1. Introduction

Ongoing development in isolated attosecond pulse generation pushes forward the field of attosecond science where ultrafast dynamics in atoms molecules and solids are investigated and controlled in the time domain [1, 2]. Isolated attosecond pulses are produced by temporal gating and spectral filtering of high-order harmonics. Several gating processes have been demonstrated, including amplitude gating using one [3,4] or multiple colors driving fields [5,6], polarization gating [7, 8], ionization-induced phase-mismatch gating [9, 10], and ionization-induced saturation [11] as well as combinations of the schemes [12]. Research in attosecond pulse generation has been focusing on decreasing the pulse-width of the generated pulse [13]. While a scheme for generation of temporally-shaped attosecond pulses has been recently proposed [14], the control over spatial properties of attosecond pulses has not been investigated yet. In the femtosecond regime, on the other hand, there has been a lot of research on shaping the spatial properties of the beam, which facilitates new applications of femtosecond pulses. A fascinating example is the recently discovered optical Airy beam in which the transverse electric field profile is described by an Airy function [15]. Airy beams exhibit intrigue properties including non-diffraction, self-bending and self-healing [15–17] that facilitate new applications [18]. For example, curved plasma filaments that are induced by the self-bending beams allow longitudinal resolution in remote spectroscopy applications [19]. To-date, Airy beams were produced by using non-trivial phase masks [16] or through second harmonic generation with intricate quasi-phase matching pattern in the visible and infrared spectral regions [20].

Here, we propose a scheme for producing attosecond pulses with sophisticated spatio-spectral waveforms. Spatial and spectral profiles of a seed attosecond pulse are shaped and its central frequency is up-converted through interaction with an infrared pump laser pulse in a gas of atoms or ions with large binding potential. The pump field, which is too weak to release electrons to the continuum, amplifies the energy of electrons that were ionized by the seed pulse through single photon absorption. An x-ray attosecond pulse is emitted when the energetic electrons recombine with their parent ions. At maximum amplification, the x-ray emission displays Airy spectrum with huge bandwidth which corresponds to an attosecond pulse with a flat-top pulse-shape. Generation of attosecond pulses with fast leading and trailing edges are also shown. We also present settings for generation of attosecond pulse beams with sophisticated spatial waveforms. When the source pulse is focused to a region in which the intensity of the pump beam has a linear slope, the produced pulse exhibits Airy profiles in both spectrum and the transverse coordinates. This newly discovered “spatio-spectral Airy beam” displays prismatic effect where each spectral component propagates along a different curved path. This effect can be utilized for fine spectral tuning of the attosecond pulse by a slit that is located up stream of the nonlinear medium. In addition, we show that a pump beam with a trough intensity profile induces a spatial parabolic phase profile in the produced attosecond pulse beam. As a result the beam auto focuses to a nano-scale hot spot at a controlled propagation distance.

Isolated attosecond pulses are produced through the process of high harmonic generation (HHG). In HHG, an electron is first excited to the continuum through tunneling ionization, then propagates under the influence of the driving laser field, and finally re-combines with its parent ion while emitting a high-energy photon [21]. The source for the short-wavelength radiation is associated with the electron acceleration to high kinetic energy during its re-collision path. The re-collision energy is proportional to the ponderomotive energy of the driving laser U_p ∝ Iλ², where I and λ are the laser intensity and wavelength, respectively [21]. In most HHG experiments, the ionization and propagation steps are driven by a single laser field. As a result, the high frequency emission is dominantly produced by the spatio-temporal maxima in the field of the driving laser pulse. This coupling complicates any attempt to engineer the waveform of the produced attosecond pulses.

