Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

The X-Ray Emission Effectiveness of Plasma Mirrors: Reexamining Power-Law Scaling for Relativistic High-Order Harmonic Generation

Abstract

Ultrashort pulsed lasers provide uniquely detailed access to the ultrafast dynamics of physical, chemical, and biological systems, but only a handful of wavelengths are directly produced by solid-state lasers, necessitating efficient high-power frequency conversion. Relativistic plasma mirrors generate broadband power-law spectra, that may span the gap between petawatt-class infrared laser facilities and x-ray free-electron lasers; despite substantial theoretical work the ultimate efficiency of this relativistic high-order-harmonic generation remains unclear. We show that the coherent radiation emitted by plasma mirrors follows a power-law distribution of energy over frequency with an exponent that, even in the ultrarelativistic limit, strongly depends on the ratio of laser intensity to plasma density and exceeds the frequently quoted value of  −8/3 over a wide range of parameters. The coherent synchrotron emission model, when adequately corrected for the finite width of emitting electron bunches, is not just valid for p-polarized light and thin foil targets, but generally describes relativistic harmonic generation, including at normal incidence and with finite-gradient plasmas. Our numerical results support the ω−4/3 scaling of the synchrotron emission model as a limiting efficiency of the process under most conditions. The highest frequencies that can be generated with this scaling are usually restricted by the width of the emitting electron bunch rather than the Lorentz factor of the fastest electrons. The theoretical scaling relations developed here suggest, for example, that with a 20-PW 800-nm driving laser, 1 TW/harmonic can be produced for 1-keV photons.

Introduction

Plasma mirrors – microscopically small, transient reflectors created by pulsed laser beams irradiating and ionizing solid surfaces – are powerful photonic devices for manipulating light. When driven by laser pulses with relativistic intensities (\({I}_{{\rm{rel}}}=1.37\times 1{0}^{18}{\lambda }_{[{\rm{\mu }}{\rm{m}}]}^{-2}\) W/cm2 for wavelength λ) and high temporal contrast, plasma mirrors can emit short-wavelength coherent light with broad bandwidth, attosecond duration, and high pulse energy, rivaling the brightness of existing large-scale sources of coherent extreme ultraviolet and x-ray radiation1,2, and showing promise for capturing electronic dynamics on the attosecond timescale3,4. This emission, relativistic high-order harmonic generation (RHHG), has been extensively studied with analytic and computational approaches5,6,7,8,9,10,11,12,13,14,15,16,17 and observed in a number of successful experiments18,19,20,21,22,23,24,25,26,27,28,29,30,31. Here we use a large set of calculated interactions to relate the RHHG spectrum – specifically the power-law relations in the distribution of energy over frequency and the cutoff that marks the onset of steeper decay – to the dynamics of the relativistic electrons accelerated by the laser and plasma fields.

The first high-order harmonics were generated from solid-density plasmas in the sub-relativistic regime32,33,34,35,36,37,38, where the highest frequencies produced are limited by the plasma frequency39,40,41. For relativistic intensities the nonlinear frequency conversion mechanism is fundamentally different, allowing much higher efficiencies and generating harmonics well above the plasma frequency5,6; experimentally demonstrated photon energies have reached 3.8 keV ( ≈3200 harmonics)23. Advances in laser technology are now making attosecond sources based on RHHG feasible42. The relativistic harmonic generation mechanism is often described as a relativistically oscillating mirror (ROM), where the incident laser field is Doppler upshifted by the longitudinal motion of the reflective plasma surface6,10, though for strong laser-plasma interactions sub-cycle energy storage in the plasma becomes important and a more detailed treatment of the electron motion is required, e.g. the relativistic electron spring (RES)12,43,44 or coherent synchrotron emission (CSE)45,46,47 models. In the CSE model, dense nanometer-scale bunches of electrons accelerated by the laser and plasma fields in relativistic trajectories radiate high frequency light; perpendicular acceleration and velocity near the time of emission produce a synchrotron-like spectrum. An emitting electron bunch is created once per laser cycle – or once per half cycle for normal incidence – and the radiation from each cycle arrives as an attosecond burst of broadband light. Determining the conversion efficiency of relativistic RHHG is particular goal of both theoretical and computational work, because it directly governs the viability of RHHG as an ultraviolet and x-ray source.

Relativistic high-order harmonics are generated in the direction of specular reflection when a relativistic-intensity laser is focused on a solid density target; a schematic of the process is shown in Fig. 1 together with an experimentally measured spectrum. The driving laser, focused to intensities greater than 1018 W/cm2, produces a non-linear response from the solid-density target, creating harmonics of the initial frequency. Simulations suggest that these harmonics are phase-locked, leaving an attosecond pulse train if the low-order frequencies are filtered out of the beam. Example spectra for single-cycle driving pulses calculated via particle-in-cell (PIC) simulations are shown in Fig. 2. The spectral intensity of the reflected spectrum I(ω) decreases as frequency increases. PIC simulations and several analytic models predict that this decrease follows a power law [I(ω) (ω)p] up to a cutoff frequency (ωc), beyond which the intensity falls much more rapidly. Under some conditions more complex behavior appears, including spectral modulations19,48. Since the efficiency – and by extension the usefulness – of RHHG rests on how rapidly the spectrum decreases, a great deal of analytic effort has been devoted to predicting the slope of the power-law decay and the cutoff frequency. Early computational work by Gibbon5 found harmonics up to the water window (2.3–4.4 nm) with an intensity-dependent exponent (p) between 6 and 3.5. This was followed by Gordienko et al.8, who used the oscillating mirror model to posit universal values of p = 2.5 and p = 3 for quasi-monochromatic and broadband pulses, respectively, with a cutoff proportional to γ2, where γ is the Lorentz factor of the surface electrons. Baeva et al. adjusted the oscillating mirror model to account for spikes in the gamma factor at the reflecting surface, leading to p = 8/3 power scaling up to a cutoff proportional to γ310. With predictions of p = 4/3 or 6/5 from the CSE model45, suggestions from the RES model that the spectrum is exponential12, and debate about the validity of any universal law49, proposed values of p span a broad range, with strong implications for the usefulness of RHHG-based x-ray sources.

Figure 1
figure 1

Schematic of relativistic harmonic generation with a p-polarized ultrafast laser incident on a solid target. The spectrum was experimentally measured using the Princeton 20 TW Ti:sapphire laser system (Amplitude Technologies), with 70 mJ incident on a BK7 target in 25 fs to produce a peak intensity of 8 × 1019 W/cm2. The central fundamental wavelength is 800 nm. The reflected single-shot spectrum was passed through a 150 nm aluminum filter to remove the fundamental and imaged with a flat-field diffraction-grating spectrometer. The plasma frequency limits sub-relativistic harmonics to 40 nm. Note that the harmonic structure appears because the driving laser pulse is multi-cycle; a single-cycle driver would produce an isolated attosecond pulse and a continuum spectrum.

Figure 2
figure 2

Calculated RHHG spectra (PIC). Upper spectrum: a0 = 1000, N = 4000, p = 1.5, and ωc/ωL = 1900. Lower spectrum: a0 = 20, N = 300, p = 3.1, and ωc/ωL = 30. For both, θL = 45°, τ = 3 fs.

The limiting values of the power-law exponent and spectral cutoff are critically important for determining how effectively near-infrared light can be converted to extreme ultraviolet (XUV) and x-ray radiation by RHHG. Contemporary sources of high-peak-power radiation include solid-state lasers providing petawatt power with near-infrared light, free electron lasers producing high-flux x-rays, and high-order-harmonic generation, supporting attosecond-duration pulses. Figure 3 plots a selection of experimentally achieved peak powers against wavelength, encompassing a broad range of coherent light sources. Relativistic harmonics are normally driven by near-infrared or visible light, where the highest powers and intensities are available from solid state Ti:sapphire, OPCPA, and Nd:glass lasers, as shown by the cluster of points exceeding a petawatt in Fig. 3.

Figure 3
figure 3

Experimentally-demonstrated coherent radiation sources with high peak power compared to relativistic harmonic scaling laws. Pulse duration is indicated by the number of circles around each point. Diagonal lines mark the diffraction limited intensity (f/1 focusing) for light of particular maximum power and wavelength. The RHHG scaling is based on peak power in each integer harmonic interval; for individual harmonics the pulse duration will match the driving laser. Points are taken from results in refs. 66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140. The key distinguishes sources derived from Ti:sapphire or Nd:glass laser architectures, RHHG (high-order-harmonic generation), FEL (free electron laser), OPA (optical parametric amplification) and OPCPA (optical parametric chirped pulse amplification), plasma amplification [including stimulated Raman (SRS) and Brillouin (SBS) scattering], betatron radiation and Compton scattering from laser-accelerated electrons – for review, see141 – and laser-filamentation based frequency conversion. Solid lines indicate a nominal tuning range. Note that for RHHG sources if an attosecond pulse duration was not measured, the power was estimated from the incident pulse duration and harmonic energy. The RHHG scalings assume a 10-PW driver and neglect absorption.

The importance of the power-law exponent value for the usefulness of RHHG is illustrated by the two theoretical scaling lines drawn on Fig. 3: p = 4/3 in red and p = 8/3 in grey. In both cases the normalization is found by assuming a 10-PW driver and neglecting absorption; the integral under the spectrum equals the input 10-PW power. For light produced at 0.8 nm by RHHG driven at 800 nm, the difference between ω−4/3 and ω−8/3 scaling is four orders of magnitude in power – for a petawatt-class driver, this is the difference between an x-ray RHHG source brighter than a free electron laser and one three orders of magnitude weaker. The scaling lines in Fig. 3 show the power in particular harmonics, but the reflected beam contains all harmonics simultaneously. Since the compression of energy into an attosecond pulse can increase the reflected intensity, the peak power achievable for the full spectrum of reflected radiation can be higher than that of the driving laser, making RHHG a possible route to intensities beyond the limits of CPA12,50,51. The power of an attosecond pulse which could be achieved from a 10 PW driver based on simulation results is also marked15.

We show that the power-law exponent p of the reflected spectrum is a continuous function of the experimental parameters, even in the relativistic limit, and that p can be less than 8/3 across a wide range of conditions. Drawing on particle-in-cell simulations, we delineate scaling relationships for the power-law exponent and cutoff and relate them to observed properties of surface electrons, including the Lorentz factor, in an effort to understand both the range of validity of proposed models and the reachable efficiencies of high-order-harmonic generation.

The Power-Law Spectrum of RHHG

RHHG is primarily characterized by the spectrum of the reflected light. Most theoretical models and simulation results are consistent with a power-law spectral shape, although there is some disagreement regarding the appropriate exponent5,8,10,45,49. A power-law fit seems to reasonably approximate the computed spectra we observe. For each spectrum in Fig. 2, the initial power-law [\(I(\omega )\propto {(\omega /{\omega }_{L})}^{-p}\), where ωL is the laser frequency] eventually gives way to a steeper falloff. We will define the cuffoff frequency ωc as the frequency at which the spectral intensity has dropped by a factor of e below that predicted by continuing the power-law scaling, a definition which is reasonably robust to the variation in spectral shape that can occur near the cutoff; ωc is marked for the two spectra in Fig. 2. The value of p is important because it can be used to calculate an approximate value for the energy in each harmonic with ω < ωc. In this section, we consider how p varies due to the laser intensity and plasma density and relate this variation to the microscopic motion of plasma electrons.

Without atomic effects, the interaction physics may be scaled by the laser frequency, removing explicit dependence on ωL. The strength of the incident laser is described by its normalized field amplitude: a0 = eE/meωLc, where E is the maximum value of the electric field envelope, me and e are the mass and charge of an electron, and c is the speed of light. The plasma density (ne) is normalized by the critical density (\({n}_{c}={m}_{e}{\omega }_{L}^{2}\)/4πe2) to give N = ne/nc. Since the two forces of interest in the problem come from the laser and the plasma, the dynamics of relativistic RHHG are governed by the balance between the laser driving force (a0) and the plasma restoring force (N)15. The similarity parameter (S = N/a0)52,53,54 arises from nondimensionalization of the Vlasov-Maxwell equations in the relativistic limit (a0 1) and generally plays an important role in relativistic laser-plasma interactions, although its exact relevance for RHHG has been questioned49,55,56,57. The generation of high frequency light via relativistic RHHG is a subcycle process, so to isolate the detailed mechanism we will primarily show results for single-cycle (Gaussian) driving pulses.

