ILLUSTRATION BY JEFF KOEGEL
Let's take the model one step further and imagine that we have thousands of such atoms, all emitting radiation in phase like a phased-array radar. Although still weak, the radiation should now be detectable. But how do we pull the electron away from the atom and then drive it back? We use an intense femtosecond laser pulse.
The intensity achieved by focusing today's standard tabletop amplified femtosecond laser systems can easily exceed 1016 W/cm2. At these intensities, the electric field associated with the laser pulse can exceed the field that binds electrons to atoms. Quantum mechanically, the electron does not wait until the laser field overwhelms the binding potentialthe electron tunnels through the potential barrier.
Once the electron is free of the attractive field of the atom, its motion is determined entirely by the oscillating laser field. Many trajectories could bring the electron back to the parent atom, but not without additional kinetic energy that has been transferred from the laser field. The electron can recombine with its parent atom, and in doing so it emits a single photon in the soft x-ray range.
This process is known as high-harmonic generation (see figure 1).1 A gas irradiated by an intense femtosecond laser pulse will radiate a collimated, coherent beam of extreme UV (EUV) radiation in a spectrum of lines at odd multiples of the driving laser frequency. This spectrum can extend to the biologically significant "water window" at 4.4 nm (280 eV). High-harmonic radiation is emitted in a series of attosecond EUV bursts, and it is responsible for the extension of laser technology from the femtosecond to the attosecond regime.
Figure 1. In high-harmonic generation, an intense terawatt femtosecond laser pulse is focused into a gas jet, which emits a collimated beam of EUV radiation collinear to the laser beam. A spectrometer records the EUV spectrum, which consists of a set of evenly spaced lines corresponding to the odd harmonics of the incident laser frequency.
We have considered high-harmonic generation as a source of coherent EUV radiation with attosecond pulse duration. Buried in the spectrum, however, is information about the atoms that produce the radiation. We will show that we can use this seemingly unlikely technique to image a single molecular orbital wave function.
The process of tunnel ionization using an IR laser is very nonlinear. Only the electron with the highest orbital energy will be removed from the atom. This is different from photoionization, which uses higher-energy photons and may remove inner-shell electrons.
We can describe the high-harmonic process using a three-step semi-classical model (see figure 2). First, an electron is tunnel ionized by the laser field. Second, the laser field accelerates the electron, and then drives it back to its parent. Finally, the electron recombines, emitting the kinetic energy that it acquired as an EUV photon.
It is the third step that enables us to image the orbital. Because of the shape and symmetry of the orbital (1s, 2p, 3d, etc.), electrons with different kinetic energies are more or less likely to recombine and radiate. The high-harmonic spectrum is modulated gently, depending on the orbital that produced it.
Mathematically, the radiated spectrum depends on the overlap integral between the bound-state wave function and a set of plane waves in the continuum. The plane waves, which cover a range of wavelengths, correspond to the kinetic energies of the returning electron. In effect, the high-harmonic spectrum contains the 1-D spatial Fourier transform of the bound-state orbital wave function.
If we could then rotate the atom in space, we could take projections of the orbital shape from different directions and then build up its 3-D structure. Atoms have no "handle" that lets us rotate them, however. Molecules, in contrast, have a spatial structure that makes it possible to align them in a particular direction, even in the gas phase.
Laser alignment of molecules has been perfected for many linear molecules using a process related to optical tweezers (see oemagazine, Jan. 2003, p. 42). A molecule irradiated by an IR laser tends to align its long axis parallel to the laser polarization axis. This technique can align certain molecules in 3-D and can even orient asymmetric molecules.2
We work with a technique called impulsive alignment that uses a short laser pulse that does not align the molecules immediately. The laser pulse creates a coherent super-position of quantum rotational states in the molecules. A quantum-mechanical "revival" occurs after a number of picoseconds, at which time the molecules briefly align parallel to the polarization vector of the laser field that has long since turned off. This is like a photon echo.
Using a set of beamsplitters, we can cause our femtosecond laser pulse to produce a pump pulse that creates the molecular alignment. We use a delayed probe pulse to ionize the aligned molecular sample and generate the high-harmonic spectrum. Wave plates allow us to rotate the polarization vector of both pulses at will.
If the third step of the three-step model contains information about the orbital shape, how can we remove the influence of the first two steps? We repeat the experiment, using a reference atom with a known orbital shape. We then divide the spectra from the unknown molecular sample by the reference spectrum. This reference process removes experimental factors such as the EUV spectrometer calibration, revealing only the pure molecular response. We then record the high-harmonic spectra for a number of alignment angles.
We use a 1-TW titanium-doped sapphire laser generating 27-fs pulses at 50 Hz. In experiments, we have obtained harmonic spectra from nitrogen molecules at a range of angles, normalized to the harmonic spectrum from argon, the reference atom (see figure 3).3 We can recover the shape of the nitrogen orbital by a procedure based on computed tomography. Medical tomography involves recording a series of 1-D radiographic projections through a body and then building a 2-D image slice by manipulating the 1-D data. In the most basic computed-axial-tomography algorithm, a spatial Fourier transform is taken of the 1-D projections. The inverse Fourier transform in 2-D spatial frequency space then gives the 2-D slice.
Figure 3. Obtaining this polar plot of harmonic intensities at various angles for a nitrogen molecule (top) involved recording the high-harmonic spectra for a set of 19 different angles of the molecular axis relative to the laser polarization axis, then normalizing that data to a spectrum from argon atoms to remove the calibration factors. Using a computed tomography algorithm, we converted the data to a 3-D image that represents the wave function of a single electron orbital of the highest occupied state of nitrogen (bottom). Red represents positive values of the wave function, and blue represents negative values.
As mentioned above, the high-harmonic spectra already contain the spatial Fourier transform of the molecular orbital projection. Applying an inverse Fourier transform to all the projections thus will give the 3-D shape of the molecular orbital. In the case of nitrogen, the shape features a central lobe and two outer lobes of opposite sign; the shape is very close to the calculated shape of the 3σg orbital of nitrogen (see figure 3). The nodes in between the lobes correspond to the location of the nitrogen atoms in N2. This shape is very close to the calculated shape of the highest occupied molecular orbital in N2, based on Hartree-Fock self-consistent-field calculations.
It is surprising that a technique based on using an intense femtosecond laser to ionize a molecule can yield the shape of a single-electron orbital wave function. Not only have we isolated a single orbital from among the 13 other electrons in N2, but we also have measured the wave function itself, not its square. The square of the wave function gives the probability distribution of the electron, but the wave function itself is usually not considered to be observable.
The ability to observe a single orbital wave function relies on several fortuitous properties of high-harmonic generation. First, tunnel ionization selectively chooses the highest occupied orbital, thus making it selective to the orbital that is most important for the chemical properties of molecules. Second, the process repeats coherently at each of the thousands of molecules in the laser focus, and the process of phase-matching means that only coherent processes will add constructively to give the strong EUV emission.
The entire measurement process occurs with a 30-fs laser pulse. Using standard pump-probe techniques, it is possible to initiate a uni-molecular reaction and then observe how the molecule evolves over a period of time with a temporal resolution of 30 fs and a spatial resolution of 0.1 nm. It should be possible, for example, to see the orbital change as a molecule rearranges itself. It might even be possible to observe the motion of a single electron within an atom. Whereas the atoms within molecules move on a time scale of tens of femtoseconds, the electrons within an atom move on an attosecond time scale. It has been proposed that the high-harmonic spectrum from an atom produced by a carrier-envelope, phase-stabilized few-cycle laser pulse of 5-fs duration can reveal attosecond electron motion.
As with many things in science, you often find something of interest when you are looking for something entirely different. It is unlikely that one would think of using an intense femtosecond laser pulse to ionize a molecule as a means of observing a single molecular orbital. This is what fundamental research is about. oe
David Villeneuve is a senior research scientist at the National Research Council of Canada, Ottawa.
1. M. Lewenstein et al., Phys. Rev. A 49, p. 2117 (1994).
2. H. Stapelfeldt and T. Seideman, Rev. Mod. Phys., 75, p. 543 (2003).
3. J. Itatani et al., Nature 432, p. 867 (2004).