Unlocking Coherent Control of Ultrafast Plasmonic Interaction

Striking a metallic nanostructure with a short and intense pulse of light excites a complex out‐of‐equilibrium distribution of electrons that rapidly interact and lose their mutual coherent motion. Due to the highly nonlinear dynamics, the photo‐excited nanostructures can generate energetic photons beyond the spectrum of the incident beam, where the shortest pulse duration is traditionally expected to induce the greatest nonlinear emission. Here, these photo‐induced extreme ultrafast dynamics are coherently controlled by spectrally shaping a sub‐10 fs pulse within the timescale of coherent plasmon excitations. Contrary to the common perception, it is shown that stretching the pulse to match its internal phase with the plasmon‐resonance increases the second‐order nonlinear emission by >25%. The enhancement is observed only when shaping extreme‐ultrashort pulses (<20 fs), thus signifying the coherent electronic nature as a crucial source of the effect. A detailed theoretical framework that reveals the optimal pulse shapes for enhanced nonlinear emission regarding the nanostructures’ plasmonic‐resonances is provided. The demonstrated truly‐coherent plasma control paves the way to engineer rapid out‐of‐equilibrium response in solids state systems and light‐harvesting applications.


Introduction
Coherent control (CC) is an interference-based approach where the phase-correlations of electromagnetic fields are mutually manipulated to steer dynamical processes in matter. It offers many capabilities across scientific disciplines that include control of chemical reactions, [1][2][3] study of biological dynamics, [4,5] enhancement of attosecond physics, [6] and steering of quantum The four stages of the photo-excited electronic dynamics 1) collective coherent electronic motion, plasmons, that de-phase in few femtoseconds into a 2) nonthermal electronic state where electron-electron relaxation occurs by the energetic carriers, 3) the electrons become hot and interact with the cold lattice to reach thermal electron-lattice equilibrium, and the 4) lattice-environment equilibration. During carrier excitation, collective electronic multiphoton excitations are induced and can produce second harmonic emission. Out-of-equilibrium dynamics spans up to the third stage. b) The nanostructure's resonance frequency and linewidth, denoted as r and int , can be illuminated by (Upper panel) a sub-10 fs pulse, with much broader than the nanostructure's bandwidth (BW), or (Lower panel) pulse with narrower BW than the nanostructure's linewidth. c) The experimental setup. A sub-10 fs pulse passes through a 4f pulse shaper consisting of a spatial light modulator (SLM, See Inset). The shaped pulses interact with an array of gold nanostructures, producing SHG emission. The mirror-based setup frees the SLM from dispersion management of the pulse and is used only for CC. d) The interaction can be coherently controlled externally by manipulating the pulse's spectral phase to reroute the accumulative excitation-pathways to constructively interfere. A properly induced stretch pulse, despite having a lower intensity than the transform-limited, may lead to a stronger nonlinear plasmonic excitation and produces more SHG emission.
requires pulse shaping within the short-lived plasmonic coherent evolution, a challenging task so far due to the metals' dephasing time (∼20 fs for Gold). During the past decade, unexpected enhancement in plasmonic interaction by pulse stretching have been studied, raising many fundamental questions about the origin and mechanism of such enhancements in plasmonic hot spots. [46] In recent research from the Van Hulst's group, [31,47] closed loop based coherent control has shown the importance of pure phase control in optimizing the optical nonlinear response for two-photon photoluminescence (TPPL) in plasmonic nanostructures. However, while such TPPL enhancement, mediated by photon absorption, has been observed, [47] an enhancement of a parametric second harmonic generation with direct resonant based interference of pure photon-plasmon interaction is still lacking. Such coherent control, which may allow unprecedented nonlinear optical enhancement in plasmonic systems, as it has in the atomic realm, has however remained elusive.
