Single-cycle infrared waveform control

Tailoring the electric-field waveform of ultrashort light pulses forms the basis for controlling nonlinear optical phenomena on their genuine, attosecond timescale. Here we extend waveform control from the visible and near-infrared—where it was previously demonstrated—to the mid-infrared spectral range. Our approach yields single-cycle infrared pulses over several octaves for the first time. Sub-10-fs pulses from a carrier-envelope-phase-stabilized, Kerr-lens-mode-locked, diode-pumped Cr:ZnS laser drive cascaded intrapulse difference-frequency generation and control the electric-field evolution of the resulting coherent emission over 0.9–12.0 μm. Sub-cycle field control in this wavelength range will be instrumental for launching and steering few-femtosecond electron/hole wavepackets in low-gap materials, extending the bandwidth of electronic signal processing to multi-terahertz frequencies, as well as for electric-field-resolved molecular fingerprinting of biological systems. Continuously adjustable single-cycle waveform spanning from 0.9 to 12.0 μm is obtained by cascaded intrapulse difference-frequency generation in a ZnGeP2 crystal. The cascade-associated phase response—distinct for different spectral bands—provides a new tuning parameter for waveform adjustment.

(1.8-4.4 µm) single-cycle mid-IR pulses with a pulse envelope that can be arbitrarily shaped 39 . None of these techniques have provided pulse-to-pulse reproducible waveforms so far.
In this Article, we present a new approach for coherent multioctave mid-IR generation with an intrinsic capability of changing the relative phase of different spectral regions and thereby shaping the emerging waveform. Specifically, we create mid-IR waveforms in several spectral bands by cascaded intrapulse difference-frequency generation (IPDFG) 31,40 and manipulate their spectral phases by taking advantage of their different dependence characteristics on the carrier-envelope phase (CEP) of the driving pulse. The adjustment of the relative phase of different spectral components-across over 3.7 optical octaves-is achieved without the need for spatial separation and subsequent interferometric recombination 14 . Our approach yields the first multi-octave synthesis of single-cycle mid-IR waveforms, with a continuously adjustable, highly reproducible electric-field evolution. The concept is generalizable towards the synthesis of a wider variety of infrared waveforms by increasing the number of adjustable parameters, such as the control of spectral phase and amplitude of the driving pulse before the cascaded IPDFG.

Cascaded IPDFG
The basic principle of cascaded IPDFG is illustrated in Fig. 1, based on a simplified modelling of broadband light propagating through a nonlinear crystal (Supplementary Section 7). To isolate and visualize the relevant frequency-mixing processes, the driving laser is represented by two isolated narrowband spectral components (centred at 1.9 and 2.3 µm) of its entire spectrum, sharing the same CEP ( Fig. 1, dashed black curves).
Difference-frequency mixing of the two driving components initially creates a peak at 30 THz. Since the two driving components share the same CEP value, the mixing product's CEP is invariant to CEP changes of the driver. This fundamental (zeroth order) IPDFG field, on further propagation in the nonlinear medium, mixes with one of the narrowband drivers to create components at 100 THz

Single-cycle infrared waveform control
Philipp Steinleitner 1,6 , Nathalie Nagl 1,2,6 ✉ , Maciej Kowalczyk 1,2,3,6 ✉ , Jinwei Zhang 1,4 , Vladimir Pervak 2 , Christina Hofer 1,2,3 , Arkadiusz Hudzikowski 5 , Jarosław Sotor 5 , Alexander Weigel 1,3 , Ferenc Krausz 1,2,3 and Ka Fai Mak 1 ✉ Tailoring the electric-field waveform of ultrashort light pulses forms the basis for controlling nonlinear optical phenomena on their genuine, attosecond timescale. Here we extend waveform control from the visible and near-infrared-where it was previously demonstrated-to the mid-infrared spectral range. Our approach yields single-cycle infrared pulses over several octaves for the first time. Sub-10-fs pulses from a carrier-envelope-phase-stabilized, Kerr-lens-mode-locked, diode-pumped Cr:ZnS laser drive cascaded intrapulse difference-frequency generation and control the electric-field evolution of the resulting coherent emission over 0.9-12.0 μm. Sub-cycle field control in this wavelength range will be instrumental for launching and steering few-femtosecond electron/hole wavepackets in low-gap materials, extending the bandwidth of electronic signal processing to multi-terahertz frequencies, as well as for electric-field-resolved molecular fingerprinting of biological systems.
