- Email: [email protected]
Modeling of ultra-short-pulse laser pumped X-ray lasers
Pergamon
J. Quant. Spectrosc. Rodiat. Transfer Vol. 58, Nos 46, pp 879-885, 1997 0 1997 Elsewer Science Ltd. All nghts reserved
Printed in Great Britain
PII: s00224@73(97)oooY44
MODELING
OF ULTRA-SHORT-PULSE X-RAY LASERS
0022~4073/97$19.00 + 0.00
LASER PUMPED
AKIRA SASAKI Advanced Photon Research Center, Kansai Research Establishment, Japan Atomic Energy Research Institute, 25-1 Miiminami-cho, Neyagawa-shi, Osaka, 572 Japan (Received
Abstract-Simple models of ultra-short-pulse have been developed. Conditions to obtain
9 July 1997)
laser pumped short gain in the collisional
wavelength x-ray lasers excited Ni-like Ta laser
have been estimated for a preformed plasma irradiated by intense laser pulses. In this scheme, preproduction of a high density plasma is essential. The density should be optimized for fast collisional ionization and excitation, minimum radiation trapping, and stable propagation of the ultra-short-pulse laser. Use of a plasma confined in a cylinder is proposed. 0 1997 Elsevier Science Ltd. All rights reserved
1. INTRODUCTION
X-ray lasers have been studied for more than 10 years toward shorter wavel!ngth. X-ray lasers in water window wavelength, between absorption edges of C and 0 (23-44 A), are particularly interesting for biological applications. Gain inside water window wavelength has been demonstrated.’ Application of the x-ray laser to interferometric measurements of a laser produced plasma has also been demonstrated.2 However, in the conventional design, water window lasers require kJ of pump energies. For scientific and industrial applications reduction of the pump energy and achievement of laser oscillation with a table-top size facility have primarily importance. The objective of our research is to develop a theoretical model of ultra-short-pulse laser pumpet water window lasers. We are also interested in modeling of soft x-ray lasers in longer (~130 A) wavelengths as a source of x-ray lithography. Recently, soft x-ray gain in ultra-shortpulse laser pumped plasmas has been successfully obtained for both recombination3 and electron collisional excitation4 schemes. In these schemes the lasing level was excited through optical field ionization (OFI). In the recombination Li Lyman-a laser (1 = 135 A)3, a linearly polarized ultra-short-pulse laser was focused into a preformed Li plasma to completely strip Li ions. Population inversion of the Ly-cr line was produced after rapid recombination of a fully stripped Li ion with a cold electron into excited states of a H-like Li ion. Recent development of ultra-short-pulse lasers has enabled us to achieve the maximum intensity of more than lOI W/cm2 at the target, which can produce a fully ionized carbon. The wavelength of C Ly-cr line is 39 A. The key technology in the OFI-recombination scheme is to avoid non-linear interaction such as stimulated Raman scattering (SRS) to keep plasma cool before recombination. In the electron collisional excitation Pd-like Xe laser (A = 418 A),4 a circularly polarized laser was focused into a gas cell filled with 10 Torr of Xe. The OF1 produced energetic electrons through AT1 heating, which excited Pd-like Xe ions to the lasing level. Unfortunately, scaling of this scheme to shorter wavelength using Ni-like ions is not straightforward. The near water window laser is expected for Ni-like Ta(A = 44 A), however, the ionization energy of Ni-like Ta is more than 2 keV, which cannot be accessible by the direct optical field ionization. We are particularly interested in electron collisional excited lasers using Ni-like ions. The Nilike system is robust and operates in short wavelength with a higher efficiency than Ne-like system. If we could produce a dense plasma with a high temperature, which is sufficient to produce Ni-like ions, steady state gain would be achieved. In the case of Ni-like Ta, estimated tempera879
880
Akira
Sasakl
ture was more than 1.5 keV’. Using solid targets irradiated by a long laser pulse ( > 100 ps), more than I kJ of pump energy would be required. To excite plasma efficiently, use of a weak prepulse followed by a short intense main laser pulse has been proposed.’ The result suggested that if we had optimized the pump laser, the total energy would be reduced to k -100 J. Experimentally, in the Ne-like system, a large transient gain was observed in Ne-like Ti and V excited by a combination of a long (xl ns) plasma production laser and a subpicosecond pump laser pulsen6 To improve efficiency, the optimization of targets is also essential. Applications of capillary targets to various schemes have been reported. The capillary target can sustain plasmas with an appropriate density and temperature to produce the population inversion. In the OFI-recombination Li Lyman-a laser, a gain length product of 5.5 was obtained using a LiF capillary target.7 Capillary targets were also used in discharge pumped Ne-like Ar laser.8 Furthermore, we can produce plasmas with an density profile which guide the pump laser and the x-ray laser beam to propagate long distance’ using capillary targets. We have developed a simple model of Ni-like Ta laser to determine temperature and density of the plasma required to produce gain. Preliminary target design has also been made by combining those results with that from HYADES 1-D hydrodynamics code. 2. ATOMIC
KINETICS
