Short-pulse laser–plasma interactions.pdf

Plasma Phys. Control. Fusion38 (1996) 769–793. Printed in the UK

REVIEW ARTICLE

Short-pulse laser–plasma interactions

P Gibbon and E ForsterMax Planck Society, Research Unit ‘X-Ray Optics’ at the Friedrich Schiller University Jena,Max-Wien-Platz 1, D-07743 Jena, Germany

Received 14 March 1996

Abstract. Recent theoretical and experimental research with short-pulse, high-intensity lasersis surveyed with particular emphasis on new physical processes that occur in interactionswith low- and high-density plasmas. Basic models of femtosecond laser–solid interaction aredescribed including collisional absorption, transport, hydrodynamics, fast electron and hard x-raygeneration, together with recently predicted phenomena at extreme intensities, such as gigagaussmagnetic fields and induced transparency. New developments in the complementary field ofnonlinear propagation in ionized gases are reviewed, including field ionization, relativistic self-focusing, wakefield generation and scattering instabilities. Applications in the areas of x-raygeneration for medical and biological imaging, new coherent light sources, nonlinear waveguiding and particle acceleration are also examined.

1. Introduction

There are few technological advances which can lay claim to the number and variety ofnew research fields as the arrival of the femtosecond laser. In less than a decade, ‘short-pulse’ lasers have found applications in the physical sciences, medicine and engineering,and dozens of new research groups have been created under the rubric of ultrafast science.

So what is so special about short-pulse lasers? First, their duration, reduced from a fewtens of picoseconds in the mid-1980s to a state-of-the-art 10 fs, allows phenomena to beinvestigated which were simply too ‘fast’ for the old generation of lasers. Second, the abilityto generate coherent light pulses 1000 times shorter means that for the same energy and cost,a laser beam can be focused to 1000 times greater intensity than was previously possible.Thus, whereas one used to speak of ‘high intensities’ of 1014–1015 W cm−2, nowadaysfluxes of 1018 W cm−2 are routinely achieved with benchtop lasers, and systems capable ofreaching a dizzy 1021 W cm−2 are already under construction in several laboratories aroundthe world.

The electric field strengths of such lasers are orders of magnitude higher than thosebinding electrons to atoms, which means that a gaseous or solid target placed at the laserfocus will undergo rapid ionization. The plasma formed in this manner will comprisethe usual fluid-like mixture of electrons and ions, but many of its basic properties willbe essentially controlled by the laser field, rather than by its own density and temperature.Under these conditions, many of the old rules of laser–plasma interaction must be rewritten; afact which has prompted a number of popular articles heralding new physics and applications(Burgess and Hutchinson 1993, Perry and Mourou 1994, Joshi and Corkum 1995). Ofcourse, it is not essential to focus to these extremes to do science with such lasers: there areplenty of applications, such as high-harmonic generation in gases (L’Huillier and Balcou

0741-3335/96/060769+25$19.50c© 1996 IOP Publishing Ltd 769

770 P Gibbon and E F¨orster

1993), which need a large number of pump photons delivered in a short time, but notnecessarily squeezed into a small focal spot. A high-power laser system therefore has theversatility to access completely different physics depending upon the optical arrangementand the nature of the target. Consider, for example, the standard T3 (‘Table-Top-Terawatt’)system now becoming standard equipment in laser and physics laboratories alike. Witha few changes to the focusing optics and target area, such a system can be used forharmonic generation at 1015 W cm−2, while at the same time providing a source for studyinginstabilities (Darrowet al 1992) and relativistic propagation (Borisovet al l992, Monotetal l995b) of intense beams in tenuous (underdense) plasmas at 1018 W cm−2, and forgenerating hard x-rays from solid targets (Murnaneet al 1991).

This new regime of laser–matter interaction has been made possible thanks to a techniqueknown as chirped-pulse-amplification (CPA) (Strickland and Mourou 1985, Maineet al1988). In CPA, a short pulse is first stretched to a nanosecond duration, amplified by afactor > 108, and then recompressed to its original duration. While it is not our intentionto provide a comprehensive review of state-of-the-art technology here, we list some of thelarger multi-terawatt systems in table 1.

Table 1. Multi-terawatt laser systems worldwide

Laboratory Name Type λ Power Pulse length Rep. rate(nm) (TW) (fs) (Hz)

LLNL, USA Petawatta Ti-Sa 1053 > 1000 500–104 –UCSD, USA –a Ti-Sa 800 10–100 10 10LULI, FR –a Nd:glass 1053 > 100 400–104 –Limeil, FR P102 Nd:glass 1053 55 400 10−3

RAL, UK VULCAN Nd:glass 1053 35 400–2500 –RAL, UK TITANIA a KrF 248 > 15 150 –ILE, JP – Nd:glass 1053 30 1000LOA, FR LIF Ti-Sa 800 10 50 10Saclay, FR a Ti-Sa 800 10 30 10

a under construction/upgrade

In view of the impending escalation in technology towards a new class of petawattlasers with pulse length of only a few optical cycles, it is perhaps an appropriate time totake stock of the progress which has been made so far in understanding how high-intensityshort-pulse lasers interact with ionized matter, and to review the areas in which this newinteraction physics is already being put to practical use in other fields. In so doing, we shallrestrict ourselves to laser–plasma interaction and we shall not attempt to cover the relatedfield of atomic physics and nonlinear optics of bound electrons, on which the reader canfind excellent reviews elsewhere (L’Huillieret al 1995).

The article is organized as follows. In section 2 we first present a brief overview of thephysical processes that can play a role in short-pulse laser interactions with both high- andlow-density plasmas. The underlying theory behind the ‘laser–solid’ phenomena, includingabsorption, thermal transport, hydrodynamics, generation of fast particles, hard x-rays andmagnetic fields, is then described in section 3 together with a broadly chronological reviewof related experiments. In section 4 we consider interactions with underdense plasmas,including propagation effects, relativistic focusing, parametric instabilities, novel x-ray laserschemes and particle acceleration. In section 5, we describe a number of applicationsbased on the physical ideas introduced in the two previous sections, and indicate the likelydirections of future research. Our philosophy here is to try and provide a logical link between

Short-pulse laser–plasma interactions 771

the basic physics of laser–plasma interaction and the end-user who is mainly interested inthe characteristics of the photons or particles emitted and, in particular, in whether thefemtosecond laser–plasma source has advantages over conventional sources.

2. Basics of short-pulse plasma physics

Whether the target medium placed at the focus is gaseous or solid, short-pulse, high-intensityinteractions with matter generally involve a number of physical processes: ionization,propagation and refraction, generation of plasma waves, and the subsequent thermal andhydrodynamic evolution of the target material. The importance of any one of theseprocesses depends heavily upon the laser parameters, and we shall shortly see how theevolution in laser technology towards shorter pulses and higher intensities has shifted theresearch emphasis from atomic physics and linear laser–plasma wave coupling, to extremelynonlinear collective phenomena. Likewise, on the application side, we have seen a shifttowards harder, brighter x-ray sources and production of more energetic particles. This haspresented a challenge both to theorists, who have to solve more complex equations to modelthe physics, and to experimentalists, who have to devise more sophisticated diagnostics inorder to isolate or exploit these effects.

Figure 1. Physics of fs laser interaction with (a) solids and (b) gases.


We have attempted to summarize the main interaction physics in figure 1, which indicatesthe intensity range in which various phenomena predominantly occur. Above intensities of,say, 1016 W cm−2, many of the effects actually depend upon the laserirradianceIλ2, whichmeans that the ‘threshold’ intensity for a given phenomenon can vary depending upon thelaser wavelength. This applies, for example, to any effect which involves the quiver velocityof an electron in the electric field of the laser,

vos

c= 0.84

(I

1018 W cm−2

λ2

µm2

)1/2

.

Thus, to give an electron an oscillation energyUos = 1/2mev2os of 1 keV, we need a 1µm

laser withI = 1016 W cm−2, or a 0.248 µm KrF laser withI = 1.6 × 1017 W cm−2.As with laser–plasma interactions generally, this simple scaling applies to many of

the phenomena to be considered here. Indeed, there are many instances in which thephysics overlaps with earlier work using nanosecond glass and CO2 lasers. The pulseduration, though not necessarily a prerequisite for the study or existence of the effectsdepicted in figure 1, is of central importance for applications that exploit the ‘ultrafast’aspect of the physics. For example, an early motivation for studying femtosecond laser–solid interactions was the widespread interest in hard x-ray sources for ‘real-time’ probingof chemical reactions that take place on a sub-picosecond timescale (Murnaneet al 1991).

3. Femtosecond laser–solid interactions

The question of what happens to a solid target when it is subjected to irradiation by ashort-pulse laser is a simple one to pose, but actually quite complicated to answer. Morethan one physical picture is possible depending on whether the material is treated as a denseconductor, or a ‘sandwich’ of cold solid plus a hot, thin layer of plasma in the region ofthe laser’s focal spot. This division still persists in the modelling and interpretation ofcontemporary experiments at high intensities. To date, there isno single modelwhich canadequately describe all the main pieces of interaction physics.

In order to see why, it is initially helpful to look back at the early ideas put forwardat the end of the 1980s in anticipation of the outcome of such experiments. Althoughthe subject of laser–plasma interaction has been established in the context of laser fusionfor at least 25 years prior to the arrival of short pulses, a number of authors (Gamaliy andTikhonchuk 1988, Moreet al 1988a, Mulseret al 1989, Milchberg and Freeman 1989) werequick to point out that much of the traditional physics would not apply to sub-picosecondinteractions for essentially two reasons. First, the short pulse duration means that there isnot enough time for a substantial region of ‘coronal’ plasma to form in front of the targetduring the interaction. Second, owing to the steep density gradient, laser energy can bedeposited at much higher densities than in nanosecond interactions, where it is absorbed ator below the critical densityNc. This is the density at which the plasma becomes overdensefor an electromagnetic wave with frequencyω0, and is defined byω2

0 = 4πeNc/me, wheree andme are the electron charge and mass, respectively.

3.1. Collisional heating

Not surprisingly, the first theoretical works appearing at the end of the 1980s concentratedon the issue of laser energy absorption in step-like density profiles. Moreet al (1988a)proposed a simple model for normal incidence interactions, in which the laser energy isabsorbed in a small skin layer of depthδ at the target surface, see figure 2.


Figure 2. Schematic of short-pulse heating of solid-density plasma.

In this picture, the plasma is assumed to be ionized to some degree and can be treatedas a conductor with a permittivity:

ε = 1 − ω2p

ω0(ω0 + iνei)(1)

whereνei is the electron–ion collision frequency, given by:

νei ' 3 × 10−6ZNeT−3/2

e log3s−1

where Z is the number of free electrons,Ne is the electron density in cm−3, Te is thetemperature in eV and log3 is the Coulomb logarithm. By matching the electric andmagnetic fields at the vacuum–plasma boundary, one can obtain the absorption coefficientfor highly overdense plasmas(ωp/ω0 � 1) in two limits (Moreet al 1988a, Gamaliy 1994):

A =

2νei

ωpνei � ω0

2ω0

ωp(νei/ωp)

1/2 νei > ω0.(2)

This result can be generalized to arbitrary angles of incidence and polarization (s or p)using the Fresnel equations (Born and Wolf 1980). The absorbed energy is then used todetermine the temperature from the equation of state for the plasma. This can be calculatedusing the ideal gas law at low densities, or the Thomas–Fermi statistical model at highdensities (Pfalzner 1991). Further refinements can be made by including non-equilibriumand low-temperature effects (Moreet al 1988b). The Thomas–Fermi model is also oftenused to determine the number of free electrons, or ionization degree, but for very shortpulses it is more appropriate to solve the atomic populations explicitly (Edwards and Rose1993). Although one can include most collisional and radiative effects this way, a fully self-consistent ionization model, including field-ionization (Pfalzner 1992) and non-Maxwellianeffects still remains a challenge.

Given the temperature and ionization state, the next step is to determine how the energyis transported into the target. Moreet al (1988a) did this by solving the diffusion equationwith conductivities based on the usual Spitzer–Harm heat flow (Zel’dovich and Raizer 1966)but modified for high densities (Lee and More 1984). This can be made more sophisticatedby using a Fokker–Planck treatment of collisions which explicitly takes into account non-local transport (Rozmus and Tikhonchuk 1990). With either approach, a self-similar solutioncan be found which describes the heat-front penetration into the target and sets an upperlimit on the pulse duration above which hydrodynamics becomes significant.


3.2. Hydrodynamic models

Until now we have only considered the idealized step-profile target. In reality, the plasmawill have a finite gradient, either because the pulse length is longer than the thermalexpansion time (which decreases with increasing intensity), or because a prepulse is presentwhich is sufficiently long and intense to ionize and ablate material from the target surfaceprior to the arrival of the main pulse. At low intensities (e.g.I < 1014 W cm−2), this isnot such a problem even with contrast ratios of 104–105, because the prepulse will remainbelow the threshold for plasma formation. On the other hand, typical pulse lengths inearly experiments were around 1 ps, so some thermal expansion was inevitable (Milchbergand Freeman 1989, Fedosejevset al 1990a). In this case, the Fresnel equations are nolonger valid and, to make quantitative comparison with experiments, it becomes necessaryto calculate the absorption in a finite density profile.

This can be done by solving the Helmholtz equations for the electromagnetic wavenumerically (Milchberg and Freeman 1989, Kiefferet al 1989a, Fedosejevs 1990b). Asbefore, the absorption model can be coupled to a set of coupled equations for the heat flowand ionization, but for self-consistency it is better to solve the hydrodynamics as well.

Figure 3. Modelling of laser–solid interaction.

These ingredients, ionization, collisions, wave propagation, thermal transport andhydrodynamics, form the basis for many standard ‘short-pulse’ simulation codes now incommon use (Nget al 1994, Daviset al 1995, Teubneret al 1996a). More sophisticatedcodes also exist which solve the Fokker–Planck (FP) equation to include non-local heatflow (Town et al 1994, 1995, Matteet al 1994, Limpouchet al 1994). The self-consistenttreatment of ionization dynamics and FP heat flow leads to a strongly non-Maxwellianelectron distribution function, which modifies the heat-flow penetration into the target in


comparison with the classical Spitzer–Harm theory of thermal transport. These workhorsemodels are depicted schematically in figure 3.

3.3. Short-pulse x-rays

One of the main motivations behind the development of femtosecond laser–plasma modelsis to understand and optimize the generation of short x-ray pulses (Kuhlke et al 1987,Stearnset al 1988, Murnaneet al 1989, Teubneret al 1995). In particular, one would liketo be able to predict the yield and duration of x-rays emitted from the plasma, which dependupon the interplay of a number of effects: absorption, heating, ionization, recombinationand transport. By treating the plasma as a high-density, blackbody radiator, Rosen (1990)and Milchberget al (1991) demonstrated the possibility of generating sub-picosecond x-raysvia rapid cooling of the target. More sophisticated treatments including hydrodynamics anddetailed atomic modelling have been able to determine the pulse duration of particular lines(Audebertet al 1994). Experimental measurements of x-ray pulse duration are currentlyinstrument-limited to around 1 ps (Kiefferet al 1993, Shepherdet al 1994) and newtechniques will have to be developed, for example, using atomic excitation and ionization(Barty et al 1995), to measure pulses in the sub-100 fs regime.

3.4. Collisionless absorption

Early experiments performed at modest intensities (Milchberget al 1988, Kieffer et al1989b, Murnaneet al 1989, Landenet al 1989, Fedosejevset al 1990a) were in goodagreement with hydrodynamic models, in terms of both the measured reflectivity, and thetypical plasma parameters inferred from atomic x-ray spectra. However, a problem soonbecame apparent as lasers were upgraded and intensities increased. First, for intensitiesabove 1015 W cm−2 or so, the plasma temperature rises so quickly that collisions becomeineffective during the interaction:Te/eV ∼ I 1/2ω0t ; νei ∼ I−3/4t−3/2 (Rozmus andTikhonchuk 1990). Second, the electron quiver velocity is comparable to the thermalvelocity, thus reducing the effective collision frequency further (Pert 1995):

νeff ' νeiv3

te

(v2os + v2

te)3/2

. (3)

In other words, collisional absorption starts to turn off for irradiancesIλ2 >1015 W cm−2 µm2, and therefore could not account for the high absorption observed,for example, by Chakeret al (1991) and Meyerhoferet al (1993).

Given this discrepancy, alternative absorption mechanisms were sought which did notrely on collisions between electrons and ions. In fact, there are a number of collisionlessprocesses which can couple laser energy to the plasma. The best known and most studiedof these is resonance absorption, although it is not immediately clear how effective this is insteep density gradients. In the standard picture (Ginzburg 1964), ap-polarized light wavetunnels through to the critical surface (Ne = Nc), where it excites a plasma wave. Thiswave grows over a number of laser periods and is eventually damped either by collisions atlow intensities or by particle trapping and wave breaking at high intensities (Kruer 1988).

For long density scale lengths, defined byk0L � 1, wherek0 = 2π/λ is the laser wavevector andL−1 ≡ |d(logNe)/dx|x=xc, the absorption rate has a self-similar dependence uponthe parameter(k0L)2/3 sin2 θ (Ginzburg 1964). This behaviour is more-or-less independentof the damping mechanism provided the pump amplitude is small, a condition which weshall quantify shortly. ‘Collisionless’ resonance absorption can therefore be modelled in a


hydro-code even if the collision rate is too small to give significant inverse-bremsstrahlung(Rae and Burnett 1991). This is usually done by introducing a phenomenological collisionfrequency in the vicinity of the critical surface (Forslundet al 1975a) such that one recoversthe ∼ 50% optimum absorption rate in the long scale-length limit.

Although the overall energy balance will be taken care of in this manner, the way inwhich it is divided into thermal and suprathermal electron heating can only be determinedself-consistently using a kinetic approach such as particle-in-cell (PIC) (Birdsall andLangdon 1985) or Vlasov simulation.

Resonance absorption was studied extensively in the 1970s and 1980s with two-dimensional PIC codes in order to understand the origin of fast electrons generated innanosecond laser–plasma interactions (Estabrooket al 1975, Forslundet al 1977, Estabrookand Kruer 1978, Adam and Heron 1988, Kruer 1988). Ironically, fast electron generationis highly undesirablein the ICF context because it leads to preheating of the fuel, thuspreventing targets from being compressed to the necessary densities for high thermonucleargain. With the advent of short pulses, however, fast electrons are very much back in fashionbecause they generate hard x-rays as they travel through the cold part of the target behindthe hot plasma where they are generated.

As hinted at earlier, resonance absorption ceases to work in its usual form in verysteep density gradients. To see this, consider a resonantly driven plasma wave at the criticaldensity with a field amplitudeEp. In a sharp-edged profile, there will be little field swelling,and Ep will be roughly the same as the incident laser fieldE0. Electrons will thereforeundergo oscillations along the density gradient with an amplitude

Xp ' eE0/meω20 = vos/ω0.

The resonance breaks down if this amplitude exceeds the density scale lengthL, i.e. ifvos/ω0L > 1.

Under these conditions, it is no longer useful to speak of electrons being heated bya plasma wave, since this wave is destroyed and rebuilt afresh each cycle. This simplefact was pointed out by Brunel (1987), who proposed an alternative mechanism in whichelectrons are directly heated by the p-polarized component of the laser field. Accordingto Brunel’s model, electrons are dragged away from the target surface, turned around andaccelerated back into the solid all within half a laser cycle. These electrons are simplyabsorbed because the field only penetrates to a skin depth or so. Assuming electrons gain avelocity vos during their vacuum orbit, the absorption fraction can be estimated as (Bonnaudet al 1991)

A =

3f (θ)

vos

cA � 1

3

8f (θ)

vos

cA ∼ 1

(4)

wheref (θ) = sin3 θ/ cosθ .The hot electron temperature inferred from this model and from electrostatic PIC

simulations is just

Thot ∝ v2os ' 3.6I16λ

2µ (5)

whereI16 is the intensity in 1016 W cm−2 andλµ the wavelength in microns.While this capacitor model predicts both high absorption and a strong hot temperature

scaling, subsequent studies withelectromagneticcodes showed that the mechanism saturatesat high intensity due to deflection of the electron orbits by thev ∧ B force (Estabrook and


Kruer 1986, Brunel 1988). On the other hand, the saturation effect can be partially overcomeby using two incident beams at±45◦.

Just as for collisional absorption, the situation becomes more complicated for realisticprofiles with finite density gradients. This case was considered by Gibbon and Bell(1992), who found a highly complex transition between resonance absorption and vacuumheating depending upon the irradiance and scale length. For high irradiances and shortscale lengths, the absorption saturates at around 10–15%, but for intermediate values (e.g.Iλ2 = 1016 W cm−2 µm2, L/λ ∼ 0.1), the absorption can be as high as 70%. Theseresults are in good agreement with a more recent analytical study of absorption in shortscale-length profiles (Andreevet al 1994). The hot electron temperature scales accordingto

Thot ' 8(I16λ

2N

)1/3keV (6)

which is somewhat lower than the scaling from simulations of resonance absorption insteepened density profiles (Forslundet al 1977, Estabrook and Kruer 1978, Kruer 1988).

In simulations with mobile ions, however, a rather different picture emerges. The strongspace charge created by electrons circulating outside the surface pulls out an underdenseion shelf which can drastically alter the absorption (Brunel 1988, Gibbon 1994). Aftera characteristic timets ' 100

√A/Zλµ fs, the absorption and hot electron distribution

resemble those seen in early simulations; the main difference being that the pressurebalance assumed and imposed in long-pulse (ns) interactions (Forslundet al 1977) isunlikely to be achieved for femtosecond pulses. This lack of hydrodynamic equilibrium canstrongly influence the hot electron fraction and temperature scaling. For extreme intensities(I > 1020 W cm−2µm2) at normal incidence, energy is transferred directly to the ionsthrough the formation of a collisionless shock (Denavit 1992).

A third collisionless mechanism which is closely related to vacuum heating is theanomalous skin effect. This is actually a well known effect in solid-state physics (Ziman1969), and was originally studied for step-like vacuum–plasma interfaces by Weibel (1967).Its potential significance for short-pulse interaction was therefore recognized quite earlyby a number of workers (Gamaliy and Tikhonchuk 1988, Mulseret al 1989, Rozmus andTikhonchuk 1990, Andreevet al 1992).

Physically, the anomalous skin effect is not as mysterious as it sounds. Consider firstthe situation for the normal skin effect. Electrons within the skin layerδ = c/ωp oscillatein the laser field and dissipate energy through collisions with ions. The oscillation energyis thus locally thermalized provided that the electron mean-free pathla = vte/νei is smallerthan the skin depth. Now imagine that the temperature is increased so thatla > δ, and thatthe mean thermal excursion lengthvte/ω0 > δ. Under these conditions, the laser field iscarried further into the plasma and the effective collision frequency is given by the excursiontime in the anomalous skin layerδa, i.e. νeff = vte/δa, whereδa = (c2vte/ω0ω

2p)

1/3 (Weibel1967). For normal incidence in the overdense limit, one finds (Rozmus and Tikhonchuk1990, Andreevet al 1992):

A ' ω0

cδa =

(vteω

20

cω2p

)1/3

'(

Te

511 keV

)1/6 (Nc

Ne

)1/3

. (7)

Self-consistent theories of the anomalous skin effect solve the full Vlasov equation inorder to correctly describe the non-local relationship between the current and the electricfield (Weibel 1967, Mulseret al 1989, Gamaliy 1994, Matte and Aguenaou 1992, Andreevet al 1992). None the less, just as with resonance absorption, the effect can also be includedin a hydrodynamic model by replacingνei with an effective collision frequency.


The Fresnel equations can again be used to obtain the angular absorption dependence inthe step-profile limit (Andreevet al 1992). The maximum absorption for p-polarized lightis nominallyA ∼ 2/3 at grazing incidence angles,independentof density and temperature,but can be enhanced if the distribution function is anisotropic. A more complete study ofanomalous skin absorption including relativistic effects has been made by Ruhl and Mulser(1995).

3.5. Ultrahigh intensities

As we saw earlier, ion motion can alter the electron dynamics by changing the densityprofile near the critical surface. As long as this motion remains normal to the gradient, theabsorption and hydrodynamics can still be modelled in one dimension, even for obliquelyincident light. This picture becomes inadequate if a hole is formed, or if the surface developsripples. Both of these situations can occur for finite focal spot sizes (which are typicallydiffraction-limited to 2–10µm) and at extreme irradiances (Iλ2 > 1018 W cm−2 µm2).This regime was studied by Wilkset al (l992) using two-dimensional PIC simulation. Theyfound that tightly focused, normally incident light can bore a hole several wavelengths deepthrough moderately overdense plasma on the sub-picosecond timescale.

‘Hole boring’ results from a combination of three effects. First, the light pressure,PL =2I/c ' 600I18 Mbar, vastly exceeds the thermal plasma pressure,Pe = 160N23Te Mbar(whereN23 is the density in 1023 cm−3 andI18 is the intensity in 1018 W cm−2). This willcause the plasma to be pushed inwards preferentially at the centre of the focal spot. Second,a radial ponderomotive force due to the transverse intensity gradient∇rI pushes electronsaway from the centre of the beam, creating a charge separation which pulls the ions out.Third, the skin depth is enhanced where the laser intensity is greatest due to relativisticdecrease in the effective plasma frequency:ω′

p = ωp/γ , where γ = (1 + v2os/2c2)1/2.

At sufficiently extreme intensities this could lead to ‘induced transparency’, where thelaser beam is transmitted through a nominally overdense plasma instead of being reflected(Lefebvre and Bonnaud 1995). As a hole is formed, the absorption and hot electrontemperature both increase because density gradients are formed parallel to the laser electricfield (Wilks 1993).

Another two-dimensional effect is magnetic field generation, which has long been asubject of fascination in the field of laser–plasma interactions. Of particular interest are thelarge DC fields which can arise from electron transport around the focal spot. In short-pulseinteractions, there are at least three mechanisms which can generate B-fields:

(i) radial transport where the electron temperature and density gradients are not parallel(Stamperet al 1971), giving a source term∂B/∂t ∝ ∇Ne ∧ ∇Te;

(ii) DC currents in steep density gradients driven by temporal variations in theponderomotive force (Sudan 1993, Wilkset al 1992);

(iii) hot electron surface currents (Brunel 1988, Gibbon 1994, Ruhl and Mulser 1995).

The first of these mechanisms, which occurs on the hydrodynamic timescale, persists longafter the laser pulse and can cause strong pinching of the ablated plasma (Bellet al 1993). Incontrast, the other two mechanisms will occur predominantly at early times (10 fs< t 6 τp)in interactions at normal and oblique incidence, respectively. For intensities of 1019 W cm−2,the magnitude of the B-field can be 109 G and above.


3.6. Hot electrons and hard x-ray generation

There are essentially three signatures of high-intensity collective effects in laser–plasmainteractions: high angular-dependent absorption, hard x-rays and fast ions. All three ofthese have since been verified experimentally for sub-picosecond pulses. Absorption of50–60% for p-polarized light has been found for intensities of 1016 and above (Kiefferetal 1989b, Klemet al 1993, Meyerhoferet al 1993, Teubneret al 1993, Sauerbreyet al1994), x-rays in the keV–MeV range have been measured (Audebertet al 1992, Kmetecet al 1992, Klemet al 1993, Chenet al 1993, Rousseet al 1994, Schnurer et al 1995,Teubneret al 1996b), and fast ion blow-off has been seen (Meyerhoferet al 1993, Fewsetal 1994).

Unlike their softer cousins (see section 3.3), hard x-rays are adirect result of hotelectrons produced in the vicinity of the focal spot. By virtue its long mean-free path, afast electron can penetrate into the cold region of the target beyond the heat front, where iteither emits bremsstrahlung via collisions with ions, or produces line radiation by knockingout a bound K-shell electron.

Figure 4. Hot electron temperature measurements in short-pulse experiments (squares) comparedwith PIC simulations (filled circles). Experimental data are taken from: Meyerhoferet al (1993)–LLE; Rousseet al (1994)–LULI; Teubneret al (1996b)–Jena/Sprite; Jianget al (1995)–INRS;Schnurer et al (1995)–MBI; Kmetecet al (1992)–Stanford; Fewset al (1994)–LLNL/IC/RAL.

Bremsstrahlung appears as a continuum anywhere in the 0.1 keV–MeV range dependingon the laser intensity and plasma parameters (Kmetecet al 1992, Schnurer et al 1995),whereas inner-shell line emission can be 1–100 keV depending upon the atomic numberof the target material (Soomet al 1993, Rousseet al 1994, Jianget al 1995). ‘Cold’line radiation is presently the more interesting of these for applications because of itstypically narrow bandwidth and high potential brightness. Kα radiation is also an importantdiagnostic tool. The spectral intensity of photons emitted is directly dependent upon theelectron energy. This can be exploited by observing that electrons slow down and eventually


stop in cold material, so that the total line emission will in general depend upon the targetthickness. The hot electron energy can therefore be inferred from a ‘sandwich’ experiment(e.g. Al on Si) in which the thickness of the front layer is varied (Hareset al 1979). Thechange in the Kα line ratios with thickness can then be fitted to a characteristic temperatureor distribution function.

Although experimental hot temperature measurements in the sub-picosecond regime arestill scarce, a picture is gradually emerging which suggests that

(i) temperatures are lower than for long-pulse experiments at the same intensities,,(ii) the scaling law isThot ' (Iλ2)1/3−1/2.

A summary of these experiments is shown in figure 4 together with results from recent(one-dimensional) PIC simulations.

3.7. High-harmonic generation

Thanks to short-pulse technology, the subject of harmonic generation by ultra-intense laser–solid interaction has also enjoyed a resurgence of interest as a means of producing short-wavelength, coherent light (Kohlweyeret al 1995, Von der Lindeet al 1995). Physically,harmonics are generated by the highly nonlinear plasma oscillations at the surface of thetarget as described in section 3.4. This effect was first demonstrated with long-pulse CO2

lasers by Carman and co-workers at Los Alamos in the early 1980s (Carmanet al 1981a,b) and was suggested as a means of inferring the plasma density from the highest harmonicobserved.

More recent work by Gibbon (1996) has demonstrated that for intensitiesI >

1018 W cm−2, the harmonic ‘cutoff’ predicted by earlier theories (Bezzerideset al 1982,Grebogi et al 1983), ωn = √

Ne/Ncω0 no longer appears in the reflected spectrum.Theoretically, over 60 harmonics can be generated with efficiencies> 10−6 for modestplasma densitiesNe/Nc ∼ 10− 30. This prediction has been verified in an experiment withthe Vulcan CPA system at RAL, UK, in which over 70 harmonics were observed using2.5 ps pulses at intensities up to 1019 W cm−2 (Norreys et al 1996). In terms of rawpower and scalability with wavelength and intensity, this mechanism is hard to beat: theefficiency essentially depends uponIλ2, whereas the minimum wavelength is justλ/nmax.For example, Norreyset al estimate that the power converted into the 38th harmonic at28 nm was 24 MW. On the other hand, it was also found that the harmonics were emittedmore-or-less isotropically, an effect which was attributed to surface dimpling. This is incontrast to the near-specular emission in the preceding experiments using 100 fs pulses(Kohlweyer et al 1995, Von der Lindeet al 1995), and it will be a challenge to optimizethe coherence of harmonic high-order emission for these ultrashort-pulse durations.

4. Laser interaction with low-density (tenuous) plasmas

Focusing a high-intensity laser pulse into a gas elicits a completely different characterof interaction from that considered in section 3. With solids, the pulse interacts with afew microns of essentially mirror-like material; with gas targets, the pulse propagates overmillimetres, during which time it can ionize, distort, refract and accelerate particles, and soon. The interaction physics is therefore determined not only by the spot sizeσ0 and intensityI at the focus, but also by the focusing geometry or the Rayleigh diffraction length:

ZR = πσ 20

λ(8)


whereσ0 is the 1/e2 spot size, defined for a Gaussian beam by:I (r) = I0 exp(−r2/σ 20 ). In

this section we describe some of these new effects in nonlinear optics and how they mightbe usefully exploited.

4.1. Multiphoton ionization and heating

One of the first obstacles to studying laser–plasma interactions at high intensities is theproduction of a plasma with accurately known properties. The simplest way to do this is tofill a target chamber with a gas at atmospheric pressure and ionize it by focusing the laserinside the chamber. Rapid ionization occurs by virtue of the fact that the laser field perturbsthe Coulomb barrier of the atom, allowing electrons to tunnel free. For hydrogen-like ions,this process can be modelled according to a theory by Keldysh (1965), who derived aDC ionization probability:

W = 4ωa

(Ei

Eh

)5/2Ea

ELexp

[−2

3

(Ei

Eh

)3/2Ea

EL

](9)

where Ei and Eh, are the ionization potentials of the atom and hydrogen respectively,Ea = m2e5/h4 is the atomic electric field,EL is the laser field, andωa is the atomicfrequency (4× 1016 s−1). This formula is particularly useful in the regimee2E2

L/4Mω20 �

Ei � hν, which is precisely the operating regime of femtosecond laser systems, wherethe photoelectric effect (that is, single-photon ionization) cannot account for the observedionization rates. In fact (9) and its generalizations for oscillating fields and more complexatoms (Ammosovet al 1986) have proved remarkably effective in describing the ionizationphysics of femtosecond interactions. This theory of so-called ‘above-threshold-ionization’(ATI) has been verified experimentally by measuring the energies of emitted electrons(Freemanet al 1987, Augstet al 1989) and multiply-charged ions (Augusteet al 1992b).

An important implication of ATI is that the kinetic energy of electrons pulled out fromatoms by the laser field is typically smaller than both the ionization potential and the quiverenergy (Burnett and Corkum 1987). This low ‘residual’ energy arises because within anoptical cycle, ionization occurs at the peak of the electric field where the quiver velocity iszero. On the other hand, any departure from linear polarization will increase the randomenergy acquired by ‘newly-born’ electrons, and ultimately increase the final temperature ofthe plasma created.

This issue is of central importance to novel x-ray laser schemes using short pulses(Amendt et al 1991), which rely on the rapid creation of a highly ionized, cold plasmawhich subsequently recombines and lases back to the ground state. If the plasma is too hot,the scheme will saturate and the x-ray efficiency will be low. A number of theoretical(Penetrante and Bardsley 1991, Pert 1995) and experimental (Offenbergeret al 1993,Dunne et al 1994, Blyth et al 1995) efforts have therefore concentrated on determiningand minimizing the plasma temperature. In particular, inverse-bremsstrahlung heatingand collective heating due to parametric instabilities (see section 4.3) could impair theeffectiveness of such schemes. Consequently, short wavelengths (λ = 1/4 µm) and pulselengths shorter than 100 fs should improve the chances of success for this type of scheme.

4.2. Nonlinear refraction

The subject of nonlinear propagation of electromagnetic (EM) waves in plasmas is too vastto do justice to in this review. Interest in this field predates the use of short pulses by severaldecades, in the context of ICF, ionospheric physics and astrophysics. Nevertheless, there


were again a number of works appearing at the end of the 1980s pointing out the ways inwhich femtosecond pulses should behave differently from longer (pico–nanosecond) pulsesin underdense plasmas.

At first sight, one might think that for laser intensities such thatEL � Ei , a plasmawould be instantly created by the leading foot of the pulse, leaving the main part propagatingthrough a fully ionized, uniform plasma. Unfortunately, the situation is complicated by aphenomenon known as ionization-induced defocusing (Augusteet al 1992a, Leemansetal 1992, Rae 1993). Near the front of an intense pulse, where the field is close to theionization threshold, the gas at the centre of the beam will be ionized more, giving rise toa steep radial density gradient. The refractive index of the plasma, given by:

η(r, z, t) =(

1 − Ne(r, z, t)

Nc

)1/2

(10)

will therefore have a minimum on axis and act as a defocusing lens for the rear portion ofthe beam. The result is that for high gas pressures, the laser beam is deflected well beforeit can reach its nominal focus (Augusteet al 1992a).

To circumvent this problem, experiments requiring high intensities are usually performedeither with a preformed plasma (Durfee III and Milchberg 1993, Mackinnonet al 1995), orusing a gas-jet configuration in which the beam is focused in vacuumbeforeit actually entersthe gas (Augusteet al 1994). Interaction at the maximum intensity is then guaranteed andone can essentially neglect the ionization physics. It is then possible to study the interactionof the laser fields with free electrons at intensities of 1018 W cm−2 and above. To see whatnew effects can be expected in this regime, it is helpful to examine the wave equation foran electromagnetic wave in a plasma:(

∂2

∂t2− c2∇2

)A = JNL = NeA

γ. (11)

The nonlinear currentJNL on the right-hand side of (11) contains both the coupling ofthe laser field to the plasma and high-intensity effects such as relativistic self-focusing. Thelatter effect arises due to a change in the refractive index via electrons quivering in the laserfield at velocities close to the speed of light. This phenomenon has been known for sometime (Litvak 1968, Maxet al 1974), but it is only through short-pulse technology that ithas become possible to study it experimentally. The reason for this is that there is a powerthreshold (Sprangleet al 1988):

Pc = 17Ne

NcGW (12)

at which beam diffraction is balanced by self-focusing. For typical electron densitiesavailable from a gas jet, namely 1018–1020 cm−3, one needs a multi-terawatt laser to havea reasonable chance of seeing the effect. On the other hand, theoretical and computationalstudies have demonstrated that self-focusing should be accompanied by partial or completeexpulsion of electrons from the beam centre (Sunet al 1987, Mori et al 1988, Borisovetal 1990, Chen and Sudan l993, Pukhov and Meyer-ter-Vehn 1996), forming a kind of self-sustained optical fibre. Recent experiments have reported evidence of extended propagationover several Rayleigh lengths (Borisovet al 1992, Sullivanet al 1994, Monotet al 1995b,Mackinnonet al 1995). However, the interpretation of these results has been complicatedby the fact that the diagnostics used to image the focused beam rely on scattering of thelaser light from plasma electrons (Gibbonet al 1995).


4.3. Wakefield excitation and instabilities

Another important effect which modifies beam propagation is the excitation of plasmawaves. These can be generated either by Raman-type instabilities (Kruer 1988) or by anappropriate choice of parameters so that a ‘wake’ is produced behind the pulse. In bothcases, large-amplitude electrostatic fields can be generated which are able to accelerateelectrons to very high energies over short distances (< 1 m), potentially to GeV levels.Not surprisingly, therefore, much of the motivation for studying short-pulse interactions hascome from the ‘plasma accelerator’ community, and we review some of the issues involvedin this application later in section 5.5.

On the physics side, a topic which has attracted some controversy is the interplay ofrelativistic focusing and plasma wave generation. Several groups have tackled this problemby solving the two-dimensional envelope equations for the laser beam together with thenonlinear fluid response of the plasma (Sprangleet al 1990, 1992, Andreevet al 1992,Abramyanet al 1992, Antonsen and Mora 1992, Krallet al 1994). Simulations using thisapproach typically predict that for pulse lengths a few times longer than the plasma period,the envelope breaks up into beamlets of lengthλp (Andreevet al 1992, Sprangleet al 1992).This effect, known as ‘self-modulation’ was interpreted by these authors as follows: owingto the finite pulse shape, a small wake plasma wave is excited non-resonantly, which resultsin an oscillating density perturbation within the pulse envelope. The EM waves thereforesee a refractive index which is alternately peaked and dented at intervals ofλp/2. Theseportions of the pulse will therefore focus and diffract, respectively, leading to modulationsin the envelope with periodλp. These modulations subsequently enhance the plasma wake.

The net result is that a large amount of energy can be scattered outside the originalfocal cone (Antonsen and Mora 1992), leaving a large-amplitude plasma wave close to thebeam axis. On the other hand, one is tempted to conclude that the nominal requirementfor wakefield excitation can be relaxed, and one can take pulse lengthsτp � ω−1

p .Unfortunately, this picture is not complete: envelope models neglect a number of importantphysical processes which turn out to be just as important. First, parametric instabilities(Drake et al 1974, Forslundet al 1975b) such as Raman backscatter (RBS) and Ramanforward scatter (RFS) are explicitly excluded by the paraxial-ray approximation used inthese models. The growth rate for RBS with a relativistic pump is (Sakharov and Kirsanov1994, Guerin et al 1995):

0B =√

3

2

(ω0ω

2p

2

)1/3 (a0

γ0

)2/3

(13)

whereas for RFS, we have (Moriet al 1994):

0F = ω2p√

8ω0

a0

(1 + a20)

1/2. (14)

In RBS, the pump wave decays into a plasma wave plus an EM wave travelling backtowards the focal lens. The instability therefore grows from the foot of the pulse towardsthe rear, and the number ofe-foldings depends upon the pulse length. For RFS, however,the instability growth depends upon thepropagationlength as well, and is thus potentiallymore damaging for applications using short, high-intensity pulses.

The second important effect excluded from all fluid models, whether paraxial or not, iswave breaking (Koch and Albritton 1974), which causes the plasma wave to lose coherenceand heat electrons. As in laser–solid interactions, a kinetic model is essential to treat thisprocess self-consistently. Several groups have already presented PIC simulations which


largely confirm the initial growth scaling of the RFS and RBS instabilities but which alsofollow them to saturation (Bulanovet al 1992, Deckeret al 1994). The state-of-the-art in this area is currently claimed by the UCLA/LLNL groups, who have been able tomodel actual experimental parameters with two-dimensional PIC simulations comprisingover 107 particles (Tzenget al 1996). These studies have shown that RFS can alsoexcite large-amplitude plasma waves and induce modulations in the pulse in a mannerindistinguishable from the self-modulational instability observed with fluid models.

Experiments at Livermore and RAL, UK largely corroborate these findings, but theoverall understanding of propagation effects is far from complete. In an experiment usinga 600 fs pulse and a 1 mmhelium gas jet, Coverdaleet al (1995) demonstrated that up to50% of the light is scattered out of the focal cone, a result that is apparently at odds with anexperiment performed under very similar conditions at Saclay (France), wherecollimationof the exiting beam was observed (Monotet al 1995a).

5. Applications of short-pulse laser-plasma sources

So far in this review, we have concentrated mainly on the basic physical issues offemtosecond laser–plasma interactions. While it is true that much current research iscuriosity driven, an equally important motivating factor is the extent to which laser-plasmascan be used as primary sources of photons and electrons for other purposes. From thepreceding sections, it should by now be clear that there are basically three main areas ofapplication:

(i) generation of hard and soft incoherent x-rays,(ii) coherent short-wavelength light sources,(iii) particle acceleration.

In this section we consider some specific applications in a little more detail, where possiblecomparing laser-plasma sources with more traditional ones. As far as the x-ray sourcesare concerned, a vital component of any successful application will be the development ofsuitable optics for the x-ray photons, which in many cases is a technological challenge initself (Forsteret al 1992, 1994, Attwood 1992).

5.1. Medical imaging

The possibility of creating ultrafast x-ray flashlamps has been one of the major motivationsbehind short-pulse technological developments (Murnaneet al 1991). As we saw insections 3.3 and 3.6, a laser-produced plasma can be crudely regarded as a polychromaticcontinuum source with peaks of line radiation characteristic of the target material. Thelatter component is generally regarded as the more interesting because of its well definedemission wavelength, narrow bandwidth and high intensity.

An application which is presently under serious evaluation by several groups is medicaland biological imaging. Since their discovery a century ago (Rontgen 1895), x-rays havebeen exploited for this purpose with increasing sophistication, so it is natural to ask whatimprovements can be offered by laser-plasma sources. The requirements of medical imagingare essentially threefold: first, x-ray photon energies need to be 20–100 keV to allowtransmission through the body; second, the bandwidth should be narrow to minimize thedose from unwanted soft x-rays; third, a high degree of tunability is needed to distinguishbetween different types of tissue. Although the traditional x-ray tube meets these general


specifications, a laser-plasma source may offer some additional advantages (Herrlinet al1993):

• First, a short-pulse x-ray source delivers a high yield in a picosecond duration, asopposed to micro–milliseconds for conventional sources. This offers the possibility ofsubstantial dose reduction by using a time-gating technique to eliminate scattered x-rayswhich degrade the image contrast (Gordon IIIet al 1995).

• Second, differential imaging with rapid, simultaneous exposure is possible (Tillmanetal 1996). This technique, traditionally implemented using large synchrotron sources,requires rapid exposure by two x-ray lines with photon energies above and belowthe K-absorption edge of a ‘contrast agent’. Subtraction of the two resulting imagesenhances the parts of the sample containing the contrast agent and suppresses unwantedinformation.

• Third, the small target size makes it possible to conceive novel image projections wherethe x-ray source is placed inside the object of investigation (Tillmanet al 1995).

While preliminary experiments by the Lund group are very encouraging, a number ofissues remain to be clarified, particularly concerning dosage. For instance, although theoverall x-ray dose may be reduced by using laser-plasma sources, it is not yet clear whatkind of damage the higherintensitiesmay cause to the molecular structure of living tissue.Furthermore, the optimization of these sources, in brightness, pulse length and size, willdepend upon the ability to control the interaction physics.

5.2. Microscopy, holography and interferometry

The realization of a tunable, coherent light source in the 1–100 nm wavelength rangepromises to open up as many new possibilities as the development of the laser in the1960s. In a tutorial review of XUV sources, Attwood (1992) gives three basic definitionswhich characterize coherent radiation: brightness, coherence length and transverse resolvingpower. The brightnessB is defined as:

B = 8/(1A 1� BW) (15)

where8 is the number of photons per second,1A is the source area,1� is the solid angleof the emitted radiation, and BW is the bandwidthδλ/λ. The temporal coherence length,

lcoh = λ2/21λ (16)

determines the detail in which information can be resolved along the propagation path. Theresolving power is set by the diffraction limit:

dθ > λ/2π (17)

whered is the minimum resolvable distance on the object andθ is the observation half-angle.Currently, there are two major high-brightness light sources with this wavelength range:

synchrotron undulators (Winick 1994) and kilojoule-class x-ray lasers (XRL) (Elton 1990).Short-pulse lasers offer two alternative routes: optical-field- or inner-shell photoionized XRLschemes and harmonic generation in gases or plasmas. At the time of writing, none of thesenovel schemes offers a water-window source, but their compactness, scaling properties, highefficiency and superior time resolution compared with long-pulse XRL schemes make themwell worth pursuing.

For applications where coherence is not essential, such as standard microscopy (seefor example, (Rochow and Tucker 1994)), electron beams can deliver both resolution andcontrast right down to atomic (sub-Angstrom) levels. The main advantage of photons over


electrons is their ability to pass less destructively through aqueous solutions. Moreover,scanning electron microscopy normally requires carefully prepared biological samples, eitherfreeze-dried or treated with hydrophobic agents, a process which can alter the cellularstructure. With soft x-ray pulses from laser-produced plasmas (section 3.3), on the otherhand, one can imagine studyingliving cells with a time resolution sufficient to capturedynamical processes on a sub-nanosecond timescale.

Apart from good temporal coherence, a prerequisite for biological holography is awavelength within the so-called ‘water-window’ between the absorption K-edges of oxygen(23.2A) and carbon (43.7A) (Solem and Chapline 1984). This choice allows transmissionof the probe beam through the sample while providing natural contrast between proteins(i.e. carbon) and water (oxygen). For example, this should yield information on proteinstructures in their natural (aqueous) environment.

An important application of coherent XUV sources which is quite widespread is plasmadensity diagnosis. In ICF and astrophysical plasmas, densities can be well above the criticaldensityNc ' 1021λ−2

µ cm−2, which make them difficult or impossible to probe with visibleor UV lasers. A soft x-ray laser withλ < 20 nm, on the other hand, has a critical density of1024 cm−3 or above, and can be used to obtain the plasma density using interferometry (DaSilva et al 1995). An added advantage of ultrafast XUV schemes would be an improvementin spatial resolution to sub-micron levels, by freezing hydrodynamic motion.

5.3. Ultrafast probing of atomic structure

Two techniques which have been used for some time to probe the structure of matter atthe atomic and molecular level arein situ x-ray diffraction and x-ray spectroscopy. Thesemethods have quite different source requirements and optical arrangements, see figure 5,but share the exciting new possibilities offered by ultrafast time resolution. Consider, forexample, the typical timescales and length scales of protein motions shown in table 2 (Petskoand Ringe 1984).

Table 2. Timescales and length scales of protein motions.

Time (s) Deflection (A)

Atomic vibration 10−15–10−11 0.01–1Collective 10−12–10−3 0.01–5Bond breaking/joining 10−9 –10−3 0.5–10

Diffractometry exploits the interference effect created by adjacent atomic planes (Braggscattering) to obtain global structural information about fluid or crystal samples. Since x-ray diffraction measurements can be directly inverted to atomic positions or bond lengths,it is conceivable that ultrafast exposures on the 100 fs timescale would ultimately allow‘filming’ of dynamic processes such as phase changes or chemical reactions (Bartyet al1995). Progress towards this goal has been recently achieved by Tomovet al (1995) using ascheme similar to that shown in figure 5(b). They demonstrated a pump–probe experimentto observe changes in the lattice temperature of a gold crystal on a 10 ps timescale.

Spectroscopy can also reveal information on atomic structure, but its interpretationis generally complicated by uncertainties in bulk properties. An exception to this isextended x-ray absorption fine structure (EXAFS), which yields direct information on thenear neighbours of a given atom. Soft x-rays from laser-plasmas have been successfully


(a)

(b)

Figure 5. X-ray optical arrangements for (a) pump–probe diffractometry and (b) ultrafastabsorption spectroscopy.

used as EXAFS sources for some time owing to their high brightness and sub-nanosecondrecording capability (Easonet al 1984). Again, short-pulse sources have been proposed asa means of extending the time resolution down to the sub-picosecond regime (Tallentset al1990). An advantage of spectroscopic techniques over diffractometry is that the requiredx-ray photon flux is several orders of magnitude lower. Preliminary proof-of-principleexperiments have none the less concentrated on the near-edge spectrum (XANES), wherethe source and detection requirements can be relaxed even further (Raksi et al 1995).


5.4. Lithography

While lithography is often cited as a potential application for laser-plasma x-ray sources,it is not obvious that ultrashort-pulse lengths bring any real advantage. To the authors’knowledge, short-pulse systems have not yet been seriously evaluated in the context. Inorder to use lasers for x-ray exposure of resists, one needs a short, tunable wavelength(around 10A) for high resolution, combined with high average power to meet throughputrequirements. This does not rule out short-pulse x-ray sources in special cases, butnanosecond lasers currently appear to represent the most promising option in this field(Chakeret al 1990, Maldonado 1995).

5.5. Bench-top particle accelerators

The demise of the superconducting super collider (SSC) has remotivated the search foralternatives to conventional particle accelerator technology. It has been realized for sometime that plasma could form the basis for a new generation of compact accelerators thanksto their ability to support much larger electric fields. Conventional synchrotrons and linacsoperate with field gradients limited to around 100 MV m−1. A plasma, on the other hand,which is already ionized, can theoretically sustain a field 104 times larger, given by:

Ep = mecωp

e'

(Ne

1018 cm−3

)1/2

GV cm−1. (18)

To accelerate particles, these fields must propagate with velocities approaching the speedof light. In a seminal paper, Tajima and Dawson (1979) proposed two methods of couplingthe transverse electromagnetic energy of a high-power laser into longitudinal plasma waveswith high phase velocity. The first requires a pulse length matched to the plasma periodsuch thatτp ' π/ωp (Gorbunov and Kirsanov 1987, Sprangleet al 1988), which translatesinto a technical requirement:

tfwhm > 50N−1/218 fs. (19)

This condition could not be met with the technology available at that time, so theyproposed an alternative ‘beat-wave’ scheme, in which two lasers are used with frequencieschosen so thatω1 − ω0 = ωp. In contrast to the wakefield scheme, where a plasma waveis forcibly driven up by the pulse, the beat-wave method relies on a more gentle build-upover tens or even hundreds of picoseconds. In both cases, the plasma wave has a phasevelocity:

Vph = c

(1 − ω2

p

ω20

)1/2

. (20)

An electron trapped in such a wave will be accelerated over at most half a wavelength (afterwhich it starts to be decelerated), giving an effective acceleration length

La = λp

2(c − vph)' 3.2

(Ne

1018 cm−3

)−3/2 (λ

µm

)−2

cm. (21)

Combining (18) and (21), we obtain the maximum energy gain of an electron acceleratedby the plasma wave:

1U = EpLa ' 3.2N−118 λ−2

µ GeV. (22)

Thus, a multi-terawatt Ti-Sa laser is in principle capable of accelerating an electron to 5 GeVin a distance of 5 cm through a plasma with density 1018 cm−3. In practice, acceleration will


be limited by other factors, such as laser diffraction or instabilities. For example, comparingthe Rayleigh length (8) with (21), we typically haveZR � La, so some means of guidingthe laser beam over the dephasing length must be found to optimize the energy coupling.Whether relativistic or channel guiding (section 4.2) can be combined with large-amplitudeplasma wave generation has yet to be proven experimentally, but this will be one of thegoals of ‘second generation’ plasma-based accelerators in the near future, see Katsouleas andBingham (1996). To date, there have been some notable experiments demonstrating particleacceleration with a beat-wave scheme (Kitagawaet al 1992, Claytonet al 1993, Everettet al 1994, Amiranoffet al 1995). First experiments with short-pulse lasers (Nakajimaetal 1995, Modenaet al 1995) have also achieved acceleration of thermal electrons to over40 MeV, but the underlying mechanism has been attributed to Raman instabilities ratherthan to ‘clean’ wakefield excitation. In an important step towards the latter, radial plasmawaves generated by ‘cigar-shaped’ pulses have been observed with both temporal and spatialresolution by the LULI group (Marqueset al 1996).

5.6. Novel fusion concepts

One of the most exciting alternative schemes to conventional ICF to emerge recently isthe so-called ‘fast igniter’ concept (Tabaket al 1994). In the standard inertial confinementscenario, ignition is achieved by compressing a deuterium–tritium pellet to high density insuch a way that a hot spot is created at the centre. This hot spot ignites first, providinga spark which then propagates outwards to burn the surrounding higher density fuel. Todo this, laser energies of more than 1 MJ are needed for significant gain and implosionsymmetry has to be controlled to better than 1%.

In the fast igniter scheme, the hot spot is replaced by an external energy source, thustheoretically relaxing the driver, compression and uniformity requirements by up to an orderof magnitude. Maxet al (1974) proposed using a short-pulse, high-intensity laser to deliverenergy to the compressed fuel via fast electrons. The scheme works in three stages. First,the fuel is compressed to a radius of around 10µm, the equivalent of anα-particle rangeat an areal density of 0.4 g cm−2. Next, a ‘prepulse’ several hundred picoseconds longwith a tailored intensity profile is used to create an optical channel through the underdensecorona and push the critical surface closer to the centre of the target. Finally, a short pulsewith intensity of around 1020 W cm−2 is sent through the channel to the high-density core,where it heats electrons to energies up to 1 MeV. These fast electrons penetrate the fuel,where they thermalize and, ultimately, heat the ions to fusion temperatures (5–10 keV).

Although the scheme is highly speculative, it has none the less sparked a revivalof interest in the energy coupling and transport physics of high-intensity laser–plasmainteractions. A large number of experiments are therefore being planned and carried outto investigate the issues outlined in sections 3.4–3.6. As our understanding of short-pulsephysics matures, we shall no doubt see improvements on the original Livermore schemeand perhaps even more radical ideas for laser fusion.

6. Summary

The first decade of research on femtosecond laser–plasma interactions has provedenormously rich both in variety of physics and in the potential for new and sometimesunexpected applications. Each improvement in technology, whether in power, shorterpulse duration, sharper focusing optics, or higher contrast ratios, sparks a fresh round ofexperiments with a ‘look-and-see’ spirit. In this review we have tried to document the


changing emphasis in physics, from weakly nonlinear optics to extreme relativistic, kineticprocesses, which has accompanied these developments. In addition, we have explored someof the areas in which femtosecond laser-produced plasmas show great promise as sourcesof fast particles and short-wavelength radiation. With a new generation of high repetition-rate, multi-terawatt lasers coming online this year, we can expect many of these ultrafastapplications to be realized in the near future.

Acknowledgments

Thanks are due to a number of colleagues who have provided valuable advice, commentsand, in some cases, unpublished material, notably: A Andreev, A R Bell, C B Darrow,G Holzer, J C Gauthier, J C Kieffer, T Missalla, S Svanberg, U Teubner,R P J Town andI Uschmann.

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