ULTRAFAST OPTICS Generation of extreme-ultraviolet ...Emilio Pisanty, Maciej Lewenstein, Luis Plaja, Henry C. Kapteyn, Margaret M. Murnane, Carlos Hernández-García INTRODUCTION:

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ULTRAFAST OPTICS

Generation of extreme-ultravioletbeams with time-varying orbitalangular momentumLaura Rego*†, Kevin M. Dorney*†, Nathan J. Brooks, Quynh L. Nguyen,Chen-Ting Liao, Julio San Román, David E. Couch, Allison Liu,Emilio Pisanty, Maciej Lewenstein, Luis Plaja, Henry C. Kapteyn,Margaret M. Murnane, Carlos Hernández-García

INTRODUCTION: Light beams carry both en-ergy andmomentum, which can exert a smallbut detectable pressure on objects they illumi-nate. In 1992, it was realized that light can alsopossess orbital angularmomentum(OAM)whenthe spatial shape of the beam of light rotates (ortwists) around its own axis. Although not visibleto the naked eye, the presence of OAM can berevealed when the light beam interacts withmatter. OAM beams are enabling new applica-tions in optical communications, microscopy,quantum optics, and microparticle manipula-tion. To date, however, all OAM beams—also

known as vortex beams—have been static; thatis, the OAM does not vary in time. Here weintroduce and experimentally validate a newproperty of light beams, manifested as a time-varying OAM along the light pulse; we termthis property the self-torque of light.

RATIONALE: Although self-torque is found indiverse physical systems (e.g., electrodynamicsand general relativity), to date itwasnot realizedthat light could possess such a property, whereno external forces are involved. Self-torque isan inherent property of light, distinguished from

the mechanical torque exerted on matter bystatic-OAM beams. Extreme-ultraviolet (EUV)self-torqued beams naturally arise when theextreme nonlinear process of high harmonic gen-eration (HHG) is driven by two ultrafast laserpulses with different OAM and time delayedwith respect to each other. HHG imprints atime-varying OAM along the EUV pulses, whereall subsequent OAM components are physicallypresent. In the future, this new class of dynamic-OAM beams could be used for manipulatingthe fastest magnetic, topological, molecular, andquantum excitations at the nanoscale.

RESULTS: Self-torqued beams are naturallyproduced byHHG, a process in which an ultra-fast laser pulse is coherently upconverted into theEUVand x-ray regions of the spectrum.Bydriving

the HHG process with twotime-delayed, infrared vor-tex pulses possessing dif-ferent OAM, ‘1 and ‘2, thegenerated high harmonicsemerge as EUV beamswith a self-torque, ℏxq ≃ℏqð‘2� ‘1Þ=td, thatdepends

on the properties of the driving fields—that is,their OAMcontent and their relative time delay(td)—and on the harmonic order (q). Notably, theself-torque of light also manifests as a frequencychirp along their azimuthal coordinate, whichenables its experimental characterization. Thisultrafast, continuous, temporal OAM variationthat spans fromq‘1 toq‘2 ismuch smaller thanthe driving laser pulse duration and changes onfemtosecond (10−15 s) andeven subfemtosecondtime scales for high values of self-torque. Thepresence of self-torque in the experimentallygenerated EUV beams is confirmed by measur-ing their azimuthal frequency chirp, which iscontrolled by adjusting the time delay betweenthe driving pulses. In addition, if driven by few-cycle pulses, the large amount of frequency chirpresults in a supercontinuum EUV spectrum.

CONCLUSION:Wehave theoretically predictedand experimentally generated light beamswitha new property that we call the self-torque oflight, where theOAMcontent varies extremelyrapidly in time, along the pulse itself. This in-herent property of light opens additional routesfor creating structured light beams. In addition,because the OAM value is changing on femto-second time scales, atwavelengthsmuch shorterthan those of visible light, self-torqued HHGbeams can be extraordinary tools for laser-matter manipulation on attosecond time andnanometer spatial scales.▪

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Rego et al., Science 364, 1253 (2019) 28 June 2019 1 of 1

The list of author affiliations is available in the full article online.*Corresponding author. Email: [email protected] (L.R.);[email protected] (K.M.D.)†These authors contributed equally to this work.Cite this article as L. Rego et al., Science 364, eaaw9486(2019). DOI: 10.1126/science.aaw9486

Generation of EUV beams with self-torque. (A) Two time-delayed, femtosecond infrared(IR) pulses with different OAM are focused into a gas target to produce self-torqued EUVbeams through HHG. The distinctive signature of self-torqued beams is their time-dependentOAM, as shown in (B) for the 17th harmonic (47 nm, with self-torque x17 = 1.32 fs

−1). (C) Theself-torque imprints an azimuthal frequency chirp, which enables its experimentalmeasurement.

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RESEARCH ARTICLE◥

ULTRAFAST OPTICS

Generation of extreme-ultravioletbeams with time-varying orbitalangular momentumLaura Rego1*†, Kevin M. Dorney2*†, Nathan J. Brooks2, Quynh L. Nguyen2,Chen-Ting Liao2, Julio San Román1, David E. Couch2, Allison Liu2,Emilio Pisanty3, Maciej Lewenstein3,4, Luis Plaja1, Henry C. Kapteyn2,5,Margaret M. Murnane2, Carlos Hernández-García1

Light fields carrying orbital angular momentum (OAM) provide powerful capabilities forapplications in optical communications, microscopy, quantum optics, and microparticlemanipulation.We introduce a property of light beams, manifested as a temporal OAM variationalong a pulse: the self-torque of light. Although self-torque is found in diverse physical systems(i.e., electrodynamics and general relativity), it was not realized that light could possess such aproperty.We demonstrate that extreme-ultraviolet self-torqued beams arise in high-harmonicgeneration driven by time-delayed pulses with different OAM.We monitor the self-torque ofextreme-ultraviolet beams through their azimuthal frequency chirp.This class of dynamic-OAMbeams provides the ability for controlling magnetic, topological, and quantum excitations andfor manipulating molecules and nanostructures on their natural time and length scales.

Structured light is critical for a host of ap-plications in imaging and spectroscopy, aswell as for enhancing our ability to opti-cally manipulate macro- to nanoscale ob-jects such as particles, molecules, atoms,

and electrons. The distinctive phase and inten-sity properties of structured light beams achievedby exploiting the angular momentum of lighthave garnered renewed interest in optical manip-ulation and control (1). One of the most relevantstructured light beams are those carrying orbitalangular momentum (OAM), also known as vortexbeams (2). The OAM of light manifests from aspatially dependent wavefront rotation of thelight beam, which is characterized by the phasewinding number, or topological charge, ‘. OAMbeams have been harnessed for applications indiverse fields (3) such as laser communication(4, 5), phase-contrast (6, 7) and superresolutionmicroscopy (8), kinematic micromanipulation (9),quantum information (10), and lithography (10).Spurred by these exciting technologies, a paral-leled interest in the ability to control andmanipu-late the OAM of ultrafast light pulses has also

emerged, resulting in numerous techniques thatcan imprint OAMdirectly onto an arbitrary wave-form. Diffractive and refractive optics (e.g.,q-plates, spiral-phase plates, and holographictechniques) (11–13) can impart OAM onto wavesfrom radio, to optical, and even x-ray (14) fre-quencies, and recent advances in high harmonicgeneration (HHG) have produced attosecondextreme-ultraviolet (EUV) pulses with designerOAM (15–28).One of the most exciting capabilities enabled

