Abstract

High-order harmonic generation (HHG) in gases is known to benefit from using mid-infrared driving laser fields, since, due to a favorable wavelength scaling of the electron ponderomotive energy, higher-energy photon production becomes feasible with longer-wavelength drivers. On the other hand, recent studies have revealed a number of physical effects whose importance for HHG increases with increasing laser wavelength. These effects, as a rule, result not only in a general decrease of the harmonic yield but also in a reshaping of the emission spectrum. Therefore, detailed study of the dependence of HHG yield on the laser wavelength has become an important issue for producing intense extremely short extreme ultraviolet (XUV) and x-ray pulses using HHG driven by long-wavelength laser fields. Here we address this issue by calculating the HHG spectra for laser wavelengths ranging from 2 to 20 µm. This study has been carried out in a frame of strong-field approximation modified properly to take into account the effect of the magnetic field of a laser pulse on the dynamics of the field-ionized electron and the atomic bound-state depletion. We show that different regions of the HHG spectrum behave differently with the laser wavelength and discuss the origins of this behavior. In particular, we show that in a weak ionization regime, the dipole-approximation scaling law for the harmonic yield, which is calculated as the integral over the spectral interval of fixed width and relative position with respect to the cutoff energy, obeys the power law, where the absolute value of the exponent is an integer equal to $\mu = {7}$ for the cutoff and $\mu = {8}$ for the plateau harmonics. Above a certain critical wavelength, due to the nondipole effects, the efficiency of HHG decreases more strongly than according to a power law, and this decrease is different for different regions of the spectrum. The analytical formulas are derived that match well the calculated wavelength scalings.

© 2019 Optical Society of America

1. INTRODUCTION

Recently, significant progress has been achieved in the development of millijoule-level femtosecond laser sources whose central wavelengths $\lambda $ range from 1.5 to 4 µm and beyond [18]. These advances have great impact on the research of strong-field phenomena [917], in particular, of high-order harmonic generation (HHG) in gases. HHG is known to be the three-step process in which, according to the semi-classical model [18], the electron is optical field ionized, accelerated by the oscillating electric field, and driven back to the parent ion to emit a photon carrying the sum of the electron’s kinetic energy plus the atomic ionization potential ${I_p}$. For HHG driven by visible or near-infrared lasers, the harmonic spectrum is typically plateau-like, where the maximum energy of the photon to be emitted (high-energy cutoff of the plateau) is given by $ \hbar {\Omega _{\max }} \approx {I_{\rm p}} + 3.17\;{U_{\rm p}} $. Since the electron quiver energy ${U_p}$ is proportional to ${\lambda ^2}$, a dramatic extension of the plateau in the high-harmonic spectrum can be achieved using mid-IR sources [19]. This was demonstrated convincingly by producing the harmonics with photon energies exceeding 1 keV in a helium gas irradiated by the pulses delivered by a laser source with a central wavelength of 3.9 µm [10]. The other side of the coin is that the efficiency of HHG by individual atoms turned out to scale unfavorably with the laser wavelength (typically, the harmonic yield within a fixed spectral window scales as ${\lambda ^{ - \mu }}$ with $\mu = {5 - 6}$ for ${0.8}\,\,\unicode{x00B5}{\rm m} \le \lambda \le {4}\,\,\unicode{x00B5}{\rm m}$ [2022]). This decrease in the efficiency of single-atom nonlinear response can be, however, substantially compensated for by implementing phase-matched HHG at high gas pressures and large interaction lengths using, e.g., gas-filled waveguides [10]. Moreover, novel schemes were proposed recently to produce bright sub-kiloelectron volt (keV) or even keV harmonic pulses using HHG driven by mid-infrared lasers [14,23,24]. Furthermore, novel intense ultrafast sources at wavelengths in the long-wave infrared (LWIR, 8–15 µm) regime have been recently demonstrated, which can provide tools to produce coherent x rays in the multi-keV range via HHG [25,26]. It is therefore topical to explore the scaling laws for HHG yield for laser wavelengths beyond 4 µm.

The above-mentioned decrease of the HHG yield with the increase of the driving laser wavelength is understood to be largely due to the spreading of the continuum electron wave packet. This spreading, which is due to the nonlinear ($\omega = h{{\rm k}^2}/{2m}$) relationship between frequency $\omega $ and wavenumber ${\rm k}$ for particle de Broglie waves (see, e.g., [27]) strongly reduces the photon emission by the electrons interacting with their parent ions after being detached from the atoms by a strong laser field [20,28,29]. In addition, high-order harmonic generation in the long-wavelength regime has a number of specific features resulting in a significant change of the spectral properties of the generated radiation in comparison with the case of shorter-wavelength drivers. First, as the laser wavelength increases, the harmonic power gets increasingly localized around the cutoff position instead of being more or less evenly distributed between all harmonics throughout the plateau [14,20,22,30]. Second, the electron magnetic drift effect [3135] becomes more important for longer wavelengths, leading both to the overall decrease of the HHG yield and, moreover, to the reshaping of the emission spectrum, which now loses a familiar structure, becoming arcuate rather than plateau-like [3537]. In addition, for the laser pulse containing a given number of cycles, the bound-state depletion grows with $\lambda $, which may result in a significant decrease of the harmonic yield. It is therefore expected that the scaling laws for high-harmonic yield in the long-wavelength regime can differ significantly from those reported earlier for shorter wavelengths.

Here, we address the above issue by calculating the HHG spectra for $\lambda $ ranging from 2 to 20 µm. The computational cost of the direct numerical integration of the time-dependent Schrödinger equation (TDSE) dramatically increases with increasing laser wavelength, and hence an analytical approach is necessary. In what follows, we used the quantum-mechanical treatment of HHG within the strong-field approximation (SFA) [38] modified properly to take into account the atomic bound-state depletion and the effect of the magnetic field of a laser pulse on the dynamics of the released electron [36,37]. An adequate performance of the modified SFA was demonstrated earlier by applying it to a number of examples, for which the full 3D numerical TDSE simulations are feasible [37]. The use of plane-wave approximation, which underlies the approach developed in Refs. [3638], can be also justified by the fact that, with such long laser wavelengths as those addressed here, the continuum electrons acquire very large kinetic energy and can therefore be very well approximated as plane waves.

2. ANALYTICAL APPROACH

According to [36], the induced dipole moment of the atom in a strong low-frequency laser field, which is linearly polarized along the $x$ axis and propagates along the $z$ axis, can be represented as (hereinafter, atomic units are used)

