## Abstract

We study semi-analytically and numerically the photon yield of high harmonic generation (HHG) on the level of the single atom response under ideal conditions: no initial depletion of the ground state prior to the main peak of the pump pulse. We show that the yield decreases exponentially as function of the cutoff energy starting at about 0.5keV in the case of a Ti:sapphire source and a helium target. We show that the yield in helium beyond the 1keV energy range can be increased by orders of magnitude when long wavelength driver sources in the range from 1.5*µ*m to 3*µ*m are used. This finding leads to the conclusion that significant HHG beyond 1keV is possible through long wavelength driver pulses.

©2005 Optical Society of America

## 1. Introduction

High harmonic generation (HHG) is currently the only technique to provide fully coherent sub-femtosecond extreme ultraviolet and soft x-ray pulses from a tabletop source and, therefore, has attracted increasing attention in recent years. Progress in ultrafast and ultraintense laser technology over the last decade has led to the landmark experimental achievement of building HHG-based soft x-ray sources reaching the water window [1, 2].

An excellent qualitative and quantitative theoretical understanding of the physics of HHG was gained by the development of the semiclassical picture or three step model [3, 4]. It explains that the spectral cutoff of HHG is approximately given by *I _{p}*+3.17

*U*, where

_{p}*I*is the ionization potential and

_{p}*U*is the ponderomotive energy [3, 4].

_{p}Although the ponderomotive energy linearly increases with the pump intensity, it soon became clear that as the latter increases, HHG becomes limited by other mechanisms. First is the depletion of the ground state: If the pulses consist of many cycles, the electron that is supposed to generate the harmonics is detached from the atom before the peak of the pulse is reached, and therefore the harmonics are generated by a weaker field than at the maximum of the pulse intensity [5]. It has been recognized [6] that reaching higher HHG photon energies requires short pump pulses, which triggered the generation of high energy few-cycle laser pulses [7].

Other mechanisms that limit HHG photon energies are propagation effects, such as dephasing and absorption, which become more pronounced at higher harmonics. Dephasing can be coped with by employing quasi phase matching techniques [8], or by exploiting nonadiabatic phase matching mechanisms [9]. Some studies predict that using the latter mechanism HHG photon energies up to 2keV can be reached using 800nm driver pulses and a neutral helium target.

Indeed over the years the record HHG photon energy is being pushed further and further. In 2004 photon energies of 700eV with yields of 2×10 ^{5} photons per second in 5% of bandwidth were obtained [10], and recently the 1keV “barrier” was crossed [11], at yields of 10^{2}–10^{3} photons per second in a 10% bandwidth. All these records were achieved using a Ti:sapphire source and a target of neutral helium. The decrease in yield as the cutoff is pushed from 500eV to 700eV and further to 1.3keV [11] is attributed to various difficulties that are hoped to be overcome, and photons up to 2.5keV are expected to be obtained [11] from this configuration.

In this work, we point towards a fundamental limitation on the HHG efficiency, on the level of the single atom response. We show that as the HHG cutoff is increased by applying higher laser intensities, the conversion efficiency on the level of the single atom drops *exponentially*. For a Ti:sapphire source and a neutral helium target, the exponential drop starts at about 0.5keV, and the yield decreases by about 5.5 orders of magnitude per keV as the cutoff energy is pushed upwards. Since helium is the best neutral atom candidate for achieving high HHG energy, having the highest ionization potential among them, we conclude that the potential of HHG in terms of photon energy with a Ti:sapphire source and a neutral atom target is nearly exhausted.

The future of HHG lies either in using ions instead of neutral atoms, or by using driver pulses with longer wavelength, since the ponderomotive energy quadratically increases with the latter. Although HHG from argon ions was recently demonstrated employing quasi phase matching techniques [12], neutral atoms are much more favorable to control propagation effects, since the density of free electrons generated during the process is then much lower and even more so once longer wavelength drive pulses are employed. We conclude that longer wavelength drives are the most promising direction for HHG at high photon energies.

