Fine control of time-varying currents enables precise sculpting of electromagnetic radiation. However, technical challenges in scaling conventional electronic oscillators to terahertz (THz) frequencies call for alternative approaches to generate THz radiation. When femtosecond laser pulses are applied to optoelectronic processes, currents can be modulated on femtosecond time scales. Transient currents can be optically excited using photoconductive switches [1–4], quantum interference current injection in semiconductors [5–8], strong-field current injection in dielectrics and monolayers [9–11], or in a laser-ignited plasma channel or filament [12–18]. Depending on the configuration, the spectral content of the resulting THz radiation can extend from approximately 0.1 THz up to the inverse of the laser pulse duration, $1/{\tau }_p$.

Over the last two decades, several solid-state platforms for generating intense THz pulses have emerged, including tilted-pulse-front excitation of ${\text{LiNbO}}_3$ crystals [19–21], optical rectification in large-aperture ZnTe or organic crystals [22,23], and large-aperture photoconductive switches [24]. Scaling the energy of THz currents and radiation using solid-state platforms is ultimately restricted by electrical or optical breakdown, leading to permanent damage, and by the availability of crystals with ever-larger apertures. In contrast, plasma channels and filaments excited in gases or liquids rely on optical breakdown and are continuously replenished, providing a highly scalable platform for THz generation [25,26]. Although THz radiation can be observed when a plasma channel is driven by a single femtosecond laser pulse, stronger electric-field asymmetry is possible using a bichromatic or few-cycle pulse excitation, leading to higher peak currents and stronger THz radiation.

In two-color plasma formation, a fundamental laser pulse [$\omega$, typically carried at a wavelength lying within the near- or mid-infrared (mid-IR) spectral region] and its second harmonic ($2\omega$) are focused collinearly into a medium. The bichromatic field is intense enough to ionize the medium and to impart the resulting electron wave packets with significant momentum. Both the time-dependent ionization rate and the subsequent electron wave packet dynamics can be controlled by adjusting the relative phase between the two pulses, ${\mathrm{\Delta }\phi }_{\omega ,2\omega }=2{\phi }_{\omega }-{\phi }_{2\omega }$. Coherent control on a microscopic scale leads to macroscopic dynamic currents, which radiate a THz pulse.

Through a quantum mechanical treatment and complementary experiments, Karpowicz and Zhang have demonstrated two linked processes that contribute to the radiated THz fields [27]. The first arises from the initial acceleration of the electron wave packets by the optical fields. Subsequent to acceleration, each electron wave packet propagates freely until it undergoes a collision with a neighboring atom or molecule, giving rise to bremsstrahlung, the second contribution to the radiated THz fields. Using ${\mathrm{\Delta }\phi }_{\omega ,2\omega }$ to precisely manipulate the electron wave packet dynamics provides complementary control over the amplitude and polarity of the THz radiation. Applying two-color mid-IR fields to plasma formation in ambient air has enabled the generation of broadband THz fields with amplitudes above $100{\mathrm{MV}·\mathrm{cm}}^{-1}$ [25].

Freedom to optimize the focusing geometry, energy, polarization, duration, and spatial structure of each of the two pulses is gained when they are diverted into separate arms of an interferometer [16,28,29]. Independent control over the pulse properties of $\omega$ and $2\omega$ in the interferometer is necessary to generate, for example, THz fields with a circular or more complex polarization state, or by mid-IR driver pulses, where zeroth-order half- and quarter- wave plates are not readily available. This provides additional opportunities for improving THz conversion efficiencies, as demonstrated with circularly polarized $\omega$ and $2\omega$ laser pulses in Ref. [29]. At the same time, a two-color interferometer introduces the need for extreme mechanical stability. Critically, any interferometric drift will change ${\mathrm{\Delta }\phi }_{\omega ,2\omega }$, which will influence the plasma currents and the radiated THz waveform.

Due to the reliance of laser-excited plasma channels and filaments on strong-field ionization, the peak plasma density and length of the channel can be sensitive to instabilities in the driving light or the surrounding environment. Notably, a plasma channel constitutes a highly dispersive medium, leading to a phase slip in $\mathrm{\Delta }{\phi }_{\omega ,2\omega }$ as the bichromatic fields propagate through it. The total change in $\mathrm{\Delta }{\phi }_{\omega ,2\omega }$ that accumulates in the plasma channel will be sensitive to instabilities in the laser source (i.e., the $\omega$ and $2\omega$ fields themselves) or surrounding environment.

