Helicity asymmetry in strong-field ionization of atoms ... - OSA Publishing

Vol. 26, No. 10 | 14 May 2018 | OPTICS EXPRESS 12684

Helicity asymmetry in strong-field ionization of atoms by a bicircular laser field A. G AZIBEGOVI C´ -B USULADŽI C´ , 1 W. B ECKER , 2,3 M ILOŠEVI C´ 1,2,4,*

AND

D. B.

1 Faculty

of Science, University of Sarajevo, Zmaja od Bosne 35, 71000 Sarajevo, Bosnia and Herzegovina Max-Born-Str. 2a, 12489 Berlin, Germany 3 National Research Nuclear University MEPhI, Kashirskoe Shosse 31, 115409, Moscow, Russia 4 Academy of Sciences and Arts of Bosnia and Herzegovina, Bistrik 7, 71000 Sarajevo, Bosnia and Herzegovina 2 Max-Born-Institut,

* [email protected]

Abstract: Ionization of atoms by an intense bicircular laser field is considered, which consists of two coplanar corotating or counterrotating circularly polarized field components with frequencies that are integer multiples of a fundamental frequency. Emphasis is on the effect of a reversal of the helicities of the two field components on the photoelectron spectra. The velocity maps of the liberated electrons are calculated using the direct strong-field approximation (SFA) and its improved version (ISFA), which takes into account rescattering off the parent ion. Under the SFA all symmetries of the driving field are preserved in the velocity map while the ISFA violates certain reflection symmetries. This allows one to assess the significance of rescattering in actual data obtained from an experiment or a numerical simulation. © 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement OCIS codes: (020.2649) Strong field laser physics; (020.4180) Multiphoton processes; (270.4180) Multiphoton processes; (270.6620) Strong-field processes.

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#319373 Journal © 2018

https://doi.org/10.1364/OE.26.012684 Received 10 Jan 2018; revised 7 Apr 2018; accepted 12 Apr 2018; published 3 May 2018


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1.

Introduction

Above-threshold ionization (ATI) refers to the process where an intense laser field ionizes an atom, with the field being so intense that a description via perturbation theory of any reasonable finite order is completely inadequate [1]. In other words, the atom absorbs many more photons than the minimum required for ionization. Loosely speaking, ATI can be considered as a two- (or more)-step process such that in the first step the electron becomes free from the atom while in the second step it moves in the continuum driven by the force exerted by the laser field. If it is driven back and rescatters off its parent ion one speaks of a three-step process [2–4]. Perturbation theory may be adequate for the first step and not for the second, but, generally, for an intense field it is insufficient for either. This so-called three-step model was crucial for the development of strong-field physics and attoscience [5–20]. Experimentally, the most complete analysis of ionization is by recording the velocity map of the liberated electron at a detector outside the laser field, i.e., by plotting the yield of electrons as a function of their (vector) velocity. For most laser fields that are currently employed we may disregard the fact that the laser field is a propagating plane wave and just describe it by a purely time-dependent electric field E(t) with vector potential A(t) so that E(t) = −dA(t)/dt. The shape of the velocity map is related to the temporal evolution of the laser field, i.e., to the curve that the vector potential traces in space. The backbone of the analytical description of intense-field ionization processes is the strong-field approximation (SFA) in its various realizations, dating back to Keldysh’s seminal paper [21–23]. In this theory, the ionization amplitude is given by an integral over a parameter t, which can be interpreted as the time at which the electron becomes free from the atomic binding potential. The integrand only depends on the vector potential A(t) at this time t. This is often referred to as an “adiabatic” theory insomuch as the electron when it becomes free is unaware of its history, i.e., the history of the electric field or the vector potential prior to the time t. The SFA is the lowest-order term in an expansion of the ionization amplitude in terms of the interaction of the electron with the binding potential V(r) [23]. The higher-order terms of this Born-like expansion describe one or more interactions of the liberated electron with V(r). Of these, only the lowest-order term is normally considered, which contains exactly one interaction with the field and is sometimes called the improved SFA (ISFA). It is evident that this term is no longer adiabatic, since it depends both on the time of ionization and the time of rescattering. Symmetries of the driving field may be reflected in symmetries of the electron’s velocity map [24, 25]. Such symmetries can be backward-forward symmetry of the curve traced by the vector potential, discrete rotational symmetry with respect to certain angles, reflection symmetry with respect to certain axes, or symmetry upon time reversal. (Discrete translational symmetry in time generates the discrete ATI peaks.) For the adiabatic case (SFA, “direct electrons”), owing to the fact that the spectrum only depends on the vector potential at the ionization time t, one expects that the velocity map exhibits all the symmetries of a parametric plot of the vector potential A(t). In the nonadiabatic case (ISFA, rescattered electrons) where the ATI spectrum depends on the vector potential at two different times, this is no longer so. The discrete rotational symmetries are still obeyed by the ISFA but the reflection symmetries, which are related to time inversion, are not [26–28]. Hence, inspection of the symmetries of the velocity map allows one to assess quantitatively to what extent the underlying ionization mechanism is adiabatic or not or to what extent rescattering is important. Effects due to the propagation of the released electron in the Coulomb field of its parent ion are rescattering effects in this sense. In this paper, we will focus on a bicircular field, which is the superposition of two circularly polarized fields with different frequencies (typically but not necessarily, the frequency ratio is 2:1) that rotate in the same plane in the same (corotating case) or in opposite directions (counterrotating case). A change of sign of both helicities is easily accomplished experimentally; it corresponds to time reversal. The direct electrons (described by the SFA) are unaffected by this