Figure 4a shows the variation in p for 1 ≤ a0 ≤ 1000 and 0.01 ≤ a0/N ≤ 5 at angle of incidence θL = 0. The value of p is found from a linear fit to the logarithm of the spectrum vs the logarithm of the normalized frequency in the range 1 < ω/ωL < 100; the fitting procedure is described in more detail in the Methods section. There are two key points of interest in Fig. 4a. First, for a0 →  at fixed a0/N, p approaches a constant value that can be approximated as proportional to the logarithm of a0/N, i.e. \({{\rm{lim}}}_{a\to \infty }p({a}_{0}/N)\propto {\rm{ln}}\,({a}_{0}/N)\), for a wide range of values. Second, for a0/N > 0.15, p exceeds the predicted maximum scaling at normal incidence (p = 8/3)10. This corresponds to where the maximum reflected intensity exceeds the incident intensity. Note in particular that as a0/N is varied, a very large range of values for p can be reached even in the limit a0 → , from p larger than 6 to less than 2. The example spectra at right show how the spectrum of reflected radiation varies at fixed a0/N and varied a0 for selected values of a0/N. As a0 increases, the spectra follow the power law over a larger frequency range; the cutoff increases with a0.

Figure 4
figure 4

(a) Spectral scaling of the power-law exponent p where \(I(\omega )\propto {(\omega /{\omega }_{L})}^{-p}\), for high-order harmonics generated at normal incidence with varied a0 and N. The exponent is found via a fit to the harmonic range 1 < ω/ωL < 100. Inset: Only those points which satisfy a0/\(\sqrt{N} > 2\). (b–d) Spectra recorded for selected a0 at (b) a0/N = 0.32, (c) a0/N = 0.2, and (d) a0/N = 0.079. The dashed red lines are power-law fits to the a0 = 1000 spectra.

At moderate values of a0 the fitted value of p can be far from the asymptotic limit. Neglecting deformation of the plasma surface, the electric field at the vacuum-plasma interface for normal incidence and a0 N can be approximated as: ES,0 = 2EI,0/\((1+\sqrt{1-N})\), so that a0,eff ≈ 2a0/\(\sqrt{N}\) for N 1. To derive this, note that at the plasma-vacuum boundary, the complex valued incident EI,0, reflected ER,0 and transmitted (ET,0) electric fields must satisfy EI,0 + ER,0 = ET,0 and the magnetic fields satisfy BI,0 + BR,0 = BT,0. In vacuum, E0 = B0, and the plasma dispersion relation gives \({E}_{T,0}=\sqrt{1-N}{B}_{T,0}\). The surface field ES,0 is simply ET,0 at the interface. Since the transverse Lorentz factor of electrons at the surface will be proportional to the surface field, the condition for relativistic motion is a0/\(\sqrt{N}\gg 1\), not simply a0 1. When we filter the plotted points by this condition, and fit p in the range 0 < ω < 20 to avoid the spectral cutoffs, we find that p is a function of only a0/N (Fig. 4a inset); there is no variation with a0 for lower-order harmonics. This suggests the relevance of the similarity model to RHHG and means we can attribute the deviations in p observed for smaller a0, particularly at smaller values of a0/N, to fact that the motion is not fully relativistic.

The same basic dependence of the spectral power law on a0 and N persists at non-zero angle of incidence for P-polarized light, shown, for example, at θL = 30° in Fig. 5b. As for θL = 0°, p approaches a constant value for a0 →  at fixed a0/N and decreases for a0/N < 0.5. Although for both θL = 0° and θL = 30° the fitted value of p does not quickly increase for a0/N > 1, Fig. 5a shows that the attosecond pulse generation efficiency (Ia/IL) reaches a maximum near a0/N ≈ 0.5 and falls for larger a0/N. This decrease occurs for all spectral filters, e.g. 0 < ω/ωL (red), 4 < ω/ωL < 100 (black, solid), 50 < ω/ωL (black, dashed), as well as for the total reflected energy (solid blue) and reflected fundamental energy (dashed blue). This corresponds to the relativistic transparency regime, where a0 N and the high-intensity laser pulse will propagate into plasma which is overdense for less-intense light. In this regime, the laser force overpowers the plasma restoring forces, leading to inefficient RHHG. The relatively shallow values of p for a0/N > 1 do not capture the total decrease in the spectrum as a result of the transparency of the plasma in this regime. The most efficient attosecond pulse generation is achieved when the laser and plasma forces are close to balanced.

Figure 5
figure 5

Attosecond pulse generation efficiency (a) and power-law spectrum scaling coefficient (b) for varied a0/N at θL = 30°. In (a), Ia for the points is calculated using a window 4 < ωωL < 100. The solid black line shows Ia/IL in the large a0 limit, i.e. a0 = 1000. The solid red line shows the total reflected intensity, and the dashed black line shows the attosecond pulse efficiency for 50 < ω/ωL in the same limit. The solid blue line shows the reflected energy and the dashed blue line shows the reflected energy in the fundamental, (ω/ωL < 1.5), both for a0 = 1000. In (b), the solid black line marks p for a0 = 1000, and the dashed lines correspond to p = 4/345 and p = 8/310. The incident pulses are single cycle.

As the angle of incidence (θL) changes, the dependence of p on a0/N as a0 →  remains relatively consistent. Some variation in the exact values of p that are found at a given a0/N can be seen in Fig. 6, where the a0 →  limit is approximated by a0 = 1000. For a0/N < 0.5, p can be very roughly estimated, to within  ±1, by \(p\approx 2{\rm{ln}}\,(N/{a}_{0})+C\), shown by the diagonal dashed line. More importantly, at all angles considered (θL = 0°, 30°, 45°, and 60°) the shallowest values of p lie in the interval 0.1 < a0/N < 1, with a limit near p = 4/3. Apart from a few points for θL = 45°, this  −4/3 line appears to be a limit for the power-law exponent across all of the angle of incidence, intensities, and densities considered here.

Figure 6
figure 6

Variation in the spectral power-law exponent p with a0/N for varied angle at a0 = 1000, an approximation for the limit a0 → . All points were calculated with separate PIC simulations using single-cycle driving pulses. The horizontal lines mark p = 8/3 and 4/3. The diagonal line shows \(p=2{\rm{ln}}\,(N/{a}_{0})+C\).

Electron Trajectories

The  −4/3 spectral scaling arises from the CSE model of high-order harmonic generation45 by assuming that the radiation of interest is emitted by a single bunch of electrons over a time when its position in the direction of specular reflection can be approximated for small t as:

$$x(t)={\beta }_{x}t{\omega }_{L}+\frac{{\alpha }_{1}}{3}{(t{\omega }_{L})}^{3}$$
(1)

and its transverse current varies linearly as:

$${j}_{y}(t)={\alpha }_{0}t{\omega }_{L}$$
(2)

where t = 0 is the time of emission, βx = vx/c is the normalized velocity component in the x direction, and α0 and α1 are constants. Using the stationary phase approximation, this leads to45:

$$I(\omega )\propto |\,\widetilde{f}(\omega ){|}^{2}{\omega }^{-4/3}{\{{{\rm{A}}{\rm{i}}}^{{\rm{{\prime} }}}[{(\omega /{\omega }_{\gamma })}^{2/3}]\}}^{2}$$
(3)

where Ai′ is the derivative of the Airy function of the first kind, \({\omega }_{\gamma }=\sqrt{8{\alpha }_{1}}{\gamma }^{3}\), and γ is the Lorentz factor at the time of emission \((\gamma =1/\sqrt{1-{v}^{2}/{c}^{2}})\). \(\widetilde{f}(\omega )\) is the Fourier transform of the shape function f describing the finite width of the electron bunch. For ω ωγ and an electron bunch much narrower than 1/ω, we recover the power law spectrum:

$$I(\omega )\propto {\omega }^{-4/3}$$
(4)

For this to be valid, ω/ωL must also be sufficiently large that t/TL is small in the expansions of x(t) and jy(t) and higher order terms can be safely neglected.

The mean and individual longitudinal and transverse velocities of an XUV-emitting electron bunch are shown in Fig. 7, along with the individual Lorentz factors in the direction of emission and the advanced time [ta = t + x(t)/c] for each electron. To reach Eq. (3) we must fit the longitudinal velocity with a parabola and the transverse velocity with a line, as marked by the dashed curves. The width of the region where these approximate the underlying trajectories sets the lowest harmonic frequency which will follow Eq. (3); 4/3-scaling comes from the third-order approximation of x and it is in principle possible to reach shallower power laws for a single electron if higher-order terms cannot be neglected. However, under most observed conditions, the third order term in x dominates even for relatively large t/TL (low-order harmonics) and the spectra produced by individual electrons tends to follow Eq. (3) with f = 1.

Figure 7
figure 7

Velocity, Lorentz factor, and advanced time for emitting electron bunch at a0/N = 0.33, a0 = 100, and θ = 30°. The dashed lines show parabolic and linear fits to the relevant components of the velocity.

If the spectra associated with single electrons are well-predicted by Eq. (3) with \(| \widetilde{f}(\omega ){| }^{2}=1\), deviations of the total reflected spectrum from this shape can be attributed to either the finite distribution in time and space of the electrons as they emit or the range in maximum Lorentz factors, and therefore ωγ, of the radiating electrons. Appropriate treatment of f allows the finite distribution of electrons to be included. The curves in Fig. 7 represent a selection of particles that reach the highest velocities, rather than all macroparticles in the simulation. These are the electrons that will dominate the emitted radiation, particularly at high frequencies, but the contribution of slower electrons to the total spectrum can result in relatively higher emission in low-order harmonics and a steeper power-law slope.

The electron-bunch kinematics shown in Fig. 7 produce characteristic synchrotron-like trajectories in the x-y plane14,46. The key attribute of these trajectories is that velocity and acceleration are perpendicular when the electrons are traveling in the specular-reflection direction, a characteristic of synchrotron-like motion that can be found under many variations of the laser and plasma conditions. Figure 8 shows synchrotron-like trajectories from PIC simulations of a thin foil at normal incidence in one (a) and two (b) dimensions, (c) of a semi-infinite target at oblique incidence, and (d) of a finite plasma density gradient at oblique incidence. In the oblique incidence examples, the direction of emission is 30° above the y = 0 axis. In each of the subplots, the specular direction Lorentz factor peaks while the transverse acceleration is maximized, producing instantaneously circular trajectories. These characteristic synchrotron-like trajectories appear under a broad range of interaction conditions, so understanding their dynamics is critical to quantifying relativistic RHHG.

Figure 8
figure 8

Synchrotron-like trajectories of emitting electron bunches across a range of conditions. Selected trajectories are marked in grey. The thick black-blue line shows the average trajectory, with the color indicating the Lorentz factor in the direction of specular reflection. For (a,b) this is along the x axis and for (c,d) the reflected direction is angled up by 30°. (a) Thin foil interaction with D = 2 nm, N = 500, a0 = 10, θ = 0°. (b) Trajectories extracted from a 2D PIC simulation for a thin foil target with D = 4 nm, a0 = 10, N = 500, and θ = 0°. (c) Semi-infinite interaction with D = λ/2, a0 = 40, N = 200, θ = 30°. (d) Interaction with a density gradient, with L = 0.1λ, a0 = 20 and θ = 30°. Synchrotron-like instantaneous motion appears under all these conditions around the time of emission.

Similarity theory suggests that in the limit a0 →  at fixed a0/N the electron trajectories will approach an asymptote. Figure 9 shows the (a) electron bunch velocity components, (b) trajectories, (c) Lorentz factors for varied a0 at fixed a0/N, and (d) the maximum Lorentz factors over a range of a0 and N. From these plots it is apparent that the electron kinematics and trajectories for fixed a0/N approach a limit; the velocities and displacements for a0 = 80 and a0 = 100 are almost identical. This is in agreement with the collapse in Figs. 4 and 5b of p, which depends on lower-order harmonics and thus the shape of the electron trajectories over a relatively long time, to an asymptotic value in the same limit. For example, since the shape of the electron bunch velocity curves are close to identical for a0 = 80 and a0 = 100 over the interval 0.05 < t/tL < 0.15, the tenth harmonic should be the same proportion of the fundamental for both.

