1. INTRODUCTION

In recent years, significant progress has been made in the development of parametric femtosecond light sources of a millijoule level with wavelengths in the mid- and long-wave infrared ranges [1–9]. The use of these sources for high-order harmonic generation (HHG) in gases makes it possible to dramatically widen the plateau in the spectrum of the generated radiation [10,11]. The major improvements in generation of high-brightness spectrally broad high-harmonic radiation have been achieved by using combined driving pulses (e.g., near-infrared and mid-infrared fields in different combinations) [12–14]. This possibility is due to the favorable scaling of the electron ponderomotive energy (${U_p}\sim I\lambda _0^2$, where $ I $ and $ \lambda_{0} $ are the intensity and wavelength of the laser field, respectively), which determines, in accordance with the three-step model of the HHG process in gases [15,16], the plateau cutoff energy according to the relation ${E_{{\max }}} = 3.17{U_p} + {I_p}$ [17], where $ I_p $ is the atomic ionization potential. However, there are a number of physical effects that limit the brightness of the generated high harmonics with increasing intensity and wavelength of the laser field.

The limitations associated with the driving laser intensity are dictated primarily by the process of atomic ionization, the increasing rate of which with an increase of the laser intensity leads to depletion of the non-linear medium and to macroscopic effects (phase mismatch, refraction, etc.), which reduce the efficiency of conversion of laser radiation into high harmonics [18,19]. The critical intensity determined by this factor is rather weakly dependent on the laser wavelength.

Another factor that can limit the efficiency of harmonic generation and their maximum photon energy at high laser intensities is the effect of the magnetic field of the laser pulse. The influence of this factor rapidly increases with an increase of the laser wavelength and, accordingly, the ponderomotive energy of the electron. This influence is expressed in the deviation (“magnetic drift”) of the trajectories of electrons moving with subrelativistic velocities from rectilinear ones and, as a consequence [20–24], in a decrease in the efficiency of the HHG mechanism, which is based on the electron recollisions with the parent ion.

The relative role of the above limiting factors varies with the laser wavelength. For sufficiently short laser wavelengths, ionization of the medium plays a dominant role, while in the long-wavelength regime, the effect of electron magnetic drift prevails. A detailed theoretical consideration taking into account the above factors showed the fundamental possibility of generating ultrahigh-order harmonics with photon energies up to about 30 keV, or with wavelengths down to 0.4 Å [25–27].

A significant drawback of HHG in gases as a method for producing coherent x-ray radiation is the low efficiency of the process of frequency conversion of optical radiation into high harmonics, which decreases even more with an increase of the fundamental wavelength. While for a Ti:sapphire laser (${\lambda _0} = 800\;{\rm nm}$) the conversion efficiency can reach ${{1}}{{{0}}^{- 4}}$, the harmonic yield decreases with increasing laser wavelength as $\lambda _0^{-\mu}$, where $\mu \approx {{5 {-} 6}}$ [28–30]. The main contribution to this decrease in the yield of harmonics is due to the spreading of the electron wave packet in the continuum, leading to a rapid (proportional to $\lambda _0^{- 3}$) decrease of the dipole moment induced by its interference with the wave function of the atomic bound state during the recollision.

Despite the above limitations, HHG remains in many ways a unique and attractive method for generating coherent x-ray radiation. One of the most important advantages of this approach is the possibility of phasing the spectral components of the generated radiation in a rather wide frequency range, which makes it possible to produce attosecond pulses [31–33]. Another important property of high-harmonic radiation is a high degree of its temporal coherence. Due to this, HHG-based sources are of great practical interest as generators of seed radiation for injection into free electron lasers (FEL), which makes it possible to dramatically improve the temporal properties of radiation generated by FELs [34–36]. Among various methods for improving the coherence of FEL radiation [37], this approach stands out for its ability to provide synchronization between a FEL pulse and an external signal, which is important for studying the dynamics of ultrafast processes in matter by the pump–probe method. Another advantage of using HHG is the possibility to significantly improve the stability of the FEL output energy from shot to shot. For both of the above-mentioned HHG applications, important tasks are to move from the harmonic wavelengths of tens of nanometers to the nanometer and subnanometer ranges, as well as to achieve high brightness of the harmonics generated in these ranges. All this motivates further studies of the generation of ultrahigh-order harmonics in the x-ray range.