$$\begin{split}x(t) = &i\int_0^t {d\tau {{\left( {\frac{\pi }{{\varepsilon + i\tau /2}}} \right)}^{{{3} {/ {\vphantom {3 2}}}2}}}} d_x^*\left( {{\textbf{p}_{{\rm st}}}(t,\tau ) - \frac{{\textbf{A}(t)}}{c}} \right)\\&\times\, {d_x}\left( {{\textbf{p}_{{\rm st}}}(t,\tau ) - \frac{{\textbf{A}(t - \tau )}}{c}} \right)F(t - \tau ) \\&\times \exp \left[ { - iS({p_{st,x}},t,\tau ) - i{S_m}({\textbf{p}_{st}},t,\tau )} \right]\\& \times \exp \left[ { - \int_0^t {\frac{{\Gamma \left( {t^\prime } \right)}}{2}dt^\prime } - \int_0^{t - \tau } {\frac{{\Gamma \left( {t^\prime } \right)}}{2}dt^\prime } } \right] + c.c.\end{split}$$
Here c.c. denotes complex conjugation; $ F( t ) $ and $ \textbf{A}( t ) $ are the magnitude of the electric field and the vector potential of the laser pulse, respectively; $ \tau $ is the time of the electron’s free motion in the laser field after release from the atom; ɛ is the regularization parameter, which can be chosen to be small; and $ {d_x}( \textbf{p} ) $ is the $x$ component of the dipole matrix element corresponding to the transition from the ground state to the continuum. For the hydrogen atom whose highly energetic continuum states can be treated as plane waves, this matrix element can be written as
$${d_x}(\textbf{p}) = i\frac{{{2^{{7 \mathord{\left/ {\vphantom {7 2}} \right. } 2}}}}}{\pi }\frac{{{p_x}}}{{{{\left( {{\textbf{p}^2} + 1} \right)}^3}}},$$
$ S( {{p_x},t,\tau } ) $ is the quasi-classical action that describes the free motion of the electron in the laser field when the magnetic field effects are neglected, written as
$$S({p_x},t,\tau ) = \int_{t - \tau }^t {\left[ {\frac{1}{2}{{\left( {{p_x} - \frac{{A(t^\prime )}}{c}} \right)}^2} + {I_{\rm p}}} \right]{\rm d}t^\prime } ,$$
$ {S_m}( {\textbf{p},t,\tau } ) $ is the correction to the quasi-classical action resulting from the influence of the magnetic field of a laser pulse, written as
$$\begin{split}{S_m}\left( {\textbf{p},t,\tau } \right) = \int_{t - \tau }^t {\frac{1}{2}{{\left\{ {{p_z} - \left[ {\frac{{{p_x}A\left( {t^\prime } \right)}}{{{c^2}}} - \frac{{{A^2}\left( {t^\prime } \right)}}{{2{c^3}}}} \right]} \right\}}^2}} {\rm d}t^\prime ,\end{split}$$
$ {\textbf{p}_{st}}( {t,\tau } ) $ is the stationary value of the electron canonical momentum that corresponds to the electron trajectory starting at the nucleus at $ t - \tau $ and returning to the same position at $ t $, written as
$${p_{{\rm st,x}}}(t,\tau ) = \frac{1}{\tau }\int_{t - \tau }^t {\frac{{A(t^\prime )}}{c}{\rm d}t^\prime } ,$$
$${p_{st,z}}\left( {t,\tau } \right) = \frac{{p_{st,x}^2}}{c} - \frac{1}{{2c\tau }}\int_{t - \tau }^t {\frac{{{A^2}\left( {t^\prime } \right)}}{{{c^2}}}} {\rm d}t^\prime ,$$
$ \Gamma ( t ) $ is the time-dependent atomic ionization rate calculated using the analytical formula for the rate of tunneling ionization in a static field, adjusted for the barrier-suppression regime [39] as follows:
$$\Gamma \left( t \right) = \frac{4}{{\left| {F\left( t \right)} \right|}}\exp \left( { - \frac{2}{{3\left| {F\left( t \right)} \right|}}} \right)\exp \left( { - 12\left| {F\left( t \right)} \right|} \right).$$

From the equations given above, it is clearly seen that, if necessary, we can take into account or, conversely, neglect the effects of the electron magnetic drift and atomic bound-state depletion both together and separately.

3. RESULTS

All the calculations presented below were carried out for a hydrogen atom exposed to a linearly polarized trapezoidal laser pulse with 2-cycle linear ramps and a 10-cycle interval of constant amplitude; the laser peak intensity is ${I_0} = {{10}^{14}}\,\,{\rm W}/{{\rm cm}^2}$. The calculations were made using Eqs. (1)–(7). The spectral intensities calculated below are defined as

$$I\left( {{E_\Omega }} \right) \equiv I\left( \Omega \right) = {\left| {\frac{1}{T}\int_0^T {\ddot x\left( t \right)\exp \left( { - i\Omega t} \right){\rm d}t} } \right|^2}.$$
Here $T$ is the total pulse duration; ${E_\Omega }=\Omega $ is the energy of a photon with frequency $\Omega $.

First, we consider the general behavior of the HHG spectrum depending on the laser wavelength. Figure 1 shows the envelopes of the spectra of harmonics generated from different sources with wavelengths ranging from 2 to 16 µm. The calculations were carried out in the simplest approximation, i.e., without taking into account the electron magnetic drift and the atomic bound-state depletion. For convenience, the photon energy on the horizontal axis is normalized to the cutoff energy.

From this figure, it is clearly seen that, as the laser wavelength increases, there is a general decrease in the frequency conversion efficiency according to a power law. Moreover, in agreement with what was mentioned in the Introduction, there is an increasing localization of harmonic power near the cutoff position. While at a laser wavelength of 2 µm, the spectral intensity is, on average, uniformly distributed between all harmonics on a plateau, for a laser source with a wavelength of 16 µm the spectral peak at the cutoff overtops the plateau level by more than an order of magnitude. It is known that anomalously sharpened structures of this type are observed in classical calculations of the energy and angular distributions of electrons rescattered by the parent ions after ejection from atoms in an intense oscillating field [18,40]. These structures were concluded to be essentially attributed to the classical kinematics of electrons in laser fields. In terms of the quantum-mechanical theory of ionization developed by Keldysh [41], these classical photoelectron distributions correspond to the tunneling regime ($\gamma \ll {1}$, where $\gamma = {({I_p}/{2}{U_p})^{1/2}}$ is the Keldysh’s adiabaticity parameter), since the underlying concept of sub-cycle ionization in an oscillating field is central for linking the tunneling instant with a certain phase of the field, while on the contrary, the phase is not defined in the multiphoton limit ($\gamma \gg {1}$) [9]. Since an increase in laser wavelength means a decrease in the Keldysh parameter, the harmonic intensity localization around the cutoff seen in Fig. 1 can be interpreted as one more demonstration that the photoelectron behavior tends to the classical limit, whereas ionization evolves towards pure tunneling. Because of this localization effect, one can expect a significant difference in the wavelength scaling of emission power between the harmonics at the cutoff and on the plateau.

 

Fig. 1. Envelopes of the spectra of harmonics generated from different sources with wavelengths from 2 to 16 µm (see legend). Calculations were carried out without taking into account the electron magnetic drift and the atomic bound-state depletion.

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Fig. 2. Envelopes of the spectra of harmonics generated from the sources with different wavelengths: (a) 8 µm; (b) 14 µm; (c) 20 µm. Calculations were carried out taking into account the atomic bound-state depletion, both in the dipole approximation (i.e., without taking into account the electron magnetic drift) and beyond it (see legend).

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In Fig. 2, the spectra of harmonics generated from different sources are calculated taking into account the electron magnetic drift (“nondipole” case) and the atomic bound-state depletion. For comparison, the results obtained by neglecting the electron magnetic drift (“dipole” case) are also presented. Comparing the behavior of the spectra in Figs. 1 and 2, it can be noted that for laser intensity used in the calculations, the depletion of the atomic bound states does not lead to a visible change in the harmonic spectra. Indeed, it is known that electron tunnelling occurs in sub-cycle ionization steps of very short duration near the field oscillation maxima [42]. Each of these ionization steps causes an ever-increasing depletion of the nonlinear medium, ultimately leading to a decrease of the HHG efficiency [43]. If the laser pulse peak intensity is high, i.e., it significantly exceeds the level at which ionization begins, these ionization steps occur already at the front of the pulse. In this case, the progressive degradation of the HHG efficiency from one half-cycle of the field to the next leads to the decrease in the intensity of the consecutive radiation bursts produced at the front of the pulse and, therefore, to the occurrence of pronounced spectrally resolved half-cycle plateaus decreasing in intensity towards higher photon energies [36,37]. In the results presented in Fig. 2, the spectral reshaping effects of this kind are not observed, which indicates that the depletion of atomic levels does not play a crucial role in this case. As a matter of fact, ionization in the latter case occurs in small steps across the most intense part of the pulse, which can be adequately described in terms of the cycle-averaged ionization rate (see Section 4). At the same time, the magnetic drift of an electron does have a significant effect on the shape of the harmonic spectra presented in Fig. 2; namely, with increasing wavelength, the spectrum acquires an arcuate shape instead of remaining plateau-like. The origin of this behavior can be explained in terms of ionization rates for different electron trajectories contributing to the HHG signal. All electron trajectories, according to the semiclassical analysis [18,38], can be divided into short (with excursion time ${0}\lt\tau\lt {0.65}{T_c}$, where ${T_c}= {2}\pi {/}\omega $ is an optical cycle) and long (with excursion time exceeding ${0.65}{T_c}$) ones. In the long-wavelength regime, due to the electron wave-packet spreading effect, the contribution of the trajectories with $\tau \lt {T_c}$ dominates. For these trajectories, in the dipole approximation, the longer the trajectory, the higher the instantaneous field intensity at the time of its launch; therefore, the corresponding ionization rate grows monotonically towards longer paths. This effect, combined with the competing effect of the electron wave-packet spreading, gives a plateau in the HHG spectrum as a sum of the contributions from all the trajectories. If the magnetic-field effect is significant (nondipole case), the electron’s initial velocity should have a nonzero transverse component to compensate for the magnetic drift. Because of the exponential decrease of the ionization amplitude with increasing the electron’s initial transverse velocity, the balance of contributions of different electron trajectories changes in the nondipole case in favor of a group of returning electron trajectories with highest probability to be launched under these conditions [35]; the contribution of this group gives a maximum in the spectrum of harmonics, which now takes on an arcuate shape. The time-frequency analysis of the harmonic signal confirms this picture [36,37]. Figure 2 indicates that the magnetic drift of an electron results in a substantial correction of the dependence of harmonic yield on the laser wavelength and that this dependence differs strongly between different regions of the spectrum.

