Acta Polytechnica Acta Polytechnica 53(2):79–87, 2013 © Czech Technical University in Prague, 2013 available online at http://ctn.cvut.cz/ap/ A STUDY OF COHERENT RADIATION GENERATED IN AN ABLATIVE CAPILLARY DISCHARGE Jakub Hübnera,b,∗, Pavel Vrbab a Dept. of Physical Electronics, Faculty of Nuclear Science and Physical Engineering, Czech Technical University, Břehová 7, 115 19 Prague 1, Czech Republic b Institute of Plasma Physics, Academy of Sciences, 182 00 Prague 8, Za Slovankou, Czech Republic ∗ corresponding author: bukajus@centrum.cz Abstract. Feasible soft-X-ray amplification in the CVI and NVII Balmer α transition is investigated in a capillary discharge. The best conditions and parameters for the experimental set-up are found for an ablative capillary. The most optimistic results have shown that the gain would be greater than one, which is the condition for successful ASE (Amplified spontaneous emission) in capillary discharges. The capillary discharge evolution is modeled using the NPINCH program, employing a one-dimensional physical model based on MHD equations. The information about the capillary discharge evolution is processed in the FLY, FLYPAPER, FLYSPEC programs, enabling the population to be modeled on specific levels during capillary discharge. Keywords: capillary discharge, XUV or soft X-ray laser, plasma modeling, ablation. 1. Introduction Non-stationary plasma of a fast capillary electrical discharge was studied as a potential active medium for a soft X-ray laser. Two types of experiments can be performed. The first type is capillary discharge, which is used to generate plasma in a channel filled with an initially preionized gas [1, 2]. The second experiment involves plasma being created by a discharge by ion- izing material ablated from capillary walls [3]. The detailed physical principle for creating the population inversion in the second type of experiment has not yet been well understood. Hence, the modified Shin experiment was simulated by our software equipment (NPINCH, FLY). In fact, the capillary pinch dynamics is determined by many selected parameters: the capillary geometry (radius and capillary length), the substance of the cap- illary (alumina — Al2O3, bornitrid — BN), the filling substance (carbon, nitrogen), the initial filling density (pressure), and the electric current time dependence. This dependence, in particular, is given by an electric circuit that is joined to the capillary. A capillary discharge Z-pinch acting as a medium for a soft X-ray laser uses ASE, the “Amplified Spon- taneous Emission” effect, and electron-collisional re- combination pumping. The main variable for ASE is the gain [4, 13], and the most important goal of this paper is to understand the impact of each capillary parameter on the maximum gain value. 2. Plasma modeling The capillary discharge dynamics is modeled by means of the NPINCH code [5]. An approximation of two- temperature (ion and electron), one-fluid magneto- hydrodynamics is used. It is assumed that the length of the capillary is much greater than its diameter; hence the one dimensional approximation is relevant. It is also assumed that the current pulse profile is known and that the capillary is prefilled with weakly ionized gas. All the simulations presented here were obtained under the presumption of wall ablation. The plasma-wall interaction was modeled by considering the evaporated material from the wall as a cold neutral gas of high density and sufficiently high total mass [6]. 