Few-cycle optical pulses with high peak electric fields are versatile tools to investigate elementary processes at the ultrafast timescale [1–8]. Owing to the brief temporal window of the nonresonant perturbation, they create extreme conditions far from equilibrium without destroying the sample under study. In this context, single-cycle pulses [9,10] set the limit for the minimum duration that can be obtained with a specific sinusoidal carrier frequency. In addition, for many experiments, not only the duration, but the exact profile, of the electric field is relevant [11,12], sparking the development of a variety of pulse-shaping techniques acting in the spectral domain. In this context, optical synthesizers are able to generate short pulses by controlling the spectral phase in a broad frequency range [13,14]. The bandwidth of these transients usually spans one to two octaves with respect to their carrier frequency. Due to the ultimately short temporal resolution they offer, such single-cycle pulses at high optical frequencies seem to be an ideal tool for the study of processes occurring upon an impulsive change of the electric field. However, their fast rise is followed by an equally fast decay, leaving little time for the system under study to evolve under a constant bias. To overcome this limitation, one has to manipulate the pulse envelope directly in the time domain, providing a sharp upsurge of the electric field followed by a long-lived plateau. This approach establishes the optical analog to acoustic shockwaves, where gas pressure changes abruptly (e.g., upon passing the Mach cone from an object moving with hypersonic speed). Similarly, our synthetic transients at multi-terahertz frequencies provide a sudden onset of quasi-static bias at extreme electric field amplitudes. Such peculiar optical waveforms are required to target phenomena like, for example, the buildup of the effective electron mass following the abrupt biasing of electrons within the crystal lattice of a semiconductor [15]. This fundamental process has not yet been observed in solid-state systems [16] due to the lack of suitable optical sources.

Generation of step-function field transients requires a new strategy for nonperturbative nonlinear optical interaction. To achieve ultrafast shaping of terahertz transients, we gate the reflectivity of a germanium (Ge) surface employed as an ultrafast active mirror [17]. A control pulse much shorter than one half-cycle of the terahertz field excites a dense electron-hole plasma, abruptly modifying the dielectric response of the semiconductor to behave like a metallic surface with high reflectivity. Previously, photoconductive switching has been applied to the generation of multicycle pulses from ${\mathrm{CO}}_2$-lasers [18,19], free electron lasers [20], and single-cycle pulses from phase-stable multicycle pulses in the mid-infrared [17]. Spectral shaping of strongly chirped terahertz transients also was achieved [21]. We now ask the question: Which waveform results when attempting to cut an electromagnetic field with a precision that is much faster than half an oscillation period? Owing to its high electron mobility and lack of infrared-active phonons that ensure a flat spectral response in the far- and mid-infrared, the group IV semiconductor Ge represents the ideal material for such a subcycle active mirror. We employ a $⟨100⟩$-cut Czochralski-grown undoped Ge wafer with $\text{resistivity}>45\mathrm{\Omega }\mathrm{cm}$. Optical excitation of an electron-hole plasma by visible photons takes advantage of the high absorption of Ge that is due to the giant joint density of states around the $L$ point of the Brillouin zone, where conduction and valence bands are parallel. This fact enables extremely high plasma densities without the two-photon absorption pathways exploited in previous approaches.

The optical system consists of a hybrid Er:doped fiber laser system followed by a Ti:sapphire regenerative amplifier operating at a 1 kHz repetition rate. This source is used to drive the experiments with 3 mJ pump pulses centered at 775 nm with 100 fs duration [10]. The experimental setup is sketched in Fig. 1. The fundamental beam undergoes optical rectification (OR) in 500 μm thick $⟨110⟩$ ZnTe, resulting in single-cycle electric field transients centered around a frequency of 2 THz. After rejecting the residual pump light with a silicon window, the $p$-polarized waveform is focused onto a Ge mirror under Brewster’s angle of incidence by an off-axis paraboloid with an effective focal length (EFL) of 20 mm. In parallel, a two-stage noncollinear optical parametric amplifier (NOPA) seeded by a broadband supercontinuum generates 4 μJ pulses at 500–700 nm wavelength [22]. These control pulses are compressed to 8 fs duration with a pair of chirped mirrors. The central photon energy of 2 eV is tuned to the maximum optical absorption of Ge [23]. The visible beam is co-focused onto the semiconductor surface through a small hole in the parabolic mirror to create a dense electron-hole plasma. Importantly, the Brewster angle of incidence allows for high contrast between the dielectric transmission and photoconductive reflection. Adjustment of the relative time delay $T$ between the terahertz waveform and the control pulse allows precise timing of the plasma buildup. The low repetition rate of 1 kHz ensures complete relaxation of the photoexcited carriers before the arrival of the following pulse pair [24]. After recollimation, the manipulated terahertz waveform is collinearly overlapped with a broadband near-infrared pulse by a silicon filter. This gate is employed for electro-optic sampling (EOS) after focusing with an off-axis paraboloid into a 17 μm thick ZnTe crystal oriented in $⟨110⟩$ direction. A quarter-wave plate ($\lambda /4$) and a Wollaston prism enable sensitive balanced detection of the ellipticity of the gate pulse, which is proportional to the electric field of the terahertz waveform convoluted with the instrument response function [25,26]. The 9 fs duration at the carrier wavelength of 1200 nm and the thin detection crystal provide a frequency bandwidth up to 60 THz.