helicity switch while the rescattered electrons (described by the ISFA) are. The magnitude of the corresponding relative asymmetry of the velocity map allows one quantitatively to assess the significance of nonadiabatic effects. Counterrotating bicircular fields have raised significant recent interest because they enable the generation of bright circularly polarized high-order harmonics [29, 30]. In fact, as early as in 1995 [31], it was observed that strong high-order harmonic generation (HHG) can be achieved using a counterrotating bicircular field having the frequencies ω and 2ω. HHG by a counterrotating bicircular field having frequencies rω and sω, with r and s integers, was analyzed in detail in [32]. The recent experimental confirmation that the pertinent high-order harmonics are really circularly polarized [33] was crucial for triggering many further investigations [29, 30] (see recent article [34] and references therein). The superposition of a continuous-wave circularly polarized field and a few-cycle counterrotating circularly polarized pulse, having the same frequency, was considered in [35]. It was shown that high-energy electrons, generated in the high-order above-threshold ionization (HATI) process by such a combination of pulses, are emitted in a direction correlated with the carrier-envelope phase of the pulse. Above-threshold detachment of electrons from negative ions induced by a counterrotating bicircular field was analyzed in [36, 37]. The three-lobed shape of the photoelectron velocity maps was confirmed experimentally for the ATI process in [38]. The corresponding HATI process has recently been considered in [26–28, 39–42]. Electron vortices in photoionization by two oppositely circularly polarized time-delayed pulses have also recently attracted attention [43–45]. From the experimental point of view a bicircular field is convenient in two aspects. First, a change of the relative phase between the bicircular field components, which can easily be controlled in the experiment, is equivalent to a rotation of the field in the polarization plane. This allows an easier recording of the (H)ATI electron velocity map. Second, it is straightforward to change the helicity of one or both bicircular-field components, keeping the other experimental parameters unchanged. In the present paper we examine the influence of this helicity change on the photoelectron velocity map. In Sec. 2 we briefly present our (I)SFA theory, define the bicircular field, consider the consequences of the helicity switching and introduce the relevant asymmetry parameter. Our numerical results are shown in Sec. 3. Finally, conclusions and comments are given in Sec. 4. We use atomic units (a.u.) throughout. 2.

Theory, definition of the bicircular field and symmetry considerations

Our theory of strong-field ionization by a bicircular laser field was presented in [27]. Here we repeat only the results necessary for reading the present paper. Within the single-active-electron approximation, using the atomic wavefunctions ψi`m , obtained by the Roothaan-Hartree-Fock method, and the appropriate rescattering potential V(r), the averaged differential ionization rate with absorption of the energy nω and emission of an electron with momentum p is given by w¯ pi` (n) ∝

` Õ

2 SFA ISFA p Tpi`m (n) + Tpi`m (n) ,

(1)

m=−`

where m is the magnetic quantum number and ` the orbital quantum number of the initial atomic bound state. In our examples of inert gases we use an expansion of the initial ground state in terms of the Slater-type orbitals Ri` (r)Y`m (ˆr), where Ri` (r) are radial wave functions, Y`m (ˆr) are spherical harmonics and ` = 1. In Eq. (1) the SFA and ISFA T-matrix elements, which, respectively, describe the direct ATI and the rescattering HATI process, are given by ∫ T dt0 SFA Tpi`m (n) = hp + A(t0 )|r · E(t0 )|ψi`m iei[Sp (t0 )+Ip t0 ], (2) T 0

Ey(t) Ay(t) αy(t)


1+2

1

1-2

0

-1 0 -1 1 Ex(t) Ax(t) αx(t)

0 -1 1 Ex(t) Ax(t) αx(t)

Fig. 1. Electric-field vector E(t), 0 ≤ t ≤ T, (solid lines) and the corresponding vector potential A(t) [dashed (red) lines] and the vector α(t) [dot-dashed (cyan) line] of the ω–2ω bicircular laser field. The left (right) panel corresponds to corotating (counterrotating) bicircular field. The intensities of the field components are equal and the results are presented in arbitrary units.

ISFA Tpi`m (n) = −i

∫ 0

T

dt T

∫

t

dt0

−∞

2π i(t − t0 )

3/2 hp|V(r)|kst i

×hkst + A(t0 )|r · E(t0 )|ψi`m iei[Sp (t)−Skst (t)+Skst (t0 )+Ip t0 ] .

(3)

The integrals over the ionization time t0 and the rescattering time t are calculated numerically. In Eqs. (2) and (3), E(t) = −dA(t)/dt is the electric-field vector and the interaction with the T-periodic laser field is in length gauge and dipole approximation. The energy-conservation condition has the form nω = Ep + Ip + Up , where Ep = p2 /2, Ip is the atomic ionization potential and Up the ponderomotive energy. We also introduced the stationary momentum ∫t ∫t 1 0 A(t 0 ) and the quantity S (t) = kst = − t−t dt dt 0 [k + A(t 0)]2 /2. The plane-wave ket k 0 t 0

vector |qi is such that hr|qi = (2π)−3/2 exp(iq · r). The exponent in Eq. (2)∫can be rewritten as t Sp (t0 ) + Ip t0 = p · α(t0 ) + U1 (t0 ) + nωt0 , where A(t) = dα(t)/dt and U1 (t) = dt 0A(t 0)2 /2 −Up t. The bicircular counterrotating laser field having the angular frequencies rω and sω, which are integer multiples of the same fundamental frequency ω = 2π/T, is given by i i h E(t) = E1 eˆ + e−irωt + E2 eˆ − e−i(sωt+ϕ) + c.c., 2 √ Ex (t) = [E1 sin(rωt) + E2 sin(sωt + ϕ)] / 2, √ Ey (t) = [−E1 cos(rωt) + E2 cos(sωt + ϕ)] / 2, (4) √ where eˆ ± = (ˆex ± iˆey )/ 2, with eˆ x and eˆ y the real unit vectors along the x and y axes, and E j and I j = E j2 are the electric-field vector amplitude and the intensity of the jth field component with the helicities h j (h1 = 1, h2 = −1). Denoting A1 = E1 /(rω) and A2 = E2 /(sω), for the ponderomotive energy we obtain Up = Up 1 + Up 2 = A21 /4 + A22 /4. A corotating bicircular field is obtained by replacing eˆ − by eˆ + in Eq. (4), i.e., h2 → −h2 = +1. In Fig. 1 we present the polar diagrams of the electric-field vector E(t), the pertinent vector potential A(t) and the vector α(t) in the x y plane for (r, s) = (1, 2) and for the corotating and counterrotating cases. A change of the relative phase ϕ by ∆ϕ corresponds to a rotation of the field about the z axis by the angle α = −∆ϕ/(r + s) [46]. This can be useful in experiments since it is easier to change the relative phase than to rotate the detector. We want to find the direct SFA matrix element for the bicircular field with the components r and s interchanged (this is equivalent to switching the helicities). The transformed field is