3. An analysis of the modified Shin experiment An experiment performed by Hyun-Joon Shin et al. [3] investigates soft X-ray amplification of the Balmer α transition (for five times ionized carbon ion C5+ at the line λ = 18.2nm) observed in a polyethylene capillary with a 1.2mm bore diameter. There the measured gain coefficient was 2.8 cm−1. The inner capillary plasma was created only by the ablation of the wall material. In our “modified Shin experiment” it is proposed to use a different ablative substance for the capillary wall (alumina — Al2O3 or boron nitride — BN) externally filled by preionized carbon or gas nitrogen at various pressures. The measured electric current [3] approximated by the damped sinus curve I(t) = I0 sin πt 2t1 e −t/t2 , where I0 = 70 kA, t1 = 112.5ns and t2 = 310ns, is intro- duced into the code. The radial-time dependencies of capillary plasma quantities such as plasma den- sity, electron temperature etc. are evaluated. The results for an ablating capillary of r0 = 0.6mm in- ner radius prefilled with carbon to an initial den- sity N0 = 3 · 1017 cm−3, and alumina wall material 79 http://ctn.cvut.cz/ap/ Jakub Hübner, Pavel Vrba Acta Polytechnica 0 50 100 150 200 250 0,0 0,2 0,4 0,6 0 50 100 150 200 250 -20 -10 0 10 20 30 40 50 Time (ns) C ap ill ar y ra d iu s (m m ) C u rr en t (k A ) Figure 1. Results of computer simulations of capillary discharge dynamics for an ablating capillary of inner radius r0 = 0.6mm prefilled with carbon to an initial density N0 = 3 · 1017 cm−3, and alumina wall mate- rial 0.05mm in thickness with density ρ = 3.96 g/cm3. The 1st frame from the top shows radial motion of plasma mass elements. Black lines correspond to the motion of the carbon mass elements, blue lines cor- respond to the motion of the wall ablated material (alumina), and the red line plots the current profile (Imax = 50 kA, t1 = 115.5ns, t2 = 310ns). The 2nd frame shows the mass density compression ratio in log10 scale. The 3rd frame shows electron density in log10 scale and units of cm−3. The 4th frame shows electron temperature in units of eV. 0.05mm in thickness with density ρ = 3.96 g/cm3 are shown in Figure 1. The plasma trajectories (the dependences of the radial Lagrangian coordinates of selected plasma elements on time) are depicted in the 1st frame from the top. In the 2nd frame it can be 200 210 220 230 240 250 260 270 0,0 0,1 0,2 0,3 0,4 200 210 220 230 240 250 260 270 -20 -10 0 10 20 30 40 50 Time (ns) C ap ill ar y ra d iu s (m m ) C u rr en t (k A ) Figure 2. Zoom of Figure 1 into the area where gain is being reached. The 1st frame from the top shows radial motion of plasma mass elements. Black lines correspond to the motion of the carbon mass elements, blue lines correspond to the motion of the wall ablated material, and the red line plots a current profile. The 2nd frame shows the mass density compression ratio in log10 scale. The 3rd frame shows electron density in log10 scale and units of cm−3. The 4th frame shows the electron temperature in units of eV. The 5th frame shows the gain profile in units of cm−1. 80 vol. 53 no. 2/2013 A Study of Coherent Radiation 0 50 100 150 200 250 0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 6+ 5+ 4+ P o p u la ti o n s* 10 18 ( cm -3 ) Time (ns) Figure 3. Showing the time history of C4+, C5+ and C6+ ions populations for the capillary parameters: N0 = 3 · 1017 cm−3, r = 0.6mm, Imax = 50 kA, t1 = 112.5ns, t2 = 310ns, carbon inner filling, and wall material alumina. 