We propose to generate sophisticated attosecond pulses by employing a temporal or spatiotemporal gating in the ionization step of the HHG process. The gating can be realized by a seed attosecond pulse [22–24]. In this process, the seed pulse induces a transition to an excited level from which tunneling ionization by the optical field becomes significant. Subsequently, the motion of the electrons in the continuum is dominantly determined by the optical field. The electrons may recombine with their parent ions while emitting HHG radiation in the same fashion as in HHG that is driven by a single field. Importantly, the enhancement in the ionization rate also dramatically increases the efficiency of HHG as was demonstrated experimentally in several works [23, 25–27]. In this work, we implement the ionization gating by a seed attosecond pulse for generating spatially, temporally and spectrally sophisticated attosecond pulses. In our technique, a seed attosecond pulse confines the ionization step, and therefore also the generation process, into a limited region in space and time. In this way, the properties of the generated attosecond pulse are controlled by the spatiotemporal profile of the mid-IR beam at that region. For example, the phase front of the generated pulse beam resembles the transverse intensity profile of the mid-IR beam. This correspondence results from the fact that in HHG, the emitted phase of a high-order harmonic is approximately linear with the intensity of the laser that accelerates the electron during its trajectory in the continuum [28, 29]. This relation was used for manipulating the quadratic phase fronts of diverging HHG beams [30, 31]. The ionization by a single photon from the EUV attosecond pulse induces de-coupling between the first and second steps of the HHG process which enables us to significantly extend this manipulation. The usage of mid-IR pulses with non-standard (non bell-shape) intensity profiles in the region of significant ionization can lead to generation of attosecond pulse beams with sophisticated waveforms.

2. Scheme for producing sophisticated attosecond pulses

We first present an example for generation of temporally-sophisticated attosecond pulses. We consider a mid-IR driving field $E^{\text{IR}}=f^{\text{IR}}\left(t\right)\text{cos}\left(\frac{2\pi c}{{\lambda }^{\text{IR}}}t\right)$ and an extreme ultraviolet (EUV) isolated attosecond pulse $E^{\text{EUV}}=f^{\text{EUV}}\left(t-\tau \right)\text{cos}\left(\frac{2\pi c}{{\lambda }^{\text{EUV}}}t\right)$ that are jointly focused onto a medium of singly ionized helium ions (ionization potential is 54.4 eV) (Fig. 1a). Here λ^IR = 2μm, λ^atto = 30 nm and c is the speed of light. The temporal envelopes, f^IR (t) and f^EUV (t – τ) are Gaussians with full width at half maximum (FWHM) of 50 fs and 250 as, respectively, and peak intensities of 7.4 · 10¹⁴ W · cm⁻² and 10¹² W · cm⁻², respectively. The parameter τ denotes the time delay between the pulses which in the first example is set to −3.06 fs (+15° after a peak of the mid-IR field). This time delay was chosen so electrons that are ionized by the attosecond pulse would emit the highest photon-energy upon recombination (the cut-off radiation). The motion of the electron under the influence of the mid-IR and EUV fields is calculated by solving the one dimensional time dependent Schrödinger equation for the potential V (x,t) = V_bind(x) + x (E^IR + E^EUV). The ion binding potential is modeled by $V_{\text{bind}}\left(x\right)=-54.4/\sqrt{{\left(x/a_0\right)}^2+0.5}\mathit{eV}$, having an ionization potential of 54.4 eV (a₀ is Bohrs radius) [32, 33]. Ionization rate is obtained by calculating the low frequency outgoing flux. Importantly, the parameters are chosen such that ionization yield by the mid-IR field alone is very low (10⁻⁷) whereas ionization is enhanced by the EUV field (Fig. 1a). The high-order radiation is calculated by the acceleration expectation value, using Ernfest theorem $a\left(t\right)\propto 〈\psi \left(t\right)\left|\left(\frac{d}{\mathit{dx}}V\left(x,t\right)\right)\right|\psi \left(t\right)〉$, where ψ is the electronic wavefunction. As shown in Fig. 1b, the radiated spectral field highly resembles an Airy function, $\tilde{E}\left(\omega \right)\sim \mathit{Ai}\left[\left(\omega -{\omega }_{\text{cut}-\text{off}}\right)/\sqrt[3]{\alpha /2}\right]$ with α = 1600 fs⁻³ is the chirp quadratic coefficient and ω_{cut off} = 930 eV is the cut-off frequency as predicted by the classical model [21]. While it is known that the spectral field at the cut-off region exhibits an Airy profile [34], the ionization gating by a seed attosecond pulse leads to generation of an Airy spectral field over a very large bandwidth. The corresponded emitted field exhibits a flat-top amplitude and cubic phase at some time interval $E\left(t\right)={ℱ}^{-1}\left[\tilde{E}\left(\omega \right)\right]{\tilde{E}}_0\text{exp}\left[i\left({\omega }_{\text{cut}-\text{off}}\left(t-t_{\text{cut}-\text{off}}\right)-\frac{\alpha }{6}{\left(t-t_{\text{cut}-\text{off}}\right)}^3\right)\right]$. t_cut-off corresponds to the emission time of the cut-off radiation, and the amplitude, E₀, is time-independent because the seed EUV pulse populates uniformly the quantum trajectories that recombine around the cut-off time. Notably, the spectral field, Ẽ(ω), is real (Fig. 1b) because the emitted field in time domain obeys E (t – t_cut-off) = E^* (t_cut-off – t) in the vicinity of t_cut-off.

