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Electron rescattering in a bicircular laser field

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Abstract

Above-threshold ionization of rare-gas atoms by a bicircular field with its two components counterrotating is theoretically investigated by means of the improved strong-field approximation. Both direct and rescattered electrons are considered and the quantum orbits that lead into a specific final state are calculated and depicted. The angle-dependent spectrum reflects the discrete rotational symmetry of the bicircular field. The backward-scattering contributions are very similar to those generated by a linearly polarized field; several such contributions are rotated one versus the other by the symmetry angle of the discrete rotational symmetry. The forward-scattering contributions dramatically affect the velocity map at comparatively low momenta. The direct-electron spectrum observes reflection symmetry about several symmetry axes determined by the field symmetry. This is broken by rescattering.

© 2016 Optical Society of America

1. Introduction

Electron rescattering is at the core of strong-field physics [1]. Unlike high-order harmonic generation (HHG) [2], which is complicated by collective effects of various origins, in the form of above-threshold ionization (ATI) it is observable as a one-atom effect under conditions such that not even space-charge effects need be considered [3]. For linear polarization, electron rescattering has been observed with high precision and theoretically analyzed with various methods; for reviews, see [4–7]. It is now very well understood. For circular polarization, there is no rescattering in so much as a classical electron released in the laser-field environment with zero initial velocity will never return to the site of its release [1]. Hence, rescattering processes such as HHG were expected not to play any role; see, however, [8, 9]. For elliptical polarization, a classical electron will not exactly return either. However, it will get closer to the site of its release than for circular polarization. The quantitative significance of rescattering quickly decreases with increasing ellipticity. Quantum-mechanically, the electron is born in the continuum with a distribution of transverse momenta. There is always some suitable momentum such that the electron will exactly return [10]. Comparison between theory and experiment is difficult because of the low experimental count rates in the rescattering plateau [11, 12].

In this paper, we will consider rescattering contributions to ATI of rare-gas atoms by a bicircular field. A bicircular field is a superposition of two circularly polarized fields with different frequencies with an integer frequency ratio (usually ω vs. 2ω), which rotate in the same plane with the same or with opposite helicities [13–20]. Owing to the highly nonlinear character of the laser-atom interaction, the afore-mentioned absence of rescattering for one circularly polarized field does not extend to the superposition of two such fields. Opposite helicities lead to more interesting effects, so we will be concerned with this case. In [21] it was argued that such a field (with equal intensities) would allow for recollisions, especially in the presence of an additional magnetic field perpendicular on the plane of the circular polarizations [21,22]. Bicircular fields are of recent interest [23–32] because they generate circularly polarized and surprisingly bright harmonics. They open up the possibility of helicity-dependent spectroscopy, for example, the investigation of x-ray magnetic circular dichroism [31, 33] or time-resolved imaging of magnetic structures [34–36]. We calculate ATI in terms of the improved strong-field approximation (ISFA) [4]. The ISFA affords, in principle, analytical results and is, moreover, quite accurate. Most importantly, it allows unmatched physical insight into the details and the physical mechanism of ATI into a final state with given electron momentum p. This is based on the expansion of the ATI amplitude in terms of quantum orbits [4, 37]. The quantum orbits are very similar for HHG and for ATI [10]. Hence, better understanding of ATI also supports understanding of HHG. However, the analysis of ATI is easier than that of HHG, owing to the afore-mentioned absence of collective effects.

Here, we will present spectra and velocity maps of ATI, especially its rescattering contributions, based on the ISFA and its quantum-orbit expansion [4, 37–40]. The paper is oriented towards explicit results and their interpretation. The underlying formalism has been explained earlier, though not all of the details that are relevant for bicircular fields have been exposed yet. Still, we will refrain from a complete presentation of such details in the current paper.

In the next section, we briefly present the formalism underlying the results of the paper. In section 3, we display various results in the form of velocity maps and compare with recent data mostly for a frequency ratio of ω1/ω2 = 1/2 but including an example for 1/3. In section 4, we analyze our results in terms of quantum orbits and identify those that make the most significant contributions. At the end of the paper we present a brief summary.

2. Transition amplitudes for direct and for rescattering ionization

We adopt a single-active-electron description even though the presence of more than one electron may enter via the ground-state configuration [27–29]. The electronic transition matrix element from the initial bound state |ψi(t′)〉 to the final scattering state |ψp(t)〉 with asymptotic momentum p and energy Ep = p2/2 is

Mpi(t,t)=ψp(t)|U(t,t)|ψi(t),
where U(t,t′) denotes the time-evolution operator of the Hamiltonian H(t) = T + V + VL(t) with the kinetic energy operator T, the binding potential V(r), and the laser-electron interaction VL(t) = r·E(t). We employ length gauge and atomic units. The times t and t′ are after and before the passage of the laser pulse. In the process of evaluating the matrix element (1) we expand the time-evolution operator into its Born series in terms of the binding potential V. The result is [4, 5, 37]
Mpi(t,t)=MpiD(t,t)+MpiR(t,t),
MpiD(t,t)=ittdτψp(t)|UL(t,τ)VL(τ)|ψi(τ),
MpiR(t,t)=(i)2ttdττtdτψp(t)|UL(t,τ)×V(r)U(τ,τ)VL(τ)|ψi(τ).

The amplitude MpiD(t,t) describes the direct electrons in the strong-field approximation (SFA), which after the ionization at the time τ do not interact with the binding potential V again. The second term MpiR(t,t) incorporates at least one such interaction at the time τ′. From here on, in Eqs. (3) and (4) we shall approximate the state 〈ψp(t)|UL(t,τ) by the Volkov state 〈χp(τ)|. In the second term, we replace the exact time-evolution operator U(t,τ′) by the Volkov time-evolution operator UL(t,t′) = ∫ d3k|χk(t)〉〈χk(t′)|. This yields the ISFA: we restrict ourselves to once-rescattered electrons and ignore the higher-order terms, which describe multiple rescattering.

We are interested in ionization by a bicircular field, which is the superposition of two circularly polarized fields with angular frequencies and , which are integer multiples of the same fundamental frequency ω = 2π/T. The two fields rotate in the same plane in opposite directions. Hence we have [13–19]

E(t)=i(E1e^+eirωt+E2e^eisωt)/2+c.c.,
where e^±=(e^x±ie^y)/2, with êx and êy the real unit polarization vectors along the x and y axis. Note that e^2=e^+2=0 while ê+·ê = 1. The quantities Ej and Ij=Ej2(j=1,2) are the amplitudes and intensities of the two circularly polarized fields with the helicities h1 = 1 and h2 = −1. The corresponding ponderomotive potential is Up=0TA2(τ)dτ/(2T)=Up1+Up2=A12/4+A22/4, with E(t) = −dA(t)/dt, A1 = E1/() and A2 = E2/(). An illustration of the ω–2ω case (r = 1, s = 2) can be found in Fig. 1.

 figure: Fig. 1

Fig. 1 Electric field vector E(t) (black solid line) and the corresponding vectors A(t) (red dashed line) and its integral α(t) (cyan dot-dashed line) of the ω–2ω bicircular laser field (5). The various symbols and their colors refer to the instants of ionization (I) and rescattering (R) of certain quantum orbits that are identified in Fig. 6 by the solid lines and discussed in this context. The arrows indicate the temporal evolution of the field vectors.

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The rescattering amplitude (4) is a five-dimensional integral over the rescattering time τ′, the ionization time τ, and the drift momentum k between ionization and rescattering. The integral can be computed in different ways. We will refer to straightforward numerical execution of the integrals over τ and τ′ by “numerical integration” [41]. While sometimes less accurate, a saddle-point (stationary-phase) evaluation provides a maximum of physical insight [4]. To this end, the integrand of Eq. (4) is rewritten in the form Api exp(iSpi) with the action Spi(tr,t0,k)=trdt[p+A(t)]2/2t0trdt[k+A(t)]2/2+Ipt0, where Ip > 0 is the ionization potential. The stationarity conditions for the integral over the intermediate electron momentum k, the ionization time t0, and the rescattering time t are:

(trt0)k=t0trdtA(t),
[k+A(t0)]2/2=Ip,
[k+A(tr)]2/2=[p+A(tr)]2/2,
which, physically, are the condition (6) that the electron returns to the origin, and the energy-conservation conditions (7) and (8) at the times t0 and tr, respectively. The five-dimensional integration is then reduced to finding the complex solutions ks, t0s, and trs of the saddle-point equations (6)(8). The subscript s indicates that there are more than one (in general infinitely many) solutions. A subset s˜ of them is relevant for the construction of the integral: MpiR=s˜MpiR|s˜. For linear polarization of the laser field, the solutions were analyzed in detail in [4, 39, 40, 42, 43]. For elliptical polarization, this was carried out in [10]. In the present paper, we will derive and illustrate analogous solutions for the bicircular field (5) without, however, going into the details of the derivation.

Any complex solution (ks,t0s,trs) defines a quantum orbit [4, 5, 37–40, 42]. For given p, the real parts of each solution define a classical orbit of the type discussed in the so-called simple-man model

r(s)(t)={(tt0s)ks+α(t)α(t0s),(ttrs)p+α(t)α(trs),
if Re t0st ≤ Re trs and if t > Re trs, respectively, where α(t) = ∫ dtA(t) is chosen so that its time average over a fundamental period is zero.

3. Results

The direct SFA for the case of above-threshold detachment by a bicircular field was already analyzed in [44, 45]. For the field the spectrum is invariant with respect to a rotation by the angle 2πj/(r + s), j = 1,…,r + s. Therefore, for the ω–2ω field we observe a three-fold symmetry of the differential ionization rate. This is obvious in Fig. 2, where we present the results obtained using Eq. (3) (one-dimensional numerical integration over the ionization time and the bound state of the Ne atom modeled by a linear combination of four 2p Slater-type orbitals). The threefold symmetry also dominates the velocity map observed experimentally in [46]. The central circle in Fig. 2 is related to the minimum electron kinetic energy determined by the energy-conserving condition Epmin=nminωIpUp>0.

 figure: Fig. 2

Fig. 2 The differential ionization rate (in a.u.) of Ne atoms, presented in false colors in the electron momentum plane for ionization by a bicircular ω–2ω laser field with equal intensity of both components I1 = I2 = 2×1014 W/cm2 and the fundamental wavelength of 800 nm (Up1 = 0.4392 a.u., 20Up1=2.964a.u.). The results are obtained by numerical integration of the direct SFA matrix element. The false color scale is logarithmic for the right panel and linear and normalized to 1 for the left panel.

