All-Optically Reconfigurable Plasmonic Metagrating for Ultrafast Diffraction Management

Hot-electron dynamics taking place in nanostructured materials upon irradiation with fs-laser pulses has been the subject of intensive research, leading to the emerging field of ultrafast nanophotonics. However, the most common description of nonlinear interaction with ultrashort laser pulses assumes a homogeneous spatial distribution for the photogenerated carriers. Here we theoretically show that the inhomogeneous evolution of the hot carriers at the nanoscale can disclose unprecedented opportunities for ultrafast diffraction management. In particular, we design a highly symmetric plasmonic metagrating capable of a transient symmetry breaking driven by hot electrons. The subsequent power imbalance between symmetrical diffraction orders is calculated to exceed 20% under moderate (∼2 mJ/cm2) laser fluence. Our theoretical investigation also indicates that the recovery time of the symmetric configuration can be controlled by tuning the geometry of the metaatom, and can be as fast as 2 ps for electrically connected configurations.


Design of the dimeric metaatom
The hybridization theory design of the unit cell of the plasmonic metagrating was supported

Hybridization scheme validation
To discuss the behaviour of the dimeric metaatom in its disconnected configuration, a hybridization scheme based on the response of the single nanostrip monomers (refer to Fig. 1c from the main text) has been employed. To validate this view, numerical simulations have been performed according to above section 1 in order to determine the spatial distribution of the induced charge density. The main results of the computation at the two resonance wavelengths are presented in Fig. S1, showing the norm of the scattered electric field E S (normalised to the incident field, E 0 ) across the structure. Based of the electric field spatial patterns, charge density can be straightforwardly deduced by invoking the Gauss' law: regions where the divergence of E S is higher correspond to higher induced charge density.
Appropriately, circles in Fig. S1 highlight such regions. In particular, red circles surround portions of the dimers with positive charge density, whereas white circles indicate regions with negative charges. Note that the distribution of induced charge density in the dimer retrieved by full-wave numerical analysis well compares, qualitatively, with the scheme of charges reported in Fig. 1c, apart from unavoidable quantitative discrepancy due to retardation effects and flip of sign (having no relevance). Hence, the numerical investigation endorses the hybridization scheme of the dimeric metaatom and ascertains the origin of the two resonant peaks observed in the optical response of the dimeric structure. Figure S1: Spatial pattern of the dimeric metaatom optical excitation. a-b, Spatial map of the scattered electric field enhancement E S /E 0 (in modulus) across the 2D dimeric nanostrips metaatom, evaluated at the excitation wavelengths λ = 930 nm (a) and λ = 575 nm (b). Black arrows indicate the scattered field lines of force. Circles highlight the regions where electric field divergence is higher (red for positive, white for negative sign), corresponding to a higher induced charge density.

Ultrafast photoexcitation modelling
As concisely outlined in the main text, the modelling of the photoinduced inhomogeneous distribution of hot carriers, which results in the symmetry breaking enabling the diffraction management in plasmonic metagratings, consists of several aspects. To simulate the interaction of the plasmonic metagrating with both the control pulse (according to the I3TM), and the signal pulse, a suitable finite element method (FEM)-based 2D model has been developed employing COMSOL Multiphysics, and a segregated approach was pursued.

Pump absorption pattern calculation
First, the scattering problem in the frequency domain is solved in static conditions to determine the absorption spatial pattern of the control pulse. A monochromatic p-polarized plane wave at λ c = 600 nm impinging at 45 • is assumed (Fig. 3a). Unperturbed Au permittivity ε 0 Au (λ) has been considered for the plasmonic metaatoms. Periodic BCs have been set to simulate the array optical behaviour and ports formalism is employed to compute the near fields and far fields of the structure. This step provides us with the key quantity A( r) to evaluate the space dependence of the absorbed power density (Eq. (1) in the main text) and to formulate the I3TM from its source term.
Starting from the latter result, the drive term of the ultrashort pulse illumination dynamical model, P abs ( r, t), can be addressed. Note that such a formulation for the absorbed power holds in a linear regime, where the control pulse fluence is such as not to self-induce nonlinear interactions. This may lead to a quantitative underestimation of the absorbed power, 6 although with no essential modifications in the photoexcitation dynamics for the relatively low fluence here considered.

