the gain distribution of the transient collisional excited x-ray lasers

Journal of Quantitative Spectroscopy &Radiative Transfer 71 (2001) 665–674

www.elsevier.com/locate/jqsrt

The gain distribution of the transient collisional excitedX-ray lasers

Akira Sasaki ∗, Takayuki Utsumi, Kengo Moribayashi, Masataka Kado,Momoko Tanaka, Noboru Hasegawa, Tetsuya Kawachi, Hiroyuki Daido

Advanced Photon Research Center, Kansai Research Establishment, Japan Atomic Energy Research Institute25-1 Miiminami-cho, Neyagawa-shi, Osaka 572-0019 Japan

Abstract

An atomic kinetics model of an electron collisional excited X-ray laser is developed, and the spatialand temporal evolution of the soft X-ray gain is investigated. The calculation of the gain agrees withexperiment for the transient collisional excited (TCE) Ni-like Ag laser (�=139 ;A) pumped by two 100 pslaser pulses. The mechanism of producing gain in the ionizing plasma is discussed. The calculation isapplied to the optimization of the gain. It is found that higher gain can be obtained by pumping a thinfoil target with 2 ps laser pulses. The saturation intensity of the X-ray lasers is also investigated throughthe analysis of the detailed atomic processes of the upper laser level. ? 2001 Elsevier Science Ltd. Allrights reserved.

Keywords: X-ray lasers; Laser produced plasma; Multiple charged ion; Collisional radiative model

1. Introduction

X-ray lasers have been studied theoretically and experimentally in various laboratories toachieve lasing at wavelengths less than 100 ;A. The soft X-ray gain is produced through complexionization and excitation processes of the multiply charged ions in the hot dense plasma [1,2].In the case of high-Z elements, the atomic kinetics is very complex as the multiple chargedions consist of a large array of levels. The complexity in the ion kinetic models are particularlydiBcult when the plasma is irradiated by an intense short laser pulse on the atomic kineticcouples with the non-equilibrium state of the electron distribution function, while radiativetransfer is important. Thus, a sophisticated atomic kinetics model is critical to the study of

∗ Corresponding author. Tel.: +81-720-31-0709; fax: +81-7210-31-0596.E-mail address: [email protected] (A. Sasaki).

0022-4073/01/$ - see front matter ? 2001 Elsevier Science Ltd. All rights reserved.PII: S 0022-4073(01)00107-8

666 A. Sasaki et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 71 (2001) 665–674

X-ray lasers. In addition, atomic processes play an important role in the generation of incoherentX-ray and energetic particle sources when short intense laser pulses interact with a plasma.

We have developed a collisional radiative model for an X-ray laser using Ni-like ion as thegain medium [3]. We have developed an atomic model, which includes the detailed populationdistribution in the Jne structure levels to calculate the soft X-ray gain for the transition betweenlevels in the Ni-like ion. On the other hand, we have included a wide range of charge stateswithin the M- and N-shell ions by considering averaged level structure. Prior to irradiationby the laser pulse of the medium of interest the matter is the solid state. The solid materialis ablated by the Jrst pulse to produce the plasma that is heated, after a delay varying from100 ps to 1 ns, to high temperature by a second laser pulse. The temperature and density of theplasma changes drastically at the time of the laser irradiation. Using the present model, we cancalculate the rapid change of the population, ion fraction, and charge state, z∗, in the plasma.We have investigated the gain of TCE Ni-like Ag laser using this model. We have simulatedthe atomic kinetics in the experimental conditions, by using the temperature and density of theplasma simulated by HYADES, one dimensional hydrodynamics code [4].

2. The simulation model

Fig. 1 shows the atomic model of the Ni-like Xe. The soft X-ray gain occurs between the3d94d(3=2; 3=2) J=0 and 3d94p(5=2; 3=2) J=1 levels. The upper laser level is populated throughthe strong electron collisional excitation, while the lower level decays rapidly to the ground state.In order to determine the detailed population distribution to the Jne structure levels, we usethe HULLAC code [5] to calculate the energy levels belonging to the 3d94l conJgurations ofthe Ni-like ion, along with the rate coeBcients of spontaneous emission, collisional excitation,ionization, and radiative recombination.

Fig. 1. Schematic level diagram of the Ni-like Xe.