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If we neglect the ionization potential Ip in the saddle-point equation (7), we obtain that the electrons are predominantly emitted opposite to the direction of the vector potential at the ionization time, i.e. p = −A(t0). The shape of A(t), which can be seen in Fig. 1, implies that emission in the directions θ = 60°, 180°, and 300° is the dominant feature of the spectrum. This is apparent in Fig. 2 where we plot the ionization rate obtained from the SFA matrix element (3). The pronounced structures that are visible are due to interference of the two dominant orbits that are the analogs of the long and the short orbit in direct ionization by a linearly polarized field [47]. The two panels of Fig. 2 have a linear vs. a logarithmic color code, which illuminates different features of the ionization rate. For example, the left panel shows that the three approximate boundaries of the momentum distribution are slightly indented, in agreement with the shape of the vector potential A(t) in Fig. 1. Moreover, it emphasizes that, unlike the case of linear polarization, electron emission peaks at nonzero momentum. This is due to the fact that the vector potential stays substantially away from zero for all times.

Figure 3 shows an example of the rescattering differential ionization rate obtained by two-dimensional numerical integration of Eq. (4). It is obvious that the three-fold symmetry of the ionization spectrum generated by the bicircular ω–2ω field also holds for the rescattered electrons. The velocity map looks like a superposition of the three velocity maps individually generated by three linearly polarized laser fields that are rotated with respect to each other by the angle 120°. Namely, the rescattering ATI angle-dependent spectrum generated by a linearly polarized laser field has the shape of concentric circles with centers at ±(E0/ω)êx, where E0 is the electric-field-vector amplitude and êx its polarization direction [48]. More precisely, the just-mentioned circles actually have a dropletlike shape (see Fig. 3 in [48] and Fig. 3 in [43]). The ATI spectra for the linearly polarized field were explained in detail in [40,42,43,48,49] in terms of backward- and forward-scattered electrons. Backscattering is responsible for the high-energy electrons, which can reach the energy 10Uplin, p with the corresponding ponderomotive energy Uplin=E02/(4ω2). The forward-scattered electrons are responsible for the low-energy structure (LES) [50–52] and the off-axis LES [48] (see also [40, 42, 43, 49, 53]). Figure 3 suggests that the rescattering ATI spectrum generated by the ω–2ω bicircular field approximately consists of three such sets of almost concentric circles, which reach up to energies of 10Up1. On top of these, we observe three prominent blobs around the angles 50°, 170°, and 290°, from which spiral-shaped tails emerge that extend mostly in the counterclockwise direction (as the time evolution of the vector potential goes, Fig. 1). The tails appear finally to connect with the rescattering circles.

 figure: Fig. 3

Fig. 3 The logarithm of the differential ionization rate for the same parameters as in Fig. 2. The ISFA result (4) is obtained by two-dimensional numerical integration and using four 2p Slater-type orbitals to represent the ground state of the Ne atom.

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As noted, without or with the presence of rescattering the velocity maps reflect the discrete rotational symmetry of the field (5). In addition, the direct-electron angle-dependent spectrum of Fig. 2 obeys reflection symmetry about three axes at the angles of 60°, 180°, and 300° with respect to the positive px axis. The reflection symmetries are broken by rescattering, as is evident from Fig. 3. This is because the direct-electron spectrum only depends on the vector potential A(t) at one time while the rescattering spectrum depends on the ionization and the rescattering times. The presence or absence of the reflection symmetry allows one to assess whether or not rescattering plays a role in a given spectrum. It is interesting to note that the experimental angle-dependent spectrum in [46] slightly violates these symmetries, which attests to the presence of rescattering contributions. In a semiclassical simulation, the effect of the Coulomb field on the released electron also generates a violation of the reflection symmetry. Such Coulomb-propagation effects are at least partly contained in the quantum-mechanical first-order rescattering contribution, which we are discussing here.

Finally, in Fig. 4 we show the spectra for the same laser parameters as in Fig. 2, but for the ω–3ω field. The left panel presents the coherent sum of direct and rescattered electron contributions, while the right panel displays the velocity map only for the rescattered electrons. From the left panel we see that the direct electrons dominate the low-energy spectrum. The rescattering results are qualitatively similar to those of the ω–2ω case in Fig. 3, except that they exhibit four-fold instead of three-fold symmetry. In the ω–3ω case, both the horizontal and the vertical axes are reflection-symmetry axes. These symmetries are broken by the rescattered electrons. This has a small effect on the coherent sum, but dominates the appearance of the rescattering-only velocity map.

 figure: Fig. 4

Fig. 4 The logarithm of the differential ionization rate for the same parameters as in Fig. 2, but for the ω–3ω field. In the left panel the coherent sum of the SFA and ISFA results is shown, while in the right panel only the ISFA result (4) is presented. The scale is logarithmic and the results are normalized to 1 in each panel separately.

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4. Quantum-orbit analysis

In order better to understand the complex spectrum presented in Fig. 3, we will analyze the saddle-point solutions of Eqs. (6)(8) and the corresponding spectra, for a fixed angle θ of the emitted-electron direction with respect to the px axis. For the determination of the saddle-point solutions, we exploit the fact that the bicircular field (5) almost looks like the superposition of three half cycles of a linearly polarized field, each rotated versus the other by 120°, so that we can start from the known solutions for the linearly polarized field.

In past work we classified the backward-scattering solutions of the saddle-point equations for a linearly polarized field by a multi-index (α,β,m) [40] where, briefly, the index m = [Re(trt0)/T] (m = 0,1,2,…) characterizes the length of the travel time in multiples of the laser period (with [x] the largest integer ≤ x), the double-valued index β the number of pairs of solutions for each m (note that solutions to the saddle-point solutions come in pairs), and for each m and β the index α = ±1 discriminates the “long” and the “short” orbits (as it does in the Lewenstein model of HHG [54], where only the shortest pair of solutions is considered). In the present case, we can largely take over this classification, except that the index β will run from 1 to r + s rather than β = ±1 for linear polarization. That is, the range of β agrees with the number of lobes of a parametric plot of the field E(t). We shall not further dwell on this classification in the present paper, except for using it to identify certain orbits, which we shall discuss in detail.

We will first concentrate on the backward-scattering solutions, which contribute to the high-energy electrons. Figure 5 demonstrates the quantitative significance of the various contributing orbits (denoted as just discussed). We observe that throughout the plateau the orbit (1,1,0) is dominant. The two orbits (±1,3,0) are the next important ones. Their contributions are smaller by a factor of 3 or 4, except near their cutoff above 7Up where their magnitude is the same as that of (1,1,0). However, due to interference an orbit may make a larger contribution than its magnitude suggests (for example, the pronounced dip in the spectrum around 8Up1 is likely due to the destructive interference of the afore-mentioned orbits). All the other orbits contribute only insignificantly. We should note that after the respective cutoff one of the two orbits of each pair has to be dropped from consideration. The respective orbit is characterized by an exponentially increasing partial contribution. In the figures, it is represented by a dashed line.

 figure: Fig. 5

Fig. 5 Comparison of the differential ionization rates as function of the electron energy (in units of the ponderomotive energy of the ω field component) for the same parameters as in Fig. 3 and emission into the angle θ = 50°. The results obtained by numerical integration are presented by the black solid line with the circles, while the coherent sum of the contributions of 14 saddle-point solutions is represented by the solid red line with squares. The contributions of the divergent solutions (dashed lines) are neglected after the corresponding cutoff values. The partial contributions of each of the 14 solutions (α,β,m) are presented separately, as denoted. The spectrum that includes both the direct and the rescattered electron, obtained by numerical integration, is represented by small diamonds and denoted by D + R.

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In Fig. 6 we exhibit the real parts of selected quantum orbits, especially the just discussed orbits (1,0) and (3,0). Each such curve can be envisioned as an optional path for an electron while it makes the transition from the initial state i to the final state p in the same sense as the optional paths of an electron passing a double slit setup [55]. Let us consider these orbits in more detail.

 figure: Fig. 6

Fig. 6 Electron trajectories for some of the quantum-orbit solutions presented in Fig. 5. The electron energies for the presented pairs α = ±1 of solutions are 10Up1 for solutions (β,m) = (1,0) and (2,0), 7Up1 for solutions (3, 0), and 9Up1 for solutions (1, 1). The ionization and rescattering times of the depicted orbits can be identified in Fig. 1.

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For the ω–2ω case, the electric field (5) consists of three segments or lobes per cycle, cf. Fig. 1. It is defined such that right after t = 0 the field E(t) points in the positive x direction. We denote the three lobes for 0 < t < T/3, T/3 < t < 2T/3, and 2T/3 < t < T, which extend, respectively, into the lower right, upwards, and the lower left, by lobe 1, lobe 2, and lobe 3. Note that the electron charge is negative. Hence, the force on the electron is opposite to the field. The tunnel exit is in the direction of −E(t0), if the electron exits into the continuum at the time t0, which is near an extremum of the field. We can then from Fig. 6 roughly read off the ionization times of the various orbits by locating the positions where they start (this is analogous to reading the attoclock [56]). The orbits (β,m) = (1,0) and (3,0) start opposite lobe 1 and lobe 2, respectively, each at a distance of about 5 a.u. off the origin. Since for a bicircular field (in contrast to a linearly polarized field) the electron usually starts its orbit with a substantially nonzero momentum, its initial trajectory is not exactly in the direction of −E(t0).

