Divergence gating towards far-field isolated attosecond pulses

Isolated attosecond pulses (IAPs) have attracted broad interests during decades [1–3] for their rich applications among the resolution and control of ultrafast electron movement in atoms [4], molecules [5], materials [6, 7], and high-energy-density plasma [8, 9]. Getting rid of intensity limitations from the ionization threshold, relativistic high-order harmonic generation (RHHG) [10–15] has been regarded as the most promising way towards ultraintense attosecond pulses (APs) for attosecond-pump-attosecond-probe experiments [16], which are challenging for those produced from gaseous high-order harmonic generation [17] due to low photon flux as well as pulse energy in the magnitude of 10⁻⁹ J [16]. However, such APs are obtained generally in form of trains as the harmonic emission is driven by periodical interactions between lasers and overdense plasmas [13].

Series of methods [18–28] have been proposed to generate IAPs through RHHG, which can be classified as 'temporal gating' and 'spatial gating'. Different from temporal gating where APs can only be generated efficiently once because of few-cycle incident lasers [18], well-controlled laser polarization [19–21] or specially designed targets [22] as well as their transparency [23, 24], spatial gating separates APs in space through different reflected directions of them due to attosecond lighthouses [25–27] or tightly focused lasers [28]. For temporal gating, not only do some of methods require ultrathin targets [22–24], but also most of them have to filter out low-order harmonics (photon energy E_ph ≲ 30 eV typically) which are usually produced efficiently for several optical cycles. While attosecond lighthouse effects used in spatial gating can relax the requirement of filters to E_ph ≳ 20 eV, experimentally challenging preplasma scales L ≲ 10⁻² λ₀ is required [25], which greatly limits the generation efficiency at the same time [29, 30]. Here, λ₀ represents the central wavelength of incident lasers. The introduction of high-pass spectral filters greatly suppresses the intensity of IAPs due to fast decay of harmonic intensities [13]. What's more, the removed spectra ranging from near ultraviolet (NUV) to vacuum ultraviolet (VUV) are critical for attosecond science in valence-shell phenomena [31, 32], molecule excitation [33] and nonlinear cluster dynamics [9, 34].

To generate IAPs with spectra and energy preserved as much as possible, the idea of divergence gating is proposed in this paper. Divergence gating separates APs through the modulation of divergences, where APs emitted in the peak optical cycles of incident lasers have minimum divergences and experience much lower energy density decay during propagation than other pulses. As all harmonics including fundamental frequency waves are influenced, divergence gating has the potential to generate near-axis IAPs in the far field with lower requirements of filters, even preserving all harmonics. Recently, the optically-curved plasma surfaces due to laser–plasma interactions have been proved to act like plasma mirrors (PMs) [35] which significantly affect the divergences of harmonics from RHHG and can be controlled by the convex wavefronts of defocused quasi-monochromatic incident lasers [27]. However, as the wavefronts are identical for different cycles, divergences of generated harmonics almost keep the same during interactions [36], forbidding the realization of divergence gating. Fortunately, defocused chirped lasers provide different wavefronts for each cycle due to the variational wavelength, i.e., variational Rayleigh length. To the best of our knowledge, chirped lasers, which are becoming accessible with relative bandwidth as high as 60% [37] and power approaching 100 TW [38] and even PW [39] in the future, are never used to generate IAPs based on RHHG although they have been investigated widely in particle acceleration [40, 41], pair production [42], and even gaseous high-order harmonic generation [43, 44].

In this paper, divergence gating as a method to generate isolated far-field APs is realized through relativistic chirped laser–plasma interactions, verified by theoretical analysis and three-dimensional (3D) numerical simulations. It is proposed that as the divergent incident wavefronts of chirped lasers change for different optical cycles due to varying wavelengths, the focusing effect of concave PMs can only be offset exactly in the peak incident cycle when overdense plasma targets are placed an optimal distance behind the focal planes. Therefore, the minimum divergence is achieved only for harmonics emitted from the peak cycle, which is critical for divergence gating. Based on this, the minimum energy threshold of filters ${E}_{\text{ph}}^{\mathrm{min}}\enspace$ is provided in theory, consistent with 3D numerical simulations where intense far-field IAPs containing millijoule (mJ) energy are obtained with broad bandwidth including low-order harmonics preserved (≳10 eV). Such brilliant sub-50 as IAPs significantly benefit attosecond-pump-attosecond-probe experiments, especially those requiring NUV or VUV photons.

