A waveform is synthesized by coherently combining multiple single-color sinusoidal near-infrared or mid-infrared laser pulses. For example, combining three such linearly polarized pulses with the same polarization direction will give a new electric field

The strong field approximation (SFA) formulated by Lewenstein et al.^[107] (see its extension for the HHG with mid-infrared lasers in Ref. [108]) is an efficient method to calculate the single-atom HHG response, in which the induced dipole moment in the time domain is

The quantitative rescattering (QRS) model,^[109,110] as a semi-empirical method, extending and modifying the SFA, which relies on the approximate factorization of the HHG induced dipole, has been “justified” based on the numerical solution of the TDSE for atoms^[111] and molecular ion ,^[112] and has also been tested for a two-color laser field.^[113] Taking an atomic target as an example, according to the QRS model, the induced dipole moment (complex and in the frequency domain) in a linearly polarized laser can be written as

For a multi-color laser waveform composed of orthogonal linearly polarized waves, the SFA formulation of Eq. (2) can be straightforwardly written in two orthogonal polarization directions.^[114] So far the QRS model has been applied to linearly polarized lights only.

Genetic algorithm (GA)^[115–117] as one of the evolutionary algorithms has been widely used in the search and optimization in a multi-parameter space. It also has been applied to study strong-field phenomena, such as retrieving atomic potentials using elastic differential cross sections between free electrons and atomic ions,^[118] coherently controlling induced-dipole emission by optimizing laser-pulse amplitude and phase,^[119] maximizing pulse energy and minimizing temporal coherence of XUV radiation by optimizing laser beam and nonlinear medium,^[106] synthesizing SAP by optimizing short intense excitation laser pulses,^[103] tailoring the temporal profile of laser pulse to identify separately the role of microscopic and macroscopic phenomena in the HHG,^[120] studying specific harmonic order by formulating laser pulse in frequency domain,^[105] and so on.

GA deals with highly nonlinear response functions, which would produce a predefined result after searching the parameter set. In GA, each candidate solution (called individual) can mutate and alter, and a population of candidate solutions evolves toward better ones. The evolution is an iterative process. The population in each iteration is defined as a generation. Here is how the algorithm works. It usually starts with a population of randomly generated individuals. Then the fitness of every individual is evaluated. The better fit individuals are selected from the current population. Their genomes are modified to form a new generation, which is then used in the next iteration. Note that the fitness is usually the value of the objective function in the optimization problem. The evolution process can be terminated if a satisfactory fitness level has been reached, or a maximum number of generations has been produced.

Before carrying out the GA, there are two essential problems: (i) choose an optimization parameter set; and (ii) determine a fitness function which is a real function using optimization parameters as variables plus additional constraints in the optimization process.

However, GA also has limitations. For instance, the evaluation of the fitness function for high-dimensional problems is very expensive, the termination criterion is varied in every problem since GA always gives the “better” solution in comparison to other solutions, and optimized solutions in the GA may converge to local optima rather than to the global optimum in many optimization problems. Such solutions are still acceptable for practical purposes once the optimization goal has been largely achieved. Alternative optimization algorithms may be applied to HHG optimization problems, such as new unconstrained optimization algorithm (NEWUOA)^[121] and Broyden–Fletcher–Goldfarb–Shannon (BFGS) algorithm.^[122]

We start with a waveform synthesized by using two or three commensurate lasers, in which the wavelength of each color has been predetermined. The optimization is simplified by the need of considering only one optical cycle of the longest wavelength color in the waveform.^[97,98]

3.1.2. Three-color combinations: 1600 + 800, 533, and 400 nm

The same optimization procedure has been carried out for three-color synthesis, in which the fundamental laser was fixed at 1600 nm with ϕ₁ = 0. The other two colors were chosen from 800, 533, or 400 nm. The optimization parameter space was enlarged to {E₁, E₂, E₃, ϕ₂, ϕ₃}. The optimized results have been shown in Ref. [97]. Since there were more parameters, the optimization tended to converge to local optima and returned with several different sets of laser parameters. They would result in nearly identical waveform and generate more or less identical HHG spectra. With the ionization level fixed, single-atom HHG yields from such three-color synthesized waveforms were only about two times higher than the previous two-color waveform.^[97] The three-color waveform obviously would increase the difficulty in the laboratories, but it would provide more degrees of freedom for manipulating laser parameters.

Fig. 1.

