2.1 Advanced focusing

The maximal attainable field strength for a given power of focused radiation is limited by the so-called dipole wave^[ Reference Bassett^43] that can also be extended to a time-limited solution known as the dipole pulse^[ Reference Gonoskov, Aiello, Heugel and Leuchs^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^[ Reference Gonoskov, Aiello, Heugel and Leuchs^44, Reference Jeong, Bulanov, Sasorov, Bulanov, Koga and Korn^45]. Let us start from considering the benefits of using tight focusing of the laser radiation, characterised by small values of $f$ -number ${f}_{\mathrm{N}}$ or, equivalently, by large values of the divergence angle $\theta =\arctan \left({f}_{\mathrm{N}}^{-1}/2\right)$ , where we for simplicity use the expression that implies $\theta <\pi /2$ . To numerically ascertain the potential gain of using tight focusing, we set the initial EM field to be a two-cycle optical pulse propagating from a spherical surface of radius ${r}_0\,{=}\,16\lambda$ , which can be considered large compared to the radiation wavelength $\lambda$ (the far-field zone):

(1) $$\begin{align} \mathbf{E}=\frac{\mathbf{r}\times \left[\hat{\mathbf{z}}\times \mathbf{r}\right]}{\left|\mathbf{r}\times \left[\hat{\mathbf{z}}\times \mathbf{r}\right]\right|}{S}_r\left(r+ ct\right){S}_{\alpha}\left(\alpha \right), \end{align}$$

(2) $$\begin{align}\mathbf{B}=\frac{1}{c}\frac{\hat{\mathbf{z}}\times \mathbf{r}}{\left|\hat{\mathbf{z}}\times \mathbf{r}\right|}{S}_r\left(r+ ct\right){S}_{\alpha}\left(\alpha \right),\end{align}$$

where the radial ${S}_r(r)$ and angular ${S}_{\alpha}\left(\alpha \right)$ shape functions are defined by the following:

(3) $$\begin{align}{S}_r(r)=\sin \left[2\pi \left(r-{r}_0\right)/\lambda \right]\!\left\{\begin{array}{@{}ll}{\cos}^2\left[\frac{\pi }{2}\left(r-{r}_0\right)/\lambda \right],& \left|r-{r}_0\right|\le \lambda, \\ {}0,& \left|r-{r}_0\right|>\lambda, \end{array}\right.\end{align}$$

(4) $$\begin{align}{S}_{\alpha}\left(\alpha \right)=\left\{\begin{array}{@{}ll}1,& \alpha \le \theta -{\theta}_{\mathrm{s}}/2,\\ {}{\sin}^2\left[\frac{\pi }{2}\left(\alpha -\theta \right)/{\theta}_\mathrm{s}\right],& \theta -{\theta}_{\mathrm{s}}/2<\alpha \le \theta +{\theta}_{\mathrm{s}}/2,\\ {}0,& \alpha >\theta +{\theta}_{\mathrm{s}}/2,\end{array}\right.\end{align}$$

(5) $$\begin{align}\alpha =\arctan \left(\sqrt{z^2+{y}^2}/\left|x\right|\right).\end{align}$$

In our setup, the smoothing angle ${\theta}_{\mathrm{s}}=0.3$ eliminates the sharp edges of the concave pulse within our model. We advance this field to the vicinity of the focal point using a spectral solver of Maxwell’s equations within the open-source package hi- $\chi$ ^{[ 46]}. To reduce the amount of needed computational resources we also employ the module of contracting a spherical window that maps the concave region of the pulse to a thin layer of space with periodic boundary conditions^[ Reference Panova, Volokitin, Efimenko, Ferri, Blackburn, Marklund, Muschet, De Andres Gonzalez, Fischer, Veisz, Meyerov and Gonoskov^47].

The radiation intensity at focus is proportional to the power $\mathcal{P}$ and inversely proportional to the focal spot area, which in turn scales as ${\lambda}^2$ , with $\lambda$ being the radiation wavelength. It is thus possible to express the peak field strength at focus for arbitrary power $\mathcal{P}$ and $\lambda$ :

(6) $$\begin{align}\frac{E}{E_{\mathrm{cr}}}=\frac{\delta }{4.1\times {10}^5}\left(\frac{\lambda }{1\;\unicode{x3bc} \mathrm{m}}\right){\left(\frac{\mathcal{P}}{1\kern0.1em \mathrm{PW}}\right)}^{1/2},\end{align}$$

where a wavelength-agnostic, dimensionless parameter $\delta$ solely characterises the efficiency of focusing. Note that we define it so that $\delta \sqrt{\mathcal{P}/\left(1\kern0.24em \mathrm{PW}\right)}$ gives field amplitude $E$ in relativistic units, that is, in units of $mc\omega /e$ , where $\omega$ is the radiation frequency. According to our simulations, the focusing with f/2 and f/1 provides $\delta \approx 170$ and $\delta \approx 230$ , respectively.