Figure 9
figure 9

Kinematics of highest-γ emitting electrons: (a) the longitudinal (vx) and transverse (vy) velocities in the boosted reference frame, (b) the trajectories of the electrons in x-y space, and (c) the Lorentz factor of the electrons as a function of time. In (b), the gray region marks the ion density, and the blue arrow marks the direction of emission. For (a,b) the thick lines are the average values and thin lines are for individual particles. In (c) the thick lines mark the maximum γ factor. (d) The maximum Lorentz factor in the direction of emission for varied a0 and a0/N. Colored points correspond to those in (c). For fixed a0/N, γ is close to proportional to a0. For (a–c), a0/N = 0.33. For (a–d), θL = 30°.

However, the Lorentz factor of the electron bunch (γ = 1/\(\sqrt{1-{v}^{2}/{c}^{2}}\) or, when calculated only for the velocity component in the direction of reflection, γr = 1/\(\sqrt{1-{v}_{r}^{2}/{c}^{2}}\)) does not collapse and instead scales linearly with a0 (Fig. 9c,d), which is consistent with the prediction from similarity theory that \(\gamma \propto {S}^{{\alpha }_{S}}{a}_{0}\)53. Note that because emission occurs when the velocity points in the specular reflection direction, γ = γr at the time of emission. To be consistent with the literature, where not otherwise noted the variable γ will indicate the value of the Lorentz factor at the emission time (which is generally the largest value of γr). To be valid for moderate a0, we write this as:

$$\frac{\gamma -1}{{a}_{0}}\propto {\left(\frac{{a}_{0}}{N}\right)}^{-{\alpha }_{S}}$$
(5)

The cutoff frequency is dependent on γ so the efficiency of the highest frequencies may depend on a0 independent of a0/N, Both the ROM10 and CSE45,46 models predict scaling of the cutoff with γ3. The electron trajectories are therefore in agreement with the observation in Fig. 4 that p approaches an asymptotic limit for large a0, but the point at which the spectra deviate from p continues to vary with a0.

Since we have shown that γ − 1 a0 at fixed a0/N, the final question is how (γ − 1)/a0 varies with a0/N. In Fig. 10 the normalized emitting-direction maximum Lorentz factor for electrons is plotted against a0/N for varied a0 and N at θL = 30°. The quantity (γ − 1)/a0 collapses to a single line with a double power-law shape, with (γ − 1)/a0 approximately proportional to \({({a}_{0}/N)}^{3}\) in the region a0/N < 0.3 and to a0/N in the region a0/N > 0.3. Note that due the transition of αs, other normalizations of γ will not produce a single collapsed curve. If the values of γ are compared to Ia/IL in Fig. 5, we can see that γ continuously increases with a0/N over the entirety of the optimal harmonic efficiency region, so the peak in attosecond pulse generation efficiency cannot be tied to a peak in γ. This suggests that an analysis that includes only γ in harmonic generation is missing key physics; in the next section we will address this gap by considering the spectral cutoff.

Figure 10
figure 10

Peak Lorentz factor in the direction of emission for the emitting electron bunch for varied a0 and N at θL = 30°. Since γ − 1 a0, the normalized quantity \(({\gamma }^{\max }-1)\)/a0 collapses to a single curve f(a0/N) asymptotically for a0 → , with two distinct power-law regimes. For low a0/N, the Lorentz factor approximately scales with \({({a}_{0}/N)}^{3}\) and for a0/N > 0.3, corresponding to the initial onset of the relativistic transparency regime, it scales proportional to a0/N. Each point is extracted from a separate PIC simulation with a single-cycle driving laser.

The Harmonic Cutoff

To determine the efficiency of harmonic conversion, especially for x rays, we must know the highest frequency for which power-law scaling with p is valid. The spectral cutoff sets the limit of p-scaling and is thus critical for determining the x-ray generation efficiencies that can be reached. We have chosen here to define the observed cutoff frequency ωc as the point where the spectrum has dropped below ωp by a factor of 1/e, in part because \({[{{\rm{Ai}}}^{{\prime} }(0)/{{\rm{Ai}}}^{{\prime} }(1)]}^{2}\approx 1\)/e, so that when ω = ωγ, the value of I(ω) from Eq. (3) is 1/e times that from the  −4/3 scaling law. Therefore, if ωγ governs the cutoff, ωγ will approximately equal ωc.

We have previously established that γ a0 at fixed a0/N, and \({\omega }_{\gamma }=\sqrt{8{\alpha }_{1}}{\gamma }^{3}\), so for a0 1:

$${\omega }_{\gamma }={a}_{0}^{3}\cdot f({a}_{0}/N,...)$$
(6)

The similarity-scaling of electron trajectories suggests that because α1 is derived from the shape of the electron velocity, it is a function of a0/N and does not strongly depend on a0 independently.

Consider Fig. 11, where reflected spectra calculated for a0/N = 0.33, θ = 30°, and selected a0 between 5 and 100 are compared to Eq. (3) (gray line). We calculate the predicted cutoff ωγ from the maximum γ of the emitting electron bunch observed in the PIC simulations and fitted α1 to the longitudinal velocity traces, which the insets show for randomly-selected particles as a function of the advanced time ta. The predicted cutoff ωγ scales favorably with a0: at a0 = 100, we find ωγ = 22300, implying that for p = 4/3 the process would produce 20 keV photons with 10−6 efficiency in each harmonic, or greater than 10−3 energy conversion efficiency to the spectral band 20–22 keV. However, the a0 = 100 spectrum actually begins to deviate from the  −4/3 law around ω/ωL = 200, and the 10 keV efficiency is 10−11 per harmonic, five orders of magnitude below the optimistic power-law-ωγ prediction. Though this still makes RHHG an extremely bright x-ray source, the substantial discrepancy requires explanation. The spectral cutoffs ωc are much smaller than ωγ in each of these plots, and, like the cutoffs in Fig. 4, appear to scale close to linearly with a0, not as \({a}_{0}^{3}\). Since ωγ is often assumed to predict the maximum frequencies that can be efficiently generated, it is crucial to understand why RHHG spectra deviate from  −4/3 scaling at much lower frequencies.

Figure 11
figure 11

HHG spectra with CSE model fits and comparison to bunch width (ωb) and Lorentz factor (ωγ) cutoffs at a0/N = 0.33 and θ = 30°. Spectra (black) for a0 varied between 5 and 100 driven by single-cycle laser pulses. The CSE Airy-function spectrum neglecting bunch shape is shown in gray. In red, the analytic spectrum is corrected for the observed spread of emission times by assuming that only electrons within a half wavelength of the bunch leading edge emit constructively and that electron emission events are evenly distributed in time with a sharp initial edge. This last assumption is confirmed by the inset plots of electron γ-factor against the advanced time of the individual electrons, radiation from the fastest electrons arrives first, with minimal interference, after which the bunch slowly trails off. The inset plots contain a random selection of electron trajectories. The incoherent combination of higher-order frequencies results in an additional ω−2 factor in the spectral scaling above ωb = (TL/2τa)ω0. ωγ is the cutoff calculated from the Lorentz factor: \({\omega }_{\gamma }=\sqrt{8{\alpha }_{1}}{\gamma }^{3}\). The bunch width is the time between the peak γ-factor of the first and last electrons to emit. For a0 = 40, an additional line (orange) shows the spectrum for a Gaussian emitting bunch shape, which leads to a factor \(| \widetilde{f}(\omega ){| }^{2}=\exp [-{(\omega /{\omega }_{b})}^{2}]\) in Eq. (3)45.

The trajectory analysis in Fig. 7 substantiates Eqs. (1) and (2) for the average bunch trajectory, and it has been established elsewhere that Eq. (3) accurately predicts the spectra of individual high-γ electrons58, so it is initially surprising that ωγ fails to match ωc so dramatically. To address the discrepancy, we consider the width of the emitting electron bunch by accurately including the shape factor term \(| \widetilde{f}(\omega ){| }^{2}\) that appears in the original formulation of Eq. (3). There are two important differences with how we treat the emitting electrons in comparison to the previous literature. First, we must consider the emission width as a function of the advanced time, not just approximate the width of the electron nanobunch in space at a fixed laboratory time. Since the electrons move relativistically and emission time in the laboratory frame varies with position in the bunch, the dependence of advanced time (ta) on time (t) is complicated. Second, the distribution is not Gaussian; the electron emission distribution has a sharp leading edge, so a better approximation is the Heaviside step function, resulting in a power-law decay rather than an exponential cutoff at ωb. This sharp leading edge can be seen in the insets of Fig. 11, which show the individual particle Lorentz factors as a function of advanced time. The distribution of emission events – which is related to, but not exactly the same as, any instantaneous electron density profile – more directly affects the total emitted radiation than the electron number density. The number density of electrons is proportional to the density of drawn lines in these plots. In each plot, emission from the fastest electrons arrives at the observation point first. After the peak, the rate of emission events remains relatively constant as the maximum Lorentz factor of the involved electrons slowly decays, which is due to electrons further from the surface feeling weaker fields. The step function is an appropriate approximation because the rate of emission events does not slowly ramp up before the peak arrives; in each of these plots no emission peak is produced before tA = 0. The difference between the orange line (Gaussian) and red line (step function) spectra for a0 = 40 in Fig. 11 illustrates the enormous impact the actual distribution of charge in the electron nanobunch has on the high-frequency components of the spectrum.

The distributions of peak γ-factors, which indicate when the highest frequency radiation is emitted by each electron, can have widths that are a substantial fraction of the laser period. For a given harmonic (n) with frequency ωn = nωL, only electrons within about half a harmonic period (Tn = TL/n) of the leading edge will contribute. The advanced time electron bunch width τa therefore sets a frequency cutoff ωb approximated as:

$$\frac{{\omega }_{b}}{{\omega }_{L}}\approx \frac{{T}_{L}}{2{\tau }_{a}}$$
(7)

At a0 = 10, for example, the emission lasts for around 2% of the laser period, which implies that all of the electrons in the bunch only contribute constructively to the first 25 harmonics. From a physical perspective, if we assume that the electron emission times are uniformly distributed after the leading edge, the condition that only those within Tn/2 constructively add means the fraction of the electron bunch that contributes to the emission of frequency ωn scales as 1/n. For coherent radiation, the radiated power will be proportional to 1/n2, or, equivalently, \({\omega }_{n}^{-2}\). Mathematically, the square of the Fourier transform of the Heaviside step function will lead to an additional ω−2 factor for ω > ωb, i.e.:

$$I(\omega )\approx \{\begin{array}{cc}{C}_{1}{\omega }^{-4/3}{\{{{\rm{A}}{\rm{i}}}^{{\rm{{\prime} }}}[{(\omega /{\omega }_{\gamma })}^{2/3}]\}}^{2} & \,{\rm{i}}{\rm{f}}\,\omega < {\omega }_{b},\\ \\ {C}_{2}{\omega }^{-10/3}{\{{{\rm{A}}{\rm{i}}}^{{\rm{{\prime} }}}[{(\omega /{\omega }_{\gamma })}^{2/3}]\}}^{2} & \,{\rm{i}}{\rm{f}}\,\omega \ge {\omega }_{b}.\end{array}$$
(8)

where C1 is a constant and \({C}_{2}={\omega }_{b}^{2}{C}_{1}\). We apply this correction to the spectra in Fig. 11 to plot the red curves, using detailed particle trajectories to calculate τa and ωb. The corrected predictions much more closely approximate the actual spectra, and the correct treatment of the bunch width becomes critically important for keV photon energies. Given that much of the detailed structure of the electron bunches has been approximated away without much justification, the analytic spectra match the PIC results remarkably well. Note that although the analytic fit has a sharp angle between the two scaling regimes, we expect both experimental and simulation spectra to exhibit a smoother transition because the onset of destructive interference is gradual.