After analyzing the general behavior of the harmonic spectra, we proceed directly to the quantitative analysis of the scaling of harmonic yield with the laser wavelength in various regions of the spectrum. To minimize the role of possible fluctuations associated with the quasi-discreteness of the spectrum of the generated harmonics, we will carry out this analysis in terms of averaged values as follows:

$${I_a} = \frac{1}{{\Delta E}}\int_{a\left( {3.173{U_p} + {I_p}} \right)}^{a\left( {3.173{U_p} + {I_p}} \right) + \Delta E} {I\left( {{E_\Omega }} \right)d{E_\Omega }} ,$$
where the spectral intensity is given by Eq. (8), ${I_p} = {0.5}$ is the ionization potential of a hydrogen atom, ${U_p} = F_0^2/{4}{\omega ^2}$ is the electron quiver energy in a laser field with frequency ${\omega}$ and amplitude ${F_0}$, $\Delta E = {5}\,\,{\rm eV}$ is the interval of photon energies over which the averaging is carried out. The width of the averaging interval was chosen to be sufficiently small, but at the same time such that for the minimum laser wavelength addressed in the calculations, several harmonics fall into this interval. We would like to emphasize that there are different kinds of wavelength scaling for the HHG process, depending on which parameters are kept fixed (see [22] and references therein). For example, one may integrate the HHG power spectrum over an interval $\Delta E = {E_f} - {E_i}$ between the fixed harmonic energies ${E_i}$ and ${E_f}$, while also maintaining a fixed laser intensity; for this case, the above-mentioned power law was reported with $\mu = {5 - 6}$ [20,21]. As another option, the laser intensity and the width of the integration interval remain fixed again, but the parameter that is also fixed is not the absolute position of this interval on the energy scale but its relative position with respect to the cutoff energy. In Refs. [44], we studied this kind of wavelength scaling for high-order harmonic yield in the cutoff region of the HHG plateau. In contrast, in the present work we address the harmonic yield not only near the cutoff but also in other regions of the HHG spectrum. Parameter $a$ in Eq. (9) determines the position of the selected interval relative to the cutoff energy. In particular, for $a = {1}$, the quantity ${I_a}$ ($\lambda $) gives the wavelength dependence of the harmonic yield on the cutoff. It is also worth noting that in order to satisfactorily determine the position of the cutoff, greater accuracy is required in the coefficient before ${U_p}$. If, as usual, this coefficient is put equal to 3.17, for large laser wavelengths, the calculated cutoff position is located a few hundred harmonics away from its actually observed position in the harmonic spectrum. Therefore, we use a more accurate value of this coefficient (3.173 [45]), which can be obtained by solving the equations of the classical model [18] with an accuracy of more decimal places.

Figure 3 shows the calculated dependence ${I_a}$ ($\lambda $) for different values of parameter $a$; both the electron magnetic drift and the atomic bound-state depletion are taken into account. First, it should be noted that up to some critical wavelength (${\lambda _c} \approx {8}\,\,\unicode{x00B5}{\rm m}$), all the curves ${I_a}$ ($\lambda $) fit well to the ${\lambda ^{ - \mu }}$ scaling law with the integer value of parameter $\mu $ ($\mu = {7}$ and 8 for the cutoff and plateau harmonics, respectively). Note that for the cutoff harmonics ($a = {1}$), the value of parameter $\mu $ is one unity less than for other points on the plateau addressed here, for which the value of parameter $\mu $ is independent of $a$. For longer wavelengths ($\lambda \gt {\lambda _c}$), the dependences ${I_a}$ ($\lambda $) become more complicated, demonstrating different behavior of the harmonic yield for different regions of the spectrum. In all cases, the observed decrease in the spectral intensity with the laser wavelength is faster than for shorter wavelengths. This behavior can be easily understood if one refers to Fig. 2, which clearly demonstrates that at large laser wavelengths the magnetic drift of an electron leads to a significant decrease in the harmonic yield and to a change in the shape of the harmonic spectrum. The strongest decline of the harmonic yield occurs in the low-energy part of the plateau and at the cutoff. It is this effect that manifests itself in the dependences ${I_a}$ ($\lambda $) at large $\lambda $ in Fig. 3, from which it is also seen that for $a = {0.4}$ and 1, the decrease in the spectral intensity is much faster than for $a = {0.6}$ and 0.8. However, what is not clear from Fig. 3 is whether the steep drop of the harmonic yield in the long-wavelength regime is caused solely by the electron magnetic drift or if there are other effects that make a significant contribution to this decline.

 

Fig. 3. HHG yield versus $\lambda $ for different positions of the integration window (see legend). Both the electron magnetic drift and the atomic bound-state depletion are taken into account. Dashed lines are the approximations by the ${\lambda ^{ - \mu }}$ scaling law.

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Figure 4 shows the same ${I_a}$ ($\lambda $) curves for different values of parameter $a$ but calculated in the dipole approximation (without taking into account the magnetic drift of the electron); the calculations additionally took into account only the effect of the atomic bound-state depletion. It can be seen from the figure that now the dependences ${I_a}$ ($\lambda $) fit much better to the ${\lambda ^{ - \mu }}$ scaling law with integer $\mu $; however, for large laser wavelengths, some discrepancy still remains.

 

Fig. 4. Same as in Fig. 3 but calculated in the dipole approximation.

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Figure 5 shows the dependences ${I_a}$ ($\lambda $) obtained by neglecting both the electron magnetic drift and the atomic bound-state depletion. As mentioned in the Introduction, our calculations were performed for a laser pulse with a fixed number of cycles, hence, the absolute pulse duration increases with a laser wavelength. As a result, with an increase of the laser wavelength, the atomic ionization probability increases, which causes an additional reduction of the harmonic yield. When the depletion effect is excluded from consideration, the dependences ${I_a}$ ($\lambda $) fit perfectly to the ${\lambda ^{ - \mu }}$ scaling law with integer $\mu $ over the entire wavelength range. Thus, all the above-mentioned deviations from this scaling law are completely due to the effects of the electron magnetic drift and atomic bound-state depletion.

 

Fig. 5. Same as in Fig. 4 but calculated by neglecting the atomic bound-state depletion.

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4. ANALYTICAL FORMULAS FOR WAVELENGTH SCALINGS

To explain the difference between the dipole-approximation scaling laws for the harmonic yield on the plateau and on the cutoff and, moreover, the origin of specific values of parameter $\mu $, we refer to the analytical theory developed in Refs. [44,46]. We will focus on the case of the hydrogen atom. According to the factorization formula substantiated in Ref. [44], the harmonic spectral intensity can be represented as