Here, we demonstrate coherent control of collective electronic dynamics and its direct effect on generated optical nonlinearities. We use sub-10 fs pulse-shaping to match our few-cycle pulse with the internal dynamics of ultrashort plasmonics excitation. We show that, contrary to the common perception, stretching (chirping) the pulse leads to a significant enhancement in the generated second-order optical response exceeding by >25% of the generation at the maximally compressed ("transform-limited") pulse. We further measure the second-order generation (SHG) as a function of the chirp parameter and find that the optimal pulse chirp is asymmetrical in time and depends on the nanostructure's geometrical configuration. We devise a theoretical framework that focuses on the dynamics of broad spectral width pulses to efficiently match the broad electronic resonance (Figure 1b). The theory suggests that the coherently enhanced nonlinearity stems from the constructive interference of excitation pathways driven by the shaped pulse. We present a direct connection  Enhanced nonlinear second harmonic generation by coherent control. a) Measured laser spectrum (colorful, logarithmic) and measured extinction cross-section of the LPR (black line). Inset: A SEM image of a single nanostructure. b-I) Normalized SHG emission for a positive chirp (dashed blue), unshaped transform-limited pulse (yellow) and negative chirp (red) showing an increase of more than 25% compared to the unshaped pulse and enhanced by 50% compared to the opposite-sign chirped pulse. The inset illustrates the pulses' temporal intensity profiles that is maximal for an unshaped pulse and is equal for both chirp signs. b-II) The SHG emission of plasmonic nanostructure as a function of the linear chirp parameter 2 , controlled via the SLM. In stark contrast to the nonlinear instantaneous interaction, maximal nonlinear emission is obtained 2 = −16 f s 2 , and not at the highest pulse's intensity. The horizontal dashed lines are cross-sections presented in (b-I). c) Experimental (black dots) measurements of the optimal chirp parameter as a function of the nanostructures resonant wavelengths. Simulation results (dashed) are plotted for several plasmonic decoherence times, NS . As seen, the simulated value of NS ∼ 23 fs best match the experimental results. d) Comparison of SHG emitted by plasmonic nanostructures (circles) and by a BBO crystal (squares) as a function of the chirp parameter. The maximal nonlinear response in plasmonic excitation is negatively shifted due to a non-instantaneous response to the interaction, in contrast to the symmetrically centered instantaneous response measured for a BBO crystal. The experimental results coincide with a numerical simulation based on a 3 level model, our theoretical approach to predict dynamical nonlinear plasmonic excitation.
between the optimal pulse durations and the extreme ultrafast plasmonic decoherence. In particular, as our theory relies on the coherent interference of the multiphoton interaction with a coherent resonant mode, it is based on the individual nanostructures localized plasmonic resonance (LPR)'s frequency and linewidth in a perturbative regime. In our simulations, we show that our model is sensitive to variation in plasmonic decoherence on the few-femtosecond scale. Such nonlinear optical enhancement not only provides a clear demonstration for its coherent manipulation but also expands the current understanding of nonlinear plasmonics enhancement, which has been relaying so far on geometrical manipulation of the nanostructures. [15][16][17][18]27,48] In our experiments, we use ultrashort pulses with a temporal resolution of sub-10 fs pulse, spanning a bandwidth of more than 300 nm (Venteon Dual, Laser-Quantum, measured linear spectrum in Figure 2a). The pulses illuminate arrays of U-shaped gold nanostructures, each with different localized plasmon resonance (LPR) (spectral extinction cross-section in Figure 2a, see Supporting Information for more details). Normal incidence second harmonic generation (SHG) was measured for each spectral phase applied to the spatial light modulator (SLM). In our experiments, the ultrashort pulses' spectral bandwidth exceeds the resonance linewidth of the nanostructures' LPR, (∼70 nm Full width-half-max, FWHM). This allows us to simultaneously excite the entire bandwidth of the metallic nanostructure's LPR response, a requirement for coherent control, in addition to enable to reach shorter timescales. This is unlike most previous ultrafast or nonlinear plasmonic experiments that operate with much narrower pulse bandwidths.
We coherently alter the driving pulse shape, using a homemade dispersionless pulse shaper based SLM. The pulse shaper was carefully designed to effectively free the SLM from dispersion management of few-cycle pulses, critically needed for high-fidelity shaping within the plasmonic lifetimes (Supporting Information). We have verified that the pulse at the illumination plane is maximally compressed in terms of its linear chirp. Any additional spectral phase applied by the SLM results in a shaped (chirped) pulse with broader temporal width and decreased intensity. Figure 2b shows the measured SHG emission from a plasmonic nanostructure with a resonant wavelength of 760 nm as a function of the quadratic spectral phase parameter, 2 (a.k.a linear chirp). While the pulse spectrum is centered at 850 nm, the plasmonic mediated SHG is slightly blue shifted. This effect, also observed in previous works, is caused by the plasmonic spectral shape of absorption and scattering cross-section. [31] The SHG spectrum induced by the transform-limited pulse ( 2 = 0) is shown in yellow.