The importance of ωb extends further than the single value of a0/N presented in Fig. 11 and is starkly apparent in Fig. 12, where spectra and electron γ-factors are shown for selected a0 at a0/N = 1.1 and θL = 45°. Higher values of a0/N produce higher γ factors (see Fig. 10), and a0/N = 1.1 gives γ ≈ 0.3a0, producing ωγ at MeV levels for a0 = 100 and at 5 GeV for the, admittedly unrealistic, a0 = 4000. However, ωb is even smaller for this larger value of a0/N, leading to bunch-width-dependent cutoffs at much lower frequencies. Note that for this a0/N the fundamental and lower-order harmonics are not strongly generated, so the first few harmonics appear to scale above the  −4/3 power law. The advanced time γ-factor shapes for individual electrons are much narrower than the total time over which the emission arrives, which can be viewed as the physical interpretation of ωγ ωb.

Figure 12
figure 12

HHG Spectra with CSE model fits and comparison to bunch width and Lorentz factor cutoffs at a0/N = 1.1 and θ = 45°. As in Fig. 11, the PIC-derived spectra are compared to analytic fits based on the CSE model with and without bunch-width correction, with the cutoffs associated with both the bunch width and the electron Lorentz factor marked. Inset: the electron Lorentz factors in the emission direction as a function of the advanced time. Note that for these conditions the width in γr for an individual electron is even narrower in comparison to the total width of the bunch than for the trajectories in Fig. 11.

To present a more general picture of how the different measures of cutoff vary, the variation in ωb, ωγ and ωc with a0 is presented in Fig. 13 for a0/N = 0.33 (red) and a0/N = 0.2 (green) at θL = 30° and a0/N = 1.1 at θL = 45° (blue), which are representative of the spectral behavior in the efficient regime of harmonic generation. To allow comparison of ωc and ωb we define \({\omega }_{\widetilde{b}}=\sqrt{e}\cdot {\omega }_{b}\), so that \({\omega }_{\widetilde{b}}\) is the point where the ωb-corrected spectrum has dropped by 1/e below the original p scaling. If the deviation from p-scaling is determined by ωγ, we should have ωc ≈ ωγ. Instead, we have \({\omega }_{c}\approx {\omega }_{\widetilde{b}}\) for the full range of representative conditions shown. Although ωγ reasonably well follows \(\sqrt{8{\alpha }_{1}}{({C}_{\gamma }{a}_{0}+1)}^{3}\) for Cγ found from Fig. 10 and α1 found from the average velocity (dashed lines), both \({\omega }_{\widetilde{b}}\) and ωc increase much less rapidly, with the solid lines linearly proportional to γ. The linear γ scaling of ωb is simply an approximate empirical fit to this data and a derived relationship between the two may point to a slightly different functional relationship. The linear fit may also not be general. We can, however, say with some confidence that neither ωb nor ωc is proportional to γ3. In plots where ωγ lies below ω/ωL = 1000 the exponential cutoff can be identified directly from PIC spectra in addition to the ωb cutoff; for each of these simulations, ωγ calculated from the electron γ-factors closely agrees with \({\omega }_{\gamma }^{{\rm{PIC}}}\) found by direct fitting to the observed spectrum. Note that the Doppler mirror cutoff8 at 4γ2 does not appear as a feature in any of the spectra which contribute to this plot.

Figure 13
figure 13

Variation with a0 of the observed spectral cutoff (ωc), the modified bunch-width-dependent cutoff (\({\omega }_{\widetilde{b}}\)), the Lorentz-factor-dependent cutoff (ωγ) and the Lorentz factor cutoff observed in the PIC spectra (\({\omega }_{\gamma }^{{\rm{PIC}}}\)) at a0/N = 0.33 (red) and a0/N = 0.2 (green) with θL = 30° and a0/N = 1.1 with θL = 45° (blue). The predicted γ3 dependence of ωγ is represented by the dashed lines, calculated from \(\sqrt{8{\alpha }_{1}}{({C}_{\gamma }{a}_{0}+1)}^{3}\), where both Cγ and α1 are estimated from the scaling of γ with a0/N. The discrepancy between the dashed line and ωγ points can be tied to γ not being exactly proportional to a0; the values of ωγ found from the electron trajectories and \({\omega }_{\gamma }^{{\rm{PIC}}}\) found directly from the spectra are almost indistinguishable. The solid lines are approximations based on ωb γ. Inset: ωγ for large values of a0.

We must also consider whether these relationships hold under less-efficient conditions, where p is substantially larger than 4/3. In Fig. 14, where θ = 30° and a0/N = 0.1, p lies around 2.5 for both a0 = 40 and a0 = 100. For these parameters, a significant fraction of the electrons which contribute to the reflected radiation do not reach the peak Lorentz factor associated with the overall interaction. Since the γ achieved by each electron determines the highest frequencies it will emit, the number of electrons which contribute to a particular harmonic decreases for higher harmonic. This causes a difference between the 4/3 scaling of the CSE model and the steeper scaling of the observed spectrum for a less ideal ratio of a0 to N. In these interactions ωb and ωγ are comparable, so although the spectral fit for the bunch-width correction appears to better match the data the difference is relatively small.

Figure 14
figure 14

HHG spectra with CSE model fits and comparison to bunch width and Lorentz factor cutoffs at a0/N = 0.1 and θ = 30°. In this lower efficiency regime, the slope p is not close to 4/3. The difference between the 4/3 power law scaling (red) and the actual scaling (dashed blue) results from the substantial contribution in these spectra of electrons which have low maximum velocities compared to the peak of the electron bunch. Here, ωb ≈ ωγ.

The interpretation of the steep power-law-like behavior of spectra in the regime ω > ωb as due to destructive interference of electrons emitting more than λn/2c after the sharp leading edge can be checked by filtering the reflected radiation and examining it in the time domain. Figure 15 shows the intensity of reflected radiation as a function of time for three different filters. Even for the lowest frequency filter (10 < ω/ωL < 50), we have created an attosecond pulse: the approximate duration produced by an 800 nm driver is 100 as. For 600 < ω/ωL < 3000, the process produces an attosecond pulse with 4 as duration. For filters with ωLF > ωb, the attosecond pulse is much shorter than the duration of the emitting electron bunch in the advanced time (Fig. 15d); the circles in (b) and (c) indicate the times at which electrons with individual ωγ greater than ωLF reach their maximum γr. Without destructive interference between electrons after the leading edge of the bunch, this would be the width of the attosecond pulse for this filter. The inset in Fig. 15d shows the full-width-half-maximum duration (Ta) of the attosecond pulse as a function of the filter frequency ωLF, where ωUF = 5ωLF, which follows the scaling Ta/TL = ωL/ωLF closely. ωLF and ωUF define the filter passband; the filter is transparent for ωLF < ω < ωUF. Therefore, although the efficiency of harmonic generation drops for ω > ωb, the duration of the attosecond pulses that are produced continues to shorten.

Figure 15
figure 15

Instantaneous intensity of reflected radiation in the lab frame after filtering to include only frequencies in the ranges (a) 10 < ω/ωL < 50, (b) 75 < ω/ωL < 375, and (c) 600 < ω/ωL < 3000 and (d) the corresponding γ-factors in the advanced time. Inset: the attosecond pulse duration Ta (full-width-half-maximum of the envelope) for varied ωLF where ωUF = 5ωLF. The line marks Ta/TL = 1/(ωLF/ω0), with the agreement between the line and points suggesting the attosecond pulse duration depends directly on the wavelength of its component light. In (b) and (c), the circles indicate the emitting time of all electron trajectories where ωγ > ωLF. Without destructive interference between electrons, we would expect the attosecond pulse in (c) to have a duration on the scale of the distribution of circles. These plots are taken at a0/N = 1.1, a0 = 200, and θL = 45° and correspond to that shown in Fig. 12c. For frequencies higher than 1000ωL numerical dispersion of the propagating radiation has a measurable widening effect on the duration of the attosecond pulse. Note that for (ac) we consider light passing the observation point, so the advanced time is equivalent to the lab frame.

Beyond Single-Cycle Semi-Infinite Interactions

Although the use of single-cycle interactions on plasma targets with steep density transitions between vacuum and solid density allows study of the underlying RHHG dynamics, real interactions often involve multi-cycle laser pulse on targets which have developed finite exponential density gradients. We consider here how both gradients and multiple-cycle pulses affect the scaling of the reflected spectrum.

Finite plasma density gradients

Over a picosecond to nanosecond timescale after ionization of a solid density target by a laser pulse, the plasma formed at the surface will expand outwards self-similarly, forming an exponential plasma density gradient [\(N(x)={N}_{\max }{e}^{x/L}\)]59,60. For relativistic surface interactions, even prepulses orders-of-magnitude weaker than the main pulse arriving tens-to-hundreds of picoseconds earlier can perturb the target enough to create a substantial plasma density gradient. The effect of pre-plasma gradients on RHHG has been studied28,29, and though there is some disagreement over the optimal scale length, most experimental and theoretical results point to enhanced harmonic generation for L/λ up to a few tenths of the laser wavelength. We must therefore establish how the analysis developed for plasma targets with sharp edges extends to the plasma density gradient case: here we show that (1) small-to-moderate gradients increase efficiency for a0/\({N}_{\max }\ll 1\), (2) that optimal gradient scale lengths have efficiencies similar to semi-infinite targets with ideal a0/N, and (3) that for longer gradients it becomes more difficult to create a narrow emitting electron bunch.

Let us first examine the effect of a gradient on attosecond pulse generation efficiency. In Fig. 16a, the ratio of attosecond intensity to incident intensity is plotted for a range of a0 values and selected L/λ between 0 and 0.5. The target maximum density is \({N}_{\max }=200\) and θL = 30°. For L/λ = 0, this means a0/N = a0/200 and the peak efficiency of Ia/IL ≈ 30 occurs around a0 = 70. The attosecond pulses are filtered from the reflected field in the interval 20 < ω/ωL < 100. As the gradient is increased from 0 to L/λ = 0.1, the efficiency increases for a0/\({N}_{\max } < 0.2\), which can be attributed to the interaction happening at a much lower density (N ≈ a0) than N = 200. Under these conditions, for L/λ > 0.1 the efficiency does not continue to increase; for gradients larger than 0.5λ the attosecond pulse become weaker. In the regime a0 > 100 the gradient has little effect, since the interaction is in the transparency regime for the maximum plasma density. These simulations are configured so that the gradient decreases exponentially to a minimum value of 0.01nc, where it drops to zero. Weighted particles are used to provide adequate resolution across the entire range of densities with reasonable computational expense.

Figure 16
figure 16

(a) Attosecond pulse generation efficiency for varied a0 and gradient scale length L. The plasma target has a maximum density of N = 200, so for a0 > 100, a0/N > 0.5 and the interaction enters the less-efficient transparency regime. θL = 30°. The filter applied to the reflected field selects 20 < ω/ωL < 100. (b) Spectra produced for a0 = 100 and varied L, with N = 200 in the L = 0 case, and \({N}_{\max }=500\) in the finite gradient cases. The gray solid line shows the bunch-width-corrected spectral prediction for the no-gradient case. For all, θL = 45°. (c,d) Harmonic frequency conversion ηω to the interval 30 < ω/ωL < 50 for (c) a finite exponential gradient scale length L and (d) varied a0/N without a gradient. The color scale is set so that blue regions correspond to efficiencies greater than 50% of that produced by  −4/3 scaling. For (c,d), θL = 30°.

The key point of Fig. 16a is that although a small (L/λ ≈ 0.1) gradient can improve RHHG efficiency by orders of magnitude for a0/N < 0.2 – which is an important regime experimentally – the peak efficiencies reached with a gradient are similar to or less than the maximum efficiencies reached for the optimal values of a0/N without a gradient. In this sense, gradients are simply a way of correcting target densities that are too high for currently reachable laser intensities. This, along with the observation that electron trajectories are similar for both efficient flat targets and gradients (Fig. 8), suggests that the corrected synchrotron model is also an adequate method to treat targets with moderate gradients.

