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For an electron returning with a kinetic energy of 2 U P , the ionization time and the recombination time for the long- open circles and short-trajectory electrons solid circles are indicated for each waveform. The inset gives the electric fields at ionization times versus the returning electron energies. Comparing the two waveforms, it is clear that the optimized one has higher electric fields at ionization times that lead to more returning electrons, and there are more short-trajectory electrons than long ones.

Figure 1b,c show the time-frequency wavelet analysis of HHG calculated using quantitative rescattering theory 22 , 23 ; see Supplementary Methods for the quantitative rescattering theory and the discussion of its validity. The target was Ne.

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The optimization returned the peak intensity for the fundamental as 1. Optimization was performed such that the cutoff and total laser power for the single-colour wave and the optimized wave were about the same. The details of the optimization are given in the Supplementary Information ; see Supplementary Figs 2—9 for two-colour optimization, Supplementary Figs 10—12 for three-colour optimization, Supplementary Methods and Supplementary Discussion.

Laser parameters of waveforms obtained from the optimization are given in Supplementary Tables 1— We comment that optimization of harmonics only from short-trajectory electrons was not considered in the previous study The optimized waveform Opt. WF is synthesized from a fundamental and its third harmonic. These two constraints guarantee that harmonics generated from each atom in the gas media are favourably phase-matched. On the waveform curves, shown in open and filled circles, are the times when an electron is tunnel-ionized and when it recombines to emit a photon, for an electron that returns with kinetic energy of 2 U P.

Here U P is defined in terms of the single-colour sinusoidal wave. The open circles are for long-trajectory electrons that have longer excursion time, and the filled circles are for short-trajectory electrons that have smaller excursion time. The inset depicts the electric fields at ionization time versus the kinetic energies of the returning electrons for short- and long-trajectory electrons.

For the optimized wave, the field strengths at the ionization times are higher than the single-colour sinusoidal wave, and the fields for the short-trajectory electrons are higher than the long ones.

G Shive Wave Machine - Superposition Of Pulses | Physics Lab Demo

To confirm that the one-cycle result in Fig. We observed an enhancement of about two orders in the optimized waveform over the single-colour case. In Fig. The additional enhancement reflects that there are more short-trajectory electrons contributing to HHG for the optimized wave. Time-frequency wavelet analysis of the harmonic spectra is shown in Fig.

Clearly, at the single-atom level the optimized pulse has much stronger contributions from short-trajectory electrons Fig. After propagation, all the long-trajectory contributions vanish Fig. In the single-colour case, at the single-atom level the long-trajectory electrons dominate Fig.

This example demonstrates the importance of optimizing harmonics from short-trajectory electrons, not from both long- and short-trajectory electrons that coexist in the total single-atom harmonics. Inspections of Fig. The secondary pulse is its third harmonic.

Waves 2: Superposition of Waves

For the single-colour wave, the peak intensity is taken at 3. The laser parameters of the two colours for the optimized wave are given in the text. Blue curves in a and b show the smoothed spectra. All the harmonics from long-trajectory electrons and from multiple returns disappear. This figure illustrates that enhancing short-trajectory electrons from each single-atom response is essential for optimizing the yields of macroscopic harmonics. This figure shows that most of the harmonics from long-trajectory electrons and multiple returns do not survive phase-matching.

Only harmonics from short-trajectory electrons are phase-matched. Can HHG yields be substantially further enhanced by synthesizing three sinusoidal waves instead of only two? The top three sets of laser parameters obtained from optimization Supplementary Table 1 resulted in nearly identical waveforms see Fig. The single-atom HHG yields from this three-colour synthesized waveform are only about two times higher than the yields from the two-colour synthesized waveform examined in Fig. In this example, the prediction from the simple classical three-step model is incorrect.

That model fails to include quantum diffusion of electrons between ionization and recombination. The diffusion effect is roughly inversely proportional to the third power of the excursion time, thus reducing more yields for long-trajectory electrons than for short-trajectory electrons Time-frequency analysis of the calculated HHG, which includes quantum diffusion, indeed indicates that short-trajectory electrons dominate the harmonic yields Fig. All of them are two orders stronger than the single colour one. The peak intensity for the single-colour laser is 3.

The total peak intensity for the optimized wave is about 3.

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The same pulse durations are used for the two other waves. The second emission within each cycle is much weaker.

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One important consequence of three-colour synthesis is that, in each cycle, the optimized waveform generates only one high energy harmonic burst, since the peak electric field of this waveform in the second half cycle see Fig. Figure 3c shows a single dominant burst of harmonic emission near the centre of the pulse.

The harmonics after propagation in the gas medium are shown in Fig. This is caused by plasma defocusing since electron density in the medium increases with gas pressure. For other examples of synthesizing three commensurate lasers, see Supplementary Fig.

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The optimization of a waveform by synthesizing two- or three-colour fields usually returns with a few sets of laser parameters that give comparable enhancement of HHG, more so for three- than for two-colour fields see Supplementary Tables 1 and 2. For enhancing HHG, any one of these sets of laser parameters can be implemented in the laboratory. For two-colour synthesis, we observed that the wavelength of the second colour has to be close to the third harmonic of the fundamental. If the second harmonic is used as the second colour, the HHG enhancement is only about 10 times instead of times stronger Supplementary Fig.

The wavelengths of fundamental and secondary waves in the synthesis do not have to be commensurate, nor do their bandwidths have to overlap. For commensurate wavelengths, say, for three colours, the synthesized wave repeats with a new optical period T , which is the lowest integer multiple of the periods of all the three waves. Optimization of waveform synthesis can then be carried out only within this optical cycle T. If the wavelengths are incommensurate, optimization has to be carried out for the whole duration of the wave.

In some laboratories, the wavelengths of the wave fields to be synthesized are already fixed in their set-up. Clearly changing the wavelengths in the experiment is harder than changing the intensity and phase of each wave. In Supplementary Information, we have performed optimization for lasers reported in Huang et al. Enhancement of HHG by two orders is obtained if the waveforms in these experiments had been optimized.

For each set of optimized laser parameters, small variations in intensity of each wave have only small effects on the enhanced HHG.

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However, the synthesized waveform depends critically on the relative phases among the input waves. Simulation indicates that if the relative phase deviates by more than 0. In present-day laser technology, the jitter of the carrier-envelope phase is about 0. Additional jitter may come from the time delay between the pulses. Phase stability of each wave in the synthesis will be the most important controlling factor in the experimental realization of waveform optimization. The optimized waveforms reported here are for optimization of HHG with the constraint that the cutoff is not extended beyond the limit set by the fundamental laser.

If different criteria are chosen for the fitness functions in the optimization, different waveforms will be generated. For example, a very different optimized waveform is obtained if long-trajectory electrons are to be enhanced Supplementary Fig. The present optimization algorithm is easily extended to include other criteria as well as constraints in the simulation. When the synthesis with more colours or direct manipulation of component phases of a supercontinuum 26 becomes practical, it would be interesting to investigate whether further waveform control can significantly enhance the HHG.

Harmonic synthesis and superposition

Finally, to generate usable lights with HHG the effect of phase matching of the harmonics in the gas medium has to be further studied. In a focused laser beam, the peak intensity of the pulse varies in space, thus only the waveform near the laser focus is optimized. In the Supplementary Information, we have carried out a number of simulations to show that enhancement of macroscopic HHG for an optimized wave versus the single-colour one is equal to or higher than that at the single-atom level Supplementary Fig. However, a more efficient optimization of phase-matching is best done experimentally at individual laboratories.

With single-colour mid-infrared lasers, experiments have shown that significant enhancement can be reached if the harmonics are generated in a high-pressure capillary 2. By optimizing the gas pressure independently, harmonics generated from a 2. Unless proven otherwise, there is good reason to believe that such additional enhancement can also be accomplished by the synthesized waves with mid-infrared lasers.

Waveform synthesis using two- or three-colour fields provides efficient modification of the electric field that enhances harmonic yields from each atom by one to two orders the factor is larger for smaller total laser power without much increase in the total laser power. This result has an immediate impact on three research areas in strong field physics.

First, applications of mid-infrared lasers for HHG from polyatomic molecules have always been plagued by weak signals, such that data were taken near saturation intensities 28 , 29 where excessive ionization in the medium will reshape the spectra. With waveform-optimized lasers, HHG data can be taken at lower intensities such that the experimental HHG spectra can be used to extract the target structure of a polyatomic molecule directly.

Similarly, waveform-optimized lasers also can help laser-induced electron diffraction experiments where the signal is proportional to the flux of laser-driven returning electrons Laser-induced electron diffraction offers potential for probing molecules with femtosecond temporal and sub-Angstrom spatial resolution. Second, the one- to two-order enhancement of HHG yields from waveform-optimized mid-infrared lasers, if implemented with the emerging intense high-repetition MHz lasers 31 , will increase photon yields per second from what is available today by several orders, thus making it possible to generate usable broadband tabletop light sources in the laboratory.

Since the optimized waves also readily generate supercontinuum spectra, intense isolated attosecond pulses are also generated after spectral filtering. Such isolated attosecond pulses, covering photon energies from the extreme ultraviolet to X-rays, can find many applications for probing electron dynamics of atoms, molecules and condensed materials.

Finally, waveform-optimized lasers are expected to have great impact on any intense laser-driven processes, including THz generation 32 , laser-plasma interactions 33 and particle acceleration. Clearly, progress in waveform synthesis and optimization will have far-reaching effects on all aspects of strong field physics in the coming years. How to cite this article: Jin, C. Waveforms for optimal sub-keV high-order harmonics with synthesized two- or three-colour laser fields.

Dudley, J. Supercontinuum light. Today 66 , 29—34 Popmintchev, T. Bright coherent ultrahigh harmonics in the keV X-ray regime from mid-infrared femtosecond lasers. Science , — Tate, J. Scaling of wave-packet dynamics in an intense midinfrared field.

Kazamias, S. Global optimization of high harmonic generation.