To be published in Science Attosecond Science on a Tabletop -

The Quest for Coherent X-Rays

Henry Kapteyn, Oren Cohen, Ivan Christov, and Margaret Murnane JILA, University of Colorado at Boulder

Boulder, CO

Introduction-- what can we learn from attosecond science? It has been 50 years since in laser was first conceived, in a paper in Physical Review by

Schawlow and Townes. This paper, which made the case for the basic physical feasibility of the laser, was a direct extension of concepts first demonstrated in the microwave region of the spectrum, in the maser. The very high frequency and coherence of laser light also made it possible to access extremely short nanosecond, picosecond, and even femtosecond time-scales as pulsed laser techniques were developed. Given the origin of the laser, it was natural for researchers to attempt to push laser techniques to shorter wavelengths. However, this proved less than straightforward because of the punishing power scaling inherent in the laser process.

However, in recent years, researchers have found a way to make rapid progress by using nonlinear optics to coherently upconvert light into the x-ray region of the spectrum. This work has provided us with a new coherent source of light that spans such a large region of the spectrum that, for the first time, researchers now have access to processes that occur on attosecond (1 as=10-18 s) time-scales. “Attosecond” science is a direct outgrowth of small-scale research in laser science.

To understand what new science might be uncovered on such unimaginably-short timescales, we can start with the Heisenberg uncertainty principle that relates energy to time:

!

"E"t # h . This relationship—a direct consequence of the wave nature of matter prescribed by quantum theory—corresponds in useful units to (time[fs])*(energy[eV])≈1. Thus a 1 fs time-scale process involves an energy uncertainty, or intrinsic spectral bandwidth, of ~1 eV. In terms of physical processes in matter, electronic excited states -- corresponding to electron motion in atoms and molecules -- are typically in the range of 1-10 eV. Vibrational states on the other hand— i.e. the motion of atoms within a molecule — involve much lower energies and longer, femtosecond-picosecond, time scales. Attosecond science’s goal is thus to capture and control the motion of electrons in atoms, molecules and materials.

Heisenberg’s relationship also highlights the fact that many areas of research, such as high energy nuclear and particle physics, have already routinely dealt with attosecond—and faster— time scales. As a broad generalization, slow processes can usually best be understood by making observations directly in the time domain, since their energy bandwidth is quite small, and other effects (“inhomogenous” broadening) often complicate the spectrum. For very fast processes, on the other hand, the uncertainty limit is often dominant, making it easy to determine very short lifetimes just by looking at the spectrum (energy) of the process. For example, the lifetime of the W boson that mediates the weak interaction (2 GeV energy width, or ~10-25 sec)— the basis for the 1984 Nobel prize in physics— is known from spectral measurements.

So the obvious question is – what is truly new about attosecond science? The answer lies in the ability of the laser to interact coherently with matter. These concepts of coherence, control,

and coherent control have become increasingly pervasive in modern physics. Bose condensation, quantum computing, and coherent control of chemical systems are all examples of active areas of research where new science originates from an unprecedented level of control over the coherent quantum evolution of matter. Attosecond science emerged from a new understanding of the coherent interaction of atoms and molecules with intense, femtosecond duration, laser fields. Furthermore, the richness and complexity of attosecond science and technology is only just beginning to be uncovered. As well as capturing the complex dance of electrons in molecules and materials, attosecond science shows great promise for developing new ultrasensitive molecular imaging and spectroscopic techniques. Attosecond science also represents the most promising avenue to achieve what had seemed hopelessly impractical until now -- the generation of bright coherent hard x-rays using tabletop-scale apparatus. The promise of these tools and techniques to impact a variety of areas of science and technology is one of the most exciting aspects of this new field.

The birth of attosecond science - attosecond electron and x-ray bursts Attosecond science can mark its origins with the discovery of the process of high-order

harmonic generation (HHG) twenty years ago. In HHG, atoms exposed to a strong laser field emit coherent light at frequencies much higher than that of the incident laser. HHG is a result of the process of rescattering (see Fig. 1). Electrons can be ripped away from atom or molecule when the electric field strength of an incident laser becomes comparable to that binding the electron to the atom. The intensities required— 1013-1015 W cm-2— are easily accessible with tabletop femtosecond lasers. Once free, the electron will follow a trajectory controlled by the laser field – first moving away from the parent ion, and then reversing in direction as the field oscillates. These electrons can re-encounter their parent ion, recombining and giving-up their excess kinetic energy as photon, with energy corresponding to dozens or hundred of photons from the laser. Each time this happens, a burst of attosecond-duration x-rays is emitted. This generally occurs twice during each cycle of the driving laser field, or every 1.2 fs for a driving laser at 800 nm wavelength, because the ionizing laser field peaks twice each optical cycle. This high harmonic emission is coherent because each atom starts in the ground quantum state, and is exposed to the same coherent laser field. Therefore, the evolution of the electron wave function that re-radiates the light is identical for each atom and adds coherently.

A classical picture of HHG gives us a simple expression for the maximum energy photons that can be generated, which is simply the energy that the electron possesses at recollision:

!

h"max

= Ip + 3.2Up # Ilaser$2 (1)

where Ip is the ionization potential of the atom, Up is the ponderomotive potential, or the average kinetic energy of an electron oscillating in response to the driving laser field, and Ilaser and λ are the intensity and wavelength of the driving laser at the time of the ionization.(1, 2) However, much of the richness and complexity that has driven long-term interest originates from its quantum physics aspects.(1, 3) During its trajectory as a “free” particle, the electron evolves with a deBroglie wavelength corresponding to

!

" = hp

. If one calculates the total phase advance of

the electron during its free trajectory, it is quite large—dozens to hundreds of radians. The optical phase of the emitted harmonic light depends on this electron’s phase, and thus is not rigidly related to the phase of the driving laser. It depends on the exact shape of the electromagnetic field of the laser during the sub-optical-cycle time of the rescattering trajectory.

This is very different from any other type of nonlinear-optical process, and the exciting consequences of this physics are twofold. First, the relationship between the time-structure of the laser and the spectral characteristics of the harmonic emission represent the best example to-date of a non-trivial attosecond process that cannot be understood solely through spectral measurements. Particularly, the use of intense light pulses only a few optical cycles in duration (so that the field is changing from one cycle to the next(4-6)) has given us a direct window into these attosecond dynamics.(7, 8) And second, during this attosecond time-interval when the electron is ionized, we can manipulate the wave function in useful ways.(9-11)

Several remarkable scientific and technological opportunities have emerged as a result of this new fundamental property in the interaction of light with matter—a property that was first

Figure 1 – Attosecond Coherent Electron Rescattering from Atoms and Molecules Classical (top) and quantum (bottom) pictures of high harmonic generation. A strong laser field plucks an electron from the atom/molecule. After evolving as a free electron for a fraction of a femtosecond, the electron can recombine with the ion, generating a coherent x-ray.

discovered only 20 years ago.*(12, 13) Most current research in attosecond science falls into three broad categories: I. To understand how to control coherent electron rescattering, to manipulate the electron in useful ways and on attosecond timescales. II. To learn how to use the rescattering electrons as a probe of molecular dynamics. III. To use the attosecond time-structure of x-rays generated by high harmonic generation as a probe of complex electron dynamics in molecules and materials. We briefly discuss the exciting scientific opportunities in each area below.

The promise of attosecond science

I. Controlling attosecond trajectories High harmonics were originally discovered as an outgrowth of nonlinear optics. However,

high harmonic generation is a purely electronic process that nevertheless does not respond instantly, but rather has a sub-femtosecond response time. The quantum phase resulting from the recollision process can be influenced by the driving laser field - or any other field that is simultaneously applied to the atom or molecule.(14) This presents an intriguing possibility for controlling electrons on angstrom spatial dimensions and on attosecond timescales. Initial experiments validated the electron rescattering picture by observing how the harmonic beam characteristics (spectrum and beam shape) changed with the driving laser characteristics.(3, 8, 15)

The obvious next step was to manipulate the rescattering process in useful ways. The first experiment to achieve coherent attosecond electron manipulation adjusted the shape of the driving laser pulse using pulse shaping.(16) This made it possible to control the phase of the recolliding electrons, on a cycle-to-cycle basis of the driving laser field, to a precision of ≈ 12 attoseconds. Using a learning algorithm, an optimal laser pulse shape was found so that so that for a selected harmonic, each attosecond burst of x-rays interfered constructively. Because x-rays emitted in adjacent harmonics interfered destructively, a single harmonic order was selectively enhanced.(9, 10)

In an exciting series of more recent developments, the concept of manipulating coherent attosecond electrons has been used to overcome one of the major outstanding problems in nonlinear optics—the efficient generation of coherent high-energy coherent x-rays from lasers. The HHG process corresponds to the coherent version of the x-ray tube first demonstrated by Roentgen, where electrons accelerated by an electric field collide with atoms in a target. In the case of HHG, the incoherent electron impact of the x-ray tube is replaced by the coherent impact

* One aspect of this field that is attractive to many of its researchers, and that distinguishes it from other areas of quantum physics such as quantum information science, is the strong interplay between experiment and theory. HHG was an experimental discovery, completely unanticipated before it was observed. This discovery took effort to understand, leading to new theory, new understanding, and new experiments that have taken these ideas further. The drawback of such discovery-driven research is that the significance of many new developments only becomes apparent years afterward.

of an electron driven by a coherent laser field and originating from the same atom (and thus coherent with the atom). In high harmonic generation, the linear intensity scaling law given by the cutoff relationship of Eqn. 1 is remarkably favorable. Indeed, the intensities required to generate hard x-rays of ~1-10 keV from HHG are readily accessible using current tabletop lasers. In comparison, basic laser physics dictate that the energy required to implement a laser scales roughly as 1/λ5 i.e a laser at 10x shorter wavelength requires ~10,000x the input power.†

However, the major challenge in generating a usable flux of x-rays from high-order harmonic generation is in phase-matching the process. As the pump laser beam propagates in the medium, the harmonic signal will build-up constructively over a long distance only if the driving wave and the generated nonlinear signal travel with the same crest or phase velocity through the medium. This is achieved in conventional nonlinear optics by using birefringence - i.e. the

structure of the medium—as a means to control the propagation velocities of two disparate colors. This approach will not work in HHG, since we ionize the medium. The dispersion of the ionized electrons reduces the distance over which x-ray signal builds up constructively (called the coherence length) to mm or only microns for photon energies between 150 eV and keV.‡

† Indeed, proposals to make an x-ray laser pumped by nuclear weapons were the basis for Reagan’s “Star Wars” Strategic Defense Initiative program. ‡ At lower energies, however, of ~50-100 eV, HHG can be phase matched well in a waveguide, by balancing dispersion in a weakly-ionized regime.17. A. Rundquist et al., Science 280, 1412 (1998). The HHG then emerge as a fully coherent output, useful for holographic, nano-

Figure 2 – Crystals made from Light - Attosecond Manipulation of Coherent Electron Rescattering Shining a light pattern on a medium while electrons are being plucked from an atom by a strong field changes the phase that the electron wave accumulates while it is free, because this phase depends on the intensity of the light. This change in electron phase is directly written onto the generated x-ray phase. This relationship provides a direct way to manipulate x-ray waves to guarantee that they interfere constructively everywhere in a medium. In this example, a light pattern is used to scramble and disrupt the electron and x-ray phase in regions that would have contributed destructively to the x-ray signal. Even more complicated light patterns could be used to manipulate the x-ray wavefronts and focus them, or to extend bright emission to hard-x-ray energies.

The phase-matching challenge for HHG can be overcome by structuring the laser field

driving the process, rather than the medium itself. This approach manipulates the quantum phase of the harmonic emission using a patterned light field to correct for the phase slip. A simple example that has already been demonstrated experimentally is quasi phase matching (QPM). In this approach, the generated x-rays are modulated periodically, so that the “in-phase” spatial regions contribute most to the x-ray signal, while the out-of-phase regions are suppressed. Controlling where in a medium the bright x-rays are generated was first accomplished using a structured, periodically modulated, gas-filled, waveguide.(19) In this case the x-ray emission is brightest where the waveguide diameter is smallest and the laser intensity is highest. More recently, a patterned light field was imposed on the gas by colliding a single forward-moving pulse with a sequence of backward-going light pulses.(11) As illustrated in Fig. 2, the interferences between the forward and backward-going pulses scrambles the recolliding electron phase, and hence the x-ray phase, in any region where the pulses intersect. Such sequences were recently used to selectively enhance a single harmonic order by almost three orders of magnitude. Remarkably, very small changes in the light field can result in large changes in the phase of the recolliding electron, and this electron phase shift is directly mapped into the phase of the x-ray emission. This scheme represents true attosecond quantum engineering.

So-- how far can we go with these attosecond manipulation techniques? Is it possible to generate bright, coherent, hard x-rays using high harmonic generation - which could revolutionize crystallography, biological, materials and medical imaging? In theory the answer is yes. Instead of using sequences of pulses to eliminate harmonic emission from wide (fraction of a mm) regions of the medium that would contribute destructively, the crests and troughs of a continuous wave laser field can be used to adjust the recolliding electron and x-ray phase continually.(20) This approach shows great promise for making bright coherent beams even at very high photon energies well above 1 keV—where otherwise the phase slip distance is extremely short, on order of microns. This prospect would have seemed absurd—science fiction at-best—to laser scientists in the 20th century.

II. Attosecond electron recollisions with molecules for in-situ spectroscopy and imaging

Another exciting frontier of attosecond science is to exploit attosecond electron recollisions with molecules. These experiments take advantage of the fact that the ionized electron is coherent with its parent ion. In a molecule, as the electron accelerates and recollides, it gains ~10’s to 100’s of eV energy, corresponding to an electron deBroglie wavelength of ~1Å. This electron wavelength is well matched to interatomic spacings in a molecule. Harmonics generated from molecules are thus very sensitive to the orientation,(21) structure,(22) and dynamic motion(23, 24) of the electrons and atoms in a molecule. As a method for observing molecular dynamics, high harmonic generation is particularly interesting in that it also conveniently probes motions of the molecule in their ground electronic state, which is particularly relevant to chemistry. The time resolution is also sufficiently fast to decouple the electronic and nuclear motions. In the future, high harmonic generation from molecules could

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become a broadly-applicable probe of chemical dynamics, combining ultrahigh time-resolution with the potential for obtaining structural information complimentary to techniques such as femtosecond electron diffraction.

III. Attosecond pulses - probing electron dynamics in atoms, molecules and materials

High harmonics driven by a femtosecond laser are emitted as a series of attosecond bursts, as has been verifed direcly through experiments that combine laser and x-ray pulses.(25) Both theoretical predictions and recent experiments have confirmed that single, isolated, attosecond x-ray bursts can be produced provided that the laser pulse consists only of a few periods of the carrier, ~5 fs or shorter pulse.(26, 27) In this case, the time-varying few cycle field ensures that the highest harmonics are emitted only during one half-cycle of the laser field. These ultrashort attosecond bursts of x-rays (whether isolated, or in a “train” depending on the experiment) are ideal probes of complex, correlated electron dynamics in atoms, molecules and materials. Recently, it has been possible to follow some of the fastest electron dynamics for the first time—such as Auger decay in atoms,(28) or laser-assisted photoemission from solids,(29) or even x-ray-induced chemistry.(30) In some cases these experiments have yielded information similar to spectral studies; however, current experiments are beginning to yield new information that cannot be obtained in other ways. The tools developed with the understanding developed to-date promise to be useful for a variety of applications, from industrial applications in support of future generation lithography and nanotechnology, to fundamental studies in chemical physics.

The future—zeptoseconds? Looking to the future, are we approaching any physical limits? Quite possibly not. Other

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