by OAM beams is their ability to exert photo-mechanical torques (2, 29, 30). Whereas thelinear momentum of light can be employed tocontrol and manipulate microscopic objects viathe gradient and scattering forces associatedwithits intensity profile, optically induced torquemanifests from angular momentum transfer be-tween an object and a light field. This enablesfundamental capabilities in advanced classicaland quantum optical control and manipulationtechniques, such as optical tweezers, lattices, andcentrifuges (9, 31–34), allowing for the realiza-tion of molecular and micromechanical rotors,single-particle trafficking, and fundamental studiesof atomic motion in liquids and Bose-Einsteincondensates (35, 36).We theoretically predict and experimentally

validate the generation of light beams that carrytime-dependent OAM, thus presenting a self-torque. This inherent property of structured light,the self-torque, ℏx, is defined as ℏx ¼ ℏd‘ðtÞ=dt ,where ℏ‘ðtÞ is the time-dependent OAM con-tent of the light pulse. After being generated, thetime-dependent OAM remains as a structuralproperty of the light beam propagating in free

space, where no interaction with external agentsis present. Thus, the term self-torque refers tothe inherent angular acceleration of the lightbeam, in an analogy with other physical systemsthat possess a self-induced time variation of theangular momentum—such as the radiation reac-tion of charged particles (37) or gravitationalself-fields (38). Although OAM is well understoodas a spatial property of light beams, to date, lightpulses with time-dependent OAM have not beenproposed or observed. We demonstrate that theself-torque arises as a necessary consequenceof angular momentum conservation during theextreme nonlinear optical process of HHG. InHHG, the interaction of an intense field with anatom or molecule leads to the ionization of anelectronic wave packet, which acquires energyfrom the laser field before being driven backto its parent ion, emitting a high-frequencyphoton upon recollision (39, 40). The emittedharmonic radiation can extend from the EUV tothe soft x-ray regime if the emissions from manyatoms add together in phase (41–44). The result-ing comb of fully coherent harmonics of thedriving field in turn yields trains of phase-lockedattosecond pulses (45, 46).Self-torqued light beams naturally emerge when

HHG is driven by two time-delayed infrared (IR)pulses that differ by one unit of OAM (Fig. 1). Thedynamical process of HHG makes it possible toimprint a continuous time-varying OAM, whereall OAM components are present—thus creatingself-torqued EUV beams. Intuitively, these exoticpulses can be understood as being composed oftime-ordered photons carrying consecutively in-creasing OAM.The self-torque of light translates to an azi-

muthal frequency chirp (i.e., a spectral shiftalong the azimuthal coordinate) on the radia-tion emission—and vice versa, which allows us toquantify the self-torque by an experimental mea-surement of the azimuthal frequency chirp. Inaddition, the degree of self-torque of EUV har-monic beams can be precisely controlled throughthe time delay and pulse duration of the driving,IR laser pulses. The generation of light beamswith self-torque opens up a route for the investi-gation of systems with time-varying OAM thatspontaneously appear in nature (47) as macro-scopic dynamical vortices or—owing to the highfrequency of the beams—microscopic ultrafastsystems. For example, because short-wavelengthlight can capture the fastest dynamics in mate-rials (48, 49), self-torqued EUV beams can beexpected to be used for imaging magnetic andtopological excitations, launching selective andchiral excitation of quantummatter (50), imprint-ing OAM centrifuges (32), switching superposi-tions of adiabatic charge migration in aromaticor biological molecules (51, 52), or manipulatingthe OAM dichroism of nanostructures (53) onattosecond time scales.

Theory underlying the self-torqueof light

To create light beams with self-torque, we drivethe HHG process with two linearly polarized IR

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1Grupo de Investigación en Aplicaciones del Láser y Fotónica,Departamento de Física Aplicada, University of Salamanca,Salamanca E-37008, Spain. 2JILA, Department ofPhysics, University of Colorado and NIST, Boulder, CO80309, USA. 3ICFO, Institut de Ciencies Fotoniques, TheBarcelona Institute of Science and Technology, Av. CarlFriedrich Gauss 3, 08860 Castelldefels (Barcelona), Spain.4ICREA, Pg. Lluís Companys 23, 08010 Barcelona, Spain.5Kapteyn-Murnane Laboratories Inc. (KMLabs Inc.), 4775Walnut Street no. 102, Boulder, CO 80301, USA.*Corresponding author. Email: [email protected] (L.R.);[email protected] (K.M.D.)†These authors contributed equally to this work.

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pulses exhibiting the same frequency content(centered atw0 ¼ 2pc=l0), butwith differentOAM,‘1 and ‘2 , where j‘1 � ‘2j ¼ 1 . The two laserpulses are separated by a variable time delay, td,which is on the order of the individual pulsewidths (Fig. 1A) [see also supplementary textsection S1 in (54)]. These two collinear IR vortexbeams are then focused into an atomic gas tar-get, such that the transverse intensity distri-bution of the two drivers exhibits maximumoverlap. We model the HHG process using fullquantum simulations in the strong-field ap-proximation (SFA) that include propagation viathe electromagnetic field propagator (55), amethod that was used in several previous cal-culations of HHG involving structured pulses(16, 18, 20, 21, 26, 28, 44, 56). We consider thedriving vortex pulses possessing ‘1 and ‘2 , de-scribed by a sin2 envelope with t = 10 fs fullwidth at half-maximum (FWHM) in intensity,centered at l0 = 800 nm, and delayed by td = t =10 fs (see materials and methods for furtherdetails). Figure 1A shows a schematic of thetemporal envelopes of each pulse (red), as wellas their superposition (blue). Figure 1C showsthe time-dependent OAM of the 17th harmonicobtained from our simulations (color scale),whereas in Fig. 1B the spatial intensity distri-bution of the 17th harmonic is sketched at threeinstants of time during the emission process.To extract the temporal variation of the OAM,we first select the HHG spectrum in the fre-quency range ðq� 1Þw0 to ðqþ 1Þw0 (where q isthe harmonic order to explore, being q = 17 inFig. 1), and then we perform a Fourier transformalong the azimuthal coordinate (20) at each timeinstant along the harmonic pulse. Notably, the

temporal variation of the OAM ismonotonic andcontinuous, spanning over an entire octave ofconsecutive topological charges—i.e., it includesall OAM components from q‘1 ¼ 17 to q‘2 ¼ 34.The nature of self-torqued beams can be un-

derstood through a simple theoretical analysis.Previous works in OAM-HHG have demonstratedthat an IR vortex beam can be coherently con-verted into high-frequency vortex beams (15–28).When HHG is driven by a single, linearly polar-ized, IR vortex beam with integer topologicalcharge, ‘1, the OAM of the qth-order harmonicfollows a simple scaling rule, ‘q ¼ q‘1 (16, 17).This scaling reflects the nature of OAM conser-vation in HHG, where q IR-photons combineto produce the qth-order harmonic. If HHG isdriven by the combination of two collinear andtemporally overlapped IR vortices with differ-ent OAM, ‘1 and ‘2 , each harmonic order willspan over a wide OAM spectrum, given by ‘q ¼n1‘1 þ n2‘2 (20), where n1 and n2 are the num-ber of photons absorbed from each driverðn1 þ n2 ¼ q;whose total must be odd due toparity restrictionsÞ. Each channel, ðn1;n2Þ, isweighted according to a binomial distribution,associated with the different combinations ofabsorbing n1 photons with ‘1 and n2 photonswith ‘2 . The effect of the harmonic intrinsicphase in the OAM spectrum, also explored in(20), is second order, and negligible for the re-sults presented here.In this work, we consider the HHG fields that

