Phase-dependent asymmetries in strong-field photoionization by few-cycle laser pulses

Using numerical solutions of the time-dependent Schrödinger equation for a hydrogen and a helium atom in a linearly polarized laser field, we calculate the photoelectron signal $P_{+}$, $P_{-}$ measured by two opposing detectors placed along the laser polarization vector, with laser focus in the center. Our calculations show a significant sensitivity of the normalized asymmetry coefficient $a=(P_{+}-P_{-})∕(P_{+}+P_{-})$ to the carrier-envelope (CE) phase for few-cycle laser pulses. We find a very simple dependence of this coefficient on the CE phase $\varphi$ and on pulse duration for laser intensities slightly below the tunneling regime, i.e., for the intensities range between the perturbative-multiphoton and tunneling regime, for laser pulses shorter than two laser cycles. In particular, we find that in this intensity regime, the asymmetry is zero for a fixed, particular value of $\varphi ={\varphi }_0\simeq -0.3\pi$ and is maximum for $\varphi ={\varphi }_M={\varphi }_0+\pi ∕2$. These regularities would allow one to measure the CE phase. In particular, the condition of zero asymmetry for few-cycle pulses can be useful for stabilizing the CE phase.

Figure 1

Electric field $E\left(t\right)∕E_0$ and its envelope ${\epsilon }_0\left(t\right)∕E_0$ as a function of time in cycles for CE phases: (a) $\varphi =0$; (b) $\varphi =\pi ∕2$; (c) $\varphi =\pi$. The central wavelength is $\lambda =800\mathrm{nm}$ and the intensity profile half-width is ${\tau }_p=3.9\mathrm{fs}$.

Figure 2

Asymmetry parameter $a=(P_{+}-P_{-})∕(P_{+}-P_{-})$ as a function of the laser intensity $I$ for (a) the CE phase $\varphi =0$, see $E\left(t\right)$ in Fig. 1a and for (b) $\varphi =\pi ∕2$, see $E\left(t\right)$ in Fig. 1b. Angular distributions were integrated in the range $0<\theta <{\theta }_0$. Four values of ${\theta }_0=15°,{\theta }_0=30°,{\theta }_0=60°$, and ${\theta }_0=90°$ were used; see the legend in (b).

Figure 3

Same as in Fig. 2 but for ${\theta }_0=15°$ and for various wavelengths: $\lambda =778$, 800, 840, and $1064\mathrm{nm}$. Here and in Fig. 2, symbols do not correspond to the intensity values at which the coefficient $a$ was calculated. The following intensity values were used: $I_{k+1}=1.2I_k$, $k=1,2,\dots ,$ with $I_1=2\times {10}^{12}\mathrm{W}∕{\mathrm{cm}}^2$.

Figure 4

Asymmetry coefficient $a$ as a function of the CE phase $\varphi$ for laser intensity $I=3.1\times {10}^{13}\mathrm{W}∕{\mathrm{cm}}^2$, $\lambda =800\mathrm{nm}$, ${\tau }_p=3.9\mathrm{fs}$. The solid line is for ${\theta }_0=15°$, squares are for ${\theta }_0=15°$, and the dotted line is a plot of function $0.55\times \sin (\varphi +\pi ∕3)$.

Figure 5

Asymmetry coefficient $a$ as function of the CE phase for laser intensity $I=3.1\times {10}^{13}\mathrm{W}∕{\mathrm{cm}}^2$, $\lambda =800\mathrm{nm}$, for various pulse durations ranging from ${\tau }_p=4.0\mathrm{fs}$ to ${\tau }_p=5.4\mathrm{fs}$.

Figure 6

Asymmetry coefficient $a$ as a function of the CE phase for four central wavelengths $\lambda =778\mathrm{nm}$ $\left(I=3.7\times {10}^{13}\mathrm{W}∕{\mathrm{cm}}^2\right)$, $840\mathrm{nm}$ $\left(I=3.1\times {10}^{13}\mathrm{W}∕{\mathrm{cm}}^2\right)$, $800\mathrm{nm}$ $\left(2.6\times {10}^{13}\mathrm{W}∕{\mathrm{cm}}^2\right)$, and $1064\mathrm{nm}$ $\left(I=3.1\times {10}^{13}\mathrm{W}∕{\mathrm{cm}}^2\right)$. ${\tau }_p=3.9\mathrm{fs}$ was used for all central wavelengths except $\lambda =1064\mathrm{nm}$, for which ${\tau }_p=5.2\mathrm{fs}$ was used to compensate for a longer period.

Figure 7

Asymmetry coefficient $a$ as a function of the CE phase for $\lambda =800\mathrm{nm}$, ${\tau }_p=3.9\mathrm{fs}$, for various intensities ranging from $I=3.1\times {10}^{13}$ to $3\times {10}^{14}\mathrm{W}∕{\mathrm{cm}}^2$.

Figure 8

Zero-asymmetry CE phase ${\varphi }_0$ as a function of laser intensity.

Figure 9

Electric field $E\left(t\right)∕E_0$ and its envelope ${\epsilon }_0\left(t\right)∕E_0$ as a function of time in cycles for CE phases: (a) $\varphi ={\varphi }_M=0.188\pi$; (b) $\varphi ={\varphi }_0=0.683\pi$. The central wavelength is $\lambda =800\mathrm{nm}$ and the intensity profile half-width is ${\tau }_p=3.9\mathrm{fs}$.

Figure 10

Asymmetry parameter $a$ as a function of the pulse duration ${\tau }_p$ (FWHM) at $I=3.1\times {10}^{13}\mathrm{W}∕{\mathrm{cm}}^2$ calculated using ${\theta }_0=15°$ (solid line) and using ${\theta }_0=30°$ (dotted line) for (a) $\varphi =0$ and (b) $\varphi =\pi ∕2$.

Figure 11

Same as Fig. 10 but for higher laser intensity $I=8\times {10}^{13}\mathrm{W}∕{\mathrm{cm}}^2$.

Figure 12

Same as Fig. 2 but for a helium atom.

Figure 13

Same as Fig. 7 but for a helium atom. In (a) ${\theta }_0<15°$; in (b) ${\theta }_0<30°$.

Figure 14

Total ionization probability $P_{+}+P_{-}$ as a function of laser intensity for a hydrogen atom (solid line) and for a helium atom for the CE phase $\varphi =0$.