0 50 100 150 200 250 300 0 1x1014 2x1014 3x1014 4x1014 5x1014 6x1014 7x1014 8x1014 9x1014 1x1015 N 2 N 3 Time (ns) P o p u la ti o n ( cm -3 ) Figure 4. Showing the time history of populations at excited energy levels N3 and N2 of hydrogen-like ions C5+ for the capillary parameters: N0 = 3 · 1017 cm−3, r = 0.6mm, Imax = 50 kA, t1 = 112.5ns, t2 = 310ns, carbon inner filling, and wall material alumina. seen that the first compression is achieved very early at the pinch time tp1 = 20ns and the compression ratio is log10 ρ(tp1) ρ0 = 1.4. At this time, the plasma electron temperature value is ∼ 90 eV (on the capil- lary axis, 4th frame) and the ion temperature is ∼ 3× higher, the plasma is non-isothermal. The second com- pression is achieved at tp2 = 98ns, when the electric current reaches its maximum value (1st frame), and the compression ratio is log10 ρ(tp2) ρ0 = 1.3 (2nd frame). The plasma becomes isothermal. The plasma electron and ion temperature values are practically the same with values of ∼ 260 eV. The plasma electron density achieves its maximum value Ne,max = 1.75 ·1019 cm−3 (3rd frame). Further time development of the carbon plasma column demonstrates enlargement of its vol- ume and the maximum radius value is rmax = 0.24mm at time te1 = 190ns. The plasma is still quasi-iso- thermal Te ∼ Ti ∼ 120 eV. After the time t > te1 the 200 210 220 230 240 250 260 270 0,00 0,05 0,10 0,15 0,20 0,25 N 0 = 1e17 cm-3 N 0 = 2e17 cm-3 N 0 = 3e17 cm-3 N 0 = 4e17 cm-3 N 0 = 5e17 cm-3 N 0 = 6e17 cm-3 N 0 = 7e17 cm-3 ga in ( cm -3 ) time (ns) Figure 5. Showing the gain behavior for various initial densities N0 in units of cm−3. For the capillary parameters: r0 = 0.6mm, Imax = 50 kA, t1 = 112.5 ns, t2 = 310ns, carbon inner filling, and alumina wall material. 200 210 220 230 240 250 260 270 0,00 0,05 0,10 0,15 0,20 0,25 N 0 = 1e17 cm-3 N 0 = 2e17 cm-3 N 0 = 3e17 cm-3 N 0 = 4e17 cm-3 N 0 = 5e17 cm-3 N 0 = 6e17 cm-3 N 0 = 7e17 cm-3 ga in ( cm -3 ) time (ns) Figure 6. Gain behavior for various current max- imum Imax = 35 kA, 45 kA, and 50 kA. For capil- lary parameters: N0 = 3 · 1017 cm−3, r0 = 0.6mm, t1 = 112.5 ns and t2 = 310 ns, carbon inner filling, and the alumina wall material. plasma column is again compressed. During compres- sion the plasma is exactly isothermal. We can declare that plasma occurred at the LTE state. The kinetic behavior of plasma near the capillary axis is evaluated by the FLY code, used as a post- processor along the plasma trajectory. The time de- pendences of plasma ion populations are shown in Fig. 3. Helium-like ions C4+ occurred at the very beginning (at pinch time tp1 = 15 ns) of plasma devel- opment. After that, the population of hydrogen-like ions C5+ culminated. The maximum value for bare C6+ is reached at the time of second compression. No inversion population between the excited state n = 2 and 3 of hydrogen-like C5+ ions occurred at this time. The situation changes remarkably when the population of bared ions drops lower than the population of hydrogen-like ions (at time t ∼ 210 ns). From this time, the hydrogen-like C5+ ions prevail 81 Jakub Hübner, Pavel Vrba Acta Polytechnica 50 100 150 200 250 0,0 0,1 0,2 0,3 0,4 0,5 0,6 t 1 = 120 ns t 1 = 112 ns t 1 = 100 ns t 1 = 90 ns t 1 = 80 ns t 1 = 70 ns t 1 = 60 ns t 1 = 50 ns t 1 = 40 ns t 1 = 30 ns t 1 = 20 ns ga in ( cm -1 ) time (ns) Figure 7. Gain behavior for various time periods t1. For capillary parameters N0 = 3 · 1017 cm−3, r0 = 0.6mm, Imax = 35 kA, t2 = 310 ns, carbon inner filling, and the alumina wall material. over both bared and helium -like C4+ ions. This is the best situation when a recombination scheme for the laser transition between an upper level n = 3 and a lower level n = 2 of hydrogen-like carbon ion operates (see Fig. 4). This process ends when the population of the excited hydrogen-like carbon ion in the upper level Nu (n = 3) becomes lower than the lower Nl (n = 2) one. According to the Elton formulae [5], we calculated the inversion function F = 1 − 2.25Nl/Nu and the gain G = σstimNuF (see Fig 5), where σstim is the cross-section for stimulated emission, Nu is the upper state density and Nl is the lower state density. 