Recently, laser-excited currents measured in ambient air have been used to characterize the carrier-envelope phase of laser pulses [30] and to sample the vector potential of visible-to-mid-IR laser fields [31,32]. Direct optoelectronic detection of currents excited in a plasma channel also provides an in situ probe of the plasma currents that ultimately radiate a THz impulse, and it can be expected that plasma currents and THz radiation share a mutual dependency on $\mathrm{\Delta }{\phi }_{\omega ,2\omega }$ and its fluctuations. In situ current detection would also provide a means to probe the coherent evolution of the bichromatic fields within the plasma channel and could potentially provide insight into decoherence mechanisms in laser-excited plasmas.

In this work, we simultaneously measure the phase dependency of optically excited currents in a plasma channel and the peak electric field of the emitted THz waveform. We demonstrate that under certain conditions, both signals are strongly correlated and are influenced similarly by instabilities in $\mathrm{\Delta }{\phi }_{\omega ,2\omega }$. Using the measured current signal as a two-color phase detector, we correct for instabilities in the THz waveform arising from both the two-color interferometer and the plasma environment in real time. This approach demonstrates a potential scheme for generating energetic THz pulses derived from plasma channels with extreme waveform stability.

The experimental configuration is shown in Fig. 1(a). One output beam from a Legend Elite Duo amplifier (wavelength, $\lambda =800\mathrm{nm}$, pulse energy, $E_p=4.5\text{mJ}$, pulse duration, ${\tau }_p=30\mathrm{fs}$, repetition rate, $f_{\mathrm{rep}}=1\mathrm{kHz}$) is first split into a strong (95%) pulse for THz generation and a weak (5%) probe pulse for electro-optic sampling (EOS). The strong pulse ($\omega$) is transmitted through a β-barium-borate (BBO) crystal for generating the second-harmonic beam ($2\omega$), and the two colors are split into separate optical paths using a dichroic mirror (DM). The $\omega$ and $2\omega$ pulses each pass through a half-wave plate (HWP) and a wire-grid polarizer (WGP) to enable independent optimization of the pulse energy and polarization of each beam. In addition, the $\omega$ beam is reflected from two mirrors mounted to a delay stage with a piezoelectric actuator, enabling precise control over $\mathrm{\Delta }{\phi }_{\omega ,2\omega }$.

The two pulses are then recombined collinearly using a DM and focused into ambient air with an $f=250\mathrm{mm}$ spherical mirror. A 150-μm-thick glass coverslip is introduced into the optical path approximately halfway between the focusing mirror and the focal plane. The pulses transmitted through the glass coverslip excite a plasma channel in ambient air. Typical pulse energies used to generate the plasma channel are $E_{p,\omega }=0.44\mathrm{mJ}$ for the $\omega$ beam and $E_{p,2\omega }=0.28\mathrm{mJ}$ for the $2\omega$ beam. Notably, the corresponding peak powers ($P_{p,\omega }\sim 13\mathrm{GW}$ for $\omega$ and $P_{p,2\omega }\sim 4.9\mathrm{GW}$ for $2\omega$) exceed the critical power for self-focusing in ambient air, and the plasma formation is in the filamentation regime. Although we have not side-imaged visible light emission from the plasma channel, we estimate the length of the plasma channel to be approximately $6.5\pm 0.5\mathrm{mm}$. Femtosecond laser pulses with similar peak power and focusing geometry undergoing filamentation typically produce plasma columns with $\sim 100\mathrm{\mu m}$ diameters [33]. THz emission arising from the plasma currents is collected and collimated using an off-axis parabolic mirror (effective focal length, 150 mm). The THz pulse energy is measured to be approximately $E_{\mathrm{THz}}=25\mathrm{nJ}$ using a pyroelectric detector (Gentec-EO, THZ9B-BL-DZ-DO), with a corresponding conversion efficiency, $\eta =3.5\times {10}^{-5}$.

The energy of the probe beam is adjusted using a neutral density filter wheel (ND), and its polarization is corrected with a WGP. Its arrival time relative to the THz pulse is controlled using a motorized delay stage. The THz pulse and the probe pulse are combined collinearly with a bare silicon wafer (Si), which simultaneously absorbs the $\omega$ and $2\omega$ beams, and are subsequently focused through a 1-mm-thick ZnTe crystal for EOS. A standard balanced photodetection setup consisting of a quarter-wave plate (QWP), Wollaston prism (WP), and a pair of photodiodes connected to a lock-in amplifier (Stanford Research Systems, SR830) is used to read out the polarization rotation of the probe in response to the THz electric fields. The measurements presented in this work are acquired using a lock-in time constant of $\tau =100\mathrm{ms}$. The THz waveform that is measured can be viewed as a probe for $\mathrm{\Delta }{\phi }_{\omega ,2\omega }$ after it has been influenced by all sources of instability in the optical paths.

