140572077-short-pulse-laser-plasma-interactions-pdf.pdf

Plasma Phys. Control. Fusion 38 (1996) 769–793. Printed in the UK

REVIEW ARTICLE

Short-pulse laser–plasma interactions P Gibbon and E F¨orster Max 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 lasers is surveyed with particular emphasis on new physical processes that occur in interactions with low- and high-density plasmas. Basic models of femtosecond laser–solid interaction are described including collisional absorption, transport, hydrodynamics, fast electron and hard x-ray generation, together with recently predicted phenomena at extreme intensities, such as gigagauss magnetic fields and induced transparency. New developments in the complementary field of nonlinear propagation in ionized gases are reviewed, including field ionization, relativistic selffocusing, wakefield generation and scattering instabilities. Applications in the areas of x-ray generation for medical and biological imaging, new coherent light sources, nonlinear wave guiding and particle acceleration are also examined.

1. Introduction There are few technological advances which can lay claim to the number and variety of new research fields as the arrival of the femtosecond laser. In less than a decade, ‘shortpulse’ 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 few tens of picoseconds in the mid-1980s to a state-of-the-art 10 fs, allows phenomena to be investigated which were simply too ‘fast’ for the old generation of lasers. Second, the ability to 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 , nowadays fluxes of 1018 W cm−2 are routinely achieved with benchtop lasers, and systems capable of reaching a dizzy 1021 W cm−2 are already under construction in several laboratories around the world. The electric field strengths of such lasers are orders of magnitude higher than those binding electrons to atoms, which means that a gaseous or solid target placed at the laser focus will undergo rapid ionization. The plasma formed in this manner will comprise the usual fluid-like mixture of electrons and ions, but many of its basic properties will be 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; a fact 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). Of course, it is not essential to focus to these extremes to do science with such lasers: there are plenty of applications, such as high-harmonic generation in gases (L’Huillier and Balcou c 1996 IOP Publishing Ltd 0741-3335/96/060769+25$19.50

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1993), which need a large number of pump photons delivered in a short time, but not necessarily squeezed into a small focal spot. A high-power laser system therefore has the versatility to access completely different physics depending upon the optical arrangement and 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. With a few changes to the focusing optics and target area, such a system can be used for harmonic generation at 1015 W cm−2 , while at the same time providing a source for studying instabilities (Darrow et al 1992) and relativistic propagation (Borisov et al l992, Monot et al l995b) of intense beams in tenuous (underdense) plasmas at 1018 W cm−2 , and for generating hard x-rays from solid targets (Murnane et al 1991). This new regime of laser–matter interaction has been made possible thanks to a technique known as chirped-pulse-amplification (CPA) (Strickland and Mourou 1985, Maine et al 1988). In CPA, a short pulse is first stretched to a nanosecond duration, amplified by a factor > 108 , and then recompressed to its original duration. While it is not our intention to provide a comprehensive review of state-of-the-art technology here, we list some of the larger multi-terawatt systems in table 1. Table 1. Multi-terawatt laser systems worldwide Laboratory

Name

Type

λ (nm)

LLNL, USA UCSD, USA LULI, FR Limeil, FR RAL, UK RAL, UK ILE, JP LOA, FR Saclay, FR

Petawatta –a –a P102 VULCAN TITANIAa – LIF

Ti-Sa Ti-Sa Nd:glass Nd:glass Nd:glass KrF Nd:glass Ti-Sa Ti-Sa

1053 800 1053 1053 1053 248 1053 800 800

a

a

Power (TW) > 1000 10–100 > 100 55 35 > 15 30 10 10

Pulse length (fs)

Rep. rate (Hz)

500–104 10 400–104 400 400–2500 150 1000 50 30

– 10 – 10−3 – – 10 10

under construction/upgrade

In view of the impending escalation in technology towards a new class of petawatt lasers with pulse length of only a few optical cycles, it is perhaps an appropriate time to take stock of the progress which has been made so far in understanding how high-intensity short-pulse lasers interact with ionized matter, and to review the areas in which this new interaction physics is already being put to practical use in other fields. In so doing, we shall restrict ourselves to laser–plasma interaction and we shall not attempt to cover the related field of atomic physics and nonlinear optics of bound electrons, on which the reader can find excellent reviews elsewhere (L’Huillier et al 1995). The article is organized as follows. In section 2 we first present a brief overview of the physical processes that can play a role in short-pulse laser interactions with both high- and low-density plasmas. The underlying theory behind the ‘laser–solid’ phenomena, including absorption, thermal transport, hydrodynamics, generation of fast particles, hard x-rays and magnetic fields, is then described in section 3 together with a broadly chronological review of related experiments. In section 4 we consider interactions with underdense plasmas, including propagation effects, relativistic focusing, parametric instabilities, novel x-ray laser schemes and particle acceleration. In section 5, we describe a number of applications based on the physical ideas introduced in the two previous sections, and indicate the likely directions of future research. Our philosophy here is to try and provide a logical link between

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the basic physics of laser–plasma interaction and the end-user who is mainly interested in the characteristics of the photons or particles emitted and, in particular, in whether the femtosecond 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-intensity interactions with matter generally involve a number of physical processes: ionization, propagation and refraction, generation of plasma waves, and the subsequent thermal and hydrodynamic evolution of the target material. The importance of any one of these processes depends heavily upon the laser parameters, and we shall shortly see how the evolution in laser technology towards shorter pulses and higher intensities has shifted the research emphasis from atomic physics and linear laser–plasma wave coupling, to extremely nonlinear collective phenomena. Likewise, on the application side, we have seen a shift towards harder, brighter x-ray sources and production of more energetic particles. This has presented a challenge both to theorists, who have to solve more complex equations to model the physics, and to experimentalists, who have to devise more sophisticated diagnostics in order to isolate or exploit these effects.

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

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We have attempted to summarize the main interaction physics in figure 1, which indicates the intensity range in which various phenomena predominantly occur. Above intensities of, say, 1016 W cm−2 , many of the effects actually depend upon the laser irradiance I λ2 , which means that the ‘threshold’ intensity for a given phenomenon can vary depending upon the laser wavelength. This applies, for example, to any effect which involves the quiver velocity of an electron in the electric field of the laser,  1/2 I λ2 vos . = 0.84 c 1018 W cm−2 µm2 2 Thus, to give an electron an oscillation energy Uos = 1/2me vos of 1 keV, we need a 1 µm 16 −2 laser with I = 10 W cm , or a 0.248 µm KrF laser with I = 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 the physics overlaps with earlier work using nanosecond glass and CO2 lasers. The pulse duration, though not necessarily a prerequisite for the study or existence of the effects depicted 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’ probing of chemical reactions that take place on a sub-picosecond timescale (Murnane et al 1991).