A significant improvement can be achieved by splitting the power into six pulses and focusing them with ${f}_{\mathrm{N}}=1$ ( $2\theta \approx 0.93<2\pi /6$ ) to the same point symmetrically from different directions in the $x$ – $y$ plane, so that the polarisation vector for each pulse is orientated along the $z$ -axis. For each pulse the power is then reduced by a factor of $6$ and the field strength reduced by $\sqrt{6}$ , but the strength of the combined field multiplies this with a factor of 6 due to coherent summation of the field. As a result we have an increase by factor $\sqrt{6}$ : $\delta \left(6\times f/1.0\right)\approx 560$ , which is relatively close to the theoretical maximum ${\delta}_{\mathrm{max}}\approx 780$ provided by the dipole wave^[ Reference Bassett^43, Reference Gonoskov, Aiello, Heugel and Leuchs^44]. We will use this six-beam configuration as the main reference for future setups, whereas configurations with a larger number of beams can better sample the dipole wave and bring the value of $\delta$ even closer to ${\delta}_{\mathrm{max}}$ .

The maximal field strength is achieved either for the electric or magnetic field component (pointing along the $z$ -axis), whereas the other field component is close to zero in the centre. The maximisation of the electric field with the so-called electric dipole wave provides a strong, oscillating electric field that is especially interesting for enhancing the production of electrons and positrons, as well as for trapping them by anomalous radiative trapping^[ Reference Gonoskov, Bashinov, Gonoskov, Harvey, Ilderton, Kim, Marklund, Mourou and Sergeev^11], which in combination provides a unique condition for the creation of radiation sources^[ Reference Gonoskov, Bashinov, Bastrakov, Efimenko, Ilderton, Kim, Marklund, Meyerov, Muraviev and Sergeev^14] and extreme plasma states^[ Reference Efimenko, Bashinov, Bastrakov, Gonoskov, Muraviev, Meyerov, Kim and Sergeev^22, Reference Efimenko, Bashinov, Gonoskov, Bastrakov, Muraviev, Meyerov, Kim and Sergeev^23]. The maximisation of the magnetic field by the magnetic dipole wave can also be of interest for initiating extreme plasma dynamics^[ Reference Bashinov, Efimenko, Muraviev, Volokitin, Meyerov, Leuchs, Sergeev and Kim^48] as well as for reaching strong fields with suppressed EM cascades in the centre. The interaction of an optical dipole wave with a high-energy electron beam leads to the generation of multi-GeV photon sources and can be used as a platform for the study of EM cascades, of both shower and avalanche types^[ Reference Magnusson, Gonoskov, Marklund, Esirkepov, Koga, Kondo, Kando, Bulanov, Korn and Bulanov^20, Reference Magnusson, Gonoskov, Marklund, Esirkepov, Koga, Kondo, Kando, Bulanov, Korn, Geddes, Schroeder, Esarey and Bulanov^21]. Finally, a symmetric mixture of electric and magnetic dipole waves provides the optimal setup for attaining the highest possible $\chi$ value for a given external beam of high-energy electrons^[ Reference Olofsson and Gonoskov^49]. Here we proceed with our analysis for the electric dipole wave.

