1. INTRODUCTION

The generation and characterization of isolated attosecond pulses have revolutionized time-resolved studies of ultrafast processes such as photo-ionization [1], Auger decay [2], or charge migration [3,4]. Today, the synthesis of attosecond pulses relies mainly on high-harmonic generation in gases driven by femtosecond laser pulses [5,6]. In this process the valence electron of an atom is ionized and driven back to the parent ion periodically by the laser pulse. Upon recombination with the parent ion, it emits high-harmonic radiation [7–9]. As a result of this process, an attosecond pulse is generated in every half-cycle of the driving laser pulse, forming an attosecond pulse train.

Conventional time-resolved spectroscopy techniques rely on the availability of single pump and probe pulses, rather than pulse trains. In the past decade significant work has been devoted to obtaining isolated attosecond pulses, and many different approaches have been developed [6]. These techniques have in common that they require a driving pulse with a very large spectral bandwidth [10], or the combination of two pulses or even more [11,12]. Isolated attosecond pulse generation by such a field is based on at least one quickly varying parameter during the process, which can be the field’s intensity [13], polarization [14], or the ionization of the medium [15]. These temporal gating techniques microscopically apply for all atoms in the medium, thus individual atoms generate isolated attosecond pulses.

The attosecond lighthouse also requires a broad bandwidth pulse, but it differs from most temporal gating techniques as it is based on a macroscopic property—the wavefront rotation of a laser pulse—for separating attosecond pulses generated in each half optical cycle of the laser pulse [16–24]. Although the harmonic radiations form an attosecond pulse train when generated in the medium, separate pulses form distinct beamlets that propagate in different directions due to the wavefront rotation of the laser pulse. Because of this unique approach, the attosecond lighthouse offers two advantageous aspects. First, it provides multiple isolated attosecond pulses, which can be used in attosecond pump and attosecond probe experiments [16,17]. Second, it provides a direct experimental access to the ultrafast temporal dynamics in successive half-cycles [18]. To take these advantages in practice, a clean angular separation of the attosecond pulses is an essential prerequisite.

In the implementation of the attosecond lighthouse, angular dispersion (and pulse front tilt) is introduced into the laser beam. After focusing, this is converted into a spatial chirp, which produces a rotating wavefront in the focal plane. Around the focus, a combination of pulse front tilt and spatial and temporal chirps appears, as shown in Fig. 1. High-order harmonic generation is known to inherit the coherence properties of the driving laser pulse [25]. As a result, harmonics generated with this spatially chirped pulse contain a series of attosecond pulses, one from each half-cycle of the laser pulse, each propagating in the direction perpendicular to the wavefront of the driving field at the moment of generation [17]. This creates the angular separation of attosecond pulses and makes possible the mapping of the far-field spatial profile of the harmonic beam to the near-field temporal structure of the attosecond pulse train [18]. The isolation of a single attosecond pulses can be, then, achieved by simple spatial filtering.

 

Fig. 1. Scheme of spatial chirp generation in a focusing laser beam, and spatiotemporal profiles of the pulse around the focal plane. Note that the negative time (leading edge of the pulse) is shown in the right side of the figures throughout the paper to present the pulses propagating from left to right. The instantaneous intensity of the pulse near the focus is shown in the bottom panels [(a) $-2.5\mathrm{mm}$, (b) $-1.3\mathrm{mm}$, and (c) $+1.3\mathrm{mm}$, where -/+ indicates the position before/after the geometrical focus.]. The pulse is slightly chirped ($-4{\mathrm{fs}}^2$). Results for the gas jet placed in these three selected positions will be discussed throughout the paper. The intensities in the three subfigures are normalized independently.

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To obtain a clean isolated attosecond pulse, the angular separation ${\delta }_{1/2}$ between two consecutive attosecond pulses has to be larger than their divergence ${\delta }_{\mathrm{XUV}}$ (i.e., ${\delta }_{1/2}>{\delta }_{\mathrm{XUV}}$). The angular separation is related to the spatial chirp as ${\delta }_{1/2}=\mathrm{\Delta }\lambda (y)/(2\mathrm{\Delta }y)$ [where $\mathrm{\Delta }\lambda (y)$ denotes the wavelength variation over the beam size $\mathrm{\Delta }y$ along the $y$ direction shown in Fig. 1] [18]. When the wavefronts of the attosecond pulse are optimized to be flat, which may be achieved by properly choosing the position of the gas jet with respect to the focus, the divergence of the attosecond pulses is inversely proportional to the laser beam size (i.e., ${\delta }_{\mathrm{XUV}}$∝$1/\mathrm{\Delta }y$). Consequently, using the linear diffraction theory, the ratio of angular separation and attosecond pulse divergence (${\delta }_{1/2}/{\delta }_{\mathrm{XUV}}$) should be limited by the bandwidth of the laser pulse.