The scheme of divergence gating is illustrated in figure 1(a). A chirped Gaussian incident laser (orange hollow column), which is linear-polarized in the y direction, propagates along X-axis and irradiates on the overdense plasma target (grey block) obliquely with angle α and central wavelength λ₀, where exponential preplasma density profile with scale L is assumed. Keeping a distance x_f between the plasma bulk and focal plane of incident laser, the transverse incident electric field E acting on the target is given as [41, 45, 46]:

$$\begin{equation}E={E}_{0}\frac{{w}_{0}}{{w}_{\mathrm{s}}}\mathrm{exp}\left(-2\,\mathrm{ln}\,2\frac{{\xi }^{2}}{{\tau }_{\mathrm{L}}^{2}}-\frac{{r}^{2}}{{w}_{\mathrm{s}}^{2}}\right)\mathrm{sin}\left[{\omega }_{0}\left(\xi +\frac{s{\xi }^{2}}{2}\right)-\frac{k{r}^{2}}{2{R}_{\mathrm{i}}}+{\phi }_{0}\right],\end{equation} \tag{ 1 }$$

where $r\equiv \sqrt{{y}^{2}+{z}^{2}}$ and ξ ≡ t − x_f/c. Here, ${w}_{\mathrm{s}}={w}_{0}\sqrt{1+{({x}_{\mathrm{f}}/{x}_{\mathrm{R}})}^{2}}$, ${R}_{\mathrm{i}}={x}_{\mathrm{f}}+{x}_{\mathrm{R}}^{2}/{x}_{\mathrm{f}}$, ${x}_{\mathrm{R}}=\pi {w}_{0}^{2}/\lambda$, k = 2π/λ, λ = λ₀/(1 + sξ), ω₀ = 2πc/λ₀ and $s\equiv \pm \sqrt{{\left({\Delta}\omega /{\omega }_{0}{\tau }_{\mathrm{L}}\right)}^{2}-{\left(4\,\mathrm{ln}\,2/{\omega }_{0}{\tau }_{\mathrm{L}}^{2}\right)}^{2}}$ [41] represent the beam radius at which the field amplitude is 1/e times of the axial value, radius of curvature, Rayleigh length, wave number, wavelength, central angular frequency and chirp parameter ($> \;0$ for positive chirp and $< \;0$ for negative chirp) of the incident laser on plasma surface, respectively. Besides, ${w}_{0},$ Δω, τ_L, c, y and z represent the beam waist, band width, pulse duration (full-width at half-maximum), speed of light in vacuum and the positions in two orthogonal transverse directions, respectively. The phase ϕ₀ ≡ ϕ_C + ϕ_G includes the contribution of carrier envelope phase ϕ_C (CEP) and Gouy phase ${\phi }_{\mathrm{G}}\equiv \mathrm{arctan}\left(x/{x}_{\mathrm{R}}\right)$. We note that ξ = 0 when the temporal envelope of the laser field achieves its maximum. Besides, as illustrated in figure 2(b), values of s increase quickly with the increase of Δω or decrease of τ_L for most of situations (${\tau }_{\mathrm{L}} > 4\sqrt{2}\,\mathrm{ln}\,2/{\Delta}\omega$ in specifically), while the minimum of the second one is limited by the first due to the transform limit (white area). The relationship between s and τ_L for a given Δω is non-monotonical as both the Gaussian envelope and frequency chirping contribute to the bandwidth. Therefore, for fixed Δω, the chirp parameter s, which characterizes the changing rate of frequency, changes nonlinearly with different τ_L which determines the bandwidth of Gaussian envelope by 0.44/τ_L [46] and equals 0 for transform-limited (TL) and infinite-duration beams.

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Reflected along X'-axis, the reflected laser (purple hollow column) is focused by the optically-curved dense electron layer, which is compressed by the driving laser and acts like a PM. Due to the spatial distribution of the light pressure ∝E², the displacement of surface electrons varies in transverse directions, resulting in parabolic PMs formed naturally [35], whose focal length ${f}_{\mathrm{p}}\approx {w}_{\mathrm{s}}^{2}(1+{{\epsilon}}^{-1})/4L\,{\mathrm{cos}}^{2}\,\alpha$ and profile illustrated in figure 1(b) (solid). Here, ≡ a_L λ₀(1 − sin α)/πL accounts for the electron dynamics of plasma [35]. Its reciprocal ⁻¹ describes the influence of laser intensity on the optical properties of curved PMs, which is negligible when a_L ≫ 1 as ⁻¹ ∼ 0. The dimensionless parameter a_L ≡ eE/m_e cω₀ is the normalized amplitude of electric field strength with m_e and e representing the mass and absolute charge of electrons respectively. Assuming the reflected harmonics have Gaussian spatial distributions either, their radii of curvature R_r on PMs satisfy 1/R_i − 1/R_r = 1/f_p cos α on the meridional plane [47] and are given as

