1. Introduction

When we expose atoms to an intense laser field various nonlinear processes can occur [1]. One of the most important of these processes is the emission of extreme ultraviolet and soft x-ray radiation. This process was discovered in the seminal experimental works [2,3] and is named high-order harmonic generation (HHG). The energy of the emitted soft x-rays can be a few thousand times larger than the energy of the generating fundamental laser field photons. The HHG provides a table-top source of coherent and ultrashort soft x-ray radiation and has found many applications [4–10].

On the single-atom or microscopic level the HHG process is well understood. It is described by the semiclassical three-step model [11–14]. In the first step the bound atomic electron tunnels through the atomic potential barrier which is lowered under the influence of the strong laser field. This electron is then driven by the laser field and, since the laser field is oscillatory and changes its sign, can return to the parent ion (second step). In the final third step, this electron recombines with the parent ion and a high harmonic photon is emitted. The maximal energy of this harmonic photon is equal to the ionization potential $I_p$ plus the maximal kinetic energy accumulated during the electron’s travel in the laser field from ionization to recombination [for a linearly polarized laser field this energy is $3.17U_p$, where $U_p=I/(4\omega ^{2})$ is the ponderomotive energy (cycle-averaged electron kinetic energy in the laser field; $I$ and $\omega$ are the laser field intensity and frequency, respectively)]. In the microscopic theory the intensity of the emitted high harmonics can be calculated using solutions of the time-dependent Schrödinger equation (see, for example, [7] and references therein). A simpler and less time consuming way to calculate the microscopic HHG spectra is to use the strong-field approximation, an approximate theory which neglects the influence of the atomic potential in the propagation step of the HHG process [14]. In our paper we will use a version of the strong-field approximation presented in [1].

Macroscopic effects in HHG are usually considered solving Maxwell’s wave equations for the fundamental and harmonic fields [4–10]. In the cited references the Maxwell wave equation is used to describe the macroscopic propagation and phase matching of the radiation. The microscopic quantities enter Maxwell’s wave equation as source terms. There are many different attempts to evaluate macroscopic effects in HHG. They can be divided into two groups. One is based on differential equations and the corresponding propagation. The three-dimensional propagation is described in detail in [8,9]. The relevant differential equations are solved using a Crank-Nicolson routine. The second group is based on integral solutions of the wave equation. We will use this approach. Such integral solutions were considered at the beginning of the development of the theory of macroscopic HHG [5] and were used to estimate the effect of phase matching (see also the more recent Refs. [15,16]; however, the HHG spectra are not presented in these references). It should be mentioned that in [17] a different propagation technique, based on the discrete dipole approximation and an integral solution of the wave equation, was used. Finally, in spite of the fact that macroscopic HHG is a well explored problem, its complexity and its dependence on many different parameters make this problem difficult and open for new discoveries. In this context, we mention here a recent study of the coherence in macroscopic HHG [18]. Our approach, which is based on integral solutions of the wave equation for the harmonic field, is particularly suitable for longer wavelengths and higher intensities of the driving laser field for which other computational methods may be too computer demanding.

In Sec. 2 we present formulas for the microscopic HHG theory and introduce our method of macroscopic focal averaging. Numerical results are presented in Sec. 3 and the conclusions are given in Sec. 4. We use atomic units.

2. Theory

2.1 Microscopic HHG

For a laser field defined in the $xy$ plane, spanned by the unit vectors $\hat {\mathbf {e}}_x$ and $\hat {\mathbf {e}}_y$, the $T$-matrix element is the vector $\mathbf {T}_n=|T_n|\mathbf {e}_n=T_{nx}\hat {\mathbf {e}}_x+T_{ny}\hat {\mathbf {e}}_y$, with $\mathbf {e}_n$ the complex unit polarization vector of the $n$th harmonic and $|T_n|=\mathbf {e}^{*}_n\cdot \mathbf {T}_n$. The vector $\mathbf {T}_n$ is the Fourier component of the time-dependent dipole matrix element between the initial and final laser-dressed atomic states. It can be approximated by [1,19]