In this work we show that propagation effects in a highly ionizing medium can dynamically increase the angular separation of individual attosecond pulses. The attosecond lighthouse has been featured in a number of previous works [17,18,24,26], but the mechanism of the angular separation in a highly ionizing medium has not been explored, to the best of our knowledge. Here we present experimental and numerical results of attosecond lighthouse experiments in Ne and ${\mathrm{N}}_2$, and show that the choice of a generating medium, focusing geometry, and laser intensity affects the angular separation of produced attosecond pulses. This dynamic wavefront rotation enables the separation of harmonic beamlets even in the case when the conditions derived from linear diffraction theory do not allow the angular separation.

2. METHODS

A. Experimental Setup

In order to demonstrate how the attosecond pulses are produced in the attosecond lighthouse, we generated high-order harmonics in Ne and ${\mathrm{N}}_2$, using a driver laser pulse with rotating wavefront [18]. We have used 120 μJ, 5 fs pulses with a spectrum centered at 750 nm, generated in a gas-filled hollow-core fiber and compressed with chirped mirrors. Angular dispersion was introduced into the beam by tilting a 7° fused silica wedge by 20° before focusing with a spherical mirror ($f/45$) into a gas jet released from a nozzle with 250 μm opening diameter. The backing pressure of the gas jet was 4 bar for both Ne and ${\mathrm{N}}_2$. The harmonic spectrum was recorded by an XUV spectrometer, consisting of a diffraction grating with variable line spacing, a microchannel-plate–phosphor-screen assembly, and a charge-coupled device. The vertical entrance slit of the spectrometer was placed 0.3 m after the focus of the laser beam. All spectra are recorded in a single shot to avoid any average effects of the CEP and the intensity fluctuation of the laser pulses.

B. Numerical Model

In our model we used a laser beam with super-Gaussian spatial profile before the focusing mirror and with a super-Gaussian spectrum centered at 1.67 eV with 0.81 eV full width at half-maximum (FWHM), corresponding to a bandwidth of 380 nm centered at 740 nm wavelength. An angular dispersion of $-0.45\mathrm{\mu rad}/\mathrm{nm}$ was introduced into the beam at $f=0.3\mathrm{m}$ before the focusing mirror ($f/40$). The beam was propagated to the entrance of the gas medium using the 2D Huygens–Fresnel integral.

To model the processes in the gas medium we solve the forward Maxwell equation for both the laser and harmonic fields in one transversal dimension [27]. The gas jet was approximated by a medium with Gaussian density distribution having 250 μm FWHM and 66 mbar peak pressure. Nonlinear propagation and harmonic generation were calculated inside a region of 400 μm total length.

In our model the nonlinear source terms for the laser propagation include the optical Kerr effect and plasma dispersion. The source term of the harmonic field is calculated in the frame of strong field approximation [28]. We filtered out long trajectories by applying a temporal window on the generated dipole. This was necessary to get a clear signal, because the lack of diffraction in the second transversal dimension in our model can artificially enhance the relative contribution of long trajectory components in the far field (long trajectories usually possess a large divergence angle). We used the Yudin–Ivanov model to calculate the ionization rate [29]. In the case of ${\mathrm{N}}_2$ the calculated ionization rate is reduced by a factor of 3.3 to account for random molecular orientation [30].

3. RESULTS

A. Optimization of Laser Wavefront

The divergence of the attosecond pulse depends on the wavefront curvature of the generating laser pulse and also on its intensity gradient in a transverse direction. The intensity gradient increases the divergence of the generated beam due to the intensity-dependent harmonic phase, known as atomic phase [31]. This can be compensated for by the converging wavefront of the laser beam, which can be obtained by placing the target before the focus. There is one more aspect to consider in the case of the spatially chirped laser beam. Because of the space–time coupling, the temporal property of the initial beam also affects its spatial property around the focus. By introducing a small amount of negative group delay dispersion (GDD), the laser beam becomes more converging, while the pulse front tilt is reduced before the focus [Fig. 1(b)] and increased after the focus [Fig. 1(c)]. We could optimize the divergence of the harmonic beam by changing the GDD of the laser beam. However, this control enables only fine adjustments. One would still expect a good separation of the beamlets by placing the target before the geometrical focus [Fig. 1(a)].

The angular separation of the beamlets is proportional to the spatial chirp rate, which was maximized by optimizing the initial angular dispersion of the laser beam [26]. This was achieved at 20° tilt of the fused silica wedge, producing $-0.45\mathrm{\mu rad}/\mathrm{nm}$ angular dispersion around the 750 nm central wavelength.

B. Attosecond Lighthouse Under Low-to-Moderate Ionization Condition

To see how the convergence of the laser beam affects the attosecond pulse generation in the attosecond lighthouse, we first generated harmonics in Ne at different target positions. The intensity was $5.9\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$ at the focus. The Ne atoms can withstand the strong field without significant ionization. Based on our model, we estimate that the ionization of Ne stayed below 10% at the end of the laser pulse, regardless of the target position. At the three selected positions we are discussing, our calculations show that the coherence lengths along the medium and across the harmonic spectrum are comparable to or longer than the medium length (250 μm FWHM). Therefore, we do not expect the presence of free electrons to have a dramatic effect on the harmonic generation process.

In this case, the experimental and theoretical results indeed show a good separation when the target is placed before the focus [Figs. 2(a) and 2(d)], making possible the isolation of a single attosecond pulse, as shown in the inset of Fig. 2(d). Closer to the focus [Figs. 2(b) and 2(e)], the divergence of the beamlets slightly increases, and modulations appear in the spectra of the harmonic beamlets, suggesting a slight overlap of the attosecond pulses. Nonetheless, the filtering of a single pulse with good contrast is still possible, as shown in the inset of Fig. 2(e). After the focus [Figs. 2(c) and 2(f)], however, the diverging driver beam and the effect of intensity gradient added up and resulted in diverging beamlets, preventing the effective spatial separation of individual attosecond pulses. These results confirm that the angular separation is clear for harmonics generated in the converging wavefront of the laser beam.

 

Fig. 2. Spatially resolved high-harmonic far-field spectra generated in neon. In (a)–(c) experimental and in (d)–(f) numerical results are shown. The gas jet position is: (a) and (d) 2.5 mm before, (b) and (e) 1.3 mm before, and (c) and (f) 1.3 mm after the geometrical focus. The insets show the temporal intensity profile of the spectral components from the indicated region. This is calculated by the Fourier transform of the complex spectral components encompassing harmonics above 45 eV, and then integrated over 1 mrad propagation angle. The estimated peak intensities were $4.2\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$ in the case shown in (a), and $5.4\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$ in the cases shown in (b) and (c). The GDD of the laser pulse was $-10{\mathrm{fs}}^2$ in the experiment. The experimental spectra were reasonably well reproduced when the GDD of the laser pulse was $-4{\mathrm{fs}}^2$ in the calculation.

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The harmonic spectra generated from Ne are obtained under the condition in which linear effects dominate the laser pulse propagation, not free electron generation. If there is significant ionization, the laser beam can be modified during the propagation in the medium. The nonlinear propagation effects in HHG are usually dominated by plasma dispersion, resulting in phase mismatch and also in the plasma defocusing of a propagating laser pulse and, hence, of the harmonic beam [32,33]. Thus, the condition for spatial separation of harmonic beamlets may change drastically [26]. As a consequence, a systematic investigation on propagation effects has to be performed to properly understand the physical processes in the attosecond lighthouse.

