Application of optimized waveforms for enhancing high-harmonic yields in a three-color laser-field synthesizer

Xiaoyong Li; Juan Fan; Jinyu Ma; Guoli Wang; Cheng Jin

doi:10.1364/OE.27.000841

1. Introduction

In the past three decades, high-order harmonic generation (HHG), which is a up-conversion process resulted from the extremely high nonlinear interaction of an intense laser pulse with gas atoms or molecules, or other materials, has attracted tremendous interests [1–6] mainly because of its potential to produce new table-top coherent light sources from the extreme ultraviolet (XUV) to the X-rays [7–10]. Such ultrashort light sources have been engaged in many research fields, such as ultrafast electronic processes probing [11–13], chemical reaction tracing [14], molecular orbital imaging and probing [15,16], and the generation of attosecond pulses [17–20]. However, the low conversion efficiency of HHG is still the biggest obstacle for preventing its wide applications in science and technology [21,22].

HHG is a coherent process in which an intense laser interacts with all atoms in the gas medium, so the final harmonic yield is determined not only by the interaction strength of each atom with a driving laser, i.e., single-atom response, but also by the coherent summation of individual harmonic emissions, i.e., macroscopic response. Many efforts have been directed at increasing either single-atom or macroscopic response to enhance the conversion efficiency of HHG. One way is to make the macroscopic harmonic field grow up coherently by creating the favorable phase-matching conditions, which can be achieved by adjusting laser and gas conditions, such as laser intensity, wavelength, focusing parameters, gas properties, and spatial beam [23–27]. Another way is to microscopically control the ionization of the atom and steer the electron dynamics in the driving field to increase the single-atom harmonic emission, which can be accomplished by modifying the sub-cycle laser waveform (see review in [28]). This approach has attracted a lot of attentions recently due to the development of optical parametric amplification (OPA) and optical parametric chirped-pulse amplification (OPCPA) technology, with which it is possible to generate arbitrary optical waveforms by synthesizing multi-color laser pulses [29–34]. One of synthesizing schemes has demonstrated by Wirth et al. [35] that a sub-cycle attosecond pulse can be formed with a 1.5-octave three-channel optical field synthesizer, and it was able to tract the nonlinear response of bound electrons [36]. Furthermore, the optimization of such synthesized pulse has been performed for attosecond-pulse generation and shaping [37,38]. In this paper, we will discuss the optimization of optical waveform synthesized with commensurate long-duration lasers for enhancing the harmonic yields.

With the aim of enhancing the harmonic yield, we have recently proposed a general scheme of optimizing a multi-color laser waveform based on a genetic algorithm (GA) by taking phase-matching effects in a macroscopic medium into account in the optimization procedure [39]. Compared to the single-color laser with the same pulse energy, the optimized two- or three-color waveform was able to extensively increase the harmonic yields by one to two orders of magnitude. We then showed that the optimization method could be applied to get the best harmonic yields under different scenarios. For example, the optimal harmonic cutoff can be extended to the water-window region either with a waveform consisting of a strong mid-infrared laser and a few percent of its third harmonic [40], or with a waveform synthesized by a strong 800-nm laser and a relatively weak mid-infrared one [41]. However, these studies only provide with the general guidance for achieving optimal harmonic yield, and the optimized waveforms cannot be directly implemented in a specific experiment. It is necessary to investigate whether the optimization method is applicable in some available experimental setups, and how this method should be modified with the consideration of real experimental situations.

One experimental setup of the multi-color synthesizer has become available recently by Burger et al. [42]. They employed a Ti: sapphire laser system with the central wavelength of 790 nm to efficiently generate its second and third harmonic fields, and then a compact and versatile three-color synthesizer was accessible. It has been used to measure the momentum distribution of Ne⁺ ion triggered by multi-color strong-field ionization. However, such the three-color synthesizer hasn’t been reported to be engineered for the generation of high harmonics, and its waveform optimization has not been demonstrated either.

Our goal in this work is to optimize the three-color synthesizer exhibited by Burger et al. [42] for the optimal harmonic generation. We will consider laser parameters in the experimental setup in our optimization program, and compare the harmonic yields of the single fundamental laser and optimized three-color waveform interacting with He and Ne gases. For this comparison, the three-dimensional (3D) Maxwell’s wave equations of both the driving laser and the harmonic field will be solved to account for the macroscopic emission of HHG in the medium. Note that the pulse energy of the single fundamental laser is larger than the three-color pulse because partial energy of the fundamental was lost in the generation of second and third harmonics. The advantages of the optimized waveforms over the single-color field will be carefully analyzed by inspecting the electron trajectories and single-atom HHG spectra.

2. Theoretical models

2.1 Multi-color waveform optimization by genetic algorithm

Generally, an optical waveform with arbitrary shape can be generated by coherently combining multi-color sinusoidal laser pulses. We take the three-color waveform as an example. The electric field of the waveform can be written as:

E (t) = E_{1} A_{1} (t) \cos (ω_{1} t + φ_{1}) + E_{2} A_{2} (t) \cos (ω_{2} t + φ_{2}) + E_{3} A_{3} (t) \cos (ω_{3} t + φ_{3}),

where

E_{1, 2, 3}

are the amplitudes of electric fields,

ω_{1, 2, 3}

are the angular frequencies, and

φ_{1, 2, 3}

are the relative phases for individual lasers.

ω_{1}

is chosen as the fundamental frequency.

A_{1, 2, 3} (t)

are functions to describe the pulse envelopes, which are all assumed to take a flat-top envelope in one optical cycle of the fundamental laser, with half-cycle ramp in the envelope when the pulse is switched “on” and “off”. In the optimization,

ω_{1}

is fixed,

ω_{2} = 2 ω_{1}

, and

ω_{3} = 3 ω_{1}

. For simplicity,

φ_{1}

is set as 0, and the waveform is expressed only in one optical cycle of the fundamental. The laser parameters {

E_{1}

,

E_{2}

,

E_{3}

,

φ_{2}

,

φ_{3}

} need to be determined in the optimization by using the genetic algorithm (GA) [43].

GA has been widely used in the search and optimization in a multi-parameter space, which mimics the process of natural evolution. 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. First, a population of randomly generated individuals is given. Next the fitness of every individual from the current population is evaluated, and the better fit individuals are selected. Then a new generation is formed by modifying their genomes, which is then used in the next iteration. Finally, the evolution process can be terminated when a satisfactory fitness level has been reached, or a maximum number of generations has been produced. In this work, we use micro-GA. The population size is chosen to be 5 to cut down the number of fitness function evaluations and the computer time. The key ingredient is the selection of fitness function, which is usually the value of the objective function in the optimization problem.

Our goal is to optimize the yields of HHG spectrum with a preset cutoff energy. So the fitness function is taken as the single-atom harmonic yield at the cutoff energy, which is calculated by the quantitative rescattering (QRS) model [44]. Since the tunneling ionization usually occurs in a narrow time window, the enhancement of one harmonic would automatically enhance a broad range of high harmonics. We also impose some general principles proposed in our previous works [39] as additional constrains to generate single-atom harmonics which could efficiently suffer from the macroscopic propagation. These constrains include: (i) The ionization is adjusted by modulating the laser waveform to release more electrons for the HHG, but the ionization level should be constrained less than 1% to avoid the excessive plasma in the gas jet that would defocus the laser beam and cause the big phase mismatch; (ii) The short-trajectory emission must be dominated than the long-trajectory one because long-trajectory emission is hard to be phase-matched in the macroscopic medium; (iii) The cutoff energy should be more or less maintained at the pre-determined value.

In this work our optimization is performed for a particular three-color synthesizer demonstrated in [42], the search parameter space in GA could be truncated according to the experimental laser values. Most likely the genetic optimization returns the local extremum rather than the global one in our previous study [39].

Once the optimized parameters for the synthesized waveform in a single optical cycle are obtained, they are directly applied to individual laser with a realistic envelope in the three-color synthesizer. In the simulations, the pulse envelopes $A_{1, 2, 3} (t)$ in Eq. (1) are replaced by Gaussian ones with different pulse durations. This manipulation has been performed for the short-duration lasers in previous studies [39–41], and the characteristics of optimized waveform in the single cycle are remained over several optical cycles, see Fig. 2 in [39]. This would become more effective for the relatively long-duration lasers in this work.

2.2 Macroscopic propagation equations

To fully describe the macroscopic harmonic generation in a gas medium, both the propagation equations of the driving field of optimized waveform and high harmonic field in an ionizing medium are solved to take into account of phase-matching and propagation effects. The details of these equations have been given in [23]. We only recall the main equations here. The evolution of the driving laser field $E (r, z, t)$ is described in the following [45–47]:

\nabla^{2} E (r, z, t) - \frac{1}{c^{2}} \frac{\partial^{2} E (r . z, t)}{\partial t^{2}} = μ_{0} \frac{\partial J_{a b s} (r, z, t)}{\partial t} + \frac{ω_{0}^{2}}{c^{2}} (1 - η_{e f f}^{2}) E (r, z, t),

where

J_{a b s} (r, z, t)

is the absorption term due to the ionization of the medium, and the effective refractive index of the gas medium can be written as:

η_{e f f} (r, z, t) = η_{0} (r, z, t) + η_{2} I (r, z, t) - \frac{ω_{p}^{2} (r, z, t)}{2 ω_{0}^{2}} .

The linear term

η_{0} = 1 + δ_{1} - i β_{1}

accounts for refraction

(δ_{1})

and absorption

(β_{1})

by the neutral atoms, the second term describes the optical Kerr nonlinearity, and the third term is determined by the plasma frequency

ω_{p} = {[e^{2} n_{e} (t) / (ε_{0} m_{e})]}^{1 / 2}

, where

m_{e}

and

e

are the mass and charge of an electron, respectively, and

n_{e} (t)

is the density of free electrons.

The 3D propagation equation for the harmonic field $E_{h} (r, z, t)$ is [25,47,48]:

\nabla^{2} E_{h} (r, z, t) - \frac{1}{c^{2}} \frac{\partial^{2} E_{h} (r, z, t)}{\partial t^{2}} = μ_{0} \frac{\partial^{2} P (r, z, t)}{\partial t^{2}},

where

P (r, z, t)

is the polarization depending on the applied fundamental field

E (r, z, t)

. In general, it can be separated into linear and nonlinear components, and the linear susceptibility

χ^{(1)} (ω)

includes both linear dispersion and absorption effects of the harmonics. The nonlinear polarization term

P_{n l} (r, z, t)

can be expressed as:

P_{n l} (r, z, t) = [n_{0} - n_{e} (r, z, t)] D (r, z, t),

where

n_{0} - n_{e} (r, z, t)

gives the density of the remaining neutral atoms, and

D (r, z, t)

is the single-atom induced dipole moment calculated in terms of the QRS model. The harmonic field at the exit face (near field) of the medium can be obtained by solving Eqs. (2) and (4) with the Crank-Nicholson routine in the frequency domain.

2.3 Time-frequency wavelet analysis

Time-frequency wavelet analysis of HHG emission can be implemented by using the wavelet transform [49–53]:

A (t^{'}, ω) = \int E_{h} (τ) w_{t^{'}, ω} (τ) d τ,

where $w_{t^{'}, ω} (τ) = \sqrt{ω} W [ω (τ - t^{'})]$ is the wavelet kernel. We use the Morlet wavelet [45]:

W (x) = (1 / \sqrt{ν}) e^{i x} e^{- x^{2} / 2 ν^{2}},

in which

ν

is a turnable parameter. To avoid the complexity of analyzing the full spatial distribution of the harmonics in the near field, we calculate

A (t^{'}, ω)

at each radial point and then integrate over the radial coordinate [51]:

| A_{n e a r} (t^{'}, ω) |^{2} = \int_{0}^{\infty} 2 π r d r | \int E_{h} (r, τ) w_{t^{'}, ω} (τ) d τ |^{2} .

3. Results and discussion

3.1 Optimized three-color waveform for He

In Burger et al.’s experiment [42], the fundamental laser of 790 nm with the pulse energy of 650 $μJ$ was split into two beams. One was used to generate its second and third harmonic fields with pulse energies of 90 µJ (395 nm) and 80 µJ (263 nm). The other one with pulse energy of 350 µJ kept the wavelength of 790 nm, and was coherently combined with newly generated harmonic fields to form the three-color synthesizer. So its total pulse energy became 520 µJ, decreased by 20% compared to the initial fundamental laser. The main issue we would like to address is whether the three-color laser pulse with less pulse energy is able to generate more harmonic yields than the fundamental laser with the help of waveform optimization. Our strategy is to first optimize the three-color waveform in the single-atom response level using the procedure suggested in [39]. The search space should be restricted within the parameters available in the experiment. And then it applies to the realistic pulses, so laser beam waist and peak intensity of individual wavelengths should be adjusted accordingly. Finally, the macroscopic propagation simulations are performed for the optimized three-color waveform and the fundamental laser.

We choose He as the target atom, and the harmonic cutoff is fixed at 154 eV for an example. The intensity of fundamental 790 nm laser thus is $7.0 \times 10^{14} W / {cm}^{2}$ . In the optimization, $λ_{1}$ , $λ_{2}$ and $λ_{3}$ are fixed at 790 nm, 395 nm, and 263 nm, respectively, and the relative phase of the first color is $φ_{1} = 0$ . The search parameters are ${E_{1}, E_{2}, E_{3}, φ_{2}, φ_{3}}$ in Eq. (1). $φ_{2}, φ_{3}$ are in the range of [0, 2π]. Since the pulse energies and durations of three colors in [42] are fixed, the ranges of intensities $I_{1} = | E_{1} |^{2}, I_{2} = | E_{2} |^{2}, I_{3} = | E_{3} |^{2}$ can be defined according to the general size of beam waist from 15 to 100 μm. The single-atom harmonic yield at 154 eV is the optimization goal. With constrains given in Sec. 2.1, the GA returns the optimized parameters after thousands of iteration as following: for intensities, $I_{1} = 6.12 \times 10^{14}$ W/cm², $I_{2} = 0.56 \times 10^{14}$ W/cm² and $I_{3} = 0.55 \times 10^{14}$ W/cm², and for relative phases, $φ_{2} = 0.24 π$ and $φ_{3} = 1.25 π$ (listed as Opt. WF1 in Table 1).

Table 1. Laser parameters available in Burger et al. [42] and optimized parameters for three-color waveforms. Three colors are $λ_{1}$ = 790 nm, $λ_{2}$ = 395 nm, and $λ_{3}$ = 263 nm. B stands the beam waist at the focus. Laser intensities $I_{1, 2, 3}$ and $I$ are in the unit of $I_{0} = 1 \times 10^{14} W / c m^{2}$ . The target atom is He for the optimization.

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The waveform (red line) obtained by these optimized parameters is shown in Fig. 1(a) in one optical cycle of the fundamental laser. The single-color sinusoidal wave (black line) is also plotted for comparison. We then apply this waveform to the realistic pulse envelopes in [42], with pulse durations of 35, 61, and 59 fs for 790, 395, and 263 nm, respectively. The single-atom harmonic spectrum calculated by using the QRS model is shown in Fig. 1(b). The fundamental single-color spectrum with the pulse duration of 35 fs is shown in the same figure. One can see the yields of HHG generated by the optimized waveform are enhanced more than ten times over a broad spectral range in comparison with the single-color field. To understand such enhancement, we show the return energy of electron as a function of ionization time and emission (or recombination) time calculated by solving classical Newton equations in Fig. 1(c). The actual vertical label is the photon energy of HHG, which equals the return energy of electron plus ionization potential of atom. We can see that for the optimized waveform, the excursion times (between ionization and recombination times) of electron are shorten for both short- and long-trajectories compared to the single-color field. This would significantly reduce the spreading of electron wave packet in the continuum, and increase the probability of electron recombination. We then depict the electric fields at ionization versus the returning electron energies (or the photon energy of HHG) in Fig. 1(d). Comparing with two waveforms, it is obvious that the optimized one has higher electric fields at ionization for short-trajectory electrons, which are preferable in the harmonic generation. Benefiting from the reduced excursion time and enhanced ionization field, the optimized waveform is able to greatly enhance the harmonic emission of short-trajectory electrons. Time-frequency analysis of single-atom harmonic emissions by using Eq. (6) for both single-color field and optimized three-color waveform are shown in Figs. 1(e) and (f). In one optical cycle of fundamental laser, compared to the single-color results in Fig. 1(e), there is only one strong harmonic emission burst due to short-trajectory electrons because the half-cycle periodicity is broken up in the three-color waveform in Fig. 1(f). The working mechanism of the optimized waveform for enhancing the single-atom HHG yield is consistent with our findings in [39].

Fig. 1 (a) Comparison of the electric fields over one optical cycle of the fundamental for a single-color (SC) sinusoidal wave and an optimally synthesized three-color field. The optimized waveform (Opt. WF) is synthesized by 790-nm, 395-nm, and 263-nm laser fields. (b) HHG spectra of single-atom response for SC (black curve) and Opt. WF (red curve). Blue lines are smoothed results of two spectra by using adjacent average method. Note that the Opt. WF can generate harmonic yields about more than one order stronger than the SC one. (c) Photon energy of the HHG as a function of electron ionization time and emission time. The purple and green lines are for SC and Opt. WF, respectively. For each waveform, the left line is for the ionization time, and the right line is for the emission time. (d) The electric fields at ionization versus the photon energies for short- and long-trajectories. The photon energy is the kinetic energy of the returning electron plus ionization potential of He atom. Time-frequency wavelet analysis of the simulated single-atom harmonics are shown for SC (e) and Opt. WF. (f). He atom is applied, and “S” and “L” in the figures stand for short- and long-trajectory emissions, respectively. o.c. means the optical cycle of the fundamental 790-nm laser.

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Since the optimization by the GA searches the local extreme values, the returned optimized parameters starting from different random numbers may be different. There are multiple solutions existing for the problem. It is needed to compare with different solutions, and their resultant waveforms and HHG spectra. The three optimized waveforms with different solutions (or different optimized parameters) are shown in Fig. 2(a). The optimized parameters of intensities and relative phases are listed in Table 1. Opt. WF1 in this figure has been plotted in Fig. 1(a). Similar to Fig. 1(b), these waveforms are applied to realistic pulse envelopes to obtain the single-atom HHG spectra in Fig. 2(b). For easy comparison, the spectra have been smoothed to remove the fast oscillation structures. One can see that three optimized waveforms generate about the same harmonic yields in the plateau and cutoff regions. There are only some discrepancies appear at far-beyond cutoff region, i.e., at 160 to 180 eV, in which the harmonic yields drop quickly. Thus these waveforms can be used equivalently for the enhancement of harmonic yields.

Fig. 2 (a) Comparison of optimized three-color waveforms starting from different random numbers. The optimized parameters for plotting waveforms are listed in Table 1. (b) Single-atom HHG spectra (smoothed) of He for optimized waveforms in (a).

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3.2 Macroscopic HHG spectra of He by optimized three-color waveform and fundamental single-color field

We next examine the macroscopic HHG spectra by applying the optimized three-color waveform in the real experimental system [42]. We assume that a 1 mm-long He gas jet with uniform density distribution is placed 1.5 mm after laser focus to achieve the good phase-matching conditions, and the waveform is formed only at the center of gas jet. The gas pressure is chosen as 10 Torr. Using the available pulse energy and pulse duration, we can calculate the beam waist at the focus for each wavelength component by using the formula (D. 14) in [54]. The calculated values are tabulated in Table 1. The beam waist in principle can be adjusted by varying the focal length of focusing mirror in the experiment. Similarly, the beam waist of fundamental laser at laser focus is shown in Table 1 as well.

In Fig. 3(a), we show the macroscopic HHG spectra of optimized waveforms and fundamental laser. Three optimized waveforms generate almost identical harmonic yields after macroscopic propagation in a gas medium, and they are about two orders at the low photon energies and about one order at high photon energies higher than those generated by the fundamental laser. The enhancement factor is much bigger than or at least same as the single-atom response depending on the photon energies concerned. If the gas pressure increases from 10 to 100 Torr, as shown in Fig. 3(b), the HHG yields of optimized waveform are increased about one hundred times over the entire spectral region, indicating the harmonic enhancement can also be achieved at high pressure. Time-frequency analysis of the macroscopic harmonic emissions by using Eq. (8) is shown in Fig. 3(c) and 3(d). For fundamental single-color field, after macroscopic propagation, the long-trajectory emissions are vanished, and only short-trajectory emissions are survived because the gas jet is put after laser focus [25]. For the optimized waveform (Opt. WF1), we can only see the short-trajectory emissions, which take place once in one optical cycle. This shows the importance of optimizing short-trajectory emissions in the single-atom response, and confirms the validity of constrains used in the optimization. It should be emphasized that the optimized three-color waveform with only 80% pulse energy is able to generate harmonic yields one to two order higher than the single-color field.

Fig. 3 (a) Comparison of macroscopic HHG spectra (smoothed) of single-color (SC) field and optimized waveform (Opt. WF). The gas pressure is 10 Torr. See text for other parameters. (b) Macroscopic HHG spectra of Opt. WF1 at different gas pressures: 10 and 100 Torr. Time-frequency analysis of the macroscopic harmonics for SC field (c) and Opt. WF1 (d). He atom is used in the simulations.

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We further show the on-axis electric fields of Opt. WF1 at different propagation positions in Fig. 4. Along the propagation axis, the amplitudes of two peaks in each optical cycle are reduced due to the diffraction effect, for example, the peaks at −0.4 and 0.1 o.c. within one cycle in Fig. 4(a). However, the main shapes of waveform around these peaks are not altered much, which are responsible for efficient harmonic emissions due to “short”-trajectory electrons as shown in Fig. 1(c). Therefore, the function of optimized waveform is remained after propagation in the medium except for reducing the cutoff energy a little bit. Comparing with the electric fields in Figs. 4(a) and 4(b), they are very similar along the propagation direction, which confirms that the harmonic yields at 100 Torr should be 100 times higher than 10 Torr. Thus, we can conclude that due to the controlled ionization level at a few percentages, the optimized waveforms can be maintained macroscopically over a range of gas pressures.

Fig. 4 On-axis electric fields of Opt. WF1 at the entrance, middle, and exit of 1-mm long He gas medium at pressures of (a) 10 Torr and (b) 100 Torr. The gas medium is put 1.5 mm after the focus. Other parameters can be found in Table 1.

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3.3 Three-color vs two-color waveform

The combination of the fundamental laser and its third harmonic field has been proposed to synthesize the waveform for optimal HHG under different conditions [39,40]. It will be interesting to check the capability of such two-color waveform in the experiment of Burger et al. [42]. We use the same optimization method to search the parameters ${E_{1}, E_{3}, φ_{3}}$ . The optimized parameters are listed in Table 2. The resulted waveform is plotted in Fig. 5(a) to compare with the previous three-color waveform. The two-color waveform indeed shows some differences with the three-color one. Time-frequency analysis of harmonic emission by the two-color optimized waveform in Fig. 5(b) clearly presents the dominant short-trajectory emissions, meaning that the optimization goal has been successfully reached. Single-atom HHG spectra in Fig. 5(c) tells that the two-color waveform with less pulse energy can generate harmonic yields only a few times weaker than the three-color one, but still more than one order higher than the single-color field. Macroscopic HHG spectra in Fig. 5(d) shows that the difference between the two- and three-color waveforms is almost kept. Note that the two-color waveform has the pulse energy about 83% of the three-color waveform, and about 66% of the fundamental laser. In reality, the two-color waveform is not too much worse than the three-color one, and it is easy to operate in the experiment. So it is also a good option for enhancing harmonic yields in the experiment of Burger et al. [42].

Table 2. Optimized laser parameters for two-color waveforms. Two colors are $λ_{1}$ = 790 nm and $λ_{3}$ = 263 nm. B stands the beam waist at the focus. Laser intensities $I_{1, 3}$ and $I$ are in the unit of $I_{0} = 1 \times 10^{14} W / c m^{2}$ . The target atom is He for the optimization.

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Fig. 5 (a) Waveforms of single-color (SC) sinusoidal wave, optimized waveforms of three-color field (Opt. 3C) and two-color field (Opt. 2C). (b) Time-frequency wavelet analysis of single-atom harmonic emission for the Opt. 2C. Comparison of single-atom (c) and macroscopic (d) HHG spectra by the waveforms displayed in (a). The target is He atom, and the gas pressure is 10 Torr in (d).

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3.4 Optimization of three-color waveform for Ne

To check the applicability of the optimized waveform for different gas targets and different cutoff energies, we choose another example of using Ne as the target atom and fixing the cutoff energy at 95 eV. The optimized parameters for the three-color waveform are given in Table. 3. The intensity of fundamental laser is also given in the same table. The gas jet is put 3 mm after the laser focus. And the gas pressure and length are 10 Torr and 1 mm, respectively. The same procedure as in Fig. 3 was carried out to calculate the beam waists of individual lasers for macroscopic HHG, which are listed in Table 3.

Table 3. Optimized parameters of three-color waveforms using Ne as target atom. The wavelengths of three colors are fixed as: $λ_{1}$ = 790 nm, $λ_{2}$ = 395 nm, and $λ_{3}$ = 263 nm. B is the calculated beam waist at laser focus. Laser intensities $I_{1, 2, 3}$ and $I$ are expressed by $I_{0} = 1 \times 10^{14} W / c m^{2}$ .

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The results of optimized waveforms and HHG spectra are summarized in Fig. 6. Figure 6(a) shows the three optimized waveforms in comparison with the single-color field. The single-atom HHG spectra by these waves are presented in Fig. 6(b). Similar to Fig. 1, we show the photon energy of HHG emission as a function of electron ionization time and emission time in Fig. 6(c), and the electric fields at the ionization vs photon energies in Fig. 6(d). They show that the short-trajectory emissions are greatly increased due to the excursion time and the electric fields at the ionization are both optimized. Macroscopic HHG spectra in Fig. 6(e) show the enhancement of harmonic yields by using optimized waveforms is about one to two orders over the single-color field, which is better than or about same as the single-atom response in Fig. 6(b). Macroscopic response is further examined by increasing the gas pressure from 10 to 100 Torr in Fig. 6(f), in which the increase of about one hundred times is gained implying that the enhancement factor by the optimized waveform at low gas pressure can be achieved at high gas pressure. This example in Fig. 6 illustrates that our optimization of multi-color waveform can be easily migrated to other cases for enhancing the harmonic yields.

Fig. 6 (a) Comparison of waveforms for a single-color (SC) field and optimized three-color (790 nm, 395 nm, and 263nm) fields with different random numbers. Ne atom is applied in the optimization. (c) Ionization time and emission time of returning electrons with different returning energies. (Photon energy is indicated in the figure, which is the kinetic energy of returning electron plus ionization potential of Ne.) For each photon energy, the ionization times appear earlier than the emission times. (d) The strengths of electric fields at ionization for short- and long-trajectory electrons. Single-atom (b) and macroscopic (e) HHG spectra of Ne by using waveforms shown in (a). The gas pressure in (e) is 10 Torr. (f) Macroscopic HHG spectra of Opt. WF1 in (e) at two gas pressures: 10 Torr (red curve) and 100 Torr (purple curve).

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4. Conclusion

In summary, we proposed to optimize a three-color synthesizer (a 790-nm laser with its second and third harmonic fields) available in a recent experiment of Burger et al. [42] to greatly enhance the harmonic yields. We modified the optimization method in the single-atom response level suggested previously [39] to consider the real laser parameters in the experiment. For two cases with He or Ne gas and different cutoff energies, we obtained the optimized three-color waveforms, which were able to generate the macroscopic harmonic yields about one to two orders of magnitude higher than the fundamental single-color laser with only 80% pulse energy. This enhancement could be achieved at both low and high gas pressures. Several different approaches were applied to show the validity and effective of the waveform optimization. The classical calculations were carried out to show that the excursion time and the electric field at the ionization for short-trajectory electrons were both tuned in the optimized waveform to reduce the spreading of the electron wave packet. Time-frequency analysis of the HHG was performed to demonstrate that short-trajectory emissions became dominant after the waveform optimization and survived after the macroscopic propagation. We also showed that the three-color waveform was better than the two-color one consisting of the fundamental and its third harmonic fields since it could generate the harmonic yields a few times higher.

How to improve the efficiency of the HHG is still the tough task in the community, especially for the soft X-ray HHG, which has been intensively focused recently [8,9,55–59]. Multi-color synthesis may become a practical solution by possibly controlling laser parameters precisely with the development of the laser technology. This work already provides with an accessible means to realize the optimized waveforms in a real experimental setup. We hope it can further inspire some experimental interests to test and verify the waveforms suggested theoretically [39,60–64] with the ultimate goal of generating intense HHG served as a table-top light source in a broad spectral region.

Funding

National Natural Science Foundation of China (NSFC) (11564033; 11774175); Fundamental Research Funds for the Central Universities of China (30916011207; 31920170032).

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	$λ_{1}$		$λ_{2}$			$λ_{3}$			Fundamental laser
Wavelength (nm)	790		395			263			790
Pulse energy (μJ)	350		90			80			650
Pulse duration (fs)	35.0		61.0			59.0			35.0
	$I_{1}$	B₁ (μm)	$I_{2}$	B₂ (μm)	$φ_{2} (π)$	$I_{3}$	B₃ (μm)	$φ_{3} (π)$	$I$	B (μm)
Opt. WF1	6.12 $I_{0}$	28.26	0.56 $I_{0}$	39.32	0.24	0.55 $I_{0}$	38.42	1.25	7.0 $I_{0}$	38.62
Opt. WF2	6.11 $I_{0}$	28.28	0.53 $I_{0}$	40.73	0.25	0.52 $I_{0}$	39.19	1.27
Opt. WF3	5.72 $I_{0}$	29.76	0.77 $I_{0}$	33.34	0.19	0.67 $I_{0}$	34.51	1.25

	$λ_{1}$		$λ_{3}$			Fundamental laser
Wavelength (nm)	790		263			790
Pulse energy (μJ)	350		80			650
Pulse duration (fs)	35.0		59.0			35.0
	$I_{1}$	B₁ (μm)	$I_{3}$	B₃ (μm)	$φ_{3} (π)$	$I$	B (μm)
Opt. WF	5.96 $I_{0}$	28.87	0.68 $I_{0}$	34.3	1.39	7.0 $I_{0}$	38.62

	$λ_{1}$		$λ_{2}$			$λ_{3}$			Fundamental laser
Wavelength (nm)	790		395			263			790
Pulse energy (μJ)	350		90			80			650
Pulse duration (fs)	35.0		61.0			59.0			35.0
	$I_{1}$	B₁ (μm)	$I_{2}$	B₂ (μm)	$φ_{2} (π)$	$I_{3}$	B₃ (μm)	$φ_{3} (π)$	$I$	B (μm)
Opt. WF1	3.07 $I_{0}$	39.85	0.63 $I_{0}$	36.09	0.16	0.57 $I_{0}$	37.10	1.19	4.0 $I_{0}$	50.54
Opt. WF2	3.19 $I_{0}$	38.62	0.58 $I_{0}$	37.65	0.18	0.49 $I_{0}$	40.31	1.19
Opt. WF3	3.29 $I_{0}$	37.64	0.52 $I_{0}$	40.29	0.20	0.44 $I_{0}$	42.64	1.20

	$λ_{1}$		$λ_{2}$			$λ_{3}$			Fundamental laser
Wavelength (nm)	790		395			263			790
Pulse energy (μJ)	350		90			80			650
Pulse duration (fs)	35.0		61.0			59.0			35.0
	$I_{1}$	B₁ (μm)	$I_{2}$	B₂ (μm)	$φ_{2} (π)$	$I_{3}$	B₃ (μm)	$φ_{3} (π)$	$I$	B (μm)
Opt. WF1	6.12 $I_{0}$	28.26	0.56 $I_{0}$	39.32	0.24	0.55 $I_{0}$	38.42	1.25	7.0 $I_{0}$	38.62
Opt. WF2	6.11 $I_{0}$	28.28	0.53 $I_{0}$	40.73	0.25	0.52 $I_{0}$	39.19	1.27
Opt. WF3	5.72 $I_{0}$	29.76	0.77 $I_{0}$	33.34	0.19	0.67 $I_{0}$	34.51	1.25

	$λ_{1}$		$λ_{3}$			Fundamental laser
Wavelength (nm)	790		263			790
Pulse energy (μJ)	350		80			650
Pulse duration (fs)	35.0		59.0			35.0
	$I_{1}$	B₁ (μm)	$I_{3}$	B₃ (μm)	$φ_{3} (π)$	$I$	B (μm)
Opt. WF	5.96 $I_{0}$	28.87	0.68 $I_{0}$	34.3	1.39	7.0 $I_{0}$	38.62

Application of optimized waveforms for enhancing high-harmonic yields in a three-color laser-field synthesizer

Abstract

1. Introduction

2. Theoretical models

2.1 Multi-color waveform optimization by genetic algorithm

2.2 Macroscopic propagation equations

2.3 Time-frequency wavelet analysis

3. Results and discussion

3.1 Optimized three-color waveform for He

3.2 Macroscopic HHG spectra of He by optimized three-color waveform and fundamental single-color field

3.3 Three-color vs two-color waveform

3.4 Optimization of three-color waveform for Ne

4. Conclusion

Funding

References

Cited By

Figures (6)

Tables (3)

Equations (8)

Optics Express