Figure 1: Principle of waveform synthesis for optimal high-harmonic generation.

(a) Comparison of the electric fields over one optical cycle of the fundamental for a single-colour (SC) sinusoidal wave and an optimally synthesized two-colour field. The optimized waveform (Opt. WF) is synthesized from a fundamental and its third harmonic. The amplitude and phase of each colour field are chosen such that harmonics generated from the synthesized wave are optimized with two constraints: (1) the emissions are dominated by the recombination of the so-called short-trajectory electrons; (2) the total ionization fraction is restricted to below 2–5%. 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. (b) The yields of harmonics emitted from the short- and long-trajectory returning electrons versus the recombination times for the optimized wave, from time-frequency wavelet analysis of the calculated harmonics. (c) Same as b but for the single-colour wave.