Intensity-dependent enhancements in high-order above-threshold ionization

The very pronounced intensity-dependent enhancements of groups of peaks of high-order above-threshold-ionization spectra of rare-gas atoms are investigated using an improved version of the strong-field approximation, which realistically models the respective atom. Two types of enhancements are found and explained in terms of constructive interference of the contributions of a large number of long quantum orbits. The first type is observed for intensities slightly below channel closings. Its intensity dependence is comparatively smooth and it is generated by comparatively few (of the order of 20) orbits. The second type occurs precisely at channel closings and exhibits an extremely sharp intensity dependence. It requires constructive interference of a very large number of long orbits (several hundreds) and generates cusps in the electron spectrum at integer multiples of the laser-photon energy. An interpretation of these enhancements as a threshold phenomenon is also given. An interplay of different types of the threshold anomalies is observed. The position of both types of enhancements, in the photoelectron-energy—laser-intensity plane, shifts to the next channel closing intensity with the change of the ground-state parity. The enhancements gradually disappear with decreasing laser pulse duration. This confirms the interpretation of enhancements as a consequence of the interference of long strong-laser-field-induced quantum orbits.

Figure 1

(Color online) Examples of the notation $\alpha \beta m$ used to label the solutions of the saddle-point equations. The solid, dotted, long-dashed, dot-dashed, and double-dot-dashed curves in the right-hand part of the figure $(0⩽t⩽T)$ specify the rescattering times for the five pairs of orbits with the shortest travel times. In the left-hand part of the figure, the counterpart of each curve identifies the corresponding ionization times. The emitted electron energy in multiples of $U_{\mathrm{P}}$ is plotted on the ordinate, and horizontal lines (at constant energy) relate ionization times and rescattering times for the respective orbits. There are infinitely many further solutions that have ionization times $t_0$ beyond the left-hand margin of the figure. Their maximal return energies converge to $8U_{\mathrm{P}}$. The curves have been calculated for He, for emission in the direction $\theta =0°$, and for a linearly polarized laser field having the intensity ${10}^{15}\mathrm{W}∕{\mathrm{cm}}^2$ and the wavelength $760\mathrm{nm}$.

Figure 2

(Color online) Differential ionization rate of He as a function of the electron kinetic energy $E_{\mathbf{p}}$ in units of the ponderomotive energy $U_{\mathrm{P}}$, for emission in the direction $\theta =0°$, for a linearly polarized laser field having the intensity ${10}^{15}\mathrm{W}∕{\mathrm{cm}}^2$ and the wavelength $760\mathrm{nm}$. Comparison of the uniform approximation (UNA) with the results obtained by numerical integration (SFA).

Figure 3

(Color online) Maximal classical electron kinetic energy as a function of the angle between the emitted electron and the linearly polarized laser field direction.

Figure 4

(Color online) The $\theta$-dependence of the cutoff values for the saddle-point solutions characterized by the index $\beta m$ (see the text for more explanations).

Figure 5

(Color online) Differential ionization rate of Kr as a function of the parameter $(I_{\mathrm{P}}+U_{\mathrm{P}})∕\omega$, for emission of electrons with kinetic energy $E_{\mathbf{p}}=23\mathrm{eV}$ in the direction $\theta =0°$. The laser field is linearly polarized having the wavelength $760\mathrm{nm}$. Its intensity changes from $0.67\times {10}^{14}\text{to}1.76\times {10}^{14}\mathrm{W}∕{\mathrm{cm}}^2$. The uniform approximation for high-order ATI is used with 4 (black dotted curve), 20 (red long-dashed curve), 40 (green dot-dashed curve), and 200 (blue solid curve) quantum orbits.

Figure 6

(Color online) The relative phase ${\Phi }_{\beta =-1}-{\Phi }_{\beta =+1}$ of the $m\text{th}$ partial contribution to the $T$-matrix element in the uniform approximation (24) as a function of the parameter $(I_{\mathrm{P}}+U_{\mathrm{P}})∕\omega$, for the atomic and laser field parameters of Fig. 5. The results presented are (a) for $1⩽m⩽5$ and (b) for $6⩽m⩽22$ in steps of 4.

Figure 7

(Color online) Differential ionization rate of He as a function of the parameter $(I_{\mathrm{P}}+U_{\mathrm{P}})∕\omega$, for emission of electrons with kinetic energy $E_{\mathbf{p}}=199\mathrm{eV}$ in the direction $\theta =0°$. The laser field is linearly polarized having the wavelength $760\mathrm{nm}$. Its intensity changes from $1.04\times {10}^{15}\text{to}1.16\times {10}^{15}\mathrm{W}∕{\mathrm{cm}}^2$. The uniform approximation for high-order ATI is used with 4 (red dashed curve), 20 (blue dot-dashed curve), 40 (green long dashed curve), and 200 (black solid curve) quantum orbits.

Figure 8

(Color online) The relative phase ${\Phi }_{\beta =-1}-{\Phi }_{\beta =+1}$ of the $m\text{th}$ partial contribution to the $T$-matrix element in the uniform approximation (24) as a function of the parameter $(I_{\mathrm{P}}+U_{\mathrm{P}})∕\omega$, for the atomic and laser field parameters of Fig. 7. The results presented are (a) for $1⩽m⩽5$, (b) $6⩽m⩽11$, and (c) for even $12⩽m⩽24$.

Figure 9

(Color online) Natural logarithm of the differential ionization rate of Xe as a function of the parameter $(I_{\mathrm{P}}+U_{\mathrm{P}})∕\omega$, for emission of the electrons having kinetic energy $E_{\mathbf{p}}=90\mathrm{eV}$ in the direction $\theta =0°$. The laser field is linearly polarized having the wavelength $760\mathrm{nm}$. Its intensity changes from $2.5\times {10}^{14}\text{to}8\times {10}^{14}\mathrm{W}∕{\mathrm{cm}}^2$. The uniform approximation for high-order ATI is used with 100 quantum orbits (black solid curve). The results with only the orbits having $\beta =1$ $(\beta =-1)$ included are presented by the red dot-dashed (green long dashed) curve.

Figure 10

(Color online) The same as in Fig. 5 but for 400 orbits. The derivative of the ionization rate (divided by the factor 10) over the parameter $n_{\mathrm{c}}=(I_{\mathrm{P}}+U_{\mathrm{P}})∕\omega$ as a function of $n_{\mathrm{c}}$ is also presented by a red solid line.

Figure 11

(Color online) The same as Fig. 7 but for 400 orbits. The derivative of the ionization rate (divided by the factor 20) over the parameter $n_{\mathrm{c}}=(I_{\mathrm{P}}+U_{\mathrm{P}})∕\omega$ as a function of $n_{\mathrm{c}}$ is also presented by the red solid line.

Figure 12

(Color online) Enlarged part of Fig. 11 near $n_{\mathrm{c}}=50$.

Figure 13

(Color) High-order ATI photoelectron energy spectrum of Xe ($I_{\mathrm{P}}=12.13\mathrm{eV}$, $p$ ground state [11]), presented in false colors, as a function of the laser intensity expressed in units $(I_{\mathrm{P}}+U_{\mathrm{P}})∕\omega$. Electron emission is in the direction $\theta =0°$, the laser field is linearly polarized having the wavelength $760\mathrm{nm}$, and the laser intensity changes from $4\times {10}^{13}\text{to}2.4\times {10}^{14}\mathrm{W}∕{\mathrm{cm}}^2$. Only the rescattering contribution to the $T$ matrix is taken into account, with the rescattering potential modeled as in [11].

Figure 14

(Color) Same as in Fig. 13, but for the Xe atom modeled by a $1s$ ground state. The ionization energy is $I_{\mathrm{P}}=12.13\mathrm{eV}$.

Figure 15

(Color online) Logarithm of the focal-averaged differential ionization rates of Kr as functions of the electron kinetic energy for emission in the direction $\theta =0°$. The laser field is linearly polarized having the wavelength $760\mathrm{nm}$. Its intensity is $a\times {10}^{14}\mathrm{W}∕{\mathrm{cm}}^2$, where $a=0.75$ (bottom black curve), 1.06 (green curve), 1.37 (blue curve), and 1.68 (upper red curve). The results are obtained using the improved SFA [11] and each upper curve is shifted up by 0.5 units with respect to the lower one.

Figure 16

(Color online) Same as Fig. 15, but for the intensities $1.06\times {10}^{14}\mathrm{W}∕{\mathrm{cm}}^2$ (lower red curve) and $1.68\times {10}^{14}\mathrm{W}∕{\mathrm{cm}}^2$ (upper black curve).

Figure 17

(Color online) The logarithm of the focal-averaged yield of Xe as a function of the energy of a photoelectron emitted in the polarization direction. The laser field is linearly polarized having the wavelength $760\mathrm{nm}$ and the intensity $1.1\times {10}^{14}\mathrm{W}∕{\mathrm{cm}}^2$. The bottom green solid curve represents the results for a flat envelope and infinitely long pulse. For the other curves presented, the pulse envelope is sine square with zero carrier-envelope phase, as defined in [5]. The number of optical cycles is $n_{\mathrm{p}}=6$ (black dotted curve), 14 (blue dashed curve), and 22 (red long dashed curve). The curves are shifted by two orders of magnitude with respect to each other for better visibility.

Figure 18

(Color) Same as Fig. 13, but for a wider range of electron energies and laser intensities.