The $n$th-harmonic intensity (power) is

2.2 Gaussian laser beam and the gas cell

In order to take into account the macroscopic effects we should define the macroscopic spatial dependence of the laser field. We suppose that the laser field can be described by a Gaussian beam, which depends on the spatial cylindrical coordinates $(\rho,z)$ [due to cylindrical symmetry the laser field does not depend on the azimuth angle $\varphi$ of cylindrical coordinates $\mathbf {r}_m =(\rho,\varphi,z)$; the subscript “$m$” stands for the medium in which high harmonics are generated]. The minimum beam waist is $w_0$ and the Rayleigh length is $z_0=\pi w_0^{2}/\lambda$ ($\lambda$ is the laser wavelength). For the peak intensity $I_0=E_0^{2}$, the laser intensity is [21]

The corresponding electric field is

The above formula for the laser intensity can be generalized to include the laser pulse shape. For a Gaussian pulse with the temporal width $\tau _g$, we have $I(\rho,z,t_p)=I(\rho,z)f_p(t_p-|\mathbf {r}_m|/c)$ where $f_p(t)=\exp (-t^{2}/\tau _g^{2})$. In this case, the electric field is defined by $E(\rho,z,t_p,t)=|E(\rho,z,t_p)|\mathrm {Re}\left (\exp \{i[\varphi _E(\rho,z)-\omega t]\}\right )$, with $|E(\rho,z,t_p)|^{2}=I(\rho,z,t_p)$. In the experiment the pulse duration time is defined as FWHM (full width at the half maximum) of the intensity profile: $\tau _p=2\sqrt {\ln 2}\tau _g$.

In Fig. 1 we present the gas cell of the length $L$ (blue rectangle) and the focused Gaussian laser beam (area between the red curves). The focus F is at the point $z=0$, while the center of the gas cell is shifted by $\Delta z$ with respect to the focus ($\Delta z$ is negative for the presented example). The gas cell is positioned between the points $z_1=-L/2 +\Delta z$ and $z_2=L/2 +\Delta z$. We also indicated examples of a point $\mathbf {r}_m=(\rho,z)$ in the medium where a high harmonic is emitted and a point $\mathbf {r}_d=(\rho _d,z_d)$ in the detector plane D where high harmonics are detected. We suppose that the harmonic photons are detected in the far field, i.e., $z_d$ is much larger than the coordinates in the medium. Another important experimental parameter is the gas pressure $p$. It is not explicitly taken into account in our simulations. But we expect that the gas density increases linearly with the increase of this pressure. Higher is this density, more atomic centers are available for harmonic generation and we expect that the high harmonic yield increases proportionally to $p^{2}$. For higher pressures this dependence becomes nonlinear [24], or even constant [25], since various effects, like plasma effects, saturation, excitation, incoherent processes [26], etc., influence the HHG process.

 

Fig. 1. Schematic presentation of the gas cell, focused Gaussian beam, and the detector plane in the HHG experiment.

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2.3 Macroscopic effects

The $T$-matrix element $\mathbf {T}_n(I(\rho,z),\varphi _E(\rho,z))$ is calculated for the laser field $E(\rho,z,t)=|E(\rho,z)|\cos [\omega t-\varphi _E(\rho,z)]$ at a fixed point $(\rho,z)$ in the focal region of the laser Gaussian beam. Here we have taken into account that $\mathbf {T}_n$ depends on ${\mathbf {r}_m}$ via the local laser intensity $I(\rho,z)$ and phase $\varphi _E(\rho,z)$, Eqs. (4) and (6). According to Eqs. (1) and (2), the $T$-matrix element $\mathbf {T}_n(I,\varphi _E)$ is proportional to

The subintegral function does not depend on the phase $\varphi _E$ and only the interval of integration over the recombination time is changed from $[0,T]$ to $[-\varphi _E,T-\varphi _E]$. In fact, the integration interval $[0,T]$ was obtained from the interval $(-\infty,\infty )$ using the $T$-periodicity of the subintegral functions. Therefore, we can approximate $\mathbf {T}_n(I,\varphi _E)$ with $e^{in\varphi _E}\mathbf {T}_n(I)$, where $\mathbf {T}_n(I)\equiv \mathbf {T}_n(I,\varphi _E=0)$.

