Wavelength scaling of high-order-harmonic-generation efficiency by few-cycle laser pulses: Influence of carrier-envelope phase

We investigate the wavelength scaling of high-order-harmonic-generation (HHG) efficiency by few-cycle laser pulses where the influence of the carrier-envelope phase (CEP) becomes prominent. Based on the numerical solution of time-dependent Schrödinger equation, in the infrared region of $\lambda$ (0.8–1.8 $\mu$m), we observe that the power law of the form ${\lambda }^{-x}$ ($x\approx 5$–6) is still present; but for constant intensity and a fixed energy interval, it differs markedly in the CEP. We find that for a specific value of CEP the wavelength dependence follows a ${\lambda }^{-4.6}$ scaling in high-harmonic yield. We also perform classical trajectory calculations to gain some understanding of the CEP dependence of the wavelength scaling.

Figure 1

HHG from hydrogen atom for $\lambda =0.8$ and $\lambda =1.8\mu$m. For the laser pulse, a Gaussian envelope function with two cycles at FWHM and $I=160$ TW/cm${}^2$ of intensity, ${\phi }_{\mathrm{CEP}}=0$ is used. Note that when $\lambda =1.8\mu$m is used, although the cutoff is extended by $\sim 125$ eV its efficiency is lowered by roughly two orders of magnitude.

Figure 2

Wavelength dependence of the high-harmonic yield $\Delta I$ for two different CEP values. The intensity of the pulse is $I=320$ TW/cm${}^2$. The harmonic yield is taken over 20–50 eV energies.

Figure 3

High-harmonic yield $\Delta I$ for $\lambda =0.8$ and $\lambda =1.8\mu$m as a function of the CEP of the laser pulse. The intensity $I$ of the pulse is (a) 160 and (b) 320 TW/cm${}^2$. The high-harmonic yield is taken over 20–50 eV energies. $\Delta I$ values are multiplied by (a) 15 and (b) 35 for $\lambda =1.8\mu$m for a better view.

Figure 4

Wavelength dependence of the travel time of the classical electron trajectories that give 35 eV of energy upon recombination.