The availability of ultraintense lasers has paved the way to studying physics in extremely strong electromagnetic fields and under high-energy-density1 conditions. Laser intensity is the primary parameter characterizing distinct realms of physical behavior.2 At intensities above 10¹⁸ W/cm², electrons can be instantaneously accelerated to near the speed of light within one laser cycle. Their collective motion and the resulting charge separation in a plasma can lead to acceleration gradients orders of magnitude higher than those in conventional accelerators,3,4 which have been utilized to obtain GeV-level electrons and heavy ions, and >100 MeV protons.5 There are theoretical predictions that at higher intensities, rich new physics is waiting to be explored. For example, quantum radiation reaction6 will become non-negligible above 10²⁴ W/cm², nuclear structure will be influenced above 10²⁶ W/cm², and electron–position pairs will be produced from the vacuum at 10²⁹ W/cm².7 To enter these realms, a number of petawatt (PW) and 10 PW laser facilities have been built and utilized in experiments.8 The intensity record to date is 10²³ W/cm²,9 and intensities exceeding 10²⁰ W/cm² have been widely achieved worldwide. However, it should be noted that the reported intensity values—obtained by dividing the laser energy by the focal area and pulse duration—warrant revisiting, since the focal spots at relativistic intensity have not been characterized in real time and in situ. Typically, they have been characterized offline using front-end pulses, assuming these to be stable and identical to the full-power pulses. This approach is problematic, because the thermal lens effect and laser jitter can bias the results. Moreover, such a method cannot take into account the spatiotemporal distortion of the pulses, which may lead to an order-of-magnitude overestimate of the intensity.10

Instead of indirect and non-in situ measurements, many schemes have been proposed to directly gauge laser intensity by exploiting laser–matter interaction.11–25 It has been proposed that the peak intensity can be inferred by observing scattered light,11–14 electrons,15–19 or protons20,21 from laser-irradiated electrons/ions. Alternatively, field ionization of high-Z atoms22,23 or electron–positron pair production24 under extremely high intensities can also be employed. All of these methods rely on the interaction of the laser with isolated electrons or atoms. They are however, unable to provide a micrometer-scale intensity distribution at the focal plane, but only an average over the Rayleigh length. Besides, other effects such as self-focusing and space charge will complicate the analysis. Owing to their intrinsic limitations, these proposed schemes have rarely been used in experiments.

The relativistic electrons generated at the laser–solid interface encapsulate valuable information about the interaction. As a result of the interplay between the pressure of light and the electrostatic force, high-energy electrons are produced in a series of thin sheets spaced by one or half of the wavelength. If we can determine the characteristics of these sheets, such as their sizes, shapes, and the amounts of charge they contain, we can deduce both the absolute value and the spatial distribution of the laser intensity on a specific plane. Previous studies have demonstrated that the amount and temperature of the electrons can be deduced by imaging the coherent transition radiation (CTR) spot on the rear surface of a laser-irradiated thin foil.26–30 However, this method is limited by the resolution of the imaging system and cannot provide more detailed information. Additionally, emissions from bremsstrahlung or blackbody radiation, which also concentrate in the focal area, will introduce false signals into the results.

In this work, we report the first demonstration that the farfield patterns of coherent radiation from laser-irradiated ultrathin foils contain sufficient information to reconstruct the intensity distribution on the surface of the foils, despite the challenges posed by the highly nonlinear relativistic interaction and noisy background signals. With the assistance of particle-in-cell simulations, we can obtain the absolute value of the peak intensity. We also demonstrate that the patterns can be utilized to optimize the focal spots for highest intensity, which is of great value for experimental studies at extreme intensities.

The experiments were conducted using an ultraintense Ti:sapphire femtosecond laser system at Peking University. Figure 1(a) shows the setup. The 30 fs laser pulses with a central wavelength of 800 nm delivered energy of ∼1.2 J on the targets after a plasma mirror. The temporal contrast was improved to 10⁻¹² at 40 ps prior to the main pulse.31 The focal spots were measured using a microscopic imaging system by attenuating the front-end laser pulse (∼60 mJ) to nanojoule level. The focal spot had a FWHM of ∼4 μm, and the laser energy portion contained in the FWHM of the focal spot was 16%, corresponding to a peak intensity of $\sim 5\times 10^{19}\text{W}/\text{c}{\text{m}}^2$. Polymer, copper, and gold foils with thicknesses of less than 500 nm were shot at normal incidence. To capture the farfield optical radiation, a Teflon scatter screen was placed 49 cm away from the laser focus in the transmission direction. The patterns on the screen were imaged using lenses and cameras after two 40 nm-bandwidth filters at central wavelengths of 800 nm (1ω_L) and 400 nm (2ω_L), respectively. The spectra of scattered light from the screen were measured using an optical fiber spectrometer. The energy spectra of laser-accelerated ions were analyzed using a Thomson parabola spectrometer.

