Optimal generation of spatially coherent soft X-ray isolated attosecond pulses in a gas-filled waveguide using two-color synthesized laser pulses

We numerically demonstrate the generation of intense, low-divergence soft X-ray isolated attosecond pulses in a gas-filled hollow waveguide using synthesized few-cycle two-color laser waveforms. The waveform is a superposition of a fundamental and its second harmonic optimized such that highest harmonic yields are emitted from each atom. We then optimize the gas pressure and the length and radius of the waveguide such that bright coherent high-order harmonics with angular divergence smaller than 1 mrad are generated, for photon energy from the extreme ultraviolet to soft X-rays. By selecting a proper spectral range enhanced isolated attosecond pulses are generated. We study how dynamic phase matching caused by the interplay among waveguide mode, neutral atomic dispersion, and plasma effect is achieved at the optimal macroscopic conditions, by performing time-frequency analysis and by analyzing the evolution of the driving laser’s electric field during the propagation. Our results, when combined with the on-going push of high-repetition-rate lasers (sub- to few MHz’s) may eventually lead to the generation of high-flux, low-divergence soft X-ray tabletop isolated attosecond pulses for applications.


Methods
Propagation equations of the driving laser pulse and harmonic field in a gas-filled waveguide. To calculate harmonic radiation generated in a macroscopic medium, time-dependent Schrödinger equations (TDSEs) and coupled Maxwell's wave equations (MWEs) [55][56][57][58][59][60] are solved to account for single-atom response to the driving pulse and the propagation of radiation in the gas medium. The solution of TDSE for the single atom response is replaced by the quantitative rescattering (QRS) model [61][62][63] . Details of propagation equations of the driving pulse and high-harmonic fields have been presented in ref. 64. Here we only discuss how to solve the key equations in a gas-filled hollow waveguide.
In a reference frame moving at the speed of light c (z′ = z and t′ = t − z/c), the MWE for the driving laser in the frequency domain is where ω ′ = ′ ′ Ê r z FE r z t ( , , ) [ ( , , )],  is the density of free electrons. The MWE for the emitted high-harmonic field can be written as where Here n 0 is the neutral atom density, and D(r, z′ , t′ ) is the single-atom induced dipole moment. δ h and β h account for the dispersion and absorption of the medium on the harmonics, respectively.
To solve Eqs (1) and (4) the operator-splitting method is used. The advance of electric field from z′ to  ′ + ′ z z is separated into two steps as shown in the following 65 : 1, stands for all linear and nonlinear terms on the right-hand sides of Eqs (1) and (4). In order to impose the boundary conditions of the hollow waveguide, is written as a superposition of eigenmodes [66][67][68] : j j j 0 where μ j are the roots of Bessel function of the first kind J 0 (μ j ) = 0, and a is the hollow-core radius. Inserting Eq. (9) into Eq. (7), and using the orthonormal relation ∫ where the propagation constant κ j is 69 j j 2 and the mode loss term is 69 Here n x is the refractive index of the cladding, and λ = 2πc/ω. Note that the minus sign before  κ ′ z j in Eq. (10) is different from that in refs 67 and 68 because of the different convention for the Fourier transform.
In the calculations, we assume the spatial distributions of the two colors at the entrance of the hollow waveguide are the lowest EH 11 mode. This can be achieved by adjusting the ratio between the beam waist of the incident laser and the radius of the waveguide to be about 65%. High harmonics emitted on the exit plane of the hollow waveguide are taken as the near-fields.
Far-field harmonic emission. Near-field high harmonics may propagate further in the vacuum till they are detected. The far-field harmonic can be obtained from the near-field through a Hankel transformation 64 where J 0 is the zero-order Bessel function, z f is the far-field position from the laser focus, r f is the transverse coordinate in the far field, and the wave vector k is given by k = ω/c. Note that the optical path difference at different radial distance r f can be eliminated by using the convex lens experimentally. Mathematically, this can be carried out by multiplying a phase factor in Eq. (13).
To avoid the complexity of analyzing the full spatial distribution of the harmonics in the near field, we calculate A(t′ , ω) at each radial point and then integrate over the radial coordinate 15,72 : A spectral filter is then used to select a range of harmonics (ω 1 -ω 2 ) to generate attosecond pulses. The total intensity of attosecond pulses in the near field is obtained from 15,75 : In the far field, a spatial filter is employed to select harmonics with a prescribed area. We assume that the filter is circular with a radius r 0 , and is perpendicular to the propagation direction. The intensity of attosecond pulses in the far field is 15

Results
In the calculations, the two colors are both assumed to have Gaussian temporal envelope with the full width at half maximum (FWHM) duration of 16 fs (3 cycles of the 1.6-μm laser). The target is neon gas and the desnity distribution is uniform within the hollow waveguide. The initial on-axis electric field is the optimized two-color waveform obtained from Supplementary Table 5  High harmonics generated with the optimized two-color waveform in the near and far field. We first show the total harmonic yield at the exit of the waveguide (near field) generated by the optimized waveform (WF) in Fig. 1(a). In the simulation, the radius of the waveguide was fixed as 125 μm (a two-color Gaussian beam with the pulse energy of 0.62 mJ is incident into the waveguide). We varied the length of the waveguide and the gas pressure to achieve highest yield at the cutoff energy of about 250 eV (close to the cutoff of single-atom response). In other words, the optimization goal here is to reach best balance between harmonic yield and cutoff energy. We found the optimal values of the waveguide length and the pressure were 5 mm and 70 Torr, respectively. In Fig. 1(a), we also show the total harmonic yields generated by the single-color (SC) 1.6-μm laser alone with the peak intensity of 3.0 × 10 14 W/cm 2 under the same macroscopic condition. The harmonics generated with the waveform is about one order higher than that by the SC field. In Fig. 1(c) and (d), we show the far field harmonics for the two cases above. (Harmonic yields in the two figures have been normalized independently). We can see high harmonics from 70 to 250 eV are well localized close to the propagation axis, their divergence is less than 1 mrad. In comparison with the SC driving laser high harmonics in the same spectral region are quite divergent, making them less useful for applications. One can use an aperture to filter out the divergent harmonics, say those outside 1 mrad. As shown in Fig. 1(b), the result is that harmonic yields from the SC would become two orders smaller than the optimized two-color waveform. Another feature of high harmonics generated by the two-color WF in Fig. 1(a) and (b) is that the spectra are quasi-continuous over a broad spectral range, implying that isolated attosecond pulses may be generated, as to be discussed in Section of "Generation of isolated attosecond pulses in the near and far field".
We next examine how the harmonic emission varies with the gas pressure. We fixed the waveguide length at 5 mm and considered the gas pressure at 10 and 150 Torr, respectively. The normalized harmonic emissions in the far field are shown in Fig. 1(e) and (f), respectively. For the 10-Torr case, the harmonics are located both near the axis and far off the axis, with the cutoff energy of 250 eV being maintained. For the 150-Torr case, harmonics from 70 to 200 eV are highly concentrated along the axis, but the cutoff energy is greatly reduced. Below we study how the gas pressure and waveguide length affect phase matching for the generation of attosecond bursts, using the method previously 45 employed for two-color laser pulses.
Evolution of the driving pulse at different gas pressures. We show the on-axis electric field in the reference frame at different propagation positions z = 1, 3, and 5 mm, for three gas pressures in Fig. 2. The evolution of the electric fields are quite different for each. At 10 Torr, the envelope of the electric field shifts monotonically to the left with increasing z. At the low pressure the mode dispersion of the waveguide is dominant and cannot be compensated by the atomic dispersion 45 alone. This leads to phase mismatch between harmonics generated at different positions of the waveguide to result in diffuse harmonic spectra, as shown in Fig. 1(e) overlapped at the three positions, while the electric fields around 0 o.c. (also shown in the inset) are better overlapped between z = 3 and 5 mm than between z = 1 and 3 mm. Around − 1 o.c., the waveguide mode dispersion is compensated only by the neutral atom dispersion. With the increase of