Recent theoretical studies have revealed that HHG from laser sources with a long wavelength (more than 2 µm) has its own peculiar properties, such as an increasing concentration of the harmonic spectral density near the cutoff energy [28,30,38,39] or a change in the shape of the harmonic spectrum from a plateau-like on an arcuate, caused by the drift of a free electron in the magnetic field of a laser pulse [24,26,27]. As a result, the aforementioned wavelength scaling laws ($\lambda _0^{-\mu}$ with $\mu \approx {{5{ -} 6}}$) for harmonic spectral intensity known for driving lasers in the visible and near-IR ranges, significantly change and become more complicated for the case of long-wavelength sources [40].

It should be noted that up to now, theoretical studies have not taken into account one more factor, which can become quite significant just when using long-wavelength sources. It has always been assumed that an atomic emitter is point-like on the scale of the wavelengths of both the laser field and all generated harmonics. While this assumption is fully justified for the laser field, especially for long-wavelength sources, it can be violated for harmonics from such sources. It is easy to estimate, for example, that the linear dimensions of an atom, which for noble gases (He, Ne, Ar, Kr, and Xe), widely used in experiments with HHG, are in the range of 0.6–2.6 Å, can be more than half the wavelength of high-frequency radiation emitted by an atom (the corresponding photon energies are greater than 10–15 keV) in the case of a laser driver with a wavelength of the order of 10–12 µm. This means that the atomic dipole in this case ceases to be point-like and becomes distributed, which should inevitably lead to a change in both the power of the emitted radiation and its directivity pattern. In this paper, we study these effects in detail using the example of a hydrogen atom.

2. THEORETICAL BACKGROUND

We will consider a hydrogen atom in the ground state and exposed to an external laser field polarized along the $ z $ axis. According to the well-known semiclassical model [15,16], one of the possible results of such an interaction is a three-step process, in the first stages of which, under the action of an alternating laser field, part of the electron wave packet is released from the atom and accelerated in free space, first in the direction away from the atom, and then back to it. During this motion, the accelerated part of the electron wave packet can recollide with the parent ion and release the acquired energy in the form of a high-energy photon. In the context of the problem considered in this work, we will be further interested only in the final part of the above three-step process.

 

Fig. 1. Schematic view of a hydrogen atom, which is hit by an electron de Broglie wave with a frequency ${\omega _f}$ and a wave vector ${\vec k_f}$. The position of the observer is characterized by the radius vector ${\vec R_0}$ relative to the center of the atom.

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We will now consider in more detail the process of emission of high-energy photons by an atom. Radiation of an atom in the far-field zone corresponds with good accuracy to the radiation of a dipole source, the electric field of which is given by

Here ${\vec R_0} = {R_0}{\vec n_0}$ is the radius vector drawn from the center of the atom to the observation point (see Fig. 1); $\vec R = {\vec R_0} - \vec r$ is the radius vector drawn from the point of location of the elementary volume of the atomic emitter, $\vec r$, to the observation point; $ c $ is the speed of light in vacuum; and $\vec d(\tau)$ is the dipole moment of the atomic emitter, which is determined by

Note that Eq. (1) takes into account the electromagnetic radiation delay stemmed from the finite speed of light. It is also worth noting that the incident part of the electron wave packet under the conditions considered in this work can be relativistic and formally Dirac bispinors should be used. However, the bound part of the electron wave packet is not relativistic, and only one component of the bispinor strongly dominates in it. In addition, it should be noted that the intensity of harmonics in the plateau region is determined only by the cross terms $\Psi _b^*({\vec r,\tau}){\Psi _f}({\vec r,\tau}) + {\Psi _b}({\vec r,\tau})\Psi _f^*({\vec r,\tau})$ present in ${| {\Psi ({\vec r,\tau})} |^2}$ (since even in the most general case, ${| {{\Psi _b}({\vec r,\tau})} |^2}$ can describe only the low-frequency components of polarization corresponding to intra-atomic transitions, and ${| {{\Psi _f}({\vec r,\tau})} |^2}$ is small due to the low degree of population of each specific continuum state). Therefore, only one component of the bispinor of the incident part of the electron wave packet will contribute to the induced dipole moment, multiplying with the only non-zero component of the bispinor of the bound part. And since the possible spin effects in the long-wavelength interaction regime are negligible [41], then for the incident part of the electron wave packet it is also justified to take into account only one component of the bispinor, which, in fact, was done in Eq. (3).

