Laser-Induced Electron Tunneling and Diffraction

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Science  13 Jun 2008:
Vol. 320, Issue 5882, pp. 1478-1482
DOI: 10.1126/science.1157980



Molecular multiphoton ionization in the tunneling limit (sketched in the upper frame of Fig. 1) is similar to tunneling in a scanning tunneling microscope (STM) (1). In both cases, electrons escape from the outer regions of the orbital to the continuum; that is, to the vacuum for multiphoton ionization of gas phase molecules or to the conduction band of the metal tip in a STM. In a STM, the sample is fixed and the tip is moved. Rotating the molecule with respect to the field direction is the analog of moving the tip. The resulting angle-dependent ionization probability (25) provides information for a molecule analogous to the position dependence of the tunneling current in a STM. However, whereas the total tunneling current is one observable, the electron wave packet that emerges into the vacuum from the tunnel retains more information about the orbital.

Fig. 1.

(Top) Tunneling creates a filtered projection of the molecular orbital. Approximately half of the electron wave packet escapes directly to the detector. (Bottom) Remaining portion is driven back to the parent ion. Here, the central portion of the recolliding wave packet is shown diffracting from the molecule. The outer portions of the wave packet (not shown) weakly interact with the ion potential. The relative strength of each component depends on the filtered projection of the molecular orbital and therefore on molecular alignment.

In contrast to the static field of a STM, the electric field in a laser pulse oscillates and forces a fraction of the tunneled electron wave packet back to the parent ion. There, the wave packet can diffract from the molecule (lower frame in Fig. 1). This phenomenon has been called laser-induced electron diffraction (LIED) (6) and relates to recent research on ultrafast electron diffraction, where a femtosecond electron bunch is created at a photocathode and accelerated in an electrostatic field onto a molecular (7) or solid state (8) target. In our case, the molecule serves as its own photocathode, whereas the laser provides the accelerating field. This situation produces extremely high current densities and attosecond timing (9).

We report the observation of molecular tunneling spectroscopy and LIED. We measured electrons produced from aligned O2 and N2, resolving their three-dimensional (3D) momentum distribution. Comparing experiment and theory, we show that, in the two dimensions perpendicular to the field direction, the momentum distribution for these direct electrons is determined by the highest occupied molecular orbital (HOMO), observed through the filter of the suppressed binding potential (Fig. 1) through which the electron tunnels. We also demonstrate LIED (6) and confirm its origin with a simulation. Selecting the wavelength of the recollision electron modifies the diffraction pattern. Thus, one set of measurements simultaneously identifies the orbital wave function of the molecule and the position of the atoms in the molecule.

Laser-induced tunneling and diffraction exploit different parts of the ionizing electron wave packet. The fraction of the electron wave packet that is created while the field strength increases within an optical half-cycle departs directly and irrevocably from its parent ion. Only tunneling stands between the orbital and the electron detector. The wave packet that is born while the field decreases returns to the ion where it can elastically scatter (diffract), inelastically scatter, or recombine to (interfere with) the orbital from which it was extracted (10). These three scattering processes offer different perspectives on the molecule.

Tunneling, the process underlying all, probes the electronic structure of the neutral molecule. The recombination radiation, known as high-harmonic radiation, also measures the orbital structure of the neutral molecule (11). However, because high-harmonic generation starts with tunneling and ends with interference, these processes must be disentangled before the techniques can be generalized to complex orbitals. In contrast, elastic and inelastic scattering occur at the molecular ion. Inelastic rescattering is closely related to field-free collision physics (12, 13). It can cause multiple ionization and subsequently lead to Coulomb explosion of small molecules. The molecular structure can then be inferred from the momentum vectors of the correlated ionic fragments. Elastic scattering is also sensitive to the potential structure of the molecular ion. However, here the molecular structure is encoded in the diffracting electron wave packet, making this imaging technique scaleable to more complex molecules.

For electron diffraction to be observable, the de Broglie wavelength of the electron needs to be on the order of the dimensions of the molecule. Small diatomic molecules have a bond length of ≈1 Å [1.9 atomic units (au)]. To obtain this wavelength, an electron would need a kinetic energy of 150 eV, corresponding to a momentum of 3.3 au. Electrons that are accelerated in the laser field and recollide with the parent molecule can easily reach this range of kinetic energies (10).