C. Attosecond Lighthouse in a Highly Ionizing Medium

To study the attosecond pulse generation in a highly ionizing medium, we generated high-order harmonics in ${\mathrm{N}}_2$. We have chosen ${\mathrm{N}}_2$ molecules as a target because they produce a significantly higher ionization without the need to change the generating laser intensity too much. Also, they do not exhibit strong spectral dependence on harmonic generation efficiency (such as the Cooper minimum in Ar) that might hinder the spectral features [34]. The estimated ionization is around 65% at the end of the laser pulse when the peak intensity is $4.4\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$ at the focus. Well before the focus [Figs. 3(a) and 3(d)], we observed that the divergence of the harmonic beamlets already increases and the separation is not as clear as in the Ne case [Figs. 2(a) and 2(d)]. Just before the focus [Figs. 3(b) and 3(e)], where the intensity is already high and the macroscopic effect dominates, the angular separation increases due to propagation effects. The first few beamlets shown in the upper side of the spectrum remain strong, and the intensity of the following ones is reduced, while their propagation angle is significantly altered. Even more striking is the fact that narrow beamlets with large angular separation appear after the focus, both in the experiment and in the calculation [Figs. 3(c) and 3(f)]. This is a surprising result because the laser propagation in the highly ionizing medium causes plasma defocusing. Then, the harmonic beam should diverge more, and, thus, the angular separation should be worse than the results obtained without the nonlinear propagation effects.

 

Fig. 3. Spatially resolved far-field high-harmonic spectra generated in ${\mathrm{N}}_2$. In (a)–(c) experimental and in (d)–(f) numerical results are shown. The gas jet position is: (a) and (d) 2.5 mm before, (b) and (e) 1.3 mm before, and (c) and (f) 1.3 mm after the geometrical focus. The insets show the temporal intensity profiles of the spectral components from the indicated regions. This is calculated by the Fourier transform of the complex spectral components encompassing harmonics above 45 eV, and then integrated over a propagation angle of 3.7 mrad. The estimated peak intensities were $3.1\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$ in the case shown in (a) and $4.0\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$ in the cases shown in (b) and (c).

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D. Dynamic Wavefront Rotation

To understand the processes in the highly ionizing target, we analyze in detail the results shown in Figs. 3(d) and 3(f). We first compare the wavefront of the laser beam at the entrance and at the exit of the gas medium, as shown in Figs. 4(a) and 4(b). When the gas jet is placed well before the focus [Fig. 4(a)], the wavefront of the beam becomes slightly more divergent due to the plasma defocusing, resulting in more diverging beamlets in the trailing edge of the pulse, as shown in Fig. 4(e).

 

Fig. 4. Evolution of the laser wavefronts and generated attosecond pulses in a highly ionizing medium. (a) and (b) Wavefronts of the laser pulse at the entrance (black, dashed line) and at the exit (colored solid line) of the ${\mathrm{N}}_2$ gas medium are shown, with parameters corresponding to those used in Figs. 3(d) and 3(f), respectively. The colors in (a) and (b) represent the ionization level at the end of the medium. (c) and (d) Intensity profile of attosecond pulses at the end of the gas medium, synthesized from harmonics above 45 eV. (e) and (f) Corresponding attosecond pulses in the far field. In (b) the dashed black arrows are perpendicular to the initial wavefronts of the field in the corresponding half-cycles. The solid red arrows in (b) and (d) are perpendicular to the laser wavefront at the end of the gas medium. The arrows in (d) are at the same relative positions as in (b). The dotted lines in (e) and (f) illustrate the same region of propagation angles as selected in Figs. 3(d) and 3(f), respectively.

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With the gas jet placed slightly after the focus [Fig. 4(b)], the wavefront curvature is significantly altered due to the plasma defocusing. This makes the first few harmonic beams more divergent, while they still keep their original propagation direction, as seen on the vertically elongated attosecond pulses in the far field, in Fig. 4(f). In the next half-cycles, as the ionization increases, the wavefront gets even more curved. At this point, harmonics are generated under a high ionization condition, which causes phase mismatch, and prevents the buildup of the signal during propagation in the middle of the laser beam, resulting in the splitting of the harmonic beam into two main parts [see the third pulse in Fig. 4(d)]. The upper part of these attosecond pulses is rotated upward, interfering with the preceding ones in the far field, as seen in Fig. 4(f). This creates the strong modulation of the harmonic spectrum shown in Figs. 3(c) and 3(f) (from 1 to 8 mrad of the propagation angle). The lower part, on the other hand, is rotated downward, significantly increasing the apparent angular separation, as illustrated by the arrows in Figs. 4(b) and 4(d). This dynamically induced wavefront rotation explains the enhanced angular separation of the harmonic beamlets shown in Figs. 3(b) and 3(c). This increased separation is a direct signature of the half-cycle variation of the ionization.