Taking into account the above discussion of the contributions to the radiation of various terms contained in ${| {\Psi ({\vec r,\tau})} |^2}$, the field of an atomic emitter in the far-field zone can be represented as

As a result of the procedure for calculating the volume integral in Eq. (5) (see Supplementary material), we obtain

3. ATOMIC DIPOLE NON-POINTNESS FACTOR

Before proceeding to the analysis of the main properties of radiation from a distributed atomic emitter under the conditions formulated above, we write down an expression similar to Eq. (7) for the case of a point dipole. In the case of small size of the dipole, the phase delay between the radiation of its different parts can be neglected, and in Eq. (5) we can put $\exp ({ikR}) \simeq \exp ({ {{ikR}_0}})$. As a result of calculating similar integrals, the final expression for the electric field of an atomic point dipole in the far-field zone is similar to Eq. (7) up to replacing $1 + {| {\vec k - {{\vec k}_f}} |^2}a_0^2$ with $1 + {({{k_f}{a_0}})^2}$:

Dividing Eq. (7) by Eq. (8), we obtain a factor that takes into account the non-point nature of the atomic dipole:

Thus, having calculated the harmonic spectrum for a hydrogen atom in the point dipole approximation, it suffices to multiply it by a factor ${K_\omega}^2({k{a_0},{\theta _0}})$ to obtain a spectrum of harmonics taking into account the non-pointness of the emitter. Dependences of ${K_\omega}^2({k{a_0},{\theta _0}})$ on ${a_0}/\lambda$ for different ${\theta _0}$ are shown in Fig. 2.

 

Fig. 2. Dependences of the factor ${K_\omega}^2({k{a_0},{\theta _0}})$ on ${a_0}/\lambda$ for different ${\theta _0}$.

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For $k{a_0} \to \alpha /2$ (or, which is the same, for $\hbar \omega \to {I_p}$), i.e., near the low-frequency boundary of the plateau of high harmonics, Eq. (9) can be approximately rewritten as

Note that Eq. (10) coincides with good accuracy with Eq. (9) up to $k{a_0} \approx 1$, that is, this expression can be used not only for low-order harmonics. It can be seen from Eq. (10) that for direction ${\theta _0} = \pi /2$ corresponding to the absolute maximum of the radiation directivity pattern of a point dipole described by Eq. (8), the non-pointness of the emitter should lead to a decrease in the efficiency of emission of high harmonics with an increase of their frequency, which is exactly what is observed in Fig. 2. In addition, it follows from Eq. (10) that for a non-point emitter, the symmetry of the directivity pattern relative to plane ${\theta _0} = \pi /2$ is broken. For $\pi /2 \lt {\theta _0} \le \pi$, the factor ${K_\omega}$ decreases with increasing $k{a_0}$, whereas for $0 \le {\theta _0} \lt \pi /2$, it increases over a wide range of $k{a_0}$. This suggests that with an increase of the frequency of the generated high harmonic, the lobes of the directivity pattern of the atomic dipole are shifted toward the ${\theta _0} = 0$ direction, i.e., the direction of motion of the incident electron. This effect will be discussed in more detail in the next section.

4. ANALYSIS OF THE RADIATION PROPERTIES OF A DISTRIBUTED ATOMIC DIPOLE

In this section, we will analyze Eq. (7) and its consequences in more detail. We will start by calculating the total radiated power. We define it as an integral of the radiation intensity Eq. (7) over the surface of a sphere of radius ${R_0}$:

Substituting Eq. (7) into Eq. (11), after some transformations we obtain

 

Fig. 3. Dependence of the normalized radiated power of an atomic dipole in the far-field zone Eq. (14) on ${a_0}/\lambda$ (red solid line). Black dashed line shows the asymptotic dependence of the same quantity in the limit of large values of ${a_0}/\lambda$ Eq. (15).

