Field-resolved detection of the temporal response of a single plasmonic antenna in the mid-infrared

Unveiling the spatial and temporal dynamics of a light pulse interacting with nanosized objects is of extreme importance to widen our understanding of how photons interact with matter at the nanoscale and trigger physical and photo-chemical phenomena. An ideal platform to study light–matter interactions with an unprecedented spatial resolution is represented by plasmonics, which enables an extreme conﬁnement of optical energy into sub-wavelength volumes. The ability to resolve and control the dynamics of this energy conﬁnement on the time scale of a single optical cycle is at the ultimate frontier towards a full control of nanoscale phenomena. Here, we resolve in the time domain the linear behavior of a single germanium plasmonic antenna in the mid-infrared by measuring the complex optical ﬁeld response in amplitude and phase with sub-optical-cycle precision, with the promise to extend the observation of light–matter interactions in the time domain to single quantum objects. Accessing this fundamental information opens a plethora of opportunities in a variety of research areas based on plasmon-mediated photonic processes and their coherent control, such as plasmon-enhanced chemical reactions and energy harvesting. © 2021 Optical Society of America under the terms of


INTRODUCTION
A full understanding of light-matter interactions at the nanoscale can be enabled by resolving and correlating both the spatial and temporal structure of a light pulse interacting with nanoscale materials and molecules.From a spatial point of view, we have already reached a wide understanding of many nanoscale light-matter coupling phenomena in their ground state.On the contrary, unveiling the temporal evolution related to excited states or to coherent dynamics of many light-matter interactions is still quite challenging.Furthermore, the manipulation of highly intense and ultrashort light pulses plays a crucial role in energy harvesting and chemical reactions and, in general, will impact on new technologies exploiting ultrafast phenomena in condensed matter.In recent years plasmonics [1,2], and especially plasmonic nanoantennas [3][4][5][6][7], has received broad attention in nanophotonics, opening up valuable opportunities to enhance light-matter interactions for sensing [8][9][10][11], energy harvesting [12,13], and opto-electronics [14][15][16].Moreover, owing to the large field enhancement over sub-diffraction limited volumes, various applications ranging from near-field microscopy [17,18] to ultra-sensitive material detection [19,20] and to the observation of higher-order optical nonlinearities [21][22][23][24] are heavily investigated.First studies were performed in the visible and near-infrared spectral range, generally based on noble-metal subwavelength structures.These investigations enabled an unprecedented understanding, along with the active manipulation, of many physical phenomena occurring at the nanoscale and usually hidden below the noise level of standard experimental techniques.
With the exploitation of heavily doped semiconductors as a new class of plasmonic materials [25][26][27][28][29][30][31], the research expanded into the mid-infrared spectral region, which is an extremely appealing frequency range because it coincides with the fingerprint region of many molecular vibrations.Plasmonic antennas in this range can then help, for instance, to resolve in amplitude and phase specific molecular dynamics in both the frequency and time domains.While classic intensity [7] or polarization [32,33] detection schemes allow one to study the resonance behavior in its steady state, the underlying temporal dynamics, as well as the extremely fast (few fs) response of a resonant antenna remain hidden.Nonlinear femtosecond spectroscopy [21,[34][35][36][37][38] in combination with electromagnetic simulations [36,39,40] can be employed to gain insight into these dynamics and to extract the temporal information that underlies both linear and nonlinear processes.In particular, knowledge of the exact field evolution with sub-optical-cycle precision allows one to observe coherent dephasing mechanisms, an aspect that is crucial for the understanding of nonlinear effects in nanostructures coupled to molecular systems supporting a plethora of resonant modes (e.g., plasmons, photons, or excitons).Current approaches, although very powerful, require a complex implementation and none of the standard methodologies based on linear detection schemes allows for a complete resolution of the optical field in both amplitude and phase.In the visible and near-infrared spectral ranges, this information is very difficult to retrieve since the carrier frequency is too high to allow a direct observation of sub-optical-cycle dynamics on a single and isolated nanostructure.In fact, attosecond spectroscopy is in its infancy, and the sensitivity necessary for this kind of measurement is still out of reach.There have been many efforts to use nonlinear optics and electron emission to fit the temporal near-field electromagnetic response of nanostructures in this spectral range, although not providing a direct measurement of the amplitude and phase of the optical field [34,[41][42][43][44].In the mid-infrared spectral range, however, the full characterization in amplitude and phase of optical pulses becomes possible at the high level of sensitivity required to observe the complex response of an isolated antenna, with a huge potential to expand our knowledge of light-matter interactions in the THz spectral range, beyond what is offered by the classical pump-probe techniques accessing only the charge dynamics.First attempts in this direction have been proposed by several groups, although they were limited to the sub-THz spectral region [45][46][47][48][49][50][51].Here, we demonstrate that the oscillating optical field E 0 of a light pulse interacting with a single plasmonic antenna and shaped by its resonance can be measured by femtosecond multi-THz (or mid-infrared) time-domain spectroscopy (TDS) [52][53][54] combined with high-resolution confocal microscopy.Our experiments promise to extend the observation of field-resolved light-matter interaction in the time domain to single quantum objects in the proximity of plasmonic nanostructures at higher energies, that is at tens of THz.

RESULTS
Sensitive electro-optic sampling TDS (EOS-TDS) is employed for the detection of transmitted fields in a confocal transmission geometry with a high numerical aperture (see Fig. 1 and a detailed description in the Methods section).All the double-rod antennas are fabricated from heavily doped germanium (see Methods) and feature arms with a thickness of 1.4 µm and a width of 1.0 µm and a gap width of 300 nm, while the arm length and thus the aspect ratio and resonance frequency are varied within the sample.With this approach, we can follow the non-trivial response of the plasmonic antenna represented by a clear phase jump of the   For completeness and for testing the robustness of our approach, we performed the same experiment on antennas where we varied the arm length to tune the plasmonic resonance position, and extracted the spectral response of all the antennas in the frequency domain [Fig.4(a)].Furthermore, FDTD simulations were performed to reproduce the observed differential transmission spectra (see also Supplement 1, Fig. S1 for the absorption spectra and near-field profiles).The calculations reproduce well the spectral dependence of the differential transmission as a function of the wavelength [Fig.4(b)].By increasing the arm length, a redshift of the main spectral features is observed.The slight disagreement between the measured and experimental curves is due to the complexity of a 3D problem with a strongly focused Gaussian, which is difficult to tackle numerically.In more detail, although in the simulation we considered as approximation an impinging Gaussian beam, in the experiment, we use Cassegrain objectives to focus the beam.Thus, the beam impinging on the structures deviates from a perfect Gaussian.Moreover, some wave vectors are not contributing to the excitation of light emission from the antenna due to the hollowness of both the illumination and the collection beams.Another cause of discrepancy can be that in the experiment, the beam might not be focused on the same plane as it is in the simulation, thus having a slightly different field distribution because of the field gradients in the focal volume.Finally, also the Gouy phase shift, when combined with the actual placement of the antenna with respect to the ideal focal plane, might contribute to the small discrepancy between measured and calculated curves.
The small redshift of the experimental response compared to the calculated one, can be also inferred to the aforementioned reasons.
It is worth mentioning here that, with our approach, it is clearly possible to experimentally access both the complex frequencyand time-resolved response of a single plasmonic antenna at tens of THz, until now possible only by using theoretical approaches such as the FDTD method.Moreover, by knowledge of either the spectral response at a certain time delay or of the temporal response at a specific frequency, we can reconstruct the transient response as a function of both time and frequency via the Wigner distribution function (WDF) [55], which allows us to describe the nanostructure response during the interaction with an external light pulse.Thus, we can estimate the frequency-time distribution of the antenna response, since we can directly measure the field response in the time domain.In our case, the temporal profile of the incident light pulse interacting with the substrate only or with the antenna with an arm length of 5.5 µm has been measured [Figs.2(a), blue and red curve, respectively].By subtracting the two signals and correcting for the offset phase of the background probe signal introduced by the measurement, we finally obtained the WDF for the response of the antenna [Fig.5(a)].As inferred by the color map, the response of the antenna shows a secondary peak at positive delays centered at 32.5 THz.This effect represents a time-frequency manifestation of the phase jump that occurs in an oscillator between frequencies above and below resonance [Figs.2(b) and 2(d), orange line].The overall response is additionally enriched by interference effects, possibly due to the finite thickness of the antenna, that determine an increase of the extinction at approximately 40 THz.These effects are observable by the modulation of the frequency response at 38 THz that occurs approximately at a delay of 25 fs.We have also calculated the WDF from the data plotted in Fig. 2(c), and the result is presented in Fig. 5(b).As can be seen, the main features of the time-frequency response of the antenna are well reproduced.It is worth mentioning here that although the matching between the calculated WDF and the experimental one is remarkable, there are also small discrepancies between experiment and calculation.This means that even if numerically one can predict well the antenna response, it is very important to experimentally access the full-field response of the real system, in particular in view of applications where other  objects, e.g., molecules or quantum emitters, are coupled with the antenna, and discrepancies with the numerical calculations might play a pivotal role in eventual coupling mechanisms.

CONCLUSIONS
In conclusion, we have demonstrated that with EOS-TDS it is possible to acquire experimentally the complete response of the linear interaction between a light pulse and a single semiconductor plasmonic antenna resonant in the mid-infrared by resolving the time-dependent optical field with a sub-cycle temporal resolution.We have shown that the direct extraction of the complex spectral response (in amplitude and phase) of a single antenna is possible beyond the current state of the art in EOS-TDS.Since we can observe the resonance in amplitude and phase, we can tailor an optical waveform to drive the plasmonic oscillation by enhancing it or switching it off completely by a suitable interference between plasmons and external fields.While the mid-infrared spectral window is especially appealing because it contains the fingerprint of vibrational resonances of complex molecules, we foresee that this approach might be pushed to resolve the full response of nanostructures also in the near-infrared and visible ranges, owing to emerging attosecond technologies [56] and to coherent control of localized plasmons by tailored waveforms.Moreover, this method can be applied to enhance light-matter interactions with the plasmonic resonance and shine new light on unexplored molecular dynamics in the mid-infrared by experimentally accessing the effect of the plasmon-enhanced electric field on coherent dynamics and the manipulation of photochemical reactions, for instance, by using one-shot EOS-TDS schemes [57,58].Additionally, new degrees of freedom will be enabled by the engineering of antenna structures with the possibility to shape the field response of the antenna.This perspective is particularly appealing since it will be possible to shape the optical field in the proximity of the antennas with the aim to control the coherent dynamics of individual molecules or quantized nanosystems.

Fabrication.
Heavily doped germanium antenna structures with resonance in the mid-infrared were fabricated via electron beam lithography and reactive ion etching from epitaxial films on a silicon substrate [20,31,59].By means of low-energy plasma-enhanced chemical vapor deposition, high-quality single crystalline growth with in situ phosphorous doping was performed, with an activated donator concentration of about 2.5 × 10 19 cm −3 [30].
Optical setup and antenna characterization.Isolated doublerod antenna structures with varying arm lengths were measured in a confocal microscopy setup based on two Cassegrain-geometry all-reflective objectives (36×, NA = 0.5).To probe the antenna response, broadband phase-stable femtosecond pulses in the mid-infrared/multi-THz frequency range were generated via phase-matched difference frequency generation in GaSe bulk crystals [60].At a bandwidth of 10.7 THz, the pulse duration equals 58 fs.The maximum peak field is 5 MV/cm, but the peak amplitude is attenuated to be lower than 100 kV/cm in the experiments.An Yb:KGW high-power laser system (Light Conversion, Pharos 10 W ) with a repetition rate of 50 kHz was used to guarantee large acquisition rates to reduce statistical noise [61].The antenna response was detected in transmission geometry via sensitive EOS-TDS in thin GaSe crystals (thickness 80 µm) [60].As EOS-TDS gating pulses, a broadband white light supercontinuum was generated in a bulk YAG crystal and compressed to sub-15 fs durations [61,62].To avoid phase fluctuations within the acquisition time, transients were recorded in fast-scan mode at an acquisition rate of 5 Hz using a piezo delay scanner.Low-noise detection was accomplished by employing a boxcar amplifier (Zurich Instruments, UHF-BOX ) on a shot-to-shot basis in combination with an optimized ellipsometry balanced detection scheme (Thorlabs, PDB440C).The presented data were averaged from 1250 recorded transients each.A cross correlation algorithm was employed to compensate for the timing jitter between single measurements before averaging was undertaken.
FDTD simulations.Electromagnetic simulations for the linear optical response of Ge antennas were performed with the FDTD method using a commercial software [63].The frequency-dependent dielectric constant of heavily doped Ge was characterized experimentally by Fourier-transform infrared spectroscopy [30] (see also Supplement 1, Fig. S2), while for Si, we adopt a constant real-valued refractive index of n = 3.4.High-accuracy conformal meshing was used for the air and substrate discretization, while a finer cell size of 30 nm was forced in the region of the antenna.For all the simulations, a Gaussian illumination resembling the experimental numerical aperture (0.5) was employed.The convergence criterion was set to halt the simulations once the electromagnetic energy initially injected into the simulation volume by the source was reduced by a factor 10 −5 .The time-domain evolution of the electric field was recorded with a point field monitor located 5 µm below the sample surface, i.e., inside the Si substrate.This monitor provides directly the time evolution of the complex (amplitude and phase) electric field, as plotted in Fig. 2(c).The same quantity is then processed with the same procedure as for the experimental data to obtain the map in Fig. 5(b).Intensity spectra were obtained by collecting spatially resolved field spectra (amplitude and phase), as obtained by Fourier-transforming the respective time evolutions, over an ideally infinite plane located 5 nm below the antenna-substrate interface.A numerical far-field projection of such near fields is then performed with the software built-in function to obtain the intensity distribution over a sphere of 1 m radius centered on the antenna position, which is then integrated over the angular acceptance of the collection objective.Finally, absorption spectra were extracted from the net flux of the Poynting vector entering the antenna volume, as measured by a closed box monitor surrounding the whole antenna.

Figure 2 (
a) shows the experimental transient real part of the impinging electric field.We plot both the electric field of the probe pulse after propagation through the bare silicon substrate (Re[E 0x ], blue graph) and the electric field of the probe pulse interacting with both substrate and a Ge double-rod antenna of arm length 5.5 µm in the microscope focal area (Re[E x ], red graph).The two curves are normalized to the maximum value of Re[E 0x ].A linear subtraction between Re[E 0x ] [Fig.2(a), blue graph] and Re[E x ] [Fig.2(a), red graph] provides a time-resolved differential signature of the interaction between the optical waveform and the antenna [Fig.2(b), green graph].

Fig. 1 .
Fig. 1.Scanning electron microscope image of a single double-rod antenna structure with an arm length of 3.5 µm fabricated from heavily doped germanium on a silicon substrate.The red transient represents the incoming incident pulse field E 0 , while the pink one is the pulse after the interaction with the Ge antenna.The light propagates along the z direction, while the beam is polarized along the x direction parallel to the long axis of the antenna.The bottom panel shows a sketch of the experimental setup used for the electro-optical sampling time-domain spectroscopy (EOS-TDS) of the transient signal after interaction with the antenna.LP, low pass filter; LED, white light emitting diode; CCD, charge-coupled device.LED and CCD are used to align the microscope.

Fig. 2 .
Fig. 2. Multi-THz TDS on a heavily doped Ge antenna with arm lengths of 5.5 µm: (a) Re[E x ] after interaction with the antenna (red), in comparison to the probe pulse Re[E 0x ] not interacting with the antenna (blue), normalized to maximum of Re[E 0x ]; (b) TDS antenna response field amplitude by subtraction of the curves plotted in (a); (c) FDTD calculated Re[E 0x ] |E 0 | (blue) and Re[E x ] (red) normalized to the maximum of Re[E 0x ]; (d) calculated antenna response field amplitude by subtraction of the curves plotted in (c).The colored dashed lines are eye-guides to highlight the in-phase (blue) and phase jump (orange) cases.

Fig. 3 .
Fig. 3. (a) Experimental intensity spectra with and without antenna interaction obtained via fast Fourier transform (FFT; red and blue, respectively) normalized to the intensity I 0 of the unperturbed incident electric field, and experimental differential transmission I/I 0 (green); (b) spectral response function in amplitude (green) and phase (blue) of a single antenna obtained via FFT of the temporal response difference signal plotted in Fig. 2(b).

Fig. 4 .
Fig. 4. (a) Experimental differential transmissions of single antennas for various arm lengths.(b) FDTD simulated differential transmissions in the frequency domain.

Fig. 5 .
Fig. 5. (a) WDF of the experimental normalized electric field amplitude of a Ge antenna with an arm length of 5.5 µm.(b) WDF of the simulated normalized electric field amplitude of a Ge antenna with an arm length of 5.5 µm.