 

Fig. 1 (color online) Frequency up-conversion of attosecond pulses. (a) Driving field (solid line) which consists of a mid-IR (λ_IR = 2μm) laser field and an EUV pulse of 250 attosecond duration (λ_EUV = 30 nm). When the joint field interacts with He⁺ medium, ionization (dashed blue) is gated by the attosecond pulse since the mid-IR field is too weak to independently ionize the medium. (b) Emitted spectral field amplitude (blue) and phase (green) as a function of photon energy. The spectral field almost perfectly matches an Airy function (dashed black). x-ray attosecond pulses are produced when the lower part of the spectrum is filtered out. (c) Pulse width of the produced x-ray attosecond pulse versus the bottom limit of the high pass filter. Blue, green and red circles mark 920 eV, 900 eV and 800 eV spectral filters respectively. (d) x-ray attosecond pulses produced by high-pass filtration at 920 eV (blue), 900 eV (green) and 800 eV (red), where multiple spectral lobes result in a flat-top pulse (phase is marked by dashed red). (e–f) Delaying the EUV pulse by +5° and −5° with respect to the mid-IR driver, result in a trailing edge and leading edge pulses, respectively.

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Figures 1c and 1d present results concerning the temporal shape of the generated pulse when the spectral Airy field (Fig. 1b) is filtered by a high-pass filter. A 250 attoseconds (FWHM) transform limited pulse corresponds to the single (and largest) lobe of the Airy spectrum at the cut-off (∼ 930 eV). Figure 1c shows that the pulse-width increases when more lobs are included. This results from the π phase jump between adjacent lobs of the Airy field. Moreover, as the number of spectral lobs increases, the pulse temporal shape becomes squarish (flat-top) (Fig. 1d). Finally, figures 1e and 1f show the pulse shapes for additional +0.1 and −0.1 femtoseconds time-delays, respectively (+5° and −5° shift in the phase of the mid-IR wave). For the large bandwidth case, the pulse shape exhibit a triangle (leading or trailing edges) pulses. The asymmetry emerges from uneven population of the quantum trajectories set to recombine before and after t_cut-off. The square and triangle pulses demonstrate a simple form for attosecond pulse shaping. More complicated pulses can be produced by using complex structures of ionizing pulses.

Next, we present production of spatially-sophisticated attosecond pulses that are controlled by the intensity profile of the mid-IR beam. We calculated the time and transverse (x-coordinate) dependence of the emitted field by solving the Schrödinger equation separately in each point of the x-grid. The accuracy of the transverse sampling of the generated radiation is guaranteed by using a grid pixel size of 100 nm which is much smaller than the transverse variation scales of the incoming laser fields (several microns). Subsequently, we simulated the propagation of the produced pulse beam along the propagation axis, z, by solving the linear (refractive index is 1), paraxial, and absorption-free Helmholtz equation for each narrow spectral component (intensity is too weak for nonlinear effects and propagation distance is much smaller than the absorption length). Coherent super position of the spectral components gives the space-time dependence of the pulse at propagation distance z: $E\left(x,t,z\right)=\sum _n\tilde{E}\left({\omega }_n,x,z\right)\text{exp}\left[i{\omega }_n\cdot t\right]\Delta \omega$.

