ARTICLES Quasi-phase-matching and quantum-path control of high-harmonic generation using counterpropagating light XIAOSHI ZHANG*, AMY L. LYTLE, TENIO POPMINTCHEV, XIBIN ZHOU, HENRY C. KAPTEYN, MARGARET M. MURNANE* AND OREN COHEN JILA and Department of Physics, University of Colorado, Boulder, Colorado 80309, USA *e-mail: xiaoshi@jilau1.colorado.edu; murnane@jila.colorado.edu Published online: 25 February 2007; doi:10.1038/nphys541 High-order harmonic upconversion of femtosecond lasers produces a unique source of short-wavelength light with femtosecond-to- attosecond pulse duration. However, because the involved nonlinear medium is a partially ionized gas, traditional approaches for phase-matching the conversion process are not applicable. This severely limits the flux from this source. Here, we demonstrate the first use of a train of counterpropagating light pulses to enhance high-harmonic emission. This all-optical quasi-phase-matching technique uses interfering beams to scramble the quantum phase of the generated short-wavelength light, to suppress emission from out-of-phase regions. Selective enhancement of more than 300 is observed at photon energies around 70 eV in argon gas. Finally, we show that by adjusting the intensity of the counterpropagating light, different electron quantum trajectories can be selectively enhanced, demonstrating attosecond-timescale coherent control of the radiating electron wavefunction. High-order harmonic generation (HHG) driven by ultrashort laser pulses is a source of extreme-ultraviolet and soft X-ray light with the unique properties of ultrashort pulse duration and high spatial and temporal coherence 1 . This source has made possible new ultrafast spectroscopic probes of atoms, molecules and materials. So far, however, most applications have used relatively long wavelengths, because the conversion efficiency rapidly decreases at shorter wavelengths. This decrease is not due primarily to the very high-order nonlinearity of the process—in fact, the atomic physics of HHG is non-perturbative 2 , and has favourable scaling to shorter wavelengths. The major challenge is that, unlike low-order nonlinear processes such as second- harmonic generation, HHG is inherently associated with ionization of the nonlinear medium 3–5 . In HHG, an electron is first ionized by the field of an intense femtosecond laser. Once free, the electron begins to oscillate in response to the laser field. A small fraction of the ionized electron can recollide with its parent ion, recombining and liberating the excess energy as a short- wavelength photon. Dispersion of the free-electron plasma creates a phase mismatch that speeds up the phase velocity of the driving laser with respect to the generated harmonics, preventing efficient frequency conversion. As in all nonlinear parametric processes in nature, efficient conversion of light from one frequency to another using nonlinear optics requires that the process be phase-matched. As the pump beam propagates, the nonlinear response of the medium coherently adds to the harmonic signal. The generated field continues to add constructively if the two waves travel with the same phase velocity through the medium, leading to a bright, phase-matched beam at the new wavelength. If the process is not phase-matched, coherent build-up is limited to a propagation distance over which the relative phase of the fundamental and the harmonic light slip by 180 ◦ . This distance is the coherence length L c = π/k, where k is the phase mismatch between the polarization wave and the harmonic wave. For HHG, dispersion of the free-electron plasma reduces L c to tens of micrometres for upconversion to very short wavelengths, which are only generated when the laser is very intense and thus the medium is already highly ionized 6–8 . As a result, efficient harmonic generation is possible only at relatively low levels of ionization, below a ‘critical’ ionization level of ∼5% for argon, or ∼0.5% for helium, corresponding to photon energies of ∼50 eV and ∼130 eV respectively. Thus, new methods that can correct for this phase mismatch in ionized media are a ‘grand challenge’ in this area of laser science. In the absence of phase-matching, quasi-phase-matching (QPM) techniques have been successfully demonstrated to compensate for this phase slip by periodically readjusting the relative phase of the fundamental and nonlinear response at an interval corresponding to the coherence length 9,10 . In the visible region, this is achieved by periodically reversing the polarization of a non-centrosymmetric nonlinear-optical material. However, this implementation cannot be used for HHG, because HHG uses a low-pressure gas as the nonlinear medium. Past experimental work used a periodically modulated hollow waveguide to modulate the intensity of the driving laser to implement QPM for high-harmonic generation 11–13 . Even a small modulation (∼1%) of the driving laser results in significant modulation in both the amplitude and phase of the harmonics. Although this past work succeeded in enhancing conversion efficiency into the soft X-ray region of the spectrum by about one order of magnitude, further optimization will require a more sophisticated approach. This is because optical loss of the driving laser, refraction, mode beating and group- velocity dispersion all result in a continuous variation of the coherence length along the direction of propagation, making it difficult to optimize the modulation period. Finally, modulation periods shorter than the waveguide diameter will not significantly 270 nature physics VOL 3 APRIL 2007 www.nature.com/naturephysics