Figure 1(b) shows the farfield patterns observed for 300-nm polymer foils irradiated by laser pulses with linear polarization (LP) and circular polarization (CP). A doughnut-shaped 2ω_L pattern is clearly observed for LP, with an intensity approximately ten times higher than that for CP. The 1ω_L patterns for both polarizations are nearly identical. Figure 1(c) displays the typical spectrum of a 100-nm polymer foil for LP laser. Narrowband peaks at 400 and 800 nm are clearly seen, indicating that the optical signal is coherent.

The on-target laser spot size and intensity were tuned by moving the focusing off-axis parabolic (OAP) mirror along the light axis. As the defocus distance $\left|\mathit{\Delta }\right|$ increased, the on-target spot size increased and the intensity decreased, while the laser power remained constant. Figure 1(d) shows the yield of 2ω_L and 1ω_L within a polar angle θ < 15° and the maximum proton energy ɛ_p as functions of Δ for 300-nm polymer foils. The OAP position at which ɛ_p peaks indicates the optimal focus (Δ = 0). One can see that the 2ω_L yield W_2ω is considerably higher for LP than CP, and varies with Δ in a manner analogous to ɛ_p. By contrast, the 1ω_L yield W_1ω exhibits no obvious dependence on Δ for either polarization.

The 2ω_L signal is contributed largely from CTR of laser-driven relativistic electron sheets as they traverse the rear foil–vacuum interface. The 100s-nm-thin foils employed here can minimize the disturbance of transport on electron sheets in the targets. At normal incidence, the v × B force of a LP laser field periodically accelerates sheets with 400-nm longitudinal spacing,32 which is absent for a CP field. Thus, the 2ω_L radiation is significantly enhanced for LP lasers. As far as the recorded 1ω_L patterns that show no obvious dependence on Δ for either polarization are concerned, we believe that they are dominated by the leaked laser pulses, since the other mechanisms that can produce 1ω_L electron sheets, such as vacuum heating33 and resonant absorption,34 are not prominent at normal incidence for high-contrast lasers. In this work, we focus on the 2ω_L signals from LP lasers.

Figure 2(a) displays the 2ω_L farfield patterns at different Δ for 500-nm gold and 100-nm polymer foils. It is clear that the diameters of the doughnut-shaped patterns decrease for larger Δ. The divergence angle θ_2ω is employed to depict the size of the doughnut patterns (see the supplementary material for its definition). Figure 2(b) shows that θ_2ω decreases with increasing Δ for both targets. On the contrary, the maximum energy of laser-accelerated protons increases with increasing θ_2ω, as shown in Fig. 2(c). Further experimental results (see the supplementary material) indicate that the doughnut-shaped patterns exist widely for foils of various thicknesses and materials.

Three-dimensional particle-in-cell simulations were performed using the EPOCH code35 to reproduce the experimental observations. A 300-nm polymer (CH₂) foil was normally irradiated by a 800-nm, 30-fs, linear polarized Gaussian laser pulse with a normalized amplitude $a_0=e{E}_0/m_{e}c\approx 0.855{\lambda }_L(\mu \text{m})\sqrt{I_0(\text{W}/\text{c}{\text{m}}^2)/{10}^{18}}=5$, where e, m_e, c, E₀, λ_L, and I₀ are the charge and rest mass of the electron, the speed of light in vacuum, the maximum amplitude of the laser field, the laser wavelength, and the laser peak intensity, respectively. The laser propagates in the +x direction. The laser focal spot size (FWHM) is D_L = 4 μm. The foil’s initial electron density is 48n_c, where $n_c=m_e{\omega }_L^2/4\pi e^2$ is the critical density. The simulation box is 10 × 20 × 20 μm³, discretized by 1000 × 500 × 500.

To facilitate comparison with experiment, the farfield pattern on the plane x = 49 cm is calculated using the Kirchhoff diffraction integral from the 2ω_L electric nearfield, which is extracted from the temporal Fourier transform to the nearfield evolution on the plane x = 5 μm. As illustrated in Fig. 3(a), the calculated farfield pattern closely resembles the doughnut-shaped pattern observed in the experiment.

