Expand this Topic clickable element to expand a topic
Skip to content
Optica Publishing Group
Journal Home About Issues in Progress Current Issue All Issues Feature Issues

Time-frequency analysis of high harmonic generation using a probe XUV pulse

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

Open Access Open Access

Abstract

Interpretation of strong-field phenomena is mostly based on the analysis of classical electron trajectories in an intense laser field, whose specific properties determine general features of nonlinear laser-matter interaction. Currently, the visualization of closed electron trajectories contributing to high harmonic generation (HHG) of the laser field is the prerogative of a theoretical analysis based on the time-frequency spectrogram of the induced dipole acceleration. Here, we propose a method for direct reconstruction of the HHG time-frequency spectrogram using a time-delayed probe XUV pulse. Our analytical theory and ab initio numerical simulations demonstrate that the XUV-assisted HHG yield as a function of time delay and harmonic energy mimics the short-time Fourier transform of the dipole acceleration induced by the laser field, thereby providing possible in-situ experimental access for tracing electron dynamics in strong-field phenomena.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Valuable macroscopic analysis of nonlinear laser-matter interaction is based on the microscopic study of fundamental processes in an intense laser field. Interaction of an intense laser field with a single atomic or molecular system may lead to ionization/dissociation decay of a target and induce the time-dependent dipole moment $\boldsymbol{d}(t)$. The latter makes possible the generation of secondary broadband radiation, whose characteristic frequencies are much higher than the frequency of an incident strong field, i.e., so-called high harmonic generation (HHG) process [1–3]. The spectrum of generated harmonics [4] is presented by two distinguished regions: a short “slope” region and long “plateau” region, which ends up by abrupt cutoff, whose position is given by the product of intensity and square of the wavelength of a laser pulse [1]. Within the quasiclassical picture of HHG, the appearance of the plateau region in the HHG spectrum is caused by the fundamental contribution of closed electron classical trajectories in an intense laser field. The properties of these closed trajectories are used for interpretation of the general features of the HHG spectrum: e.g., the cutoff position is given by maximal gained energy along a closed trajectory; interference phenomena are attributed to the interference of quantum amplitudes, whose phases are given by the classical action gained along closed trajectories [5,6].

The access to the classical trajectories contributing to HHG is achieved by applying a short-time Fourier transformation (STFT) to the laser-induced dipole acceleration, $\ddot{\boldsymbol{d}}(t) \equiv d^{2} \boldsymbol{d}(t)/dt^{2}$:

$${\boldsymbol{\mathcal{S}}}[\ddot{\boldsymbol{d}}](\Omega_w,t_w)=\int_{-\infty}^{\infty} \ddot{\boldsymbol{d}}(t) w(t-t_w)e^{i\Omega_w (t-t_w)} dt,$$
which specifies a distribution of signal in the frame of time, $t_w$, and frequency, $\Omega _w$ [6]. Here, $w(t-t_w)$ is a smooth window function which tends to zero at $t = \pm \infty$ and whose width and form affect the time-frequency resolution. A wide window gives high frequency resolution, but low time resolution. As the window size becomes narrower, the frequency resolution becomes worse, but the time resolution improves. The most demanded STFT for physical problems uses Gaussian window function [6–8]: the corresponding STFT is called the Gabor transform (and with modifications for multiresolution becomes the Morlet wavelet transform) [9]. The STFT (1) of the dipole acceleration informs about contributed trajectories [6–8,10–13], the chirp of the generated harmonics [14,15], and electron dynamics [16,17]. Moreover, the trajectory analysis is very useful for studying the inner dynamics of a target: contribution of inner electron for HHG [18], migration of electron between different bands during the harmonic generation in solids [19,20]. Trajectory analysis becomes important when studying propagation (or medium) effects in harmonic generation [21]. Notwithstanding the wide range of applications of the time-frequency analysis, one is achievable only theoretically since it requires precise knowledge of the phase and magnitude of the dipole $\boldsymbol{d}(t)$, which cannot be obtained accurately from an experiment.

In this work, we propose a method based on the XUV-assisted HHG [22–25], whose realization makes possible experimental access to the time-frequency analysis of the IR-field-induced dipole moment, thereby providing in-situ study of electron-trajectory dynamics. The proposed method consists of the measurement of harmonics yield as a function of the time delay between IR and probe XUV pulses in the high-energy region of HHG spectra induced by XUV pulse [24,25]. Our theoretical estimations and ab initio numerical simulation demonstrate that the map of these spectra in the frame "time delay vs. harmonic frequency" mimics the STFT (1) with the window function corresponding to the envelope of XUV pulse. Varying the XUV pulse duration provides control of frequency and time resolution of the obtained STFT. Thus, the proposed method belongs to IR+XUV pump-probe techniques [26–29]. In our case, an intense IR field (pump) liberates a valence electron from the target and accelerates it, while the probe XUV field traces the electron recollisions and contribution of different electron trajectories to HHG. In the next, we shall use atomic units ($\hbar =m=|e|=1$) unless specified otherwise.

2. Theoretical background

Let us consider the simplest case of two linearly polarized IR and XUV pulses interacting with an atomic target:

$$\boldsymbol{F}(t)= \hat{\boldsymbol{z}}[F_{\textrm{IR}}(t)+F_{\textrm{XUV}}(t-\tau)],$$
where $\hat {\boldsymbol{z}}$ is the unit vector along the $z$-axis, $\tau$ is the time delay between IR pulse, $F_{\textrm{IR}}(t)$, and XUV pulse, $F_{\textrm{XUV}}(t)$. The IR and XUV pulses are parameterized as
$$F_{\textrm{IR}/{\textrm {XUV}}}(t)=F_{\textrm{IR}/{\textrm {XUV}}}f_{\textrm{IR}/{\textrm {XUV}}}(t)\cos\omega_{\textrm{IR}/{\textrm {XUV}}} t,$$
where $F_{{\textrm{IR}}/ {\textrm{XUV}}}$, $f_{\textrm{IR}/{\textrm {XUV}}}(t)$ and $\omega _{\textrm{IR}/{\textrm {XUV}}}$ are strength, envelope, and the carrier frequency of IR and XUV fields, respectively. The electric field $\boldsymbol{F}(t)$ induces the time-dependent dipole moment $\boldsymbol{d}(t)$ (directed along the $z$-axis), which determines the microscopic HHG amplitude.

The dipole moment can be found analytically using the approach suggested in the Ref. [25]. The key idea is to consider the interaction of XUV pulse with an atomic system within perturbation theory, while interaction with an intense IR field is described in terms of quasiclassical approximation [30]. The main ingredients of this approach are real times of ionization, $t_i$, and recombination, $t_f$, which determine the classical motion of the liberated electron along a closed trajectory in a strong IR field. These times can be found from the system of equations:

$$A_{\textrm{IR}}(t_i)-\frac{1}{\Delta t}\int_{t_i}^{t_f}A_{\textrm{IR}}(t)dt=0,\quad \Delta t=t_f-t_i,$$
$$\begin{aligned} &\left(A_{\textrm{IR}}(t_f)-\frac{1}{\Delta t}\int_{t_i}^{t_f}A_{\textrm{IR}}(t)dt\right)^{2}=\Omega-I_p-\delta E,\\ &\delta E=\frac{I_p}{\Delta t F_{\textrm{IR}}(t_i)}\left(A_{\textrm{IR}}(t_f)-\frac{1}{\Delta t}\int_{t_i}^{t_f}A_{\textrm{IR}}(t)dt\right), \end{aligned}$$
where $A_{\textrm{IR}}(t)=-\int ^{t} F_{\textrm{IR}}(t') dt'$ is the projection of vector potential on the $z$-axis, and $\delta E$ is so-called quantum correction [31]. The system (3) can be solved for real times only in the limited range of frequency $\Omega$ and the upper limit of this range gives cutoff frequency of the HHG spectrum $\Omega _{\textrm {cut}} \approx I_p+E_{\max}$, where $I_p$ is the ionization potential and $E_{\max} \approx 0.8I_{\textrm{IR}}/\omega _ {\textrm{IR}}^{2}$ is the maximal gained energy of a classical electron in the IR field of intensity $I_{\textrm{IR}}=F_{\textrm{IR}}^{2}$. For frequency $\Omega <\Omega _{\textrm {cut}}$ the system (3) gives two or more roots for pair $\{t_i, t_f\}$, thereby determine different branches for $t_i$ and $t_f$ as a function of frequency $\Omega$: different branches $t_f=t_f(\Omega )$ are presented by parabolic-like curves in Figs. 1 and 2 (see blue lines in Figs. 1 and 2).

 figure: Fig. 1.

Fig. 1. (a, b) The color-coded HHG spectrum induced by IR and probe XUV pulses in atomic hydrogen as a function of the time delay between pulses. The IR pulse has the intensity $I_{\mathrm {IR}}=2\times 10^{14}$ W/cm$^{2}$, frequency $\omega _{\mathrm {IR}}=1$ eV, and FWHM duration $T_{\mathrm {IR}} = 7.5$ fs; the XUV pulse has the intensity $I_{\mathrm {XUV}}=2\times 10^{14}$ W/cm$^{2}$, frequency $\omega _{\mathrm {XUV}}=80$ eV (a), $40$ eV (b), and FWHM duration $T_{\mathrm {XUV}}=200$ as. (c, d) Gabor transform of the dipole acceleration induced by the single IR field for the window function equal to the XUV-pulse envelope $f_{\mathrm {XUV}}$ (12)(b) with parameters from panels (a) and (b), respectively. Blue lines in panels show the dependence of the classical electron energy on the returning time moment in the single IR field. Solid and dashed lines correspond to trajectories with the first and second returns, respectively.

Download Full Size | PDF

 figure: Fig. 2.

Fig. 2. The same as in Fig. 1, but for neon atom and $I_{\mathrm {IR}}=6\times 10^{14}$ W/cm$^{2}$, $\omega _{\mathrm {IR}}=1.55$ eV, $T_{\mathrm {IR}} = 4.9$ fs, $I_{\mathrm {XUV}}=2\times 10^{13}$ W/cm$^{2}$, $\omega _{\mathrm {XUV}}=80$ eV, and $T_{\mathrm {XUV}}=100$ as.