Simplified atomic kinetics of N&like Ta is shown in Fig. 1. The upper laser level is 3d$z4d3,2 (J = 0) excited by strong monopole excitation from 3d” ground state. Since the collisional excitation rate to 3d$24d3,2 (J = 0) is one order of magnitude greater than to any other terms in 4d configuration,” and the transition from 3d&4d3,2 (J = 0) to 3dzj24p1,2 (J = 1) has the largest oscillator strength among possible transition to 4p, 3d$z4d3i2 (J = 0) + 3d$24pl,z(J = 1) has the largest gain coefficient. Calculation of the population inversion and gain can be divided into two parts, the calculation of population to excited states of N&like ions, and the calculation of ionization balance and abundance of Ni-like ions. In high-z atoms, both parts have complexities. Excited levels of the Ni-like ion have a complex structure. Based on a J-J coupling scheme, 3d94d configuration consists of 20 spectral terms. Other one-electron excited states from 3s and 3p subshell and Rydberg states (3d9nl) should be considered because the population of each states are coupled each other through collisional and radiative processes. Number of levels in Ni-like ions should be included in the kinetics model is at least 100. A large database of energy levels and rate coefficients is required for kinetics calculations. For calculation of ionization balance, we have to consider populations to multiple excited states of Ni-like and neighboring ions. Not like low-z ions, one electron excited states of the Nilike ion captures an electron to form doubly excited states of Cu-like ion. Zn-like triplly excited states can be formed in a similar manner. Moreover, especially at high densities, inner shell 4d (3/2,3/2) J=O
4p (l/2,3/2)
laser
J=l
transition
4s, J=2
1670eV
4
collision al excitation
15ooev radiative decay
ground state WO
w Fig. 1. Energy level diagram
for the Ni-like Ta electron
collisional
excited laser.
Modeling of ultra-short-pulse laser pumped x-ray lasers
881
excited states of Co-like and Fe-like ions which have holes in n = 3 shell may have significant population, In the present study, firstly, plasma temperature and density required to produce Ni-like Ta was estimated using a simple model based on the average ion model. We used I-dependent model developed by Perrot” to calculate energy levels of a multiple charged Ta ion. Population of each shell was calculated by a collisional radiative model using rate coefficients given by the screened hydrogenic model.” Collisional ionization, three-body recombination, radiative recombination, collisional excitation and deexcitation, and spontaneous emission were included. This method gave averaged charge of multiple charged ions with a reasonable accuracy without detailed knowledge of their structure. Figure 2a shows the averaged charge of Ta in collisional radiative equilibrium(CRE) as a function of temperature. For ion densities of 1019 to 102’ l/cm3, which was considered as a xray laser medium, the temperature required to produce Ni-like ions (Ta4’+) was found to be 1.0-1.5 keV. Figure 2b shows the temporal development of the averaged charge. This condition corresponds to the situation where an ultra-short-pulse laser irradiates a preformed plasma of various ion densities to heat the plasma instantaneously. Ionization time decreased as the inverse of the density. It was found that to complete ionization within the duration of excitation, the ion density must be greater than 102’ l/cm3. Increasing temperature up to 4 keV was not effective to
60t
. . . . . . . .. -..- ................
. . _.-
. . . . . . . .. .........
i--KZ
t
..............
40 .---4..... .I ......_._ ....._. :jlda lTY77-l
lO-l3
lo-'2
time [s]
lo"
_ ....
_._.
.._ .._._....._
lO-‘0
Fig. 2. Averaged charge of Ta ion calculated using the average ion model. (a) Averaged charge as a function of electron temperature for ion densities from lOI* to 102’ l/cm’. (b) Temporal development of averaged charge number for an electron temperature of 1.6 keV and ion densities from lOI to 102’ l/cm’.
882
Akira Sasaki Table 1. Rate coefficients used in the simple kinetics model of the N&like Ta laser. Rate coefficients
Source
Monopole excitation 3dt0-+4s, J = 2 k = 3.7 x lo-l2 cm3/s 3d”-+4p(l/2,3/2) J = 1 k = 3.7 x lO@ cm’/s 3dio+d(3/2,3/2) J = 0 k = 2.1 x IO-” cm3/s Radiative decay of lower state (4~): k,=2.9 x lOI l/s k,=0.6 x lOI l/s Laser transition: 4d(3/2, 3/2) J = 0 -+ 4p(1/2, 312) J = 1 k, = 4.0 x lo-” cm3/s
IO IO IO I3 5.14
reduce ionization time. In short pulse excitation, higher electron densities were required due to finite ionization time than in the case of conventional long pulse excitation. Secondly, we have calculated Ni-like Ta 4d-4p gain for a simplified system. Rate coefficients used in the model are summarized in Table 1. Branching ratio from the upper laser level to the laser transition was assumed to be 0.3. Figure 3a shows the calculated steady-state gain as a function of electron density. The solid line corresponds to a condition for a pure Ta plasma where the electron temperature is 1.5 keV. Ion density was set to l/45 of electron density. In the present analysis, to take into account the population into neighboring ions, such as Cu-like and Co-like ions, the density of Ni-like ground state was set to be 30% of the ion density using estimations by Maxon’ and Goldstein.” The gain coefficient increased as the electron density until
idO
lo*'
lo**
electron density[l/cm3]
0
0.2
0.4
0.6
0.6
1
escape probability Fig. 3. (a) Gain coefficient of the 4d-4p transition of Ni-like Ta as a function of the electron density. (b) Gain as a function of the escape probability of 3d-4p transition for r,= 1.5 keV, ni= lOI l/cm , n,=45n,, and the fraction of Ni-like ground state is 30% of the total ion density.