3. Femtosecond laser–solid interactions The question of what happens to a solid target when it is subjected to irradiation by a short-pulse laser is a simple one to pose, but actually quite complicated to answer. More than one physical picture is possible depending on whether the material is treated as a dense conductor, or a ‘sandwich’ of cold solid plus a hot, thin layer of plasma in the region of the laser’s focal spot. This division still persists in the modelling and interpretation of contemporary experiments at high intensities. To date, there is no single model which can adequately 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 forward at the end of the 1980s in anticipation of the outcome of such experiments. Although the subject of laser–plasma interaction has been established in the context of laser fusion for at least 25 years prior to the arrival of short pulses, a number of authors (Gamaliy and Tikhonchuk 1988, More et al 1988a, Mulser et al 1989, Milchberg and Freeman 1989) were quick to point out that much of the traditional physics would not apply to sub-picosecond interactions for essentially two reasons. First, the short pulse duration means that there is not enough time for a substantial region of ‘coronal’ plasma to form in front of the target during the interaction. Second, owing to the steep density gradient, laser energy can be deposited at much higher densities than in nanosecond interactions, where it is absorbed at or below the critical density Nc . This is the density at which the plasma becomes overdense for an electromagnetic wave with frequency ω0 , and is defined by ω02 = 4π eNc /me , where e and me are the electron charge and mass, respectively. 3.1. Collisional heating Not surprisingly, the first theoretical works appearing at the end of the 1980s concentrated on the issue of laser energy absorption in step-like density profiles. More et al (1988a) proposed a simple model for normal incidence interactions, in which the laser energy is absorbed in a small skin layer of depth δ at the target surface, see figure 2.

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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 treated as a conductor with a permittivity: ε =1−

ωp2 ω0 (ω0 + iνei )

(1)

where νei is the electron–ion collision frequency, given by: νei ' 3 × 10−6 ZNe Te−3/2 log 3s −1 where Z is the number of free electrons, Ne is the electron density in cm−3 , Te is the temperature in eV and log 3 is the Coulomb logarithm. By matching the electric and magnetic fields at the vacuum–plasma boundary, one can obtain the absorption coefficient for highly overdense plasmas (ωp /ω0  1) in two limits (More et al 1988a, Gamaliy 1994):  2ν ei  νei  ω0   ω p (2) A= 2ω    0 (νei /ωp )1/2 νei > ω0 . ωp 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 to determine the temperature from the equation of state for the plasma. This can be calculated using the ideal gas law at low densities, or the Thomas–Fermi statistical model at high densities (Pfalzner 1991). Further refinements can be made by including non-equilibrium and low-temperature effects (More et al 1988b). The Thomas–Fermi model is also often used to determine the number of free electrons, or ionization degree, but for very short pulses it is more appropriate to solve the atomic populations explicitly (Edwards and Rose 1993). Although one can include most collisional and radiative effects this way, a fully selfconsistent ionization model, including field-ionization (Pfalzner 1992) and non-Maxwellian effects still remains a challenge. Given the temperature and ionization state, the next step is to determine how the energy is transported into the target. More et al (1988a) did this by solving the diffusion equation with conductivities based on the usual Spitzer–H¨arm heat flow (Zel’dovich and Raizer 1966) but modified for high densities (Lee and More 1984). This can be made more sophisticated by using a Fokker–Planck treatment of collisions which explicitly takes into account nonlocal transport (Rozmus and Tikhonchuk 1990). With either approach, a self-similar solution can be found which describes the heat-front penetration into the target and sets an upper limit on the pulse duration above which hydrodynamics becomes significant.

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3.2. Hydrodynamic models Until now we have only considered the idealized step-profile target. In reality, the plasma will have a finite gradient, either because the pulse length is longer than the thermal expansion time (which decreases with increasing intensity), or because a prepulse is present which is sufficiently long and intense to ionize and ablate material from the target surface prior to the arrival of the main pulse. At low intensities (e.g. I < 1014 W cm−2 ), this is not such a problem even with contrast ratios of 104 –105 , because the prepulse will remain below the threshold for plasma formation. On the other hand, typical pulse lengths in early experiments were around 1 ps, so some thermal expansion was inevitable (Milchberg and Freeman 1989, Fedosejevs et al 1990a). In this case, the Fresnel equations are no longer valid and, to make quantitative comparison with experiments, it becomes necessary to calculate the absorption in a finite density profile. This can be done by solving the Helmholtz equations for the electromagnetic wave numerically (Milchberg and Freeman 1989, Kieffer et al 1989a, Fedosejevs 1990b). As before, the absorption model can be coupled to a set of coupled equations for the heat flow and 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 and hydrodynamics, form the basis for many standard ‘short-pulse’ simulation codes now in common use (Ng et al 1994, Davis et al 1995, Teubner et al 1996a). More sophisticated codes also exist which solve the Fokker–Planck (FP) equation to include non-local heat flow (Town et al 1994, 1995, Matte et al 1994, Limpouch et al 1994). The self-consistent treatment of ionization dynamics and FP heat flow leads to a strongly non-Maxwellian electron distribution function, which modifies the heat-flow penetration into the target in

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comparison with the classical Spitzer–H¨arm theory of thermal transport. These workhorse models are depicted schematically in figure 3. 3.3. Short-pulse x-rays One of the main motivations behind the development of femtosecond laser–plasma models is to understand and optimize the generation of short x-ray pulses (K¨uhlke et al 1987, Stearns et al 1988, Murnane et al 1989, Teubner et al 1995). In particular, one would like to be able to predict the yield and duration of x-rays emitted from the plasma, which depend upon the interplay of a number of effects: absorption, heating, ionization, recombination and transport. By treating the plasma as a high-density, blackbody radiator, Rosen (1990) and Milchberg et al (1991) demonstrated the possibility of generating sub-picosecond x-rays via rapid cooling of the target. More sophisticated treatments including hydrodynamics and detailed atomic modelling have been able to determine the pulse duration of particular lines (Audebert et al 1994). Experimental measurements of x-ray pulse duration are currently instrument-limited to around 1 ps (Kieffer et al 1993, Shepherd et al 1994) and new techniques 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 (Milchberg et al 1988, Kieffer et al 1989b, Murnane et al 1989, Landen et al 1989, Fedosejevs et al 1990a) were in good agreement with hydrodynamic models, in terms of both the measured reflectivity, and the typical plasma parameters inferred from atomic x-ray spectra. However, a problem soon became apparent as lasers were upgraded and intensities increased. First, for intensities above 1015 W cm−2 or so, the plasma temperature rises so quickly that collisions become ineffective during the interaction: Te /eV ∼ I 1/2 ω0 t; νei ∼ I −3/4 t −3/2 (Rozmus and Tikhonchuk 1990). Second, the electron quiver velocity is comparable to the thermal velocity, thus reducing the effective collision frequency further (Pert 1995): νeff ' νei

vte3 . 2 + v 2 )3/2 (vos te

(3)