The (1,0) orbit in Fig. 6 looks close to an orbit typical of linear polarization as it is essentially restricted to one spatial direction, here the y direction. Figure 1 shows that its electron is liberated in lobe 1 right after the field has become maximal and rescatters in lobe 2 shortly before a zero of the field. This means that in the y direction the electron is first accelerated to positive momenta, then decelerated and bent around such that it recollides with substantial momentum. All this is rather parallel to orbits in a linearly polarized field. The force exerted in the negative x direction mostly by the field during lobe 1 is largely compensated by the initial momentum in this direction. This is parallel to ATI by an elliptically polarized field [10].

The orbit (3,0) is more complicated. Its length in time is almost four thirds of a period; so it experiences all segments of the field. Figure 1 shows that it starts almost at the maximum of the field in segment 2; so it exits the tunnel almost exactly on the negative y axis with some negative initial momentum in the x direction. Figure 6 then illustrates the action of the electric field of half the segment 2, and subsequently the full segments 3, 1, and 2, before it rescatters shortly after a zero of the field in the beginning of segment 3. The resulting orbit is approximately triangular. Its three rather straight sections are due to the action of the field in the half segment 2, and thereafter segments 3, 1, and 2. In general, after rescattering and, for a long enough orbit, already between ionization and rescattering, the electron orbits will be given by α(t) on top of a fixed drift momentum. We can observe the corresponding triangles in Fig. 6.

All orbits presented in Fig. 6 rescatter into the same direction, at an angle of 50°, and the associated drift momenta are −A(tr). The return times tr are different for different orbits. However, the red dashed curve in Fig. 1 shows that the corresponding vector potentials are quite close. The situation is then rather similar to linear polarization where the velocity map of the rescattered electrons is shaped by circles about the vector potentials −A(tr) with radii that are specified by the energy of the respective recolliding electron. The bicircular field (5) generates three such groups of almost concentric circles as illustrated in Figs. 3 and 7.

The upper left panel of Fig. 7 displays the velocity map generated by the (±1,1,0) backward-scattering orbits. Their contributions are strongest on those parts of the rescattering circles that extend upward from the px axis between 0 ≤ px ≤ 2 a.u. and the corresponding parts rotated by ±120°. Comparison with Fig. 3 then suggests that the (±1,1,0) backward-scattering orbits are responsible for the spiral structures in the complete velocity map.

The quantum-orbit formalism is simplest for high energies. Only backward-scattering orbits contribute, and their imaginary parts are small, i.e., they are very close to classical trajectories. In addition, for low energies forward-scattering solutions of the saddle-point equations must be taken into consideration; cf. [40]. Namely, the saddle-point equation (8) allows for rescattering into any direction. So far, we have investigated those solutions that are connected with the backward-scattering case, that is, the velocities k + A(tr) just before and p + A(tr+) just after rescattering point in approximately opposite directions. Only these solutions contribute to the high-energy spectrum. For energies below about 4Up1, however, we also need to consider forward-scattering solutions where these velocities are approximately in the same direction [40]. For linear polarization, we classified these solutions by the double index (ν,μ) where again the index μ is related to the travel time, −μ < Re t0/T < −μ + 1, μ = 0,1,2,…, while ν distinguishes between the long and the short orbit. This notation can be taken over to the bicircular case. More detailed explanations will be given elsewhere. The velocity maps of the most important contributing forward-scattering orbits are exhibited in Fig. 7. A comparison with the complete velocity map of Fig. 3 allows us to retrieve their contributions: the thick blobs in Fig. 3 near the centers of the three rescattering rings are generated by the forward-scattering orbits (ν,μ) = (1,0) (upper right panel of Fig. 7). The (±1,1) orbits (lower left panel of Fig. 7) can also be identified in Fig. 3. The upshot is that we are able to associate certain forward-scattering quantum orbits with specific features in the velocity map. The coherent sum of these two forward-scattering-orbit contributions and the contribution of backward-scattering orbits (α,β,m) = (±1,1,0), which is presented in the lower right panel of Fig. 7, shows that with only five orbits we are able to reproduce the velocity map of Fig. 3, which was obtained by numerical integration.

 figure: Fig. 7

Fig. 7 Same as in Fig. 3 but obtained using the saddle-point method with the solutions (α,β,m) = (±1,1,0) (upper left panel), and the forward-scattering solutions (ν,μ) = (1,0) (upper right panel) and (ν,μ) = (±1,1) (lower left panel). In the lower right panel the coherent sum of all above-mentioned contributions is presented.

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5. Summary

Using the improved strong-field approximation and the quantum-orbit expansion, we investigated above-threshold ionization generated by a bicircular counterrotating field, both for the direct and the rescattered electrons. Bicircular fields have become of great recent interest, since they can be used to generate circularly polarized high-order harmonics in the extreme UV. Since a bicircular field unfolds in a plane, the electron trajectories are two-dimensional and the rescattering structures (backward and forward) are well separated. Moreover, compared with a linearly polarized field many more control parameters are available (bicircular-field-component intensities, frequencies, and ellipticities). The combination of attoclock features with the discrete rotational symmetry of the bicircular field should open up unprecedented prospects for attosecond spectroscopy of atoms and molecules [57] and laser-induced electron diffraction [58, 59], especially of planar molecules.

Acknowledgments

We gratefully acknowledge support by the Federal Ministry of Education and Science, Bosnia and Herzegovina.

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16. D. B. Milošević and W. Sandner, “Extreme-ultraviolet harmonic generation near 13 nm with a two-color elliptically polarized laser field,” Opt. Lett. 25, 1532–1534 (2000). [CrossRef]  

17. D. B. Milošević, W. Becker, R. Kopold, and W. Sandner, “High-harmonic generation by a bichromatic bicircular laser field,” Laser Phys. 11, 165–168 (2001).

18. D. B. Milošević and W. Becker, “Attosecond pulse trains with unusual nonlinear polarization,” Phys. Rev. A. 62011403 (2000). [CrossRef]  

19. D. B. Milošević and W. Becker, “Attosecond pulse generation by bicircular fields: from pulse trains to a single pulse,” J. Mod. Opt. 52, 233–241 (2005). [CrossRef]  

20. F. Ceccherini, D. Bauer, and F. Cornolti, “Harmonic generation by atoms in circularly polarized two-color laser fields with coplanar polarizations and commensurate frequencies,” Phys. Rev. A. 68, 053402 (2003). [CrossRef]  

21. T. Zuo and A. D. Bandrauk, “High-order harmonic generation in intense laser and magnetic fields,” J. Nonlinear Opt. Phys. Mat. 4, 533–546 (1995). [CrossRef]  

22. A. D. Bandrauk and H. Z. Lu, “Controlling harmonic generation in molecules with intense laser and static magnetic fields: orientation effects,” Phys. Rev. A. 68, 043408 (2003). [CrossRef]  

23. A. Fleischer, O. Kfir, T. Diskin, P. Sidorenko, and O. Cohen, “Spin angular momentum and tunable polarization in high-harmonic generation,” Nat. Photonics 8, 543–549 (2014). [CrossRef]  

24. E. Pisanty and M. Ivanov, “High-harmonic generation: taking control of polarization,” Nat. Photonics 8, 501–503 (2014). [CrossRef]  

25. E. Pisanty, S. Sukiasyan, and M. Ivanov, “Spin conservation in high-order-harmonic generation using bicircular fields,” Phys. Rev. A. 90, 043829 (2014). [CrossRef]  

26. O. Kfir, P. Grychtol, E. Turgut, R. Knut, D. Zusin, D. Popmintchev, T. Popmintchev, H. Nembach, J. M. Shaw, A. Fleischer, H. Kapteyn, M. Murnane, and O. Cohen, “Generation of bright phase-matched circularly-polarized extreme ultraviolet high harmonics,” Nat. Photonics 9, 99–105 (2015). [CrossRef]  

27. D. B. Milošević, “Generation of elliptically polarized attosecond pulse trains,” Opt. Lett. 40, 2381–2384 (2015). [CrossRef]  

28. D. B. Milošević, “High-order harmonic generation by a bichromatic elliptically polarized field: conservation of angular momentum,” J. Phys. B. 48, 171001 (2015). [CrossRef]  

29. D. B. Milošević, “Circularly polarized high harmonics generated by a bicircular field from inert atomic gases in the p state: A tool for exploring chirality-sensitive processes,” Phys. Rev. A. 92, 043827 (2015). [CrossRef]  

30. L. Medišauskas, J. Wragg, H. van der Hart, and M. Yu. Ivanov, “Generating isolated elliptically polarized attosecond pulses using bichromatic counterrotating circularly polarized laser fields,” Phys. Rev. Lett. 115, 153001 (2015). [CrossRef]  

31. T. Fan, P. Grychtol, R. Knut, C. Hernández-García, D. D. Hickstein, D. Zusin, C. Gentry, F. J. Dollar, C. A. Mancuso, C. Hogle, O. Kfir, D. Legut, K. Carva, J. L. Ellis, K. Dorney, C. Chen, O. Shpyrko, E. E. Fullerton, O. Cohen, P. M. Oppeneer, D. B. Milošević, A. Becker, A. A. Jaroń-Becker, T. Popmintchev, M. M. Murnane, and H. C. Kapteyn, “Bright circularly polarized soft X-ray high harmonics for X-ray magnetic circular dichroism,” Proc. Natl. Acad. Sci. USA 112, 14206–14211 (2015). [CrossRef]   [PubMed]  

32. S. Odžak and D. B. Milošević, “Bicircular-laser-field-assisted electron-ion radiative recombination,” Phys. Rev. A. 92, 053416 (2015). [CrossRef]  

33. G. Lambert, B. Vodungbo, J. Gautier, B. Mahieu, V. Malka, S. Sebban, P. Zeitoun, J. Luning, J. Perron, A. Andreev, S. Stremoukhov, F. Ardana-Lamas, A. Dax, C. P. Hauri, A. Sardinha, and M. Fajardo, “Towards enabling femtosecond helicity-dependent spectroscopy with high-harmonic sources,” Nat. Commun. 6, 6167 (2015). [CrossRef]   [PubMed]  

34. S. Eisebitt, J. Lüning, W. F. Schlotter, M. Lörgen, O. Hellwig, W. Eberhardt, and J. Stöhr, “Lensless imaging of magnetic nanostructures by x-ray spectro-holography,” Nature (London) 432, 885–888 (2004). [CrossRef]  

35. C. von KorffSchmising, B. Pfau, M. Schneider, C. M. Günther, M. Giovanella, J. Perron, B. Vodungho, L. Müller, F. Capotondi, E. Pedersoli, N. Mahne, J. Lüning, and S. Eisebitt, “Imaging ultrafast demagnetization dynamics after a spatially localized optical excitation,” Phys. Rev. Lett. 112, 217203 (2014). [CrossRef]  

36. F. Willems, C. T. L. Smeenk, N. Zhavoronkov, O. Kornilov, I. Radu, M. Schmidhauer, M. Hanke, C. von KorffSchmising, M. J. J. Vrakking, and S. Eisebitt, “Probing ultrafast spin dynamics with high-harmonic magnetic circular dichroism spectroscopy,” Phys. Rev. A. 92, 220405 (2015).

37. R. Kopold, W. Becker, and M. Kleber, “Quantum path analysis of high-order above-threshold ionization,” Opt. Commun. 179, 39–50 (2000). [CrossRef]  

38. D. B. Milošević and W. Becker, “Role of long quantum orbits in high-order harmonic generation,” Phys. Rev. A. 66, 063417 (2002). [CrossRef]  

39. D. B. Milošević, E. Hasović, M. Busuladžić, A. Gazibegović-Busuladžić, and W. Becker, “Intensity-dependent enhancements in high-order above-threshold ionization,” Phys. Rev. A. 76, 053410 (2007). [CrossRef]  

40. D. B. Milošević, “Forward- and backward-scattering quantum orbits in above-threshold ionization,” Phys. Rev. A. 90, 063414 (2014). [CrossRef]  

41. D. B. Milošević, “Reexamination of the improved strong-field approximation: Low-energy structures in the above-threshold-ionization spectra for short-range potentials,” Phys. Rev. A. 88, 023417 (2013). [CrossRef]  

42. W. Becker, S. P. Goreslavski, D. B. Milošević, and G. G. Paulus, “Low-energy rescattering in laser-induced ionization,” J. Phys. B. 47, 204022 (2014). [CrossRef]  

43. W. Becker and D. B. Milošević, “Above-threshold ionization for very low electron energy,” J. Phys. B. 48, 151001 (2015). [CrossRef]  

44. A. Kramo, E. Hasović, D. B. Milošević, and W. Becker, “Above-threshold detachment by a two-color bicircular laser field,” Laser Phys. Lett. 4, 279–286 (2007). [CrossRef]  

45. E. Hasović, A. Kramo, and D. B. Milošević, “Energy- and angle-resolved photoelectron spectra of above-threshold ionization and detachment,” Eur. Phys. J. Spec. Top. 160, 205–216 (2008). [CrossRef]  

46. C. A. Mancuso, D. D. Hickstein, P. Grychtol, R. Knut, O. Kfir, X. M. Tong, F. Dollar, D. Zusin, M. Gopalakrishnan, C. Gentry, E. Turgut, J. L. Ellis, M.-C. Chen, A. Fleischer, O. Cohen, H. C. Kapteyn, and M. M. Murnane, “Strong-field ionization with two-color circularly polarized laser fields,” Phys. Rev. A. 91, 031402 (2015). [CrossRef]  

47. P. A. Korneev, S. V. Popruzhenko, S. P. Goreslavski, W. Becker, G. G. Paulus, B. Fetić, and D. B. Milošević, “Interference structure of above-threshold ionization versus above-threshold detachment,” New. J. Phys. 14, 055019 (2012). [CrossRef]  

48. M. Möller, F. Meyer, A. M. Sayler, G. G. Paulus, M. F. Kling, B. E. Schmidt, W. Becker, and D. B. Milošević, “Off-axis low-energy structures in above-threshold ionization,” Phys. Rev. A. 90, 023412 (2014). [CrossRef]  

49. W. Becker and D. B. Milošević, “Quantum-orbit theory of low-energy above-threshold ionization on and off axis,” Chin. Opt. Lett. 13, 070006 (2015). [CrossRef]  

50. C. I. Blaga, F. Catoire, P. Colosimo, G. G. Paulus, H. G. Muller, P. Agostini, and L. F. DiMauro, “Strong-field photoionization revisited,” Nat. Phys. 5, 335–338 (2009). [CrossRef]  

51. W. Quan, Z. Lin, M. Wu, H. Kang, H. Liu, X. Liu, J. Chen, J. Liu, X. T. He, S. G. Chen, H. Xiong, L. Guo, H. Xu, Y. Fu, Y. Cheng, and Z. Z. Xu, “Classical aspects in above-threshold ionization with a midinfrared strong laser field,” Phys. Rev. Lett. 103, 093001 (2009). [CrossRef]   [PubMed]  

52. C. Y. Wu, Y. D. Yang, Y. Q. Liu, Q. H. Gong, M. Wu, X. Liu, X. L. Hao, W. D. Li, X. T. He, and J. Chen, “Characteristic spectrum of very low-energy photoelectron from above-threshold ionization in the tunneling regime,” Phys. Rev. Lett. 109, 043001 (2012). [CrossRef]   [PubMed]  

53. D. B. Milošević, “Low-frequency approximation for above-threshold ionization by a laser pulse: Low-energy forward rescattering,” Phys. Rev. A 90, 063423 (2014). [CrossRef]  

54. M. Lewenstein, Ph. Balcou, M. Yu. Ivanov, A. L’Huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A 49, 2117–2132 (1994). [CrossRef]   [PubMed]  

55. F. Lindner, M. G. Schätzel, H. Walther, A. Baltuška, E. Goulielmakis, F. Krausz, D. B. Milošević, D. Bauer, W. Becker, and G. G. Paulus, “Attosecond double-slit experiment,” Phys. Rev. Lett. 95, 040401 (2005). [CrossRef]   [PubMed]  

56. A. N. Pfeiffer, C. Cirelli, M. Smolarski, R. Dörner, and U. Keller, “Timing the release in sequential double ionization,” Nat. Phys. 7, 428–433 (2011). [CrossRef]  

57. F. Krausz and M. Ivanov, “Attosecond physics,” Rev. Mod. Phys. 81, 163–234 (2009). [CrossRef]  

58. M. Meckel, D. Comtois, D. Zeidler, A. Staudte, D. Pavičić, H. C. Bandulet, H. Pépin, J. C. Kieffer, R. Dörner, D. M. Villeneuve, and P. B. Corkum, “Laser-induced electron tunneling and diffraction,” Science 320, 1478–1482 (2008). [CrossRef]   [PubMed]  

59. C. I. Blaga, J. Xu, A. D. DiChiara, E. Sistrunk, K. Zhang, P. Agostini, T. A. Miller, L. F. DiMauro, and C. D. Lin, “Imaging ultrafast molecular dynamics with laser-induced electron diffraction,” Nature (London) 483, 194–197 (2012). [CrossRef]  

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  39. D. B. Milošević, E. Hasović, M. Busuladžić, A. Gazibegović-Busuladžić, and W. Becker, “Intensity-dependent enhancements in high-order above-threshold ionization,” Phys. Rev. A. 76, 053410 (2007).
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  41. D. B. Milošević, “Reexamination of the improved strong-field approximation: Low-energy structures in the above-threshold-ionization spectra for short-range potentials,” Phys. Rev. A. 88, 023417 (2013).
    [Crossref]
  42. W. Becker, S. P. Goreslavski, D. B. Milošević, and G. G. Paulus, “Low-energy rescattering in laser-induced ionization,” J. Phys. B. 47, 204022 (2014).
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  43. W. Becker and D. B. Milošević, “Above-threshold ionization for very low electron energy,” J. Phys. B. 48, 151001 (2015).
    [Crossref]
  44. A. Kramo, E. Hasović, D. B. Milošević, and W. Becker, “Above-threshold detachment by a two-color bicircular laser field,” Laser Phys. Lett. 4, 279–286 (2007).
    [Crossref]
  45. E. Hasović, A. Kramo, and D. B. Milošević, “Energy- and angle-resolved photoelectron spectra of above-threshold ionization and detachment,” Eur. Phys. J. Spec. Top. 160, 205–216 (2008).
    [Crossref]
  46. C. A. Mancuso, D. D. Hickstein, P. Grychtol, R. Knut, O. Kfir, X. M. Tong, F. Dollar, D. Zusin, M. Gopalakrishnan, C. Gentry, E. Turgut, J. L. Ellis, M.-C. Chen, A. Fleischer, O. Cohen, H. C. Kapteyn, and M. M. Murnane, “Strong-field ionization with two-color circularly polarized laser fields,” Phys. Rev. A. 91, 031402 (2015).
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    [Crossref]
  48. M. Möller, F. Meyer, A. M. Sayler, G. G. Paulus, M. F. Kling, B. E. Schmidt, W. Becker, and D. B. Milošević, “Off-axis low-energy structures in above-threshold ionization,” Phys. Rev. A. 90, 023412 (2014).
    [Crossref]
  49. W. Becker and D. B. Milošević, “Quantum-orbit theory of low-energy above-threshold ionization on and off axis,” Chin. Opt. Lett. 13, 070006 (2015).
    [Crossref]
  50. C. I. Blaga, F. Catoire, P. Colosimo, G. G. Paulus, H. G. Muller, P. Agostini, and L. F. DiMauro, “Strong-field photoionization revisited,” Nat. Phys. 5, 335–338 (2009).
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  51. W. Quan, Z. Lin, M. Wu, H. Kang, H. Liu, X. Liu, J. Chen, J. Liu, X. T. He, S. G. Chen, H. Xiong, L. Guo, H. Xu, Y. Fu, Y. Cheng, and Z. Z. Xu, “Classical aspects in above-threshold ionization with a midinfrared strong laser field,” Phys. Rev. Lett. 103, 093001 (2009).
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    [Crossref] [PubMed]
  53. D. B. Milošević, “Low-frequency approximation for above-threshold ionization by a laser pulse: Low-energy forward rescattering,” Phys. Rev. A 90, 063423 (2014).
    [Crossref]
  54. M. Lewenstein, Ph. Balcou, M. Yu. Ivanov, A. L’Huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A 49, 2117–2132 (1994).
    [Crossref] [PubMed]
  55. F. Lindner, M. G. Schätzel, H. Walther, A. Baltuška, E. Goulielmakis, F. Krausz, D. B. Milošević, D. Bauer, W. Becker, and G. G. Paulus, “Attosecond double-slit experiment,” Phys. Rev. Lett. 95, 040401 (2005).
    [Crossref] [PubMed]
  56. A. N. Pfeiffer, C. Cirelli, M. Smolarski, R. Dörner, and U. Keller, “Timing the release in sequential double ionization,” Nat. Phys. 7, 428–433 (2011).
    [Crossref]
  57. F. Krausz and M. Ivanov, “Attosecond physics,” Rev. Mod. Phys. 81, 163–234 (2009).
    [Crossref]
  58. M. Meckel, D. Comtois, D. Zeidler, A. Staudte, D. Pavičić, H. C. Bandulet, H. Pépin, J. C. Kieffer, R. Dörner, D. M. Villeneuve, and P. B. Corkum, “Laser-induced electron tunneling and diffraction,” Science 320, 1478–1482 (2008).
    [Crossref] [PubMed]
  59. C. I. Blaga, J. Xu, A. D. DiChiara, E. Sistrunk, K. Zhang, P. Agostini, T. A. Miller, L. F. DiMauro, and C. D. Lin, “Imaging ultrafast molecular dynamics with laser-induced electron diffraction,” Nature (London) 483, 194–197 (2012).
    [Crossref]

