Diffraction at a time grating in above-threshold ionization: The influence of the Coulomb potential

We analyze the photoelectron emission spectrum in atomic above-threshold ionization by a linearly polarized short-laser pulse. Direct electrons can be characterized by both intracycle and intercycle interferences. The former results from the coherent superposition of two different electron trajectories released in the same optical cycle, whereas the latter is the consequence of the superposition of multiple trajectories released in different cycles. In the present article, a semiclassical analytical expression for the complete (both intracycle and intercycle) interference pattern is derived. We show that the recently proposed semiclassical description in terms of a diffraction process at a time grating remains qualitatively unchanged in the presence of the long-range Coulomb potential. The latter causes only a phase shift of the intracycle interference pattern. We verify the predictions of the semiclassical model by comparison with full three-dimensional (3D) time-dependent Schrödinger equation (TDSE) solutions. One key finding is that the subcycle interference structures originating from trajectories launched within a time interval of less than 1 femtosecond should be experimentally observable also in low-resolution spectra for longer multicycle pulses.

Figure 1

Electric field $F(t)$ (left axis) and vector potential $A(t)$ (right axis) of a flat-top pulse [as defined in Eq. (19) with $N=3,m=m^{\prime }=1/2$]. The electron emission times for a given final momentum $k$ are marked by circles [$t_r^{(j,1)}$] and triangles [$t_r^{(j,2)}$]. Each pair of circles and triangles determines the structure factor $F(k)$ and leads to intracycle interference while the periodic train of such pairs gives rise to intercycle interference. Each optical cycle can be viewed as a “unit cell” of the time lattice.

Figure 2

(a) Buildup of the interference pattern following the semiclassical SMM: intracycle interference pattern given by the structure pattern $F(k)$ for a multicycle pulse [Eq. (16)]. The thin line shows the pattern for a single-cycle pulse [Eq. (17)]. (b) Intercycle interference given by the function $B(k)$ with $N=2$. (c) Total interference pattern $F(k)B(k)$ [see Eq. (27)] with $N=2$. The laser parameters are $F_0=0.05$, $\omega =0.05$, and $I_p=0.5$. Vertical grid lines correspond to multiphoton energies.

Figure 3

Semiclassical total ionization probability within the SMM as a function of the channel-closing parameter $R$ and the scaled energy $n^{*}$ (see text) calculated for (a) $N=1$, (b) $N=2$, and (c) $N=3$. In (d), the results of (a) (green stripes) with those of (c) are overlaid (blue-red islands). $F_0=0.0675$ (intensity equal to $1.6\times {10}^{14}$ W/cm${}^2$) and $I_p=0.5$. The range in $R$ corresponds to wavelengths 1000–1060 nm or frequencies 0.0456–0.0430 a.u.. The white arrow in (c) points to one secondary fringe.

Figure 4

(a) Semiclassical prediction [Eq. (32)] for the separation between consecutive intracycle interference peaks (divided by the laser frequency) as a function of the scaled momentum $\kappa$, for $\gamma =0$, $0.1$, $1,$ and $10$; the horizontal dashed line corresponds to the scaled period of the multiphoton peaks. (b) Maximum ${\kappa }_m$ as a function of the Keldysh parameter $\gamma$ [see Eq. (33)]. The tunneling and multiphoton limits are marked with dashed lines.

Figure 5

2D interferogram for the ionization probability within a cone of ${10}^{°}$ in the forward and backward direction as a function of channel-closing parameter $R$ and scaled energy $n^{*}$ (see text), for a four-cycle pulse with $F_0=0.0675$, $N=3$ and with $m=m^{\prime }=1/2$, within the (a) SFA and (b) CVA.

Figure 6

Photoelectron spectra for different values of the nuclear charge $Z_T=0,0.25,0.5,0.75,$ and $1$ within the CVA, from bottom to top calculated for the four-cycle pulse described in Fig. 5 (solid line) and the one-and-half cycle of Fig. 8 (dashed line). The carrier frequency is $\omega =0.0456$ $(R=23)$. The horizontal arrow indicates the Coulomb shift estimated within the SMM.

Figure 7

2D TDSE interferogram for electron emission of hydrogen for the same field as in Fig. 5. (a) Into a cone of ${10}^{°}$ around the polarization axis and (b) into all angles.

Figure 8

2D interferogram for emission within a cone of ${10}^{°}$ around the polarization axis as a function of channel-closing parameter $R$ and scaled energy $n^{*}$ calculated for a 1.5-cycle pulse with $F_0=0.0675$, $m=0$, and $m^{\prime }=1/2$. (a) SPA, (b) SFA, (c) CVA, and (d) TDSE (see text).

Figure 9

Energy distribution within a cone of ${10}^{°}$ around the polarization axis for the same laser parameters as in Fig. 8 for $\omega =0.0456$ ($R=23$). The data are from the SFA (dashed line), the CVA (thin solid line), and TDSE (thick solid line).

Figure 10

Energy difference between consecutive intracycle peaks as a function of the energy position of the corresponding peaks calculated with SFA (open squares), CVA (full squares), and TDSE (circles). The full line is the semiclassical prediction of Eq. (32). Same laser parameters as in Fig. 9.

Figure 11

(a) TDSE total ionization probability $P_{\mathrm{ion}}$ as a function of the channel-closing parameter $R$ (wavelengths in the range $1000$ to $1100$ nm) calculated within the TDSE. The parameters of the laser field are $F_0=0.0675$ a.u. and the total number of cycles is $N=3$ with $m=m^{\prime }=1/2$ cycle for the ramp on and off. (b) Same as for (a) but for emission in a cone of ${10}^{°}$ around the polarization axis.