1. Introduction

High-order harmonic generation (HHG) is a highly nonlinear process arising from the interaction of strong infrared laser fields with atoms or molecules. HHG has an important role in advanced time-resolved imaging science as it provides a route for the generation of coherent extreme ultra-violet (XUV) radiation and attosecond pulses [1,2]. Moreover, the coherent character of HHG has been successfully used to retrieve dynamical and structural information on atoms and molecules [3,4]. Since the start of the exploration of HHG there has been a drive to control the properties and yield of the harmonic radiation using multi-color fields. For experimental convenience this is generally carried out in a ω + 2ω (i.e. typically 800 nm + 400 nm) configuration [5–10]. Recent investigations using an optical parametric mid-IR source and a ~800 nm field have been reported [11], and here the two frequencies are incommensurate (field frequencies not a simple multiple of each other). This scheme has also been proposed using three colors [12]. In the present work we further study the enhancements in efficiency from using combinations of incommensurate frequency fields and elucidate a key part of the mechanism behind this enhancement.

The HHG process can be intuitively understood for a single atom by a semi-classical three-step process. This involves the ionization, acceleration in the continuum and recombination of an electron of the atom or molecule under study [13,14]. The laser field can accelerate the electron to a state of high momentum, such that upon recombination its energy can be many times higher than that of the ground state, leading to the emission of high frequency photons. The HHG spectrum is composed of a coherent sum over the medium of these single atom emissions which must be phase matched to achieve a significant intensity. In the case of a monochromatic field, an increase in the field intensity pushes the harmonic cut-off to higher energies, shortens the electron excursion time for a given harmonic, and increases the ionization rate of the sample enhancing the overall harmonic yield. However, an excess of ionization in the generating medium depletes the ground state population and hinders the phase matching of the harmonic, reducing the detected signal. Thus there are two factors that limit the generation of intense high-frequency harmonics: the maximum power available from the laser system in use and the ionization saturation of the sample.

The cut-off of the harmonic spectrum is known to increase linearly with the ponderomotive potential in which the continuum electron evolves. This potential scales quadratically with the laser wavelength, allowing the cut-off of the harmonic spectrum of a longer wavelength drive laser to match that of a shorter driving one whilst using a much lower intensity [15,16]. This allows the investigation of molecules with low saturation intensities to be carried out over much bigger energy ranges. The drawback of using longer wavelengths is a dramatic decrease, worse than λ⁻⁵ [17–20], of the generation efficiency mainly due to the larger spreading of the electron wave packet which spends longer in the continuum. This dramatic reduction in efficiency is prohibitive for many applications.

The majority of experiments on HHG conducted so far have used 800 nm lasers (Titanium: Sapphire), here we explore harmonic generation from a longer wavelength generating field, 1290 nm combined with the 800 nm field. Good commercial mid-IR optical parametric amplifier (OPA) sources are limited to one or a few millijoules output in as few as four or five cycles. The high cut-off energy available from those sources at relatively low intensities is appealing to a number of users but the low number of photons generated can be restrictive. Those sources are often seeded and pumped by powerful 800 nm Titanium: Sapphire lasers which can generally spare energy that can be used in combination with the OPA output.

Here we extend our understanding of this topic by showing through experiment and theoretical analysis how a superposition of multiple unrelated frequencies (SMURF) can significantly enhance the harmonic yield and cut-off energy of a modest power mid-IR source. We find that the two color field leads to increased efficiency not only through the well known effect of an enhanced ionization rate but can also control the electron trajectory so as to return the electron earlier and thus increase efficiency further.

