Photon deceleration in plasma wakes generates single-cycle relativistic tunable infrared pulses

Availability of relativistically intense, single-cycle, tunable infrared sources will open up new areas of relativistic nonlinear optics of plasmas, impulse IR spectroscopy and pump-probe experiments in the molecular fingerprint region. However, generation of such pulses is still a challenge by current methods. Recently, it has been proposed that time dependent refractive index associated with laser-produced nonlinear wakes in a suitably designed plasma density structure rapidly frequency down-converts photons. The longest wavelength photons slip backwards relative to the evolving laser pulse to form a single-cycle pulse within the nearly evacuated wake cavity. This process is called photon deceleration. Here, we demonstrate this scheme for generating high-power (~100 GW), near single-cycle, wavelength tunable (3–20 µm), infrared pulses using an 810 nm drive laser by tuning the density profile of the plasma. We also demonstrate that these pulses can be used to in-situ probe the transient and nonlinear wakes themselves.

In most wavelengths except below 2 µm, FWDFG process has larger conversion efficiency than FWSFG process.

Characterization of plasma density structures
In our experiments, the plasma density structures are produced using a round supersonic nozzle with an insertable blade that covers a portion of the gas jet. We combine online and offline measurements to fully characterize such non-axisymmetric plasma structures.
In the online measurement, the gas used is hydrogen. The delay between the ionization/wake generating pulse and the probe pulse of interferometry is around 50 ps.
In such a short time period, the evolution of the shock profile (especially in the PC section) is negligible. The interferometry images of the plasma structure in three different cases (the blade inserts 275, 400, and 525 µm into the gas jet) are shown in Supplementary Fig. 1a-c. By Abel inversion, we can retrieve the plasma density profiles in the PC section, but not in the IR-CON and OC sections, since the plasma densities in these latter two sections are non-axisymmetric due to the oblique shock. The measured plasma density profiles in the PC sections are shown in Supplementary Fig.   1d. The mean plasma density in the plateau is about 7.2 × 10 /0 cm 45 . To obtain the complete plasma structure information, we turn to the offline measurement.
In the offline measurement, hydrogen is replaced with argon to get a larger refractive index change. Using fluid simulations shown in Supplementary Fig. 2, one can see that the simulated density profiles of hydrogen and argon are similar with the same gas pressure, but the argon density is about 30% higher than the hydrogen density.
A wavefront sensor (SID-4, PHASICS) camera is used to measure the phase difference 17 of the neutral argon gas at a particular gas jet setup at ten different angles. From these measurements, two-dimensional argon density profiles can be reconstructed by using a tomographic reconstruction algorithm (Supplementary Fig. 3). Therefore, the complete plasma density profile is obtained by multiplying the measured offline density profile (normalized to the plateau density in the PC section) with the measured online plateau density in the PC section (7.2 × 10 /0 cm 45 ). (see Supplementary Fig. 4).

Correction of transport efficiencies and FWM efficiencies of IR pulses
In our experiment, the IR energy and XFROG signal (at a particular time) are measured simultaneously on every shot during the experiment. The measured IR energy (240 shots) for the case shown in Fig 2 is 133 ± 42 µJ as shown in Supplementary Fig. 6.
To correctly retrieve the actually generated IR energy, two corrections need to be considered: transport efficiencies and FWM efficiencies for different IR wavelengths.
First, the net transport efficiency of the IR pulses from Gas jet 1 to Gas jet 2 can be estimated as follows. At the exit of Gas jet 1, all the IR beams with different wavelengths have roughly the same spot size of w < = 13.5 µm (the transverse wake size), and diffract like Gaussian pulses with divergent angles proportional to the wavelengths (confirmed by 3D-PIC simulations). Due to the limited collimation aperture (~35 mm) in the optical path, the collection efficiency decreases with the increase of the IR wavelength, and can be estimated using expressions for Gaussian beam propagation (for λ < 2 µm, the collection efficiency is close to 1). The total transport efficiency can be obtained by further including the transmission of every optical component in the transport path, which is either measured directly or supplied by manufacturers.
For the FWM efficiencies of XFROG at Gas jet 2, a calculation based on the integral equation method 1 is carried out for different IR wavelengths, where the effective length of Gas jet 2 is 1.5 mm, and the argon gas density is 8 × 10 /0 cm 45 .
The focal spot size is roughly proportional to the IR wavelength for λ > 2 µm ( ∝ BC D ), where is the beam size after collimation (~35 mm), and is the focal length of the off-axis parabola. Based on the above parameters (and also the focal spot size of the reference beam 26.6 µm), the FWM phase-matching function F (3) (representing conversion efficiencies) of different IR wavelengths can be calculated, shown in Supplementary Fig. 7. Both the FWDFG and FWSFG efficiencies increase with the IR wavelength, with FWDFG more efficient than FWSFG for most wavelengths.
Combining the above two corrections, the corrected IR spectra at Gas jet 1 can be obtained. In Supplementary Fig. 8, a comparison of the normalized IR spectra before and after correction is shown to have relatively small differences, and this is mainly due to the complementary effect of the transport efficiency and the FWM efficiency.
In addition, the IR energy is simultaneously measured together with XFROG measurement. Similar transport efficiency correction is considered for this measurement. With the measurements of the IR energy and IR spectra, the full information of the generated IR pulse at Gas jet 1 can be properly retrieved.

Supplementary Note 3 Comparison of PIC simulation and experimental results
To make a direct comparison with the experimental result, full 3D PIC simulations are performed using the code OSIRIS 2,3 . The plasma density profile in the simulation is set to the measured density profile when the blade is inserted 400 µm into the gas jet. The drive laser in the simulation is an ideal Gaussian beam with the same spot size as the actual laser beam for simplicity. The laser energy in the simulation is set to 410 mJ so that it has the same enclosed energy as the actual drive laser beam.
The simulated transverse electric field of the original and filtered LWIR (6-20 µm) pulse after exiting the plasma structure (begin to diffract) are shown in Supplementary   Fig. 9a and b, respectively. The simulated LWIR pulse energy in the wavelength range of 6-20 µm is 4.1 mJ, corresponding to a conversion efficiency of 1%. The simulated LWIR pulse has a central wavelength of 9.6 µm, a pulse duration (FWHM) of 36 fs ( Supplementary Fig. 9c), and a near ideal TEM00 mode after exiting the plasma structure ( Supplementary Fig. 9d). The simulation also gives us the electric field of the LWIR pulse as shown in Supplementary Fig. 9c. A direct comparison of the measured and simulated spectrum is shown to have reasonably good agreement ( Supplementary   Fig. 10).
For further improvement of the conversion efficiency, two main factors could be optimized. First, the driving laser's beam quality can be improved: In the experiment, only 60% of the laser energy is contained within the central spot size w0=13.5±1.0 µm, corresponding to 70% of ideal Gaussian pump energy. Second, the plasma density 21 profile in the experiment is only an approximation of the ideal profile necessary for achieving the maximum conversion efficiency 4 . Specifically, the plasma downramp scale length is too long (~500 µm) in the present experiment. It needs to be shortened to ~100 µm to approach the theoretical efficiency.