It is generally accepted [4–9] that the macroscopic effects in HHG should be considered using classical electrodynamics, i.e., solving Maxwell’s wave equations for the fundamental and harmonic fields. Quantum mechanics comes into play only in the calculation of the (microscopic) time-dependent dipole moment which is related to the polarization of the medium (atomic gas) induced by the laser field. The $n$th harmonic field $\mathbf {\cal E}_n(\mathbf {r},t)$ satisfies the wave equation [4,5,15,16]

We assume that the polarization $\mathbf {\cal P}_n$ is equal to the $T$-matrix element $\mathbf {T}_n(I,\varphi _E)\approx e^{in\varphi _E}\mathbf {T}_n(I)$.

We choose $\mathbf {r}=\mathbf {r}_d$ and $\mathbf {r}'=\mathbf {r}_m\equiv (\rho,\varphi,z)$, where $\mathbf {r}_d$ is the detector position (observation point) and $\mathbf {r}_m$ is a point in the generating medium. We consider two different levels of approximations. In the simpler one, which we call the coherent intensity focal averaging, we approximate $\exp (iK_n|\mathbf {r}-\mathbf {r}'|+in\varphi _E) /|\mathbf {r}-\mathbf {r}'|$ with a constant so that the subintegral function in Eq. (11) reduces to $\mathbf {T}_n(I)$. In the second one, which we call the coherent spatio-temporal focal averaging, we integrate over all times $t_p=t$ and calculate $\mathbf {\cal E}_{n,p}(\rho _d,z_d)=\int dt_p\mathbf {\cal E}_n(\mathbf {r}_d;t_p)$.

2.3.1 Coherent intensity focal averaging

We define the macroscopic coherent intensity focal-averaged $T$-matrix element as

The factor $2\pi$ comes from the integral over the azimuthal angle, while the integrals over $\rho$ and $z$ can be replaced by the integral over the laser intensity, similarly as this has been done for the focal-averaging of the differential ionization yield in strong-field ionization [28–30]. The result is

We have already used a similar formula for the calculation of the incoherent focal-averaged high-harmonic yield in Refs. [31,32].

In the case of the geometry presented in Fig. 1, the integral over $z$ in Eq. (12) is from $z_1$ to $z_2$, i.e., it is not from $-\infty$ to $+\infty$. In this case, in Eq. (12) we introduce $1=\int _0^{I_0}dI\,\delta (I-I(\rho,z))$ and make the substitutions $r=\rho ^{2}/w_0^{2}$ and $z'=z/z_0$. Then from Eq. (4) it follows that $-z'_{I}\le z'\le z'_{I}$, $z'_{I}=\sqrt {I_0/I-1}$. Therefore, the lower and upper limits of the integral over $z'$ are

Equations (12)–(15) can be generalized to include the laser pulse shape. We multiply the Gaussian laser beam intensity $I(\rho,z)$ with the factor $f_p(t_p-z/c)$, similarly as this has been done for the strong-field ionization with strong focusing in Refs. [33,34]. The result is

After a derivation, analogous to that used to obtain Eq. (15), we get

2.3.2 Coherent spatio-temporal focal averaging

In the case of coherent spatio-temporal focal averaging the harmonic field is given by

In the far-field approximation we have $|{\mathbf {r}_m}-\mathbf {r}_d|\approx z_d-z+[\rho ^{2}+\rho _d^{2}-2\rho \rho _d\cos (\varphi -\varphi _d)]/[2(z_d-z)]$. We use this approximation for the term in the exponent, while for the denominator we take into account only the lowest order term, $|{\mathbf {r}_m}-\mathbf {r}_d|\approx z_d-z$. Since $\mathbf {T}_n$ does not depend on the azimuthal angle $\varphi$, the integral over $\varphi$ in Eq. (18) can be calculated analytically. The result is