obtained from the original rω–sω field by changing the sign of the y component of E(t) [the same ˜ and analogously is valid for the vectors A(t) and α(t)]. We denote the transformed field by E(t) ˜ for the other transformed quantities. It can be checked that time inversion leads to E(−t) = −E(t), ˜ and α(−t) = −α(t). ˜ A(−t) = A(t) The SFA matrix element for the transformed field is ∫ T ∫ dt i[nωt+p· α˜ (t)+ U˜ 1 (t)] dr ˜ SFA ˜ T˜pi`m (n) = e−i[p+A(t)]·r r · E(t)ψ (5) e i`m (r). 3/2 T (2π) 0 After the substitution (r, t) → (−r, −t), using the following relation for the spherical harmonics: Y`m (ˆr) = (−1)`+m [Y`,−m (−ˆr)]∗ , we get (∫ )∗ ∫ T dt i[nωt+p· α (t)+U1 (t)] dr SFA `+m −i[p+A(t)]·r ˜ Tpi`m (n) = (−1) e r · E(t)ψi`,−m (r) e (2π)3/2 0 T h i∗ SFA = (−1)`+m Tpi`,−m (n) . (6) Using this result and the definition of the differential ionization rate (1), we obtain that the direct ionization rate is invariant with respect to the substitution r → s, i.e., with respect to helicity switching. As soon as rescattering is involved, the fact that the rescattering time is later than the ionization time prevents one from exploiting the above time-inversion symmetry. Therefore, by inspecting a given (theoretical or experimental) velocity map with respect to helicity switching one can assess the significance of rescattering. It is important to note that Coulomb corrections act like rescattering in that they also spoil the symmetry (6). The rescattering ionization rate obeys the symmetry with respect to simultaneous helicity switching and a change of sign of the py component of the momentum. Namely, using relation (3), the invariance of the scalar product with respect to a change of sign of the y component ˜ − τ) = r˜ · E(t − τ) etc.], and the relation ˜ [for example, we have p˜ · α(t) = p · α(t), r · E(t ISFA ISFA ˜ ˜ ˆ Y`m (r) = Y`,−m (ˆr), we obtain: Tpi`m (n) = Tpi`,−m (n), which proves the symmetry in question. ˜

(h1,h2 ) We denote the differential ionization rate (1) for the component helicities (h1, h2 ) by w¯ pi (n). Then we can define the helicity asymmetry parameter by

Api` (n) =

(+1,−1) (−1,+1) w¯ pi` (n) − w¯ pi` (n) (+1,−1) (−1,+1) w¯ pi` (n) + w¯ pi` (n)

.

(7)

In the next section we will present results for this asymmetry parameter for different photoelectron momenta. √ For r = s = 1, ϕ = 0, and E1 = E2 = E0 / 2, the counterrotating bicircular field (4) reduces to linearly polarized field E(t) = E0 eˆ x sin(ωt) for which Api` (n) = 0 and the helicity switching does not have a meaning. This means that there is no difference in the symmetries that are obeyed by direct and rescattered electrons. For given experimental or calculated solving the time-dependent Schrödinger equation (TDSE) photoelectron spectra, there is practically no way to get closer insight in (degree of the adiabaticity of) the mechanism that led the electrons to the detector. But, in the case r , s, an opportunity to extract additional information about the nature of the ionization process in strong laser field from experimental and/or TDSE photoelectron spectra arises. We should mention that the symmetries of the bicircular field and their relation to the electron’s velocity map are quite reminiscent of the case of elliptical polarization. Actually, the latter is a special realization of a bicircular field, namely, the counterrotating case with r = 1 and s = 1 and unequal intensities. For elliptical polarization, it was observed early on [47] and confirmed later [48] that the experimental velocity map obeys only a twofold symmetry [(px, py ) → −(px, py ) with x and y the polarization axes], while the SFA predicts a fourfold


symmetry (px → −px or py → −py separately). The ISFA destroys the fourfold symmetry while leaving the twofold one intact. Asymmetry will appear as the ellipticity increases from zero, but, with increasing ellipticity, the probability that the electron returns and rescatters decreases. However, counterrotating bicircular field with equal intensities provides several parts of “nearly linear” electric field that increases the probability of rescattering for photoelectrons, but also provides different symmetries for direct and rescattered electrons. Including the Coulomb field approximately in the SFA, for example by employing Coulomb-Volkov instead of Volkov solutions, also destroys the fourfold symmetry. A detailed analysis can be found in [24]. Moreover, we should point out that the distinction between direct and rescattered electrons is model based. In the Born expansion that underlies Eqs. (1)–(3), the SFA describes the direct electrons and the ISFA those that rescatter. But distinguishing a direct electron that swept over the position of its parent ion (described by the SFA) from another one that followed the same trajectory but forward scattered elastically in the recollision [49, 50] (described by the ISFA) is only possible in theory. So, rigorously speaking, a nonzero value of the asymmetry parameter only says that the zeroth-order term in the Born expansion is insufficient. Yet, usually it is tacitly assumed that the SFA and the ISFA correspond to no rescattering and rescattering, respectively, in an intuitive sense. In contrast, the TDSE (or, in a fully classical context, the full Newton equations of motion) makes no distinction between direct and rescattered electrons. These two different approaches have been categorized as “step by step” vs. “all at once” [17]. 3.

Numerical results

We consider (H)ATI of Ne atoms by a bicircular ω–2ω laser field with equal intensity of both components I1 = I2 = 2 × 1014 W/cm2 , the fundamental wavelength of 800 nm, and the relative phase ϕ = 0. In Figs. 2 and 3 we present the differential ionization rate for ionization by a counterrotating bicircular field with the component helicities h1 = +1 and h2 = −1 (upper panel) and h1 = −1 and h2 = +1 (lower panel). For Fig. 2 both the direct SFA and rescattering ISFA matrix elements are taken into account while for Fig. 3 only the ISFA matrix element is considered. In both figures we observe threefold rotational symmetry of the differential ionization rate. This symmetry was found in above-threshold detachment by a bicircular field in [36] (for the direct SFA) and in HATI by a bicircular field in [26]. It was analyzed in detail in [27]. The small central circle in Figs. 2 and 3 is related to the minimal electron kinetic energy determined by the energy-conservation condition Ep min = nmin ω − Ip − Up > 0. The direct SFA rate corresponds to lower energies and, in addition to the threefold rotational symmetry, obeys reflection symmetry about the axes at angles jπ/3, with j integer, with respect to the positive px axis. Comparing the central triangular parts in the panels of Fig. 2 we confirm that the direct SFA differential ionization rate is invariant with respect to helicity switching. On the other hand, the rescattering ISFA violates this symmetry, as is obvious from Fig. 3 and already in Fig. 2. Moreover, the ISFA rates presented in the upper and lower panels are connected by reflection symmetry with respect to the px axis, i.e., the rate presented in the lower panel of Fig. 3 can be obtained from the one in the upper panel by the replacement py → −py , which is in accordance with our analytical results. [Corresponding reflection symmetries apply for the other symmetry axes of the vector potential A(t)]. Using the results of Figs. 2 and 3 we calculated the helicity asymmetry parameter Api` (n), defined in Eq. (7). The results obtained using Fig. 2 (Fig. 3) are presented in the upper (lower) panel of Fig. 4. The asymmetry parameter Api` (n) satisfies the mentioned threefold symmetry. In addition, it satisfies the relation Api` ˜ (n) = −Api` (n), i.e., the sign of the asymmetry parameter changes for py → −py (and analogously for the other symmetry axes). The most striking feature of the upper panel of Fig. 4 is that the asymmetry parameter is almost zero in the central part of figure. This agrees with the fact that the central low-energy part is dominated by the direct electron and that the asymmetry parameter is zero for these electrons. Therefore, by measurement