Results
We note that transform-limited pulse, which has zero-chirp and thus with any pulse's highest peak-intensity is usually expected to induce a maximal nonlinear optical response. Illuminating the nanostructures with positive linear chirp ( 2 = 16 fs 2 , blue) results, as expected by a stretched pulse, in reduced SHG emission (positive chirp corresponds to a pulse where its higher frequencies arrive first). However, negative linear chirp ( 2 = −16 fs 2 , red) shows an increased nonlinear emission of >25% in the accumulated SHG compared to the transform-limited pulse. In Figure 2b(II), we introduce a map of the measured SHG emission for a larger range of linear chirp (−150 fs 2 to 150 f s 2 respectively), confirming clear maxima at 2 = −16 fs 2 . Note, the pulse duration spans from sub-10 fs (at 2 = 0, maximal pulse peak intensity) to ∼155 fs at 2 = ±150 fs 2 (see Supporting Information for more details). Contrary to common perception, in plasmonic nanostructures, the maximal nonlinear generation does not occur when maximizing the peak intensity of the pulse (i.e., the shortest pulse).
We compare the results with the SHG signal from a commonly used Beta-Barium-Borate (BBO) nonlinear crystal (see Figure 2d), which has an instantaneous nonlinear response as it is lossless and dispersionless in the near-infrared and optical regime (as dictated by the Kramers-Kronig relations). As expected, in BBO, the maximal SHG is obtained with a transformlimited pulse illumination. We also observe that in distinction to the sign-symmetrical intensity profile found in the BBO crystal, in plasmonic nanostructures, the nonlinear excitation is asymmetrical in relation to the chirp parameter. As can be observed, the nonlinear response has an asymmetrical shape in relation to the chirp phase's sign. The asymmetry highlights that the plasmonic excitation process consists of a nontrivial coherent response that contributes to the multiphoton structure of the nonlinear interaction that is crucial in the first few femtoseconds of the excitation.
To deepen the understanding of the extreme ultrafast excitation, we study plasmonic nanostructures ranging in their effective lengths, from 130 to 300 nm (correspond to LPR's wavelengths 770 to 970 nm, respectively). We find a correlation between the optimal linear chirp with the LPR wavelength of the U-shaped nanostructures (see Figure 2c). Importantly, we see that the optimal linear chirp sign is inversely related to detuning, which is defined by the relative location of the plasmonic resonance to the pulse's central wavelength (See Figure 2a). Thus, in positively detuned interactions, where the plasmonic resonant frequency (wavelength) is higher (lower) than the central frequency (wavelength) of the incoming pulse, the optimal linear chirp is negative.
We utilize a theoretical and numerical approach that predicts the influence of pulse shaping on the measured optical nonlinear response in nanostructures, which allows to analyze the evolution of the interaction process capturing the inner dynamics in the multiphoton excitation. Our model aims to describe the ultrafast plasmonic dynamics of an isolated nanostructure during the effective timescale for plasmonic dephasing and ignores lattice scattering effects, strong-field effects or other effects that might be induced by the array. In a nutshell, we have considered the interaction as a three-level resonant model solved to the secondorder in a time-dependent perturbation framework. The nonlinear polarization reads as follow: where P (2) NL (Ω) is the nonlinear polarization at frequency Ω, E( ) and E(Ω − ) are the linear fields of the fundamental mode, where the generated frequency follows Ω = + (Ω − ) and (2) eff ( , NS , Γ NS ) ≈ (2) − NS +iΓ NS is the spectral-dependent local nonlinear susceptibility that depends on the nanostructures parameters NS and Γ NS . The (2) ( , NS , Γ NS ) expands the known geometrical-dependent-only (2) term, which results in a generalized nonlinear emission E SHG (Ω) ∝ ∫ P (2) NL (Ω) ⋅ E(Ω). [18] Our findings spotlight the importance of ultrafast phenomena in analyzing resonant nanostructures, appending to the ongoing efforts in understanding the crucial role of the dynamics in the interaction with singular nanostructures and collective arrays. [19,27,49,50] Our method for analysis of the multispectral interaction is captured in the schematic illustration depicted in Figure 3a, which directly reflects the theoretical model presented in Equation (1) (See Supporting Information for more information). As portrayed in Figure 3a, the induced second-order interaction can be viewed as a summation of complimentary frequency-pairs that add-up to the energy of the second harmonic frequency NL . The broadband interaction induces multiple pathways for excitation (small arrows in the diagram). Each interaction can be represented by the interference of multiple pathways, where the orientation of each arrow stems from the combined phase of both the pulse and nanostructure. The accumulated amplitude, represented by a large arrow, corresponds to the measured intensity of the nonlinear plasmonic excitation. The Feynman diagram of P (2) NL (Ω) for a maximally compressed pulse, E TL and a chirped pulse, E chirp are illustrated. The chirped pulse enhances the interaction by rerouting the accumulative pathways to constructively interfere. Marked along the trajectory, I-IV, correspond to 4 possible pathways illustrated in Figure 3a. c) The Feynman diagram for the nonlinear ultrafast electronic excitation in the case of (left) a transform limited pulse. (middle) the optimal phase function predicted by the 3-level model without the nanostructures contribution and (right) the combined spectral contribution of both pulse and nanostructure, maximally elongating the amplitude of the interaction according to the 3-level model. energy, illustrated by the excited state |e⟩, which corresponds to the second harmonic frequency generated at that energy. The broadband pulse induces various sum frequency events, varying in the composition of their commentary frequency-pairs (illustrated by vertical arrows in Figure 3a). Due to the resonant nature of the interaction, each frequency-pair that interacts with the nanostructure contributes in both in amplitude and phase according to the detuning from the plasmonic resonate level |r⟩. The accumulation of the complex-valued events coherently interferes and determines the total measured second harmonic outcome.
