An experimental method to measure the electric fields existing in a plasma wake produced by a ∼0.24 GW, ∼0.5 ns, 9.5 GHz microwave pulse traversing a plasma-filled waveguide is presented. The intensity of the second harmonic of a 30 fs 800 nm laser generated inside a gas-filled dielectric tube placed inside the waveguide is used to characterize the wakefield parameters. Three distinct decaying oscillations of the plasma wakefield, with peak amplitude of ∼20 kV/cm, were observed. The experimental results were confirmed by 3D large-scale plasma particle-in-cell simulations.

Since Tajima and Dawson's seminal proposition to utilize intense electromagnetic pulses for driving plasma wakes,1 significant interest has surged, primarily due to the generation of extremely high electric fields (∼109 V/m) in the wake of high-power lasers.2,3 Over the past decade, extensive studies of the laser-driven plasma wakefields have been conducted, yielding insight through diverse diagnostic techniques such as laser interferometry for plasma density and electric field measurements,4 time-resolved polarimetry, plasma shadowgraphy,5 and probing by relativistic electron beams on the femtosecond scale.6 While the focus has traditionally been on intense laser-induced wakes, the advent of ultra-short (sub-nanosecond), ultra-intense (gigawatt-level) coherent microwave radiation sources7 in the super-radiant regime has opened avenues for the excitation of plasma wakefields by high-power microwave (HPM) pulses. Compared to laser-wakefield excitation, the wake excitation by HPM pulse is characterized by significantly larger temporal and spatial scales and several orders of magnitude smaller wakefield intensities. Numerical studies conducted by Malik and Aria8,9 demonstrated that wakefield excitation is feasible in a plasma-filled rectangular waveguide when excited by a microwave pulse.

In recent years, the Technion Plasma Physics and Pulsed Power laboratory has carried out extensive theoretical and experimental research on HPM-driven wakefields in plasma-filled waveguides.10–12 The wakefields were reconstructed by analyzing the time and frequency evolution of the HPM pulse13 and by observing its perturbation of a probing electron beam.14 However, a quantitative measurement of the wakefields was still challenging due to the relatively low expected field amplitudes, ∼10 kV/cm, excited in a plasma of ≤1017 m−3 density at the tail of the microwave pulse.10 

In this Letter, we report the first direct measurement of the temporal evolution of wakefield following an HPM pulse by electric field induced second harmonic (EFISH) generation using a 30 fs laser beam probe.15–18 Our experimental findings are supported by large-scale plasma (LSP)19,20 3D particle-in-cell simulations.

In this experiment, a super-radiant backward-wave oscillator (SR-BWO)21–23 driven by an electron beam (∼300 keV, ∼2 kA, 5 ns), generated by a magnetically insulated foil-less diode, was used. The diode is supplied by a semiconductor opening switch based all-solid-state high-voltage pulse generator.24 This setup generates a HPM pulse [∼0.5 ns FWHM (full width half maximum), 9.5 GHz], which propagates in a 24-wire cylindrical 1.4 cm-radius waveguide located coaxially inside a tube containing a flashboard plasma source.13 The density of wires is selected to ensure that the gaps between them are large enough to allow plasma penetration into the waveguide, yet sufficiently small to ensure that only a negligible amount (a few percent) of microwave power escapes, effectively making the wire waveguide nearly equivalent to a solid structure.11 The microwave power, averaged over 300 generator shots, is 240 ± 40 MW. The plasma density inside the waveguide is controllable by varying the time delay between the microwave pulse generation and the flashboard operation. The density of the plasma as function of this time delay was studied in an earlier research.13 Calibrated loop-type couplers installed at the entrance and exit of the waveguide are used to measure the azimuthal magnetic field component of the incident and transmitted HPM pulse.

