Generation of single-cycle relativistic infrared pulses at wavelengths above 20 μ m from density-tailored plasmas

Ultra-intense short-pulse light sources are powerful tools for a wide range of applications. However, relativistic short-pulse lasers are normally generated in the near-infrared regime. Here, we present a promising and ef ﬁ cient way to generate tunable relativistic ultrashort pulses with wavelengths above 20 μ m ina density-tailored plasma. In thisapproach, inthe ﬁ rst stage,an intense drivelaser ﬁ rstexcites anonlinear wake inan underdense plasma, and its photonfrequency isthen downshifted viaphase modulation asit propagates in the plasma wake. Subsequently, inthe secondstage,the drivepulseenters alower-densityplasmaregionsothatthewake hasalargerplasmacavityinwhichlonger-wavelength infrared pulses can be produced. Numerical simulations show that the resulting near-single-cycle pulses cover a broad spectral range of 10 – 40 μ m with a conversion ef ﬁ ciency of ∼ 2.1% ( ∼ 34 mJ pulse energy). This enables the investigation of nonlinear infrared optics in the relativistic regime and offers new possibilities for the investigation of ultrafast phenomena and physics in strong ﬁ elds.


I. INTRODUCTION
Ultrashort-duration high-energy light sources in the mid-infrared (mid-IR) spectral range play key roles in a number of areas of fundamental research, such as the scaling of strong-field interactions to mid-IR wavelengths, 1,2 high-order harmonic generation, 3 and supercontinuum generation. 4 They are also of particular interest for ultrafast molecular dynamics imaging, 5 IR spectroscopy 6 for biological and medical diagnostics, molecular fingerprinting, and the generation of optical frequency combs. 7,8 The boosting of such IR light sources to relativistic intensity will open up a new realm of research, dealing, for instance, with the generation of bright hard x-ray or even gamma-ray sources, 9 next-generation laser-plasma accelerators, [10][11][12] and the study of relativistic light-matter interactions in the mid-IR domain. 13 Most of these studies would benefit significantly from intense driving optical fields with ultrashort pulse durations of a few cycles, long carrier wavelength, multi-millijoule (multi-mJ) pulse energy, and high peak intensity reaching up to the relativistic level. It is obviously an enormous challenge for current laser technology to simultaneously achieve all of these pulse parameters. So far, massive efforts have been devoted to producing few-cycle intense mid-IR pulses through various optical devices using nonlinear crystals. [14][15][16][17] However, it is still hard to expand these methods for obtaining single-cycle IR pulses at long wavelengths beyond 5 μm into the relativistic regime, since they normally suffer from optical breakdown damage and the low power-carrying capacity and limited bandwidth of optical materials.
On the other hand, plasma-based methods are of particular interest for manipulation and generation of high-intensity ultrashort laser pulses, [18][19][20][21][22][23] owing to the ability of plasmas to sustain much higher power and optical intensity without the limitations imposed by the risk of damage. In recent years, there have been a number of proposals for the generation of intense single-cycle mid-IR pulses via photon deceleration of high-power relativistic short-pulse lasers in plasmas [24][25][26] or via frequency downshifting in a two-pulse laserdriven plasma optical modulator. 27 However, the central wavelengths that are produced by these methods are usually limited to a mid-IR spectral range of less than 10 μm. This can be attributed to the use of a relatively high-density plasma as a nonlinear converter, which restricts the ability of the plasma wake to form a cavity large enough to accommodate a longer-wavelength IR pulse. Hence, it remains a challenge to extend the generation of such IR pulses into the longwavelength range beyond 20 μm with current methods.
In this work, we present a scheme to address this difficulty with the use of density-tailored plasmas as a new type of nonlinear optical medium. These plasmas have a two-stage structure, with one stage having a relatively high density for efficient pulse modulation and photon deceleration, and the other stage having moderately low density for significant frequency down-conversion to the longerwavelength spectral domain. This has the advantage of overcoming both the restrictions imposed by limited wavelength elongation in previous plasma-based methods and the disadvantages of low damage threshold and low power amplification associated with traditional optical techniques. Finally, the drive laser pulse experiences strong frequency downshifting and continuous spectral broadening as it propagates in a plasma with such a two-stage structure, and hence it can be converted into near-single-cycle IR pulses of wavelengths above 20 μm with a few percent efficiency at relativistic intensities. Figure 1(a) presents a schematic illustration of how a drive laser pulse of near-IR wavelengths can be transformed into near-singlecycle IR pulses of long wavelength by using a density-tailored plasma structure. In this scheme, a relativistically intense short-pulse laser first excites a plasma wave of density disturbance to form a nonlinear wake as it propagates in an underdense plasma, as shown in Fig. 1(b). As the drive laser pulse interacts with the plasma wave, it undergoes continuous frequency downshifting via phase variation or modulation in the wake. Subsequently, the IR photons slip backward to the central area of the wake, because they have a lower group velocity v g c 1 − ω 2 p / ω 2 ir ) 1/2 than the drive laser photons. As soon as the created IR photons arrive at the wake center, which is near the electron-free region, they are trapped there and experience a tiny frequency shift. This configuration thus acts as an ideal optical medium for producing and sustaining intense near-single-cycle IR pulses, as shown in Fig. 1(c). To broaden such IR pulses to a new spectral range of wavelengths above 20 μm, a lower-density plasma region as a pulse stretcher is used for forming a large enough plasma cavity to generate longer-wavelength IR pulses [ Fig. 1

