Soft X-ray harmonic comb from relativistic electron spikes

We demonstrate a new high-order harmonic generation mechanism reaching the `water window' spectral region in experiments with multi-terawatt femtosecond lasers irradiating gas jets. A few hundred harmonic orders are resolved, giving uJ/sr pulses. Harmonics are collectively emitted by an oscillating electron spike formed at the joint of the boundaries of a cavity and bow wave created by a relativistically self-focusing laser in underdense plasma. The spike sharpness and stability are explained by catastrophe theory. The mechanism is corroborated by particle-in-cell simulations.

In this Letter we experimentally demonstrate bright Extreme Ultraviolet (XUV) and soft X-ray harmonics resolved up to a few hundred orders from gas jets irradiated by multi-terawatt femtosecond lasers. We propose a new mechanism of high-order harmonics generation.
We have performed two experimental campaigns using J-KAREN [13] and Astra Gemini [14] lasers. The laser pulse power P 0 , duration τ , wavelength λ 0 , off-axis parabolic mirror f-number, and irradiance in vacuum are 9 TW, 27 fs, 820 nm, f /9, and 4 × 10 18 W/cm 2 in the first campaign, and 120 TW, 54 fs, 804 nm, f /20, and 4 × 10 18 W/cm 2 in the second campaign, respectively. Focusing laser pulses onto a supersonic helium gas jet, Fig. 1, we record harmonics in the 80-360 eV spectral region in the forward (laser propagation) direction, Figs. 2, 3. We use flat-field spectrographs [15], comprising a gold-coated grazing-incidence collecting mirror, spherical varied-linespace grating, and back-illuminated Charge Coupling Device (CCD), shielded by optical blocking filters -two 0.16 µm multilayer Mo/C, two 0.2 µm Pd, or one 0.2 µm Ag on 0.1 µm CH substrate. The acceptance angle is 3 × 10 −5 sr in the first and 3 × 10 −6 sr in the second campaign. In the latter case, the spectrograph has two channels for observation at 0 • (forward direction) and 0.53 • in the laser (linear) polarization plane. The wavelength calibration is performed in-place using spectra of Ar and Ne plasmas. The harmonics energy and photon numbers are conservatively estimated using idealized spectrograph throughputs, i.e. the products of the calculated [16] collecting mirror reflectivities and filter transmissions, measured diffraction grating efficiencies, and manufacturer-provided CCD efficiencies.  (c) Spectra for the shot shown in (a) (red) and for P0 = 120 TW, ne = 1.9 × 10 19 cm −3 obtained from the 2 nd (blue) and 3 rd (green) diffraction orders. Inset: harmonics in the 2 nd diffraction order; similarly for the 3 rd order. (d) The spectral region embracing the 'water window' for P0 = 9 TW, ne = 4.7 × 10 19 cm −3 . The drop at the carbon K absorption edge, KC , is due to hydrocarbon contamination. Dashed curves in (c,d) for background due to the CCD dark current, read-out noise, and scattered photons.
In both campaigns the laser power significantly exceeds the relativistic self-focusing threshold, P sf ≈ 17 GW×(n cr /n e ), where n cr = mω 2 0 /4πe 2 ≈ 1.1 × 10 21 cm −3 (µm/λ 0 ) 2 is critical density. Thus both lasers' pulses self-focus to tighter spots reaching higher irradiance determined by the plasma density and laser power.
The comb-like spectra comprising even and odd harmonic orders of similar intensity and shape are generated by both linearly and circularly polarized pulses in a broad range of plasma electron densities from n e 1.7 × 10 19 to 7 × 10 19 cm −3 . The harmonic base frequency ω f is downshifted from the laser frequency ω 0 , Fig. 2(b,c), in correlation with the transmitted laser spectra.
The data obtained with different lasers demonstrate the effect's reproducibility and robustness and the photon yield scalability with the laser power, Fig. 2(c). For the 120 TW laser, the photon number and energy in unit solid angle within one harmonic at 120 eV are 2 × 10 12 photons/sr and 40µJ/sr, respectively. These amount to 4 × 10 9 photons and 90 nJ, assuming 1.5 • angular radius inferred from our particle-in-cell (PIC) simulations, which agrees with similar spectral intensities observed at 0 and 0.53 • channels. Fig. 2(d) shows the spectrum up to the spectrograph throughput cut-off of 360 eV. The emission reaches the 'water window' region (284-543 eV), required for high-contrast femtosecond bioimaging, where the 9 TW laser produces (1.5 +0. 6 −0.4 ) × 10 10 photons/sr and 0.8 +0. 3 −0.2 µJ/sr, corresponding to 3 × 10 7 photons and 1.7 nJ for the same 1.5 • radius. The uncertainties are due to the 10% filter thickness tolerance and CCD noise.
A large number of resolved harmonics, e. g. n * H 126 in Fig. 2(a,b) and > ∼ 160 in Fig. 2(c, inset), strictly bounds the laser frequency change during the harmonic emission process, δω/ω ≤ 1/(2n * H ), otherwise the orders n * H and n * H + 1 overlap. The laser frequency decreases due to an adiabatic depletion of the laser pulse losing energy on plasma waves, while the number of photons is conserved [17][18][19]. For slow depletion, the frequency downshift rate equals the energy depletion rate, δω/ω = −δx/L dep [17][18][19]. For Fig. 2(a,b), the depletion length is L dep ≈ 2.7 mm estimated from the observed 70% energy transmission through the 0.9 mm plasma. Estimate [19] gives a similar value, L dep ≈ 8.7(n cr /n e ) 3/2 λ ≈ 3.4 mm. The condition δω/ω < 0.4% (or, alternatively, the phase error of < 25 mrad) gives the harmonic emission length of < ∼ 12µm. We note that the emission length can be much longer than the longitudinal size of the moving source.
In many shots with linear polarization (40% with the 9 TW laser), the harmonic spectrum exhibits deep equidistant modulations, Fig. 3. They are visible with discernible high orders and even with nearly unresolved individual harmonics in some shots, Fig. 3(b,d), where blur- ring can be caused by a greater downshift of the laser frequency during a longer emission of harmonics, consistently exhibiting a few times greater photon number. Since in some cases the modulation depth is ∼100%, we conclude that the modulations result from interference between two almost identical strongly localized coherent sources separated in time and/or space. In shots with circularly polarized pulses (90 shots with the 120 TW laser), large-scale spectral modulations are not observed.
The unique properties of the observed harmonics mismatch previously suggested mechanisms of high-order harmonics generation. Atomic harmonics are excluded because both linearly and circularly polarized laser pulses produce both even and odd harmonic orders with a weak sensitivity to gas pressure. Betatron radiation is not relevant because the base frequency of its harmonics is determined by the plasma frequency and electron energy, not the laser frequency. Nonlinear Thomson scattering cannot provide the observed photon numbers even under the most favorable assumptions. For the 9 TW shot, Fig. 2(a,b,c), the numerical calculation of singleelectron radiation spectra [7,20] gives at 100 eV at most 2 × 10 −10 nJ/eV sr, assuming the self-focused laser amplitude of a 0 = 7, pulse duration of 30 fs, and the observation angle of 15 • close to optimum [7], where the high-frequency nonlinear Thomson scattering component is most intense. The number of electrons encountered by the laser pulse is N e = πd 2 sf ∆xn e /4 ≈ 6 × 10 9 , where n e = 2.7×10 19 cm −3 , d sf ≈ 5µm is the self-focusing channel diameter [21], and ∆x ≈ 12µm is the harmonics emission length (see above). This gives at most 1 nJ/eV·sr, 200 times smaller than experimentally observed. For other spectra shown in Fig. 2(c,d) even less fraction corresponds to nonlinear Thomson scattering.
We performed two-(2D) and three-dimensional (3D) PIC simulations of harmonics generation using the code REMP [22]. As seen in simulations [23, 24], the laser pulse undergoes self-focusing [2,18], pushes electrons out evacuating a cavity in electron density [18,25] and generating a bow wave [26]. Fig. 4(a) represents the case when a linearly polarized (along y axis) laser pulse selffocused to the amplitude of a 0 = 6.6 and FWHM waist of 10λ 0 , reaches a point where n e = 1.8 × 10 18 cm −3 . The simulation window size is 125λ 0 × 124λ 0 × 124λ 0 with the resolution of dx = λ/32, dy = dz = λ/8; the number of quasi-particles is 2.3 × 10 10 ; ions are immobile. The electromagnetic energy density of a high-frequency (ω ≥ 4ω 0 ) field reveals the source of high-order harmonics, Fig. 4(a, red arcs): they are emitted by the oscillating electron density spikes formed at the joining of the bow wave and cavity boundaries. While emission is also seen from the cavity walls [12], at higher frequencies much stronger radiation originates from the electron spikes.
The high-order harmonics emitted by the electron spikes are seen in 2D PIC simulation, performed in the 87λ 0 ×72λ 0 window with dx = λ 0 /1024, dy = λ 0 /112 and 6×10 8 quasi-particles. In Fig. 5, the laser pulse acquiring the amplitude of a 0 = 10, the duration of 16λ 0 and the waist of 10λ 0 meets plasma with n e = 1.7 × 10 19 cm −3 . Starting from the 7 th order, harmonics are well discernible up to the 128 th order, as allowed by the simulation resolution. The emission is slightly off-axis with the angle decreasing with increasing harmonic order.
A strong localization of electron spikes, their robustness to oscillations imposed by the laser and, ultimately, their superior contribution to high-order harmonics emission is explained by catastrophe theory [27]. The laser pulse creates a multi-stream electron flow [28,29] stretching and folding an initially flat surface formed by electrons in their phase space, Fig. 4(b), [23,30]. The surface projection onto the (x, y) plane gives the electron density where outer and inner folds are mapped into singular curves outlining the bow wave and cavity boundaries, respectively. Catastrophe theory here establishes universal structurally stable singularities, insensitive to perturbations. The bow wave and cavity boundaries produce the 'fold' type singularity (A 2 , according to Arnold's classification [27]), where the density grows as (∆y) −1/2 with decreasing distance to the boundary, ∆y. At the joint of the two folds, the density grows as (∆y) −2/3 , forming a higher order singularity -the 'cusp' (A 3 ), Fig. 5(c,d), [23,31]. Stronger singularities exist [29], however they are not stable against perturbations. Located in a ring surrounding the cavity head, the cusp is seen in simulations as an electron spike. For linear polarization, the harmonic emitting ring breaks up into two segments, Fig. 4(a). The spike oscillations imposed by the laser generate high-order harmonics. The cusp singularity ensures a tight concentration of electric charge, making the emission coherent, i. e. the intensity is proportional to the particle number squared N 2 e , similarly to the coherent nonlinear Thomson scattering. However, the cusp consists of different particles at every moment of time, in contrast to a synchronous motion of the same particles. We note that for constructive interference it is sufficient that the source size is smaller than the emitted wavelength in the direction of observation only. The estimated number of electrons within the singularity ring, N e ∼ 10 6 , provides a signal level close to the experiment.
In classical electrodynamics [20], an oscillating electron emits harmonics up to critical order n Hc , proportional to the cube of the particle energy E e ≈ a 0 mc 2 , then the spectrum exponentially vanishes as seen in our experiments. This gives clues about the harmonics generation scaling. For the laser dimensionless amplitude in the stationary self-focusing [21], a 0,sf = (8πP 0 n e /P c n cr ) 1/3 , we obtain the critical harmonic order of n Hc ∼ P 0 n e /P c n cr , where P c = 2m 2 e c 5 /e 2 ≈ 17 GW. The total energy E s emitted by the electron spike [20] is proportional to N 2 e and the harmonic emission time τ H : Here γ ≈ (n cr /n e ) 1/2 is the Lorentz factor associated with the spike velocity, which is close to the laser pulse group velocity in plasma. In the 120 TW shot, Fig. 2(c), the energy emitted into a harmonic near 120 eV is 40 µJ/sr. In the 2D PIC simulation with the parameters close to this shot (Fig. 5), the energy of 100 th harmonic is 30 µJ/sr. An estimate based on Eq. (2) gives 100 µJ/sr within the 100 th harmonic, thus providing a simple analytical estimate for the expected harmonic energy.
With the detector at 1.4 m and the source size of 10 µm estimated from simulations, the spatial coherence width is 1 mm, which is large enough for many phase contrast imaging applications in a compact setup [32].
In conclusion, irradiating gas jets with multi-terawatt lasers we observe comb-like harmonic spectra reaching the 'water window' region. We propose the harmonic generation mechanism based on self-focusing, nonlinear wave formation with electron density spikes, and collective radiation by a compact electric charge. Catastrophe theory explains the electron spike sharpness and stability. The maximum harmonic order scaling with the laser intensity will allow reaching the keV range. Our results open the way to a compact coherent X-ray source built on a university laboratory scale repetitive laser and accessible, replenishable and debris-free gas jet target. This will impact many areas requiring a bright X-ray/XUV source for pumping, probing, imaging, or attosecond science.