modelling of short pulse x-ray laser experiments

C. R. Acad. Sci. Paris, t. 1, Série IV, p. 1093–1104, 2000Électromagnétisme, optique/Electromagnetism, optics(Gaz, plasmas/Gas, plasmas)

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LES DÉVELOPPEMENTS RÉCENTS DES LASERS À RAYONS X

RECENT PROGRESS IN X-RAY LASERS

Modelling of short pulse X-ray laser experimentsRobert KING, Geoff J. PERT

Department of Physics, University of York, Heslington, York, YO10 5DD, UK

(Reçu le 1 août 2000, accepté le 21 août 2000)

Abstract. The basic theory of amplified spontaneous emission (ASE) from long lived laser mediawas established several years ago enabling a clear understanding of the effects of saturationon output irradiance and line profile to be achieved. The advent of ultra-short pulse laserpumping for X-ray lasers using a travelling wave has necessitated the extension to a timedependent description. Using the simple model developed earlier, we have investigatedthe key features of time dependent ASE laser action, with particular emphasis on thematched travelling case. We have found that, initially, substantial pulse shortening occurs,but that after saturation the pulse slowly lengthens. Concomitantly the output energy isreduced below that predicted by the steady state theory and the gain pulse duration.Line width narrowing is consistent with the static case. Comparison with more detailedsimulations shows that the beam is also broadened transverse to its propagation directionafter saturation. Good agreement is found between the simulation and experimental values. 2000 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS

X-ray laser / travelling wave / transient gain / laser pumping

Modélisation des expériences laser X–UV en impulsions brèves

Résumé. Les bases théoriques de l’émission spontanée amplifiée (ASE), dans les milieux laser àlongue durée de vie, ont été établies depuis plusieurs années et elles ont permis une bonnecompréhension des effects de saturation sur l’éclairement de sortie et le profil de la raielaser. L’apparition du pompage des lasers X–UV par des impulsions laser ultrabrèvesen régime d’onde progressive a entraîné la prise en compte des effets temporels dansle formalisme. En utilisant un modèle simple que nous avons développé précédemment,nous avons étudié les caractéristiques essentielles de l’ASE dépendent du temps eninsistant plus précisément sur le cas de l’excitation progressive accordée. Nous montronsun rétrécissement substantiel de l’impulsion aux temps courts mais, après saturation,l’impulsion voit sa durée augmenter lentement. De manière concomitante, l’énergie desortie est réduite en dessous de la valeur prédite par la théorie stationnaire et la duréedu gain. Une diminution de la largeur de raie est en accord avec le résultant du régimestatique. En comparant les résultats avec des simulations plus élaborées, on montre que lefaisceau X–UV est élargi transversalement à sa direction de propagation après saturation.On trouve un bon accord entre les valeurs expérimentales et les simulations. 2000Académie des sciences/Éditions scientifiques et médicales Elsevier SAS

laser à rayonnement X / onde progressive / laser transitore / pompage laser

Note présentée par Guy LAVAL .

S1296-2147(00)01115-X/FLA 2000 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS. Tous droits réservés. 1093

R. King, G.J. Pert RECENT PROGRESS IN X-RAY LASERS

1. Introduction

X-ray lasers using amplified spontaneous emission are now well established. The general theoryunderlying such laser action was established nearly thirty years ago by Casperson [1] and related authors.At that time, most of the characteristic effects, namely the role of saturation, the distinction betweenhomogeneously and inhomogeneously broadened lines, line width narrowing and subsequent re-broadeningfor inhomogeneously broadened lines, were all identified, and in many cases observed in experiments usingprincipally xenon lasers. The simple theory used to model this behaviour based on radiation intensities andpopulation rate equations was replaced in later work by models based on the Maxwell–Bloch formulation,with a substantial increase in mathematical complexity, but little improvement in the ability to modelexperimental systems. Recently these earlier models have been applied to X-ray lasers by Pert [2] andKoch et al. [3]. In the latter study, measured line widths were compared with the predicted values based ondetailed gain calculations and the model. Somewhat unexpectedly, no re-broadening was seen although thelines were predominately Doppler broadened. This was accounted for by the inclusion of relaxation by Pert[4] in a re-working of the Casperson model.

The development of collisionally pumped lasers has seen a progressive shortening of the primary pulsedelivered by the Nd glass pump laser. As a result the lifetime of the gain and duration of the X-ray outputhave both decreased to such an extent that the pulse transit time has become comparable to the pump time.To avoid the effects of gain depletion on the amplification of the X-ray pulse, travelling wave systems havebeen exploited, where the gain pulse is synchronised with the pump.

The standard model of ASE lasers assumes that the laser is uniform and steady state, and that thebeam has achieved a stable bidirectional pattern. Clearly this is unsatisfactory for treating transient, non-uniform phases as occur in travelling wave lasers. Lin et al. [5] adapted the standard ASE model to considerunidirectional behaviour with gain, which decayed exponentially in time. The model was applied to specificexperimental systems and shown to satisfactorily account for the observed pattern, if suitable parameterswere used. Subsequently Strati and Tallents [6] have developed a generalisation of this model to includemismatched travelling wave pulses and arbitrary gain profiles.

In this paper, we will use this model to investigate the general behaviour of homogeneously broadened,unidirectional ASE lasers by a direct extension of our earlier work. In particular we investigate pulseshortening in the unsaturated phase, and subsequent lengthening once saturation sets in. We examinethe behaviour of the observed line width for both Lorentz and Gaussian profiles. The results from thesimple model are compared with calculations obtained by ray tracing, which reveal its limitations whenapplied in an experimental situation. The simulation results are shown to give a good representation of thecorresponding experiment.

2. Analytic model

As noted earlier the analytic model of the pulse development is based on earlier work by the authors [2]using the simple model of ASE pulse formation in a saturating medium due to Casperson [1]. In that workthe laser is envisaged as a large aspect ratio cylinder of lengthl and cross sectional areaA. The mediumhas a uniform and constant spontaneous emission rate per unit volumeΣ0. Only radiation within a coneof solid angleΩ is amplified to form the output beam. Thus the medium spontaneously emits radiation ata rateE0 = Σ0/4π per unit volume into the beam, and has a population inversion generating a constantline-centred, small signal gain coefficientG0. In the presence of the beam the spontaneous emission rateand gain are both reduced by saturation. The reductions in the rates are described by two characteristicsaturation irradiancesI ′s andIs: in practical casesI ′s Is and may be neglected. The saturation irradianceis given by the usual formulaIs = hν/σsτR in terms ofν the line frequency,σs the emission cross sectionandτR the recovery time. The latter is the characteristic time for the restoration of the population balanceon the lasing transition, and is a measure of the pumping time for the transition. The line has a normalisedfrequency profilef(ν), which is assumed to be identical for both spontaneous and stimulated emission.

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RECENT PROGRESS IN X-RAY LASERS Modelling of short pulse X-ray laser experiments

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The line-centre profile ratio is defined asφ(ν) = f(ν)/f(0), such thatφ(0) = 1. In general the line profileis formed by the convolution of both homogeneous and inhomogeneous components. An extension to themodel allows for relaxation of the inhomogenous component at a rate given by a characteristic timeτ0,such that ifτ0 τR the inhomogenous component behaves as homogenous. Very little experimentalor theoretical data is available concerning the relaxation time in X-ray lasers. However the line profilesmeasured by Koch [3] for the selenium transitions at 206 Å and 209 Å did not show saturation re-broadening characteristic of inhomogeneous effects. Analysis by Pert [7] showed that the results couldbe accounted for by rapid relaxation withτ0 . 0.2τR. Although the pumping conditions for this experimentwere dissimilar to those used in more recent work, the plasma parameters determining the relaxation rateare not greatly different. We may therefore, with some confidence, assume that the broadening may beconsidered entirely homogeneous, and include both Lorentz and Doppler components on an equal footingthrough their convolution, namely a Voigt profile.

This relatively simple model works well for long and multiple pulse lasers, where the gain lifetime,τ &100 ps, is much greater than optical transit time along the laser axis,l/c. However the development of‘transient’ lasers schemes, using a very short heating pulse, gives rise to the possibility that the gain willnot be coincident with the optical pulse over the entire duration of its transit. Travelling wave pumpingallows the pump and gain to be synchronised along the axis. In this case the variation of gain duringbeam generation plays an important role. The key result from the earlier study [2] is that the irradiance ofa unidirectional beam exiting the laser given by:

(1− I0/Is)X + I0/Isα(X) = x (1)

whereI0 = E0/G0f(0) is the spontaneous emission irradiance,x = G0l is the small signal gain–length

product andX =∫ l

0 G(0) dz is the line-centre gain–length product. The line irradiance is given by theamplification factorI = I0α(X), whose accurate approximation may be tabulated for arbitrary line profiles.The line-centre gain is reduced by saturation, and is given byG(0) = G0/(1 + I/Is), in terms of thefrequency averaged irradianceI =

∫I(ν)f(ν) dν/f(0) = I0β(X), whereβ(X) = d(α[X ])/dX−1. Thus

if G0, I0 andIs are known, equation (1) provides an implicit equation for the parameterX(x), hence thebeam irradiance.

When the plasma is varying, we may consider that equation (1) still applies along the optical pathprovided we make a set of further assumptions which are generally obeyed:• the population distribution at each time is quasi-steady, determined by the rates and the instantaneous

irradiance, i.e. the lasing pulse duration is long compared to the recovery time;• the saturation irradiance is constant; since the lower state decay rate is dominant, the recovery time is

principally determined by the laser transition spontaneous decay time;• the spontaneous emission irradiance is constant; since both the spontaneous emission rate and the gain

depend linearly on the upper state population,I0 is constant if the lower state population is small;• the line profile is independent of time; since the heating pulse and the gain lifetime are both very short,

the ambient plasma conditions do not change markedly during lasing. The dominant Doppler line width(ion temperature dependent) is nearly constant. The Lorentz width is principally determined by the(slowly varying) lower state spontaneous decay rate; inelastic electron Stark broadening decreases asthe electrons cool and the gain diminishes.To calculate the overall output signal, we consider a series of elemental pulses travelling along the

laser. Each is amplified appropriately to its individual value ofx, the small signal gain–length product, inaccordance with equation (1). The pumping wave propagates along the lasant with speedv, and the lasingpulse with speedc. Since the pumping is uniform, the small-signal gain seen by the element depends onlyon the relative delay between the waves. We define the element timeT as the delay between the elementand the onset of the pulse leaving the laser. If the gain wave travels faster/slower than the lasing, the onsetof the pulse originates at the exit/entry to the laser, i.e. time zero occurs when the faster wave reaches the

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end of the laser rod. Defining the distancez from the exit/entry point, the gainpulse delay for the elementT at pointz is (T − γz), whereγ = |1/c− 1/v|. If the gain has a local profileG0ψ(t/τ), wheret is thedelay after gain onset, andτ is the lifetime of the gain, the gain–length product for the ray exiting at timeT , in either case, is given by:

x(T ) =G0τ/γ[Ψ(T/τ)−

Ψ[(T − γl)/τ ] if l < T/τ0 otherwise

](2)

whereΨ(s) =∫ s

0 ψ(s) ds. Solving equation (1) we obtainX(T ) and hence the irradiance of the elementT , namelyI(T ) = I0α[X(T )]. Thus we obtain the temporal profile of the output signal. If the gain pulsedecays exponentiallyψ(s) = exp(−s), the above result is simplified:

x(T ) =G0τ/γ[exp

min[(γl− T )/τ,0

]− exp−T/τ

](3)

As γ → 0, i.e. the matched case, we obtain the obvious limitx(T )→ G0lψ(T/τ). Equivalent resultswere obtained by Strati and Tallents [6], using direct integration of the equation of radiative transfer by themethod of characteristics.

Figure 1. Output energyflux E normalised toIsτ

as a function of thesmall-signal gain–lengthproduct,x, for Gaussian(Doppler) and Lorentz

line profiles for differingvalues of the ratio of

saturation tospontaneous irradianceIs/I0 = 10−10 , 10−8,

10−6, 10−4, 10−2.

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3. Analytic results

Equations (2) and (3) are relatively easy to solve by direct numerical integration adapting the methodsdescribed earlier. The results are particularly interesting for the case of a matched travelling wave. Ifthere is no travelling wave component, the system is bidirectional. In the unlikely event that saturationwere achieved, time delayed bidirectional integration would be required to evaluate the model. Using aunidirectional model, Strati and Tallents [6] have surveyed the transition from uniform pumping (v c) tomatched wave (v = c) to stationary (v c), in particular studying the effect of mismatch on the energy andpulse length of the output beam under experimental conditions. In this work, we examine in more detail theproperties of the output pulse for matched travelling wave operation. As an example, we consider the caseof an exponential gain profile of1/e width, τ .

Figure 1 shows the variation of output energy flux normalised against the product of the saturationirradiance and1/e temporal width (Isτ ) for Lorentz and Doppler line profiles as a function of the peak smallsignal gain–length productx(0) for a range of values of the ratio of the spontaneous emission irradiance tosaturation irradiance (I0/Is). The pattern of behaviour shown mirrors that in the long pulse case, althoughthe values differ due to pulse shortening as discussed below.

Figure 2shows the variation of the output pulse1/e width as a function of the peak small-signal gain–length productG0l for a range of saturation-spontaneous irradiance ratios for Lorentz and Gaussian line

Figure 2. Pulse lengthTnormalised to the gain

lifetime τ as a function ofthe small-signal

gain–length product,x, forGaussian (Doppler) andLorentz line profiles fordiffering values of theratio of saturation to

spontaneous irradianceIs/I0 = 10−10 , 10−8,10−6 , 10−4, 10−2. For

comparison, theunsaturated approximation

is also shown.

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profiles. AsG0l initially increases the pulse shape is strongly modified and its length decreases rapidlydue to the exponentiating gain. Once saturation is achieved, the pulse length increases, due to the slowerincrease in the overall gain–length productX(t) as the early parts of the pulse are saturated. In the limit asG0l→∞, the pulseI→ IslG0ψ(T ), and the pulse length tends to the gain duration. This effect was notedearlier by Lin et al. [5].

From equation (3) it is easily seen that the maximum gain–length product, and therefore irradiance,occurs whenT = T0 = γl, and has the valuex0 = (G0τ/γ)[1− exp(γl/τ)]. Since for largeX , α(X) ∼exp(X)/

√(X), we may identify the1/e points in the absence of saturation, whereX1,2 ≈ x0 − 1, at times

T2 ≈ γl− τ ln1 + (γ/G0τ) exp(γl/τ) andT1 ≈ γl − τ ln1− γ/[G0τ(1 − exp(−γl/τ)], and hencethe1/e pulse width is approximatelyγ/[G0(1− exp(−γl/τ)]. In the matched case asγ→ 0 the1/e widthbecomesτ/(G0l).

This decrease in pulse width has a marked effect on the total output energy. The peak power will bedetermined by the peak gainG0, namelyI0α[X(T0)]. However in contrast to long pulse lasers the outputenergy is not simply the product of the peak power and gain duration.Figure 3shows the reduction in thetotal energy as a fraction of thepeak power × gain duration . It is evident that simple energy calculationsbased on the peak output and gain duration can lead to serious error.

Figure 3. Output energyflux E normalised to

Ipeakτ as a function of thesmall-signal gain–lengthproduct,x, for Gaussian(Doppler) and Lorentz

line profiles for differingvalues of the ratio of

saturation to spontaneousirradianceIs/I0 = 10−10,10−8, 10−6, 10−4, 10−2

and100 .

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It is well known that laser lines are gain narrowed due to the enhanced gain on line centre. The actualline profile recorded experimentally in a time integrated experiment is easily obtained by an extension ofthe result given earlier. The line centre ratio form of the profile is given by:

Φ(ν) =

∫ ∞0

exp[X(T )φ(ν)

]− 1

dT

/∫ ∞0

exp[X(T )

]− 1

dT (4)

Figure 4 shows the variation of the half-half line width for both Lorentz and Gaussian line profiles asa fraction of the emission half-half-width. The results show similar behaviour to the long pulse case, withstrong line narrowing whilst unsaturated, but only weak once saturation sets in.

For large peak gain–length products, we may simply estimate the line width by noting that the outputsignal is strongly peaked and that the1/e temporal width of the irradiance occurs within the rangeX0 − 1to X0. As shown in our earlier work, the line profile will be approximately Gaussian, and since the linewidth scales asX−1/2, the observed line1/e half-width will be approximately(X0φ2)−1/2, whereφ2 isthe coefficient the squared term in the expansion of the line-centre line profile function. We note that the

Figure 4. Line widthνnormalised to the emissionline widthν0 as a function

of the small-signalgain–length product,x, for(a) Gaussian (Doppler) and(b) Lorentz line profiles atdiffering values of the ratio

of saturation tospontaneous emission

irradianceIs/I0 = 10−10 ,10−8, 10−6, 10−4 , 10−2

and100. Also shown arethe values from the

unsaturated approximation.

(a)

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(b)

Figure 4.— continued.

line is narrowest at the peak of the pulse and broadens thereafter. This scaling is demonstrated infigure 4whilst the system is unsaturated.

4. Comparison with simulation

The analytical model can be compared with more detailed compuational modelling. To match theanalytical model to simulation we use two codes, EHYBRID [9,10] and RAYTRACE [11] developed atYork. The 1.5D fluid and atomic code EHYBRID models the interaction of the driving laser and the target,providing physical characteristics of the plasma in time and space. The 3D geometric RAYTRACE code,

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Figure 5. Electron temperature, ionisation and peak gain of the 196 Å line at the location of peak gain as functions oftime after main pulse onset, generated by EHYBRID modelling the experimental parameters given in the text.

which includes saturation effects, is used as a post-processor to describe the propagation of the X-ray laser(XRL) beam through the plasma. This can be used to provide information on the XRL output, includingthe power, at each time for different target lengths. The codes require data on the driving laser beam andmany other parameters, so we simulate a typical travelling wave experimental setup, used by MacPhee etal. [8] on Ne-like Germanium lasing at 196 Å. Using light of 1.06 µm wavelength radiation and a focalline width of 100 µm, a 3 ps pulse of irradiance1.5 · 1015 W·cm−2 was incident on a germanium plasmaformed by a pair of 280 ps pulses of5 · 1011 and5 · 1012 W·cm−2 irradiance separated by 2 ns. A smallsignal gain (SSG) of> 40 cm−1, and a saturation irradiance of2 · 1010 W·cm−2 were estimated from theexperiments [8].

The application of the analytical model requires knowledge of certain physical parameters of both theplasma and the X-ray laser (XRL) beam. We can use the output from the two codes to provide this data. Thegain time history is shown infigure 5and indicates that the gain has a slow build-up and a long life-time.The slow onset is seen to be due to the low ionisation (< 20) prior to the main pulse, and the tail resultingfrom the decrease in the electron temperature at optimum ionisation (≈ 22). The model generates valuesfor Is, I0 and for the area of the XRL beam to find the output power. These are taken at a time coincidingwith the peak output XRL power for a target length of about 4 mm, just before saturation. In this case, weobtain values ofIs/I0 ∼ 10−7, 1/e width ∼ 35 ps and an area of∼ 5 · 10−6 cm2. We can compare thegrowth curves produced by the model with the more detailed computational modelling.

Figure 6 shows the growth curves produced by detailed simulation with EHYBRID and RAYTRACE,and by the analytical model for a Doppler broadened line. As discussed eariler, the total output power mustbe obtained from the codes by integrating the output power over time in contrast with experiments usinglonger pulse lasers where a good approximation was to multiply the peak output power by the gain1/ewidth in the plasma.

A feature of the EHYBRID growth curve is the ‘tail’ on the curve at short target lengths∼ 2 mm.RAYTRACE predicts that at short target lengths, regions of comparatively low gain but large spontaneousemission play an important role. As a result, signals generated at late times by recombination over a wide

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Figure 6. Total outputenergy of the 196 Å line

under the standardconditions plotted as a

function of targetlength, from detailed

computationalsimulation (full line)

and from the analyticalmodel. The two

analytical curves relateto an exponential gain

profile (dotted line) andthe gain profile

produced by EHYBRID(dot–dash line).

cross section and radiated over a large solid angle contribute a major fraction of the output. The integrateddivergence of the output XRL is consequently high, and at this late stage the plasma is behaving as a‘floodlight’. The normal collisional output, produced over times of the order of the gain duration, is stillwell collimated, and the magnitude is in agreement with the analytic model, which assumes a constant solidangle for all target lengths.

The output power from RAYTRACE shows the characteristic pulse shortening discussed earlier. Thepulse width drops to∼ 0.3 of the gain duration in the plasma for a target length of 5mm, and tends to thegain duration for largegl ( figure 7). Once saturation sets in at the peak of the gain, the pulse length startsto increase as rays still unsaturated make a proportionately greater contribution.

At longer target lengths, the analytical curve for energy output (figure 6) falls significantly below theEHYBRID curve. Above saturation the rays with smaller gain lengths contribute strongly withgl valuesapproaching the saturation limit. In consequence additional parts of the plasma start to make significantcontributions to the total output signal. RAYTRACE predicts that the area of the output beam in the nearfield (i.e. on the face of the plasma) differs by a factor of∼ 6 between a 4 mm target and a 10 mm target.Figure 7 shows the variation of the area of the output XRL for different target lengths. In addition theanalytical model assumes that other physical parameters, such asIs/I0, will not change significantly intime and space. Modelling with RAYTRACE suggests that this is not the case, and consequently the modelwill begin to fail at these longer target lengths. However, general agreement between the simulation and theanalytical model is good around saturation.

Figure 8shows the experimental growth curve produced by Macphee et al. [8], as well as the simulationand analytical data. Clearly, simulation differs significantly with experiment at small target lengths, but is ingood agreement around and above saturation. As discussed above the predicted beam has a high divergenceat low target lengths; on the other hand the experimental setup typically receives radiation over a divergenceof about15 mrad radially and10 mrad transversely. A large amount of the radiation will therefore be missedby the detector. The analytical model will approximate the expermental more closely at small target lengthsas a constant beam solid angle is assumed. At longer target lengths the calculated output XRL beam hasa much lower divergence, consistent with experiment, and the problem is obviated.

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Figure 7. Output pulse duration of the 196 Å line as a function of target length, as predicted byEHYBRID/RAYTRACE for the standard case (full line) and by the model (dotted). Also shown is the area of the

output XRL beam as calculated by EHYBRID/RAYTRACE (dashed).

Figure 8. Total output energy of the 196 Å line as a function of target length calculated by modelling (dotted line) andexperiment (full line). The experimental points relate to data produced by MacPhee et al. [8].

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5. Conclusions

We have shown that the simple model of ASE laser generation can be extended to treat the situationwhere the gain has a finite lifetime, which is short compared to the transit time of a wave along the laser. Theoriginal model has been successfully used to treat the behaviour of collisionally pumped X-ray lasers, andmore recently applied to short pump pulse ‘transient’ systems. In this work we have shown that the modelis capable of giving satisfactory agreement with simulation and with experiment, provided suitable valuesof the small signal gain, saturation and spontaneous intensities, and gain lifetime are used. Discrepancieshighlight the differences and inadequacies in the models. It is shown that significant laser pulse shorteningwill occur just before saturation, but that once the beam has saturated the pulse will start to lengthen again.As in the long-pulse case line narrowing occurs with values similar in magnitude.

The model in its present quasi-analytic form is applicable only to unidirectional laser propagation.Provided the gain duration is much shorter than the wave transit time, this is a reasonable requirement.However, if this condition is not obeyed, and if the travelling wave is poorly matched, the backward wavemay start to play a significant role and reduce the output. We will investigate this behaviour in a later paper,exploring the transiton from short pulse unidirectional to bidirection amplification in ASE lasers.

Acknowledgements.We wish to acknowledge useful discussions with Dr. G.J. Tallents and F. Strati on theextension of the model to include time dependent gain, and the application to experiment. This work is supportedby by EPSRC and AWE as part of the UK X-ray laser programme.

References

[1] Casperson L.W., Threshold characteristics of mirrorless lasers, J. Appl. Phys. 48 (1977) 256.[2] Pert G.J., Output characteristics of amplified-spontaneous emission lasers, J. Opt. Soc. Am. B 11 (1994) 1425.[3] Koch J.A., MacGowan B.J., Da Siva L.B., Matthews D.L., Underwood J.H., Batson P.J., Lee R.W., London R.A.,

Mrowka S., Experimental and theoretical investigation of neonlike selenium X-ray laser spectral line widths andtheir variation with amplification, Phys. Rev. A 50 (1994) 1877.

[4] Pert G.J., Collisional cross relaxation effects in amplified-spontaneous emission lasers, Phys. Rev. A 50 (1994)4412.

[5] Lin J.Y., Tallents G.J., MacPhee A.G., Demir A., Lewis C.L.S., O’Rourke R.M.N., Pert G.J., Ros D., Zeitoun P.,Travelling wave chirped pulse amplified transient pumping for collisional excitation lasers, Opt. Commun. 166(1999) 211–218.

[6] Strati F., Travelling wave effects on short pulse X-ray lasers, in: Central Laser Facility RAL Annual Report, RAL-TR-1999-062, 1998–99, pp. 73–76.

[7] Pert G.J., Saturation and line profile modification in ASE lasers, in: Central Laser Facility RAL Annual Report,RAL-94-042, 1994, pp. 93–94.

[8] Macphee A.G., Lewis C.L.C., Keenan R., O’Rourke R.M.N., Tallents G.J., Pestehe S.J., Strati F., Pert G.J.,McCabe S.P., Simms P., Neely D., Allot R., Saturated X-ray lasers at 196 Å and 73 Å pumped with travellingwave excitation, in preparation.

[9] Pert G.J., The hybrid model and its application for studying free expansion, J. Fluid Mech. 141 (1983) 401.[10] Holden P.B., Healy S.B., Lightbody M.T.M., Pert G.J., Plowes J.A., Kingston A.E., Robertson E., Lewis C.L.S.,

Neely D., A computational investigation of the neonlike germanium collisionally pumped laser, J. Phys. B 27(1994) 341.

[11] Plowes J.A., Pert G.J., Healy S.B., Toft D.T., Beam modelling for X-ray lasers, Opt. Quantum Electron. 28 (1996)219.

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modelling of short pulse x-ray laser experiments

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