4. The role of capillary parameters In this section, each important capillary parameter and its influence on the capillary dynamic and on the gain in particular will be described in detail. First of all, these parameters should be defined. Three pa- rameters relate to the shape of the current-impulse. The shape of the current impulse is defined by an electric circuit joined to a capillary. At present, most of available sources [7–9] can be approximated by a damped sinus curve I(t) = I0 sin πt 2t1 e −t/t2 ; in- stead of I0 it is better to use Imax (range 35–50 kA), which can be associated with the amount of over- all energy deposited in the capillary; t1 (range 15– 120 ns) is a quarter period, and t2 (range 30–300 ns) is a term for exponential decay. The fourth pa- rameter is the initial density N0 (range 0.5 · 1017– 5 · 1017 cm−3) (or pressure), and the fifth is the ra- dius of the capillary r0 (range 0.25–2.5mm). The length l (range 2–20 cm) of the capillary is much greater than the diameter, so a one-dimensional ap- proximation is relevant. This parameter has not been dealt with in our research. In the past, some low- Z elements were studied as possible sources of co- herent radiation for recombination pumping in cap- illary discharges [6, 10, 11]. Nitrogen and carbon were selected as the best adepts for our investiga- tion in an ablated capillary. Capillaries are mostly 50 100 150 200 250 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 t1 = 10 ns t1 = 20 ns t1 = 30 ns t1 = 40 ns t1 = 50 ns t1 = 60 ns t1 = 70 ns t1 = 80 ns t1 = 90 ns t1 = 100 ns t1 = 112 ns t1 = 120 ns g ai n ( cm -1 ) time (ns) Figure 8. Gain behavior for various time periods t1. For capillary parameters N0 = 3 · 1017 cm−3, r0 = 0.6mm, Imax = 50 kA, t2 = 310 ns, carbon inner filling, and alumina wall material. made from alumina (Al2O3) or quartz (SiO2). Alu- mina was chosen as the material for the capillary, and was compared with boron nitride (BN), which is promising new material for this model. We at- tempted to illustrate the influence of these param- eters on the gain, and to optimize them [13]. The optimization process involves searching for the max- imum gain in the five-dimensional space of these pa- rameters. All the following figures show gain be- havior only along the capillary axis, because that is where the best conditions for gain are found (Fig- ure 2). 4.1. Initial density Among all the parameters, the initial density is the easiest to change in a real experiment. There always some optimal value, Figure 5, where the optimal value is N0 = 3 · 1017 cm−3. The gain for the recombina- tion of the pumping schema is mostly influenced by the behavior of the electron temperature and the ion density. The thinner the plasma is, the higher the electron temperature is at its maximum, but on the other hand the thinner the plasma is, the fewer ions at excited states it contains. The delay among the times when the gain maximum is reached for different initial densities, is due to the fact that the thinner plasma is more compressed (there are higher temper- atures at the maximum) and it takes slightly more time to cool down to the electron temperatures where recombination pumping works. This behavior is valid even if the other parameters (r0, Imax, t1, t2) are changing. 