Download Full Size | PDF

In the first order of perturbation theory in the XUV field, the dipole moment is given by a sum of two terms: the dipole moment induced by an intense IR field and the perturbative term induced by the XUV pulse [25]:

$$\boldsymbol{d}(t) = \hat{\boldsymbol{z}} (d_0(t) + F_{\textrm{XUV}} d_1(t, \tau)).$$
The Fourier spectrum of the laser-induced dipole moment $d_0(t)$ is parameterized in terms of a laser-induced factor, $a_0(\Omega )$, and a photorecombination amplitude $f_{\textrm {rec}}(E)$ of an electron with kinetic energy $E=\Omega - I_p$ [32–35] (see also Appendix):
$$d_{0}(\Omega) \equiv \int_{-\infty}^{\infty} d_0(t) e^{i\Omega t} dt =a_{0}(\Omega)f_{\textrm{rec}}(E).$$
The laser-induced factor $a_0(\Omega )$ contributes to the yield of harmonics with energies $I_p<\Omega <\Omega _{\textrm {cut}}$. The explicit form of $a_0(\Omega )$ shows that it is determined by the Fourier transform of $a_0(t)$, which encodes information about closed classical electron trajectories [32–35]:
$$a_0(\Omega)=\int_{-\infty}^{\infty} a_0(t)e^{i\Omega t} dt.$$
The parametrization (5) can be used for analytic evaluation of the STFT of the dipole acceleration induced by the IR field. Indeed, assuming that $|(dw/dt)/w| \ll \Omega _w$ and $|d f_{\mathrm {rec}} / d E| \ll f_{\mathrm {rec}}(E) / \Delta \Omega _w$, where $\Delta \Omega _w$ is the width of the Fourier spectrum of the window function $w$, we obtain the STFT of $\ddot d_0$ in the form:
$${\boldsymbol{\mathcal{S}}}[ \hat{\boldsymbol{z}} \ddot d_0](\Omega_w,t_w) \approx - \Omega_w^{2} f_{\textrm{rec}}(\Omega_w - I_p) {\boldsymbol{\mathcal{S}}}[ \hat{\boldsymbol{z}} a_0(t)](\Omega_w,t_w) .$$
The Fourier transform of the second term in Eq. (4) can be presented in the form [24,25] (see also Appendix):
$$d_{1}(\Omega,\tau) \equiv \int_{-\infty}^{\infty} d_1(t, \tau) e^{i\Omega t} dt =a_1(\Omega_1,\tau) f^{(1)}_{\textrm{rec}}(E),$$
$$a_{1}(\Omega_1,\tau)=e^{i\Omega\tau}\int_{-\infty}^{\infty} a_0(t) f_{\textrm{XUV}}(t-\tau)e^{i\Omega_1 (t-\tau)} dt, \quad \Omega_1=\Omega-\omega_{\textrm{XUV}},$$
where $f_{\textrm {rec}}^{(1)}(E)$ is the two-photon photorecombination amplitude corresponding to the absorption of an XUV photon and simultaneous emission of a harmonic photon of frequency $\Omega$. We note that Eqs. (8) are valid for the sufficiently smooth envelope of the XUV pulse, i.e.,
$$|f_{\textrm{XUV}}/(d f_{\textrm{XUV}}/dt)| \Omega \gg 1.$$
Comparison of Eq. (8b) with the definition (1) explicitly shows that $a_1(\Omega _1,\tau )$ can be treated as STFT of $a_0(t)$ with $\Omega _w=\Omega _1=\Omega -\omega _ {\textrm{XUV}}$, $t_w=\tau$ and window function $w(t)=f_{\textrm{XUV}}(t)$. As follows from (8), the Fourier spectrum of XUV-assisted dipole acceleration, $\mathbf {R}_1(\Omega , \tau ) = - \hat {\boldsymbol{z}} F_{\textrm{XUV}} \Omega ^{2} d_{1}(\Omega ,\tau )$, represents the frequency-shifted STFT of $d_0(t)$ with the window function, whose shape coincides with the envelope of XUV pulse:
$$\textbf{R}_1(\Omega, \tau) = e^{i \Omega \tau} F_{\textrm{XUV}} \frac{f^{(1)}_{\textrm{rec}}(E)}{f_{\textrm{rec}}(E)} \left(\frac{\Omega}{\Omega_1}\right)^{2} {\boldsymbol{\mathcal{S}}}[ \hat{\boldsymbol{z}} \ddot d_0](\Omega_1, \tau), \quad w(t) \equiv f_{\textrm{XUV}}(t).$$
The term $\mathbf {R}_{1}(\Omega ,\tau )$ gives dominant contribution to the HHG amplitude in the prescribed frequency range $\Omega _{\textrm {cut}}<\Omega <\Omega _{\textrm {cut}}+\omega _{\textrm{XUV}}$. As a result, measurement of HHG spectra in the range $\Omega _{\textrm {cut}}<\Omega <\Omega _{\textrm {cut}}+\omega _{\textrm{XUV}}$ for different time delays $\tau$ makes it possible to retrieve the spectrogram of the IR-induced dipole acceleration in the frequency interval of width $\omega _{\textrm{XUV}}$ below the cutoff, reconstruct the electron trajectories, and look "inside" the HHG process. We should note that the relation between XUV-induced dipole ${\mathbf {R}}_1(\Omega , \tau )$ and STFT of dipole acceleration in a single IR field (10) explicitly shows that returning time $t_f$ is the same for IR and IR+XUV cases if the gained energy in a single IR field coincides with difference $\Omega _1-I_p$.

3. Numerical simulations

The proposed method for reconstructing STFT of the dipole acceleration is verified on the basis of a "numerical experiment", which includes the calculation of the XUV-assisted HHG spectra with a large number of delays between the pump IR and probe XUV pulses and the comparison of the resulting map (frequency - delay) with the spectrogram (1) of the dipole acceleration induced by the single infrared field (see Figs. 1 and 2). We test the proposed method both with the hydrogen (H) atom, a pure single-electron system, and the neon (Ne) atom, a multielectron target containing ten electrons. Since the XUV pulse is the probe pulse, it should not significantly affect the atomic target dynamics induced by the pump IR field. Therefore, the XUV pulse parameters (carrier frequency and peak intensity) are chosen so that the ionization of the atom by XUV pulse (including ionization from inner subshells [36]) do not significantly affect HHG. Numerical simulation of HHG for the H atom is performed by solving the three-dimensional Schrödinger equation (3D TDSE). Simulations for the case of Ne atom utilized the 3D time-dependent Kohn-Sham equations (TDKSE), which take into account the interaction of atomic electrons with the laser pulse, nucleus, and with each other [37]. The numerical solution of TDSE and TDKSE is carried out on the basis of the expansion of wave functions in spherical harmonics and the split-step method described in Ref. [17]. The local density approximation potential with a self-interaction correction [38] is used as the exchange-correlation potential in the TDKSE. For the Ne atom, we keep frozen the electrons on the deep-lying $1s$ shell and use a nonuniform radial grid, which becomes denser towards the nucleus with the radial step smoothly varied in the range $[10^{-3}, 0.1]$ a.u. For the H atom, we use a uniform spatial grid with the radial step $0.1$ a.u. The spatial grid has the size $R_{\max} = r_{\max} + \mathcal {R}_{\mathrm {abs}}$, where $r_{\max} = 150$ a.u. is size of the simulation region and $\mathcal {R}_{\mathrm {abs}} = 50$ a.u. is the width of absorbing layer. In both cases the time step is $\Delta t = 0.02$ a.u. and the maximum orbital momentum in expansion of the wave functions in spherical harmonics is $l_{\max} = 512$. The HHG yield for frequency $\Omega$ is given by the square of the corresponding Fourier component $\mathbf {R}(\Omega )$ of the dipole acceleration $\ddot {\boldsymbol{d}}(t)$:

$$\textbf{R}(\Omega) = \int_{-\infty}^{\infty} \ddot{\boldsymbol{d}}(t) e^{i \Omega t} dt .$$
The atom dipole acceleration is calculated using the Ehrenfest theorem [17].

We choose the Gaussian envelope of the XUV field in the numerical simulations to reproduce the Gabor transform of the infrared-field-induced dipole acceleration. To set the IR field, we used $\cos ^{2}$-shape envelope:

$$f_{\textrm{IR}}(t)=\begin{cases} \cos^{2}(\pi t/T_0) & t\in (-T_0/2,T_0/2)\\ 0 & \textrm{otherwise} \end{cases},\\$$
$$f_{\textrm{XUV}}(t)=\exp\left(-2\ln 2\, t^{2}/T^{2}_{\textrm{XUV}} \right).$$
Here, $T_{\textrm{XUV}}$ is the intensity full width at half maximum (FWHM) duration of XUV pulse and $T_0 = 10 \pi /\omega _{\textrm{IR}}$ is the total (5-cycle) duration of the IR pulse related with the FWHM pulse duration, $T_{\textrm{IR}}$, as $T_{\textrm{IR}} = [2 \mathrm {cos}^{-1}(2^{-1/4})/\pi ] T_0$.