$$I\left( {{E_\Omega }} \right) = \Omega {I_{\rm atomic}}\left( {\tilde F,\omega } \right)W\left( E \right)\sigma _{\rm atomic}^{\left( r \right)}\left( E \right),$$
where $E= {E_\Omega }- {I_p}$, $ {I_{\rm atomic}}( {\tilde F,\omega } ) $ is the ionization factor
$${I_{\rm atomic}}\left( {\tilde F,\omega } \right) = \frac{{8{\omega ^2}}}{{\pi \tilde FF_0^2}}\exp \left( { - \frac{2}{{3\tilde F}}} \right),$$
where $ \tilde F = 0.951{F_0} $ is an effective static electric field, and $W(E)$ is the factor that describes the propagation of an ionized electron in the laser field between the ionization and recombination instants as follows:
$$W\left( E \right) = \frac{{p{{\left( {{\delta \mathord{\left/ {\vphantom {\delta {{I_0}}}} \right. } {{I_0}}}} \right)}^{{2 \mathord{\left/ {\vphantom {2 3}} \right. } 3}}}{{{\mathop{\rm Ai}\nolimits} }^2}\left( \xi \right)}}{{{\tau ^3}}},$$
where $ p = \sqrt {2E} $ and $\delta = {1.866}$. ${\rm Ai}(\xi )$ is the Airy function, whose argument is
$$\xi = {\left( {{\delta \mathord{\left/ {\vphantom {\delta {{I_0}}}} \right. } {{I_0}}}} \right)^{{1 \mathord{\left/ {\vphantom {1 3}} \right. } 3}}}\left( {E - {E_{\max }}} \right);$$
$ {E_{\max }} = 3.173{U_p} + 0.32{I_p} $; $ \sigma _{\rm atomic}^{( r )}( E ) $ is the atomic photorecombination cross section, written as
$$\sigma _{\rm atomic}^{\left( r \right)}\left( E \right) = \frac{{32\pi }}{{{c^3}}}\frac{{\exp \left( { - \frac{4}{p}{\rm arc tan} \left( p \right)} \right)}}{{{p^2}{{\left( {{p^2} + 1} \right)}^2}\left( {1 - \exp \left( { - \frac{{2\pi }}{p}} \right)} \right)}},$$
and $c$ is the speed of light in a vacuum.

Each point in Figs. 35 was obtained by averaging over the photon energy interval $\Delta E$, whose midpoint is ${E_\Omega }=a({3.173}{U_p} + {I_p}) + \Delta E/{2}$; hence, for corresponding points in Eqs. (10)–(14), the electron energy is $E= a({3.173}{U_p} + {I_p}) + \Delta E/{2 - }{I_p}$. For a hydrogen atom and laser parameters given by ${I_0}= {{10}^{14}}\,\,{\rm W}/{{\rm cm}^2}$, $\lambda \ge {2}\,\,\unicode{x00B5}{\rm m}$, and $\Delta E = {5}\,\,{\rm eV}$, the following relations are satisfied:

$$3.173{U_p} \gg {I_p} \gg \Delta E/2,$$
and therefore, in Eqs. (10)–(14), the proportionalities $\Omega \sim{\lambda ^2}$ and $E\sim{\lambda ^2}$ hold with good accuracy. In addition, even with the minimum values of parameter $a$ used here, the asymptotics $ \sigma _{\rm atomic}^{( r )}( E ) \sim {p^{ - 5}} $ for the recombination cross section work well for laser wavelengths of about 4 µm or more. Thus, for the factors entering Eq. (10), we have the following dependences on the laser wavelength: $\Omega \sim{\lambda ^2}$, $ {I_{\rm atomic}}( {\tilde F,\omega } ) \sim {\lambda ^{ - 2}} $, $ W( E ) \sim {\lambda ^{ - 2}}{{\mathop{\rm Ai}\nolimits} ^2}( \xi ) $, and $ \sigma _{\rm atomic}^{( r )}( E ) \sim {\lambda ^{ - 5}} $. Putting together all these factors, we ultimately get
$$I\left( {{E_\Omega }} \right) \sim {\lambda ^{ - 7}}{{\mathop{\rm Ai}\nolimits} ^2}\left( \xi \right),$$
where the argument of the Airy function is
$$\begin{split}\xi = & - {\left( {{\delta \mathord{\left/ {\vphantom {\delta {{I_0}}}} \right. } {{I_0}}}} \right)^{{1 \mathord{\left/ {\vphantom {1 3}} \right. } 3}}}\left[ \left( {1 - a} \right)3.173{U_p} \right.\\ & \left.+ (1.32 - a){I_p} - {{\Delta E} \mathord{\left/ {\vphantom {{\Delta E} 2}} \right. } 2} \right].\end{split}$$
On the cutoff ($a = {1}$), the argument of the Airy function does not depend on $\lambda $; therefore, for the cutoff we find $ I( {{E_\Omega }} ) \sim {\lambda ^{ - 7}} $, in perfect agreement with the corresponding scaling ${I_a}$ ($\lambda $) obtained in the calculations above (Fig. 5).

On the plateau ($a \lt {1}$), due to the relations from Eq. (15), the argument of the Airy function is

$$\begin{split}\xi & \approx - {\left( {{\delta \mathord{\left/ {\vphantom {\delta {{I_0}}}} \right. } {{I_0}}}} \right)^{{1 \mathord{\left/ {\vphantom {1 3}} \right. } 3}}}\left( {1 - a} \right)3.173{U_p}\\ & = - 2.65 \times {10^{ - 8}}\left( {1 - a} \right){\lambda ^2}\,({\rm a}{\rm .}\,{\rm u}{\rm .})\\& = - 9.47\left( {1 - a} \right){\lambda ^2}\,(\unicode{x00B5}{\rm m}).\end{split}$$
Thus, on the plateau, except for the close vicinity of the cutoff, the argument of the Airy function is a large negative quantity; therefore, one can use the well-known asymptotic expression for the Airy function
$${\mathop{\rm Ai}\nolimits} \left( { - \left| \xi \right|} \right) \approx \frac{{\sin \left( {\frac{2}{3}{{\left| \xi \right|}^{{3 \mathord{\left/ {\vphantom {3 2}} \right. } 2}}} + \frac{\pi }{4}} \right)}}{{\sqrt \pi {{\left| \xi \right|}^{{1 \mathord{\left/ {\vphantom {1 4}} \right. } 4}}}}}.$$
In the context of comparing the results of analytical theory discussed here with the calculations of ${I_a}$ ($\lambda $) presented in Section 3, the rapidly oscillating factor in the Airy function is of no interest, since in averaging even over a small interval $\Delta E ={5}\,\,{\rm eV}$ far from the cutoff, a large number of oscillations is summed up. Therefore, only the asymptotics of the Airy function envelope are important, that is,
$${\mathop{\rm Ai}\nolimits} \left( { - \left| \xi \right|} \right) \sim {\left| \xi \right|^{ - {1 \mathord{\left/ {\vphantom {1 4}} \right. } 4}}}$$
at large negative values of the argument. Substituting Eq. (20) into Eq. (16) and taking into account Eq. (18), we finally obtain $ I( {{E_\Omega }} ) \sim {\lambda ^{ - 8}} $ for the harmonics on a plateau, in full agreement with the corresponding scaling law for ${I_a}$ ($\lambda $) obtained above (Fig. 5).

We now turn to the results shown in Fig. 4 in order to consider how the scaling law for the harmonic yield can be corrected analytically to take into account the atomic bound-state depletion. According to Eqs. (1), (8), and (9), for a small degree of ionization of an atom over the entire duration of the pulse (allowing us to use a cycle-averaged ionization rate), the sought-for correction to the HHG yield can be reduced to the introduction of the following factor:

$${R_{\rm depl}} = \exp \left( { - \alpha \Gamma T} \right),$$
where $ \Gamma $ is the ionization rate [Eq. (7)] in the field maximum, and $ \alpha $ is a dimensionless coefficient. By adjusting the coefficient $ \alpha $ for the pulse used in the calculation, we obtain the correction factor from Eq. (21) in the form
$${R_{\rm depl}} = \exp \left( { - 1.6\Gamma \lambda /c} \right).$$
Figure 6 demonstrates a good agreement between the results of calculations using Eqs. (1)–(9), and the analytical scaling law for the HHG yield with the correction factor from Eq. (22) taken into account.
 

Fig. 6. Same as in Fig. 4 but with the correction factor Eq. (22) taken into account.

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In the following, we will consider the role of the electron’ magnetic drift effect. To derive an analytical formula that quantifies the decrease in the HHG efficiency due to this effect, we will follow an approach similar to that proposed in Ref. [47]. The magnetic part of the Lorentz force leads to the displacement of the electron wave packet in the laser wave propagation direction (see [3133] and references therein). As a result, the main contribution to the harmonic generation will be made not by the central part of the electron wave packet, but rather by its wings, which correspond to electrons having a nonzero transverse momentum when ejected from the atom. In other words, to ensure the recollision of the continuum electron, initial transverse momentum of the ionized electron is required to compensate for the displacement induced by the magnetic field. Since the ionization rate decreases if the electron is tunnel ionized with a nonzero transverse momentum, this decrease finally results in the reduction of the HHG efficiency.

