Experimental details

The laser pulses used for the experiments were from a mid-infrared laser system based on optical parametric chirped-pulse amplification (OPCPA). The details are in ref. ⁴⁵. In brief, the laser for the mid-infrared OPCPA was from a 1 µm Innoslab Yb laser system. The pulses were used to generate the seed and the pump for the OPCPA. The OPCPA system produced pulses at 2.03 µm centre wavelength with a 100 kHz repetition rate. During the experiment, the average power obtained from the laser was 5.5–6 W with a pulse duration of approximately 25 fs. The pulse duration was carefully characterized using a frequency-resolved optical gating⁴⁶ with a set-up developed in-house. The laser pulses were characterized after they had passed through the interferometer. To accurately determine the temporal characteristics of the pulses used in the experiment, they were picked up right before they entered the reflective objective of the microscope arrangement. A second-harmonic generation (SHG) frequency-resolved optical gating was used to characterize the 1 µm pulses in a beta-barium borate crystal, whereas the 2 µm pulses were characterized by a surface third-harmonic generation frequency-resolved optical gating on glass (NBK7), both in a non-collinear geometry. The results of these measurements are shown below.

A large part of the experimental set-up was a three-arm Mach–Zehnder interferometer, as shown in Extended Data Fig. ​Fig.6.6. Various aspects of the interferometer are described in detail in the following sections.

The acronyms of the components have the following meaning: WP, Wollaston prism; F1, long pass filter; F2, short pass filter; F3, l ong pass filter; PD, photodiode; TIA, trans-impedance amplifier; BS, beam splitter; DBS, dichroic beam splitter; Pol.,  polarizer; WeP, wedged pair; CM, chirped mirrors; SHG, second harmonic generation LiNbO₃ crystal; CW, continuous wave laser; LIA, lock-in amplifier. A detailed description of the setup is provided in the text.

The 2 µm wavelength pulses from the laser source were fed directly into the three-arm interferometer. First, the pulses were split in two by a 50–50 beam splitter (red beam from right to left in Extended Data Fig. ​Fig.6).6). The reflected part proceeded to the pump and the transmitted one to the probe arm. The pump beam was sent through a 1.5-mm-thick lithium niobate (LiNbO₃) crystal (SHG) cut at 45.6°. This interaction produced a second pulse at half the wavelength (frequency of 2ω) of the fundamental (frequency of ω) by sum frequency generation (SHG)⁴⁷. LiNbO₃ is suited well for a 2 µm fundamental wavelength, given its wide transparency range in the mid-infrared of up to 3.5 µm. The phase-matching conditions (type 1) led to the second-harmonic component being cross-polarized with respect to the fundamental pulse⁴⁷. The co-propagating two-colour pulses (ω and 2ω) were separated using a custom-designed dichroic beam splitter. The ω arm passed through a variable neutral-density filter, a retro-reflection stage and CaF₂ as used in the probe arm, but without the piezo motion rails, as the length of this arm was fixed with respect to the other arms. Additionally, a wire grid polarizer was placed in its path to clean its polarization state before it combined with the 2ω arm at another dichroic beam splitter. The 2ω arm was reflected off the first dichroic beam splitter at 90° and traversed the exact same arrangement as the ω arm. The only difference was a different pair of custom chirped mirrors for 2ω on the delay stage above a closed-loop stick–slip piezo nano-positioning rail with a positioning resolution of 1 nm, a different amount of dispersion-compensating material and a perpendicular axis set on the polarizer. The pump arm (co-propagating ω and 2ω) was further sent through a long-pass spectral filter (F1) to block the parametric optical signals generated at the harmonics of the two-colour pump in the LiNbO₃ crystal. Finally, the two-colour pump (ω and 2ω) pulses passed through the fused-silica wedged pair to recombine with the probe pulses (ω), which were reflected off it.

After the three-arm interferometer, the laser beam was expanded using a reflective telescope arrangement to roughly match the input diameter of the tight-focusing reflective objective and adjust the beam divergence. However, right before the focusing objective, there was a broadband quarter-wave plate (λ/4) with its optical axis at 45° with respect to the cross-polarized axis of the two-colour pump. This wave plate arrangement transformed the linear, cross-polarized, pump pulses into circularly polarized, counter-rotating pulses that look like a trefoil or a three-leaf pulse in the X–Y plane, as required in the experiment. The quarter-wave plate was intentionally placed right before the focusing element to prevent any changes in ellipticity due to a phase shift between the s- and p-polarized components upon reflection, especially from beam-folding mirrors and other coated optical elements. For the ω and 2ω pump components, the pulse durations measured just before the focusing objective were about 26 and 48 fs, respectively. Extended Data Fig. ​Fig.77 depicts the spectral and temporal characteristics of the ω and 2ω pump components. A similar kind of waveform synthesizer was also used earlier in attosecond-controlled strong-field experiments by the group³⁶. The data shown in the manuscript were obtained with the pump intensity in the range 4–7 TW cm⁻². For the intensity scaling measurements, the overall power of the pump beam was changed with a variable neutral-density filter keeping the relative power ratio between the components intact.