$$\begin{equation}{R}_{\mathrm{r}}=\frac{{x}_{\mathrm{f}}^{2}+{(1+s\xi )}^{2}{x}_{\mathrm{R}0}^{2}}{{x}_{\mathrm{f}}-{b}_{0}{(1+s\xi )}^{2}{x}_{\mathrm{R}0}},\end{equation} \tag{ 2 }$$

where b₀ ≡ 4πL cos α/λ₀(1 + ⁻¹) is actually the ratio of the optimal x_f to x_R0 to realize divergence gating as discussed later and ${x}_{\mathrm{R}0}\equiv \pi {w}_{0}^{2}/{\lambda }_{0}$ is the Rayleigh length of incident lasers for the central wavelength. It is clear that the curvature ${R}_{\mathrm{r}}^{-1}$ is independent on frequencies of harmonics, which is rational as all harmonics are generated in a region with thickness about the skin depth (sub-10 nm) causing similar shape of spatial distributions. The dependence of ${R}_{\mathrm{r}}^{-1}$ on ξ is introduced by various wavefronts of chirped lasers (dashed) and profiles of parabolic PMs (solid) as shown in figure 1(b). Then, the propagation of reflected harmonics is considered with their radii on PMs w_sn = β_n w_s (β_n ⩽ 1) assumed, where the intensity is 1/e² times of the axial value. In this paper, symbols with subscript n represent the corresponding physical quantities for harmonics whose photon energy E_ph = nɛ₀ and ɛ₀ is the energy of single photon at central wavelength λ₀. Utilizing the well-known q-parameters of Gaussian beams [46], we have ${q}_{\mathrm{s}n}^{-1}={R}_{\mathrm{r}}^{-1}-\mathrm{i}{\lambda }_{0}/n\pi {w}_{\mathrm{s}n}^{2}$ and ${q}_{0n}^{-1}=-\mathrm{i}{x}_{\mathrm{R}n}^{-1}$ for harmonics on the PM and at their focal planes respectively, where x_Rn represents the Rayleigh length of the corresponding harmonic. Considering an equivalent definition of q-parameter as q ≡ x + ix_R [46], q_sn and q_0n satisfy ${q}_{\mathrm{s}n}^{-1}={({q}_{0n}+{x}_{\mathrm{f}n})}^{-1}$ for the free propagation distance x_fn in vacuum between the two positions. After derivation, the expressions of x_fn and x_Rn are given as

$$\begin{equation}{x}_{\mathrm{f}n}=n{\beta }_{n}^{2}\left[\frac{{x}_{\mathrm{f}}}{{\left(1+s\xi \right)}^{2}{x}_{\mathrm{R}0}}-{b}_{0}\right]{x}_{\mathrm{R}n},\end{equation} \tag{ 3 }$$

$$\begin{equation}{x}_{\mathrm{R}n}=\frac{n{\beta }_{n}^{2}{x}_{\mathrm{R}0}\left[1+{x}_{\mathrm{f}}^{2}/{(1+s\xi )}^{2}{x}_{\mathrm{R}0}^{2}\right]}{1+{n}^{2}{\beta }_{n}^{4}{\left[{x}_{\mathrm{f}}/{(1+s\xi )}^{2}{x}_{\mathrm{R}0}-{b}_{0}\right]}^{2}}.\end{equation} \tag{ 4 }$$

According to equation (4) and the definition of divergences ${\theta }_{n}\equiv \sqrt{{\lambda }_{0}/n\pi {x}_{\mathrm{R}n}}$, we get the divergences of harmonics, which vary for different ξ, as

$$\begin{equation}{\theta }_{n}={\theta }_{0}\sqrt{\frac{{(n{\beta }_{n})}^{-2}+{\beta }_{n}^{2}{\left[{x}_{\mathrm{f}}/{x}_{\mathrm{R}0}{\left(1+s\xi \right)}^{2}-{b}_{0}\right]}^{2}}{1+{x}_{\mathrm{f}}^{2}/{x}_{\mathrm{R}0}^{2}{\left(1+s\xi \right)}^{2}}},\end{equation} \tag{ 5 }$$

where ${\theta }_{0}\equiv \sqrt{{\lambda }_{0}/\pi {x}_{\mathrm{R}0}}$ is the divergence of incident laser for central wavelength.

Further, as refocused harmonics propagate a distance x_far (much larger than x_Rn and x_fn ) from PMs to far-field observation points on the X'-axis, the far-field harmonic intensity ${I}_{n}^{\text{far}}\approx {I}_{n}^{\text{focus}}/[1+{({x}_{\text{far}}/{x}_{\text{R}n})}^{2}]$ where ${I}_{n}^{\text{focus}}$ represents the intensity of the reflected harmonic at its focus. Similarly, the near-field harmonic intensity on PMs ${I}_{n}^{\text{near}}={I}_{n}^{\text{focus}}/\left[1+{\left({x}_{\mathrm{f}n}/{x}_{\mathrm{R}n}\right)}^{2}\right]$. Therefore, ${I}_{n}^{\text{far}}\approx {I}_{n}^{\text{near}}[1+{({x}_{\mathrm{f}n}/{x}_{\mathrm{R}n})}^{2}]/[1+{({x}_{\text{far}}/{x}_{\mathrm{R}n})}^{2}]$. Assuming ${I}_{n}^{\text{near}}={I}_{0}{\eta }_{n}(\xi )\mathrm{exp}\left(-4\,\mathrm{ln}\,2{\xi }^{2}/{\tau }_{\mathrm{L}}^{2}\right){w}_{0}^{2}/{w}_{\mathrm{s}}^{2}$ and considering equations (3)–(5) as well as the far field condition x_far ≫ x_Rn , we obtain

$$\begin{equation}{I}_{n}^{\text{far}}\approx \frac{{I}_{0}{w}_{0}^{2}{\eta }_{n}(\xi )}{{x}_{\text{far}}^{2}}\mathrm{exp}\left(-4\,\mathrm{ln}\,2\frac{{\xi }^{2}}{{\tau }_{\mathrm{L}}^{2}}\right){\beta }_{n}^{2}(\xi ){\theta }_{n}^{-2},\end{equation} \tag{ 6 }$$

where I₀ and η_n (ξ) represent the focal intensity of incident lasers and generation efficiency of harmonics, respectively. As ${I}_{n}^{\text{far}}\propto {\theta }_{n}^{-2}$, the minimum θ_n should be achieved exactly at the peak laser cycle (ξ = 0) to take full advantage of divergence gating. Specifically, for optimal x_f = b₀ x_R0, we get ${R}_{\mathrm{r}}^{\text{opt}}$ and ${\theta }_{n}^{\text{opt}}$ as

$$\begin{equation}{R}_{\mathrm{r}}^{\text{opt}}=\frac{{b}_{0}+{(1+s\xi )}^{2}{b}_{0}^{-1}}{1-{\left(1+s\xi \right)}^{2}}{x}_{\mathrm{R}0},\end{equation} \tag{ 7 }$$

$$\begin{equation}{\theta }_{n}^{\text{opt}}={\theta }_{0}\sqrt{\frac{{\left[n{\beta }_{n}(\xi )\right]}^{-2}+{\beta }_{n}^{2}(\xi ){b}_{0}^{2}{\left[1-{\left(1+s\xi \right)}^{-2}\right]}^{2}}{1+{b}_{0}^{2}{(1+s\xi )}^{-2}}}.\end{equation} \tag{ 8 }$$

In this situation, foci of harmonics generated at the peak laser cycle (ξ = 0) are on the PMs as x_fn = 0 according to equation (3), indicating that the convex wavefront of the incident laser exactly offset the focusing effect of concave PMs within the central optical period.

As shown in equation (7), the flat near-field wavefronts $({R}_{\mathrm{r}}^{\text{opt}}=\infty )$ of reflected pulses are realized when ξ = 0, which indicates the minimum divergence and lowest decay during propagation comparing to harmonics generated at other cycles with either concave $({R}_{\mathrm{r}}^{\text{opt}}< 0)$ or convex $({R}_{\mathrm{r}}^{\text{opt}} > 0)$ wavefronts on PMs. Considering APs are generated efficiently once a cycle $\sim {T}_{0}\equiv {\lambda }_{0}/c$ for oblique incidence, we have ${\theta }_{n}^{\text{opt}}(\pm {T}_{0})\sim {\theta }_{0}\sqrt{[{n}^{-2}{\beta }_{n}^{-2}(\pm {T}_{0})+4{\beta }_{n}^{2}(\pm {T}_{0}){b}_{0}^{2}{s}^{2}{T}_{0}^{2}]/(1+{b}_{0}^{2})}$ for small sT₀ ≪ 1, which is much bigger than the divergence of harmonics at the peak cycle ${\theta }_{n}^{\text{opt}}(0)={\theta }_{0}/n{\beta }_{n}(0)\sqrt{1+{b}_{0}^{2}}$. Defining ${{\Gamma}}_{n}(\xi )\equiv {\beta }_{n}^{2}(\xi ){\theta }_{n}^{-2}$, we get the effect of divergence gating as

$$\begin{equation}\frac{{{\Gamma}}_{n}(0)}{{{\Gamma}}_{n}(\pm {T}_{0})}=1+4{n}^{2}{\beta }_{n}^{4}(0){b}_{0}^{2}{s}^{2}{T}_{0}^{2}\end{equation} \tag{ 9 }$$

which is independent on τ_L for a fixed chirp parameter.

Finally, as the satellite harmonics can be ignored compared with the one generated at the peak laser cycle when ${I}_{n}^{\text{far}}(0)/{I}_{n}^{\text{far}}(\pm {T}_{0})\sim {{\Gamma}}_{n}(0)/{{\Gamma}}_{n}(\pm {T}_{0})\;\gtrsim \;{e}^{2}$, we get the critical photon energy of filters ${E}_{\text{ph}}^{\mathrm{min}}\sim 1.26{\varepsilon }_{0}/s{b}_{0}{T}_{0}{\beta }_{n}^{2}(0)$ to produce far-field IAPs through divergence gating. Here, e is the base of the natural logarithm. Considering that influences of temporal envelope $\propto \mathrm{exp}(-4\,\mathrm{ln}\,2{\xi }^{2}/{\tau }_{\mathrm{L}}^{2})$ and generation efficiency η_n (ξ) on ${I}_{n}^{\text{far}}(0)/{I}_{n}^{\text{far}}(\pm {T}_{0})$ have been neglected during theoretical analysis for simplicity, the requirements of ${E}_{\text{ph}}^{\mathrm{min}}\enspace$ can be further relaxed as shown in simulations. We note that there is astigmatism for reflected pulses from parabolic PMs when α ≠ 0 as harmonics satisfy 1/R_i − 1/R_r = cos α/f_p on the sagittal plane [48]. As a result, the corresponding optimal defocused distance x_f = b₀ x_R0 cos²  α, whose relative error comparing to the one on the meridional plane is smaller than 0.1 when α < 18° and can be neglected. Besides, the chirp induced by plasma surface deformation hardly affects our scheme as it does not influence the focusing properties of PMs.

3D numerical simulations are carried out with time-domain near-to-far-field transformation (TDNFFT) method [49, 50] to verify our theory where particle-in-cell (PIC) code 'EPOCH' [51] is used to provide near-field data for it. In simulations, except noted in text, the Gaussian-like linear-polarized chirped laser (orange hollow column) with λ₀ = 800 nm (ɛ₀ ≈ 1.55 eV), w₀ = 2 μm, τ_L = 8 fs ∼ 3T₀, Δω = 0.5ω₀ $(s\approx 0.162{T}_{0}^{-1})$, ϕ₀ = 0 and peak intensity I₀ ≈ 1.92 × 10²¹ W cm⁻² (a_L = 30) at focus is incident along X-axis from the right boundary as shown in figure 1(a). The quadratic polynomial phase factor about ξ in equation (1) is introduced to PIC simulations to initialize the chirp. According to reference [45], the influence of temporal envelope on the formula of incident beams considered in simulations can be neglected.

Irradiated by lasers with incident angle α = 15° and defocused distance x_f, the bulk density n₀ and thickness d of the thin foil (grey block) are 100n_c $({n}_{\mathrm{c}}={m}_{\mathrm{e}}{\omega }_{0}^{2}/4\pi {e}^{2})$ and 0.5λ₀ respectively. Clearly, for the peak cycle (ξ = 0), the laser intensity acting on plasma surfaces ${I}_{\mathrm{s}0}={I}_{0}/\left(1+{x}_{\mathrm{f}}^{2}/{x}_{\mathrm{R}0}^{2}\right)$. Before the bulk, the preplasma with exponential density scale L = 0.1λ₀ is set. As the interaction between the beam and preplasma is dominant and the width of interaction region is about the skin depth, the small thickness of plasma bulk is chosen to save computational resources whose rationality has been verified [52]. In addition to computationally expensive 3D PIC simulations which are performed with limited numerical resolution (illustrated in figure 1 with simulation time step Δt ≈ T₀/300), high-precision two-dimensional (2D) PIC simulations are also carried out (Δt ≈ T₀/2263) for figures 3–5, where the 3D far-field results are calculated through TDNFFT method based on axisymmetric approximations. We note that all far-field results are calculated in 3D geometries except for figure 3(f) where the result in 2D situation is shown for the comparison. The size of 3D (2D) PIC simulation box x × y × z (x × y) is 8.5λ₀ × 25λ₀ × 20λ₀ (10λ₀ × 35.5λ₀), containing 850 × 2500 × 2000 (8000 × 28 400) cells with 2 (16) and 1 (8) superparticles per cell for electrons and immobile ions, respectively. The hole-boring effect is negligible in our scheme. All of far-field results are calculated after reflected pulses propagating a distance x_far = 10⁵ λ₀.

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In figure 1(a) where x_f = b₀ x_R0 ∼ 23.31λ₀, I_s0 ≈ 7.97 × 10²⁰ W cm⁻², curvatures of chirped incident laser change for different optical cycles due to the time-varying wavelengths (colorful solid), which is indicated by its spatial–temporal distribution of intensities (orange shadow). Such various spatial distributions cause variation in the focal length f_p of the optically-curved PM, resulting in wavefronts (green solid) of near-field reflected laser change significantly for different optical cycles. As shown in the intensity distribution (purple shadow), the flat reflected wavefront is obtained only at the peak cycle ξ = 0, which meets our expectation and indicates the realization of divergence gating. Clearly, the IAP (red solid) is obtained in the far field after passing through a proper filter (brown cylinder, ${E}_{\text{ph}}^{\mathrm{min}}=10\enspace \mathrm{e}\mathrm{V}$) while the attosecond pulse train (APT, purple solid in figure 1(d)) is obtained in near field with the same filter used for the near-field reflected pulses (purple solid in figure 1(a)). Quantitatively, as illustrated in figure 1(c), the numerical results (red cycles) of curvature $1/{R}_{\mathrm{r}}^{\text{opt}}$ are consistent with the theoretical predictions (black solid) for different cycles, calculated from wavefronts (green solid) in figure 1(a) and equation (7) respectively. We note the varying distances between circles are introduced by the chirp.

Figure 2(a) illustrates the theoretical θ_n for the same case as figure 1(a), where β_n is assumed to be 0.9 for n = 5, 10 and 20 (black, blue and red). Because the approximate values of ${\theta }_{n}^{\text{opt}}(\xi )\sim {\theta }_{0}\sqrt{[{n}^{-2}{\beta }_{n}^{-2}(\xi )+4{\beta }_{n}^{2}(\xi ){b}_{0}^{2}{s}^{2}{\xi }^{2}]/(1+{b}_{0}^{2})}$ represented by crosses are close to the corresponding precise theoretical values calculated from equation (8) (solids) when |ξ| ⩽ T₀, equation (9) is valid and capable to evaluate ${E}_{\text{ph}}^{\mathrm{min}}\enspace$ required by divergence gating. Besides, the lines and crosses confirm that the minimum divergences are obtained at the peak cycle (ξ = 0) as expected. Here, the chirped parameter $s\approx 0.162{T}_{0}^{-1}$ is determined by Δω and τ_L as shown in figure 2(b). The influence of incident angle α on divergence gating is considered in figure 2(c). To obtain far-field IAPs, the requirement of ${E}_{\text{ph}}^{\mathrm{min}}\enspace$ is increased for bigger α which means more narrow spectra as well as less energy are preserved. Therefore, although the most efficient RHHG is achieved when α ∼ 55° [53], α = 15° is chosen to balance the generation efficiency and isolation of APs with astigmatism considered at the same time. Further, figure 2(d) illustrates the dependence of Γ_n (0)/Γ_n (±T₀) on E_ph and s for α = 15°. Clearly, IAPs are predicted to be produced when ${E}_{\text{ph}}^{\mathrm{min}}\sim 10\enspace \mathrm{e}\mathrm{V}$ for $s\approx 0.162{T}_{0}^{-1}$ and β_n = 0.9. The same β_n is set for all harmonics to show the effects of divergence gating briefly in theory and the value 0.9 is chosen based on the simulation result of harmonics contained in the peak AP.