From the theory, we find that the nonlinear interaction's coherent structure consists of two contributions: the spectral phase of the ultrafast pulse and the nanostructure's inherent resonant phase. The pulse contribution, controlled at will by the SLM, serves as a probe to study dynamical interactions. The nanostructure's contribution is composed of the amplitude and phase of the spectrally dependent plasmonic response, stemming from the collective electron dynamics, which also incorporate effects of geometry and environment.
To further gain an intuitive physical understanding of the interaction process, we use the pictorial representation in the complex plane along with the numerical simulations. The complex plane representation portrays all spectral components' accumulative contributions in the interaction and is calculated based on a three-level system approach, illustrated in Figure 3a and further described in Figure 3b (see more details in the Supporting Information). Our coherent control demonstrations can be Laser Photonics Rev. 2022, 16, 2100467 www.advancedsciencenews.com www.lpr-journal.org intuitively explained by the interfering pathways picture on the complex plane (see Figure 3b). Each excited nonlinear frequency is generated by the coherent accumulation of multiple individual pathways. Each pathway is set by a frequency pair in the driving pulse { , Ω SHG − }, dictating Ω SHG = + (Ω SHG − ). An example for four pathways, set by four frequency pairs, illustrated as paths I to IV.
Inherent to the nanostructures' excitation dynamics, the plasmonic response induces self-interfering pathways to the excitation (Figure 3c, left). However, a properly shaped pulse will induce constructive interference between the accumulated pathways, rearranging the trajectory to maximize the nonlinear excitation (Figure 3c, right). Since the unaltered pathways consist of the intrinsic destructive interference inherent to the interaction, a reciprocal manipulation of the ultrashort pulse leads to constructive interference in the interaction and to an enhanced excitation beyond any other pulse shapes, including the maximally compressed, transform-limited pulse. Our simulations that agree well with the experimental results provide a complimentary and intuitive understanding for the observed coherently controlled interaction.

Discussion
We utilize our numerical simulations and compare them with the experimental results of the optimal linear chirp for a variety of nanostructures across the spectral landscape. Such simulations offer a window to variations in the ultrafast plasmonic response and offer a way to determine the lifetime of the coherent excitation, NS , which is inversely related to the effective nanostructures linewidth Γ NS . We assume that Γ NS , which takes an inherent part in the coherent excitation process, is based on the individual plasmonic nanostructure (see Refs. [51,52,53] for more information on the complexity of linewidths in plasmonic metasurfaces). It can differ from the experimental linewidth value extracted by far-field measurements, which might be broader as a result of inhomogeneity in the array. Γ NS is related to the plasmonic damping factor derived for localized plasmonic resonance taking into consideration the shortened mean free path of electrons in gold nanostructures (see Section 2.3 for more information [19] ). We also note that the contribution of electronic loss to the coherent collective dynamics in ultrafast timescales is yet to be fully understood. It consists of processes such as electron-electron scattering, nonthermalized electron formation, and evolution, which are not trivially contributing to the fewfemtosecond regime. [51,52] A summary of our simulations is presented in Figure 2c. The optimal chirp parameter is plotted as a function of the nanostructure's LPR for three plasmonic decoherence timescales, 17, 23, and 30 fs, which can be derived for plasmonic nanostructures based on common literature or measured in similar configurations, [19,48,54,55] viewing the plasmonic lifetime based on free electron absorption and geometry. As seen, the numerical simulations provide very high sensitivity to variations in decoherence times. Worth noting that in the numerical analysis, the effective interaction linewidth Γ NS is the only parameter that is not determined experimentally.
Using our theoretical analysis, which allows predicting the nonlinear excitation leading to SHG for any pulse shape, detuning, and coherence time parameters, we find the optimal pulse shape that globally maximizes the nonlinear interaction. More specifically, we find that for plasmonic nanostructures, the optimal spectral phase is tan −1 ( Γ atan − atan ), where Γ atan and atan determine the width and the central frequency of the inverse tangent function, accordingly. Maximal enhancement of SHG for a specific design is obtained when the characteristics of the plasmonic nanostructure set the parameters Γ atan = Γ NS and atan = NS (see Figure 4). Since the optical nonlinearity is produced separately for each frequency in the manifold of target second harmonic excitations, the optimal pulse shape prediction requires analysis of a range of frequencies (see Supporting Information for more details).
We have verified our predictions experimentally and consistently observe maximal enhancement for the predicted optimal phases. Interestingly, by setting different parameters to the inverse tangent spectral phase, we can uniformly suppress the SHG emission for the central emission profile, maintaining constant profiles of the emission spectra in the generated signal. Taking advantage of this quality, we are able to continuously vary the inverse tangent function to distinctly yet simultaneously facilitate coherent control capabilities in a single-shot. Such control allows contrastingly manipulating to enhance, maintain, or suppress specific components in the induced nonlinear spectrum. In order to demonstrate these effects, we selectively manipulate the central spectral profile of the nonlinear emission while effectively maintaining a general form of the emission profile. For example, by setting the parameters to atan = NS and Γ atan = −Γ NS , we demonstrate variations in the intensity of up to 200% over the intensity of the SHG emission for particular frequencies while maintaining a constant emission profile for other frequencies.
The ability to induce differential control, which is shown to suppress only the central profile of the nonlinear emission, stems from the interference of pathways that correspond to the excited frequency Ω SHG . Therefore, as the interference for each excited nonlinear frequency has a different accumulated composition, the interaction process is effectively manipulated separately for each of the second harmonic frequency components.

Conclusion
To conclude, by ultrafast pulse shaping within the coherence time of the plasmonic evolutions, we unlock fundamental coherent control capabilities and enable the steering and enhancement of the nonlinear optical generation in plasmonic nanostructures. We demonstrate a fundamental effect of coherent control in resonant media [13] for localized plasmonic nanostructures. Such realization, which requires pulse shaping capabilities in the single-cycle regime, expands the concept to extreme limits. While coherent excitation in systems with high loss, such as LPRs, are commonly overlooked due to their short-lived, out-of-equilibrium nature, we show that by properly shaping the incoming pulse, the induced excitation can be coherently manipulated throughout the interaction. Notably, such control substantially modifies the nonlinear electronic dynamics with only subtle modifications to the temporal pulse shape and width. Also, the experimental results show that competing processes in the interaction, such as the formation of highly energetic non-thermalized electrons, do not play a significant role in  induced by the plasmonic excitation with LSP resonance of r = 890 nm for the three pulse shapes: (red) The optimal pulse shape, which is obtained by a spectral phase of an inverse tangent phase function-as predicted by the 3-level model, (black) an unshaped, maximally compressed pulse, and (blue) an example of differentialy suppressing pulse shape, diminishing the central profile of the generated spectrum while maintaining a constant emission frame. The spectral phase of the inverse tangent function, which is applied by the SLM, can be continuously tuned by changing the value of Γ atan , to produce pulse shapes that vary the generated nonlinear plasmonic excitation spectrum. As dictated by Fourier-transform, the pulse shapes are only slightly deformed in their temporal profiles (lower right) yet have a dramatic effect in the generation of the nonlinear signal.
hindering the initial coherence of the plasmonic state. Based on the accumulative interference of the individual spectral components, our theoretical framework is found suitable to explain the experimental results and predict the optimal pulse for maximal global nonlinear enhancement. We believe that our coherent control demonstration will promote a paradigm shift in the view of nanoscale nonlinearities, where the origin of optical nonlinearity in nanostructures does not stem only from the enhanced absorption but can rather be dominated by coherent interference.
Furthermore, our fundamental demonstration links the manipulation of photo-excited nanostructure evolution to the vast coherent control schemes explored in atomic and molecular systems. Thus, opening a doorway to implement ultrafast control based on intrinsic plasmonic dynamics is expected to unlock entirely new capabilities desirable in active spatiotemporal metasurfaces at extreme ultrafast speeds. [50] Beyond the significant contribution to fundamental research, we expect that coherent control at extreme ultrafast timescales will provide novel control methods in metamaterials-based applications in nonlinear microscopy, electronic and excitonic dynamics in 2D metamaterials, cancer phototherapy as well as artificial and natural lightharvesting complexes.

Supporting Information
Supporting Information is available from the Wiley Online Library or from the author.