Next, in the so obtained equation we introduce $1=\int _0^{I_0}dI'\,\delta (I'-I(\rho,z,t_p))$ and make the substitutions $\rho '=\rho /w_0$, $z'=z/z_0$, and $t'=[t_p-(z_d-z)/c]/\tau _g$. We get a four-dimensional integral in the form

Here we used the fact that, in the case of the geometry presented in Fig. 1, the integral over $z$ is from $z_1$ to $z_2$, i.e., it is not from $-\infty$ to $+\infty$, so that the lower and upper limits of the integral over $z'$ are given by Eq. (14). Making another substitution: $\sqrt {2}\rho '/\sqrt {1+z'^{2}}=r\cos \beta$, $t'=r\sin \beta$, $d\rho ' dt'=\sqrt {(1+z'^{2})/2} \,rdr d\beta$, $r\in [0,\infty )$, $\beta \in [-\pi /2,\pi /2]$, we obtain

The final result for the coherent spatio-temporal focal-averaged harmonic field is given by Eqs. (21), (23), and (24). Supposing an infinite gas cell, for $\varepsilon _d=0$, $\varepsilon _\rho =0$, $\Delta K_n=0$, we can approximate this by

3. Numerical results

3.1 Wavelength and intensity dependence of the macroscopic HHG spectra

In order to get an impression how the HHG spectra change with the change of the central wavelength and the peak intensity of the laser field, in Fig. 2 we present the spectra for the He atom (${I_p}=24.59$ eV) obtained using coherent intensity focal averaging for an infinitely long cell and a flat pulse, Eqs. (3) and (13). The laser field is linearly polarized having the peak intensity $I_0=9.4\times 10^{14}\;\mathrm {W/cm}^{2}$ and the wavelengths 800, 1300, 1600, and 2100 nm (upper panel), and the wavelength 800 nm and the peak intensities $I_0=9.4\times 10^{14}\;\mathrm {W/cm}^{2}$, $6\times 10^{14}\;\mathrm {W/cm}^{2}$, and $3\times 10^{14}\;\mathrm {W/cm}^{2}$ (lower panel). In all cases, after a maximum in the region $45-50$ eV, the spectrum has the form of a declining plateau, with a clear cutoff at the energy $3.173{{U_p}}_0+1.325{I_p}$. The plateau of the microscopic HHG spectrum has the same cutoff. However, it is rather flat (see Fig. 5). The reason for this discrepancy is the fact that HHG is a coherent process. Namely, for incoherent processes we sum the contributions of the yields (i.e., the probabilities, not the amplitudes) from all points in the laser focus and the corresponding plateau is flat. An example is the plateau in high-order above-threshold ionization [1,36], with the observed cutoff at $10{{U_p}}_0$. For HHG the coherent focal averaging leads to the declining plateaus as those presented in Fig. 2. We see that the harmonic intensity is higher for shorter wavelengths and higher intensities. The maximum harmonic photon energy scales as ${{U_p}}_0\propto I_0\lambda ^{2}$ in accordance with the mentioned cutoff law. In most experiments the high-energy photons in the upper part of the plateau region are below the detection threshold and this cutoff cannot be observed. For example, even in the early HHG experiments, it was found that the experimental cutoff energy is lower, approximately at ${I_p}+2{{U_p}}_0$ [37]. The detection threshold problem is particularly pronounced for longer wavelengths for which the yield is lower (compare the upper panel of Fig. 2). In the next two subsections we will show, applying our coherent spatio-temporal focal-averaging method, that it is possible, changing the macroscopic conditions, to obtain an observable peak in the spectrum which agrees with the experimental findings.

 

Fig. 2. Harmonic intensities for the He atom obtained using the coherent intensity focal averaging for an infinitely long cell, a flat laser pulse, and various wavelengths and peak intensities of the linearly polarized laser field. Upper panel: $I_0=9.4\times 10^{14}\;\mathrm {W/cm}^{2}$ and $\lambda =800$ nm, 1300 nm, 1600 nm, 2100 nm. Lower panel: $\lambda =800$ nm and $I_0=9.4\times 10^{14}\;\mathrm {W/cm}^{2}$, $6\times 10^{14}\;\mathrm {W/cm}^{2}$, $3\times 10^{14}\;\mathrm {W/cm}^{2}$.