can be produced by two IR laser vortex pulsesseparated by some time delay. The superpositionof the delayed envelopes turns into a temporaldependence in the relative weights of the drivingfields—thus introducing time as an additional

parameter. To show how this influences theOAM structure of the EUV harmonics, we con-sider two time-delayed, collinear, linearly polar-ized, IR driving pulses with different OAM, ‘1and ‘2 . We denote, in cylindrical coordinatesðr; f; zÞ, the complex amplitudes of the drivingfields at the focus position (z = 0) as U1ðr; f; tÞand U2ðr; f; tÞ. For simplicity, we consider thefield amplitudes at the ring of maximum inten-sity at the target—where the HHG efficiency ishighest—and the resulting field can be written asUðf; tÞ ¼ U0ðtÞf½1� hðtÞ�ei‘1f þ hðtÞei‘2fg,whereU0ðtÞ ¼ U1ðtÞ þ U2ðtÞ and hðtÞ ¼ U2ðtÞ=U0ðtÞis the relative amplitude of the second beam.According to the strong-field description of HHG,the amplitude of the qth-order harmonic, Aqðf; tÞ,scales nonperturbatively with that of the drivinglaser, with an exponent p < q [p ≃ 4 for our laserparameters (20)], whereas the qth-order harmonicphase is considered to be q times that of thedriver (see supplementary text section S1 for thecomplete derivation); thus

Aqðf; tÞºUp0 ðtÞ

� Ppr¼0

pr

� �ð1� �hðtÞÞreir‘1f�hðp�rÞðtÞeiðp�rÞ‘2f

� �

� eiðq�pÞ½ð1��hðtÞÞ‘1þ�hðtÞ‘2 �f ð1Þ

where r is an integer and �hðtÞ is the average ofhðt) over the time it takes the ionized electronto complete the rescattering trajectory that con-tributes to the generation of a particular har-monic. For this average, we have considered theso-called short trajectories (57, 58), whose excur-sion time can be approximated to half a cycle.The contribution of long trajectories to the OAMcontent is two orders of magnitude weaker thanthat of the short ones (20). The summation inEq. 1 is carried over p different OAM channels,eachweighted by a binomial distribution in accord-ance with the combinatory nature of the HHGup-conversion process. Parity conservation inHHG demands that the total number of photonsabsorbed from each driving field, n1 þ n2 , mustbe odd, which implies that to generate all inter-mediate OAM states between q‘1 and q‘2 , theOAM of the drivers must differ by one unit, i.e.,j‘1 � ‘2j =1. The mean OAM of the qth-orderharmonic at any instant of time along the har-monic pulse is given by [see (54)]

�‘qðtÞ ¼ qh1� �hðtÞð Þ‘1 þ �hðtÞ‘2

ið2Þ

and the width of the OAM distribution is

s‘q ¼ j‘2 � ‘1jffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip�hðtÞ

�1� �hðtÞ

�rð3Þ

In analogy with mechanical systems, we char-acterize the time-varying OAM spectrum of theqth-order harmonic via the self-torque

xq ¼ d�‘qðtÞ=dt ð4Þ

As the OAM of light is defined as ℏ‘, the self-torque is given by ℏx. For simplicity we factor

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Fig. 1. Generation of EUV beams with self-torque. (A) Two time-delayed, collinear IR pulseswith the same wavelength (800 nm), but different OAM values, are focused into an argon gas target(HHG medium) to produce harmonic beams with self-torque. The spatial profile of the complete,time-integrated, HHG beam from full quantum simulations is shown on the EUV CCD. (B) Predictedevolution of the intensity profile of the 17th harmonic at three instants in time during the emissionprocess. (C) Temporal evolution of the OAM of the 17th harmonic, for two driving pulses with thesame duration t ¼ 10 fs, at a relative time delay of td ¼ t. The average OAM, �‘17 (solid green), andthe width of the OAM distribution, s‘17 (distance between the solid and dashed-green lines), areobtained from Eqs. 2 and 3. The self-torque associated with this pulse, x17 = 1.32 fs

−1, is obtainedfrom the slope of the smooth and continuous time-dependent OAM.

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out ℏ and denote the self-torque by x, in unitsof fs−1. It is worth mentioning that s‘q de-pends weakly on the harmonic order, as theparameter p remains almost constant along thenonperturbative spectral plateau. The nonper-turbative nature of the HHG process reducesthe number of available channels to generatethe qth-order harmonic from q (perturbative)to p~4 (nonperturbative). As typically p

dashed lines). This result shows that the spectralbandwidth of the harmonics can be preciselycontrolled via the temporal and OAM propertiesof the driving pulses. Moreover, it provides adirect, experimentally measurable parameter toextract the self-torque, without measuring theOAM of each harmonic at each instant of timewith subfemtosecond resolution, which is cur-rently unfeasible. This reasoning implies that abeamwith azimuthal frequency chirp would alsoexhibit self-torque. Up to now, however, HHGbeams have only been driven either by spatiallychirped pulses [such as the so-called “attosecondlighthouse” technique (64, 65)], or angularlychirped pulses through simultaneous spatialand temporal focusing, which (in theory) yieldspatially chirped harmonics (66). However, tothe best of our knowledge, azimuthal chirp—and thus, self-torque—has not been imprinted

into EUV harmonics or in any other spectralregime.

Experimental confirmation of theself-torque of EUV beams

Light beams possessing a self-torque were exper-imentally generated by driving the HHG pro-cess in argon gas using two collinear, IR vortexbeams with topological charges ‘1= 1 and ‘2= 2that are derived from a high-power, ultrafast re-generative amplifier (Fig. 3A). Briefly (see mate-rials and methods for full details), the two vortexbeams are spatiotemporally overlapped to yielda mixed OAM driving mode, which is then di-rected onto a supersonic expansion of argon gasto generate self-torqued EUV beams (q= 13 to 23,~20 to 36 eV). The presence of self-torque in theemitted high harmonics is confirmed by using acylindrical mirror–flat-grating EUV spectrometer

that serves to transform the self-torque-inducedazimuthal chirp into a spatial chirp, which isthen spectrally resolved as the (1D) focusing har-monic beam is dispersed (Fig. 3B). This simulta-neous mapping of the azimuthal frequency chirpand high-harmonic comb to the same spectralaxis is achieved by aligning the intensity crescentof the EUV beam [see materials and methodsand (54)] such that its intensity-weighted centerof mass (COM) is orthogonal to the mutuallyparallel focusing and dispersion axes of the EUVspectrometer. In this configuration, the azimuthalfrequency chirp is mapped to a linear spatialchirp by the cylindrical mirror, and this resultingspatial chirp in each harmonic is then resolvedby the grating. The resulting spatial-spectral dis-tribution is then imaged via a high–pixel density,EUV charge-coupled device (CCD) camera, whichallows for the simultaneous measurement ofthe azimuthal angular extent of the self-torquedbeams (54) and the induced azimuthal frequencychirp with a high precision. High-resolutionHHGspectra are collected as a function of time delaybetween the driving pulses by scanning the rela-tive time delay between the two beams in two-cycle increments (i.e., 5.272 fs), which ensuresthat the HHG beam remains aligned to the spec-trometer at each experimentally sampled timedelay. Such exquisite control (fig. S6) allows usto simultaneously measure both the self-torque-induced frequency chirp of the HHG beams andthe azimuthal angular range over a large range ofrelative time delays.Figure 4 shows the comparison between ex-

perimental and theoretical results. Panels (A)and (B) show the experimental and theoreticalspatial profile of the high harmonic beams, re-spectively. The crescent shape of the measuredspatial profile already gives a clear indicationof the presence of all intermediate OAM con-tributions from q‘1 to q‘2, and thus, of the crea-tion of self-torqued beams. Panels (C) to (F) show


Fig. 2. Azimuthal frequency chirp of self-torqued beams. Simulated spatial HHG spectrumalong the azimuthal coordinate ðfÞ when the time delay between the driving pulses is (A) 10 fsand (B) −10 fs. The self-torque of light imprints an azimuthal frequency chirp, which is differentfor each harmonic, as indicated by the gray dashed lines (obtained from Eq. 6). The azimuthal fre-quency chirp serves as a direct measurement of the self-torque of each harmonic beam. The insetof (A) shows the intensity profile of the HHG beam, as well as the definition of the azimuth, f.