If we compare the reflected spectra produced by targets with gradient scale lengths in the interval 0 ≤ L/λ ≤ 0.2 (Fig. 16b), we find that where the zero-gradient case is close to optimum (a0/N = 0.4), the gradient does not substantially change RHHG efficiency. For larger gradients, the reflectivity in the interval 2 < ω/ωL < 30 drops, which flattens the apparent initial scaling law, but does not increase \({\eta }_{\omega }^{[n]}\) for n > 30, where \({\eta }_{\omega }^{[n]}\) is the fraction of incident energy reflected in harmonic n and \({\eta }_{\omega }^{[{n}_{1},{n}_{2}]}\) is the fraction of energy in the between harmonics n1 and n2, inclusive. In all finite-gradient cases significant modulations appear in the spectrum for higher frequencies, indicating complex plasma dynamics. These modulations produce significant apparent fluctuations in efficiency for particular intervals of the harmonic spectrum, but the large-scale behavior of each of the spectra shown here is a decrease at a similar rate.

To consider a broader range of parameters, we plot \({\eta }_{\omega }^{[30,50]}\) for both varied gradient scale length with \({N}_{\max }=1000\) (Fig. 16c) and varied a0/N without a gradient (Fig. 16d). The color scheme for both plots is chosen so that blue regions represent efficiencies greater than 50% of those predicted by the  −4/3 power law. Although the zero-gradient case reaches its optimum around a0/N ≈ 0.5 and a0 > 100, the finite gradient case is restricted to lower efficiencies, even for a0 up to 200. This suggests that although a gradient is useful for increasing efficiency for moderate intensity lasers, the ideal condition for RHHG is a semi-infinite target with density appropriately balanced to the laser field strength.

The trajectories of emitting electron bunches in finite gradients are similar to those for a high-efficiency flat target: as shown in Fig. 17, the width of the emitting electron bunch in the advanced time frame is much longer than the attosecond pulses that are produced for high frequencies. The destructive interference in the trailing edge of the electron bunch means that attosecond pulses are still produced, but the harmonic conversion efficiency drops below the ideal power-law scaling. Provided the gradient is reasonably short (L/λ < 0.2), the corrected synchrotron emission model is still applicable. The most efficient gradients are those with L/λ < 0.2, so the breakdown of the synchrotron model for very long gradients is less important for high-order harmonic generation.

Figure 17
figure 17

Intensity of reflected radiation after filtering to include only frequencies in the ranges (a) 10 < ω/ωL < 50, (b) 75 < ω/ωL < 375, and (c) 600 < ω/ωL < 3000 and (d) the corresponding γr in the advanced time, for L/λ = 0.1, a0 = 20, and θL = 45°. \({N}_{\max }=1000\). Inset: the attosecond pulse duration Ta (full-width-half-maximum of the envelope) for varied ωLF where ωUF = 5ωLF. The line marks Ta/TL = 1/(ωLF/ω0). In (b) and (c), the circles indicate the emitting time of all electron trajectories where ωγ > ωLF.

Multi-cycle driving pulses

The emphasis on single-cycle driving pulses in the previous sections has been justified by the fact that the frequency conversion process is intrinsically sub-cycle; the emission occurs within a fraction of the driving laser period. Here we briefly suggest how the single-cycle analysis extends to multi-cycle pulses, arguing that a long driving laser can be represented as a sequence of single cycle interactions provided that the initial conditions of each interaction can be treated appropriately.

The carrier envelope phase (CEP), represented here by ϕ, describes the alignment of the envelope and underlying phase. We take the convention ϕ = 0 for a sine-like pulse. Since the generation of attosecond pulses is associated with transitions of the laser electric field during particular optical half-cycles15, we can assign each attosecond pulse a phase ϕa based on the CEP of the pulse that produced it and the optical cycle from which it was generated. For example, an attosecond pulse generated during the central optical cycle for a ϕ = 0 incident beam has phase ϕa = 0 and additional pulses will be generated during different cycles of the same interaction with ϕa = nπ for normal incidence. We can in general write ϕa = πn + ϕ, noting that for oblique incidence, attosecond pulses at odd values of n will be suppressed, a similar role to that fulfilled by the differently-defined ψg used by Ma et al.61.

In the limit of short driving pulses (full-width-half-maximum duration τ), where the parasitic effects of preceding cycles can be neglected, the strength an emitted attosecond pulse will depend on the effective normalized potential \( {\tilde{a}} \) during the optical half-cycle which leads to its emission. A Gaussian incident laser [\( {\tilde{a}} (t)={a}_{0}\exp \{-b{\left(t/\tau \right)}^{2}\}\) where \(b=2{\rm{ln}}\,2\)] produces an attosecond pulse with phase ϕa has \( {\tilde{a}} ({\phi }_{a})={a}_{0}\exp \{-b{\left({\phi }_{a}/{\omega }_{L}\tau \right)}^{2}\}\). For isolated interactions with steep surfaces, we may often write – as can be seen in Fig. 5aIa/\({I}_{L}\propto {({a}_{0}/N)}^{q}\) where \({I}_{L}\propto {a}_{0}^{2}\), so that for a Gaussian driving laser the attosecond pulse intensity scales with a0, ϕa and τ as \({I}_{a}({\phi }_{a})\propto {[ {\tilde{a}} ({\phi }_{a})]}^{q+2}={a}_{0}^{q+2}\exp \{-b(q+2){\left({\phi }_{a}/{\omega }_{L}\tau \right)}^{2}\}\). This equation predicts the attosecond pulse train shapes in Fig. 18a,b, where the relative intensities of attosecond pulses are presented for  −0.8 < t/τ < 0.8. In Fig. 18a,b, the attosecond pulse trains produced by τ = 3.5 fs and τ = 21.2 fs incident pulses agree with the predicted attosecond pulse envelope (red and blue dashed lines), based on q = − 6.6 calculated from a0/N scaling for single-cycle pulses under these conditions.

Figure 18
figure 18

Dependence of attosecond pulse intensity (IaIL) on τ and ϕ at a0 = 100, N = 1000, and θL = 0°. (a) Attosecond pulse train for τ = 3.5 fs and (b) τ = 21.2 fs incident Gaussian pulses, plotted with incident intensity envelope (black dashed lines) and predicted attosecond pulse train envelope (red/blue dashed lines) for 10 < ω/ωL < 100 based on a0/N scaling (p = 6.6) found for single-cycle pulses. (c) Dependence of attosecond pulse intensity (Ia/IL) on τ and ϕ at a0 = 20, N = 80, and θL = 30°. (a) Attosecond pulse train generated by incident pulse at τ = 21.2 fs, and incident pulse intensity envelope (black dashed line). (d–g) Intensity of attosecond pulse trains produced by relativistic high harmonic generation (a0 = 100, N = 1000), for varied carrier envelope phase and pulse envelope intensity FWHM (τ), at (a) θL = 15° and L/λ = 0, (b) θL = 45° and L/λ = 0, (c) θL = 0° and L/λ = 0.05, and (d) θL = 45° and L/λ = 0.01 The attosecond pulses are calculated by filtering the reflected signal to the range 10 < ω/ωL < 100. The x-axis is the attosecond phase normalized by the pulse duration and the y-axis is the attosecond pulse intensity normalized by the maximum incident laser intensity. The dashed lines indicate the intensity envelope of the incident signal; in these units the FWHM of the incident pulse (τ) is 1.

Figure 18d  summarizes attosecond pulse intensities calculated for a range of incident τ and ϕ at a0 = 100, N = 1000, and θL = 15°, showing that for these parameters the attosecond pulse intensity depends only on position under the envelope ϕa/ωLτ, and that the relative strength of pulses in the train matches the expectation from the single-cycle scaling. In Fig.18e–g the attosecond pulse intensity (Ia/IL) is plotted against the envelope position ϕa/ωLτ for varied θL and L/λ, showing a reasonable collapse of ηa to a single function of ϕa/ωLτ for varied τ across a range of conditions. The second line for L/λ = 0.1 and θL = 45° are from attosecond pulses emitted on alternative half-cycles from the main pulse.

Generation efficiency becomes more complicated for multi-cylce pulses under the most efficient conditions, where, for example at a0/N = 0.25, the maximum attosecond pulse intensity decreases for longer pulse duration (Fig. 18c). Since intensities for different values of τ coincide at early times on our normalized scale, the decrease in maximum intensity for longer τ can be attributed to the parasitic effect of strong early pulses on the achievable intensity in later cycles, through disruption of the plasma surface. This effect is not observed for smaller values of a0/N because the plasma response time is much faster than the laser period, so the disruption due to the generation of an attosecond pulse is damped before the next pulse arrives. In contrast, at higher a0/N, the relativistic plasma frequency is closer to the laser frequency and plasma disruption may persist for a full optical cycle.

The total spectrum will be the sum of the spectrum contributions the individual pulses. In all multi-cycle pulse cases, a range of intensities under the envelope drive different attosecond pulses, so the individual spectral contributions vary under the envelope. The total spectrum can therefore appear more complex, even if the individual attosecond pulse generation effects are well described by an analytic CSE model.

Conclusion

In this article we have presented results from a large number of particle-in-cell simulations, focusing on how the reflected high-order harmonic spectra varies with plasma density, laser intensity, and angle of incidence, and we have explained key features in terms of a bunch-width corrected coherent synchrotron emission model. Specifically, we have shown that under most conditions the energy spectrum decays as a power of frequency with exponent  −p up to a rolloff set by the width of the emitting electron bunch. The spectrum then approximately decreases as ωp−2, often with substantial modulation due to the interference between different parts of the electron bunch, until an exponential falloff at a frequency proportional to the cube of the electron Lorentz factor. The initial power-law exponent p appears limited to around 4/3, the prediction of the CSE model, so the maximum harmonic peak powers reachable in the 0.1-10 keV range are likely \(0.33{(\omega /{\omega }_{L})}^{-4/3}{P}_{0}\) in each harmonic, where P0 is the initial power and the 0.33 factor arises from normalizing the total reflected power to the incident power. for a 10-PW driver at 800 nm, this leads to 13 TW/harmonic at 100 eV, 0.5 TW/harmonic at 1 keV, and 25 GW/harmonic at 10 keV.

Outside of the most efficient conditions, we have found that p is a continuous function of a0 and N which approaches an asymptotic limit for a0 →  at fixed a0/N and decreases monotonically for a0/N < 0.5. The bunch-width cutoff ωb normally lies at smaller frequencies than the Lorentz-factor dependent cutoff ωγ, leading to a steep power-law decay for ωb < ω < ωγ before the exponential falloff above ωγ. Since ωb approximately scales linearly with a0 rather than with the \({a}_{0}^{3}\) dependence of ωγ, the increase in the range of high-order harmonics that can be efficiently generated is not as rapid as the naive CSE model predicts. Nonetheless, for experimentally reachable conditions, efficient keV photon production is possible.

Although previous work has limited CSE models to thin foils in reflection46 or transmission30,47,62, or highly-oblique p-polarized incidence45, we have found that when the bunch width is treated appropriately, the CSE model is valid across a wide range of conditions and that both ωb and ωγ are important for many reachable parameters. In short, we have generalized the CSE model to explain the evolution of the reflected harmonic spectrum observed in a large number of particle-in-cell simulations spanning a comprehensive set of interaction parameters, developing scaling relationships between the interaction parameters and the efficiency of harmonic and attosecond pulse generation. The scaling laws developed here point to relativistic high-order harmonic generation driven by petawatt-class lasers as the brightest extreme-ultraviolet and soft x-ray source, and competitive with free-electron lasers for peak power at few-keV photon energies.