According to the theory of atomic tunneling ionization, the dependence of the ionization rate on the ejected electron momentum $ {p_ \bot } $, which is perpendicular to the polarization axis of the field, is given by [48]

$${\Gamma _{{p_ \bot }}}(t) = {\Gamma _0}\exp \left( { - \frac{2}{{3\left| {F(t)} \right|}}{{\left( {2{I_p} + p_ \bot ^2} \right)}^{3/2}}} \right).$$
The induced dipole moment is determined by the amplitude of the returning electron wave packet; therefore, the modification of the scaling law ${I_a}$ ($\lambda $) due to the electron magnetic drift effect can be represented as the factor
$$\begin{split}{R_{ND}}(a) = \exp \left\{ {\frac{2}{{3\left| {F(a)} \right|}}\left[ {{{\left( {2{I_p}} \right)}^{3/2}} - {{\left( {2{I_p} + p_ \bot ^2(a)} \right)}^{3/2}}} \right]} \right\},\end{split}$$
where parameter $a$ determines the instant of the field-induced electron ejection, the magnitude of the field at this instant, and also the $ {p_ \bot } $ value.

For the case of a monochromatic field $ F(t) = {F_0}\sin (\omega t) $, one can obtain a simple expression for $ {p_ \bot } $ as follows:

$$\begin{split}\!\!\!\!\!{p_ \bot } &= \frac{1}{\tau }\int_{t - \tau }^t {\dot z(t^\prime )dt^\prime } \\ & = - \frac{{F_0^2}}{{c\omega \tau }}\int_{t - \tau }^t {dt^\prime \int_{t - \tau }^{t^\prime } {\left[ {\cos \left( {\omega t^{\prime \prime} } \right) - \cos \varphi } \right]} \sin \left( {\omega t^{\prime \prime} } \right)dt^{\prime \prime} } \\ & = \beta \frac{{{U_p}}}{c},\end{split}$$
where
$$\beta = \frac{1}{{2\omega \tau }}\left[ {\sin (2\omega \tau + 2\varphi ) - \sin (2\varphi )} \right] - \cos (2\varphi ),$$
$${\rm tg}\varphi = \frac{{\omega \tau - \sin (\omega \tau )}}{{\cos (\omega \tau ) - 1}},$$
with $ \varphi = \omega (t - \tau ) $ the phase of the field at the ionization instant. In this case, parameter $a$ can be represented as
$$\begin{split}a & = \frac{{E + {I_p}}}{{3.173{U_p} + {I_p}}} \approx \frac{E}{{3.173{U_p}}}\\ &= \frac{2}{{3.173}}{\left[ {\cos (\omega \tau + \varphi ) - \cos \varphi } \right]^2}.\end{split}$$
Equations (25)–(28) parametrically represent the dependence $ {p_ \bot }(a) $ shown in Fig. 7. Figure 8 shows the dependence $ F(a) = {F_0}\sin (\varphi ) $. From Figs. 7 and 8 it can be seen that the magnitude of the nondipole correction to the harmonic yield strongly depends on the position on the plateau. In addition, this correction turns out to be very different for short and long trajectories. Since, for the above-mentioned reasons, it is difficult to deduce a simple formula to describe the nondipole corrections to the harmonic intensity through the entire broad HHG spectrum, a simplified approach was proposed in Ref. [47]. This approach assumes an “effective” initial transverse momentum, which, like in Eq. (25), is proportional to ${{U}_p}/{ c}$, whereas the coefficient of proportionality is assumed to be constant, i.e., trajectory independent; this constant is treated as a fitting parameter whose value is found based on the criterion of the best agreement between the analytical formula for the harmonic yield and the results of modified SFA calculations. The fitting curves presented in Ref. [47] show that this simple model does allow one to evaluate nondipole effects for a limited range of laser parameters and harmonic orders. However, as we have checked, this approach does not allow one to satisfactorily evaluate the contribution of the nondipole effects for entire set of data presented in Fig. 3. In the following, we go beyond the scope of the above simplified approach by calculating the trajectory-dependent transverse momentum according to Eqs. (25)–(28).
 

Fig. 7. Relation between parameters $a$ and β, which, according to Eqs. (25),(28), gives the $ {p_ \bot } $ value as a function of the relative position of the harmonic frequency within the plateau, determined by parameter $a$, for the electron short and long trajectories contributing predominantly to the harmonic photon emission.

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Fig. 8. Normalized magnitude of the electric field at the instants of launching the electron short and long trajectories as a function of the relative position of the harmonic frequency within the plateau.

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The results presented in Figs. 7 and 8 allow one, by calculating the factors in Eq. (24) separately for the short and long trajectory, to find the corrected weights of these trajectories in the total signal at a given harmonic frequency. However, in order to find the corrected total harmonic yield, one needs to know a priori the initial weight coefficients for these trajectories (determined in the dipole approximation) that are to be corrected. The only point on the plateau, which does not require additional a priori information, is the cutoff, where the two trajectories coalesce. For $a ={1}$, the magnetic-drift-induced correction factor for the case of a hydrogen atom takes the form

$$\begin{split}{R_{ND}}(1) = &\exp \left\{ \frac{2}{{3 \times 0.951{F_0}}}\right.\\&\times\left.\left[ {1 - {{\left( {1 + {{\left( {0.809\frac{{{U_p}}}{c}} \right)}^2}} \right)}^{\frac{3}{2}}}} \right] \right\}.\end{split}$$

Figure 9 provides a comparison of the scaling laws for HHG yield obtained analytically, taking into account the correction factors from Eqs. (22) and (24) with the results of SFA calculations using Eqs. (1)–(9). For the cutoff ($a= {1}$), one can see excellent agreement between the analytical scaling and the SFA results. For $a ={0.8}$, if the electron magnetic drift is neglected, the contributions of the short and long trajectories to the harmonics signal are comparable. However, if the electron magnetic drift is taken into account, it affects the long trajectories much more strongly, and as $\lambda $ increases, the harmonic signal in this region of the plateau is almost entirely governed by the short electron trajectory. Therefore, in the case of $a= {0.8}$, the analytical scaling just for the short trajectory was chosen for comparison with the SFA calculations (see Fig. 9). From Fig. 9 it can be seen that this scaling gives good agreement with the results of SFA calculations. In the case of $a ={0.2}$, once the electron magnetic drift is ignored, the harmonic signal is almost entirely governed by the long electron trajectories. This is confirmed by a good agreement between the results of SFA calculations and the analytical scaling for long-trajectory contributions up to wavelengths of 12–13 µm. For larger $\lambda $, the figure shows a strong difference between these curves. This discrepancy can be explained by the fact that at these wavelengths, the harmonic signal due to long trajectories is so strongly suppressed because of the electron magnetic drift that it becomes comparable to the harmonic signal due to the short trajectory. In the limit of infinitely large $\lambda $, the harmonic signal across the entire plateau is dominated by a short electron path.

 

Fig. 9. Same as in Fig. 3 but with correction factors Eqs. (22) and (24) taken into account.

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5. DISCUSSION

The study presented above has been performed for the particular case of a hydrogen atom driven by a multicycle laser field with peak intensity ${I_0} = {{10}^{14}}\,\,{\rm W}/{{\rm cm}^2}$. It would be interesting to discuss how the scaling laws for the HHG yield will depend on the generation conditions, for example, on the peak intensity and duration of a laser pulse or on the choice of an atomic target.