Fig. 3. Experimental scheme for generating and measuring light beams with a self-torque. (A) Two time-delayed, collinear IR pulses with the samewavelength (790 nm), but different OAM values, are focused into an argon gas target to produce harmonic beams with self-torque. (B) An EUVspectrometer, composed of a cylindrical mirror and flat-grating pair, collapses the HHG beam in the vertical dimension (lab frame y axis), while preservingspatial information, and thus the azimuthal extent in the transverse dimension (lab frame x axis). (Lower-right inset) The cylindrical mirror effectivelymaps the azimuthal frequency chirp into a spatial chirp along the lab frame x axis (i), which is then dispersed by the grating (ii).

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the azimuthal chirp of the high harmonics fortime delays of td = 50.4 (C and D) and −50.4 fs(E and F), respectively. The different slope ofthe azimuthal chirp, and the excellent agree-ment with the analytical theory given by Eq. 6(gray dashed lines), and the full quantum sim-ulations, confirm the presence of self-torque inthe retrieved harmonic beams. Driving pulses oft ¼ 52 fs have been used in our full quantum sim-ulations to mimic the experimental parameters.In Fig. 5, we plot the experimental (solid lines)

and theoretical (dashed lines) self-torques ob-tained for the 17th (A), 19th (B), 21st (C), and23rd (D) harmonics as a function of the timedelay between the IR drivers, for the same pa-rameters as in Fig. 4. As the time delay is varied,so too is the degree of azimuthal frequency chirpacross the entire harmonic spectrum (accordingto Eqs. 2 and 6), verifying the dynamical build-upof OAM in the self-torqued beams. The self-torque is extracted from themeasured azimuthalspectral shift (see Fig. 4F) and the azimuthalextent of the HHG beam [see (54) for details],using Eq. 6. The excellent agreement and, es-pecially, the overall trend, unequivocally dem-onstrate the presence of a temporally evolvingOAM content and, thus, a self-torque, in all theEUV harmonics generated.

Self-torque versus time duration andEUV supercontinuum generation

EUV beams with self-torque can be generatedand controlled via the properties of the drivingIR vortex beams, with optimal self-torque pro-duced when the laser pulse separation is equal totheir duration (i.e., td ¼ t), where all intermed-iate OAM contributions appear with a similarweight (fig. S1). To illustrate this concept, Fig. 6Ashows the simulated self-torque obtained for dif-ferent IR driving pulse durations.In particular, if driven by few-cycle pulses, the

self-torque—and thus the azimuthal chirp—ishigh, with large amounts of OAM building up onan attosecond time scale (Fig. 6B, where t ¼ 4 fs).If the torque is high enough, the harmonic fre-quency comb sweeps along the azimuth, encap-sulating all the intermediate frequencies betweenthe teeth of the harmonic comb. Thus, the fre-quency chirp of time-dependent OAM beamsnot only is useful to measure the self-torque butalso represents an approach to obtain an EUVsupercontinuum, as shown in the right inset ofFig. 6B. This allows for the creation of a veryprecise, azimuthally tunable frequency comb inthe EUV and a supercontinuum spectrum that iscomplementary, yet distinct, from that of otherapproaches (67–69).

Conclusions

We have demonstrated that light beams withtime-dependent OAM can be created, thus carry-ing optical self-torque. This property spans theapplications of structured light beams (1) by ad-ding a new degree of freedom, the self-torque,and thus introducing a new route to controllight-matter interactions. In particular, ultrafast,short-wavelength, high harmonic beams with


Fig. 4. Azimuthal frequency chirp and experimental measurement of the self-torque of EUVbeams. (A and B) Experimental and theoretical spatial intensities of the HHG beams, after passingthrough an Al filter, comprising harmonics q = 13 to 23. (C to F) Spatial HHG spectrum along theazimuthal coordinate ðfÞ from experiment [(C) and (E)] and quantum simulations [(D) and (F)],when the time delay between the driving pulses is [(C) and (D)] 50.4 fs and [(E) and (F)] −50.4 fs.The self-torque of light imprints an azimuthal frequency chirp, which is different for eachharmonic, as indicated by the gray dashed lines (obtained from Eq. 6). (G and H) Theoreticaland experimental harmonic lineouts obtained at f = –0.8 rad (green), f = 0.0 rad (yellow), and f =0.8 rad (blue) for td= 50.4 fs.The azimuthal frequency chirp serves as a direct measurement of theself-torque of each harmonic beam. Differences in mode size of the theoretical and experimental EUVbeam are due to slight differences in the fundamental beam mode sizes (see materials and methods).



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self-torque can be naturally produced by takingadvantage of the conservation laws inherent toextreme nonlinear optics. This capability can yielddistinctively structured light beams that can de-liver optical torque on the natural time and

length scales of charge and spin ordering, e.g.,femtosecond and nanometer. Finally, the self-torque of light imprints an azimuthal frequencychirp, which allows a way to experimentallymeasure and control it. Moreover, if the self-

torque is high enough, the harmonic frequencycomb sweeps smoothly along the azimuth, andif integrated, a high-frequency supercontinuumis obtained, thus presenting exciting perspectivesin EUV and ultrafast spectroscopies of angularmomentum dynamics.

Materials and methodsTheoretical approach for full quantumsimulations describing the self-torqueof OAM high harmonic beams

To calculate the HHG driven by two time-delayedOAM pulses, we use a theoretical method thatcomputes both the full quantum single-atomHHG response and subsequent propagation (55).The propagation is based on the electromagneticfield propagator, in which we discretize the tar-get (gas jet) into elementary radiators (55). Thedipole acceleration of each elementary source iscomputed using the full quantumSFA, instead ofsolving directly the time-dependent Schrödingerequation, yielding a performance gain in com-putational time when computing HHG over theentire target (55). At the microscopic single-atomlevel, and for the parameters considered in thiswork, the spatial phase of the electric field can bewell approximated as homogeneous in the vici-nity of the atom where the wave packet dynam-ics take place. We assume that the harmonicradiation propagates with the vacuum phasevelocity, which is a reasonable assumption forhigh-order harmonics. Propagation effects inthe fundamental field, such as the productionof free charges, the refractive index of the neu-trals, and the group velocity walk-off, as well asabsorption in the propagation of the harmonics,are taken into account. Although we account forthe time-dependent nonlinear phase shifts in thedriving fields, nonlinear spatial effects are nottaken into account.We consider two vortex beamswith ‘1 ¼ 1 and ‘2 ¼ 2, whose spatial structureis represented by a Laguerre-Gaussian beam [see


Fig. 5. Experimental confirmation of the self-torque of light in EUV beams. Self-torquesobtained as a function of the time delay between the IR laser drivers for the 17th (A), 19th (B), 21st(C), and 23rd (D) harmonics. The experimental data are shown in solid-color lines, the results fromfull quantum simulations in dashed lines, and the analytical estimation given by Eq. 2 in solid blacklines. The shaded regions depict the experimental uncertainty in the retrieved self-torque for eachharmonic order, which themselves comprise the standard “one sigma” deviation of the measuredself-torque (i.e., 68% of the measured self-torque values will fall within this uncertainty range).

Fig. 6. Manifestation of self-torque for EUV supercontinuum generation. (A) Self-torque as a function of pulse duration for the 17th and 23rd harmonics,for time delays equal to their pulse duration. Solid lines are calculated from Eq. 2, and the squares correspond to results from full quantum simulations.(B) Spatiospectral HHG distributions when driven by two 800-nm, 4-fs pulses with ‘1=1 and ‘2=2, delayed by 4 fs with respect to each other.The opticalself-torque imprints an azimuthal frequency chirp, which is different for each harmonic order, as indicated by the gray dashed lines (obtained from Eqs. 5 and 6).The right panel shows the HHG yield at p/2 rad (blue line, and white vertical dashed line in B) and the spatially integrated supercontinuum (red line).



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eq. S13 in (54)]. The laser pulses are modeledwith a sin2 envelopewhose FWHM in intensity ist , and centered at 800 nm in wavelength. Theamplitudes of the two fields are chosen to obtainthe same peak intensity (1:4� 1014 W/cm2) at fo-cus for each driver at the radii ofmaximum super-position (i.e., the brightest intensity rings overlapspatially). The driving beam waists are chosen tooverlap at the focal plane (beingw1 ¼ 30:0mmfor‘1, and w2 ¼ w1=

ffiffiffi2

p ¼ 21:4 mm for ‘2) where a10-mm-wide Ar gas jet flows along the directionperpendicular to the beam propagation, with apeak pressure of 667 Pa (5 torr). The low thick-ness of the gas jet is due to computational timelimitations; however, on the basis of our previousresults of OAM-HHG (18), we do not foresee anyfundamental deviationwhen considering thickergas jets closer to the experimental jet used in thiswork (a diameter of 150 mm).

Experimental setup for the generationand characterization of self-torquedEUV beams

The generation of self-torquedhigh-harmonics isachieved by impinging a pair of collinear, lin-early polarized, nondegenerate IR-vortex beamsonto a supersonic expansion of argon gas. TheIR vortex beams (with topological charges of‘1 ¼ 1; ‘2 ¼ 2) are derived from a high-power,ultrafast regenerative amplifier (790 nm, 40 fs,9 mJ, 1 kHz, KMLabs Wyvern HE). The near fulloutput of the amplifier is sent into a frequency-degenerate Mach-Zehndertype interferometer,which separates and later recombines the twodriving pulses to form the dual-vortex IR driver.In each spatially separated arm of the interfer-ometer, a combination of half-waveplates, facetedspiral phase plates (16 steps per phase ramp,HoloOr), and independent focusing lenses resultin each beam possessing linear polarization, non-degenerate topological charges, and similarlysized intensity rings at focus. Independent irisesin each beam path allow for fine tuning of thetransverse mode size at focus and are used tomatch the size of themaximum-intensity ring foreach driver. Using this strategy, the two drivingbeams possessed a full diameter of the intensityof ∼65 mm—corresponding to waists sizes ofw‘1≈45 mm andw‘2 ≈ 33 mm (70). The driving lasermodes themselves, both individually and com-bined, are characterized by amodified Gerchberg-Saxton phase retrieval algorithm, which solvesfor the phase of a propagating light beam andallows extraction of the OAM content of the IRvortices (see supplementary text section S4), thusensuring high-quality vortex beams for driving theHHGprocess (movie S1). Thismodified Gerchberg-Saxton method acquires and retrieves OAMcontent much faster than our previous charac-terization method using ptychography (71), butit is limited to nonmultiplexed (i.e., single-color)beams. A high-precision, high-accuracy, and high-repeatability delay stage (Newport, XMS-160S) isused to control the relative time delay betweenthe two driving pulses, with subfemtosecond pre-cision. The pulses are recombined at the outputof the interferometer using a low-dispersionbeam-

splitter and then directed onto the supersonicexpansion of argon gas in a vacuum chamber.We take extreme care to ensure that the twoarms experience similar dispersion by usingthe same thickness and design of optics in eacharm of the interferometer, which helps to reduceeffects from carrier-to-envelope phase variationin the separate beam paths, while also ensur-ing similar pulse widths. Finally, the use of afrequency-degenerate Mach-Zehnder interfer-ometer results in a 50% intensity loss of eachdriver when combined at the interferometer’sexit; however, this configuration proved ideal tominimize pulse dispersion, while also allowingfor independent control of the polarization andtopological charge of the driving beams.Self-torqued high harmonics are generated via

the HHG up-conversion process, then dispersedin 1D via a cylindrical mirror–flat-grating EUVspectrometer and finally collected by a CCDcamera (Andor Newton 940). A 200-nm-thickaluminum filter blocks the residual driving lightbefore entering the spectrometer—while passingharmonics over its transmission range, ~17 to72 eV—and all harmonic spectra are corrected forthe transmission of the EUV beamline. To alignthe resulting HHG crescent to the spectrometer,we exploit the natural physics of time-delayedOAM beams. When two vortex beams with ‘1= 1and ‘2 = 2 are superposed, such that theiramplitudes and intensity rings are equal, theresulting intensity distribution exhibits a char-acteristic crescent shape. The azimuthal orien-tation of the COM of the intensity crescent canbe controlled via a relative phase delay betweenthe two single-mode OAM drivers, such that afull-cycle phase delay (i.e., 2.635 fs for the 790-nmpulses used here) returns the intensity crescentto its initial position. By carefully adjusting thetime delay between the two single-mode IRvortex beams, we can control the alignment ofthe intensity crescent of the driving beam [see(54)], and so to the resulting crescent-shapedharmonic beam (because, to first order, theHHG beam profile mimics the intensity distri-bution of the driving beam). Once the harmonicbeam is aligned to the spectrometer, the rel-ative phase delay between the driving beams isscanned in two-cycle increments (i.e., 5.272 fs),which ensures that theHHGbeam remains alignedto the spectrometer at each experimentally sam-pled time delay. Such exquisite control (see fig.S6) allows us to simultaneously measure boththe self-torque-induced frequency chirp of theHHG beam and the azimuthal angular range(see supplementary text section S7) with a highresolution.

REFERENCES AND NOTES

1. H. Rubinsztein-Dunlop et al., Roadmap on structured light.J. Opt. 19, 013001 (2017). doi: 10.1088/2040-8978/19/1/013001

2. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman,Orbital angular momentum of light and the transformation ofLaguerre-Gaussian laser modes. Phys. Rev. A 45, 8185–8189(1992). doi: 10.1103/PhysRevA.45.8185; pmid: 9906912

3. A. M. Yao, M. J. Padgett, Orbital angular momentum: Origins,behavior and applications. Adv. Opt. Photonics 3, 161–204(2011). doi: 10.1364/AOP.3.000161

4. A. E. Willner et al., Optical communication using orbital angularmomentum beams. Adv. Opt. Photonics 7, 66–106 (2015).doi: 10.1364/AOP.7.000066

5. A. Trichili et al., Optical communication beyond orbital angularmomentum. Sci. Rep. 6, 27674 (2016). doi: 10.1038/srep27674; pmid: 27283799

6. S. Fürhapter, A. Jesacher, S. Bernet, M. Ritsch-Marte,Spiral phase contrast imaging in microscopy. Opt. Express13, 689–694 (2005). doi: 10.1364/OPEX.13.000689;pmid: 19494929

7. M. A. Lauterbach, M. Guillon, A. Soltani, V. Emiliani, STEDmicroscope with spiral phase contrast. Sci. Rep. 3, 2050(2013). doi: 10.1038/srep02050; pmid: 23787399

8. G. Vicidomini, P. Bianchini, A. Diaspro, STED super-resolvedmicroscopy. Nat. Methods 15, 173–182 (2018). doi: 10.1038/nmeth.4593; pmid: 29377014

9. M. Padgett, R. Bowman, Tweezers with a twist. Nat. Photonics5, 343–348 (2011). doi: 10.1038/nphoton.2011.81

10. J. P. Torres, L. Torner, Twisted Photons: Applications of Lightwith Orbital Angular Momentum. (Bristol: Wiley-VCH, 2011).

11. L. Marrucci, C. Manzo, D. Paparo, Optical spin-to-orbitalangular momentum conversion in inhomogeneous anisotropicmedia. Phys. Rev. Lett. 96, 163905 (2006). doi: 10.1103/PhysRevLett.96.163905; pmid: 16712234

12. M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen,J. P. Woerdman, Helical-wavefront laser beams produced witha spiral phase plate. Opt. Commun. 112, 321–327 (1994).doi: 10.1016/0030-4018(94)90638-6

13. J. Atencia, M.-V. Collados, M. Quintanilla, J. Marín-Sáez,Í. J. Sola, Holographic optical element to generate achromaticvortices. Opt. Express 21, 21056–21061 (2013). doi: 10.1364/OE.21.021056; pmid: 24103978

14. J. C. T. Lee, S. J. Alexander, S. D. Kevan, S. Roy,B. J. McMorran, Laguerre-Gauss and Hermite-Gauss soft x-raystates generated using diffractive optics. Nat. Photonics 13,205–209 (2019). doi: 10.1038/s41566-018-0328-8

15. M. Zürch, C. Kern, P. Hansinger, A. Dreischuh, C. Spielmann,Strong-field physics with singular light beams. Nat. Phys. 8,743–746 (2012). doi: 10.1038/nphys2397

16. C. Hernández-García, A. Picón, J. San Román, L. Plaja,Attosecond extreme ultraviolet vortices from high-orderharmonic generation. Phys. Rev. Lett. 111, 083602 (2013).doi: 10.1103/PhysRevLett.111.083602; pmid: 24010438

17. G. Gariepy et al., Creating high-harmonic beams with controlledorbital angular momentum. Phys. Rev. Lett. 113, 153901 (2014).doi: 10.1103/PhysRevLett.113.153901; pmid: 25375710

18. C. Hernández-García, J. San Román, L. Plaja, A. Picón,Quantum-path signatures in attosecond helical beams drivenby optical vortices. New J. Phys. 17, 093029 (2015).doi: 10.1088/1367-2630/17/9/093029

19. R. Géneaux et al., Synthesis and characterization of attosecondlight vortices in the extreme ultraviolet. Nat. Commun. 7,12583 (2016). doi: 10.1038/ncomms12583; pmid: 27573787

20. L. Rego, J. San Román, A. Picón, L. Plaja, C. Hernández-García,Nonperturbative twist in the generation of extreme-ultravioletvortex beams. Phys. Rev. Lett. 117, 163202 (2016). doi: 10.1103/PhysRevLett.117.163202; pmid: 27792355

21. A. Turpin, L. Rego, A. Picón, J. San Román,C. Hernández-García, Extreme ultraviolet fractional orbitalangular momentum beams from high harmonic generation.Sci. Rep. 7, 43888 (2017). doi: 10.1038/srep43888;pmid: 28281655

22. F. Kong et al., Controlling the orbital angular momentum ofhigh harmonic vortices. Nat. Commun. 8, 14970 (2017).doi: 10.1038/ncomms14970; pmid: 28378823

23. D. Gauthier et al., Tunable orbital angular momentum inhigh-harmonic generation. Nat. Commun. 8, 14971 (2017).doi: 10.1038/ncomms14971; pmid: 28378741

24. C. Hernández-García, A twist in coherent X-rays. Nat. Phys. 13,327–329 (2017). doi: 10.1038/nphys4088

25. R. Géneaux et al., Radial index of Laguerre-Gaussian modes inhigh-order-harmonic generation. Phys. Rev. A 95, 051801(2017). doi: 10.1103/PhysRevA.95.051801

26. K. M. Dorney et al., Controlling the polarization and vortexcharge of attosecond high-harmonic beams via simultaneousspin-orbit momentum conservation. Nat. Photonics 13,123–130 (2019). doi: 10.1038/s41566-018-0304-3

27. W. Paufler, B. Böning, S. Fritzsche, Tailored orbital angularmomentum in high-order harmonic generation with bicircularLaguerre-Gaussian beams. Phys. Rev. A 98, 011401 (2018).doi: 10.1103/PhysRevA.98.011401

28. E. Pisanty et al., Conservation of torus-knot angularmomentum in high-order harmonic generation. Phys. Rev. Lett.




Dow

nloaded from

http://dx.doi.org/10.1088/2040-8978/19/1/013001http://dx.doi.org/10.1103/PhysRevA.45.8185http://www.ncbi.nlm.nih.gov/pubmed/9906912http://dx.doi.org/10.1364/AOP.3.000161http://dx.doi.org/10.1364/AOP.7.000066http://dx.doi.org/10.1038/srep27674http://dx.doi.org/10.1038/srep27674http://www.ncbi.nlm.nih.gov/pubmed/27283799http://dx.doi.org/10.1364/OPEX.13.000689http://www.ncbi.nlm.nih.gov/pubmed/19494929http://dx.doi.org/10.1038/srep02050http://www.ncbi.nlm.nih.gov/pubmed/23787399http://dx.doi.org/10.1038/nmeth.4593http://dx.doi.org/10.1038/nmeth.4593http://www.ncbi.nlm.nih.gov/pubmed/29377014http://dx.doi.org/10.1038/nphoton.2011.81http://dx.doi.org/10.1103/PhysRevLett.96.163905http://dx.doi.org/10.1103/PhysRevLett.96.163905http://www.ncbi.nlm.nih.gov/pubmed/16712234http://dx.doi.org/10.1016/0030-4018(94)90638-6http://dx.doi.org/10.1364/OE.21.021056http://dx.doi.org/10.1364/OE.21.021056http://www.ncbi.nlm.nih.gov/pubmed/24103978http://dx.doi.org/10.1038/s41566-018-0328-8http://dx.doi.org/10.1038/nphys2397http://dx.doi.org/10.1103/PhysRevLett.111.083602http://www.ncbi.nlm.nih.gov/pubmed/24010438http://dx.doi.org/10.1103/PhysRevLett.113.153901http://www.ncbi.nlm.nih.gov/pubmed/25375710http://dx.doi.org/10.1088/1367-2630/17/9/093029http://dx.doi.org/10.1038/ncomms12583http://www.ncbi.nlm.nih.gov/pubmed/27573787http://dx.doi.org/10.1103/PhysRevLett.117.163202http://dx.doi.org/10.1103/PhysRevLett.117.163202http://www.ncbi.nlm.nih.gov/pubmed/27792355http://dx.doi.org/10.1038/srep43888http://www.ncbi.nlm.nih.gov/pubmed/28281655http://dx.doi.org/10.1038/ncomms14970http://www.ncbi.nlm.nih.gov/pubmed/28378823http://dx.doi.org/10.1038/ncomms14971http://www.ncbi.nlm.nih.gov/pubmed/28378741http://dx.doi.org/10.1038/nphys4088http://dx.doi.org/10.1103/PhysRevA.95.051801http://dx.doi.org/10.1038/s41566-018-0304-3http://dx.doi.org/10.1103/PhysRevA.98.011401http://science.sciencemag.org/

122, 203201 (2019). doi: 10.1103/PhysRevLett.122.203201;pmid: 31172784

29. M. Babiker, W. L. Power, L. Allen, Light-induced torque onmoving atoms. Phys. Rev. Lett. 73, 1239–1242 (1994).doi: 10.1103/PhysRevLett.73.1239; pmid: 10057660

30. H. He, M. E. Friese, N. R. Heckenberg, H. Rubinsztein-Dunlop,Direct observation of transfer of angular momentum toabsorptive particles from a laser beam with a phase singularity.Phys. Rev. Lett. 75, 826–829 (1995). doi: 10.1103/PhysRevLett.75.826; pmid: 10060128

31. A. T. O’Neil, M. J. Padgett, Three-dimensional opticalconfinement of micron-sized metal particles and thedecoupling of the spin and orbital angular momentum withinan optical spanner. Opt. Commun. 185, 139–143 (2000).doi: 10.1016/S0030-4018(00)00989-5

32. D. M. Villeneuve et al., Forced molecular rotation in anoptical centrifuge. Phys. Rev. Lett. 85, 542–545 (2000).doi: 10.1103/PhysRevLett.85.542; pmid: 10991335

33. D. G. Grier, A revolution in optical manipulation. Nature 424,810–816 (2003). pmid: 12917694

34. M. E. J. Friese, H. Rubinsztein-Dunlop, J. Gold, P. Hagberg,D. Hanstorp, Optically driven micromachine elements.Appl. Phys. Lett. 78, 547–549 (2001). doi: 10.1063/1.1339995

35. E. M. Wright, J. Arlt, K. Dholakia, Toroidal optical dipoletraps for atomic Bose-Einstein condensates usingLaguerre-Gaussian beams. Phys. Rev. A 63, 013608 (2000).doi: 10.1103/PhysRevA.63.013608

36. A. Turpin et al., Blue-detuned optical ring trap for Bose-Einstein condensates based on conical refraction. Opt. Express23, 1638–1650 (2015). doi: 10.1364/OE.23.001638;pmid: 25835921

37. A. F. Rañada, L. Vázquez, On the self-torque on an extendedclassical charged particle. J. Phys. Math. Gen. 17, 2011–2016(1984). doi: 10.1088/0305-4470/17/10/013

38. S. R. Dolan et al., Gravitational self-torque and spin precessionin compact binaries. Phys. Rev. D Part. Fields Gravit. Cosmol.89, 064011 (2014). doi: 10.1103/PhysRevD.89.064011

39. K. J. Schafer, B. Yang, L. F. DiMauro, K. C. Kulander, Abovethreshold ionization beyond the high harmonic cutoff.Phys. Rev. Lett. 70, 1599–1602 (1993). doi: 10.1103/PhysRevLett.70.1599; pmid: 10053336

40. P. B. Corkum, Plasma perspective on strong field multiphotonionization. Phys. Rev. Lett. 71, 1994–1997 (1993). doi: 10.1103/PhysRevLett.71.1994; pmid: 10054556

41. A. Rundquist et al., Phase-matched generation of coherent softx-rays. Science 280, 1412–1415 (1998). doi: 10.1126/science.280.5368.1412; pmid: 9603725

42. R. A. Bartels et al., Generation of spatially coherent light atextreme ultraviolet wavelengths. Science 297, 376–378(2002). doi: 10.1126/science.1071718; pmid: 12130779

43. T. Popmintchev et al., Bright coherent ultrahigh harmonics inthe keV x-ray regime from mid-infrared femtosecond lasers.Science 336, 1287–1291 (2012). doi: 10.1126/science.1218497;pmid: 22679093

44. T. Fan et al., Bright circularly polarized soft X-ray highharmonics for X-ray magnetic circular dichroism. Proc. Natl.Acad. Sci. U.S.A. 112, 14206–14211 (2015). doi: 10.1073/pnas.1519666112; pmid: 26534992

45. P. M. Paul et al., Observation of a train of attosecond pulsesfrom high harmonic generation. Science 292, 1689–1692(2001). doi: 10.1126/science.1059413; pmid: 11387467

46. M. Hentschel et al., Attosecond metrology. Nature 414,509–513 (2001). doi: 10.1038/35107000; pmid: 11734845

47. B. Rodenburg et al., Influence of atmospheric turbulenceon states of light carrying orbital angular momentum.Opt. Lett. 37, 3735–3737 (2012). doi: 10.1364/OL.37.003735;pmid: 22941007

48. C. Chen et al., Distinguishing attosecond electron-electronscattering and screening in transition metals. Proc. Natl. Acad.

Sci. U.S.A. 114, E5300–E5307 (2017). doi: 10.1073/pnas.1706466114; pmid: 28630331

49. P. Tengdin et al., Critical behavior within 20 fs drives theout-of-equilibrium laser-induced magnetic phase transition innickel. Sci. Adv. 4, eaap9744 (2018). doi: 10.1126/sciadv.aap9744; pmid: 29511738

50. A. Picón et al., Transferring orbital and spin angularmomenta of light to atoms. New J. Phys. 12, 083053 (2010).doi: 10.1088/1367-2630/12/8/083053

51. F. Calegari et al., Charge migration induced by attosecondpulses in bio-relevant molecules. J. Phys. B At. Mol. Opt. Phys.49, 142001 (2016). doi: 10.1088/0953-4075/49/14/142001

52. G. Hermann et al., Attosecond angular flux of partial chargeson the carbon atoms of benzene in non-aromatic excited state.Chem. Phys. Lett. 683, 553–558 (2017). doi: 10.1016/j.cplett.2017.01.030

53. R. M. Kerber, J. M. Fitzgerald, S. S. Oh, D. E. Reiter, O. Hess,Orbital angular momentum dichroism in nanoantennas.Commun. Phys. 1, 87 (2018). doi: 10.1038/s42005-018-0088-2

54. See supplementary materials.55. C. Hernández-García et al., High-order harmonic propagation in

gases within the discrete dipole approximation. Phys. Rev. A82, 033432 (2010). doi: 10.1103/PhysRevA.82.033432

56. C. Hernández-García et al., Extreme ultraviolet vector beamsdriven by infrared lasers. Optica 4, 520 (2017). doi: 10.1364/OPTICA.4.000520

57. M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L’Huillier,P. B. Corkum, Theory of high-harmonic generation bylow-frequency laser fields. Phys. Rev. A 49, 2117–2132 (1994).doi: 10.1103/PhysRevA.49.2117; pmid: 9910464

58. A. Zaïr et al., Quantum path interferences in high-orderharmonic generation. Phys. Rev. Lett. 100, 143902 (2008).doi: 10.1103/PhysRevLett.100.143902; pmid: 18518033

59. M. V. Berry, Optical vortices evolving from helicoidal integerand fractional phase steps. J. Opt. A, Pure Appl. Opt. 6,259–268 (2004). doi: 10.1088/1464-4258/6/2/018

60. J. Leach, E. Yao, M. Padgett, Observation of the vortexstructure of non-integer vortex beam. New J. Phys. 6, 71(2004). doi: 10.1088/1367-2630/6/1/071

61. S. N. Alperin, M. E. Siemens, Angular Momentum ofTopologically Structured Darkness. Phys. Rev. Lett. 119,203902 (2017). doi: 10.1103/PhysRevLett.119.203902;pmid: 29219346

62. E. Karimi, L. Marrucci, C. de Lisio, E. Santamato, Time-divisionmultiplexing of the orbital angular momentum of light.Opt. Lett. 37, 127–129 (2012). doi: 10.1364/OL.37.000127;pmid: 22854442

63. I. J. Sola et al., Controlling attosecond electron dynamics byphase-stabilized polarization gating. Nat. Phys. 2, 319–322(2006). doi: 10.1038/nphys281

64. H. Vincenti, F. Quéré, Attosecond lighthouses: How to usespatiotemporally coupled light fields to generate isolatedattosecond pulses. Phys. Rev. Lett. 108, 113904 (2012).doi: 10.1103/PhysRevLett.108.113904; pmid: 22540475

65. J. A. Wheeler et al., Attosecond lighthouses from plasmamirrors. Nat. Photonics 6, 829–833 (2012). doi: 10.1038/nphoton.2012.284

66. C. Hernández-García, A. Jaron-Becker, D. D. Hickstein,A. Becker, C. G. Durfee, High-order-harmonic generation drivenby pulses with angular spatial chirp. Phys. Rev. A (Coll. Park)93, 023825 (2016). doi: 10.1103/PhysRevA.93.023825

67. I. P. Christov, M. M. Murnane, H. C. Kapteyn, High-harmonicgeneration of attosecond pulses in the “single cycle” regime.Phys. Rev. Lett. 78, 1251–1254 (1997). doi: 10.1103/PhysRevLett.78.1251

68. M. Chini, K. Zhao, Z. Chang, The generation, characterizationand applications of broadband isolated attosecond pulses.Nat. Photonics 8, 178–186 (2014). doi: 10.1038/nphoton.2013.362

69. W. Holgado et al., Continuous spectra in high-harmonicgeneration driven by multicycle laser pulses. Phys. Rev. A 93,013816 (2016). doi: 10.1103/PhysRevA.93.013816

70. J. Zhang, S.-J. Huang, F.-Q. Zhu, W. Shao, M.-S. Chen,Dimensional properties of Laguerre-Gaussian vortex beams.Appl. Opt. 56, 3556–3561 (2017). doi: 10.1364/AO.56.003556;pmid: 28430234

71. Y. Esashi et al., Ptychographic amplitude and phase reconstructionof bichromatic vortex beams. Opt. Express 26, 34007–34015(2018). doi: 10.1364/OE.26.034007; pmid: 30650831

ACKNOWLEDGMENTS

Funding: C.H.-G., J.S.R., and L.P. acknowledge support fromJunta de Castilla y León (SA046U16), Ministerio de Economía yCompetitividad (FIS2016-75652-P), FEDER funds, and Ministerio deCiencia, Innovación y Universidades (Eq. C2018-0041 17-P). L.R.acknowledges support fromMinisterio de Educación, Cultura y Deporte(FPU16/02591). C.H.-G. acknowledges support from a 2017 LeonardoGrant for Researchers and Cultural Creators, BBVA Foundation, andMinisterio de Ciencia, Innovación y Universidades for a Ramón y Cajalcontract (RYC-2017-22745), cofunded by the European Social Fund.H.K. and M.M. acknowledge support from the Department of EnergyBES Award no. DE-FG02-99ER14982 for the experimentalimplementation, the DARPA TEE Program Award. no. D18AC00017 forthe new experimental characterization methods developed, as well asa MURI grant from the Air Force Office of Scientific Research underAward no. FA9550-16-1-0121 for the theory. Q.L.N. acknowledgessupport from National Science Foundation Graduate ResearchFellowships (grant no. DGE-1144083). N.J.B. and D.C. acknowledgesupport from National Science Foundation Graduate ResearchFellowships (grant no. DGE-1650115). E.P. acknowledges Cellex-ICFO-MPQ fellowship funding; E.P. and M.L. acknowledge the SpanishMinistry MINECO (National Plan 15 Grant: FISICATEAMO no. FIS2016-79508-P, SEVERO OCHOA no. SEV-2015-0522, FPI), European SocialFund, Fundació Cellex, Generalitat de Catalunya (AGAUR grant no. 2017SGR 1341 and CERCA/Program), ERC AdG OSYRIS, EU FETPROQUIC, and the National Science Centre, Poland-Symfonia grantno. 2016/20/W/ST4/00314. We acknowledge the computer resourcesat MareNostrum and the technical support provided by BarcelonaSupercomputing Center (RES-AECT-2014-2-0085). This researchmade use of the high-performance computing resources of the Castillay León Supercomputing Center (SCAYLE, www.scayle.es/), financedby the European Regional Development Fund (ERDF). Authorcontributions: L.R., J.S.R, L.P., and C.H.-G. conceived the idea ofself-torqued beams. K.M.D., L.R., H.C.K., M.M.M., and C.H.-G.designed the experiment. K.M.D., N.J.B., Q.L.N., C.-T.L., D.E.C.,and A.L. conducted the experiment. K.M.D. and N.J.B. analyzedthe experimental data. L.R., J.S.R., L.P., E.P., and C.H.-G.performed the theoretical simulations and analyzed the resultingdata. C.H.-G., L.P., M.L., M.M.M., and H.C.K. supervised thetheoretical simulations, experimental work, and developed thefacilities and measurement capabilities. L.R., K.M.D., J.S.R.,M.M.M., L.P., and C.H.-G. wrote and prepared the manuscript. Allauthors provided constructive improvements and feedback tothis work. Competing interests: M.M.M. and H.C.K. have afinancial interest in KMLabs. All other authors declare nocompeting financial interests. Data and materials availability:All data needed to evaluate the conclusions in the paper arepresent in the paper or the supplementary materials.

SUPPLEMENTARY MATERIALS

science.sciencemag.org/content/364/6447/eaaw9486/suppl/DC1Supplementary TextFigs. S1 to S8References (72–79)Movie S1

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Generation of extreme-ultraviolet beams with time-varying orbital angular momentum

Hernández-GarcíaAllison Liu, Emilio Pisanty, Maciej Lewenstein, Luis Plaja, Henry C. Kapteyn, Margaret M. Murnane and Carlos Laura Rego, Kevin M. Dorney, Nathan J. Brooks, Quynh L. Nguyen, Chen-Ting Liao, Julio San Román, David E. Couch,

DOI: 10.1126/science.aaw9486 (6447), eaaw9486.364Science

, this issue p. eaaw9486Sciencemanipulate nanostructures and atoms on ultrafast time scales.time-dependent angular momentum, called self-torque. Such dynamic vortex pulses could potentially be used to controlled time delay between the pulses allowed the high harmonic extreme-ultraviolet vortex beam to exhibit atime-delayed vortex beams with different orbital angular momenta through the process of high harmonic generation. A

generated dynamic vortex pulses by interfering two incidentet al.enhance optical communications and imaging. Rego Structured light beams can serve as vortex beams carrying optical angular momentum and have been used to

Pulses with a twist and torque

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REFERENCES

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ULTRAFAST OPTICS Generation of extreme-ultraviolet ...Emilio Pisanty, Maciej Lewenstein, Luis Plaja, Henry C. Kapteyn, Margaret M. Murnane, Carlos Hernández-García INTRODUCTION: