Abstract
High harmonics extending to Xrays have been generated from gases by intense lasers. To establish these coherent broadband radiations as an allpurpose tabletop light source for general applications in science and technology, new methods are needed to overcome the present low conversion efficiencies. Here we show that the conversion efficiency may be drastically increased with an optimized twocolor pulse. By employing an optimally synthesized 2µm midinfrared laser and a small amount of its third harmonic, we show that harmonic yields from sub to fewkeV energy can be increased typically by tenfold over the optimized singlecolor one. By combining with favorable phasematching and together with the emerging highrepetition MHz midinfrared lasers, we anticipate efficiency of harmonic yields can be increased by four to five orders in the near future, thus paving the way for employing high harmonics as useful broadband tabletop light sources from the extreme ultraviolet to the Xrays, as well as providing new tools for interrogating ultrafast dynamics of matter at attosecond timescales.
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Introduction
In science and technology, light sources in specific spectral regions are often needed for different applications. Some large national facilities have been built to provide powerful broadband tunable lights, for synchrotron radiations or freeelectron Xray lasers. On the other hand, for wide applications it is desirable that small tabletop light sources be available in individual laboratories. Today such broadband coherent lights have been generated as highorder harmonics in gases by intense femtosecond lasers, ranging from extreme ultraviolet (XUV) to soft Xrays, but their intensities are still too weak for most applications.
Highorder harmonic generation (HHG) is an extremely nonlinear process in which ultrashort coherent lights are produced when atoms or molecules are exposed to an intense laser pulse. This is usually achieved with the conventional Ti:sapphire lasers operating at the wavelength of 0.8 µm^{1}. In recent years advance of midinfrared (MIR) lasers with wavelength of a few microns has made it possible to extend high harmonics to the soft Xray region^{2,3,4}. It is wellknown that the cutoff photon energy by a laser with intensity I_{L} and wavelength λ_{L} is given by , where the ponderomotive energy , with I_{p} the ionization potential of the target and the reduced Planck's constant. However, HHG yield from each atom has been shown^{5,6,7,8} to drop roughly like and further reduction occurs if the propagation of harmonics in the gas medium is accounted for^{9}, resulting in very unfavorable scaling law for harmonics^{10,11} as the laser wavelength is increased. Thus, in spite of keV harmonics have been generated with MIR driving lasers, the low conversion efficiency has prevented them from becoming useful light sources for many applications in the laboratories.
To increase harmonic yields, many efforts have been directed at creating favorable phasematching conditions such that high harmonics are efficiently built up in the gas medium^{12,13,14}, or by temporal and spatial pulse shaping of the driving lasers^{15,16}. An alternative approach is to synthesize multicolor sinusoidal fields to create waveforms that maximize harmonic emission from each atom. This approach has been explored often in the last two decades^{17,18,19,20,21,22,23,24,25,26,27}, usually by combining the 2nd or 3rd harmonic with the fundamental laser. More recently, advance in optical parametric amplification (OPA) and optical parametric chirped pulse amplification (OPCPA) technology^{28,29} has made it possible to generate practically any optical waveforms by coherent wavelength multiplexing of ultrabroadband (over two or more octaves) pulses^{30,31,32,33,34,35,36,37,38}. Despite these efforts, waveform synthesis has rarely been systematically investigated with the aim of achieving optimal harmonic yields^{17,39}. In view of the increasing availability of MIR driving lasers^{3,4,28} for generating subkeV or keV harmonics, it is timely to investigate this optimization issue, particularly for harmonics near and above the water window, with the ultimate goal of generating useful tabletop coherent light sources.
Recently we proposed^{39} a general scheme for optimizing a waveform with synthesized multicolor laser pulses that can enhance harmonic yield by one to two orders over the singlecolor one without the increase of the total laser power. In that work, we investigated how to optimize harmonic yields by adding one or two more color fields to a fixed midinfrared laser. In the present work, we have another specific goal: we ask what are the best midinfrared wavelength and target atom to use in order to generate the highest harmonic yields up to a given cutoff energy near the water window region or up to about 1 keV? Since our earlier work^{39} has shown that an optimized twocolor (fundamental and its third harmonic) waveform was already capable of generating nearly the same maximal harmonic yield as compared to the optimized threecolor waveform, we restrict the simulation to twocolor fields which would be easier to achieve in the laboratories than the multicolor ones^{19}. As mentioned above harmonic cutoff is extended by increasing wavelength or laser intensity, but the harmonic yield drops very unfavorably with increasing wavelength. On the other hand, the increase of laser intensity may result in excessive ionization which reduces neutral atom density and excessive free electrons that are detrimental to good phase matching. These competing factors make the present simulations essential in order to identify optimal conditions for efficient harmonic generation in the sub to fewkeV region. Our simulations show that only a few percent of the third harmonic intensity of the fundamental is needed to achieve the enhancement reported here, thus they are doable with the present laser technology^{19,37}.
Methods
In a twocolor field, we define the longer wavelength (λ_{1}) of the MIR laser as the fundamental and its third harmonic as the complementary one. As shown in Ref. 39 as well as in Supplementary Fig. S1, Supplementary Table S1 and Supplementary Note, the third harmonic is the best in a twocolor synthesis. In the optimization we consider the synthesized waveform in one optical cycle of the fundamental given by
Here E_{i}, ω_{i} and φ_{i} (i = 1, 2) are the respective amplitudes, angular frequencies and phases of the two pulses. The carrierenvelope phase (CEP) and the phase caused by the timedelay between the two pulses^{31,34} are all included in φ_{i}. In the optimization, φ_{1} is set to 0 for simplicity. We search parameters {ω_{1}, E_{1}, E_{2}, φ_{2}} to maximize the singleatom HHG yield. Since optimization takes tens of thousands of iteration and we are studying harmonics up to keV energies, an efficient algorithm of generating harmonics is desirable. Instead of the standard strongfield approximation (SFA)^{40} used in our earlier paper^{39}, a simplified version of SFA^{41,42} is used to calculate harmonics at each iteration. In this method the induced dipole is expressed as D(t) = a(t) exp[iS(t)] where
Here g(t) is the groundstate amplitude, t_{b}(t), τ(t) and v(t) are the born time, excursion time and velocity at time t for the returning electron. The transition dipole moment from the ground state to the continuum state is d[v(t)]. The action S(t) is given as
The launched wave packet with zero transverse momentum component, f(t), in Eq. (2), can be written as^{42}
where I_{p} is the ionization potential, E(t) the laser's electric field and γ(t) the ionization rate.
We define the returning electron wave packet as
and assume that the ground state amplitude g[t_{b}(t)] = g*(t) = 1. Note that Eq. (5) in the time domain is similar to the frequencydomain returning electron wave packet W(ω) = D(ω)/d(ω) used in the quantitative rescattering (QRS) model^{43}. Only solutions that satisfy the classical recollision equations enter into Eq. (5).
For optimization we use the standard genetic algorithm (GA) by D. L. Carroll (FORTRAN genetic algorithm driver, version 1.7a, 2001, available at http://cuaerospace.com/carroll/ga.html). We are interested in HHG over a broad energy region. In the tunnel ionization regime, ionization occurs in a very narrow time window, thus enhancement of one harmonic would automatically also enhance a broad range of harmonics. Since the optimization does not affect the recombination step in the harmonic generation, it is convenient to talk about returning electron wave packet (REWP)^{44,45}. Thus we choose the W′(t) defined in Eq. (5) that gives the optimal yield for the cutoff harmonic as the fitness function.
In addition to the fitness function, other constraints are imposed: (a) The cutoff energy should be more or less maintained at the predetermined value. (b) The ionization level at the end of a singlecycle waveform should be less than a few percent. (c) In the plateau region, harmonics generated from “short”trajectory electrons should be stronger than those from the “long” ones. These constraints are the same as in the previous work^{39}, but the parameter space {ω_{1}, E_{1}, E_{2}, φ_{2}} (mentioned earlier) to be optimized is different. They are assumed not limited in the laboratory. We have checked that the present method of optimization agrees well with the one used in Ref. 39, but the calculation is much faster for harmonics up to keV's.
The constraints listed above are important since it marks the main difference between our approach from the work of Chipperfield et al.^{17}. Constraint (b) is set to avoid excessive free electrons in the gas medium as they will cause phasemismatch of harmonics and plasma defocusing of the laser beam. The ionization rate is calculated by the standard AmmosovDeloneKrainov (ADK) formula^{46,47}. Constraint (c) is imposed since only harmonics from “short”trajectory electrons can be efficiently phasematched in the gas medium. Constraint (a) is used since we are interested in maximal harmonic yields in a preset energy region, while Chipperfield et al.^{17} was interested in the extension of harmonic cutoff energy. Our additional constraints incorporated effects of phase matching were absent in Ref. 17.
Once the waveform in one optical cycle of the fundamental is obtained, it is used to “stitch” a realistic synthesized twocolor fewcycle pulse for a given pulse envelope. For simplicity, in our simulation below the pulse duration (full width at half maximum, or FWHM) for both colors are chosen to be three cycles of the fundamental. Each singlecolor laser is assumed to have Gaussian envelope and the time delay between them is set at zero. We then insert the dipole matrix element d(ω) to REWP in the context of the QRS model^{43}, to obtain laserinduced dipole D(ω) for each atom. The latter is then fed into threedimensional (3D) Maxwell's wave equation to simulate propagation of harmonics, where the copropagation of the full driving laser beam is also included^{9,48,49,50,51}. Since both the intensity and the geometric phase of a focused laser beam have spatial dependence, the optimized waveform is not maintained away from the center of the gas jet. Note that proper focusing conditions, such as beam waists and focusing positions, gas pressure, medium length, are all important factors that will determine the quality of phasematching. Such optimization is better performed in the laboratory as numerical simulation would be too timeconsuming.
Results
Figure 1 summarizes the main results of our simulations for cutoff energies from 0.2 to 1 keV. It shows the minimum (also the optimal) fundamental wavelength that can be used to generate maximal harmonic yields versus the desired cutoff energy for the three targets of Ar, Ne and He, respectively. While it is always possible to use longer wavelength to achieve the same cutoff energy, the harmonic yields reached would be weaker. The results are compared to the optimal singlecolor driving lasers. In both cases the degree of ionization is set at the same 2% level.
Note that single and twocolor optimized waveforms both have about the same minimum wavelength for a given cutoff energy, but the harmonic yields for the synthesized wave are much stronger (see below). This implies that harmonic cutoff energy is governed by the longwavelength laser, but the third harmonic is needed to enhance the harmonic yield. This enhancement was explained in our early paper^{39} as due to the strong increase of electric field during the time interval where “short”trajectory electrons were ionized (see Fig. 1a of Ref. 39). The slope for Ar is the steepest because tunnel ionization rate depends on the ionization potential exponentially. The laser parameters for the optimized waveforms at different cutoff energies shown in the figure are given in the Tables 1–23. This figure clearly shows that by using 2µm laser and its third harmonic, keVphotons can be produced using He as the generating gas.
We next consider how the results of Fig. 1 are derived. If one is to reach cutoff harmonics up to, say, 350 eV (the actual cutoff is 350 eV plus I_{p} of the target) which target and what fundamental wavelength are the best for reaching the highest yields? In Fig. 2(a) we compare singleatom HHG (∝ ω^{4}D(ω)^{2}) spectra of three targets calculated by the QRS model. The spectrum for Ne has the highest yields over a photon energy range of about 330 eV, followed by He and then by Ar. For the Ne target, the wavelengths of two lasers used are 1625 nm and its third harmonic, at a total intensity of 4.1 × 10^{14} W cm^{−2}. For the He target, the wavelengths are 1350 nm and its third harmonic, with total intensity of 6.2 × 10^{14} W cm^{−2} (for other parameters, see Tables 2 and 3). To generate harmonics up to a given cutoff energy with optimal yields, it is desirable to use the shortest fundamental wavelength due to unfavorable scaling of harmonic yields with increasing wavelength. However, a shorter fundamental wavelength has to be accompanied by a higher laser intensity in order to reach the desired cutoff energy. On the other hand, higher laser intensity cannot be running unrestrained since the total ionization yield is fixed at 2%. This can be controlled by choosing a target with higher binding energy, for atoms the highest one is helium. Still further complicated is the fact that harmonic yield also depends on photorecombination (or photoionization) cross section. The interplay of these factors would give the waveform as well as the favored target for generating maximal harmonic yields at a desired cutoff energy. The minimal fundamental wavelength for a given cutoff energy for each target shown in Fig. 1 was thus derived.