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3.2 Examples of HHG by a linearly polarized field

As a check of our focal-averaging methods, we now calculate the HHG spectra of Ar and He atoms exposed to a linearly polarized laser field having the central wavelength 2100 nm and the peak intensity $I_0=4.1\times 10^{14}\;\mathrm {W/cm}^{2}$ for Ar and $I_0=9.4\times 10^{14}\;\mathrm {W/cm}^{2}$ for He. These are the parameters of the experiment [38]. In the experiment with Ar the laser beam was focused with an $f=750$ mm lens into a 6 mm long gas cell, which led to a beam with $w_0=80\;{\mathrm{\mu} \mathrm{m}}$, so that $z_0\approx 9.6$ mm. The above parameters for the experiment with He are $f=500$ mm, $w_0=53.5\;{\mathrm{\mu} \mathrm{m}}$, and $z_0\approx 4.3$ mm. For these parameters we estimate that the parameter $\varepsilon _d\le 0.01$. The estimated beam divergence in [38] is 1 mrad which corresponds to $\varepsilon _\rho \le 0.001$ so that we expect that the influence of this parameter can be neglected. Furthermore, we expect that the experimental conditions are such that phase matching is achieved and $\Delta K_n=0$.

The experimentally observed maximum harmonic photon energy for HHG by Ar atoms and the mentioned laser-field parameters was 170 eV in [38]. However, the (theoretical) cutoff law for high-order harmonic generation [14] gives $n_{\max }\omega =3.173{{U_p}}_0+1.325{I_p}=557$ eV, where the ponderomotive energy for the peak intensity $I_0=4.1\times 10^{14}\;\mathrm {W/cm}^{2}$ is ${{U_p}}_0=169$ eV and the ionization potential of Ar is ${I_p}=15.76$ eV. Since in the experiment there is no abrupt cutoff at 557 eV as expected from the microscopic theory, it is obvious that, for a successful simulation, the macroscopic effects should be taken into account.

 

Fig. 3. Comparison of the microscopic single-atom harmonic intensity (top curve) and the macroscopic harmonic intensities obtained using coherent intensity focal averaging (denoted as “intensity, $\infty$” for an infinitely long cell and a flat pulse) and using coherent spatio-temporal focal averaging for $\varepsilon _d=0$, $\varepsilon _\rho =0$, $\Delta K_n=0$, cell length $L=6$ mm, and its shift with respect to the laser focus $\Delta z=0$ mm and $\Delta z=-2.5$ mm, as denoted. HHG is by argon atoms and a linearly polarized laser field with the wavelength 2100 nm, the peak intensity $4.1\times 10^{14}\;\mathrm {W/cm}^{2}$, and the minimum beam waist $w_0=80\; \mu \mathrm {m}$.

