We start reporting the results of simulation A (see Section 2). We use this simulation to describe the main aspects of the laser–DLT interaction and the consequent NICS emission. Figure 1 helps visualize the behavior of the laser and hot electrons during the simulation.

Before the interaction, the laser pulse focused on the target left boundary undergoes free-propagation (Figure 1A laser snapshot at 38 fs). When the interaction with the low-density layer starts, the pulse experiences a progressive alteration of its spatial and temporal shape, and a gradual absorption [56]. The near-critical plasma behaves as a lens causing relativistic self-focusing [57–59]. This phenomenon induces the reduction of the laser spot size, the increase of the laser pulse intensity, and, consequently, the pulse confinement in a channel in which the electron density is depleted because of the ponderomotive force. As shown by the blue line in Figure 1A, the laser waist is decreased symmetrically inside the foam from 3 µm down to nearly 1 µm. During the interaction with the near-critical layer, a = e|E|/(m _e ω ₀ c) increases from 50 to 70 and goes above 100 when the laser is reflected on the substrate (Figure 1A laser snapshot at 108 fs). As a consequence of reflection, the electromagnetic field strength is further enhanced due to the superposition of the incident and reflected pulse. A temporary standing wave builds in front of the substrate, and the peaks of the normalized electric field e|E|/(m _e ω ₀ c) and magnetic field e|B|/(m _e ω ₀) alternatively grow up to 150 at the nodes and antinodes of this wave. Self-focusing and superposition can thus significantly boost NICS. Another phenomenon is the magnetization of the channel [60]: following the laser pulse propagation, the B _z component of the field fills the channel in two symmetric regions of opposite signs (Figure 1B).

The channel dug by the laser inside the plasma has a variable diameter that follows the reduction of the laser size during propagation. Hot electrons are generated in this channel with a broad energy spectrum with maximum energies up to 460 MeV. Indeed, when the interaction starts, electrons are rapidly accelerated via direct laser acceleration [61]. The trajectories of the most energetic electrons, black and blue lines in Figure 1B, show that, at the beginning of the interaction, the electrons are pushed backwards and then are accelerated into the channel following the laser pulse and co-moving with it. Simultaneously, the electrons experience betatron oscillations in the transverse direction driven by the magnetization of the channel, which, in addition, constrains electrons to be confined in the channel itself [61].

Some examples of the higher-energy emission are presented in Figure 1B, where the emitted photons are shown in green, and the emitting electron appears in blue. In general, we can distinguish two emission phases. Photons are produced firstly during the laser propagation through the foam; we call this the ramp phase [43] because emission occurs at a constant rate and the energy radiated away grows linearly. Then, a peak of emission happens in front of the substrate (burst phase [43]) when electrons see the counterpropagating reflected laser field. In Figure 2A, the number density of all emitted photons is reported. The emission occurs predominantly in the channel region of the foam, where most energetic electrons are confined. In Figure 2B, all the emitted photons are represented as arrows directed according to their momentum and colored according to their energy. We can recognize low-energy backward emitted photons, especially in the early phase of interaction (left side of the channel), and higher-energy forward emission due to oscillating electrons in the channel. Most photons and the most energetic ones are emitted before the substrate during the burst phase. The distributions in Figure 2C describe the directionality of photon emission in more detail. Although a background of emission is present at all angles due to the chaotic motion of electrons, photons are mainly emitted backwards during the ramp phase (blue curve) and in a cone of 30° around the forward direction during the burst phase (green curve). The energy distribution in the ramp phase (yellow curve) shows a more relevant contribution in energy by the forward emission due to electrons undergoing oscillations and following the laser pulse, as recognized in Figure 2B. During the burst phase, the emission shows two lobes peaked at 15°. This bilobal structure is typically observed when emitting electrons accelerated by a linearly polarized laser undergo oscillations in a near-critical plasma channel [23, 25, 27, 29, 62, 63]. The peak angle of the bilobal structure is determined by the angular deviation of betatron oscillations at the instant of copious burst emission. This angle is given by ϕ ∼ r _β ω _β /c [63, 64], where r _β is the amplitude of the oscillations which is approximately given by the channel radius, and ω β = ω p / 2 γ with ω _p the plasma frequency. Approximating r _β to the minimum waist achieved by the laser, that is 1 µm, and γ to 800, which is a value achieved by the most energetic electrons in the simulation, the angular deviation of oscillations results in 15,9°, which is in good agreement with what we see in Figure 2C.

The maximum values of χ reached by electrons are around 0.04 in the ramp phase and up to 0.33 in the burst phase. Since electron χ goes above 0.1 in this phase, we expect to see the impact of quantum effects [3, 48]. Using the approximation for χ mentioned in Section 1 for the counterpropagation condition, we can simply estimate the maximum value for this parameter in the burst phase using χ ≈ 2γa _f ℏω ₀/(m _e c ²), where a _f is the maximum normalized vector potential achieved by self-focusing. We can check that using γ ≈ 800 and a _f ≈ 75, which are reasonable values for simulation A, this formula gives a χ value around 0.33. More insights are gained by discussing a more specific approximation for χ. Since the most relevant fields in the physical system are the laser ones, E _y and B _z , and the electrons are mainly accelerated in the forward direction, we can evaluate χ using this approximation of formula 1: χ ≃ γ a s | E y − β x B z | , (2) where a _s = mc ²/(ℏω ₀), β _x = v _x /c with v _x the electron velocity component in the laser propagation direction, and E _y and B _z are normalized by m _e ω ₀ c/e and m _e ω ₀/e, respectively. As shown in Figure 3A by the red line, this approximated formula describes well the maximum value of χ achieved during the simulation. An improvement in the agreement is obtained considering also the term β y 2 ( B z 2 − E y 2 ) due to the transversal motion of electrons (Figure 3A blue line). This panel helps understand which fields are determining the emission. Electric and magnetic fields’ contributions appear with opposite signs in formula 2, meaning that if the two fields are in phase, they cancel each other, and the emission is low unless β _x is negative (counterpropagating electron). This case occurs in the ramp phase in which emission is mainly due to low-energy electrons travelling against the laser (see backward emission in Figure 2). Furthermore, the rising quasi-electrostatic fields and a relevant y component of electron velocity lead to further emission during the ramp phase (Figure 3A blue line). In the burst phase, when reflection occurs, superposition leads to alternatively maximizing one field amplitude while lowering the other, and the E _y and B _z fields are phase-shifted. This effect is reported in Figure 3B. In the region between 110 and 130 fs, E _y and B _z values on the macroelectron with maximum χ at each timestep (represented by connected dots) are alternatively large and small, and their signs end up being opposite. The electric and magnetic fields’ contributions are not cancelling out, and χ reaches the maximum possible values as long as hot forward-propagating electrons are available to experience these fields.

We compare some properties of emitted photons in simulation A with analytical results. Since Smilei simulates emission according to the LCFA and, thus, using the results for NICS in constant crossed fields, that is, when E⊥B and |E| = c|B|, we can use the results of this theory [3, 65–68] to estimate the evolution of the emitted power and the final spectrum of all emitted photons. We rearrange the formulas of this theory as functions of macro-electron properties: parameter χ, kinetic energy E _e , and numerical weight w _e . The instantaneous emitted power by all electrons is given by the summation of their contributions at a given time: P rad = 2 α 2 m e c 2 3 τ e ∑ e χ 2 g χ w e , (3) where g χ is equal to 1 in the classical limit and given by the following expression in the general case: g χ = 9 3 8 π ∫ 0 ∞ 2 ξ 2 2 + 3 χ ξ 2 K 5 / 3 ξ + 36 χ 2 ξ 3 2 + 3 χ ξ 4 K 2 / 3 ξ d ξ , (4) where K _v are the modified Bessel functions of the second kind, and ξ is the variable of integration. In Figure 4A, we have plotted the evolution of the emitted power in the simulation (black) compared with the same quantity obtained from formula 3 in the classical case (red) and with the quantum correction of formula 4 (blue). More precisely, the plotted quantities are power densities due to the reduced dimensionality of the simulations. The evolution of emitted power shows the two distinct phases we have highlighted in Section 3.1. The emitted power is nearly constant from 60 fs to 110 fs; thus, the energy converted in photons linearly increases: ramp phase. Instead, around 120 fs, the peak of emission characteristic of the burst phase occurs. While emission in the ramp phase is well-described in a classical framework of synchrotron emission, the emission peak in front of the substrate needs quantum corrections to be estimated correctly. This fact indicates that the quantum regime of synchrotron emission is observed here. The plot confirms that the quantum effects arising when χ approaches one reduce the emitted power against the classical case.

The energy spectrum of all emitted photons during the simulation is given analytically by a summation of electrons contributions integrated in time: d N d E p = α 2 m e c 2 3 π τ e ∫ t 0 t 1 ∑ E e ≥ E p w e 1 E e 2 E p 2 E e E e − E p K 2 / 3 y + ∫ y ∞ K 5 / 3 x d x d t , (5) where t ₀ and t ₁ are the starting and ending time of the simulation, respectively, E _p is the photon energy, and y is given by the following equation: y = 2 E p 3 E e − E p χ . (6)

It is worth noting that E _e , χ, and, consequently, y are time-dependent quantities. The classical limit, corresponding to the usual synchrotron emission formula [54], is obtained neglecting the recoil of the radiating electrons, that is, by taking E _p ≪ E _e in Eqs. 5, 6: d N d E p = α 2 m e c 2 3 π τ e ∫ t 0 t 1 d t ∑ E e ≥ E p w e 1 E e 2 ∫ y ∞ K 5 / 3 x d x , (7) y = 2 E p 3 E e χ . (8)

In Figure 4B, we compare the spectrum obtained in the simulation (solid black line) with those analytically estimated in the classical case (dashed red line) and with quantum corrections (dashed blue line). The NICS spectrum is broad with an exponential shape. This shape is expected considering the typically broad, nearly exponential spectra obtained for electrons in DLTs [69–71]. The quantum corrections change the slope and cut-off of the estimated spectrum at high energy since the most energetic emission is expected to come from electrons with higher χ and thus more affected by these corrections. The cut-off of the simulated spectrum is not present in the estimated ones. This effect is due to the reduced number of particles per cell used for macro-electrons in our simulations. If this number is low, reduced sampling affects the high-energy electrons and the photons they emit. We have tested that a visible improvement in the high-energy sampling in this plot occurs only by strongly increasing the number of macro-electrons by at least two orders of magnitude, making the simulation more expensive in computational resources.

The procedure described in this section is similar to some strategies reported in literature [72, 73] evaluating photon emission without using a Monte Carlo approach. Our estimates were performed after the simulation using the macro-electron properties saved by the diagnostics with a high time resolution. Even if the stochasticity of emission is not considered, formulas 3 and 5 can be valuable tools to check and estimate emission after PIC simulations in any case of interest and not only in the DLT case. Furthermore, this procedure can be used to recognize and evaluate the impact of quantum effects of NICS emission according to the laser and target configurations, as we have demonstrated in the DLT case.

Using analytical formulas and simulations, we can compare NICS and bremsstrahlung in the range of parameters explored in this work. Similarly to Eq. 5, we can estimate bremsstrahlung generated in the substrate, which is the target region with higher density and atomic number, thus the most relevant for this type of emission. We use a generalization of Eq. 7 of Ref. [46]: d N d E p = n i ∫ t 0 t 1 d t ∫ E p ∞ d E e d N d E e v e a Z 2 E p 1 − b E p E e , (9) where n _i is the ion density of the substrate, dN/dE _e is the time-dependent spectrum in energy of electrons inside the substrate, a = 11 ⋅ 10^–31 m² and b = 0.83. In Figure 5, we report NICS-bremsstrahlung comparisons in the case of foam thickness 25 µm and foam density 1 n _c varying the parameter a ₀ (20–40-50–60). We show the NICS spectra obtained with Smilei (green), the bremsstrahlung spectra from analogous simulations using the PIC code EPOCH [74] that allows for Monte Carlo evaluation of bremsstrahlung [46] (blue), and the analytical estimates of bremsstrahlung using formula 9 (red). Due to the lower efficiencies of the two processes, we consider them independent, and we do not activate NICS emission in EPOCH. When a ₀ = 40, 50, 60, bremsstrahlung is lower than NICS by at least two orders of magnitudes on the whole range of the spectrum. Instead, at a ₀ = 20, NICS contribution is strongly reduced due to the relatively low laser intensity and becomes closer to the bremsstrahlung contribution. In this case, the two processes contribute similarly: the high-energy spectrum is dominated by bremsstrahlung, while in the low-energy portion, synchrotron radiation prevails. Overall, we deduce that the impact of NICS in our DLTs in non-extreme laser regimes is remarkably relevant. The results of Figure 5 can be discussed considering that NICS and bremsstrahlung have a different dependence on laser intensity. Bremsstrahlung depends on the electron energy achieved during the laser-driven acceleration of electrons and, thus, indirectly on laser intensity. On the other hand, NICS has an explicit dependence on intensity and the production yield goes approximately as a 0 3 as demonstrated for different target cases [25, 40, 75] and approximately valid also in our parametric scan (see Figure 7A). Increasing the intensity at a fixed target thickness, NICS will ultimately prevail on bremsstrahlung. Even if we are considering a foam-attached target, our results are consistent with the bare-aluminum case of [76]: in a micrometric target of aluminum, bremsstrahlung can be ignored for intensities > 1 0 21 W/cm². Practically, only in the case a ₀ = 20, among those presented, NICS and bremsstrahlung are comparable. In this case, the competition between the two processes could be discussed in detail with more accurate simulations, for example, in 3D geometry. If the thickness and/or the atomic number of the solid substrate are changed, the bremsstrahlung contribution becomes much more important [17, 76, 77], and the threshold to ignore bremsstrahlung shifts at higher intensities.

