Waveform retrieving of an isolated attosecond pulse using high-order harmonics generation of the superimposed infrared field

T. S. Sarantseva; T. S. Sarantseva; A. A. Romanov; A. A. Romanov; A. A. Silaev; A. A. Silaev; N. V. Vvedenskii; N. V. Vvedenskii; N. V. Vvedenskii; M. V. Frolov; M. V. Frolov; M. V. Frolov

doi:10.1364/OE.440811

1. Introduction

Attosecond physics remains one of the most actively developing research areas for the past two decades [1,2]. Significant progress in the attosecond pulse generation was achieved in both shortening of the attosecond pulse [3,4] and increasing attosecond pulse intensity [5–8], which makes possible to use such pulses for tracing ultrafast dynamics in atomic, molecular and solid systems [9–14]. It should be emphasized that for practical applications of the isolated attosecond pulse (IAP) the stabilization and control of its waveform are crucial [2].

The temporal characterization of an IAP usually employs attosecond streaking [15,16], i.e., measurement of the photoelectron spectrum produced by the IAP and a few-cycle mid-infrared pulse as a function of time delay between these two pulses. This method provides direct visualization of the shape of the mid-infrared pulse from the time delay–energy (or streaking) spectrogram of the photoelectron yield, while the iterative analysis is required for the complete reconstruction of the IAP [4,17–23]. As an alternative to the photoelectron-based methods, a number of all-optical methods for attosecond pulse reconstruction were introduced in the past few years [24–27]. One of these methods presents the ability to directly measure the attosecond pulse envelope produced by an arbitrary external source. This method is based on the analysis of high-order harmonic generation (HHG) spectra produced by an atomic system subjected to an intense infrared (IR) field and desired IAP, whose waveform should be retrieved [27]. Despite the significantly larger time scale, the IR field plays a role of measuring source that helps extract the amplitude of the attosecond pulse at different times.

It should be noted that laser-atom interaction in the presence of IAP with a carrier frequency in extreme ultraviolet (XUV) region can be considered within the perturbation theory up to a comparatively high IAP intensity $\sim 10^{15}~$W/cm$^{2}$. Indeed, for such IAP parameters, the application of the perturbation theory is ensured by the smallness of the parameter $\beta _{\textrm {XUV}}= F_{\textrm {XUV}}/(\omega _{\textrm {XUV}}^{2} a)$ (atomic units are used), where $F_{\textrm {XUV}}$ and $\omega _{\textrm {XUV}}$ are the strength and carrier frequency of IAP, respectively, and $a$ is the length scale of an atomic system [28–30]. Interaction of IAP with an atomic system subjected to an intense laser field initiates new channels for HHG resulting from the modification of the three-step scenario of HHG [31]. For an intense IR pulse, this scenario is realized as: in the first step, the active electron tunnels into the continuum; in the second step, it propagates in the continuum by getting extra energy from the acceleration in an intense IR field; in the third step, it recombines to the initial state with harmonic emission.

In the first order of the perturbation theory in the XUV field, two alternative XUV-induced channels lead to noticeable modification of HHG spectra. The first channel consists of the following steps: the electron being in an initial bound state interacts with the XUV field and liberates into the continuum through the photoionization, then it propagates into the IR-dressed continuum and recombines to the initial state with harmonic emission. This channel is attributed to the XUV-initiated HHG, which currently attracts some attention [32–35]. The interest in the XUV-initiated HHG was stimulated by new possibilities for controlling electron motion in the continuum and probing XUV field polarization through the interference of direct and XUV-initiated HHG channels [36]. We note that if the carrier frequency of the XUV field is high enough, the inner-shell electrons may contribute to the XUV-initiated HHG by leading the extension of high-energy plateau [32,37,38]. Indeed, absorption of XUV photon by an inner electron liberates electron in the continuum with a small initial velocity, so that electron can be efficiently accelerated by an intense IR field with further emission of harmonic. The energy of emitted harmonic may exceed cutoff of IR-induced plateau due to the large binding energy of an inner electron.

In the second XUV-induced channel, the XUV field modifies the recombination step in the three-step scenario, i.e., initially bound electron populates some continuum states in the IR-dressed continuum through the nonlinear interaction with an intense IR field and recombines to the initial state with simultaneous harmonic emission assisted by absorption of XUV photon of IAP. This XUV-assisted channel leads to an additional high-energy plateau structure in HHG spectra [39,40]. The harmonic signal on the additional plateau can be utilized for several practical applications. The HHG yield from the additional plateau is proportional to the two-photon recombination/ionization cross-section in the XUV range and can be used to measure it in the same way that HHG spectroscopy is used to measure single-photon cross-sections [40]. Moreover, the additional plateau keeps encoded information about the return trajectories in the single IR field, which can be used to implement trajectory analysis in an experiment [41]. Finally, the harmonic yield on the additional plateau holds a piece of information about XUV pulse parameters and can be used for attosecond pulse reconstruction [27]. However, since the yield of the corresponding harmonic is governed by the square of the perturbation theory parameter $\sim \beta _{\textrm {XUV}}^{2}$, it is smaller by few orders of magnitude than HHG yield on IR-induced plateau resulting in some difficulties in experimental detection of the harmonic signal.

In this work, we propose a method for retrieving attosecond pulse waveform based on the analysis of HHG yield just below the IR-induced plateau cutoff. Harmonics for this frequency range originate from a specific interference between direct IR-induced HHG channel and XUV-assisted HHG channel corresponding to the XUV-modified recombination step. Since the information about IAP waveform is encoded in the interference term, which is linear in perturbation parameter $\sim \beta _{\textrm {XUV}}$, then the useful harmonic signal should be much intense than the signal in previously suggested retrieving procedure [27], thereby making a new retrieving procedure more robust. The direct application of the suggested method can restore the carrier-envelope (or absolute) phase of IAP up to some constant independent of the IAP parameters, which can be further utilized for monitoring the phase difference between reference and controlled pulses.

This article is organized as follows. In section 2, we present the theoretical background of HHG for the two-component laser field just beyond the IR-induced plateau. We point out those channels of HHG, which contribute to the total harmonic yield in the desired frequency range, and study the dependence of HHG yield on parameters of IR and XUV fields, including time delay between two pulses. In section 3, we discuss the details of our retrieving procedure for IAP. In section 4, we discuss our numerical calculations of the HHG yield based on the solution of time-dependent Kohn-Sham (TDKS) equations [42,43]. We summarize our results in section 5. Equations for ionization and recombination times and details of numerical integration of the TDKS equations are discussed in Appendixes A and B, respectively. Atomic units are used through the paper unless specified otherwise.

2. Theoretical background

We consider an atomic system subjected to an intense IR field, ${\boldsymbol {F}}_{\textrm {IR}}(t)$, and perturbative time-delayed XUV pulse, ${\boldsymbol {F}}_{\textrm {XUV}}(t)$:

(1)$${\boldsymbol{F}}(t)={\boldsymbol{F}}_{\textrm{IR}}(t)+{\boldsymbol{F}}_{\textrm{XUV}}(t-\tau),$$

where $\tau$ is the time delay between IR and XUV pulses. Electric components of both pulses are linearly polarized and parameterized in the form:

(2)$${\boldsymbol{F}}_{\textrm{IR}/{\textrm{XUV}}}(t)=\boldsymbol{e}_z F_{\textrm{IR}/{\textrm{XUV}}}f_{\textrm{IR}/{\textrm{XUV}}}(t)\cos(\omega_{\textrm{IR}/{\textrm{XUV}}} t+\phi_{\textrm{IR}/{\textrm{XUV}}}),$$

where $F_{\textrm {IR}/{\textrm {XUV}}}$, $\omega _{\textrm {IR}/{\textrm {XUV}}}$, $f_{\textrm {IR}/{\textrm {XUV}}}(t)$ are strength, carrier frequency, and envelope of $\textrm {IR}/\textrm {XUV}$ pulse, respectively, $\phi _{\textrm {IR}/{\textrm {XUV}}}$ is the carrier-envelope phase (CEP) of the $\textrm {IR}/\textrm {XUV}$ pulse. In our analysis we assume that $\beta _{\textrm {XUV}}\ll 1$ and $\gamma _{\textrm {IR}}=\omega _{\textrm {IR}}\kappa /F_{\textrm {IR}}\ll 1$ ($\kappa =\sqrt {2I_p}$, where $I_p$ is the ionization potential), so that the interaction with XUV pulse can be treated within the perturbation theory, while nonlinear effects from the IR field should be treated nonperturbatively.

In this work, we consider the XUV field in the first order of the perturbation theory. In contrast, the nonlinear response of an atomic system caused by IR-field is considered with quasiclassical accuracy, i.e., in terms of classical electron trajectories [44]. The HHG amplitude in an intense IR and perturbative XUV pulse, $\mathcal {A}(\Omega )$, is presented by a sum of two terms:

(3)$$\mathcal{A}(\Omega)\approx\mathcal{A}_0(\Omega)+\mathcal{A}_1(\Omega),$$

where $\mathcal {A}_0(\Omega )$ is the harmonic amplitude in the IR field, $\mathcal {A}_1(\Omega )\propto F_{\textrm {XUV}}$ is the XUV-induced HHG amplitude in the first order of the perturbation theory in $F_{\textrm {XUV}}$, $\Omega$ is the harmonic frequency. In our further analysis, we assume that $\mathcal {A}_1(\Omega )$ is determined by the channel associated with the XUV-assisted recombination. We neglect contribution from the XUV-initiated channel because: (i) we consider harmonics on the slope of IR-induced plateau (beyond the cutoff of the IR-induced plateau, but before second XUV-induced plateau [39,40]); (ii) we consider Ne atom as the atomic target, for which contribution of inner electrons is suppressed due to small ionization and recombination cross sections. The last channel may be important for Kr, Ar, and Xe atoms, for which our primary density functional calculations (see also section 4 for details) show a significant contribution from inner electrons [42,43].

Amplitudes $\mathcal {A}_0(\Omega )$ and $\mathcal {A}_1(\Omega )$ can be factorized in terms of laser parameters [$a_0(\Omega )$ and $a_1(\Omega,\omega _{\textrm {XUV}})$] and transition amplitudes corresponding to a single-photon, $f_{\mathrm {rec}}^{(0)}(E_0)$, and two-photon, $f_{\mathrm {rec}}^{(1)}(E_1)$, recombination [27,41]:

(4a)$$\mathcal{A}_0(\Omega)=a_0(\Omega)f_{\mathrm{rec}}^{(0)}(E_0),$$

(4b)$$\mathcal{A}_1(\Omega)=F_{\textrm{XUV}}e^{i\omega_{\textrm{XUV}}\tau-i\phi_{\textrm{XUV}}}a_1(\Omega,\omega_{\textrm{XUV}})f_{\mathrm{rec}}^{(1)}(E_1),$$

where $E_0=\Omega -I_p$ and $E_1=\Omega -\omega _{\textrm {XUV}}-I_p$ are returning electron energy in IR-induced and XUV-assisted channels. Photorecombination amplitudes are given by one- and two-photon transition matrix elements:

(5a)$$f^{(0)}_\textrm{rec}(E_0)=\langle\psi_0(\boldsymbol{r})|z|\psi_{{\boldsymbol{k}}_0}(\boldsymbol{r})\rangle,\quad {\boldsymbol{k}}_0=\boldsymbol{e}_z\sqrt{2E_0},$$

(5b)$$\begin{aligned} f^{(1)}_\textrm{rec}(E_1)&=\frac{1}{2}\langle \psi_0(\boldsymbol{r})\vert z G_{k_1^{2}/2+\omega_{\textrm{XUV}}}^{(0)}(\boldsymbol{r}, \boldsymbol{r}') z' \vert \psi_{{\boldsymbol{k}}_1}(\boldsymbol{r}') \rangle \\ &+\frac{1}{2}\langle \psi_0(\boldsymbol{r})\vert z G_{k_1^{2}/2-\Omega}^{(0)}(\boldsymbol{r}, \boldsymbol{r}')z'\vert \psi_{{\boldsymbol{k}}_1}(\boldsymbol{r}') \rangle,\quad {{\boldsymbol{k}}}_1=\boldsymbol{e}_z\sqrt{2 E_1},\end{aligned}$$

where $\psi _{{\boldsymbol {k}}}(\boldsymbol {r})$ is the scattering state with outgoing spherical-wave asymptotics, $\psi _0(\boldsymbol {r})$ is the bound state, and $G_{E}^{(0)}(\boldsymbol {r}, \boldsymbol {r}')$ is the atomic Green’s function. Although the explicit expressions for laser factors $a_0(\Omega )$ and $a_1(\Omega,\omega _{\textrm {XUV}})$ have been obtained for short-range atomic potential [27], we use this result for further parametrization of $a_0(\Omega )$ and $a_1(\Omega,\omega _{\textrm {XUV}})$ in terms of closed electron trajectories, assuming that general analytical parametrization does not depend on the shape of an atomic potential.

