In femtochemistry, the generation of controlled ultrashort light pulses from femtosecond lasers (1 fs = 10−15 s) plays a vital role to study the breaking and forming of bonds and movements of atoms in molecules e.g. during chemical reactions or dissociation processes in real time [1]. The genuine time scale on which atoms move on Angstrom length scales, are typically ∼100 femtoseconds. Time-resolved experiments are using well established pump-probe techniques with two synchronized femtosecond light pulses, where the pump pulse is preparing a coherent molecular wavepacket and triggering a change of the atomic configuration of the target, while the probe pulse measures that change. The temporal evolution of the system can then be recorded as a series of temporal snapshots by varying the delay between both pulses (“stroboscopic movie”). In such experiments, the temporal resolution is to first order limited by the time duration of the probe pulse itself. Today, state-of the art laser systems provide light pulses with a temporal duration approaching the limit of only one cycle of the electromagnetic field (period time ∼2.6 fs at 800 nm wavelength).

Even shorter electromagnetic pulses can be generated in a coherent highly non-linear conversion process dubbed High Harmonic Generation (HHG), when an intense femtosecond laser light pulse (I > 1013 W/cm2) interacting with a (usually) noble gas is converted into a broadband harmonic spectrum spanning tens to hundreds of harmonics and ranging from the fundamental laser wavelength into the extreme ultraviolet or even soft x-ray spectral range (see Fig. 1 showing a photograph of a HHG gas nozzle plus incident laser beam). This process can be understood semi-classically within the famous three-step model developed by Paul Corkum [2], describing the tunnel ionization of a bound electron induced by the instantaneous laser field (step 1), followed by the acceleration of the free electron along the polarization direction of the laser field (step 2) and eventually the recombination with the parent ion, giving rise to the emission of a high harmonic spectrum (step 3).


Figure 1. Photographic image of a High Harmonic Generation (HHG) gas nozzle illuminated by a few cycle laser pulse line (image: courtesy of Thorsten Naeser (LMU Munich)).

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In this semi-classical model the high-energy cut-off of the HHG spectrum can be calculated from the ionization potential Ip of the gas target (e.g. in He Ip = 24.6 eV for the ground state) and the ponderomotive energy Up of the free electron in the laser field (Up = e2Elaser2/4meω2) as hνmax = Ip + 3.17 Up.

By spectrally selecting and coherently superimposing the HHG spectrum over a broad bandwidth, trains of attosecond pulses [3] or even single isolated attosecond pulses [4] can be generated (1as = 10−18s). The laser pulse duration as well as (for few-cycle pulses) the phase of the electrical field under the pulse envelope (carrier envelope offset phase) determines the number of ionization events and thus the number of emitted attosecond pulses per laser pulse. Currently, the shortest single isolated attosecond pulses being experimentally achieved exhibit a pulse duration of about 67 as [5] to 80 as [6]. Such pulses have been applied to study the movement of electrons (better the dynamics of electron waves and wave packets) in atoms, molecules and from solid surfaces (for an in depth review of attosecond physics see [7, 8]).

A big challenge of High Harmonic Generation lays in the improvement of the typically very low conversion efficiencies for even brighter pulses (e.g. by phase matching [9]) as well as in the extension of the HHG spectrum towards even smaller wavelength into the soft X-ray range. The extension of the HHG spectrum into the soft X-ray range is also a prerequisite for achieving broader plateau areas and thus shorter isolated pulses, eventually approaching the atomic unit of time (defined as h/13.6 eV = 24 as).

A number of research papers have been published over the last few years discussing the possible extension of the HHG spectra, e.g. by the use of longer wavelength driver laser pulses [10] (because the spectral HHG cutoff scales with the laser frequency as ω−2) or by shortening the effective driver laser pulses by polarization gating techniques [11]. Mashiko et al. [12] have presented an idea based on double optical gating to realize HHG supercontinua with a bandwidth even supporting attosecond pulses down to 16 attoseconds, however the technique is very inefficient and requires very high driver laser intensities (>1016 W/cm2) as well as a compensation of the spectral chirp (phase nonlinearity) over the HHG spectral range.

