Moiré patterns in doubly differential electron-momentum distributions in atomic ionization by mid-infrared lasers

We analyze the doubly differential electron momentum distribution in above-threshold ionization of atomic hydrogen by a linearly polarized mid-infrared laser pulse. We reproduce side rings in the momentum distribution with forward-backward symmetry previously observed by Lemell et al. [Phys. Rev. A 87, 013421 (2013)], whose origin, as far as we know, has not been explained so far. By developing a Fourier theory of moiré patterns, we demonstrate that such structures stem from the interplay between intra- and intercycle interference patterns which work as two separate grids in the two-dimensional momentum domain. We use a three-dimensional (3D) description based on the saddle-point approximation (SPA) to unravel the nature of these structures. When the periods of the two grids (intra- and intercycle) are similar, principal moiré patterns arise symmetrically as concentric rings in the forward and backward directions at high electron kinetic energy. Higher order moiré patterns are observed and characterized when the period of one grid is multiple of the other. We find a scale law for the position (in momentum space) of the center of the moiré rings in the tunneling regime. We verify the SPA predictions by comparison with time-dependent distorted-wave strong-field approximation calculations and the solutions of the full 3D time-dependent Schrödinger equation.

Figure 1

Momentum distributions (linear gray scale) after interaction of a mid-infrared laser pulse with a hydrogen atom. (a) SFA and (b) TDSE [28, 46]. The cosinelike pulse has a peak field $F_0=0.0533$ ($I={10}^{14}{\mathrm{W}/\mathrm{cm}}^2$), frequency $\omega =0.01424$ ($\lambda =3200$ nm) and a ${\sin }^2$ envelope with total pulse duration of eight cycles. Both distributions have been normalized.

Figure 2

SPA doubly differential momentum distribution (linear gray scale) of Eq. (25). (a) Intercycle interference: $4\mathrm{\Gamma }\left(\vec{k}\right)F\left(\vec{k}\right)$; (b) intracycle interference: $4\mathrm{\Gamma }\left(\vec{k}\right)B\left(k\right)$ for $N=2$ cycles; and (c) total (intra- and intercycle) interference $4\mathrm{\Gamma }\left(\vec{k}\right)F\left(\vec{k}\right)B\left(k\right)$ for $N=2$ cycles. The laser parameters are $F_0=0.0533$ ($I={10}^{14}{\mathrm{W}/\mathrm{cm}}^2$) and frequency $\omega =0.01424$ ($\lambda =3200$ nm). Distributions have been normalized.

Figure 3

Magnification of SPA doubly differential momentum distribution in Fig. 2 (linear gray scale). On top of it we have drawn the different intercycle maxima, i.e., $\tilde{S}=2n\pi$ (with integer $n$) in green (gray) and the different intracycle interference maxima, i.e., $\mathrm{\Delta }S=2m\pi$ (with integer $m$) in blue (black). The local maxima of the doubly differential momentum distribution coincide with the intersection of the inter- and intracycle maxima.

Figure 4

SPA doubly differential momentum distribution (linear gray scale) of Eq. (25) [(a) and (c)] and the corresponding $(1,-1)$ moiré pattern [(b) and (d)]. The laser frequency is $\omega =0.01424$ for (a) and (b) and $\omega =0.02$ for (c) and (d). The rest of the laser parameters are the same as in Figs. 2 and 3. Distributions have been normalized.

Figure 5

Inter- and intracycle spacings with their first harmonic (twice the spacings) of Eqs. (26) and (28), respectively [(a) and (f)]. Main moiré $(1,-1)$ pattern [(b) and (g)] and secondary moiré $(1,-2)$ [in (c) and (h)] and $(2,-1)$ [in (d) and (i)] patterns. SPA doubly differential momentum distribution (linear gray scale) of Eq. (25) in (e) and (j). For the first column (a)–(e) $\omega =0.01424$ and for the second column (f)–(j) $\omega =0.02$. The rest of the laser parameters are the same as in Figs. 2, 3, 4. Distributions have been normalized.

Figure 6

Position of the center of the main and secondary moiré patterns in units of the scaled parallel momentum ${\kappa }_z$ (a) and the parallel momentum $k_z$ (b) as a function of the Keldysh parameter $\gamma$.