In this issue NATURE PHOTONICS FOCUS: X-rays DECEMBER 2010 VOL 4 NO 12 www.nature.com/naturephotonics A new era for X-rays QUANTUM OPTICS Robust entanglement NANOSCALE CIRCUITS Rewritable photodetectors PHOTONIC CRYSTALS Soliton compression nphoton.2010.CoverDec.indd 1 16/11/10 17:19:49 REVIEW ARTICLE | FOCUS published online: 30 noveMber 2010 | doi: 10.1038/nphoton.2010.256 the attosecond nonlinear optics of bright coherent X-ray generation tenio popmintchev, Ming-chang chen, paul arpin, Margaret M. Murnane and henry c. Kapteyn* The frequency doubling of laser light was one of the first new phenomena observed following the invention of the laser over 50 years ago. Since then, the quest to extend nonlinear optical upconversion to ever-shorter wavelengths has been a grand challenge in laser science. Two decades of research into high-order harmonic generation has recently uncovered several feasible routes for generating bright coherent X-ray beams using small-scale femtosecond lasers. The physics of this technique combines the microscopic attosecond science of atoms driven by intense laser fields with the macroscopic extreme nonlinear optics of phase matching, thus essentially realizing a coherent, tabletop version of the Roentgen X-ray tube. T he promise of nonlinear optics became clear soon ater the invention of the laser 1, when Peter Franken focused light from a ruby laser into a quartz crystal and observed how a small amount of laser light was converted to the second harmonic2. However, the full impact and realization of nonlinear optics came only with an understanding of how a signal wave generated through a nonlinear optical polarization grows as it propagates through a nonlinear medium. he beautiful and elegant perturbation treatment of nonlinear optics developed by Nicolaas Bloembergen and co-workers made it possible to develop both birefringent-phase-matching and quasi-phasematching (QPM) techniques for eiciently converting laser light from one wavelength to another 3. In the early days, upconverting laser light to very short wavelengths was considered in the context of perturbative nonlinear optics. In this regime, the nonlinear medium is not damaged, and the nonlinearity of the medium is a weak perturbation such that higher orders of nonlinearity become successively weaker. As a result, the most practical approach for generating vacuum ultraviolet light of around 100 nm was to use excimer lasers or conventional harmonic generation to make near-ultraviolet light, followed by phase matching using excited-state atomic resonances4. his situation changed completely in 1987, when Mcpherson et al. discovered high-harmonic generation (HHG)5. In experiments seeking to characterize perturbative low-order harmonic generation, they unexpectedly observed a large number of odd harmonics of the fundamental driving laser, corresponding to the coherent combination of an odd number of photons (3rd, 5th, 7th, 9th, 11th, 13th, 15th and 17th harmonics). Moreover, the intensity of successively higher harmonic orders did not decrease signiicantly, as was expected in perturbative nonlinear optics. hese exciting observations were rapidly reproduced using infrared driving lasers6 that produced an even more remarkable comb of harmonic frequencies. Basic symmetry considerations can easily explain why only odd harmonics are generated — a gas medium has an inversion symmetry that precludes the generation of even-order harmonics. However, a completely new non-perturbative understanding of nonlinear optics was necessary to explain why many harmonics — all of comparable intensity — were generated simultaneously. he unique physics of the HHG process also necessitated a new perspective, and a new formalism, of phase matching in dynamic media that could not have been foreseen from the perspective of perturbative nonlinear optics. Microscopic attosecond science of hhG he key to uncovering the attosecond physics underlying HHG is to understand how atoms respond to strong laser ields. his was a topic of great interest in the 1980s, when above-threshold multiphoton ionization was observed and studied7. he time-dependent Schrödinger equation simulation of how a single electron is ionized from an atom by a strong laser ield, which was successfully used to explain abovethreshold multiphoton ionization, was used by Kulander and coworkers to reproduce the characteristic multiple harmonic spectra of HHG8. Other work discussed the basic HHG process in as early as 19829,10, considering it to be a Raman-like process via continuum states and describing the recollision process in classical terms in the context of above-threshold multiphoton ionization11. he next few years saw an intuitive picture of HHG emerge that is now known as the three-step model. In this picture, an electron is irst liberated from an atom through strong ield ionization, is then accelerated by the laser ield, and inally recombines with the parent ion, emitting any excess energy as a high-energy photon (Fig. 1a). he explicit relation between the energy of the emitted HHG photon and the ponderomotive energy (the oscillation energy) of the free electron in the laser ield can be used to write a simple cut-of rule for the maximum HHG photon energy that can be generated from a single atom or ion: hνSA cut-of = Ip + 3.17 Up ≈ ILλL2 (1) where Ip is the ionization potential of the gas atom and Up is the ponderomotive energy of a free electron in an intense ield (that is, the average energy of an electron driven by the oscillating electric ield of a laser of intensity IL and wavelength λL). his expression was irst identiied through quantum simulations12,13, and can also be derived from Newton’s second law (F = mea = −eEL) applied to a free electron of charge e, mass me and acceleration a in an electric ield of magnitude EL (ref. 14). hese models were developed to be consistent with a variety of observations at the time, and later rigorous systematic tests15 bore them out. he linear single-atom scaling of harmonic cut-of against laser intensity means that harmonics can in-principle be generated at very high photon energies, simply by increasing either the intensity or wavelength of the driving laser 16. In 1994, the development of a full semi-classical theory 17 established the microscopic quantum picture of HHG. his theory applied a path-integral approximation to the time-dependent Schrödinger solution for the electron motion, resulting in an expression for the time-dependent harmonic dipole that identiied the electron pathway, JILA, the Department of Physics and the NSF Engineering Research Center in Extreme Ultraviolet Science and Technology, University of Colorado at Boulder, Boulder, Colorado 80309, USA. *e-mail: henry.kapteyn@colorado.edu 822 nature photonics | VOL 4 | DECEMBER 2010 | www.nature.com/naturephotonics © 2010 Macmillan Publishers Limited. All rights reserved. FOCUS | REVIEW ARTICLE NATURE PHOTONICS doi: 10.1038/nphoton.2010.256 a Laser field X-ray a Macroscopic phase matching Laser field Coulomb potential e– Constructive addition of X-ray fields b Tight focus c Plane wave b Gas-filled waveguide Laser Gas jet Electron wavefunction Ion X-ray beam Lmedium independent of confocal parameter Gas-filled cell ~50 Å Lmedium < confocal parameter ~1–1 ,00 0 X-ray field Å Lmedium < confocal parameter X-ray wavefront Wavefronts Laser wavefront X-ray Laser wavefront wavefront c Tunnel ionization Laser field Laser beam X-ray beam Atomic gas Atomic gas Quantum ionization Geometrical Guoy phase and CEP ΔϕGuoy ΔϕGuoy = 0 Cos Sin Cos –Cos ~Cos Wavevector picture Katomic kX-ray qkL kX-ray Time (cycles) Figure 1 | the microscopic single-atom physics of hhG. a, Classical schematic of HHG. The electric field of an intense laser extracts an electron from an atom through tunnel ionization. The laser field then accelerates the electron, with a small fraction of the electron returning back to the ground state of the same atom, liberating its excess energy as a high-energy photon. b, The quantum nature of HHG. The electron’s wavefunction is driven by the laser field, giving rise to quantum interference between the bound and free portions, as well as to transverse spreading. The rapidly varying time-dependent dipole modulation gives rise to short-wavelength radiation (courtesy of I. Christov). c, The total fractional ionization of the medium grows stepwise with each laser cycle. The more realistic quantum model includes the modification in ionization due to the recolliding electron. including ionization, propagation and recombination, while preserving the de Broglie-wave quantum nature of the free electron driven by the laser ield: t – – – – * [p– – A (t)]e–iS(p, t, t')EL(t')dx [p– – AL(t')] + c.c. x(t) = i∫ dt'∫d3pd x L 0 (2) where x(t) is the electron displacement, p is the momentum, AL is the vector potential of the driving laser ield, EL(t') is the applied electric ield and c.c. denotes the complex conjugate. In equation (2), the irst Katomic qkL Figure 2 | Macroscopic phase-matching of hhG in the spatial domain. a, Phase-matched signal growth ensures that harmonic emission from many atoms over an extended medium (of length Lmedium) adds together coherently. b, Tight focusing geometry with associated curved laser and X-ray wavefronts. The geometrical Gouy phase shift (ΔφGuoy) results in a dynamical phase change between the carrier wave and envelope of the laser pulse. In a wave vector picture, the strong intensity gradient leads to a radial atomic wave vector Katomic that favours non-collinear phase matching before the focus and collinear phase matching after the focus. c, Plane-wave propagation in a waveguide or cell minimizes the curvature of the laser and HHG wavefront. The result is a well-directed, fully phasematched HHG output. The carrier–envelope phase change is absent in vacuum. The wave vector picture is also significantly simplified in this geometry, in which only near-collinear phase matching is possible. factor corresponds to ionization of the atom; that is, a transition from a bound state into the continuum. he second factor, e−iS(p-,t,t'), where S is the action integral, takes into account how the quantum phase of the electron wavefunction evolves over the ~1 fs timescale during which the electron resides in the continuum. he inal factor represents the transition from the continuum back to the ground state. Physically, the origin of high harmonics is the rapidly spatially and temporally modulated electron wavefunction that evolves as an atom nature photonics | VOL 4 | DECEMBER 2010 | www.nature.com/naturephotonics © 2010 Macmillan Publishers Limited. All rights reserved. 823 REVIEW ARTICLE | FOCUS a b t’r e– NATURE PHOTONICS doi: 10.1038/nphoton.2010.256 X-ray amplitude tr Snapshots in frame moving at speed c vL > c ELinitial 2Lcoh Lcoh + – 3Lcoh + 4Lcoh – Lmedium Ion Lmedium ELfinal t’b tb EX-ray initial EX-ray final EX-ray EL vX-ray = c Out of phase c d Δk(t) = 0 Lcoh→ ∞ (Lcoh >> Lmedium) + vL = c + + + vX-ray = c In phase Figure 3 | Macroscopic phase-matching of hhG in the temporal domain. a,b, Non-phase-matched build-up. Without phase matching, the evolution of the laser electric field due to geometrical and dynamically changing dispersion leads to a variation in the moment of birth and recombination of the rescattering electron (for any given energy) (a). The superimposed laser and HHG fields are plotted at the input (ELinitial and E ) and output (ELfinal and E ) of the nonlinear medium, shot in a frame moving with the speed of the X-rays (that is, the speed of light, c). The X-ray amplitude oscillates over a characteristic length called coherence length (Lcoh), which prevents strong HHG signal build-up (b). c,d, Phase-matched build-up. In dynamically changing conditions, phase-matching Δk(t) = 0 manifests itself through a minimal phase variation of the driving laser field for a fraction of a laser cycle, within which the electron can return (c). Under such conditions, the HHG amplitude (that is, the quadratic HHG intensity) grows linearly with distance (d). ionizes in a strong laser ield (Fig. 1b). his leads to very high order harmonics in the radiated dipole ield, even when driven by visiblewavelength lasers. he expression for x(t) in equation (2) can immediately be translated into a polarization response P(t) of the medium with atomic density n, P(t) = −nex(t), which can be inserted into Maxwell’s wave equation3. However, the non-instantaneous nature of HHG as a nonlinear process, in which the portion of the electron wavefunction that is free acquires a quantum phase, means that there is a phase diference between the driving laser and the HHG dipole that directly appears in the high-frequency polarization response driving the harmonic emission. his quantum phase diference can be quite large (>>π), because the electron, when free, can oscillate with a large excursion at a short de Broglie wavelength. As we discuss below, this quantum or intrinsic phase can be exploited to manipulate the HHG phase, spectrum and pulse duration, and also to implement powerful and general QPM schemes using additional light ields. Macroscopic phase-matching of hhG For a nonlinear optical frequency conversion process to be eicient, the phase velocities of the fundamental driving laser ield, vL, and the harmonic light, vX-ray, must ideally be matched so that the harmonic emissions from many atoms in the medium add coherently (Figs 2,3). If the HHG conversion process is not phase-matched, the high-harmonic signal builds up only over a propagation distance 824 in which the relative phase of the driving laser and harmonic ields slip by π radians (Fig. 3b). his distance is the coherence length Lcoh(q) = π/Δkq, where Δkq is the phase mismatch between the driving laser and harmonic wave vectors for harmonic order q, denoted kω (or kL) and kqω (or kX-ray), respectively. Moreover, in HHG the nonlinear medium is usually far from transparent and is also being ionized as the harmonics are emitted (Fig. 1c). hese efects, combined with the quantum phase shits between the driving laser and the harmonics, result in a dynamically and spatially varying index of refraction (and dispersion). Finally, HHG uses a gas medium with no birefringence (precluding most phase-matching techniques). As a result, the prospects of achieving full phase-matching look challenging. he story of how these challenges were overcome — by developing a new ield of extreme nonlinear optics to generate a useful lux of coherent X-rays — is one of the most exciting chapters in the recent progress of nonlinear optics and quantum physics. he importance of phase matching to HHG was recognized very early on. Initial experimental work required a tight focus to reach an intensity capable of ionizing a noble gas atom. his in turn caused a large Gouy phase shit through the focus, which is particularly problematic for HHG because of the large diference in wavelength and divergence of the harmonics compared with those of the driving laser (Fig. 2b)18. Furthermore, the Gouy phase shit interacts with the intrinsic quantum phase of the electron19–21. However, pioneering work by Balcou et al. treated this problem elegantly by nature photonics | VOL 4 | DECEMBER 2010 | www.nature.com/naturephotonics © 2010 Macmillan Publishers Limited. All rights reserved. FOCUS | REVIEW ARTICLE NATURE PHOTONICS doi: 10.1038/nphoton.2010.256 a of phase matching emerged from experiments using short-duration (~25 fs, or <10 optical cycle) driving laser pulses with suicient energy (~1 mJ) such that a tight focusing geometry was not required24–26. By propagating the driving laser in a gas-illed hollow waveguide, a non-diverging plane-wave phase-matching condition could be maintained over an extended distance, making it possible to buildup a fully spatially coherent and fully phase-matched high-harmonic beam (Figs 2–4). he waveguide also makes it straightforward to create a region of controlled, conined and well-characterized gas pressure, with optimal diferential pumping at the entrance and exit of the waveguide. Plane-wave propagation also means that, neglecting loss, the laser peak intensity is constant along the direction of propagation, thus avoiding any variation in the intrinsic quantum phase (Fig. 2c). In such a one-dimensional geometry, the major sources of phase mismatch are due to both a pressure-dependent (neutral and free-electron plasma dispersion) term, and a constant, pressure-independent waveguide propagation term. Full phase-matching can be achieved within some fraction of the optical cycles of the laser pulse by eliminating the phase mismatch, which is given by: HHG beam profile b ∆k(t) = kqω ‒ qkω 30 nm 13 nm 3 nm ≈q 4 4 3 3 Intensity (a.u.) y (mm) c 2 1 2 1 0 0 0 1 2 x (mm) 3 4 0 1 2 3 4 x (mm) Figure 4 | spatially coherent hhG emission for applications in imaging. a, Under phase-matched conditions, a near-perfect Gaussian HHG beam can be generated. b, Young’s double-pinhole interferograms show near-100% fringe visibility and full spatial coherence of HHG beams at 30 nm, 13 nm and 3 nm (refs 32,41,93). c, Also shown is a point-difraction interferometry measurement to characterize the HHG wavefront, where a pinhole in a thin-film filter leads to interferences between the residual beam transmitted through the filter and the diverging wavefront from the pinhole94. Figure c reproduced with permission from ref. 94, © 2003 OSA. considering the intrinsic phase as an efective additional wave vector added to the overall phase matching of the HHG process22. his interplay between geometrical and atomic phase means, for example, that the maximum HHG yield with a laser focused into a gas jet favours a geometry in which the laser is slowly diverging from a focus as it propagates through the gas (Fig. 2b)23. In this case, the Gouy-induced phase mismatch is compensated for by a decreasing laser intensity, which results in a quantum wave vector opposite to that of the Gouy phase. he phase mismatch can therefore be minimized over distances corresponding to a small fraction of the confocal parameter. his method is far from ideal, however, because the intensity of the driving laser will diverge and eventually fall below the threshold for generating harmonics. Even more importantly, early work did not consider any phase mismatch contributions due to the index of refraction of the gas and the generated plasma, which are typically the dominant factors for eicient phase matching. he complete picture { u211λL 2π ‒P [1 ‒ η(t)] δn ‒ η(t)NatmreλL λL 4πa2 } + ∆kquantum(t) (3) In equation (3), q is the harmonic order, u11 is the lowest-order waveguide mode factor, λL is the centre wavelength of the driving laser, a is the inner radius of the hollow waveguide, P is the pressure, η(t) is the ionization fraction, re is the classical electron radius, Natm is the number density of atoms at 1 atm, δn is the diference between the indices of refraction at the fundamental and harmonic wavelengths, and Δkquantum is the quantum phase mismatch associated with the intrinsic phase of the rescattering electron. Physically, the phase velocity of the high harmonics is close to the speed of light in vacuum, c, because the refractive index for high photon frequencies (far above most atomic resonances) is close to unity. he phase velocity of the driving laser, however, is altered through both the static and dynamic terms in equation (3). Although waveguide propagation increases the phase velocity of the driving laser, the medium dispersion is dynamic; that is, the neutral gas index slows the phase velocity, but the phase velocity rapidly increases as free electrons are created through ionization. Phase matching occurs during ionization of the pulse (typically on the leading edge or ideally at the peak), for a fractional ionization η, when Δk = 0. In a waveguide, this occurs at a level of ionization that can be controlled by the pressure and laser intensity in the waveguide. Tuning the pressure in this geometry therefore provides a clear signature of phase matching 24–27. A near-plane-wave geometry can also be realized using a large focus and high laser energy when the confocal parameter of the beam is much longer than the gas medium. Such a high-energy HHG geometry has been efectively used in experiments that beneit from high peak power rather than high repetition rate28–30. In conventional nonlinear optics, the medium is usually transparent, and relatively long crystal lengths can be used to saturate the conversion. For HHG, particularly in the extreme-ultraviolet region of the spectrum, the medium is absorbing, dispersive and refractive. Re-absorption limits the useful medium length to ~5–10 absorption lengths31. Under these conditions, conversion eiciencies of laser light to individual harmonic orders are ~10–5 per harmonic at photon energies of around 45 eV (using argon gas as a nonlinear medium; ref. 26) and 100 eV (in helium). hus, a useful lux of harmonics (~microwatts) can be produced by few-wattlevel laser systems. When phase matched, the harmonics naturally nature photonics | VOL 4 | DECEMBER 2010 | www.nature.com/naturephotonics © 2010 Macmillan Publishers Limited. All rights reserved. 825 REVIEW ARTICLE | FOCUS NATURE PHOTONICS doi: 10.1038/nphoton.2010.256 a λ L = 0.8 µm λ L = 1.3 µm EX-ray(t) EX-ray(t) e– e– 20 20 0 oh rr 40 adiu s) 0 60 6 0 0 0 20 L 20 (B L( ) ius rad 4 0 B r oh EL(t) 40 80 4 0 8 0 EL(t) Phase-matching cut-ofs 5 He hν PM cut-of ∝ λ 1.7 L Extreme-UV Soft X-ray Ne 1 0.5 Ar Water window 0.05 0.8 1 2 3 4 5 6 Water window 0.4 0.3 He 15 as Ne 30 as 0.2 Region I Perfect phase matching 0.1 λ L = 1.3 µm 0.5 Photon energy (kev) Region II QPM 0.1 7 8 9 10 Ar 0 Laser wavelength λL (µm) Transform-limited pulse duration 0.6 10 Attosecond pulses Photon energy (keV) c Phase-matched HHG spectra Zeptosecond pulses Hard X-ray b λ L = 2.0 µm 10 as 20 as 65 as 80 as 0 1 Normalized HHG intensity 1 Figure 5 | phase matching of hhG using mid-infrared lasers. a, As the driving laser wavelength (period) is increased, the electron’s wavefunction spreads transversely due to quantum difusion, which rapidly reduces the single-atom HHG yield. (courtesy of I. Christov). 1 Bohr radius = 0.53 Å. b, Theoretical HHG phase matching cut-of as a function of driving laser wavelength for three- (dashed lines) and eight-cycle (solid lines) pulses. This global phasematching picture has been validated experimentally (solid circles) using several laser wavelengths and nonlinear media demonstrating that bright, phasematched HHG emission is possible over broad regions of the X-ray spectrum38–41,43. c, Experimental phase-matched soft-X-ray supercontinua that support very short isolated attosecond pulses38,39,41,43. emerge as a fully coherent beam (Fig. 4)32, with pulses in the femtosecond-to-attosecond regime that are perfectly synchronized to the driving laser. he phase-matched bandwidth can also be broad, simultaneously spanning many harmonic orders (Fig. 5). Finally, for driving laser pulses that are 15 fs or shorter, phase matching can occur only over a single half-cycle of the laser, providing a straightforward way to generate sub-optical-cycle attosecond pulses using gated phase matching (Fig. 6)33,34. Other approaches for attosecond pulse generation shown in Fig. 6 require very short driving laser pulses. he cut-of rule of equation (1) shows that very high harmonic orders are generated at higher laser intensities and, as a result, at 826 higher ionization levels (Fig. 1c). Moreover, from equation (3) it is also clear that for ionization levels η higher than some critical ionization fraction ηcr, there is no way to satisfy Δk = 0. his critical ionization level is given by: ηcr = λL2 reNatm 1 1 – 2 +1 q 2πδn –1 and corresponds to the ionization level at which phase matching is achieved for large focal diameters (that is, plane-wave propagation). Above this critical ionization level, the phase mismatch due to the nature photonics | VOL 4 | DECEMBER 2010 | www.nature.com/naturephotonics © 2010 Macmillan Publishers Limited. All rights reserved. FOCUS | REVIEW ARTICLE NATURE PHOTONICS doi: 10.1038/nphoton.2010.256 b 100 Circularly polarized laser field 80 60 Linearly polarized laser field 1.0 40 20 –4 –2 0 2 4 Time delay (fs) Ne 0.4 3 80 as 2 1 0.2 0 –200 0 5 130 as 0 0.2 0 λL = 0.8 µm 10 0.4 4 0.8 0.6 0.8 Contributing subcycle Contributing subcycle EUV intensity 1.0 Phase (rad) λL = 0.8 µm Ar 0.6 –300 –150 0 150 Phase (rad) Ionization Laser field Single HHG burst EUV intensity Electron energy (eV) a –5 300 Time (as) 200 Time (as) 0 163 as –4 EUV intensity ELinitial 20 10 0 initial EX-ray final EX-ray vX-ray = c 30 20 10 0 2 4 1 λL = 2.0 µm 1 He 210 as 20 10 30 2 –1 0 1 2 Time (fs) Theory, multiple pulses 20 10 –1 Time delay (fs) e Ar 30 0 4 Time delay (fs) Theory, isolated pulse ELfinal 0 200 400 Time (as) 2 Photoelectron energy (eV) Bi-colour field with shaped polarization Single HHG burst 4 Photoelectron energy (eV) 1.0 Ar 0.8 0.6 0.4 0.2 0 –200 Phase (rad) EUV intensity vL = c λL = 0.8 µm 1 Experiment 30 Phase (rad) λL = 0.8 µm HHG bursts Laser field Photoelectron energy (eV) d c 0 1 Time delay (fs) λ L = 1.3 µm He 10 as 15 as 1.0 0.5 1 0 X-ray intensity X-ray intensit ntensity teensi (a.u.) intensity X-ray intensity (a.u.) 0 Ne 20 as 0 1 1 30 as 0 1 Ar Ne Ar 65 as 80 as 0.0 0.2 0.3 0.4 0.5 0.6 0 Photon energy (keV) 0 –100 0 Time (as) 100 –100 0 Time (as) 100 Figure 6 | attosecond pulse generation. a, Isolated attosecond pulse generation driven by a few-cycle laser field. The cut-of photon energy varies significantly on a cycle-by-cycle basis, so that simple spectral filtering can isolate HHG emission from a single electron recollision at the peak of the laser field95. This approach has yielded isolated extreme-ultraviolet (EUV) pulses as short as 80 as (ref. 96). b, Isolated 130 as EUV pulse generation using polarization gating97,98. HHG emission is suppressed except during the most intense central cycle of the field, where the laser polarization is linear. c, If a second laser field of a diferent colour is added to this polarization gating method, even shorter EUV pulses can be generated99. d, Isolated attosecond pulse generation using gated phase-matching confined to a narrow temporal window near the critical ionization level. This approach results in isolated 210 as EUV pulses at photon energies around 50 eV, using multicycle driving lasers33. e, Isolated attosecond pulses through gated infrared phase-matching in the soft-X-ray region of the spectrum. Bandwidths suicient to support 10 ± 1 as pulses are observed at around 400 eV. Figure reproduced with permission from: a (right), ref. 96, © 2008 AAAS; b (right), ref. 97, © 2006 AAAS; c (left), ref. 100, © 2010 NPG; c (right), ref. 99, © 2010 APS. plasma cannot be eliminated, even by using very short driving laser pulses. As higher harmonics are generated at higher levels of ionization, the phase-matching cut-of is always less than the single-atom cut-of given by equation (1), and is <150 eV even for helium driven by Ti:Sapphire lasers35. For harmonics generated at ionization levels of >ηcr, the inite phase mismatch due to uncompensated plasma dispersion reduces the coherent build-up length to the micrometre range, which reduces the yield by several orders of magnitude. Overcoming this phase-matching limit to generate bright harmonics in the sot- and hard-X-ray regions (required for many applications in spectroscopy and imaging) has therefore been a grand challenge in extreme nonlinear optics. nature photonics | VOL 4 | DECEMBER 2010 | www.nature.com/naturephotonics © 2010 Macmillan Publishers Limited. All rights reserved. 827 REVIEW ARTICLE | FOCUS NATURE PHOTONICS doi: 10.1038/nphoton.2010.256 c a Periodically modulated hollow waveguide Successive gas jets 2Lcoh Lcoh + Harmonic beam Laser beam Elaser 150 μm 0.25 mm 3Lcoh 4Lcoh + 0 0 ECP1 ECP2 Lmedium EX-ray 2Lcoh Lcoh + E laser 0 3Lcoh + 4Lcoh 0 + Lmedium λCP << Lcoh cτ CP ~ Lcoh EX-ray b d X-ray-generating laser pulse + Argon-filled waveguide 2Lcoh Lcoh 3Lcoh + — 4Lcoh — + EQCW laser CP laser pulses Lmedium Pulse collision point in plasma 41 45 0 2 cτQCW laser ~ Lmedium lse 1 pu 29 33 37 Harmonic order CP 25 λ QCW laser ~ Lcoh s + 4 Figure 7 | QpM techniques for extreme nonlinear optics. QPM relies on switching of or phase shifting the HHG emission in zones that contribute destructively to the signal. a, Geometrical QPM schemes. A modulated waveguide can periodically modulate the peak intensity of the driving laser beam and hence the HHG phase. A sequence of gas jets can also be used, in which the spacing and density can be varied. b,c, All-optical QPM approaches are superior to their geometrical predecessors because a versatile periodic light structure can adaptively compensate for any laser propagation dynamics and can be implemented over many coherence zones. A sequence of weak counter-propagating (CP) pulses (of wavelength λCP, pulse duration τCP and electric field strength ECP) can create a light structure inside a waveguide that is matched to the coherence length of a particular harmonic, in order to selectively enhance a quasi-monochromatic HHG bandwidth59,60. d, All-optical grating-assisted phase matching. At higher photon energies, the coherence length reduces significantly, and the laser electric field itself can be used to modulate the recolliding electron and HHG phase. QCW, quasi-continuous-wave. Figure a (right) reproduced with permission from ref. 56, © 2007 NPG. phase-matching of hhG in the kev region A variety of approaches for phase matching the HHG process at higher photon energies have been explored, including the QPM schemes discussed in the next section. However, over the past two years, a simple method of extending phase matching to the keV region and beyond has been developed, based on the same phase matching physics described above but using longer-wavelength driving lasers. Somewhat paradoxically, by increasing the wavelength of the driving laser, the phase-matching cut-of shits to shorter wavelengths (Fig. 5). It had long been understood from equation (1) that because the HHG cut-of photon energy is proportional to Up ≈ ILλL2, using a longer-wavelength driving laser should result in a higher singleatom cut-of photon energy. Experimentally, however, most highharmonic experiments used Ti:Sapphire ultrafast lasers operating at a wavelength of ~0.8 μm. Direct veriication of the wavelength scaling of the single-atom cut-of was therefore irst tested in 2002, ater Ti:Sapphire laser and optical parametric ampliier technology had advanced to the point at which mid-infrared (mid-IR) pulses of suicient energy were practical36. Furthermore, it was also known that the longer excursion of the oscillating electron means that the probability of rescattering — and thus the efective 828 nonlinear susceptibility of the process — would decrease rapidly with increasing driver wavelength (Fig. 5a). Recent theoretical and experimental studies have veriied a detrimental scaling in single-atom HHG eiciency of between λL–5 and λL–9, depending on how the scaling is deined14,37,39,43. his severe eiciency penalty, in addition to an already painful 10–20% conversion eiciency of the Ti:Sapphire driver energy to mid-IR optical parametric ampliier light, has limited the prospects for mid-IR laser drivers for practical HHG sources. However, this picture changed dramatically when phase matching was taken into account 38,39. As explained previously, phase matching is possible only up to a critical ionization level that depends on the gas species and driving laser wavelength39. However, the maximum cut-of photon energy of HHG scales as ILλL2, meaning that for a longer wavelength driving laser, the laser intensity required to generate a given high-harmonic energy rapidly decreases. Because the tunnel ionization rate (the irst step of HHG) is a quasi-d.c. process, it depends exponentially on the laser intensity but is reasonably independent of laser wavelength. herefore, even for very high harmonic photon energies, the medium can be less ionized when longer-wavelength lasers are used, making full phase-matching possible. Taking all the nature photonics | VOL 4 | DECEMBER 2010 | www.nature.com/naturephotonics © 2010 Macmillan Publishers Limited. All rights reserved. NATURE PHOTONICS doi: 10.1038/nphoton.2010.256 contributions to the phase mismatch into account (equation (3)) a favourable and strong scaling of the phase matching cut-of — the maximum photon energy that can be generated under ideal phase matching conditions at a given λL — is predicted to vary as hνPM cut-of ~ λL1.6–1.7 throughout the X-ray region39,43, which is almost as favourable as the single-atom cut-of hνSA cut-of ~ λL2. his prediction was recently veriied by several experiments in diferent laboratories, as shown in Fig. 5b, including the recent demonstration of a fully phase-matched, spatially coherent source with a bandwidth spanning the entire water window 38–43. Finally, this generalized picture of phase matching as a function of the laser wavelength is valid not only for infrared lasers but also for λL in the visible and ultraviolet regions. Fortunately, several factors work to countervail the low singleatom yield for HHG driven by mid-IR lasers. he required phase matching pressures are very high (multi-atmosphere) for pressuretuned phase matching in waveguides. Moreover, gases generally become more transparent at higher photon energies, except just above their inner shell electron ionization thresholds. his means that although each atom is generating less HHG light, there is also less re-absorption of the generated harmonics by the nonlinear medium, resulting in a rapidly increasing optimum density length product (around several absorption lengths). When all the scaling parameters are included, in the case of helium the predicted conversion eiciency actually increases with increasing HHG photon energy 39,43. hus, there is a very real prospect of generating bright harmonic beams in the hard-X-ray region of the spectrum, with averages powers of tens of microwatts, from a tabletop setup. Another interesting aspect of mid-IR phase matching is the large phase-matched bandwidths available39,41,43, spanning an energy range of more than 300 eV for HHG driven by 2.0 μm light 41,43. he bright X-ray supercontinuum shown in Fig. 5c is the broadest coherent bandwidth generated so far using any light source. Moreover, past work has shown that because phase matching is conined to only a few half-cycles of the laser — even when using relatively long six-cycle driving laser pulses — the harmonics emerge as a single attosecond burst 33,34. Such macroscopic phase-matching gating functions as a femtosecond Pockels cell. hus, when combined with the reduced attochirp (1/λL) of longer-wavelength driving lasers, and together with previously demonstrated chirp-compensation techniques44, it is likely that pulses as short as 11 ± 1 as can be supported (Fig. 6e). hese phase-matched bandwidths, ΔEX-ray, are related to the transformlimited X-ray pulse duration by an approximate rule of thumb: τ [as] ~ 1.8/ΔEX-ray [keV] (ref. 43). hus, for harmonics driven by λL > 4 μm light, the phase-matching bandwidth for HHG in helium is predicted to span over >1.8 keV, providing a coherent X-ray supercontinuum that is useful for elemental-speciic spectroscopy and imaging, capable of supporting zeptosecond-duration pulses (Fig. 5b,c). Experiments employing this technique will probably use the phase coherence and synchronization of diferent parts of the X-ray continuum emission to excite and probe complex materials and molecular systems. Finally, we note that the phase matching of HHG at high photon energies using mid-IR driving lasers generally requires a waveguide geometry, for several reasons. First, the waveguide conines the multi-atmosphere-pressure gas through the ends of the waveguide, with optimal diferential pumping and extended plane-wave propagation of the driving laser. Second, a longer driving laser wavelength results in a shorter Rayleigh length for the focused beam. Although in principle one can always choose the laser focal spot size such that a long enough confocal parameter is maintained and optimal conversion eiciency is reached, this results in a pulse energy requirement that quickly becomes prohibitive without a guided geometry. FOCUS | REVIEW ARTICLE controlling the quantum phase on attosecond timescales A remarkable property of HHG is the existence of the large intrinsic quantum phase that arises from the phase advance of the electron wavefunction in the continuum. Although phase matching of the HHG process can occur regardless of this phase, the fact that it depends on the laser ield provides a beautiful means of manipulating the HHG process in several useful ways. Understanding how to take full advantage of the quantum phase for optimizing high-harmonic emission emerged with a deeper understanding of the HHG process itself. he irst proposal for quantum manipulation was suggested in 199519, using a geometry that intersected laser pulses. A few years later, Chang et al.45 took advantage of the interaction between the time-varying chirp of the laser and the intrinsic chirp in the HHG output due to the quantum phase to manipulate the spectral bandwidth of the HHG emission. When harmonics are driven by a positively chirped laser pulse, the laser and quantum phase subtract, leading to narrow-bandwidth harmonic peaks. However, for negatively chirped laser driving pulses, the quantum and laser chirps add, leading to near-continuum HHG emission. hese simple experiments represented some of the irst experiments in attosecond science, in which time-domain manipulation of the laser electromagnetic ield incident on a single atom enabled timedomain control of the radiating electron wave function. he next step in complexity for exploiting the quantum phase in HHG was to shape the driving laser pulse itself, thus providing better control of the HHG spectrum. By controlling the nonlinear chirp of the driving laser pulse, it became possible to sculpt the nonlinear response of the atom (that is, the electron wavefunction) to selectively emit a preferred frequency harmonic46. he mechanism for this selective control can be understood in the context of the recollision picture of HHG47. he intensity of the driving laser couples quite strongly to the intrinsic quantum phase, as the de Broglie phase advance of the electron during its excursion is quite large, increasing with the energy of the emitted harmonics (an approximate rule-of-thumb is ~1 radian per harmonic order). herefore, an optimal nonlinear chirp on the driving laser can interact with the intrinsic quantum phase to allow emission from a speciic harmonic order from diferent cycles in the laser ield to add constructively for that harmonic order, while suppressing adjacent orders. his single-atom efect is known as intra-atom phase matching, to distinguish it from macroscopic phase matching 47, and clearly shows that the HHG spectrum is in fact a single, broad-bandwidth structured spectrum — not a series of discrete peaks representing independent conversion processes48. QpM in the hhG process In visible nonlinear optics, QPM is a powerful and widely employed scheme for regions in which phase matching is not possible. Instead of matching the phase velocities of the driving laser and nonlinear polarization throughout a medium, QPM periodically corrects the phase mismatch every coherence length (Fig. 7), thus preventing back-conversion of the harmonic light 3. QPM schemes in the visible region of the spectrum periodically vary the crystalline structure of the nonlinear material (by periodic polling, for example). he original proposals for QPM also considered alternating linear and nonlinear materials, so that the harmonic signal is always generated in phase. Finally, other QPM schemes considered using graded-index materials to modulate the harmonic phase. All of these visible-wavelength schemes can be adapted to the extreme nonlinear optics of HHG. Moreover, the dependence of the quantum phase of HHG emission on the total light ield present in the medium makes all-optical QPM schemes practical and very promising. Many diferent ways of implementing QPM for HHG have been proposed. As early as 199449, calculations by Shkolnikov et al. nature photonics | VOL 4 | DECEMBER 2010 | www.nature.com/naturephotonics © 2010 Macmillan Publishers Limited. All rights reserved. 829 REVIEW ARTICLE | FOCUS proposed that periodically modulating the medium density could achieve QPM. By 2003, experiments by Paul et al. and Gibson et al. demonstrated the proposal of Christov, which suggested that by periodically modulating the diameter of a hollow waveguide, the resulting laser intensity modulations could impose the necessary phase modulation on the HHG ield50–54. Other experiments by Zepf et al.55 used HHG in a very tight-focus geometry in a simple waveguide, thereby exciting very high order modes in the waveguide that interfere to create an on-axis modulation of the laser intensity. he idea of achieving QPM through a periodic modulation of density was also demonstrated by Seres et al.56. However, laser-induced defocusing, laser mode scrambling and other geometric limitations together make such QPM schemes diicult to scale to the very long interaction lengths needed for good eiciency. So far, the most powerful approach for achieving QPM in the HHG process is to superimpose additional light ields on the nonlinear medium (Fig. 7b,c). As explained above, the quantum phase shit acquired by the electron during its trajectory in the continuum is linear in laser intensity, and can reach tens to hundreds of radians. hus, by generating harmonics in the presence of a pair of interfering laser pulses of wavelength λL, spatiotemporal interferences over distances of ~λL/2 (≪Lcoh) then lead to laser intensity and phase variations that can easily scramble the quantum phase of the HHG emission, even to the extent that HHG is essentially turned of, as demonstrated by Voronov et al.57 (based on theoretical suggestions19,58). When HHG in a certain region can be successfully turned of, the next step is to implement QPM by sending the correct light pattern or sequence of counter-propagating pulses through the medium in the direction opposite to the driving laser (see Fig. 7b,c and work of Zhang et al.59–61). A gas-illed waveguide is an ideal geometry for these experiments because it creates an approximately plane-wave interaction with explicit control over the phase mismatch, and because it is straightforward to align the forwards- and backwards-travelling beams. Lytle and co-workers demonstrated that the correct light pattern can be experimentally measured by sending a single counter-propagating pulse through the medium, as the harmonic output oscillates when the single pulse travels through the diferent coherent zones in the medium61,62. A sequence of only four pulses (Fig. 7b) will signiicantly and selectively enhance a particular narrow range of harmonics (700-fold), as each successive harmonic is generated at a slightly higher ionization level corresponding to a slightly shorter coherence length and pulse sequence. Using a longer pulse sequence should allow for even greater spectral selectivity. he idea that the phase of the high-harmonic ield can be engineered with an additional light ield opens up many new possibilities beyond using a simple counter-propagating pulse sequence to establish discrete zones in the medium. For example, the HHG phase can be modulated by the oscillating ield of a single long-duration counter-propagating pulse, as proposed by Cohen et al.63. In this scheme (Fig. 7d), the electric ield of the backwards-travelling pulse, when added to the driving pulse, can shit the phase of the HHG emission, ideally with a periodic phase shit of π between the peak and trough of the backwards-travelling wave — which is then matched to the coherence length of the forwards-travelling pulse. his grating-assisted phase-matching concept will ideally be implemented with a mid-IR backwards-travelling pulse to match coherence lengths in the range of micrometres, ideally suited for multi-keV generation (with very short coherence lengths in the micrometre range). Development of these all-optical phase-matching concepts also led to the realization that they represent special cases of spatiotemporal QPM64. In standard QPM, energy conservation is assumed, 830 NATURE PHOTONICS doi: 10.1038/nphoton.2010.256 whereby the energies of the input and the output photons sum to zero for the nonlinear process, and the spatial structure of the medium represents a virtual k-vector that can be included in the conservation of momentum. In-fact, QPM can also be implemented by matching k-vectors and compensating for any energy mismatch by a temporal modulation of the medium. In the case of QPM of HHG, the backwards-travelling wave corrects for both the energy and momentum mismatch. his means that the period of the backwards-travelling wave does not strictly match the coherence length of the forwards-travelling wave. hese new concepts of spatiotemporal QPM are quite general, and can also in principle be applied to low-order nonlinear optics. his makes it clear that the topic of high-order, non-perturbative nonlinear optics is a non-trivial extension of nonlinear optics that promises to contribute broadly to nonlinear optics and nonlinear science. QPM schemes are very attractive because they can use any wavelength for the driving laser, and therefore do not sufer from the low λL–5–λL–9 dependence of the single-atom yield for mid-IR driving laser wavelengths. hey remain more complex to implement experimentally than the perfect phase-matching schemes described above, but promise to play a major role as the ield of extreme nonlinear optics develops further. applications of ultrafast coherent X-rays When considering attosecond timescales, it is important to start by bearing in mind the Heisenberg uncertainty principle. he timeenergy uncertainty — one form of this relation — gives ΔEΔt ≥ ħ/2, which in physical units is ΔE [eV] Δt [fs] > 1. In other words, dynamics on a subfemtosecond timescale will always involve large energy bandwidths of >1 eV. In this context, the HHG process itself is the premier example of new physics on an attosecond timescale. A typical spectrum of high-harmonic emission spans tens-to-hundreds of electron volts in the extreme-ultraviolet/sotX-ray regions — in a single coherent spectrum — with a structure that indicates complex dynamics and that can be manipulated to understand the HHG process45–47. In terms of its application, one can argue that HHG is following a trajectory similar to that of NMR. NMR started as a basic physical observation, progressed to spectroscopic applications, then to very basic imaging, and is now a staple technique for both basic science and medical imaging. Advances in X-ray science and technology have resulted in breakthrough discoveries ranging from unravelling the structure of DNA and proteins to visualizing molecular and material structure. Ultrashort coherent pulses of X-rays are powerful probes of the nanoworld; X-rays can penetrate and (by virtue of their short wavelength) image very small objects. By using elemental absorption edges, X-rays can provide information that is speciic to each element or chemical species. Ultrashort pulses of X-rays can also be used to capture even the fastest dynamics of electrons, giving a tabletop source of coherent X-rays the potential for a broad range of applications in medicine and nanotechnology. he many emerging applications of HHG sources are bearing-out this promise, making it possible to capture dynamics in atoms65–68, molecules69–77, surfaces78–80 and materials81,82, even at the fastest timescales, and making coherent X-ray imaging 83 with nanometre resolution possible on a tabletop84–86. Moreover, as the useful photon energy range for high-harmonic sources expands considerably over the next few years, a wealth of new applications will open up in the ields of nano-, bio-, magnetic-, molecular- and materials science and technology. Finally, the ultrafast time durations of high harmonics — in the femtosecond-to-attosecond (and soon possibly zeptosecond) regime — makes this source the ultimate strobe light, perfectly synchronized to the driving laser and capable of capturing the fastest motion in our natural world, even at the electron level. nature photonics | VOL 4 | DECEMBER 2010 | www.nature.com/naturephotonics © 2010 Macmillan Publishers Limited. All rights reserved. NATURE PHOTONICS doi: 10.1038/nphoton.2010.256 summary and perspective he past two decades have seen a revolution in optical science as a new region of the spectrum has become available to coherent light. he physics of HHG involves previously inaccessible regimes of energy and timescales, and despite the fact that attosecond timescales might be considered to be fundamentally incompatible with dynamics in atoms and molecules, the high-order harmonic process shows that we can observe — and usefully manipulate — dynamic processes in these systems on such timescales. hese developments open up a wide range of studies that will make use of ultrafast, laser-like X-ray beams for a multitude of applications across all areas of science and technology. he useful wavelength range of tabletop HHG sources is poised to expand signiicantly in the near future. 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Zhang and J. Zhou. his work was funded by the National Science Foundation, a National Security Science and Engineering Faculty Fellowship, the US Department of Energy (DOE) and the DOE National Nuclear Security Agency. additional information he authors declare competing inancial interests: details accompany the paper at www.nature.com/naturephotonics. nature photonics | VOL 4 | DECEMBER 2010 | www.nature.com/naturephotonics © 2010 Macmillan Publishers Limited. All rights reserved.