4.2. Electric current First, we investigate the influence of Imax on the gain. The range for Imax is chosen between Imax = 35 kA, because there might be no ablation at the lower Imax, and Imax = 50 kA. The upper limit could definitely be even higher, but the higher Imax is, the more difficult it is to build such a powerful source from technical point of view. Figure 6 shows the gain behavior for 82 vol. 53 no. 2/2013 A Study of Coherent Radiation 190 200 210 220 230 240 250 260 270 280 0,00 0,05 0,10 0,15 0,20 0,25 0,30 r = 0.40 (mm) r = 0.45 (mm) r = 0.50 (mm) r = 0.55 (mm) r = 0.60 (mm) r = 0.65 (mm) r = 0.70 (mm) g ai n ( cm -1 ) time (ns) Figure 9. Gain behavior for various capillary inner radius r. For capillary parameters: N0 = 3 ·1017 cm−3, Imax = 50 kA, t1 = 112.5ns and t2 = 310ns, carbon inner filling, and wall material alumina. Imax = 35 kA, 45 kA and 50 kA. For higher Imax the gain increased only slightly. The optimal N0 was the same for all currents. The reason for the differences among the pulse shapes (tG,max — time where G reached its maximum, tF+ — time where G firstly appeared and where inversion function F becomes nonzero, tdiff — difference between tG,max and tF+) are the same as for the initial density, see above. Plasma is more compressed at higher Imax and it takes slightly more time to cool down. Secondly, the influence of t1 on the gain is investi- gated. The behavior of the gain for various time peri- ods t1 and two Imax parameters are shown in Figures 7 and 8, respectively. For each Imax there exists opti- mal time period t1 = 17.5ns for Imax = 35 kA where Gmax ∼ 0.56 cm−1 and t1 = 25ns for Imax = 50 kA where Gmax ∼ 0.78 cm−1. As t1 becomes smaller (cur- rent pulse quicker) the profile of the gain changes, and tdiff is smaller. 4.3. Capillary radius The influence of the capillary radius is also investi- gated, for Imax = 50 kA. Results for longer time period t1 = 112.5ns and shorter time period t2 = 25ns are presented in Figures 9 and 10. For both periods there is a certain optimal capillary radius. The gain is influ- enced only slightly with the longer time period, the optimal radius is r0 = 0.5mm and Gmax ∼ 0.27 cm−1. However, the situation for the shorter time period is promising. Gmax reached value 1.64 cm−1 for a very narrow capillary with radius r0 = 0.3mm. It should be mentioned that these results are only for initial density N0 = 3 ·1017 cm−3 and are not pressure optimized. 4.4. Optimal set-up for Carbon filling Now we can take a closer look at a particular case. The best parameters for an alumina capillary filled with Carbon and Imax = 50 kA are r0 = 0.3mm, t1 = 25 ns, t2 = 310 ns. The results are shown in Figure 11. It is 50 55 60 65 70 75 0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8 2,0 r = 0.25 (mm) r = 0.30 (mm) r = 0.35 (mm) r = 0.40 (mm) r = 0.45 (mm) r = 0.50 (mm) r = 0.55 (mm) r = 0.60 (mm) r = 0.65 (mm) g ai n ( cm -1 ) time (ns) Figure 10. Gain behavior for various capillary inner radius r. For capillary parameters: N0 = 3 ·1017 cm−3, Imax = 50 kA, t1 = 25 ns and t2 = 310 ns, carbon inner filling, and wall material alumina. shown that a better set of parameters would increase the gain from Gmax = 0.25 cm−1 to Gmax = 1.6 cm−1. If the r–t profile of the gain in Figure 11 is compared with the initial settings in Figure 1 and 2, it can be seen that the area (r–t) of gain for optimal settings is smaller r = 0.1mm, and a possible laser pulse would be about 4 ns long against r = 0.2mm and length of pulse about 30 ns. Shortening the electric current pulse leads to quicker cooling, the value of the gain increases, and the laser pulse is shortened. Reducing the capillary radius to R0 = 0.3mm also increases the gain. The narrower the capillary is, the more the area along the capillary axis is influenced by ablation of the capillary wall, which consequently increases the plasma density near the axis due to wall ablation. 