Solving the I3TM
The core of the energy transfer dynamics is then the I3TM of Eqs. (2)-(4) from the main text, that we developed and used to describe the hot-electron driven symmetry breaking and predict the diffraction management effect. The model is a generalization of the wellestablished and widely employed 3TM, 7 where mainly two aspects have been introduced.
First of all, the source term P abs ( r, t) has been considered as a space-dependent quantity, hence inducing an inhomogeneous distribution, via Eq. (2), of the energy density N , which relates to the variation of the occupation probability of electrons. 8,9 Further, spatial Fourier-like diffusion terms for both the electronic and the lattice temperatures have been included in the model, Eqs. (3)-(4), so to describe the propagation in space of the thermal non-uniform perturbation in electron and phonon populations in the plasmonic nanostructures.
All the coefficients of the rate coupled partial differential equations of the I3TM are deduced, straightforwardly or by direct comparison, from more formal and fundamental models of ultrafast photoexcitation of metallic nanostructures (see e.g. Refs. 8-12 for further details, at the basis of the formal foundation of our I3TM, not discussed in the present work for the sake of conciseness).
In particular, regarding the parameters in the model Eq. In terms of the model numerical implementation, with a subsequent time-domain analysis we solved Eqs. (2)-(4) to determine the local temporal dynamics of the three energetic degrees of freedom in each point of the plasmonic structure. Both time (by fixing the time step dt = ∆t/30) and space (by setting the mesh maximum element size to r/2 = 5 nm) have been appropriately discretized to resolve the photoexcitation spatio-temporal evolution. This step retrieves the spatial distribution of N, Θ e and Θ l in time, which is then used to compute the corresponding Au permittivity modulation pattern, ∆ε( r, λ, t), that presides over the optical symmetry breaking of the plasmonic metaatoms.

Calculation of photoinduced permittivity changes
Regarding the calculation of the photogenerated inhomogeneous permittivity modulation, as mentioned in the main text, contributions from N, Θ e and Θ l have been considered in describing both the intra-and interband terms of Au permittivity, based on well-established models of the metal nonlinear optical properties upon ultrashort pulse illumination.
For the intraband Drude-like contribution, this term has been considered to be modified by a change in Θ e and Θ l via the Drude damping Γ and the metal plasma angular frequency ω p . In particular, Γ has been written 11 as the sum of an electron-electron Γ e−e and an electron-phonon Γ e−ph damping factor. The former is derived in the framework of Landau's Fermi liquid theory 18 and exhibits a quadratic dependence on Θ e , the latter can be evaluated from the Fermi golden rule at the second order 19 and, due to the limited ranges of ∆Θ e predicted, approximated 20 to be directly proportional to ∆Θ l and quadratically to the control pulse photon energy. The plasma pulsation modification instead has been considered to be driven by the volume expansion of the metal nanostructure. Its modification has been formulated in accord with Refs. 21-23.
On the other hand, the interband modulation is driven by both nonthermal and ther-

Modulation of metagrating diffraction
To finally determine the effect on the metagrating optical diffraction orders of the photoinduced inhomogeneous perturbation experienced by a signal pulse, a frequency-domain simulation is then performed over a broad spectral range for a given control-signal delay τ , which acts as a parameter. Numerical implementation is similar to the first step (Section 3.1), however Au permittivity is modified according to the previous step (3.3) and given by ε Au ( r, λ; τ ) = ε 0 Au (λ) + ∆ε( r, λ; τ ) and the p-polarized plane wave modelling the signal pulse impinges on the structure at normal incidence. All the array diffraction orders are then computed via port formalism at different time delays.
Note that the linear and segregated approach we pursued in modelling control-signal pulses interaction with the plasmonic metagrating is licit as long as the control pulse has a time duration such that several optical cycles are contained in the pulse (that is the so-   Figure S3: Transient modulation of the metagrating optical response. Spectra of the all-optical modulation of each of the diffraction orders supported by the periodic structure, here shown for the disconnected metagrating configuration at a fixed control-signal time delay of τ = 100 fs. Variations are shown for the ±1 orders both in transmission and reflection (a), and for direct transmission T 0 and reflection R 0 , along with absorption A (b). c, Spectra of the sum of the modulations either in the ±1 orders shown in (a) or in T 0 , R 0 and A from (b) are compared to show power redistribution and conservation.
Although the present work has been mostly focused on the modulation of ±1 diffraction orders, since those are the ones responsible for a symmetry breaking in the metagrating optical response, it is worth highlighting that the control pulse induces in fact an all-optical modulation for each of the aforementioned orders. Throughout photoexcitation, a variation evolving in time occurs for each of them, since ultrashort radiation entails a modification of Au optical properties, regardless of the symmetry arguments which hold for the ±1 orders.
Interestingly, the fact that the control pulse can indeed change dynamically the spectra of T 0 , R 0 and A introduces some extra channels in the framework of the interaction between the signal pulse and the photoexcited metagrating at a certain delay τ . In other terms, the power of the signal pulse can be redistributed not only between reflection and transmission involving exclusively the ±1 diffraction orders. Therefore, for a given ε Au ( r, λ; τ ) = ε 0 Au (λ) + ∆ε( r, λ; τ ), absorption as well as transmission and reflection 0-th orders are also modified, upon the constraint of total power conservation, that is the optical theorem, which requires that the sum of all orders and losses equals one, corresponding to all order variations summing to zero. This is clearly shown in Fig. S3, where modulation spectra of all the orders involved in the structure optical response are reported in the case of the disconnected configuration at an exemplary time delay of τ = 100 fs. While Fig. S3a shows the spectra of the ±1 diffraction orders, more extensively discussed in the work, the spectrally-dispersed variations for absorption losses and direct transmission and reflection are reported in Fig. S3b. According to power conservation (or, equivalently, the optical theorem written for periodic structures), all these quantities sum to zero, as confirmed in Fig. S3c, where perfectly specular spectra, computed as the sum of curves in Figs. S3a and S3b respectively, are depicted (purple and green curves, respectively). Note that there is a slight mismatch, at short wavelengths, in the zero-crossing of curves in Fig. S3c. This difference is due to the contribution of the T ±2 orders, accounted for in numerical simulations, but not included in the sum here presented for the sake of clarity. This couple of orders is in fact non-zero only in the blue region of the spectrum, with an output power which is more than one order of magnitude weaker if compared to any other.
Generally speaking, full-wave simulations of the metagrating optical response reveal a complex dynamics of diffraction from the periodic structure. Spectrally-dispersed variations of all the diffraction orders are photoinduced by the ultrashort control pulse, which modifies the optical properties of the metaatom. As a result, throughout the interaction between the signal pulse and the illuminated metagrating, the all-optical diffraction modulation involves all the available channels for power redistribution. In particular, the R ±1 and T ±1 orders compete with modulation of direct transmission and reflection as well as absorption losses, experiencing a non-zero modulation as well, although with no role in the symmetry-breaking mechanism reported in this work.