Modeling of ultra-short-pulse laser pumped x-ray lasers
883
excitation to the lower laser level destroyed the population inversion in a very high density (r1023 l/cm3). We can expect high gain for electron density of more than lo*’ l/cm’. On the other hand, the maximum ion density is limited by radiation trapping of the lower laser level (4~). Importance of the radiation trapping in Ni-like Nd laser was pointed out by Goldstein. I5 We have made an estimation of the escape probability of 3d-4p transition. Assuming Doppler line broadening, the absorption coefficient a l/cm of this line for an ion temperature of 100 eV is, (2.657 x 10-*)4Ns&,,
a =
as a function of the population ening function,
z 1.6 x 10-‘6N3d
of Ni-like ground state N3d, where 4 is the Doppler line broadf#)= c/vd_
3b shows the gain as a function of the escape probability. It was found that in the case of the density of Ni-like ground state, N3d=0.3 x 1019 l/cm3 and a plasma size of 10 urn, estimated escape probability is P,= exp(-al) x 0.6, where the effect of radiation trapping was not so pronounced. However in higher density gain would decrease significantly. Available beam size of the x-ray laser will be limited by the effect of radiation trapping to a few tens of micrometers. Requirements of the high electron density from ionization and production of gain, and limitations of the ion density from radiation trapping should be accommodated. One possibility is the use of composite targets. Using a mixture of Ta with low-z materials, we can increase the electron density beyond lO*l l/cm3 while keeping the ion density to moderate value (~10’~ l/ cm3). In dotted lines in Fig. 3a the electron density was increased while the Ta ion density was fixed at lo’* and 1OL9l/cm3. The graph shows increased gain up to the electron density of %lO** l/cm3, for increased electron collisional excitation. Ionization time of Ta ion will also reduced by increasing the electron density. Figure
3. MODELING
OF TARGET
High density plasmas for the Ni-like Ta laser can be produced using several target designs. Solid targets are used in previous experiments. However, even with a large irradiation energy, the plasma had a steep density gradient so that the gain region was too thin. This fact might make it difficult to obtain a large gain length product because the x-ray laser beam might be refracted away from the gain region. It also made the gain to be sensitive to the spatial non-uniformity of the pump laser irradiation. Moreover, using solid targets, the energy efficiency was low, because a large fraction of the pump energy was consumed for ablation of the solid surface, conduction into the cold material, and hydrodynamics motion. Several methods which have been proposed for interaction experiments between lasers and long scale length, sub-critical density plasmas may be useful. Using high pressure gas filled targets we can produce moderately high density plasmas (n,-,lO*l l/cm3). The plasma confined in a cavity is also useful. Schematics of a cylinder target is illustrated in Fig. 4. Firstly, a plasma production laser irradiates and ablates the inner surface of the cylinder. Consider a cylinder of 0.3 mm diameter and 3 mm long, irradiated by a laser with an energy of 100 J and a pulse duration of 0.5 ns. The intensity of the laser at the solid surface is ~10’~ W/cm*. The ablated Ta cylinder
excited region
&
plasma production laser pump(USP) laser X-ray laser
Fig. 4. Schematic diagram of the cylinder target
884
Akira
Sasakl
plasma converges to the center of the cylinder. Secondly, a short-pulse intense laser pulse is focused to the central region to heat the plasma to ionize Ta ions to Ni-like stage, and to excite them to the upper laser level. Using this two-step procedure, we expect we can pump the Ni-like Ta laser with an energy of the short-pulse laser of 10 J, because we can irradiate the only small gain region before the energy is dissipated by heat conduction. However, the length of plasma will be limited to less than few mm due to longitudinal non-uniformity. In order to obtain large gain length product, much higher gain (~-10 l/cm3) than that obtained in the present experiments should be achieved. The density and temperature of the plasma confined in the cylinder has been estimated using hydrodynamics code HYADES as shown in Fig. 5. HYADES is a one-dimensional, three-temperature, three-geometry Lagrangian hydrodynamics code. l6 In the present calculations for Ta, SESAME table was used to evaluate the equation of state. Intensity, wavelength, and pulse duration of the laser were lOI W/cm2, 1.06 urn, and 0.5 ns, respectively. About 25% of the laser energy was absorbed to produce a plasma. After 5 ns from the beginning of the laser pulse, the plasma converged at the center of the cylinder. After 10 ns, the central region was filled with rather uniform plasma with a density of 0.05 g/cm3, and a temperature of 35 eV, behind the outgoing shock front. The Thomas-Fermi ionization model gave averaged charge of 12. Density and temperature decreased slowly in the next 10 ns until the reflected shock wave reconverged at the center of the cylinder. 4.