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As seen in Fig. 3, the dependence of the normalized power of the atomic dipole on ${a_0}/\lambda$ can be divided into three sections. The first section corresponds to condition ${a_0}/\lambda \lt 0.00116$. Using the dispersion relation Eq. (6) between $k$ and ${k_f}$, it is easy to show that the first extremum in Fig. 3 corresponds to condition ${k_f}{a_0} = 1$. Up to this point, the atomic dipole is small on the scale not only of the wavelength of the emitted field, but also of the de Broglie wavelength of the incident part of the electron wave packet, which characterizes the spatial scale of the excited intra-atomic currents. In other words, in this wavelength range, all parts of the atomic dipole emit in phase; therefore, the efficiency of the emitter increases until its size becomes close to the de Broglie wavelength of the incident part of the electron wave packet; the radiated power in this range increases with increasing ${a_0}/\lambda$ ratio. At even higher values of ${a_0}/\lambda$, the atomic dipole remains point-like on the scale of the emitted photon wavelength, but the spatial scale of the excited currents becomes smaller than the size of the atom. As a result, the atomic dipole becomes divided into regions that emit in different phases. The shorter the de Broglie wavelength of the incident part of the electron wave packet, the smaller the spatial scale of modulation of the atomic dipole becomes. As a result of destructive interference of fields generated by different parts of the atomic dipole, the total radiated power decreases. The decrease in radiated power is observed approximately up to ${a_0}/\lambda = 9.033$, which corresponds to condition ${k_f}{a_0} = 1/\alpha$. At this point, with a non-relativistic approach, the velocity of the incident electron would become equal to the speed of light. At higher values of ${a_0}/\lambda$, the electron becomes ultrarelativistic. As seen in Fig. 3, in this region the power of the atomic dipole becomes an increasing function of ${a_0}/\lambda$. It is easy to show from Eq. (14) that in the limit ${a_0}/\lambda \to \infty$ the quantity $P_\omega$ can be represented as

The proportionality ${P_\omega} \sim {({k{a_0}})^2}$ indicates that, in this generation regime, the electron releases all of its energy over a time interval inversely proportional to the energy of the emitted photon.

We now turn to the consideration of the radiation directivity pattern of the atomic dipole. We will characterize it by the quantity

Figure 4(a) shows the radiation directivity patterns of an atomic dipole (in the plane of polarization of the laser field) in the far-field zone for different values of ${a_0}/\lambda$. It can be seen that with increasing ${a_0}/\lambda$, i.e., with decreasing wavelength of the generated field, the lobes of the directivity pattern tend to tilt closer to direction ${\theta _0} = 0$ (the direction of motion of the electron returning to the parent ion), and their width tends to zero. Differentiating Eq. (17) with respect to ${\theta _0}$ and equating the resulting expression to zero, it is easy to obtain an expression for the angle corresponding to the maximum of the radiation directivity pattern:

From Eq. (19) it follows that in the last of the above cases, for the angle at which the atomic dipole radiates most efficiently, the condition $\theta _0^{({\max})} \sim 1/k \approx 1/{k_f} \sim 1/\gamma$ is satisfied (where $\gamma = 1/\sqrt {1 - {v^2}/{c^2}}$ is the Lorentz factor of the incident electron and $v$ is its velocity), as it should be in the case of radiation from ultrarelativistic particle.

 

Fig. 4. (a) Directivity pattern of a single burst of radiation from an atomic dipole (in the plane of polarization of the laser field) in the far-field zone at the ratio ${a_0}/\lambda$ equal to 0.00116 (blue line), 1 (green line), 9.033 (red line), and ${{1}}{{{0}}^3}$ (black line). Dashed line represents the radiation pattern of the atomic dipole for the entire cycle of the laser field at ${a_0}/\lambda = 1$. (b) Dependence of the angle corresponding to the maximum of the radiation directivity pattern of the atomic dipole on ${a_0}/\lambda$ Eq. (18) (red solid line). Black dotted line corresponds to asymptotics Eq. (19) at ${a_0}/\lambda \to \infty$.