Obviously, the critical wavelength ${\lambda _c}$ that draws the boundary between the dipole and nondipole regimes should vary with a change of the laser intensity. The value of ${\lambda _c}$ is determined mainly by the dynamics of the electron in the continuum and can be estimated from a comparison of the oscillatory velocity (${v_{osc}}= {F_0}/\omega $) of the electron with the speed of light. The critical ratio ${F_0}/\omega $ deduced from our calculations is ${F_0}/\omega {\approx}{9.3}$, which corresponds to ${v_{osc}}\approx{0.07}{\rm c}$, in excellent agreement with the estimate given in Ref. [28]. Further, a change of the peak intensity of the laser pulse should vary the rate of depletion of the nonlinear medium. For instance, as mentioned in Section 3, in a rapidly growing field at the front of an intense laser pulse, the population of atomic bound states can decrease stepwise near each maximum of the oscillating driving field, giving rise to a series of multiple half-cycle plateaus decreasing in intensity towards higher photon energies. It should be noted, however, that the cutoff positions of these plateaus in relative units will not depend on the driver wavelength, i.e., in a sense, the HHG spectra for different $\lambda $ will be self-similar. Consequently, the wavelength dependence of the integrated yield, determined according to Eq. (9), will not contain any sharp features associated with jumps from one step to another. The transformation of a single wide plateau in the HHG spectrum into several plateaus of different intensities should lead to splitting of the families of curves, which, in the dipole regime, are merged in the case of a weak field as presented above. In other aspects, the wavelength scalings for different half-cycle plateaus in the dipole regime are not expected to differ strongly from those shown in Fig. 5 for the multicycle case. The same, in general, is true for the case of a few-cycle driving pulse, for which the spectrally resolved half-cycle plateaus can also arise [49]. An additional complication in the latter case is, however, associated with a strong carrier-envelope-phase (CEP) dependence of the HHG spectra [50] and, moreover, with a CEP-dependent interference between electron wave packets originating from ionization at neighboring half-cycles [51]. In order to get the full picture for this case, in general, special consideration is required. However, it should be noted that a generalization of the theory [44,46] to the case of few-cycle pulses [49,51] shows that the analytical structure of the spectral dependences of the HHG that yields near half-cycle cutoffs is generally the same as in the case of a multicycle pulse. In particular, the spectral interference of the contributions of short and long electron trajectories is described by the same Airy function; therefore, if half-cycle plateaus are spectrally well separated, one should expect the same difference in wavelength scalings for the regions near the cutoffs and far from them, as for the case of a multicycle pulse considered in this work.

The difference in wavelength scalings for different atomic species is expected to stem, to a large extent, from the difference in the ionization potentials ${I_p}$. In particular, the boundary between the weak- and strong-field regimes, in terms of the atomic bound-state depletion, will shift depending on ${I_p}$. Furthermore, the wavelength scalings of the kind considered in this work may be significantly modified due to atomic structure effects [44].

6. CONCLUSION

In conclusion, in this work, by the example of atomic hydrogen, we theoretically studied the process of high-order harmonic generation in gases for a wide interval of laser wavelengths ranging from 2 to 20 µm. The wavelength scaling laws were found for the harmonic yield in different spectral intervals having a fixed width and relative position with respect to the cutoff energy. The frequency conversion efficiency calculated without taking into account the atomic bound-state depletion and the electron magnetic drift is shown to decrease in the entire wavelength range as ${\lambda ^{ - \mu }}$, in agreement with the previous studies for the wavelength range from 0.8 to 4 µm. It was found that the absolute value of the exponent in this scaling law is an integer equal to $\mu = {7}$ for the cutoff and $\mu = {8}$ for all parts of the plateau, except for a close vicinity of the cutoff. Using an analytical theory, it was explained why the decline of the harmonic yield on the cutoff and on the plateau occurs at different rates, and the origin of the observed values of µ has been clarified. Calculations taking into account the electron magnetic drift and the atomic bound-state depletion showed that both effects accelerate the drop of the conversion efficiency, starting at wavelengths of about 8 µm for the hydrogen atom with the laser pulse parameters chosen for this study. In this regime, the wavelength scaling of the harmonic yield no more obeys the power law and, moreover, the decline of the conversion efficiency with the laser wavelength becomes different for different regions of the spectrum. While for relatively short laser wavelengths the harmonics are most efficiently generated around the cutoff, at wavelengths longer than 15 µm, due to the magnetic-field-induced reshaping of the harmonic spectrum, the region of the most efficient harmonic generation shifts to the central part of the spectrum. The analytical formulas are derived that provide an adequate description of the contribution of different physical effects to the overall wavelength scaling of the harmonic intensities in a wide range of parameters. The results of our study are useful to optimize the conditions for producing extremely short extreme ultraviolet (XUV) and x-ray pulses using ultrahigh-order harmonic generation driven by mid- and long-wave infrared lasers.

Funding

Russian Academy of Sciences (0035-2018-0005); Russian Foundation for Basic Research (18-02-00924, 18-41-520006); Russian Science Foundation (16-12-10279).

Acknowledgment

The authors are grateful to the Joint Supercomputer Center of RAS for the provided supercomputer resources.

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References

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  1. C. Vozzi, F. Calegari, E. Benedetti, S. Gasilov, G. Sansone, G. Cerullo, M. Nisoli, S. De Silvestri, and S. Stagira, “Millijoule-level phase-stabilized few-optical-cycle infrared parametric source,” Opt. Lett. 32, 2957–2959 (2007).
    [Crossref]
  2. G. Andriukaitis, T. Balciunas, S. Ališauskas, A. Pugžlys, A. Baltuška, T. Popmintchev, M. C. Chen, M. M. Murnane, and H. C. Kapteyn, “90 GW peak power few-cycle mid-infrared pulses from an optical parametric amplifier,” Opt. Lett. 36, 2755–2757 (2011).
    [Crossref]
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2019 (2)

D. J. Wilson, A. M. Summers, S. Zigo, B. Davis, S.-J. Robatjazi, J. A. Powell, D. Rolles, A. Rudenko, and C. A. Trallero-Herrero, “An intense, few-cycle source in the long-wave infrared,” Sci. Rep. 9, 6002 (2019).
[Crossref]

E. Migal, A. Pushkin, B. Bravy, V. Gordienko, N. Minaev, A. Sirotkin, and F. Potemkin, “3.5 mJ 150 fs Fe:ZnSe hybrid mid-IR femtosecond laser at 4.4 µm for driving extreme nonlinear optics,” Opt. Lett. 44, 2550–2553 (2019).
[Crossref]

2018 (4)

2017 (1)

2016 (3)

2015 (4)

M. V. Frolov, N. L. Manakov, W.-H. Xiong, L.-Y. Peng, J. Burgdörfer, and A. F. Starace, “Scaling laws for high-order-harmonic generation with midinfrared laser pulses,” Phys. Rev. A 92, 023409 (2015).
[Crossref]

A. S. Emelina, M. Yu. Emelin, and M. Yu. Ryabikin, “Multi-keV ultrahigh-order harmonics produced in gases with subrelativistically intense mid-IR laser pulses,” J. Opt. Soc. Am. B 32, 2478–2487 (2015).
[Crossref]

A. V. Mitrofanov, A. A. Voronin, D. A. Sidorov-Biryukov, A. Pugžlys, E. A. Stepanov, G. Andriukaitis, T. Flöry, S. Ališauskas, A. B. Fedotov, A. Baltuška, and A. M. Zheltikov, “Mid-infrared laser filaments in the atmosphere,” Sci. Rep. 5, 8368 (2015).
[Crossref]

B. Wolter, M. G. Pullen, M. Baudisch, M. Sclafani, M. Hemmer, A. Senftleben, C. D. Schröter, J. Ullrich, R. Moshammer, and J. Biegert, “Strong-field physics with mid-IR fields,” Phys. Rev. X 5, 021034 (2015).
[Crossref]

2014 (3)

A.-T. Le, H. Wei, C. Jin, V. N. Tuoc, T. Morishita, and C. D. Lin, “Universality of returning electron wave packet in high-order harmonic generation with midinfrared laser pulses,” Phys. Rev. Lett. 113, 033001 (2014).
[Crossref]

K.-H. Hong, C.-J. Lai, J. P. Siqueira, P. Krogen, J. Moses, C.-L. Chang, G. J. Stein, L. E. Zapata, and F. X. Kärtner, “Multi-mJ, kHz, 2.1 µm optical parametric chirped-pulse amplifier and high-flux soft x-ray high-harmonic generation,” Opt. Lett. 39, 3145–3148 (2014).
[Crossref]

A. S. Emelina, M. Y. Emelin, and M. Y. Ryabikin, “On the possibility of the generation of high harmonics with photon energies greater than 10 keV upon interaction of intense mid-IR radiation with neutral gases,” Quantum Electron. 44, 470–477 (2014).
[Crossref]