DISCUSSION
Simple atomic kinetics and hydrodynamics calculations showed possibility to produce gain by pumping a preformed plasma confined in a cylinder using an ultra-short-pulse laser. Development of a detailed model is projected for more quantitative estimation and optimization of the gain coefficient. Detailed structures of multiple excited states and inner shell excited states of Cu-like, Zn-like and Co-like ions along with rates of dielectronic recombination, inner shell excitation, and autoionization should be considered to calculate the ionization balance. However the kinetics model based on full detailed term accounting (DTA)model will consist of more than lo4 levels as in the case of Ne-like system,” which may be beyond the capacity of present computers. To develop a more practical model which can be coupled with a multidimensional hydrodynamics code, simplification of the model is required. Populations of not all of those levels have effects on the gain. Levels have similar behavior should be grouped together. The methods originally developed for opacity calculations such as the super configuration transition array (STA) method,i8 and other methods which assume statistical distribution of population over detailed levels” may be useful.
0.01
0.005
0.015
r km1 Fig. 5. Profiles of density (solid line) and temperature (dotted line) of the Ta plasma confined in the cylinder calculated by HYADES I-D hydrodynamics code. Profiles were taken at 2, 10 and 20 ns from the beginning of the calculation.
Modeling of ultra-short-pulse laser pumped x-ray lasers
885
Finally, propagation of the ultra-short-pulse laser through dense plasmas should also be examined. In rarefied plasmas, channel formation and stable pulse propagation have been theoretically predicted in the presence of ponderomotive filamentation and relativistic focusing.20 In x-ray laser plasmas consist of high-z ions, rapid increase of the electron density will cause decreased index of refraction at the center of the laser beam. The pump laser beam may be refracted by negative lens effect, and the maximum available intensity in the target may be significantly decreased. The ionization refraction should be controlled by preionizing the plasma to appropriate ionization stage and by tailoring the density profile of the preformed plasma.’ REFERENCES I. MacGowan, B. J., Maxon, S., Hagelstein, P. L., Keane, C. J., London, R. A., Matthews, D. L., Rosen, M. D., Scofield, J. H. and Whelan, D. A., Phys. Rev. Left., 1987, 59, 2157. 2. Cauble, R., DaSilva, L. B., Barbee, T. W. Jr., Celliers, P., Libby, S., Moreno, J. C., Ress, D., Trebes, J., Wan, A. S. and Weber, F., J. Quant. Spectrosc. Radiat. Transf., 1995, 54, 97. 3. Nagata, Y., Midorikawa, K., Kubodera, S., Obara, M., Tashiro, H. and Toyoda, K., Phys. Rev. Left., 1993,71, 3774. 4. Lemoff, B. E., Yin, G. Y., Gordon III, C. L., Barty, C. P. J. and Harris, S. E., Phys. Rev. Left., 1995, 74, 1574. 5. Maxon, S., Estabrook, K. G., Prasad, M. K., Osterheld, A. L., London, R. A. and Eder, D. C., Phys. Rev. Lett., 1993, 70, 2285. 6. P. V. Nickles, V. N. Shlyaptsev, M. Shntirer, M. Kalachnikov, T. Schlegel and W. Sandner, in Proceedings of the 5th International Colloquium on X-ray lasers, p. 84, S. Svanberg and C.-G. Wahlstriirm, eds, Lund, Sweden, IOP conference series 151 (1986). 7. Korobkin, D. V., Nam, C. H. and Suckewer, S., Phys. Rev. Let?., 1996, 77, 5206. 8. Rocca, J. J., Shlyaptsev, V., Tomasel, F. G., Dcotazar, O., Harthon, D. and Chilla, J. L. A., Phys. Rev. Lett., 1994, 73, 2192. 9. Milchberg, H. M., Durfee III, C. G. and Lynch, J., J. Opt. Sot. Am., 1995, B12, 731. 10. Chen, M. H. and Osterheld, A. L., Phys. Rev., 1995, A52, 3790. 11. Perrot, F., Phys. Scripta, 1989, 39, 332. 12. W. A. Lokke and W. H. Grasberger, XSNQ-U, A Non-LTE Emission and Absorption Coefficient Subroutine, Lawrence Livermore National Laboratory Internal Report UCRL-52276, 1997 (unpublished). 13. Hagelstem, P. L., Phys. Rev., 1986, A34, 874. 14. Hagelstein, P. L., Phys. Rev., 1988, A37, 1357. 15. Goldstein, W. H., Oreg, J., Zigler, A., Bar-Shalom, A. and Klapisch, M., Phys. Rev., 1988, A38, 1797. 16 Abdallah, J. Jr., Clark, R. E. H., Peek, J. M. and Fontes, C. J., J. Quant. Spectrosc. Radiut. Transf, 1994, 51, 1. 17. Larsen, J. T. and Lane, S. M., J. Quant. Spectrosc. Radiat. Transf., 1994, 51, 179. 18. Bar-Shalom, A., Oreg, J. and Goldstein, W. H., J. Quant. Spectrosc. Radiut. Transf, 1994, 51, 27. 19. Takabe, H. and Nishikawa, T., J. Quant. Spectrosc. Radiut. Transf, 1994, 51, 379. 20. Borisov, A. B., Borovskiy, A. V., Shiryaev, 0. B., Korobkin, V. V., Prokhorov, A. M., Solem, J. C., Luk, T. S., Boyer, K. and Rhodes, C. K., Phys. Rev., 1992, A45, 5830.
J. Quant. Spectrosc. Rodiat. Transfer Vol. 58, Nos 46, pp 879-885, 1997 0 1997 Elsewer Science Ltd. All nghts reserved
Printed in Great Britain
PII: s00224@73(97)oooY44
MODELING
OF ULTRA-SHORT-PULSE X-RAY LASERS
0022~4073/97$19.00 + 0.00
LASER PUMPED
AKIRA SASAKI Advanced Photon Research Center, Kansai Research Establishment, Japan Atomic Energy Research Institute, 25-1 Miiminami-cho, Neyagawa-shi, Osaka, 572 Japan (Received
Abstract-Simple models of ultra-short-pulse have been developed. Conditions to obtain
9 July 1997)
laser pumped short gain in the collisional
wavelength x-ray lasers excited Ni-like Ta laser
have been estimated for a preformed plasma irradiated by intense laser pulses. In this scheme, preproduction of a high density plasma is essential. The density should be optimized for fast collisional ionization and excitation, minimum radiation trapping, and stable propagation of the ultra-short-pulse laser. Use of a plasma confined in a cylinder is proposed. 0 1997 Elsevier Science Ltd. All rights reserved
1. INTRODUCTION
X-ray lasers have been studied for more than 10 years toward shorter wavel!ngth. X-ray lasers in water window wavelength, between absorption edges of C and 0 (23-44 A), are particularly interesting for biological applications. Gain inside water window wavelength has been demonstrated.’ Application of the x-ray laser to interferometric measurements of a laser produced plasma has also been demonstrated.2 However, in the conventional design, water window lasers require kJ of pump energies. For scientific and industrial applications reduction of the pump energy and achievement of laser oscillation with a table-top size facility have primarily importance. The objective of our research is to develop a theoretical model of ultra-short-pulse laser pumpet water window lasers. We are also interested in modeling of soft x-ray lasers in longer (~130 A) wavelengths as a source of x-ray lithography. Recently, soft x-ray gain in ultra-shortpulse laser pumped plasmas has been successfully obtained for both recombination3 and electron collisional excitation4 schemes. In these schemes the lasing level was excited through optical field ionization (OFI). In the recombination Li Lyman-a laser (1 = 135 A)3, a linearly polarized ultra-short-pulse laser was focused into a preformed Li plasma to completely strip Li ions. Population inversion of the Ly-cr line was produced after rapid recombination of a fully stripped Li ion with a cold electron into excited states of a H-like Li ion. Recent development of ultra-short-pulse lasers has enabled us to achieve the maximum intensity of more than lOI W/cm2 at the target, which can produce a fully ionized carbon. The wavelength of C Ly-cr line is 39 A. The key technology in the OFI-recombination scheme is to avoid non-linear interaction such as stimulated Raman scattering (SRS) to keep plasma cool before recombination. In the electron collisional excitation Pd-like Xe laser (A = 418 A),4 a circularly polarized laser was focused into a gas cell filled with 10 Torr of Xe. The OF1 produced energetic electrons through AT1 heating, which excited Pd-like Xe ions to the lasing level. Unfortunately, scaling of this scheme to shorter wavelength using Ni-like ions is not straightforward. The near water window laser is expected for Ni-like Ta(A = 44 A), however, the ionization energy of Ni-like Ta is more than 2 keV, which cannot be accessible by the direct optical field ionization. We are particularly interested in electron collisional excited lasers using Ni-like ions. The Nilike system is robust and operates in short wavelength with a higher efficiency than Ne-like system. If we could produce a dense plasma with a high temperature, which is sufficient to produce Ni-like ions, steady state gain would be achieved. In the case of Ni-like Ta, estimated tempera879
880
Akira
Sasakl
ture was more than 1.5 keV’. Using solid targets irradiated by a long laser pulse ( > 100 ps), more than I kJ of pump energy would be required. To excite plasma efficiently, use of a weak prepulse followed by a short intense main laser pulse has been proposed.’ The result suggested that if we had optimized the pump laser, the total energy would be reduced to k -100 J. Experimentally, in the Ne-like system, a large transient gain was observed in Ne-like Ti and V excited by a combination of a long (xl ns) plasma production laser and a subpicosecond pump laser pulsen6 To improve efficiency, the optimization of targets is also essential. Applications of capillary targets to various schemes have been reported. The capillary target can sustain plasmas with an appropriate density and temperature to produce the population inversion. In the OFI-recombination Li Lyman-a laser, a gain length product of 5.5 was obtained using a LiF capillary target.7 Capillary targets were also used in discharge pumped Ne-like Ar laser.8 Furthermore, we can produce plasmas with an density profile which guide the pump laser and the x-ray laser beam to propagate long distance’ using capillary targets. We have developed a simple model of Ni-like Ta laser to determine temperature and density of the plasma required to produce gain. Preliminary target design has also been made by combining those results with that from HYADES 1-D hydrodynamics code. 2. ATOMIC
KINETICS
Simplified atomic kinetics of N&like Ta is shown in Fig. 1. The upper laser level is 3d$z4d3,2 (J = 0) excited by strong monopole excitation from 3d” ground state. Since the collisional excitation rate to 3d$24d3,2 (J = 0) is one order of magnitude greater than to any other terms in 4d configuration,” and the transition from 3d&4d3,2 (J = 0) to 3dzj24p1,2 (J = 1) has the largest oscillator strength among possible transition to 4p, 3d$z4d3i2 (J = 0) + 3d$24pl,z(J = 1) has the largest gain coefficient. Calculation of the population inversion and gain can be divided into two parts, the calculation of population to excited states of N&like ions, and the calculation of ionization balance and abundance of Ni-like ions. In high-z atoms, both parts have complexities. Excited levels of the Ni-like ion have a complex structure. Based on a J-J coupling scheme, 3d94d configuration consists of 20 spectral terms. Other one-electron excited states from 3s and 3p subshell and Rydberg states (3d9nl) should be considered because the population of each states are coupled each other through collisional and radiative processes. Number of levels in Ni-like ions should be included in the kinetics model is at least 100. A large database of energy levels and rate coefficients is required for kinetics calculations. For calculation of ionization balance, we have to consider populations to multiple excited states of Ni-like and neighboring ions. Not like low-z ions, one electron excited states of the Nilike ion captures an electron to form doubly excited states of Cu-like ion. Zn-like triplly excited states can be formed in a similar manner. Moreover, especially at high densities, inner shell 4d (3/2,3/2) J=O
4p (l/2,3/2)
laser
J=l
transition
4s, J=2
1670eV
4
collision al excitation
15ooev radiative decay
ground state WO
w Fig. 1. Energy level diagram
for the Ni-like Ta electron
collisional
excited laser.
Modeling of ultra-short-pulse laser pumped x-ray lasers
881
excited states of Co-like and Fe-like ions which have holes in n = 3 shell may have significant population, In the present study, firstly, plasma temperature and density required to produce Ni-like Ta was estimated using a simple model based on the average ion model. We used I-dependent model developed by Perrot” to calculate energy levels of a multiple charged Ta ion. Population of each shell was calculated by a collisional radiative model using rate coefficients given by the screened hydrogenic model.” Collisional ionization, three-body recombination, radiative recombination, collisional excitation and deexcitation, and spontaneous emission were included. This method gave averaged charge of multiple charged ions with a reasonable accuracy without detailed knowledge of their structure. Figure 2a shows the averaged charge of Ta in collisional radiative equilibrium(CRE) as a function of temperature. For ion densities of 1019 to 102’ l/cm3, which was considered as a xray laser medium, the temperature required to produce Ni-like ions (Ta4’+) was found to be 1.0-1.5 keV. Figure 2b shows the temporal development of the averaged charge. This condition corresponds to the situation where an ultra-short-pulse laser irradiates a preformed plasma of various ion densities to heat the plasma instantaneously. Ionization time decreased as the inverse of the density. It was found that to complete ionization within the duration of excitation, the ion density must be greater than 102’ l/cm3. Increasing temperature up to 4 keV was not effective to
60t
. . . . . . . .. -..- ................
. . _.-
. . . . . . . .. .........
i--KZ
t
..............
40 .---4..... .I ......_._ ....._. :jlda lTY77-l
lO-l3
lo-'2
time [s]
lo"
_ ....
_._.
.._ .._._....._
lO-‘0
Fig. 2. Averaged charge of Ta ion calculated using the average ion model. (a) Averaged charge as a function of electron temperature for ion densities from lOI* to 102’ l/cm’. (b) Temporal development of averaged charge number for an electron temperature of 1.6 keV and ion densities from lOI to 102’ l/cm’.
882
Akira Sasaki Table 1. Rate coefficients used in the simple kinetics model of the N&like Ta laser. Rate coefficients
Source
Monopole excitation 3dt0-+4s, J = 2 k = 3.7 x lo-l2 cm3/s 3d”-+4p(l/2,3/2) J = 1 k = 3.7 x lO@ cm’/s 3dio+d(3/2,3/2) J = 0 k = 2.1 x IO-” cm3/s Radiative decay of lower state (4~): k,=2.9 x lOI l/s k,=0.6 x lOI l/s Laser transition: 4d(3/2, 3/2) J = 0 -+ 4p(1/2, 312) J = 1 k, = 4.0 x lo-” cm3/s
IO IO IO I3 5.14
reduce ionization time. In short pulse excitation, higher electron densities were required due to finite ionization time than in the case of conventional long pulse excitation. Secondly, we have calculated Ni-like Ta 4d-4p gain for a simplified system. Rate coefficients used in the model are summarized in Table 1. Branching ratio from the upper laser level to the laser transition was assumed to be 0.3. Figure 3a shows the calculated steady-state gain as a function of electron density. The solid line corresponds to a condition for a pure Ta plasma where the electron temperature is 1.5 keV. Ion density was set to l/45 of electron density. In the present analysis, to take into account the population into neighboring ions, such as Cu-like and Co-like ions, the density of Ni-like ground state was set to be 30% of the ion density using estimations by Maxon’ and Goldstein.” The gain coefficient increased as the electron density until
idO
lo*'
lo**
electron density[l/cm3]
0
0.2
0.4
0.6
0.6
1
escape probability Fig. 3. (a) Gain coefficient of the 4d-4p transition of Ni-like Ta as a function of the electron density. (b) Gain as a function of the escape probability of 3d-4p transition for r,= 1.5 keV, ni= lOI l/cm , n,=45n,, and the fraction of Ni-like ground state is 30% of the total ion density.
Modeling of ultra-short-pulse laser pumped x-ray lasers
883
excitation to the lower laser level destroyed the population inversion in a very high density (r1023 l/cm3). We can expect high gain for electron density of more than lo*’ l/cm’. On the other hand, the maximum ion density is limited by radiation trapping of the lower laser level (4~). Importance of the radiation trapping in Ni-like Nd laser was pointed out by Goldstein. I5 We have made an estimation of the escape probability of 3d-4p transition. Assuming Doppler line broadening, the absorption coefficient a l/cm of this line for an ion temperature of 100 eV is, (2.657 x 10-*)4Ns&,,
a =
as a function of the population ening function,
z 1.6 x 10-‘6N3d
of Ni-like ground state N3d, where 4 is the Doppler line broadf#)= c/vd_
3b shows the gain as a function of the escape probability. It was found that in the case of the density of Ni-like ground state, N3d=0.3 x 1019 l/cm3 and a plasma size of 10 urn, estimated escape probability is P,= exp(-al) x 0.6, where the effect of radiation trapping was not so pronounced. However in higher density gain would decrease significantly. Available beam size of the x-ray laser will be limited by the effect of radiation trapping to a few tens of micrometers. Requirements of the high electron density from ionization and production of gain, and limitations of the ion density from radiation trapping should be accommodated. One possibility is the use of composite targets. Using a mixture of Ta with low-z materials, we can increase the electron density beyond lO*l l/cm3 while keeping the ion density to moderate value (~10’~ l/ cm3). In dotted lines in Fig. 3a the electron density was increased while the Ta ion density was fixed at lo’* and 1OL9l/cm3. The graph shows increased gain up to the electron density of %lO** l/cm3, for increased electron collisional excitation. Ionization time of Ta ion will also reduced by increasing the electron density. Figure
3. MODELING
OF TARGET
High density plasmas for the Ni-like Ta laser can be produced using several target designs. Solid targets are used in previous experiments. However, even with a large irradiation energy, the plasma had a steep density gradient so that the gain region was too thin. This fact might make it difficult to obtain a large gain length product because the x-ray laser beam might be refracted away from the gain region. It also made the gain to be sensitive to the spatial non-uniformity of the pump laser irradiation. Moreover, using solid targets, the energy efficiency was low, because a large fraction of the pump energy was consumed for ablation of the solid surface, conduction into the cold material, and hydrodynamics motion. Several methods which have been proposed for interaction experiments between lasers and long scale length, sub-critical density plasmas may be useful. Using high pressure gas filled targets we can produce moderately high density plasmas (n,-,lO*l l/cm3). The plasma confined in a cavity is also useful. Schematics of a cylinder target is illustrated in Fig. 4. Firstly, a plasma production laser irradiates and ablates the inner surface of the cylinder. Consider a cylinder of 0.3 mm diameter and 3 mm long, irradiated by a laser with an energy of 100 J and a pulse duration of 0.5 ns. The intensity of the laser at the solid surface is ~10’~ W/cm*. The ablated Ta cylinder
excited region
&
plasma production laser pump(USP) laser X-ray laser
Fig. 4. Schematic diagram of the cylinder target
884
Akira
Sasakl
plasma converges to the center of the cylinder. Secondly, a short-pulse intense laser pulse is focused to the central region to heat the plasma to ionize Ta ions to Ni-like stage, and to excite them to the upper laser level. Using this two-step procedure, we expect we can pump the Ni-like Ta laser with an energy of the short-pulse laser of 10 J, because we can irradiate the only small gain region before the energy is dissipated by heat conduction. However, the length of plasma will be limited to less than few mm due to longitudinal non-uniformity. In order to obtain large gain length product, much higher gain (~-10 l/cm3) than that obtained in the present experiments should be achieved. The density and temperature of the plasma confined in the cylinder has been estimated using hydrodynamics code HYADES as shown in Fig. 5. HYADES is a one-dimensional, three-temperature, three-geometry Lagrangian hydrodynamics code. l6 In the present calculations for Ta, SESAME table was used to evaluate the equation of state. Intensity, wavelength, and pulse duration of the laser were lOI W/cm2, 1.06 urn, and 0.5 ns, respectively. About 25% of the laser energy was absorbed to produce a plasma. After 5 ns from the beginning of the laser pulse, the plasma converged at the center of the cylinder. After 10 ns, the central region was filled with rather uniform plasma with a density of 0.05 g/cm3, and a temperature of 35 eV, behind the outgoing shock front. The Thomas-Fermi ionization model gave averaged charge of 12. Density and temperature decreased slowly in the next 10 ns until the reflected shock wave reconverged at the center of the cylinder. 4.