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Thus, in the case of generation of high harmonics with wavelengths of the atomic or subatomic scale (which, in principle, is possible when using a sufficiently long-wavelength laser source), one should expect that the atom will emit the maximum power not in the plane perpendicular to the electric field of the laser pulse, but along the generatrix of the cone with an opening angle decreasing with increasing harmonic order. In this case, the highest-order harmonics will be emitted in the form of the shortest bursts, and the repetition period of such high-frequency bursts in each particular direction will be equal to one cycle of the laser field.

The fan-like structure observed in Fig. 4(a) can be regarded as an example of space–time coupling of the generated harmonic field. However, it should be emphasized that the effect discussed above is fundamentally different in nature and manifestation from those observed in the attosecond lighthouse phenomenon [42] or in [43], where the spatial or spectral separation of consecutive bursts of the harmonic field is due to, respectively, the laser pulse wavefront rotation or the propagation in a macroscopic medium. In our case, the unusual spatial and spectral distributions of the harmonic signal are purely of single-atom nature and stem from the non-pointness of the atomic system.

If in the experiment the efficiency of generation of ultrahigh-order harmonics is sufficiently high, so that at the same time a large number of atomic emitters contributes to the total field at the frequency of a particular harmonic (which makes it possible to describe the radiation of harmonics in terms of macroscopic fields), then the angular distributions of harmonic radiation emitted by the macroscopic volume of the medium will depend on the intensity distribution in the laser beam, the geometry of the target, and on the phase matching between the driving and generated fields in the medium, i.e., will strongly depend on the specific conditions of the experiment. However, in practice, the efficiency of generation of harmonics with a photon energy of ${\sim}{{25 {-} 30}}\;{\rm keV}$ (which corresponds to ${a_0}/\lambda \approx 1$) from a laser source with a wavelength longer than 10 µm should be expected to be extremely low (at the level of a few tens of photons per shot). At the same time, it is just this low generation efficiency that makes it possible to carry out a rather simple experiment to observe manifestations of the non-pointness of an atomic dipole. If the number of photons is small, then in the experiment one can simply count the number of photons of a certain energy emitted at each elementary angle in the plane of polarization of the laser pulse, summing up the data over a large number of shots. As a result of this counting, the radiation directivity pattern for photons of a particular energy will be retrieved. If for a photon of a given energy the atom is essentially a point-like source, then the radiation pattern will be a typical dipole one, with two maxima [see, for example, the blue line in Fig. 4(a)]. If, in contrast, the condition ${a_0}/\lambda \approx 1$ is satisfied for the chosen harmonic photon energy, then the radiation pattern will already have four maxima [see dashed line in Fig. 4(a)]. Furthermore, as the ratio ${a_0}/\lambda$ increases, the picture with four maxima becomes more and more pronounced.

It is also worth noting that, as in any distributed system, the angular distribution of a harmonic field from a non-point atomic or molecular system will be extremely sensitive to the specific amplitude distribution of the emitter (electron orbital). This fact allows, in principle, to arrange the reconstruction of the electron orbital, similar, in a sense, to the tomographic imaging of molecular orbitals using HHG [44].

5. RELATIONSHIP BETWEEN DISTRIBUTED ATOMIC DIPOLE AND CLASSICAL ANTENNA

Methodologically, it is interesting to trace the transition from a distributed atomic dipole to a classical antenna. It is known from electrodynamics that the radiation of an antenna system, characterized by the spatial distribution of the current ${{\vec {\tilde j}}_\omega}({\vec r})$ at a frequency $\omega$, in the far-field zone is determined by the complex amplitude of the electric field:

Comparing Eq. (5) for the electric field of an atomic dipole in the far-field zone at a frequency $\omega$ with Eq. (20), one can note that a quantum-mechanical atomic dipole is equivalent to a classical antenna emitter with a current density