The explicit form of laser factors $a_0$ and $a_1$ can be presented in terms of the integral transform:

(6a)$$a_0(\Omega)=\mathcal{D}_{\Omega}\left[1\right],$$

(6b)$$a_1(\Omega,\omega_{\textrm{XUV}})=\mathcal{D}_{\Omega-\omega_{\textrm{XUV}}}\left[f_{\textrm{XUV}}(t-\tau)\right],$$

where

(7a)$$\mathcal{D}_{\alpha}[\varphi(t)]\equiv{\mathcal{C}} \int\limits_{-\infty}^{\infty} dt\int\limits_{-\infty}^{t} dt'\frac{e^{i\alpha t-i{\mathcal S}(t,t')}}{(t-t')^{3/2}}\varphi(t),$$

(7b)$$\mathcal{S}(t,t')=I_p(t-t')+\frac{1}{2}\int\limits_{t'}^{t}P^{2}(\xi;t,t')d\xi,$$

(7c)$$\begin{aligned}P(\xi;t,t')&=\left(A_{\textrm{IR}}(\xi)-\frac{1}{t-t'}\int\limits_{t'}^{t}A_{\textrm{IR}}(\xi')d\xi'\right), \\ A_{\textrm{IR}}(t)&={-}\int^{t}F_{\textrm{IR}}(\xi) d\xi,\end{aligned}$$

and $\mathcal {C}$ is some laser-field-free constant. For an intense IR field, the result of the integral transform (7a) can be estimated within the saddle point method [45]. The saddle points of the integrand (7a) can be associated with starting and ending times of free electron motion along closed trajectories (or quantum orbits) in an intense IR field, which formally obey Newton equations [46,47]. The number of contributed saddle points depends on the shape of IR pulse and magnitude of parameter $\alpha$ in Eq. (7a). For instance, for few-cycle IR pulse with zero carrier-envelope phase [see Eq. (2) with $\phi _{\textrm {IR}}=0$] only single closed trajectory contributes for $\alpha \gtrsim \Omega _\textrm {cut}$ and two ones for $\alpha \lesssim \Omega _\textrm {cut}$, where $\Omega _\textrm {cut}$ is the cutoff frequency for IR-induced plateau [48].

Let us consider harmonics just beyond the IR-induced plateau produced by the field (1). For this case, the laser factor $a_0(\Omega )$ is determined by a single extreme closed trajectory with corresponding return time $t_0$ [48] (equation for $t_0$ is discussed in Appendix A):

(8)$$a_0(\Omega)=\tilde a_0(\Omega,t_0).$$

The electron, moving along the extreme closed trajectory, returns to the origin at the time $t_0$ with maximal kinetic energy $E_{\max }=\Omega _\textrm {cut}-I_p$. On the contrary, the laser factor $a_1(\Omega,\omega _{\textrm {XUV}})$ is determined by a bunch of saddle points, since $\alpha =\Omega -\omega _{\textrm {XUV}}<\Omega _\textrm {cut}$ [27,40,41] (equation for $t_j$ is given in Appendix A):

(9)$$a_1(\Omega,\omega_{\textrm{XUV}})=\sum_{j\ne 0} \tilde a_0(\Omega-\omega_{\textrm{XUV}};t_j)f_{\textrm{XUV}}(t_j-\tau),$$

where $t_j$ is the returning moment for the $j$-th electron closed trajectory. These saddle points may be associated with starting and returning moments of electron motion along short (with shortest returning time) and long (with longest returning time) classical closed trajectories. For further convenience in notation, we use negative $j$ for short trajectories and positive $j$ for long trajectories. Utilization of the saddle points method for evaluation of the integrals in (6) imposes some restrictions for the IAP parameters [27]:

(i) the XUV pulse envelope $f_{\textrm {XUV}}(t)$ should be a smooth function of time, i.e. $(10)$$\left|\frac{1}{f_{\textrm{XUV}}(t)}\frac{\partial f_{\textrm{XUV}}(t)}{\partial t}\right|\ll\omega_{\textrm{XUV}};$$$
(ii) the duration of the IAP should exceed the vicinity of the saddle point giving a dominant contribution to the integral. This vicinity is determined by the second derivative of the classical action (7b), $(11)$$\delta t_j=\left[\frac{1}{2}\frac{\partial^{2} \mathcal{S}(t_j,t'_j)}{\left(\partial t_j\right)^{2}}\right]^{{-}1/2}=\alpha_j\frac{\sqrt{\omega_{\textrm{IR}}}}{F_{\textrm{IR}}},$$$ where $\alpha _j$ is a function of the energy of returning electron. We note that this function rapidly behaves near zero and maximum of gained energy, while for other energies, it can be well approximated by a constant: $\alpha _j\sim 2$.

The total HHG yield is given by the square of the amplitude (3) and up to a linear term in $F_{\textrm {XUV}}$ can be presented in the form:

(12)$${\mathcal Y}(\Omega,\tau)\propto |{\mathcal A}(\Omega)|^{2} \approx|{\mathcal A}_0(\Omega)|^{2}+F_{\textrm{XUV}}\sum_{j\ne 0} \mathcal{A}^{(j)}_\textrm{int}\; f_{\textrm{XUV}}(t_j-\tau)\cos[\omega_{\textrm{XUV}}(t_j-\tau)-\phi_\textrm{tot}^{(j)}],$$

where amplitude $\mathcal {A}^{(j)}_\textrm {int}$ and phase $\phi _\textrm {tot}^{(j)}$ are:

(13a)$$\mathcal{A}^{(j)}_\textrm{int}=|\tilde a_0(\Omega;t_0)f_{\mathrm{rec}}^{(0)}(E_0)||\tilde a_0(\Omega-\omega_{\textrm{XUV}};t_j)f_{\mathrm{rec}}^{(1)}(E_1)|,$$

(13b)$$\phi_\textrm{tot}^{(j)}=\Delta\phi_\textrm{at}+\Delta\phi_{\textrm{IR}}^{(j)}+\omega_{\textrm{XUV}}t_j-\phi_{\textrm{XUV}}.$$