(first cascading order). This cascading effect continues with the re-mixing of the first order and one of the driving components to generate new frequencies near 60 THz (second order), which, in turn, generate the 70 THz band (third order) 40 . Depending on the cascading order, the CEP of the newly generated wavelength components will either be invariant to (even orders) or follow the changes in the CEP of the driver (odd orders). Consequently, any changes in the driver CEP modifies the combined electric-field waveform in the time domain. This also implies that the electric-field evolution of radiation emerging from cascaded nonlinear frequency mixing varies from pulse to pulse in the absence of CEP stabilization.
For a broadband driving pulse (Fig. 1, dashed grey curve), the fundamental and cascaded components merge into a continuum ( Fig. 1, light-grey and light-red curves). Importantly, the spectral region where the even and odd orders overlap exhibit pronounced interference effects on tuning the CEP of the driving laser (Supplementary Section 6). Shifting the CEP by Δφ = ±π with respect to its reference that yielded the maximum mid-IR intensity ( Fig. 1, light-red curve; Δφ = 0) results in the maximum suppression of spectral intensity in this region by destructive interference (Fig. 1, light-grey curve). This change in the spectral distribution of mid-IR radiation underlines the necessity of CEP-stabilized driver light for reproducible multi-octave infrared waveform generation via cascaded second-order nonlinearities.
Here we demonstrate a CEP-stabilized femtosecond Cr:II-VI laser oscillator, with the Cr:ZnS active medium being directly pumped by laser diodes. Femtosecond pulses are generated by Kerr-lens mode locking, yielding 24 nJ pulses with a duration of 28 fs (full-width at half-maximum (FWHM)) at a repetition rate of approximately 23 MHz 44 .
The output pulses undergo further spectral broadening via self-phase modulation (SPM) in a highly nonlinear bulk dielectric medium (Fig. 2a, TiO 2 ; Methods and Supplementary Section 2) to generate a spectrum spanning from 1.1 µm (272 THz) to 3.2 µm (94 THz) at the −20 dB level (Fig. 2b). Furthermore, despite the strong spectral broadening, the spatial intensity profile of the laser beam remains smooth and Gaussian-like (Fig. 2c, inset) with a laser-beam quality parameter M 2 < 1.4 ( Supplementary Fig. 2). This value indicates excellent focusability for downstream experiments. The spectrally broadened pulses are temporally compressed in a set of specially designed chirped dielectric multilayer mirrors to a duration of 7.7 fs (FWHM; Fig. 2c), which is equivalent to one optical cycle at the spectral centroid position of 2.24 µm.
The spectrally broadened pulses are subsequently split with an uncoated wedged ZnS beam splitter. The two main reflections off the wedge's front (48 mW) and back (36 mW) surface are used as a gate pulse for characterizing the mid-IR waveform with EOS (Methods and Supplementary Section 5) and for carrier-envelope offset

Fig. 1 | Waveform synthesis via cascaded nonlinear processes.