## 2. The intracycle depletion of the ground state

The results presented in this work are based on three step model of HHG [4], and supported by numerical simulations of a single-atom and a single-electron response. Many electron effects are not considered, and the back reaction of the xray photons on the medium is neglected.

The origin of the fundamental limitation on the photon yield is then very simple. The three step model starts with the following ansatz for the wave function |*ψ*〉, which in atomic units (*h*̄=*m*=*e*=1) has the form

where *I _{p}* is the ionization potential of the atom, |0〉 is the ground state, and |

*φ*〉 is the part of the wavefunction describing the freed electron [4]. Here,

*a*(

*t*) is the amplitude of the ground state, whose modulus decreases in time due to ionization by the external field. The three step model associates HHG with the interference term

in the dipole moment 〈*ψ*|*x*|*ψ*〉, due to the recollision of the freed electron with the parent ion.

The high harmonics emitted by this process are determined by the dipole acceleration [5]. The latter is proportional to *a*(*t*), the amplitude of the ground state, and the amplitude of *ψ*. According to the three step model, the main contribution to harmonic emission near the spectral cutoff (the “plateau harmonics”) comes from electrons that were released around the time *t*
_{1}≈(*π*/2+0.313)ω^{-1}, and return to the nucleus around the time *t*
_{2}≈5.97ω^{-1} [3].

Even under ideal conditions, when *a*(*t*)=1 at the beginning of the optical cycle, that is, *even if no initial depletion of the ground state has occurred before the largest electric field in the pulse acts on the atom*, the amplitude of the dipole acceleration is proportional to |*a*(*t*
_{1})| and |*a*(*t*
_{2})|, the amplitude remaining in the ground state upon ionization, and upon the reencounter. These two factors are always smaller then one and may significantly suppress the HHG photon yield. It is crucial that the depletion of the ground state affects the HHG yield “twice”: Once at *t*
_{1} and once at *t*
_{2}, as pointed out in various publications [4, 13].

Intuitively, the high frequencies in the dipole moment (or acceleration) come from the beating between the recolliding electron and the wavefunction remaining in the ground state. The former oscillates with a frequency related to the energy v^{2}/2 (v is the velocity of the electron), and the latter oscillates with the frequency related to -*I _{p}* (neglecting Stark shifts). The beat frequency is related to v

^{2}/2+I

_{p}, the time derivative of the well known quasi-classical action [4]. The amplitude of this beating is proportional to the amplitudes of the two interfering waves.

These arguments can be easily cast into a quantitative formula, using ionization rates. In order to isolate the above described effect, indeed we consider a single cycle “pulse” whose electric field amplitude is given by

We assume the ideal conditions of the atom being fully in the ground state at *t*=0. For a more realistic pulse |*a*(0)|<1, and the photon yield is obviously further suppressed by an additional factor of |*a*(0)|^{4}.

Let *w*(*E*) be the static ionization rate as function of the electric field. keV HHG occurs mostly in the barrier suppression regime at the wavelengths we are considering here. Therefore the quasi-static approximation for the ionization rate is excellent [14]. Within this approximation, *a*(*t*
_{1}) and *a*(*t*
_{2}) satisfy

*a*(*t*
_{1})*a*(*t*
_{2}) enters as a prefactor to the dipole moment amplitude, and therefore the HHG photon yield will have |*a*(*t*
_{1})*a*(*t*
_{2})|^{2} as a prefactor. While the HHG photon yield obviously has other factors in it [15], |*a*(*t*
_{1})*a*(*t*
_{2})|^{2} is the only factor that exponentially decreases with increasing *E*
_{0}. Since we intend to compare the HHG efficiency for different drive wavelengths, we define the normalized efficiency

that is proportional to the overall photon yield on the level of the single atom response. The *ω*
^{3} factor comes from quantum diffusion, which degrades the yield [4] with increasing wavelength. η_{0} enters as a prefactor to the HHG yield, and we wish to study its scaling as a function of the drive wavelength and intensity.