The weak reflections of the two beams from the glass CS are transmitted through a second BBO crystal, which frequency-doubles the $\omega$ light. Interference fringes between the two sources of $2\omega$ light, originating from the two arms of the interferometer, are then observed using a spectrometer. In this arrangement, shifts in the position of the interference maxima and minima over time relate directly to phase instabilities in the two-color interferometer. This detection scheme serves as a probe for coherence in the bichromatic fields that is established in the two-color interferometer.

In addition, a pair of gold electrodes deposited onto separate quartz substrates is mounted to a printed circuit board, such that they are separated by an air gap approximately 1-mm wide, shown schematically in Fig. 1(b). When this detector is positioned near the focal plane of the optical beams, neither the plasma generation nor the measured THz radiation is altered in any way. Currents excited in the plasma channel can easily be measured with this detector using a second lock-in amplifier. In situ detection of phase-dependent currents provides a means to relate the THz waveform emitted from the plasma channel to coherence in the bichromatic fields emerging from the two-color interferometer.

To investigate the feasibility of generating stable THz waveforms using this configuration, we record 20 sequential EOS scans, a measurement that lasts a total of approximately 45 min. The waveforms obtained in scans 1, 10, and 20 are shown in Figs. 2(a)–2(c), respectively. To quantify drifts in the waveform over time, we calculate the normalized root mean square error (RMSE) between the initial waveform and each subsequent scan, $${\mathrm{RMSE}}_i=\frac{1}{E_{1,\max }}\sqrt{\frac{{\sum }_{n=1}^N{(E_{i,n}-E_{1,n})}^2}{N}}.$$(1)

In this expression, $N$ is the number of data points that make up each EOS scan, $i$ is the scan number, $n$ is the data point number within a scan, $E_1$ is the initial waveform that was measured, $E_i$ is the $i$th waveform that was recorded, and $E_{1,\mathrm{m}\mathrm{a}\mathrm{x}}$ is the normalization constant, which we take to be the peak amplitude of the initial THz waveform.

Figure 2(d) shows the calculated RMSE for each of the 20 scans. In this data set, the waveform is relatively stable for the first six scans, but experiences considerable drifts thereafter. From these measurements, it is clear that using this configuration makes long-term THz waveform stability a challenge.

Next, we control the relative-phase-dependent THz waveform using the piezoelectric actuator and record its peak electric-field amplitude at a fixed probe delay. We limit the extent to which the plasma environment influences the current measurement by positioning the current detector near the beginning of the plasma channel. Then we simultaneously record the $2\omega$ interference spectrum, current, and the EOS signal as $\mathrm{\Delta }{\phi }_{\omega ,2\omega }$ is adjusted over a range of approximately 375 rad. Here, we define $\mathrm{\Delta }{\phi }_{\omega ,2\omega }$ as the relative phase introduced to the two-color interferometer by the piezo-actuated delay stage; it does not account for any drifts in the interferometer. Therefore, analyzing changes in the scans over time provides insight into interferometric instabilities. An exemplary scan is plotted in Fig. 3(a) and, as expected, each parameter undergoes periodic oscillations at the $2\omega$ frequency as $\mathrm{\Delta }{\phi }_{\omega ,2\omega }$ is increased. Figure 3(b) displays the envelopes extracted from the current and THz measurements, and it is determined that the center of mass of the THz envelope is advanced by 25.7 rad with respect to that of the current envelope.

Due to plasma dispersion, $\mathrm{\Delta }{\phi }_{\omega ,2\omega }$ evolves along the length of the plasma channel. Positioning the current detector at a certain plane, $z=z_0$, within the plasma channel enables the local $\mathrm{\Delta }{\phi }_{\omega ,2\omega }$ to be sampled. Conversely, the collection parabolic mirror collimates the THz waveform emitted from the region of the plasma channel near its focal plane. In general, what is detected is a weighted average of the waveforms emitted from each element along the plasma channel, where strong electric fields radiated from a position close to the focal plane of the parabolic mirror contribute substantially to the measured THz waveform. The envelope offset shown in Fig. 3(b) arises from plasma dispersion that accumulates between $z_0$ and the region from which the strongest THz emission is collected. We take the envelope offset to be a measure of the position of the current detector within the plasma channel.