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In Fig. 3 we compare, on a logarithmic scale, the results for the harmonic intensity as a function of the harmonic photon energy obtained using the microscopic theory (solid black line) and using coherent intensity focal averaging with an infinite cell, Eq. (13) (dashed green line). We see that the microscopic spectrum is rather flat having its cutoff really at 557 eV. On the other hand, the spectrum obtained using the coherent intensity focal averaging, with an infinite cell and for a flat pulse, exhibits a maximum near 35 eV and then decreases for many orders of magnitude, still having the cutoff at 557 eV. We also present the results obtained using coherent spatio-temporal focal averaging, Eqs. (21)–(24), for the minimum beam waist $w_0=80\; \mu \mathrm {m}$ and the shift of the gas cell, having the length $L=6$ mm, with respect to the laser focus $\Delta z=0$ mm (solid red line) and $\Delta z=-2.5$ mm (dotted blue line). All spectra exhibit a minimum of the harmonic yield at 367 eV. This minumum is a single-atom effect and related to the dipole matrix element with the $p$ ground state of the Ar atom. We have checked this by calculating the harmonic spectra (keeping the same laser parameters) for He atom, which has the $s$ ground state. In this case, the minimum at 367 eV does not appear and the spectrum is rather flat (see Fig. 5). The minimum near 367 eV should not be mixture with the Cooper minimum which appears for Ar atoms at much lower energies near 46 eV. The Cooper minimum does not appear in the strong-field approximation, used in the present paper, but it can be obtained in the low-frequency approximation considered in [31]. The low-energy part of the high-harmonic plateau, for energies below the mentioned minimum at 367 eV, is flat for the microscopic spectrum. However, for the macroscopic spectra, after a sharp decrease for the first few harmonics, the harmonic intensity increases, having a maximum in the region from 30 eV to 150 eV, depending on the geometry of the experiment, and then decreases toward the mentioned minimum at 367 eV. It seems that the macroscopic high-harmonic yield is affected by the destructive interference of contributions from various positions in the laser focus and is dominated by laser pulse regions with lower intensities. Instead of the flat plateau which is characteristic of the microscopic HHG, or the declining plateau in accordance with the coherent intensity focal averaging, a maximum is formed at lower energies; the position of this maximum depends on the macroscopic conditions. This is interesting since it allows manipulation of the harmonic yield by changing the macroscopic condition. Such kind of manipulation is not possible for incoherent processes such as above-threshold ionization. The constructive interference in macroscopic HHG leads to the formation of a maximum which is observed in the experiment. The position of this maximum strongly depends on the shift of the gas cell $\Delta z$.

 

Fig. 4. Harmonic intensities as functions of the harmonic photon energy, calculated using coherent spatio-temporal focal averaging, for HHG by argon atoms and the laser parameters as in Fig. 3. The parameter $\varepsilon _d$ is zero, except for the dashed green curve (denoted by a star) for which it is $\varepsilon _d=0.01$. The spectrum denoted by a double star (solid maroon line) is calculated for $\Delta z=-3.75$ mm taking into account the saturation effects.

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In Fig. 4 we present the results obtained using the coherent spatio-temporal focal averaging for various shifts $\Delta z$, as denoted in the legend. For $\Delta z=-2.5$ mm we also present the results for $\varepsilon _d=0.01$ (dashed green line denoted by a star). The results are now presented on a linear scale. We see that the harmonic photon energy for which the harmonic intensity has a maximum shifts from the lowest value of 60 eV for $\Delta z=0$ mm to 160 eV for $\Delta z=-5$ mm. The intensity is maximal for $\Delta z=-2.5$ mm. For the geometrical parameter $\varepsilon _d=z_0/z_d=0.01$ it is at 105 eV, while for $\varepsilon _d=0$ the maximum is slightly lower and is shifted to 110 eV. We conclude that the influence of the parameter $\varepsilon _d$ is small for this experiment. With a further shift of the focus to the negative values of $\Delta z$, the harmonic photon peak energy increases but the corresponding maximum harmonic intensity becomes lower. For $\Delta z=-3.75$ mm the peak is near 140 eV, which is very close to that observed in the experiment [38,39]. The harmonic peak energy is even higher for $\Delta z=-5$ mm: the peak is near 160 eV, but the harmonic intensity is ten times lower than that for $\Delta z=-2.5$ mm.

As the next example, in Fig. 5 we present our results for the He atom. The microscopic plateau is rather flat, with strong interference oscillations due to the used long wavelength and high intensity, and a sharp cutoff appears at 1260 eV. However, such high energies cannot be observed in the experiment since the macroscopic effects drastically change the spectrum. The spectrum obtained using the coherent intensity focal averaging exhibits a maximum near 50 eV and then continuously decreases by four orders of magnitude, having a sharp cutoff at the same energy as in the microscopic case. On the other hand, the results obtained using coherent spatio-temporal focal averaging show that it is possible to manipulate the spectrum changing the position of the focus, i.e., the shift $\Delta z$ (notice that the results in Fig. 5 are in arbitrary units and that the results for different $\Delta z$ are shifted down with respect to the remaining results for visual convenience). The spectra for $\Delta z=0$ mm (blue line), $-2.5$ mm (green line), and $-5$ mm (magenta line), exhibit the maxima at 310 eV, 410 eV, and 180 eV, respectively. These results show that it is possible, by changing the geometry of the experiment, to generate harmonic photons with energies above 400 eV, which is in accordance with the experiment [38]. However, for the used parameters it is not possible to reach the energy 1260 eV predicted by the microscopic theory.

