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770 J. Opt. Soc. Am. B/Vol. 20, No. 4 /April 2003 Lee et al.

Finite temperature dense matter studies onnext-generation light sources

Richard W. Lee, Stephen J. Moon, and Hyun-Kyung Chung

Lawrence Livermore National Laboratory, Mail Stop L-411, P.O. Box 808, Livermore, California 94550

Wojciech Rozmus

Department of Physics, University of Alberta, Edmonton, Alberta T6G 2J1, Canada

Hector A. Baldis

Institute for Laser Science and Applications, University of California, Davis, Davis, California 95616

Gianluca Gregori, Robert C. Cauble, and Otto L. Landen

Lawrence Livermore National Laboratory, P.O. Box 808, Livermore, California 94550

Justin S. Wark

Department of Physics, Clarendon Laboratory, University of Oxford, Parks Road, Oxford OX1 3PU, UK

Andrew Ng

Department of Physics and Astronomy, University of British Columbia, 6224 Agricultural Road, Vancouver,B.C. V6T 1Z1, Canada

Steven J. Rose

Atomic Weapons Establishment, Aldermaston, Reading, Berkshire RG7 4PR, UK

Ciaran L. Lewis and Dave Riley

Department of Physics, Queen’s University of Belfast, Belfast BT7 1NN, UK

Jean-Claude Gauthier and Patrick Audebert

Laboratoire pour I’Utilisation des Lasers Intenses, Unite Mixte de Recherche 7605, Ecole Polytechnique, CentreNational de la Recherche Scientifique, Commissariat a l’Energie Atomique, Palaiseau Cedex 91128, France

Received April 1, 2002; revised manuscript received December 6, 2002

The construction of short-pulse tunable soft x-ray free electron laser sources based on the self-amplified spon-taneous emission process will provide a major advance in capability for dense plasma-related and warm densematter (WDM) research. The sources will provide 1013 photons in a 200-fs duration pulse that is tunable fromapproximately 6 to 100 nm. Here we discuss only two of the many applications made possible for WDM thathas been severely hampered by the fact that laser-based methods have been unavailable because visible lightwill not propagate at electron densities of ne > 1022 cm23. The next-generation light sources will removethese restrictions. © 2003 Optical Society of America

OCIS codes: 140.7240, 340.6720, 350.5400, 300.6560, 340.0340.

1. INTRODUCTIONThe dynamics of dense matter at finite temperature hasbeen important for researchers interested in the creationof extreme states of matter. This includes research inhigh-pressure physics, applied material studies, plan-etary interiors, inertial fusion, and all forms of plasmaproduction in which energy is deposited in a solid. Theprimary difficulties in studying these states of matter are

0740-3224/2003/040770-09$15.00 ©

that the time scales for changes are rapid (<1 ps)whereas the matter, being dense, cannot be probed by theusual short-pulse visible laser techniques. Both impedi-ments will be removed when the next-generation lightsources, based on the self-amplified spontaneous emission(SASE) free electron laser (FEL) process, create lasingfrom the VUV to the XUV. These sources will provide animportant advance in our understanding of these dense fi-

2003 Optical Society of America

Lee et al. Vol. 20, No. 4 /April 2003 /J. Opt. Soc. Am. B 771

nite temperature regimes. Of interest here are two ex-amples, the first is in the area of warm dense matter re-search, the second concerns the development ofdiagnostics for dense plasmas. Whether we are inter-ested in creating warm dense matter, performing Thom-son scattering as a diagnostic, or probing a plasma, thenext-generation light sources will provide a major ad-vance of any capability that exists in the current, thirdgeneration. The key to the advance is a tunable, narrow-band x-ray source with very short pulse duration. Theindividual bunch photon number is the essential quantityfor all the plasma-based research. Using the figure ofmerit for standard light sources one can compare thenext-generation FEL sources to current synchrotronsources by comparing peak spectral brightness, i.e., thenumber of photons/mm2/mrad2/s/0.1% bandwidth. Thefourth generation SASE FELs will have an enhancementof more than 10 orders of magnitude in peak spectralbrightness, making these promising sources for finite-temperature dense matter research. Further, we notethat the high repetition rate of other sources, e.g., the Ad-vanced Photon Source (APS) at Argonne National Labora-tory or the European Synchrotron Radiation Facility(ESRF) at Grenoble, are not that useful as we require asingle photon pulse to either heat, scatter, or probe mat-ter that is transient. Indeed, to create solid matter thatis at temperatures greater than 1 eV but does not expandrequires a FEL-based source. Moreover, to measure theThomson scattering signal from a dense finite tempera-ture medium requires a probe with temporal durationthat is short compared to the evolution of the system andhas sufficient intensity to exceed the background.

A. Warm Dense MatterWith a short duration pulse that contains a substantialnumber of high-energy photons, one can generate solidmatter at temperatures of <10 eV, i.e., warm dense mat-ter. The interest in the warm dense matter regime arisesas the atoms and/or ions behave in a manner that is in-trinsically coupled to the dense plasma. That is, theplasma starts to exhibit long- and short-range orders thatare due to the correlating effects of the atoms or ions.This intriguing regime, in which the plasma can no longerbe considered a thermal bath and the atoms are no longerwell described by their isolated atom behavior, provides atremendous challenge to researchers. Within the limit ofdense cool plasmas, one obviously arrives at the thresholdof condensed matter. Here the problem has changedfrom a perturbative approach to ground-state methods inwhich complete renormalization of the atom or ion and itsenvironment is essential.

From the perspective of plasma studies the definingquantity is the coupling parameter G, i.e., the ratio of theinteratomic potential energy to the thermal energy givenby the equation

G 5Z2e2

r0kTwith r0 5 S 3Z

4pneD 1/3

, (1)

where Z is the ion charge and r0 is the interparticle spac-ing given in terms of electron density ne . The regions ofinterest span the density-temperature phase space from

modestly coupled (G < 1) to strongly coupled (G . 1) re-gimes, while bridging the transition regimes from solid toliquid to plasma.

For all cases discussed here, the ratio of the thermal deBroglie wavelength to ion spacing is smaller than one sothat the ion–ion system is classical and the expression forG is accurate. However, for the electron–electron inter-actions there are cases for which the extension to the de-generate plasma domain must be considered.1 In thesecases the quantum diffraction effect prevents the elec-trons from getting arbitrarily close, so that in the degen-erate limit G is defined as the ratio of the potential energyto the Fermi energy, Ef 5 kTf 5 h2(3p2ne)

2/3/2me .Thus, in the degenerate regime, as ne increases the cou-pling constant decreases, as G is proportional to 1/ne

1/3 .To span the space of interest, for the electron–electronsubsystem we define a representation of G using an effec-tive temperature defined as Teff 5 (Te 1 Tq)

1/2, where Tq5 Tf /(1.3251 2 0.1779rs

1/2) with rs 5 r0 /a0 , where a0 isthe Bohr orbit.2 It has been shown by Perrot andDharma-Wardana2 that, with the use of Teff , one is ableto reproduce the exact quantum statistical treatment atkinetic temperatures well below the Fermi temperature,i.e., Te ! Tf .

In Fig. 1 we show the region of the temperature-densityplane for which warm dense matter studies are impor-tant. Here we show the temperature (T) in electron voltsversus the density (r) in grams per cubic centimeter bothfor hydrogen, a low-Z element, and aluminum, a moder-ate Z element. The region in which the theoretical un-certainties are largest are those in which the standardtheoretical approaches fail and experiments are exceed-ingly difficult. The difficulty arises theoretically fromthe fact that this is a regime for which there are no obvi-ous expansion parameters, as the usual perturbation ex-pansions in small parameters used in plasma phase theo-ries are no longer valid. Further, there becomes anincreased importance on density-dependent effects, e.g.,pressure ionization, as the surroundings start to impingeon the internal structure of the ion or atom. Experimen-tally the study of warm dense matter is difficult, as theisolation of samples in this regime is complicated. In-deed, although every r –T path for the evolution of anyplasma that starts from the solid phase goes through thisregime and therefore plays an important role, trying toisolate warm dense matter remains a major challenge.

The exceedingly difficult task of performing experi-ments in the warm dense matter regime is one importantfactor that contributes to the general lack of informationwith regard to warm dense matter. As a first step, onemust create a well-characterized warm dense matterstate; the second step is to gain information about thestate through experiments. The first step has been aproblem: warm dense matter is a transient state of mat-ter. When created in a laboratory environment, it doesnot tend to remain in a specified thermodynamic state forlong, making characterization difficult. The only otherimaginable way to produce the kind of warm dense mat-ter of interest here might be the use of sub-30-fs laserpulses irradiating sub-10-nm-thick foils and to performthermodynamic measurements on a few-femtosecondtime scale over extremely small spatial dimensions. To


be able to do this on comparatively macroscopic sampleswith the next-generation sources would be a boon.

B. Plasma Diagnostic DevelopmentThere is great interest in the higher-temperature denseplasma regime. Here problems arise from the productionof high-temperature plasmas at electron densities in ex-cess of 1022 cm23. In any experiment in which a high-intensity, e.g., I > 1012 W/cm2, laser irradiates a solidtarget there will be a region of the target that is hot andat near-solid densities. Lasers with wavelengths .0.25mm do not directly heat the solid as they cannot propagatebeyond the critical electron density3:

ncr 5 1029 cm23/l2~Å!, (2)

however, heat flow from the surface efficiently generatesthe hot dense medium.3 The spectroscopic informationderived from these plasmas provides, on the one hand, di-agnostic information about the plasma itself, whereas onthe other hand by using spectroscopy, we can investigateour understanding of the mechanism at play in the cre-ation of the plasma and the interaction of the atoms orions with the plasma in which it is embedded. Here

Fig. 1. Temperature-density phase diagram for hydrogen andaluminum. The relevant regimes are noted, as are the variousvalues of the coupling G. The regions of greatest uncertainty arebroadly noted by the black outlined areas and are marked withWDM for warm dense matter. Also indicated is the region inwhich degeneracy will become important: it is the region to theright of the line at which the chemical potential is m 5 0.

next-generation light sources will provide two related andintriguing possibilities. First, there is the possibility toperform Thomson scattering on plasmas at near-soliddensity.4,5 Second, we can explore laser pump–probetechniques for high density plasmas that have been usedin low density plasmas to measure line shapes, observeradiation redistribution, and determine the kineticsprocesses.6–10 Here we discuss the diagnostic develop-ment of Thomson scattering in dense plasmas.

Thomson scattering provides an in situ measurementof the temperature, density, charge state, and collectivebehavior of the plasma. Indeed, the Thomson scatteringdiagnostic is directly related to the dynamic structure fac-tor S(k, v) of the plasma and thus provides insight intothe theoretical predictions from different theories. It isfair to say that in recent years each effort at diagnosing ahigher density plasma, i.e., higher than 1020 cm23, by useof Thomson scattering has led to new and importantdiscoveries.11 These experiments have, of course, beenfew since the constraints on the experiments are substan-tial. Here we believe that the next-generation lightsources will provide a major advance in diagnosing denseplasmas. This is clearly a complement to the concept ofcreating warm dense matter, as Thomson scattering canprovide a diagnostic of the warm dense matter conditions.However, the preconditions for the interpretation of thescattering data are that there is a valid theoretical modelfor the S(k, v) in the high density regime, and this in it-self will be a challenge. The tunable nature of the x-raysource, the high energy, bandwidth, the short pulse dura-tion and, most importantly, the very high peak photonflux make this source, to the best of our knowledge, theonly one that can address the Thomson scattering of tran-sient plasmas.

In both areas the next-generation light source providesinformation that would not be obtainable with othersources. A combination of the short pulse length, thetunable wavelength, the repetition rate, and the energyper pulse makes the data derived from these plasma-based experiments an important facet for advancing ourknowledge in this area.

2. EXPERIMENTSHere we discuss two experiments, one in the warm densematter regime and the second in the plasma-related re-gime, that can be performed with an XUV FEL. In par-ticular we use parameters that are consistent with theTera Electron Volt Superconducting Linear Accelerator(TESLA) Test Facility (TTF) upgrade that will become op-erational in 2004. The motivation for research in thewarm dense matter and plasma-related regimes on x-rayFELs has been made for the two currently proposed facili-ties, i.e., the linear coherent light source (LCLS) at theStanford Linear Accelerator Center (SLAC)12 and theTESLA at the Deutsches Elektronen-Synchrotron(DESY).13 Here we restrict ourselves to the XUV FELcase and, by doing this, show clearly that a facility withXUV capability will provide a critical step toward under-standing the physical processes that occur in a broad areaof finite-temperature near-solid density matter. The twogeneric experiments are


A. Warm dense matter: isochoric heating of a solidand observation of the isentropic expansion.

B. Diagnostics: possibility of Thomson scattering.

We note that some aspects of these two experimentsoverlap and are complementary, even though they will bepursued by different researchers with distinct motiva-tions.

A. Creation of Warm Dense Matter: Isochoric Heatingof a Thin FoilWarm dense matter is accessed in all laboratory experi-ments in which one creates a plasma from solid or near-solid density targets; however, it is difficult to study thispart of the plasma creation process in isolation. Rapidtemporal variations, steep spatial gradients, and uncer-tain energy sources lead to indecipherable complexity.Indeed, although there has been much interest in this re-gime, witnessed by the literature on strongly coupledplasmas, there has been little progress.14 The interestgenerated in laboratory experiments is mirrored in theastrophysical literature in which the warm dense matterregime is found; for example, in the structural formationof large planets and brown dwarfs.15–23

The creation of accessible warm dense matter states atFEL-based light sources, as discussed briefly below, willprovide data that will spark interest in the field. Theidea is simple, but its effect will be vast, as the data ob-tained in the generation of the warm dense matter alongan isochore, i.e., a track of constant density, with subse-quent probing along the release isentrope, i.e., a track ofconstant entropy, will be unique and critically importantfor progress in the field. The importance of these dataderives from the fact that the only practical currentmethod of generating warm dense matter is exposure ofthe material to a shock. The shock method provides in-formation along the principal Hugoniot, that is, the locusof points in the pressure–density space that are accessedby a single shock—one point for each shock strength. Al-though this has been an exceedingly fertile technique, itprovides a limited set of data, yielding little informationon material behavior in other parts of the phase space.Indeed, the amount of data currently available cannotprovide constraints on theoretical development.

In contrast to the idea of shocking the material we il-lustrate that one will be able to use the capability of anXUV FEL. Here the XUV FEL rapidly heats a thin Alfoil, ;500 Å, to temperatures between 1 and 10 eV. Us-ing the 200-fs pulse of the FEL provides heating beforehydrodynamic motion occurs. Operation of the tunableXUV FEL at 60 Å can provide uniform volumetric heatingby means of photoabsorption, as the absorption length for60-Å light is ;500 Å.24 For uniform single-sided heatingthe thickness of the foil must be much less than this ab-sorption length. If the dimension of the foil is of the or-der of the absorption length, as it is in the example we il-lustrate, double-sided heating can be used to improveheating uniformity. Figure 2 indicates that the couplingis high and the majority of the photons are absorbed. Toensure uniform heating of a 500-Å foil requires illumina-tion of both sides of the target, as shown at top in Fig. 2,indicating that the development of high-quality XUV FELoptical components will be necessary. The bottom graph

of Fig. 2 shows the calculated deposition in kilojoules pergram for both beams and the total absorbed energy.

We can estimate temperature, Te and ionization of thesample by equating energy E, deposited in volume V tothe internal and kinetic energy of the sample. The en-ergy per unit volume goes into electron kinetic energy andionization and thus is given by the formula

E/V 5 3/2neTe 1 SniIPi , (3)

where IPi is the ionization potential of ion stage i and niis the number in that ion stage. For Al there are threeelectrons in the conduction band, so that when we heatthese electrons without ionization, which will occur, forexample, at 1 eV, we assume that the deposited energy,with sufficient rapidity, goes into these conduction bandelectrons.

We note that this is not correct in detail as the photo-absorption process frees primary electrons from the innershell to produce vacancies that, in turn, give rise to sec-ondary electrons that are due to Auger processes. Thus,on a time scale of 10 fs we have three distributions, thehigh-energy photoelectrons, the mid-energy Auger elec-trons, and lower-energy electrons produced by collisionalionization processes. However, on the time scale of a

Fig. 2. Experimental setup for isochoric heating of a thin Al foil.The XUV FEL with a 60-Å wavelength is incident from bothsides, uniformly heating a 500-Å Al foil. The XUV FEL is fo-cused to 140 mm to heat the sample to 1 eV or focused to 30 mmto obtain 10 eV. The bottom graph shows the energy depositionin the foil versus distance. The light dashed curve representsthe exponential deposition of the laser energy from the left-handside, the darker dashed curve represents the laser energy depos-ited from the right-hand side, and the solid curve represents thetotal energy deposited in the foil over the 200-fs time that thepulse is on. This energy deposition is used in the simulationsshown in Figs. 3 and 4.


hundred femtoseconds, simple estimates indicate thatthese electron velocity distributions should come into asingle Maxwellian distribution.25 On the other hand,when the deposited energy per unit volume becomeslarger, as in the 10-eV case, we find that the ionization isfinite and its contribution to the internal energy must betaken into account. In both cases, the electron–ionequilibration time will be greater than 100 fs.

For solid density Al the number of atoms is ; 63 1022 cm23 and we therefore have ; 1.8 3 1023-cm23

electrons. Using the ;60-Å XUV FEL output, i.e., effec-tively 200 eV/photon for the total 1012 photons availableafter splitting the beam, we find that we can heat theelectrons in the conduction band to 1 eV. We can solveEq. (3) to determine the volume of the irradiated sample:

volume 5 1012 photons 3 200 eV/~1.5 3 1.8

3 1023 cm23 3 1 eV! ; 1029 cm3.

So that, for a 500-Å-thick foil, we can heat an area of140-mm diameter. To heat this same 500-Å-thick foil to10 eV requires a smaller spot of 30 mm. The estimate ofthe energy deposited assumes that the photoabsorptioncross sections do not significantly change on the timesscale of the deposition. Given that we find, a posteriori,from the simulations that the aluminum is only weaklyionized indicates the assumption is acceptable.

Figure 3 shows a simulation performed with a radia-

Fig. 3. Two cases of XUV FEL heating of a 500-Å thick foil. In(a) the XUV beam heats a 140-mm-diameter spot with1012 photons with an energy of 200 eV. The density and electronand ion temperatures are shown immediately after the heatingpulse was turned off at 200 fs. The electron temperaturereached ;1 eV at solid density. In (b) we show the case for a30-mm-diameter spot that reaches an electron temperature of 10eV.

tion hydrodynamics code26 of the response of the sampleto the energy deposition indicated in Fig. 2. Here the en-ergy is deposited by photoabsorption into the electronsubsystem in a thin foil with a thickness of the order ofthe absorption length. The ion subsystem is initially atroom temperature and because of the slow electron–ionenergy transfer does not equilibrate by the end of the200-fs deposition. We have performed the calculation de-positing this energy of 200 fs into a 500-Å-thick Al foilover a 140-mm spot as shown in Fig. 3(a). The figureshows the electron temperature of 1 eV at 200 fs at theend of the 200-fs laser pulse, indicating there is consis-tency between the estimate and the simulation. Alsoshown are the ion temperature and the electron density.It is important to note that the entire sample remains es-sentially at solid density. In Fig. 3(b) we show the casefor a 30-mm spot at which we estimated that a tempera-ture of 10 eV was obtained. The uniformity of the samplefor the 10-eV case is similar to that of the 1-eV case andthis provides encouragement that a reasonable portion ofthe warm dense matter region can be accessed.

To address the temporal evolution of the sample onceheated, in Fig. 4(a) we show electron density contours ona distance versus time plot. As can be seen from the con-tours, the uniformity is excellent up to 1 ps and reason-able thereafter. In Fig. 4(b) we show the density profile

Fig. 4. Simulation for the deposition of 32 mJ over 200 fs in a140-mm spot. The electron density contours are shown in (a) asa distance versus time graph. Note that the original sample is500 Å thick. In (b) we show the density profiles at times equalto 1, 5, and 10 ps.


for 1, 5, and 10 ps. These indicate that the sample gra-dient will be amenable to probing and that a smallamount of overlaid material on the sample surface couldmitigate the gradient in the probed region if, for example,element-specific diagnostics such as spectroscopy wereemployed. The calculation shown in Fig. 4 indicates thatfor the first few picoseconds one could obtain data on theexpansion of the approximately uniform sample. Thesimulation indicates that indeed, as hoped, the foil ex-pands isentropically. Using a Fourier domain interfero-metric (FDI) probe allows measurement of the expansionvelocity, whereas the use of a three-color FDI probe pro-vides both the expansion velocity and the profile. In Fig.5 the electron density versus distance is shown for 1, 2,and 3 ps after the start of the heating pulse. In the fig-ure the critical surfaces for 1v (lL 5 1 mm), 2v (lL5 0.5 mm), and 3v (lL 5 0.33 mm) probe wavelengthsare indicated and correspond to electron densities of 13 1021, 4 3 1021, and 9 3 1021 cm23, respectively. InTable 1 we show the expansion velocities for each of thesedensities. Measuring the isentropic expansion gives usthe sound velocity and in turn yields equation of state in-formation.

B. Diagnostic DevelopmentTo develop techniques to probe warm dense matter and todiagnose the plasmas of interest we use an extension ofthe visible laser-based techniques employed on lower den-sity plasmas. Here the extension of Thomson scatteringtechniques into the XUV could provide a significant ad-vance on the current state of the art. The experiment to

Fig. 5. Simulation showing the density profile at times of 1, 2,and 3 ps. The expansion velocities for the three critical surfaceslabeled as 1v(lL 5 1 mm), 2v(lL 5 0.5 mm), and 3v(lL5 0.33 mm) corresponding to densities of 1021, 4 3 1021, and 93 1021 cm23 are shown in Table 1.

Table 1. Velocities at the Critical Surface forThree Probe Wavelengthsa

lprobe (mm) ne/1021 cm23 1 ps 2 ps 3 ps 4 ps 5 ps

1 1 2.85 6.27 7.16 7.54 7.741/2 4 2.80 5.98 6.69 6.90 6.921/3 9 2.74 5.49 5.90 5.83 5.56

a For various times in the evolution of the case of a 1-eV XUV FELheated 500-Å foil extracted from the simulations shown in Fig. 5.

be performed will use a short-pulse optical laser to heat asmall dot of material deposited on a CH backing. The la-ser spot is larger than the dot, creating a plasma blowofffrom the material of interest that is constrained laterally,i.e., in the direction along the target surface, by the morerapidly expanding, usually lower Z, surrounding mate-rial. See Fig. 6 for the experimental arrangement. Theimportance of development of an XUV probe is that it canpotentially propagate through solid density matter, be-cause critical density ncr of Eq. (2) for a 100-Å laser cantheoretically probe plasmas of solid densities and above.

Figure 7 shows the regions in the electron temperatureand density space that can be probed by various wave-length probes. In Fig. 7 the temperature and densityspace is divided into two regions depending on whetherthe number of particles per Debye sphere is larger or

Fig. 6. Experimental arrangement shows the laser incident onan Al dot with a CH backing. The backing constrains the Alplasma and provides a one-dimensional expanding plasma.

Fig. 7. Temperature and density phase space indicating thenonideal plasma regime by the gray area. Nonideal is definedas a plasma for which there is less than one particle per Debyesphere. The heavy dashed line indicates the temperature-density contour at which there is one particle per Debye sphere.We also indicate by solid curve the temperature and density atwhich the scaling parameter a 5 2 for backscatter (u 5 180°)for three probe wavelengths. Wavelengths 60 and 190 Å,achievable with the XUV FEL, can probe the strongly coupledplasma regime whereas a standard visible laser at, e.g., 5000 Å,cannot access this region. The cases contained in Table 2 areindicated by the symbol X.


smaller than unity. The region above this demarcationline is essentially the region in which the strong couplingparameter G of Eq. (1) is greater than one. For the studyof strongly coupled plasmas or, more generally, warmdense matter, one would be able to probe these regionswith the XUV FEL. In addition, in Fig. 7 we show linesof constant scattering parameter a 5 2, given a 180°backscattered signal, for three probe wavelengths of 60,190, and 5000 Å. Parameter a is proportional to the ratioof the probe wavelength to the screening length, i.e.,

a 5lprobe

4plscreening sin~u/2!, (4)

where lscreening 5 (kTeff/4pnee2)1/2 and u is the angle be-

tween the incoming and the observed scattered probe ra-diation; see Fig. 8. Note that the definition of screeninglength in Eq. (4) takes into account the transition be-tween the nondegenerate and the degenerate plasma re-gimes. As a becomes larger than 1 the probe samples thecollective modes, whereas for probe wavelengths less thanthe screening length the individual particles are sampled.Figure 7 shows that both the 60- and the 190-Å wave-lengths probe the collective modes, which are of impor-tance in the strongly coupled plasma regime.

Although the critical density is well above the normalsolid density for probe wavelengths less than 200 Å, theabsorption of the probe that is due to inverse bremsstrah-lung sufficiently diminishes the signal to render the probe

Fig. 8. Thomson scattering arrangement. The angle of obser-vation u is shown for both backscattered and forward-scatteredradiation.

useless for experimentally realizable plasma lengths.Thus, we need to determine the plasma parameters andprobe wavelengths that provide appropriate conditionsfor experiments. In Table 2 we present the results of cal-culations of the scattering signal for various plasma con-ditions. We also include in Table 2 the number of par-ticles per Debye sphere, which when less than unityindicates a transition away from ideal plasma behavior,and the strong coupling parameter G. The values ofthese parameters indicate that the conditions discussedhere are only weakly nonideal and not strongly coupled.

In Table 2 we have included one case that makes con-tact with recent calculations performed to illustrate thepotential utility of a tabletop XUV laser probe of near-solid density plasmas.27 In Ref. 27 the probe was a147-Å laser-produced Ni-like Pd laser, and it was foundthat the threshold intensity for observation correspondedto 100 MW of XUV probe power incident on the plasma,i.e., 500 mJ in a 5-ps pulse duration. However, since only50 mJ has been achieved by optimization of the irradia-tion conditions for the tabletop Ni-like Pd laser,28 more re-search is required to achieve further enhancements to.100 mJ of XUV laser energy. On the other hand, largerlaser facilities routinely produce 300 mJ to .1 mJ of XUVlaser output.29,30 In this case, shown in bold in Table 2,we make contact with previous experimental estimates,and it is clear that the XUV FEL will provide a practicalhigh repetition rate Thomson scattering source. We notethat the XUV FEL has more than 500 mJ in a pulse with200-fs duration, which will simplify the interpretation ofthe Thomson scattering signal from rapidly evolving sys-tems.

The results shown in Table 2 provide the Thomson scat-tered signal in photons for several temperatures and den-sity plasmas. We also assume the incident XUV beam isdepleted by inverse bremsstrahlung over a plasma lengthappropriate for the type of plasma created. That is, forhigh density systems the path length becomes smaller fortwo reasons: first, one cannot produce large volumes ofhigh density relatively uniform plasmas; second, were one

Table 2. Thomson Scattering Parameters for Several Plasma Conditions

Te

(eV)Log ne

(cm23)l laser(Å)

Number of Particles/DebyeSpherea Gb ac

Labsd

(mm)Lengthe

(mm)Fractionf

Scattered (#)

1 19 190 0.54 0.5 1.0 5.00 3 105 5000 1.22 3 1026

1 20 190 0.17 1.08 2.0 1.90 3 104 5000 1.31 3 1026

2 21 190 1.50 1.16 4.6 3.47 3 102 500 2.83 3 1028

50 23 147 1.90 0.22 7.1 1.84 3 100 10 3.04 3 1029

1 19 60 0.54 0.5 0.2 5.00 3 106 5000 1.68 3 1026

1 20 60 0.17 1.08 0.6 1.09 3 105 5000 1.12 3 1025

2 21 60 0.15 1.16 1.4 4.50 3 103 500 4.07 3 1026

10 22 60 0.54 0.5 2.0 1.58 3 102 500 1.43 3 1026

10 23 60 0.17 0.67 6.5 3.45 3 100 10 2.12 3 1029

100 22 60 17.0 0.03 0.6 1.90 3 103 10 2.46 3 1026

100 23 60 5.37 0.07 2.0 2.40 3 101 10 4.47 3 1026

a When less than 1 the plasma can be considered nonideal.b Coupling parameter for electrons defined by Eq. (1) when Z 5 1.c For the 180° backscattered signal, which is defined in Eq. (4). For a.1 the collective regime is probed.d Absorption length that is due to inverse bremsstrahlung (IB) absorption.e Length probed is the linear distance in the plasma that the XUV probe beam traverses.f Symbol # represents the fraction of the incident photons Thomson backscattered after IB absorption.


able to produce the large volume, absorption would ren-der the signal smaller than the background emission.The scattered fraction of incident photons is calculated bythe formula

Fractionscattered 5 exp~2L/Labs!sneL,

s 5sThomson

2~1 1 a2!. (5)

It is clear that the number of scattered photons will besubstantial as the incident number of photons will be be-tween 1012 and 1013 depending on the wavelength. How-ever, we point out that the finite solid angle of the detec-tor, ; 1023 sr, and the finite efficiency of the detector,;10%, indicates that the number detected might be rela-tively small. On the other hand, the emission from anequivalent blackbody source will suffer the same degrada-tion so that the signal-to-background noise will be large.Finally, we find that the required spectral resolution nec-essary to resolve the electron collective feature, i.e.,

Dl

l'

1

a2Ane

ncr, (6)

can be satisfied as the XUV FEL source will have a reso-lution of 0.003 or better.

3. SUMMARYIn the two examples chosen we have emphasized theshort wavelength of the next generation of light sources tobe based on FEL technology. However, the high-field as-pects of these short-wavelength sources could also be pur-sued. Focusing the XUV FEL should be possible to reachintensities well in excess of 1015 W/cm2 with 200-fs pulselengths and spot sizes of >10 mm. Studies of this naturewill extend laser–solid matter interaction research intonew areas. We leave these topics for others to pursue.

We should make clear the position we believe that theXUV FEL has with respect to research on laboratory XUVlasers and high-order harmonics generated by short-pulselaser–gas interactions,31–33 both of which are actuallyXUV laser systems. The latter field has proceeded rap-idly to the point at which usable tabletop systems havebeen proposed for various proof-of-principle applications,such as, microscopy, interferometry, absorptionmeasurements,34–38 and Thomson scattering.27 Al-though it is clear that the development of a tabletop XUVlaser has many advantages, it is also clear that an XUVlight source-based laser capability should be utilized toperfect applications and, indeed, to test various experi-mental schemes. In the latter case it is clear that thestudy of gain is a most sensitive measure of our ability toperform plasma spectroscopic modeling. Thus, XUVtabletop laser development and the XUV FEL-basedsources will be complementary.

It is clear from the two cases outlined above that thepotential for the next-generation FEL-based light sourceswill provide a major advance in our study of finite-temperature dense matter. These next-generation lightsources bring the convergence of two experimental capa-bilities. On the one hand, we will have a light source

with a high repetition rate and its ability to serve a largeuser community; on the other hand, we will have a laserwith a high peak brightness, coherence, and extremelyshort pulse duration. The confluence of these two capa-bilities will open the door for an important set of experi-ments in the plasma and warm dense matter regimes. Inthe discussion above we have briefly touched on two as-pects of a large array of possible experiments. Therefore,we end by noting that much more research is requiredand hope that this will encourage study in these areas.

The e-mail address for R. W. Lee is [email protected].

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