In the probe arm, the pulses went through a variable neutral-density filter followed by a delay stage, which, along with a pair of silver retro-reflecting mirrors, hosted two customized chirped mirrors for simultaneous positive dispersion and spectral filtering. Spectral filtering is crucial given the existence of weak optical signals at lower wavelengths arising from the parametric processes in various crystals in the laser system. The delay stage was mounted on a closed-loop stick–slip piezo rail (Smaract SLC-24 series) with a positioning resolution of 1 nm. Further down, the pulses went through a defined thickness of material (CaF₂) to compensate for excess positive dispersion. Additionally, another long-pass spectral filter was placed in the beam path to further suppress the unwanted optical signals at lower wavelengths. Finally, the probe pulses went through a half-wave (λ/2) plate and a quarter-wave (λ/4) plate, after which they were reflected off the wedge plate and recombined with the pump beam. This wave-plate combination allowed us to control the polarization state of the probe pulses, and the mechanism is described in greater detail later. This wedge pair arrangement not only acted as a beam recombination element but also as a power attenuator for the probe pulses, as only 4% of the power was reflected. After being recombined, the probe beam followed the collinear path with the pump beam. Just before the focusing objective, a pulse duration of about 26 fs was achieved for the probe pulses.

As shown in Extended Data Fig. ​Fig.6,6, a λ/4 plate was the last optical element before the pulses entered the reflective objective. This led to a major problem in which the third arm (or the linearly polarized probe as in this experiment) cannot remain linear once it has passed through the λ/4 plate unless it is along the optical axis at 45°. Also, a linear polarization launched at 45° with respect to the s- or p-polarized states would lose its linear contrast, as it would become elliptical on acquiring a different phase shift in the s and p components on every reflective optic in its beam path. To have full flexibility over the polarization state of the probe pulses after the λ/4 plate, a scheme was implemented such that a combination of λ/2 and λ/4 wave plates were additionally placed in the probe arm, as depicted in Extended Data Fig. ​Fig.6.6. Intuitively, one can think of these additional plates as inducing perfectly opposing elliptical polarization, which cancels out in the final λ/4 before the reflective objective to produce linearly polarized light with a high extinction ratio. This scheme was numerically tested using a Jones matrix approach. It was observed that any arbitrary shifts in phase between the s and p components in the beam path between the two λ/4 wave plates can be compensated. However, changes in magnitude between the s and p components lead to a deviation from linearity and cannot be compensated for by this scheme.

A movable silver mirror was placed at 45° right before the reflective objective intercepting the probe pulses to optimize and characterize the polarization extinction ratio or linearity. The pulses were then guided to an InGaAs photodiode with a polarizer attached to it at a fixed angle. The fixed angle was such that the probe polarization was aligned along s or p to prevent any additional ellipticity induced by the intercepting mirror, which would not be present otherwise during the experiment.

The data shown in the main manuscript were obtained at a pump–probe delay of about 60–110 fs.

When the bicircular ω and 2ω components of the pump were combined in the interferometer such that the E field ratio at the focus was 2:1, the coherent sum of their electric fields in the X–Y plane (the plane perpendicular to propagation) was transformed to that of a trefoil waveform. The rotation of the trefoil waveform was controlled by the subcycle phase delay between the ω and 2ω components. Experimentally, this was achieved by introducing an optical path difference between the ω and 2ω arms. When the central wavelength (λ₀) of the ω arm was 2 µm, a rotation of 360° was induced by delaying the piezo stage by 3 µm. This information was used to convert the stage delay to angular rotation, which was recorded as the experimental data.

During the experiment, it was critical that the angle of the three-leaf or trefoil pump remained stable, as this was directly linked to the delay between the two colours in the pump arm. To characterize the delay stability, an additional temperature-stabilized continuous-wave diode laser (Thorlabs L785P090 with LDMT9) was sent through the two pump arms to interferometrically measure its path difference over time.

Using the above-mentioned scheme, the interferometric stability was found to be highest around 2 h after switching on the driving laser system. Additionally, tests were carried out to measure the stability when the laser was going through the interferometer and the piezo delay stage being scanned, as during the experiment, as illustrated in Extended Data Fig. ​Fig.8.8. Over a period of 10 min, the standard deviation of the position generated by the closed-loop piezo stages from the position extracted from the continuous-wave interferometer was close to 38.1 nm, which roughly translates to about 4.6° in rotation error of the ω and 2ω bicircular trefoil structure. A similar scheme with active stabilization was used earlier and achieved few tens of attosecond interferometric stability⁴⁸.

The relative displacement between the ω − 2ω pump arms is recorded while a piezo closed-loop stage controlling the displacement is swept over a period of ten minutes. The position read out by the stage is compared to that extracted from an additional CW interferometer to estimate the error in displacement mapping. A standard deviation in drift of 38.1 nm corresponds to an angular jitter of 4.6° in the trefoil pump structure.