For certainty, we will focus on the case of a classical linear antenna, which is a segment of a thin wire of length ${a_0}$ aligned parallel to the $ z $ axis, with a running (or standing) excitation wave of constant amplitude. A quantum-mechanical analog of such a linear antenna can be obtained from Eq. (21) by setting

For a running excitation wave, the spatial distribution of current density Eq. (21) in the case of interest to us can be rewritten as

Substituting Eq. (24) into Eq. (11), we obtain an expression for the total power of the linear emitter considered by us:

For the case of pumping by a standing wave, the corresponding expression for the total power of a linear emitter can be obtained by replacing ${e^{i{{\vec k}_f}\vec r}}$ in Eq. (23) with $\cos ({\vec k_f}\vec r)$. As a result, we have

Note that when an atomic dipole is excited by the de Broglie wave of the recolliding electron during the process of HHG [16], pumping by a standing wave is never the case, since the recolliding electron wave packet always strikes the parent ion from only one direction. In contrast, for a classical antenna, standing wave pumping is the most common in practice. Furthermore, there is another obvious difference between an atomic dipole and a classical antenna. It can be seen from Eq. (21) that for the case of an atomic dipole, the amplitude of the current density is proportional to the frequency of the emitted field. This is due to the fact that the energy of the emitted photon directly depends on the energy of the incident electron. In addition, as shown above, the electron loses its energy in a time inversely proportional to the energy of the emitted photon. Thus, the pumping power of the atomic dipole ($\sim{| {{{{\vec {\tilde j}}}_\omega}({\vec r})} |^2}$) turns out to be proportional to ${({k{a_0}})^2}$. As for the case of a classical antenna, its pumping power can be kept constant. Therefore, for comparison with a classical antenna, Eqs. (25) and (26) for an atomic dipole should be normalized to the pumping power, that is, divided by ${({k{a_0}})^2}$. The corresponding curves of the normalized radiated power versus ${a_0}/\lambda$ are shown in Fig. 5.

 

Fig. 5. Dependence of the normalized power of an atomic linear emitter on ${a_0}/\lambda$ in the cases of a running [Eq. (25)] (red lines) and standing [Eq. (26)] (blue lines) excitation wave. The velocity factor $k/{k_f}$ for the emitter is (a) determined from Eq. (6); (b) set equal to 1/1.1.

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Figure 5(a) presents the dependences of the normalized power of a linear atomic emitter Eqs. (25), (26) on ${a_0}/\lambda$ for both a running and a standing excitation wave. The relation between $k$ and ${k_f}$ was used, given by Eq. (6), which is a consequence of the law of conservation of energy during the generation of high harmonics in the electron recollision regime. As for the classical antenna, in the range of ${a_0}/\lambda$ presented here, the velocity factor is usually approximately constant and less than one [45]. This is one more difference between an atomic dipole and a classical antenna. Figure 5(b) shows the same dependences of the normalized power of a linear emitter obtained from Eqs. (25) and (26), but with a velocity factor $k/{k_f} = 1/1.1$. Dependences of the power of a linear emitter on ${a_0}/\lambda$, shown in Fig. 5(b), are in good agreement with the dependences known in radio-frequency engineering for a linear antenna.

6. CONCLUSIONS

In this work, the properties of an atomic dipole responsible for the generation of high-order harmonics in the regime of electron recollision during ionization of an atom were studied, taking into account real physical dimensions of an atom, which, under certain conditions, may be comparable to the wavelength of the emitted photon or even exceed it. The dependences of the radiated power of the atomic dipole and its radiation directivity pattern on the wavelength of the emitted photon are investigated. A transition in the short-wavelength limit to the well-known case of radiation of an ultrarelativistic electron is demonstrated. The fundamental differences between an atomic emitter and a classical antenna are also demonstrated, and a sequential transition from a quantum emitter to a classical antenna system is shown. One of the most important results of this study is that a factor has been obtained that describes the non-point properties of an atomic emitter. This makes it possible, if necessary, to easily correct the results obtained in the point dipole approximation by introducing an additional factor obtained in this work.