In Figs. 1(a) and (b) we present the color-coded distribution of the HHG yield induced by IR and probe XUV pulses in atomic hydrogen as a function of harmonic frequency and time delay. The IR pulse has the intensity $I_{\textrm{IR}}=2\times 10^{14}$ W/cm$^{2}$, the frequency $\omega _{\textrm{IR}}=1$ eV, and the XUV pulse has the intensity $I_{\textrm{XUV}}=2\times 10^{14}$ W/cm$^{2}$, the FWHM duration $T_{\textrm{XUV}}=200$ as, and two different central frequencies, $\omega _{\textrm{XUV}}=80$ eV [panel (a)] and $\omega _{\textrm{XUV}}=40$ eV [panel (b)]. In Figs. 1(c) and (d) we show the computed Gabor transform of the dipole acceleration induced by the single IR field; the shape of the window function in Gabor transform coincides with the envelope of probe XUV pulse considered in panels (a), (b). It is seen that HHG yield in the frame "time delay vs. harmonic frequency" in Figs. 1(a) and ( b) reproduces with high accuracy the Gabor transform for the frequencies of the dipole acceleration in the range $\Omega _{\mathrm {th}} - \omega _{\textrm{XUV}} < \Omega _w < \Omega _{\mathrm {th}}$, where $\Omega _{\mathrm {th}} \approx 120$ eV is the threshold frequency below which the IR-field-induced HHG yield is higher than that induced by the probe XUV field for any time delay. This threshold frequency is slightly higher than the classical cutoff frequency $\Omega _{\textrm {cut}} \approx 110$ eV. Thus, for the XUV frequency $\omega _{\textrm{XUV}} = 40$ eV, the HHG spectrogram is retrieved in the range $80\textrm {~eV}<\Omega _w<120\textrm {~eV}$, while for $\omega _{\textrm{XUV}}=80$ eV this range is enlarged to $40\textrm {~eV}<\Omega _w<120\textrm {~eV}$. In the latter case (Figs. 1(a) and (c)), the time range of the retrieved spectrogram covers the FWHM pulse duration FWHM $T_{\textrm{IR}} = 7.5$ fs, while for $\omega _{\textrm{XUV}} = 40$ eV (Figs. 1(b) and (d)) this range is limited by the half-period of IR field after the envelope maximum.

The color-coded HHG spectrograms are allocated around classical dependences of emitted photon energy $\omega _c = E + I_p$ on the recombination time $t_f$, calculated in accordance with the system (3), where $E$ is the kinetic energy of photoelectron at the moment of recollision. The closed trajectories with the same energy of the first return $E$, but different excursion time in the continuum are called short and long trajectories [39]. Both real and reconstructed HHG spectrograms in Fig. 1 demonstrate the predominance of the contribution of short trajectories to the HHG for a single-color infrared laser field, as well as trace the contribution to HHG from individual field half-periods. Moreover, for large probe XUV pulse frequency (Fig. 1(a)), the reconstructed spectrogram also tracks the contribution of trajectories with multiple returns that have smaller return energy and minor weight in the overall harmonic spectrum [6–8,40] (see two parabolic-formed distributions in Fig. 1(a) contoured by dashed lines for the time delay in the range 0$<\tau <$4 fs and 40 eV$<\Omega -\omega _ {\textrm{XUV}}<$60 eV). In the retrieved spectrogram in Fig. 1(a) we also observe some interference pattern near $\Omega -\omega _ {\textrm{XUV}}\approx 80$ eV, which is well pronounced at time delays $\tau =$−2 fs, 0 fs, 2 fs and disappears at delays −1 fs and 1 fs. This interference pattern is the result of interference of two amplitudes: the amplitude of the second harmonic of the XUV field generated in the presence of an intense IR field and XUV-assisted HHG amplitude, analytically approximated by the vector $\textbf {R}_1$. Indeed, in a two-color IR+XUV field, the second harmonic of the XUV field is generated and maximized (minimized) for those time delays, which correspond to maximum (minimum) in the IR electric-field strength absolute value $|F_{\textrm{IR}}(t)|$. For used laser parameters in Fig. 1, maxima of $|F_{\textrm{IR}}(t)|$ are reached at $t=-2$ fs, 0 fs, 2 fs, and minima are reached at $t=-1$ fs and 1 fs, respectively. The interference strip (with the spectral width defined by the XUV pulse bandwidth) is allocated near $\Omega - \omega _{\textrm{XUV}} = \omega _{\textrm{XUV}} = 80\textrm {~eV}$. Therefore, IR-induced second-harmonic generation of XUV can create some difficulties in implementing the proposed method for reconstructing the spectrogram in a certain frequency range near $\Omega _w = \omega _{\textrm{XUV}}$.

Figure 2 presents the reconstruction of spectrogram the for neon atoms. The parameters of the IR field used in the calculations correspond to the higher intensity of the IR field, compared to the hydrogen atom, $I_{\textrm{IR}} = 6 \times 10^{14}$ W/cm$^{2}$, and the higher frequency, $\omega _{\textrm{IR}} = 1.55$ eV, corresponding to the wavelength of 800 nm Ti:Sa laser. An increase in the driving frequency leads to decreasing in the field period, as well as the time intervals between short and long trajectories in HHG. Therefore, the higher temporal resolution in the HHG spectrogram is necessary to resolve the trajectories near the caustic energies; this can be achieved by using a shorter probe XUV pulse. In our calculations, we used the following parameters of the XUV field: $\omega _{\textrm{XUV}}=80$ eV, $T_{\textrm{XUV}} = 100$ as, and $I_{\textrm{XUV}}= 2\times 10^{13}$ W/cm$^{2}$. As can be seen from the comparison of Figs. 2(a) and (b), the time delay map of the XUV-assisted HHG yield with high accuracy reconstructs the calculated spectrogram obtained within the Gabor transformation for the dipole acceleration of Ne atom subjected into the IR field in the frequency range $90\textrm {~eV}<\Omega _w<170\textrm {~eV}$.

Two numerical examples above explicitly show that measuring the intensity of generated harmonics as a function of a time delay can be used to separate the contribution of different trajectories in the time-frequency domain, i.e., perform the time-frequency analysis of the dipole acceleration. However, in some cases, access to the phases of partial amplitudes corresponding to short and long trajectories is a goal of importance. As follows from (10), the XUV-field-induced spectral component of the dipole acceleration, $\mathbf {R}_1(\Omega ,\tau )$, contains the phase of the STFT of the dipole acceleration induced by the single IR field. This provides access to the phases of partial amplitudes corresponding to short and long trajectories using measured XUV-field-induced harmonic signal as a function of time delay. In order to demonstrate this, Fig. 3 shows retrieved and original phase of the IR-induced dipole corresponding to the extreme short and long trajectories given by the solution of Eqs. (3). This solution provide the emitted photon energy $\omega _c(t_f) = E(t_f) +I_p$, where $E(t_f)$ is the gained electron kinetic energy, and $t_f$ is the recombination time moment (see solid blue lines in Figs. 1(a) and (c) and Fig. 3(b)). All parameters for IR and XUV pulses are the same as in Figs. 1(a) and (c). In particular, the blue line in Fig. 3(a) presents a phase of STFT ${\boldsymbol{\mathcal{S}}}[\ddot{\boldsymbol{d}}](\omega _c(t_f), t_f)$ of dipole acceleration induced by a single IR-field as a function of $t_f$, which can be considered as the exact phase along the extreme trajectory. The blue lines in Fig. 3(a) present a phase of spectral component of the dipole acceleration induced by IR+XUV field (see Eq. (11)) as a function of time delay $\tau$ at $\Omega (\tau ) = \omega _c(t_f)|_{t_f=\tau} + \omega _{\textrm{XUV}}$. As is seen from Fig. 3(a), both dependencies qualitatively agree with each other; however, discrepancies increase as the time delay increased. Thus, having the ability to measure both the power spectrum and the phase of the XUV-assisted HHG, it is possible to reconstruct the phases of individual trajectories in the single IR field.

 figure: Fig. 3.

Fig. 3. (a) The time dependence of the phase of partial HHG amplitudes corresponding to short (green area) and long (white area) trajectories and shifted by $- \omega _c t_f$. Blue solid lines: calculated phase; orange solid lines: retrieved phase (see text). (b) The classical dependence of the emitted photon energy $\omega _c$ on the recombination time $t_f$ (see solid lines in Figs. 1(a) and (c)). All parameters for IR and XUV pulses are the same as in Figs. 1(a) and (c). The hydrogen atom is considered as an atomic target.

Download Full Size | PDF

4. Conclusion and outlook

We have developed a method that allows experimental time-frequency analysis of high-harmonic generation in the infrared laser field, which can be utilized for tracing the trajectories contributed to HHG. Our analytical theory and ab initio numerical simulation for hydrogen and neon atoms show that the time-delay dependence of the XUV-assisted HHG spectrum reproduces the short-time Fourier transform (1) of the dipole acceleration induced by a single IR field. As it was pointed out in the Introduction, the duration of the probe XUV pulse ($T_{\textrm{XUV}}$) determines the time-frequency resolution of the retrieved spectrogram since the window function in the STFT is equal to XUV pulse envelope [the resolution in time/frequency domain decreases/increases with $T_{\textrm{XUV}}$]. Thus, by changing $T_{\textrm{XUV}}$, one can obtain the desired optimal resolution. Note that linearly polarized probe XUV pulse is an "ideal" measuring tool since the XUV-induced dipole $d_1$ keeps analytical structure similar to the structure of IR-induced dipole $d_0$ [see Eqs. (5), (8)]. Such a probe pulse can also be used for tracing trajectories in the IR field with a complex spatial configuration: elliptically polarized IR field, IR field tailored from many components, with arbitrary polarizations, etc. The extension of the proposed method to the case of arbitrary IR field polarizations and waveforms can be fulfilled based on two issues: (i) XUV field gives a linear contribution to the polarization response of an atomic target; (ii) Each recombination event is accompanied by the emission of a photon whose polarization properties are given by corresponding recombination amplitude in the XUV field, while the IR field determines the electron dynamics in the continuum. The first issue is ensured by the application of perturbation theory to the description of the interaction of an intense XUV field with the atomic target (see e.g., [41]). The second issue is ensured by the factorization of XUV-induced dipole on the IR- and XUV-related parts [24,25]. Based on these two assumptions, the XUV-induced dipole moment for arbitrary waveform of the IR-field can be presented in the form

$$\boldsymbol{d}_{1}(\Omega,\tau)=a_1(\Omega_1,\tau)\boldsymbol{f}^{(1)}_{\textrm{rec}}(E),$$
where $\boldsymbol{f}^{(1)}_{\textrm {rec}}$ is the vector, which determines the XUV-assisted recombination amplitude. Indeed, the recombination amplitude with absorption of XUV photon is given by the expression
$${\cal A}_{\textrm{r}}(\boldsymbol{e}_\Omega, \boldsymbol{K})\propto \boldsymbol{e}^{*}_\Omega\cdot \boldsymbol{f}^{(1)}_{\textrm{rec}}(E),$$
$$\begin{aligned} & \boldsymbol{f}^{(1)}_{\textrm{rec}}(E)= \frac{1}{2}\langle \psi_0({\boldsymbol{r}})\vert \boldsymbol{r} G_{E+\omega_ {\textrm{XUV}}}^{(0)}(\boldsymbol{r}, \boldsymbol{r}') {(\hat{\boldsymbol{z}}\cdot\boldsymbol{r}')} \vert \psi_{\boldsymbol{K}}({\boldsymbol{r}}') \rangle\\ &+ \frac{1}{2}\langle \psi_0({\boldsymbol{r}})\vert {(\hat{\boldsymbol{z}}\cdot \boldsymbol{r})} G_{E-\Omega}^{(0)}(\boldsymbol{r}, \boldsymbol{r}') \boldsymbol{r}' \vert \psi_{\boldsymbol{K}}({\boldsymbol{r}}') \rangle, \end{aligned}$$
where $\vert \psi _0({\boldsymbol{r}}) \rangle$ is the initial state of an atomic target, $\vert \psi _{\boldsymbol{K}}({\boldsymbol{r}}) \rangle$ is the continuum state of an electron with momentum $\boldsymbol{K}$, $G_{E}^{(0)}(\boldsymbol{r}, \boldsymbol{r}')$ is the Green function of the field-free atomic system, $E=\boldsymbol{K}^{2}/2$ is the kinetic energy of electron into the continuum, $\Omega$ is the frequency of the spontaneous photon, $\boldsymbol{e}_\Omega$ is the polarization vector of an emitted photon, the scalar product $(\hat{\boldsymbol{z}}\cdot \boldsymbol{r}')$ describes interaction with a linearly polarized XUV field (polarization vector coincides with unit vector $\hat{\boldsymbol{z}}$ and the dipole operator $\boldsymbol{r}$ is the operator of spontaneous transition. The integration in the matrix element (14b) is performed in vectors $\boldsymbol{r}$ and $\boldsymbol{r}'$, so that vector $\boldsymbol{f}^{(1)}_{\textrm {rec}}$ is determined by two vectors $\boldsymbol{K}$ and $\hat{\boldsymbol{z}}$ and its structure for unpolarized target has a form
$$\boldsymbol{f}^{(1)}_{\textrm{rec}} = \hat{\boldsymbol{K}}( \hat{\boldsymbol{K}}\cdot \hat{\boldsymbol{z}})\alpha_1+\hat{\boldsymbol{z}}\alpha_2,$$
where $\alpha _1\equiv \alpha _1(K)$ and $\alpha _2\equiv \alpha _2(K)$ are independent atomic scalars depending on the energy of recombining electron, $\hat{\boldsymbol{K}}$ is the unit vector directed along the velocity of the electron at the moment of recombination. The square of the vector $\boldsymbol{f}^{(1)}_{\textrm {rec}}$ is a smooth function of energy and does not significantly affect the time delay map of the XUV-assisted HHG yield. Thus, the proposed method for tracing electron trajectories in HHG spectra within a linearly polarized XUV field is robust and can be applied for any tailored IR field.

Appendix

Let us consider a nonlinear interaction of an atomic target with an intense IR field and perturbative XUV pulse, which initiates the generation of a harmonic having frequency $\Omega$ and polarization $\boldsymbol{e}_\Omega$. The amplitude of this process can be presented in the form [42]:

$${\mathcal{A}}(\Omega, \boldsymbol{e}_\Omega)=\int_{-\infty}^{\infty} \mathcal{A}(\Omega,{\boldsymbol{e}}_\Omega;t) dt,$$
$$\begin{aligned} &\mathcal{A}(\Omega,{\boldsymbol{e}}_\Omega; t)=\int_{-\infty}^{\infty}\langle\psi_0({\boldsymbol{r}},t)|V_\Omega({\boldsymbol{r}},t) G({\boldsymbol{r}},t,{\boldsymbol{r}}',t')V({\boldsymbol{r}}',t')|\psi_0({\boldsymbol{r}}',t')\rangle dt'\\ &+\int_{-\infty}^{\infty}\langle\psi_0({\boldsymbol{r}},t)|V({\boldsymbol{r}},t)G({\boldsymbol{r}},t,{\boldsymbol{r}}',t')V_\Omega({\boldsymbol{r}}',t')|\psi_0({\boldsymbol{r}}',t')\rangle dt'\\ &+\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\langle\psi_0({\boldsymbol{r}},t)|V({\boldsymbol{r}},t)G({\boldsymbol{r}},t,{\boldsymbol{r}}',t')\\ &\times V_\Omega({\boldsymbol{r}}',t')G({\boldsymbol{r}}',t',{\boldsymbol{r}}^{\prime\prime},t^{\prime\prime})V({\boldsymbol{r}}^{\prime\prime}, t^{\prime\prime})|\psi_0({\boldsymbol{r}}^{\prime\prime},t^{\prime\prime})\rangle dt^{\prime\prime}dt', \end{aligned}$$
where $V_\Omega ({\boldsymbol{r}},t)\equiv ({\boldsymbol{r}}\cdot {\boldsymbol{e}}^{*}_\Omega ) e^{i\Omega t}$ is the operator of the dipole interaction with the generated photon, $V({\boldsymbol{r}},t)$ is the operator of total interaction of an atomic electron with IR and XUV field, $G({\boldsymbol{r}},t,{\boldsymbol{r}}',t')$ is the retarded Green function of an atomic system in both IR and XUV fields, $\psi _0({\boldsymbol{r}}, t)=\psi _0({\boldsymbol{r}}) e^{iI_p t}$ is the wave function of an atomic target in an initial state, $I_p$ is the ionization potential. In Eq. (16), the symbol $\langle \ldots |\ldots |\ldots \rangle$ means the integration for all repeated spatial variables.

For the case of a single IR field, only the first term in Eq. (16b) gives a contribution to the HHG amplitude [42]:

$$\mathcal{A}^{(0)}(\Omega,{\boldsymbol{e}}_\Omega;t)=\int_{-\infty}^{\infty}\langle\psi_0({\boldsymbol{r}},t)|V_\Omega({\boldsymbol{r}},t) G_{\textrm{IR}}({\boldsymbol{r}},t,{\boldsymbol{r}}',t')V_{IR}({\boldsymbol{r}}',t')|\psi_0({\boldsymbol{r}}',t')\rangle dt',$$
where $V_{\textrm{IR}}({\boldsymbol{r}},t)$ is the operator of the dipole interaction of an atomic electron with an intense IR field, $G_{\textrm{IR}}({\boldsymbol{r}},t,{\boldsymbol{r}}',t')$ is the retarded Green function of an atomic system in a single IR field. In the quasiclassical limit (which is ensured by the smallness of the Keldysh parameter [43] and the smallness of the IR field carrier frequency with respect to the ionization potential), the result of integration in ${\boldsymbol{r}}'$ and $t'$ in Eq. (17) can be presented in the universal form [44,45]
$$\int_{-\infty}^{\infty} G_{\textrm{IR}}({\boldsymbol{r}},t,{\boldsymbol{r}}',t')V_{\textrm{IR}}({\boldsymbol{r}}',t')|\psi_0({\boldsymbol{r}}',t')\rangle dt'=a(t)e^{-i\int^{t} {\cal E} dt}|\psi_{{\boldsymbol{K}}}({\boldsymbol{r}})\rangle ,$$
where $a(t)$ is a time-dependent factor, ${\boldsymbol{K}}\equiv {\boldsymbol{K}}(t)$ is the instantaneous momentum of the electron in the IR field, and $\vert \psi _{\boldsymbol{K}}({\boldsymbol{r}})\rangle$ is the field-free continuum state of an atomic system with instant energy ${\cal E}\equiv {\cal E}(t)={\boldsymbol{K}}(t)^{2}/2$. HHG amplitude (17) in the quasiclassical limit can be presented in the form:
$${\mathcal{A}}^{(0)}(\Omega,{\boldsymbol{e}}_\Omega)=\int_{-\infty}^{\infty} \langle\psi_0({\boldsymbol{r}})|({\boldsymbol{r}}\cdot \boldsymbol{e}_\Omega^{*})|\psi_{{\boldsymbol{K}}}({\boldsymbol{r}})\rangle a(t)e^{-i\int^{t} ({\cal E}+I_p-\Omega)dt} dt.$$
Estimating the temporal integral (19) by the steepest descent method, we can present amplitude ${\mathcal {A}}^{(0)}(\Omega ,{\boldsymbol{e}}_\Omega )$ in the factorized form (5) if we assume that IR field is linearly polarized:
$${\mathcal{A}}^{(0)}(\Omega,{\boldsymbol{e}}_\Omega)=({\boldsymbol{e}}_\Omega^{*}\cdot \hat{\boldsymbol{z}}) d_{0}(\Omega),\quad d_{0}(\Omega)=a_{0}(\Omega)f_{\textrm{rec}}(E),$$
$$a_0(\Omega)=\int_{-\infty}^{\infty} a_0(t)e^{i\Omega t} dt,\quad a_0(t)=a(t)e^{-i\int^{t} ({\cal E}+I_p)dt},$$
where $a_0(\Omega )$ is so-called laser-induced factor [25] and $f_{\textrm {rec}}(E)$ is the photorecombination amplitude:
$$f_{\textrm{rec}}(E)=\langle\psi_0({\boldsymbol{r}})|(\boldsymbol{e}_\Omega^{*}\cdot {\boldsymbol{r}})|\psi_{{\boldsymbol{K}}}({\boldsymbol{r}})\rangle, \quad E=\frac{{\boldsymbol{K}}^{2}}{2}=\Omega-I_p.$$
For the case of an intense IR field and perturbative XUV field, we expand amplitude (16) in series over the strength of the XUV field up to the first order. For this expansion, we decompose the operator $V({\boldsymbol{r}},t)$ into two parts
$$\begin{aligned}& V({\boldsymbol{r}},t)=V_{\textrm{IR}}({\boldsymbol{r}},t)+V_{\textrm{XUV}}({\boldsymbol{r}},t),\\ & V_{\textrm{XUV}}({\boldsymbol{r}},t)=V_{\textrm{XUV}}^{(+)}({\boldsymbol{r}})f_ {\textrm{XUV}}(t-\tau)e^{-i\omega_ {\textrm{XUV}} (t-\tau)}+ V^{(-)}_{\textrm{XUV}}({\boldsymbol{r}}) f_ {\textrm{XUV}}(t-\tau) e^{i\omega_{\textrm {XUV}} (t-\tau)},\\ & V_{\textrm{XUV}}^{(+)}({\boldsymbol{r}})=F_{\textrm{XUV}} ({\boldsymbol{e}}_{\textrm{XUV}}\cdot{\boldsymbol{r}})/2,\quad V_{\textrm{XUV}}^{(-)}({\boldsymbol{r}})=F_{\textrm{XUV}}({\boldsymbol{e}}^{*}_{\textrm{XUV}}\cdot{\boldsymbol{r}})/2, \end{aligned}$$
where $F_{\textrm{XUV}}$ is the strength of the XUV field, $f_ {\textrm{XUV}}(t)$ is the envelope of the XUV pulse, ${\boldsymbol{e}}_{\textrm{XUV}}$ is the polarization vector of the XUV pulse and $\tau$ is the time delay between XUV and IR fields. Moreover, for the total Green function, we use the approximate expression, which is based on the Dayson equality:
$$\begin{aligned}&G({\boldsymbol{r}},t,{\boldsymbol{r}}',t')\\ &=G_{\textrm{IR}}({\boldsymbol{r}},t,{\boldsymbol{r}}',t')+\int_{-\infty}^{\infty}\int G_{\textrm{IR}}({\boldsymbol{r}},t,{\boldsymbol{r}}^{\prime\prime},t^{\prime\prime})V_{\textrm{XUV}}({\boldsymbol{r}}^{\prime\prime},t^{\prime\prime})G({\boldsymbol{r}}^{\prime\prime},t^{\prime\prime},{\boldsymbol{r}}',t') dt^{\prime\prime} d{\boldsymbol{r}}^{\prime\prime}\\ &\approx G_{\textrm{IR}}({\boldsymbol{r}},t,{\boldsymbol{r}}',t')+\int_{-\infty}^{\infty}\int G_{\textrm{IR}}({\boldsymbol{r}},t,{\boldsymbol{r}}^{\prime\prime},t^{\prime\prime})V_{\textrm{XUV}}({\boldsymbol{r}}^{\prime\prime},t^{\prime\prime})G_{\textrm{IR}}({\boldsymbol{r}}^{\prime\prime},t^{\prime\prime},{\boldsymbol{r}}',t')dt^{\prime\prime} d{\boldsymbol{r}}^{\prime\prime}. \end{aligned}$$
The final expansion of the XUV-induced part of the HHG amplitude in the first order in the XUV field is presented by a sum of sixteen terms: eight terms correspond to the emission of XUV photon in the generation high energy harmonic, and eight terms correspond to the absorption of XUV photon in HHG process. Let us focus on the XUV-induced channel of HHG with the absorption of the XUV photon in the continuum. The HHG amplitude in this channel is determined by a sum of two terms:
$$\mathcal{A}^{(1)}(\Omega,{\boldsymbol{e}}_\Omega)=\int_{-\infty}^{\infty}\mathcal{A}^{(1)}(\Omega,{\boldsymbol{e}}_\Omega; t)dt,$$
$$\begin{aligned} &\mathcal{A}^{(1)}(\Omega,{\boldsymbol{e}}_\Omega;t)\\ &=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\langle\psi_0({\boldsymbol{r}},t)|V_\Omega({\boldsymbol{r}},t)G_ {\textrm{IR}}({\boldsymbol{r}},t,{\boldsymbol{r}}',t')V^{(+)}_{\textrm{XUV}}({\boldsymbol{r}}') f_ {\textrm{XUV}}(t'-\tau)e^{-i\omega_ {\textrm{XUV}} (t'-\tau)}\\ & \times G_ {\textrm{IR}}({\boldsymbol{r}}',t',{\boldsymbol{r}}^{\prime\prime},t^{\prime\prime})V_{\textrm{IR}}({\boldsymbol{r}}^{\prime\prime},t^{\prime\prime})|\psi_0({\boldsymbol{r}}^{\prime\prime},t^{\prime\prime})\rangle dt^{\prime\prime}dt'\\ &+\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} \langle\psi_0({\boldsymbol{r}},t)|V^{(+)}_ {\textrm{XUV}}({\boldsymbol{r}}) f_{\textrm{XUV}}(t-\tau) e^{-i\omega_{\textrm{XUV}} (t-\tau)}G_ {\textrm{IR}}({\boldsymbol{r}},t,{\boldsymbol{r}}',t')V_\Omega({\boldsymbol{r}}',t')\\ &\times G_{\textrm{IR}}({\boldsymbol{r}}',t',{\boldsymbol{r}}^{\prime\prime},t^{\prime\prime})V_{\textrm{IR}}({\boldsymbol{r}}^{\prime\prime},t^{\prime\prime})|\psi_0({\boldsymbol{r}}^{\prime\prime},t^{\prime\prime}) \rangle dt'dt^{\prime\prime}.\end{aligned}$$
We simplify expression (25) by applying expression (18) and taking into account that $G_{\textrm{IR}}({\boldsymbol{r}},t,{\boldsymbol{r}}',t')\equiv 0$ for $t<t'$, and $G_{\textrm{IR}}({\boldsymbol{r}},t,{\boldsymbol{r}}',t')\ne 0$ for $t>t'$:
$$\begin{aligned} &\mathcal{A}^{(1)}(\Omega,{\boldsymbol{e}}_\Omega;t)=\int_{-\infty}^{t}\langle\psi_0({\boldsymbol{r}},t)|({\boldsymbol{e}}^{*}_\Omega\cdot{\boldsymbol{r}})e^{i\Omega t}G_{\textrm{IR}}({\boldsymbol{r}},t,{\boldsymbol{r}}',t')\\ &\times V^{(+)}_{\textrm{XUV}}({\boldsymbol{r}}')f_{\textrm{XUV}}(t'-\tau)e^{-i\omega_{\textrm{XUV}} (t'-\tau)}a(t')e^{-i\int^{t'} {\cal E} dt}|\psi_{{\boldsymbol{K}}}({\boldsymbol{r}}')\rangle dt'\\ &+\int_{-\infty}^{t}\langle\psi_0({\boldsymbol{r}},t)|V^{(+)}_{\textrm{XUV}}({\boldsymbol{r}})f_{\textrm{XUV}}(t-\tau) e^{-i\omega_{\textrm{XUV}} (t-\tau)}G_{\textrm{IR}}({\boldsymbol{r}},t,{\boldsymbol{r}}',t')\\ & \times({\boldsymbol{e}}^{*}_\Omega\cdot{\boldsymbol{r}}' )e^{i\Omega t'}a(t')e^{-i\int^{t'} {\cal E} dt}|\psi_{{\boldsymbol{K}}}({\boldsymbol{r}}')\rangle dt' . \end{aligned}$$
Integration in $t'$ is performed for a rapidly oscillating function; therefore, the contribution to the integral is given by the vicinity of the upper integration limit: $t'\approx t$. We note that the Green function $G_{\textrm{IR}}({\boldsymbol{r}},t,{\boldsymbol{r}}',t')$ at $t'\approx t$ tends to the Green function of an atomic system in the DC field, whose strength coincides with the strength of IR field at the time $t$. Since the intensity of the IR field is few orders of magnitude less than the corresponding atomic intensity, we neglect all effects form the static field (Stark effect, the polarization of core, etc.) and approximate $G_{\textrm{IR}}({\boldsymbol{r}},t,{\boldsymbol{r}}',t')$ at $t'\to t$ by the field-free nonstationary atomic Green function:
$$G_{\textrm{IR}}({\boldsymbol{r}}',t',{\boldsymbol{r}},t)\approx G_0({\boldsymbol{r}}',t',{\boldsymbol{r}},t)=\int_{-\infty}^{\infty} G_E^{(0)}({\boldsymbol{r}},{\boldsymbol{r}}')e^{-iE(t-t')}dE,$$
where $G_E^{(0)}({\boldsymbol{r}},{\boldsymbol{r}}')$ is the stationary atomic Green function. With given assumptions, the integration of the first and second terms in expression (26) results:
$$\begin{aligned}&\int_{-\infty}^{t} G_ {\textrm{IR}}({\boldsymbol{r}},t,{\boldsymbol{r}}',t')V^{(+)}_{\textrm{XUV}}({\boldsymbol{r}}')f_{\textrm{XUV}}(t'-\tau)e^{-i\omega_{\textrm{XUV}} (t'-\tau)}a(t')e^{-i\int^{t'} {\cal E} dt}|\psi_{{\boldsymbol{K}}}({\boldsymbol{r}}')\rangle dt'\\ &\approx f_{\textrm{XUV}}(t-\tau)e^{i\omega_{\textrm{XUV}}\tau} a(t)e^{-i\int^{t} ({\cal E}+\omega_{\textrm{XUV}}) dt} G_{{\cal E}+\omega_{\textrm{XUV}}}^{(0)}({\boldsymbol{r}},{\boldsymbol{r}}') ({\boldsymbol{e}}_ {\textrm{XUV}}\cdot{\boldsymbol{r}}')|\psi_{{\boldsymbol{K}}}({\boldsymbol{r}}')\rangle, \end{aligned}$$
$$\begin{aligned}&\int_{-\infty}^{t} G_{\textrm{IR}}({\boldsymbol{r}},t,{\boldsymbol{r}}',t')({\boldsymbol{e}}^{*}_\Omega\cdot{\boldsymbol{r}} )e^{i\Omega t'}a(t')e^{-i\int^{t'} {\cal E} dt}|\psi_{{\boldsymbol{K}}}({\boldsymbol{r}}')\rangle dt'\\ &\approx a(t)e^{-i\int^{t} ({\cal E}-\Omega) dt} G_{{\cal E}-\Omega}^{(0)}({\boldsymbol{r}},{\boldsymbol{r}}')({\boldsymbol{e}}_\Omega^{*}\cdot{\boldsymbol{r}}')|\psi_{{\boldsymbol{K}}}({\boldsymbol{r}}')\rangle. \end{aligned}$$
Taking into account (28) and assuming ${\boldsymbol{e}}_{\textrm{XUV}}=\hat{\boldsymbol{z}}$, we find $\mathcal {A}^{(1)}(\Omega ,{\boldsymbol{e}}_\Omega )$ in the form:
$$\mathcal{A}^{(1)}(\Omega,{\boldsymbol{e}}_\Omega)=F_{\textrm{XUV}} e^{i\Omega\tau}\int_{-\infty}^{\infty} f_{\textrm{XUV}}(t-\tau) a_0(t) e^{i\Omega_1 (t-\tau)} f_{\textrm{rec}}^{(1)}({\cal E})dt,$$
where $\Omega _1=\Omega -\omega _{\textrm{XUV}}$ and
$$\begin{aligned}&f_{\textrm{rec}}^{(1)}({\cal E})= \frac{1}{2}\langle \psi_0({\boldsymbol{r}})\vert ({\boldsymbol{e}}^{*}_\Omega \cdot \boldsymbol{r}) G_{{\cal E}+\omega_{\textrm{XUV}}}^{(0)}(\boldsymbol{r}, \boldsymbol{r}') (\hat{\boldsymbol{z}}\cdot\boldsymbol{r}') \vert \psi_{\boldsymbol{K}}({\boldsymbol{r}}') \rangle+\\ &\frac{1}{2}\langle \psi_0({\boldsymbol{r}})\vert (\hat{\boldsymbol{z}}\cdot\boldsymbol{r}) G_{{\cal E}-\Omega}^{(0)}(\boldsymbol{r}, \boldsymbol{r}') (\boldsymbol{r}'\cdot{\boldsymbol{e}}^{*}_\Omega) \vert \psi_{\boldsymbol{K}}({\boldsymbol{r}}') \rangle. \end{aligned}$$
Finally, assuming that $f_{\textrm {rec}}^{(1)}({\cal E})$ is the smooth function, it can be factored out from integrand and HHG amplitude for the case of linearly polarized IR field can be presented in the form which coincides with Eq. (8) of the main text:
$$\mathcal{A}^{(1)}(\Omega,{\boldsymbol{e}}_\Omega)=F_{\textrm{XUV}}({\boldsymbol{e}}^{*}_\Omega \cdot \hat{\boldsymbol{z}}) {d}_1(\Omega,\tau),\quad {d}_1(\Omega,\tau)= a_1(\Omega_1,\tau)f_{\textrm{rec}}^{(1)}(E),$$
$$a_1(\Omega_1,\tau)= e^{i\Omega\tau}\int_{-\infty}^{\infty} f_ {\textrm{XUV}}(t-\tau) a_0(t) e^{i\Omega_1 (t-\tau)} dt.$$