4.5. Optimal set-up for Nitrogen filling On the basis of recent research on non-ablated capil- laries [12], it was supposed that nitrogen would also be suitable for this ablative model of pumping. At normal conditions, nitrogen is a gas, so there are none of problems with filling it into the capillary that there are with carbon filling. The results for nitrogen with the parameters discussed here (N0, t1, r0, Imax) are shown in the following tables. Similar patterns were obtained for each parameter as for carbon. The re- sults have been entered into Tables 4 to 7. The best settings are highlighted with red color. Furthermore, all the results for nitrogen were pressure optimized and it was shown that there exists optimal r0 only for constant N0. The thinner the capillary is, the higher the gain that can be obtained. However, there will be some technological limits that have to be taken into consideration in a real experiment. Bornitrid as a capillary material is investigated for some cases. The results are highlighted in green. 83 Jakub Hübner, Pavel Vrba Acta Polytechnica t1 (ns) t2 (ns) N0 (cm−3) Gmax (cm−1) tGmax (ns) tF+ (ns) tdiff (ns) 12.5 310 3.0 · 1017 0.3 39 36 3 15 310 3.0 · 1017 0.55 43.5 39 4.5 17.5 310 3.0 · 1017 0.56 48.8 43.5 5.3 20 310 3.0 · 1017 0.53 54 47.3 6.7 30 310 3.0 · 1017 0.44 73.5 64.5 9 Table 1. Capillary parameters: Imax = 35 kA, r0 = 0.6mm, carbon, alumina. t1 (ns) t2 (ns) N0 (cm−3) Gmax (cm−1) tGmax (ns) tF+ (ns) tdiff (ns) 10 310 3.0 · 1017 0.36 43.5 39.8 3.7 20 310 3.0 · 1017 0.66 56.3 52.5 3.8 22.5 310 3.0 · 1017 0.74 61.5 57 4.5 25 310 3.0 · 1017 0.78 66 61.5 5.5 27.5 310 3.0 · 1017 0.77 72 66 6 30 310 3.0 · 1017 0.75 76.5 70.5 6 40 310 3.0 · 1017 0.66 96 87 9 Table 2. Capillary parameters: Imax = 50 kA, r0 = 0.6mm, carbon, alumina. radius (mm) N0 (cm−3) Gmax (cm−1) tGmax (ns) tF+ (ns) tdiff (ns) 0.65 3.0 · 1017 0.66 68.3 63 5.3 0.6 3.0 · 1017 0.78 66 61.5 5.5 0.55 3.0 · 1017 0.90 64.5 59.3 5.2 0.5 3.0 · 1017 1.04 63 57.8 5.2 0.45 3.0 · 1017 1.20 60.8 56.3 4.5 0.4 3.0 · 1017 1.35 59.3 54.8 4.5 0.35 3.0 · 1017 1.58 57 53.3 3.7 0.3 3.0 · 1017 1.66 54.8 51.8 3.0 0.25 3.0 · 1017 1.38 53.3 50.3 3.0 Table 3. Capillary parameters: Imax = 50 kA, t1 = 25ns, t2 = 310ns, carbon, alumina. t1 (ns) t2 (ns) N0 (cm−3) Gmax (cm−1) tGmax (ns) tF+ (ns) tdiff (ns) 15 30 1.0 · 1017 0.09 46 39.8 6.2 17.5 35 1.0 · 1017 0.11 51 43.5 7.5 20 40 1.0 · 1017 0.26 54.8 48 6.8 22.5 45 1.0 · 1017 0.13 59.3 50.3 9 25 50 1.0 · 1017 0.12 63 54 9 30 60 1.0 · 1017 0.002 84.5 74.8 9.7 Table 4. Capillary parameters: Imax = 35 kA, r0 = 0.6mm, nitrogen, alumina. 84 vol. 53 no. 2/2013 A Study of Coherent Radiation t1 (ns) t2 (ns) N0 (cm−3) Gmax (cm−1) tGmax (ns) tF+ (ns) tdiff (ns) 15 30 1.0 · 1017 0.34 50.3 44.3 6 17.5 35 1.0 · 1017 0.36 54 47.3 6.7 17.5 35 1.0 · 1017 0.08 50 43.5 6.5 20 40 1.0 · 1017 0.37 58.5 47.3 6.7 20 40 1.0 · 1017 0.16 54 47.3 6.7 22.5 45 1.0 · 1017 0.32 63 54.8 8.2 22.5 45 1.0 · 1017 0.1 59.3 49.5 9.8 25 50 1.0 · 1017 0.29 67 58.5 8.5 30 60 1.0 · 1017 0.24 76 64 10 40 80 1.0 · 1017 0.16 93 81 12 Table 5. Capillary parameters: Imax = 50 kA, r0 = 0.6mm, nitrogen, alumina, bornitrid. radius (mm) N0 (cm−3) Gmax (cm−1) tGmax (ns) tF+ (ns) tdiff (ns) 0.6 0.5 · 1017 0.29 56.3 49.5 6.8 0.55 0.5 · 1017 0.37 54 47.3 6.7 0.5 0.5 · 1017 0.46 52.5 45.8 6.7 0.45 1.0 · 1017 