 

Fig. 1. Scheme of experimental setup. Single-cycle pulses centered at 2 THz are generated by optical rectification (OR) and colinearly focused with 8 fs short control pulses onto an intrinsic germanium surface. A photoexcited carrier plasma reflects part of the terahertz transient. The resulting waveform is detected by electro-optic sampling (EOS) with a 9 fs short gating pulse. PM, parabolic mirror; EFL, effective focal length; WP, Wollaston prism.

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Waveforms resulting from the EOS measurements are shown in Fig. 2(a). The entire terahertz transient is reflected by the semiconductor surface when the switching pulse arrives earlier, e.g., for $T=-5\mathrm{ps}$ [red line in Fig. 2(a)]. The maximum field amplitude exceeds 50 kV/cm, corresponding to 70% of the maximum field amplitude upon reflection on a gold surface. The field transient obtained under optimum timing of the control with respect to the central optical cycle of the terahertz pulse ($T=0$) is depicted as a blue line in Fig. 2(a). Remarkably, at EOS delay times $t<0$ the optical cycles are almost completely suppressed, while for $t>0$ the signal remains close to the original transient. Minor deviations result from the fact that the re-emitted waveform is directly related to the current induced in the Ge surface, which exhibits a retarded response after generation of real photocarriers. The finite lifetime of these currents also ensures that the time integral of the field remains equal to zero, despite the fact that the cutting is performed on a subcycle scale (i.e., at a position with well-defined polarity of the field).

 

Fig. 2. Temporal analysis of terahertz shockwaves. (a) Electric field of the reflected waveform as a function of EOS time $t$. Red line: Control pulse arrives 5000 fs before terahertz transient; blue line: Both pulses arrive with optimum timing, creating a step in the transient. (b) Electric field (color coded) as function of EOS time $t$ and switching delay $T$. The light blue line denotes $T=0$ as in (a). Dashed gray line: guide to the eye.

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The new waveform indeed corresponds to a shockwave with a sharp onset of the electric field where 50% of the rise to the global maximum is achieved within less than 20 fs. Taking into account the 9 fs gating pulse for EOS, this result corresponds to a field onset of the actual shockwave within approximately 15 fs. Continuous variation of the arrival time of the control pulse sections the terahertz transient at arbitrary times. Figure 2(b) shows a map of the electric field (color coded) as a function of the EOS delay time $t$ and the switching delay $T$. The position of the sharp onset of the shockwave is visible as diagonal line (gray dashed: guide to the eye), denoting the arrival of the control pulse. The quasi-instantaneous electron-hole plasma readily slices the terahertz wave on a subcycle timescale. Here, the total response function for EOS has been taken into account to also emphasize the step function at delays $T$ where it is becoming small compared to the main extremum. To this end, we first Hann-filter the measured transients to avoid artifacts from limiting the data at positive times $t$. We then calculate the Fourier spectrum of this output and divide it by the complex-valued frequency response due to the dispersion of the optical properties and finite thickness of the detector crystal [26]. Transforming back into the time domain then yields the corrected profile of the electric field.

The few-femtosecond rise time of the terahertz shockwaves is incompatible with the sinusoidal carrier frequency that characterizes the transient incident on the Ge surface. Consequently, a drastic change of the spectral content of the reflected pulse has to result. Intensity spectra of the original (red line) and manipulated (blue) waveforms are depicted in Fig. 3 with identical scaling. The shockwave exhibits an increased spectral content up to a frequency of 60 THz. This finding represents a broadening of nearly a factor of 10 with respect to the bandwidth of the incident spectrum. Between 15 THz, where the unmodulated spectrum reaches the noise floor, and 35 THz, the shockwave possesses a spectral intensity almost four orders of magnitude higher than that of the unmodulated transient. This frequency conversion originates from the inherently nonperturbative nature of the action of the plasma generated by the visible control pulse on the terahertz waveform. At 45 THz, the broadband spectrum of the shockwave exhibits a dip. This feature is explained by optically induced absorption in highly excited Ge. In fact, transitions between the heavy and light hole band arise upon significant depletion of valence electrons due to interband excitation [27]. Heavy and light hole bands are degenerate at the $\mathrm{\Gamma }$ point and start with different curvatures before turning essentially parallel for most of their way toward the $L$ point. A spin-orbit splitting of ${\mathrm{\Delta }}_1=187\mathrm{meV}$ exists between the valence band $L$ valleys [23], corresponding to a frequency of 45.2 THz. Consequently, a spectrally sharp inter-valence-band absorption emerges around a frequency of 45 THz when the photoexcited hole population covers a sufficiently extended section of the Brillouin zone.

