1. Introduction

Photoionization of atoms and molecules is the pioneering process in strong-field physics, where the above-threshold ionization (ATI) illustrates a well-known scenario that the electrons therein can absorb multiple photons from strong laser fields, even beyond the minimum number required to overcome the ionization potential. The ATI photoelectron kinetic energy spectrum was first observed by Agostini et al. in 1979 [1] and has been utilized to investigate photoelectron circular dichroism [2], few-cycle carrier-envelope phase characterization [3], laser-assisted electron scattering [4,5], spin-orbit coupling of noble-gas atoms [6–8], and laser-induced molecular dissociation [9,10] over the past four decades.

Beyond the energy-resolved spectra, photoelectron angular distributions (PADs) provide abundant dynamical information on the resonant intermediate and continuum states. The fine structures in the PAD, e.g., the multiple concentric circles shaped with regular nodes, have been observed in previous experiments [11–15]. The regular nodal structure is related to the significant angular momentum of the freed electron [16–20] in the frequency domain, which can also be understood as the intracycle interference of the released electronic wave packets [21–25] in the time domain.

As the bound electron is liberated in intense laser fields, it experiences a quiver motion driven by the oscillating electric field, gaining an average quiver energy known as ponderomotive energy (U_p). The ponderomotive potential gives rise to the energy up-shift of continuum states of the electron, which results in a field-dressed ionization potential compared to the field-free one. The laser-intensity-dependent effect can be decoupled in single ionization of atoms by considering the peak of the laser pulse at the ionization instant approximately. However, considering subsequent laser interactions towards different reaction pathways, it is hard to identify the ionization instant and thus the absorbed photon number in molecules due to the synergy of the field-strength-dependent ionization-potential rise and the chemical-bond stretching. The two-fold modulation on the photoelectron kinetic energy spectra hinders further investigations on the laser-molecule interaction, such as the attosecond time delay of molecular photoionization [26], electron-nuclear energy sharing of molecules [27–29], and charge-resonance-enhanced ionization of molecules [30–33].

In this article, we experimentally investigate the photoelectron momentum distributions (PMDs) of different reaction channels in H₂ molecules driven by an orthogonally polarized two-color (OTC) laser pulse. The OTC scheme with its sculptured waveform was employed to control the emission direction of photoelectrons and thus the momentum distributions by altering the relative phase [34–40]. Here, the two-dimensional photoelectron spectroscopy upon the OTC scheme takes advantage of different photon energies and transition selection rules, providing a powerful tool to analyze the involved electron continuum partial waves and absorbed photon numbers in the molecule. The comparison between the dissociative and non-dissociative channels shows that an energy shift of ATI peaks and different ionization pathways are deduced from the observed PADs. It is attributed to the fact that the photoionization of the dissociative channel with an extra time advance invoked by the bond stretching occurs at the rising edge of the laser pulse, earlier than that of the non-dissociative channel which mainly occurs at the pulse peak.

2. Experimental method

Experimentally, a linearly polarized fundamental wave (FW) pulse (25 fs, 790 nm, 10 kHz, polarized along the z axis) derived from a multipass Ti:sapphire amplifier was down-collimated into a 150-µm-thick β-barium borate (BBO) crystal to generate a second harmonic (SH, polarized along the y axis) pulse centered at 395 nm. We produced the phase-locked orthogonally polarized two-color laser field in a collinear scheme [41]. The time lag between the FW and SH pulses was compensated by a birefringent α-BBO crystal. A pair of fused silica wedges was used to tune the relative phase between the two-color pulses finely. The experimental results shown throughout the article are integrated over the relative phase of the two colors, which actually does not alter the positions of the discrete ATI peaks in the photoelectron kinetic energy spectrum and the number of nodes in the angular distributions. As shown in Fig. 1(a), the generated OTC laser pulse was tightly focused onto a supersonic gas jet of H₂ by a concave silver mirror (f = 7.5 cm) inside an ultrahigh vacuum chamber of the cold-target recoil ion momentum spectrometer [42,43]. The produced photoelectron and nuclear fragments were accelerated and guided by a static electric field (∼ 3.8 V/cm) and a magnetic field (∼ 6.1 G). The positions of the charged particles were detected by two time- and position-sensitive microchannel plate detectors at the opposite ends of the spectrometer, from which the three-dimensional momenta of the electrons and nuclear fragments were reconstructed. The peak intensities of the FW and SH laser pulses in the interaction region were estimated to be I_FW ∼ 8.3 × 10¹³ W/cm² and I_SH ∼ 6.0 × 10¹² W/cm², respectively.