A. Sasaki et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 71 (2001) 665–674 667

Fig. 2. The class objects represent the atomic model of the X-ray lasers. Ni-like ion object contains class ob-jects for conJgurations determined by diNerent averaging methods. Fine structure levels are considered for 3d94lconJgurations, 3d95l are averaged, and 3d96l are averaged over conJguration to form a super conJguration (SC).

For the higher excited conJgurations (3d95l; 3d96l), inner shell conJgurations (3s4l; 3p54l),and double excited conJgurations (3d84l4l′), the energy levels and rates are calculated once bythe HULLAC code, and then the rates are averaged for each conJguration. If we included theJne structure levels, the size of the atomic model would become prohibitively large (¿ 104).However, we found the detailed population distribution into these levels does not make signif-icant diNerence in population inversion or gain. The principle of the averaging method and itsveriJcation is found in [6] and references therein.

Some of the levels in the doubly-excited conJgurations, 3d94l4l’ of the Cu-like ion and3d84l4l’ of the Ni-like ion are above the Jrst ionization potential. For these levels, the conJg-uration averaged dielectronic recombination rates from the ground state of the Co-like (Ni-like)ion to the excited states of Ni-like (Cu-like) ion are considered. For the Rydberg conJgurationsand most of the states in ions other than Ni-like, the energy levels and rates are calculatedusing the empirical formulae based on the screened hydrogenic approximation [7].

DiBculty in calculating the atomic kinetics may arise from the fact that we require diNerentmodels and averaging methods in a single atomic model. This complex atomic model canbe handled more easily using modern programming methods. For instance a conJguration hasvarious kinds of attributes, such as the energy, statistical weight, and population. In conventionalkinetic codes, each state is numbered and the attributes are stored in arrays separately. Toimprove on this situation we Jrst introduced structures to combine these level quantities so thatthese can be manipulated as a single variable. We also included the number of electrons in eachsubshell for a conJguration into the structure, which is useful for calculating energy levels andtransition probabilities with an in-line atomic model.

Use of object oriented programming (OOP) method makes it possible to reproduce a realisticatomic level structure within the code. An object consists of the structured data accompanied


by the programs. For example, an atom is deJned as an object class, which contains ion objectscorresponding to each charge states, as illustrated in Fig. 2. The ion object contains within itobjects for the conJgurations, which further contains objects for the many Jne structure levels.The internal data structure is not necessarily the same for each object. For example, the numberof levels may be diNerent for each conJguration. One ion may have conJgurations with bothdetailed and averaged level structure at a same time.

For each conJguration one can calculate its attribute irrespective of the type of averagingmethod is used for the internal structure. For instance, to calculate the energy of the conJgu-ration, the program Jrst examines the internal structure of the conJguration, if detailed levelsare deJned inside the conJguration the program accesses the level objects for the appropri-ate information and takes average over them. For the conJguration object that are part of asuper conJguration (SC), the program accesses conJgurations’ data and recursively obtain theconJguration averaged energy.

3. Results and discussion

3.1. Calculation of the gain for the two 100 ps laser pumped Ni-like Ag laser

First, we have carried out hydrodynamic simulations using HYADES for a silver slab targetirradiated by two 100 ps laser pulses with a Gaussian temporal pulse proJles that correspond toexperiments performed by Sebban et al. [8]. In these experiments the laser wavelength was 1 �mwith intensities of 4× 1012 and 8× 1013 W=cm2 for the prepulse and main pulse, respectively,and 3 ns interval between pulses.

As shown in Fig. 3, the low intensity prepulse ablates the surface of the solid target toproduce a preplasma. During the interval between the two pulses, the electron temperaturedrops due to expansion. At the time of arrival of the second pulse, the electron temperatureand scale length of the plasma are approximately 20 eV and 50 �m, respectively. The energy of

Fig. 3. ProJle of the electron temperature (a) and electron density (b) calculated by the 1D hydrodynamics codefor a plasma irradiated by two 100 ps laser pulses. 0ns corresponds to the peak of the main pulse.


Fig. 4. ProJle of the soft X-ray gain (a) and averaged charge (b) for a silver plasma irradiated by two 100 ps laserpulses. 0ns corresponds to the peak of the main pulse.

the main pulse is absorbed in the underdense region, 50–100 �m from the surface of the target,where the electron density is approximately 1020 cm−3. As the energy of the laser is deposited,the temperature rises quickly to 1 keV, causing rapid ionization to form a localized increasein the electron density proJle. At the time of the peak of the main pulse, the energy of thelaser is mainly absorbed near the critical density surface. A strong pressure front occurs at thecritical density surface and the pressure pushes the electron density increase toward the targetsurface. The laser energy is transferred into the plasma by electron heat conduction. The heatfront, which conforms to the ionization front, is formed in front of the absorption region. Asthe conductivity scales as T 5=2

e , the heat front propagates rather slowly into the low temperaturepreplasma. It reaches the target surface ∼ 200 ps after the peak of the laser pulse. After thepeak of the heating pulse, the temperature gradually decreases to 200 eV in 1 ns.

Second, we have carried out the atomic kinetics calculation by postprocessing the temporaland spatial evolution of each Lagrangian cell used in the one dimensional hydrodynamics cal-culation. Fig. 4(a) shows the temporal and spatial evolution of the soft X-ray gain. At Jrst,gain appears 20 ps after the peak of the laser pulse at around 40 �m from the target surface,which increases to 100 cm−1 rapidly in the subsequent 20 ps. Then the proJle splits havingtwo peaks, one becomes a narrow high gain peak that moves toward the target surface and theother, which lasts longer time (¿ 100 ps), is a broad peak that moves away from the target.

This feature indicates that the gain is produced in the ionizing plasma (see [9,10]). Thedistribution of the z∗ is shown in Fig. 4(b). Before the arrival of the main pulse, the plasmais in a low charge state, z∗¡ 10, because of the low temperature. Ionization occurs as thetemperature becomes high due to the absorption of the laser. Since the rate of the electroncollisional ionization is proportional to the density, it is fastest at the critical density surface.As shown in Fig. 4(a), the initial gain occurs in the ionizing plasma close to the heat frontwhen z∗ has exceeded 15. In the inner region of the plasma where the density increases rapidlytoward the target surface, the gain occurs immediately at the point where the heat front occursbecause the ionization rate is very high. This corresponds to the narrow peak of the gain inthe proJle. In contrast, in the outer region of the plasma, even though the temperature is high,the ionization is so slow due to the low density that gain occurs at later time. Gain for both


Fig. 5. ProJle of the electron temperature (a) and electron density (b) calculated for a plasma produced from 0:1 �mthick foil irradiated by two 2 ps; 1015 W=cm2 laser pulses, with a temporal separation of 100 ps.

peaks are almost proportional to the local ion density. In the region between the two peaks, theplasma is overionized and the gain ceases as the abundance of the Ni-like ion decreases.

The experiment shows the temporally and spatially integrated gain of 19 cm−1 averaged overthe plasma length of 23 mm. The steep peak shown in the calculation may not be observedin the presence of the refraction of the X-ray laser beam. Thus, the measured gain may rathercorrespond to that in the plateau region of 30 �m at around 60 ps. The experiment also showedemission of 4f–3d transitions from various M-shell ions beyond the Ni-like state that indicatesthe over ionization of the plasma yielding z∗ up to 25, as seen in Fig. 4(b).

3.2. Calculation of the gain from a short pulse laser irradiated foil target

In order to determine the condition to pump the X-ray laser more eBciently, we have per-formed hydrodynamics and atomic kinetic calculations for plasmas produced from a thin foiltarget irradiated by short laser pulses. We showed that a short laser pulse with an intensity of1016-17 W=cm2 and pulse duration of 1 ps heats a thin foil target eBciently [11]. A thin plasmalayer, with a scale length comparable to the wavelength of the laser, is formed on the surfaceof the foil during the irradiation. Even with the rapid radiative cooling, the electron temperaturecan become higher than a few keV [12]. Furthermore, eBcient resonance absorption occursin the plasma layer to produce hot electrons. The hot electrons ionize the target while in thesolid density, and also drive expansion of the foil. In the absence of the loss of energy due tothe heat conduction into the solid target, the absorbed energy per ion may exceed 10 keV=ion,assuming a modest absorption eBciency of 10%. We have showed that high Ni-like fractioncan be maintained by arrival of the second laser pulse. The second laser pulse does not haveto ionize the plasma to produce the Ni-like ion [13], the X-ray gain is then expected with asmaller pump energy.

Fig. 5 shows the electron temperature and density of the plasma produced by irradiating a100 nm thick Ag foil with two 1015 W=cm2 2 ps laser pulses with an interval of 100 ps between


Fig. 6. Spatial and temporal evolution of the gain of Ni-like Ag laser using a thin foil target irradiated by two 2 pslaser pulses; pulse interval =100 ps (a), 200 ps (b), and 300 ps (c).

pulses. By the time of arrival of the second laser pulse, the thickness of the plasma becomes50 �m, with a central electron density of 3 × 1021 cm−3. The second laser pulse is absorbedat the front side of the plasma forming a heat front, which propagates toward the center ofthe plasma. A peak temperature of 2 keV is achieved, which is twice the temperature producedfrom the solid target described above.

The soft X-ray gain calculated for these conditions is shown in Fig. 6(a). As a highertemperature is obtained in the dense plasma, the peak gain is ¿ 500 cm−1. However, the width


of gain region is very narrow and propagates toward the center of the foil in a few tens of ps.The X-ray laser light will not propagate through this gain region for a distance long enough toreach the saturation.

Figs. 6(b) and (c) show the calculated gain for diNerent pulse intervals. As the pulse intervalincreases, the density of the plasma decreases so that the width of the heated region becomeslarger. Although the peak gain is smaller than in the case of the short time interval the situationis more favorable for ampliJcation of the X-ray laser. It is found that for a pulse interval of300 ps that we obtain higher gain in a similar region to a solid target irradiated by 100 ps laserpulses, even the absorbed laser energy is 1=5 in the present scheme.

3.3. The saturation intensity of Ni-like collisional excited X-ray lasers

The saturation intensity of the X-ray laser is important because it determines the availableenergy in the laser beam. The saturation intensity Isat is determined from the total quenchingrate of the upper laser level. If we assume a Doppler line proJle for the lasing transition, thesaturation intensity is,

Isat =h��eN

; (1)

where � is the frequency of the X-ray laser, � is the stimulated emission cross-section deJnedas,

�=�e2

mc0�fosc; (2)

where fosc is the absorption oscillator strength. � is the Doppler line broadening functiondeJned as,

�=c0�

√Mi

kTi; (3)

which is determined from the ion temperature, Ti, and ion mass Mi.Calculation of the saturation intensity requires the eNective lifetime �eN of the upper laser

level that is calculated from the rate of all decay processes. Fig. 7 shows calculated saturationintensity as a function of the electron density of the plasma for Ag, Xe, Nd, and Dy. It isfound that the major decay processes of the upper laser level are de-excitation to the lower laserlevel through spontaneous emission, electron collisional de-excitaton to the ground state, andcollisional excitation to 3d’ 4f and other higher conJgurations. It is found that at an electrondensity below 1020 cm−3 the spontaneous emission dominates the decay process so that thesaturation intensity approaches a constant value. At density above 1021 cm−3, the collisionalprocesses dominate over radiative processes, so the saturation intensity increases linearly as theelectron density. It is also found that the saturation intensity depends weakly on the electrontemperature.

In the calculation shown in Fig. 7, the ion temperature is assumed to be the same as theelectron temperature. However, in TCE X-ray lasers where the gain occurs immediately after


Fig. 7. Comparison of the saturation intensity of Ni-like Ag, Xe, Nd, and Dy lasers, for Te = Ti = 1 keV.

heating by the short laser pulse, the ion temperature is likely to remain low even for highelectron temperature. The saturation intensity for an X-ray laser with an electron density of5× 1020 cm−3, electron temperature of 700 eV, and ion temperature of 200 eV is calculated tobe 2:1×1010 W=cm2, which is 30% greater than those reported by Lin [14] from his calculationsand experiments. This result implies the maximum intensity of the X-ray laser could become ashigh as 1011 W=cm2 and assuming a gain region of 100 �m2 and the pulse duration of 10 ps,the estimated maximum energy of the beam is 100 �J.

4. Conclusion

The atomic kinetics model of the X-ray lasers is developed and the gain of Ni-like Ag TCEX-ray lasers using solid and foil targets is investigated. The present calculation qualitativelyreproduced the experimental gain. The calculations also showed the advantage of using thinfoil targets compared to conventional solid targets as one can obtain higher gain with a smallerpump energy. Moreover, the detailed analysis of the atomic kinetics of the upper laser level isuseful for estimating the saturation intensity of the X-ray lasers.

For the future the atomic kinetics model still requires improvement to make predictionsof the X-ray lasing reliable enough to conJdently optimize the pumping conditions. For thispurpose the calculation should be compared with experimentally determined time-dependentgain and other X-ray spectral features such as from 3d–4f resonance lines from M-shell ions.These comparisons will provide detailed information on the population distribution in theNi-like excited states as well as the mechanism of successive ionization of the multiple-chargedions.


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the gain distribution of the transient collisional excited x-ray lasers

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