After repeating the scan 15 times, we examine the relative phase between the oscillations comprising each measurement and calculate their correlation over subsequent scans. The evolution of the phase-dependent EOS, current, and $2\omega$ spectral interference measurements over the duration of the 15 scans is shown in Figs. 3(c)–3(e), respectively. Figure 3(f) displays the phase of each measurement for a fixed position of the piezoelectric actuator, corresponding to $\mathrm{\Delta }{\phi }_{\omega ,2\omega }=-13.9\mathrm{rad}$. This value was chosen because it is close to the maximum of the EOS envelope shown in Fig. 3(b) and, therefore, provides THz pulses with the highest signal-to-noise ratio (SNR). For clarity, we have added a constant offset to each data set in order to make its average value zero. Each of the three measurements displays a trend with qualitative similarities. However, numerical analysis of the data reveals that the strongest correlation among any of the three measurements lies between the current and the $2\omega$ spectral interference. The differences between the current and $2\omega$ spectral interference phases, and between the current and THz phases, are calculated and plotted on the right-hand axis of Fig. 3(f). The RMSE between the current and $2\omega$ spectral interference phases is calculated to be 139 mrad. Similarly, the RMSE between the current and THz phases is found to be 400 mrad.

The close correlation between the current signal and the $2\omega$ spectral interference signal near the beginning of the plasma channel signifies that, in this detection plane, drifts in the phase-dependent plasma currents are primarily linked to interferometric instabilities. Here, the bichromatic fields have only interacted with a small fraction of the plasma channel, and $\mathrm{\Delta }{\phi }_{\omega ,2\omega }$ has not yet been significantly influenced by instabilities in the plasma. The weaker link between the current measurement and the THz signal demonstrates that significant instability to $\mathrm{\Delta }{\phi }_{\omega ,2\omega }$ accumulates as the bichromatic fields experience the plasma situated between the current detection plane and the plane from which the strongest THz emission is collected. Therefore, it is advantageous to probe currents generated in the middle or latter half of the plasma channel, where signatures of the plasma environment have been imprinted onto $\mathrm{\Delta }{\phi }_{\omega ,2\omega }$.

Given that phase drifts arising from the plasma channel can be significant and these are not detected using the $2\omega$ spectral interference signal, we now focus exclusively on using the current detector as an all-encompassing probe for instabilities in $\mathrm{\Delta }{\phi }_{\omega ,2\omega }$. The current detector is shifted into the region of the plasma channel from which the strongest side-emitted visible light is observed, where the highest current densities exist. In a similar manner, we measure the currents excited in the plasma channel and the peak amplitude of the THz waveform as $\mathrm{\Delta }{\phi }_{\omega ,2\omega }$ is adjusted. An exemplary scan showing the two measurements is plotted in Fig. 4(a). Figure 4(b) displays the envelope extracted from each measurement, and it is determined that the centers of mass of the two envelopes are now separated by only 13.4 rad.

This scan is repeated 20 times, lasting a total of approximately 130 min. By extracting the phase of each scan at a fixed $\mathrm{\Delta }{\phi }_{\omega ,2\omega }=-12.8\mathrm{rad}$, we directly examine the correlation between drifts in the phase dependency of the two signals over the duration of the 20 scans. As before, this value of $\mathrm{\Delta }{\phi }_{\omega ,2\omega }$ was chosen because it produces a THz pulse with near-optimal SNR. The phase drift of the two signals as well as their difference is shown in Fig. 4(c). The RMSE between the two phases is calculated to be 57.1 mrad. This seven-fold improvement in the RMSE between the phase sensitivity of the current and EOS measurements with respect to that measured in the previous section introduces the possibility of using one measurement as a detector for $\mathrm{\Delta }{\phi }_{\omega ,2\omega }$ that can in turn be used to stabilize the other.