 

Fig. 1. (a) Schematic diagram of the experiment apparatus. The OTC pulse propagating along the x axis is back-focused by a concave mirror onto the molecular beam propagating along the y axis. The emitting electrons and charged nuclear fragments such as H₂⁺ and H⁺ are guided by the external electric field and magnetic field and measured by detectors at the ends of the spectrometer. The polarization of FW and SH waves is along the z and y axes, respectively. Measured PMDs of (b) non-dissociative and (c) dissociative single ionization of H₂, where the white dashed rings highlight the contribution of the first- and second-order ATI.

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3. Results and discussions

To extract corresponding electron events, we investigate the following two reaction channels [44]. The first one is non-dissociative single ionization of H₂ expressed as

Figures 1(b) and 1(c) display the measured two-dimensional PMDs of the two channels in the y-z polarization plane. There are many ATI rings existing in the PMDs shaped by multiple nodes. These nodal structures imply that the released photoelectrons emit along these directions, characteristic of partial waves with specific angular momenta [45]. For the PMD of H₂⁺ channel shown in Fig. 1(b), the number of nodes from the first- to fourth-order ATI is 10, 16, 18, and 20, respectively, while the number of nodes from the first- to forth-order ATI in the PMD of H10 channel shown in Fig. 1(c) is 12, 18, 20, and 22, respectively. Both two channels demonstrate the same phenomenon that the number of nodes in each order ATI increases linearly with the order except for the first one. The counterintuitive experimental observation is beyond the previous understanding of the ATI structure, the details of which will be discussed later.

A distinct discrepancy between the two channels occurs that the PAD of each order ATI of H10 channel owns two more nodes than the one of H₂⁺ channel, which, undoubtedly, implies the participation of different partial waves. This finding breaks the well-accepted assumption that the first ionization step of H10 channel happens at the same time as the one of H₂⁺ channel, i.e., the dissociation is regarded as the subsequent process of the observed non-dissociative single ionization event, which is mainly considered as the origin of a semiclassical model [46–49].

To further investigate the underlying physics of the experimental observations, we analyze the corresponding photoelectron kinetic energy spectra, as shown in Fig. 2. From the numerical fitting via multiple Gaussian functions, we can extract equidistant ATI peaks with the FW photon energy apart, regardless of the contribution of sharp Freeman resonant peaks [50]. There is an apparent offset in the photoelectron kinetic energy spectra of the two channels, where the ATI peaks of H10 channel are 0.55 eV larger than the ones of H₂⁺ channel. The kinetic energy of the released photoelectron satisfies

 

Fig. 2. Measured photoelectron kinetic energy spectra of (a) non-dissociative and (b) dissociative single ionization of H₂. Blue circles and violet solid curves indicate the measured kinetic energy spectra and the fitting curves via multiple Gaussian functions, respectively. Violet areas denote the individual ATI regions in the fitting.

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We now discuss the underlying physics of the non-dissociative single ionization of H₂, analogous to the multiphoton ionization of atoms. The single ionization mainly happens at the peak of a femtosecond laser pulse with the Gaussian envelope. Hence, the corresponding ponderomotive energy can be calculated to be 4.9 eV via Eq. (2), taking the peak intensity of the FW wave into account. It is important to mention that the U_p of the weak SH wave (∼ 0.08 eV) is negligible compared to the one of the strong FW wave. The effective ionization potential of H₂ is 16.2 eV considering the population around the vibrationally excited state (v’ = 3) of the ionization-created H₂⁺ in the Frank-Condon region, different from the conventional ionization potential of 15.4 eV (from v = 0 of H₂ to v’ = 0 of H₂⁺) [51], and the photoelectron kinetic energy of the first-order ATI in H₂⁺ channel is 0.6 eV. According to Eq. (1), the energy of 14$\omega$_FW is required for the electron to overcome the ionization potential and the ponderomotive energy as well. After confirming the absorbed energy, we further infer the partial wave with the help of phase-dependent measurement. Figure 3 shows the polar plots of PADs of the first- and second-order ATI in two reaction channels integrated over the corresponding energy region. We found that the photoelectron emitting near 90° (along the vertical axis) undergoes a phase step of $\pi$ (not shown here), which is characteristic of the interference between odd-order and even-order partial waves [52,53]. Thus, the quantization axis is determined to be the polarization direction of the SH wave (along the horizontal axis). Based on this, we can refer to the possible partial waves of each order ATI. Taking the first-order ATI in H₂⁺ channel as an example, there are 10 nodes in the PAD shown in Fig. 3(a), sharing the analogous structure with the spherical harmonics function of Y₅₀. According to the selection rule of dipole transitions and the total energy of absorbed photons, we can deduce that the photoelectron of the first-order ATI is generated by resonant multiphoton ionization via intermediate states with the orbital angular momentum of l = 5, where a series of bound states are involved, i.e., Y₅₀, Y₅₂, Y₅₄ (represented by the typical one of Y₅₀ hereafter for convenience). After the population on the resonant state of Y₅₀, the electron further absorbs 2$\omega$_FW or 1$\omega$_SH and is released to the continuum, giving birth to the partial wave of Y₅₀ or Y₄₀, respectively. For the photoelectron with the same kinetic energy, the absorption of the SH photon with duple photon energy of the FW pulse will alter the orbital angular momentum of the emitted photoelectron and introduce multiple pathways for interference. Figure 3(b) shows the PAD of the second-order ATI in H₂⁺ channel, where the 16-node structure is considered as the interference of Y₈₁ and Y₇₁ via absorbing 3$\omega$_FW and 1$\omega$_FW + 1$\omega$_SH, respectively.

 

Fig. 3. (a), (b) Polar plots of the PADs of the first- and second-order ATI in non-dissociative ionization integrated over the corresponding energy region. (c), (d) The same as (a), (b) but for dissociative ionization.

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While for the ionization process of H10 channel, it is more complicated to determine how much photon energy is absorbed considering different ionization instants. According to the PAD of the first-order ATI shown in Fig. 3(c), there are 12 regular nodes, which is related to the partial wave of Y₆₁. Thus, the PADs of H10 channel demonstrate opposite polarity compared to the ones of H₂⁺ channel shown in Fig. 3(a). Regarding the minimum energy around 14$\omega$_FW absorbed in H₂⁺ channel, the minimum absorbed energy in the first-step ionization of H10 channel should be an odd number of $\omega$_FW to satisfy the polarity, such as 11$\omega$_FW, 13$\omega$_FW, and 15$\omega$_FW. The electron kinetic energy of the first-order ATI is extracted to be 1.15 eV from the fitting in Fig. 2(b). If the electron of H10 channel absorbs the energy less than 11$\omega$_FW, a negative ponderomotive energy is deduced according to Eq. (1), which is impossible for the photoelectron occurring in strong laser fields. If a photon energy of more than 15$\omega$_FW is absorbed during the ionization process, the corresponding ponderomotive energy is 5.8 eV, which exceeds the maximal ponderomotive energy at the peak intensity in our experimental conditions. Therefore, we can determine that the minimal energy absorbed by the photoelectron in H10 channel is 13$\omega$_FW. Hence the ponderomotive energy is deduced to be 2.8 eV, which means that the ionization process in dissociative ionization happens at a lower laser intensity instead of the peak intensity. According to Eq. (2), we can infer two ionization instants from certain ponderomotive energy except for the maximum, i.e., one at the leading edge of the pulse envelope and the other at the falling edge. In addition, since the subsequent dissociation process in H10 channel demands a specific bond-stretching time and enough laser intensity to provide additional photons, the ionization instant mainly populates at the leading edge instead of the falling edge. Considering the 2.1-eV energy difference of the ponderomotive shift of the ionization threshold between the dissociative and non-dissociative single ionization channels, which exceeds the one-FW-photon energy of 1.55 eV, it is hardly possible to observe the same Freeman resonance peak in these two channels.

After determining the photon energy absorbed in H10 channel, we can further investigate the specific ionization pathways. The photoelectron of the first-order ATI is produced via the excitation of the resonant intermediate state with the orbital angular momentum l = 6, where Y₆₁, Y₆₃, and Y₆₅ are involved and represented by the typical one of Y₆₁ hereafter for convenience. After the population on the intermediate state, the ionization pathway is similar to that of the non-dissociative ionization. The electron can absorb 2$\omega$_FW or 1$\omega$_SH and transit to the continuum, giving birth to the partial wave of Y₆₁ or Y₅₁, respectively. Figure 3(d) shows the PAD of the second-order ATI in H10 channel, where the 18-node structure (not apparent in the energy-integrated PAD) is considered as the interference of Y₉₀ or Y₈₀ via absorbing 3$\omega$_FW and 1$\omega$_FW + 1$\omega$_SH, respectively.

According to the photoelectron kinetic energy spectra and corresponding PMDs, we obtain the possible ionization pathways of the two reaction channels, providing new insight into these ultrafast processes in H₂. The schematic illustration of the two channels is depicted in Fig. 4. Figure 4(a) shows that the ionization of H₂⁺ channel happens at the peak intensity of the laser pulse via the resonance of an intermediate state with the orbital angular momentum l = 5. Here the photon energy absorbed from the ground state to the resonant intermediate state is deduced to be 12$\omega$_FW, while the excitation pathways with different FW and SH photons are coherently interfered with and thus unable to be determined. The excited electron further absorbs 2$\omega$_FW or 1$\omega$_SH to overcome the field-dressed ionization potential (including the field-free ionization potential and the ponderomotive energy), being released to the continuum and forming the first-order ATI composed of Y₅₀ and Y₄₀. When the excited electron absorbs 3$\omega$_FW or 1$\omega$_FW + 1$\omega$_SH consecutively, the second-order ATI will be formed via the interference between Y₈₁ and Y₇₁. It is worth mentioning that the second-order ATI cannot be obtained by merely absorbing one more $\omega$_FW from the first-order ATI due to the nonlinear variation of the node number.

 

Fig. 4. Schematic illustration of the ionization and subsequent dissociation of H₂ molecules. The highlighted ionization pathway diagram of (a) non-dissociative and (b) dissociative single ionization, where the orange shadow indicates the continuum state. (c) The potential energy curves of neutral H₂ and ionic $\textrm{H}_2^ + $ molecules, where the 1$\omega$_SH − 1$\omega$_FW dissociation pathway is depicted. The potential energy curve is adapted from [51], and the dissociation limit of H⁺ + H is set to zero. (d) Measured nuclear kinetic energy spectrum for the dissociative single ionization of H₂. The orange shadow highlights the energy region of the 1$\omega$_SH − 1$\omega$_FW dissociation pathways. Multiple peaks highlighted with yellow indicate the net-2$\omega$_FW dissociation pathway jointly contributed by adjacent vibrational states.

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However, the bound electron in H10 channel absorbs 11$\omega$_FW and transits to an intermediate state with the orbital angular momentum l = 6 at the leading edge of the laser pulse with a smaller ponderomotive energy, as shown in Fig. 4(b). The subsequent ionization pathway is similar to the one of H₂⁺ channel, and here the photoelectrons of the first- and second-order ATI demonstrate the same nonlinear characteristic. For the photoionization-created NWP in H10 channel, it can move outwards along the 1s$\sigma$_g curve, absorb extra photons from the OTC field, and generate various dissociation pathways, where the major one is schematically shown in Fig. 4(c). In this 1$\omega$_SH − 1$\omega$_FW dissociation pathway highlighted with orange in the nuclear kinetic energy spectrum in Fig. 4(d), the stretching H₂⁺ absorbs one SH photon at the resonant internuclear distance, transits to the 2p$\sigma$_u state, and moves outwards along the 2p$\sigma$_u curve, followed by the emission of one FW photon, transition back to the 1s$\sigma$_g state and dissociation along the 1s$\sigma$_g curve. It is important to mention that we have examined the minor net-2$\omega$_FW dissociation pathway, i.e., 3$\omega$_FW − 1$\omega$_FW pathway with the three-FW-photon absorption and subsequent one-FW-photon emission, followed by dissociation along the 1s$\sigma$_g curve [46–48,54], by selecting the energy region around 1.2 eV and only found a slight photoelectron energy shift due to different initial vibrational states according to the well-known electron-nuclear energy sharing.

4. Conclusion

In summary, we experimentally investigate the photoelectron momentum distributions of dissociative and non-dissociative ionization channels of H₂ using an OTC laser pulse. Based on the two-dimensional photoelectron spectroscopy and the kinetic energy spectrum, we have distinguished the photoelectron releasing order of two reaction channels in molecular systems and confirmed the participation of partial waves with specific angular momenta. Our results verify that the OTC scheme can be utilized to accurately determine the number of absorbed photons during the ATI and the field-dressed ionization potential in the multiphoton ionization of molecules. The approach presented here is applicable to other atomic and molecular experiments, which helps investigate further strong-field electronic dynamics.