Methods

Using the PIC codes EPOCH63 and BOPS64, we examined the interaction of single-cycle pulses with fully-ionized overdense (N > 1) semi-infinite plasma at relativistic intensities (a0 > 1) to find p and ωc. To determine the underlying scaling behavior, target densities and laser intensities were allowed to vary somewhat outside of realistic parameters. Here we neglect quantum electrodynamic (QED) effects, even for light fields with very large nominal a0. Although the neglect of QED for large a0 is not physical, it allows us to determine the "relativistic limit” of a particular set of conditions; if we fix a0/N, electron trajectories approach an asymptote as a0 → . This limit provides an idealized case which more rigorously validates the underlying mechanism. At more realistic conditions, a0 ≈ 1 effects can obscure the relativistic mechanism and simulations results may not unambiguously distinguish models, even when those models make different predictions. With a mechanism validated by the relativistic limit, we can treat more realistic values of a0 – where there are no QED effects – as small deviations from the ideal relativistic case. The key physics of relativistic RHHG can be captured with a one-dimensional treatment; for non-zero laser angle of incidence (θL) a relativistic transform to a boosted reference frame (Bourdier method65) removed dependence on the second spatial dimension. Although the spatial axis is one-dimensional, all three components of the velocity are calculated, and two-dimensional trajectories are found by integration of the transverse velocity components. The simulations presented here used spatial resolutions between 3,000 and 50,000 cells per laser wavelength, with at least 10 cells/wavelength for the shortest harmonic wavelength from which any conclusions were drawn. The number of particles/cell was varied based on noise requirements and ensuring that the critical density was resolved; up to 5000 particles/cell were used. The reflected spectra were found by taking the Fourier transform of the reflected fields. Filtered attosecond pulses were found in a spectral range ωLF < ω < ωUF from the inverse Fourier transform. In all cases, the pulse envelopes were taken as Gaussian.

Calculation of power law exponent

The exponent p of the spectral power law [I(ω) ωp] is found by a linear fit of the PIC-calculated spectrum. The slope of the logarithm of the intensity vs the logarithm of the frequency (\({{\rm{\log }}}_{10}I=-\,p{{\rm{\log }}}_{10}\omega +C\)) is the exponent associated with the power-law. The coefficient of proportionality becomes the additive constant C. The fit is based on a particular interval of the spectrum: e.g. in Fig. 4  p is calculated based on the range 1 < ω/ωL < 100. Since the entire spectrum – not just the harmonic peaks – is used, spectra with distinct harmonic structure will be fit by a line that passes below their harmonic maxima. Note that the range of the fit also means that the fundamental frequency does not dominate or constrain the fit, and in some cases there may be a noticeable discrepancy between the power-law fits and the calculated spectra at ω/ωL ≈ 1.

References

  1. 1.

    Tanaka, H. et al. A compact x-ray free-electron laser emitting in the sub-angstrom region. Nat. Photon 6, 540–544 (2012).

    ADS  Article  CAS  Google Scholar 

  2. 2.

    Bostedt, C. et al. Linac Coherent Light Source: The first five years. Rev. Mod. Phys. 88, 015007 (2016).

    ADS  Article  Google Scholar 

  3. 3.

    Krausz, F. & Ivanov, M. Attosecond physics. Rev. Mod. Phys. 81, 163 (2009).

    ADS  Article  Google Scholar 

  4. 4.

    Teubner, U. and Gibbon, P. High-order harmonics from laser-irradiated plasma surfaces. Rev. Mod. Phys. 81, 445 (2009).

    ADS  CAS  Article  Google Scholar 

  5. 5.

    Gibbon, P. Harmonic generation by femtosecond laser-solid interaction: A coherent "water-window” light source? Phys. Rev. Lett. 76, 50 (1996).

    ADS  CAS  PubMed  Article  Google Scholar 

  6. 6.

    Lichters, R., Meyer-ter Vehn, J. & Pukhov, A. Short-pulse laser harmonics from oscillating plasma surfaces driven at relativistic intensity. Phys. Plasmas 3, 3425–3437 (1996).

    ADS  CAS  Article  Google Scholar 

  7. 7.

    Naumova, N. M., Nees, J. A., Sokolov, I. V., Hou, B. & Mourou, G. A. Relativistic generation of isolated attosecond pulses in a λ 3 focal volume. Phys. Rev. Lett. 92, 063902 (2004).

    ADS  CAS  PubMed  Article  Google Scholar 

  8. 8.

    Gordienko, S., Pukhov, A., Shorokhov, O. & Baeva, T. Relativistic Doppler effect: Universal spectra and zeptosecond pulses. Phys. Rev. Lett. 93, 115002 (2004).

    ADS  CAS  PubMed  Article  Google Scholar 

  9. 9.

    Mikhailova, Yu. M., Platonenko, V. T. & Rykovanov, S. Generation of an attosecond x-ray pulse in a thin film irradiated by an ultrashort ultrarelativistic laser pulse. JETP Lett. 81, 571–574 (2005).

    ADS  CAS  Article  Google Scholar 

  10. 10.

    Baeva, T., Gordienko, S. & Pukhov, A. Theory of high-order harmonic generation in relativistic laser interaction with overdense plasma. Phys. Rev. E 74, 046404 (2006).

    ADS  CAS  Article  Google Scholar 

  11. 11.

    Naumova, N. et al. Towards efficient generation of attosecond pulses from overdense plasma targets. New J. Phys. 10, 025022 (2008).

    ADS  Article  Google Scholar 

  12. 12.

    Gonoskov, A. A., Korzhimanov, A. V., Kim, A. V., Marklund, M. & Sergeev, A. M. Ultrarelativistic nanoplasmonics as a route towards extreme-intensity attosecond pulses. Phys. Rev. E 84, 046403 (2011).

    ADS  CAS  Article  Google Scholar 

  13. 13.

    Edwards, M. R., Platonenko, V. T. & Mikhailova, J. M. Enhanced attosecond bursts of relativistic high-order harmonics driven by two-color fields. Opt. Lett. 39, 6823–6826 (2014).

    ADS  PubMed  Article  Google Scholar 

  14. 14.

    Edwards, M. R. & Mikhailova, J. M. Multipass relativistic high-order harmonic generation for intense attosecond pulses. Phys. Rev. A 93, 023836 (2016).

    ADS  Article  CAS  Google Scholar 

  15. 15.

    Edwards, M. R. & Mikhailova, J. M. Waveform-controlled relativistic high-order-harmonic generation. Phys. Rev. Lett. 117, 125001 (2016).

    ADS  PubMed  Article  Google Scholar 

  16. 16.

    Cherednychek, M. & Pukhov, A. Analytical approach to high harmonics spectrum in the nanobunching regime. Phys. Plasmas 23, 103301 (2016).

    ADS  Article  CAS  Google Scholar 

  17. 17.

    Nayak,  A.  et al.  Saddle point approaches in strong field physics and generation of attosecond pulses. Phys. Rep. (2019).

  18. 18.

    Norreys, P. et al. Efficient extreme UV harmonics generated from picosecond laser pulse interactions with solid targets. Phys. Rev. Lett. 76, 1832 (1996).

    ADS  CAS  PubMed  Article  Google Scholar 

  19. 19.

    Watts, I. et al. Dynamics of the critical surface in high-intensity laser-solid interactions: Modulation of the XUV harmonic spectra. Phys. Rev. Lett. 88, 155001 (2002).

    ADS  CAS  PubMed  Article  Google Scholar 

  20. 20.

    Teubner, U. et al. Anomalies in high-order harmonic generation at relativistic intensities. Phys. Rev. A 67, 013816 (2003).

    ADS  Article  CAS  Google Scholar 

  21. 21.

    Dromey, B. et al. High harmonic generation in the relativistic limit. Nat. Phys. 2, 456–459 (2006).

    CAS  Article  Google Scholar 

  22. 22.

    Tarasevitch, A., Lobov, K., Wünsche, C. & von der Linde, D. Transition to the relativistic regime in high order harmonic generation. Phys. Rev. Lett. 98, 103902 (2007).

    ADS  PubMed  Article  CAS  Google Scholar 

  23. 23.

    Dromey, B. et al. Bright multi-keV harmonic generation from relativistically oscillating plasma surfaces. Phys. Rev. Lett. 99, 085001 (2007).

    ADS  CAS  PubMed  Article  Google Scholar 

  24. 24.

    Dromey, B. et al. Diffraction-limited performance and focusing of high harmonics from relativistic plasmas. Nat. Phys. 5, 146 (2009).

    CAS  Article  Google Scholar 

  25. 25.

    Behmke, M. et al. Controlling the spacing of attosecond pulse trains from relativistic surface plasmas. Phys. Rev. Lett. 106, 185002 (2011).

    ADS  CAS  PubMed  Article  Google Scholar 

  26. 26.

    Heissler, P. et al. Few-cycle driven relativistically oscillating plasma mirrors: a source of intense isolated attosecond pulses. Phys. Rev. Lett. 108, 235003 (2012).

    ADS  CAS  PubMed  Article  Google Scholar 

  27. 27.

    Rödel, C. et al. Harmonic generation from relativistic plasma surfaces in ultrasteep plasma density gradients. Phys. Rev. Lett. 109, 125002 (2012).

    ADS  PubMed  Article  CAS  Google Scholar 

  28. 28.

    Dollar, F. et al. Scaling high-order harmonic generation from laser-solid interactions to ultrahigh intensity. Phys. Rev. Lett. 110, 175002 (2013).

    ADS  CAS  PubMed  Article  Google Scholar 

  29. 29.

    Kahaly, S. et al. Direct observation of density-gradient effects in harmonic generation from plasma mirrors. Phys. Rev. Lett. 110, 175001 (2013).

    ADS  CAS  PubMed  Article  Google Scholar 

  30. 30.

    Dromey, B. et al. Coherent synchrotron emission in transmission from ultrathin relativistic laser plasmas. New J. Phys. 15, 015025 (2013).

    ADS  Article  CAS  Google Scholar 

  31. 31.

    Yeung, M. et al. Dependence of laser-driven coherent synchrotron emission efficiency on pulse ellipticity and implications for polarization gating. Phys. Rev. Lett. 112, 123902 (2014).

    ADS  CAS  PubMed  Article  Google Scholar 

  32. 32.

    Burnett, N. H., Baldis, H. A., Richardson, M. C. & Enright, G. D. Harmonic generation in CO2 laser target interaction. Appl. Phys. Lett. 31, 172–174 (1977).

    ADS  CAS  Article  Google Scholar 

  33. 33.

    McLean, E. A. et al. Harmonic generation in Nd: laser-produced plasmas. Appl. Phys. Lett. 31, 825–827 (1977).

    ADS  CAS  Article  Google Scholar 

  34. 34.

    Carman, R. L., Rhodes, C. K. & Benjamin, R. F. Observation of harmonics in the visible and ultraviolet created in CO2 -laser-produced plasmas. Phys. Rev. A 24, 2649 (1981).

    ADS  CAS  Article  Google Scholar 

  35. 35.

    Von der Linde, D. et al. Generation of high-order harmonics from solid surfaces by intense femtosecond laser pulses. Phys. Rev. A 52, R25 (1995).

    ADS  PubMed  Article  Google Scholar 

  36. 36.

    Zepf, M. et al. Role of the plasma scale length in the harmonic generation from solid targets. Phys. Rev. E 58, R5253 (1998).

    ADS  CAS  Article  Google Scholar 

  37. 37.

    Tarasevitch, A. et al. Generation of high-order spatially coherent harmonics from solid targets by femtosecond laser pulses. Phys. Rev. A 62, 023816 (2000).

    ADS  Article  Google Scholar 

  38. 38.

    Teubner, U. et al. Harmonic emission from the rear side of thin overdense foils irradiated with intense ultrashort laser pulses. Phys. Rev. Lett. 92, 185001 (2004).

    ADS  CAS  PubMed  Article  Google Scholar 

  39. 39.

    Gibbon, P. High-order harmonic generation in plasmas. IEEE J. Quantum Electron. 33, 1915–1924 (1997).

    ADS  CAS  Article  Google Scholar 

  40. 40.

    Bezzerides, B., Jones, R. D. & Forslund, D. W. Plasma mechanism for ultraviolet harmonic radiation due to intense CO2 light. Phys. Rev. Lett. 49, 202 (1982).

    ADS  CAS  Article  Google Scholar 

  41. 41.

    Quéré, F. et al. Coherent wake emission of high-order harmonics from overdense plasmas. Phys. Rev. Lett. 96, 125004 (2006).

    ADS  PubMed  Article  CAS  Google Scholar 

  42. 42.

    Mondal, S. et al. Surface plasma attosource beamlines at ELI-ALPS. J. Opt. Soc. Am. B 35, A93–A102 (2018).

    CAS  Article  Google Scholar 

  43. 43.