The spatiotemporal couplings in the pulse also play a significant role in this process. The sign of the pulse-front tilt is different before and after the focus. Before the focus, the lower part of the beam is more intense at the beginning of the pulse [Fig. 1(a)]. This pulse front tilt causes a higher ionization and produces phase mismatch at the lower part of the beam in the following half-cycles [Fig. 4(c)]. After the focus, however, the upper part of the beam is stronger in the leading edge [Fig. 1(c)], eliminating the emissions in the following half-cycles. In the lower part of the beam, phase-matching is better, and, thus, a stronger signal builds up. This causes the asymmetry observed in the intensity and number of beamlets generated before and after the geometrical focus [compare Figs. 3(b) and 3(c)]. This also shows that the increase of the angular separation of attosecond pulses by dynamic wavefront rotation is effective only at or after the focus due to the direction of the pulse front tilt. It should also be noted that the efficiency of the attosecond pulse generation has not been affected too much. The peak spectral intensity (integrated over 1 eV and 1 mrad range) of the strongest beamlet with the dynamic wavefront rotation [Fig. 3(c)] is only 25% smaller than that obtained without the dynamic wavefront rotation [Fig. 3(a)].

4. DISCUSSION

The angular separation of the attosecond pulses generated at each half-cycle should be large for clear separation. In the experiment, the angular separation of the beamlets was around ${\delta }_{1/2}=2.4\mathrm{mrad}$ in neon, and also in ${\mathrm{N}}_2$ when using a lower intensity pulse (results not shown). At high intensities, separations up to ${\delta }_{1/2}=3.2\mathrm{mrad}$ were observed in ${\mathrm{N}}_2$ when the gas jet was put slightly after the focus, as shown in Fig. 3(c). In neon the separation remained at ${\delta }_{1/2}=2.4\mathrm{mrad}$ even at higher laser intensities, as shown in Fig. 2. This shows that the proper choice of intensity, focusing geometry, and target gas enables the control of the angular separation of attosecond pulses.

In the calculations, an isolated attosecond pulse with less than 10% intense side wings could be generated in ${\mathrm{N}}_2$ before the focus, by using a restricted portion of the beam within 2.35 mrad divergence [case of Fig. 3(d)]. Using harmonics generated after the focus, the same contrast could be achieved with an enlarged area encompassing 3.7 mrad divergence [Fig. 3(f)]. Moreover, in this case the wings appear due to the tilting of the beamlets, showing a slight angular dispersion of the harmonics. This, however, was not observed in the experiment, suggesting that experimentally an even larger portion of the beam could be used for clean attosecond pulse synthesis. The angular dispersion of the beamlets in the lower part of Figs. 3(b), 3(e), and 3(f) is a consequence of the attosecond pulses being generated around the “edge” of the laser beam where the intensity gradient is high, and the intensity dependence of the generated harmonics’ atomic phase tilts the harmonic wavefront. The atomic phase’s sensitivity to the intensity is different for different harmonics, thus angular dispersion appears. The discrepancies in angular dispersion and cut-off energy between the experiment and calculation in case of Figs. 3(c) and 3(f) shows the limitation of our 2D model to reproduce the results accurately under strong nonlinear propagation effects.

In summary, the dynamically induced wavefront rotation increases the angular separation of the attosecond pulse generated in successive half-cycles, relaxing the requirements for the pulse duration that can be used in the attosecond lighthouse. When combined with two-color gating [23], femtosecond laser pulses without compression may be possible to use in the attosecond lighthouse.

Our work reveals the physical mechanism of the increased angular separation that is important for the study of time-resolved processes on the optical half-cycle level in a highly ionizing medium. This is an important step toward the understanding of ultrafast processes under high ionization levels (like the dynamics of the ionization itself within the laser pulse), and toward the reliable application of angular streaking at high laser intensities.

In conclusion, the dynamic wavefront rotation has been investigated to find a way to enhance the lighthouse effect in the generation of isolated attosecond pulses. We have shown that propagation effects could modify the wavefront of a laser pulse in the medium and also the phase-matching conditions in high-harmonic generation, increasing the angular separation in the far field. This dynamically induced wavefront rotation relaxes the condition for generating isolated attosecond pulses in the attosecond lighthouse without sacrificing the generation efficiency much. Our results would facilitate the applications of attosecond lighthouse, to the study of ultrafast dynamics that varies one half-cycle to the next.