Figure 2 shows η_{0} computed using the tabulated static ionization rates for helium [14]. The normalized efficiency η_{0} is calculated for a range of electric field amplitudes *E*
_{0}, and for convenience we plot it as function of the HHG cutoff, obtained by the semiclassical formula, associated with each *E*
_{0}. η_{0} decays exponentially as function of the cutoff energy, which follows simply from the fact that ionization in the barrier suppression regime [5] *w*(*E*) is nearly linearly growing with *E*.

The main conclusion from Fig. 2 is that above 1keV 1.5–3*µ*m pumps outperform a 0.8*µ*m one, and as the cutoff increases, the ratio grows by orders of magnitude. Indeed the opposite is true at low energies due to quantum diffusion. However, the intracycle depletion becomes a much more dominant limitation for higher cutoff energies. Another interesting observation is that η_{0} for a 0.8*µ*m decreases by 3 orders of magnitude as the cutoff is pushed from 0.7 to 1.3keV, which agrees with recently reported experiments [10, 11].

In order to test the predictive power of the normalized efficiency η_{0} we compare it with the overall HHG single-atom photon yield. We numerically solved the Schrödinger equation of a hydrogen atom in the field given by Eq. (3). In order to isolate the effect discussed here, we constructed the spectrum of the harmonics by Fourier-transforming the dipole acceleration in the time interval [0,2.04*πω*
^{-1}]. At the end of this interval the most energetic trajectories, responsible for the highest harmonics, have reached the nucleus. Terminating the interval later, but not reaching the region *ωt*≈2*π*·1.5, makes the spectrum more complicated. It adds interference fringes, but the last peak before the cutoff varies very little, as expected from the three step model.

Figure 3 shows three examples of simulated HHG spectra. The plotted quantity is the “number of photons” emitted per atom during the interval [0,2.04*πω*
^{-1}]. These spectra are obtained using the dipole radiation formula, which in atomic units reads

*c* is the speed of light, which equals the reciprocal of the fine structure constant in atomic units. The prefactor 1/20 is because the number of photons refers to a 5% bandwidth. This measure is in commonly used [10]. Fig. 4 summarizes the magnitude of the last lobe in the spectrum, before the cutoff, as function of the cutoff energy. The exponential drop of the simulated yields well agrees with that of η_{0}.

## 3. Discussion

We studied the limitation on the HHG photon yield due to the intracycle depletion of the ground state. We have shown this phenomenon results in an exponential decay of the HHG photon yield as function of the cutoff. The decay rate agrees with simulations for hydrogen. The limitation posed by intracycle depletion is fundamental and cannot be overcome by shortening the pump pulses, since it exists even in the ideal condition of a single cycle sinusoidal pulse. We have shown that under these ideal conditions above 1keV a 1.5µmor longerwavelength driver pulses are more efficient on the level of a single atom response.

For more realistic driving pulses, the single atom efficiency advantage of longer wavelength over the traditional 0.8*µ*m is even greater. For a given cutoff energy longer wavelength means a weaker field, and therefore weaker input intensity and weaker depletion of the ground state by the leading edge of the pulse, before its main peak hits the atom.

The macroscopic photon yield will also critically depend on the phase mismatch building up during propagation. On one hand, for a given cutoff energy much less plasma is released as the wavelength of the driver pulses gets longer. On the other hand, the phase mismatch due to plasma at a given plasma density is larger for longer wavelength. A full simulation including phase matching and other propagation effects is required to find the overall conversion efficiency.

Several studies of HHG with long wavelength drive pulses have been recently reported [17, 18]. The pump intensities in these studies were not sufficient for providing ponderomotive energies that compete with those achievable from Ti:sapphire sources. The conclusion of this study is that longer wavelength drive pulses are a necessity for significant HHG above the 1keV barrier.

## Acknowledgments

We thank Christian Jirauschek for many helpful discussions. We thank Reviewer 1 of this paper for his or her suggestions that improved the work. We acknowledge support by AFOSR under contract FA9550-04-01-0011.

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