Scaling of the generation of high-order harmonics in large gas media with focal length

We present theoretical and experimental results on high-order harmonic generation in a low-density few-centimeter-long gas medium ($L_{\mathrm{med}}⩽$ 10 cm). We study the dependence with focal length of harmonic efficiency. Theoretically, we consider in detail the generation of the $25\mathrm{th}$ harmonic of a short pulse Ti:sapphire laser in argon. Within the strong-field approximation for the atomic dipole, and a complete account of the macroscopic propagation, we compute the number of photons produced as a function of the medium parameters and the focusing conditions. The simulations show that, at constant intensity, the emission of the $25\mathrm{th}$ harmonic scales with the focal length as $\sim$$f$${}^4$ at low pressure ($P=2$ Torr) and as $\sim$$f$${}^6$ at higher pressure ($P=5$ Torr). At constant laser energy, we find that the harmonic signal scales approximately as $f$${}^2$ at low pressure and as $f$${}^4$ at higher pressure. Those numerical results are compared with experimental data.

Figure 1

Variation of the number of photons with medium length calculated for H25 and different focal lengths. The argon gas pressure is 2 Torr and the laser is focused in the middle of the medium at a peak vacuum intensity of $2.0\times {10}^{14}$ W/cm${}^2$.

Figure 2

Photon number as a function of gas pressure calculated for different focal lengths. The medium length is 5 cm. The laser is focused in the middle of the medium at a peak vacuum intensity of $2.0\times {10}^{14}$ W/cm${}^2$.

Figure 3

Number of photons as a function of laser focus position calculated for H25 and for several focal lengths and argon gas pressures: (a) $P=2$ Torr, (b) $P=3$ Torr, (c) $P=4$ Torr, and (d) $P=5$ Torr. The laser peak intensity in vacuum is $2.0\times {10}^{14}$ W/cm${}^2$ and the medium length is 5 cm.

Figure 4

Maximum number of photons as a function of focal length calculated for H25 and for several argon gas pressures. The laser peak intensity in vacuum is $2.0\times {10}^{14}$ W/cm${}^2$ and the medium length is 5 cm.

Figure 5

Maximum number of photons as a function of the focal length calculated for H25, different argon gas pressures, and laser energies. The medium length is 5 cm. (a) $P=2$ Torr, (b) $P=3$ Torr, (c) $P=4$ Torr, and (d) $P=5$ Torr. The thick red curves in (a) and (d) correspond to $f$${}^{2.0\pm 0.08}$ and $f$${}^{4.3\pm 0.03}$ scalings, respectively.

Figure 6

Variation of the harmonic intensity (right scale) and the beam section (left scale) with focal length for (a) H25, (b) H21, and (c) H19 generated in argon. The driving laser energy is between 20 and 38 mJ, the gas cell is 5-cm long, and the gas pressure is 2 Torr. The beam cross sections were measured by imaging the laser beam at the cell position. Data are normalized to the case $f=4$ m.

Figure 7

Comparison between numerical results (black curves and points) of Fig. 6a and experimental data (blue curves and points) for H25. Both experimental and numerical results are normalized to the case $f=5$ m. The dotted line shows an $f$${}^2$ scaling.