In the interest of generating THz fields with long-term waveform stability, we continuously sample the current, and use a software-based feedback loop to stabilize the current value by adjusting $\mathrm{\Delta }{\phi }_{\omega ,2\omega }$ using the piezoelectric-actuated delay stage. The detection and feedback scheme is illustrated schematically in Fig. 5(a). Prior to engaging the feedback loop, the program conducts a scan of $\mathrm{\Delta }{\phi }_{\omega ,2\omega }$ over approximately $4\pi \mathrm{rad}$ in order to measure the maximum and minimum current amplitudes. Upon entering the feedback loop, it reads out the current once per second and compares it with the value measured in the initial position. If the error exceeds a certain threshold, it steps the piezoelectric actuator in the appropriate direction until the error value is reduced below the threshold. Due to the relatively slow response time of this feedback loop, it is primarily effective at stabilizing slow drifts in the two-color interferometer or in the laser parameters. Corrections to impulsive instabilities, such as vibrational shocks introduced to the optical table, are slower, and these events can introduce significant distortions to the measured waveforms.

We perform 40 scans of the THz waveform, lasting a total of approximately 90 min, and plot the mean waveform together with error margins representing the $1-\sigma$ standard deviation of the measurements in Fig. 5(b). As before, we calculate the normalized RMSE between the first scan and each subsequent scan, which is shown in Fig. 5(c). With the exception of one waveform, the RMSE is maintained at approximately 0.05. Even when exposed to vibrational perturbations by the work of another researcher, the RMSE never exceeds 0.15.

Overall, the feedback scheme is successful in stabilizing the THz waveform over the duration of the scans, and it is expected that this could be extended to considerably longer time intervals. Here, we have focused on demonstrating the utility of this approach through optical measurements. We note that improving the feedback scheme using analog electronics and more advanced algorithms could lead to a significant reduction in the error margins and an improved response to impulsive instabilities.

We have presented a straightforward and robust technique for monitoring fluctuations in $\mathrm{\Delta }{\phi }_{\omega ,2\omega }$ when bichromatic laser fields are applied to the formation of plasma channels in air and THz generation. Since contributions to the observed waveform instabilities arise from both the two-color interferometer and the plasma channel, it is anticipated that this scheme will be useful for both “in-line” and interferometric approaches to generating THz radiation from plasma channels. A direct outcome of this stabilization scheme will be energetic THz pulses with highly stable waveforms. Using shorter $\omega$ and $2\omega$ pulses will enable generation and sampling of broadband light spanning the entire infrared spectral region, expanding the scope of applications of the radiated electromagnetic waves [34,35].

A robust scheme enabling independent manipulation of the $\omega$ and $2\omega$ beams opens up the possibility to spatially structure each color and to drive laser-ignited plasmas using structured light pulses. The ability to do so would enable ambient air to be used as a platform for three-dimensional active metaoptical current structures. Controlling intricate current arrangements contained in the plasma channel would facilitate vectorization, spatial structuring, beam steering, and phase front control of the resulting THz radiation without requiring sophisticated nanostructures, in a manner that is similar to what has recently been demonstrated using two-color-injected semiconductor currents [36–38]. Implementing a two-color scheme using ultraviolet light could provide a route to current control on 100-nm scales.

Spatial structuring of THz fields could also be exploited to launch exotic electromagnetic pulses, such as a “flying torus” [39]. Furthermore, the direct link between currents and magnetic fields would enable intricate control of magnetic fields within the plasma channel and of the radiated THz pulses. While magnetic fields on the order of 100 T have been observed in dense plasmas excited by relativistic laser pulses, these fields have conventionally been challenging to access and use [40,41]. Applying intense structured light beams to similar experiments could enable current densities in the form of ring currents to be driven, providing spatial separation between the currents and magnetic fields. These magnetic fields could potentially be used to guide relativistic electron beams.

Conventionally, information about plasma channels and filaments can only be extracted using external measurements. This in situ current detection scheme provides direct access to currents within plasma channels and filaments. We anticipate that this approach could be used to spatially map current densities within filaments in order to provide fundamental insight into the processes that govern their formation. Finally, for sufficiently long or dense plasma channels, the detection of currents along the length of the plasma channel would enable direct visualization of the buildup of decoherence along its length. Measuring the conditions under which decoherence accumulates rapidly would provide an understanding of the ultimate limits of coherent control as applied to filamentation.

Acknowledgment. The authors are grateful for stimulating discussions with Aleksey Korobenko, Mathew Britton, Shima Gholam-Mirzaei, Mojtaba Taheri, Andrei Naumov, David Villeneuve, Saroj Tripathi, Wei Cui, and Jean-Michel Ménard. Y. M. acknowledges support from the Deutsche Forschungsgemeinschaft (German Research Foundation).