time, dispersion due to the plasma is increased, thus around 0 o.c, atomic dispersion is also compensated by the plasma effect 45 . For a single-color laser pulse, this results in a phase mismatch 76,77   where q is the harmonic order, Δ t is the shift in time of the peak electric field over a propagation distance Δ z, and the second term is dependent of the laser intensity variation Δ I and the type of electron trajectory (α i ). We use Eq. (19) to estimate the phase mismatch between a two-color driving laser and the generated high harmonics, to decide when the second term can be neglected if only "short"-trajectory emission is considered 77 and when the laser intensity variation is small, see Table 1. For electrons born around − 1 o.c., Δ t = − 0.01 (− 0.006) fs between z = 1 and 3 mm (z = 3 and 5 mm) in Table 1, the first term in Eq. (19) gives Δ k ≈ 1.90 (1.14) rad/mm for the 250-eV harmonic. The coherence length L coh = π/|Δ k| for the 250-eV harmonic between z = 1 and 3 mm (z = 3 and 5 mm) is thus 1.65 (2.76) mm, which is comparable to or longer than the medium length. For electrons born around 0 o.c., we can similarly calculate the coherence length for the 100-eV harmonic between z = 1 and 3 mm (z = 3 and 5 mm) as 0.86 (2.07) mm. The former (latter) value is smaller (bigger) than the medium length. The behavior of electric fields at different time moments demonstrates the dynamic nature of phase matching. This explains the buildup of high harmonics due to electrons ionized around − 1 o.c. is from z = 1 to 5 mm, while high harmonics due to electrons ionized around 0 o.c. only efficiently grow from z = 3 to 5 mm (this will be further discussed in the next Section). If gas pressure is increased to 150 Torr, the peak electric fields around − 0.5 and 0 o.c. are reduced by the plasma effect, resulting in reduction of cutoff energy, as seen in Fig. 1(f). The overlap of electric fields occurs only around 0 o.c. between z = 3 and 5 mm. From such analysis the buildup of harmonics inside the waveguide can be understood even though it is tedious.
Time-frequency analysis of the evolution of harmonic emission along z at the optimal gas pressure. In this section we carry out wavelet time-frequency analysis of the evolution of harmonic emission along the waveguide at the optimal gas pressure of 70 Torr. Figure 3 shows the near-field time-frequency harmonic emission, |A near (t′ , ω)| 2 , calculated using Eq. (16), vs the propagation distance z. Harmonic yields in each figure were normalized to the maximum value independently. At z = 0 mm, the harmonics in Fig. 3(a) reflect features of the single-atom response. From − 0.5 to 1.5 o.c., there are four major bursts consisting of "short"-or "long"-trajectory harmonics with positive or negative chirp, respectively. By increasing z to 1 mm, see Fig. 3(b), the picture changes dramatically. The two "long"-trajectory bursts from − 0.5 to 0.5 o.c. disappear after the propagation. We further look at the two emission bursts: one is between − 0.5 and 0 o.c., mostly due to electrons ionized around − 1 o.c.; the other is between 0.5 and 1 o.c., caused by electrons ionized around 0 o.c. The first burst has only "short"-trajectory emission at z = 1 mm. This burst is maintained in the time domain all the way to z = 5 mm,  −1 and 0 o.c.) and the strength (E p ) of the peak electric field of the laser pulse read from Fig. 2(b) for the optimal pressure of 70 Torr at three propagation positions in the moving frame: z = 1, 3 and 5 mm. Laser intensity variation is small and can be neglected in the phase mismatch analysis in Eq. (19). (o.c. stands for the optical cycle of the 1.6-μm laser). while its intensity grows. This is understood by the good overlap of electric fields around − 1 o.c. from z = 1 to 5 mm seen in Fig. 2(b), showing good phase matching is achieved. The second burst changes rapidly from z = 1 to 3 mm. It only has "long"-trajectory emission at z = 1 mm, has both "short"-and "long"-trajectory emissions at z = 2 mm, and only has "short"-trajectory emission at z = 3 mm. From z = 3 to 5 mm, the burst structure is very stable, and its intensity continues to increase. The behaviors for z = 1 to 3 mm and for z = 3 to 5 mm are totally different but can be explained by the electric fields around − 1 o.c. in Fig. 2(b) that good phase matching is only accomplished in the second half of the gas medium. In Fig. 3(f), at the exit face the results show two features: (i) only "short"-trajectory emission is presented, thus the far field harmonics have low divergence in Fig. 1(c), and (ii) three emission bursts are dominant for high harmonics in different photon energy ranges for the generation of isolated attosecond pulses in the near field by proper spectral filtering.