2. Experimental setup

We used a 1 kHz Ti:Sapphire laser system that provided pulses of up to ~14 mJ and 80 fs duration at 780 nm. Approximately 6 mJ of the beam were reflected off a beam splitter to seed an OPA that produced pulses of 1 mJ and 50 fs duration (determined by auto-correlation) at 1290 nm. The remaining few mJ of the 780 nm beam were sent through a high precision motorized translation stage in order to control the synchronization of the 1290 nm and 780 nm pulses. A half-wave plate inserted in the 780 nm beam was used to control the relative polarization of the two fields. The beams were combined at a normal incidence dichroic mirror and sent collinearly into a vacuum chamber where they were focused onto the target by a 30 cm focal length lens. A telescope inserted in the 1290 nm beam was used to compensate for the chromatic aberration of the lens. The target consisted of a continuous flow of argon diffused into the chamber through a 100 μm diameter aperture with 2 bar backing pressure, the laser is focused just below this aperture (within ~50 μm). The harmonics generated in the gas jet were spectrally resolved by a 1200 lines/mm curved grating which focused them onto a microchannel plate (MCP) detector. Each spectrum was averaged over 500 laser shots at 1 kHz. The shot-to-shot energy fluctuation of each beam was less than 4% RMS.

The laser intensities were controlled using variable apertures in the beam and were estimated from measurements of the pulse energy and the focal spot size. These intensities scaled in agreement with those obtained from the position of the harmonic cut-off (when this was below our detection limit of 80 eV) indicating that we were operating below the ionization saturation limit. This was confirmed by the symmetry of the experimental time delay scans that we show in the next section. The generating medium was ~100 μm long, much shorter than the Rayleigh range of the focus (1 cm) so re-absorption effects can also be neglected.

Data was also obtained for a combination of 1200 nm and 780 nm pulses, producing results qualitatively similar to the ones obtained with 1290 nm and 780 nm. These are not presented here but indicate that the chosen wavelength is not critical in this range.

3. Experimental results

We investigated the effect of multiple frequency fields on HHG by recording the harmonic spectra as a function of the time delay between the two pulses for different relative polarizations. We performed the measurements in three different regimes: (1) generating harmonics with a strong 1290 nm pulse and using the 780 nm as a weaker control field (I₁₂₉₀ > 5×I₇₈₀, where I_λ denotes the intensity of the field of wavelength λ nm); (2) vice versa (I₇₈₀ > 5×I₁₂₉₀); and (3) generating harmonics with both beams at approximately the same intensity.

Common to all three cases was the appearance of non-integer order harmonics when the two pulses were overlapped. These intermediate harmonics were seen as two peaks between the harmonics of 1290 nm or four peaks between those of 780 nm. These are the result of frequency mixing, which induces a sub-cycle reshaping of the electric field. Their spacing relates to odd harmonics of the approximate minimum common multiple of the two wavelengths, which is 3885 nm. The exact energy of the peak of those intermediate harmonics is observed to shift very slightly when the relative intensities are varied since these two sets of harmonics do not have the same periodicity. These photon energies, E_ih, correspond exactly to a linear combination of photons energies from both fields: E_ih = nE₁₂₉₀ + mE₇₈₀ with (n + m) being equal to an odd integer due to conservation of angular momentum [21]. In cases (1) and (2), n>>m and m>>n respectively. In the matched intensity case, (3), their relative values are comparable.

Figure 1a shows the harmonic spectra obtained as a function of the time delay between the two pulses with parallel polarizations when the intensities of the 1290 nm and 780 nm beams were of the same order (I₁₂₉₀ = I₇₈₀ = 0.5×10¹⁴ Wcm⁻²). The intensities were matched by adjusting the cut-offs of the harmonic spectra obtained with each field alone. This configuration produced an enhancement factor in the 1290 nm harmonics signal of more than two orders of magnitude. Here too the intermediate harmonics rose to a signal level equal to the enhanced single color harmonic signal further increasing the total number of XUV photons emitted. When the pulses were temporally overlapped to better than 30 fs, the cut-off moves to higher energies: from 40 eV when the pulses are temporally separated (the 780 nm cut-off was down at 25 eV) to at least 80 eV. This was the highest energy which could be recorded by the detection system. This large range of delays, where both enhancement and cut-off extension are produced with no fine sensitivity to relative delay highlights the stability of the spectrum against delay jitter. In contrast if commensurate frequencies are used any enhancements are known to be very sensitive to the relative phase of the two fields [8].