The total phase contains atomic phase, $\Delta \phi _\textrm {at}$, laser phase, $\Delta \phi _{\textrm {IR}}^{(j)}$, CEP of IAP, and gained phase, $\omega _{\textrm {XUV}} t_j$, at the moment $t_j$. The atomic phase originates from the phase difference between recombination amplitudes $f_{\mathrm {rec}}^{(0)}(E_0)$ and $f_{\mathrm {rec}}^{(1)}(E_1)$ and does not depend on the returning time $t_j$:

(14)$$\Delta\phi_\textrm{at}=\arg[f_{\mathrm{rec}}^{(1)}(E_1)]-\arg[f_{\mathrm{rec}}^{(0)}(E_0)].$$

The laser phase is determined by the difference of laser factors $a_0$ and $a_1$:

(15)$$\Delta\phi_{\textrm{IR}}^{(j)}=\arg[\tilde a_0(\Omega-\omega_{\textrm{XUV}};t_j)]-\arg[\tilde a_0(\Omega;t_0)].$$

3. Retrieving procedure for IAP waveform

The expression for HHG yield (12) gives a solid background for our retrieving procedure. Indeed, the interference term in Eq. (12) is given by a coherent sum of partial terms, which replicate (up to phase) the waveform of IAP as a function of the time delay. Partial terms of this sum do not overlap with each other if the duration of IAP is smaller than the difference in returning times for short and long trajectories corresponding to the energy $\Omega -\omega _{\textrm {XUV}}$ [see Eqs. (9) and (13a)]. In the absence of overlapping, the dependence of HHG yield on the time delay mimics the waveform of IAP. For a simple estimation of the difference between two recombination instants, we use a quadratic approximation for the gained in the IR-field energy near its maxima (this approximation is led by a cubic polynomial approximation of the classical action of a free electron in the IR field [48,49]):

(16)$$E=E_{\max}-\delta(t-t_0)^{2},$$

where $\delta \sim 1$ is the dimensionless parameter. From Eq. (16) we can easily find the time difference between short and long trajectories [49]:

(17)$$\Delta t_{|j|}=t_{+|j|}-t_{-|j|}=(t_{+|j|}-t_0)-(t_{-|j|}-t_0)\propto \frac{\sqrt{E_{\max} -E}}{F_{\textrm{IR}}}=\frac{\sqrt{\omega_{\textrm{XUV}}}}{F_{\textrm{IR}}},$$

where we assume that $\Omega \sim E_{\max }+I_p$ and $E$ is the returning electron energy in the XUV-assisted HHG channel, which has the order of $E_{\max }-\omega _{\textrm {XUV}}$. In our theoretical analysis, we assume that IAP minimally affects the ionization of an atomic system in comparison with the IR field, whose contribution should be dominant. Indeed, if the ionization of an initial state by XUV pulse dominates over IR-induced tunneling, the laser factor $a_0(\Omega )$ becomes sensitive to the time delay, and it distorts the desired dependence of HHG yield (12) on the time delay. In practical numerical calculations or experimental measurements, this XUV-related ionization channel cannot be completed excluded, so that it may add a low-frequency noisy signal to the desired “useful” HHG signal. In order to minimize this useless noisy signal, we suggest performing two measurements: one with IR CEP $\phi _{\textrm {IR}}$ and second with IR CEP $\phi _{\textrm {IR}}+\pi$. For these two CEP’s, the corresponding HHG amplitudes in the IR pulse (${\mathcal A}_0$) are different by a common sign, while the XUV-induced amplitudes ${\mathcal A}_1$ have the same sign, so that interference term in the Eq. (12) changes the sign for $\phi _{\textrm {IR}}\to \phi _{\textrm {IR}}+\pi$. We note that such CEP’s symmetry properties of ${\mathcal A}_0$ and ${\mathcal A}_1$ are caused by dependencies of the photorecombination amplitudes on the electron momentum at the recombination instant [41]. Indeed, for $\phi _{\textrm {IR}}\to \phi _{\textrm {IR}}+\pi$, the vector potential of the IR field changes sign, so that it reverses the electron momentum at the instant of recombination, ${\boldsymbol {k}}_0\to -{\boldsymbol {k}}_0$, ${\boldsymbol {k}}_1\to -{\boldsymbol {k}}_1$ [see Eqs. (4), (5)], while it does not change the sign of laser factors $a_0$ and $a_1$. Since $f_\textrm {rec}^{(0)}\propto {\boldsymbol {k}}_0\cdot \boldsymbol {e}_z$ and $f_\textrm {rec}^{(1)}$ is some function of ${\boldsymbol {k}}_1^{2}$, the changing $\phi _{\textrm {IR}}\to \phi _{\textrm {IR}}+\pi$ leads to the sign changing in the amplitude $f_\textrm {rec}^{(0)}$, and does not affect the amplitude $f_\textrm {rec}^{(1)}$. The half of sum of two HHG signals for $\phi _{\textrm {IR}}$ and $\phi _{\textrm {IR}}+\pi$ gives a noisy signal, which should be extracted from these two signals:

(18a)$${\mathcal N}=\frac{1}{2}\left({\mathcal Y}_{\phi_{\textrm{IR}}}+{\mathcal Y}_{\phi_{\textrm{IR}}+\pi} \right),$$

(18b)$$\overline{\mathcal Y}=Y_{\phi_{\textrm{IR}}}-{\mathcal N},$$

where ${\mathcal N}$ is the noisy signal, ${\mathcal Y}_{\phi_{\textrm{IR}}}$ and ${\mathcal Y}_{\phi_{\textrm{IR}}}$ are the HHG signals for $\phi_{\textrm{IR}}$ and $\phi_{\textrm{IR}} +\pi$, respectively, and $\overline {\mathcal Y}$ is the “cleaned” HHG signal for operating frequency $\Omega$. (We note that cleaning procedure can be applied for any harmonic in operating range of frequencies, i.e. just beyond IR-induced plateau.) In addition, this procedure excludes from the signal the contribution of term $|\mathcal {A}_1(\Omega )|^{2}$ and the other terms $\sim F_{\textrm {XUV}}^{2}$, which we neglect in our theory.

