266 J. Opt. Soc. Am. B/Vol. 7, No. 3/March 1990

Simulation of recombination-pumped soft-x-ray lasers inwall-confined laser-produced plasmas

Hiroyuki Daido, Katsunobu Nishihara, Eisuke Miura, and Sadao Nakai

Institute of Laser Engineering, Osaka University, 2-6 Yamada-Oka Suita, Osaka 565, Japan

Received May 22, 1989; accepted November 17, 1989

We describe the physical mechanisms of the recombination-pumped laser action of the carbon VI Balmer-a line(18.2 nm), which is expected to have a gain duration of as long as 10 nsec in the cylindrical-wall-confinement plasmasproduced by a C02 laser. We present the modeling of the laser action with the combination of the one-dimensionalhydrodynamic code and the multilevel atomic rate equation code. The gain coefficient derived from the calculateddynamics of the level populations is somewhat smaller than the experimental data. Additional expansion coolingmay account for the discrepancy. The higher gain coefficient of the order of 5 cm-' is expected when we irradiate ahigh-Z doped cylinder-type target.

INTRODUCTION

A significant amount of gain and intense amplified sponta-neous emissions at soft-x-ray wavelengths have been suc-cessfully demonstrated with several approaches.' Ne-like2 3

and Ni-like4 ions produced by line-focused, powerful lasersare excited by heated electrons, and large gain-length prod-ucts of the order of 10 have been achieved.2 Recombina-tion-pumped lasers, such as those using H-like5' 0 and Li-likell 2 schemes, have also demonstrated significant gain-length products. The key factor in achieving high gain in arecombination scheme is rapid cooling of highly ionizedplasma through expansion, radiation, and thermal conduc-tion. Many of the results were achieved with multistage,powerful laser fusion systems, except by the Princetongroup, who used other means.57 ,12 We describe recombina-tion-pumped soft-x-ray lasers produced by an intense C02laser oscillator, which is much simpler than the usual multi-stage laser fusion systems.

In previous publications, we proposed a cylindrical-wall-confinement carbon plasma as the H-like Balmer-a (18.2-nm) soft-x-ray laser medium,' 3"14 using a large aperture C02laser oscillator. We demonstrated experimentally a gain-length product of up to 2.4 with a 3-mm-diameter, 10-mm-long cylinder target.'5 Similar configurations of the x-raylaser targets have been proposed by others,16'8 and someexperimental results have been published. 7"8 However,the direct observation of lasing transitions has not beenachieved, except in our own research.'5 Although the pa-rameters of the targets and excitation lasers, such as pulseduration, wavelength, and energy, differ between our ap-proach and that of others, the most important common fea-tures, such as sustaining and controlling the plasma densityby the wall instead of freely expanding a high-temperatureplasma, are similar.

In this paper, we describe recombination-pumped soft-x-ray-laser action in the proposed scheme. We present themodeling of the laser action in the wall-confinement plasmawith the combination of the one-dimensional hydrodynamiccode and the multilevel atomic rate equation code. Then wecalculate the dynamics of the level populations of H-like C

atoms and propose a high-Z impurity-doped soft-x-ray lasertarget.

SCENARIO OF LASER ACTION

The general features of the expected laser action are asfollows. The high-power C02 laser'3 (typically 400 J in 50nsec) irradiates a 0.2-,um-thick thin foil attached at the endof the cylinder with /10 focusing optics as shown in Fig. 1.The thin foil is completely ablated, and the laser light isabsorbed. Then the plasma expands from the thin foil.The rest of the laser light is scattered back to the innersurface of the cylinder wall. The focal-spot diameter at thethin foil is 500 ,gm, giving a peak irradiance of 3 X 1012 W/cm2. An almost uniform plasma with a density below thecritical density of the laser is produced inside the cylinderduring the early stage of the irradiation pulse. The mainpart of the laser pulse continues to heat the plasma, and analmost fully ionized C plasma is generated in a long, cylindri-cal heated region. In this region the electron density issmaller than that near the cylinder wall owing to the radialpressure balance, which is in agreement with the numericalcalculation described below. The radial density profile maysatisfy the waveguiding profile.'9 With the scale length ofthe collisional absorption2 0 of a C-H mixture plasma, theheated length is estimated to be of the order of 1 cm underthe plasma conditions of an electron temperature greaterthan 200 eV and an electron density less than 1019 cm-3.The f-number of focusing optics is greater than 10. To avoida rapid decrease of the electron density, the mass of thecylinder wall should be adjusted not only to sustain theplasma pressure but also to control the plasma density dur-ing the ionization and the recombination phases. When thelaser irradiation stops or the plasma density exceeds thecritical density of the C02 laser, the plasma cools rapidlythrough radiative transport and heat conduction to the cyl-inder wall. Population inversion is produced by the three-body recombination process. In addition to the main cool-ing mechanisms, plasma expansion through both ends of thecylinder and through the observation windows in the cylin-der wall is expected to cool the plasma further. Typically,

0740-3224/90/030266-06$02.00 © 1990 Optical Society of America

Daido et al.

Vol. 7, No. 3/March 1990/J. Opt. Soc. Am. B 267

I, - 10mm - 1 M

CO2 Lasr I400J/50nsec Scattered Light

Fig. 1. Schematic diagram of the cylinder target and laser irradia-tion geometry. A 0.2-1um-thick parylene (C8H8 ) thin foil is attachedat the end of the 30-um-thick parylene hollow cylinder of 3-mmdiameter. The focal-spot diameter is 500 Atm, giving a peak irradi-ance of 3 X 1012 W/cm2 on the target.

we choose the diameter and the length of the cylinder to be 3and 10 mm, respectively, determined mainly by the f-num-ber of focusing optics. Thus the ratio of the hole area,through which the plasma escapes, and the wall area, whichconfines the plasma, is 0.075. Additional holes, such as anobservation window in the wall, contribute additional cool-ing while also decreasing the plasma density. High-atomic-number material seeded in the C plasma also contributes tothe plasma cooling. However, too many impurity atomsproduce too many electrons, which reduce population inver-sion because of collisional deexcitation. Furthermore, alarge amount of absorbed laser energy goes into the ioniza-tion energy of the impurity atoms. Therefore one needs tochoose the atomic number and concentration of impuritiescarefully.

In the expansion cooling schemes, such as with a C fibertarget,8 the temperature drop is proportional to (Ro/R)413,where Ro and R are the initial and the final radii of theplasma, respectively.2 ' On the other hand, the electron den-sity, which is proportional to (RO/R)2, drops more rapidlythan temperature. Generally speaking, a longer coolingtime, which reduces the peak Balmer-a gain, is expected inour scheme than would be expected in the expansionscheme. Practically, we should design the target consider-ing the above advantages and disadvantages.

COMPUTATIONAL RESULTS

A number of papers contain the computational analysis of arecombination-pumped laser action.2 21-28 Most discussthe expansion cooling type; Refs. 12 and 25 describe thepopulation inversion in a magnetically confined plasma col-umn produced by a long-pulse (50-nsec) CO2 laser.

In this section we show the computational modeling andthe calculated results of the wall-confined C recombinationlasers. First we calculate the plasma dynamics, includingradiation transport, with the hydrodynamic simulationcode. The results of the hydrodynamic code, such as iondensity and electron temperature, are fed into the rate equa-tion code. Then we obtain population distributions of Cions and gain coefficients.

The plasma dynamics is calculated with the one-dimen-sional Lagrangian hydrodynamic simulation code HISHO,29

in which axial symmetry is assumed. The physical modelused in HISHO is based on a two-temperature fluid modelwith a radiation package. The radiation transport is calcu-lated by a multigroup diffusion model that includes free-free, free-bound, and bound-bound emissions and absorp-tions. The ionization states and the electron level popula-tions are calculated by the average atom model, which isdescribed precisely in App. C of Ref. 29. In our simulation,the hot electrons produced by resonance absorption and by

the parametric processes are negligible because of the low-intensity laser irradiation.3 0 The model of the flux-limitedthermal conduction inhibition is included in this code. Inthe vicinity of the wall, where the electron temperature isless than 10 eV, the ordinary Spitzer-Harm formula yieldsseveral orders of magnitude less thermal conductivity be-cause the material is a partially ionized gas rather than aplasma. We examine the cooling time of the hot plasma,using the gas-phase heat conductivity3l in the region wherethe temperature is below 10 eV. However, this conductivitydoes not affect the dynamics of the hot plasma because thetemperature gradient in the low-temperature region is toosmall to cause any change of the heat flux in the high-temperature region. What is the dominant cooling mecha-nism in our scheme? Figure 2 shows the plasma dynamics inthe cylindrical wall, which is initially 1.5 mm from the centerof the cylinder. Each solid curve shows the Lagrangiantrajectory (fluid element) of the cylindrical-wall-confine-ment plasma. The radiation cooling, including the effect ofradiation transport, is taken into account in Fig. 2(a), where-as the radiation package is turned off in Fig. 2(b). Otherconditions of the computation are the same. The numbersin the flow diagrams indicate the electron temperature (inelectron volts) of the nearest time and space points. Thetemporal shape of the irradiation pulse is Gaussian; its peaktime and width are 50 nsec, as shown on the right-hand sideof Fig. 2. The irradiation pulse is terminated at 60 nsecthroughout our simulation because the laser light does notenter the cylinder target after that interval. The irradiationenergy is 270 J/cm. We note that entire energy of theGaussian-shaped laser pulse is 400 J, which corresponds to

100

Ew50

100

c50EP

1.182 134

i96 \10 :47

!1114 10'7

K 9N

i-209 167

105 125

-48 14,He&0 1 2 Laser

Radius(mm) Intensity(Arb. Unit)

Fig. 2. Lagrangian trajectories (solid curves) of the cylinder targetmade of parylene (C8H8) with oxygen impurities. The waveforms ofthe CO2 laser are also shown on the right-hand side. The initial wallposition is at 1.5mm from the center of the cylinder. The radiationpackage in the hydrodynamic code is turned on in (a) and turned offin (b). The laser energy is 270 J/cm (irradiation energy per 1-cmcylinder target), and the absorption is 0.5. The numbers in the flowdiagrams show the electron temperature (in electron volts) of thenearest time and space points. The darker curves show criticalpoints.

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268 J. Opt. Soc. Am. B/Vol. 7, No. 3/March 1990

the irradiation energy in the experiment. The darkercurves show the critical density point at which the laserenergy is converted into plasma thermal energy (50% ab-sorption is assumed). The wall material, which becomesablated into the center of the cylinder, is parylene (C8H8)with oxygen impurities, which are observed in our experi-ments.15 The ratio of the impurities to the total number ofatoms is assumed to be 0.13. Possible candidates for La-grangian meshes (fluid elements) for producing H-likeBalmer-a gain are only the first and second meshes from thecenter of the cylinder, because those meshes were seen tohave a significant amount of fully ionized C ions. In Fig.2(a), when radiation cooling is taken into account, the plas-ma temperatures at the first and second meshes decreaserapidly to 50 eV; in Fig. 2(b) these hot plasma temperaturesbarely fall below 100 eV, because heat conductivity greatlydecreases when the temperature decreases. In Fig. 2(a)energy loss from the hot plasma of first and second meshes isalso calculated in the code. The radiation energy loss ismore than 100 times greater than that from heat conductionand expansion. Obviously the hot plasma cools predomi-nantly by radiation. The cooling power of the radiation ishigher when the temperature is higher, and the radial pres-sure balance is violated. The cold plasma, which is pro-duced mainly by the radiation-driven ablation, compressesthe hot plasma, as shown in Fig. 2(a). Figure 2(b) showsclearly that thermal conduction alone does not create thepopulation inversion in the cylinder, whose radius is of theorder of 1 mm.

The profiles of the electron temperature, the ion density,and the pressure during and after the laser injection areshown in Fig. 3, which is derived from Fig. 2(a) at time = 50nsec, corresponding to the laser peak in Fig. 2(a), and at time= 80 nsec, when the population inversion occurs in Fig. 2(b).The temperature is higher and the density is lower in thehot-plasma region, which may give waveguiding for soft x

1U -

CU 0.1_ .

1 0-2

10-3

10-4

lor

I

- 0.1

0.10 -2

I n-3

1 .

. 1021

Z 102°

1019

1018

1024

1023

1W

Zu-1019 ( b)1019 ~ ~ 10IQ-4 1018 0 1 0.1

Radius(mm)

Fig. 3. Profiles of the electron temperature, the ion density, andthe pressure during and after the laser injection; (a) time 50 nsec and(b) time 80 nsec. The temperature is higher and the density is lowerin the hot-plasma region, which may give the waveguiding profile forsoft x rays.

rays. Nearly uniform pressure inside the cylindrical wall isclearly visible in Fig. 3. Note that we take into account theradiation energy losses through both ends of the cylinderand the observation window, modifying the energy equa-tions of the hydrodynamic code, but we do not take intoaccount the additional expansion cooling and particle losses;thus the calculated density, temperature, and cooling timeare somewhat overestimated, leading to lower populationinversion. The output of the hydrodynamic simulation isfed into a postprocessor that calculates the level populationsbased on the atomic rate equation model.

The level populations of H-like atoms are calculated fromthe time-dependent rate equations, i.e.,

dNn/dt= -N. {Ne[K(f c) + EK in)]

+ E E(n, m)A(n, m) + Ne NmK(m, n)m <n m m n

+ E NmE(m, n)A(m, n)m>n

+ N0 NZ[NeK(c, n) + R(n)], (1)

where Nn, Nz, and Ne are the population density of H-like Cwith the principal quantum number n, that of fully ionizedC, and electron density, respectively. K(n, c), K(n, m), K(c,n), R(n), A(n, m) are the collisional ionization, collisionalexcitation and deexcitation, the three-body recombination,the radiative recombination, and the radiative decay rates,respectively. E(n, m) is the escape factor,3 2 which is a func-tion of the opacity of the line transition between the nth andmth levels. Opacity for the Lyman-a line, which pumps thelower level of the Balmer-a line, is one of the crucial prob-lems for producing Balmer-a gain and is also a difficultproblem to solve. For a cylindrically expanding medium,the F Balmer-a gain has been calculated including accurateescape probabilities. However, the experimentally ob-tained gain coefficient is closer to the results calculatedwithout opacity effects than to those with them.3 3 We test-ed the escape-factor dependence of Balmer-a gains. Forexample, the Lyman-a opacity of 1.0, which corresponds tothe escape factor of 0.514 on the basis of Doppler broaden-ing, yields a gain coefficient of only 0.416 cm-', whereas thegain coefficient is 2.6 cm-' in optically thin plasma (asshown in curve d of Fig. 5 below). An opacity of 4.5, whichcorresponds to an escape factor of 0.104, yields negative gain.However, an opacity of 0.1 yields an escape factor of 0.93,giving a gain coefficient almost similar to that in an opticallythin plasma. In the plasma parameters of interest here, theopacity is estimated to be less than unity when the H-likeground-state population is 1018 cm-3 and the effective Ly-man-c absorption length is of the order of 10 m, wheremotional Doppler decoupling is more dominant than ther-mal Doppler broadening.2 ' In our simulation, optically thinapproximation is assumed. In the ground state of the H-likeatom, additional recombination and ionization terms be-tween this level and the He-like level are included. All theserates are calculated step by step in the Runge-Kutta calcula-tion loop according to their temperature dependence (fromRef. 34 and references therein).

Twenty-five levels of the H-like atom and the ground-

Daido et al.

Vol. 7, No. 3/March 1990/J. Opt. Soc. Am. B 269

state levels in the other ionization states are taken intoaccount. In order to save computer time, we usually calcu-late their populations on the basis of a steady-state approxi-mation above n = 2 levels, whose relaxation times are typi-cally of the order of several picoseconds. This leads to littleloss of accuracy. The other levels, including the fully ion-ized state, are calculated in a time-dependent fashion. Thepopulations of the sublevels in the H-like ions are distribut-ed according to their statistical weights. The above as-sumptions are valid if the electron density is sufficientlyhigh,35 36 a condition that is satisfied throughout our calcula-tions. Thus the line-center gain coefficient between level nand m, namely, G(n, m) is given by the following equation:

G(n, m) = X 2 A(n, m)(Nn - gnN./gm)SI(47rAv), (2)

where , Av, gn, gm are the wavelength and the broadening ofthe transition and the statistical weights of nth and mthlevels, respectively. S is a constant that depends on theshape of the line profile. For example, S takes a value of (ln2/7r1/2 = 0.47 for a Gaussian profile and 1/hr = 0.32 for aLorentzian profile.2 4 In our calculation when the tempera-ture is above 20 eV, the expression AvD = 7.6 X 10- 5 v(Ti/M)1/2

for the Doppler width2 5 is used, where the ion temperatureT is in electron volts and M and v are the atomic weight andthe frequency of the transition, respectively. S is taken tobe 0.47. We note that Stark broadening that includes ion-dynamic effects given by the model calculation becomescomparable with the Doppler broadening when the electrondensity is greater than 1019 cm-3.37 If we take the Starkbroadening effect into account, the calculated gain coeffi-cients may sometimes be smaller (of order 50/o-20%).

In terms of plasma parameters of interests, such as iondensity, electron temperature, and cooling time (the mostimportant parameters for determining gain coefficients), wecalculate the C Balmer-a gain without calculating hydrody-namics. First we investigate the ion density dependence ofgain coefficients. We also investigate the cooling time de-pendence. These calculations provide plasma parametersthat yield significant gain. The initial population distribu-tion is calculated based on the assumption that all the levelsare in steady state at the initial temperature. Figure 4shows the time evolution of the gain for various ion densitiesunder rapid cooling conditions. The C ion density is keptconstant in time. The electron temperature cools exponen-tially, as shown in Fig. 4, and the cooling time is defined asthe exponential decay constant, 5 nsec. C ion densities aregiven in Fig. 4. The collisional deexcitation quenches theupper level as the ion density becomes higher; hence, thegain disappears at an ion density of -9 X 1018 cm- 3. Thepeak gain coefficient gradually increases as the ion densitydecreases when the density is greater than 2 X 1018 cm- 3. Inthe lower-density region, the peak gain coefficient graduallydecreases because the level populations gradually decrease.The optimum ion density for producing the gain is 2 X 1018cm 3.

Figure 5 shows the time evolution of the gain for severaldifferent cooling times. The ion density is kept constant ata value of 2 X 1018 cm- 3 throughout the calculations. Theelectron temperature decreases exponentially. Curves a, b,c, d, e, and f denote cooling times of 0.5, 1, 2, 5, 10, and 20

nsec, respectively. The gain histories, corresponding tocurves a-f, of temperature variations are shown. The peakgain is higher as the cooling time becomes shorter, while the

auaxiu,C 0b:5x10'6

2s 10 c:10 1 200b)d2x10 16

*:1x1018

0 C~~~~~~~~~~~~~100 16

ow

00 ~5 10 15 20Tnme(nsec)

Fig. 4. Calculated time evolution of the C Balmer-a gain coeffi-cient at 18.2-nm wavelength as a parameter of ion densities underrapidly cooling conditions. The exponential decay time of the tem-perature is 5 nsec. The maximum gain is obtained at an ion densityof 2 x 1018 cm 3.

102 N

I

I-C

010 .g

UJ

1

Time(nsec)

Fig. 5. Time evolution of gain as a parameter of the cooling time.The ion density is kept constant at 2 X 1018 cm- 3. The electrontemperature decreases exponentially. Curves a, b, c, d, e, and fdenote cooling times of 0.5, 1, 2, 5, 10, and 20 nsec, respectively.The peak gain is higher as the cooling time becomes shorter, whereasthe gain duration is longer as it becomes longer.

gain duration is longer as the cooling time becomes longer.To use a resonator as a high-quality soft-x-ray coherentsource, one should choose the optimized lasing condition,i.e., the gain should at least overcome resonator loss and thegain duration should be much longer than the cavity round-trip time. At a cooling time of 5 nsec, the peak gain is 2.5cm-1 and the gain duration is 4.1 nsec full width at half-maximum (FWHM).

Figure 6 shows the gain histories, including the ionizationphase. Curves a, b, and c denote the electron temperaturerise times of 0.5, 2, and 5 nsec from 20 to 150 eV. The initialpopulation distribution is given by the steady-state approxi-mation. The ion density is 2 X 1018 cm- 3, and the coolingtime is 5 nsec, as shown in Fig. 6. The peak gain is smaller asthe pumping time becomes shorter owing to insufficientionization in spite of constant peak temperature. The risetime of 5 nsec gives a peak gain equal to that calculated fromthe assumption of initially steady-state conditions at a tem-

Daido et al.

270 J. Opt. Soc. Am. B/Vol. 7, No. 3/March 1990

perature of 150 eV. In our scheme, the pumping time is longenough to ionize the C atoms.

The output of the hydrodynamic simulation, such as C iondensity and electron temperature, is fed into the atomic rateequation code as a postprocessor. Then we obtain the gainhistories in the cylindrical-wall-confinement plasma. Thecalculated results, with the same irradiation condition andtarget parameters as in the experiment,'5 are shown in Fig.2(a). Population inversion occurs within the second meshfrom the center of the cylinder, where the peak electrontemperature exceeds 100 eV. Figure 7 shows the simplifiedtime histories of C-ion density, the electron temperature forthe first (solid curves) and the second (dashed curves)meshes from the center of the cylinder in Fig. 2(a), andresultant gain histories just after the laser absorption isterminated. At 60 nsec, heating stops, and the plasma,which was initially in steady state (see Fig. 6), goes into anonequilibrium recombination phase, and then the popula-tion inversion between n = 3 and n = 2 occurs. The calculat-ed peak gain coefficient is approximately half the experi-mental value. The reason for the discrepancy has not beenidentified, but the plasma expansion in the axial direction ofthe cylinder and through the observation window, which arenot taken into account in the calculation, may give a largergain coefficient in the experiment. Actually, shorter-lengthcylinder targets tend to give larger gain coefficients in theexperiment.' 5 The gain duration is 10 nsec, which is longenough to operate a resonator if one has high-reflectivitymirrors at a wavelength of 18.2 nm. However, the radius ofthe gain region is only -0.5 mm, in spite of an initial cylinderradius of 1.5 mm. Spatially inhomogeneous gain may re-strict resonator operation.

In order to increase the gain coefficient, we performedsimulations for targets containing high-atomic-number im-purities. Figure 8 shows the time histories of the electrontemperatures and the C-ion densities at the first (solidcurves) and the second (dashed curves) Lagrangian meshesfrom the center of the cylinder. The target material isC8H6CI2. The temporal shapes of the irradiation pulse and

3 150ab C

C,b

a

so

00 5 10 15llme(rsec)

Fig. 6. Gain histories including the ionization phase. Curves a, b,and c denote electron temperature rise times of 0.5, 2 and 5 nsec,respectively. The ion density is 2 X 1018 cm-3, and the cooling timeis 5 nsec. The peak gain is smaller as the pumping time becomesshorter owing to insufficient ionization, in spite of the constant peaktemperature.

_200.

; 150 .So

~15E 1 0:0

C

To so:.0

LU .0

-1.!

.2-C

00

0.o

c0)

E0

C.X1O'8

C

0

C0

-

0

(U0'c

C.

Tnme(nsec)

Fig. 7. Time histories of the ion density, the electron temperaturefor the first and the second meshes from the center of the cylinder inFig. 2(a), and resultant gain histories just after the laser absorptionis terminated. The laser energy and the absorption are 270 J/cmand 0.5. The peak gain coefficient is approximately half that of theexperimental data. The duration of the gain is 10 nsec, which islong enough to operate a resonator mode if we were to use a high-reflectivity mirror at 18.2-nm wavelength.

250

I,200

0

! 1500

E-1 00C

,, 50w

I

0

0

OL

'0

0D

C0

.0

a.,

c

Time(nsec)

Fig. 8. Time histories of the electron temperature, the C ions, andthe resultant gain coefficients at the first and the second meshesfrom the center of the cylinder just after the laser absorption isterminated. The laser energy and the absorption are 404 J/cm and0.5. The target material is C8H6Cl2. At the first mesh, the coolingtime is as short as 5 nsec, giving a peak gain coefficient of nearly 4cm- 1 with a duration of 5 nsec (FWHM).

the absorption are the same as those in Fig. 7, but a largerirradiation energy of 404 J/cm is required because of thelarge amount of absorbed laser energy that goes into ioniza-tion of the Cl atoms. The cooling time is much shorter thanthat shown in Fig. 7. The Cl ions play a significant role inradiative cooling. At the first mesh, the cooling time is asshort as 5 nsec, giving a peak gain coefficient of nearly 4 cm-'with a duration of 5 nsec (FWHM). The radius of the gainregion is also 0.5 mm. To optimize the peak gain coefficientand the gain duration for a soft-x-ray resonator, we shouldsystematically investigate the concentration as well as theatomic number of high-Z impurities.

CONCLUSION

We describe the expected scenario of the recombination-pumped C VI Balmer-a (18.2 nm) soft-x-ray-laser action inwall-confined plasmas. We model the laser dynamics with acombination of a one-dimensional hydrodynamic code and amultilevel atomic rate equation code. We calculate the dy-

Daido et al.

Vol. 7, No. 3/March 1990/J. Opt. Soc. Am. B 271

namics of the level populations of H-like C atoms. Thecalculated gain coefficient of 1 cm-' is half of the experimen-tal value. The difference may be due to the additionalexpansion cooling in the experiment. High-atomic-numberimpurity ions play a significant role in producing a highergain coefficient, of the order of 5 cm-1 .

ACKNOWLEDGMENTS

We thank H. Takabe of the Institute of Laser Engineering,T. Fujimoto of Kyoto University, and the reviewers of thispaper for their helpful discussions. We also thank Y. Ki-tagawa, Y. Kato, and C. Yamanaka for their valuable sugges-tions and encouragement throughout the study.

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