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^[ Reference Bulanov, Esirkepov and Tajima^38, Reference Martins, Santos, Fonseca and Silva^50, Reference Matlis, Reed, Bulanov, Chvykov, Kalintchenko, Matsuoka, Rousseau, Yanovsky, Maksimchuk, Kalmykov, Shvets and Downer^51] or by the ejection of electrons from thin plasma layers^[ Reference Kulagin, Cherepenin, Hur and Suk^52– Reference Esirkepov, Bulanov, Kando, Pirozhkov and Zhidkov^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^[ Reference Bulanov, Naumova and Pegoraro^57– Reference von der Linde and Rzàzewski^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^[ Reference Gordienko, Pukhov, Shorokhov and Baeva^60, Reference Baeva, Gordienko and Pukhov^61] that can also be shaped for harmonic focusing by tailoring the pulse intensity shape^[ Reference Naumova, Nees, Sokolov, Hou and Mourou^39, Reference Dromey, Adams, Hörlein, Nomura, Rykovanov, Carroll, Foster, Kar, Markey, McKenna, Neely, Geissler, Tsakiris and Zepf^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^[ Reference Der Brügge and Pukhov^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^[ Reference Gonoskov, Korzhimanov, Kim, Marklund and Sergeev^41, Reference Gonoskov^64, Reference Blanco, Flores-Arias and Gonoskov^65]. Further studies^[ Reference Blackburn, Gonoskov and Marklund^66] showed that optimal conversion is achievable with an incidence angle of $50{}^{\circ}{-}62{}^{\circ}$ and the density ramps are achievable via tailored pulse contrast^[ Reference Rödel, an der Brügge, Bierbach, Yeung, Hahn, Dromey, Herzer, Fuchs, Pour, Eckner, Behmke, Cerchez, Jäckel, Hemmers, Toncian, Kaluza, Belyanin, Pretzler, Willi, Pukhov, Zepf and Paulus^67]. The latest numerical studies exploiting plasma denting in combination with oblique incidence indicate the possibility of significant field intensification^[ Reference Baumann, Nerush, Pukhov and Kostyukov^33, Reference Vincenti^34]. Some of the reported numerical results are summarised in Table 1. As a way to estimate future prospects, we consider the conversion described in Refs. [Reference Gonoskov, Korzhimanov, Kim, Marklund and Sergeev41,Reference Der Brügge and Pukhov63].

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^[ Reference Gonoskov^68] with the quantum radiation reaction accounted for via the QED event generator described in Ref. [Reference Gonoskov, Bastrakov, Efimenko, Ilderton, Marklund, Meyerov, Muraviev, Sergeev, Surmin and Wallin69]. (The oblique incidence is transformed to normal in a moving reference frame^[ Reference Bourdier^70].) We assumed a single-cycle laser pulse ( $\lambda =0.81\kern0.22em \unicode{x3bc} \mathrm{m}$ ) interacting with a steep-front plasma surface with immobile ions. Many factors, including, for example, the motion of ions, plasma spreading due to limited contrast and the pulse shape, can significantly affect both the increase of the amplitude and the optimal conditions for achieving it. However, the physics of this process has been shown to be sufficiently robust to justify the considerations here as a good starting point for further studies^[ Reference Gonoskov^64, Reference Blackburn, Gonoskov and Marklund^66, Reference Bhadoria, Blackburn, Gonoskov and Marklund^71].

The amplitude increase becomes larger with the increase of incident wave amplitude ${a}_{\mathrm{in}}$ , which we express in relativistic units^[ Reference Bashinov, Gonoskov, Kim, Mourou and Sergeev^72]. We consider two cases ${a}_{\mathrm{in}}\approx 70$ ( ${I={10}^{22}\kern0.1em \mathrm{W}/{\mathrm{cm}}^2}$ ) and ${a}_{\mathrm{in}}\approx 220$ ( $I={10}^{23}\kern0.1em \mathrm{W}/{\mathrm{cm}}^2$ ). For each case we fine tune the incidence angle $\alpha$ and the plasma density $n$ expressed in units of plasma critical density. For $I={10}^{22}\kern0.1em \mathrm{W}/{\mathrm{cm}}^2$ we find that the maximal amplitude increase of $8.4$ is achieved for $\alpha =61.43{}^{\circ}$ and $n=0.4125{a}_{\mathrm{in}}$ , whereas for $I={10}^{23}\kern0.1em \mathrm{W}/{\mathrm{cm}}^2$ the maximal amplitude increase of $16.1$ is achieved for the same incidence angle but for $n=0.397{a}_{\mathrm{in}}$ . For the latter, the resulting field distribution is shown in Figure 2(b). The length of the generated pulses in these cases is less than 1 nm, which corresponds to the XUV range.