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.

References

1. P. B. Corkum, “Plasma perspective on strong field multiphoton ionization,” Phys. Rev. Lett. 71(13), 1994–1997 (1993). [CrossRef]  

2. F. Krausz and M. Ivanov, “Attosecond physics,” Rev. Mod. Phys. 81(1), 163–234 (2009). [CrossRef]  

3. T. Kroh, C. Jin, P. Krogen, P. D. Keathley, A.-L. Calendron, J. P. Siqueira, H. Liang, E. L. Falcão-Filho, C. D. Lin, F. X. Kärtner, and K.-H. Hong, “Enhanced high-harmonic generation up to the soft x-ray region driven by mid-infrared pulses mixed with their third harmonic,” Opt. Express 26(13), 16955–16969 (2018). [CrossRef]  

4. We should notice that terminology “harmonic generation” may be strictly applied for the case of a long laser pulse producing a spectrum with narrow peaks separated by an integer of a carrier frequency. However, the same terminology is historically applied for the case of a few-cycle laser pulse generating continuum spectrum.

5. A. Zaïr, M. Holler, A. Guandalini, F. Schapper, J. Biegert, L. Gallmann, U. Keller, A. S. Wyatt, A. Monmayrant, I. A. Walmsley, E. Cormier, T. Auguste, J. P. Caumes, and P. Salières, “Quantum path interferences in high-order harmonic generation,” Phys. Rev. Lett. 100(14), 143902 (2008). [CrossRef]  

6. Yae-lin Sheu, H.-t. Wu, and L.-Y. Hsu, “Exploring laser-driven quantum phenomena from a time-frequency analysis perspective: a comprehensive study,” Opt. Express 23(23), 30459–30482 (2015). [CrossRef]  

7. C. Hernández-García and L. Plaja, “Resolving multiple rescatterings in high-order-harmonic generation,” Phys. Rev. A 93(2), 023402 (2016). [CrossRef]  

8. P.-C. Li, Y.-L. Sheu, H. Z. Jooya, X.-X. Zhou, and S.-I. Chu, “Exploration of laser-driven electron-multirescattering dynamics in high-order harmonic generation,” Sci. Rep. 6(1), 1–9 (2016). [CrossRef]  

9. I. Daubechies, Ten Lectures on Wavelets (SIAM, Philadelphia, Pennsylvania, 1992).

10. C. C. Chirilă, I. Dreissigacker, E. V. van der Zwan, and M. Lein, “Emission times in high-order harmonic generation,” Phys. Rev. A 81(3), 033412 (2010). [CrossRef]  

11. M. C. Kohler, C. Ott, P. Raith, R. Heck, I. Schlegel, C. H. Keitel, and T. Pfeifer, “High harmonic generation via continuum wave-packet interference,” Phys. Rev. Lett. 105(20), 203902 (2010). [CrossRef]  

12. F. Risoud, J. Caillat, A. Maquet, R. Taïeb, and C. Lévêque, “Quantitative extraction of the emission times of high-order harmonics via the determination of instantaneous frequencies,” Phys. Rev. A 88(4), 043415 (2013). [CrossRef]  

13. Y. Sheu, L.-Y. Hsu, H. Yu, P.-C. Li, and S.-I. Chu, “A new time-frequency method to reveal quantum dynamics of atomic hydrogen in intense laser pulses: Synchrosqueezing transform,” AIP Adv. 4(11), 117138 (2014). [CrossRef]  

14. K. Varjú, Y. Mairesse, B. Carré, M. B. Gaarde, P. Johnsson, S. Kazamias, R. López-Martens, J. Mauritsson, K. J. Schafer, P. Balcou, A. L’huillier, and P. Salières, “Frequency chirp of harmonic and attosecond pulses,” J. Mod. Opt. 52(2-3), 379–394 (2005). [CrossRef]  

15. Z. Abdelrahman, M. A. Khokhlova, D. J. Walke, T. Witting, A. Zair, V. V. Strelkov, J. P. Marangos, and J. W. G. Tisch, “Chirp-control of resonant high-order harmonic generation in indium ablation plumes driven by intense few-cycle laser pulses,” Opt. Express 26(12), 15745 (2018). [CrossRef]  

16. D. Faccialà, S. Pabst, B. D. Bruner, A. G. Ciriolo, S. De Silvestri, M. Devetta, M. Negro, H. Soifer, S. Stagira, N. Dudovich, and C. Vozzi, “Probe of multielectron dynamics in xenon by caustics in high-order harmonic generation,” Phys. Rev. Lett. 117(9), 093902 (2016). [CrossRef]  

17. A. A. Romanov, A. A. Silaev, M. V. Frolov, and N. V. Vvedenskii, “Influence of the polarization of a multielectron atom in a strong laser field on high-order harmonic generation,” Phys. Rev. A 101(1), 013435 (2020). [CrossRef]  

18. Y. Li, T. Sato, and K. L. Ishikawa, “High-order harmonic generation enhanced by laser-induced electron recollision,” Phys. Rev. A 99(4), 043401 (2019). [CrossRef]  

19. G. Vampa and T. Brabec, “Merge of high harmonic generation from gases and solids and its implications for attosecond science,” J. Phys. B: At., Mol. Opt. Phys. 50(8), 083001 (2017). [CrossRef]  

20. T. Ikemachi, Y. Shinohara, T. Sato, J. Yumoto, M. Kuwata-Gonokami, and K. L. Ishikawa, “Trajectory analysis of high-order-harmonic generation from periodic crystals,” Phys. Rev. A 95(4), 043416 (2017). [CrossRef]  

21. M. B. Gaarde, J. L. Tate, and K. J. Schafer, “Macroscopic aspects of attosecond pulse generation,” J. Phys. B: At., Mol. Opt. Phys. 41(13), 132001 (2008). [CrossRef]  

22. A. Fleischer and N. Moiseyev, “Amplification of high-order harmonics using weak perturbative high-frequency radiation,” Phys. Rev. A 77(1), 010102 (2008). [CrossRef]  

23. A. Fleischer, “Generation of higher-order harmonics upon the addition of high-frequency XUV radiation to IR radiation: Generalization of the three-step model,” Phys. Rev. A 78(5), 053413 (2008). [CrossRef]  

24. T. S. Sarantseva, M. V. Frolov, N. L. Manakov, A. A. Silaev, N. V. Vvedenskii, and A. F. Starace, “XUV-assisted high-order-harmonic-generation spectroscopy,” Phys. Rev. A 98(6), 063433 (2018). [CrossRef]  

25. T. S. Sarantseva, M. V. Frolov, N. L. Manakov, A. A. Silaev, A. A. Romanov, N. V. Vvedenskii, and A. F. Starace, “Attosecond-pulse metrology based on high-order harmonic generation,” Phys. Rev. A 101(1), 013402 (2020). [CrossRef]  

26. M. Uiberacker, T. Uphues, M. Schultze, A. J. Verhoef, V. Yakovlev, M. F. Kling, J. Rauschenberger, N. M. Kabachnik, H. Schröder, M. Lezius, K. L. Kompa, H. G. Muller, M. J. J. Vrakking, S. Hendel, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, “Attosecond real-time observation of electron tunnelling in atoms,” Nature (London) 446(7136), 627–632 (2007). [CrossRef]  

27. G. Sansone, F. Kelkensberg, J. F. Pérez-Torres, F. Morales, M. F. Kling, W. Siu, O. Ghafur, P. Johnsson, M. Swoboda, E. Benedetti, F. Ferrari, F. Lépine, J. L. Sanz-Vicario, S. Zherebtsov, I. Znakovskaya, A. L’Huillier, M. Y. Ivanov, M. Nisoli, F. Martín, and M. J. J. Vrakking, “Electron localization following attosecond molecular photoionization,” Nature (London) 465(7299), 763–766 (2010). [CrossRef]  

28. F. Lépine, M. Y. Ivanov, and M. J. J. Vrakking, “Attosecond molecular dynamics: fact or fiction?” Nat. Photonics 8(3), 195–204 (2014). [CrossRef]  

29. T. Schultz and M. Vrakking, eds., Attosecond and XUV Spectroscopy: Ultrafast Dynamics and Spectroscopy (Wiley, New-York, 2014).