Simulated mid-IR intensity spectrum generated in a 0.5-mm-thick ZGP crystal by two narrowband laser pulses centred at 130 THz (2.3 µm) and 160 THz (1.9 µm) (dashed black curves), simulated by using commercial nonlinear pulse propagation software. Downconverted radiation results from the difference-frequency mixing of differing orders (0-3). The most intense, fundamental component (zeroth order) is created by the mixing of the two narrowband inputs from the driving laser, whereas the higher orders (1-3) arise from further difference-frequency mixing of the preceding orders (0-2) with the driving laser. Further cascading does occur, but their progressively lower intensities mean that they play a negligible role in the dynamics. The CEP of the odd orders (shaded in blue) follow the changes in the CEP of the driving laser, whereas even orders are invariant (shaded in orange). By varying the CEP of the driving laser, the phase relation between the odd and even orders can be changed, resulting in a change in the total waveform. For a single driving pulse with a broad spectral width (simulated here with a spectral width of 44 THz and centred at 145 THz (dashed grey curve)), the pronounced spectral broadening of the different orders overlap to form a broad continuum (grey and light-red curves). In the region of the strongest overlap between odd and even mixing orders, at around 60 THz (light-red-shaded area), the interference between waveform components of differing CEP values leads to the modulation of spectral intensity with respect to the driving field's CEP. The red and grey curves depict the intensity distribution for the driving field's CEP set to yield constructive (Δφ = 0) and destructive (Δφ = ±π) interference, respectively. The same (second-order) susceptibility also implies sum-frequency generation (SFG) at about 190 THz, by mixing the 160 THz driver with the zeroth-order IPDFG band. This, however, is likely to be shrouded by the more efficient self-phase-modulation-induced spectral broadening of the driver in the ZGP crystal.  The output of the broadening stage (16 nJ) is temporally compressed by chirped multilayer mirrors and split into three branches using a ZnS wedge.
The first reflection is used as a gate pulse for EOS in a 0.1-mm-thick GaSe crystal (θ = 35°, φ = 30°). The second reflection is used for f ceo detection and stabilization. The transmitted beam (9 nJ) is used for mid-IR generation in a 1-mm-thick ZGP crystal (θ = 51°, φ = 0°). To characterize the generated mid-IR pulses (cyan waveform), they are combined with the gate pulse (blue waveform) using a second ZnS wedge for EOS. b, Measured intensity spectrum after the broadening stage (grey curve) compared with the retrieved spectrum (blue curve) from a second-harmonic frequency-resolved optical gating (SHG-FROG) measurement. The corresponding spectral phase is shown in orange. c, Temporal intensity profile (blue curve) and temporal phase (orange curve) of the compressed pulse as retrieved from the SHG-FROG measurement, together with the Fourier-transform limit (FTL) of the measured intensity spectrum (grey curve). The inset shows a spatial beam profile of the broadband pulses, with M 2 < 1.4. d, CEP noise power spectral density (PSD) of the 7.7- fs pulses (blue curve) and the corresponding r.m.s. phase noise integrated between 0.001 Hz and 11.43 MHz (orange curve). e, Relative intensity noise (RIN) of the 7.7 fs pulses (blue curve) and the corresponding r.m.s. noise integrated between 10 Hz and 1 MHz (orange curve). The frequency span differs from the one in d due to limitations of the employed measurement device.
frequency (f ceo ) detection, respectively. The f ceo beat note is detected in an f-2f interferometer with a 60 dB signal-to-noise ratio (100 kHz resolution bandwidth) and locked to zero via pump-power modulation 45 . This results in a CEP identical for each laser pulse and continuously tunable by adjusting the feedback loop-a crucial feature for the controlled shaping of mid-IR waveforms generated by cascaded frequency-mixing processes. The CEP noise, measured at the output of the broadening stage (out of the feedback control loop), exhibits a root mean square (r.m.s.) jitter of 11 mrad integrated from 0.001 Hz up to the Nyquist frequency of 11.43 MHz (Fig. 2d, Methods and Supplementary Section 3). This represents one of the highest CEP stabilities ever reported 46,47 , owing, among others, to the high amplitude stability of the diode-pumped Cr:ZnS laser driver, which yields an integrated relative intensity noise of 0.036% (r.m.s. value, measured between 10 Hz and 1 MHz) after the broadening stage (Fig. 2e). The outstanding stability of the single-cycle near-IR waveforms indicated by these noise figures is the key prerequisite for highly reproducible mid-IR waveform generation.