0.49 50.3 43.5 6.8 0.4 2.0 · 1017 0.55 48 41.3 6.7 0.35 2.0 · 1017 0.68 46.5 40.5 6 0.3 5.0 · 1017 0.86 45 39 6 0.3 2.0 · 1017 1.05 45 38.8 6.7 0.25 5.0 · 1017 1.24 42.8 37.5 5.3 0.25 5.0 · 1017 1.29 42 36.8 5.2 Table 6. Capillary parameters: Imax = 35 kA, t1 = 20 ns, t2 = 40 ns, nitrogen, alumina, bornitrid. radius (mm) N0 (cm−3) Gmax (cm−1) tGmax (ns) tF+ (ns) tdiff (ns) 0.6 1 · 1017 0.36 58.5 51 7.5 0.55 1 · 1017 0.44 56.3 48.8 7.5 0.5 1 · 1017 0.55 54 47.3 6.7 0.45 1 · 1017 0.66 51 45 6.7 0.4 2 · 1017 0.76 48.8 42.8 6 0.35 5 · 1017 0.98 45.8 40.5 5.3 0.3 5 · 1017 1.24 43.5 39 4.5 0.25 10 · 1017 1.54 41.3 37.5 3.8 Table 7. Capillary parameters: Imax = 50 kA, t1 = 20 ns, t2 = 40 ns, nitrogen, alumina. 85 Jakub Hübner, Pavel Vrba Acta Polytechnica 0 10 20 30 40 50 60 70 0,0 0,1 0,2 0,3 0 10 20 30 40 50 60 70 -40 -30 -20 -10 0 10 20 30 40 50 Time (ns) C ap ill ar y ra d iu s (m m ) C u rr en t (k A ) Figure 11. Results of computer simulations of the capillary discharge dynamics for an ablating capillary of 0.3mm inner radius prefilled with carbon to an initial density N0 = 3 ·1017 cm−3. The 1st frame from the top shows the radial motion of the plasma mass elements. Black lines correspond to the motion of the carbon mass elements, blue lines correspond to the motion of wall ablated material (alumina), and the red line plots a current profile (Imax = 50 kA, t1 = 25ns, t2 = 310ns). The 2nd frame shows the mass density compression ratio in log10 scale. The 3rd frame shows the electron density in log10 scale and units of cm−3. The 4th frame shows the electron temperature in units of eV. The 5th frame shows the gain profile in units of cm−1. 0 10 20 30 40 50 60 70 0,0 0,1 0,2 0,3 0 10 20 30 40 50 60 70 -20 -10 0 10 20 30 40 50 N 0 = 0.5e17 cm-3 N 0 = 1.0e17 cm-3 N 0 = 2.0e17 cm-3 N 0 = 5.0e17 cm-3 N 0 = 10e17 cm-3 N 0 = 20e17 cm-3 Time (ns) C ap ill ar y ra d iu s (m m ) C u rr en t (k A ) Figure 12. Motion of the border between the inner filling (nitrogen) and the wall ablated material plasma mass elements for various different initial densities in units of g/cm3 in a capillary of 0.3mm inner radius, the thick red line plots the current profile (Imax = 50 kA, t1 = 20 ns, t2 = 40 ns). 40 45 50 55 0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8 40 45 50 55 -40 -30 -20 -10 0 10 t 2 = 40 ns t 2 = 80 ns t 2 = 160 ns t 2 = 320 ns G ai n ( cm -1 ) Time (ns) I (k A ) Figure 13. Gain profile (thick lines) at the axis in the unit of cm−1 for different exponential decays (t2 = 40– 320 ns) at current profiles (dotted lines, Imax = 50 kA, t1 = 20ns). The other capillary parameters were: inner radius r0 = 0.3mm, initial density N0 = 5 · 1017 g/cm3, inner filling nitrogen, and alumina wall material. 4.6. Initial density and the border between inner and wall ablated material The initial density N0 influences not only the gain but also the sizes of the areas that belong to the inner and ablated plasma. Figure 12 shows how the boundary between these two nonmixing areas evolves. The information about the maximum gain at the Gmax axis and the gain at the boundary line Gmax,b for each initial density N0 is embedded into Table 8. The best conditions for Gmax at the axis are reached with N0 = 5 · 1017 cm−3 but if the information about the boundary is taken into account, N0 = 10 · 1017 cm−3 would be better for the total gain from the whole area. 86 vol. 53 no. 2/2013 A Study of Coherent Radiation N0 (cm−3) Gainmax (cm−1) tGmax (ns) tF+ (ns) tdiff (ns) GainB,max (cm−1) 1 · 1017 1.14 44.3 39 5.3 1.08 2 · 1017 1.19 44.3 39 5.3 1.08 5 · 1017 1.24 43.5 39 4.5 0.89 10 · 1017 1.18 42.8 38.3 4.5 0.52 20 · 1017 0.82 41.3 37.5 3.8 0.072 Table 8. Capillary parameters: Imax = 50 kA, t1 = 20 ns, t2 = 40 ns, r0 = 0.3mm, nitrogen, alumina. 