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 ${10}^{23}\kern0.1em \mathrm{W}/{\mathrm{cm}}^2$ at $7\kern0.22em \unicode{x3bc} \mathrm{m}$ from the focus, where the plasma converters provide an amplitude boost by a factor of 15 and frequency upshift by a factor of approximately ${10}^4$ . The conversion is followed by the MCLP (e-dipole) focusing using six beams at $f/1.0$ . (c) The dependency of the field strength on the $x$ -coordinate (green curve), $z$ -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. [Reference Nerush, Kostyukov, Ji and Pukhov73] and Fig. 2 in Ref. [Reference Gonoskov, Bastrakov, Efimenko, Ilderton, Marklund, Meyerov, Muraviev, Sergeev, Surmin and Wallin69]). 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^[ Reference Rivas, Borot, Cardenas, Marcus, Gu, Herrmann, Xu, Tan, Kormin, Ma, Dallari, Tsakiris, Földes, Chou, Weidman, Bergues, Wittmann, Schröder, Tzallas, Charalambidis, Razskazovskaya, Pervak, Krausz and Veisz^74], exploiting pulse steepening^[ Reference Gustafson, Taran, Haus, Lifsitz and Kelley^75– Reference Wang, Lin, Sheng, Liu, Zhao, Guo, Lu, He, Chen and Yan^79] and possibilities to use high- $Z$ , that is, heavy nuclei, material (note that a higher intensity can cause almost complete ionisation, making the charge-to-mass ratio close to $e/2{m}_p$ independent of the nuclei type). Finally, for a given incident power, higher intensity at the converter means a smaller area of conversion. Once the characteristic size of the conversion area ${r}_0/\left(1\;\unicode{x3bc} \mathrm{m}\right)\sim {\left(P/\mathrm{PW}\right)}^{1/2}{\left[I/\left({10}^{22}\;{\mathrm{W\,cm}}^{-2}\right)\right]}^{-1/2}$ becomes comparable to the wavelength $\lambda \sim 1\;\unicode{x3bc}$ m, multidimensional effects start to affect both the conversion process itself and the collimation and synchronisation of the XUV pulses generated across the focal area (for 2D numerical studies of the plasma conversion by the considered mechanism see Refs. [Reference Blackburn, Gonoskov and Marklund66,Reference Bhadoria, Blackburn, Gonoskov and Marklund71]). In this context and for $I={10}^{22}\kern0.1em \mathrm{W}/{\mathrm{cm}}^2$ the minimal power is a few PW, whereas for $I={10}^{23}\kern0.1em \mathrm{W}/{\mathrm{cm}}^2$ , it is preferable to have a few tens of PW.

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 ${f}_{\mathrm{N}}=1$ . We assume that the laser radiation is split into six beams, pre-focused and delivered so that the optimal conditions for the plasma converters are achieved at the plasma surfaces and the generated XUV pulses become focused at the central point. We assume that the conversion happens at the distance of 6 μm from the centre. We consider two cases: the total power $\mathcal{P}$ is 20 and 200 PW, which results in the intensities of ${10}^{22}\kern0.1em$ and ${10}^{23}\kern0.1em \mathrm{W}/{\mathrm{cm}}^2$ at the plasma surfaces, respectively. A rough estimate for the peak field strength achievable in this configuration suggests that for $\mathcal{P}=20\kern0.22em \mathrm{PW}$ (200 PW) we can reach ${a}_{\mathrm{out}}\sim 2500$ (8000) given in relativistic units for the wavelength $\lambda \sim 1\kern0.1em \mathrm{nm}$ , which is well above the Schwinger field strength in both cases. However, this estimate is not sufficient because different spectral fractions are focused to different diffraction-limited volumes. That is why we need to perform numerical calculation to realize estimations for these cases.

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 $\mathcal{P}=200\kern0.22em \mathrm{PW}$ is shown in Figure 2. The peak field of 130 ${E}_{\mathrm{cr}}$ is achieved in the centre within a volume of about a few nanometres in size. The following fit can be used for estimates and calculations:

(7) $$\begin{align}\frac{E\left(r,t\right)}{E_{\mathrm{cr}}}\approx \mathcal{A}{\left({\left|\frac{r+ ct}{R}\right|}^{3/2}+1\right)}^{-1}{\left(\left|\frac{ct}{D}\right|+1\right)}^{-1},\end{align}$$

where $\mathcal{A}=130$ , $R=0.2\kern0.22em \mathrm{nm}$ and $D=0.3\kern0.22em \mathrm{nm}$ in this case (see the solid black curves in Figure 2(c)). A similar result is obtained for the case of $\mathcal{P}=20\kern0.22em \mathrm{PW}$ , for which we got best fit for $\mathcal{A}=10$ , $R=0.5\kern0.22em \mathrm{nm}$ and $D=0.5\kern0.22em \mathrm{nm}$ .

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 ${E}_{\mathrm{s}}$ is too small for the occurrence of the cascade based on the Breit–Wheeler process.