An 800 nm (fundamental wavelength), ≤3.3 mJ, ∼30 fs Ti:sapphire femtosecond laser (Avesta REUS-3m20) is used for EFISH generation in a gas-filled glass tube placed transversely to the plasma-filled waveguide axis at a distance of 10 cm from input. The synchronization between the HPM pulse and the femtosecond laser utilizes the fixed 20 Hz laser pulse train. The scattered light of the 800 nm laser pulse captured by a fast photodiode is used to trigger a BNC trigger generator. This triggers the operation of the SR-BWO's solenoid pulse power supply and the high-voltage generator at a time, which determines the timing of the next laser pulse. The synchronization between the HPM pulse and the laser carries a jitter of ∼±20 ns due to a jitter in the operation of the high voltage generator. Because of this, approximately one in 50 shots falls within the desired time interval for wakefield measurements. The appearance time of the HPM pulse at the location of the gas-filled region is defined relative to its arrival time at coupler #1. Synchronization of the second harmonic (SH) signal with the HPM pulse is achieved by a signal from a high voltage divider with accuracy of ±0.2 ns, which measures the voltage at the HV generator output. Considering the time delays between the signals from the voltage divider, SH and coupler #1, and accounting for the propagation time in the optical fiber (for the SH), in the waveguide (for the HPM), and in the coaxial cables, the actual time delay between the SH signal and the HPM pulse was obtained.

The efficiency of the generation of the 400 nm SH reaches maximum when the external field direction aligns with the laser field polarization, namely, along the axial direction of the waveguide. Thus, to measure the wake field, the laser beam polarization is aligned in parallel with the propagation path of the HPM pulse using a half-wave plate. The laser beam is directed through a 3 × 105 Pa air-filled 7 mm diameter, 1 mm thick glass tube. The high-pressure gas in the glass tube increases EFISH sensitivity and prevents its ionization by the microwave pulse. The optical setup for the EFISH experiments is drawn schematically in Fig. 1(a) relative to the experimental chamber enlarged in Fig. 1(b).

Ultra-fast, enhanced silver-coated mirrors (Thorlabs, UM10-AG) are used to minimize group delay dispersion (GVD), thereby avoiding the potential decrease in seed laser intensity due to pulse broadening. This reduction in intensity can lead to diminished SH measurements. We found that 800 nm, 30 fs laser produces a SH by interacting with mirrors and lenses, which in fact is typical of such powerful laser. Thus, to mitigate the effects of SH generation from interactions between the femtosecond laser and optical components, two long-pass filters (Thorlabs, FELH0750) are used (see Fig. 1). These filters effectively eliminate SH light before the laser beam enters the flashboard tube. The 400 nm SH photons and 800 nm laser beams exiting the gas tube are filtered using a short-pass filter (Thorlabs, FELH0650). The remaining SH light is then directed to a collimator via a blazed grating (Thorlabs, GR25-1204) and cleaned further by two narrowband filters (Thorlabs, FBH400-40) to eliminate any potential measurement errors. Finally, the SH photons, collected by a 1-mm-diameter optical fiber (Thorlabs FP1000URT), are transmitted to a photo-multiplier tube (PMT) (Hamamatsu R9880U-210) located inside a Faraday room to minimize the electromagnetic noise levels to <20 mV.

For a uniform external electric field E ext in the plane wave approximation, the SH signal intensity is given as25,26
I i 2 ω χ ijkl N E ext j E k ω E l ω 2 R R exp i Δ k r d r 2 ,
(1)
where χ ijkl is the third-order susceptibility, i , j , k , and l are the x or y components of the electric field of the SH beam, external electric field, and the incident laser beam, and R is the waveguide radius. In the experiment, with air at 3 × 105 Pa,27 the coherence length of the fundamental and SH beams is L c = π / Δ k, approximately 1 cm. This length is close to the dimension of both wakefield and HPM's axial electric field. Here, Δ k = k 2 ω 2 k ω, where k ω is the wave number of the fundamental harmonic.

The EFISH signal is generated not only by the external electric field of the plasma wake but also by the electric field of the 9.5 GHz HPM pulse. The time it takes the laser beam to propagate through the waveguide is approximately equal to the period of the HPM oscillations. Consequently, the intensity of the EFISH signal, generated by the HPM pulse, is to be considered as an average over its propagation time through the entire waveguide. This average is over all wave phases encountered by the laser beam as it propagates inside the glass tube as illustrated in Fig. 2. The average over all encountered phases in Fig. 2 results in a normalization factor fc = 6.62. i.e., the SH electric fields measured during the presence of the HPM pulse is fc times smaller than one would obtain for a constant maximal electric field of the microwave pulse. Introducing this normalization factor, fc, enables the calibration of the absolute field value measured by the EFISH signal through comparison with the HPM field values obtained from calibrated couplers. This calibration agrees well with a separate calibration, conducted using two parallel square electrodes (1 × 0.5 cm2) with an applied DC field.

Figure 3 highlights the main results obtained in this research. One can see calibrated EFISH amplitudes obtained during and following the HPM pulse propagation through the 3 × 105 Pa air-filled glass tube in the absence (Fig. 3, black squares) and in the presence of ρ e 3 × 10 10 c m 3 plasma (Fig. 3, red circles). The microwave field, registered by coupler #1 and fc allow the calculation of the corresponding SH signal, which is shown in Fig. 3. Here, zero-time represents the time of arrival of the HPM pulse's peak at the center of the glass tube. In the presence of plasma, following the main pulse, one can distinguish three distinct tail peaks with decaying maxima. Such tail peaks are not observed when the plasma density is decreased or increased by a factor of up to two. The reason is that the wake excitation is a resonant process, which takes place when the Langmuir wave period, which depends on the plasma density, and the HPM pulse duration match.13 The maximum amplitude of the first peak, which appears at the tail of the HPM pulse, is ∼22 kV/cm, and the second pulse appears after a time delay of ∼2.2. ns. This corresponds to a frequency of ∼1.2 GHz, which agrees with the plasma electron frequency in our experiment.

To confirm the experimental results, LSP 3D particle-in-cell simulations were performed. These simulations were carried out for an HPM pulse with a peak power of 250 MW, a pulse duration of 0.5 ns at FWHM, and a central frequency of 9.5 GHz, propagating through both an empty waveguide and a plasma-filled waveguide with a plasma density of n e = 3 × 10 10 c m 3. The absolute value of the axial electric field, measured on the waveguide axis inside an empty glass tube of the same dimensions as in the experiment and a relative dielectric constant of 3.6, is depicted in Fig. 4(a). One can see two distinct peaked regions along the envelope of the absolute value of the LSP calculated oscillating electric field [highlighted in Fig. 4(a)] up to 4 ns at the tail of the main propagating HPM pulse in the plasma-filled waveguide, like those observed in the experiment (Fig. 3) but with smaller amplitudes. These do not appear in simulations conducted in vacuum, and the experiment cannot resolve the underlying oscillations. These are similar to the peaks observed by the EFISH measurements but with smaller amplitudes.

In Figs. 4(b) and 4(c), the plasma electron density distribution snapshots illustrate the highest density separation around the glass tube displayed within the time interval of the first SH peak's appearance at the tail of the HPM. Note the two earlier wakes appearing in the charge density distribution, which have propagated beyond the glass tube, are not seen in Fig. 4(a) as their amplitudes are probably too small to perturb the main pulse. The simulations also show that the presence of the glass tube perturbs the charge density distribution and the corresponding wakefield but not to the extent that the structure of the wakes and wakefields changes considerably relative to simulations without it.10 

In conclusion, this study presents the first direct observation of a wakefield excited by an HPM pulse in a plasma-filled waveguide by the EFISH method, which measures the SH signal generated from a 30-fs probe laser beam. Our findings reveal three distinct decaying oscillations of the plasma wakefield, with peak amplitude of ∼20 kV/cm. These experimentally measured wakefields were confirmed by 3D LSP particle-in-cell simulations.

The authors acknowledge E. Flyat for his technical support and Z. Chen for assistance in experiments. This study was supported by the PAZY Foundation, Grant No. 2032056.

The authors have no conflicts to disclose.

Y. Cao: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal). V. Maksimov: Data curation (equal); Formal analysis (equal); Investigation (equal). A. Haim: Data curation (equal); Formal analysis (equal); Investigation (equal). J. G. Leopold: Software (lead); Validation (equal); Writing – review & editing (equal). A. Kostinskiy: Data curation (supporting); Investigation (equal). Y. P. Bliokh: Methodology (supporting); Validation (supporting). Y. Hadas: Funding acquisition (supporting); Project administration (supporting). Ya. E. Krasik: Conceptualization (equal); Funding acquisition (lead); Project administration (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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