frame lengths and then numerically using the spectral solver on the grid 128 × 5122.
The result of our numerical calculation for
where
The threshold for the cascade can be estimated as the equality of the volume size (distance to the centre) to the mean scale length of pair production. This estimate is shown in Figure 2(c) with a dashed black line and indicates that the region where the field reaches
We conclude this section by showing schematically the potential of reaching strong fields with different strategies based on a given value of the total laser power of a laser facility (see Figure 3). One can see that using tight focusing or, better still, multiple colliding laser pulses (MCLPs)[80], provides a substantial increase of the peak field, which is, however, well below the Schwinger field even in case of 1 EW total power. The plasma converter can give a significant increase once the intensity of
Figure 3.The prospects of reaching high field strength using tight focusing, multiple laser colliding pulses, the plasma conversion and their combination on the map of the attainable field strength and total power of the laser facility. The two outlined options correspond to the use of the plasma conversion at and , respectively. The labels show the results of simulations by Gonoskov
Certainly, this analysis is performed under the assumption of the best-case scenario and the implementation of such a concept requires many technological advances. Among them, driving plasma conversion and reaching spatial-temporal synchronisation of the generated XUV pulses appear to the central difficulties. However, from our results we can draw a conclusion that achieving the needed spatio-temporal control in the domain of nanometre-attosecond could provide a pathway towards reaching the Schwinger field strength using the outlined concept based on high-intensity laser facilities.
We estimate that delivering 10 GeV electrons to the strong-field region of the outlined setup would result in a
3 Physical processes in critical and supercritical fields
Critical and supercritical fields open up the possibility to perform experiments in regimes that traditionally have not been available to light sources. For the purpose of illustration, we briefly discuss a number of possible studies that could be performed using the extreme-field source we have outlined.
3.1 Nuclear dynamics
Studies of nuclear photonics have been strongly motivated by using high-brilliance gamma sources for, for example, excitation of nuclei. However, strong EM fields can also reach the scales relevant for probing internal nuclear dynamics. Before going further with some examples, we point out that these experiments would give rise to significant challenges, such as the alignment of the target with the focal spot, as well as timing issues. It is still of interest to consider these possibilities, because solid-density plasma can be transparent for the high-intensity XUV pulses. It might also be possible to deliver accelerated nuclei to focus along the dipole pulse axis, where the field strength is greatly suppressed[20]. In this case, it might be necessary to accumulate the signal of repeated experiments to compensate for the small size of the strong-field region. To motivate further studies on possible experimental arrangements we discuss some of the possibilities that arise in nuclear physics through the development of strong-field sources.
Electric fields of the strength discussed in Section 2 are sufficient to strip atoms of their electrons; the field strength necessary for barrier-suppression ionisation of the deepest lying electron is
and
Stronger electric fields affect even the internal dynamics of the nucleus, by modifying the Coulomb barrier through which daughter particles tunnel. For example, the characteristic electric field required to modify the
where
The same logic can be applied to
where
3.2 Electron dynamics
A supercritical field structure of this type is a platform for investigating nonlinear QED in a completely unexplored regime, either by probing it with externally injected electrons or by exploiting the nonlinear dynamics of virtual particles from the quantum vacuum.
Based on an analysis of quantum loop corrections to physical processes in constant, crossed fields, it has been conjectured that the relevant expansion parameter is not the fine structure constant
Higher-order corrections, normally thought of as
Nonlinear quantum dynamics are evident for pure EM fields as well, driven by virtual electron loops that modify the classical linearity of Maxwell’s equations. The nonlinear behaviour of pure magnetic field of strength,
Although this growth is slower than the power-law behaviour of higher-order corrections at ultralarge quantum parameter
4 Summary
We have outlined how optimal configurations of laser systems and/or secondary sources could give us the opportunity to approach, or even exceed, the critical field of QED. The configurations presented certainly constitute immense engineering challenges. Realising an optical-XUV plasma converter demands sophisticated material engineering, high laser contrast and spatial uniformity, as well as timing and pointing stability. The quality of the vacuum is important if observations are to be made of Schwinger pair creation, or of the nuclear physics processes we have considered. These feasibility questions should be addressed in future work. Our results nevertheless indicate that the presented concepts are promising and warrant further analysis. Reaching such critical fields could give an opportunity to probe some of the most extreme environments in the universe, and investigate the behaviour of electrons, nuclei and the quantum vacuum under such conditions. We have given several examples of the use of such new photon sources for probing physical laws, ranging from electron and nuclear physics to probing the quantum vacuum.
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