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INSTITUTE OF PHYSICS PUBLISHING REPORTS ON PROGRESS IN PHYSICS

Rep. Prog. Phys. 69 (2006) 443505 doi:10.1088/0034-4885/69/2/R04

Femtosecond x-ray science

T Pfeifer, C Spielmann and G Gerber

Physikalisches Institut, Universitat Wurzburg, Am Hubland, 97074 Wurzburg, Germany

Received 11 November 2005Published 18 January 2006Online at stacks.iop.org/RoPP/69/443

Abstract

We present theadvances in x-ray femtosecond pulse generation and themost recentdiscoveriesin the field of ultrashort (femtosecond) x-ray science. Nowadays x-rays show their potentialnot only when it comes to resolving atomic spatial scale but also the inherent temporal scaleof quantum dynamics in atoms, molecules and solids. We discuss ultrafast x-ray sources thatare currently used to generate femtosecond duration pulses of soft and hard x-ray radiation.Several techniques of x-ray pulse characterization arepresented along with a method to controlthe shape of coherent soft x-rays. A large number of experiments using femtosecond x-raypulses have been conducted recently and we review some of them. The field of ultrafast x-rayscience draws its strength from the large variety of different sources of femtosecond durationx-ray pulses that are complementary rather than competing.

(Some figures in this article are in colour only in the electronic version)

0034-4885/06/020443+63$90.00 2006 IOP Publishing Ltd Printed in the UK 443
http://dx.doi.org/10.1088/0034-4885/69/2/R04http://stacks.iop.org/RoPP/69/443http://dx.doi.org/10.1088/0034-4885/69/2/R04


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Contents

Page1. Introduction 4452. Femtosecond x-ray sources 446

2.1. High-harmonic generation 4482.1.1. Single particle response 4492.1.2. Propagation effects 457

2.2. Laser-induced plasma 4652.3. Ultrafast x-ray tube 4692.4. Accelerator-based sources 469

2.4.1. Electron-bunch slicing 4692.4.2. Electron-bunch compression 4702.4.3. Free-electron laser 471

3. Pulse characterization and control 4713.1. Direct measurement (streak cameras) 4733.2. Correlation techniques 474

3.2.1. Autocorrelation 4753.2.2. X-FROG measurements 4753.2.3. Crosscorrelation 4763.2.4. Ponderomotive streaking 4783.2.5. Laser-induced absorption gating 478

3.3. Control of pulse parameters 4794. Experiments 481

4.1. Gas-phase 4834.1.1. Probing of molecular dynamics 4834.1.2. Auger lifetime 4854.1.3. Adaptive control with spectrally shaped soft x-rays 486

4.2. Clusters 4874.3. Surfaces 489

4.3.1. Surface electron dynamics 4894.3.2. Surface photochemistry 490

4.4. Solid state 4914.4.1. Lattice dynamics 4924.4.2. Nonthermal melting 4954.4.3. Solid-state inner-shell electron dynamics 496

4.4.4. Electronlattice interaction dynamics 4974.5. Dense plasmas 4984.6. Nonlinear x-ray optics 498

5. Conclusion 499Acknowledgments 500References 501


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Femtosecond x-ray science 445

1. Introduction

The past 20th century has witnessed tremendous progress of science in many different fieldsand disciplines. Without question, the discovery of the x-rays [1] has been an important if notthe most importantdriving force to push forward our understanding andknowledge in all kindsof scientific areas. Applications range from structure analysis in solid-state as well as atomicphysics and molecular chemistry via imaging applications in medicine and the life sciences tothe discovery of the basic building blocks of life, in particular the DNA and more generally thestructure of proteins and other macromolecules. Uber eine neue Art von Strahlen (About anew kind of rays) was the title that Wilhelm Conrad Roentgen gave to his seminal report aboutthe x-rays that he discovered in 1895 at the University of Wurzurg. This discovery openedup the door to a new era of science where it should be and indeed is possible to examine themicroscopic structural details of molecules, liquids and solids.

Today at the beginning of the 21st century, we are once again in the situation where a newkind of rays begins to shed light on unexplored regions of science. Having accessto the spatialresolution of molecular/crystal structure and electron orbitals is only one side of the coin to

be illuminated by x-rays. It took about one century to flip the coin to the other side showingthe temporal resolution of the atomic and molecular motion, making it possible to monitorthe dynamics of molecules (such as rotation, vibration and dissociation) and the dynamics ofelectrons on their natural time scale, which is in the femtosecond to picosecond range and evenmuch shorter for electron dynamics. Early x-ray generation devices and techniques such asx-ray tubes, electrical discharges or even the first synchrotron sources have not been able todeliver x-ray pulses that had a duration of less than several nanoseconds and thus could not beused to gather both types of information, spatial and temporal, simultaneously.

Interestingly enough, insight into the temporal dynamics of quantum systems was firstgained by using probes of much larger wavelength than the one of x-rays: infrared (IR),visible and ultraviolet (UV) laser pulses. In this respect, the invention of the laser in 1960started a development in which the accessible time scale for monitoring dynamic processescould be steadily decreased to shorter and shorter events. Today, the fundamental limit ofthese ultrashort femtosecond laser pulses has almost been reached [2, 3], which is the singleoptical cycle, lasting 1 fs in the near UV to several femtoseconds in the IR spectral region.Having at hand the possibility of monitoring molecular and electronic motion, it is of coursenot possible to use lasers to directly image molecular structure at atomic resolution due to thelarge wavelength of the laser photons.

The solution to this problem seems easy but it took several decades to become feasiblein practice: combine x-rays and lasers to take advantage of both the short wavelength and thetemporal coherence properties to create ultrashort (femtosecond) pulses of x-ray radiation.This can be done in many ways and different experimental approaches have been carried outin the past.

This paper reviews the development in the field of femtosecond x-ray science. It coversboth the development of sources/instrumentation and experimental applications of ultrafast

x-rays to monitor and control quantum dynamical processes. To monitor fast events in time,two approaches can be taken: in one of them, the detector has to be fast enough to resolve thedetails of the dynamical processes (just as the opening time of a photo camera has to be shortenough in order not to blur the picture). The other approach is to use a short pulse of radiationto illuminate the dynamical process (working in the same way as a strobe flash for resolvingfast mechanical motion by eye or another slow detector). At the moment, the latter approach(using ultrashort, femtosecond duration pulses) is the more common one. Usually, an intensefemtosecond laser (pump)pulse interactswith a sample in order to start a dynamicalprocess for


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Synchrotron

lightsources

Laser-

induced

plasmageneration

Lasers

High-harmonicgeneration

Free-electron

lasers

Electronbunchslicin

g/compression

andproposedfree-e

lectronlasers

1 m

100 nm

10 nm

1 nm

1

1 ns 1 ps 1 fs1 eV

10 eV

100 eV

1 keV

10 eV

wavelength

photonenergy

pulse duration

Figure 1. Temporal and wavelength/photon-energy ranges covered by ultrafast x-ray sources. The

different sources of pulsed x-rays cover different ranges of photon energy and pulse duration inthe x-ray spectrum. It is thus not possible to identify an optimal source of ultrafast x-rays; thedifferent techniques are rather complementary. The dashed line in the lower right of the graphdenotes the single-optical-cycle limit for electromagnetic radiation. High-harmonic generationdriven by few-cycle laser pulses is currently the most efficient method to produce x-ray pulses thatare closest to this limit.

instance by populating an excited state of the system, which is then analysed by the ultrashortx-ray (probe) pulse, which arrives at a selectable time delay afterwards and delivers structuralinformation about the system. In some cases, for instance to study inner-shell dynamics [4,5],it is necessary to use an x-ray pump pulse.

This review is structured as follows: in section 2, the most important sources offemtosecond-duration x-raypulses will be presented. Section 3 will outline ways of measuring

and characterizing the temporal shape and duration of these short pulses. Experimental resultsin the field of ultrafast x-ray science will be discussed in section 4, before we conclude thisreview in section 5.

2. Femtosecond x-ray sources

Presently, a variety of methods is available to produce femtosecond-duration flashes of x-rayradiation. In particular, there are many different sources to cover the large extent of theelectromagnetic spectrum in the frequency region which we refer to as x-rays. In addition,each of these sources has its own characteristics in terms of the minimum achievable pulseduration or spatial and temporal coherence properties (figure 1). Virtually for any of theseultrafast x-ray sources, femtosecond laser pulses have to be employed in some way. One

method is the direct conversion of the laser pulse into the x-ray region by a nonlinear opticalprocess called high-harmonic generation (section 2.1). In another method, the laser pulsegenerates a hot and dense plasma emitting incoherent x-rays (laser-induced plasma x-rays,section 2.2). A third way of ultrashort x-ray production is to use the laser pulse in order tocreate an ultrashort electron pulse, which is accelerated to kiloelectronvolt energies by a dcelectric field (ultrafast x-ray tube, section 2.3) or to gigaelectronvolt energies at large scaleelectron accelerator facilities (synchrotrons, free-electron lasers, section 2.4) before x-rays areemitted as bremsstrahlung radiation. Another common aspect of all ultrafast x-ray sources is


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Photon Energy [eV]

1 10 100 1000 10000

AverageBrilliance

photons/(smm

2mr

ad2

0.1

%BW)

106

109

1012

1015

1018

1021

1024

Photon Energy [eV]

1 10 100 1000 10000

PeakBrillian

ce

photons/(shotmm

2 m

rad2

0.1

%BW)

1017

1020

1023

1026

1029

1032

Ti:S laser

Undulator Bessy II

X-ray FEL(proposed)

Laser plasmax-ray

line source

Laser pumpedx-ray laser

High harmonicsoft x-rays

Discharge pumpedx-ray laser

Laser plasmax-ray

line source

Laser pumpedx-ray laser

X-ray FEL(proposed)Ti:S laser

High harmonicsoft x-rays

Undulator Bessy II

(a) (b)

Figure 2. Comparison of (a) average and (b) peak brilliance of different light sources. As canbe seen, the most promising x-ray light source in terms of both average and peak brilliance arethe proposed x-ray free-electron lasers (FELs) that are currently under construction. The rescalingfrom average to peak brilliance can be understood in terms of two parameters: pulse duration and

repetition rate. This is why theundulator sourcefallsback behind the high-harmonic sources, sincethe high average flux at synchrotron undulators typically comes with high repetition rate (MHz)and long (ns) pulse duration as compared with high-harmonic sources where repetition rates are onthe order of kilohertz and pulse durations on the femtosecond to attosecond time-scale.

the fact that accelerated electrons are used in the conversion process. This acceleration stepis most evident in the cases of the ultrafast x-ray tube or the synchrotron/free-electron laserbased sources. For laser-induced plasma x-rays, a hot electron plasma is produced whichemits black-body radiation or characteristic lines (during hot-electron plasma interaction withcoreshell bound electronic states in the atoms) at x-ray photon energies. Laser-driven high-harmonic generation uses acceleration of electrons on time-scales that are on the order of andsmaller than the optical cycle of the driving laser field itself. Currently this latter techniquegives rise to the shortest flashes of (coherent soft x-ray) light which are on the order of a fewhundreds of attoseconds.

Thenumber ofphotons produced ineach pulsealso differs for thevarioussources. Figure2shows the brightness of radiation that is generated by different x-ray light sources. In general,thenumberof photons (intensity) of thegeneratedx-ray light perpulse depends on three crucialfactors: (1) the density of accelerated electrons, (2) the number of bremsstrahlung emissionacts and (3) the mutually spatial and temporal coherence of the electrons that contribute to thegeneration of x-ray light. The more the electrons and the emission acts, the more photons willbe produced. However, the intensity scales differently with electron density and the numberof emission acts for the coherent and the incoherent generation regime. If spatial coherenceprevails between electrons across the x-ray pulse beam (all electrons across the beam oscillatein the same phase), the intensity depends on the density squared, since all electric dipoles

add up coherently (intensity is proportional to the square of the electric field). If in additiontemporal coherence is present, each subsequent bremsstrahlung emission act contributes anequal amount of field strength that adds coherently to the already generated x-ray field. Thisleads to the x-ray intensity depending quadratically on the number of acceleration acts inthe temporally coherent case. For incoherent generation, the dependence is only linear sinceaveraging over the phase occurs. From a different point of view, to obtain high peak intensitiesit is beneficial to have high spatial and temporal coherence. Coherent radiation can be focusedon very small spot sizes andhaveshort pulse durations, both of which give rise to high intensity.


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448 T Pfeifer et al

Both the temporal and spatial coherence are largely absent in the cases of x-ray tubesor laser-induced plasma x-ray generation. High-harmonic generation provides a maximumamount of both the temporal and spatial coherence. The newly developed free-electron x-raylaserswill also feature particular spatial andtemporal coherenceproperties, whereas the typical

insertion devices in beamlines (wigglers, undulators) exhibit only limited spatial coherence.In the following, we will discuss different sources of ultrafast x-ray radiation. We will

startout with femtosecond-laser driven high-harmonic generation, continue with laser-inducedplasma sources, before we discuss ultrafast x-ray tubes, the large-facility synchrotron sourcesand the x-ray free-electron lasers. A recent review of pulsed x-ray sources can also befound in [6].

2.1. High-harmonic generation

High-harmonic generation describes the process in which laser light at a given frequencyis converted to integer multiples of this fundamental frequency during the highly nonlinearinteraction with a conversion medium. The first report of high-harmonic generation dates back

to the late 1980s [7, 8]. A plateau of equally intense harmonics of high order was observed,which was not immediately understood. Applying perturbation theory one would expect arapid decrease in efficiency with increasing harmonic order: in the framework of n-photonexcitation, the excitation probability decreases exponentially with n. However, since therewas a large number of equally strong harmonic peaks, another mechanism had to be found toexplain the results. In 1993 Corkum [9] and Kulander et al [10] published a quasiclassicaltheory,whichreproducedtheplateaubehaviourofharmonicemissionfoundintheexperiments.According to these works, the electron cannot be treated as a bound particle in the high electricfields at work in the experiments, which had so far been assumed. In fact, the electron isionized (freed from the binding force of the nucleus) when the absolute electric field of thelaser is close to its peak during an optical cycle and is driven away from the parent ion. Sincethe laser electric field changes its sign about a quarter of a period later, the electron will slowdown, stop at a position far from the ion and start to re-accelerate towards it. When it returnsto the ion, it can possess a significant amount of kinetic energy, much larger than the photonenergy. This energy plus the ionization potential will be transferred into photon energy assoon as the electron recombines with its parent ion, which gives rise to the very high harmonicorders observed in the experiment. It is thus three steps which make up the model: ionization,propagation in the laser field and recombination. The model has therefore been named three-step model or also simple-mans model due to its striking simplicity. As this three-stepprocessand therefore also high-energy photon emissionusually occurs every half-cycle ofthe laser field it is immediately clear that the spectrum of the produced radiation has to consistof peaks at odd integer multiples of the laser frequency. Inspired by Kulanders and Corkumsidea, a fully quantum-mechanical treatmentof the three-step model hasalso been found shortlyafter its introduction [11, 12] which confirmed the validity of the classical approximation. Itis a very intriguing aspect of nonlinear physics that highly nonlinear effects (as the process

of high-harmonic generation indeed is) can be treated with high accuracy considering simplemodels.Typical high-harmonic spectra are shown in figure 3. As was mentioned, the harmonic

intensity decreases rapidly for low harmonic orders (the so-called perturbative region). Forhigher orders, a plateau of equally intense harmonics is found, which extends up to the cut-off harmonic order. In the following, the mechanism of high-harmonic generation will bediscussed in more detail (reviews can also be found in [14,15]). First of all the single particleresponse is considered in section 2.1.1, since it is this microscopic process outlined above


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plateaucut-off

perturbative regime

40 80 120 160 200 240 2800.0

0.2

0.4

0.6

0.8

1.0

spectralintensity

[arb.u.]

35791113

10 11 12 13 140.0

0.2

0.4

0.6

0.8

1.0

83 79 75 71 67 63 59 55

wavelength [nm]wavelength [nm]

(b)(a) harmonic orderharmonic order

Figure 3. Typical spectra produced in high harmonic generation. (a) A full harmonic spectrumgenerated in xenon gas shows the characteristic spectral shape (from [13]). For low orders,the harmonic intensity is rapidly decreasing. A plateau is visible for the higher orders whichis terminated at the cut-off, the highest harmonics that are generated. The spectral position ofthe plateau depends on both, the gas species used and the intensity of the driving light. Anothermeasurement in neon reveals very high harmonic orders (b). Lower orders are less intense due tothe zirconium filter used to separate fundamental and harmonic light.

which generates the harmonic frequencies in the first place. High-harmonic generation is nota process involving only one single atom but many of them which are coherently stimulatedby the laser. For that reason, the importance of propagation effects, including phase-matchingand distortions, is addressed in section 2.1.2. High-harmonic generation can be employed toproduce the shortest man-made electromagnetic pulses. The duration of these pulses is on theorder of hundreds of attoseconds, far shorter than the optical cycle of the driving laser light.

2.1.1. Single particle response. As mentioned, the process of high-harmonic generation

can be broken up into three steps: ionization, propagation and recombination. Each of thesefundamental processes will now be discussed separately.

Ionizationstep I. If the intensity of light interacting with matter is steadily increased,the electric field of the electromagnetic wave, E(t) = E0 cos(t), at some point becomescomparable to the intratomic field strength (i.e. the average field strength sensed by theelectron). The electron can escape the binding potential of the atom by tunnelling. Theelectric field of the laser will produce a potential, e E(t)r , in addition to the Coulomb potentialof the ion:

V (r , t ) = e2

4 0r+ e E(t)r. (1)

As can be seen in figure 4(b)) the Coulomb potential is significantly distorted for high enoughlaser field strengths. A barrier is created for the bound electronic state, which can be overcomeby tunnelling, upon which the electron can be considered free, affected only by the electricfield of the laser (note the proximity of the asymptote to the real potential at the appearanceposition of the electron in figure 4(b)). This process of tunnel ionization has been treated indetail in the literature. The first notion by Keldysh [16] dates back to 1965. He calculated ananalytical formula for the ionization rate, w, of the hydrogen atom exposed to a strong electric


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450 T Pfeifer et al

(a) (b) (c)

Figure 4. Different ionization scenarios. If the ionization potential is low compared with thefrequency of the light and large compared with the electric field of the laser (all in atomic units)the absorption of multiple photons is the dominant ionization mechanism (multiphoton ionization(a)). If the electric field strength is increased, the Coulomb potential is significantly modified. Ifthe frequency of light is low enough such that the electron can respond to this changing potential,ionization can be understood as the tunnelling of the electron through a static potential wall (tunnelionization (b)). If the electric field is yet higher, the barrier is completely suppressed and theelectron will be classically ripped off the ion (barrier-suppressed ionization (c)).

field, E, in the quasi-static limit:

wK = 6 Ip4h eEh

m1/2I3/2p

1/2exp

42mI3/2p

ehE

1 m2Ip

5e2E2

, (2)

where e and m are the unit charge and the electron mass, respectively, and Ip denotes theionization potential.

Much later, in 1986, Ammosov, Delone and Krainov published a generalized analyticaltheory [17] extending Keldyshs approach to arbitrary atoms and initial electronic states. Theircalculated ionization rate, which agreed very well with experimental findings, is now knownas the ADK ionization rate. It reads (atomic units, h = m = e = 1, are used here)

wADK =

3E(2Ip)3/2

|Cnl |2f(l,m)Ip

2(2Ip)3/2

E

((2Z/2Ip )|m|1)exp

2(2Ip)

3/2

3E

,

(3)with the (time dependent) electric field strength, E, the ionization potential, Ip, the ioncharge, Z (once the electron is detached) and l and m represent the angular momentum andmagnetic quantum number, respectively. Further, the factor f(l,m) is given by

f(l,m) = (2l + 1)(l + |m|)!2|m|(|m|)!(l |m|)! ,

and the constant Cnl is on the order of 2, but more precisely

|Cnl |2 = 22n

n(n + l + 1)(n l) .

Theeffectiveprincipalandangular momentum quantum numberaregiven by n

=Z(2Ip)1/2

and l = n 1, respectively.The work of Keldysh contained another important finding. He determined for which

range of laser field strengths, E0, and angular frequency, together defining the so-calledponderomotive potential, Upin combination with a particular ionization potential, Ip, thetunnelling description is valid. He introduced a parameter

=

Ip

2Up, Up = e2E20 /(4me2), (4)


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which isnow calledtheKeldyshparameter. Theunit charge is denoted by e, me istheelectronmass and E0 is the amplitude of the laser electric field oscillating at frequency, . By the useof this quantity we are able to determine whether the atom is ionized in the tunnel ( 1) orthe multiphoton regime (

1). To understand this in a qualitative way we can imagine the

laser-distorted Coulomb potential to oscillate with the laser frequency. For higher and higherfrequency (larger and larger ), the quasi-static approximation breaks down and the electrondoes not have enough time to accommodate to the fast changes in the potential. Its motion willbe governed by an average over many cycles of the laser field rather than tunnelling in a singlecycle, since the tunnelling timedepending on E0 and Ipis larger than the optical period.The steady nonlinear interaction with the laser field (absorption of many photons) will finallylead to an electronic state with an energy larger than zero, thus a free electron (figure 4(a)).In the opposite limiting case, the field of the laser can get high enough to fully suppress thebarrier (figure 4(c)). The electron is then classically ripped off the atom. This ionizationscenario is called the barrier-suppression ionization [18]. It will now be discussed why thetunnel-ionization regime is best suited to high-harmonic generation.

PropagationStep II. After the electron is ionized, it can approximately be regarded as afree electron whose motion is just governed by the laser field. To understand the motion ofthe electron, we first regard an initially free classical electron interacting with a laser fieldE(t) = E0 cos(t) and calculate its velocity

v(t) =t

0 e

mE(t) dt + v0 = E0e

msin(t) + v0, (5)

where vector arrows are omitted due to the one-dimensional motion of the electron. If weconsider only electrons possessing a zero drift velocity, v0 = 0, their average kinetic energy,Ekin, defines the ponderomotive potential, Up, mentioned above (equation (4)). Note that Upis proportional to E20 , which makes the average kinetic energy of the electrons grow linearlywith laser intensity. It can also be written as

Up[eV] = 0.97 1013

I[Wcm2

]2

[m2

]. (6)Let us now consider an electron which is initially bound to an atom at x = 0 to appear in

the continuum (i.e. to be ionized) at time zero with initial velocity, v0 = 0. It is ionized at anarbitrary phase of the electric field E(t) = E0 cos(t + ). Velocity v(t) and position x(t)can then be calculated to be

v(t) =t

0 e

mE(t) dt = E0e

m(sin(t + ) sin()), (7)

x(t) =t

0v(t) dt = E0e

m2(cos(t + ) cos()) + sin()t. (8)

The time-independent term in the velocity can be understood as drift velocity. If this term iszero, the electron will oscillate around a fixed reference position. If it is nonzero, the reference

position will be moving in time. The maximum kinetic energy of an electron can also beextracted from equation (7) to be 8Up. The amplitude, E0e/m2 = a0, of the oscillatoryposition is sometimes referred to as the ponderomotive radius, a0. Typical values (at laserintensities 1014 W cm2) are on the order of some nanometres, thus much larger than theatomic radius, demonstrating the validity to treat the electron as a freely moving particle inthe laser field. Note that the ponderomotive radius is limited to a quarter of the wavelengthfor electrons moving close to the speed of light. These relativistically moving electrons, ofcourse, need to be treated differently from above and do not play a role in the high-harmonic


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0 20 40 60 80

0

1

2

3

0.0 0.1 0.2 0.3 0.4 0.5

kineticenergy

[Up

]

cos[t+] electric field phase []

cos[t+] electric field phase [ rad]

18

3.17 Up

Figure 6. Kinetic energy of the electrons at the moment of re-encounter for various phases, , ofthe driving laser field. A maximum exists at a phase of 18.

model [9] with experimental data. Before the idea of the semi-classical three-step model was

published, Kulander and co-workers empirically introduced this cut-off law formula [19].It can now also be understood why the tunnelling regime supports the generation of high-harmonic radiation. The highest photon energies are produced when electrons ionize at aphase of 18, which is close to the peak of the electric field in the optical wave. In thetunnel-ionization regime, the ionization rate given by equations (2) and (3) increases with theelectric field. Therefore, many ionized electrons will contribute to high-harmonic generation.On the other hand, in the multiphoton-ionization regime electrons are continuously produced,depending only on the intensity of the field and not on its phase. It is also clear considering thecut-off formula, equation (10), that in order to generate the highest harmonic orders we needto provide a large ponderomotive potential, Up. This is also fulfilled in the tunnelling regimeof ionization, where the Keldysh parameter defined in equation (4) has to be much smallerthan unity. It is clear from the dependence of the high-harmonic process on the phase of thelaser field that harmonic radiation is intrinsically coherent to the fundamental field.

In figure 7 the three-step model is summarized. What has not been discussed so far is thefact that harmonic peaks exist at odd integermultiples of the fundamental frequency, f. Wecan understand this considering the temporal structure of the high-harmonic emission. Thethree-step process repeats every half-cycle T /2 of the laser field. The Fourier transform willthus be discrete, with a separation corresponding to 1/(T/2) = 2f, which is what we observein the spectrum. If the conversion medium exhibits a broken inversion symmetry, harmonicemission will not occur in the same way at every half-cycle but at every full cycle of the laserfield. In the harmonic spectrum this gives rise to a harmonic peak spacing of one times thelaser frequency; thus odd and even harmonic orders are produced. Another way to breakinversion symmetry is to drive a plasma at very high intensities, where the magnetic field ofthe laser becomes high enough to break the inversion symmetry. This occurs in the process ofnonlinear Thompson scattering [20]. In this case the laser intensities are on the order of more

than 1018 W cm2 and the free-electron motion is not harmonic anymore as described by theclassical propagation equation (8). It is strongly influenced by the relativistic mass increaseof the electron and the high magnetic fields. This leads to the well-known nonlinear figure ofeight-motion of theelectron, which again radiates harmonicsat even andodd frequencies. Yetanother possibility to generate even harmonics is the use of few-optical-cycle duration laserpulses. In this regime, where the carrier-envelope (absolute) phase of the laser pulse becomesimportant, the situation for the atom is also different for successive negative and positivehalf-cycles, and even harmonic orders and coherent continuum radiation can be generated.


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tunnel ionization acceleration inthe laser field

Ec~ Ip + 3.17Up

h

recombinationand photoemission

E

t1

2

3x

x

x

step 1 step 2

step 3

Figure 7. Summary of the three-step model. The electron is ionized in step 1 at some particularphase of the electric field. It is then driven away from the parent ion in the laser field (step 2). Aftersign reversal of the ac-field, the electron stops far from the atom, possibly returns and recombinesto emit a photon carrying the kinetic energy of the electron plus its ionization potential (step 3).The kinetic energy of the returning electron can be as high as 3.17Up , defining the so-called cut-offphoton energy in the harmonic spectrum.

It should be stated here that high-harmonic generation represents only one class of strong-field nonlinear processes. Other important processes are nonsequential double ionization andabove-threshold ionization (ATI). Any of these processes can be understood in the frameworkof the three-step model that was introduced above.

Innonsequentialdoubleionization[21,22], a characteristicintensitydependence of doublyionized atomic ions was reported that followed the intensity dependence of single ionization.A knee was obtained in the intensity dependence exactly at the point where single ionizationsaturated, indicatingthatbothprocessesarecoupled. Onepossibleexplanation for thecouplingcan again be given in terms of the three-step model [9]. The first electron is tunnel ionizedand accelerated in the laser field. When it returns to the ion, the high kinetic energy can be

sufficient to knock out a second electron.In ATI [23,24] high-energy photo electrons are detected at integer multiples of the photon

energy. Again, the photo-electron spectra show a characteristic plateau structure, reminiscentof the plateau of photon energies obtained in high-harmonic generation. The high kineticenergy and the peaked structure of the electrons can again be explained by the three-stepmodel [9]: the tunnel-ionized electron that is accelerated to high kinetic energies by the laserfield can scatter off its parent-ion. This scattering event changes the motion of the electronin such a way that it can reach very high energies after further acceleration by the laserfield.

The common aspect in all these effects is the interaction of a tunnel-ionized free electron,accelerated in the laser field, with its ion core. Only the particular kind of interaction withthe core differentiates these processes. Elastic or inelastic scattering of the electron with the

ion core leads to above-threshold ionization or nonsequential double ionization, respectively.Recombination of the electron with the core produces high-harmonic radiation.

Strong-field process Electron-core interactionHigh-harmonic generation RecombinationNonsequential double ionization Inelastic scatteringAbove-threshold ionization Elastic scattering


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The electron always has to revisit the core for any of these processes to occur. Thereforethe term rescattering scenario is nowadays frequently used to describe these strong-fieldeffects.

Quantum-mechanical description. We now turn towards the quantum-mechanicalformulation of the three-step model, which has been found by Lewenstein et al [11, 12].The source of additional frequencies to be generated besides the fundamental incoming laserfrequency is the nonlinear dipole oscillation of the medium. We thus have to calculate thedipole response for the special case of high-harmonic generation. The Schr odinger equationin this case is given by (in atomic units)

i

t|(r , t ) =

1

22 + V (r) + r E cos(t)

|(r , t ) . (11)

We need to calculate the time-dependent dipole moment,

(t) = (r , t )| r|(r , t ), (12)

from which we can then extract the harmonic spectrum by Fourier transformation. Thecalculation presented in [11] expands the time-dependent electron wavefunction (where onlyone electron is considered to be responsible for harmonic generation, which is often referredto as single-active electron approximation (SAE) in the literature [10, 25]) in terms of thebound electron ground state of the atom and the continuum states |v where v stands for thekinetic momentum

|(r , t ) = eiIp t

a(t) |0 +

b(v , t )|v d3v

, (13)

with time-dependent ground-state, a(t), and continuum-state amplitudes, b(v , t ). By doingso, we do not take into account excited bound statesof theelectron. Since thecontinuum statesare defined to be solutions to the free electron Schrodinger equation,

Ekin|v = 122|v, (14)thepotentialofthenucleusisneglectedassoonastheelectronisionized. Usingthedefinitionofthe time-dependentelectron wavefunction in equation (13), the time-dependent dipolemomentcan be calculated to be

(t) = it

0dt

d3 pE cos(t) ( d( p A(t)) a

exp(iS( p , t , t )) b

d( p A(t))) c

+c.c.

(15)

where A(t) is the vector potential of the laser field. The canonical momentum, p, is given byp = v + A(t). (16)

S denotes the so-called quasi-classical action and is written as

S( p , t , t ) =t

tdt

( p A(t))2

2+ Ip

. (17)

It contains the phase advance of the electron during the time it spends in the continuum.The atomic potential only enters as a constant ionization potential, Ip. The expression d(v)in equation (15) stands for the transition probability from the bound electronic state |0 to a


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continuum statev| (describing ionization), where the complex conjugate describes the inverseprocess, i.e. recombination of the free electron to the ground state,

d(v) = v|r|0, (18)

d(v) = 0|r|v. (19)This helps us to find a very intuitive interpretation of the formula of the time-dependent dipolemoment (t) in equation (15). We can now straightforwardly identify the different parts ofthe formula:

a: ionization of the ground state at time t,b: propagation in the continuum in the time interval t t,c: recombination to the ground state at time t.

Thus, the classical three-step model discussed before is contained in the quantum-mechanical description as well. It also resembles Feynmans path integral description ofquantum-mechanical processes [26]. To calculate (t), we do not need to calculate the integralover all p. We only need to consider the p for which the action becomes stationary,

pS( p , t , t ) = 0 (20)This relation can again be interpreted in terms of the classical three-step model. Since

pS( p , t , t ) = x(t) x(t), (21)thestationary phaseconditionequation (20) yields theinformation that we only need to accountfor those electron trajectoriesthat returnat time ttothesamepointtheyleftattime t wheretheywere ionized. Most importantly, the quantum-mechanical treatment also yields the classicalcut-off law equation (10) up to a small correction. By Fourier transforming the time-dependentdipolemoment, theharmonic spectra canbe calculated andanalysed; thecut-off photonenergycan now be found to be

hc = 3.17Up + f

Ip

Up

Ip, (22)

where f(x) is a slowly varying function on the order of 1, which assumes the valuesf (0) 1.32 at x = 0 (Up Ip) and f (3) 1.25. The physical origin of this correctionlies in purely quantum-mechanical effects such as tunnelling and the spreading of the electronwavepacket in the continuum that have not been included in the purely classical treatment.These effects enable the electron to collect more energy on its trajectory than the amountpredicted by the classical equations of motion.

According to the formula for the dipole moment equation (15), different electrontrajectories contained in the integral acquire different phases, at = S(p,t,t ) (called theatomic dipole phases), during their propagation in the continuum. The shape of the electronicwavepacket at the moment of recombination will be governed by the interference betweenthese separate quantum paths. In particular, different trajectories leading to the same photonenergy (having the same kinetic energy at the time of recombination) will interfere with each

other. Re-examinationof the classically calculated kinetic energy of the electron (which agreesvery well with the quantum-mechanical result) at the moment of return to the nucleus revealsthat each kinetic energy in the plateau region of an electron can be produced by two distinctparticular phases of the electric field at the moment of ionization (see figure 8). Therefore,there are two electron trajectories which are most important for the generation of a particularphoton energy. Since one of them spends a longer time in the continuum, we call it the longtrajectory (the one ionizing at a smaller phase of the electric field), the other one is called theshort trajectory. These trajectories will interfere constructively or destructively, depending


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Plasma dispersion. High-harmonicsaregenerated when thelaser electric field is high enoughto generate free electrons that are accelerated in the field and recombine. As we discussed insection 2.1.1, only a very small part of electrons really recombine or interact with the coreto emit a harmonic photon. The vast majority of electrons miss the core and become free

for a long time compared with the duration of the laser pulse (several ns fs). The plasmaresonance frequency is given by

p =

e2Ne

0me, (31)

where e is the unit charge, Ne the free-electron density, 0 is the dielectric constant and me isthe electron mass. This resonance leads to a polarizability of the plasma that in turn causes arefractive index of the form

nplasma() =

1 p

2 =

1

Ne

Nc()

, (32)

where

Nc = 0me2e2

(33)

is the critical plasma density. It is the free-electron density at which the plasma becomescompletely absorbing for electromagnetic radiation of frequency . Typically, in high-harmonic-generation experiments the plasma densities generated are very small comparedwith the critical density (Nc = 1.75 1021 cm3) of the fundamental 800nm laser radiation(and even more so for the high-harmonics), we can linearly approximate the refractive index,which gives

nplasma() 1 12

p

2. (34)

The plasma contribution to the wavevector then yields

kplasma() = (nplasma() 1) c

= 2p

2c. (35)

For the wavevector mismatch that means

kplasma = mkplasma(f) kplasma(mf) =2p(1 m2)

2mcf. (36)

We are left with a negative contribution of the plasma to the phase mismatch,

kplasma < 0. (37)

Geometric dispersion. Now let us discuss the geometrical contribution to the wavevector.This term is only present if the electromagnetic light wave is confined to a small region inspace. For a plane wave, this contribution would be zero. Since we need to generate high

light intensity in order to drive the process of high-harmonic generation, we always have totake this term into account. There are basically two geometries that play a role in practice: theregion around the waist of a focused laser beam and propagation in a waveguide. The initiallycollimated gaussian laser beammeaning the radial intensity distribution is gaussian, whichis also known as TEM00 modeis sent through a lens, which creates a converging beam. Theradius, w, of this beam is continuously decreasing down to the waist size:

w0 = fw

, (38)


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460 T Pfeifer et al

detector

gas-filled capillary

laser pulse

gas jet

Figure 9. High-harmonic generation can be conducted in the free focusing and the waveguidegeometry. A specific geometricphase is imparted onto the femtosecond laser pulse as it propagatesthrough a focus or through a capillary.

where f is the focal length of the lens and w denotes the radius of the beam at the lens. Thisformula is derived from the more general formula for the propagation of a gaussian beam [38]:

w(z) = w0

1 +

z

zR

2, (39)

where w(z) is the radius of the beam at some point z along the propagation direction (opticalaxis). The Rayleigh length is given by

zR = w20

. (40)

A focusing beam with beam size w(z) transports the same integrated electromagnetic power

everywhere along the optical axis. Thus its intensity increases with 1/w(z)2. However, dueto equation (39), the focal spot size, w0, is only sustained over approximately one Rayleighlength, after which the beam size increases approximately linearly with z.

To overcome this problem and sustain high intensity over an extended propagation length,hollow waveguides (capillaries) can be used for high-harmonic generation (figure 9). Whena laser pulse is focused into a capillary having about the same radius, a, as w0 (optimallyw0 = 0.64a), the beam radius is bound to be constant over the length of the capillary, dueto (partial) reflection of the light at the boundaries. The capillary geometry also affects thewavevector, due to the particular boundary conditions at the capillary walls. The smallerthe diameter of the waveguide compared with the wavelength of the guided light, the larger isthe modification of the wavevector.

Free space propagation. In analogy to the temporal phase, (t), of a short laser pulse, wecan define the spatial phase of a laser beam (r):

E(r) eikr + c.c. = ei(r) + c.c. (41)Also analogous to the definition of the instantaneous frequency (t), we can define the localwavevector:

k(r) = (r). (42)


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The total phase mismatch for the focusing geometry can be written as

k = kdisp

+ kplasma

+ kfoc

>0 0=

(n(f) n(mf)) fc

+

2p(1 m2)

2qcf+

2(m 1)

b.

(50)

Sinceweneedtoachieve k = 0foramostefficientconversionoflaserlightinto mthharmonicradiation, the positive contribution of neutral dispersion and the focusing geometry have to bebalanced by the plasma dispersion. The above expression contains only the formula for kfocat the position of the waist. However this term in general depends on z, the distance from thefocus on the optical axis, as can be seen by looking at equations (43) and (42). In additionto this geometric contribution, we need to keep in mind that the intensity also changes alongthe propagation direction z as the beam is focused. Since the atomic dipole phase dependson intensity with equation (23) for one particular trajectory, this also affects the wavevectorof the harmonic radiation [31, 44]. This behaviour leads to different spatial distributions of

harmonic intensity in the generated beam depending on whether the gas jet is placed before orbehind the focus [26,31]. The beam shape itself can be used in order to optimize the harmonicproduction process in the free focusing case [45].

The density of the gas can also be varied in order to arrive at perfect phase matching.This can be done either by increasing the backing pressure of the gas jet or by increasing thepressure in the gas cell.

The plasmadensity depends on thegas species (ionization potential) andcanbe controlledby means of the intensity or the duration of the laser pulse. Note, however, that a changein intensity will naturally cause a change in the maximum harmonic order (cut-off lawequations (10) and (22)) that can be produced.

Forthecase of a hollow waveguide filled with theconversionmedium, the total wavevectormismatch readsk

=k

disp + kplasma + kcap >0 >0 >0

=

(n(f) n(mf)) fc

+

2p(1 m2)

2mcf+

u2nl c(1 m2)

2ma2f

(51)

Now, the geometric contribution, kcap, has the opposite sign as in the focusing geometry.This means that only neutral gas dispersion has to balance both the plasma and the waveguidecontribution. Free parameters in this case are plasma density (controlled by the intensity andduration of the laser pulse) and the density of the neutral medium. In practice, we are oftenbound to work at a given laser pulse duration and intensity to produce harmonics up to a certainharmonic photon energy. Thus varying the neutral density is the easiest way to experimentallyaccomplish phase matching. In figure 10 the harmonic spectrum is plotted versus pressure in

the capillary. Obviously, the experimental finding matches the simulation based on the aboveformula quite well. In particular, note the shift of the optimum pressure to higher values withincreasing harmonic order m, which is visible in both experiment and simulation results. Thisis due to the inherent dependence of the wavevector mismatch equation (51) on the harmonicorder m. The refractive indices of the gas in the soft x-ray region used for the simulation weretaken from the literature [46].

The high-harmonic capillary output has also been shown to have extraordinary spatialcoherence properties [47]. This is because the fibre acts as a spatial filter for the driving laser


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464 T Pfeifer et al

-60 -40 -20 0 20 40 60

0.0

0.2

0.4

0.6

0.8

1.0

-0.1

0.0

0.1

ionization

probability

time [fs]

-60 -40 -20 0 20 40 60

0.0

0.2

0.4

0.6

0.8

1.0

-0.1

0.0

0.1

electricfield[a.u.]

time [fs]

(a) (b)

Figure 11. Plasmabuild-up in a long 25fs FWHM(a)andashort8fsFWHM(b) laser pulse of thesame peak electric field. The probability of ionization reaches 1 for the long pulse before the peakelectric field is reached. Therefore, no harmonic photonscorrespondingto the highest intensitycanbe produced in this case. For the shorter pulse, the electron has about 20% probability to survivein the atom when the peak electric field is reached. This is the reason why very short pulses arebeneficial for the production of highest harmonic photon energies.

a confined beam diameter without an externally applied guiding structure. Thus, an effectivewave-guide scenario is created, and harmonic generation can take place over a long interactionlength [59, 60].

Absorption. A limiting factor to phase-matched high-harmonic generation is reabsorption ofthe generated harmonic light in the conversion medium. The high-harmonic photon energiesare typically in the soft x-ray spectral region (for 800 nm driving laser pulses), where light isextremely well absorbed by matter. This is because the outermost electrons in many materialsare bound with energies corresponding to 10100 eV. This results in large photoionizationcross-sections, , of any system in this spectral range, explaining the strong absorption. Theabsorptionlength,La (thedistanceafterwhichtheintensityoflightpropagatinginanabsorbingmedium drops to 1/e), is given by

La = , (52)where denotes the particle density. As the driving laser pulse propagates throughthe conversion medium, it continuously generates new harmonic light that adds to thecopropagating harmonic radiation generated earlier. This earlier-generated light however isaffected by absorption. By summing over all contributions to the mth-harmonic radiation atthe point where the laser pulse leaves the conversion medium, we find for the mth-harmonicyield [61]

Im

L0

Am(z) expL z

2La

exp(im(z)) dz

2

, (53)

with the harmonic amplitude, Am(z), of the single-particle response and phase m(z), at the

exit of the conversion medium of length, L. If we consider Am(z) does not depend on theposition z along the optical axisas is approximately the case in the waveguide or loosefocusing geometry where the laser intensity stays roughly constantequation (53) becomes

Im 2A2m4L2a

1 + 42(L2a /L2c )

1 + exp

L

La

2cos

L

Lc

exp

L

2La

, (54)

whereagain Lc = /k is thecoherencelength that canbe calculated from thetotalwavevectormismatch, k, introduced above. It can be seen that unlike perfect phase matching without


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0 2 4 6 8 10 12 140.0

0.2

0.4

0.6

0.8

1.0

1.2

high-harmonicyield

medium length [in units of La]

no absorption

Lc >> La

Lc = 10 La

Lc = 5 La

Lc = La

Figure 12. The effect of reabsorption in the conversion medium during high-harmonic generation.If absorption is not included in the consideration, the harmonic yield increases quadratically with

the medium length (- - - -). If absorption is present (absorption length La) there is a limit ( )to the maximum harmonic yield even in the case of perfect phase matching, Lc L. For smallercoherence lengths, Lc , the maximum achievable harmonic yield decreases.

absorption (where the converted signal increases with L2) equation (54) converges to a finitevalue for the case ofL . In figure 12, equation (54) is evaluated for different ratios ofLc/La . In order to generate about half the asymptotic high-harmonic yield obtained for a longcoherence and propagation length (indicated as dotted line in figure 12), we have to fulfill theconditions [61]:

L > 3La , (55)

Lc > 5La. (56)

In this case, we generate close to the maximum amount of harmonic radiation that is possiblegiven the absorption of the gas. This absorption limit has been attained in a number ofexperiments [6163].

At the end of this section, we would like to note that it has been possible to use high-harmonic radiation as a seed for an x-ray laser [64]. In the future, this approach could possiblylead to the generation of very high-intensity soft-x-ray pulses by making use of the excellentspatial andcoherenceproperties of x-raysproduced by high-harmonic generation of laser light.

2.2. Laser-induced plasma

When a highly intense femtosecond laser pulse interacts with high-density material (solid-state), a hot and dense plasma is created. Electrons are accelerated to very high kinetic

energies. The energy transfer mechanism from laser light intensity to the electrons is mediatedby collisions of the electrons with the ions they left behind and other electrons in the plasma.This mechanism is also known as inverse bremsstrahlung absorption, since electrons in thiscase do not emitbut absorb photons when colliding with other particles. These high kineticenergies acquired by theelectronscan then be released as bremsstrahlung emission in thex-raywhentheelectronsscatterofftheioniccores. Thetimeittakestheveryhotplasmatocooldown(by expanding) is typically in the sub-picosecond regime. Therefore, the emitted x-ray pulseshave femtosecond time durations. The first laser-induced plasma femtosecond x-ray source


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in the beam-path of the laser (such as the focusing lens or vacuum chamber window). Alongthese lines, droplets have been shown to provide an excellent conversion medium [70, 71].Another advantage is the automatic replacement of the material with each laser shot.

Microdroplets can even be used to monitor the transition between two different regimes

of laser-driven ultrafast x-ray generation, namely high harmonic generation (section 2.1) andthe just discussed laser-induced plasma emission. A microdroplet serves as a high-densityconversion medium for nonlinear lasermatter interaction, but it can also be forced to expandandthus to decrease itsdensity to arbitrary levels. This way, it is possible to study thedensity ofthe conversion medium as an optimization parameter for efficient conversion of laser light intosoft coherent or hard incoherent x-rays. An intense enough prepulse can be used to drive themicrodroplet into explosion, allowing the second laser pulse to interact with a target mediumof variable density between the condensed and gaseous phases.

In our experiments [72], we used a water droplet jet of droplet size 20 m. Since therepetition rate of the droplet jet was 1 MHz compared with the repetition rate of our laser at1 kHz, each laser pulse encountered a fresh droplet. The laser we used was a regenerativelyamplified Ti : sapphire laser system, delivering 80fs, 800nm central wavelength, 0.8mJ laser

pulses at an intensity of 5 1014

W cm2

at the interaction region after focusing with a 20cmfocal length lens. When a single intense laser pulse is used for excitation, emission of brightfrequency converted light is observed from the droplet under a variety of observation angles.When this light is analysed by an XUV spectrometer, we can clearly identify plasma emissionlines of highly charged oxygen states, ranging up to O5+. However, no high-harmonic linesat integer multiples of the fundamental (1.5eV) laser photon energy could be measured. Thefrequency conversion mechanism in this case is thus the laser-induced plasma generationprocess.

If the intensity of the laser pulse is split into a double pulse of 2 1014 W cm2 each, andthe time delay between the pulses is chosen larger than 1 ns, no plasma line emission occursand high-harmonic generation takes place with harmonic lines at odd integer multiples of thelaser frequency. The highest harmonic order to be observed was the 27th, which is the highestone ever recorded to date in an aqueous conversion medium to the best of our knowledge.

Since the temporal delay seemed to play a crucial role in the process, we set up an opticaldelay line to create a variable delay between our pump pulse (exploding the droplet) and thedriver pulse(generation harmonics from an expandedwater droplet). This pumpdrive schemecan be considered a general proposal to enhance the high-harmonic conversion efficiency bysuitably preparing the target medium in a state that most efficiently produces high-harmonicradiation [73]. In our case, variation of the pumpdrive delay resulted in high harmonicsactually being produced at all, since for an unprepared droplet irradiated by a single pulseharmonic generation could not be observed. This shows the potential of sample preparationby a pumpdrive approach to possibly increase the conversion efficiency by many orders ofmagnitude.

The results from our experiment (figure 14) can be interpreted in the following way: thefirst laser pulse produces a confined overcriticalplasmainside thedroplet (due to a small region

of high intensity caused by additional focusing at the curved droplet surface) that creates ahigh-pressure region within the droplet. The laser pulse is thus completely absorbed by theovercritical plasma and no harmonics are produced since the laser pulse experiences severedistortion in the plasma region and high-density water prevents produced harmonics fromescaping the droplet. The droplet expands immediately after the interaction with the firstpulse. Most of the expanding droplet, however, will be composed of neutral water clusters andmolecules instead of ions (which were only efficiently created in the confined plasma region).For small time delays, the high density of the neutral expanding target material will be enough


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468 T Pfeifer et al

17

19

21

23

25

harmonicorder

driver pulse delay [ps]

wavelength[nm]

00.5>1.0

intensity [arb. units]

Figure 14. TransientXUV-emissionspectrum acquired in the pumpdriveexperiment. A transitionoccursat 650ps froma regimewhere only plasmaluminescenceis detected (inseconddiffractionorder) into a different one with high-harmonic generation being the dominant contribution to thespectrum.

to create overcritial plasma density again upon interaction with the second (driver) laser pulse.As a result, for short time delays the same as for a single laser pulse happens, where anylaser energy only feeds into plasma build-up and no harmonics can be observed. Anotherpossible explanation for the absence of high-harmonic generation has also been worked out intheory [74]. For large enough time-delays, the density of the droplet is low enough to permitpropagation of the driver pulse without creating the critical plasma density. Spatio-temporaldistortion of the pulse is now very low, and the density of the water cluster is sufficiently lowto allow for substantial transmission in the harmonic spectral region.

Apart from demonstrating high-harmonic generation in water droplets and clusters, thisstudy pointsout a way towardsefficienthigh-harmonic generation by theapplication of pumpdrive schemes. Using this approach, it has been possible to determine and to explore the finallimit of high-density high-harmonic generation: plasma breakdown. Although a higher andhigher target pressure in the case of perfect phase-matching generally leads to a rapid increaseof conversion efficiency (scaling quadratically with density), we have to find a means ofovercoming the plasma threshold for high-efficiency high-harmonic generation in the future.

Recently, another way of producing ultrafast x-ray radiation from laser-generatedplasmaswas pioneered. Rousse et al [75] used a laser pulse focused to relativistic intensities

(3 1018 W cm2) into a He gas jet. The intense laser pulse generates a wakefield by itsponderomotive force, the resulting plasma wave of which can break and accelerate electrons(in the propagation direction of the laser pulse) to high kinetic energies. These energies aretypically on the order of mega-electron-volts and can attain hundreds of mega-electron-voltseven for relatively small millimetre-size interaction lengths compared with several metresof length needed for conventional particle accelerators. These laser-accelerated electronscan undergo betatron oscillations caused by transverse fields in the plasma and emit highlydirectional hard x-ray synchrotron radiation up to several kilo-electron-volts. This scheme is


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thus similar to the production of x-ray radiation in large-scale synchrotron facilities by the useof wiggler insertion devices to deflect the electron beam. Since the x-ray radiation in this caseis generated by a highly directional electron beam, its divergence is very small (100 mrad [75])and directed along the original laser path.

2.3. Ultrafast x-ray tube

The very first technique to generate x-rays, as employed by Roentgen himself, is to usex-ray tubes. An evacuated tube contains a cathode and an anode at an electrostatic potentialdifference, U, on the order of a few kilovolts. The cathode is heated by an additional voltage inorder to emit electrons according to the Richardson equation. These electrons are acceleratedfrom cathode to anode, where they will lose all their energy in collisions with the anodematerials atoms. Bremsstrahlung radiation is emitted, which has a typical high-energy cutoffat U e (e is the unit charge). Since electrons are continuously emitted from the cathode, thex-ray radiation produced is also continuous.

In order to create ultrashort flashes of x-rays with this setup, another mechanism forelectron release was used in the group of Rentzepis [7678]. Instead ofthermally ejectingelectrons from the cathode, a femtosecond laser pulse can be employed [79, 80] to createphotoelectrons just as long as the laser pulse lasts. This electron bunch produces an ultrashortx-ray burst by the time it interacts with the anode. Again, as in the case with laser-drivenplasma x-rays (section 2.2), the duration of the x-ray pulse is governed by the penetrationdepth of the electrons inside the anode material. Care has also to be taken in order notto release too many electrons from the cathode since the Coulomb repulsion also increasesthe duration of the electron pulse on its way to the anode. This sets an upper limit on thephoton number achievable with this technique, when sub-picosecond x-ray pulses are to begenerated. It was experimentally shown that increasing the acceleration voltage betweencathode and anode decreases the pulse duration and could eventually enable femtosecondx-ray pulse generation [81]. A number of experiments on structural dynamics of solids andliquids with nanosecond and picosecond resolution have already been carried out using this

approach [82,83].

2.4. Accelerator-based sources

In thefollowing, electron-beam x-ray sources will be discussed, which also feature good spatialcoherence. However, the pulse length of standard synchrotron pulses is on the order of 100 ps.With the ongoing progress in the field of high-energy free-electron lasers, pulse durations of10100 fs are expected in the near future. An advantage of these sources is the high photonenergy that is achievable, which can be up to more than 10 keV. This enables applications suchas time-resolved structure determination in crystals. X-ray free-electron lasers are expected todeliver enough x-ray photons in one shot to resolve the structure even of single molecules.

2.4.1. Electron-bunch slicing. For many scientific applications, in particular the monitoringof atomic and electronic dynamics, the typical pulse duration of synchrotron sources poses afundamental limit. To overcome these limitations in the future, interesting new techniques toshorten the synchrotron x-ray pulses were developed and implemented experimentally. Theso-called slicing scheme was experimentally realized by Schoenlein et al [84]: before theelectron pulse passes through the undulator or bend magnet to produce x-ray radiation, itinteracts with an intense femtosecond laser pulse in another undulator section (see figure 15).This leads to twofemtosecondduration electron subpulsesof lowerand higherenergyelectrons


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470 T Pfeifer et al

Figure 15. Schematic setup of femtosecond electron-bunch slicing. An intense femtosecond laserpulse interacts with an electron bunch in a wiggler device. The laserelectron interaction generatesa femtosecond slice of electronswith slightly higherand lowerkinetic energies than themain bunchenergy. These femtosecond electron pulse slices (higher and lower energy) can be separated fromthe main bunch by using bend magnets. The photons produced by each of these pulses in anotherbend magnet or wiggler/undulator will then be separated in angle as well, allowing to single outa femtosecond x-ray pulse. (Reprinted figure with permission from [84]. Copyright (2000) byAAAS.)

in the main electron bunch, which can be extracted. This femtosecond subpulse of electronswill generate a femtosecond pulse of x-rays along its passage through the bend magnet orundulator. These ultrashort x-ray pulses were already used for experimental applications.Direct evidence of ultrafast disordering in laser-perturbed InSb on a sub-picosecond time scalecould be observed for the first time by time-resolved x-ray diffraction [85].

Another way to shorten the duration of synchrotron light pulses is to use ultrafast shutters.It was shown that laser-induced acoustic pulses in crystals can modulate and switch the x-raytransmission propertiesby making useof theBorrmanneffect. This techniquecouldpotentiallybe employed to generate subpicosecond pulses [86].

Common to both of these shortening approaches just discussed is the fact that a shorterpulse is only cut out of the longer synchrotron pulse, leading to a decrease in the photon

number per shot. For that reason, tremendous effort is put into the development andconstruction of free-electron lasers, which would supply both ultrashort pulses of hard x-raysand large photon numbers per shot.

A different route to ultrashort x-ray pulse production by combining lasers andhigh-energyelectronbeamshasalsobeendevelopedbySchoenlein etal [87]. Whenanintensefemtosecondlaser pulse scatters off a high-energy electron beam at an incidence angle of 90, the forwardscattered photons (photons that after scattering travel in the same direction as the electrons)can be promoted to several kilo-electron-volts photon energy due to momentum transfer fromelectrons to photons. In addition, the emission of these photons is highly directed.

2.4.2. Electron-bunch compression. Instead of only using a short temporal slice of anoriginally longer electron pulse to generate x-rays, there are possibilities of compressing

the original electron pulses to ultrashort time-scales prior to x-ray generation. In anexperiment currently performed at Stanfords Linear Accelerator Center (SLAC), magneticdeflecting chicanes are used to perform the electron-pulse analogue of optical laser pulsecompression [88]. If a femtosecond laser pulse is used to create a femtosecond electron pulsein an electron injector section of an accelerator, electron pulse broadening typically arises dueto an energy spread of the electrons in the bunch. By having the slower electrons in the bunchrun a shorter distance as compared with the faster ones the magnetic chicane recompressesthe electron bunch to sub-picosecond durations. This compressed electron bunch accordingly


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produces sub-picosecond x-ray pulses after sending it through a bend magnet or on insertiondevice such as an undulator or a wiggler. At the Subpicosecond-pulse source (SPPS) [89]at SLAC, it has been possible to produce x-ray pulses as short as 80fs and to use themin measurements of ultrafast crystalline-to-liquid phase transitions [90]. The disadvantage

of this approach lies in the fact that the produced ultrashort x-ray pulse is not inherentlylocked and synchronized to a femtosecond laser pulse, which has to be used for a pumpprobeexperiment unlessthe x-ray intensity is high enough to inducea measurable two-photon opticalnonlinearity by itself. The problem of temporal jitter arises between the x-ray pulse and anartificially (technically) synchronized femtosecond laser pulse. Recently it has been possibleto measure the jitter of a linear-accelerator-based x-ray source with respect to a femtosecondlaser with an accuracy of 60 fs [91].

2.4.3. Free-electron laser. The free-electron laser (FEL) working principle is an extensionof the undulator technique. In an undulator, the electron beam is deflected by a periodicmagnetic structure along the propagation path where the modulation period is chosen suchthat the bremsstrahlung emitted by each electron constructively interferes with the radiation

produced by one modulation period further downstream. The radiation produced by differentelectrons, however, is not coherent and leads to a linear growth of x-ray radiation with electrondensity. In the proposed x-ray FEL sources, the electron density in the beam is increased (forinstance by compressing the electron bunch prior to its entry into the undulator), such that themore intense radiation produced by the electrons interacts with the electron cloud itself. In thiscase, microbunches are formed in the electron beam (figure 16) which now lead to a coherentaddition of the radiation produced by each electron. We therefore obtain an increase in x-rayflux that is proportional to the electron density squared.

If the microbunches have to form spontaneously this is called self-amplified spontaneousemission (SASE) [93]. If the microbunching is created externally by interaction with a laserfield, for example, the term seededfree-electron laser is used [94, 95].

RecentlythefirstexperimentsonnonlinearinteractionofVUVlightfromaSASEFEL[96]

with xenon atoms and clusters were carried out [97] and will be described in section 4.2. Thevery high charge states of xenon ions detected in the experiment led to fundamentally newinsights into lightmatter interaction [98].

3. Pulse characterization and control

It has been and still is a big challenge in the field of ultrafast x-ray science to obtain reliablemeasurementsoftheshapeofthex-raypulsethatisproducedbyacertaingenerationtechnique.There is only one direct measurement technique, which is able to acquire a snapshot of thex-ray pulse intensity as a function of time with a resolution of some hundreds of femtoseconds:the streak camera, which is essentially no more than a combination of a fast photodiodeand an oscilloscope. Almost any other technique (except the autocorrelation method) that is

available today relies on a crosscorrelation of the x-ray pulse with an optical femtosecond laserpulse, raising two important implications: first, the measurement process is a nonlinear opticalprocess, and second, we need a laser pulse that is perfectly (or at least sufficiently, dependingon the desired temporal resolution) synchronized to the pulsed x-ray source. The fact that theprocess is an optical nonlinear interaction of light fields requires us to use high intensities ofthe pulses involved. Since optically-nonlinear polarizabilities rapidly decrease for photons ofhigher and higher energy, we are usually bound to use highly intense laser pulses (instead ofhighly intense x-ray pulses, where it is also much harder to produce intensities comparable


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a)

b) c) d)

Figure 16. Proposed x-ray free-electron-laser working principle (SASE) with simulation results.(a) Schematic setup: a high-density high-energy electron beam enters an undulator device fromthe left-hand side. If the electron density is sufficiently high, enough electromagnetic radiation isgenerated in order to affect the electron beam itself. This leads to the longitudinal formation ofstructure (microbunches) in the electron beam (shown in (b), (c), and (d) at positions close to thebeginning, the centre, and theend of theundulator, respectively) that will coherently emit radiation.(Reprinted figure with permission from [92].)

to the ones achievable with lasers). The fact that the laser pulse has to be synchronized tothe x-ray pulse is certainly of great concern if the x-ray pulse production proceeds withoutdirect employment of an optical laser. For instance, for measuring the pulse duration of afree-electron laser (section 2.4.3) we need to actively synchronize and lock an optical laser tothe electron bunches that produce the x-ray pulses. This synchronization is also essential ifwe wish to perform experiments to record the temporal behaviour of systems in a pumpprobeapproach. Usually, the femtosecond pulse then serves as a pump-pulse (trigger) to createsufficient population of an excited state or a coherent superposition of such states in a samplesystem and the x-ray pulse is used to monitor the ensuing structural or chemical changes.Measuring or monitoring the pulse width is essential in order to evaluate the meaning of therecorded data from such experiments: when we observe a smooth/slowpumpprobe behaviour,

isitduetothefactthatthedynamicsisreallyslowordoesitmerelymeanthatthepulsedurationof the pump or the probe or the temporal stability of the pump and probe pulse with respect toeach other are not sufficient? Monitoring the pulse width is also helpful to improve the x-raysource in terms of obtaining shorter and more stable x-ray pulses.

Additionally, as soon as we canproduce ultrashort pulsesof radiation topassively monitorthe evolution of a system under study, we are immediately thinking of using these pulses inorder to actively control its dynamics. The progress in the field of femtosecond laser pulseshaping [99102] and coherent/optimal control [103105] showed that it is nowadays possible


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to use specifically tailored, shaped light fields in order to drive molecular dynamics alongcertain pathways in molecular configuration space. Using shaped intense x-ray light fields,one day we will be able to control the quantum dynamics of electrons, even in low-lying statesof atoms and molecules.

This section is structured as follows: streak cameras will be presented in section 3.1 as away to directly measure the intensity of the x-ray pulse versus time. Afterwards in section 3.2,we will turn to thediscussion of correlation techniques, where intense femtosecond laser pulsesor the x-ray pulse itself (autocorrelation) are used to acquire information about the temporalshape of the x-ray pulses. Finally, we will present the first steps towards controlling the shapeof coherent x-ray light fields in section 3.3.

3.1. Direct measurement (streak cameras)

A common approach to measure a short temporal phenomenon is to convert it to an electricalsignal that can be recorded for instance with an oscilloscope. To measure the duration of alight pulse (how long is the light switched on), fast photodetectors such as photodiodes or

photomultipliers can be employed. However, this approach is only feasible down to pulsedurations that are longer or at least on the order of the response time of the detector. Theresponse time is typicallygiven by thetemporaldurationof thedetected pulseof photoelectronsproduced in the detector for an infinitesimally short pulse of incident light. The photoelectronsthus convert the photonic signal into an electric signal that can be monitored or recorded bymeans of a (fast) oscilloscope. The electronic response time of these devices is not onlygoverned mainly by the size, geometry and material of the photosensitive material but also bythe intensity of the incident light. The intensity dependence can be understood from the factthat many photoelectrons create a significant space charge and thus repel each other enoughto lengthen the electronic pulse duration. Typically, the shortest response times that are easilyattainable from fast photodiodes andphotomultipliers are larger than tens of picoseconds. Thismakes these devices unsuitable for the detection of femtosecond laser or x-ray pulses.

In the early 1970s, a method of recording very short flashes of light directly in the timedomain [106, 107] with picosecond resolution was found. The idea was to convert the lightpulse into an electron beam pulse by photoemission and ramp up a transverse electric fieldquickly enoughthat electronsproduced at different times hita photo plate at differentpositions.The electric field ramp has to be fast enough to allow electrons produced at temporal delaysof only picoseconds by the light pulse to be clearly separable at the photo plate. This way,a temporal structure in the ultrashort light pulse can be recorded on the photo plate with aresolution of a few picoseconds. It is the temporal image of a pulse that resembles a streakon the photo plate which gives its name to the device.

Streak cameras canbe operated either in thesingle-shot or themulti-shotmode, dependingon whether a single x-ray pulse already produces enough electrons for an image or whetheraveraging over many shots is necessary to obtain enough signal. In both cases, it is necessaryto use a trigger in order to start the transverse electric field ramp. In the multi-shot mode, it is

also necessary to maintain a good shot-to-shot stability of the timing between the trigger andthe peak of the x-ray pulse, which is highly nontrivial. The need for this stability is relaxedin the single-shot operation mode, since in this case a (slight) offset of trigger timing wouldmerely lead to a shift of the streaked temporal pulse image along the time axis.

When using streak cameras, particular care has to be taken in order not to lose temporalresolution due to a variety of effects. If the x-ray intensity is too high, space charge effectsat the photocathode will lead to a temporal broadening of the electron pulse. Another severelimitation is the production of secondary electrons that are produced when the inner-shell


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vacancies, produced by theimpingingx-ray pulse, refill by releasingAuger electrons. Electronbeam focusing on the streaking plane is also crucial to obtain a high temporal resolution. Todate, these limitations in streak cameras allow for a temporal resolution of600fs [108,109].For measuring shorter pulses, we have to resort to a different approach, which will be outlined

in the next section.Combininga streakcamerawith an x-ray spectrometercanprovide valuable spectroscopic

information about time-dependent processes [110]. Since for a standard streak camera, onlyone dimension of the two-dimensional photo plate is identified as the time axis, we have aleft-over degree of freedom along the second dimension to encode another information, forinstance, the spectrum of the incoming x-ray light. As the streak camera image is typicallyindependent on the energy of the incident photons, we have to use a spectrometer beforephotoelectron production to spatially separate the x-ray spectrum on the photocathode of thestreak camera in a direction perpendicular to both, the photoelectron beam and the electricfield of the ramp. Using electron optics, the photocathode can be imaged onto the photo plate,maintaining the spectral resolution in the streaked image.

3.2. Correlation techniques

In the optical frequency domain, the use of correlation techniques has allowed for thecomplete characterization of the electric field of ultrashort laser pulses down to the few cycleslimit [111113]. Correlation refers to scanning the temporal shape of the ultrashort pulse to becharacterized by another (if possible even shorter) reference pulse. In the ideal case, we coulduse an infinitesimally short pulse to directly sample the electric field of the ultrashort pulseto be characterized. This, in fact, is the way how femtosecond laser pulses in the terahertzfrequency range are characterized [114, 115]. However, most of the times there is no shorterreference pulse available. In this case, the ultrashort pulse can be split into two copies whereone copy serves as the reference. This most simple correlation technique is therefore termedautocorrelation. Whenever a different pulse is used as the reference, the term crosscorrelationis used.

How does the sampling work? If we know the exact temporal pulse shape of the referencepulse, it is enough to have both reference and unknown pulse co-propagating with a smalltemporal offset. In analogy to the spatial double slit interference pattern in the k-vectordomain, the temporal interferenceof twopulseswill create a fringed interference pattern in thefrequency domain (spectrum) where the two pulses overlap. The frequency dependent fringespacing canbe used in order to completely characterizethe unknownpulsefrom theinformationwe have about the reference pulse. This technique is known as spectral interferometry [116].If the reference pulse is unknown, this method cannot be directly employed. The crucialinformation determining the pulse shape is the spectral phase of the pulse. Together withthe spectral amplitude, which is simply the spectrum of the laser pulse, the exact laser pulseshape in frequency and thus also time (which is the Fourier transform) can be retrieved. Sincein linear interferometry only the interference of identical frequencies can be monitored with

a slow detector (where slow refers to the response time being much longer than the pulseduration), we cannot straightforwardly record the relative phase between different spectralcomponents constitutingthepulse. To do that, wehavetoartificially shift thespectral regions infrequency, whichmeansweareboundto usea nonlinear(sum-frequency, difference-frequency)mixing technique. Autocorrelation for example works by creating two copies of the sameunknown laser pulse and recording the light produced by second-harmonic generation (SHG)in a nonlinear optical crystal versus the time delay. The signal will be maximum where thepulses overlap completely in time and the dependence on time delay carries information about


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the temporal structure. While the complete information about the laser pulse shape cannotbe retrieved from this simple autocorrelation measurement, an extension of the technique,where the spectrum of the SHG light is recorded for each time delay, is able to do so. Thelatter approach is termedfrequency-resolved optical gating (FROG)andhasbeen pioneeredby

Trebino et al [117,118]. Another way of retrieving the spectral phase from an unknown pulsewithout the help of an external known reference is called spectral interferometry for directelectric-field reconstruction (SPIDER) and was developed by Iaconis and Walmsley [119].

Here we will present the first steps in the field of ultrafast x-ray science to implementthesemethods of pulsecharacterization thatwereoriginallydeveloped for theoptical frequencyregion. However, a number of newcharacterization techniquesemployingthe cross-correlationidea have been developed specifically for x-ray applications and will also be presented in thefollowing.

3.2.1. Autocorrelation. An important issue in the transfer of autocorrelation to the x-rayspectral region lies in the fact that nonlinear susceptibilities and in general the interactioncross section of two-photon processes are rapidly decreasing with increasing photon energyused for excitation. In order to observe the nonlinear process that is vital for the performance

of autocorrelation, very high intensities of x-ray light have to be used. So far, it has beenpossible to create these highly intensive x-ray pulses by only two methods: free-electron lasers(section 2.4.3) and high-harmonic generation (section 2.1). The first autocorrelation in the softx-ray range was reported by Kobayashi et al [120] and later by Tzallas et al [121, 122] andwe will review the latter in the following paragraph. In these works, two-photon ionization ofHe has been employed as the nonlinear detection signal.

The fundamental laser pulse used for high-harmonic generation was an 790 nm, 130 fsduration, 10 mJ pulse energy laser pulse produced by a Ti:sapphire laser operating at 10 Hz.The (odd) harmonic orders 7 through 15 were selected by an 0.2 m thickness indium filter.From earlier theoretical studies it is known that the two-photon response to the selectedharmonic orders is rather flat, which is important for second-order autocorrelation. It wasalso verified that the amount of He+ produced scaled quadratically with the harmonic intensity.

The intensity of the harmonic pulse at the focus was 1011

W cm2

and roughly 90 ions perlaser pulse were produced in agreement with the known two-photon absorption cross-sectionof He. The harmonic pulse to be measured was focused and split with a special mirror, wherehalf of the mirror surface could be moved with respect to the other half by means of a piezoelement. Controlling the offset of the mirror halves allowed us to control the temporal delaybetween the two XUV pulses. After accounting for the effect that the two identical pulsesdo not co-propagate it has been possible to extract the autocorrelation trace as the amountof He+ ions created versus time delay. From the measurement, a pulse duration for each ofthe attosecond pulses constituting the XUV attosecond pulse train could be determined to be800as.

Usinga differentregimeofhigh-harmonic generation andthe second harmonic (400 nm)ofa Ti:sapphire laser as the fundamental radiation, an autocorrelation in a higher photon energyregime could recently be carried out [123] (section 4.6). In this case, a single attosecondpulse could be observed as a result of the autocorrelation measurement (figure 17), which wasconducted by looking at the first ATI-photoelectron peak (two-photon absorption) producedby the XUV light in He gas.

3.2.2. X-FROG measurements. Whileitispossibletoobtainthedurationoreventhetemporalintensity of the x-ray pulses by performing a cross-correlation measurement, it is surely betterto record the shape of the pulse in a more general way, extracting its temporal intensity andphase. Ifthisisdone,itispossibletoreconstructtheelectricfieldofthepulse,exceptforaphase

femtosecond x-ray science.pdf

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