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Abstract
Rather than being freed to the continuum, the strong-field tunneled electrons can make a trajectory driven by the remaining laser fields and have certain probability to be captured by the high lying Rydberg states of the parent atoms or molecules. To explore the effect of molecular orbital on Rydberg state excitation, the ellipticity dependence of Rydberg state yields of N2 and O2 molecules are experimentally investigated using cold target recoil ion momentum spectroscopy and are compared with their counterpart atoms Ar and Xe with comparable ionization potentials. We found the generation probability of the neutral Rydberg fragment O2* was orders of magnitude higher than that of Xe* due to the butterfly-shaped highest occupied molecular orbital of O2. Meanwhile, our experimental and simulation results reveal that it is the initial momentum distribution (determined by the detailed characteristics of orbitals) that finally leads to the tendency that the Rydberg state yield of O2 (Ar) decreased slower than that obtained for Xe (N2) when the ellipticity of the excitation laser field is increased.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
When atoms and molecules are exposed to an intense laser field, the bound electrons can be released to the continuum though tunneling or multiphoton ionization. Rather than directly escaping, it was demonstrated that an initially freed electron may revisit its parent nucleus due to the combined influence of the Coulomb attraction and the laser field. This gives rise to many interesting phenomena, such as high harmonic generation (HHG) [1,2], non-sequential double ionization [3–5], laser-induced electron diffraction [6,7], etc. Moreover, the freed electron can be recaptured by the high lying Rydberg states below ionization threshold thus forming excited neutral particles. The Rydberg state excitation (RSE) yield has been observed using, e.g. the reaction spectroscopy [8–10]. The frustrated tunneling ionization (FTI) and multiphoton excitation scenarios have been experimentally and theoretically demonstrated as the underlying mechanisms in the strong-field RSE of the atoms and molecules [11–14]. From the FTI perspective, the RSE process can be simply understood by a semi-classical picture. The electrons tunnel near the maxima of the oscillating optical field and gain nearly zero drift momentum from the remaining laser field so that the electrons may finally stay in Rydberg orbitals by the Coulomb attraction of the parental ion. Since the scenario of FTI is similar to the well-known three-step model [2], the underlying relationship between them is also a fascinating topic. In the multiphoton scenario, the electron directly populates the Rydberg states via resonant multiphoton excitation thus the process can take place under circularly polarized laser fields, which serves as the main distinct feature compared to the situation in the FTI scenario.
A comparative study of a molecule and its companion atom (the atom with a comparable ionization potential (IP) value to that of the molecule) provides an important way to reveal the orbital or structure effect of molecule in its response to the external laser field. Recent studies proved that the molecular orbitals significantly influence the strong-field process and lead to some fascinating phenomena [15–17]. In this work, we experimentally investigated the ellipticity dependence of Rydberg excitation yields for two pairs of targets, i.e. O2/Xe and N2/Ar, where the ionization potentials of the atomic and molecular species in each pair are similar. Our experimental results showed that when excited with 25 fs laser pulses at an intensity around 1×1014 W/cm2 the RSE yields were maximized at linear polarization and the yields decreased gradually when the ellipticity of the excitation laser field was increased. However, the descending yield as a function of ellipticity exhibited dramatic difference for different targets, which could be attributed to the effect of the detailed molecular orbitals. Phenomena of ellipticity dependence has also been observed in HHG with similar descending behavior [18]. Our experiment can help to improve the understanding of the underlying physics of electron recombination processes, including RSE and HHG.
2. Experimental method
The measurements were carried out in an ultrahigh-vacuum reaction microscope of cold target recoil ion momentum spectroscopy (COLTRIMS) [19,20]. A combination of a quarter-wave plate (QWP) and a half-wave plate (HWP) were employed to adjust the ellipticity of the femtosecond pulses (25 fs, 790 nm, 10 kHz) from a multipass Ti:sapphire laser system. The pulses were afterwards focused onto a supersonic gas jet (1:1 mixture of Xe/O2 and N2/Ar) by a concave mirror (f = 75 mm) in the COLTRIMS. Inside the spectroscope, the created ions and electrons (guided by a weak homogeneous dc electric field (Es∼18.5 V/cm) and a magnetic field (B∼12G), respectively) were detected by two time- and position-sensitive micro-channel plate (MCP) detectors at the opposite ends of the spectrometer. In our experiment the laser pulses were elliptically polarized in the y-z plane with the major axes along the z axis and the ellipticity was finely controlled by rotating the half-wave plate so that the major axis of the elliptical polarization was kept unchanged. The electric field of the laser can be given by F(t) = F0f(t)[cos($\omega $t)$\overrightarrow {{e_z}} $ +$\varepsilon $sin($\omega $t)$\overrightarrow {{e_y}} $]/$\sqrt {1 + {\varepsilon ^2}} $, where ω is the frequency of the laser, ɛ is the ellipticity, and f (t) is the slowly varying pulse envelope. The laser intensity in the interaction region was calibrated to be 8×1013 W/cm2 for the Xe/O2 measurement and 1.2×1014 W/cm2 for the Ar/N2 measurement by tracing the intensity-dependent time-of-flight spectrum of protons from the dissociative ionization of H2 [21]. The Keldysh parameters were calculated to be about 1 for each pair of atoms and molecules under the laser intensity used in the present work. Utilizing the coincidence detection techniques, not only the electron and ion fragments could be detected, but also the generated excited neutral atoms or molecules in Rydberg states could be identified on the detector if they were post-pulse ionized by the static electric field of the spectrometer or the black body radiation (BBR).
3. Results and discussions
To identify the neutral atomic and molecular fragments populated to the Rydberg states, typical photoelectron-photoion-coincidence (PEPICO) spectra for O2/Xe and N2/Ar were employed as displayed in Fig. 1. The neutral fragments excited to Rydberg states were further ionized by the spectrometer DC electric field (∼18.5V/cm) or through BBR process during their flight towards the detector. The corresponding signal featured several long lines at long flying times which gradually disappear for increasing ellipticity. Here the observed signals with long time of flights (TOF > 60 ns) were interpreted as postpulse ionization of high-lying Rydberg states generated by the recapture of the freed electrons [8,13]. In the PEPICO measurement, the obtained signal can cover a wide range of the principal quantum number (n) for the Rydberg states. Time delayed ionization of Rydberg states has been studied in detail previously [9,22], showing that the process can be divided into two main parts as indicated in Figs. 1 and 2(b) by the red dashed lines at the TOF of around 200 ns which is mainly affected by the quantum number distribution of the RSE and the strength of the DC field [22]. The yield deceased rapidly from about 60 ns to 200 ns due to the DC field ionization (n>74) [10,22] and the descending became slower for the TOF beyond 200 ns which is attributed to the BBR (n>10) [22]. Here we counted the excited atoms from about TOF = 60 ns (the border of the strong field ionization and the RSE, and was illustrated in the inset in Fig. 1(a)) where a clear statistics for the RSE events could be obtained. The experimental results (shown in Fig. 2(b)) showed that the high lying Rydberg fragments obtained in our experiment are mainly detected in the first 200 ns time window. As can be seen in Figs. 1 and 2(a), the yields denoted by Y(M*) (where M* represents the neutral Rydberg particle of the species M) and the generation probability Y(M*) / Y(M+) (where M+ represents the singly ionized fragment of the species M) of N2* and Ar* were comparable which agrees with the results reported in Ref. 15. However, we found that the generation probability of the neutral Rydberg fragment O2* was orders of magnitude higher than that of Xe* which is significantly different to the previous results [15]. This could be rooted in the different detection methods used in our work compared to that used in Ref. 15. In principle, two reasons may contribute to the orders of magnitude higher ratio of O2*/O2+ than that of Xe*/Xe+. Firstly, the ionization suppression in O2 lead to a relatively lower O2+ population than that of Xe+[23–27]. Secondly, the occupation of Rydberg orbitals with higher principle quantum numbers is larger for O2 than Xe. Due to the butterfly-shaped molecular orbital, the electrons emitted from O2 tend to have a broader initial momentum distribution thus are easier to diffuse away from the nuclei compared to electrons emitted from Xe [15]. It is reasonable to infer that the recaptured electrons from O2 are more likely to occupy Rydberg orbits with higher angular momentum. As mentioned above, the excited fragments in our experiments are mainly generated from DC field ionization of the spectrometer, which is associated with higher principal quantum numbers (n>74) than that of Ref. 15 (20<n< 30).
Fig. 1. PEPICO spectra obtained in linearly polarized femtosecond laser pulses for (a) O2/Xe at the intensity of 8×1013 W/cm2 and for (b) N2/Ar at the intensity of 1.2×1014 W/cm2. The inserted image in Fig. 1(a) represents the enlarged spectra at around TOF = 60 ns. The yellow dashed line indicates the division from strong field ionization and RSE. The red dashed curves in both figures indicate the separation between DC electric field ionization and photon ionization due to BBR of the Rydberg atoms and molecules.
Fig. 2. (a) The measured ellipticity dependence of the electron recapture probability for the four indicated species. (b) The normalized yield of N2* as a function of time of flight (TOF) for two laser ellipticities, i.e. ɛ = 0 and 0.14, respectively.
Interestingly, previous theoretical studies predicted that the principle quantum number distributions of the Rydberg state occupation did not show clear differences for the linearly and the circularly polarized laser fields [28], which was yet to be demonstrated by experimental evidences. Recently, an alternative method [22] to detect the Rydberg states was proposed and showed that the postpulse ionized Rydberg atoms or molecules as a function of the time of flight was sensitive to the external DC field strength. This was due to the fact that the ionization rate in a static field obeys the saddle-point model as F = Z3/9n4 by incorporating the effect of linear Stark shift, where F is the external static field and Z represents the charge of the state [10,15]. Here, as shown in Fig. 2(b) there was almost no difference between the normalized yields of the ionized Rydberg molecules in the external DC field at two different laser ellipticities, i.e. ɛ = 0 (the red curve) and 0.14 (the blue curve), respectively. This implies that the n distribution of N2* does not exhibit apparent ellipticity dependence for laser fields of small elliticity (similar results can be seen in Xe*, O2* and Ar*) since different n distributions will lead to noticeable change of the ionization ratio by the DC field at short TOFs.
Through integration of the PEPICO spectra, the yield of high lying Rydberg states of N2*, O2*, Ar* and Xe* from weak DC-field and BBR induced ionization can be obtained. Figures 3(a) and (b) show the normalized yield of the RSE for two pairs of ionic species as a function of laser ellipticity. The normalization is done by dividing the RSE yield of different ellipticities by the maximum yield obtained in linearly polarized field. The normalized yields here include the events from both the DC-field ionization and the BBR process. As can be seen in Fig. 3(a), the RSE yield of O2 (red squares) and Xe (black circles) exhibited a maximum value in the linearly polarized laser fields, i.e. at ɛ = 0, and decreases quickly for increasing ellipticities. We fit the ellipticity dependence data with a Gaussian profile, as shown by the solid curves in Fig. 3(a). The full width at half maximum (FWHM) of the Gaussian distribution of O2* is wider than that of Xe*, indicating a weaker ellipticity dependence of the RSE for O2 than for Xe. Similarly, the ellipticity dependence of N2* and Ar* are shown in Fig. 3(b). The situation of N2/Ar was different compared to the case of the O2/Xe where the FWHM of the atomic species Ar* is larger than that of the N2*. These results indicate that besides the ionization potential of the targets, there must be other factors associating with the inherent property of the targets that influenced the RSE process.
Fig. 3. The measured ellipticity dependence of the normalized RSE yields of (a) O2* and Xe* and (b) N2* and Ar*. The scattered data are experimental results and the solid lines are the fitting curves using Gaussian functions. (c) and (d) are the corresponding simulation results based on the trajectory model. (e) and (f) are the initial transverse momentum distributions for the O2/Xe and N2/Ar pair at certain molecular orientations, i.e., ϕO2 = 0° and ϕN2 = 90° with respect to the linear polarization of the driven field along 0°.
To gain a deeper understanding of the mechanisms of the atomic and molecular RSE, a simple semi-classical model was employed. The wave functions of the highest occupied molecular orbitals (HOMOs) of N2 and O2 were described as linear superposition of the atomic orbitals [29–32],
Here R is the vector that points from one atomic center to the other, p is the momentum of electron. The molecular orbital will affect the initial conditions of the ejected electron at tunnel exit [33,34]. As shown in Eq. (1), a cosine term is in the wave function of N2. For molecular orientation perpendicular to the polarization direction, it is easy to realize that the ejected electron from N2 tends to have a zero initial transverse momentum along the polarization direction, i.e. cos(p·R/2) = 1 for p⊥R. But the electron from O2 prefer to have a nonzero initial transverse momentum because of the sine term in its wave function (sin(p·R/2) =0 for p⊥R). Recently, the molecular quantum trajectory Monte Carlo model (MO-QTMC) has been proposed under the framework of strong field approximation (SFA) [31,32] in which the static Ammosov-Delone-Krainov (ADK) ionization rate has been extent to the molecular frame though multiplying a molecular dependent modification factor. Such ionization rate shares the similar concept of molecular orbital (MO)-ADK theory that the initial molecular wave function can be expressed as a linear combination of atomic wave functions, which was widely-used nowadays [35]. Unlike the traditional ADK rate, the ADK rate derived from the SFA not only has a modulation on the initial transverse momentum distribution but also proved to be orientation dependent. Based on this, the ionization rate for molecules was given by W(t0,py) = a2W0(t0)W⊥(py) [31,32]. Linearly polarized laser field is applied in our simulation. The initial momentum distribution is calculated using a quasi-static formula where the laser field ellipticity only affect the strength of the electric vector along the major axis and does not have obvious influences on the initial momentum distribution of electrons. Figures 3(c) and 3(d) show simulation results of the ellipticity dependence of strong-field Rydberg state excitation of N2, Ar, O2 and Xe averaged over various molecular orientations which are in good agreement with the experimental results. Specific molecular orientations were selected in the calculation to show the main difference of the orbital character induced initial transverse momentum distributions in various molecules. Within the SFA [36], the direct transition amplitude from field-free bound state to Volkov state can be expressed asMoreover, we conducted a simple trajectory simulation according to Landsman’s work [38]. The succeeding particle dynamics was simulated in the FTI picture. The electrons launched with a certain initial momentum distribution at the tunnel exit are afterwards driven by the action from the laser field. The normalized yields of the RSE events under different laser ellipticities could thus be calculated and are presented in Figs. 3(c) and (d) for the O2/Xe and N2/Ar pairs, respectively. The simulation results are qualitatively consistent with the measured data as shown in Figs. 3(a) and (b). Although in the present simulation the Coulomb effect and the n distribution were not considered, the resulting tendency could already show the main physical picture of the RSE processes. With the present simulation, we show that the RSE could take place when the transverse momentum at the tunneling exit is compensated by the action of the oscillating laser field, such that the electron could be decelerated and recaptured by the Coulomb field.
Figure 4(a) presented the ellipticity dependence of the recapture probability of O2 at two different laser intensities, i.e. at 8×1013 W/cm2 and 3×1014 W/cm2 W/cm2, respectively. The ellipticity dependence of the Rydberg yields at higher intensity showed a narrower distribution than that obtained at lower intensity. Such effect agreed with the mechanism that the initial transverse momentum and the drift momentum obtained from the laser field are compensated in the FTI process. As the intensity increases, the component of the laser electric field in the transverse direction will increase, therefore the initial momentum would not be large enough to cancel the drift momentum caused by the oscillating laser field. The corresponding RSE yield would decrease faster at higher ellipticities. These would result in a narrower width for the ellipticity dependence of the RSE at higher laser intensity.
Fig. 4. The (a) experimentally measured (-0.3 < ɛ < 0.3) and (b) numerically simulated (-0.15 < ɛ < 0.15) ellipticity dependence of the RSE yield of O2* at two laser intensities, i.e. 0.8 and 3.0×1014 W/cm2.
4. Conclusion
In summary, we experimentally and theoretically investigated the ellipticity dependence of the Rydberg yields of two pairs of targets, in which the atomic and the molecular species exhibit similar ionization potentials [15]. We found the generation probability of the neutral Rydberg fragment O2* was orders of magnitude higher than that of Xe* due to the butterfly-shaped molecular orbitals of O2 in which the strong field ionization was highly suppressed. The PEPICO measurements showed distinct ellipticity dependence of the RSE yields for different molecular species at relatively low laser intensities. For the O2 molecule featuring butterfly-shaped orbitals, the ellipticity-dependent distributions of the RSE yield showed a larger width comparing to that obtained for its counterpart atomic target Xe. While the distribution width of the ellipticity dependent yield of N2* was somehow smaller than Ar*. These results could be explained by simple classical simulations considering the effect of molecular orbitals. It was indicated that the compensation of the initial transverse momentum of the freed electron and the drift momentum obtained from the external optical field could lead to the observed ellipticity dependence. Similar behavior has also been found in the HHG process in previous work [18]. Together with it, our experiments could serve as a powerful proof for exploring the FTI processes in the tunneling regime. Our work revealed the effect of molecular orbitals in the RSE process for various atomic and molecular species. Theoretical models involving Coulomb effects would be required to improve the simulation results.
Funding
National Key Research and Development Program of China (2018YFA0306303); National Natural Science Foundation of China (11621404, 11704124, 11834004, 11904103, 92050105); the Project supported by Science and Technology Commission of Shanghai Municipality (19JC1412200, 19ZR1473900, 21ZR1420100).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
Data availability
The data underlying the results presented herein are not publicly available currently but can be obtained from the authors upon reasonable request.
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