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Fig. 1. (a) Comparison of a single-color (SC) sinusoidal wave and an optimized waveform (Opt. WF) over one optical cycle (o.c.) of the fundamental. Open and filled circles show the tunnel ionization and recombination times, for an electron with kinetic energy of 2Up following the “long” or “short” trajectories, respectively. The inset depicts the electric fields at ionization time versus the kinetic energies of the returning electrons for “short”- and “long”-trajectory electrons, labeled as “S” and “L” in the figure, respectively. The yields of harmonic emission versus electron recombination times are shown for (b) the two-color and (c) the single-color waves, respectively. Adapted from Ref. [97].

The development of mid-infrared laser technology has increased the central wavelength up to 7 μm,^[24] which promises to greatly extend the HHG cutoff energy. However, harmonic yield drops very unfavorably with increasing laser wavelength.^[46–50] Consider a specific question: what are the best mid-infrared wavelength and target atom for the generation of the highest harmonic yields for a given cutoff energy up to about 1 keV? An optimized two-color (the mid-infrared fundamental laser and its third harmonic) waveform has been identified in Section 3.1.1. So the optimization parameter space was {λ₁, E₁, E₂, ϕ₂}, in which ϕ₁ = 0 and λ₂ = λ₁/3, and the same optimization method in Section 3.1 was applied.^[98]

In Ref. [98], for Ar, Ne, and He, the optimization has been carried out to search the shortest (also the optimal) fundamental wavelength λ₁ for the cutoff energy ranging from 0.2 to 1.0 keV. Compared to optimal single-color driving lasers, adding a third harmonic with a few percent intensity of the fundamental would give about the same given cutoff energy, but would generate much higher harmonic yields.^[98]

We demonstrate the optimized results by taking one example with the cutoff energy at about 350 eV (the actual energy is 350 eV plus I_p of the target). Single-atom HHG (∝ ω⁴|D(ω)|²) spectra of the three targets calculated by the QRS model are shown in Fig. 2(a). The spectrum for Ne has the highest yields over a photon energy range of about 330 eV, followed by He, and then by Ar, when the total ionization probability is fixed at 2%. To understand these results, the HHG is separated into two parts in the spirit of the QRS model. The returning electron wave packets [defined as the modulus square of W(ω)] for three targets are compared in Fig. 2(b), where He is a factor of two or three higher than Ne. However, as shown in Fig. 2(d), the photorecombination cross sections (PRCSs) for Ne are about ten times larger than He, thus ending up that Ne is more favorable for generating high harmonics up to the 350-eV region. The propagated results are shown in Fig. 2(c). The macroscopic conditions are: the laser beam waist w₀ for each color is 50 μm, 1-mm long gas jet is positioned at z_R/2 after the laser focus where the Rayleigh range of the fundamental laser is defined by , and gas pressure of 10 Torr is uniformly distributed within the jet. From Fig. 2(c), macroscopic HHG spectrum from Ne is still the strongest one.

Fig. 2.

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Fig. 2. (a) Single-atom and (c) macroscopic HHG spectra calculated by using the QRS model, and (b) returning electron wave packet calculated by the SFA. The maximum returning electron energy is fixed at 350 eV. The fundamental optimal wavelengths in the two-color waveforms are 2831, 1625 and 1350 nm for Ar, Ne and He, respectively. (d) Photorecombination cross sections (PRCSs) for the three targets over a broad photon energy region are shown. Adapted from Ref. [98].

In Section 3.2, the two-color waveform consisting of a strong mid-infrared laser and a few percent of its third harmonic is able to extend the harmonic cutoff energy to the “water-window” and keV region with the optimal harmonic yield. To realize this scheme in the laboratories, one needs to wait for a high-power mid-infrared laser and a strong phase-controlled third harmonic. However, current wavelength-tunable, CEP-stabilized mid-infrared lasers are mostly pumped by 800-nm Ti:sapphire lasers, thus their pulse energies are smaller than pump lasers.^[123,124] An alternative method is to synthesize an intense 800-nm Ti:sapphire laser with a relatively weaker long-wavelength mid-infrared laser. The wavelength of the latter was optimized, and it could be incommensurate with 800 nm and the optimization was performed over the whole laser pulse (the full-width-half-maximum or FWHM duration was assumed as 21.3 fs, eight optical cycles of the 800-nm laser) instead of a single cycle only as in Section 3.1 and Section 3.2. λ₁ was fixed at 800 nm, ϕ₁ = 0, and E₁ was larger than E₂. The search parameter space was {E₁, E₂, λ₂, ϕ₂}. The optimization goal was to maximize the single-atom Ne HHG yield and to achieve the given cutoff energy with the ionization level of 4% over the whole pulse.^[100]

Fig. 3.

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Fig. 3. Macroscopic HHG spectra of Ne calculated by the QRS model. (a) Two-color (800 + 2126 nm) waveform in the current scheme is compared with the two-color waveform (mid-infrared laser and its 3rd harmonic) in Fig. 2. One-color results are shown for reference. The parameters are indicated in the figures, I0 is in the units of 1014 W/cm2. (b) Comparison of the harmonic spectra by optimized two-(800 + 2126 nm) and three-color (800 + 2356 + 400 nm) waveforms. Adapted from Ref. [100].

The optimized laser parameters were obtained for the cutoff energy from 200 to 550 eV covering the “water-window” region: the peak intensity of the 800-nm laser was always about 2.0 × 10¹⁴ W/cm², the wavelength and intensity of the mid-infrared laser increased monotonically with the increase of the cutoff energy, and the phase ϕ₂ was quite stable. The macroscopic calculations were applied to test the single-atom optical waveform. The laser beam waist w₀ for each color was 50 μm, and the gas jet (1 mm long) was put at 2 mm after the focus. Gas pressure with uniform distribution in the jet was 10 Torr. For the cutoff energy of 350 eV, four different simulations with nearly identical total pulse energy are shown in Fig. 3(a). Clearly, a single 1925-nm laser can reach the required cutoff, but with the yield about 1000 times weaker than the 800-nm one. With the two-color waveform, by combining 800-nm and 2126-nm laser pulses, where the latter has about 50% of the energy of the former, the HHG yields over the same spectral range can be increased by about 30 to 100 times with the comparable total laser energy. The one with 1625 nm (3.88 × 10¹⁴ W/cm²) plus its third harmonic gives the highest yields, however, they are only a factor of two or four larger than those in the current scheme, while the total power is slightly larger.

The three-color scheme has also been checked. With all other conditions staying more or less the same, a 400-nm laser was added as the third color, thus the search parameter space became {E₁, E₂, E₃, λ₂, ϕ₂, ϕ₃}. In Fig. 3(b), the macroscopic harmonic spectrum by optimally synthesizing 800-nm, 2356-nm, and 400-nm laser pulses is shown. Two-color (800 + 2126 nm) spectrum in Fig. 3(a) is also displayed for comparison. The two waveforms generate about the same harmonic yields, thus adding one more color does not help further enhancement of the harmonic yield.

According to the three-step model,^[21,22] the maximum photon energy achieved in harmonic generation process is related to the maximum returning electron kinetic energy. The synthesis of multi-color laser pulses offers a potential way to extend the harmonic emission to high photon energies by shaping the waveform at the single-cycle level in which the electron motion in the laser field can be steered. Chipperfield et al.^[95] have derived the so-called “perfect wave” for HHG, which generates returning electrons with the maximum possible energy for any given oscillation period T and per period fluence F. This ideal waveform consists of a linear ramp with a dc offset, and is expressed as

Fig. 4.

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Fig. 4. The driving electric field (red lines) of (a) a “perfect wave” and (b) an optimized waveform consisting of the first four harmonics of 1/T and a field with a period of 2T. Trajectories of the returning electrons with the maximum kinetic energies are also shown (green lines). εopt = (e2/m)(FT/4ɛ0c) is the maximum returning electron energy generated by the “perfect wave”, where e and m are the electron charge and mass, respectively, T is the oscillation period, and F is per period fluence. Reprinted with permission from Ref. [95]. © (2009) by the American Physical Society.

In reality it is not possible to create the “perfect wave”, because the generation of a linear ramp requires a large comb of frequencies and a high dc field is required for maintaining the asymmetric field amplitude. However, these can be ignored if one is only interested in generating a returning electron kinetic energy very close to the optimal value. A practically achievable approach has been proposed by Chipperfield et al.^[95] to use the GA searching for the optimum waveform by coherently combining only a finite number of frequency components. In such optimization process, the fitness function was taken to be the returning electron kinetic energy, in which consideration of the electric field at the ionization time was also incorporated. The first four harmonics of 1/T and a field at half the fundamental frequency 1/2T, limited to half the power of the fundamental, was used to obtain the optimized waveform. The search parameter space consisted of amplitudes and phases of five colors. The five-color optimal waveform is displayed in the red line of Fig. 4(b), which is capable of generating the returning electron energy as high as the “perfect wave”. The calculations of high harmonics at typical macroscopic conditions have demonstrated that this optimized waveform retained the same harmonic yield as a pure sinusoidal pulse with the same periodicity.^[95]

In 2011, Wirth et al.^[85] reported the optical field synthesizer by dividing a white light super-continuum into three channels (spanning the infrared, visible, and ultraviolet frequency regimes, and covering 1.5 octaves) for the generation of sub-cycle laser pulses. Based on light field synthesis, Balogh et al.^[96] have demonstrated in a recent theoretical work that attosecond pulse generation including the SAP generation could be controlled by optimally shaping the driver waveform. They added a fourth (UV) channel, together with three channels in Wirth et al.’s experimental spectra, to extend the total usable bandwidth to 1.88 octaves such that there are more freedom in the simulation. The optimization parameters in each channel included a time delay, the CEP, and the amplitude. SFA was applied to calculate the single-atom HHG response of Ne gas, and GA was used for the optimization. A fitness function was defined for different simulation goals, such as single shortest attosecond pulse generation or double-attosecond-pulse generation.

Here, only the waveform optimization for the SAP generation is discussed. Specifically, the fitness function was the shortest possible SAP with at least 10:1 contrast ratio by varying all free parameters. In the meanwhile, the integration time in the SFA calculation was limited to the short-time region such that harmonic emission from “long”-trajectory returning electrons can be eliminated. These integration times were obtained by classical calculations. After 50–80 generations, the GA was converged and returned the optimized parameters for generating a 55-as SAP. The corresponding laser field is shown in Fig. 5(a) including its electric field (solid black line) and the half-cycle period (red dashed line) defined as π/|dφ/dt|, where φ is the phase of the laser field. The time-frequency analysis of the harmonic energies from 120 to 275 eV is presented in Fig. 5(b). The single-atom harmonic emission was calculated by the SFA, in which the integration time was not limited. The optimization determined that the best harmonic region for the shortest SAP production was 118- to 195-eV bandwidth, which gives a 55-as SAP in Fig. 5(c). If spectral filtering was used to select only “short” trajectory electrons, the SAP duration was reduced to 51 as, and a second peak corresponding to “long” electron trajectories vanished as indicated in Fig. 5(c). Macroscopic calculations were performed to test the single-atom optimized generating field, which predicted a slightly longer 72-as pulse in Fig. 5(d) by filtering out harmonic emissions with the divergence angle > 1 mrad.

Fig. 5.

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Fig. 5. (a) Electric field (black solid line) and instantaneous half-cycle period (red dashed line) of the optimized laser field. (b) The time-frequency picture of harmonic emission calculated by the full SFA. (c) Attosecond pulses by synthesizing single-atom harmonics between 118 and 195 eV. (d) Attosecond pulses generated after macroscopic propagation, in the near field and in the far field including small divergent (< 1 mrad) harmonic emissions only. Reprinted with permission from Ref. [96]. © (2014) by the American Physical Society.

In 2013, Wei et al.^[90] reported selective enhancement of a single harmonic by synthesizing an 800-nm laser and its second and third harmonics with perpendicular polarizations. The intensity contrast ratio between the target harmonic and adjacent harmonics was more than one order of magnitude. Other earlier studies have demonstrated similar enhancement of selective harmonics with quite different experimental scheme by shaping an 800-nm femtosecond pulse in the frequency domain.^{[119,125–127]} A narrow frequency bandwidth around the central frequency was divided into dozens of small components and the phases of individual components were adjusted. Inspired by these works, Wang et al.^[99] theoretically discussed the possibilities of combining the two techniques, i.e., pulse shaping and multi-color laser synthesizer, to achieve better harmonic control.

Fig. 6.

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	Fig. 6. Single-atom harmonic spectrum showing selectively enhancement of H26. The inset zooms in H26 with its nearest neighbors. A two-color field (800 + 267 nm) has been used with the optimized relative phase and chirp rates. Adapted from Ref. [99].

We will only review the simple case of two-color field combination with a linear chirp and the same polarization for both colors. Linear chirp rates of two colors and the relative phase between them were composed of the parameter search space. The fitness function was the yield of the highest harmonic peak with all other harmonics suppressed for a given fundamental intensity while the intensity ratio between the fundamental and its third harmonic was fixed.^[99] One example of optimized single-atom harmonic spectra is shown in Fig. 6. The harmonic order 26 (H26) stands out prominently from the background. The inset zooms in H26 with its neighbor harmonics, and the contrast ratio between them is about ten times. In the simulation, the intensity of the fundamental 800-nm laser is 3.0 × 10¹⁴ W/cm², the intensity ratio is 5%, and the FWHM duration of the laser pulse is ten optical cycles of the fundamental. Theoretical analysis indicated that for the chirp-optimized waveform, target harmonics emitted at different times (or optical cycles) have good temporal coherence, thus they add up constructively. The single-atom result has been tested by averaging harmonic spectra over a range of intensities to partially include macroscopic effects.^[128] The enhancement of the target harmonic was maintained. It also has been shown that selectively enhancing two harmonic orders of interest was possible by a simple two-color-linear-chirp combination.^[99]

The hollow waveguide is a commonly used experimental setup for generating high harmonics, which has been implemented for creating quasi-phase matching (QPM) conditions,^[130,131] selecting electron trajectory,^[132] temporally and spatially shaping the driving laser pulse,^[125,133] combining with a UV laser to efficiently producing soft X-ray harmonics,^[134] and generating keV harmonics with mid-infrared lasers.^[25,135,136] In a hollow waveguide, diffraction can be eliminated such that a spatially constant laser intensity can be maintained over an extended distance and the geometric phase does not depend on the radial distance. This is considered to be an ideal setup to keep a nearly constant waveform as the laser pulse propagates. We have proposed to combine two advanced technologies: multi-color laser synthesizer and laser guiding in the waveguide, to generate extremely bright high harmonics in the soft X-ray region. This goal has been achieved by optimizing the waveguide parameters (radius and length) and gas pressure.^[129]

Fig. 7.

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Fig. 7. (a) Total harmonic yield emitted at the exit of the hollow waveguide and (b) harmonic yield integrated within 1 mrad using an aperture in the far field for two-color (1.6 + 0.533 μm) waveform (WF) and single-color (SC) laser. The corresponding harmonic divergence in the far field for WF (c) and for SC (d). Length and radius of the waveguide are 5 mm and 125 μm, respectively, and gas pressure is 50 Torr. Adapted from Ref. [129].

Figure 7(a) shows the total harmonic yield (integrated over the exit plane of the hollow waveguide) generated by the two-color (1.6 + 0.555 μm) waveform at the optimal conditions (the radius and length of the waveguide are 125 μm and 5 mm, respectively, and the gas pressure is 50 Torr). These conditions ensure that the highest cutoff energy and the highest harmonic yields are achieved simultaneously. The two-color waveform used here has been optimized at the single-atom level favorable for the phase matching, as shown in Fig. 1(a). The harmonic yield from a reference 1.6-μm laser with the intensity of 3.0 × 10¹⁴ W/cm² is also shown, which is about 1 to 2 orders of magnitude smaller than the two-color pulse. The harmonic emission in the far field for the two cases are shown in Figs. 7(c) and 7(d). Two-color high harmonics in the spectral range of 70 to 250 eV are well localized along the propagation axis with divergence angle smaller than 1 mrad. If one uses an aperture to filter out divergent harmonics (> 1 mrad), results in Fig. 7(b) show that two-color high harmonics have more than three orders of magnitude than the single-color ones. Further theoretical analysis uncovered that under the optimal waveguide parameters and gas pressure, the balance between waveguide mode, atomic dispersion, and plasma effect could be reached to achieve dynamic phase matching and the optimized two-color waveform was maintained throughout the hollow waveguide.^[129]

This work demonstrates the advantage of a predetermined two-color laser waveform, such as enhancing the harmonic yield, suppressing the “long”-trajectory electron emission, maintaining the given cutoff energy, and minimizing the HHG phase-mismatch and laser defocusing, can be fully realized when macroscopic conditions are properly chosen. It also provides a scheme to limit the number of macroscopic parameters for the ultimate optimization.

A gas jet or gas cell is widely used in HHG experiments. A laser beam is usually focused to reach the required intensity to interact with a gas to initiate the HHG process. The distributions of intensity and phase of the laser beam are not uniform in space. By simply focusing multi-color laser beams into the same spot, the optimized waveform is maintained only in a very small spatial range. This method has been applied in the simulations in Section 3 to check macroscopic propagation effects for single-atom optimized waveforms.

Here we suggest two possible ways to optimize macroscopic conditions for a given multi-color laser waveform for the geometry of a gas jet or a gas cell. First, we assume the length of a gas cell is so short that the effects of nonuniform laser intensity and phase can be minimized and that defocusing of the laser beam is not serious. In this case, the gas pressure can be greatly increased and optimized to compensate the short gas-cell length. Second, for a typical 1–2 mm long gas jet, the combination of a multi-color laser beam can be optimized such that laser waveforms in the interaction region with the gas jet are the most closest to the given waveform. This involves adjustment of a large number of parameters, such as the gas-jet position, and the parameters determining laser focusing conditions and input energy for each color, and a proper definition of the best waveform referring to the given one. This is better accomplished by using an optimization algorithm, GA for example. Gas pressure can be treated as an additional free parameter. The macroscopic propagation calculations by solving the three-dimensional Maxwell’s equations of the driving laser and high-harmonic fields should be used to check the outcome of these approaches, thus modifying them accordingly.