778 OPTICS LETTERS / Vol. 29, No. 7 / April 1, 2004

Isolated attosecond pulses generated by relativistic effects in awavelength-cubed focal volume

Natalia M. Naumova, John A. Nees, Bixue Hou, and Gerard A. Mourou

Center for Ultrafast Optical Science, University of Michigan, Ann Arbor, Michigan 48109-2099

Igor V. Sokolov

Space Physics Research Laboratory, University of Michigan, Ann Arbor, Michigan 48109-2143

Received October 13, 2003

Lasers that provide an energy encompassed in a focal volume of a few cubic wavelengths (l3) can create rela-tivistic intensity with maximal gradients using minimal energy. With particle-in-cell simulations we foundthat single 200-as pulses could be produced efficiently in a l3 laser pulse ref lection by means of def lection andphase compression caused by the coherent motion of the plasma electrons that emit these pulses. This noveltechnique is efficient (�10%) and can produce single attosecond pulses from the millijoule to the joule level.© 2004 Optical Society of America

OCIS codes: 320.7110, 320.5520, 350.5400.

Nonlinear generation of subfemtosecond pulses hasbeen proposed1 and demonstrated2 using laser–atominteraction in the nonperturbative regime in gasesat intensities of the order of 1014 W�cm2. In thisdomain attosecond pulses may be generated, but, evenwith quasi-periodic phase matching,3 they produceefficiency far below 1%. It may, however, be possiblethat higher intensity coupled with dense plasma andrelativistic effects could generate attosecond pulseswith very high eff iciency (�10%) in the visible andultraviolet domains, where optics are readily available.

We have recently produced relativistic intensity inthe kilohertz regime with short pulses of less than10 fs,4 composed of only a few cycles, precisely focusedon a single wavelength spot size.5 In this case theentire laser pulse energy is contained within a focalvolume of a few l3. This has become known as thel3 regime. It can be beneficial to operate in the l3

regime for the following reasons: (1) The drivingbeam consists of well-corrected and tightly focusedfundamental frequency light, which limits wave-frontdistortions and produces spatially coherent ref lectedradiation. (2) Using only a few optical cycles makespossible a better discrimination between the tempo-rally coherent initial collective action of electrons andtheir complex response to successive cycles. (3) Thenarrow focus impresses maximal spatial gradients.

Driven relativistically, electrons acquire a quiverenergy exceeding their rest-mass energy mec2. Whenconsidering relativistic laser interactions, it is conve-nient to express intensity in terms of dimensionlessamplitude a0 � eE0�mvc. With a0 . 1 correspond-ing to intensities greater than 2 3 1018 W�cm2 for0.8-mm light, the plasma dielectric constant musttake relativistic effects into account. Accordingly,e � 1 2 v

2p0�gv2, where g � �1 1 a2

0�1�2 andvp0 � �4pne0e2�me�1�2.

Nonlinear optics based on the relativistic effect on e

has been studied mainly in subcritical-density plasma

0146-9592/04/070778-03$15.00/0

conditions, that is, in refraction—commonly known asrelativistic self-focusing—where the laser wave front ismodified as a result of the dielectric constant’s depen-dence on a0. This effect was discussed by Litvak6 andMax et al.7 It was observed with excimer lasers8 andwith chirped-pulse amplifier lasers.9 Mourou et al.10

proposed that this effect could be studied in the l3

regime, provided that the numerical aperture of therelativistic filament is matched with that of the fo-cusing optics. This regime has the advantage that itrequires only millijoule energies, which are easily pro-duced at kilohertz repetition rates.2 – 5 Relativistic ef-fects in supercritical plasmas have been discussed intheir application to the generation of harmonics11 – 14

and attosecond pulse trains15 by weakly and tightly14

focused long pulses.We have also suggested16 that the concept of rela-

tivistic self-focusing be extended from refraction to re-f lection for the case when the l3 laser pulse interactswith an overcritical-density plasma. For a0 . 1 thelight pressure can significantly modify the shape andmotion of the ref lecting surface of the plasma, whichin turn will change the ref lected wave front.

We then discovered by particle-in-cell (PIC) simu-lations that the plasma mirror produced by the laserpulse can also be def lective, enabling us to isolatesingle attosecond pulses. This def lection can make asubperiod cut from the driving pulse and produce iso-lated subcycle pulses traveling in different directions.This occurs because the deformation of the plasmaprofile results in significant changes of the localincidence angle (def ined to be the angle between theoriginal wave front and the deformed ref lecting sur-face) on a subcycle time scale. To enhance this effect,the plasma density n0 must be slightly overcritical,making the critical-density surface more responsiveto light pressure.

Furthermore, the driving f ield also produces anelectric f ield component that is normal to the plasma

© 2004 Optical Society of America

April 1, 2004 / Vol. 29, No. 7 / OPTICS LETTERS 779

ref lecting surface. Through the longitudinal in-f luence of the driving f ield we come to one morerelativistic effect, which is the most important for us:Along with the intense and coherent relativistic elec-tron motion in the direction perpendicular to the wavevector of the ref lected wave kr, which is responsiblefor the magnitude of the ref lected wave, there mustalso be a coherent motion parallel to kr , which, ac-cording to the Doppler effect, should be responsible fordramatic compression or elongation of the separatedfractions of the ref lected pulse, depending on the signof the parallel velocity. Thus, in the relativistic l3

regime, ref lection, def lection, and compression act inconcert to produce isolated attosecond pulses insteadof yielding a quasi-periodic wave propagating in onedirection.

To demonstrate this, we perform fully relativistictwo-dimensional (2D) PIC simulations and study thehighly nonlinear regime of the ref lection. The PICcode integrates, self-consistently, Maxwell’s equationsand relativistic equations of motion for electrons andions.17 The computation box is 20l 3 20l, withspatial resolution as high as 100 cells per l. Toresolve the density gradient, we take 16 electrons and16 protons per cell.

A linearly polarized laser pulse with its electric f ieldalong the y direction has been initiated at the leftboundary (x � 214) in vacuum and focused to a 1lspot normal to the plasma layer. The laser pulse has aGaussian profile and a duration of 5 fs (�2 cycles, fullwidth at half-maximum). The maximal intensity inthe focus is I � 2 3 1019 W�cm2. For l � 0.8 mm thiscorresponds to the dimensionless amplitude a0 � 3.The preionized collisionless plasma layer has a uni-form profile of thickness 2l and density 1.5ncr. Wechoose t � 0 to be the instant when the peak of thepulse envelope reaches the plasma boundary at x � 0.Space coordinates are measured in laser wavelengths,and time in cycles.

In Fig. 1(a) the electromagnetic energy density isshown at t � 11 for x , 21. We see that the ref lectedradiation has been split into separate impulses, eachmoving in its own direction. The most intense impulse(3) is directed toward the upper left-hand corner ofthe box, the next most intense impulse (1) is closer tothe middle of the box, and the least intense impulseof the three (2) is located toward the lower left-handcorner.

We plot the ref lected radiation at the half-intensitylevel [Fig. 1(b)] and find that only a subperiod pulsehas been ref lected in the upward direction. Tracingthis impulse, we find that it has a divergence of �20±.The maximum intensity values of this impulse followan �40± direction with respect to normal incidence.

The evolution of the electron density (Fig. 2) showsthe motion of the ref lecting surface driven by the laserpulse. Electron layers, pushed by the pulse, def lecteach half-cycle into a new direction depending on thephase and amplitude of the particular half-cycle. Inaddition, the ref lecting layer focuses these impulsesin the backward direction with varying focal lengths.Thus each cycle has its own divergence and its ownvirtual source point within the �l3 volume. Because

of the �f�1 focus and short duration of the incidentpulse this discrimination in the direction and curva-ture between separate cycles is possible. The thirdframe in Fig. 2, for t � 0.9, shows the surface fromwhich impulse (3) has been ref lected and def lected.The surface that redirects this impulse in the upwarddirection (indicated by an arrow) has the minimumfocal length.

Our simulations show that, as in photoionization ex-periments,18,19 the laser–plasma interaction is phasesensitive because of the short incident pulse duration.Changing the carrier-envelope phase offset of the pulseby p, we obtain a symmetrically mapped distributionfor the ref lected light, the most intense subperiod go-ing to the lower left-hand corner.

We plot the time dependence of the electromagneticenergy density at the point x � 23.5, y � 3 in Fig. 3[along the maximal intensity path of impulse (3)]. Wefind that the duration of this impulse is only 200 as,yet it contains 10% of the incident pulse energy. Thepulse has been compressed under the motion of theref lective layer of electrons in the direction parallelto kr .

Analytical modeling of coherent radiation from arelativistically driven charge layer as given in Ref. 16confirms that a Doppler-based compression occurs andthat it is in good agreement with PIC simulations. Al-though such analysis is beyond the scope of this Letter,

Fig. 1. (a) Electromagnetic energy density of the ref lectedradiation (E2 1 B2). Numbers (1), (2), and (3) indicatethe most intense impulses in the ref lected radiation.(b) Half-intensity level of the ref lected radiation.

Fig. 2. Snapshots of the electron density. The arrows in-dicate the instantaneous orientation of the surfaces fromwhich impulses (1), (2), and (3) were ref lected at the ap-proximate times of ref lection.

780 OPTICS LETTERS / Vol. 29, No. 7 / April 1, 2004

Fig. 3. Time evolution of E2 1 B2 at the point x � 23.5,y � 3. The arrows indicate the half-intensity level of anisolated impulse that contains 10% of the incident pulseenergy.

it has been used to show that the action of an opti-cal field with oblique incidence is capable of driving asheet of electrons relativistically with velocity compo-nents both with and against the ref lected wave vec-tor. The radiation from this sheet will form a shorterpulse while the charge motion is in the direction ofthe ref lected wave. Conversely, if the charge motionis receding, the opposite effect, Doppler stretching, oc-curs. Consequently the self-induced relativistic mo-tion of charges in the l3 regime directly generatesisolated attosecond pulses by the triple effect of ref lec-tion, def lection, and compression.

We performed three-dimensional (3D) simulationsin addition to 2D runs. The 3D results agreed withthose in 2D geometry, with some natural differencesbecause of the anisotropy caused by the polarizationeffects.20 On the other hand, in 3D simulations it iscurrently hard to achieve higher spatial resolution, sothe 2D result is more reliable.

The process of attosecond pulse formation is robust,scaling to joule energies and working with exponentialplasma profiles. However, in scaling to larger spotsizes we would expect to lose the high slopes that leadto angular isolation of adjacent half-cycles.

The suggested attosecond pulse generation schemeusing supercritical plasma is several orders of mag-nitude more eff icient than those involving recollisionsin gas targets. This is because the ensemble effect ofcritical surface ref lection, being coherent in nature, isnear unity, whereas the cross section for recollision ingases is quite small.

With PIC simulation and analytical modeling wehave shown that ref lections in the relativistic l3

regime are accompanied by def lection and compres-sion, leading to the generation of isolated attosecondpulses with �10% efficiency. This technique can bescaled from the millijoule to the joule level.

This work was supported by National Science Foun-dation grant 0114336. N. Naumova’s e-mail addressis naumova@umich.edu.

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