    Blackburn, T., Gonoskov, A. & Marklund, M. Relativistically intense XUV radiation from laser-illuminated near-critical plasmas. Phys. Rev. A 98, 023421 (2018).

    ADS  CAS  Article  Google Scholar 

  44. 44.

    Gonoskov, A. Theory of relativistic radiation reflection from plasmas. Phys. Plasmas 25, 013108 (2018).

    ADS  Article  CAS  Google Scholar 

  45. 45.

    an der Brügge, D. & Pukhov, A. Enhanced relativistic harmonics by electron nanobunching. Phys. Plasmas 17, 033110 (2010).

    ADS  Article  CAS  Google Scholar 

  46. 46.

    Mikhailova, J. M. et al. Isolated attosecond pulses from laser-driven synchrotron radiation. Phys. Rev. Lett. 109, 245005 (2012).

    ADS  CAS  PubMed  Article  Google Scholar 

  47. 47.

    Dromey, B. et al. Coherent synchrotron emission from electron nanobunches formed in relativistic laser-plasma interactions. Nat. Phys 8, 804–808 (2012).

    CAS  Article  Google Scholar 

  48. 48.

    Boyd, T. & Ondarza-Rovira, R. Plasma modulation of harmonic emission spectra from laser-plasma interactions. Phys. Rev. Lett. 98, 105001 (2007).

    ADS  CAS  PubMed  Article  Google Scholar 

  49. 49.

    Boyd, T. J. M. & Ondarza-Rovira, R. Anomalies in Universal Intensity Scaling in Ultrarelativistic Laser-Plasma Interactions. Phys. Rev. Lett. 101, 125004 (2008).

    ADS  CAS  PubMed  Article  Google Scholar 

  50. 50.

    Gordienko, S., Pukhov, A., Shorokhov, O. & Baeva, T. Coherent focusing of high harmonics: A new way towards the extreme intensities. Phys. Rev. Lett. 94, 103903 (2005).

    ADS  CAS  PubMed  Article  Google Scholar 

  51. 51.

    Vincenti, H. Achieving extreme light intensities using optically curved relativistic plasma mirrors. Phys. Rev. Lett. 123, 105001 (2019).

    ADS  CAS  PubMed  Article  Google Scholar 

  52. 52.

    Gordienko, S. & Pukhov, A. Scalings for ultrarelativistic laser plasma and quasimonoenergetic electrons. Phys. Plasmas 12, 043109 (2005).

    ADS  Article  CAS  Google Scholar 

  53. 53.

    Pukhov, A. & Gordienko, S. Bubble regime of wake field acceleration: similarity theory and optimal scalings. Phil. Trans. R. Soc. A 364, 623–33 (2006).

    ADS  CAS  PubMed  Article  Google Scholar 

  54. 54.

    Ryutov, D. D. & Remington, B. A. Similarity laws for collisionless interaction of superstrong electromagnetic fields with a plasma. Plasma Phys. Contr. F 48, L23–L31 (2006).

    ADS  CAS  Article  Google Scholar 

  55. 55.

    Pukhov, A., Baeva, T., AnderBrügge, D. & Münster, S. Relativistic high harmonics and (sub-) attosecond pulses: relativistic spikes and relativistic mirror. Eur. Phys. J. D 55, 407–414 (2009).

    ADS  CAS  Article  Google Scholar 

  56. 56.

    Boyd, T. & Ondarza-Rovira, R. J. D 58, 137–138 (2010). Comment on "relativistic high harmonics and (sub-) attosecond pulses: relativistic spikes and relativistic mirror. Eur. Phys.

    CAS  Google Scholar 

  57. 57.

    Ondarza-Rovira, R. & Boyd, T. Radiation from Brunel-induced Langmuir waves in ultra-relativistic laser-plasma interactions. Laser Part. Beams 33, 157–162 (2015).

    ADS  CAS  Article  Google Scholar 

  58. 58.

    Edwards, M. R., Fasano, N. M. & Mikhailova, J. M.  Electron-nanobunch-width-dominated spectral power law for relativistic harmonic generation from ultrathin foils. Under Review (2019).

  59. 59.

    Gibbon, P.  Short Pulse Laser Interactions with Matter (Imperial College Press, 2005).

  60. 60.

    Kruer, W. L.  The Physics of Laser Plasma Interactions (Westview Press, 2003).

  61. 61.

    Ma, G. et al. Intense isolated attosecond pulse generation from relativistic laser plasmas using few-cycle laser pulses. Phys. Plasmas 22, 033105 (2015).

    ADS  Article  CAS  Google Scholar 

  62. 62.

    Cousens, S., Reville, B., Dromey, B. & Zepf, M. Temporal structure of attosecond pulses from laser-driven coherent synchrotron emission. Phys. Rev. Lett. 116, 083901 (2016).

    ADS  CAS  PubMed  Article  Google Scholar 

  63. 63.

    Arber, T. D. et al. Contemporary particle-in-cell approach to laser-plasma modelling. Plasma Phys. Contr. F 57, 113001 (2015).

    ADS  Article  CAS  Google Scholar 

  64. 64.

    Gibbon, P. et al. Calibration of one-dimensional boosted kinetic codes for modeling high-intensity laser-solid interactions. Phys. Plasmas 6, 947–953 (1999).

    ADS  CAS  Article  Google Scholar 

  65. 65.

    Bourdier, A. Oblique incidence of a strong electromagnetic wave on a cold inhomogeneous electron plasma. relativistic effects. Phys. Fluids 26, 1804–1807 (1983).

    ADS  MATH  Article  Google Scholar 

  66. 66.

    Gan, Z. et al. 200 J high efficiency Ti:sapphire chirped pulse amplifier pumped by temporal dual-pulse. Opt. Express 25, 5169–5178 (2017).

    ADS  CAS  PubMed  Article  Google Scholar 

  67. 67.

    Aoyama, M. et al. 0.85-PW, 33-fs Ti:sapphire laser. Opt. Lett. 28, 1594–1596 (2003).

    ADS  CAS  PubMed  Article  Google Scholar 

  68. 68.

    Wang, Y. et al. 0.85 PW laser operation at 3.3 Hz and high-contrast ultrahigh-intensity λ = 400 nm second-harmonic beamline. Opt. Lett 42, 3828–3831 (2017).

    ADS  PubMed  Article  Google Scholar 

  69. 69.

    Sung, J. H. et al. 4.2 PW, 20 fs Ti:sapphire laser at 0.1 Hz. Opt. Lett. 42, 2058–2061 (2017).

    ADS  PubMed  Article  Google Scholar 

  70. 70.

    Scarlet Laser Facility, The Ohio State University. Hedp.osu.edu/facilities/scarlet.

  71. 71.

    Gales, S. et al. The extreme light infrastructure–nuclear physics (ELI-NP) facility: new horizons in physics with 10 PW ultra-intense lasers and 20 MeV brilliant gamma beams. Rep. Prog. Phys. 81, 094301 (2018).

    ADS  CAS  PubMed  Article  Google Scholar 

  72. 72.

    Spaeth, M. L. et al. National Ignition Facility laser system performance. Fusion Sci. Tech 69, 366–394 (2016).

    Article  Google Scholar 

  73. 73.

    Bowers,  M.  et al.  Status of NIF laser and high power laser research at LLNL. In High Power Lasers for Fusion Research IV, vol. 10084, 1008403 (International Society for Optics and Photonics, 2017).

  74. 74.

    Texas Petawatt. Texaspetawatt.ph.utexas.edu/overview.

  75. 75.

    OMEGA Facility, Laboratory for Laser Energetics, University of Rochester. Lle.rochester.edu/omega facility.

  76. 76.

    Takahashi, E. J., Lan, P., Mücke, O. D., Nabekawa, Y. & Midorikawa, K. Attosecond nonlinear optics using gigawatt-scale isolated attosecond pulses. Nat. Commun. 4, 2691 (2013).

    ADS  PubMed  PubMed Central  Article  CAS  Google Scholar 

  77. 77.

    Heissler, P. et al. Multi-μ J harmonic emission energy from laser-driven plasma. Appl. Phys. B 118, 195–201 (2015).

    ADS  CAS  Article  Google Scholar 

  78. 78.

    Nomura, Y. et al. Attosecond phase locking of harmonics emitted from laser-produced plasmas. Nat. Phys. 5, 124–128 (2009).

    CAS  Article  Google Scholar 

  79. 79.

    Takahashi, E., Nabekawa, Y. & Midorikawa, K. Generation of 10-μ j coherent extreme-ultraviolet light by use of high-order harmonics. Opt. Lett. 27, 1920–1922 (2002).

    ADS  CAS  PubMed  Article  Google Scholar 

  80. 80.

    Popmintchev, D. et al. Ultraviolet surprise: Efficient soft x-ray high-harmonic generation in multiply ionized plasmas. Science 350, 1225–1231 (2015).

    ADS  MathSciNet  CAS  PubMed  Article  Google Scholar 

  81. 81.

    Ferrari, F. et al. High-energy isolated attosecond pulses generated by above-saturation few-cycle fields. Nat. Photon. 4, 875 (2010).

    ADS  CAS  Article  Google Scholar 

  82. 82.

    Skantzakis, E., Tzallas, P., Kruse, J., Kalpouzos, C. & Charalambidis, D. Coherent continuum extreme ultraviolet radiation in the sub-100-nJ range generated by a high-power many-cycle laser field. Opt. Lett. 34, 1732–1734 (2009).

    ADS  CAS  PubMed  Article  Google Scholar 

  83. 83.

    Goulielmakis, E. et al. Single-cycle nonlinear optics. Science 320, 1614–1617 (2008).

    ADS  CAS  PubMed  Article  Google Scholar 

  84. 84.

    Bierbach, J. et al. Generation of 10 μ W relativistic surface high-harmonic radiation at a repetition rate of 10 Hz. New J. Phys. 14, 065005 (2012).

    ADS  Article  CAS  Google Scholar 

  85. 85.

    Jahn, O. et al. Towards intense isolated attosecond pulses from relativistic surface high harmonics. Optica 6, 280–287 (2019).

    ADS  Article  Google Scholar 

  86. 86.

    Yeung, M. et al. Experimental observation of attosecond control over relativistic electron bunches with two-colour fields. Nat. Photon. 11, 32–35 (2017).

    ADS  CAS  Article  Google Scholar 

  87. 87.

    Rus, B. et al. Efficient, high-brightness soft-x-ray laser at 21.2 nm. Phys. Rev. A 55, 3858 (1997).

    ADS  CAS  Article  Google Scholar 

  88. 88.

    Sebban, S. et al. Full characterization of a high-gain saturated x-ray laser at 13.9 nm. Phys. Rev. A 61, 043810 (2000).

    ADS  Article  Google Scholar 

  89. 89.

    Rus,  B.  et al.  Multi-millijoule, highly coherent x-ray laser at 21 nm as a routine tool for applications. In AIP Conference Proceedings, vol. 641, 182–190 (2002).

  90. 90.

    Zeitoun, P. et al. A high-intensity highly coherent soft X-ray femtosecond laser seeded by a high harmonic beam. Nature 431, 426 (2004).

    ADS  CAS  PubMed  Article  Google Scholar 

  91. 91.

    Wang, Y. et al. High-brightness injection-seeded soft-x-ray-laser amplifier using a solid target. Phys. Rev. Lett. 97, 123901 (2006).

    ADS  CAS  PubMed  Article  Google Scholar 

  92. 92.

    Rohringer, N. et al. Atomic inner-shell X-ray laser at 1.46 nanometres pumped by an X-ray free-electron laser. Nature 481, 488 (2012).

    ADS  CAS  PubMed  Article  Google Scholar 

  93. 93.

    FLASH Accelerator. Flash.desy.de/accelerator.

  94. 94.

    Emma, P. et al. First lasing and operation of an Ångstrom-wavelength free-electron laser. Nat. Photon. 4, 641–647 (2010).

    ADS  CAS  Article  Google Scholar 

  95. 95.

    Ackermann, W. et al. Operation of a free-electron laser from the extreme ultraviolet to the water window. Nat. Photon 1, 336–342 (2007).

    ADS  Article  CAS  Google Scholar 

  96. 96.

