Moir\'e patterns in doubly differential electron momentum distributions in atomic ionization by midinfrared 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 \textit{et al.} in Phys. Rev. A \textbf{87}, 013421(2013), whose origin, as far as we know, has not been explained so far. By developing a Fourier theory of moir\'{e} patterns, we demonstrate that such structures stems 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 moir\'{e} patterns arise as concentric rings symmetrically in the forward and backward directions at high electron kinetic energy. Higher order moir\'{e} 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 moir\'{e} rings in the tunneling regime. We verify the SPA predictions by comparison with time-dependent distorted wave strong-field approximation (SFA) calculations and the solutions of the full 3D time-dependent Schr\"{o}dinger equation (TDSE).


I. INTRODUCTION
In a typical photoionization process in the tunneling regime, electrons are emitted by tunneling through the potential barrier formed by the combination of the atomic potential and the external strong field. Tunneling occurs within each optical cycle predominantly around the maxima of the absolute value of the electric field. According to the well-known three-step model, photoelectrons can be classified into direct and rescattered electrons [1][2][3].
After ionization, direct electrons can escape without being strongly affected by the residual core potential. The classical cutoff energy for this process is twice the ponderomotive energy.
After being accelerated back by the laser field, a small portion of electrons are rescattered by the parent ion and can achieve a kinetic energy of up to ten times de ponderomotive energy.
Trajectories that correspond to direct ionization are crucial in the formation of interference patterns in photoelectron spectra. Quantum interference within an optical cycle was firstly reported (as far as we know) in Ref. [4] and theoretically analyzed and experimentally observed by Paulus et al in [5] both for negative ions. A thorough saddle-point analysis with the strong field approximation can be found in Becker's review [6]. Non-equidistant peaks in the photoelectron spectrum were firstly calculated for neutral atoms by Chirila et al [7]. A temporal double-slit interference pattern has been studied in near-single cycle pulses both experimentally [8,9] and theoretically [6,10]. Near threshold oscillations in angular distribution were explained as interferences of electron trajectories [11] and measured by [12]. Diffraction fringes have been experimentally observed in photoionization of He [9,13] and Ne atoms [13] and photodetachment in H − and [14] F − ions by femtosecond pulses for fixed frequency [15] and theoretically analyzed [16][17][18][19]. Diffraction patterns were also found in spectra of laser-assisted XUV ionization, whose gross structure of sidebands were explained as the interference between electrons emitted within one period [20][21][22][23][24]. The interference pattern in multi-cycle photoelectron spectra can be identified as a diffraction pattern at a time grating composed of intra-and intercycle interferences [16][17][18][19]. While the latter gives rise to the well-known ATI peaks [25][26][27], the former leads to a modulation of the ATI spectrum in the near infrered regime offering information on the subcycle ionization dynamics.
In previous papers we analyzed how the interplay between the intercycle interference [factor B(k) in Eq. (25)] and the intracycle interference [factor F ( k) in Eq. (25)] controls the doubly differential distribution of direct ATI electrons for lasers in the NIR [17][18][19].
In a theoretical study about the quantum-classical correspondence in atomic ionization by midinfrared pulses, Lemell et al. calculated the doubly differential momentum distribution after the interaction of a strong midinfrared laser pulse with a hydrogen atom, which shows multiple peaks and interference structures (see Fig. 1 of [28]). At both sides of the wellknown intercycle ATI rings, two distinct ring-like structures appear (symmetrically) in the forward and backward directions. As far as we know, the origin of these structures has not been identified so far. In this paper, we extend the analysis of the SPA to the midinfrared regime.
Large scale interference patterns can be produced when a small scale grid is overlaid on another similar grid [29,30]. These patterns are named moiré [29,30] and appear in art, physics, mathematics, etc.. They show up in everyday life such as a striped shirt seen on television, in the folds of a moving curtain, when looking through parallel wiremesh fences, etc. More than a rareness, moiré is widely used in projection interferometry complementing conventional holographic interferometry, especially for testing optics used at long wavelength. The use of moiré for reduced sensitivity testing was introduced by Lord Rayleigh in 1874 to determine the quality of two identical gratings even though each individual grating could not be resolved under a microscope [31]. Moiré patterns have been extremely useful to help the understanding of basic interferometry and interferometric test results [32][33][34].
In the present communication, we theoretically investigate on the origin of side ring structures that appear in the doubly differential momentum distribution for atomic ionization by laser pulses in the midinfrared spectral region [28]. We demonstrate that such structures stems from the interplay between intra-and intercycle interference patterns which work as two separate grids in the two-dimensional momentum domain. When the periods of the two grids (intra-and intercycle) are similar, principal moiré patterns arise as concentric rings at high electron kinetic energy in the forward and backward directions symmetrically. Besides, we show that a whole family of secondary moiré patterns with less visibility of the principal one is also present. We characterize these structures within the Fourier theory of the moiré patterns finding simple scale laws for the position of their center in the momentum distribution. In order to do that, we previously discard the formation of spurious (non-physical) moiré patterns due to the presence of the numerical grid of the momentum map. We use a three dimensional (3D) description based on the saddle-point approximation (SPA) [17][18][19] to unravel the nature of these structures. Our SPA predictions are corroborated by comparison with time-dependent distorted wave strong-field approximation (SFA) [3,7,[35][36][37] calculations and the solutions of the full time-dependent Schrödinger equation (TDSE).
The paper is organized as follows. In the first part of Sec. II A, we develop the Fourier theory of moiré patterns. We continue by scheming the semiclassical model for atomic ionization by laser pulses showing that the separation of intracycle and intercycle interferences can be interpreted in terms of diffraction at a time grating when studying the doubly differential distributions within the SPA. In the last part of the section we show how moiré patterns are formed from inter-and intracycle interferences in view of this Fourier theory.
In Sec. III, we analyze the ring-like structures in the doubly differential momentum distribution within the SPA and compare with the SFA and TDSE ab initio calculations. We also characterize the moiré structure by analyzing the dependence of the position of the center as a function of laser parameters finding a scale law. Atomic units are used throughout the paper, except when otherwise stated.

A. Fourier theory of moiré patterns
We define a 1D grating (vertical stripes) as a periodic function G(x ), of period p is the period of the grating. Due to its periodicity, the function G(x ) can be thought as a sum of different harmonic terms of discrete frequency, where a n is the Fourier coefficient and Gratings with a general geometrical layout can be considered as extended coordinatetransformed structures which can be obtained by applying geometric transformations to a standard 1D-grating. By replacing x with a certain function T (x, y), the 1D grating of Eq.
(1)] can be transformed into another curvilinear grating G T (x, y) = G[T (x, y)]. Therefore, in the same way, the latter can be expressed as geometrical layout of two (or more) superposed structures. The two gratings with the extended layout can be obtained by applying the transformations T 1 (x, y) and T 2 (x, y) to two 1D gratings of frequencies f 1 and f 2 , respectively. The generalized gratings can be expressed as in Eq. (2), The two superimposed gratings can be written as the multiplication of the two general gratings G 1 and G 2 , in respective equations (3a) and (3b), From Eq. (4), we can extract the partial sum ∞ , with k 1 and k 2 integer numbers different from zero. Then, we express this partial sum in the same way as in Eq. (2), namely, where f is a standardized frequency. In this way, the twofold sum of Eq. (4) can be decomposed into many partial sums. Eq. (5) can be regarded as the transformation of a 1D grating with a compound transformation function applied to the 1D gratingG For every pair (k 1 , k 2 ), the partial sum in Eq. (7) converges to a periodic-distributed pattern similar to the layout of standard 1D gratings. By transforming the partial sum of Eq. (7) with the transformation function of Eq. (6), we get the (k 1 , k 2 )-order moiré pattern of Eq. (5). Summing up, we can say that two geometrically transformed 1D gratings exhibit equivalent patterns to the one obtained by application of a compound transformation to a certain 1D-distributed moiré pattern.
In general, moiré fringes generated by two superposed gratings are transformed from two standard 1D gratings with different frequencies by different transformations. However, in the following section, we restrict to the special case of moiré fringes generated from 1D gratings with the same frequency, i.e., f 1 = f 2 = f , and different transformations, i.e., . Therefore, the moiré pattern of Eq. (5) can be written as The lowest frequency pattern corresponds to the pair (k 1 , k 2 ) = (1, −1), which is usually the most visible one. We name the pair (1, −1) as the principal moiré pattern with the Higher order or secondary moiré patterns are also present with less visibility. Later, we will see how the side-ring structure can be thought of as the principal moiré pattern arising from the superposition of intra-and intercycle interferences, each considered as a separate grid G 1 and G 2 . But before, in the next subsection, we pose the semiclassical theory of inter-and intracycle interference in the electron yield after atomic ionization by a short laser pulse.

B. Semiclassical model
In this subsection we repeat the theory of the semiclassical model for atomic ionization in the single active electron approximation interacting with a linearly polarized laser field F (t) firstly posed in [17][18][19]. The reader familiar with the semiclassical model can skip this subsection and go directly to the analysis of the formation of the moiré patterns in the next subsection.
The Hamiltonian of the system in the length gauge is where V (r) is the atomic central potential and p and r are the momentum and position of the electron, respectively. The term r · F (t) couples the initial state |φ i to the continuum final state |φ f with momentum k and energy E = k 2 /2. The TDSE for the Hamiltonian of momentum distributions as dP where T if is the T-matrix element corresponding to the transition φ i → φ f .
The transition amplitude within the time-dependent distorted wave theory in the strong field approximation (SFA) in the post form is expressed as [38] T where χ − f (t) is the final distorted-wave function and the initial state φ i (t) is an eigenstate of the atomic Hamiltonian without perturbation with eigenenergy equal to minus the ionization potential I p . If we choose the Hamiltonian of a free electron in the time-dependent electric field as the exit-channel distorted Hamiltonian, i.e., , the solutions are the Volkov states [39] where S denotes the Volkov action In equations (12) and (13), is the vector potential of the laser field divided by the speed of light. Eq. (11) together with Eq. (12) leads to the SFA transition matrix. Accordingly, the influence of the atomic core potential on the continuum state of the receding electron is neglected and, therefore, the momentum distribution is a constant of motion after conclusion of the laser pulse [3,40].
To solve the time integral in Eq. (11), we closely follow the "saddle-point approximation" (SPA) [3,7,37,41], which considers the transition amplitude as a coherent superposition of electron trajectories Here, M is the number of trajectories born at ionization times t (i) r reaching a given final momentum k, and G(t where d * ( v) is the dipole element of the bound-continuum transition.
The release time t (i) r of trajectory i is determined by the saddle-point equation, yielding complex values since I p > 0. The condition for different trajectories to interfere is to reach the same final momentum k to satisfy Eq. (16) with release times t (i) . Whereas the interference condition involves the vector potential A, the electron trajectory is governed by the electrical field F . We now consider a periodic laser linearly polarized along the z axis whose laser field is F (t) = F 0ẑ sin(ωt), where F 0 is the field amplitude.
Accordingly, the vector potential is given by A(t) = F 0 ωẑ cos(ωt). There are two solutions of Eq. (16) per optical cycle. The first solution in the j-th cycle is given by whereκ denotes the complex final momentum defined bỹ and κ z and κ ⊥ are the respective longitudinal and transversal components of the dimensionless scaled final momentum of the electron κ = ω k/F 0 . In Eq. (18) γ = 2I p ω/F 0 is the Keldysh parameter. The second solution fulfills In equations (17) and (19), t (j,α) r with α = 1(2) denotes the early (late) release times within the j-th cycle.
For a given value of k, the field strength for ionization at t (j,α) r is independent of j and identical for all subsequent ionization bursts (or trajectories) and, therefore, only a function of the time-independent final momentum k provided the ground-state depletion is negligible.
As there are two interfering trajectories per cycle, the total number of interfering trajectories with final momentum k is M = 2N , with N being the number of cycles involved in the laser pulse. Hence, the sum over interfering trajectories [Eq. (14)] can be decomposed into those associated with two release times within the same cycle and those associated with release times in different cycles [17][18][19]. Consequently, the momentum distribution [Eq. (10)] can be written within the SPA as where the second factor on the right hand side of Eq. (20) describes the interference of 2N trajectories with final momentum k, where t (j,α) r is a function of k through equations (17) and (19).
In turn, due to discrete translation invariance in the time domain (t → t + 2jπ/ω), the difference of the action ∆S j is independent of the cycle number j, which can be expressed (dropping the subscript j) as where sgn denotes the sign function that accounts for positive and negative longitudinal momentum k z .
After some algebra, Eq. (20) can be rewritten as an equation of a diffraction grating of the form [17][18][19], where the interference pattern can be factorized into two contributions: (i) the interference stemming from a pair of trajectories within the same cycle (intracycle interference), governed by F ( k), and (ii) the interference stemming from trajectories released at different cycles (intercycle interference) resulting in the well-known ATI peaks given by B(k) (see Ref. [42]).
The intracycle interference arises from the superposition of pairs of trajectories separated by a time slit ∆t = t

C. Formation of moiré patterns from inter-and intracycle interference
The intercycle principal maxima fulfill the equationS = 2nπ, leading to the ATI energies E n = nω− U p − I p in agreement with the conservation of energy for the absorption of n photons. Therefore, in the doubly differential momentum distribution, the 2D intercycle grid follows the relation between the parallel and perpendicular momenta k inter The spacing between two consecutive maxima can be easily calculated for E n = k 2 z /2, (provided k ⊥ = 0) as [k inter where α = 4U p /F 0 = F 0 /ω 2 is the quiver amplitude of the escaping electron, κ = (ω/F 0 ) k; and in the last line we have used that E n+1 − E n = ω.
The intracycle maxima correspond to the equation ∆S = 2mπ with integer m. Equivalently to the intercycle case, the intracycle spacing can be calculated as After a bit of algebra, the derivative of the accumulated action with respect to the parallel momentum can be written in a close form and, thus, the intracycle spacing reads In Eq. (28) we have considered forward emission, i.e., k z ≥ 0. We have an analogous result for backward emission.
According to Eq. (25), the transformations from the 1D grating to the inter-and intracycle 2D grating are T 1 (k z , k ⊥ ) =S/2 given by Eq. (23) and T 2 (k z , k ⊥ ) = ∆S/2 given by Eq. (24). Therefore, we can write the (k 1 , k 2 )−order compound transformation By eye inspection [at least for the lowest orders (k 1 , k 2 ) = (1, −1), (2, −1), and (1, −2)] the function T (k z , k ⊥ ) exhibits one global minimum for forward (and backward) emission, which corresponds to the center of the side ring. The minimum can be easily found as The derivative in the first term of right hand side of Eq. (31) can be easily written as where we have used equations (23) and (26). Doing the same with the derivative in the second term of Eq. (31), we get that ∂∆S/∂k z = 2π/∆k intra z . Therefore, Eq. (31) can be written as which is equivalent to The principal moiré rings is given by the lowest order (k 1 , k 2 ) = (1, −1), which means that the intra-and intercycle spacings should be the same, i.e., ∆k intra

III. RESULTS AND DISCUSSION
In Fig. 1 (a) and (b) we show the doubly differential electron momentum distribution within the SFA [equations (10) and (11)], and TDSE [28,43] after ionization of atomic hydrogen by an intense (I = 10 14 W/cm 2 ) midinfrared (λ = 3200 nm or equivalently ω = 0.001424 a.u.) sine-pulse of eight-cycle of total duration with a sin 2 envelope. The intercycle pattern appear as concentric (ATI) rings centered at threshold. In the TDSE momentum distribution, the characteristic bouquet-shape structure due to interference of electron trajectories oscillating about the Kepler trajectory is clearly observed [11,44]. The bouquet-shape structure is absent in the SFA since it lacks of the effect of the Coulomb potential on the escaping trajectories. At both sides of the ATI rings, two symmetrical annular structures at |k z | 0.82 are observed in both (SFA and TDSE) approaches. As far as we know, these side rings has not been studied. As the SFA does not consider rescattering electrons, we must discard this effect as a possible explanation for the formation of the side rings. In the rest of the paper we identify the origin of these rings with the aid of the semiclassical model and the theory of moiré patterns.
The interplay between the intercycle interference [factor B(k) in Eq. (25)] and the intracycle interference [factor F ( k) in Eq. (25)] controls the doubly differential distribution of direct ATI electrons for lasers [17][18][19]. Firstly, we examine the intercycle interference within the SPA by setting the intracycle factor to be F ( k) = 1 and N = 2 in Eq. (25) for the same laser parameters as in Fig. 1, except the duration and envelope, we use N = 2 cycles of duration. The factor B(k) reduces to the two-slit Young interference expression B(k) = 4 cos 2 π/ω S /2 , whereS is given by Eq. (23). We plot the corresponding SPA doubly differential momentum distribution in Fig. 2 (a), where we can observe concentric rings with radii k n = √ 2E n . The intracycle interference arises from the superposition of two trajectories released within the same optical cycle, i.e., α = 1, 2 and N = 1 in Eq. (25) or, equivalently, 4 Γ( k)F ( k), since B(k) = 1 in this case. In Fig. 2 (b), we see that the SPA intracycle interference pattern gives approximately vertical thin stripes which bend  Fig. 2 (c). We can see that a grosser structure emerges as two side rings centered at k z ±0.83 and k ⊥ 0, and two less visible rings centered at k z ±0.5 and k ⊥ 0. If we consider longer pulses, the contrast of intercycle factor B(k) will increase as N increases.
For example, the ATI rings will become narrower and N − 2 secondary rings will appear between two consecutive principal ATI rings. On the other side, the intracycle factor F ( k) is independent of the number of cycles N involved in the laser pulse and, in consequence, the intracycle interference pattern remains unchanged. This is strictly valid provided we consider a flattop pulse in the SPA. Moreover, we have checked that the position of the side rings is independent of the pulse duration (not shown).
In the Fig. 3, we show in red the maxima of the intercycle interference pattern, i.e., S = 2nπ with n integer, given by the conservation of energy relation and in blue the maxima Contrarily to the intercycle grid, the intracycle grid does not have an explicit form and must be solved numerically.
In Fig. 4 (a) we show a closeup of the SPA doubly differential momentum distribution for the same laser parameters as in figures 2 and 3. The side ring centered at (k z , k ⊥ ) (0.83, 0) is clearly seen. In Fig. 3 (b) we plot the principal moiré ring, i.e., cos 2 [(T (k z , k ⊥ )] , where the transformation T (k z , k ⊥ ) is given by Eq. (29) for (k 1 , k 2 ) = (1, −1). We see that the shape and position of the moiré pattern in Fig. 4 (b) coincide with the side ring of the doubly differential momentum distribution in Fig. 4 (a). When the laser frequency is increased to ω = 0.2, the principal side ring shifts horizontally towards less energetic domains and is centered at (k z , k ⊥ ) (0.62, 0), as can be observed in Fig. 4 (c). The corresponding moiré pattern in Fig. 4 (d), also shifts accurately reproducing the side ring. This is a confirmation of the application of the theory of the principal moiré patterns posed in the last section to atomic ionization in the midinfrared range.
To fully confirm the theory of the moiré patterns, we plot the inter-and intracycle spacings for ω = 0.01424 in Fig. 5 (a) and ω = 0.2 in Fig. 5  for ω = 0.01424 in Fig. 5 (d), whereas it is at k z = 0.97 for ω = 0.02 in Fig. 5 (i). For the sake of completeness, in the last row, we show the complete doubly differential momentum distribution within the SPA. We clearly observe how the principal (1, −1) and secondary From equations (26) and (28) where we have used that κ = (ω/F 0 ) k. This means that the position of the center of the moiré rings scale as the inverse of the Keldysh parameter γ −1 , which is observed in Fig. 6 (b) in the tunneling regime (γ < 1). In the multiphoton regime (γ > 1), we see in Fig. 6 (a), that the center of the moiré rings follows an approximate linear behavior with γ, i.e., κ (1,−1) zc 0.345γ, which is consistent with the asymptotic values for the center of the moiré rings in the multiphoton limit (γ 1), observed in Fig. 6 (b). The proportionality coefficients were calculated as the average slope of curves in Fig. 6 (a) between γ = 3 and 4. The approximate agreement between the asymptotic values for the center of the moiré fringes in the multiphoton limit and the coefficients of Eq.
(34) in the tunneling regime is very suspicious to say that it is pure coincidence and deserves more investigation.

IV. CONCLUSIONS
We have presented a study of interference effects observed in the direct ionization of atoms subject to multicycle laser pulses with wavelength in the range of the midinfrared. In the framework of the SPA we describe the full differential electron momentum distribution and identify side rings calculated within the SFA and TDSE [28] as the moiré fringes due to the interplay between the intra-and intercycle interferences of electron trajectories in photoelectron 3D momentum distribution. A whole family of moiré fringes of varying visibility was characterized. An analytical expression for the moiré patterns within a Fourier analysis is presented showing an excellent agreement with the numerical calculations. The principal (secondary) side rings are centered along the parallel momentum axis (k ⊥ = 0) with k z values where the spacing of the intracycle pattern is equal to (multiple of or one over a multiple of) the intercycle spacing. The position of the center of the side rings follows