The first example, which we term spatio-spectral Airy pulse beam, is presented in figure 2. In this example, the mid-IR and EUV fields of Fig 1a possess transverse (x) profiles of Gaussians with FWHM of 10μm and 150μm respectively. A lateral shift of x₀ = 63.7μm between the peaks of the pulses was introduced such that the EUV pulse resides on an approximately linear mid-IR intensity slope (zero 2^nd derivative) (inset of Fig. 2a). Consequently, in this geometry the local cut-off frequency also grows linearly with x. Thus, the spectral Airy of Fig. 1 becomes x-dependent: $\tilde{E}\left(\omega ,x\right)=Ai\left[\left(\omega -{\omega }_0+\beta \left(x-x_0\right)\right)/\sqrt[3]{\alpha /2}\right]$. Here ω₀ is the cut-off at x₀ and β is the cut-off lateral slope. Indeed, the calculated intensity exhibits spatial and spectral Airy profile (Fig. 2a). In this pulse, each spectral component forms a spatial Airy beam (Fig. 2b) that propagates along a curved path (Fig. 2c). The dependences of the curvature, initial propagation angle and initial position of the peak on the photon-energy (Fig. 2d) disperse the spectral components in space. The consequence of this prismatic effect is shown in figure 3. Figures 3(a–c) show spatiotemporal profiles after 5 mm propagation, formed by a 10 eV band pass filter centered at 608 eV, 578 eV and 548 eV, respectively. As shown, each spectral region undergoes a different lateral shift. As a result, a narrow slit that is located downstream can be used as a tunable spectral filter of the attosecond pulse (Fig. 3d). A linear dependence of the spectral intensity vs. the slit central-position is shown in Fig. 3e. Figure 3 shows that spatio-spectral Airy pulse beam may be used for tunable manipulation and control of the spectrum of attosecond pulses or for spectrometer-less attosecond spectroscopy, where different segments of a sample are irradiated by an attosecond pulse of varying spectrum.

 

Fig. 2 (color online) Spatio-spectral Airy beam. (Inset of plot a) The source EUV attosecond pulse is transversely shifted relative to the peak intensity of the mid-IR beam. (a) Emission exhibits Airy spectrum and Airy intensity profile in the transverse axis. (b) Single spectral component of the emitted field, $E_{\left(x\right)}^{578eV}$, versus the transverse coordinate (solid blue) compared to a Gaussian enveloped Airy function (dash red). (c) Propagation of $E_{\left(x\right)}^{578\mathit{eV}}$. The peak of the beam exhibits parabolic propagation path. (d) Initial propagation angle (θ), parabolic curvature coefficient, and initial position of the main lobe as a function of photon-energy.

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Fig. 3 (color online) Prism effect of spatio-spectral Airy beam. Propagation of attosecond pulses of 10 eV bandwidth centered at (a) 608 eV (b) 578 eV and (c) 548 eV. Plots show the spatio-temporal intensity after 5 mm propagation is of attosecond scale duration and μm width. (d) Schematic illustration for enhanced prism effect utilization by a slit. (e) A 1μm slit located 5 mm after the emission plane act as a source of 7.6 eV bandwidth attosecond pulse, with linear x(ω) dependence.

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We now move to a second example of isolated attosecond pulse production that exhibit sophisticated spatial characteristic. In this example, an isolated attosecond pulse with parabolic phase front is generated where the source EUV pulse overlaps a trough profile of the mid-IR pulse. Consequently, the produced isolated attosecond pulse beam gets focused downstream. Figure 4 shows a concrete example where the EUV (30 nm) source pulse is 10μm wide and centered at the mid-IR (2μm) trough (inset of Fig. 4a). The mid-IR intensity is given by $I_{\text{trough}}^{\text{mid}-\text{IR}}\left(x\right)=I^{\mathit{IR}}\left[A_w\text{exp}\left(-{\left(x/{\sigma }_w\right)}^2\right)-A_n\text{exp}\left(-{\left(x/{\sigma }_n\right)}^2\right)\right]$ where A_w = 1.4, A_n = 0.4, σ_w = 90μm, σ_w = 30μm, and I^IR is the mid-IR intensity previously discussed. Figures 4a and 4b show the intensity and phase of the radiation, clearly exhibiting the parabolic profiles. The parabolic phase front leads to focusing of the produced pulse after 5 mm. Figure 4c shows the propagation and focusing of the 929 eV photon-energy spectral component. The intricate focusing spot details and the extended Rayleigh range result from the π phase jumps at the beam margins (see inset in Fig. 4b). Figure 4d shows the spatio-temporal profile of the focused pulse at the focal point. The pulse consists of a 250 eV spectral bandwidth that is centered at 929 eV and exhibit a sub-micron attosecond focal spot. The focusing distance and width can be manipulated by the profile of the mid-IR beam.

 

Fig. 4 (color online) Generation of auto-focusing x-ray attosecond pulses. (Inset of plot a) The source attosecond pulse beam (blue) is located at a trough of the mid-IR beam (red). Emitted spatial-spectral (a) intensity, and (b) phase exhibiting Airy spectrum with parabolic variation in the transverse coordinate. (b-inset show the phase profile along the dashed line). (c) Propagation of the 929 eV spectral component of the generated radiation demonstrates focusing effect due to spatial quadratic phase. (d) Spatio-temporal spot of generated attosecond pulse at the focal point. The calculation consists of 250 eV bandwidth centered at ĥω = 929 eV.

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We emphasize that the results presented in this paper are a direct consequence of universal features of HHG at the single-atom level, namely, the cut-off relation, the quadratic chirp at cutoff, and the fact that the intrinsic phase grows linearly with the intensity of the driving laser. Nonetheless, propagation effects can influence the process. First, phase matching is crucial for obtaining efficient macroscopic generation. When the wave-mixing process is not phase matched, the generated attosecond pulse builds up only over a propagation distance where the relative phase between the high-order polarization and attosecond pulse slip by radians; the so-called coherence length. Coherence lengths of very high-order harmonics in pre-ionized plasma, as are the numerical examples in this paper, are in the order of several microns [35]. In this regime, grating assisted phase matching can be used for significantly extending the effective region that contributes to the produced signal [36,37]. A potential approach for phase-matching the process is to implement pressure tuning phase matching in initially neutral or partially pre-ionized nonlinear medium [38]. As shown in Ref. 38, mid-IR pump can be phase matched with very short-wavelength HHG light through detailed balance between the atomic, plasma and modal dispersions. The second important propagation effect is volume averaging which, in principle, can average out the sophisticated details of the produced isolated attosecond pulse. Attosecond pulses with sophisticated transvers profiles can exhibit non-trivial dynamics (e.g. the parabolic propagation path of Airy beams). That is, the transverse profiles of these pulses are approximately conserved for only a limited propagation distance (this “shape preserved” length is in the order of 0.4 millimeter for the spatio-spectral Airy beam of Fig. 2). Thus, in order to obtain the aimed sophisticated pulse at the exit of the nonlinear medium, its effective thickness (i.e. coherence length) should not exceed the “shape preserved” propagation length.

3. Conclusions

In this work we propose the first scheme for producing attosecond pulses with sophisticated spatio-temporal and spatio-spectral waveforms. In this scheme, a seed attosecond pulse gates the ionization step of high harmonic generation process, temporally and spatially, and a shaped mid-IR field which accelerates the electrons in the continuum, controls the spatial dependence of the emitted attosecond pulses. Using the proposed scheme, we demonstrated numerically, all for the first time, i) HHG emission with very broad Airy spectrum which corresponds to a flat-top attosecond pulse, ii) spatio-spectral Airy beam which exhibits prismatic self-bending propagation, and iii) an auto-focusing isolated attosecond pulse which focuses to a sub-micron spot without the need of a focusing lens or nonlinearity.