Optical static response of the connected configuration
The plasmonic 1D array with metaatoms made of pairs of connected nanostrips was first analysed by calculating the static optical response of the resulting metagrating. The same numerical tools presented in above sections and employed to discuss the disconnected metaatom configuration (refer to Fig. 2) have been used by consistently changing the modelled geometry.  165 nm) to keep the single dimer metallic volume constant between the two configurations (see Fig. S4a). In such a way, we ensured a fair comparison between the two topological variants of the metaatom.

Diffraction management in the connected configuration
As for the disjointed dimeric nanoantenna array detailed in the main text, all-optical diffraction management of the connected variant has been predicted and investigated by exciting the array with a control pulse impinging at 45 • on the structure and by probing the resulting optical perturbation with a signal pulse, arriving at normal incidence at delay τ .
The ultrafast symmetry breaking effect, as for the disconnected configuration, is then evaluated by considering the evolution in time of the spectra for the ±1 reflection (transmission) orders, as well as the figure of merit D R (D T ) introduced in the main text, Eq. (5), and measuring the optically-induced asymmetry between originally degenerate diffraction orders.
In particular, Figure S5

Dynamics of the internal degrees of freedom
In comparing the two, disconnected and connected, configurations of the metagrating under investigation, it is apparent that the key feature governing the optical symmetry breaking is thermal equilibration between the two strips in the metaatom. In particular, the recovery of the power balance between ±1 diffraction orders is accomplished when electron temperature Θ e and, although to a lesser degree, Au lattice temperature Θ l , recover a symmetric spatial distribution. Equilibration of electronic temperature in the two strips of the metaatom occurs via two channels, namely electron-phonon relaxation processes and hot-carrier diffusion.
Although both present in the disconnected configuration as well, the gap between strips inhibits the energy flow, whereas the behaviour of the electrically connected metaatoms is indeed mostly dominated by the latter. Moreover, since according to the I3TM their distribution in space inherits the spatial pattern of the control pulse absorption 30 (A( r) in Eq. (1)), which is similar for the two configurations at λ c , the dynamics of N across the disconnected and the connected metaatoms are comparable. Therefore, in terms of contribution to the optical perturbation arising from nonthermal carriers (refer to Section 3.3) and corresponding transient symmetry breaking, the structures are expected to behave alike. On the other hand, the considered metaatom geometries exhibit deeply different behaviours when both electronic and lattice temperatures are considered. In fact, Θ e follows qualitatively comparable dynamics in the four considered points, showing that a full relaxation of the electronic excitation requires some tens of ps. The major difference between connected and disconnected structures is, as highlighted in commenting on Fig. 4c in the main text, given by homogenisation, achieved in the connected structure only. Besides and, most importantly, before relaxation, (solid) curves referring to distinct points in the connected geometry follow an ultrafast dynamics causing their overlap. Such behaviour is governed by thermal diffusion and its characteristic time is dictated by electron thermal conductivity and heat capacity. On the contrary, a long-lasting asymmetry of Θ e in the disconnected metaatom is observed, namely curves will overlap only when vanishing, i.e. when electron excess energy is dissipated. Interestingly, this affects the dynamics of the lattice temperature as well, since its source term (see Eq. (4) from the main text) remains asymmetric over a long time interval. This explains the dynamical behaviour of Θ l shown in  here presented in the two cases respectively. Indeed, an interplay of N and Θ e dominates the response over the first ps, with the contribution arising from thermal hot carriers which affects the symmetry breaking to a lesser degree and a short time interval in the connected structure (Fig. S6b, solid). On the other hand, electron and phonon temperatures govern the signal decay and entail, in the disconnected structure, a long-lasting asymmetry effect.