DISCUSSION
Simple atomic kinetics and hydrodynamics calculations showed possibility to produce gain by pumping a preformed plasma confined in a cylinder using an ultra-short-pulse laser. Development of a detailed model is projected for more quantitative estimation and optimization of the gain coefficient. Detailed structures of multiple excited states and inner shell excited states of Cu-like, Zn-like and Co-like ions along with rates of dielectronic recombination, inner shell excitation, and autoionization should be considered to calculate the ionization balance. However the kinetics model based on full detailed term accounting (DTA)model will consist of more than lo4 levels as in the case of Ne-like system,” which may be beyond the capacity of present computers. To develop a more practical model which can be coupled with a multidimensional hydrodynamics code, simplification of the model is required. Populations of not all of those levels have effects on the gain. Levels have similar behavior should be grouped together. The methods originally developed for opacity calculations such as the super configuration transition array (STA) method,i8 and other methods which assume statistical distribution of population over detailed levels” may be useful.
0.01
0.005
0.015
r km1 Fig. 5. Profiles of density (solid line) and temperature (dotted line) of the Ta plasma confined in the cylinder calculated by HYADES I-D hydrodynamics code. Profiles were taken at 2, 10 and 20 ns from the beginning of the calculation.
Modeling of ultra-short-pulse laser pumped x-ray lasers
885
Finally, propagation of the ultra-short-pulse laser through dense plasmas should also be examined. In rarefied plasmas, channel formation and stable pulse propagation have been theoretically predicted in the presence of ponderomotive filamentation and relativistic focusing.20 In x-ray laser plasmas consist of high-z ions, rapid increase of the electron density will cause decreased index of refraction at the center of the laser beam. The pump laser beam may be refracted by negative lens effect, and the maximum available intensity in the target may be significantly decreased. The ionization refraction should be controlled by preionizing the plasma to appropriate ionization stage and by tailoring the density profile of the preformed plasma.’ REFERENCES I. MacGowan, B. J., Maxon, S., Hagelstein, P. L., Keane, C. J., London, R. A., Matthews, D. L., Rosen, M. D., Scofield, J. H. and Whelan, D. A., Phys. Rev. Left., 1987, 59, 2157. 2. Cauble, R., DaSilva, L. B., Barbee, T. W. Jr., Celliers, P., Libby, S., Moreno, J. C., Ress, D., Trebes, J., Wan, A. S. and Weber, F., J. Quant. Spectrosc. Radiat. Transf., 1995, 54, 97. 3. Nagata, Y., Midorikawa, K., Kubodera, S., Obara, M., Tashiro, H. and Toyoda, K., Phys. Rev. Left., 1993,71, 3774. 4. Lemoff, B. E., Yin, G. Y., Gordon III, C. L., Barty, C. P. J. and Harris, S. E., Phys. Rev. Left., 1995, 74, 1574. 5. Maxon, S., Estabrook, K. G., Prasad, M. K., Osterheld, A. L., London, R. A. and Eder, D. C., Phys. Rev. Lett., 1993, 70, 2285. 6. P. V. Nickles, V. N. Shlyaptsev, M. Shntirer, M. Kalachnikov, T. Schlegel and W. Sandner, in Proceedings of the 5th International Colloquium on X-ray lasers, p. 84, S. Svanberg and C.-G. Wahlstriirm, eds, Lund, Sweden, IOP conference series 151 (1986). 7. Korobkin, D. V., Nam, C. H. and Suckewer, S., Phys. Rev. Let?., 1996, 77, 5206. 8. Rocca, J. J., Shlyaptsev, V., Tomasel, F. G., Dcotazar, O., Harthon, D. and Chilla, J. L. A., Phys. Rev. Lett., 1994, 73, 2192. 9. Milchberg, H. M., Durfee III, C. G. and Lynch, J., J. Opt. Sot. Am., 1995, B12, 731. 10. Chen, M. H. and Osterheld, A. L., Phys. Rev., 1995, A52, 3790. 11. Perrot, F., Phys. Scripta, 1989, 39, 332. 12. W. A. Lokke and W. H. Grasberger, XSNQ-U, A Non-LTE Emission and Absorption Coefficient Subroutine, Lawrence Livermore National Laboratory Internal Report UCRL-52276, 1997 (unpublished). 13. Hagelstem, P. L., Phys. Rev., 1986, A34, 874. 14. Hagelstein, P. L., Phys. Rev., 1988, A37, 1357. 15. Goldstein, W. H., Oreg, J., Zigler, A., Bar-Shalom, A. and Klapisch, M., Phys. Rev., 1988, A38, 1797. 16 Abdallah, J. Jr., Clark, R. E. H., Peek, J. M. and Fontes, C. J., J. Quant. Spectrosc. Radiut. Transf, 1994, 51, 1. 17. Larsen, J. T. and Lane, S. M., J. Quant. Spectrosc. Radiat. Transf., 1994, 51, 179. 18. Bar-Shalom, A., Oreg, J. and Goldstein, W. H., J. Quant. Spectrosc. Radiut. Transf, 1994, 51, 27. 19. Takabe, H. and Nishikawa, T., J. Quant. Spectrosc. Radiut. Transf, 1994, 51, 379. 20. Borisov, A. B., Borovskiy, A. V., Shiryaev, 0. B., Korobkin, V. V., Prokhorov, A. M., Solem, J. C., Luk, T. S., Boyer, K. and Rhodes, C. K., Phys. Rev., 1992, A45, 5830.