The dependence of the “cleaned” HHG signal on the time delay can be further handled within Hilbert transform, which allows extracting from the retrieved data the envelope of IAP as well as carrier frequency and phase [50]. Indeed, the Hilbert transform of the real signal $s(t)$ gives an imaginary part of an analytic function $z(t)$, $\overline {s}(t)$:

(19)$$\overline{s}(t)={\mathcal H}[s],\quad {\mathcal H}[s]=\frac{1}{\pi}{\mathcal P}\int\nolimits_{-\infty}^{\infty}\frac{s(t')}{t-t'}dt',$$

(20)$$z(t)=s(t)+i\overline{s}(t)=A(t)e^{i\varphi(t)},$$

where ${\mathcal H}[s]$ is the Hilbert transform of a signal $s(t)$, $A(t)$ and $\varphi (t)$ are the envelope and phase of the signal $s(t)$. Once function $\overline {s}(t)$ was obtained, the envelope and phase can be simply calculated:

(21)$$A(t)=\sqrt{ s^{2}(t)+\overline{s}^{2}(t)},\quad \varphi(t)=\arctan \frac{\overline{s}(t)}{s(t)}.$$

Phase $\varphi (t)$ can be further approximated by the linear polynomial:

(22)$$\varphi(t)\approx \overline{\omega}_{\textrm{XUV}}t+\overline{\phi}_{\textrm{XUV}},$$

where $\overline {\omega }_{\textrm {XUV}}$ and $\overline {\phi }_{\textrm {XUV}}$ are fitting parameters, which has meaning of the retrieved carrier frequency and CEP.

We emphasize that the proposed method requires pulse-to-pulse stability of XUV CEP. Indeed, if the carrier-envelope phase of the XUV pulse is not stable, the HHG yield (12) should be averaged in phase $\phi _{\textrm {XUV}}$ [see Eq. (13b)]. This averaging results in the disappearing of the interference term in Eq. (12), so that in the case of large fluctuation of XUV CEP the HHG yield is governed by a single channel induced by the IR field. (We note that high-order effects in the XUV field retain the dependence on the time delay [27]). The appearance of a clear interference pattern in the dependence of the harmonic yield (on the slope of IR-induced plateau) on the time delay characterizes the degree of pulse-to-pulse stability of XUV CEP.

4. Numerical results

In this section, we demonstrate the afore discussed retrieval procedure by considering HHG from Ne atom. Two reasons dictated the choice of an atomic target. (i) Our density functional calculations show that the contribution from inner electrons to the HHG yield on the slope of the IR-induced plateau is negligible for Ne atom. In contrast, for Kr, Ar, and Xe, the contribution of inner electrons is significant in the XUV-initiated channel. (ii) The large ionization potential of Ne allows to increase the intensity of the IR field and thus decrease the magnitude of $\delta t_j$ [see Eq. (11)].

The nonlinear dynamic of Ne atom in an intense IR field and perturbative XUV pulse was considered within TDKS, which were previously utilized by us for description of HHG from Ar [42] and Xe [43] atoms. The details of numerical solution of TDKS equations are presented in Appendix B. For our numerical calculations, we parameterize the envelope of IR pulse [see also Eq. (2)] by $\cos ^{2}$-profile with total duration $T_{\textrm {IR}}$:

(23)$$f_{\textrm{IR}}(t)=\begin{cases} \cos^{2}(\pi t/T_{\textrm{IR}}) & t\in [{-}T_{\textrm{IR}}/2,T_{\textrm{IR}}/2]\\ 0 & \textrm{otherwise}, \end{cases},$$

while the IAP envelope is presented by the gaussian profile:

(24)$$f_{\textrm{XUV}}(t)=\exp\left({-}2\ln 2\, t^{2}/T^{2}_{\textrm{XUV}} \right),$$

where $T_{\textrm {XUV}}$ is the full width at half maximum of the intensity profile. The results of numerical calculations demonstrating the proposed method of XUV waveform retrieval are shown in Figs. 1–3.

Fig. 1. (a,b) The HHG spectra for Ne atom in the two-color laser field containing IR and XUV pulse components. The IR laser pulse parameters are: the peak intensity is $I_\textrm {IR}=6\times 10^{14}$ W/cm$^{2}$, the carrier frequency is $\omega _\textrm {IR}=1$eV (wavelength $\lambda _\textrm {IR} \simeq 1.2~\mu$m), $\phi _{\textrm {IR}}=0$ and duration is $T_\textrm {IR}=10\pi /\omega _\textrm {IR}=20.7$ fs. The XUV pulse parameters are: the peak intensity is $I_\textrm {XUV}=2\times 10^{13}$ W/cm$^{2}$, the time delay is $\tau =0.5$ fs, $\phi _{\textrm {XUV}}=0$. Two sets of carrier frequency and duration are used: (a,c,e) $\omega _\textrm {XUV}=41$ eV and $T_\textrm {XUV}=200$ as; (b,d,f) $\omega _\textrm {XUV}=80$ eV and $T_\textrm {XUV}=100$ as. (c,d) The Gabor transform of the dipole acceleration producing HHG spectra. Solid lines show the classical emitted photon energy in the single IR field, $\omega _c = E_0 + I_p$, (black line) and in the XUV-assisted channel, $\omega _c + \omega _{\mathrm {XUV}}$ (white line) as functions of the recombination time calculated in accordance with the system of Eqs. (28), where $E_0$ is the kinetic energy of photoelectron at the moment of recombination. The vertical dashed lines mark the "operating" harmonic of the frequency $\Omega =310.5$ eV. Vertical solid line in (a,c) marks the harmonic shifted from the "operating" one by the energy of XUV photon. Horizontal lines denote time delay $\tau$, extreme recombination time moment $\tau _0=0.81$ fs, and recombination time moments $t_{\pm 1}$, corresponding to return energy $\Omega - \omega _{\textrm {XUV}} - I_p$ for long and short trajectories. (e,f) The time delay dependencies of the spectral intensity of "operating" harmonic. Solid blue points are the results of numerical solution of TDKS equations; red and orange solid lines denote the time-dependence of XUV pulse electric field with the retrieved XUV CEP; dotted gray lines show the temporal dependence of the original XUV pulse, i.e., with $\phi _{\textrm {XUV}}=0$.

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In Figs. 1(a) and (b) we present HHG spectra for Ne atom subjected to the IR pulse with peak intensity $I_{\textrm {IR}}=6\times 10^{14}~$W/cm$^{2}$, carrier frequency $\omega _{\textrm {IR}}=1~$eV (wavelength $\lambda \simeq 1.2~\mu$m), duration $T_{\textrm {IR}}=20.7$ fs (five-cycle pulse) and XUV pulse with peak intensity $I_{\textrm {IR}}=2\times 10^{13}$ W/cm$^{2}$, CEP $\phi _\textrm {XUV}=0$, time delay $\tau = 0.5$ fs and two sets of carrier frequency and duration: (a) $\omega _{\textrm {XUV}}=41$ eV and $T_{\textrm {XUV}}=200$ as; (b) $\omega _{\textrm {XUV}}=80$ eV and $T_{\textrm {XUV}}=100$ as. HHG spectra show a broad HHG plateau with abrupt cutoff near 300 eV. Besides the complicated interference structure on the high-energy plateau, we observe a wide and intense peak centered at the carrier frequency of the XUV pulse, associated with the re-emission of the XUV pulse. The time-frequency analysis of the HHG process (performed within the Gabor transform as in [42]) is presented in Figs. 1(c) and (d). The channel related to the XUV re-emission is shown up by an eye-catching ellipse, whose center in the time-frequency coordinate is determined by the magnitude of the time delay and carrier frequency. The extension of this ellipse in the time-frequency coordinates is determined by the duration of the XUV pulse. Besides of the well-known distribution pattern, which visualizes contributions of different closed trajectories in the IR field, the Gabor transform also explicitly shows the XUV-assisted channel, associated with the amplitude ${\mathcal A}_1$. The XUV-assisted channel can be seen in Figs. 1(c) and (d) as a satellited cloud distributed in time of the same order as XUV pulse duration and centered at the time delay. For the trajectory visualization, we plot the dependence of returning time on the harmonic frequency in accordance with the system of Eq. (28) (black solid lines for pure IR pulse and white lines for XUV-assisted channel) and we mark the position of recombination times for the long ($t_{+1}$), short ($t_{-1}$), and extreme ($\tau _0 = 0.81$ fs) trajectories.

Our "operating" harmonic of frequency $\Omega =310.5$ eV [vertical dashed line in Figs. 1(a) and (b)] is placed on the slope of the IR plateau between the IR-induced cutoff and onset of the XUV-induced plateau, thereby minimize the high-order effects in the interaction of an atomic system in the XUV field. In Figs. 1(e) and (f), we present the dependence of the HHG yield for operating harmonic on the time delay (blue solid points) and the time dependence of the XUV pulse field with the retrieved XUV CEP (red and orange lines for retrieving based on short and long trajectories, respectively). For both sets of parameters, we observe that dependence of the HHG yield on time delay mimics the original waveform of the XUV pulse with retrieved CEP $\overline {\phi }_{\textrm {XUV}}$. Results in Figs. 1(e) and (f) also explicitly illustrate the separation of two neighboring bursts with increasing IAP carrier frequency.

In Fig. 2, we present the dependence of the noisy signal ${\mathcal N}$ [(a) and (b)] and “cleaned” HHG signal $\overline {\mathcal Y}$ [(c) and (d)] on the time delay for parameters as in the Fig. 1. We use the cleaned signal as an input for the Hilbert transform, which allows one to reconstruct the signal envelope, its central frequency, and CEP. The retrieved values of central frequencies $\overline {\omega }_{\textrm {XUV}}=38.9$ eV and $\overline {\omega }_{\textrm {XUV}}=78.8$ eV agree well with the initial pulses frequencies $\omega _{\textrm {XUV}}=41$ eV and $\omega _{\textrm {XUV}}=80$ eV. We note that the accuracy of the retrieved values of carrier frequencies ($5\%$ and $1.5\%$, respectively for 41 eV and 80 eV pulses) may be insufficient for some applications. The more accurate value for the carrier frequency of the XUV pulse can be obtained by collimating the position of the maximum of the hump in the measured spectrum associated with (i) direct measurement of the IAP and (ii) re-emission of IAP (see Fig. 1(a) and (b)). On panels (e) and (f), the retrieved envelopes of the attosecond pulses (solid points) are presented. Our comparison shows excellent agreement between original and retrieved (within the suggested procedure from section 3) time dependence of the XUV envelope.

Fig. 2. The time delay dependence of the noisy signal (a,b) and “cleaned” HHG signal (c,d). (e,f) The time dependence of the retrieved envelope of the attosecond pulse (solid points) and time profile of the original envelope $f_{\textrm {XUV}}(t)$ from the Eq. (1). Vertical dashed lines indicate the positions of the retrieved envelope maxima. For panels (a,c,e), parameters are the same as in Figs. 1(a), (c), (e). For panels (b), (d), (f), parameters are the same as in Fig. 1(b), (d), (f).

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The Hilbert transform also allows to reconstruct CEPs, $\overline {\phi }_{\textrm {XUV}}^{(j)}$, for the signals originating from short and long trajectories. These phases, in turn, relate to the total phase $\phi _\textrm {tot}^{(j)}$:

(25)$$\overline{\phi}_{\textrm{XUV}}^{(j)}=\phi_\textrm{tot}^{(j)}-\omega_{\textrm{XUV}}t_{j}.$$

The explicit expression for $\phi _\textrm {tot}^{(j)}$ [see Eq. (13b)] shows that retrieved CEPs $\overline {\phi }_{\textrm {XUV}}^{(j)}$ are given by few constituents ($\Delta \phi _\textrm {at}$ and $\Delta \phi _{\textrm {IR}}^{(j)}$), whose magnitude can hardly be obtained from the experiment or theory. Indeed, calculation of $\Delta \phi _\textrm {at}$ requires spectroscopy precise multielectron calculations of multiphoton transition matrix elements, which cannot be performed for a complex atom, while estimation of $\Delta \phi _{\textrm {IR}}^{(j)}$ is difficult due to laser field fluctuation along the interaction volume. Within these circumstances, the retrieved CEPs, $\overline {\phi }_{\textrm {XUV}}^{(j)}$, cannot be used for retrieving absolute CEP, $\phi _{\textrm {XUV}}$. However, they can be utilized (i) for checking the accuracy of the suggested retrieval scheme and (ii) for monitoring the phase difference between the “reference” and controlled pulses. Below we consider these points in more detail.

(i) If the retrieval scheme is robust, it should provide reasonable accuracy for phase $\phi _\textrm {tot}^{(j)}$, which includes the phase of partial amplitude associated with $j$th trajectory [see Eq. (13b)]. According to the Eq. (13b), the difference between $\phi _\textrm {tot}^{(|j|)}$ and $\phi _\textrm {tot}^{(-|j|)}$ gives a difference between phases of amplitudes associated with short and long trajectories: $(26)$$\begin{aligned} &\Delta\phi_\textrm{tot}^{(|j|)}=(\phi_\textrm{tot}^{(|j|)}-\omega_{\textrm{XUV}}t_{|j|})-(\phi_\textrm{tot}^{(-|j|)}-\omega_{\textrm{XUV}}t_{-|j|})\\ &=\arg[a_0(\Omega-\omega_{\textrm{XUV}};t_{|j|} )]-\arg[a_0(\Omega-\omega_{\textrm{XUV}};t_{-|j|} )]\approx \overline{\phi}_{\textrm{XUV}}^{(|j|)}-\overline{\phi}_{\textrm{XUV}}^{(-|j|)}. \end{aligned}$$$
If the phase difference (moded in $2\pi$) is zero, two amplitudes constructively interfere, giving a rise in the harmonic yield. While, if the phase difference is $\pi$, the interference of two amplitudes is destructive and harmonic yield is suppressed. According to the Eq. (9), the partial amplitude corresponding to the closed electron trajectory in the XUV-assisted channel is proportional to the partial amplitude for harmonic with shifted energy in the IR-induced channel. Thus, the phase difference for the “operating harmonic" can be estimated by a harmonic on the IR-induced plateau which is shifted by the energy of XUV photon from the “operating" one. In some cases, it provides the opportunity to check the accuracy of the retrieving procedure. For example, according to the Fig. 1(a) the shifted harmonic corresponds to the interference minimum [see the solid vertical line in the Fig. 1(a)] for IAP with the central frequency $41$ eV. Thus, the phase difference $\Delta \phi _\textrm {tot}^{(|j|)}$ (26) should be close to $(2 k +1) \pi$, where $k$ is integer. The reconstructed with the Hilbert transform phases correspond to $\overline {\phi }_{\textrm {XUV}}^{(+1)}=0.748\pi$ and $\overline {\phi }_{\textrm {XUV}}^{(-1)}=1.755\pi$. The desired phase difference stands at $\Delta \phi _\textrm {tot}^{(|j|)}=-1.007\pi$, which indicates the robustness of the presented procedure.
(ii) According to the Eq. (13b), CEP of the desired pulse is included in the total phase as an additive term, while all other terms are insensitive to the XUV pulse parameters. It gives us a way of controlling the desired pulse CEP using the “reference” pulse with the determined CEP. Since both pulses are under the same interacting conditions, the difference between two retrieved phases corresponding to the “reference” and desired pulses minimally depends on the fluctuation of the laser field and does not depend on the atomic target. In Fig. 3 we present HHG yields as a function of the time delay for two CEPs of an attosecond pulse, $\phi _\textrm {XUV} = 0$, and $\pi /2$. It is noticeable that the distance between neighboring maxima or minima of two retrieved pulses (see vertical dashed lines in Fig. 3) exactly coincides with the quarter laser pulse period corresponding to the $\pi /2$ phase difference. This explicitly shows the ability of the suggested method for retrieving the relative phase for two pulses.

Fig. 3. The dependence of the harmonic yield for $\Omega =310.5$ eV on the time delay for the same laser parameters and atomic target as in Figs. 1(a), (c), (e) and for the CEP of original XUV pulse $\varphi _\textrm {XUV} = 0$ (thick blue line) and $\varphi _\textrm {XUV} = \pi / 2$ (thin orange line).

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5. Summary

In this work, we have suggested an all-optical scheme for retrieving waveform of a CEP-stabilized attosecond pulse, whose application ability is much wider in the range of intensities and frequencies of IAP than the previous method [27]. Our scheme consists in the measurement of the HHG yield as a function of time delay between IR and desired XUV pulses for harmonics just beyond IR-induced HHG plateau. For this range of harmonics, the HHG amplitude is the sum of two terms: the first term is associated with the regular HHG amplitude for an intense IR laser field; the second term is originated from the joint electron-laser interaction of an intense IR field and perturbative IAP with an atomic system, thereby keeping desired information about attosecond pulse. These two terms can be formally considered as two waves in holography: the first term can be associated with the reference wave, which does not contain information about the target (i.e., IAP), while the second term is the signal wave, which carries (decodes) the desired information.

The interference of these two waves encodes the waveform of the IAP in the time delay dependence of the HHG yield. The Hilbert transformation of this dependence provides an envelope of IAP, its carrier frequency, as well as CEP (with an accuracy up to a constant phase independent on IAP). We note that the suggested method can be also applied for slightly chirped attosecond pulses, e.g., with the chirp much smaller than the derivative of the kinetic energy with respect to the recombination time for the given closed trajectory (known as frequency chirp of harmonics in IR field [51]). In this case, nonlinear dependence of the IAP phase (in time) does not affect drastically the recombination and ionization times in the IR field, and the IAP chirp can be retrieved within quadratic approximation for phase $\varphi (t)$ [see Eq. (22)]. Moreover, the suggested method cannot retrieve the absolute CEP of IAP, but it can be used for retrieving the phase difference between two pulses.

The successful realization of the retrieval scheme requires a burst-like structure of the HHG yield as a function of the time delay [each burst replicates (up to phase) the waveform of the IAP]. This is realized at certain parameters of the infrared laser pulse, at which the XUV pulse duration is less than the difference between the return times on the short and long paths at the return energy $\Omega _{\mathrm {cut}} - \omega _{\mathrm {XUV}} - I_p$. On the other hand, this condition can be relaxed by suppressing the contribution of either a short or long trajectory. For example, during the propagation of the HHG signal through the medium, the partial amplitudes associated with long trajectories may be suppressed [52] so that all bursts in the time delay dependence of HHG yield result from interference of the regular HHG amplitude for an intense IR laser field and XUV-assisted channel associated with the short trajectory. Suppression of the long trajectories can also be achieved on a single-atom-level by the use of two-component IR field [27,53,54]. In this case, all bursts are separated by half period of the IR field, which makes possible metrology of the XUV pulses with duration comparable with the IR field period. There is also a negative influence of uncontrollable fluctuations of intensities XUV and IR pulses, which leads to the “blurring” of the theoretically predicted curve so that an extra statistical method (e.g., as it is used in the attosecond streak camera) should be applied to retrieve the temporal waveform of the XUV pulse.

Appendix A: explicit form of equations for recombination times $t_0$ and $t_j$

Since the integrand of expression (7a) oscillates rapidly, we can use the saddle point method for the estimation of integral (7a). The system of equations for saddle points $t'$ and $t$ is given by

(27)$$\frac{\partial {\mathcal S}(t,t')}{\partial t'}=0,\quad \frac{\partial {\mathcal S}(t,t')}{\partial t}=\alpha,$$

where ${\mathcal S}(t,t')$ is given by expression (7b). In general, the system of Eqs. (27) is solved for complex times, while in the adiabatic limit the ionization and recombination times may be well approximated by the real times $t'$ and $t$ [55]. Equation for real times can be obtained from (27) within formal expansion of one in power series in small imaginary parts of the time $t'$ [55,56]:

(28a) $$P(t'_j;t_j,t'_j)=0,$$

(28b)$$\frac{P^{2}(t_j;t_j;t'_j)}{2}+\frac{I_p}{F_{\textrm{IR}}(t'_j)}\frac{P(t_j;t_j,t'_j)}{t_j-t'_j}=\alpha-I_p,$$

where $\alpha =\Omega -\omega _{\textrm {XUV}}$ for the XUV-assisted channel and the momentum $P(\xi ;t,t')$ is given by expression (7c). The system (28) can be solved for real pair of times $\{t'_j,t_j\}$, which correspond to the starting and ending times of motion along a closed real trajectory. We use index $j$ to enumerate solutions of the system (28): positive $j$ we associate with times for “long” trajectory, while negative with “short” trajectory. “Long” and “short” solutions merge to the “extreme” solution, which ensures specific starting and ending times for extremum (local maximum) of the gained energy, ${P^{2}(t_j;t_j;t'_j)}/{2}$. Neglecting second term in LHS of Eq. (28b) and differentiating (28b) in $t_j$, we can find system of equation for extreme ionization ($t'_0$) and recombination ($t_0$) times [48]:

(29a)$$P(t'_{0};t_{0},t'_{0})=0,$$

(29b)$$F_{\textrm{IR}}(t_0)+\frac{P(t_{0};t_{0};t'_{0})}{t_{0}-t'_{0}}=0.$$

Appendix B: numerical solution of the Kohn-Sham equations for Ne

Our simulation of the nonlinear interaction of Ne atom with the electric field ${\boldsymbol {F}}(t) = \boldsymbol {e}_z F(t)$ is based on the three-dimensional TDKS equations [57]:

(30)$$i \frac{\partial}{\partial t} \psi_{j} (\boldsymbol{r}, t) = \hat{H} \psi_{j} (\boldsymbol{r}, t), \quad j = 1, 2,\ldots,N/2,$$

(31)$$\hat{H} ={-}\frac{\nabla^{2}}{2} - \frac{N}{r} + z F(t) + {V_\textrm{ee}}[\rho(\boldsymbol{r}, t)].$$

Here, $\psi _{j}(\boldsymbol {r}, t)$ is the wave function of $j$-th Kohn-Sham orbital (doubly degenerate in spin), $N = 10$ is the total number of electrons in Ne atom, ${V_\textrm {ee}}[\rho ]$ is the electron-electron interaction potential, which is the functional of the electron density, $\rho (\boldsymbol {r}, t) = 2 \sum _{j=1}^{N/2} \left | \psi _{j}(\boldsymbol {r}, t) \right |^{2}$. The electron-electron interaction potential is a sum of Hartree potential and the exchange-correlation potential, which is considered in the local density approximation with a self-interaction correction [58]. The initial state corresponds to the ground state of the Ne atom, which has the electronic configuration $1s^{2} 2s^{2} 2p^{6}$. The Eqs. (30) are written taking into account the spin symmetry of the atom configuration and neglecting spin-orbit interaction and other relativistic effects. For the numerical solution of Eqs. (30), we use the split-step method with the expansion of wave functions in spherical harmonics (see Refs. [42,43]).

We use a time step $\Delta t = 0.02$ a.u. and a nonuniform radial grid, which becomes denser towards the nucleus with the radial step smoothly varied in the range from $10^{-3}$ to 0.1 a.u. The spatial grid has the size $R_{\max } = r_{\max } + R_{\mathrm {abs}}$ , where $r_{\max } = 150$ a.u. is the size of the simulation region and $R_{\mathrm {abs}} = 50$ a.u. is the width of the absorbing layer. The maximum orbital momentum in expansion of the wave functions in spherical harmonics is $l_{\max } = 512$. The HHG amplitude, $\mathcal {A}(\Omega )$, is calculated by finding the Fourier transform from the the dipole acceleration, $\mathbf {a}(t) = \boldsymbol {e}_z a(t)$:

(32)$$\mathcal{A}(\Omega) = \int\nolimits_{-\infty}^{\infty} \exp({-}i \Omega t) a(t) dt,$$

(33)$$a(t) = \sum_{j=2}^{N/2} a_{j}(t), \quad a_{j}(t) ={-} 2 F(t) -2\int |\psi_{j}|^{2} \frac{\partial}{\partial z} \left( {V_\textrm{ee}} - \frac{N}{r} \right) d\boldsymbol{r}.$$

In order to reduce the computation time, we keep frozen the deep-lying $1s$ orbital (having index $j=1$), so that the atom dipole acceleration is obtained within the Ehrenfest theorem applying to Kohn-Sham orbitals with $j \geq 2$ [43].

Funding

Russian Science Foundation (20-11-20289); Russian Foundation for Basic Research (20-32-70213).

Acknowledgments

Numerical simulations were supported by the Russian Science Foundation (20-11-20289). T.S. Sarantseva acknowledges the Foundation for the Advancement of Theoretical Physics and Mathematics "BASIS" (19-1-3-72-1) for the financial support.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Waveform retrieving of an isolated attosecond pulse using high-order harmonics generation of the superimposed infrared field

Abstract

1. Introduction

2. Theoretical background

3. Retrieving procedure for IAP waveform

4. Numerical results

5. Summary

Appendix A: explicit form of equations for recombination times $t_0$ and $t_j$

Appendix B: numerical solution of the Kohn-Sham equations for Ne

Funding

Acknowledgments

Disclosures

Data availability

References

Data availability

Cited By

Figures (3)

Equations (42)

Optics Express