Scaling of neonlike lasers using exploding foil targets

Vol. 5, No. 12/December 1988/J. Opt. Soc. Am. B 2537

Scaling of neonlike lasers using exploding foil targets

Barbara L. Whitten,* Richard A. London, and Rosemary S. Walling

Lawrence Livermore National Laboratory, University of California, Livermore, California 94550

Received June 13, 1988; accepted July 25, 1988

We present a set of calculations for laser-gain predictions in a neonlike collisional excitation scheme using laser-driven exploding foil targets. The calculation includes three steps: the ionization balance, the neonlike excited-state kinetics, and the hydrodynamics of the exploding foil target. The ionization-balance model solves steady-state rate equations, including excited states, using scaled hydrogenic atomic physics. The model for the neonlikeexcited-state kinetics is also steady state and includes the ground state and the 36 n = 3 excited states, with radiativeand collisional transitions connecting these states. The plasma conditions in the exploding foil targets arecalculated by using the similarity model of London and Rosen [Phys. Fluids 29,3813 (1986)]. For selected elementsin the range 20 < Z < 56, we predict the gain for the two most prominent 2p 53p to 2p 53s (J = 2-1) transitions seen inexperiments, the plasma conditions necessary to maximize the gain, and the specifications for the laser driver andtarget required to reach those plasma conditions. Our predicted gains are larger than those measured in experi-ments, for reasons we discuss, but our calculations agree qualitatively with the observed trends; the gain peaks forelements around selenium and falls off for both lighter and heavier ions. Neglected effects, such as time-dependentkinetics and radiation trapping, are also discussed.

1. INTRODUCTION

Neonlike ions have long been recognized' as potential candi-dates for producing soft-x-ray lasers. The 2s 22p 53s excitedstates have a fast electric-dipole radiative transition to the2s

22p

6 ground state, whereas the 2s2 2p 53p states can radiateto the ground state only by much slower electric-quadrupoletransitions. A population inversion between the 2s2 2p 53pand 2s2 2p53s excited states is natural and easy to create,provided that the correct plasma conditions can be obtained.A series of experiments using exploding foil targets driven bya high-powered laser have demonstrated amplified sponta-neous emission in selenium,2 yttrium,2 molybdenum,3 ger-manium,4 and copper.4

In this paper we present model calculations of the behav-ior of the neonlike soft-x-ray laser for a series of elementsbetween calcium and barium (20 < Z ' 56). Table 1 listssome basic information about the neonlike ions of theseelements. Our calculations predict the maximum gain to beexpected from the collisionally pumped neonlike laserscheme, the plasma conditions under which this gain can beexpected to occur, and the appropriate exploding foil targetdesign and laser-driver parameters that will achieve theseplasma conditions. We use a simple three-part model thatfirst calculates the fraction of neonlike ions for a broad rangeof plasma conditions and then determines the inversion den-sity of the laser transitions for the same set of plasma condi-tions. Combining these two sets of results gives us themaximum gain and the plasma conditions for which it isobtained. Finally, the similarity model for laser-driven ex-ploding foil targets5 is used to predict the target compositionand thickness and the wavelength, intensity, and pulselength of the laser driver necessary to produce these plasmaconditions.

The purpose of this simple model calculation is to suggest

qualitative trends in the kinetics of neonlike ions and in thedesign of targets, as a function of Z. We have calculatedgain coefficients for two 2s2 2p 53p (J = 2) to 2s2 2p 53s (J = 1)transitions (which occur at 206.4 and 209.8 A in selenium)because these two transitions show the highest gain in mostexperiments and because the modeling is relatively straight-forward. The J = 0-1 transition (at 182.4 A in selenium) isdifficult to model correctly, even when detailed atomic phys-ics and hydrodynamical models are used,6 and it is not usefulto apply simple models to this problematic transition.

Our results show that the gain of the two J = 2-1 linespeaks near Z = 34 (selenium). It is small for lighter ele-ments because of a mismatch between the plasma conditionsrequired to maximize the fraction of neonlike ions and thoseconditions necessary to produce a large excited-state popu-lation. For heavier ions the gain falls off because of therapid increase in radiative decay rates compared with colli-sional excitation rates, given the constraints on the maxi-mum density.

Exploding foil targets designed to establish the plasmaconditions giving maximum calculated gains are easy to findfor lighter elements by using pumping lasers with wave-lengths of 0.5 m or longer. Suitable designs also exist forheavier elements, particularly when shorter-wavelengthpump lasers are used to achieve high plasma densities; how-ever, these designs appear to be somewhat inferior. Thepump-laser energy required to produce optimal gain condi-tions increases sharply for heavy elements, making it diffi-cult to produce extremely short-wavelength lasers by usingthis scheme, which is in agreement with Ref. 7.

Finally, we discuss the importance of effects that havebeen neglected in this model, such as time dependence of theionization balance, population of the laser levels by recombi-nation and ionization processes, and losses due to line trap-ping and other effects.

0740-3224/88/122537-11$02.00 © 1988 Optical Society of America

Whitten et al.

2538 J. Opt. Soc. Am. B/Vol. 5, No. 12/December 1988

Table 1. Basic Information about the Neonlike Ion for Various Elementsa

3-3 Transition 3-3 Einstein A 3-2 Transition 3-2 Einstein AElement Z Energy (eV) Coefficient (sec- 1) Energy (eV) Coefficient (sec'1)

Calcium 20 22.6 1.46E9 348.5 1.49E11Iron 26 36.4 3.28E9 727.1 9.11E11Selenium 34 60.0 7.69E9 1438. 3.89E12Molybdenum 42 94.5 1.81E10 2381. 1.08E13Tin 50 147. 4.52E10 3549. 2.41E13Barium 56 204. 9.24E10 4568. 4.05E13

a The 3-3 transition shown is the 2pl 23p 3/2 (J = 2)-2p31 23sl/2 (J = 1) laser transition. The 3-2 transition is the 23/23sl/2 (J =transition, which is the main depopulation mechanism of the lower laser level.

