Generation of high-order harmonics with tunable photon energy and spectral width using double pulses

This work theoretically investigates high-order harmonic generation in rare-gas atoms driven by two temporally delayed ultrashort laser pulses. Apart from their temporal delay, the two pulses are identical. Using a single-atom model of the laser-matter interaction it is shown that the photon energy of the generated harmonics is controllable within the range of one eV—a bandwidth comparable to the photon energy of the fundamental field—by varying the time delay between the generating laser pulses. It is also demonstrated that high-order harmonics generated by double pulses have advantageous characteristics, which mimick certain properties of an extreme ultraviolet monochromator. With the proposed method, a simpler setup at a much lower cost and comparatively higher spectral yield can be implemented in contrast to other approaches.

Figure 1

(a) Spectra and (b)–(e) the corresponding temporal amplitude of the double pulses for different time delays: (b) $0\mathrm{fs}$ (solid blue), (c) $13.3\mathrm{fs}$ (dashed orange), (d) $14.7\mathrm{fs}$ (yellow open circles), and (e) $16\mathrm{fs}$ (green crosses) between the pulses. Their amplitude ratio is chosen to be 1. At delays of $0\mathrm{fs}$, $13.3\mathrm{fs}$, and $16\mathrm{fs}$ the two pulses interfere constructively, while at a delay of $14.7\mathrm{fs}$ they interfere destructively.

Figure 2

(a) Spectral intensity and (b)–(e) the corresponding temporal evolution at a fixed delay, but (b) at 0, (c) $\pi /2$, (d) $\pi$, and (e) $3\pi /2$ rad relative ${\phi }_{\mathrm{cep}}$ between the composing pulses of the double-pulse structure. The chosen delay is ${\tau }_d=16\mathrm{fs}$, where the two pulses constructively interfere in the case of 0 rad ${\phi }_{\mathrm{cep}}$. The other parameters are the same as those in Fig. 1.

Figure 3

(a) Variation of the spectral bandwidth and (b) the central angular frequency, of the driving double pulses changing the time delay between the constituting pulses. The colored map (c) shows the dependence of the high-order harmonic spectrum on the time separation of the two pulses, applying $50\mathrm{as}$ time resolution. The composing pulses have $12\mathrm{fs}$ FWHM duration and $800\mathrm{nm}$ central wavelength. Their amplitude ratio is 1. The target gas is argon. Harmonic order is defined as the higher orders of the central frequency of a single pulse. The harmonic yield (color bar) is in linear scale.

Figure 4

Harmonic spectra generated by double pulses. The time separation varies around the $6\text{th}$ constructive interference region ($\sim 13.3\mathrm{fs}$). The other simulation conditions are the same as in Fig. 3. The white curve represents the calculated cutoff positions $E_c$. The harmonic yield (color bar) is in logarithmic scale.

Figure 5

Photon energy variation of the $15\text{th}$, $25\text{th}$, and $39\text{th}$ harmonics as the time separation of the double pulses changes. In Fig. 4, these harmonics are marked with appropriately colored rectangles.

Figure 6

Variation of spectral bandwidth of the 21st harmonic at two distinct delay regions. Panels (a) and (c) show how the spectrum changes in the vicinity of this harmonic by the delay: the black dots represent the spectral position of the harmonic peak; the red and green dots show the first spectral valley on the low and high photon energy sides of the harmonic peak, respectively. To determine the bandwidth, the region between the red and green dots was used. Panels (b) and (d) demonstrate the spectral bandwidth dependence on the delay.