## 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. 2*ω*), 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 *ω*_{1}/*ω*_{2} = 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 |*ψ _{i}*(

*t*′)〉 to the final scattering state |

*ψ*

**(**

_{p}*t*)〉 with asymptotic momentum

**p**and energy

*E*

**=**

_{p}**p**

^{2}/2 is

*U*(

*t,t*′) denotes the time-evolution operator of the Hamiltonian

*H*(

*t*) =

*T*+

*V*+

*V*(

_{L}*t*) with the kinetic energy operator

*T*, the binding potential

*V*(

**r**), and the laser-electron interaction

*V*(

_{L}*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]

The amplitude
${M}_{\mathbf{p}i}^{D}(t,{t}^{\prime})$ 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
${M}_{\mathbf{p}i}^{R}(t,{t}^{\prime})$ incorporates at least one such interaction at the time *τ*′. From here on, in Eqs. (3) and (4) we shall approximate the state 〈*ψ*** _{p}**(

*t*)|

*U*(

_{L}*t,τ*) by the Volkov state 〈

*χ*

**(**

_{p}*τ*)|. In the second term, we replace the exact time-evolution operator

*U*(

*t,τ*′) by the Volkov time-evolution operator

*U*(

_{L}*t,t*′) =

*∫ d*

^{3}

**k**|

*χ*

**(**

_{k}*t*)〉〈

*χ*

**(**

_{k}*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 *rω* and *sω*, which are integer multiples of the same fundamental frequency *ω* = 2*π*/*T*. The two fields rotate in the same plane in opposite directions. Hence we have [13–19]

**ê**

*and*

_{x}**ê**

*the real unit polarization vectors along the*

_{y}*x*and

*y*axis. Note that ${\widehat{\mathbf{e}}}_{-}^{2}={\widehat{\mathbf{e}}}_{+}^{2}=0$ while

**ê**

_{+}·

**ê**

_{−}= 1. The quantities

*E*and ${I}_{j}={E}_{j}^{2}(j=1,2)$ are the amplitudes and intensities of the two circularly polarized fields with the helicities

_{j}*h*

_{1}= 1 and

*h*

_{2}= −1. The corresponding ponderomotive potential is ${U}_{p}={\int}_{0}^{T}{\mathbf{A}}^{2}(\tau )d\tau /(2T)={U}_{p1}+{U}_{p2}={A}_{1}^{2}/4+{A}_{2}^{2}/4$, with

**E**(

*t*) = −

*d*

**A**(

*t*)/

*dt*,

*A*

_{1}=

*E*

_{1}/(

*rω*) and

*A*

_{2}=

*E*

_{2}/(

*sω*). An illustration of the

*ω*–2

*ω*case (

*r*= 1,

*s*= 2) can be found in Fig. 1.

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 *A*_{p}* _{i}* exp(

*iS*

_{p}*) with the action ${S}_{\mathbf{p}i}({t}_{r},{t}_{0},\mathbf{k})=-{\displaystyle {\int}_{{t}_{r}}^{\infty}d{t}^{\prime}{\left[\mathbf{p}+\mathbf{A}({t}^{\prime})\right]}^{2}}/2-{\displaystyle {\int}_{{t}_{0}}^{{t}_{r}}d{t}^{\prime}}{\left[\mathbf{k}+\mathbf{A}({t}^{\prime})\right]}^{2}/2+{I}_{p}{t}_{0}$, where*

_{i}*I*> 0 is the ionization potential. The stationarity conditions for the integral over the intermediate electron momentum

_{p}**k**, the ionization time

*t*

_{0}, and the rescattering time

*t*are:

*t*

_{0}and

*t*, respectively. The five-dimensional integration is then reduced to finding the complex solutions

_{r}**k**

*,*

_{s}*t*

_{0}

*, and*

_{s}*t*of the saddle-point equations (6)–(8). The subscript

_{rs}*s*indicates that there are more than one (in general infinitely many) solutions. A subset $\tilde{s}$ of them is relevant for the construction of the integral: ${M}_{\mathbf{p}i}^{R}={\displaystyle {\sum}_{\tilde{s}}{M}_{\mathbf{p}i}^{R}{|}_{\tilde{s}}}$. 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 (**k**_{s},t_{0}* _{s},t_{rs}*) 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

*t*

_{0}

*≤*

_{s}*t*≤ Re

*t*and if

_{rs}*t*> Re

*t*, respectively, where

_{rs}*α*(

*t*) =

*∫ dt*

**A**(

*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 *rω*–*sω* field the spectrum is invariant with respect to a rotation by the angle 2*πj*/(*r* + *s*), *j* = 1,…,*r* + *s*. Therefore, for the *ω*–2*ω* 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 2*p* 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
${E}_{{\mathbf{p}}_{\mathrm{min}}}={n}_{\mathrm{min}}\omega -{I}_{p}-{U}_{p}>0$.

If we neglect the ionization potential *I _{p}* 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**(

*t*

_{0}). 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*–2*ω* 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 *±*(*E*_{0}/*ω*)**ê*** _{x}*, where

*E*

_{0}is the electric-field-vector amplitude and

**ê**

*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 $10{U}_{{p}_{\text{lin}}}$,*

_{x}*p*with the corresponding ponderomotive energy ${U}_{{p}_{\text{lin}}}={E}_{0}^{2}/\left(4{\omega}^{2}\right)$. 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*–2

*ω*bicircular field approximately consists of three such sets of almost concentric circles, which reach up to energies of 10

*U*

_{p}_{1}. 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.

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 *p _{x}* 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*–3*ω* 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*–2*ω* case in Fig. 3, except that they exhibit four-fold instead of three-fold symmetry. In the *ω*–3*ω* 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.

## 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 *p _{x}* 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(*t _{r}* −

*t*

_{0})/

*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 7*U _{p}* 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 8

*U*

_{p}_{1}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.

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.

For the *ω*–2*ω* 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* < 2*T*/3, and 2*T*/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**(*t*_{0}), if the electron exits into the continuum at the time *t*_{0}, 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**(*t*_{0}).

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**(*t _{r}*). The return times

*t*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 −

_{r}**A**(

*t*) 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.

_{r}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 *p _{x}* axis between 0 ≤

*p*≤ 2 a.u. and the corresponding parts rotated by

_{x}*±*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**(*t _{r}*

_{−}) just before and

**p**+

**A**(

*t*

_{r}_{+}) just after rescattering point in approximately opposite directions. Only these solutions contribute to the high-energy spectrum. For energies below about 4

*U*

_{p}_{1}, 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

*t*

_{0}/

*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.

## 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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