30. M. V. Frolov, N. L. Manakov, A. A. Minina, A. A. Silaev, N. V. Vvedenskii, M. Y. Ivanov, and A. F. Starace, “Analytic description of high-order harmonic generation in the adiabatic limit with application to an initial s state in an intense bicircular laser pulse,” Phys. Rev. A 99(5), 053403 (2019). [CrossRef]  

31. M. Lewenstein, Ph. Balcou, M. Yu. Ivanov, A. L’Huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A 49(3), 2117–2132 (1994). [CrossRef]  

32. T. Morishita, A.-T. Le, Z. Chen, and C. D. Lin, “Accurate retrieval of structural information from laser-induced photoelectron and high-order harmonic spectra by few-cycle laser pulses,” Phys. Rev. Lett. 100(1), 013903 (2008). [CrossRef]  

33. A.-T. Le, T. Morishita, and C. D. Lin, “Extraction of the species-dependent dipole amplitude and phase from high-order harmonic spectra in rare-gas atoms,” Phys. Rev. A 78(2), 023814 (2008). [CrossRef]  

34. S. Minemoto, T. Umegaki, Y. Oguchi, T. Morishita, A.-T. Le, S. Watanabe, and H. Sakai, “Retrieving photorecombination cross sections of atoms from high-order harmonic spectra,” Phys. Rev. A 78(6), 061402 (2008). [CrossRef]  

35. M. V. Frolov, N. L. Manakov, T. S. Sarantseva, and A. F. Starace, “Analytic confirmation that the factorized formula for harmonic generation involves the exact photorecombination cross section,” Phys. Rev. A 83(4), 043416 (2011). [CrossRef]  

36. A. C. Brown and H. W. van der Hart, “Extreme-ultraviolet-initated high-order harmonic generation: Driving inner-valence electrons using below-threshold-energy extreme-ultraviolet light,” Phys. Rev. Lett. 117(9), 093201 (2016). [CrossRef]  

37. C. A. Ullrich, Time-dependent density-functional theory: concepts and applications, Oxford graduate texts (Oxford University Press, 2012), reprinted, with corrections ed.

38. D. Bauer and F. Ceccherini, “Time-dependent density functional theory applied to nonsequential multiple ionization of ne at 800 nm,” Opt. Express 8(7), 377–382 (2001). [CrossRef]  

39. Y. Mairesse, A. de Bohan, L. J. Frasinski, H. Merdji, L. C. Dinu, P. Monchicourt, P. Breger, M. Kovačev, T. Auguste, B. Carré, H. G. Muller, P. Agostini, and P. Salières, “Optimization of attosecond pulse generation,” Phys. Rev. Lett. 93(16), 163901 (2004). [CrossRef]  

40. H. Z. Jooya, D. A. Telnov, and S.-I. Chu, “Exploration of the electron multiple recollision dynamics in intense laser fields with bohmian trajectories,” Phys. Rev. A 93(6), 063405 (2016). [CrossRef]  

41. E. A. Pronin, A. F. Starace, M. V. Frolov, and N. L. Manakov, “Perturbation theory analysis of attosecond photoionization,” Phys. Rev. A 80(6), 063403 (2009). [CrossRef]  

42. M. Y. Kuchiev and V. N. Ostrovsky, “Quantum theory of high harmonic generation as a three-step process,” Phys. Rev. A 60(4), 3111–3124 (1999). [CrossRef]  

43. L. V. Keldysh, “Ionization in the field of a strong electromagnetic wave,” Zh. Eksp. Teor. Fiz. 47, 1945 (1964). [J. Exp. Theor. Phys. 92 20, 1307 (1965)].

44. J. Itatani, J. Levesque, D. Zeidler, H. Niikura, H. Pépin, J. C. Kieffer, P. B. Corkum, and D. M. Villeneuve, “Tomographic imaging of molecular orbitals,” Nature (London) 432(7019), 867–871 (2004). [CrossRef]  

45. O. I. Tolstikhin and T. Morishita, “Adiabatic theory of ionization by intense laser pulses: Finite-range potentials,” Phys. Rev. A 86(4), 043417 (2012). [CrossRef]  

Cited By

Optica participates in Crossref's Cited-By Linking service. Citing articles from Optica Publishing Group journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (3)
Fig. 1.
Fig. 1. (a, b) The color-coded HHG spectrum induced by IR and probe XUV pulses in atomic hydrogen as a function of the time delay between pulses. The IR pulse has the intensity $I_{\mathrm {IR}}=2\times 10^{14}$ W/cm$^{2}$, frequency $\omega _{\mathrm {IR}}=1$ eV, and FWHM duration $T_{\mathrm {IR}} = 7.5$ fs; the XUV pulse has the intensity $I_{\mathrm {XUV}}=2\times 10^{14}$ W/cm$^{2}$, frequency $\omega _{\mathrm {XUV}}=80$ eV (a), $40$ eV (b), and FWHM duration $T_{\mathrm {XUV}}=200$ as. (c, d) Gabor transform of the dipole acceleration induced by the single IR field for the window function equal to the XUV-pulse envelope $f_{\mathrm {XUV}}$ (12)(b) with parameters from panels (a) and (b), respectively. Blue lines in panels show the dependence of the classical electron energy on the returning time moment in the single IR field. Solid and dashed lines correspond to trajectories with the first and second returns, respectively.

View in Article | Download Full Size | PDF

Fig. 2.
Fig. 2. The same as in Fig. 1, but for neon atom and $I_{\mathrm {IR}}=6\times 10^{14}$ W/cm$^{2}$, $\omega _{\mathrm {IR}}=1.55$ eV, $T_{\mathrm {IR}} = 4.9$ fs, $I_{\mathrm {XUV}}=2\times 10^{13}$ W/cm$^{2}$, $\omega _{\mathrm {XUV}}=80$ eV, and $T_{\mathrm {XUV}}=100$ as.

View in Article | Download Full Size | PDF

Fig. 3.
Fig. 3. (a) The time dependence of the phase of partial HHG amplitudes corresponding to short (green area) and long (white area) trajectories and shifted by $- \omega _c t_f$. Blue solid lines: calculated phase; orange solid lines: retrieved phase (see text). (b) The classical dependence of the emitted photon energy $\omega _c$ on the recombination time $t_f$ (see solid lines in Figs. 1(a) and (c)). All parameters for IR and XUV pulses are the same as in Figs. 1(a) and (c). The hydrogen atom is considered as an atomic target.

View in Article | Download Full Size | PDF

Equations (40)

Equations on this page are rendered with MathJax. Learn more.

S [ d ¨ ] ( Ω w , t w ) = ∫ − ∞ ∞ d ¨ ( t ) w ( t − t w ) e i Ω w ( t − t w ) d t ,
F ( t ) = z ^ [ F IR ( t ) + F XUV ( t − τ ) ] ,
F IR / XUV ( t ) = F IR / XUV f IR / XUV ( t ) cos ⁡ ω IR / XUV t ,
A IR ( t i ) − 1 Δ t ∫ t i t f A IR ( t ) d t = 0 , Δ t = t f − t i ,
( A IR ( t f ) − 1 Δ t ∫ t i t f A IR ( t ) d t ) 2 = Ω − I p − δ E , δ E = I p Δ t F IR ( t i ) ( A IR ( t f ) − 1 Δ t ∫ t i t f A IR ( t ) d t ) ,
d ( t ) = z ^ ( d 0 ( t ) + F XUV d 1 ( t , τ ) ) .
d 0 ( Ω ) ≡ ∫ − ∞ ∞ d 0 ( t ) e i Ω t d t = a 0 ( Ω ) f rec ( E ) .
a 0 ( Ω ) = ∫ − ∞ ∞ a 0 ( t ) e i Ω t d t .
S [ z ^ d ¨ 0 ] ( Ω w , t w ) ≈ − Ω w 2 f rec ( Ω w − I p ) S [ z ^ a 0 ( t ) ] ( Ω w , t w ) .
d 1 ( Ω , τ ) ≡ ∫ − ∞ ∞ d 1 ( t , τ ) e i Ω t d t = a 1 ( Ω 1 , τ ) f rec ( 1 ) ( E ) ,
a 1 ( Ω 1 , τ ) = e i Ω τ ∫ − ∞ ∞ a 0 ( t ) f XUV ( t − τ ) e i Ω 1 ( t − τ ) d t , Ω 1 = Ω − ω XUV ,
| f XUV / ( d f XUV / d t ) | Ω ≫ 1.
R 1 ( Ω , τ ) = e i Ω τ F XUV f rec ( 1 ) ( E ) f rec ( E ) ( Ω Ω 1 ) 2 S [ z ^ d ¨ 0 ] ( Ω 1 , τ ) , w ( t ) ≡ f XUV ( t ) .
R ( Ω ) = ∫ − ∞ ∞ d ¨ ( t ) e i Ω t d t .
f IR ( t ) = { cos 2 ⁡ ( π t / T 0 ) t ∈ ( − T 0 / 2 , T 0 / 2 ) 0 otherwise ,
f XUV ( t ) = exp ⁡ ( − 2 ln ⁡ 2 t 2 / T XUV 2 ) .
d 1 ( Ω , τ ) = a 1 ( Ω 1 , τ ) f rec ( 1 ) ( E ) ,
A r ( e Ω , K ) ∝ e Ω ∗ ⋅ f rec ( 1 ) ( E ) ,
f rec ( 1 ) ( E ) = 1 2 ⟨ ψ 0 ( r ) | r G E + ω XUV ( 0 ) ( r , r ′ ) ( z ^ ⋅ r ′ ) | ψ K ( r ′ ) ⟩ + 1 2 ⟨ ψ 0 ( r ) | ( z ^ ⋅ r ) G E − Ω ( 0 ) ( r , r ′ ) r ′ | ψ K ( r ′ ) ⟩ ,
f rec ( 1 ) = K ^ ( K ^ ⋅ z ^ ) α 1 + z ^ α 2 ,
A ( Ω , e Ω ) = ∫ − ∞ ∞ A ( Ω , e Ω ; t ) d t ,
A ( Ω , e Ω ; t ) = ∫ − ∞ ∞ ⟨ ψ 0 ( r , t ) | V Ω ( r , t ) G ( r , t , r ′ , t ′ ) V ( r ′ , t ′ ) | ψ 0 ( r ′ , t ′ ) ⟩ d t ′ + ∫ − ∞ ∞ ⟨ ψ 0 ( r , t ) | V ( r , t ) G ( r , t , r ′ , t ′ ) V Ω ( r ′ , t ′ ) | ψ 0 ( r ′ , t ′ ) ⟩ d t ′ + ∫ − ∞ ∞ ∫ − ∞ ∞ ⟨ ψ 0 ( r , t ) | V ( r , t ) G ( r , t , r ′ , t ′ ) × V Ω ( r ′ , t ′ ) G ( r ′ , t ′ , r ′ ′ , t ′ ′ ) V ( r ′ ′ , t ′ ′ ) | ψ 0 ( r ′ ′ , t ′ ′ ) ⟩ d t ′ ′ d t ′ ,
A ( 0 ) ( Ω , e Ω ; t ) = ∫ − ∞ ∞ ⟨ ψ 0 ( r , t ) | V Ω ( r , t ) G IR ( r , t , r ′ , t ′ ) V I R ( r ′ , t ′ ) | ψ 0 ( r ′ , t ′ ) ⟩ d t ′ ,
∫ − ∞ ∞ G IR ( r , t , r ′ , t ′ ) V IR ( r ′ , t ′ ) | ψ 0 ( r ′ , t ′ ) ⟩ d t ′ = a ( t ) e − i ∫ t E d t | ψ K ( r ) ⟩ ,
A ( 0 ) ( Ω , e Ω ) = ∫ − ∞ ∞ ⟨ ψ 0 ( r ) | ( r ⋅ e Ω ∗ ) | ψ K ( r ) ⟩ a ( t ) e − i ∫ t ( E + I p − Ω ) d t d t .
A ( 0 ) ( Ω , e Ω ) = ( e Ω ∗ ⋅ z ^ ) d 0 ( Ω ) , d 0 ( Ω ) = a 0 ( Ω ) f rec ( E ) ,
a 0 ( Ω ) = ∫ − ∞ ∞ a 0 ( t ) e i Ω t d t , a 0 ( t ) = a ( t ) e − i ∫ t ( E + I p ) d t ,
f rec ( E ) = ⟨ ψ 0 ( r ) | ( e Ω ∗ ⋅ r ) | ψ K ( r ) ⟩ , E = K 2 2 = Ω − I p .
V ( r , t ) = V IR ( r , t ) + V XUV ( r , t ) , V XUV ( r , t ) = V XUV ( + ) ( r ) f XUV ( t − τ ) e − i ω XUV ( t − τ ) + V XUV ( − ) ( r ) f XUV ( t − τ ) e i ω XUV ( t − τ ) , V XUV ( + ) ( r ) = F XUV ( e XUV ⋅ r ) / 2 , V XUV ( − ) ( r ) = F XUV ( e XUV ∗ ⋅ r ) / 2 ,
G ( r , t , r ′ , t ′ ) = G IR ( r , t , r ′ , t ′ ) + ∫ − ∞ ∞ ∫ G IR ( r , t , r ′ ′ , t ′ ′ ) V XUV ( r ′ ′ , t ′ ′ ) G ( r ′ ′ , t ′ ′ , r ′ , t ′ ) d t ′ ′ d r ′ ′ ≈ G IR ( r , t , r ′ , t ′ ) + ∫ − ∞ ∞ ∫ G IR ( r , t , r ′ ′ , t ′ ′ ) V XUV ( r ′ ′ , t ′ ′ ) G IR ( r ′ ′ , t ′ ′ , r ′ , t ′ ) d t ′ ′ d r ′ ′ .
A ( 1 ) ( Ω , e Ω ) = ∫ − ∞ ∞ A ( 1 ) ( Ω , e Ω ; t ) d t ,
A ( 1 ) ( Ω , e Ω ; t ) = ∫ − ∞ ∞ ∫ − ∞ ∞ ⟨ ψ 0 ( r , t ) | V Ω ( r , t ) G IR ( r , t , r ′ , t ′ ) V XUV ( + ) ( r ′ ) f XUV ( t ′ − τ ) e − i ω XUV ( t ′ − τ ) × G IR ( r ′ , t ′ , r ′ ′ , t ′ ′ ) V IR ( r ′ ′ , t ′ ′ ) | ψ 0 ( r ′ ′ , t ′ ′ ) ⟩ d t ′ ′ d t ′ + ∫ − ∞ ∞ ∫ − ∞ ∞ ⟨ ψ 0 ( r , t ) | V XUV ( + ) ( r ) f XUV ( t − τ ) e − i ω XUV ( t − τ ) G IR ( r , t , r ′ , t ′ ) V Ω ( r ′ , t ′ ) × G IR ( r ′ , t ′ , r ′ ′ , t ′ ′ ) V IR ( r ′ ′ , t ′ ′ ) | ψ 0 ( r ′ ′ , t ′ ′ ) ⟩ d t ′ d t ′ ′ .
A ( 1 ) ( Ω , e Ω ; t ) = ∫ − ∞ t ⟨ ψ 0 ( r , t ) | ( e Ω ∗ ⋅ r ) e i Ω t G IR ( r , t , r ′ , t ′ ) × V XUV ( + ) ( r ′ ) f XUV ( t ′ − τ ) e − i ω XUV ( t ′ − τ ) a ( t ′ ) e − i ∫ t ′ E d t | ψ K ( r ′ ) ⟩ d t ′ + ∫ − ∞ t ⟨ ψ 0 ( r , t ) | V XUV ( + ) ( r ) f XUV ( t − τ ) e − i ω XUV ( t − τ ) G IR ( r , t , r ′ , t ′ ) × ( e Ω ∗ ⋅ r ′ ) e i Ω t ′ a ( t ′ ) e − i ∫ t ′ E d t | ψ K ( r ′ ) ⟩ d t ′ .
G IR ( r ′ , t ′ , r , t ) ≈ G 0 ( r ′ , t ′ , r , t ) = ∫ − ∞ ∞ G E ( 0 ) ( r , r ′ ) e − i E ( t − t ′ ) d E ,
∫ − ∞ t G IR ( r , t , r ′ , t ′ ) V XUV ( + ) ( r ′ ) f XUV ( t ′ − τ ) e − i ω XUV ( t ′ − τ ) a ( t ′ ) e − i ∫ t ′ E d t | ψ K ( r ′ ) ⟩ d t ′ ≈ f XUV ( t − τ ) e i ω XUV τ a ( t ) e − i ∫ t ( E + ω XUV ) d t G E + ω XUV ( 0 ) ( r , r ′ ) ( e XUV ⋅ r ′ ) | ψ K ( r ′ ) ⟩ ,
∫ − ∞ t G IR ( r , t , r ′ , t ′ ) ( e Ω ∗ ⋅ r ) e i Ω t ′ a ( t ′ ) e − i ∫ t ′ E d t | ψ K ( r ′ ) ⟩ d t ′ ≈ a ( t ) e − i ∫ t ( E − Ω ) d t G E − Ω ( 0 ) ( r , r ′ ) ( e Ω ∗ ⋅ r ′ ) | ψ K ( r ′ ) ⟩ .
A ( 1 ) ( Ω , e Ω ) = F XUV e i Ω τ ∫ − ∞ ∞ f XUV ( t − τ ) a 0 ( t ) e i Ω 1 ( t − τ ) f rec ( 1 ) ( E ) d t ,
f rec ( 1 ) ( E ) = 1 2 ⟨ ψ 0 ( r ) | ( e Ω ∗ ⋅ r ) G E + ω XUV ( 0 ) ( r , r ′ ) ( z ^ ⋅ r ′ ) | ψ K ( r ′ ) ⟩ + 1 2 ⟨ ψ 0 ( r ) | ( z ^ ⋅ r ) G E − Ω ( 0 ) ( r , r ′ ) ( r ′ ⋅ e Ω ∗ ) | ψ K ( r ′ ) ⟩ .
A ( 1 ) ( Ω , e Ω ) = F XUV ( e Ω ∗ ⋅ z ^ ) d 1 ( Ω , τ ) , d 1 ( Ω , τ ) = a 1 ( Ω 1 , τ ) f rec ( 1 ) ( E ) ,
a 1 ( Ω 1 , τ ) = e i Ω τ ∫ − ∞ ∞ f XUV ( t − τ ) a 0 ( t ) e i Ω 1 ( t − τ ) d t .
Select as filters
Select Topics Cancel
Top
       
© Copyright 2024 | Optica Publishing Group. All rights reserved, including rights for text and data mining and training of artificial technologies or similar technologies.
Privacy | Terms of Use