The theoretical paper of Feng, Duan and Chu [13] now presents new simulation results, which may open a new route towards achieving the goal of attosecond pulse generation approaching the atomic unit of time.

The authors simulated a High Harmonic spectrum, which has been excited by femtosecond light pulses of different colors (wavelength 800 nm, 1200 nm, 1600 nm) instead of a single color pulse at 800 nm wavelength (which is the mostly used configuration in experiments). Additionally, the authors introduced relative spectral phases (quadratic chirp) between the three pulses and investigated the influence of that spectral chirp on the High Harmonic spectrum

The simulations where achieved by solving the time-dependent Schrödinger equation (TDSE) for a Helium atom (the HHG target atom) in the approximation of one active electron. In contrast to other publications, the authors used a quasi 3D simulation code, which is known to yield more accurate, but also more time-consuming results than the simpler 1D calculations. The time-dependent wave function is then further expanded in the laser field (under the influence of the two control pulses) by means of a second-order split operator method. The high harmonic spectrum has then been calculated from the Fourier transform of the induced time dependent dipole acceleration as sum over the partial dipole contributions between different angular momentum l-states.

The key result of their simulations has been given by comparing the HHG spectrum driven by an unchirped and intense main pulse (5fs/800 nm, I = 1015 W/cm2) with the HHG spectrum excited by the main pulse plus 2 more control pulses of different color and lower intensity (10fs/1200 nm and 10fs/1600 nm, I = 1013 W/cm2 each), where the relative spectral phases (“chirp”) between all three pulses have been controlled and optimized. While the control pulses do not play a significant role during the initial step of tunneling ionization (which is achieved by the field maxima of the 100 times more intense main pulse), they affect the trajectories of the free electrons during the acceleration process.

While the conventional single color pulse excited spectrum extends up to the ∼138th order, giving rise to a maximum photon energy of about 214 eV (in quantitative agreement with the prediction from the three step model), the three color chirped pulse excited spectrum extends to much higher photon energies up to the ∼436 harmonics without a significant drop in intensity over a large bandwidth of the higher harmonics >100 (“plateau region”).

The generated broadband supercontinuum in this case extends over a record value large bandwidth of 491 eV ranging even into the “water window” spectral range between the C-K and O-K absorption edge, which may be interesting in the near future for in-vitro studies of biomolecules with such coherent soft x-ray pulses.

While achieving a broad spectral HHG bandwidth is only a necessary pre-requisite for achieving a short attosecond pulse, a further requirement is a well-controlled and locked phase between the individual emitted harmonics. This is not necessarily the case over the whole spectral range.

From their simulations, the authors could also identify the underlying electron quantum paths in the recollision process. The main difference between the two excitation schemes (with and without control pulses) is, that the favorable “short quantum path”, which gives rise to the emission of phase-locked harmonics and can thus best be used for the generation of attosecond pulses, is much preferred in the case of the three color field excitation, while in case of the single color field excitation both, short and long quantum path, almost equally contribute to the HHG spectrum.

From these findings, the authors could identify the best spectral range within the HHG spectrum, where only contributions from short quantum path recollisions are superimposed.

Selecting this sub-range from the 120th to the 180th order, a pulse of only 32 attoseconds duration could theoretically be synthesized without any further phase compensation.

The theoretical simulations of Feng, Duan and Chu thus provide a new idea for future experimental modifications of current HHG schemes. It is particularly interesting, that the laser parameters assumed within their calculations can already nowadays be achieved by state of the art femtosecond laser systems (Ti:Sa amplifier plus optical parametric amplifier). Their work will certainly stimulate experimental efforts towards the development of new HHG attosecond sources ranging into the soft x-ray range with even larger spectral bandwidth and providing attosecond pulses close to the atomic unit of time.


  1. Top of page
  2. Acknowledgements
  3. References

U.K acknowledges support by the Deutsche Forschungsgemeinschaft (DFG) under the framework of the excellence cluster “Munich Centre for Advanced Photonics” (MAP)


  1. Top of page
  2. Acknowledgements
  3. References