As illustrated in figures 3(a) and (b) where α = 0° (I_s0 ≈ 7.60 × 10²⁰ W cm⁻²) and 15° (I_s0 ≈ 7.97 × 10²⁰ W cm⁻²) respectively, the unchirped incident laser with the same intensity as the chirped one can only generate ATPs (blue solids) on the axis when harmonics whose E_ph < 10 eV are filtered out. Corresponding to our theoretical predictions, once divergence gating is realized by chirped lasers, the intensity distributions of reflected APs (red solids, ${E}_{\text{ph}}^{\mathrm{min}}=10\enspace \mathrm{e}\mathrm{V}$) show that satellite pulses can be ignored if their emission intervals with the peak AP approach one cycle as their intensities are lower than ${I}_{\text{AP}}^{\text{peak}}/{e}^{2}$ (red dashed). Here, ${I}_{\text{AP}}^{\text{peak}}$ represents intensity of the peak AP and all far-field results are obtained when x_far = 10⁵ λ₀. However, as APs are generated each half cycle for normal incidence, intense far-field IAP is only obtained for α = 15° with intensity and duration about 10¹⁶ W cm⁻² and 40 as respectively, as shown in figure 3(b). The little higher ${I}_{\text{AP}}^{\text{peak}}$ for cases using chirped lasers is caused by slightly smaller divergences. Comparisons with cases using TL pulses with the same bandwidth Δω = 0.5ω₀ are discussed in the appendix. Besides, for the case shown in the top half of figure 3(b), its spatial–temporal distribution on z–t plane (y = 0) in the far field is illustrated in figure 3(c), where near-axis IAP is obtained with total angular spread reaching 12 mrad (|z| ≲ 600λ₀ when x_far = 10⁵ λ₀). Therefore, the energy of it is about 1.5 mJ and its peak intensity can be boosted to 10²⁰–10²¹W cm⁻² after refocused to micrometer scales. Besides, the curved wavefronts of APs indicate their arriving time varies with transverse positions of the observation plane, caused by the natural spherical far-field wavefront of the Gaussian beam whose radius approximates to x_far.

The time-frequency analysis of APs obtained from chirped (the top half) and non-chirped driving lasers (the bottom half) are presented in figures 3(d) and (e), which verify that divergence gating greatly suppresses intensities of satellite APs while hardly affect the peak one as expected. What's more, to compare with 3D far-field results shown in figure 3(b), the 2D result (red solid) for chirped laser–plasma interactions (same parameters as the case represented by the red solid in figure 3(b)) is illustrated in figure 3(f). As expected, a weaker effect of divergence gating is observed in the 2D situation where the APT instead of IAP is obtained while intensities of satellite pulses are also suppressed comparing to the 3D results for unchirped driving pulses (blue solid in figure 3(b)).

Our scheme works for a broad range of parameters as illustrated in figures 4(a)–(c) where filters with ${E}_{\text{ph}}^{\mathrm{min}}=10\enspace \mathrm{e}\mathrm{V}$ are used. Figure 4(a) shows that with the increase of band width Δω, the intensity rate between the strongest and second strong APs I_rate (red circle) improves while the duration τ_AP of peak APs (blue triangle) keeps constant $\sim 40\enspace \mathrm{a}\mathrm{s}$, consistent with our theoretical predictions shown in figure 2(d). Besides, the smallest bandwidth ∼0.35ω₀ is required to obtain IAP in the far field. Figure 4(b) shows the relations of L with I_rate and IAPs can be obtained for 0.075 ≲ L/λ₀ ≲ 0.125 which is feasible in experiments nowadays [54]. We note that the decline of the performance of divergence gating for L > 0.1λ₀ stems from the decreased harmonic generation efficiency as well as worse optical properties of PMs because of instability [55]. Furthermore, there is about 50% probability to generate IAPs for random ϕ₀, i.e. random CEP, as illustrated in figure 4(c), which is capable to perform pump-probe experiments requiring IAPs without the stabilization of CEP with the help of CE tagging technology [18, 56, 57]. We note that such result is three times of those obtained from reference [18] where unchirped three-cycle lasers and filters with ${E}_{\text{ph}}^{\mathrm{min}}\enspace$ as high as 35 eV are used.