The farfield radiation pattern is determined by the electron distribution in time t, transverse position r, and velocity v, where t is the time at which an electron arrives at the foil’s rear surface and radiates. The spectral intensity of CTR of all the relativistic electrons (v → c) in a cylindrically symmetric geometry, considering that the radiation energy is concentrated in a small angle θ, can be approximately derived as $d^{2}W/d\omega d\mathrm{\Omega }\propto {\sin }^2\theta {\left|F\right|}^2$ (see the Appendix and also Refs. 36–38), where $F=\int f\left(\mathbf{r},t\right)\exp \left(-i{\mathbf{k}}_{\perp }\cdot \mathbf{r}+i\omega t\right)d^2\mathbf{r}dt=\mathcal{F}\left[f\left(\mathit{r},t\right)\right]$ is the form factor of the electron sheet, f(r, t) is the normalized spatial distribution function of the electrons, and k_⊥ = 2π sin θ/λ.

The electron sheets bend into bow shapes instead of planar ones, owing to the spatially varying laser intensity, as shown by a snapshot of the density distribution at 35 fs of forward-moving electrons with kinetic energy greater than 0.5 MeV in x − r space, where $r=\sqrt{y^2+z^2}$. The section at r < 0 is a mirror image of that at r > 0 for better visualization. For simplicity, we model the density of the curved electron sheet as $f\left(r,t\right)\propto \exp \left(-4\ln 2r^2/D_e^2\right)\delta (t-r^2/2cR)$, where R is the radius of curvature. Therefore, the farfield intensity distribution at 2ω_L can be derived as $I_{FF}\left(\theta \right)\propto {\sin }^2\theta \exp (-4\ln 2{\sin }^2\theta /D^2)$, where $D=(2\sqrt{2}\ln 2\lambda /\pi D_e)\sqrt{1+{\left(\pi D_e^2/4\ln 2R\lambda \right)}^2}$. This naturally leads to a doughnut-shaped pattern matching that in Fig. 3(a).

Then, we extract the divergence angle as θ_2ω = sin⁻¹(D/2). For θ_2ω ≪ 1, and D_e ≈ D_L,$$\begin{array}{c}\hfill {\theta }_{2\omega }\approx D_L^{-1}{A}_0\sqrt{1+{\left(\frac{\pi }{4\ln 2}\right)}^2{\left(\frac{D_L^2}{R\lambda }\right)}^2},\hfill \end{array}$$(1)where the coefficient $A_0=(180/\pi )\sqrt{2}\ln 2\lambda /\pi =$ 7.15 μm° corresponds to a perfectly flat sheet. For curved electron sheets, ${\theta }_{2\omega }{D}_L\stackrel{def}{=}{A}_R>A_0$, and their θ_2ω will be larger. Physically, a curved electron sheet radiates with a curved wavefront, which results in a more diverging pattern.

In the simulations, we varied the defocus distance Δ for a₀ = 5, D_L = 4 μm, and we plot the on-target laser intensity distribution along with the corresponding farfield 2ω_L patterns in Fig. 3(c). It can be seen that the dependence of θ_2ω on D_Δ is quantitatively consistent with the experimental results in Fig. 2(b). It is worth noting that if planar sheets are taken (R → ∞), then the θ_2ω given by Eq. (1) is significantly smaller than the simulation results, as shown in Fig. 2(b).

In addition to the experimental parameters, the simulations were extended to include a wider range of laser intensities and focal spot sizes. It is found that θ_2ω is inversely proportional to D_L for a fixed a₀, as shown in Fig. 4(a). The scaling coefficient A_R = θ_2ωD_L decreases with a₀, as shown in Fig. 4(b).

Roughly speaking, the shape of the curved electron sheet coincides with the isosurface of equal intensity of ∼10¹⁸ W/cm², where the electrons’ velocity v → c. For a Gaussian beam with fixed D_L, a larger a₀ leads to a larger area of the isosurface, i.e., a larger R. According to Eq. (1), A_R = θ_2ωD_L decreases, as demonstrated in Fig. 4(b). It should be noted that this trend differs from that for an oscillating overdense plasma surface under high intensity.39 In the latter case, the plasma surface becomes more curved at higher intensity, as a result of the balance between radiation pressure and charge-separation field.

