High-harmonic spectroscopy of solids driven by structured light

. Understanding high-order harmonic generation (HHG) from solid targets holds the key of potential technological innovations in the field of high-frequency coherent sources. Solids present optical nonlinearities at lower driving intensities, and harmonics can be efficiently emitted due to the increased electron density in comparison with the atomic and molecular counterparts. In addition, crystalline solids introduce a new complexity, as symmetries play a role in the anisotropic character of the optical response. An extraordinary playground is, therefore, the scenario in which solids are driven by vector beams, since crystal symmetries can be directly coupled with the topology of the driving laser beam. In this contribution we analyze the topological properties of the HHG radiation emitted by a single-layer graphene sheet driven by a vector beam. We show that the harmonic field is a complex combination of vortices, whose geometrical properties hold information about the details of the non-linear response of the crystal. We demonstrate, therefore, that the analysis of the topological structure of the harmonic field can be used as a spectroscopic measurement technique, paving the way of topological spectroscopy as a new strategy for the characterization of the optical response of macroscopic targets.


Introduction
One of the most striking consequences of the interaction of matter with intense fields is the generation of highorder harmonics with efficiencies orders of magnitude above the perturbative expectation.Until recently, HHG has been mainly studied in atomic, molecular gases and plasmas.The experimental demonstration a decade ago of HHG in solids [1] sparked the interest in the study of the non-perturbative optical phenomena in crystalline solids.Due to their higher electron density, an obvious advantage of solids is the increased efficiency of the process.On the other hand, also during this last decade, there has been a considerable interest in exploiting HHG as a new route for the production of structured high-frequency harmonic radiation.Structured light presents a non-trivial, and sometimes interleaved, distribution of spin and orbital angular momentum.In this sense, HHG offers an extraordinary playground, as it is capable to map some of the structural parameters of the driving field to the harmonic field.In particular, gas targets irradiated by vector beams have been demonstrated to map the vectorial driver's symmetry to high-order harmonics [2].
In this work, we explore the topological structure of the high-order harmonic field from graphene driven by linearly polarized vector beams (LPVB).In particular, we focus on driving beams with a well-defined topology, characterized by the so-called Poincaré index , which refers to the number of complete rotations of the electric field polarization along a closed path.We show that, unlike harmonics driven in gaseous targets, the graphene's anisotropic nonlinear response -emerging from its band structure with  !rotational symmetryleads to a complex far-field spatial structure that breaks the conservation of the driver's topology.In addition, the topology of the far field encodes information about the target response, establishing topology as a novel parameter to be taken into account in high-order harmonic spectroscopy (HHS), revealing information that, up to now, was hidden in traditional HHS.

Interaction scheme and methods
We carry out simulations of HHG in a single-layer graphene, based on the resolution of the TDSE in the reciprocal space [3].Note that, due to the spatially structured nature of the driving field, the macroscopic HHG response needs to be considered.Our simulations propagate the harmonic near field, i.e. the harmonic distribution at the target plane, to the far field, where the harmonics are detected in a standard experiment.
Fig. 1.A radial mid-infrared vector beam is used to drive HHG in a single-layer graphene sheet.The harmonic field is propagated from the target to a far-field detection plane.The spatially integrated spectrum is shown in the inset.

Results: harmonic near and far field spatial distributions
The graphene's hexagonal symmetry is translated to its non-linear response by a modulation of the harmonic efficiencies with the driver's polarization tilt.The interplay between the anisotropic response and the topology of the driving field introduces an azimuthal modulation to the harmonic field, resulting in a necklace intensity distribution at the target, near-field plane.In Fig. 2, we show the intensity distribution of the 9 th harmonic near field.The beads of the necklace intensity distribution correspond to the azimuth angles where the local tilt of the polarization of the driving field maximizes the graphene's anisotropic response [4].Upon propagation, the near-field necklace structure gives rise to the complex far field shown in Fig. 3.For each polarization mode, we can distinguish two regions of different divergence: (i) a central vortex with topological charge ℓ = 1, and (ii), a ring of six vortices also with ℓ = 1.Note that, considering the superposition of both polarization modes, it is clear that the central beam corresponds to a radial vector beam with Poincaré index  = 1, i.e. it preserves the topology of the driving beam.On the other hand, the ring of vortices encodes the information about the material nonlinear response.We demonstrate, by developing an analytical model for the build-up of the harmonic far field, that the number of vortices is  for a crystal with  .rotational symmetry (N=6 in graphene) [5], therefore resulting from the coupling of the crystal N-fold symmetry with the driver's topology, i.e. the Poincaré index .Note that any phenomenon or target's characteristic that may lead to a different nonlinear anisotropic responseeither due to a change in the symmetry of the crystal or in the HHG mechanism-can be therefore traced attending to the structure of the topological cluster obtained in the harmonic far field.Some possibilities include the exploration of interband and intraband contributions or induced strains in the crystal.

Fig. 2 .
Fig. 2. Near-field total intensity and intensity and phase distributions for the left (LCP) and right (RCP) circularly polarized components of the 9 th harmonic.

Fig. 3 .
Fig. 3. Far-field total intensity and intensity and phase distributions for the left (LCP) and right (RCP) circularly polarized components of the 9 th harmonic.The dashed circumferences mark the positions of the vortices.