We report results from the whole simulation scan focusing on the dependencies of NICS emission properties on the laser and target parameters. We have chosen two properties for this analysis: the conversion efficiency from laser energy into photon energy and the average energy of all emitted photons (Figure 6). These quantities are expected to be useful in assessing the feasibility of applications and can be quantitatively evaluated in 2D simulations avoiding units altered by the reduced dimensionality. The conversion efficiencies represent the ratio between the energy in photons generated in the simulation, which have energy above 0.25 m _e c ², and the laser energy. The presence of this cut-off reduces the conversion efficiency since only the high-energy portion of the electromagnetic spectrum is considered but avoids generating a lot of low-energy photons, which can be computationally heavy to handle. Furthermore, as noted in Section 3.2, low-energy emission is not affected by quantum effects and, thus, less relevant to be simulated with a Monte Carlo approach.

In Figure 6A, we observe that the conversion efficiencies tend to show an optimum case in thickness for fixed a ₀ and the foam density. Exceptions are the low-density cases at high a ₀ where probably the scan is not covering the optimal cases corresponding to higher foam lengths, as shown by [40]. By decreasing a ₀ or increasing the foam density, this optimum tends to shift toward lower lengths, and its value decreases. This effect is due to the lower transparency of the plasma to the laser. Shorter foams grant that the laser can reach the reflection on the substrate and thus the counter-propagation condition of the burst phase that enhances the efficiency and energy of emission. Contrarily, the cases with dense and long foams tend to suppress the burst emission because of relevant laser absorption during the propagation, a phenomenon evidenced in [78]. Because of this, the conversion efficiencies in long foam cases increase by lowering the density. However, denser cases can become optimal at shorter foam lengths.

The average photon energy (Figure 6B) shows similar optimal lengths once fixed a ₀ and the density. The same comments on efficiencies are valid for the average photon energy, although dense foam cases have lower performance considering this parameter even in thinner foams.

The processes involving the laser pulse, in particular, the self-focusing and the consequent electron heating, directly impact NICS and its dependence on target parameters. 3D PIC simulations have revealed that the self-focusing effect generated by the near-critical target is representable as the action of a lens [56]. A thin-lens model can be used to obtain the focal length f of the near-critical layer and to describe the self-focusing process. This model gives f ≈ w 0 / n ̄ with the transparency factor n ̄ = n e / ( γ 0 n c ) and γ 0 = 1 + a 0 2 / 2 , where w 0 is the beam waist [59]. f gives the rule to calculate the distance inside the foam at which the laser reaches its maximum focusing before being absorbed considerably. For example, if a ₀ = 50, w 0 = 3 µm, and n _e /n _c = 2, the self-focusing length is equal to 12.6 µm. Therefore, in simulation A the foam is long enough to reach the maximum possible focalization of the laser pulse. Since absorption and filamentation prevail for propagation longer than the self-focusing length, the foam thickness must be tuned to maximize the intensity when the substrate is reached, which means roughly having a foam length equal to the self-focusing length f. Despite relativistic self-focusing being an inherent 3D process, this reasoning is still valid in our 2D simulations when comparing the optimal values with the vertical lines corresponding to the self-focusing lengths in Figure 6. In many cases, the optimal length identified when fixing foam density and laser intensity is approximately the self-focusing length with a better matching when considering the average photon energy. Indeed, conversion efficiencies take advantage of laser absorption; thus, optimal values slightly longer than the self-focusing lengths are expected due to the increased laser absorption. With n _e /n _c = 1 at high intensities, the optimal lengths go beyond the self-focusing limit. This fact is reasonable since the transparency factor n ̄ becomes small in these cases, and, consequently, the thin-lens model starts to fail [59].

Figure 7 reports the conversion efficiency, panel (A), and the average photon energy, panel (B), varying a ₀, thus making identifiable their dependence on this parameter. The simulations considered here are only those with foam thickness closest to the self-focusing length once fixed a ₀ and n _e /n _c . This plot confirms a general trend reported in the literature according to which the conversion efficiency from the laser in photon energy goes approximately as a 0 3 . Instead, the average photon energy shows a milder dependence going approximately as a 0 1.25 . Overall, the laser intensity proves to be the essential parameter deciding the order of magnitude of the emission properties we focused on.

Considering a nanostructured morphology for the target, the quantitative results reported in this section and the previous ones would be different. Alteration of the interaction processes due to the presence of the nanostructure is expected [79]. We mention that, as reported in [39, 43], including a nanostructured morphology for the foam in 2D simulations has mainly the impact of reducing the photon yield and maximum photon energy due to density dishomogeneities and more chaotic motion of electrons with respect to the uniform case, albeit remaining more efficient than the single layer. For this work, based on 2D simulations, we focus on something other than the detailed morphology expecting it to be worthy of consideration in more realistic 3D simulations. In addition, relatively high laser intensities tend to homogenize the nanostructure [79], and, if the laser contrast is not very high, the laser prepulse is expected to ionize and make the foam uniform before the interaction with the main laser peak.