In Fig. 2(b) we compare the returning electron wave packets [defined as the modulus square of W(ω) in the QRS model] for the three targets where it shows that He is higher than Ne. However, as seen in Fig. 2(d), the photorecombination cross sections (PRCSs) for He are about ten times smaller than Ne, thus ending up that Ne target is more favorable for generating harmonics up to the 350 eV region. The results of the propagated harmonics are shown in Fig. 2(c). In the simulation, the laser beam waist w_{0} for each color in the waveform is fixed at 50 µm, the gas jet (1mm long) is centered at z_{R}/2 after the focus where is the Rayleigh range of the fundamental laser. Thus for Ar, Ne and He, gas jet is at 1.4, 2.4 or 2.9 mm after the laser focus, respectively. Gas pressure with uniform distribution in the jet is 10 Torr. After propagation, the HHG spectrum from Ne is still the strongest. Note that the gas pressure used in the simulation is low. At higher pressure complications may result from phase mismatch and reshaping of the driving laser beam caused by the free electrons. When high gas pressure is used, absorption by the gas also becomes important. These additional target specific dependent properties make comparing harmonics from different targets complicated. In the absorption limited regime, helium would be the better gas for generating HHG for highenergy photons close to the keV region^{13}.
In Fig. 1 we show the minimum wavelength needed to reach a certain harmonic cutoff energy. Within the same photon energy region, can a longer wavelength (together with its third harmonic) generate more intense harmonics? In Fig. 3(a), we compare the macroscopic HHG spectra of Ne for three different cutoff energies. (Similar singlecolor results are shown in Supplementary Fig. S2 and singleatom HHG spectra of twocolor waveforms are shown in Supplementary Fig. S3.) These spectra are generated by waveforms with fundamental wavelengths of 1075, 1367 and 1625 nm in order of increasing cutoff energy, respectively. Macroscopic conditions used to generate macroscopic HHG spectra are similar to Fig. 2. It is clear that the HHG yields drop rapidly when the harmonic cutoff energy is extended, or equivalently, when longer wavelength lasers are used. With the wavelength increases from 1075 nm to 1625 nm, i.e., by a factor of 1.5, the lowenergy harmonics up to 150 eV drop by a factor of 125. Note that this result is based on the macroscopic conditions used in the simulation. If one is only interested in harmonics up to 150 eV with a longer wavelength laser, higher gas pressure can be used. For singlecolor laser an enhancement factor of 10 has been reported^{11} if the pressure is optimized. Nevertheless, it is preferable to use the shortest wavelength laser to generate harmonics for each photon energy range as depicted in Fig. 1.
In Fig. 3 (b) we show a typical example of using He as a target for an optimized singlecolor wave and a twocolor field, where the harmonic cutoff energy is over 500 eV and macroscopic propagation effect has been included. The harmonic yields from the synthesized waveform show about one order increase in the plateau region to a few times stronger near the cutoff. According to Table 3, the intensity of the third harmonic needed is only about 5% in order to achieve such an enhancement. The fundamental wavelength for the synthesized wave used is 1422 nm and 1477 nm for singlecolor wave. The peak intensity used is about 7.8 × 10^{14} W cm^{−2} for both waves. This result shows that a small amount of third harmonic can enhance HHG yield significantly for the longwavelength laser because it is able to modify the driving waveform in such way that the contribution of “short” trajectories becomes dominant, while in the singlecolor field the contribution of “long” trajectories is dominant. (Singleatom HHG spectra are shown in Supplementary Fig. S3 and the corresponding timefrequency analysis is in Supplementary Fig. S4.) These arguments can be applied to other atoms and cutoff energies in Tables 1–23.
According to Fig. 1 the highest cutoff energy can be reached by 1600 nm and its third harmonic is about 400 eV with Ne as the target. We can optimize the peak field and phase of the two waves to achieve the highest yield at a given harmonic cutoff below 400 eV. In Fig. 4(a) we show the singleatom HHG for cutoff energies at 150, 200, 250 and 300 eV plus 21.56 eV, the I_{p} of Ne, respectively. The HHG yield for the 150 eV curve is the highest in the lowenergy region, as compared to other curves where the cutoff energies are higher. All the four curves are generated with about the same total peak intensity. For the 150 eV curve, the peak intensity for the 1600 nm wave is half of the intensity of the 533 nm wave. In other words, the shorter 533 nm wavelength laser is the “fundamental” laser. For the 300 eV curve, the 533 nm wave is much weaker, with only 15% of the 1600 nm wave, i.e., the long wavelength 1600 nm one is the “fundamental” laser. The relative phase between the two waves remain about the same in all the four combinations; see laser parameters in Table 4. In conjunction with Fig. 3(a), we conclude that for the twocolor fields, it is preferable to use the minimum wavelength from Fig. 1 to reach a given cutoff energy and its third harmonic to enhance the harmonic yields.
Figure 4(b) displays the four optimized waveforms in one optical cycle of 1600 nm for the HHG shown in Fig. 4(a). The electric fields at the time of ionization (near 0.4 optical cycle in the figure) are essentially identical. On the other hand, the electric field near the return time for the “short”trajectory electrons (near 0.1 optical cycle) starts to decrease its strength, less for the 300 eV curve and much more for the 150 eV one till its direction is reversed. Thus the kinetic energy gained by the electron between the born time and return time (best understood by the difference of vector potentials between the two instants, see Supplementary Fig. S5) is bigger for the 300 eV case and smaller for the 150 eV one, thus resulting in higher and lower cutoff energies, respectively.
Finally we note that the optimal relative phase between the fundamental and its 3rd harmonic fields is quite stable and in the range of 1.2π to 1.5π as shown in Tables 1–234 regardless of targets or cutoff energies. This is in agreement with both the experimental and theoretical results in Brizuela et al.^{19}. They found that the relative phase of around 1 rad (equivalent to 1.32π if the carrier is assumed as a cosine function) led to the maximum HHG enhancement. To achieve the enhancement of a factor of 10 as predicted in this work, further optimization of the intensity of the third harmonics in the experiment is necessary. On the other hand, the precise control of CEP of each color and time delay between the two colors is also very crucial to realize the predictions in this work in experiments.
Discussion
In summary, we proposed a twocolor synthesis scheme to achieve optimal harmonic yields ranging from the extreme ultraviolet to soft Xrays. Between the two colors, one is the strong fundamental midinfrared, the other is its weak third harmonic. For targets of argon, neon and helium, we optimized the shortest wavelength of the fundamental pulse that should be used for a given cutoff energy, as shown in Fig. 1. Our results also showed that optimized twocolor field can always enhance the HHG yield by up to about ten times, when compared with the singlecolor one under the same generating condition and requiring only about 5% intensity for the third harmonic which is reachable today. Indeed, some progress in waveform synthesis and optimization has been made^{31,35,36,37}. Still much remains to be explored, including waveform control with multicolor lasers, or full phase control of a broadband supercontinuum over two octaves^{52}. With the emergence of hundreds kHz and MHz MIR lasers on the horizon^{53}, the present simple twocolor optimization, together with favorable phase matching conditions, make us to believe that highorder harmonics would soon be ready to become useful coherent tabletop light sources from the extreme ultraviolet to the Xrays. In the time domain, they also serve as intense attosecond pulses that can be used for attosecondpump and attosecondprobe experiments for studying dynamics of electrons in matter in their intrinsic timescales.
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Acknowledgements
This work was supported in part by Chemical Sciences, Geosciences and Biosciences Division, Office of Basic Energy Sciences, Office of Science, U.S. Department of Energy. Some of the computing for this project was performed on the Beocat Research Cluster at Kansas State University. G.W. was also supported by the National Science Foundation of China under Grants No. 11364038.
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C.J. and C.D.L. conceived the procedure of waveform optimization. C.J. performed all the calculations and analysis. G.W. and A.T.L. initiated the waveform optimization problem. A.T.L. provided the photoionization data. C.J. and C.D.L. wrote the manuscript. All authors discussed the results and reviewed the manuscript.
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Jin, C., Wang, G., Le, AT. et al. Route to optimal generation of soft Xray high harmonics with synthesized twocolor laser pulses. Sci Rep 4, 7067 (2014). https://doi.org/10.1038/srep07067
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DOI: https://doi.org/10.1038/srep07067
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