 

Fig. 3. Shockwave spectrum. Intensity of the reflected transients over frequency, acquired by Fourier transform of the EOS traces filtered with a Hann function. Red and blue lines correspond to identical control pulse delay times $T$, as in Fig. 2(a). ${\mathrm{\Delta }}_1$ denotes the frequency position of the literature value for the energy distance between heavy and light hole bands around the $L$ point of the Ge band structure.

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To further study the subcycle manipulation of terahertz transients by the ultrafast plasma reflection, the spot size of the control pulse on the Ge surface is varied from 150 μm to 270 μm full width at half maximum (FWHM), measured perpendicular to the beam. Absorbed energy fluences per pulse of $3.6\mathrm{mJ}/{\mathrm{cm}}^2$ and $1.1\mathrm{mJ}/{\mathrm{cm}}^2$ result, respectively. The fully reflected waveform ($T=-5000\mathrm{fs}$) has an increased field strength at a larger spot size. This finding is due to the electron-hole plasma affecting a larger area of the terahertz mode. However, the shockwave displays a slower onset at $T=0\mathrm{fs}$ for the larger spot size (lower fluence) of the control pulse. The reflected waveforms for both cases are displayed in Fig. 4(a) with solid and dotted lines depicting the high- and low-energy fluences, respectively. Since the control pulse duration remains constant, this behavior must be associated with a difference in the electron-hole density and thus in the plasma reflectivity. Corresponding intensity spectra are depicted blue in Fig. 4(b). At the lower fluence (dotted lines), generation of high frequencies is less efficient, yet still allows to reach up to 50 THz. Also, the absorption dip at 45 THz is only observed at high fluence, where a high hole density causes inter-valence-band absorption far from the $\mathrm{\Gamma }$ point. Importantly, the plasma frequency is still significantly higher than the measured frequency components in both cases, explaining the lack of a sharp spectral cutoff. We calculate the frequency $\nu$ dependent dielectric function at the Ge surface using the Drude model,

 

Fig. 4. Influence of excitation density on the generation of terahertz shockwaves. Dotted lines correspond to a 270 μm spot size of the visible pulses and a fluence of $1.1\mathrm{mJ}/{\mathrm{cm}}^2$, while the spot size is 150 μm for the full lines, resulting in a higher fluence of $3.6\mathrm{mJ}/{\mathrm{cm}}^2$. (a) Reflected waveforms measured with the control pulse set to a delay time of $T=0\mathrm{fs}$. (b) Corresponding intensity spectra normalized to the maximum intensity (blue). Hann windows are applied before Fourier transformation as well as temporal averaging outside the central region between $t=-120\mathrm{fs}$ and $+120\mathrm{fs}$ to suppress high-frequency noise components. Calculated amplitude reflectivity spectra owing to the photoinduced electron-hole plasma in Ge are shown as orange lines for corresponding carrier densities.

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The transport effective masses for electrons and holes in Ge are $m_{e,L}=0.12m_0$ and $m_{h,\mathrm{\Gamma }}=0.36m_0$ [29]. With these values, we estimate a plasma frequency at the Ge surface of 290 THz at the high fluence and ${\nu }_p=160\mathrm{THz}$ at the lower fluence, respectively. The significant difference in the response at frequencies up to 60 THz is due to the strong influence of scattering in the photoinduced electron-hole plasma. In fact, for a realistic scattering rate of $\gamma ={(10\mathrm{fs})}^{-1}$ [30], the plasma edge is smeared out over a wide spectral region, rendering the position of the plasma frequency also significant for the reflectivity at lower frequencies. The results of our calculations are depicted in Fig. 4(b) as full and dotted orange lines for the case of the two excitation fluences, respectively.

In conclusion, we report the shaping of terahertz waveforms with deeply subcycle precision. This capability results in the generation of shockwaves in which the onset of the electric field is much shorter than half a cycle of the original carrier frequency. The reflected waveform shows a strong temporal asymmetry and is fundamentally different from the incident single cycle. Previous applications [17,21] of this technique are outperformed in rise time by the use of 8 fs short control pulses centered at a 600 nm wavelength. The nonlinear interaction between the terahertz waveform and the control pulse is mediated through a time-dependent electron-hole plasma acting as a dynamical nonlinearity, giving rise to a new kind of wave mixing outside the perturbative interaction regime. Unlike well-known frequency mixing effects, this interaction is understood best in the time domain, where the sudden rise of carrier density creates a step-function-like behavior of the reflectivity. Somewhat against physical intuition, this strategy supports a clean cut in an electromagnetic field without significantly inducing spurious temporal or spectral substructures. This type of waveform represents a promising tool for the study and control of any switch-on processes on a molecular timescale. The dynamical buildup of the carrier effective mass in the periodically modulated potential of bulk solids is a prominent example of such a phenomenon.