Intracycle and intercycle interferences in above-threshold ionization: The time grating

Within a semiclassical description of above-threshold ionization (ATI) we identify the interplay between intracycle and intercycle interferences. The former is imprinted as a modulation envelope on the discrete multiphoton peaks formed by the latter. This allows one to unravel the complex interference pattern observed for the full solution of the time-dependent Schrödinger equation (TDSE) in terms of diffraction at a grating in the time domain. These modulations can be clearly seen in the dependence of the ATI spectra on the laser wavelength. Shifts in energy modulation result from the effect of the long Coulomb tail of the atomic potential.

Figure 1

Electric field $F(t)$ (left axis) and vector potential $A(t)$ (right axis) of a flat-top pulse ($N=4,m=\frac{1}{2}$). 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 circle-triangle pair 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

Semiclassical intracycle and intercycle interference pattern. (a) Intracycle interference pattern given by the structure factor $F(k)$. (b) Intercycle interferences with “Bragg” peaks given by $B(k)$ in Eq. (7). (c) Total semiclassical interference [Eq. (5)] for $N=2$. The laser wavelength and intensity are $1000$ nm and $1.6\times {10}^{14}$ W/cm${}^2$, respectively. To stress the interferences, we set $\Gamma (k)=1$ and normalize the respective maxima of $F(k)$ and $B(k)$ to unity [see Eq. (5)].

Figure 3

Semiclassical total ionization probability as a function of the laser wavelength $\lambda$ and the ATI order $n$ (see text). Shown is the spectrogram for both a one-cycle pulse as (light gray) green stripes and and a four-cycle pulse as (dark gray) red islands. $F_0=0.0675$ a.u. ($1.6\times {10}^{14}$ W/cm${}^2$).

Figure 4

Two-dimensional interferogram as a function of laser wavelength $\lambda$ and ATI order $n$ (see text) calculated by using the TDSE for hydrogen for a four-cycle pulse with $F_0=0.0675$ a.u. ($1.6\times {10}^{14}$ W/cm${}^2$). (a) Emission into a cone of ${10}^{°}$ around the polarization axis and (b) for all angles. (c) Energy- (or $n$) integrated total ionization yield as a function of $\lambda$.

Figure 5

Comparison of the photoelectron spectra within the ${10}^{°}$ cone, calculated with the full Coulomb potential (marked as “FULL”) with the truncated Coulomb potentials for varying values of $r_c$ as indicated. The laser wavelength is 1000 nm. Other laser parameters are the same as in Fig. 4. Thin solid lines are approximate peak envelopes.