Fig. 2. The logarithm of the differential ionization rate (in a.u.) of Ne atoms presented in false colors in the electron-momentum plane for ionization by a bicircular counterrotating ω–2ω field, equal intensity of both components I1 = I2 = 2 × 1014 W/cm2 , the fundamental wavelength of 800 nm, and the relative phase ϕ = 0. The results are obtained by numerical integration. Both the direct SFA and the rescattering ISFA T-matrix elements are taken into account. The false-color scale is normalized to the maximum rate, which is 1.4702 × 10−7 a.u. For the upper (lower) panel the helicities are h1 = +1 (−1), h2 = −1 (+1).


Fig. 3. Same as in Fig. 2 but with only the rescattering T-matrix element taken into account. The false-color scale is normalized to the maximum rate, which is 5.1177 × 10−10 a.u.


Fig. 4. Asymmetry parameter, Eq. (7), for the counterrotating bicircular field that corresponds to Fig. 2 (upper panel) and Fig. 3 (lower panel).


of the asymmetry parameter one can directly retrieve information about the rescattered electrons and their significance. The violation of the reflection symmetry is already evident in Fig. 3, which is for just specified helicities. However, it is difficult to extract the amount of violation. In contrast, the plot of the asymmetry parameter shown in Fig. 4 quantifies the violation by a number, which also depends on the spectral region. In other words, rescattering plays a larger role in some spectral regions than in others. The prize that had to be paid is a measurement or a simulation both for the original and for the reversed helicities. We should reemphasize that the SFA does not incorporate the effect of the Coulomb potential on the liberated electron at all, while the ISFA does, but only to first order in a Born expansion. Many attempts have been made to account for the Coulomb potential while the electron propagates in the continuum. One such approach employs the so-called Coulomb-Volkov solutions, which are a product of field-free Coulomb states multiplied with the Volkov phase [51–53]. Other approaches augment the phase of the pertinent quantum orbit by the path-dependent contribution of the Coulomb field (eikonal Volkov approximation [54, 55]; trajectory-based Coulomb SFA [56]). Whenever such Coulomb effects are included (either in a theoretical simulation or in an experimentally measured velocity map) the asymmetry parameter will no longer be zero. Let us now consider the case of a bicircular field with corotating components. Rescattering, i.e., the ISFA matrix element, is negligible in this case. This can be related to the fact that the classical trajectories of the liberated electron only very rarely revisit their parent ion [28]. The differential ionization rate for this case is presented in the upper panel of Fig. 5. The corresponding electron velocity map has a crescent shape and obeys reflection symmetry about axes at the angles r jπ/(s − r) = jπ with respect to the positive px axis [39]. It is interesting to calculate the asymmetry parameter between the corotating and counterrotating bicircular field, which is defined by Bpi` (n) =

(+1,+1) (+1,−1) w¯ pi` (n) − w¯ pi` (n) (+1,+1) (+1,−1) w¯ pi` (n) + w¯ pi` (n)

.

(8)

In an experiment this can be achieved by flipping the helicity of only the 2ω bicircular field component. The corresponding results are presented in the lower panel of Fig. 5. One can again notice that the high-energy part is dominated by the contribution of the counterrotating bicircular field since it allows for rescattering. The low-energy part of the velocity map, in the velocity region of the “crescent” part of the upper panel of Fig. 5, is dominated by the corotating-bicircular-field ATI electrons. This may not be a realistic description of an experiment, since Coulomb effects are not taken into account in the SFA. Namely, for the counterrotating bicircular field the Coulomb field may enhance the low-energy ATI rate more than in the corotating case (recall the low-energy structures in the case of linear polarization [57–60]). 4.

Conclusions and comments

We have explored the effect of switching the helicities of both components of a bicircular field on the (H)ATI photoelectron velocity map. It was shown that the helicity asymmetry parameter that we introduced is zero for the direct ATI electrons, which are described by the strong-field approximation. Therefore, analysis of a velocity map – be it the result of an experimental measurement or calculated via the time-dependent Schrödinger equation (TDSE) or some other method – with respect to the asymmetry parameter allows us to explore the significance of the rescattered HATI electrons without disturbance from the direct electrons. We have also introduced the asymmetry parameter between a corotating and a counterrotating bicircular field. This parameter also allows an investigation of the rescattered HATI electrons. In


Fig. 5. The logarithm of the differential ionization rate (in a.u.; upper panel) of Ne atoms presented in false colors in the electron-momentum plane for ionization by a corotating bicircular ω–2ω field, equal intensity of both components I1 = I2 = 2 × 1014 W/cm2 , the fundamental wavelength of 800 nm, and the relative phase ϕ = 0. The false-color scale is normalized to the maximum rate, which is 2.8446 × 10−7 a.u. In the lower panel we show the asymmetry parameter between the rates for the corotating and the counterrotating field (for the parameters of Fig. 2), defined by Eq. (8). If both ionization rates are lower than 10−13 a.u., the asymmetry parameter is set to zero.


the experiments, these asymmetry parameters can be measured by changing the helicity of one or both bicircular field components, keeping the other parameters unchanged. The effect known as strong-field photoelectron holography was observed for strong-field ionization by a linearly polarized laser field [61, 62] and for orthogonally polarized two-color fields [63]. This effect is related to the interference of direct and rescattered photoelectron wave packets and should also exist for a bicircular field. In the bicircular field case the electron trajectories develop in a plane, so that the structures that correspond to direct electrons and rescattered electrons overlap in an intricate fashion [27, 38]. The helicity asymmetry parameters A and B introduced in the present paper will provide a powerful tool for their interpretation. One should recall that the distinction between rescattering and no rescattering is a theoretical concept. Given a numerical (for example TDSE-based) simulation or experimental data it is far from evident which role rescattering played in the data. Photoelectron holography is a good example. Here, the analysis that we suggest for a bicircular field should be an excellent means to aid in the interpretation. We look forward to more examples in laser-induced electron diffraction [64], recollision induced inelastic scattering [65], and tomographic imaging of molecular orbitals [66]. Potential applications are in attosecond physics and chemistry, as well as in characterization of the dynamics and structure of atomic and molecular systems. The importance of this method is that it provides a tool to “dissect” experimental and/or TDSE photoelectron spectra and provides insight in the ionized electron dynamics in the laser and atomic field. It may also be useful as additional avenue to proper Coulomb correction inclusion in theoretical calculations. Finally, it should be mentioned that the introduction of asymmetry parameters in various problems of strong-field physics was crucial for discovery of new phenomena. The probabilities of processes are usually considered on a logarithmic scale and one can identify, for example, plateaus, cutoffs, and resonance-like enhancements. However, it is difficult to extract information about small interference effects. Examples where asymmetry parameters were successfully used are: left-right asymmetry as a function of the carrier-envelope phase for (H)ATI by few-cycle pulses [7, 67], asymmetry in probability of the strong-field processes with aligned and antialigned molecules [68], circular dichroism [30], and spin asymmetry [69–71]. We hope that the helicity asymmetry parameter, introduced in our paper, will be useful in future experiments. It would be particularly useful for molecules for which the symmetries of (H)ATI spectra of a bicircular field were recently considered in [72]. Funding Deutsche Forschungsgemeinschaft (DFG). Acknowledgments We acknowledge discussions in the frame of the DFG Priority Programme “Quantum Dynamics in Tailored Intense Fields” (QUTIF) of the German Research Foundation.