2. IONIZATION BALANCE

The problem of calculating kinetic factors that affect thegain falls into two parts: the ionization balance, which de-termines the fraction of neonlike ions, and the excited-statekinetics, which determines the excited-state populations ofthe neonlike ion and the inversion density of the x-ray-laserlines. These two parts are reasonably independent of eachother. The ionization balance of the plasma depends mostlyon the interactions between highly excited states and thenext higher continuum. It is therefore important to includein the model a large number of highly excited states. How-ever, because the excited electron is far from the core forthese states, it is not necessary to use detailed atomic phys-ics.

We have used the atomic physics code YTL,8 which setsup a model including hydrogenic states as great as n = 10(where n is the principal quantum number of the excitedelectron) for the ten most probable charge states. All othercharge states are represented by the ground state only.Scaled hydrogenic rates for collisional excitation and ioniza-tion, three-body recombination, radiative recombination,and radiative decay connect all these levels. Dielectronicrecombination is also included. For a given temperatureand density, the code solves for the steady-state populationsof all these states and sums them to calculate the fraction ofions in each charge state.

Results are shown in Figs. 1(a)-1(d), in which the fractionof neonlike ions (fNe) is plotted as a function of ne for differ-ent elements and for several temperatures scaled to theneonlike ionization potential. fNe is large over a broad rangeof plasma conditions for low-Z elements. As Z gets larger,the maximum percentage of neonlike ions gets smaller.

It is also important to note that as Z rises the electrondensity required to maximize fNe also rises rapidly. This isbecause the collisional ionization rate falls as Z increases,whereas the dielectronic recombination rate is approximate-ly constant with Z and radiative recombination rises.Therefore, for heavy ions, higher temperature and densityare required for the neonlike charge state to be reached.This result agrees with the predictions derived by Apruzeseet al.9 and by Rosen et al.

7

The scaled hydrogenic model should give qualitativelycorrect results for the ionization balance as a function of Z,ne, and Te, but it is important to recognize that it has someserious limitations. The most important is the assumptionthat the ionization balance is in the steady state; estimatesindicate that the most important ionization and recombina-tion rates are too slow to bring the ionization balance into

1)-2p6

(J = 0) resonance

the steady state during the lifetime of the plasma. Wetested this by using a time-dependent version of YTL,10

which uses the same atomic physics but solves time-depen-dent rate equations for an arbitrary temperature-densityhistory. Using a constant temperature-density profile, wecalculated the 1/e equilibration time of the ionization bal-ance for several cases. For example, for selenium at Te =1268 eV (0.5 Eo) and ne = 1020 cm-3, the plasma relaxes in 3nsec. The time scales roughly with electron density; at ne =1022 cm-3, equilibration requires 20 psec. For reasonabledensities, these times are not short compared with the pulselength of high-powered lasers, so the assumption that theionization balance is in the steady state is not a good one.This problem is particularly acute for light ions, for whichmaximum gain is achieved at low electron density.

A realistic time-dependent calculation of the ionizationbalance requires information about the temperature-densi-ty history of the plasma. There is little experimental infor.-mation available; thus a temperature-density profile re-quires a computer simulation of the interaction between thelaser driver and the exploding foil target. These simula-tions can be done (see, for example, Refs. 5, 6, and 11), butthey shift the focus of the calculation to hydrodynamics.Because we are interested primarily in the kinetics of thelaser plasma, we have considered only steady-state solu-tions. Berthier-et al.'

2 have compared steady-state to time-dependent solutions of the ionization balance for somewhatsimilar plasma conditions.

A second limitation on this model is the lack of detailedatomic physics; scaled hydrogenic rates tend to overestimatethe fraction of ions in the neonlike charge state. This isbecause the ground state of the neonlike ion has a closedshell, and therefore the first excited state (n = 3) has manymetastable levels. The scaled hydrogenic model, which per-mits no metastables, can underestimate the population ofthe first excited state. Because ionization rates out of excit-ed states are much larger than those out of the ground state,the ionization from neonlike to fluorinelike ions will be toosmall, and the neonlike ionization fraction will be too large.The large gains predicted by this model (see Section 4) aredue in part to this aspect of the ionization-balance calcula-tion.

3. NEONLIKE EXCITED-STATE KINETICS

The second half of the kinetics problem is that of the neon-like excited states, which determines the inversion density ofthe laser transitions. We treat this problem as the comple-

Whitten et al.


ment of the ionization balance problem; we consider only thelow-lying states of the neonlike ion but use detailed, high-quality atomic physics.

We calculate the kinetics of the neonlike excited states byusing the steady-state code NELI.3 This includes theneonlike ground state and the 36 n = 3 excited states, withradiative and collisional transitions among them. Energylevels and radiative decay rates are calculated by using fullyrelativistic configuration-interaction wave functions. 4 Theground-state-excited-state collision rates were calculated byusing the distorted-wave approximation. Collisionstrengths were calculated for several ions by using the par-tially relativisitic codes DSW5 and JAJOM16 for low-Z ionsand the fully relativistic code MCDW17 for heavy ions (Z >36). For intermediate ions the collision strengths () arescaled by using the dipole scaling law Z2Q = constant. Rateswere then determined by averaging over a Maxwell-Boltz-mann electron distribution for the appropriate temperature.For the dipole-allowed excited-state-excited-state collisionrates, the oscillator strength approximation' 8 is used. Non-dipole-allowed excited-state-excited-state transitions arenot included.