Controlled single-cycle waveforms in the mid-Ir region
The beam traversing the ZnS wedge (Fig. 2a), with an average power of 215 mW, is focused into a 1-mm-thick ZnGeP 2 (ZGP) crystal for frequency downconversion via cascaded IPDFG (Methods and Supplementary Section 4). The radiation exiting the nonlinear crystal consists of the field of the transmitted driver pulse as well as the field resulting from the frequency-mixing processes (Fig. 1). The driver field has two orthogonal components of equal magnitude (Methods), one perpendicular and one parallel to the linearly polarized products of the frequency-mixing processes. As a consequence, the mid-IR components originating from IPDFG and the driver components with the same polarization together result in a coherent near-IR-to-mid-IR supercontinuum. Its spectrum was characterized, behind a polarizer, using a monochromator with multiple gratings and a pyroelectric detector (Fig. 3). The generated infrared supercontinuum (Fig. 3, top) spans from 25 THz (12.0 μm) to 330 THz (0.9 μm) at the −30 dB level, equivalent to over 3.7 octaves, limited at both ends by the transmission window of ZGP. This agrees well with our simulations based on the experimental parameters (Fig. 3, bottom).
By locking the CEP of the laser to a value ensuring constructive interference (Δφ = 0) between the cascading orders of the IPDFG process, we maximized the spectral intensity in the region of spectral overlap between the odd and even cascading orders (Fig. 1) from 45 THz (6.7 μm) to about 75 THz (4.0 μm) (Fig. 3, blue solid curve). On changing the CEP of the driver pulse with respect to the (unknown) reference that yielded the maximum mid-IR intensity by Δφ = π, a pronounced dip developed in the overlap region (Fig. 3, orange solid line) due to destructive interference. This effect is also verified by numerical simulations (Fig. 3, bottom). With the beam path purged with nitrogen to reduce ambient absorption, the power of the mid-IR part of the supercontinuum (λ > 3.6 µm, Δφ = 0) is 31 mW (1.3 nJ; Fig. 2a), corresponding to an optical-to-optical conversion efficiency of 14% for the IPDFG process. Furthermore, the corresponding spatial beam profile (Fig. 3, inset) exhibits an excellent mid-IR beam quality. EOS 48 has recently been advanced to optical frequencies of several hundred terahertz 49 , and hence, it is ideally suited for the direct measurement of electric-field waveforms 15,[17][18][19]50,51 . Using part of the 7.7 fs laser pulses (Fig. 2c) as the gate pulse, we electro-optically sampled the mid-IR part of the supercontinuum generated by cascaded IPDFG in a 0.1-mm-thick GaSe crystal (Methods and Supplementary Fig. 3). The EOS trace-indicative of the actual electric waveform 51 -exhibits a clean cosine-pulse-like temporal profile for Δφ = 0 (Fig. 4a, blue curve). The main peak coincides with the central peak of the calculated intensity envelope (Fig. 4b, blue curve) and indicates an FWHM pulse duration of 20 fs-equivalent to 0.9 optical cycles at the spectral centroid of 6.6 μm. The full supercontinuum exiting the nonlinear crystal, as measured by the monochromator (Fig. 3), can support even shorter pulses. If needed, appropriate spectral filters will be used to separate and isolate the controllable mid-IR waveforms from their driving fields for spectroscopic applications. Figure 4c (blue curve) shows the spectral intensity obtained by Fourier transforming the measured EOS trace. The resulting spectrum is, as expected, narrower than that measured using a monochromator (Fig. 3), given the EOS detection limit of approximately 100 THz for a gate-pulse duration of 7.7 fs (Fig. 2c).