Generation of isolated attosecond pulses in the near and far field. In Fig. 4(a), we show three near-field soft X-ray attosecond pulses that can be synthesized by high harmonics from 116 to 155 eV, 190 to 221 eV, and 229 to 283 eV. They would be able to support isolated attosecond pulses with FWHM durations of 100, 270, and 160 as, respectively. The near-field IAP was calculated by using Eq. (17), and high harmonics were generated by the two-color waveform at the optimal conditions in Fig. 1(a). The three IAPs correspond to three emission bursts in Fig. 3(f). To check the spatial quality of the IAP in the far field and the effect of harmonic propagation in the vacuum, we calculate the far-field IAPs by using Eq. (18) and show the results in Fig. 4(b). The spectral range used to synthesize the IAP is the same as the near-field one. For IAPs centered around − 0.3, 0.1, and 0.6 o.c., the angular divergence in the far field was chosen to be 2, 1, and 1 mrad, respectively. We can see from this figure that (i) the intensity of the IAP in the far field is comparable to or even greater than that in the near field, and (ii) the duration of the IAP is close to or even shorter than that in the near field. This example demonstrates that high-intensity and low-divergence soft X-ray IAPs are generated in the far field.
Generation of high harmonics and isolated attosecond pulses by varying the waveguide radius. In the discussion above, the radius of the waveguide was fixed at 125 μm. Could soft X-ray high harmonics and IAPs be generated if the radius of the waveguide is changed? To answer this question, we chose two radii: 75 and 200 μm (the pulse energy of the incident Gaussian beam is 0.22 and 1.58 mJ, respectively), and searched for the optimal waveguide length and gas pressure to obtain the best cutoff energy and the harmonic yield. For 75 (200) μm, the optimal values of length and pressure are 2 mm and 200 Torr (10 mm and 30 Torr). The incident two-color beam waist is adjusted to ensure that the EH 11 mode is guided. The input laser pulse energy required for the 200-μm case is much higher than that for the 75-μm one. In Fig. 5, we show the time-frequency harmonic emission in the near field, harmonic divergence in the far field, and isolated attosecond pulses in the near and far field for the waveguide radius of 75 μm. Three emission bursts with only "short"-trajectory emission are observed at the exit of the waveguide (z = 2 mm) in Fig. 5(c), leading to low divergent harmonics (divergence angle smaller than 1 mrad) in the far field in Fig. 5(d). The low divergent soft X-ray IAPs in the far field [shown in Fig. 5(b)] preserve the intensities and pulse durations of the IAPs in the near field in Fig. 5(a). Similar results are shown in Fig. 6 for the waveguide radius of 200 μm.

Discussion
In summary, we proposed a method to generate bright soft X-ray isolated attosecond pulses using optimized two-color laser pulses in a hollow waveguide filled with gas. The laser waveform consists of the fundamental laser and its second harmonic has been optimized 42 for maximal single-atom harmonic yields. In the hollow waveguide its length and radius, as well as the gas pressure, are further optimized to achieve optimal dynamic phase matching. The combined effort allows the optimal generation of intense soft X-ray attosecond pulses. Note that our optimization procedure was performed in a reduced parameter space in which the most important macroscopic parameters have been varied. The scaling of macroscopic parameters may indeed likely 78 , but it needs extensive numerical calculations to verify in the future.
Our simulations demonstrated that high harmonics from XUV to soft X-rays with divergence smaller than 1 mrad can be generated at the optimal macroscopic conditions. In comparison with single-color lasers, the yields were greatly enhanced. The harmonics can be used to produce spatially coherent soft X-ray isolated attosecond pulses with proper spectral filtering. We illustrated the dynamic phase-matching conditions resulting from the interplay among waveguide mode, neutral atomic dispersion, and plasma effect by comparing the time-dependent  electric fields at different propagation distances and gas pressures. Time-frequency pictures at optimal conditions showed that the behavior of harmonic emission bursts as a function of propagation distance at different time intervals were quite different, further indicating the dynamic nature of phase matching.
Experimentally, attempts have been made to reach favorite optical waveforms by controlling the multi-color lasers, especially controlling the full phase of a broadband supercontinuum over two octaves 79 . The maturity of waveform-synthesis technology has also been shown by other examples 16,[36][37][38] . At the same time, high-repetition-rate lasers of hundreds kHz and MHz lasers [80][81][82] are becoming available in the laboratory. By combing with the method suggested in this paper, the increase of harmonic yield or attosecond pulse intensity by several orders is possible. Such all-purpose tabletop isolated attosecond light sources in the soft X-ray region can then be used for time-resolved X-ray spectroscopy in atoms, molecules, and condensed matters.