Using this model, we can calculate the fractional inversiondensity, which is related to the total inversion density by

(nu - gUnLlgL) = NCNZ(YU - gUyL/gL)I (1)

1.0

c 0.8C0

._d-c 0.60

a 0.40

LL 0.2

0.0 1=1016 o21 1 o221o°" 10o18 10°'9 1 o20

Electron density (cm3)

(a)

where nL(U) and gL(U) are the population and multiplicity,respectively, of the lower (upper) laser level, fNe is the frac-tion of neonlike ions, Nz is the total density of all ions of thelasing element, and (Yu - 9UYL/L) is the fractional inversiondensity.

The results of our calculations are shown in Figs. 2(a)-2(c), in which the fractional inversion density for the2p3/23p3/2 (J = 2)-2p3/ 23sl/2 (J = 1) transition (the analog ofthe 206.4-A transition in neonlike selenium) is plotted as afunction of electron density for several temperatures andelements. Note that the electron temperature and densityrequired to create a large fractional inversion density riserapidly with Z because the collisional excitation rates thatcreate the inversion get smaller. The absolute size of thefractional inversion density tends to rise with Z, creating thepossibility of high gain for heavy ions.

Although collisional excitation from the neonlike groundstate is the most important mechanism for populating the2s22p53p (J = 2) upper laser states, we have neglected sever-al other processes that more detailed calculations haveshown to have an effect on the kinetics of neonlike selenium.Collisional cascade from higher states, for example, tends topopulate the J = 2 upper laser states more than the J = 1lower states. Including the n = 4 excited states in thekinetic model increases the inversion density of the J = 2-1transitions by 15-20% in selenium.19

1.0

c 0.80

a

c 0.60a,

o 0.4a0

u.. 0.2

0.0 L...1 o16 1021 10221 o17 1 018 10°19 1 o20


(b)

1.0

£ 0.8

a,

= 0.60

o 0.40

ui. 0.2

0.0 L16 1 o21 1 o22

1.,

a

0

0

0

.0

LL

0.

0.

O.

O.:

1o16 1017 1018 1019 1 o20 1 o21 1 o22

Electron density (cm3)(d)

Fig. 1. Fraction of neonlike ions as a function of electron density for various ions: (a) Te = 0.25Eo, where Eo is the ionization potential of theneonlike ion; (b) Te = 0.50Eo; (c) Te = 0.75Eo; (d) Te = 1.00Eo.

Whitten et al.

U I lI I ITe = 1.00 Eo

18

6

.4

Sn

2 Ba -

n Fe I I I \

1017 1018 1019 1 o20


(c)

I



(a)

16 1i0t7 1018 90'9 1o20 1o21 1o22

Electron density (cm3)(b)

Z)

C~0C

U)C

C

0

LI.

a)C

0

a)C

0

U:Cu


(c)

1o16 1 oi7 1018 1019 1 o20 1 o21 1 o22


(d)

Fig. 2. Fractional inversion density of the 2p3/23p3/2 (J = 2)-2p5/23s1/2 (J = 1) laser transition as a function of electron density: (a) Te =0.25Eo, where E0 is the ionization potential of the neonlike ion; (b) Te = 0.50Eo; (c) Te = 0.75Eo; (d) Te = 1.OOEo.

Table 2. Comparison of Rates Populating the 2P5/23p3/2 (J = 2) Upper Laser LevelaCollisional Dielectronic Inner-shellExcitation F/Ne Recombination Na/Ne Ionization

Z Rate (cm3/sec) Ratio Rate (cm 3/sec)b Ratio Rate (cm3/sec)

20 4.78E-12 1.93 4.50E-11 0.012 2.97E-1226 1.25E-12 0.088 5.92E-11 0.613 7.66E-1134 1.03E-12 0.906 4.55E-11 0.200 2.51E-1242 6.23E-13 0.636 2.49E-11 0.569 2.31E-1250 3.76E-13 0.699 0.662 1.67E-1256 2.63E-13 0.536 1.12 1.11E-12

Plasma conditions are those maximizing the gain (see Table 3).b Total to all neonlike levels.

We have also neglected the effect of dielectronic recombi-nation from fluorinelike ions' 9' 20 and of inner-shell ioniza-tion from sodiumlike ions,2 ' both of which directly populatethe neonlike excited states and can therefore affect the in-version density. These effects can be significant; in seleni-um more detailed calculations'9 have shown that, when thefluorinelike-to-neonlike ion ratio is taken to be 0.8, includingdielectronic recombination increases the inversion densityof the J = 2-1 transitions by 20-25%.

How much the inversion density of a particular transitionis affected by these processes depends on the size of the raterelative to other population mechanisms, the ionization bal-ance, and the inversion density (a smaller inversion densityis more sensitive to small changes in the level populations).

Some of this information is provided in Table 2 for the2p3/23p3/2 (J = 2)-2p'/ 23sl/2 (J = 1) transition. Collisionalexcitation from the neonlike ground state"3 is the principalpopulation mechanism in our model. The dielectronic re-combination rates are interpolated from the calculations ofRoszman.2 2 Note that these are total dielectronic recombi-nation rates into all neonlike states, not state-specific ratesas in the other two cases. The inner-shell ionization ratesare taken from the scaled Coulomb-Born calculation ofSampson and Zhang.23 The ion ratios are calculated asdescribed in Section 2. There is some variation in the ratios(most notably for iron) because of the finite scaled tempera-ture grid, but on the whole the plasma is less ionized forheavy ions.

0.012

0.010

0.008

0.006

0.004

0.002

.

CC)

C

0._

0

C._

0.012

0.010

0.008

0.006

0.004

0.002

.0

C._

0)

._2

U)

C

C0

.5

0

Whitten et al.


We can infer from these results that the effect of dielec-tronic recombination on gain should be large for low-Z ions,both because the rate is large and because there are morefluorinelike ions present in the plasma. Inner-shell ioniza-tion, on the other hand, is more important for heavy ionsbecause the rate is larger relative to the collisional excitationrate and because the ionization balance is more favorable.

The use of a steady-state model to calculate the excited-state kinetics is an excellent approximation because the col-lisional excitation and radiative decay rates are fast andpermit the excited-state kinetics to relax much faster thanthe plasma conditions change.' 2

4. GAIN

The gain coefficient of a Doppler-broadened line can bewritten as

X3A / Mi 1/2

87r 2rkTigUnL

1. 9L )

where X and A are the wavelength and radiative decay rate,

Table 3. Maximum Gain of the Two Strongest 3p-3s(J = 2-1) Transitions for Several Neonlike Ions and

the Plasma Conditions for Which They Are Obtaineda

Z Te (eV) ne (cm-3) Lambda (A)b Gain (cm-')

20 147 (0.25Eo) 4.64 X 1018 547. 0.54566. 0.56

26 315 (0.25Eo) 1020 341. 9.9348. 6.7

34 1268 (0.5Eo) 1021 206.3 38.209.6 24.

42 3189 (0.75Eo) 1021 131.0 18.132.7 13.

50 6416 (1.OEo) 1021 84.3 3.584.8 2.3

56 8332 (1.OEo) 1021 60.7 1.262.0 0.5

a The electron temperature is chosen to be a multiple of one quarter of theionization potential of the neonlike ion (E0). The electron density is con-strainted to be no higher than 1021 cm-3.

b For selenium and molybdenum, wavelengths are taken from Refs. 2 and 3,respectively; all others are calculated using fully relativistic configuration-interaction wave functions. 4

respectively, of the lasing transition; Mi and Ti are the massand temperature, respectively, of the ion; and the total in-version density is defined in Eq. (1).

The results of the gain calculation are shown in Table 3, inwhich we list the maximum gain of the two most prominent(J = 2-1) lasing transitions and the plasma conditions forwhich they are obtained. We have assumed that the iontemperature is always 40% of the electron temperature, as ispredicted for the selenium plasma.1 ' We have also made theassumption that Nz = ne/(Z), where (Z) is the averagecharge state of the lasing material. In practice, Nz can bereduced by diluting the lasing material in the target. Theelectron density is not permitted to rise higher than 1021cm-3 for these calculations because higher densities are dif-ficult to achieve experimentally with currently available la-sers and are generally accompanied by refractive problemsfor the x-ray propagation (see Section 5). We see that thetemperature for which the gain is maximized rises fasterthan Z2, although not as fast as the temperature that maxi-mizes fNe-

The gain is largest for elements around selenium (Z = 34)and falls off for both lighter and heavier elements. For low-Z ions, the gain is small because of a mismatch between theplasma conditions required to produce a large inversion den-sity and those needed to produce a large density of neonlikeions. A comparison of Figs. 1 and 2 shows that, for calcium,the most neonlike ions are produced when the electron tem-perature is one quarter or less of the ionization potential.The inversion density is largest at higher temperatures,when few neonlike ions are present. The match improves asZ increases; for barium the two conditions are met at ap-proximately the same plasma conditions. This interplaybetween the ionization balance and inversion density wasnot taken into account by Feldman et al.,2

4 who calculatedonly the inversion density and assumed a constant ioniza-tion balance. They predict much higher gain for low-Z ions.

For heavy ions, the gain falls off because the unfavorablescaling of collisional excitation and ionization rates make itharder to create neonlike ions and to produce a populationinversion. This means that the density required to producelarge gain grows rapidly with Z and quickly becomes un-reachable experimentally.

Table 4 summarizes the results of experimental measure-ments of gain in neonlike ions. The results are qualitatively

Table 4. Summary of Experimental Results

Gain Laser Driver PulseElement Z Lambda (A) (cm-') Target Designa Frequency (rim) Length (psec)

Copperb 29 279.31 1.7 1ooo-A Foil 1.06 2000284.67 1.7

Germaniumb 32 232.24 4.1 Solid 1.06 2000236.26 -

Seleniumc 34 206.3 5 750-A Foil 0.53 450209.6 5

Yttriumc 39 155.0 (observed, 750-A Foil 0.53 450157.1 not measured)

Molybdenumd 42 131.0 4.1 iooo-A Foil 0.53 500132.7 4.2

a All foils have a 1000-1500-A CH substrate.b From Ref. 4.' From Ref. 2.d From Ref. 3.

Whitten et al.


similar to our predictions, although the measured gains aremuch smaller. Measured gain declines from germanium (Z= 32) to copper (Z = 29) in similar experiments, as predict-ed. Experimental gain also declines in similar experimentsfor selenium, yttrium (Z = 39), and molybdenum (Z = 42),although it does so more slowly than is predicted by thiscalculation.

Although the calculated and measured gains show similartrends, the calculated gains are much larger. There are anumber of reasons for this. First, the experimental gainswere not maximized and, in particular, were measured in alower-density plasma. The measured gains are space andtime integrated over a range of plasma conditions. Thesteady-state hydrogenic ionization-balance model overesti-mates the fraction of neonlike ions, as discussed in Section 2.Finally, this simple model does not include any loss process-es. Mixing between excited states, radiation trapping of the2p5 3s-2p6 radiative decay line, additional mechanisms forpopulating the 2p 53s lower laser levels, and refraction fromdensity gradients will all tend to reduce the gain.

It is also interesting to note that, although the experi-ments measure nearly identical gains for the two lines inevery case, our model predicts significantly larger gains forthe shorter-wavelength J = 2-1 line. This is due mostly tothe absence in this model of 3p-3p collisions, which have theeffect of more evenly distributing population among the2p53p levels and thus tend to equalize the gain of the twolines. More-detailed calculations in selenium show that in-cluding these processes changes the ratio of the gains forthese two lines from 1.5 to 0.95.

Reabsorption of the neonlike 3-2 resonance lines can alterthe 3p-3s gains, mainly by repopulating the lower laser lev-els and thereby decreasing the inversion density [see Eq.(2)]. To study this effect, we use the escape probabilitymethod, in which each radiative decay rate used in the kinet-ics equations is multiplied by an escape probability in orderto estimate how much reabsorption will occur in the plasma.For exploding foil plasmas the escape of 3-2 line radiation isdominated by the gradient in the flow velocity, which causesthe line-absorption profile to Doppler shift in frequencyacross the plasma. It has been shown that the Sobolevescape probability theory2 5 is an accurate approximation ofthe line-transfer problem in the limit of foils with largevelocity gradients.26

In the examples discussed below, we use a formula for theescape probability in a plane-parallel slab with constantvelocity gradient2 7:

P = (3r,)' 1 + exp(-r)(2T, - 1) + 2T. 3/2 [erf(Tr 1

2 ) - 1,

(3)

where r, is the Sobolev optical depth:

TS = (f L orMc , (L -g~nlguc dx) (4)

where fosc is the absorption oscillator strength for the transi-tion, v is the line frequency, and dv/dx is the velocity gradi-ent in the plasma.

The Sobolev optical depth is approximately the opticaldepth of a distance in which the change in Doppler shiftowing to the gradient in the flow velocity is equal to theintrinsic linewidth. It is typically an order of magnitude less

than the optical depth would be if the plasma were notexpanding. Therefore the degree of trapping is greatly re-duced by the plasma expansion.

The effect of trapping on gain has been calculated for twoexamples: selenium at 1268 eV and tin at 6416 eV. Weshow the results in Figs. 3(a) and 3(b). The gains shown arefor the 2p3/ 2 3P3 /2 (J = 2)-2p3/2 3sl/2 (J = 1) transition plottedversus the optical depth for the 2p3/2 3sl/2 (J = 1)-2p6 (J = 0)

E

,., 1 0

Coc'4

(a)

102

E

Cut

0 .011 I I I I I I I I I I I I I I I I I0.01 0.1 1 10

(b)

Fig. 3. Gain of the 2p3/2 3p3/2 (J = 2)-2p3/2 3sl/2 (J = 1) laser transi-tion versus optical depth of the 2p3/23sl/2 (J = l)-2p6 (J = 0) line,which connects the lower laser level to the ground state for severalvalues of electron density. (a) Selenium at Te =0.50Eo (=1268 eV).The laser transition is at 206 A. The electron-density values (incm- 3) are 1, 4.6 X 1019; 2,1.0 X 1020; 3, 2.2 X 1020; 4, 4.6 X 1020; 5, 1.0X 1021; 6, 2.2 X 1021; 7, 4.6 X 1021; 8,1.0 X 1022. (b) Tin at Te =1.00Eo (=6416 eV). The laser transition is at 84.2 A. The electron-density values (in cm- 3) are 1, 1.0 X 1020; 2, 2.2 X 1020; 3, 4.6 X 1020; 4,1.0 X 1021; 5, 2.2 X 1021; 6, 4.6 X 1021; 7, 1.0 X1022.

Whitten et al.


transition, which connects the lower laser state to the groundstate. The optical depths of the other 3-2 lines are includedin the calculation, in a manner proportional to the specifiedline. It is clear from the figures that optical depths of theorder of unity are necessary to affect the gain. At fixedoptical depth, trapping has a larger influence for higherdensities. This is because at higher density 3s-3p collisionsreduce the inversion density, making the gain more sensitiveto the additional increase in lower-state population causedby trapping. The calculation of rs for particular target de-signs is discussed in Section 5.

5. TARGET DESIGN

Now we look at how to achieve the desired plasma conditionsby using an exploding foil target driven by a high-powerlaser. Figure 4 shows a schematic diagram of such a target.The thin foil is irradiated by one or more cylindrically fo-cused beams of pulsed optical laser light. The incident lightis along the x direction, perpendicular to the foil surface. Aplasma is created that is elongated in the z direction. Theadvantage of the exploding foil design is that the plasmaproduced is relatively homogeneous, with a gentle densityprofile."1 The density and temperature of the plasma can beadjusted by varying the thickness of the target as well as theintensity and pulse length of the driver laser.

We use a simple model for the hydrodynamics of theexploding foil to determine the target and laser parametersrequired to achieve the temperature and density calculatedin Part 4 to produce maximum gain. The model is based on asimilar solution of the hydrodynamic equations and hasbeen validated by comparisons with detailed numerical sim-ulations using the LASNEX code. The details of the modelare given in Ref. 5. Assumptions of the model are that theexpansion is one dimensional (in the x direction perpendicu-lar to the foil surface in Fig. 4), that the expansion velocity isa linear function of position, and that the plasma is spatiallyisothermal. With these assumptions, we find that the densi-ty has a Gaussian spatial profile and that the hydrodynamicequations can be reduced to time-dependent ordinary dif-ferential equations for the electron temperature, electrondensity, and scale length.

We use the analytical solutions for the mid-time period(after the foil becomes optically thin to the driver laser lightbut before the laser is turned off) for a constant intensity(flat-top) pulse. The following scaling laws for the electrontemperature, the electron density, and the scale length L ofthe plasma are taken from Eqs. (16), (19a), and (19b) of Ref.

Te = (1.74 keV)Il/3X2/3ml/3A-1 /6Q1 /6, (5)

ne = (1.78 X 1020 m-3)-/3X-1/6 m7 /6 A- 17 /2 4 Q5 /8t-5/4, (6)

L = (4.2 X 10-2 cm)1/3X/ 3m-1/6A-7/24Q3 /8t5 /4, (7)

where I and X are the flux and wavelength, respectively, ofthe driving laser, t is the time measured from the beginningof the driver laser pulse, m is the mass column density(thickness) of the target, and A and Q are the atomic massnumber and ion charge, respectively, of the target element.These six quantities are scaled to a convenient set of values,

optical laserx-ray laser

x-ray laser optical lasery L

" XI

Fig. 4. Schematic diagram of an exploding foil target in which aplasma is created by optical laser heating using a cylindrically fo-cused beam. Typical dimensions of the foil and laser focus areindicated. The foil is initially located in the y-z plane, with theoptical laser beams incident along the x axis. The foil plasmaexpands mainly in the x direction, perpendicular to the foil surface.

Table 5. Normalizing Values for Scaled Variables

Physical Variable Symbol Normalizing Value

Time t 1 nsecLaser intensity I 1014 W/cm 2

Laser wavelength X 0.53 ,umFoil Column density m 10-4 g/cm2

Ion charge Q 25Atomic mass A 80Laser focal spot width W 200 timElectron density ne 1021 cm-3

X-ray laser wavelength Ax 200 AScale length L 100 m

listed in Table 5. In Eqs. (5)-(7) we have assumed a con-stant Coulomb logarithm of 5.

We would like to establish the desired temperature anddensity at the peak of the laser pulse, so we evaluate Eqs.(5)-(7) at t = tL/2, where tL is the duration (FWHM) of thelaser pulse. For a laser with a different pulse shape, Eqs.(3)-(5) should give a reasonable estimate of the conditions atthe peak of the pulse. For example, the errors for Gaussianpulses found by comparing numerical solutions of the simi-larity equations with the analytical solutions are less than40%.

With this simple description of the hydrodynamics, wecan design targets to achieve the desired temperature anddensity by inverting Eqs. (5) and (6). Treating the laserintensity and target thickness as dependent variables andleaving the other experimental parameters independent, wearrive at the following equations:

I = (8.69 X 1012 W cm-2)T 7/3ne-2/3A-1/2Q-

3/4X-5/

3t -

5/6,

(8)

m = (2.18 x 10-4 g cm-2)Te2 /3n 23A7/12Q- 3/4X- 1/3tL5/6 I

(9)

Therefore, for a fixed laser wavelength, a range of pulselengths and corresponding laser intensities and targetmasses gives the desired density and temperature. We mayalso vary the laser wavelength somewhat, but there are onlya few values at which high-powered lasers exist.

Whitten et al.


101

FE0)

0)n

E

0

c5

100

lo-1

Pulse length (FWHM, nsec)(a)

Pulse length (FWHM, nsec)

(c)

101

CD)

E'D

EU

100

10-1

10-2


(b)

10-1 100


(d)(figure continued)

The range of allowed pulse lengths is further limited byseveral constraints, which are due to the conditions neces-sary for the validity of the simple hydrodynamical modeland to requirements on the plasma to ensure x-ray-laseramplification and propagation. We describe each con-straint briefly.

A. Burn-Through ConstraintThe similarlity model is valid only after the laser heat fronthas burned through the foil. This condition is also desirablein order to produce a relatively uniform plasma conducive tolaser amplification. We therefore require burn-through be-fore the peak of the pulse. Using the estimate for burn-through time given by London and Rosen [Ref. 5, Eq. (21)]and substituting Eqs. (8) and (9) for I and m, we get thefollowing constraint:

tL < tbt (0.10 nsec)Ten,-2 2A 2 Q-3/2 .

This constraint gives an upper limit on the pulse lengthbecause the burn-through time actually increases fasterthan linearly with the pulse length, given the dependences ofintensity and mass on pulse length in Eqs. (8) and (9).

B. One-Dimensional Expansion ConstraintA second criterion for the validity of the model is that theexpansion still be approximately one dimensional at thepeak of the pulse. This requires the density scale length tobe less than the transverse width (W) of the focal spot of thedriving laser. Using Eq. (5) for L gives us another upperlimit on the pulse length:

tL < td = (1.27 nsec)Te-45ne 2I5 2/5A1/ 2Q-3lOW3/2. (11)

This constraint on the applicability of the model is alsodesirable because the density and temperature drop rapidlyas the foil goes into two-dimensional expansion. Becausewe would like to achieve high density and temperature with

Whitten et al.

E

0

ECc

E

E

0

co

010

E

*0zC:

100

(10)


10.1 100


(e)

t

EE£0

E11

10-2 10- 100


(f)

Fig. 5. Diagram of target design parameter space. We plot I, the required laser intensity, and m, target thickness, versus laser pulse length.

Plasma conditions are from Table 3, chosen to create maximum gain, except where indicated. The constraints are labeled 2D for two-

dimensional expansion, B-T for burn-through, and R for refraction. The half-space excluded by each constraint is indicated by the hatch

marks. (a) Calcium with 10-ttm laser illumination. (b) Iron with 1.06-Mm laser illumination. (c) Selenium with 0.53-Mm laser illumination for

n, = 5 X 1020 cm-3. (d) Selenium with 0.35-Mum laser illumination for n1 = 1021 cm-3. (e) Molybdenum with 0.35-Mm laser illumination. (f) Tinwith 0.35-Am laser illumination.

the least expenditure of driver energy, it would be best toavoid this regime. This can be done by making the spotwidth sufficiently large, with the penalty of requiring addi-tional laser power.

C. Refraction ConstraintA third condition comes from the requirement that the las-ing x rays be able to propagate the length of the targetwithout being refracted out of the plasma by gradients in theelectron density. We can estimate the maximum propaga-tion length permitted by refraction (LR) if we assume thatthe density profile has a linear gradient with scale length L:

LR = (0.8 cm)neL/Xx, (12)

where Xx is the wavelength of the x-ray laser.Requiring LR to be larger than the length of the x-ray-laser

target (Lz) and using Eqs. (6) and (7), we get the followingconstraint:

tL > tR = (0.27 nsec)Te4/5neX2/5A1/2Q3/loxx6I5 Lz5I6. (13)

We can now apply this model to the design of targets forspecific elements. We wish to achieve the plasma condi-tions (temperature, density, and ionization balance) thatwill maximize the gain, as listed in Table 3. The problem oftarget design may now be studied by plotting the range ofintensity and target thickness versus pulse length, as speci-fied by Eqs. (6) and (7), together with the constraints givenby Eqs. (8)-(10). From such plots we can see whether anallowed region exists and, if so, what laser and target param-eters are called for. We show in Figs. 5(a)-5(f) such plots forseveral elements and laser drivers of interest.

Figures 5(a) and 5(b) show that for low-Z materials it iseasy to design a target that will be effective with long-wave-length lasers. The requirements become more severe for

higher-Z materials, however. Figure 5(c) shows that aneffective selenium target can be designed for a frequency-doubled neodymium:glass laser, although at a less than opti-mum density. To reach ne = 1021 cm-3 , the electron densitypredicted to produce maximum gain, it is necessary to usefrequency-tripled light, as seen in Fig. 5(d). Using the samelaser conditions to produce an appropriate plasma for mo-lybdenum is barely possible; Fig. 5(e) shows only a smallallowed region. For tin there is no allowed region at all. Aneffective tin target can be produced in several ways: byincreasing the lateral size of the line focus (which furtherincreases the power requirements of the driver laser), bydecreasing the desired electron density (which reduces thegain), or by further decreasing the wavelength of the laserdriver (which is difficult to do at high power). Clearly thedesign of high-gain targets using high-Z materials placesextreme requirements on the power of the driver laser.

We have examined the degree to which trapping diminish-es the gain for the target designs for selenium and tin shownin Figs. 5(d)-5(f). We require that the degree of trapping ofthe lower laser level decay line be small enough that thepopulation inversion is not destroyed. We use the escapeprobability method to describe the radiative transfer for thetrapped lines, as discussed in Section 4. For the velocitygradient we use the linear velocity assumption of the similar-ity model, noting that the characteristic velocities are of theorder of the isothermal sound speed. We write

dv/dx = 4C/L, (14)

where Cs = (kTeQ/Mi) 1 2 is the isothermal sound speed and ,= (C)-' dL/dt is the Mach number of the expansion. UsingEq. (14) in Eq. (4) and assuming that nL >> nu, we have

Tr = (57.1)m(AQTeY)"- 2(tVY)'ylfosc, (15)

E0)0

Eaco

EU7

Whitten et al.

2546 J. Opt. Soc. Am. B/Vol. 5, No. 12/December 1988 Whitten et al.

where yi is the fraction of ions in the neonlike ground stateand v is in thousands of electron volts.

For the ne = 5 X 1020 cm-3 selenium design [Fig. 5(c)], theoptical depth is approximately 1 in the worst case, which iswithin the acceptable range of pulse lengths. From Fig. 3(a)we infer that the gain will be diminished by less than 15% asa result of trapping. For the higher-density (ne = 1021 cm- 3)case shown in Fig. 5(d), trapping lowers the gain by approxi-mately 30%. In the case of tin [Fig. 5(f)] the optical depthsare low, the maximum effect being a 4% decrease in gain,according to Fig. 3(b).

6. CONCLUSIONS

The philosophy of this paper is to use simple models tocalculate the ionization balance, inversion density, and gainof the neonlike collisionally pumped soft-x-ray laser and topredict the laser and target parameters of a laser-drivenexploding foil target. We do not expect such a simple modelto produce quantitatively correct gains, but we do expect itto predict general trends. We see that the neonlike colli-sional excitation scheme is a robust scheme for producinggain over a broad range of plasma conditions, laser-driverparameters, and x-ray-laser wavelengths. The gain is pre-dicted to be highest for elements near Z = 34 and to besmaller for both lighter and heavier ions. This trend isconfirmed by experiments in elements ranging from copper(Z = 29) to molybdenum (Z = 42).

We have also attempted to address issues that are notincluded in this simple model but that we expect to have asignificant effect on the performance of these lasers. Addi-tional kinetic processes, line trapping, and time-dependenteffects are all discussed and assessed regarding their effecton this laser.

A major goal of soft-x-ray laser research at present is toproduce a laser at a wavelength of less than 44 A. This isbecause of potential medical and biological applications; thecontrast between the x-ray transmission of water and pro-tein is largest just below the carbon K edge. Unfortunately,this research suggests that the collisionally pumped soft-x-ray laser scheme is unlikely to be useful below 44 A forseveral reasons. Table 3 shows that the gain decreases sig-nificantly as Z increases because the collision rates thatmaintain the inversion grow smaller, and the plasma densityrequired to optimize the gain grows rapidly with Z andquickly becomes experimentally unreachable. Even if thesehigh densities were possible to reach, refraction would great-ly decrease the gain predicted by this model.

We have also shown, in Fig. 5, that it is difficult to meet allthe requirements of a desirable x-ray-laser target for a heavyion. The designer must produce a plasma with less thanoptimal density, increase the lateral size of the line focus ofthe driver laser, or decrease the wavelength of the driverlaser. The first of these choices further reduces the gain,and the other two place unrealistic demands on the power ofthe driver laser. We can therefore conclude that the neon-like scheme involving collisional excitation by thermal elec-trons is not a promising candidate for a sub-44-A laser. Thisdoes not, however, preclude the possibility that a collisionalexcitation scheme involving suprathermal electrons will bemore efficient. Such schemes have been proposed2 8-30 andare showing experimental promise.31

ACKNOWLEDGMENTS

The authors are pleased to acknowledge useful conversa-tions with M. D. Rosen, A. U. Hazi, D. L. Matthews, and B. J.MacGowan of Lawrence Livermore National Laboratoryand with R. C. Elton and J. P. Apruzese of the Naval Re-search Laboratory. This research was performed under theauspices of the U.S. Department of Energy by LawrenceLivermore National Laboratory under contract W-7405-Eng-48.

* Present address, Department of Physics, Colorado Col-lege, Colorado Springs, Colorado 80903.

REFERENCES

1. A. N. Zherikhin, K. N. Koshelev, and V. S. Letokhov, Kvanto-vaya Elektron. (Moscow) 3, 152 (1976) [Sov. J. Quantum Elec-tron. 6, 82 (1976)].

2. D. L. Matthews, P. L. Hagelstein, M. D. Rosen, M. J. Eckart, N.M. Ceglio, A. U. Hazi, H. Medecki, B. J. MacGowan, J. E.Trebes, B. L. Whitten, E. M. Campbell, C. W. Hatcher, A. M.Hawrylvk, R. L. Kauffman, L. D. Pleasance, G. Ramback, J. H.Scofield, G. Stone, and T. A. Weaver, Phys. Rev. Lett. 54, 110(1985).

3. B. J. MacGowan, M. D. Rosen, M. J. Eckart, P. L. Hagelstein, D.L. Matthews, D. G. Nilson, T. W. Phillips, J. H. Scofield, G.Shimkaveg, J. E. Trebes, R. S. Walling, B. L. Whitten, and J. G.Woodworth, J. Appl. Phys. 61, 5243 (1987).

4. T. N. Lee, E. A. McLean, and R. C. Elton, Phys. Rev. Lett. 59,1185 (1987).

5. R. A. London and M. D. Rosen, Phys. Fluids 29, 3813 (1986).6. M. D. Rosen, R. A. London, P. L. Hagelstein, M. S. Maxon, D. C.

Eder, B. L. Whitten, M. H. Chen, J. K. Nash, J. H. Scofield, A.U. Hazi, R. Minner, R. E. Stewart, T. W. Phillips, H. E. Dalhed,B. J. MacGowan, J. E. Trebes, C. J. Keane, and D. L. Matthews,in Proceedings of 6th Conference on Atomic Processes in HighTemperature Plasmas, A. Hauer and A. Merts, eds. (AmericanInstitute of Physics, New York, 1988).

7. M. D. Rosen, R. A. London, and P. L. Hagelstein, Phys. Fluids31, 666 (1988).

8. Y. T. Lee, J. Quant. Spectrosc. Radiat. Transfer 38, 131 (1987).9. J. P. Apruzese, P. C. Kepple, and M. Blaha, J. Phys. (Paris) 47,

C6-15 (1986).10. Y. T. Lee, G. B. Zimmerman, D. S. Bailey, D. Dickson, and D.

Kim, in Proceedings of the Third International Conference onRadiative Properties of Hot Dense Matter, B. Rozsnyai, C.Hooper, R. Cauble, R. Lee, and J. Davis, eds. (World Scientific,Singapore, 1987).

11. M. D. Rosen, P. L. Hagelstein, D. L. Matthews, E. M. Campbell,R. E. Turner, R. W. Lee, G. Charatis, G. E. Busch, C. L. Shep-ard, and P. D. Rocket, Phys. Rev. Lett. 54, 110 (1985).

12. E. Berthier, J.-F. Delpech, and M. Vuillemin, J. Phys. (Paris)47, C6-327 (1986).

13. R. S. Walling, Ph.D. dissertation (University of California, Da-vis, Davis, Calif., 1988)

14. J. H. Scofield, Lawrence Livermore National Laboratory, P.O.Box 5508, Livermore, California 94550 (personal communica-tion).

15. W. Eissner and M. J. Seaton, J. Phys. B 5, 2187 (1972).16. H. E. Saraph, Comput. Phys. Commun. 1, 232 (1970).17. P. L. Hagelstein and R. K. Jung, At. Data Nucl. Data Tables 37,

121 (1987).18. H. van Regemorter, Astrophys. J. 136, 906 (1962).19. B. L. Whitten, A. U. Hazi, M. H. Chen, and P. L. Hagelstein,

Phys. Rev. A 33, 2171 (1986).20. P. L. Hagelstein, M. D. Rosen, and V. L. Jacobs, Phys. Rev. A 34,

1931 (1986).21. W. H. Goldstein, B. L. Whitten, A. U. Hazi, and M. H. Chen,

Phys. Rev. A 36, 3607 (1987).22. L. J. Roszman, Phys. Rev. A 35, 2138 (1987); "Revised dielec-

tronic-recombination coefficients for the fluorine isoelectronicsequence," submitted to Phys. Rev.


23. D. H. Sampson and H. Zhang, Phys. Rev. A 36, 3590 (1987).24. U. Feldman, J. F. Seely, and A. K. Bhatia, J. Appl. Phys. 56,

2475 (1984).25. V. Sobolev, Moving Envelopes of Stars (Harvard U. Press,

Cambridge, Mass., 1960).26. R. A. London, in Laser Program Annual Report 86, M. L. Rufer

and P. W. Murphy, eds., Lawrence Livermore National Labora-tory Rep. No. UCRL-50021-86 (National Technical Informa-tion Service, Springfield, Va., 1988).

27. D. G. Hummer and G. R. Rybicki, Astrophys. J. 254,767 (1982).28. R. C. Carman and G. Chapline, in Proceedings of the Interna-

tional Conference on Lasers '81 (STS, McLean, Va., 1982).29. J. P. Apruzese and J. Davis, Phys. Rev. A 28, 3686 (1983).30. W. H. Goldstein and R. S. Walling, Phys. Rev. A 36,3482 (1987).31. J. P. Apruzese, P. G. Burkhalter, J. E. Rogerson, J. Davis, J. F.

Seely, C. M. Brown, D. A. Newman, R. W. Clark, J. P. Knauer,and D. K. Bradley, Bull. Am. Phys. Soc. 33, 1998 (1988).

Whitten et al.

Documents

Scaling of neonlike lasers using exploding foil targets