The experimental set-up and methodology

The experiment was performed with a 20 terawatt, chirped-pulse-amplified titanium-doped sapphire-based intense laser. The main interaction pulse of duration 30 fs and peaked around a central wavelength of 800 nm was focused with an off-axis parabolic mirror to a focal spot size of 12 μm × 15 μm on a millimetre-thick aluminium-coated BK7-glass slab to produce peak irradiances of 3 × 1018 W cm–2. A time-delayed probe laser pulse, derived from the main interaction pulse, was converted into its third harmonic at a central wavelength of 266 nm using a pair of beta-barium-borate (BBO) crystals. A third-harmonic probe can penetrate to near-solid densities of ∼1022 cm–3 in the plasma at near-normal incidence, nearly an order of magnitude higher than the critical density of the main interaction pulse (Fig. 1a). The incident probe was focused loosely to a spot-size diameter of ∼75 μm on the target, while the reflected probe was channelled through ultraviolet-sensitive high-extinction-ratio Glan-Taylor polarizers into a set of ultraviolet-sensitive charge-coupled-devices coupled with narrow-bandpass interference filters allowing radiation at 266 nm. The optical resolution of the imaging system, calibrated with a standard USAF-1951 target, was measured as 3.1 μm, which, along with the probe focal-spot diameter on the target plane (∼75 μm), specified the spectral range for our observations. Further details on the methodology of mapping the spatial and temporal evolution of the megagauss magnetic fields by pump-probe Cotton-Mouton polarimetry can be found in our previous work25.

The temporal delay of the probe with respect to the main interaction pulse was initially varied until 8 ps with a 200-fs resolution, and the experiment was then repeated until a temporal delay of 75 ps with a 1-ps resolution. In addition, the experiment was also repeated for bulk aluminium and bulk copper targets employing a second-harmonic probe, peaked at a central wavelength of 400 nm, under otherwise identical experimental conditions. Qualitatively similar turbulent features could be observed, although the data were more prominent for the third-harmonic probe rather than the second-harmonic one. This is because the strong self-generated second harmonic of the laser from the plasma amplifies the background noise in the detection system, while using a second-harmonic probe. This effect is even more prominent at longer timescales, when the reflected probe is weak. In contrast, the third-harmonic background noise was found to be non-perturbative and negligible for all timescales, while using a third-harmonic probe.

The magnetic field polarigrams obtained by the measurement of the Stokes’ parameters of the incident and the reflected probe25 were fast-Fourier-transformed with a rectangular windowing function. Other standard windowing functions (such as Hann, Hamming, Gaussian and triangular) were found to produce qualitatively similar results with the spectral kink even more pronounced.

Determination of the effective degree of ionization

One-dimensional MULTI-fs26 hydro simulations were used to estimate the effective degree of ionization Z of the target. The effect of a 1-ns prepulse at an intensity of 3 × 1013 W cm–2 (consistent with an experimentally measured, laser-intensity contrast of 10−5) was simulated on a bulk (510-μm thick) aluminium target. The main interaction pulse with a pulsewidth of 30 fs and at an irradiance of 3 × 1018 W cm–2 was made to interact with the preplasma generated by the prepulse. A temperature of ∼750 eV was obtained in the underdense plasma, rapidly decreasing with increasing density while approaching the bulk of the initially cold solid target. According to FLYCHK27 simulations, temperatures of ≲750 eV are consistent with a degree of ionization of Z≲12. The expansion speed c s of the plasma calculated from the above parameters was found to be consistent with time-resolved shadowgraphy measurements of the expanding plasma under identical experimental conditions. The simplistic model described here is aimed at obtaining only an approximate order-of-magnitude estimate for the ion cyclotron frequency ω ci and the ion gyroradius ρ i .

Model for spectral scaling

The experimental observation of an initially single spectral index (α≈2), followed by the evolution of a spectral break separating two distinct turbulent regimes with disparate spectral indices (α<2 and α>2), may be understood on the basis of the dynamic roles played by the electrons and the ions at various stages of evolution of the magnetic turbulence.

At initial timescales , the laser energy is fed in to the electrons, which are responsible for driving the magnetic turbulence. This turbulence is driven both at the electron skin-depth scale d e due to the Weibel instability, as well as at the long spatial scale-lengths corresponding to the transverse extent of the hot electron beam due to velocity shear at the edges. The latter has been corroborated by PIC simulations (Supplementary Fig. 2 and Supplementary Discussion). These long scale-length magnetic fields act as the ambient background for the whistler-mediated EMHD cascade, which produces the broad turbulent spectra of magnetic excitation. This leads to the α≈2 scaling23, close to our experimental observations (as shown in Fig. 2a). It should be noted that the ion-beam-mediated Weibel instabilities and shocks play no role in our experiment. This is because the estimated growth time for these instabilities, given our experimental parameters, is more than the duration of the experiment.

At later timescales , the ion response becomes significant. Furthermore, if the spatial scale-length under consideration includes the ion gyroradius ρ i , the physics in the two regions separated by this scale-length is significantly different. For a scale-length longer than the ion gyroradius , Alfven-like MHD perturbations may be expected in the magnetic-field spectrum. The typical power spectrum for strong turbulence in this regime1 scales as α≈5/3, consistent with our experimental observations, as shown in Fig. 2c. The slope in this regime is shallower than that of the α≈2 scaling, which was observed throughout during the electron-mediated regime in our experiments (as shown in Fig. 2a).

For a scale-length shorter than the ion gyroradius , the equations describing Alfvenic perturbations are significantly altered because warm plasma effects have to be retained. Here, the kinetic effects in the ion dynamics become crucial and the regime is aptly termed KAW regime. The turbulent spectra in this regime is believed to have a scaling of α≈7/3 or α≈8/3 (see, for instance, refs 24, 28 and references therein). Both these scalings are steeper than α≈2, consistent with our experimental observations (as shown in Fig. 2c). It may be noted that the resistive scale falls beyond the maximum value of k depicted in Fig. 2. The spectral curve for follows a power-law and can be easily distinguished from an exponential resistive decay , where k d denotes the resistive wave-number and should take a value in the range 0.5–2 μm−1, if the resistive scale is to lie within our spectral range of observation.

A conservative estimate for the magnetic Reynolds number for our experimental parameters is , whereas that of the solar wind29 is typically . The inertial range of turbulence, therefore, shows similar properties in systems with magnetic Reynolds number greater than unity. In systems with , the number of decades in the wave-number over which the inertial range can operate is very large, which obviously imposes rather impractical demands on laboratory experiments.

Data availability

All relevant data are available from the authors on request.