1. INTRODUCTION

Ultrafast ionization of atoms and molecules subject to strong laser fields has been intensively studied for decades (see, e.g., [1–4]). In this field, several experimental procedures with temporal resolution on the attosecond scale have been developed, such as attosecond streaking [5,6], reconstruction of attosecond beating by interfering two-photon transitions [7,8], and attoclock [9,10]. In contrast to the other experimental schemes mentioned above, in the attoclock procedure, the attosecond scale temporal resolution of the electron dynamics can be achieved by a relatively simpler optical scheme, i.e., a near-circularly polarized near-infrared femtosecond laser field [9,10], and this feature makes attoclock very popular. The attoclock procedure relies on a one-to-one relationship between the photoelectron tunneling instant and the measurable photoelectron emission angle. Specifically, as well accepted (see, e.g., [11,12]), in a circularly or elliptically polarized laser field, if the Coulomb potential is ignored and the adiabatic approximation can be applied, the electrons ionized at the instant when the laser electric field is along the minor (major) axis of the laser ellipse will drift in the major (minor) axis. This one-to-one correspondence between the tunneling time and the final emission angle of the photoelectron (${ t} - \theta$ correspondence) establishes the basis of the attoclock procedure.

Since it was exploited in 2008, the attoclock scheme was developed to investigate many intriguing issues and, in these studies, the ${t} - \theta$ correspondence plays a key role. It is worth mentioning that the long standing tunneling delay time problem [13–15], i.e., how long does it take for a particle (usually an electron) to tunnel from one side of the barrier to the other, has been explored by the attoclock procedure [9,10] and has caused intensive debate (see, e.g., [16–19]). Moreover, the attoclock has been extended to experimentally time the release of two electrons in sequential double ionization of noble-gas atom [20], to obtain information on the electron tunneling geometry and exit point [21], and to confirm the up-field atom ionization in the enhanced ionization of dimer [22]. By application of bicircularly polarized two-color laser fields, attosecond angular streaking with photoelectron interferometric metrology has been applied to reveal electron sub-Coulomb-barrier dynamics [23]. A phase-resolved two-electron-angular-streaking method is developed to image the shape of the hole density and motion of a wave packet in noble-gas cations [24]. More recently, the attoclock procedure was combined with a two-color phase of phase spectroscopy (POP attoclock) to realize the time-resolved ability in the full momentum space [25,26], in contrast to the traditional attoclock where only the tunneling instant of momentum distribution peak can be obtained [9,10]. It was shown that the mapping relation between the final photoelectron momentum and its ionization instant is not uniform, especially when the photoelectron is emitted in a direction close to the minimum yield [25]. Nevertheless, the physics behind has not been revealed.

In this paper, based on the strong-field approximation (SFA) theory [2,27], we demonstrate that there could be two tunneling instants, instead of one, for a photoelectron emitted in a direction close to the minimum yield, indicating breakdown of the ${ t} - \theta$ correspondence in the attoclock procedure for the typical laser parameters and photoelectron kinetic energy range explored in this work. Specifically, two trajectories with different tunneling instants and different initial velocities are found to correspond to one specific final momentum of photoelectron, and the interference between these two trajectories results in a multi-peak structure in the photoelectron kinetic energy spectrum. This multi-peak structure can be validated by our experimental measurements. Furthermore, we show that the jump of the mapping curve in the low yield zone observed in [25] can be attributed to the existence of the two tunneling instants corresponding to one specific final photoelectron kinetic energy.

2. RESULTS

In our calculation, the transition amplitude is given in the length gauge as follows:

To obtain the relationship between the ionization instant and the photoelectron kinetic energies or emission angles, a Wigner distribution-like function (WDL) [28–30] is introduced,

Our calculations are performed for Ar subject to a multi-cycle elliptically polarized laser field. A calculated angle-resolved photoelectron kinetic energy distribution of Ar has been presented in Fig. 1(d), where the definition of $\theta$ is given. From Fig. 1(d), it can be found that the photoelectron yield maximum (minimum) appears at $\theta ={ 0^ \circ}$ or 180° ($\theta = { 90^ \circ}$ or 270°). For clarity, our analysis based on WDL distributions is performed in the temporal range of $7.9\;{\rm o.c}. \lt t \lt 8.5\,{\rm o.c}.$, which locates around the peak of the pulse envelop employed in our calculations. The false-color WDL distributions for photoelectrons emitted at the angles of $\theta = {180^ \circ},{93^ \circ},{90^ \circ}$ are presented in Figs. 1(a)–1(c), respectively. As shown in Fig. 1(a), at $\theta ={ 180^ \circ}$, the WDL distributions concentrated on a small area around ${t_0} = 8.0\,{\rm o.c}.$ and ${E_k} \approx 5\;{\rm eV} $, indicating that there is only one tunneling instant, unambiguously corresponding to ${t_0} = 8.0\,{\rm o.c}$, when the laser electric field strength becomes maximum [see red curve in Fig. 1(a)]. On the other hand, at $\theta ={ 90^ \circ}$ [see Fig. 1(c)], below a critical photoelectron kinetic energy of ${E_c} \approx 30\;{\rm eV} $, there are two symmetric yellow stripes, corresponding to two tunneling instants, which vary consecutively with respect to ${E_k}$. Moreover, a prominent colorful pattern [29], which originates from the interference between wave packets from the two yellow stripes, can be identified. In fact, these kinds of interferences have already been experimentally observed [2,31] and theoretically discussed [32]. The pattern is equally distant from the two tunneling instants and becomes especially pronounced around ${t_0} = 8.25\,{\rm o.c}.$ (local minimum of laser electric field strength). For $\theta ={ 93^ \circ}$, one of the stripes becomes less discernible, denoting that the contributions from two tunneling instants becomes uneven, as shown in Fig. 1(b). Nevertheless, the interference pattern is still obvious at $\theta ={ 93^ \circ}$, indicating that the contributions of both tunneling instants are significant. The photoelectron kinetic energy spectra at the above-mentioned $\theta$ are presented in Figs. 1(a)–1(c). In Fig. 1(a), an above-threshold ionization spectrum can be identified at $\theta ={ 180^ \circ}$ when there is no contribution of the interference pattern at all. On the other hand, if the interference effect becomes important, the photoelectron kinetic energy spectrum is modulated significantly, and a peculiar multi-peak structure below ${E_k} \approx 20\;{\rm eV} $ appears at both $\theta = { 90^ \circ}$ and $\theta = { 93^ \circ}$. In addition, the contrast of the modulation at $\theta = { 93^ \circ}$ is much less than that of $\theta ={ 90^ \circ}$ due to uneven contributions from two tunneling instants as shown in Fig. 1(b).

 

Fig. 1. Typical WDL distributions for the photoelectron emitted at the angles of (a) $\theta ={ 180^ \circ}$, (b) 93°, and (c) 90° are presented. The red curves in (a)–(c) represent the strength (absolute value) of laser electric field. (d) Calculated angle-resolved photoelectron kinetic energy distribution for Ar. ${E_k}$ dependence of initial photoelectron velocities of the saddle-point solutions in the direction (e) parallel and (f) perpendicular to the instantaneous laser electric field for photoelectrons emitted at $\theta ={ 90^ \circ}$ are also pictured. The laser pulse with a wavelength of 800 nm, an intensity of $1.2 \times {10^{14}}\,\,{\rm W/cm^2}$, an ellipticity of 0.6, and a pulse duration of ${n} = {16}$ optical cycles has been applied in the calculations. The major (minor) axis of the laser polarization ellipse is along the horizontal (vertical) axis, as shown in the inset of (d), where the definition of the angle $\theta$ and the direction of ${v_\parallel}$ (${v_ \bot}$) are also presented. See the text for details.

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To shed more light on the WDL distributions presented above, for an electron with final momentum p, the ionization time ${t_s}$ is calculated by solving the saddle-point equation ${[{\textbf p} + {\textbf A}({t_s})]^2}/2 + {I_p} = 0$ (for review, see, e.g., [2]). The real part of the complex time ${t_s}$ is generally taken as the tunneling instant, which is shown in Figs. 1(a)–1(c). In Fig. 1(a), the saddle-point solution reveals a horizontal line, which means that the tunneling instant of the photoelectron can be determined unambiguously to be ${t_0} = 8.0\,{\rm o.c}$. This result is consistent with the WDL distribution. In Figs. 1(b) and 1(c), two saddle-point solutions clearly show up for ${E_k} \lt {E_c}$ and qualitatively match the corresponding WDL distributions, although ${E_c}$ determined by the saddle-point solutions might be a little larger than that in the case of WDL distributions. In Fig. 1(c), above ${E_c}$, the two saddle-point solutions merge into one. It is worthwhile mentioning that, although the saddle-point solutions are less accurate compared to the numerical calculations adopted in the WDL distribution, they can be used to qualitatively comprehend the results.

The WDL distributions and saddle-point solutions shown in Figs. 1(b) and 1(c) reveal that, for the photoelectron with kinetic energy of ${E_k} \lt {E_c}$, there are two tunneling instants at each ${E_k}$. This result clearly demonstrates that, if ${E_k} \lt {E_c}$, the one-to-one correspondence between the photoelectron emission angle and the tunneling instant breaks down and the attoclock experimental scheme becomes not viable any more.

To comprehend the physical origin behind the results described above, the ${E_k}$ dependence of the initial photoelectron momenta of the saddle-point solutions in the direction parallel (${v_\parallel} = {{\textbf p}_\parallel} + {{\textbf A}_\parallel}({\rm Re}[{t_s}])$) and perpendicular (${v_ \bot} = {{\textbf p}_ \bot} + {{\textbf A}_ \bot}({\rm Re}[{t_s}])$) to the instantaneous laser electric field are presented in Figs. 1(e) and 1(f), respectively. As shown in Figs. 1(e) and 1(f), although the value of ${v_ \bot}$ can be unambiguously determined at each ${E_k}$, we can find two non-zero values of ${v_\parallel}$, corresponding to two tunneling instants at each ${E_k}$ if ${E_k} \lt {E_c}$. Moreover, for ${E_k} \ge {E_c}$, ${v_\parallel}$ becomes zero. Considering that, for ${E_k} \ge {E_c}$, the two tunneling instants merge into one and, in the meantime, ${v_\parallel}$ becomes zero, the breakdown of the one-to-one correspondence between the photoelectron emission angle and the tunneling instant can be attributed to the existence of two non-zero ${v_\parallel}$ at each ${E_k}$, which can be further ascribed to the nonadiabatic photoelectron sub-barrier dynamics [33].

In this work, experiments have been performed to confirm our numerical calculation results. Laser pulses with a center wavelength of 800 nm, a repetition rate of 5 kHz, and a pulse duration of 30 fs are generated by a commercial Ti:sapphire femtosecond laser system (FEMTOPOWER compact PRO CE-Phase HP/HR). The single pulse energy is up to 0.8 mJ, and the output energy is controlled with a neutral density filter. The combination of a broadband thin film polarizer and a $\lambda /4$ waveplate is applied to change the polarization from linear to elliptical. The laser beam is directed into the vacuum chamber of cold target recoil ion momentum spectroscopy (COLTRIMS) [34,35] and focused by an on-axis spherical mirror ($f = 75\;{\rm mm} $) onto a cold supersonic atomic beam. With a constant and uniform DC electric field, the photoions and photoelectrons are extracted to the ion and electron position-sensitive microchannel plate detectors with delay line anodes equipped, respectively. A constant and uniform magnetic field generated by a pair of Helmholtz coils is further applied to confine the photoelectron movement in the plane perpendicular to the DC electric field. The 3D momenta of photoelectrons and photoions can be retrieved from their time of flights (TOFs) and the impact positions on the corresponding detectors. The laser intensity is calibrated with a procedure utilizing the photoelectron momentum distribution in a close-to-circularly polarized laser field [36]. More details on our COLTRIMS apparatus can be found in [37–40].

The measured typical angle-resolved photoelectron kinetic energy spectra at the laser parameters identical to those of Fig. 1(d) are presented in Fig. 2(a). Due to the influence of Coulomb potential and the possible tunneling delay, the distribution rotates a little bit compared to the results in Fig. 1(d). Furthermore, the clear above-threshold ionization rings in Fig. 1(d) disappears in the distributions in Fig. 2(a), and the reason relies on the influence of focal averaging and the finite momentum resolution of the COLTRIMS. The angle and energy range of photoelectrons in both figures are close to each other.

 

Fig. 2. (a) Measured angle-resolved photoelectron kinetic energy distribution for Ar. The laser parameters are identical to those in Fig. 1. (b)–(d) The measured photoelectron kinetic energy distribution (black open rectangles) extracted from (a) at the photoelectron emission angle of (b) $\theta = \alpha \pm {3^ \circ}$, (c) $\beta \pm {8^ \circ}$, and (d) $\gamma \pm {3^ \circ}$. The corresponding results of SFA calculations (red solid curves) are also given.

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Fig. 3. (a)–(c) WDL distributions for photoelectrons emitted at the angles of $\theta ={ 90^ \circ}$ for (a) $\varepsilon = 0.65$ and (b) $\varepsilon = 0.7$, and 98° for (c) $\varepsilon = 0.6$. (d)–(f) WDL distributions for $\varepsilon = 0.6$ at ${E_k} = 10$, 20, and 35 eV, respectively. In panels (d)–(f), the red squared lines (indicated by WDL) correspond to the highest WDL amplitude at each $\theta$, and the black dots (indicated by SP) denote the $t - \theta$ correspondences calculated by the saddle-point method. (g)–(i) $\theta$ dependence of the yields of photoelectrons emitted from two instants obtained by the saddle-point method at ${E_k} = 10$, 20, and 35 eV, respectively. The other laser parameters are identical to those of Fig. 1.

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The photoelectron kinetic energy spectra extracted from Fig. 2(a) at the photoelectron emission angles of $\alpha$, $\beta$, and $\gamma$ are presented in Figs. 2(b)–2(d), respectively. These three photoelectron emission angles correspond to the three $\theta$ values in Figs. 1(a)–1(c), i.e., photoelectron yield maximum ($\alpha$), minimum yield ($\beta$), and 3 deg away from the minimum yield ($\gamma$). As can be found in Figs. 2(b)–2(d), the measured photoelectron kinetic energy spectrum at $\alpha$ shows a single hump structure, while a multi-peak structure extends to ${E_k} = 0$ at both $\beta$ and $\gamma$. The corresponding calculations by the numerical SFA method are also given in Figs. 2(b)–2(d). Note that the influence of focal averaging has been considered in all the calculations. The SFA calculations are consistent with the measurements, including the multi-peak structures in the photoelectron kinetic energy spectra. The existence of the multi-peak structure is an unambiguous evidence that there are two tunneling instants instead of one. Moreover, the agreement between the measurements and SFA calculations also indicates that the absence of Coulomb potential in the numerical model will not influence the conclusion of our work.

In Fig. 3, the WDL distributions at several typical photoelectron emission angles and laser ellipticities have been presented. With a comparison of Figs. 3(a) and 3(b) [and also Fig. 1(c)], it can be found that, at a larger ellipticity, the interference structure becomes less obvious and, especially, the low energy part of the interference structure for ${E_k} \lt 10\;{\rm eV} $ almost disappears at $\varepsilon = 0.7$ [see Fig. 3(b)]. On the other hand, the interference structure at ${t_0} = 8.25\;{\rm o.c}$. becomes less obvious at a photoelectron emission angle deviating more from $\theta = { 90^ \circ}$, as shown in Figs. 1(b), 1(c), and 3(c). At $\theta = { 98^ \circ}$, the interference structure is almost invisible, and the contributions of one of the tunneling instants become much more significant compared to those of the other. Thus, the existence of two tunneling instants corresponding to one emission angle for ${E_k} \lt {E_c}$ becomes less relevant at a larger laser ellipticity or for a photoelectron emission angle deviating more from $\theta ={ 90^ \circ}$ at otherwise identical parameters.

To further understand how the existence of the two tunneling instants affects the $t - \theta$ correspondence, the WDL distributions as a function of $\theta$ and $t$ have been presented at the photoelectron energies of ${E_k} = 10\;{\rm eV}$ [Fig. 3(d)], 20 eV [Fig. 3(e)], and 35 eV [Fig. 3(f)]. In Figs. 3(d)–3(f), the red squared lines correspond to the highest WDL amplitude at each $\theta$. For comparison, the $\theta$ dependence of tunneling instant calculated by the saddle-point method is also given (black dots), which shows two branches located before and after the local minimum of laser electric field strength (8.25 o.c.). For both branches, the tunneling instant shifts toward the local minimum of laser electric field strength when $\theta$ approaches 90° and the distance between two branches decreases with increasing photoelectron energy when ${E_k} \lt {E_c}$. When ${E_k} \ge {E_c}$, two branches intersect at $\theta ={ 90^ \circ}$. In fact, this result shows that there are always two tunneling instants for each $\theta$ except when ${E_k} \ge {E_c}$, where the two instants merge into one at 90°. However, as shown in Figs. 3(g)–3(i), the upper and lower branches dominate in the regions of $\theta { \lt 90^ \circ}$ and $\theta { \gt 90^ \circ}$, respectively. The yields of two branches intersect at $\theta = { 90^ \circ}$ and decrease fast when $\theta$ deviates from 90°. Therefore, as shown in Figs. 3(d)–3(e), the dominant tunneling instant in the $t - \theta$ distribution will shift from the upper one for $\theta { \lt 90^ \circ}$ to the lower one for $\theta { \gt 90^ \circ}$, giving rise to an apparent break of the $t - \theta$ curve around $\theta ={ 90^ \circ}$ when ${E_k} \lt {E_c}$. Further, when these two instants merge into one, the $t - \theta$ curve becomes continuous [see Fig. 3(f)]. It is worthwhile mentioning that the saddle-point results and the WDL distributions [Figs. 3(d)–3(i)] are consistent with the measured ${E_k}$ dependence of the jump amplitude of the mapping curve in [25], which were experimentally obtained by the POP attoclock procedure.

3. CONCLUSION

In conclusion, the ionization dynamics of noble-gas atoms subject to strong elliptically polarized laser fields have been investigated. Our calculations demonstrate that the one-to-one $t - \theta$ correspondence ceases to exist if the photoelectron emission angle is close to $\theta ={ 90^ \circ}$ and, meanwhile, the photoelectron energy (${E_k}$) is smaller than a critical value of ${E_c}$. The calculation results are consistent with the experimental data. Our work is essential for the further development of the attoclock procedure.