The relationship of θ_2ω to D_L and a₀ can be used to diagnose the focal spot at relativistic intensities in experiments. Figure 4(c) plots contours of a₀ and D_L as functions of θ_2ω and laser power P (Gaussian pulses) according to the simulation results. We can use this figure to speculate about the on-target a₀. As an example, θ_2ω = 3.42° for the LP laser in Fig. 1(b). Since the on-target laser power P = ɛ_L/τ_L ∼ 40 TW, if the focal spot were an ideal Gaussian, then Fig. 4(c) would give a₀ = 15.1 (the black star). Considering that the real focal spot is not an ideal Gaussian, the actual value of a₀ will be smaller. Figure 4(d) shows the laser spot measured with the microscopic imaging system at low intensity. The encircled energy contained within the FWHM diameters of the ellipse is 16%, which is far less than the 50% of a Gaussian spot. If we use the corrected power P = 12.8 TW, then the corresponding a₀ is 7.7 (the black square).

By virtue of the coherent diffraction imaging algorithm,40 it is possible to retrieve the delicate laser intensity distribution, including all the details of the farfield pattern, and obtain the accurate peak intensity directly. This is worth some endeavor in the future work. On the other hand, even without such information, the farfield pattern provides an immediate and in situ assessment of the actual laser focusing quality in the experiments, which can guide the adjustment of adaptive optical systems or OAPs for optimal results. As an example, in the experiment, when we horizontally rotated the OAP in the range of −0.008° to 0.008°, which induces astigmatic aberrations to the focal spots, the farfield 2ω_L patterns were tilted and deteriorated as seen in Fig. 4(e). The optimal maximum proton energy was obtained when the pattern was close to a perfect doughnut.

In experiments at relativistic intensities, normal incidence on a solid foil is dangerous for the laser system if it lacks the protection of a plasma mirror as in our case. The allowed minimal tilting angle is about 4°. We also studied the situations where the foils are slightly tilted. For a tilted ultrathin foil target, the flying direction of the sheets is still along the laser axis. The TR in the direction away from the target normal is stronger, which will lead to asymmetric farfield patterns. Fortunately, such an asymmetry becomes less prominent with the rise in intensity due to the greater longitudinal momentum of the electrons.36 Simulations indicate that the doughnut pattern for a₀ = 5 is still distinguishable at an incident angle θ_i = 5°, but becomes obscure at θ_i = 10°, as shown in Fig. 5(a). When a₀ is increased to 15, the doughnut is still clear even at θ_i = 10°. In the experiments, we tilted the targets and found that the changes in the patterns were consistent with the simulation results.

For any intensity gauging method, its range of applicability is an important issue. 3D simulations indicate that the dependence of θ_2ω on intensity becomes less prominent for a₀ > 30 [Fig. 5(b)]. Instead, the total energy of the 2ω_L emission contained in the pattern W_2ω increases steeply with rising a₀, which can be utilized to assist in determining the intensity in combination with the doughnut patterns.

We also considered the influence of the preplasma on the farfield pattern. Figure 5(c) illustrates that if the scale length is small, then W_2ω increases owing to enhanced laser absorption. However, too long a scale length leads to electron generation far away from the foil, which makes the electrons less coherent, and therefore W_2ω decreases. Moreover, the farfield pattern becomes more complicated as the electrons gain more transverse momentum along the laser polarization direction.

We have found that the coherent radiation farfield patterns from laser–ultrathin foil interaction depend on the shape and curvature of the accelerated relativistic electron sheets, which is a result of the spatially varying laser intensity distribution on the focal plane. This relationship can be employed to perform a real-time, in situ, and easy-to-implement diagnosis of the relativistic laser focus, from which it is possible to infer the on-target peak intensity and distribution. Despite the deterioration of farfield patterns in the presence of a very large preplasma and the less pronounced variation of divergence angle at very high intensity, this method offers rich information about highly condition-sensitive interactions and can greatly aid experimentalists in their research.

See the supplementary material for further experimental data on the farfield patterns.

Acknowledgment. The authors would like to express their gratitude to J. Schreiber for a fruitful discussion. This work was supported by the Guangdong High Level Innovation Research Institute (Grant No. 2021B0909050006), the National Grand Instrument Project (Grant No. 2019YFF01014402), and the National Natural Science Foundation of China (Grant No. 12205008). W. Ma acknowledges support from the National Science Fund for Distinguished Young Scholars (Grant No. 12225501). The particle-in-cell simulations were carried out on the High-Performance Computing Platform of Peking University.