More interesting, the far-field IAP (blue solid) is obtained without filters when ϕ₀ = −0.1π, τ_L = 8 fs and Δω = 0.5ω₀ $(s\approx 0.162{T}_{0}^{-1})$ as shown in figure 4(d). Due to all harmonics are preserved, the intensity and energy of the IAP (τ_AP = 49 as) reach 1.8 × 10¹⁶ W cm⁻² (1.1 × 10¹⁸ W sr⁻¹) and mJ, respectively. Finally, in the bottom half of the same figure, we present that the divergence gating is valid for lasers with long pulse duration as τ_L ≈ 16 fs whose bandwidth Δω = 0.5ω₀ $(s\approx 0.084{T}_{0}^{-1})$. Through increasing the ${E}_{\text{ph}}^{\mathrm{min}}\enspace$ to 20 eV based on the prediction in figure 2(d), the far-field IAP (red solid, ∼4 × 10¹⁴ W cm⁻² or 2.5 × 10¹⁶ W sr⁻¹) is generated with little longer τ_AP ≈ 72 as. Here, intensities of incident beams are the same as the one used in figure 1 and ϕ₀ = 0. Further simulations (not shown) confirm that the divergence gating also works for incident laser intensities in 10²⁰ W cm⁻² which is achievable for 100 TW class facilities. What's more, according to the theoretical prediction of equation (9) and the estimation based on high-precision 1D simulations, IAPs in the water-window regime (282–533 eV) can be obtained through divergence gating with such laser intensities irradiated on the PM.

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In summary, we have proposed a scheme called divergence gating realized by defocused chirped lasers to obtain IAPs in far field. Theoretical analysis and numerical simulations show that the divergence of the reflected peak AP is much smaller than others when solid targets are placed at a position behind the focal plane of the incident laser with distance x_f = [4πL cos α/λ₀(1 + ⁻¹)]x_R0. As the much lower decrease of on-axis intensity, far-field IAPs can be obtained after propagation, if filters satisfy ${E}_{\text{ph}}^{\mathrm{min}}\sim 1.26{\varepsilon }_{0}/s{b}_{0}{T}_{0}{\beta }_{n}^{2}(0)$. 3D simulation results show that our scheme is robust and bright sub-50 as mJ-level IAPs are obtained in the far field without the requirement of stable CEP by chirped lasers with intensities of 1.92 × 10²¹ W cm⁻². Besides, our scheme can produce intense IAPs even without filters, whose intensities reach 1.8 × 10¹⁶ W cm⁻² (1.1 × 10¹⁸ W sr⁻¹). Through divergence gating, generated mJ-level IAPs with broad spectrum and intense photon flux benefit the metrology and utilization of IAPs [58] greatly, especially for attosecond-pump-attosecond-probe experiments.

Y Zhang thanks L Li for the helpful discussion. This work is supported by the Science Challenge Project, No. TZ2018005; the National Natural Science Foundation of China, Grant Nos. 12135001, 11921006 and 11825502; the Strategic Priority Research Program of the Chinese Academy of Sciences, Grant No. XDA25050900. B Qiao acknowledges support from the National Natural Science Funds for Distinguished Young Scholars, Grant No. 11825502. Y Zhang acknowledges support from the China Scholarship Council for Visiting PhD Students, File No. 202106010198. The simulations are carried out on the Tianhe-2 supercomputer at the National Supercomputer Center in Guangzhou.

The data generated and/or analysed during the current study are not publicly available for legal/ethical reasons but are available from the corresponding author on reasonable request.

For the typical chirped incident laser used in this paper (I₀ ≈ 1.92 × 10²¹ W cm⁻², τ_L = 8 fs, Δω = 0.5ω₀), its intensity can be increased to 6.53 × 10²¹ W cm⁻² with duration decreased to 2.3 fs after compressed to the TL beam [46]. Utilizing such TL pluses, 2D PIC simulations are carried out with far-field results calculated in 3D geometries. The far-field on-axis intensity distributions of cases where TL pulses interact with targets away from their focal planes (x_f = b₀ x_R0 ≈ 23.68λ₀, I_s0 ≈ 2.66 × 10²¹ W cm⁻²) or at foci are illustrated in figures 5(a) and (b), respectively. Other simulation parameters are the same as those used in the case illustrated by the red line of figure 3(b).

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Clearly, because of larger divergences caused by optically-curved PMs, the case using the focused TL pulse (figure 5(b)) generates much weaker far-field APs than those using proper defocused pulses no matter they are TL beams (figure 5(a)) or not (figure 3(b)) [36]. Besides, for defocused situations, although the IAP generated by the TL beam (figure 5(a)) is intenser due to higher incident intensity, its isolation property (I_rate ≈ 12) is worse than the one obtained through the divergence gating, illustrated by red solids (I_rate ≈ 16) in figure 3(b) and it is very challenging to compress such incident lasers to the TL situation (as short as 2.3 fs) in experiments.