II. THEORETICAL MODEL AND NUMERICAL SIMULATION
We now discuss the physical processes involved in plasma-based optical modulation for intense IR pulse generation. As an intense laser pulse travels in a plasma, its ponderomotive force is large enough to push away almost all the plasma electrons, leaving the massive ions behind and thus creating a nonlinear plasma wave wake field. This gives rise to rapid responses in the plasma frequency ω p (ξ,τ) and electron density n e (ξ,τ). [28][29][30] These lead to a variation in the local phase velocity of the laser pulse in the wake, which can be estimated by dv p /dξ } z[n e (ξ, τ)]/zξ, where z[n e (ξ, τ)]/zξ is the plasma density gradient, thereby causing an increase or a decrease of the laser wavelength or photon frequency. Here, the local phase velocity is given by the ap- where ω(ξ, τ) is the instantaneous frequency of laser photons in the plasma, c is the speed of light in vacuum, τ t, and ξ x − ct is the variable in the light frame. The local change in the light wavelength within a time dτ can be expressed as dλ Δv p dτ, where Δv p ≈ λzv p /zξ is the difference in phase velocity between two adjoining wave peaks of the light pulse, n c m e ω 2 0 /4πe 2 is the critical plasma density, ω 0 2πc/λ 0 is the initial laser photon frequency, e is the elementary charge, and m e is the electron rest mass. Therefore, one can obtain the local variation in the radiation wavelength as which can be further written in integral form as 1 photons with plasma waves. Therefore, the resulting radiation wavelength of the light pulse in the plasma can be approximated by Equation (2) predicts that the radiation photons are frequencyredshifted in the front of the wake, where zn e (ξ, τ)/zξ > 0, while photons are frequency-blueshifted when they arrive at the tail of the wake, where zn e (ξ, τ)/zξ < 0, and only a very small frequency change occurs in the central area of the wake (or near the electron-free zone), where zn e (ξ, τ)/zξ ∼ 0. As a consequence, the drive laser undergoes significant wavelength widening via phase variation, and the modulated IR photons slip backward relative to the drive pulse owing to group velocity dispersion in the plasma, and they are thus potentially converted into intense long-wavelength IR pulses. This prediction is confirmed by our electromagnetic relativistic particle-in-cell (PIC) simulations detailed below.
To quantitatively investigate the underlying physics of the IR pulse generation scheme, we carry out a series of two-dimensional (2D) PIC simulations using the EPOCH code, 31 which allows selfconsistent modeling of laser-plasma interactions in the relativistic regime. A simulation window moving at the speed of light along the x-axis direction is used to save computing resources, and absorbing boundary conditions are assumed for both particles and electromagnetic fields. The size of the window is 80λ 0 (x) 3 100λ 0 (y), with grid cells of 5600 3 600, sampled by eight macroparticles per cell. The linearly polarized laser pulse has normalized amplitude a 0 2.5 (corresponding to a peak intensity of 8.5 3 10 18 W/cm 2 ), with a spatial profile exp(−r 2 /r 2 0 ), pulse duration 30 fs full width at halfmaximum (FWHM), r 0 20λ 0 , wavelength λ 0 1 μm, oscillation frequency ω 0 2πc/λ 0 , and pulse energy 1.6 J. With the goal of generating relativistic-intensity near-single-cycle light pulses at long wavelengths as discussed earlier, the plasma structure is longitudinally tailored to form two contiguous stages that serve respectively as a photon decelerator with relatively high density and as a pulse stretcher with moderately low density, as shown in Fig. 2(a). Such a density-tailored plasma structure can be formed by inserting a blade at the entrance of the gas jet 32,33 or by using dual-stage gas jets. 34 Figures 2(b) and 2(c) illustrate the evolutions of the electric field and the radiation wavelength of the laser pulse modulated by the wake as functions of the interaction distance in the density-tailored plasma. In the first stage, a relatively high density is employed to excite a plasma wave of large density gradient that is very beneficial for strong pulse modulation and phase variation in the plasma. It can transform the wavelength of the light pulse induced by the plasma wave to the mid-IR range as Δλ } zn e (ξ, τ)/zξ > 0 [see Eq. (1)] when it resides in the region of increasing density (i.e., the front of the wake). Although this is an efficient process, the spectrum of the IR pulse that is produced is limited by the wake cavity to a central wavelength range of less than 15 μm. To overcome this bottleneck, we introduce a second stage with moderately low density as a stretcher, making the plasma wake much larger and thus enlarging the radiation pulse wavelength up to 40 μm. The resulting IR pulses fill almost the entire plasma cavity [see Fig. 1(d)] and in the long-wavelength portion have a nearly 20 μm central wavelength with a near single optical cycle and a relativistic field strength a ir eE ir /m e cω ir ≈ 3.5, as shown in Fig. 2. The total energy conversion efficiency from the drive laser pulse to an IR pulse of wavelength over 10 μm is about 2.1% (i.e., about 34 mJ pulse energy).

III. TUNABILITY OF IR PULSES
The radiation wavelength, peak intensity, optical cycle number, and energy conversion efficiency of the generated IR pulses are flexibly controlled by altering the plasma parameters. Here, we primarily investigate the effect of the stretcher on IR pulse generation, since the it can vary the photon frequency downshift to a much greater degree than the decelerator. We first consider the effect of the length L of the uniform-density part of the stretcher on the IR photon radiation, where L is varied from 100 to 350 μm while all other parameters are kept unchanged, as shown in Fig. 3(a). It should be noted that a suitably long plasma is advantageous to transform high-energy intense IR pulses with wavelengths beyond 20 μm. This is because it allows a long interaction distance for increasing the radiation wavelength or downshifting the photon frequency, which is consistent with our analytical model since the Δλ ≈ λ 3 Δcτ } ΔL. However, for lengths above an optimal value of L ≈ 300 μm, the IR radiation pulse reaches saturation and gradually attenuates owing to pulse absorption and pump depletion in the plasma.
In Fig. 3(b), we investigate the influence of the density of the plasma stretcher on the IR radiation. With the increasing plasma density in the region of the stretcher, the size of the wake cavity shrinks, since the l w ≈ 2π/k p } a 0 /n e √ , and so it may not be able to contain more IR photons and thereby form longer-wavelength light pulses. This leads to a rapid decrease in the radiation wavelength of the IR pulses produced in the plasma, which is verified by our simulations. For example, the IR central wavelength is shortened to λ c ≈ 16 μm when a higher density n e 4.5 3 10 −3 n c is used. However, the plasma density should not be too low; otherwise, the density gradient of the plasma wake may be reduced so much such that it will not have enough strength to produce intense IR radiation, even though it may create a central wavelength as long as 21 μm. A lowerdensity plasma also requires a longer interaction distance for photon frequency downshifting [as suggested by Eq. (2)], which will cause pulse diffraction or absorption and thus weaken the generated IR pulse. In addition, a very low density can induce a steep density ramp between the decelerator and the stretcher, which may result in injection of massive plasma electrons and so consume a lot of the drive laser energy in accelerating these electrons. As a consequence, the energy conversion efficiency of the generated IR pulses will be reduced.
We now examine the robustness of this IR radiation scheme with regard to the carrier-envelope phase (CEP), which is important for light-matter interactions with few-cycle pulses. 35,36 Figure 3(c) shows the CEP of the output IR pulse as a function of the CEP of the initial laser pulse. When an intense short-pulse laser propagates in a plasma, this gives rise to a variation in the time-dependent plasma frequency ω p (τ) or density n e (τ) and thus triggers a shift in the CEP of the light pulse due to the difference between the group velocity (determining the pulse envelope) v g ≈ c[1 − ω 2 p (τ)/2ω 2 ir ] < c and the phase velocity (determining the pulse phase) v p ≈ c[1 + ω 2 p (τ)/2ω 2 ir ] > c. The resulting shift in the carrier phase φ ir of the IR pulse relative to the initial phase φ 0 of the drive laser pulse can be approximated by where ω ir (τ) is the instantaneous frequency of the IR photons produced in the plasma. This implies that the phase shift Δφ will be determined when a specified plasma is used, because both the ω ir (τ) and n e (τ) given by pulse modulation are fixed. This results in an almost-linear phase relationship between the IR pulse and the initial laser pulse, which is validated by our simulations, as can be seen in Fig. 3(c). Consequently, the CEP of the output IR pulse is locked and controlled by the drive pulse, which is very useful for few-cycle pulse-based applications.

IV. CONCLUSION
An efficient scheme has been proposed for generating tunable intense single-cycle IR pulses with central wavelengths above 20 μm, based on photon frequency down-conversion induced by phase modulation in a two-stage density-tailored plasma. The operation of the scheme has been demonstrated by PIC simulations. It is found that such IR pulses can reach relativistically high intensities and pulse energies of tens of millijoules. These pulse parameters can be fully controlled by varying the plasma parameters. Moreover, the carrier envelope phase of the IR pulses is locked and controlled by that of the drive laser pulse in a specified plasma structure. The proposed scheme allows high-intensity ultrashort IR pulses to be produced over a broad spectral range of wavelengths extending up to 40 μm (reaching the THz range), which may extend the investigation of light-matter interactions into an unprecedented range of relativistic intensity and long wavelength, and with single-cycle pulses. Such powerful light sources provide an exciting new tool for exploring the frontiers of ultrafast phenomena and strong-field physics.

Conflict of Interest
The authors have no conflicts to disclose.

Author Contributions
X.-L.Z. and W.-Y.L. contributed equally to this work.

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