We recorded the momentum of electrons arising from tunneling ionization of aligned O2 and N2, employing a Cold Target Recoil Ion Momentum Spectroscopy (COLTRIMS) system (14). COLTRIMS allows us to measure the 3D momentum vectors of single electrons and ions in coincidence. This coincidence capability is used to measure both O2 and N2 simultaneously under identical conditions by means of a gas mixture. It also allows us to eliminate any contribution to the electron spectra from inelastic scattering resulting in fragmentation or multiple ionization of the molecules. However, with monomolecular targets, we achieve very similar results even without the ion-electron coincidence requirement. Finally, the coincidence measurement permits us to pick up weak fragmentation channels that provide information about the quality of our alignment [see supporting online material (SOM) text B].

Our laser pulse (800 nm) had a total energy of 5 μJ (split and used for both alignment and probing) and a repetition rate of 30 kHz. We align the molecules (15) using a slightly stretched (60 fs), moderately intense (≤8 × 1013 W/cm2) laser pulse. The intense (2.5 × 1014 W/cm2), short (40 fs), ionizing pulse, which is applied after a delay, catches the molecule at any alignment that can be achieved with polarized light. Experimentally, aligning (pump) and ionizing (probe) pulses are perpendicularly polarized (y axis and z axis in the inset of Fig. 2A, respectively). The pump pulse creates a rotational wave packet (15) that results in molecules aligned primarily along the y axis at a well-defined time delay (sketch in Fig. 2A). At a different time delay, the alignment distribution will be confined in the xz plane (sketch in Fig. 2B). We refer to this complementary case as anti-alignment. A computerized delay stage moved between the respective pump-probe delays every 10 s (details in SOM text A and B).

Fig. 2.

Measured 3D electron momentum distributions projected onto three orthogonal planes. (A and B) Raw electron distributions for aligned and anti-aligned O2 molecules, respectively. (Insets) Sketches of the molecular alignment distributions. The aligning pulse (green) is polarized along the y axis, whereas the ionizing laser polarization (red) lies along the z axis. Spectra are not symmetrized, and data points in the white area at pz = –2 au are removed because there the spectrometer resolution deteriorates. Each figure represents ≈5 × 108 laser shots, yielding a dynamic range of 105. (C) Difference between (A) and (B) normalized to their sum (normalized difference). (D) Same as (C) but for N2.

The electron momentum distributions (Fig. 2) obtained for aligned (A) and anti-aligned (B) O2 molecules have been correlated with singly charged O +2 ions. For display purposes, the distribution is projected onto three planes. Most prominent in the figure is the large dynamic range in the distribution. The color code covers five orders of magnitude. The rapid fall-off of the signal is so prominent that it is hard to discern anything else from the figure. The spectra for N2 (fig. S1) are qualitatively indistinguishable from those of O2.

Molecular specific structure arising from tunneling and diffraction is revealed when we employ normalized differences. The anti-aligned projections are subtracted from the aligned projections and then divided by their sum. The normalized differences for O2 and N2 are plotted in Fig. 2, C and D. Because the data are not symmetrized, the quadrant-to-quadrant reproducibility is a measure of the quality of the image.

The low lateral-momentum electrons (Embedded Image, where p is momentum) exhibit clear patterns in the projection perpendicular to the ionizing laser field. The vast majority of these electrons drift directly to the detector without further interaction with the parent ion. They provide information on the ionizing orbital, filtered by the tunnel. These electrons are equally evident in the other two projections. They are spread along the z direction between |pz| < 1.5 au, reflecting the fact that ionization can occur over a range of times within a field cycle.

Figure 3 presents an expanded view of the px-py projections of the electron momenta from N2 and O2 and focuses on the central part of the distribution. The very different distributions created by tunneling from O2 and N2 reflect the very different structures of the respective ionizing orbitals. In tunneling theory, the tunnel serves as a filter for the perpendicular component p of the orbital wave function. The tunneled wave packet then can be expressed as (in atomic units) (1618) Embedded Image(1) where the exponential factor is the filter function. It contains the ionization potential Ip and the electric field E of the ionizing laser pulse at the moment of tunneling. The exponential factor depends only weakly on the alignment of the molecule (through a weakly angular-dependent Stark-shift). Therefore, it is approximately cancelled in any ratio of two different alignments. However, the electronic wave function |Ψi〉 is locked to the molecular frame, whereas p is defined with respect to the laser-field direction. Therefore, the alignment dependence of the projected wave function 〈pi〉 is accentuated by taking ratios. We now show that this basic tunneling concept agrees with our experimental measurements.

Fig. 3.

px versus py projections of the normalized differences for O2 and N2. (Top) Experimental data (from Fig. 2). The color scale has been changed to emphasize the low lateral-momentum structure near the center of each figure enclosed in the dashed circle. The low lateral momentum retains the symmetry of the HOMO (shown above). (Bottom) Magnified results of our simulation (see text).

Using the wave function for the HOMOs of N2 and O2, we calculated the lateral electron momentum distributions (both shape and amplitude) that emerge from the molecules for each molecular alignment. The electrons are given initial conditions determined by tunneling and then classically propagated in the field of a pulse identical to the experiment, taking into account the parent ion's Coulomb potential (19, 20). Finally, each molecular alignment contributes to the predicted spectrum, according to its weight in the measured alignment or anti-alignment distribution. There are no free parameters in this model (described more fully in SOM text C). The results are shown in the bottom row of Fig. 3. The simulation reproduces the symmetry seen in the experiment. The model also shows that imperfect alignment causes the orbital's footprint to disappear in the unnormalized spectra (Fig. 2, A and B). That is, the node is filled by the alignment distribution of ionizing molecules.

We now turn to the structures at higher electron momenta (p > 0.5 au) and (|pz| > 1.5 au) in Fig. 2, C and D. This region is dominated by electrons that have rescattered. In Fig. 2, C and D, the relative probability of finding an electron in the momentum region pz = 2 au, py = 1.5 au in all quadrants and at pz = 0 au, px = 1 au passes through rather broad local maxima. There are corresponding minima at pz = 2 au, px = 1.5 au and at pz = 0 au, py = 1 au. This structure contains information on the scattering potential. That is, we experimentally observed LIED.

Interpreting an electron distribution such as this is different from interpreting conventional electron diffraction because of the presence of the laser field (17). We analyzed our data in analogy to the attosecond streak camera (21) to remove the influence of the laser field on the electron momentum after the scattering has occurred. The final, observed momentum of a scattered electron is Embedded Image (17), where Embedded Image is the vector potential defined by the electric field Embedded Image of the probe pulse (here, ω is the frequency of the light), and Embedded Image is the electron momentum at the time trec of recollision. This analysis yields a manifold of momentum spheres centered at pz = ±Arec with radii prec. The maximum radius prec,max corresponds to a recollision energy of 3.17Up [where Up = (E0/2ω)2 is the ponderomotive potential]. Slicing the momentum spheres at perpendicular planes defined by px ≈ 0 and py ≈ 0 yields diffraction circles such as those illustrated in Fig. 4 (at left). Within each diffraction circle, we can relate our results to conventional electron diffraction: φ is the scattering angle, and the radius of each circle is the electron momentum or wavelength, λ = h/prec (where h is Planck's constant), at which scattering occurs.

Fig. 4.

Left panel is reproduced from Fig. 2C. Three color-coded diffraction circles are superimposed. The radius of the circle defines the recollision momentum prec (scattering wavelength); their width defines the range of momenta that are analyzed; and φ is the scattering angle. The four plots at right show the angular distributions within the indicated diffraction circles (see SOM text D for details). The color coding is preserved. Results of the simulations are superimposed as solid lines. The curves for recollision momentum of 1.0 au have no offset. For clarity, other curves are offset in units of 0.2. The data are shown with statistical error bars (1σ).

The four panels at right in Fig. 4 show the value of the measured electron distributions along diffraction circles of radius 1.4 au (blue), 1.2 au (red), and 1.0 au (black). Each curve is offset for clarity. Electrons in the range 0° < φ < 30° or 330° < φ < 360° are direct electrons and are not plotted in the figure (details in SOM text D).

Because the molecules were aligned along the y axis, the py-pz projection is the most intuitive plane in which to analyze diffraction. At a recollision momentum prec of 1 au, the electron wavelength λ = 6.3 au is larger than twice the equilibrium bond length of O2 (d = 2.3 au). However, numerical simulations (22, 23) and experiments (24) have shown that the appropriate electron wavelength for diffraction is the wavelength at the core of the ion. That is, the recollision energy (corresponding to the radius of the diffraction circle) plus the ionization potential of the HOMO yield the appropriate diffraction wavelength. With this assumption (λ = 4.57 au), the first minimum at around 60° (and then at 300°) thus corresponds to a bond length of 2.6 au.

To help extend this interpretation, we turned to simulation (SOM text E). We used the probability of recollision as a function of the angle between the molecule and the laser electric field and averaged over the alignment distributions. The electron recollision momentum is chosen by selecting a diffraction circle for analysis. We made three final assumptions: (i) We assumed that the electron wavelength corresponds to the sum of recollision energy and the field free ionization energy. (ii) We assumed two-center diffraction. (iii) Because long trajectory electrons dominate, we concentrated on them. The resulting distributions are shown in the solid curves in Fig. 4. Comparing the calculated and measured distributions, we see that the modulation in the regions φ ≈ 50° – 150° and φ ≈ 210° – 310° arises from diffraction. The dashed lines highlight the shift of the diffraction maxima and minima with the electron wavelength. Electrons that backscatter through φ ≈ 180° are not properly described. They appear in the figure in the range φ ≈ 150° – 210°. The overall agreement in the positions of the diffraction minima and maxima between simulations and experiment is good, except for N2 in the px-pz plane, where the simulated diffraction peaks appear shifted. This peak shift probably comes from an overestimation of the recollision probability for N2 molecules oriented parallel to the laser field in the model.

There are three reasons why we obtain agreement for most angles. First, using normalized differences makes the two-center scattering contribution highly visible while making our results somewhat insensitive to details of the atomic contribution (25). Second, by concentrating on relatively high-energy electrons, we only need to consider the first recollision (9), keeping the electron trajectories relatively simple. Third, although the screened Coulomb potential is not fully treated in our diffraction analysis, we add the most important aspect by augmenting the electron energy measured from the scattering circles by Ip (17, 22, 23).

We have demonstrated that multiphoton ionization in the tunneling limit gives extensive information on the ionizing electronic orbital. If we rotate the molecule in small steps, all momenta from the wave function are sampled. We only need to select one angle for diffraction. This diffraction pattern reveals the interatomic dimensions of a molecule. Interestingly, the angle could be selected to favor recollision only from particular electronic configurations, allowing us to determine the structure of selected isomers or of excited states.

In a diffraction experiment, selecting a different electron wavelength λ changes the diffraction pattern in the same way as a change in the internuclear separation d (sinφ = nλ/d, where n is the order of the diffraction peak). By showing that a 20% change in electron wavelength makes a measurable difference in the diffraction pattern, we indirectly demonstrate that vibrational or photochemical dynamics can be combined with LIED. Coincidence imaging allows us to use the characteristics of the correlated ion to select those events that we wish to analyze. Thus, LIED provides a method to isolate structural changes of any excited state that impresses a distinct signature on the ion.

In studies where coincidence imaging is unnecessary, employing a technology such as velocity map imaging (26, 27) would allow electron spectra to be recorded using lasers with repetition rates as high as 136 MHz (28). Data of the same quality that we show could be obtained in milliseconds or less. Such a measurement is practical for any molecule that can be placed in the gas phase, provided that the strong field does not dislodge other electrons in an uncontrolled manner (29) and provided that the molecule can be aligned with sufficient precision.

Finally, we stand back and survey the emerging strong-field technology for molecular imaging. Tunneling, one of the most fundamental of the quantum mechanical processes, initiates all of the methods. Information about the ionizing orbital of the neutral molecule is imprinted on the tunneled electron. A second perspective on the molecular orbital is contained in the high-harmonic radiation that is produced upon recollision. High-harmonic radiation has two major advantages over tunneling. First, by measuring photons rather than electrons the problem of space-charge is avoided. Second, if the tunneled wave packet is independent of the angle between laser-field and molecular axis, tomography provides a well-developed algorithm for inverting the image (11, 30). However, there is one major disadvantage: Tunneling and recombination are entwined in the angle-dependent harmonic spectrum (31). This problem has so far limited the application of orbital tomography.

A complementary insight is achieved by laser Coulomb explosion imaging (3234) We become sensitive to the position of the nuclei in a molecular ion. LIED supplies us with the means to overcome the limitations of Coulomb explosion imaging and opens the pathway toward the imaging of larger molecules. Taken together, we can obtain a full picture of molecules. With the extension of these tools to pump-probe techniques, we will be able to simultaneously trace the temporal changes in the nuclear and electronic structure of molecules.

Supporting Online Material

SOM Text

Figs. S1 to S3


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