    Ayvazyan, V. et al. First operation of a free-electron laser generating GW power radiation at 32 nm wavelength. Eur. Phys. J. D 37, 297–303 (2006).

    ADS  CAS  Article  Google Scholar 

  97. 97.

    FELIX Accelerator. Ru.nl/felix/facility-0/overview.

  98. 98.

    Thiré, N. et al. 10 mJ 5-cycle pulses at 1.8 μm through optical parametric amplification. Appl. Phys. Lett. 106, 091110 (2015).

    ADS  Article  CAS  Google Scholar 

  99. 99.

    Andriukaitis, G. et al. 90 GW peak power few-cycle mid-infrared pulses from an optical parametric amplifier. Opt. Lett. 36, 2755–2757 (2011).

    ADS  CAS  PubMed  Article  Google Scholar 

  100. 100.

    Sell, A., Leitenstorfer, A. & Huber, R. Phase-locked generation and field-resolved detection of widely tunable terahertz pulses with amplitudes exceeding 100 MV/cm. Opt. Lett. 33, 2767–2769 (2008).

    ADS  CAS  PubMed  Article  Google Scholar 

  101. 101.

    Yang, X. et al. Multiterawatt laser system based on optical parametric chirped pulse amplification. Opt. Lett. 27, 1135–1137 (2002).

    ADS  PubMed  Article  Google Scholar 

  102. 102.

    Ross, I. N. et al. Generation of terawatt pulses by use of optical parametric chirped pulse amplification. Appl. Opt. 39, 2422–2427 (2000).

    ADS  CAS  PubMed  Article  Google Scholar 

  103. 103.

    Lozhkarev, V. et al. 200 TW 45 fs laser based on optical parametric chirped pulse amplification. Opt. Express 14, 446–454 (2006).

    ADS  CAS  PubMed  Article  Google Scholar 

  104. 104.

    Lozhkarev, V. et al. Compact 0.56 petawatt laser system based on optical parametric chirped pulse amplification in KD*P crystals. Laser Phys. Lett. 4, 421 (2007).

    ADS  CAS  Article  Google Scholar 

  105. 105.

    Witte, S. et al. A source of 2 terawatt, 2.7 cycle laser pulses based on noncollinear optical parametric chirped pulse amplification. Opt. Express 14, 8168–8177 (2006).

    ADS  CAS  PubMed  Article  Google Scholar 

  106. 106.

    Chekhlov, O. V. et al. 35 J broadband femtosecond optical parametric chirped pulse amplification system. Opt. Lett. 31, 3665–3667 (2006).

    ADS  CAS  PubMed  Article  Google Scholar 

  107. 107.

    Zhao, K. et al. Generation of 120 GW mid-infrared pulses from a widely tunable noncollinear optical parametric amplifier. Opt. Lett. 38, 2159–2161 (2013).

    ADS  CAS  PubMed  Article  Google Scholar 

  108. 108.

    Yin, Y. et al. High-efficiency optical parametric chirped-pulse amplifier in BiB3 O6 for generation of 3 mJ, two-cycle, carrier-envelope-phase-stable pulses at 1.7 μ m. Opt. Lett. 41, 1142–1145 (2016).

    ADS  CAS  PubMed  Article  Google Scholar 

  109. 109.

    Rivas, D. et al. Next generation driver for attosecond and laser-plasma physics. Sci. Rep. 7, 5224 (2017).

    ADS  CAS  PubMed  PubMed Central  Article  Google Scholar 

  110. 110.

    Herrmann, D. et al. Generation of sub-three-cycle, 16 TW light pulses by using noncollinear optical parametric chirped-pulse amplification. Opt. Lett. 34, 2459–2461 (2009).

    ADS  PubMed  Article  Google Scholar 

  111. 111.

    Ren, J. et al. A compact double-pass Raman backscattering amplifier/compressor. Phys. Plasmas 15, 056702 (2008).

    ADS  Article  CAS  Google Scholar 

  112. 112.

    Lancia, L. et al. Signatures of the self-similar regime of strongly coupled stimulated Brillouin scattering for efficient short laser pulse amplification. Phys. Rev. Lett. 116, 075001 (2016).

    ADS  CAS  PubMed  Article  Google Scholar 

  113. 113.

    Lancia, L. et al. Experimental evidence of short light pulse amplification using strong-coupling stimulated Brillouin scattering in the pump depletion regime. Phys. Rev. Lett. 104, 025001 (2010).

    ADS  CAS  PubMed  Article  Google Scholar 

  114. 114.

    Marquès, J.-R. et al. Joule-level high-efficiency energy transfer to subpicosecond laser pulses by a plasma-based amplifier. Phys. Rev. X 9, 021008 (2019).

    Google Scholar 

  115. 115.

    Rousse, A. et al. Production of a keV X-ray beam from synchrotron radiation in relativistic laser-plasma interaction. Phys. Rev. Lett. 93, 135005 (2004).

    ADS  PubMed  Article  CAS  Google Scholar 

  116. 116.

    Cole, J. et al. Laser-wakefield accelerators as hard x-ray sources for 3D medical imaging of human bone. Sci. Rep. 5, 13244 (2015).

    ADS  CAS  PubMed  PubMed Central  Article  Google Scholar 

  117. 117.

    Kneip, S. et al. Bright spatially coherent synchrotron X-rays from a table-top source. Nat. Phys 6, 980 (2010).

    CAS  Article  Google Scholar 

  118. 118.

    Wang, X. et al. Quasi-monoenergetic laser-plasma acceleration of electrons to 2 GeV. Nat. Commun. 4, 1988 (2013).

    ADS  PubMed  PubMed Central  Article  CAS  Google Scholar 

  119. 119.

    Fourmaux, S. et al. Pedestal cleaning for high laser pulse contrast ratio with a 100 TW class laser system. Opt. Express 19, 8486–8497 (2011).

    ADS  CAS  PubMed  Article  Google Scholar 

  120. 120.

    Schnell, M. et al. Optical control of hard X-ray polarization by electron injection in a laser wakefield accelerator. Nat. Commun. 4, 2421 (2013).

    ADS  PubMed  PubMed Central  Article  CAS  Google Scholar 

  121. 121.

    Ju, J. et al. Enhancement of x-rays generated by a guided laser wakefield accelerator inside capillary tubes. Appl. Phys. Lett. 100, 191106 (2012).

    ADS  Article  CAS  Google Scholar 

  122. 122.

    Chen, L. et al. Bright betatron X-ray radiation from a laser-driven-clustering gas target. Sci. Rep. 3, 1912 (2013).

    CAS  PubMed  PubMed Central  Article  Google Scholar 

  123. 123.

    Wenz, J. et al. Quantitative X-ray phase-contrast microtomography from a compact laser-driven betatron source. Nat. Commun. 6, 7568 (2015).

    ADS  CAS  PubMed  PubMed Central  Article  Google Scholar 

  124. 124.

    Kneip, S. et al. Observation of synchrotron radiation from electrons accelerated in a petawatt-laser-generated plasma cavity. Phys. Rev. Lett. 100, 105006 (2008).

    ADS  CAS  PubMed  Article  Google Scholar 

  125. 125.

    Albert, F. et al. Angular dependence of betatron x-ray spectra from a laser-wakefield accelerator. Phys. Rev. Lett. 111, 235004 (2013).

    ADS  CAS  PubMed  Article  Google Scholar 

  126. 126.

    Phuoc, K. T. et al. All-optical Compton gamma-ray source. Nat. Photon 6, 308 (2012).

    CAS  Google Scholar 

  127. 127.

    Chen, S. et al. MeV-energy X rays from inverse Compton scattering with laser-wakefield accelerated electrons. Phys. Rev. Lett. 110, 155003 (2013).

    ADS  CAS  PubMed  Article  Google Scholar 

  128. 128.

    Tsai, H.-E. et al. Compact tunable Compton x-ray source from laser-plasma accelerator and plasma mirror. Phys. Plasmas 22, 023106 (2015).

    ADS  Article  CAS  Google Scholar 

  129. 129.

    Oh, T. I. et al. Intense terahertz generation in two-color laser filamentation: energy scaling with terawatt laser systems. New J. Phys. 15, 075002 (2013).

    ADS  Article  CAS  Google Scholar 

  130. 130.

    Théberge, F., Aközbek, N., Liu, W., Becker, A. & Chin, S. L. Tunable ultrashort laser pulses generated through filamentation in gases. Phys. Rev. Lett. 97, 023904 (2006).

    ADS  PubMed  Article  CAS  Google Scholar 

  131. 131.

    Trushin, S. A., Kosma, K., Fuß, W. & Schmid, W. E. Sub-10-fs supercontinuum radiation generated by filamentation of few-cycle 800 nm pulses in argon. Opt. Lett. 32, 2432–2434 (2007).

    ADS  CAS  PubMed  Article  Google Scholar 

  132. 132.

    Horio, T., Spesyvtsev, R. & Suzuki, T. Generation of sub-17 fs vacuum ultraviolet pulses at 133 nm using cascaded four-wave mixing through filamentation in Ne. Opt. Lett. 39, 6021–6024 (2014).

    ADS  CAS  PubMed  Article  Google Scholar 

  133. 133.

    Hubba, J. D.  NRL Plasma Formulary (2013).

  134. 134.

    Polyanskiy, M. N., Pogorelsky, I. V. & Yakimenko, V. Picosecond pulse amplification in isotopic CO2 active medium. Opt. Express 19, 7717–7725 (2011).

    ADS  CAS  PubMed  Article  Google Scholar 

  135. 135.

    Haberberger, D., Tochitsky, S. & Joshi, C. Fifteen terawatt picosecond CO2 laser system. Opt. Express 18, 17865–17875 (2010).

    ADS  CAS  PubMed  Article  Google Scholar 

  136. 136.

    Hornung, M. et al. 54 J pulses with 18 nm bandwidth from a diode-pumped chirped-pulse amplification laser system. Opt. Lett. 41, 5413–5416 (2016).

    ADS  CAS  PubMed  Article  Google Scholar 

  137. 137.

    Strickland, D. & Mourou, G. Compression of amplified chirped optical pulses. Opt. Comm. 55, 447–449 (1985).

    ADS  CAS  Article  Google Scholar 

  138. 138.

    Glownia, J., Kaschke, M. & Sorokin, P. Amplification of 193-nm femtosecond seed pulses generated by third-order, nonresonant, difference-frequency mixing in xenon. Opt. Lett. 17, 337–339 (1992).

    ADS  CAS  PubMed  Article  Google Scholar 

  139. 139.

    Kando, M. et al. Enhancement of photon number reflected by the relativistic flying mirror. Phys. Rev. Lett. 103, 235003 (2009).

    ADS  CAS  PubMed  Article  Google Scholar 

  140. 140.

    Obenschain, S. et al. High-energy krypton fluoride lasers for inertial fusion. Appl. Opt. 54, F103–F122 (2015).

    CAS  PubMed  Article  Google Scholar 

  141. 141.

    Albert, F. & Thomas, A. G. Applications of laser wakefield accelerator-based light sources. Plasma Phys. Control. F 58, 103001 (2016).

    Google Scholar 

Download references

Acknowledgements

This work was partially supported by the National Science Foundation under Grants No. PHY 1506372 and No. PHY 1806911 and the Department of Energy under grant No. DE-SC0017907. M.R.E. gratefully acknowledges the support of the NSF through a Graduate Research Fellowship. Simulations were performed at the High Performance Computing Center at Princeton University. The EPOCH code was developed as part of the UK EPSRC 300 360 funded project EP/G054940/1. We would like to thank Paul Gibbon for help with the BOPS code.

Author information

Affiliations

Authors

Contributions

Both M.R.E. and J.M.M. conceived and conducted the research and wrote the manuscript.

Corresponding authors

Correspondence to Matthew R. Edwards or Julia M. Mikhailova.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Edwards, M.R., Mikhailova, J.M. The X-Ray Emission Effectiveness of Plasma Mirrors: Reexamining Power-Law Scaling for Relativistic High-Order Harmonic Generation. Sci Rep 10, 5154 (2020). https://doi.org/10.1038/s41598-020-61255-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1038/s41598-020-61255-0

Comments

By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.

Search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing