Pulse-length dependence of multiphoton ionization by phase-controlled ultrashort x-ray pulses

Adapting earlier results that are appropriate in x-ray and perturbative regimes, the double-differential probability of multiphoton ionization by a phase-controlled ultrashort pulse is given. The infinite-pulse-length limit is also discussed. The numerical calculation was carried out in the case of two-photon processes and for a hydrogen-type $1s$ initial state that describes well most inner $\left(K\right)$ shells. The numerical results indicate that the efficiency of multiphoton ionization strongly increases with the shortening of the length of a few-cycle pulse. A moderate carrier-envelope phase dependence of the ionization probability has been found in the one-cycle case only.

Figure 1

Energy [$x=E/\left(\hslash {\omega }_0\right)<2$] and carrier-envelope phase (CEP) $\left(-2\pi \le \varphi \le 0\right)$ dependence of $G\left(x,\varphi \right)=x{\left|I\left(x,\varphi \right)\right|}^2$ in the “two-photon” process $\left[x_b=\left|E_b\right|/\left(\hslash {\omega }_0\right)=1.95\right]$ at pulse length $T=1$ ($T={\omega }_0\tau$, where ${\omega }_0$ is the carrier angular frequency and $\tau$ is the pulse length) and with $L=0.01$, $L=\left(e{F}_0/\hslash \right)\sqrt{2/\left(m{c}^2\right)}=\sqrt{2{\alpha }_f{\lambda }_0^3{I}_0/\left({\pi }^{2}m{c}^3\right)}$, where $I_0=c{E}_0^2/\left(8\pi \right)$ and ${\lambda }_0$ is the carrier wavelength. For other notation, see the text.

Figure 2

The same as in Fig. 1 with $T=1.5$.

Figure 3

The same as in Fig. 1 with $T=10$.

Figure 4

The ${\text{log}}_{10}\left[G\left(x=0.05,\varphi =0\right)/T\right]$ versus the dimensionless quantity $T$ of the “two-photon” ionization process at $x_b=1.95$. For other notation, see Fig. 1.

Figure 5

The energy dependence of $G\left(x,\varphi =0\right)$ in the “two-photon” ionization process $\left(x_b=1.95\right)$ in the $0.04<x\le 0.06$ interval at pulse length $T=300$. For other notation, see Fig. 1.