 

Fig. 5. Comparison of the microscopic single-atom harmonic intensity (top curve) and the harmonic intensities obtained using coherent intensity focal averaging and coherent spatio-temporal focal averaging for $\varepsilon _d=0$, $\varepsilon _\rho =0$, and $\Delta K_n=0$. HHG is by helium atoms and a linearly polarized laser field with the wavelength 2100 nm, the peak intensity $9.4\times 10^{14}\;\mathrm {W/cm}^{2}$, and the minimum beam waist $w_0=53.5\; \mu \mathrm {m}$. The shift of the gas cell, having the length $L=6$ mm, with respect to the laser focus is $\Delta z=0$ mm, $-2.5$ mm, and $-5$ mm, as denoted.

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Fig. 6. Comparison of the microscopic single-atom harmonic intensity (top curves) and the harmonic intensities obtained using the coherent intensity and spatio-temporal focal averaging for $\varepsilon _d=0$, $\varepsilon _\rho =0$, and $\Delta K_n=0$. HHG is by neon atoms and a bicircular laser field with the component wavelengths $\lambda _1=2000$ nm and $\lambda _2=800$ nm and peak intensities $I_{10}=1.5\times 10^{14}\;\mathrm {W/cm}^{2}$ and $I_{20}=2.8\times 10^{14}\;\mathrm {W/cm}^{2}$. The minimum beam waist is $w_0=80\; \mu \mathrm {m}$ and the gas cell length is $L=6$ mm. For the coherent spatio-temporal focal averaging the shift of the gas cell with respect to the laser focus is $\Delta z=0$ mm and $\Delta z=-2.5$ mm, as denoted. The emitted harmonics are circularly polarized having the helicity $+1$ (red curves) or $-1$ (green curves).

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Saturation effects can be taken into account by multiplying the harmonic intensity with the probability that the electron remains bound in the volume element characterized by the intensity $I$ at the time $t$ during the laser pulse of duration $\tau _p$. Our calculations show that the He experiment is performed mostly outside of the saturation regime. However, for the Ar experiment we have complete saturation and only about 50% of the atoms survive up to the intensity of about $2\times 10^{14}\;\mathrm {W/cm}^{2}$. But our additional simulation shows that in this case the saturation effects only suppress the oscillations, which can be noticed in the spectra shown in Fig. 4. We confirmed this by calculating the spectra for the Ar experiment taking into account the saturation effects for the pulse duration of 30 fs (see the solid maroon line in Fig. 4).

3.3 Examples of HHG by a bicircular field

We illustrate the HHG spectra generated by a bicircular field using the example of Ne atoms (${I_p}=21.56$ eV) and recent experimental results [40]. The counter-rotating bicircular components have the wavelengths $\lambda _1=2000$ nm and $\lambda _2=800$ nm and the peak intensities $I_{10}=1.5\times 10^{14}\;\mathrm {W/cm}^{2}$ and $I_{20}=2.8\times 10^{14}\;\mathrm {W/cm}^{2}$ (these component intensities are in accordance with [41]; for the lower intensities mentioned in [40] the microscopic cutoff is at 82 eV).