2013 (1)

C. Hernández-Garsía, J. A. Pérez-Hernández, T. Popmintchev, M. M. Murnane, H. C. Kapteyn, A. Jaron-Becker, A. Becker, and L. Plaja, “Zeptosecond high harmonic keV x-ray waveforms driven by midinfrared laser pulses,” Phys. Rev. Lett. 111, 003002 (2013).
[Crossref]

2012 (6)

Y. Deng, A. Schwarz, H. Fattahi, M. Ueffing, X. Gu, M. Ossiander, T. Metzger, V. Pervak, H. Ishizuki, T. Taira, T. Kobayashi, G. Marcus, F. Krausz, R. Kienberger, and N. Karpowicz, “Carrier-envelope-phase-stable, 1.2 mJ, 1.5 cycle laser pulses at 2.1 µm,” Opt. Lett. 37, 4973–4975 (2012).
[Crossref]

C. Vozzi, M. Negro, and S. Stagira, “Strong-field phenomena driven by mid-infrared ultrafast sources,” J. Mod. Opt. 59, 1283–1302 (2012).
[Crossref]

C. I. Blaga, J. Xu, A. D. DiChiara, E. Sistrunk, K. Zhang, P. Agostini, T. A. Miller, L. F. DiMauro, and C. D. Lin, “Imaging ultrafast molecular dynamics with laser-induced electron diffraction,” Nature 483, 194–197 (2012).
[Crossref]

T. Popmintchev, M. C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Ališauskas, G. Andriukaitis, T. Balciunas, O. D. Mücke, A. Pugžlys, A. Baltuška, B. Shim, S. E. Schrauth, A. Gaeta, A. Hernández-Garsía, L. Plaja, A. Becker, A. Jaron-Becker, M. M. Murnane, and H. C. Kapteyn, “Bright coherent ultrahigh harmonics in the keV X-ray regime from mid-infrared femtosecond lasers,” Science 336, 1287–1291 (2012).
[Crossref]

A. D. DiChiara, E. Sistrunk, C. I. Blaga, U. B. Szafruga, P. Agostini, and L. F. DiMauro, “Inelastic scattering of broadband electron wave packets driven by an intense midinfrared laser field,” Phys. Rev. Lett. 108, 003002 (2012).
[Crossref]

M. V. Frolov, N. L. Manakov, A. M. Popov, O. V. Tikhonova, E. A. Volkova, A. A. Silaev, N. V. Vvedenskii, and A. F. Starace, “Analytic theory of high-order-harmonic generation by an intense few-cycle laser pulse,” Phys. Rev. A 85, 033416 (2012).
[Crossref]

2011 (2)

G. Andriukaitis, T. Balciunas, S. Ališauskas, A. Pugžlys, A. Baltuška, T. Popmintchev, M. C. Chen, M. M. Murnane, and H. C. Kapteyn, “90 GW peak power few-cycle mid-infrared pulses from an optical parametric amplifier,” Opt. Lett. 36, 2755–2757 (2011).
[Crossref]

M. V. Frolov, N. L. Manakov, A. A. Silaev, N. V. Vvedenskii, and A. F. Starace, “High-order harmonic generation by atoms in a few-cycle laser pulse: carrier-envelope phase and many-electron effects,” Phys. Rev. A 83, 021405 (2011).
[Crossref]

2009 (2)

M. V. Frolov, N. L. Manakov, T. S. Sarantseva, and A. F. Starace, “Analytic formulae for high harmonic generation,” J. Phys. B 42, 035601 (2009).
[Crossref]

M. V. Frolov, N. L. Manakov, T. S. Sarantseva, M. Yu. Emelin, M. Yu. Ryabikin, and A. F. Starace, “Analytic description of the high-energy plateau in harmonic generation by atoms: Can the harmonic power increase with increasing laser wavelengths?” Phys. Rev. Lett. 102, 243901 (2009).
[Crossref]

2008 (2)

M. V. Frolov, N. L. Manakov, and A. F. Starace, “Wavelength scaling of high-harmonic yield: threshold phenomena and bound state symmetry dependence,” Phys. Rev. Lett. 100, 173001 (2008).
[Crossref]

P. Colosimo, G. Doumy, C. I. Blaga, J. Wheeler, C. Hauri, F. Catoire, J. Tate, R. Chirla, A. M. March, G. G. Paulus, H. G. Muller, P. Agostini, and L. F. DiMauro, “Scaling strong-field interactions towards the classical limit,” Nat. Phys. 4, 386–389 (2008).
[Crossref]

2007 (3)

J. Tate, T. Auguste, H. G. Muller, P. Salières, P. Agostini, and L. F. DiMauro, “Scaling of wave-packet dynamics in an intense midinfrared field,” Phys. Rev. Lett. 98, 013901 (2007).
[Crossref]

C. Vozzi, F. Calegari, E. Benedetti, S. Gasilov, G. Sansone, G. Cerullo, M. Nisoli, S. De Silvestri, and S. Stagira, “Millijoule-level phase-stabilized few-optical-cycle infrared parametric source,” Opt. Lett. 32, 2957–2959 (2007).
[Crossref]

M. Uiberacker, Th. 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 446, 627–632 (2007).
[Crossref]

2006 (1)

V. V. Strelkov, A. F. Sterjantov, N. Y. Shubin, and V. T. Platonenko, “XUV generation with several-cycle laser pulse in barrier-suppression regime,” J. Phys. B 39, 577–589 (2006).
[Crossref]

2005 (2)

X. M. Tong and C. D. Lin, “Empirical formula for static field ionization rates of atoms and molecules by lasers in the barrier-suppression regime,” J. Phys. B 38, 2593–2600 (2005).
[Crossref]

M. Yu. Emelin, M. Yu. Ryabikin, A. M. Sergeev, M. D. Chernobrovtseva, T. Pfeifer, D. Walter, and G. Gerber, “High-efficiency generation of attosecond pulses during atomic ionization from excited electronic states,” Europhys. Lett. 69, 913–919 (2005).
[Crossref]

2003 (1)

A. Baltuška, Th. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, Ch. Gohle, R. Holzwarth, V. S. Yakovlev, A. Scrinzi, T. W. Hänsch, and F. Krausz, “Attosecond control of electronic processes by intense light fields,” Nature 421, 611–615 (2003).
[Crossref]

2002 (1)

C. C. Chirilă, N. J. Kylstra, R. M. Potvliege, and C. J. Joachain, “Nondipole effects in photon emission by laser-driven ions,” Phys. Rev. A 66, 063411 (2002).
[Crossref]

2001 (2)

N. J. Kylstra, R. M. Potvliege, and C. J. Joachain, “Photon emission by ions interacting with short intense laser pulses: beyond the dipole approximation,” J. Phys. B 34, L55–L61 (2001).
[Crossref]

B. Shan and Z. Chang, “Dramatic extension of the high-order harmonic cutoff by using a long-wavelength driving field,” Phys. Rev. A 65, 011804 (2001).
[Crossref]

2000 (2)

M. W. Walser, C. H. Keitel, A. Scrinzi, and T. Brabec, “High harmonic generation beyond the electric dipole approximation,” Phys. Rev. Lett. 85, 5082–5085 (2000).
[Crossref]

V. D. Taranukhin, “Relativistic high-order harmonic generation,” Laser Phys. 10, 330–336 (2000).
[Crossref]

1999 (1)

A. V. Kim, M. Y. Ryabikin, and A. M. Sergeev, “From femtosecond to attosecond pulses,” Phys. Usp. 42, 54–61 (1999).
[Crossref]

1995 (1)

M. V. Fedorov and J. Peatross, “Strong-field photoionization and emission of light in the wave-packet-spreading regime,” Phys. Rev. A 52, 504–513 (1995).
[Crossref]

1994 (3)

W. Becker, S. Long, and J. K. McIver, “Modeling harmonic generation by a zero-range potential,” Phys. Rev. A 50, 1540–1560 (1994).
[Crossref]

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

G. G. Paulus, W. Becker, W. Nicklich, and H. Walther, “Rescattering effects in above-threshold ionization: a classical model,” J. Phys. B 27, L703–L708 (1994).
[Crossref]

1993 (1)

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

1991 (1)

1965 (1)

L. V. Keldysh, “Ionization in the field of a strong electromagnetic wave,” Sov. Phys. JETP 20, 1307–1314 (1965).
[Crossref]

Agostini, P.