Most interestingly, the EOS trace can be continuously changed (Fig. 4a) by tuning the relative phase Δφ from 0 to π. The trace is transformed from a clean cosine-pulse-like temporal profile (Δφ = 0)   Fig. 3 | Infrared supercontinuum from cascaded IPDFG. Blue curves show the measured (solid) and simulated (dashed) spectra of the p-polarized IPDFG stage output for a relative phase of Δφ = 0 (constructive interference). The orange curves show the same settings as the blue curves, but for a relative phase of Δφ = π (destructive interference). The light-red-shaded area shows the overlap region where interference effects occur (Fig. 1). The inset shows the spatial beam profile of the mid-IR continuum (λ > 3.6 µm, Δφ = 0). The sharp features showing up near 100 and 270 THz in the simulations (both Δφ = 0 and Δφ = π) are probably artefacts arising from numerical approximations. The spectral resolution of 50 nm-chosen for higher signal-to-noise ratios-was insufficient to verify their presence.
to a sine-like profile (Δφ = 3/6π) and then to a double-pulsed structure (Δφ = π). The evaluation of the intensity envelopes (Fig. 4b) shows that all the measured pulses are temporally well compressed, featuring single to sub-cycle pulse durations. The corresponding spectra (Fig. 4c) exhibit similar spectral characteristics as observed using a monochromator (Fig. 3), however with a dip appearing to be slightly shifted in frequency by 5 THz. This is likely caused by additional nonlinear mixing of the supercontinuum (λ < 3 µm) with the gate pulse in the GaSe detection crystal. Nevertheless, this process is CEP invariant, and the observed changes in the EOS traces for different Δφ values (Fig. 4a) are reproduced by simulations ( Supplementary Fig. 4). Constructive interference occurs for a value of 0, whereas the strong modification of the mid-IR waveform for π indicates destructive interference. b, Corresponding intensity envelopes, extracted FWHM pulse durations, number of optical cycles calculated for the spectral centroid and FTL based on the derived spectral intensities. c, Corresponding spectral intensities for different relative phases, as extracted via the Fourier transformation of the time-domain data. Constructive interference occurs for Δφ = 0, whereas the strong spectral dip for Δφ = π indicates destructive interference between the spectrally overlapping components originating from even and odd IPDFG orders. The inset shows the extreme spectra (constructive and destructive) on a linear scale.

Conclusions and outlook
We have reported broadband waveform manipulation in the mid-IR spectral range. Cascaded difference-frequency mixing in a single nonlinear crystal enables waveform adjustment across multiple octaves, without the need for the spatial separation of spectral bands. This monolithic approach to waveform manipulation eliminates instabilities originating from beam-pointing fluctuations and timing jitter inherent in multiple-channel schemes for multi-octave synthesis 14 . With the control parameter being the CEP of the driver laser pulse, no additional dispersive material is required in the mid-IR beam path to manipulate the waveform. As a result, temporal confinement close to a single-field cycle can be preserved for different waveforms without demanding multi-octave dispersion management. For these reasons, the generated waveforms are inherently robust, with their reproducibility solely depending on that of the driver waveform. The fidelity of the reported mid-IR waveforms benefits from the unprecedented CEP and amplitude stability and the resultant reproducibility of single-cycle waveforms of our diode-pumped Cr:ZnS oscillator-based source. The 1.7 MW peak power of these waveforms can be boosted beyond the 10 MW level without substantial added noise by diode-pumped Cr:ZnS amplifiers 52 . The higher peak intensity allows the use of nonlinear crystals with lower nonlinear coefficients but broader window of transparency for extending the mid-IR continuum towards the low-terahertz regime 34 . It also extends the SPM-generated near-IR continuum towards the visible, yielding few-femtosecond pulses for EOS of the entire 300 THz infrared supercontinuum demonstrated 49 .
These advances offer technical capabilities unavailable to date. They include the temporal confinement of coherent multi-octave infrared light to sub-cycle transients of several femtosecond duration and-by introducing more control parameters (for example, via spectral manipulation of the driver waveform)-the sculpting of waveforms with sub-femtosecond-scale precision. Multi-octave infrared-field synthesis can probably impact research at several frontiers, including the advancement of electronic signal processing 7-10 and electric-field-resolved molecular fingerprinting of biological systems 23 .

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