The microscopic single-atom spectra generated by the bicircular field with the above-defined parameters are represented by the top curves in Fig. 6. They are in accordance with the results of Ref. [19]. If we define the fundamental wavelength $\omega$ such that it corresponds to the wavelength of 4000 nm, then our bicircular field consists of counter-rotating circularly polarized components of commensurate frequencies $\omega _1=2\omega$ and $\omega _2=5\omega$. Since the intensity of the $5\omega$ component is almost two times higher, the shape of the spectrum is as presented in Fig. 6: from 30 eV to 100 eV it is increasing while above 100 eV we have a flat plateau which finishes with the cutoff at 160 eV. There is an asymmetry in the emission of harmonics of different helicities: below 125 eV the harmonics with the helicity $+1$ are stronger, while for higher energies the harmonics with the helicity $-1$ prevail. This asymmetry is explained in [42,43] as a consequence of the $p$ ground state of Ne. The selection rules for the $2\omega$–$5\omega$ bicircular field [19] imply that harmonics with the energy $(7q\pm 2)\omega =q(\omega _1+\omega _2)\pm \omega _1$, $q$ integer, with helicity $\pm 1$ are emitted. Therefore, the distance of the harmonic peaks for the harmonics of the same helicity is $\omega _1+\omega _2=7\omega$, while for adjacent harmonics of different helicity it is $\omega _2-\omega _1=3\omega$, which is in accordance with the experimental results [40].

As in the case of the linearly polarized field, the situation changes if the macroscopic effects are taken into account. If the simplest coherent intensity focal averaging method is applied, the spectrum is inclined and the harmonic intensity decreases with increasing harmonic photon energy. The change in the dominant helicity occurs at 120 eV, while the cutoff remains at 160 eV. If one applies the coherent spatio-temporal focal averaging, the spectrum further changes. For $\Delta z=0$ mm the spectrum exhibits a maximum in the region from 50 eV to 80 eV. For $\Delta z=-2.5$ mm this maximum is shifted to higher energies and is between 70 eV and 120 eV. The harmonics with negative helicity become stronger for energies higher than 115 eV. These results are in accordance with the experiment [40,41].

4. Conclusions

The microscopic quantum-mechanical $T$-matrix element, Eqs. (1) and (2), describes the emission of high harmonics from a single atom exposed to a laser field whose amplitude and phase depend on the position of this atom in the laser focus. For a realistic simulation of the high-order harmonic generation process, the macroscopic effects should be taken into account. The fields of all emitted high harmonics should be added coherently and the harmonic field should be calculated in the far field at the detector. In the present paper we have introduced various focal-averaging methods, based on an integral solution of the wave equation, and applied them to simulate two recent experiments [38,40].

In previous publications [31,32] we used a method of incoherent intensity focal averaging, in which the focal-averaged intensity was calculated integrating the harmonic intensity $P_n(I)$ over the laser intensity distribution in the focus. However, incoherent focal averaging is inadequate for the calculation of the harmonic ellipticity. In the present paper we have introduced coherent intensity focal averaging, Eqs. (12)–(17), which is a more approximative method [see the paragraph below Eq. (11)]. We have also presented a more precise method, the coherent spatio-temporal focal averaging, Eqs. (21)–(24), which takes into account the macroscopic conditions of the HHG experiment.

In comparison with the microscopic case, the macroscopic HHG spectrum is changed qualitatively: on a linear scale a peak in the spectrum appears at energies much lower than the microscopic cutoff energy. It seems that the high-harmonic yield is affected by the destructive interference of contributions from the various positions of the atoms in the laser focus and is dominated by the contributions of laser-pulse spatio-temporal regions with lower intensities. The microscopic high-energy plateau, with its sharp cutoff, becomes blurred or may not even be visible in the experimental spectrum. Our macroscopic simulations are consistent with these experimental results: the harmonic yield in the high-energy plateau and cutoff region is orders of magnitude lower than the yield of this low-energy peak. While for our simplest coherent intensity focal averaging the position on this peak does not change much with the laser intensity, wavelength, and macroscopic conditions, for our coherent spatio-temporal focal averaging both the position and the height of this peak strongly depend on the macroscopic conditions which we can simulate with our theory. In particular, the harmonic photon energy which corresponds to this peak can be increased few times by shifting the position of the gas cell with respect to the laser focus. We have illustrated this by presenting results for the parameters of recent experiments with a linearly polarized field and Ar and He atoms [38] and with a bicircular field and Ne atoms [40].