Determination of the absolute carrier-envelope phase by angle-resolved photoelectron spectra of Ar by intense circularly polarized few-cycle pulses

The angle-resolved photoelectron spectra of Ar are recorded using intense circularly polarized near-infrared few-cycle laser pulses, and the effect of the depletion of Ar atoms by the ionization and the effect of the Coulombic potential are examined by the classical trajectory Monte Carlo simulations. On the basis of the comparison between the experimental and theoretical photoelectron spectra, a procedure for estimating the absolute carrier-envelope phase (CEP) of the few-cycle laser pulses interacting with atoms and molecules is proposed. It is confirmed that the absolute CEP can securely be estimated without any numerical calculations once the angular distribution of the yield of photoelectrons having the kinetic energy larger than 30 eV is measured with the peak laser intensity in the range between $1\times {10}^{14}$ and $5\times {10}^{14}\mathrm{W}/\mathrm{c}{\mathrm{m}}^2$.

Figure 1

The schematics of the experimental setup. TFP: thin-film polarizer; BS: beam splitter; QWP: quarter wave plate.

Figure 2

(a) The experimental CEP-resolved photoelectron spectrum recorded at the peak laser-field intensity of $5.1\times {10}^{14}\mathrm{W}/\mathrm{c}{\mathrm{m}}^2$. (b) The $E_{\mathrm{kin}}-{\varphi }_{\mathrm{abs}}$ asymmetry map constructed from (a). The values of the phase offset (${\varphi }_0$) are plotted with black dots. (c) The asymmetry parameter $A$ in the kinetic energy range (29.5–30.5 eV) between the two vertical lines in (b) plotted as a function of CEP. The red solid curve is the sinusoidal function derived by the least-squares fit to the variation of the asymmetry parameter. The CEP values in these figures are those converted from the relative CEP (${\varphi }_{\mathrm{rel}}$) to the absolute CEP (${\varphi }_{\mathrm{abs}}$). The data in the range of $\pi <{\varphi }_{\mathrm{abs}}<5\pi /2$ are the same as the data in the range of $-\pi <{\varphi }_{\mathrm{abs}}<\pi /2$, and the data in the range of $-3\pi /2<{\varphi }_{\mathrm{abs}}<-\pi$ are the same as the data in the range of $\pi /2<{\varphi }_{\mathrm{abs}}<\pi$.

Figure 3

(a) The ejection direction of a photoelectron in the $x-y$ laboratory frame in which the relative CEP (${\varphi }_{\mathrm{rel}}$) is recorded in the experiment. (b) The ejection direction of a photoelectron expressed in the $x^{\prime }-y^{\prime }$ frame in which the emitting angle is simulated by the calculation. The $x^{\prime }$ axis takes the direction of the maximum amplitude of the circularly polarized light so that CEP becomes $0^{\circ }$.

Figure 4

(a) The $E_{\mathrm{kin}}-\theta$ plot for the ionization rate $W$, represented by the color scale, obtained by solving Newton's equation of motion of a photoelectron in the laser field whose peak intensity is $5.1\times {10}^{14}\mathrm{W}/\mathrm{c}{\mathrm{m}}^2$. Neither the effect of the depletion of neutral Ar nor the effect of the Coulombic potential of $\mathrm{A}{\mathrm{r}}^{+}$ is taken into account. The values of the time of tunneling $t_0$ are also shown. (b) The folded $E_{\mathrm{kin}}-\theta$ plot in which the five angle ranges of (a) are folded into a $2\pi$ angle range: (i) $-3\pi <\theta \le -5\pi /2$, (ii) $-5\pi /2<\theta \le -\pi /2$, (iii) $-\pi /2<\theta \le 3\pi /2$, (iv) $3\pi /2<\theta \le 7\pi /2$, and (v) $7\pi /2<\theta \le 4\pi$. (c) The 3D representation of (b). (d) The kinetic-energy distribution $f(E_{\mathrm{kin}},t_0)$ derived by the initial transverse momentum distribution at the four different values of the time of tunneling: $t_0=0$ (red), $0.25T$ (green), $0.5T$ (blue), and $0.75T$ (purple).