C. I. Blaga, J. Xu, A. D. DiChiara, E. Sistrunk, K. Zhang, P. Agostini, T. A. Miller, L. F. DiMauro, and C. D. Lin, “Imaging ultrafast molecular dynamics with laser-induced electron diffraction,” Nature 483, 194–197 (2012).
[Crossref]

A. D. DiChiara, E. Sistrunk, C. I. Blaga, U. B. Szafruga, P. Agostini, and L. F. DiMauro, “Inelastic scattering of broadband electron wave packets driven by an intense midinfrared laser field,” Phys. Rev. Lett. 108, 003002 (2012).
[Crossref]

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M. V. Frolov, N. L. Manakov, A. A. Silaev, N. V. Vvedenskii, and A. F. Starace, “High-order harmonic generation by atoms in a few-cycle laser pulse: carrier-envelope phase and many-electron effects,” Phys. Rev. A 83, 021405 (2011).
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A. S. Johnson, D. R. Austin, D. A. Wood, C. Brahms, A. Gregory, K. B. Holzner, S. Jarosch, E. W. Larsen, S. Parker, C. S. Strüber, P. Ye, J. W. G. Tisch, and J. P. Marangos, “High-flux soft x-ray harmonic generation from ionization-shaped few-cycle laser pulses,” Sci. Adv. 4, eaar3761 (2018).
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D. J. Wilson, A. M. Summers, S. Zigo, B. Davis, S.-J. Robatjazi, J. A. Powell, D. Rolles, A. Rudenko, and C. A. Trallero-Herrero, “An intense, few-cycle source in the long-wave infrared,” Sci. Rep. 9, 6002 (2019).
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D. J. Wilson, A. M. Summers, S. Zigo, B. Davis, S.-J. Robatjazi, J. A. Powell, D. Rolles, A. Rudenko, and C. A. Trallero-Herrero, “An intense, few-cycle source in the long-wave infrared,” Sci. Rep. 9, 6002 (2019).
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Figures (9)

Fig. 1.
Fig. 1. Envelopes of the spectra of harmonics generated from different sources with wavelengths from 2 to 16 µm (see legend). Calculations were carried out without taking into account the electron magnetic drift and the atomic bound-state depletion.
Fig. 2.
Fig. 2. Envelopes of the spectra of harmonics generated from the sources with different wavelengths: (a) 8 µm; (b) 14 µm; (c) 20 µm. Calculations were carried out taking into account the atomic bound-state depletion, both in the dipole approximation (i.e., without taking into account the electron magnetic drift) and beyond it (see legend).
Fig. 3.
Fig. 3. HHG yield versus $\lambda $ for different positions of the integration window (see legend). Both the electron magnetic drift and the atomic bound-state depletion are taken into account. Dashed lines are the approximations by the ${\lambda ^{ - \mu }}$ scaling law.
Fig. 4.
Fig. 4. Same as in Fig. 3 but calculated in the dipole approximation.
Fig. 5.
Fig. 5. Same as in Fig. 4 but calculated by neglecting the atomic bound-state depletion.
Fig. 6.
Fig. 6. Same as in Fig. 4 but with the correction factor Eq. (22) taken into account.
Fig. 7.
Fig. 7. Relation between parameters $a$ and β, which, according to Eqs. (25),(28), gives the $ {p_ \bot } $ value as a function of the relative position of the harmonic frequency within the plateau, determined by parameter $a$ , for the electron short and long trajectories contributing predominantly to the harmonic photon emission.
Fig. 8.
Fig. 8. Normalized magnitude of the electric field at the instants of launching the electron short and long trajectories as a function of the relative position of the harmonic frequency within the plateau.
Fig. 9.
Fig. 9. Same as in Fig. 3 but with correction factors Eqs. (22) and (24) taken into account.

Equations (29)

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x ( t ) = i 0 t d τ ( π ε + i τ / 2 ) 3 / 3 2 2 d x ( p s t ( t , τ ) A ( t ) c ) × d x ( p s t ( t , τ ) A ( t τ ) c ) F ( t τ ) × exp [ i S ( p s t , x , t , τ ) i S m ( p s t , t , τ ) ] × exp [ 0 t Γ ( t ) 2 d t 0 t τ Γ ( t ) 2 d t ] + c . c .
d x ( p ) = i 2 7 / 7 2 2 π p x ( p 2 + 1 ) 3 ,
S ( p x , t , τ ) = t τ t [ 1 2 ( p x A ( t ) c ) 2 + I p ] d t ,
S m ( p , t , τ ) = t τ t 1 2 { p z [ p x A ( t ) c 2 A 2 ( t ) 2 c 3 ] } 2 d t ,
p s t , x ( t , τ ) = 1 τ t τ t A ( t ) c d t ,
p s t , z ( t , τ ) = p s t , x 2 c 1 2 c τ t τ t A 2 ( t ) c 2 d t ,
Γ ( t ) = 4 | F ( t ) | exp ( 2 3 | F ( t ) | ) exp ( 12 | F ( t ) | ) .
I ( E Ω ) I ( Ω ) = | 1 T 0 T x ¨ ( t ) exp ( i Ω t ) d t | 2 .
I a = 1 Δ E a ( 3.173 U p + I p ) a ( 3.173 U p + I p ) + Δ E I ( E Ω ) d E Ω ,
I ( E Ω ) = Ω I a t o m i c ( F ~ , ω ) W ( E ) σ a t o m i c ( r ) ( E ) ,
I a t o m i c ( F ~ , ω ) = 8 ω 2 π F ~ F 0 2 exp ( 2 3 F ~ ) ,
W ( E ) = p ( δ / δ I 0 I 0 ) 2 / 2 3 3 Ai 2 ( ξ ) τ 3 ,
ξ = ( δ / δ I 0 I 0 ) 1 / 1 3 3 ( E E max ) ;
σ a t o m i c ( r ) ( E ) = 32 π c 3 exp ( 4 p a r c t a n ( p ) ) p 2 ( p 2 + 1 ) 2 ( 1 exp ( 2 π p ) ) ,
3.173 U p I p Δ E / 2 ,
I ( E Ω ) λ 7 Ai 2 ( ξ ) ,
ξ = ( δ / δ I 0 I 0 ) 1 / 1 3 3 [ ( 1 a ) 3.173 U p + ( 1.32 a ) I p Δ E / Δ E 2 2 ] .
ξ ( δ / δ I 0 I 0 ) 1 / 1 3 3 ( 1 a ) 3.173 U p = 2.65 × 10 8 ( 1 a ) λ 2 ( a . u . ) = 9.47 ( 1 a ) λ 2 ( µ m ) .
Ai ( | ξ | ) sin ( 2 3 | ξ | 3 / 3 2 2 + π 4 ) π | ξ | 1 / 1 4 4 .
Ai ( | ξ | ) | ξ | 1 / 1 4 4
R d e p l = exp ( α Γ T ) ,
R d e p l = exp ( 1.6 Γ λ / c ) .
Γ p ( t ) = Γ 0 exp ( 2 3 | F ( t ) | ( 2 I p + p 2 ) 3 / 2 ) .
R N D ( a ) = exp { 2 3 | F ( a ) | [ ( 2 I p ) 3 / 2 ( 2 I p + p 2 ( a ) ) 3 / 2 ] } ,
p = 1 τ t τ t z ˙ ( t ) d t = F 0 2 c ω τ t τ t d t t τ t [ cos ( ω t ) cos φ ] sin ( ω t ) d t = β U p c ,
β = 1 2 ω τ [ sin ( 2 ω τ + 2 φ ) sin ( 2 φ ) ] cos ( 2 φ ) ,
t g φ = ω τ sin ( ω τ ) cos ( ω τ ) 1 ,
a = E + I p 3.173 U p + I p E 3.173 U p = 2 3.173 [ cos ( ω τ + φ ) cos φ ] 2 .
R N D ( 1 ) = exp { 2 3 × 0.951 F 0 × [ 1 ( 1 + ( 0.809 U p c ) 2 ) 3 2 ] } .

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