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)^[ Reference Bulanov, Mur, Narozhny, Nees and Popov^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 ${10}^{22}\kern0.1em \mathrm{W}/{\mathrm{cm}}^2$ is reached, which can be provided by $f/1.0$ focusing already with the total laser power of about 100 TW. The conversion at ${10}^{23}\kern0.1em \mathrm{W}/\mathrm{cm}^{2}$ provides an even larger boost. In both cases, tight focusing of the generated XUV pulses can provide a significant increase of field strength beyond ${E}_{\mathrm{cr}}$ .

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 ${10}^{22}$ and ${10}^{23}\kern0.1em \mathrm{W}/{\mathrm{cm}}^2$ , respectively. The labels show the results of simulations by Gonoskov et al. ^[ Reference Gonoskov, Korzhimanov, Kim, Marklund and Sergeev^41] (1), by Baumann et al. ^[ Reference Baumann, Nerush, Pukhov and Kostyukov^33] (2) and by Vincenti^[ Reference Vincenti^34] (3).

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 $\chi$ of order ${10}^6$ for the case of $\mathcal{P}=20$ PW (see estimates in Appendix B). We will investigate such possibilities in future work.

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^[ Reference Magnusson, Gonoskov, Marklund, Esirkepov, Koga, Kondo, Kando, Bulanov, Korn and Bulanov^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 ${E}_{\mathrm{BSI}}\simeq {\left( Z\alpha \right)}^3{E}_{\mathrm{cr}}/16$ . The bare nucleus can then be accelerated to relativistic velocity, in a single wave period, if the electric field amplitude $E>{E}_{\mathrm{rel}}$ , where

(8) $$\begin{align}\frac{E_{\mathrm{rel}}}{E_{\mathrm{cr}}}=\frac{Am_p\omega }{eZ}=3.8\times {10}^{-3}\kern0.1em \frac{A}{Z\lambda \left[\unicode{x3bc} \mathrm{m}\right]},\end{align}$$

and $Z$ and $A$ are the nucleus’ atomic and mass numbers, respectively. Thus, the source of a near-critical field could accelerate heavy nuclei from rest to normalised momentum, $p/M=225\left(Z/A\right)\left(E/{E}_{\mathrm{cr}}\right)\lambda \left[\unicode{x3bc} \mathrm{m}\right]\gg 1$ .

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 $\alpha$ -decay rate of an unstable nucleus, ${E}_{\alpha }$ , can be estimated as follows^[ Reference Pálffy and Popruzhenko^81]:

(9) $$\begin{align}\frac{E_{\alpha }}{E_{\mathrm{cr}}}=\frac{2\sqrt{2}{Q}_{\alpha}^{5/2}}{3{\pi \alpha}^2{Z}^2{Z}_{\mathrm{eff}}{m}_{\mathrm{e}}^2{m}_{\mathrm{r}}^{1/2}}\simeq 300\kern0.1em \frac{Q^{5/2}\left[\mathrm{MeV}\right]}{Z^2{Z}_{\mathrm{eff}}},\end{align}$$

where $Q$ is the energy of the $\alpha$ particle, ${Z}_{\mathrm{eff}}=\left(2A-4Z\right)/\left(A+4\right)$ , $Z$ and $A$ are the proton and mass numbers of the daughter nucleus, respectively, and ${m}_{\mathrm{r}}$ is the reduced mass of the $\alpha$ –daughter-nucleus system. For polonium- $212$ , which has a half-life of $0.3$ ms, $Q\simeq 9.0$ MeV and ${E}_{\alpha }/{E}_{\mathrm{cr}}\simeq 30$ . The correction to the decay rate $C=\exp \left[2E(t)\cos \theta /{E}_{\alpha}\right]$ , where $\theta$ is the angle between the electric field vector and the $\alpha$ -emission direction. Averaging over all $\theta$ , we obtain ${\left\langle C\right\rangle}_{\theta}=\sinh \left[2E(t)/{E}_{\alpha}\right]/\left[2E(t)/{E}_{\alpha}\right]$ . Further averaged over a single cycle, with $E(t)={E}_0\sin \omega t$ , we find that the modification to the decay rate is ${\left\langle C\right\rangle}_{\theta, t}\simeq 1.4$ for ${E}_0=30{E}_{\mathrm{cr}}$ and as much as ${\left\langle C\right\rangle}_{\theta, t}\simeq 21$ for ${E}_0=100{E}_{\mathrm{cr}}$ . We note that by exceeding ${E}_{\alpha }$ we enter a regime where the effect of the external field is no longer a small correction.