Figure 5

The results of the CTMC simulations obtained at the peak laser intensity of $5.1\times {10}^{14}\mathrm{W}/\mathrm{c}{\mathrm{m}}^2$. Neither the effect of the depletion of neutral Ar nor the effect of the Coulombic potential of $\mathrm{A}{\mathrm{r}}^{+}$ is taken into account. (a) The angle-resolved photoelectron spectrum obtained when the focal averaging effect is neglected. (b) The $E_{\mathrm{kin}}-\theta$ asymmetry map constructed from (a). (c) The $E_{\mathrm{kin}}-\theta$ asymmetry map obtained when the focal averaging is taken into account. In (b) and (c), the values of the phase offset (${\varphi }_0$) are plotted with black dots.

Figure 6

The $E_{\mathrm{kin}}-\theta$ asymmetry maps obtained by the CTMC simulations when the peak laser intensity is $5.1\times {10}^{14}\mathrm{W}/\mathrm{c}{\mathrm{m}}^2$. The values of the phase offset (${\varphi }_0$) are plotted with black dots. (a) Only the effect of the depletion of neutral Ar is taken into account. (b) Only the effect of the Coulomb potential of $\mathrm{A}{\mathrm{r}}^{+}$ is taken into account. (c) Both of the two effects are taken into account.

Figure 7

(a) The experimental CEP-resolved photoelectron spectrum recorded at the peak laser-field intensity of $1.2\times {10}^{15}\mathrm{W}/\mathrm{c}{\mathrm{m}}^2$. (b) The $E_{\mathrm{kin}}-{\varphi }_{\mathrm{abs}}$ asymmetry map constructed from (a). The values of the phase offset (${\varphi }_0$) are plotted with black dots. The CEP values in (a) and (b) are those converted from the relative CEP (${\varphi }_{\mathrm{rel}}$) to the absolute CEP (${\varphi }_{\mathrm{abs}}$).

Figure 8

The results of the CTMC simulations obtained when the peak laser intensity is $1.2\times {10}^{15}\mathrm{W}/\mathrm{c}{\mathrm{m}}^2$. Both the effect of the depletion of neutral Ar and the effect of the Coulombic potential of $\mathrm{A}{\mathrm{r}}^{+}$ are taken into account. (a) The angle-resolved photoelectron spectrum. (b) The $E_{\mathrm{kin}}-\theta$ asymmetry map constructed from (a). (c) The angle-resolved photoelectron spectrum obtained when the ionization rate $W_{\mathrm{eff}}$ is calculated using the scaled ADK rate with the scale factor of 0.27. (d) The $E_{\mathrm{kin}}-\theta$ asymmetry map constructed from (c). In (b) and (d), the values of the phase offset (${\varphi }_0$) are plotted with black dots.

Figure 9

The results of the CTMC simulations obtained when the peak laser intensity is $5.1\times {10}^{14}\mathrm{W}/\mathrm{c}{\mathrm{m}}^2$. The folded $E_{\mathrm{kin}}-\theta$ plot, their 3D representations, the corresponding angle-resolved photoelectron spectra, and the corresponding $E_{\mathrm{kin}}-\theta$ asymmetry maps are shown. The top row for (a)–(d) shows the results obtained when neither the effect of the depletion of neutral Ar nor that of the Coulomb potential of $\mathrm{A}{\mathrm{r}}^{+}$ is taken into account. The second row for (e)–(h) shows the results obtained when only the effect of the depletion of neutral Ar is taken into account. The third row for (i)–(l) shows the results obtained when only the effect of the Coulomb potential of $\mathrm{A}{\mathrm{r}}^{+}$ is taken into account. The bottom row for (m)–(p) shows the results obtained when both of the two effects are taken into account.

Figure 10

The results of the CTMC simulations obtained when the peak laser intensity is $1.2\times {10}^{15}\mathrm{W}/\mathrm{c}{\mathrm{m}}^2$. The folded $E_{\mathrm{kin}}-\theta$ plot, their 3D representations, the corresponding angle-resolved photoelectron spectra, and the corresponding $E_{\mathrm{kin}}-\theta$ asymmetry maps are shown. The top row for (a)–(d) shows the results obtained when neither the effect of the depletion of neutral Ar nor that of the Coulomb potential of $\mathrm{A}{\mathrm{r}}^{+}$ is taken into account. The second row for (e)–(h) shows the results obtained when only the effect of the depletion of neutral Ar is taken into account. The third row for (i)–(l) shows the results obtained when only the effect of the Coulomb potential of $\mathrm{A}{\mathrm{r}}^{+}$ is taken into account. The bottom row for (m)–(p) shows the results obtained when both of the two effects are taken into account.