 

Fig. 1 Harmonic intensity as a function of delay between the two pulses for a) I₁₂₉₀ = I₇₈₀ = 0.5 × 10¹⁴ Wcm⁻².We observe an enhancement of more than 2 orders of magnitude and a cut-off extension from 40 eV to beyond 80 eV b) I₁₂₉₀ = 1.5×10¹⁴ Wcm⁻² and I₇₈₀ = 0.2×10¹⁴ Wcm⁻². We observed an enhancement of a factor 4 but the cut-off of the single color field is already beyond the detection range. Both images have been normalized to their respective maxima and plotted on the same color scale

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The emergence of the intermediate harmonics contributes extra frequencies which could help tailor larger periodicity attosecond pulse trains (APT) over an energy range wider than the single color case. Moreover, the extended cut-off energy approaches the 90 eV range, so the photons are reaching sufficiently high energies to be used, for example, to excite core electrons in atoms whilst keeping the generating intensity relatively low. The significant enhancement in photon energy is of interest also for free electron laser sources using HHG seeding [22,23].

Figure 1b shows a similar scan of harmonic spectra recorded with parallel polarizations when I₁₂₉₀ >> I₇₈₀ (I₁₂₉₀ = 1.5×10¹⁴ Wcm⁻²; I₇₈₀ = 0.2×10¹⁴ Wcm⁻²). Here the cut-off of the mid-IR field alone was seen to reach the end of the MCP so it was not possible to measure the cut-off extension. The peaks of the single color harmonics were enhanced by a factor of approximately four and the intermediate harmonics rose from noise level to the same level as the enhanced single color harmonics. The combined fields’ intensity is here greater than the ionization saturation intensity for argon, and this may be reducing the harmonic yield. The symmetry of the time delay scans around t = 0 confirms that the individual intensity of each field is below the saturation intensity of argon. If one of the pulses had been causing significant ionization we would expect to see some delay time asymmetry in the harmonic signal.

The dependence of the harmonic enhancement on the angle between the polarization axes of the two fields was measured and revealed that the enhancement was always greatest when they were parallel. In fact, the combination of the 1290 nm beam with a perpendicularly polarized 780 nm beam always produced a reduction of the 1290 nm harmonic signal even though intermediate harmonics still emerged.

4. Discussion and theoretical analysis

The principal results of the experiment are that; (a) there is a large enhancement in yield of several orders of magnitude if fields at 5 x 10¹³ Wcm⁻² at 1290 nm and 780 nm are combined, (b) this enhancement is robust to changes of relative delay up to several optical cycles. To explain these observations we have performed numerical simulations of the harmonic generation process in order to understand the mechanisms behind the harmonic enhancement. The single active electron strong field approximation (SFA) [24] is a good framework to qualitatively explain the observed effects and it successfully reproduces the appearance of intermediate harmonics at the correct energies.

Figure 2 shows the variation of the intensity of the 25th harmonic of the 1290 nm field for cases (1) and (3) previously mentioned as the pulses are temporally scanned across each other. The curves on the left column are extracted from the plots in Fig. 1 and the ones on the right are calculations performed for a single atom in the laser field. The yield enhancement is seen to be extremely sensitive to the relative and absolute values of the intensity of each field. As such the simulation was run for the case with the nominal intensities worked out from the cut-off of the harmonic spectrum and for another pair of intensities within a 50% deviation from these that best matched the experimental results. It must be noticed that these calculations do not include the effects of propagation of the harmonics in the gaseous medium and are therefore likely to be an upper estimate of the potential macroscopic signal. The calculationshows that apart from a relatively small amplitude ripple in the enhancement the yield is robust to the relative delay between the fields in agreement with observations. The dips are only seen at the integer harmonic frequencies of the single color field and arise through the transfer of energy from the single frequency harmonics into generation of the additional harmonics of the mixed frequency field that lie outside of the spectral window. As the temporal overlap improves all of the harmonics are enhanced. The general agreement between the calculations and the experimental results, including the dips in harmonic yield at both sides of the enhancement peak, indicate that most of the observed effects are due to the shape of the waveform that results from the addition of two unrelated frequencies acting on the single atoms.

 

Fig. 2 Plot of the intensity of the 25th harmonic of 1290 nm field as a function of time delay between the 780 nm pulse and the 1290 nm pulse. Left column: experimental data; right column: single atom calculation. The black lines in the right column correspond to the exact intensities extracted from the cut-off measurement and the intensities for the cases drawn in red were adjusted for best match of the experimental data but were still within 50% of the retrieved values. All the other harmonics follow a similar trend but only one was plotted for clarity.

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To gain further insight into this process, classical calculations of the electron trajectories were also performed as a means to obtain an intuitive understanding of the effect of using a field synthesized from two colors. Over the duration of the pulses, the continuously slipping phase between the two fields gives rise to a whole variety of electric field shapes. These drive very different electron trajectories during each half cycle, unlike the ω + 2ω case where the synthesized field pattern is repeated every cycle. The electron trajectories have been plotted in Fig. 3 for the case where the pulses are exactly temporally overlapped with carrier envelop phases (CEPs) of zero. One can see that the pattern is irregular but contains a number of continuum electron bunches of similar energy repeated throughout the pulse. Providing the pulse envelope is sufficiently long, changing the CEP or relative phase of the two pulses only changes the times within the pulse at which these “typical” bunches are generated. Some electrons can be seen to spend a long time in the continuum before returning to the parent atom such that the transverse spread of their wave packet will reduce the harmonic yield. Others return sooner than in the single color case increasing the yield. These shortened trajectories dominate the total harmonic yield.

 

Fig. 3 Electric field amplitude (grey dashed curve) of the sum of the 1290 nm field and 780 nm field (delay and CEP fixed at zero) and the classically calculated electron trajectories driven by this field (colored lines). The red dots mark the nine ionized electron bunches which have been selected for the plot in Fig. 5 as they are the ones which will dominate the mid plateau region photon emission.

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To fully establish the advantages of using SMURF against a simple rise increase in the intensity of the 1290 nm field, a propagation model following the approach of [25–28] was used to calculate the two harmonic spectra. Experimental conditions were duplicated to compare the spectrum generated from the matched intensities case with a spectrum generated by a 1290 nm field with twice the electric field, i.e. at four times the intensity. The gas jet was simulated as a 100 microns FWHM diameter Gaussian profiled sample with a peak density of 10¹⁸ atoms cm⁻³, consistent with the gas jet geometry.

Up to energies of 60 eV the intensity of the integer harmonics of the two color case is higher than that of the fourfold more intense single color case, even though the fluence of the SMURF field is approximately 65% of the magnitude of the single color field. The number of photons in a 2 eV energy step, the spacing between two adjacent 1290 nm harmonics, is actually even greater in the two color case due to the presence of the intermediate harmonics. The single color 1290 nm spectrum at 0.5 × 10¹⁴ Wcm⁻² (not shown here) peaks just under the 10⁻⁹ mark on the intensity scale of Fig. 4a and cuts off around 40 eV. The maximum energy photon obtained by SMURF is less than that of the 1290 nm field of four times the intensity, however the pulse energies required to produce this were not attainable from our laser system which points to the usefulness of the SMURF technique when a limited amount of mid-IR light is available. The emergence of the intermediate harmonics may be used for the generation of APTs of higher peak electric field and larger periodicity, allowing for easier isolation of single attosecond pulses [29]. These features are seen in the emitted APTs of Fig. 4b, as calculated from the spectra of Fig. 4a. We note that the individual pulses in the APT are of the order of ~10 times more intense than those arising from the higher fluence single frequency field and the quasi-periodic train has approximately 3 times period (~6.6 fs) of the single color case. A 780 nm field at 2 × 10¹⁴Wcm⁻² will have a cut-off ~60 eV, with a harmonic yield that exceeds the SMURF case due to the electron returning even sooner. The total fraction of the population remaining in the ground state after the laser-medium interaction is much greater in the synthesized field case than in the single color case, leaving a larger margin for the intensity to be increased before running into ionization saturation issues.

 

Fig. 4 a) Calculated harmonic spectra after propagation for the two color case with the intensities of the 1290 nm and 780 nm fields matched at 0.5×10¹⁴ Wcm⁻² (blue curve) compared to the 1290 nm field alone with a four fold increase in intensity (red curve). b) Derived APTs from an inverse Fourier transform of the spectra in (a) normalized to the peak value of the 1290 nm APT. c) Fraction of the electronic population remaining in the ground state as a function of time for three different pulses; the two aforementioned ones and the 780 nm at 2×10¹⁴ Wcm⁻² (green curve), as calculated using the time dependent Schrödinger equation. The total ionization due to the SMURF pulse is much less than that due to the reference single color pulse.

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This intensity crossover between the SMURF and single color 1290 nm spectra around photon energies of 60 eV in Fig. 4a agrees well with a comparison (Fig. 5 ) of the ADK ionization rate with the ionized electrons’ excursion times extracted from the classical trajectories of Fig. 3. This was chosen to illustrate the relative merits of the ionization rate enhancements and the reduction in electron travel time in SMURF. Each line plotted in the Fig. 5 corresponds to the emission from one burst of electrons which has been launched into the continuum by one half-cycle of the field. The lines appear as a series of dots as the trajectory calculation was carried out for a series of discrete times of ionization separated by a 2.5 as step. Only the nine most efficient electron bunches have been plotted for each case for clarity. Many other bunches of electrons also get launched in the continuum throughout the pulse but their contribution to the harmonic generation is less significant.

 

Fig. 5 Comparison of the ADK ionization rates and electron excursion times for the same two cases as Fig. 4. The left most nine lines are obtained by SMURF and the right most nine ones with the single color. Only the nine most significant electron bursts from each case have been plotted for clarity which correspond to the red dots in Fig. 3. The grey dotted lines indicate lines of equal photon energies.

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The first striking feature of Fig. 5 is a shift of some of the two color electron bunches towards the upper left corner of the graph for photon energies up to 60 eV. This means a significant number of them combine the increased ionization rate with a shorter excursion time. The harmonic yield resulting from these sections of the pulse is significantly higher than the single-color case, such that the overall yield is greatly increased. This is despite the fact that yields from less favorable half cycles are reduced and contribute very little to the final spectrum. Above 60 eV the single color case becomes more efficient.

The dependence of the curves plotted in Fig. 5 upon the relative phase and CEP of the two pulses was also considered. The only difference between curves with differing phases is that the nine bunches selected for plotting were generated at different times in the pulse, confirming the unimportance of the phase.

5. Conclusions

A large enhancement of two orders of magnitude was observed in the yield of high harmonics generated in argon by using two parallel polarized fields of unrelated frequencies and similar intensities. The recent work of Chipperfield et al. [30] on the synthesis of an optimal waveform for HHG identified the potential benefits of waveform shaping for reducing the excursion time of the higher energy electrons as compared to a monochromatic field. Even though this work only considered commensurate frequencies, there is no reason that similar effects cannot be achieved using an incommensurate frequency implementation. Reduced return times of the electrons appears also to play a significant role in the SMURF scheme investigated experimentally in the current work. Despite the use of only two colors, our method already shows many promising features through its versatility including compatibility with long pulses, robustness over a large range of delays and the possibility to provide substantial enhancement of the XUV yield together with broadening of spectral range beyond 80 eV for an argon target.

The main practical appeal of this technique is its ease of implementation. To see a strong yield increase all that is required is to have spatial and temporal overlap between the near-IR and mid-IR pulses. There is no need for carrier envelope phase or relative phase stabilization, neither short pulses nor a specific frequency ratio. Photon energies in excess of 80 eV have been generated using just 5 × 10¹³ Wcm⁻² of both 1290 nm and 780 nm light. The utility of this may become especially important for higher repetition rate mid-IR sources where the available pulse energy is likely to be still more limited. Although our experiment is a first demonstration of this scheme, we expect that careful choice of the target species and the field parameters used will offer flexibility on the flux and bandwidth of the high harmonic source produced. The SMURF method also has the potential to produce higher flux APTs with larger temporal separations (> 5 fs). This may permit more effective interrogation by pump-probe methods of processes evolving on the several femtosecond timescale than is possible with standard more closely spaced APTs.