In other words, collisional absorption starts to turn off for irradiances I λ2 > 1015 W cm−2 µm2 , and therefore could not account for the high absorption observed, for example, by Chaker et al (1991) and Meyerhofer et al (1993). Given this discrepancy, alternative absorption mechanisms were sought which did not rely on collisions between electrons and ions. In fact, there are a number of collisionless processes which can couple laser energy to the plasma. The best known and most studied of these is resonance absorption, although it is not immediately clear how effective this is in steep density gradients. In the standard picture (Ginzburg 1964), a p-polarized light wave tunnels through to the critical surface (Ne = Nc ), where it excites a plasma wave. This wave grows over a number of laser periods and is eventually damped either by collisions at low intensities or by particle trapping and wave breaking at high intensities (Kruer 1988). For long density scale lengths, defined by k0 L  1, where k0 = 2π/λ is the laser wave vector and L−1 ≡ |d(log Ne )/dx|x=xc , the absorption rate has a self-similar dependence upon the parameter (k0 L)2/3 sin2 θ (Ginzburg 1964). This behaviour is more-or-less independent of the damping mechanism provided the pump amplitude is small, a condition which we shall quantify shortly. ‘Collisionless’ resonance absorption can therefore be modelled in a

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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 collision frequency in the vicinity of the critical surface (Forslund et al 1975a) such that one recovers the ∼ 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 in which it is divided into thermal and suprathermal electron heating can only be determined self-consistently using a kinetic approach such as particle-in-cell (PIC) (Birdsall and Langdon 1985) or Vlasov simulation. Resonance absorption was studied extensively in the 1970s and 1980s with twodimensional PIC codes in order to understand the origin of fast electrons generated in nanosecond laser–plasma interactions (Estabrook et al 1975, Forslund et al 1977, Estabrook and Kruer 1978, Adam and H´eron 1988, Kruer 1988). Ironically, fast electron generation is highly undesirable in the ICF context because it leads to preheating of the fuel, thus preventing targets from being compressed to the necessary densities for high thermonuclear gain. With the advent of short pulses, however, fast electrons are very much back in fashion because they generate hard x-rays as they travel through the cold part of the target behind the hot plasma where they are generated. As hinted at earlier, resonance absorption ceases to work in its usual form in very steep density gradients. To see this, consider a resonantly driven plasma wave at the critical density with a field amplitude Ep . In a sharp-edged profile, there will be little field swelling, and Ep will be roughly the same as the incident laser field E0 . Electrons will therefore undergo oscillations along the density gradient with an amplitude Xp ' eE0 /me ω02 = vos /ω0 . The resonance breaks down if this amplitude exceeds the density scale length L, i.e. if vos /ω0 L > 1. Under these conditions, it is no longer useful to speak of electrons being heated by a plasma wave, since this wave is destroyed and rebuilt afresh each cycle. This simple fact was pointed out by Brunel (1987), who proposed an alternative mechanism in which electrons are directly heated by the p-polarized component of the laser field. According to Brunel’s model, electrons are dragged away from the target surface, turned around and accelerated back into the solid all within half a laser cycle. These electrons are simply absorbed because the field only penetrates to a skin depth or so. Assuming electrons gain a velocity vos during their vacuum orbit, the absorption fraction can be estimated as (Bonnaud et al 1991)  vos  A1  3f (θ) c (4) A=   3 f (θ) vos A∼1 8 c where f (θ ) = sin3 θ/ cos θ. The hot electron temperature inferred from this model and from electrostatic PIC simulations is just 2 ' 3.6I16 λ2µ Thot ∝ vos

(5)

where I16 is the intensity in 1016 W cm−2 and λµ the wavelength in microns. While this capacitor model predicts both high absorption and a strong hot temperature scaling, subsequent studies with electromagnetic codes showed that the mechanism saturates at high intensity due to deflection of the electron orbits by the v ∧ B force (Estabrook and

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Kruer 1986, Brunel 1988). On the other hand, the saturation effect can be partially overcome by using two incident beams at ±45◦ . Just as for collisional absorption, the situation becomes more complicated for realistic profiles with finite density gradients. This case was considered by Gibbon and Bell (1992), who found a highly complex transition between resonance absorption and vacuum heating depending upon the irradiance and scale length. For high irradiances and short scale lengths, the absorption saturates at around 10–15%, but for intermediate values (e.g. I λ2 = 1016 W cm−2 µm2 , L/λ ∼ 0.1), the absorption can be as high as 70%. These results are in good agreement with a more recent analytical study of absorption in short scale-length profiles (Andreev et al 1994). The hot electron temperature scales according to
1/3 keV (6) Thot ' 8 I16 λ2N which is somewhat lower than the scaling from simulations of resonance absorption in steepened density profiles (Forslund et al 1977, Estabrook and Kruer 1978, Kruer 1988). In simulations with mobile ions, however, a rather different picture emerges. The strong space charge created by electrons circulating outside the surface pulls out an underdense ion shelf which can drastically√alter the absorption (Brunel 1988, Gibbon 1994). After a characteristic time ts ' 100 A/Zλµ fs, the absorption and hot electron distribution resemble those seen in early simulations; the main difference being that the pressure balance assumed and imposed in long-pulse (ns) interactions (Forslund et al 1977) is unlikely to be achieved for femtosecond pulses. This lack of hydrodynamic equilibrium can strongly influence the hot electron fraction and temperature scaling. For extreme intensities (I > 1020 W cm−2 µm2 ) at normal incidence, energy is transferred directly to the ions through the formation of a collisionless shock (Denavit 1992). A third collisionless mechanism which is closely related to vacuum heating is the anomalous skin effect. This is actually a well known effect in solid-state physics (Ziman 1969), and was originally studied for step-like vacuum–plasma interfaces by Weibel (1967). Its potential significance for short-pulse interaction was therefore recognized quite early by a number of workers (Gamaliy and Tikhonchuk 1988, Mulser et al 1989, Rozmus and Tikhonchuk 1990, Andreev et al 1992). Physically, the anomalous skin effect is not as mysterious as it sounds. Consider first the situation for the normal skin effect. Electrons within the skin layer δ = c/ωp oscillate in the laser field and dissipate energy through collisions with ions. The oscillation energy is thus locally thermalized provided that the electron mean-free path la = vte /νei is smaller than the skin depth. Now imagine that the temperature is increased so that la > δ, and that the mean thermal excursion length vte /ω0 > δ. Under these conditions, the laser field is carried further into the plasma and the effective collision frequency is given by the excursion time in the anomalous skin layer δa , i.e. νeff = vte /δa , where δa = (c2 vte /ω0 ωp2 )1/3 (Weibel 1967). For normal incidence in the overdense limit, one finds (Rozmus and Tikhonchuk 1990, Andreev et al 1992): 1/6  1/3  1/3  vte ω02 ω0 Te Nc δa = A' ' . (7) c cωp2 511 keV Ne Self-consistent theories of the anomalous skin effect solve the full Vlasov equation in order to correctly describe the non-local relationship between the current and the electric field (Weibel 1967, Mulser et al 1989, Gamaliy 1994, Matte and Aguenaou 1992, Andreev et al 1992). None the less, just as with resonance absorption, the effect can also be included in a hydrodynamic model by replacing νei with an effective collision frequency.

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The Fresnel equations can again be used to obtain the angular absorption dependence in the step-profile limit (Andreev et al 1992). The maximum absorption for p-polarized light is nominally A ∼ 2/3 at grazing incidence angles, independent of density and temperature, but can be enhanced if the distribution function is anisotropic. A more complete study of anomalous 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 density profile near the critical surface. As long as this motion remains normal to the gradient, the absorption and hydrodynamics can still be modelled in one dimension, even for obliquely incident light. This picture becomes inadequate if a hole is formed, or if the surface develops ripples. Both of these situations can occur for finite focal spot sizes (which are typically diffraction-limited to 2–10 µm) and at extreme irradiances (I λ2 > 1018 W cm−2 µm2 ). This regime was studied by Wilks et al (l992) using two-dimensional PIC simulation. They found that tightly focused, normally incident light can bore a hole several wavelengths deep through 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 = 160N23 Te Mbar (where N23 is the density in 1023 cm−3 and I18 is the intensity in 1018 W cm−2 ). This will cause 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 ∇r I pushes electrons away 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 relativistic 2 /2c2 )1/2 . decrease in the effective plasma frequency: ωp0 = ωp /γ , where γ = (1 + vos At sufficiently extreme intensities this could lead to ‘induced transparency’, where the laser 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 electron temperature both increase because density gradients are formed parallel to the laser electric field (Wilks 1993). Another two-dimensional effect is magnetic field generation, which has long been a subject of fascination in the field of laser–plasma interactions. Of particular interest are the large DC fields which can arise from electron transport around the focal spot. In short-pulse interactions, there are at least three mechanisms which can generate B-fields: (i) radial transport where the electron temperature and density gradients are not parallel (Stamper et al 1971), giving a source term ∂B/∂t ∝ ∇Ne ∧ ∇Te ; (ii) DC currents in steep density gradients driven by temporal variations in the ponderomotive force (Sudan 1993, Wilks et 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 long after the laser pulse and can cause strong pinching of the ablated plasma (Bell et al 1993). In contrast, 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.

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3.6. Hot electrons and hard x-ray generation There are essentially three signatures of high-intensity collective effects in laser–plasma interactions: high angular-dependent absorption, hard x-rays and fast ions. All three of these have since been verified experimentally for sub-picosecond pulses. Absorption of 50–60% for p-polarized light has been found for intensities of 1016 and above (Kieffer et al 1989b, Klem et al 1993, Meyerhofer et al 1993, Teubner et al 1993, Sauerbrey et al 1994), x-rays in the keV–MeV range have been measured (Audebert et al 1992, Kmetec et al 1992, Klem et al 1993, Chen et al 1993, Rousse et al 1994, Schn¨urer et al 1995, Teubner et al 1996b), and fast ion blow-off has been seen (Meyerhofer et al 1993, Fews et al 1994). Unlike their softer cousins (see section 3.3), hard x-rays are a direct result of hot electrons produced in the vicinity of the focal spot. By virtue its long mean-free path, a fast electron can penetrate into the cold region of the target beyond the heat front, where it either emits bremsstrahlung via collisions with ions, or produces line radiation by knocking out a bound K-shell electron.

Figure 4. Hot electron temperature measurements in short-pulse experiments (squares) compared with PIC simulations (filled circles). Experimental data are taken from: Meyerhofer et al (1993)– LLE; Rousse et al (1994)–LULI; Teubner et al (1996b)–Jena/Sprite; Jiang et al (1995)–INRS; Schn¨urer et al (1995)–MBI; Kmetec et al (1992)–Stanford; Fews et al (1994)–LLNL/IC/RAL.

Bremsstrahlung appears as a continuum anywhere in the 0.1 keV–MeV range depending on the laser intensity and plasma parameters (Kmetec et al 1992, Schn¨urer et al 1995), whereas inner-shell line emission can be 1–100 keV depending upon the atomic number of the target material (Soom et al 1993, Rousse et al 1994, Jiang et al 1995). ‘Cold’ line radiation is presently the more interesting of these for applications because of its typically narrow bandwidth and high potential brightness. Kα radiation is also an important diagnostic tool. The spectral intensity of photons emitted is directly dependent upon the electron energy. This can be exploited by observing that electrons slow down and eventually

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stop in cold material, so that the total line emission will in general depend upon the target thickness. 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 (Hares et al 1979). The change in the Kα line ratios with thickness can then be fitted to a characteristic temperature or distribution function. Although experimental hot temperature measurements in the sub-picosecond regime are still 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 is Thot ' (I λ2 )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 shortwavelength, coherent light (Kohlweyer et al 1995, Von der Linde et al 1995). Physically, harmonics are generated by the highly nonlinear plasma oscillations at the surface of the target 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 (Carman et al 1981a, b) and was suggested as a means of inferring the plasma density from the highest harmonic observed. More recent work by Gibbon (1996) has demonstrated that for intensities I > predicted by earlier theories (Bezzerides et al 1982, 1018 W cm−2 , the harmonic ‘cutoff’ √ Grebogi et al 1983), ωn = Ne /Nc ω0 no longer appears in the reflected spectrum. Theoretically, over 60 harmonics can be generated with efficiencies > 10−6 for modest plasma densities Ne /Nc ∼ 10 − 30. This prediction has been verified in an experiment with the Vulcan CPA system at RAL, UK, in which over 70 harmonics were observed using 2.5 ps pulses at intensities up to 1019 W cm−2 (Norreys et al 1996). In terms of raw power and scalability with wavelength and intensity, this mechanism is hard to beat: the efficiency essentially depends upon I λ2 , whereas the minimum wavelength is just λ/nmax . For example, Norreys et al estimate that the power converted into the 38th harmonic at 28 nm was 24 MW. On the other hand, it was also found that the harmonics were emitted more-or-less isotropically, an effect which was attributed to surface dimpling. This is in contrast to the near-specular emission in the preceding experiments using 100 fs pulses (Kohlweyer et al 1995, Von der Linde et al 1995), and it will be a challenge to optimize the 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 character of interaction from that considered in section 3. With solids, the pulse interacts with a few microns of essentially mirror-like material; with gas targets, the pulse propagates over millimetres, during which time it can ionize, distort, refract and accelerate particles, and so on. The interaction physics is therefore determined not only by the spot size σ0 and intensity I at the focus, but also by the focusing geometry or the Rayleigh diffraction length: ZR =

πσ02 λ

(8)

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where σ0 is the 1/e2 spot size, defined for a Gaussian beam by: I (r) = I0 exp(−r 2 /σ02 ). In this section we describe some of these new effects in nonlinear optics and how they might be usefully exploited. 4.1. Multiphoton ionization and heating One of the first obstacles to studying laser–plasma interactions at high intensities is the production of a plasma with accurately known properties. The simplest way to do this is to fill a target chamber with a gas at atmospheric pressure and ionize it by focusing the laser inside the chamber. Rapid ionization occurs by virtue of the fact that the laser field perturbs the 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 a DC ionization probability: " #    5/2 2 Ei 3/2 Ea Ea Ei exp − (9) W = 4ωa Eh EL 3 Eh EL where Ei and Eh , are the ionization potentials of the atom and hydrogen respectively, Ea = m2 e5 / h4 is the atomic electric field, EL is the laser field, and ωa is the atomic frequency (4 × 1016 s−1 ). This formula is particularly useful in the regime e2 EL2 /4Mω02  Ei  hν, which is precisely the operating regime of femtosecond laser systems, where the photoelectric effect (that is, single-photon ionization) cannot account for the observed ionization rates. In fact (9) and its generalizations for oscillating fields and more complex atoms (Ammosov et al 1986) have proved remarkably effective in describing the ionization physics of femtosecond interactions. This theory of so-called ‘above-threshold-ionization’ (ATI) has been verified experimentally by measuring the energies of emitted electrons (Freeman et al 1987, Augst et al 1989) and multiply-charged ions (Auguste et al 1992b). An important implication of ATI is that the kinetic energy of electrons pulled out from atoms by the laser field is typically smaller than both the ionization potential and the quiver energy (Burnett and Corkum 1987). This low ‘residual’ energy arises because within an optical cycle, ionization occurs at the peak of the electric field where the quiver velocity is zero. On the other hand, any departure from linear polarization will increase the random energy acquired by ‘newly-born’ electrons, and ultimately increase the final temperature of the 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 plasma which 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 (Offenberger et al 1993, Dunne et al 1994, Blyth et al 1995) efforts have therefore concentrated on determining and minimizing the plasma temperature. In particular, inverse-bremsstrahlung heating and collective heating due to parametric instabilities (see section 4.3) could impair the effectiveness of such schemes. Consequently, short wavelengths (λ = 1/4 µm) and pulse lengths 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 vast to do justice to in this review. Interest in this field predates the use of short pulses by several decades, in the context of ICF, ionospheric physics and astrophysics. Nevertheless, there

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were again a number of works appearing at the end of the 1980s pointing out the ways in which femtosecond pulses should behave differently from longer (pico–nanosecond) pulses in underdense plasmas. At first sight, one might think that for laser intensities such that EL  Ei , a plasma would be instantly created by the leading foot of the pulse, leaving the main part propagating through a fully ionized, uniform plasma. Unfortunately, the situation is complicated by a phenomenon known as ionization-induced defocusing (Auguste et al 1992a, Leemans et al 1992, Rae 1993). Near the front of an intense pulse, where the field is close to the ionization threshold, the gas at the centre of the beam will be ionized more, giving rise to a steep radial density gradient. The refractive index of the plasma, given by:   Ne (r, z, t) 1/2 (10) η(r, z, t) = 1 − Nc will therefore have a minimum on axis and act as a defocusing lens for the rear portion of the beam. The result is that for high gas pressures, the laser beam is deflected well before it can reach its nominal focus (Auguste et al 1992a). To circumvent this problem, experiments requiring high intensities are usually performed either with a preformed plasma (Durfee III and Milchberg 1993, Mackinnon et al 1995), or using a gas-jet configuration in which the beam is focused in vacuum before it actually enters the gas (Auguste et al 1994). Interaction at the maximum intensity is then guaranteed and one can essentially neglect the ionization physics. It is then possible to study the interaction of the laser fields with free electrons at intensities of 1018 W cm−2 and above. To see what new effects can be expected in this regime, it is helpful to examine the wave equation for an electromagnetic wave in a plasma:   2 Ne A ∂ 2 2 − c ∇ A = JNL = . (11) ∂t 2 γ The nonlinear current JNL on the right-hand side of (11) contains both the coupling of the laser field to the plasma and high-intensity effects such as relativistic self-focusing. The latter effect arises due to a change in the refractive index via electrons quivering in the laser field at velocities close to the speed of light. This phenomenon has been known for some time (Litvak 1968, Max et al 1974), but it is only through short-pulse technology that it has become possible to study it experimentally. The reason for this is that there is a power threshold (Sprangle et al 1988): Pc = 17

Ne GW Nc

(12)

at which beam diffraction is balanced by self-focusing. For typical electron densities available from a gas jet, namely 1018 –1020 cm−3 , one needs a multi-terawatt laser to have a reasonable chance of seeing the effect. On the other hand, theoretical and computational studies have demonstrated that self-focusing should be accompanied by partial or complete expulsion of electrons from the beam centre (Sun et al 1987, Mori et al 1988, Borisov et al 1990, Chen and Sudan l993, Pukhov and Meyer-ter-Vehn 1996), forming a kind of selfsustained optical fibre. Recent experiments have reported evidence of extended propagation over several Rayleigh lengths (Borisov et al 1992, Sullivan et al 1994, Monot et al 1995b, Mackinnon et al 1995). However, the interpretation of these results has been complicated by the fact that the diagnostics used to image the focused beam rely on scattering of the laser light from plasma electrons (Gibbon et al 1995).

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4.3. Wakefield excitation and instabilities Another important effect which modifies beam propagation is the excitation of plasma waves. These can be generated either by Raman-type instabilities (Kruer 1988) or by an appropriate choice of parameters so that a ‘wake’ is produced behind the pulse. In both cases, large-amplitude electrostatic fields can be generated which are able to accelerate electrons 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 has come from the ‘plasma accelerator’ community, and we review some of the issues involved in this application later in section 5.5. On the physics side, a topic which has attracted some controversy is the interplay of relativistic focusing and plasma wave generation. Several groups have tackled this problem by solving the two-dimensional envelope equations for the laser beam together with the nonlinear fluid response of the plasma (Sprangle et al 1990, 1992, Andreev et al 1992, Abramyan et al 1992, Antonsen and Mora 1992, Krall et al 1994). Simulations using this approach typically predict that for pulse lengths a few times longer than the plasma period, the envelope breaks up into beamlets of length λp (Andreev et al 1992, Sprangle et al 1992). This effect, known as ‘self-modulation’ was interpreted by these authors as follows: owing to the finite pulse shape, a small wake plasma wave is excited non-resonantly, which results in an oscillating density perturbation within the pulse envelope. The EM waves therefore see a refractive index which is alternately peaked and dented at intervals of λp /2. These portions of the pulse will therefore focus and diffract, respectively, leading to modulations in the envelope with period λp . These modulations subsequently enhance the plasma wake. The net result is that a large amount of energy can be scattered outside the original focal cone (Antonsen and Mora 1992), leaving a large-amplitude plasma wave close to the beam axis. On the other hand, one is tempted to conclude that the nominal requirement for wakefield excitation can be relaxed, and one can take pulse lengths τp  ωp−1 . Unfortunately, this picture is not complete: envelope models neglect a number of important physical processes which turn out to be just as important. First, parametric instabilities (Drake et al 1974, Forslund et al 1975b) such as Raman backscatter (RBS) and Raman forward scatter (RFS) are explicitly excluded by the paraxial-ray approximation used in these models. The growth rate for RBS with a relativistic pump is (Sakharov and Kirsanov 1994, Gu´erin et al 1995): √  1/3  2/3 3 ω0 ωp2 a0 (13) 0B = 2 2 γ0 whereas for RFS, we have (Mori et al 1994): ωp2 a0 . 0F = √ 8ω0 (1 + a02 )1/2

(14)

In RBS, the pump wave decays into a plasma wave plus an EM wave travelling back towards the focal lens. The instability therefore grows from the foot of the pulse towards the rear, and the number of e-foldings depends upon the pulse length. For RFS, however, the instability growth depends upon the propagation length as well, and is thus potentially more damaging for applications using short, high-intensity pulses. The second important effect excluded from all fluid models, whether paraxial or not, is wave breaking (Koch and Albritton 1974), which causes the plasma wave to lose coherence and heat electrons. As in laser–solid interactions, a kinetic model is essential to treat this process self-consistently. Several groups have already presented PIC simulations which

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largely confirm the initial growth scaling of the RFS and RBS instabilities but which also follow them to saturation (Bulanov et al 1992, Decker et al 1994). The state-of-theart in this area is currently claimed by the UCLA/LLNL groups, who have been able to model actual experimental parameters with two-dimensional PIC simulations comprising over 107 particles (Tzeng et al 1996). These studies have shown that RFS can also excite large-amplitude plasma waves and induce modulations in the pulse in a manner indistinguishable from the self-modulational instability observed with fluid models. Experiments at Livermore and RAL, UK largely corroborate these findings, but the overall understanding of propagation effects is far from complete. In an experiment using a 600 fs pulse and a 1 mm helium gas jet, Coverdale et al (1995) demonstrated that up to 50% of the light is scattered out of the focal cone, a result that is apparently at odds with an experiment performed under very similar conditions at Saclay (France), where collimation of the exiting beam was observed (Monot et al 1995a). 5. Applications of short-pulse laser-plasma sources So far in this review, we have concentrated mainly on the basic physical issues of femtosecond laser–plasma interactions. While it is true that much current research is curiosity driven, an equally important motivating factor is the extent to which laser-plasmas can be used as primary sources of photons and electrons for other purposes. From the preceding sections, it should by now be clear that there are basically three main areas of application: (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 possible comparing laser-plasma sources with more traditional ones. As far as the x-ray sources are concerned, a vital component of any successful application will be the development of suitable optics for the x-ray photons, which in many cases is a technological challenge in itself (F¨orster et al 1992, 1994, Attwood 1992). 5.1. Medical imaging The possibility of creating ultrafast x-ray flashlamps has been one of the major motivations behind short-pulse technological developments (Murnane et al 1991). As we saw in sections 3.3 and 3.6, a laser-produced plasma can be crudely regarded as a polychromatic continuum source with peaks of line radiation characteristic of the target material. The latter component is generally regarded as the more interesting because of its well defined emission wavelength, narrow bandwidth and high intensity. An application which is presently under serious evaluation by several groups is medical and biological imaging. Since their discovery a century ago (R¨ontgen 1895), x-rays have been exploited for this purpose with increasing sophistication, so it is natural to ask what improvements can be offered by laser-plasma sources. The requirements of medical imaging are essentially threefold: first, x-ray photon energies need to be 20–100 keV to allow transmission through the body; second, the bandwidth should be narrow to minimize the dose from unwanted soft x-rays; third, a high degree of tunability is needed to distinguish between different types of tissue. Although the traditional x-ray tube meets these general

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specifications, a laser-plasma source may offer some additional advantages (Herrlin et al 1993): •

•

•

First, a short-pulse x-ray source delivers a high yield in a picosecond duration, as opposed to micro–milliseconds for conventional sources. This offers the possibility of substantial dose reduction by using a time-gating technique to eliminate scattered x-rays which degrade the image contrast (Gordon III et al 1995). Second, differential imaging with rapid, simultaneous exposure is possible (Tillman et al 1996). This technique, traditionally implemented using large synchrotron sources, requires rapid exposure by two x-ray lines with photon energies above and below the K-absorption edge of a ‘contrast agent’. Subtraction of the two resulting images enhances the parts of the sample containing the contrast agent and suppresses unwanted information. Third, the small target size makes it possible to conceive novel image projections where the x-ray source is placed inside the object of investigation (Tillman et al 1995).

While preliminary experiments by the Lund group are very encouraging, a number of issues remain to be clarified, particularly concerning dosage. For instance, although the overall x-ray dose may be reduced by using laser-plasma sources, it is not yet clear what kind of damage the higher intensities may cause to the molecular structure of living tissue. Furthermore, the optimization of these sources, in brightness, pulse length and size, will depend 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 range promises to open up as many new possibilities as the development of the laser in the 1960s. In a tutorial review of XUV sources, Attwood (1992) gives three basic definitions which characterize coherent radiation: brightness, coherence length and transverse resolving power. The brightness B is defined as: B = 8/(1A 1 BW)

(15)

where 8 is the number of photons per second, 1A is the source area, 1 is the solid angle of the emitted radiation, and BW is the bandwidth δλ/λ. The temporal coherence length, lcoh = λ2 /21λ

(16)

determines the detail in which information can be resolved along the propagation path. The resolving power is set by the diffraction limit: dθ > λ/2π

(17)

where d is the minimum resolvable distance on the object and θ 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 XRL schemes and harmonic generation in gases or plasmas. At the time of writing, none of these novel schemes offers a water-window source, but their compactness, scaling properties, high efficiency and superior time resolution compared with long-pulse XRL schemes make them well worth pursuing. For applications where coherence is not essential, such as standard microscopy (see for example, (Rochow and Tucker 1994)), electron beams can deliver both resolution and contrast right down to atomic (sub-Angstrom) levels. The main advantage of photons over

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electrons is their ability to pass less destructively through aqueous solutions. Moreover, scanning electron microscopy normally requires carefully prepared biological samples, either freeze-dried or treated with hydrophobic agents, a process which can alter the cellular structure. With soft x-ray pulses from laser-produced plasmas (section 3.3), on the other hand, one can imagine studying living cells with a time resolution sufficient to capture dynamical processes on a sub-nanosecond timescale. Apart from good temporal coherence, a prerequisite for biological holography is a wavelength within the so-called ‘water-window’ between the absorption K-edges of oxygen ˚ and carbon (43.7 A) ˚ (Solem and Chapline 1984). This choice allows transmission (23.2 A) of 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 protein structures in their natural (aqueous) environment. An important application of coherent XUV sources which is quite widespread is plasma density diagnosis. In ICF and astrophysical plasmas, densities can be well above the critical −2 density Nc ' 1021 λ−2 µ cm , which make them difficult or impossible to probe with visible or UV lasers. A soft x-ray laser with λ < 20 nm, on the other hand, has a critical density of 1024 cm−3 or above, and can be used to obtain the plasma density using interferometry (Da Silva et al 1995). An added advantage of ultrafast XUV schemes would be an improvement in 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 at the atomic and molecular level are in situ x-ray diffraction and x-ray spectroscopy. These methods have quite different source requirements and optical arrangements, see figure 5, but share the exciting new possibilities offered by ultrafast time resolution. Consider, for example, the typical timescales and length scales of protein motions shown in table 2 (Petsko and Ringe 1984). Table 2. Timescales and length scales of protein motions.

Atomic vibration Collective Bond breaking/joining

Time (s)

˚ Deflection (A)

10−15 –10−11 10−12 –10−3 10−9 –10−3

0.01–1 0.01–5 0.5–10

Diffractometry exploits the interference effect created by adjacent atomic planes (Bragg scattering) to obtain global structural information about fluid or crystal samples. Since xray 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 (Barty et al 1995). Progress towards this goal has been recently achieved by Tomov et al (1995) using a scheme similar to that shown in figure 5(b). They demonstrated a pump–probe experiment to 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 interpretation is generally complicated by uncertainties in bulk properties. An exception to this is extended x-ray absorption fine structure (EXAFS), which yields direct information on the near neighbours of a given atom. Soft x-rays from laser-plasmas have been successfully

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(a)

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

used as EXAFS sources for some time owing to their high brightness and sub-nanosecond recording capability (Eason et al 1984). Again, short-pulse sources have been proposed as a means of extending the time resolution down to the sub-picosecond regime (Tallents et al 1990). An advantage of spectroscopic techniques over diffractometry is that the required x-ray photon flux is several orders of magnitude lower. Preliminary proof-of-principle experiments have none the less concentrated on the near-edge spectrum (XANES), where the source and detection requirements can be relaxed even further (R´aksi et al 1995).

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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. In order to use lasers for x-ray exposure of resists, one needs a short, tunable wavelength ˚ for high resolution, combined with high average power to meet throughput (around 10 A) requirements. This does not rule out short-pulse x-ray sources in special cases, but nanosecond lasers currently appear to represent the most promising option in this field (Chaker et al 1990, Maldonado 1995). 5.5. Bench-top particle accelerators The demise of the superconducting super collider (SSC) has remotivated the search for alternatives to conventional particle accelerator technology. It has been realized for some time that plasma could form the basis for a new generation of compact accelerators thanks to their ability to support much larger electric fields. Conventional synchrotrons and linacs operate 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:  1/2 me cωp Ne Ep = ' GV cm−1 . (18) e 1018 cm−3 To accelerate particles, these fields must propagate with velocities approaching the speed of light. In a seminal paper, Tajima and Dawson (1979) proposed two methods of coupling the transverse electromagnetic energy of a high-power laser into longitudinal plasma waves with high phase velocity. The first requires a pulse length matched to the plasma period such that τp ' π/ωp (Gorbunov and Kirsanov 1987, Sprangle et al 1988), which translates into a technical requirement: −1/2

tfwhm > 50N18

fs.

(19)

This condition could not be met with the technology available at that time, so they proposed an alternative ‘beat-wave’ scheme, in which two lasers are used with frequencies chosen so that ω1 − ω0 = ωp . In contrast to the wakefield scheme, where a plasma wave is forcibly driven up by the pulse, the beat-wave method relies on a more gentle build-up over tens or even hundreds of picoseconds. In both cases, the plasma wave has a phase velocity:   ωp2 1/2 . (20) Vph = c 1 − 2 ω0 An electron trapped in such a wave will be accelerated over at most half a wavelength (after which it starts to be decelerated), giving an effective acceleration length   −3/2  λp Ne λ −2 cm. (21) ' 3.2 La = 2(c − vph ) 1018 cm−3 µm Combining (18) and (21), we obtain the maximum energy gain of an electron accelerated by the plasma wave: −1 −2 λµ GeV. 1U = Ep La ' 3.2N18

(22)

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

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be limited by other factors, such as laser diffraction or instabilities. For example, comparing the Rayleigh length (8) with (21), we typically have ZR  La , so some means of guiding the 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-amplitude plasma wave generation has yet to be proven experimentally, but this will be one of the goals of ‘second generation’ plasma-based accelerators in the near future, see Katsouleas and Bingham (1996). To date, there have been some notable experiments demonstrating particle acceleration with a beat-wave scheme (Kitagawa et al 1992, Clayton et al 1993, Everett et al 1994, Amiranoff et al 1995). First experiments with short-pulse lasers (Nakajima et al 1995, Modena et al 1995) have also achieved acceleration of thermal electrons to over 40 MeV, but the underlying mechanism has been attributed to Raman instabilities rather than to ‘clean’ wakefield excitation. In an important step towards the latter, radial plasma waves generated by ‘cigar-shaped’ pulses have been observed with both temporal and spatial resolution by the LULI group (Marqu`es et al 1996). 5.6. Novel fusion concepts One of the most exciting alternative schemes to conventional ICF to emerge recently is the so-called ‘fast igniter’ concept (Tabak et al 1994). In the standard inertial confinement scenario, ignition is achieved by compressing a deuterium–tritium pellet to high density in such a way that a hot spot is created at the centre. This hot spot ignites first, providing a spark which then propagates outwards to burn the surrounding higher density fuel. To do this, laser energies of more than 1 MJ are needed for significant gain and implosion symmetry has to be controlled to better than 1%. In the fast igniter scheme, the hot spot is replaced by an external energy source, thus theoretically relaxing the driver, compression and uniformity requirements by up to an order of magnitude. Max et al (1974) proposed using a short-pulse, high-intensity laser to deliver energy to the compressed fuel via fast electrons. The scheme works in three stages. First, the fuel is compressed to a radius of around 10 µm, the equivalent of an α-particle range at an areal density of 0.4 g cm−2 . Next, a ‘prepulse’ several hundred picoseconds long with a tailored intensity profile is used to create an optical channel through the underdense corona and push the critical surface closer to the centre of the target. Finally, a short pulse with 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 revival of interest in the energy coupling and transport physics of high-intensity laser–plasma interactions. A large number of experiments are therefore being planned and carried out to investigate the issues outlined in sections 3.4–3.6. As our understanding of short-pulse physics matures, we shall no doubt see improvements on the original Livermore scheme and perhaps even more radical ideas for laser fusion. 6. Summary The first decade of research on femtosecond laser–plasma interactions has proved enormously rich both in variety of physics and in the potential for new and sometimes unexpected applications. Each improvement in technology, whether in power, shorter pulse duration, sharper focusing optics, or higher contrast ratios, sparks a fresh round of experiments with a ‘look-and-see’ spirit. In this review we have tried to document the

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changing emphasis in physics, from weakly nonlinear optics to extreme relativistic, kinetic processes, which has accompanied these developments. In addition, we have explored some of the areas in which femtosecond laser-produced plasmas show great promise as sources of fast particles and short-wavelength radiation. With a new generation of high repetitionrate, multi-terawatt lasers coming online this year, we can expect many of these ultrafast applications to be realized in the near future. Acknowledgments Thanks are due to a number of colleagues who have provided valuable advice, comments and, in some cases, unpublished material, notably: A Andreev, A R Bell, C B Darrow, G H¨olzer, J C Gauthier, J C Kieffer, T Missalla, S Svanberg, U Teubner, R P J Town and I Uschmann. References Abramyan L A, Litvak A G, Mironov V A and Sergeev A M 1992 Sov. Phys.–JETP 75 978–82 Adam J C and H´eron A 1988 Phys. Fluids 31 2602–14 Amendt P, Eder D C and Wilks S C 1991 Phys. Rev. Lett. 66 2589–92 Amiranoff F et al 1995 Phys. Rev. Lett. 74 5220–3 Ammosov M V, Delone N B and Krainov V P 1986 Sov. Phys.–JETP 64 1191 Andreev A A, Gamaliy E G, Novikov V N, Semakhin A N and Tikhonchuk V T 1992 Sov. Phys.–JETP 74 963–73 Andreev A A, Limpouch J and Semakhin A N 1994 Bull. Russ. Acad. Sci. 58 1056–63 Antonsen Jr T M and Mora P 1992 Phys. Rev. Lett. 69 2204–7 Attwood D 1992 Phys. Today 45 24–31 Audebert P, Geindre J P, Gauthier J C, Mysyrowicz A, Chamberet J P and Antonetti A 1992 Europhys. Lett. 19 189–94 Audebert P, Geindre J P, Rousse A, Gauthier J C, Mysyrowicz A, Grillon G and Antonetti A 1994 J. Phys. B: At. Mol. Opt. Phys. 27 3303–14 Augst S, Strickland D, Meyerhofer D, Cin S L and Eberly J 1989 Phys. Rev. Lett. 63 2212–5 Auguste T, Monot P, Lompr´e L-A, Mainfray G and Manus C 1992a Opt. Commun. 89 145–8 ——1992b J. Phys. B: At. Mol. Phys. 25 4181–94 Auguste T, Monot P, Mainfray G, Manus C, Gary S and Louis-Jacquet M 1994 Opt. Commun. 105 292–6 Barty C P J et al 1995 Time Resolved Electron and X-ray Diffraction vol 2521 (Bellingham, USA: SPIE) pp 246–57 Bell A R et al 1993 Phys. Rev. E 48 2087–93 Bezzerides B, Jones R D and Forslund D W 1982 Phys. Rev. Lett. 49 202–5 Birdsall C K and Langdon A B 1985 Plasma Physics via Computer Simulation (New York: McGraw-Hill) Blyth W J, Preston S G, Offenberger A A, Key M H, Wark J S, Najmudin Z, Modena A, Djaoui A and Dangor A E 1995 Phys. Rev. Lett. 74 554–7 Bonnaud G, Gibbon P, Kindel J and Williams E 1991 Laser Part. beams 9 339–54 Borisov A B, Borovskiy A V, Korobkin V V, Prokhorov A M, Rhodes C K and Shiryaev O B 1990 Phys. Rev. Lett. 65 1753–6 Borisov A B et al 1992 Phys. Rev. Lett. 68 2309–12 Born M and Wolf E 1980 Principles of Optics 6th edn (Oxford: Pergamon) Brunel F 1987 Phys. Rev. Lett. 59 52–5 ——1988 Phys. Fluids 31 2714–9 Bulanov S V, Inovenkov I N, Kirsanov V I, Naumova N M and Sakharov A S 1992 Phys. Fluids 4 1935–42 Burgess D and Hutchinson H 1993 New Scientist 140 28–33 Burnett N H and Corkum P B 1987 J. Opt. Soc. Am. B 6 1195–9 Carman R L, Rhodes C K and Benjamin R F 1981a Phys. Rev. A 24 2649–63 Carman R L, Forslund D W and Kindel J M 1981b Phys. Rev. Lett. 46 29–32 Chaker M, Boily S, Lafontaine B, Keffer J C, P´epin H, Toubhans I and Fabbro R 1990 Microelectron. Eng. 10 91–105 Chaker M, Kieffer J C, Matte J P, P´epin H, Audebert P, Maine P, Strickland D, Bado P and Mourou G 1991 Phys. Fluids B 3 167–75

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