2015 (12)

O. Kfir, P. Grychtol, E. Turgut, R. Knut, D. Zusin, D. Popmintchev, T. Popmintchev, H. Nembach, J. M. Shaw, A. Fleischer, H. Kapteyn, M. Murnane, and O. Cohen, “Generation of bright phase-matched circularly-polarized extreme ultraviolet high harmonics,” Nat. Photonics 9, 99–105 (2015).
[Crossref]

D. B. Milošević, “Generation of elliptically polarized attosecond pulse trains,” Opt. Lett. 40, 2381–2384 (2015).
[Crossref]

D. B. Milošević, “High-order harmonic generation by a bichromatic elliptically polarized field: conservation of angular momentum,” J. Phys. B. 48, 171001 (2015).
[Crossref]

D. B. Milošević, “Circularly polarized high harmonics generated by a bicircular field from inert atomic gases in the p state: A tool for exploring chirality-sensitive processes,” Phys. Rev. A. 92, 043827 (2015).
[Crossref]

L. Medišauskas, J. Wragg, H. van der Hart, and M. Yu. Ivanov, “Generating isolated elliptically polarized attosecond pulses using bichromatic counterrotating circularly polarized laser fields,” Phys. Rev. Lett. 115, 153001 (2015).
[Crossref]

T. Fan, P. Grychtol, R. Knut, C. Hernández-García, D. D. Hickstein, D. Zusin, C. Gentry, F. J. Dollar, C. A. Mancuso, C. Hogle, O. Kfir, D. Legut, K. Carva, J. L. Ellis, K. Dorney, C. Chen, O. Shpyrko, E. E. Fullerton, O. Cohen, P. M. Oppeneer, D. B. Milošević, A. Becker, A. A. Jaroń-Becker, T. Popmintchev, M. M. Murnane, and H. C. Kapteyn, “Bright circularly polarized soft X-ray high harmonics for X-ray magnetic circular dichroism,” Proc. Natl. Acad. Sci. USA 112, 14206–14211 (2015).
[Crossref] [PubMed]

S. Odžak and D. B. Milošević, “Bicircular-laser-field-assisted electron-ion radiative recombination,” Phys. Rev. A. 92, 053416 (2015).
[Crossref]

G. Lambert, B. Vodungbo, J. Gautier, B. Mahieu, V. Malka, S. Sebban, P. Zeitoun, J. Luning, J. Perron, A. Andreev, S. Stremoukhov, F. Ardana-Lamas, A. Dax, C. P. Hauri, A. Sardinha, and M. Fajardo, “Towards enabling femtosecond helicity-dependent spectroscopy with high-harmonic sources,” Nat. Commun. 6, 6167 (2015).
[Crossref] [PubMed]

F. Willems, C. T. L. Smeenk, N. Zhavoronkov, O. Kornilov, I. Radu, M. Schmidhauer, M. Hanke, C. von KorffSchmising, M. J. J. Vrakking, and S. Eisebitt, “Probing ultrafast spin dynamics with high-harmonic magnetic circular dichroism spectroscopy,” Phys. Rev. A. 92, 220405 (2015).

W. Becker and D. B. Milošević, “Above-threshold ionization for very low electron energy,” J. Phys. B. 48, 151001 (2015).
[Crossref]

C. A. Mancuso, D. D. Hickstein, P. Grychtol, R. Knut, O. Kfir, X. M. Tong, F. Dollar, D. Zusin, M. Gopalakrishnan, C. Gentry, E. Turgut, J. L. Ellis, M.-C. Chen, A. Fleischer, O. Cohen, H. C. Kapteyn, and M. M. Murnane, “Strong-field ionization with two-color circularly polarized laser fields,” Phys. Rev. A. 91, 031402 (2015).
[Crossref]

W. Becker and D. B. Milošević, “Quantum-orbit theory of low-energy above-threshold ionization on and off axis,” Chin. Opt. Lett. 13, 070006 (2015).
[Crossref]

2014 (8)

D. B. Milošević, “Low-frequency approximation for above-threshold ionization by a laser pulse: Low-energy forward rescattering,” Phys. Rev. A 90, 063423 (2014).
[Crossref]

M. Möller, F. Meyer, A. M. Sayler, G. G. Paulus, M. F. Kling, B. E. Schmidt, W. Becker, and D. B. Milošević, “Off-axis low-energy structures in above-threshold ionization,” Phys. Rev. A. 90, 023412 (2014).
[Crossref]

W. Becker, S. P. Goreslavski, D. B. Milošević, and G. G. Paulus, “Low-energy rescattering in laser-induced ionization,” J. Phys. B. 47, 204022 (2014).
[Crossref]

C. von KorffSchmising, B. Pfau, M. Schneider, C. M. Günther, M. Giovanella, J. Perron, B. Vodungho, L. Müller, F. Capotondi, E. Pedersoli, N. Mahne, J. Lüning, and S. Eisebitt, “Imaging ultrafast demagnetization dynamics after a spatially localized optical excitation,” Phys. Rev. Lett. 112, 217203 (2014).
[Crossref]

D. B. Milošević, “Forward- and backward-scattering quantum orbits in above-threshold ionization,” Phys. Rev. A. 90, 063414 (2014).
[Crossref]

A. Fleischer, O. Kfir, T. Diskin, P. Sidorenko, and O. Cohen, “Spin angular momentum and tunable polarization in high-harmonic generation,” Nat. Photonics 8, 543–549 (2014).
[Crossref]

E. Pisanty and M. Ivanov, “High-harmonic generation: taking control of polarization,” Nat. Photonics 8, 501–503 (2014).
[Crossref]

E. Pisanty, S. Sukiasyan, and M. Ivanov, “Spin conservation in high-order-harmonic generation using bicircular fields,” Phys. Rev. A. 90, 043829 (2014).
[Crossref]

2013 (2)

X. Y. Lai, C. L. Wang, Y. J. Chen, Z. L. Hu, W. Quan, X. J. Liu, J. Chen, Y. Cheng, Z. Z. Xu, and W. Becker, “Elliptical polarization favors long quantum orbits in high-order above-threshold ionization of noble gases,” Phys. Rev. Lett. 110, 043002 (2013).
[Crossref] [PubMed]

D. B. Milošević, “Reexamination of the improved strong-field approximation: Low-energy structures in the above-threshold-ionization spectra for short-range potentials,” Phys. Rev. A. 88, 023417 (2013).
[Crossref]

2012 (4)

P. A. Korneev, S. V. Popruzhenko, S. P. Goreslavski, W. Becker, G. G. Paulus, B. Fetić, and D. B. Milošević, “Interference structure of above-threshold ionization versus above-threshold detachment,” New. J. Phys. 14, 055019 (2012).
[Crossref]

C. Y. Wu, Y. D. Yang, Y. Q. Liu, Q. H. Gong, M. Wu, X. Liu, X. L. Hao, W. D. Li, X. T. He, and J. Chen, “Characteristic spectrum of very low-energy photoelectron from above-threshold ionization in the tunneling regime,” Phys. Rev. Lett. 109, 043001 (2012).
[Crossref] [PubMed]

C. I. Blaga, J. Xu, A. D. DiChiara, E. Sistrunk, K. Zhang, P. Agostini, T. A. Miller, L. F. DiMauro, and C. D. Lin, “Imaging ultrafast molecular dynamics with laser-induced electron diffraction,” Nature (London) 483, 194–197 (2012).
[Crossref]

P. Agostini and L. F. DiMauro, “Atomic and molecular ionization dynamics in strong laser fields: from optical to x-rays,” Adv. At. Mol. Opt. Phys. 61, 117–158 (2012).
[Crossref]

2011 (1)

A. N. Pfeiffer, C. Cirelli, M. Smolarski, R. Dörner, and U. Keller, “Timing the release in sequential double ionization,” Nat. Phys. 7, 428–433 (2011).
[Crossref]

2010 (3)

F. Mauger, C. Chandre, and T. Uzer, “Recollisions and correlated double ionization with circularly polarized light,” Phys. Rev. Lett. 105, 083002 (2010).
[Crossref] [PubMed]

X. Wang and J. H. Eberly, “Elliptical polarization and probability of double ionization,” Phys. Rev. Lett. 105, 083001 (2010).
[Crossref] [PubMed]

C. D. Lin, A.-T. Le, Z. Chen, T. Morishita, and R. Lucchese, “Strong-field rescattering physics — self-imaging of a molecule by its own electrons,” J. Phys. B. 43, 122001 (2010).
[Crossref]

2009 (3)

F. Krausz and M. Ivanov, “Attosecond physics,” Rev. Mod. Phys. 81, 163–234 (2009).
[Crossref]

C. I. Blaga, F. Catoire, P. Colosimo, G. G. Paulus, H. G. Muller, P. Agostini, and L. F. DiMauro, “Strong-field photoionization revisited,” Nat. Phys. 5, 335–338 (2009).
[Crossref]

W. Quan, Z. Lin, M. Wu, H. Kang, H. Liu, X. Liu, J. Chen, J. Liu, X. T. He, S. G. Chen, H. Xiong, L. Guo, H. Xu, Y. Fu, Y. Cheng, and Z. Z. Xu, “Classical aspects in above-threshold ionization with a midinfrared strong laser field,” Phys. Rev. Lett. 103, 093001 (2009).
[Crossref] [PubMed]

2008 (2)

E. Hasović, A. Kramo, and D. B. Milošević, “Energy- and angle-resolved photoelectron spectra of above-threshold ionization and detachment,” Eur. Phys. J. Spec. Top. 160, 205–216 (2008).
[Crossref]

M. Meckel, D. Comtois, D. Zeidler, A. Staudte, D. Pavičić, H. C. Bandulet, H. Pépin, J. C. Kieffer, R. Dörner, D. M. Villeneuve, and P. B. Corkum, “Laser-induced electron tunneling and diffraction,” Science 320, 1478–1482 (2008).
[Crossref] [PubMed]

2007 (2)

D. B. Milošević, E. Hasović, M. Busuladžić, A. Gazibegović-Busuladžić, and W. Becker, “Intensity-dependent enhancements in high-order above-threshold ionization,” Phys. Rev. A. 76, 053410 (2007).
[Crossref]

A. Kramo, E. Hasović, D. B. Milošević, and W. Becker, “Above-threshold detachment by a two-color bicircular laser field,” Laser Phys. Lett. 4, 279–286 (2007).
[Crossref]

2006 (1)

D. B. Milošević, G. G. Paulus, D. Bauer, and W. Becker, “Above-threshold ionization by few-cycle pulses,” J. Phys. B. 39, R203–R262 (2006).
[Crossref]

2005 (2)

D. B. Milošević and W. Becker, “Attosecond pulse generation by bicircular fields: from pulse trains to a single pulse,” J. Mod. Opt. 52, 233–241 (2005).
[Crossref]

F. Lindner, M. G. Schätzel, H. Walther, A. Baltuška, E. Goulielmakis, F. Krausz, D. B. Milošević, D. Bauer, W. Becker, and G. G. Paulus, “Attosecond double-slit experiment,” Phys. Rev. Lett. 95, 040401 (2005).
[Crossref] [PubMed]

2004 (1)

S. Eisebitt, J. Lüning, W. F. Schlotter, M. Lörgen, O. Hellwig, W. Eberhardt, and J. Stöhr, “Lensless imaging of magnetic nanostructures by x-ray spectro-holography,” Nature (London) 432, 885–888 (2004).
[Crossref]

2003 (2)

F. Ceccherini, D. Bauer, and F. Cornolti, “Harmonic generation by atoms in circularly polarized two-color laser fields with coplanar polarizations and commensurate frequencies,” Phys. Rev. A. 68, 053402 (2003).
[Crossref]

A. D. Bandrauk and H. Z. Lu, “Controlling harmonic generation in molecules with intense laser and static magnetic fields: orientation effects,” Phys. Rev. A. 68, 043408 (2003).
[Crossref]

2002 (2)

W. Becker, F. Grasbon, R. Kopold, D. B. Milošević, G. G. Paulus, and H. Walther, “Above-threshold ionization: from classical features to quantum effects,” Adv. At. Mol. Opt. Phys. 48, 35–98 (2002).
[Crossref]

D. B. Milošević and W. Becker, “Role of long quantum orbits in high-order harmonic generation,” Phys. Rev. A. 66, 063417 (2002).
[Crossref]

2001 (2)

D. B. Milošević, W. Becker, R. Kopold, and W. Sandner, “High-harmonic generation by a bichromatic bicircular laser field,” Laser Phys. 11, 165–168 (2001).

P. Salières, B. Carré, L. Le Déroff, F. Grasbon, G. G. Paulus, H. Walther, R. Kopold, W. Becker, D. B. Milošević, A. Sanpera, and M. Lewenstein, “Feynman’s path-integral approach for intense-laser-atom interactions,” Science 292, 902–905 (2001).
[Crossref]

2000 (5)

R. Kopold, W. Becker, and M. Kleber, “Quantum path analysis of high-order above-threshold ionization,” Opt. Commun. 179, 39–50 (2000).
[Crossref]

D. B. Milošević and W. Becker, “Attosecond pulse trains with unusual nonlinear polarization,” Phys. Rev. A. 62011403 (2000).
[Crossref]

D. B. Milošević, W. Becker, and R. Kopold, “Generation of circularly polarized high-order harmonics by two-color coplanar field mixing,” Phys. Rev. A. 61, 063403 (2000).
[Crossref]

D. B. Milošević and W. Sandner, “Extreme-ultraviolet harmonic generation near 13 nm with a two-color elliptically polarized laser field,” Opt. Lett. 25, 1532–1534 (2000).
[Crossref]

R. Kopold, D. B. Milošević, and W. Becker, “Rescattering processes for elliptical polarization: a quantum trajectory analysis,” Phys. Rev. Lett. 84, 3831–3834 (2000).
[Crossref] [PubMed]

1995 (3)

H. Eichmann, A. Egbert, S. Nolte, C. Momma, B. Wellegehausen, W. Becker, S. Long, and J. K. McIver, “Polarization-dependent high-order two-color mixing,” Phys. Rev. A. 51, R3414–R3417 (1995).
[Crossref] [PubMed]

S. Long, W. Becker, and J. K. McIver, “Model calculations of polarization-dependent two-color high-harmonic generation,” Phys. Rev. A. 52, 2262–2278 (1995).
[Crossref] [PubMed]

T. Zuo and A. D. Bandrauk, “High-order harmonic generation in intense laser and magnetic fields,” J. Nonlinear Opt. Phys. Mat. 4, 533–546 (1995).
[Crossref]

1994 (2)

G. G. Paulus, W. Nicklich, H. Xu, P. Lambropoulos, and H. Walther, “Plateau in above threshold ionization spectra,” Phys. Rev. Lett. 72, 2851–2854 (1994).
[Crossref] [PubMed]

M. Lewenstein, Ph. Balcou, M. Yu. Ivanov, A. L’Huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A 49, 2117–2132 (1994).
[Crossref] [PubMed]

1993 (1)

P. B. Corkum, “Plasma perspective on strong field multiphoton ionization,” Phys. Rev. Lett. 71, 1994–1997 (1993).
[Crossref] [PubMed]

1988 (1)

M. Ferray, A. L’Huillier, X. F. Li, L. A. Lompré, G. Mainfray, and C. Manus, “Multiple-harmonic conversion of 1064 nm radiation in rare gases,” J. Phys. B. 21, L31–L35 (1988).
[Crossref]

Agostini, P.

P. Agostini and L. F. DiMauro, “Atomic and molecular ionization dynamics in strong laser fields: from optical to x-rays,” Adv. At. Mol. Opt. Phys. 61, 117–158 (2012).
[Crossref]

C. I. Blaga, J. Xu, A. D. DiChiara, E. Sistrunk, K. Zhang, P. Agostini, T. A. Miller, L. F. DiMauro, and C. D. Lin, “Imaging ultrafast molecular dynamics with laser-induced electron diffraction,” Nature (London) 483, 194–197 (2012).
[Crossref]

C. I. Blaga, F. Catoire, P. Colosimo, G. G. Paulus, H. G. Muller, P. Agostini, and L. F. DiMauro, “Strong-field photoionization revisited,” Nat. Phys. 5, 335–338 (2009).
[Crossref]

Andreev, A.

G. Lambert, B. Vodungbo, J. Gautier, B. Mahieu, V. Malka, S. Sebban, P. Zeitoun, J. Luning, J. Perron, A. Andreev, S. Stremoukhov, F. Ardana-Lamas, A. Dax, C. P. Hauri, A. Sardinha, and M. Fajardo, “Towards enabling femtosecond helicity-dependent spectroscopy with high-harmonic sources,” Nat. Commun. 6, 6167 (2015).
[Crossref] [PubMed]

Ardana-Lamas, F.

G. Lambert, B. Vodungbo, J. Gautier, B. Mahieu, V. Malka, S. Sebban, P. Zeitoun, J. Luning, J. Perron, A. Andreev, S. Stremoukhov, F. Ardana-Lamas, A. Dax, C. P. Hauri, A. Sardinha, and M. Fajardo, “Towards enabling femtosecond helicity-dependent spectroscopy with high-harmonic sources,” Nat. Commun. 6, 6167 (2015).
[Crossref] [PubMed]

Balcou, Ph.

M. Lewenstein, Ph. Balcou, M. Yu. Ivanov, A. L’Huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A 49, 2117–2132 (1994).
[Crossref] [PubMed]

Baltuška, A.

F. Lindner, M. G. Schätzel, H. Walther, A. Baltuška, E. Goulielmakis, F. Krausz, D. B. Milošević, D. Bauer, W. Becker, and G. G. Paulus, “Attosecond double-slit experiment,” Phys. Rev. Lett. 95, 040401 (2005).
[Crossref] [PubMed]

Bandrauk, A. D.

A. D. Bandrauk and H. Z. Lu, “Controlling harmonic generation in molecules with intense laser and static magnetic fields: orientation effects,” Phys. Rev. A. 68, 043408 (2003).
[Crossref]

T. Zuo and A. D. Bandrauk, “High-order harmonic generation in intense laser and magnetic fields,” J. Nonlinear Opt. Phys. Mat. 4, 533–546 (1995).
[Crossref]

Bandulet, H. C.

M. Meckel, D. Comtois, D. Zeidler, A. Staudte, D. Pavičić, H. C. Bandulet, H. Pépin, J. C. Kieffer, R. Dörner, D. M. Villeneuve, and P. B. Corkum, “Laser-induced electron tunneling and diffraction,” Science 320, 1478–1482 (2008).
[Crossref] [PubMed]

Bauer, D.

D. B. Milošević, G. G. Paulus, D. Bauer, and W. Becker, “Above-threshold ionization by few-cycle pulses,” J. Phys. B. 39, R203–R262 (2006).
[Crossref]

F. Lindner, M. G. Schätzel, H. Walther, A. Baltuška, E. Goulielmakis, F. Krausz, D. B. Milošević, D. Bauer, W. Becker, and G. G. Paulus, “Attosecond double-slit experiment,” Phys. Rev. Lett. 95, 040401 (2005).
[Crossref] [PubMed]

F. Ceccherini, D. Bauer, and F. Cornolti, “Harmonic generation by atoms in circularly polarized two-color laser fields with coplanar polarizations and commensurate frequencies,” Phys. Rev. A. 68, 053402 (2003).
[Crossref]

Becker, A.

T. Fan, P. Grychtol, R. Knut, C. Hernández-García, D. D. Hickstein, D. Zusin, C. Gentry, F. J. Dollar, C. A. Mancuso, C. Hogle, O. Kfir, D. Legut, K. Carva, J. L. Ellis, K. Dorney, C. Chen, O. Shpyrko, E. E. Fullerton, O. Cohen, P. M. Oppeneer, D. B. Milošević, A. Becker, A. A. Jaroń-Becker, T. Popmintchev, M. M. Murnane, and H. C. Kapteyn, “Bright circularly polarized soft X-ray high harmonics for X-ray magnetic circular dichroism,” Proc. Natl. Acad. Sci. USA 112, 14206–14211 (2015).
[Crossref] [PubMed]

Becker, W.

W. Becker and D. B. Milošević, “Quantum-orbit theory of low-energy above-threshold ionization on and off axis,” Chin. Opt. Lett. 13, 070006 (2015).
[Crossref]

W. Becker and D. B. Milošević, “Above-threshold ionization for very low electron energy,” J. Phys. B. 48, 151001 (2015).
[Crossref]

W. Becker, S. P. Goreslavski, D. B. Milošević, and G. G. Paulus, “Low-energy rescattering in laser-induced ionization,” J. Phys. B. 47, 204022 (2014).
[Crossref]

M. Möller, F. Meyer, A. M. Sayler, G. G. Paulus, M. F. Kling, B. E. Schmidt, W. Becker, and D. B. Milošević, “Off-axis low-energy structures in above-threshold ionization,” Phys. Rev. A. 90, 023412 (2014).
[Crossref]

X. Y. Lai, C. L. Wang, Y. J. Chen, Z. L. Hu, W. Quan, X. J. Liu, J. Chen, Y. Cheng, Z. Z. Xu, and W. Becker, “Elliptical polarization favors long quantum orbits in high-order above-threshold ionization of noble gases,” Phys. Rev. Lett. 110, 043002 (2013).
[Crossref] [PubMed]

P. A. Korneev, S. V. Popruzhenko, S. P. Goreslavski, W. Becker, G. G. Paulus, B. Fetić, and D. B. Milošević, “Interference structure of above-threshold ionization versus above-threshold detachment,” New. J. Phys. 14, 055019 (2012).
[Crossref]

A. Kramo, E. Hasović, D. B. Milošević, and W. Becker, “Above-threshold detachment by a two-color bicircular laser field,” Laser Phys. Lett. 4, 279–286 (2007).
[Crossref]

D. B. Milošević, E. Hasović, M. Busuladžić, A. Gazibegović-Busuladžić, and W. Becker, “Intensity-dependent enhancements in high-order above-threshold ionization,” Phys. Rev. A. 76, 053410 (2007).
[Crossref]

D. B. Milošević, G. G. Paulus, D. Bauer, and W. Becker, “Above-threshold ionization by few-cycle pulses,” J. Phys. B. 39, R203–R262 (2006).
[Crossref]

D. B. Milošević and W. Becker, “Attosecond pulse generation by bicircular fields: from pulse trains to a single pulse,” J. Mod. Opt. 52, 233–241 (2005).
[Crossref]

F. Lindner, M. G. Schätzel, H. Walther, A. Baltuška, E. Goulielmakis, F. Krausz, D. B. Milošević, D. Bauer, W. Becker, and G. G. Paulus, “Attosecond double-slit experiment,” Phys. Rev. Lett. 95, 040401 (2005).
[Crossref] [PubMed]

W. Becker, F. Grasbon, R. Kopold, D. B. Milošević, G. G. Paulus, and H. Walther, “Above-threshold ionization: from classical features to quantum effects,” Adv. At. Mol. Opt. Phys. 48, 35–98 (2002).
[Crossref]

D. B. Milošević and W. Becker, “Role of long quantum orbits in high-order harmonic generation,” Phys. Rev. A. 66, 063417 (2002).
[Crossref]

P. Salières, B. Carré, L. Le Déroff, F. Grasbon, G. G. Paulus, H. Walther, R. Kopold, W. Becker, D. B. Milošević, A. Sanpera, and M. Lewenstein, “Feynman’s path-integral approach for intense-laser-atom interactions,” Science 292, 902–905 (2001).
[Crossref]

D. B. Milošević, W. Becker, R. Kopold, and W. Sandner, “High-harmonic generation by a bichromatic bicircular laser field,” Laser Phys. 11, 165–168 (2001).

D. B. Milošević, W. Becker, and R. Kopold, “Generation of circularly polarized high-order harmonics by two-color coplanar field mixing,” Phys. Rev. A. 61, 063403 (2000).
[Crossref]

R. Kopold, D. B. Milošević, and W. Becker, “Rescattering processes for elliptical polarization: a quantum trajectory analysis,” Phys. Rev. Lett. 84, 3831–3834 (2000).
[Crossref] [PubMed]

D. B. Milošević and W. Becker, “Attosecond pulse trains with unusual nonlinear polarization,” Phys. Rev. A. 62011403 (2000).
[Crossref]

R. Kopold, W. Becker, and M. Kleber, “Quantum path analysis of high-order above-threshold ionization,” Opt. Commun. 179, 39–50 (2000).
[Crossref]

S. Long, W. Becker, and J. K. McIver, “Model calculations of polarization-dependent two-color high-harmonic generation,” Phys. Rev. A. 52, 2262–2278 (1995).
[Crossref] [PubMed]

H. Eichmann, A. Egbert, S. Nolte, C. Momma, B. Wellegehausen, W. Becker, S. Long, and J. K. McIver, “Polarization-dependent high-order two-color mixing,” Phys. Rev. A. 51, R3414–R3417 (1995).
[Crossref] [PubMed]

Blaga, C. I.

C. I. Blaga, J. Xu, A. D. DiChiara, E. Sistrunk, K. Zhang, P. Agostini, T. A. Miller, L. F. DiMauro, and C. D. Lin, “Imaging ultrafast molecular dynamics with laser-induced electron diffraction,” Nature (London) 483, 194–197 (2012).
[Crossref]

C. I. Blaga, F. Catoire, P. Colosimo, G. G. Paulus, H. G. Muller, P. Agostini, and L. F. DiMauro, “Strong-field photoionization revisited,” Nat. Phys. 5, 335–338 (2009).
[Crossref]

Busuladžic, M.

D. B. Milošević, E. Hasović, M. Busuladžić, A. Gazibegović-Busuladžić, and W. Becker, “Intensity-dependent enhancements in high-order above-threshold ionization,” Phys. Rev. A. 76, 053410 (2007).
[Crossref]

Capotondi, F.

C. von KorffSchmising, B. Pfau, M. Schneider, C. M. Günther, M. Giovanella, J. Perron, B. Vodungho, L. Müller, F. Capotondi, E. Pedersoli, N. Mahne, J. Lüning, and S. Eisebitt, “Imaging ultrafast demagnetization dynamics after a spatially localized optical excitation,” Phys. Rev. Lett. 112, 217203 (2014).
[Crossref]

Carré, B.

P. Salières, B. Carré, L. Le Déroff, F. Grasbon, G. G. Paulus, H. Walther, R. Kopold, W. Becker, D. B. Milošević, A. Sanpera, and M. Lewenstein, “Feynman’s path-integral approach for intense-laser-atom interactions,” Science 292, 902–905 (2001).
[Crossref]

Carva, K.

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O. Kfir, P. Grychtol, E. Turgut, R. Knut, D. Zusin, D. Popmintchev, T. Popmintchev, H. Nembach, J. M. Shaw, A. Fleischer, H. Kapteyn, M. Murnane, and O. Cohen, “Generation of bright phase-matched circularly-polarized extreme ultraviolet high harmonics,” Nat. Photonics 9, 99–105 (2015).
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T. Fan, P. Grychtol, R. Knut, C. Hernández-García, D. D. Hickstein, D. Zusin, C. Gentry, F. J. Dollar, C. A. Mancuso, C. Hogle, O. Kfir, D. Legut, K. Carva, J. L. Ellis, K. Dorney, C. Chen, O. Shpyrko, E. E. Fullerton, O. Cohen, P. M. Oppeneer, D. B. Milošević, A. Becker, A. A. Jaroń-Becker, T. Popmintchev, M. M. Murnane, and H. C. Kapteyn, “Bright circularly polarized soft X-ray high harmonics for X-ray magnetic circular dichroism,” Proc. Natl. Acad. Sci. USA 112, 14206–14211 (2015).
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P. Salières, B. Carré, L. Le Déroff, F. Grasbon, G. G. Paulus, H. Walther, R. Kopold, W. Becker, D. B. Milošević, A. Sanpera, and M. Lewenstein, “Feynman’s path-integral approach for intense-laser-atom interactions,” Science 292, 902–905 (2001).
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G. Lambert, B. Vodungbo, J. Gautier, B. Mahieu, V. Malka, S. Sebban, P. Zeitoun, J. Luning, J. Perron, A. Andreev, S. Stremoukhov, F. Ardana-Lamas, A. Dax, C. P. Hauri, A. Sardinha, and M. Fajardo, “Towards enabling femtosecond helicity-dependent spectroscopy with high-harmonic sources,” Nat. Commun. 6, 6167 (2015).
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M. Möller, F. Meyer, A. M. Sayler, G. G. Paulus, M. F. Kling, B. E. Schmidt, W. Becker, and D. B. Milošević, “Off-axis low-energy structures in above-threshold ionization,” Phys. Rev. A. 90, 023412 (2014).
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S. Eisebitt, J. Lüning, W. F. Schlotter, M. Lörgen, O. Hellwig, W. Eberhardt, and J. Stöhr, “Lensless imaging of magnetic nanostructures by x-ray spectro-holography,” Nature (London) 432, 885–888 (2004).
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C. von KorffSchmising, B. Pfau, M. Schneider, C. M. Günther, M. Giovanella, J. Perron, B. Vodungho, L. Müller, F. Capotondi, E. Pedersoli, N. Mahne, J. Lüning, and S. Eisebitt, “Imaging ultrafast demagnetization dynamics after a spatially localized optical excitation,” Phys. Rev. Lett. 112, 217203 (2014).
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F. Willems, C. T. L. Smeenk, N. Zhavoronkov, O. Kornilov, I. Radu, M. Schmidhauer, M. Hanke, C. von KorffSchmising, M. J. J. Vrakking, and S. Eisebitt, “Probing ultrafast spin dynamics with high-harmonic magnetic circular dichroism spectroscopy,” Phys. Rev. A. 92, 220405 (2015).

C. von KorffSchmising, B. Pfau, M. Schneider, C. M. Günther, M. Giovanella, J. Perron, B. Vodungho, L. Müller, F. Capotondi, E. Pedersoli, N. Mahne, J. Lüning, and S. Eisebitt, “Imaging ultrafast demagnetization dynamics after a spatially localized optical excitation,” Phys. Rev. Lett. 112, 217203 (2014).
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F. Lindner, M. G. Schätzel, H. Walther, A. Baltuška, E. Goulielmakis, F. Krausz, D. B. Milošević, D. Bauer, W. Becker, and G. G. Paulus, “Attosecond double-slit experiment,” Phys. Rev. Lett. 95, 040401 (2005).
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L. Medišauskas, J. Wragg, H. van der Hart, and M. Yu. Ivanov, “Generating isolated elliptically polarized attosecond pulses using bichromatic counterrotating circularly polarized laser fields,” Phys. Rev. Lett. 115, 153001 (2015).
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X. Y. Lai, C. L. Wang, Y. J. Chen, Z. L. Hu, W. Quan, X. J. Liu, J. Chen, Y. Cheng, Z. Z. Xu, and W. Becker, “Elliptical polarization favors long quantum orbits in high-order above-threshold ionization of noble gases,” Phys. Rev. Lett. 110, 043002 (2013).
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C. Y. Wu, Y. D. Yang, Y. Q. Liu, Q. H. Gong, M. Wu, X. Liu, X. L. Hao, W. D. Li, X. T. He, and J. Chen, “Characteristic spectrum of very low-energy photoelectron from above-threshold ionization in the tunneling regime,” Phys. Rev. Lett. 109, 043001 (2012).
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M. Meckel, D. Comtois, D. Zeidler, A. Staudte, D. Pavičić, H. C. Bandulet, H. Pépin, J. C. Kieffer, R. Dörner, D. M. Villeneuve, and P. B. Corkum, “Laser-induced electron tunneling and diffraction,” Science 320, 1478–1482 (2008).
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G. Lambert, B. Vodungbo, J. Gautier, B. Mahieu, V. Malka, S. Sebban, P. Zeitoun, J. Luning, J. Perron, A. Andreev, S. Stremoukhov, F. Ardana-Lamas, A. Dax, C. P. Hauri, A. Sardinha, and M. Fajardo, “Towards enabling femtosecond helicity-dependent spectroscopy with high-harmonic sources,” Nat. Commun. 6, 6167 (2015).
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C. I. Blaga, J. Xu, A. D. DiChiara, E. Sistrunk, K. Zhang, P. Agostini, T. A. Miller, L. F. DiMauro, and C. D. Lin, “Imaging ultrafast molecular dynamics with laser-induced electron diffraction,” Nature (London) 483, 194–197 (2012).
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F. Willems, C. T. L. Smeenk, N. Zhavoronkov, O. Kornilov, I. Radu, M. Schmidhauer, M. Hanke, C. von KorffSchmising, M. J. J. Vrakking, and S. Eisebitt, “Probing ultrafast spin dynamics with high-harmonic magnetic circular dichroism spectroscopy,” Phys. Rev. A. 92, 220405 (2015).

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O. Kfir, P. Grychtol, E. Turgut, R. Knut, D. Zusin, D. Popmintchev, T. Popmintchev, H. Nembach, J. M. Shaw, A. Fleischer, H. Kapteyn, M. Murnane, and O. Cohen, “Generation of bright phase-matched circularly-polarized extreme ultraviolet high harmonics,” Nat. Photonics 9, 99–105 (2015).
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C. A. Mancuso, D. D. Hickstein, P. Grychtol, R. Knut, O. Kfir, X. M. Tong, F. Dollar, D. Zusin, M. Gopalakrishnan, C. Gentry, E. Turgut, J. L. Ellis, M.-C. Chen, A. Fleischer, O. Cohen, H. C. Kapteyn, and M. M. Murnane, “Strong-field ionization with two-color circularly polarized laser fields,” Phys. Rev. A. 91, 031402 (2015).
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Adv. At. Mol. Opt. Phys. (2)

W. Becker, F. Grasbon, R. Kopold, D. B. Milošević, G. G. Paulus, and H. Walther, “Above-threshold ionization: from classical features to quantum effects,” Adv. At. Mol. Opt. Phys. 48, 35–98 (2002).
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Figures (7)

Fig. 1
Fig. 1 Electric field vector E(t) (black solid line) and the corresponding vectors A(t) (red dashed line) and its integral α(t) (cyan dot-dashed line) of the ω–2ω bicircular laser field (5). The various symbols and their colors refer to the instants of ionization (I) and rescattering (R) of certain quantum orbits that are identified in Fig. 6 by the solid lines and discussed in this context. The arrows indicate the temporal evolution of the field vectors.
Fig. 2
Fig. 2 The differential ionization rate (in a.u.) of Ne atoms, presented in false colors in the electron momentum plane for ionization by a bicircular ω–2ω laser field with equal intensity of both components I1 = I2 = 2×1014 W/cm2 and the fundamental wavelength of 800 nm (Up1 = 0.4392 a.u., 20 U p 1 = 2.964 a . u .). The results are obtained by numerical integration of the direct SFA matrix element. The false color scale is logarithmic for the right panel and linear and normalized to 1 for the left panel.
Fig. 3
Fig. 3 The logarithm of the differential ionization rate for the same parameters as in Fig. 2. The ISFA result (4) is obtained by two-dimensional numerical integration and using four 2p Slater-type orbitals to represent the ground state of the Ne atom.
Fig. 4
Fig. 4 The logarithm of the differential ionization rate for the same parameters as in Fig. 2, but for the ω–3ω field. In the left panel the coherent sum of the SFA and ISFA results is shown, while in the right panel only the ISFA result (4) is presented. The scale is logarithmic and the results are normalized to 1 in each panel separately.
Fig. 5
Fig. 5 Comparison of the differential ionization rates as function of the electron energy (in units of the ponderomotive energy of the ω field component) for the same parameters as in Fig. 3 and emission into the angle θ = 50°. The results obtained by numerical integration are presented by the black solid line with the circles, while the coherent sum of the contributions of 14 saddle-point solutions is represented by the solid red line with squares. The contributions of the divergent solutions (dashed lines) are neglected after the corresponding cutoff values. The partial contributions of each of the 14 solutions (α,β,m) are presented separately, as denoted. The spectrum that includes both the direct and the rescattered electron, obtained by numerical integration, is represented by small diamonds and denoted by D + R.
Fig. 6
Fig. 6 Electron trajectories for some of the quantum-orbit solutions presented in Fig. 5. The electron energies for the presented pairs α = ±1 of solutions are 10Up1 for solutions (β,m) = (1,0) and (2,0), 7Up1 for solutions (3, 0), and 9Up1 for solutions (1, 1). The ionization and rescattering times of the depicted orbits can be identified in Fig. 1.
Fig. 7
Fig. 7 Same as in Fig. 3 but obtained using the saddle-point method with the solutions (α,β,m) = (±1,1,0) (upper left panel), and the forward-scattering solutions (ν,μ) = (1,0) (upper right panel) and (ν,μ) = (±1,1) (lower left panel). In the lower right panel the coherent sum of all above-mentioned contributions is presented.

Equations (9)

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M p i ( t , t ) = ψ p ( t ) | U ( t , t ) | ψ i ( t ) ,
M p i ( t , t ) = M p i D ( t , t ) + M p i R ( t , t ) ,
M p i D ( t , t ) = i t t d τ ψ p ( t ) | U L ( t , τ ) V L ( τ ) | ψ i ( τ ) ,
M p i R ( t , t ) = ( i ) 2 t t d τ τ t d τ ψ p ( t ) | U L ( t , τ ) × V ( r ) U ( τ , τ ) V L ( τ ) | ψ i ( τ ) .
E ( t ) = i ( E 1 e ^ + e i r ω t + E 2 e ^ e i s ω t ) / 2 + c . c . ,
( t r t 0 ) k = t 0 t r d t A ( t ) ,
[ k + A ( t 0 ) ] 2 / 2 = I p ,
[ k + A ( t r ) ] 2 / 2 = [ p + A ( t r ) ] 2 / 2 ,
r ( s ) ( t ) = { ( t t 0 s ) k s + α ( t ) α ( t 0 s ) , ( t t r s ) p + α ( t ) α ( t r s ) ,

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