High order mode structure of intense light fields generated via a laser-driven relativistic plasma aperture

The spatio-temporal and polarisation properties of intense light is important in wide-ranging topics at the forefront of extreme light-matter interactions, including ultrafast laser-driven particle acceleration, attosecond pulse generation, plasma photonics, high-field physics and laboratory astrophysics. Here, we experimentally demonstrate modifications to the polarisation and temporal properties of intense light measured at the rear of an ultrathin target foil irradiated by a relativistically intense laser pulse. The changes are shown to result from a superposition of coherent radiation, generated by a directly accelerated bipolar electron distribution, and the light transmitted due to the onset of relativistic self-induced transparency. Simulations show that the generated light has a high-order transverse electromagnetic mode structure in both the first and second laser harmonics that can evolve on intra-pulse time-scales. The mode structure and polarisation state vary with the interaction parameters, opening up the possibility of developing this approach to achieve dynamic control of structured light fields at ultrahigh intensities.


Frequency resolved optical gating (FROG) measurements
To measure the temporal-intensity profile of the light detected at the rear of the target, a fraction of the collimated beam, after the second F/2 parabola and before the wedged mirrors, was sampled using an elliptical pick-off mirror. This light was directed out of the target chamber through a thin fused silica window, into a frequency resolved optical gating (FROG) diagnostic. The particular type of FROG employed was the GRENOUILLE, described in detail in Supplementary Ref. 1 . This enabled the temporal-intensity profile and the instantaneous frequency of the light detected at the target rear, to be determined. The light entering the FROG was filtered using an (800±35) nm interference filter, and attenuated using a neutral density filter.
During full power shots, an image is obtained using the FROG (a so-called trace), with axes corresponding to time (i.e. the delay between two separate pulses, obtained by passing the incident light through a Fresnel biprism) and angular frequency. To recover the temporal intensity profile of the light, it was necessary to employ the FROG pulse-retrieval software. This is because there is no function which enables the temporal intensity profile to calculated directly from the experimentally obtained FROG trace. Instead, the pulse-retrieval software makes an initial estimate of the temporal profile of the electric field and calculates the corresponding FROG trace. The initial form of the electric field is iteratively improved until good agreement is achieved between the experimentally obtained trace and that calculated using the software. In order to determine the direction of time (and therefore whether a pulse is positively or negatively chirped), the FROG must first be calibrated. This was achieved via a series of calibration shots with no target in place. The pulse duration was stretched by adding positive chirp, via an opto-acoustic device known as a Dazzler. The presence of chirp leads to a gradient in the FROG trace; given that it is known whether the chirp is positive or negative, it is then possible to determine the direction of time. In addition, varying the pulse energy enabled the magnitude of non-linear optical effects to be determined. The FROG traces were corrected for such effects, which add group velocity dispersion to the propagating light.

Evaluating potential depolarisation
We cannot assess the extent to which the generated and transmitted light may be depolarised (i.e. with polarisation states randomly distributed) in the experiment, but we can explore this with the simulation results. We do this by evaluating the distribution of the angle θ = arctan(E Z /E Y ) between the individual spatial E Y and E Z electric field components from the 3D simulations, over a spatial extent X=6 µm to X=12 µm at t=40 fs for the d=10 nm and d=30 nm targets. The resultant magnitude of these |E| components is used to weight the contribution of each angle and the distribution is normalised to the maximum total |E| (explicitly, |E| = E 2 Y + E 2 Z ). This is shown in Figure 1a and 1b for the two target thicknesses. A perfectly linearly polarised pulse in the Y direction will have a single value at ±π/2. For the light transmitted through the d=10 nm target, there is a small degree of broadening induced by the generated light polarised in the Z direction. For the d=30 nm case, however, the distribution is broader, due to an increased ratio of total |E Z | to |E Y |, with an increase in the base noise level. This noise level arises due to a combination of numerical noise in the simulation and the degree of depolarisation of the light.
The results are compared to a theoretical model used to evaluate the superposition of a simple Gaussian TEM 00 beam polarised in the Y direction and a TEM 11 beam polarised in the Z direction, i.e. without depolarisation. A full 3D spatial grid identical to the sampled spatial extent of the simulations was used. The ratio of total |E Z | and |E Y | determined from the same spatial extent in the simulations was used to reduce the total |E Z | in the theoretical model for both the d=10 nm and d=30 nm targets. The beam waist used in each model was also determined from the averaged spatial profile of the beam across the sampled spatial extent in each simulation. -π -π/2 0 π/2 π -π -π/2 0 π/2 π θ θ . The results are sampled over a spatial extent X=6 µm to X=12 µm at t=40 fs and are normalised to each maximum total |E| in all cases.
In Figure 1a, we see good agreement between the electric field angular distributions extracted from the simulation results and that expected from the theoretical model, for d=10 nm. The slight difference in width could be due to variation in the temporal profile and/or pulse separation which was not included in the model. For the d=30 nm case, shown in Figure 1b, the simulation results also follow the model predictions, but with an off-set caused by an increase in the base level of noise. As noted above, this includes numerical noise within the simulation, which is relatively higher in thicker targets because the total detected light signal is lower, and so the contribution due to depolarisation of the light cannot easily be deconvolved and quantified. Nevertheless, the fact that the electric field distributions in the simulations are in overall very good agreement with that expected from the model indicates that very little depolarisation takes place, with negligible amounts for the thinner targets.

Further details on the analytical model for the generation of the TEM 11 mode
The complex field amplitude of the TEM n,m (or Hermite-Gauss) mode is given in Cartesian coordinates by: × exp − r 2 w 2 (X) − ik L X − ik L r 2 2R (X) + i (m + n + 1) ξ (X) where r 2 = Y 2 + Z 2 , w(X) and R(X) are the waist size and the radius of curvature, ξ is the Guoy phase and H m denotes the m th Hermite polynomial. The transmitted light is a combination of a TEM 00 mode and a TEM 02 mode, produced by the deceleration