The same logic can be applied to $\beta$ decay, where the characteristic electric field required to modify the decay rate is as follows^[ Reference Akhmedov^82]:

(10) $$\begin{align}\frac{E_{\beta }}{E_{\mathrm{cr}}}={\left(\frac{2{Q}_{\beta }}{m_{\mathrm{e}}}\right)}^{3/2},\end{align}$$

where ${Q}_{\beta }$ is the energy release associated with the decay. In the case of tritium, ${Q}_{\beta}=18.6$ keV and ${E}_{\beta}=0.02{E}_{\mathrm{cr}}$ . As this is a non-relativistic beta decay, ${Q}_{\beta }/m\ll 1$ , the modification to the decay rate is $C\simeq {\left({E}_0/{E}_{\beta}\right)}^{7/3}$ for an applied electric field ${E}_0$ , which satisfies ${E}_0/{E}_{\beta}\gg 1$ ^[ Reference Akhmedov^82]: at ${E}_0=0.1{E}_{\mathrm{cr}}$ , $C\simeq 50$ .

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 $\alpha$ , small, but rather ${\alpha \chi}^{2/3}$ , which can become large in extremely strong fields^[ Reference Ritus^83, Reference Narozhny^84]. Hence, the usual small expansion parameter of perturbative QED becomes, in principle, a large parameter for $\chi >1600$ . In a collision between an electron beam of energy $\mathrm{\mathcal{E}}$ and a supercritical electric field of magnitude $E$ , we have the following:

(11) $$\begin{align}{\alpha \chi}^{2/3}=5.3\kern0.1em {\mathrm{\mathcal{E}}}^{2/3}\left[10\,\mathrm{GeV}\right]{\left(\frac{E}{E_{\mathrm{cr}}}\right)}^{2/3}.\end{align}$$

Higher-order corrections, normally thought of as suppressed by powers of $\alpha$ , are implied by the conjecture to become larger and larger as the order increases. The technical implication is that the perturbative expansion of QED breaks down and needs (somehow) to be re-summed^[ Reference Heinzl, Ilderton and King^85]; the physical implication is that QED enters a new ‘fully nonperturbative’ regime in which it behaves as a strongly coupled theory^[ Reference Fedotov, Ilderton, Karbstein, King, Seipt, Taya and Torgrimsson^5, Reference Fedotov^86]. It is essential that large $\chi$ is reached not by simply increasing the particle energy $\mathrm{\mathcal{E}}$ at low field strength, as the Ritus–Narozhny conjecture only applies in the high-intensity (locally constant, crossed field approximation (LCFA)) regime, where ${a}^3/\chi \gg 1$ ^[ Reference Podszus and Di Piazza^87, Reference Ilderton^88]. Furthermore, the mitigation of radiative energy losses requires the field duration to be kept as short as possible; for alternative scenarios see Refs. [Reference Baumann, Nerush, Pukhov and Kostyukov33, Reference Blackburn, Ilderton, Marklund and Ridgers89–Reference Tamburini and Meuren92].

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, $B$ , is controlled by the Heisenberg–Euler interaction Lagrangian (see, e.g., Ref. [Reference Fedotov, Ilderton, Karbstein, King, Seipt, Taya and Torgrimsson5]). At the one-loop order, $\mathrm{\mathcal{L}}={m}^4{\left(B/{B}_{\mathrm{cr}}\right)}^4/\left(360{\pi}^2\right)$ for $B\ll {B}_{\mathrm{cr}}$ and $\mathrm{\mathcal{L}}={m}^4{\left(B/{B}_{\mathrm{cr}}\right)}^2\ln \left(B/{B}_{\mathrm{cr}}\right)/\left(24{\pi}^2\right)$ for $B\gg {B}_{\mathrm{cr}}$ . For supercritical magnetic fields, higher-order corrections grow logarithmically, with the following^[ Reference Karbstein^93]:

(12) $$\begin{align}\frac{{\mathrm{\mathcal{L}}}^{n-\mathrm{loop}}}{{\mathrm{\mathcal{L}}}^{1-\mathrm{loop}}}\sim {\left[\frac{\alpha }{\pi}\ln \left(\frac{B}{B_{\mathrm{cr}}}\right)\right]}^{n-1}.\end{align}$$

Although this growth is slower than the power-law behaviour of higher-order corrections at ultralarge quantum parameter $\chi$ , as predicted (above) in the Ritus–Narozhny conjecture, re-summation is still required. Investigating this nonperturbative, nonlinear regime of electrodynamics motivates the creation of ultrastrong EM fields that are not probed by ultrarelativistic external particles.