4.7. Exponential decay parameter The final parameter which has not been mentioned yet is time t2 in an exponential function which affects the oversight value of the electric current, and the position of tImax (tImax is the time when the maximum current is reached. The smaller t2 is, the sooner tImax is reached). Because the gain is situated in the third quarter period of the sinusoidal pulse, the gain is also influenced by the oversight of the current. Figure 13 shows how the gain profile changes with different decay times. This behavior can be explained exactly by shifts of tImax rather than by the value of the current oversight. The time tImax is reached slightly later with increasing t2, so the maximum for the plasma quantities is reached slightly later. The subsequent cooling is then more rapid, which leads to better gain. 5. Discussion and conclusion Optimal set-ups have been sought for a nitrogen filling and for a carbon filling in the range of the selected pa- rameters. The influence of each parameter on the gain is described in detail. Capillary radius r0 and quarter period t1 have the greatest impact on the gain. Hence, to achieve a sufficiently high gain (G > 1) it is neces- sary to use a very narrow capillary (r0 = 0.25–0.3mm) and a very fast electric current (t1 = 15–30ns). We should also point out problems that can arise. The biggest problem is the ablation process itself. The ab- lation subsequently damages the inner capillary wall and consequently changes the radius of the capillary from shot to shot. Repetition while conserving the initial parameters is not therefore possible for a larger amount of shots. Such narrow capillaries could even break very often. We should also mention the sim- plification of the plasma-wall interaction, which is considering the evaporated material from the wall as a cold neutral gas of high density and sufficiently high total mass: this simplification may not be valid for Imax < 35 kA. The rate of ablation also changes with initial density. With higher initial pressures the rate is lower than for lower pressures. Acknowledgements This research was supported by the Czech Science Foun- dation Project No. GAČR P102/12/2043. References [1] Rocca, J.J.: Table-top soft x-ray lasers. Review of Scientific Instruments, vol. 70, no. 10, 3799 (1999) [2] Hosokai, T., Nakajima, M.,Aoki, T., Ogava, M., Horioka, K.: Correlation between Soft X-Ray Emission and Dynamics of Fast Capillary Discharges. Jpn. J. Appl. Phys. 36, 2327 (1997) [3] Shin, H.-J., Kim, D.-E., Lee, T.-N.: Soft X-Ray Amplification in Capillary Discharge. Phys. Rev. E 50, 1376 (1994) [4] Elton, R. C.: X-Ray Laser. Academic Press, New York, (1990). [5] Razinkova, T.L., Sasarov, P.V.: Program NPINCH na počítání dynamiky z-pinče, Moskva 1998. [6] Vrba, P., et al.: Modeling of capillary Z-pinch recombination pumping of boron extreme ultraviolet laser. Physics of Plasmas 16, 073105 (2009) [7] Kampel, N. S., et al.: Feasibility of a nitrogen-recombination soft-x-ray laser using capillary discharge Z pinch. Phys. Rev. E 78(5), 056404 (2008) [8] Nevrkla, M., et al.: Time-resolved XUV Radiation Diagnostics from Nitrogen Discharge Z-pinching Plasma. Proceedings of SPIE 8140, 814016 (2011) [9] Nevrkla, M., Jancarek, A., et al.: Capillary Discharge Apparatus for Intense XUV Radiation Generation. IET Pulsed Power Conference (2009) [10] Hubner, J.: Capillary Discharge Parameter Assessment for X-ray Laser Pumping. Acta Polytechnica Vol. 50 No. 4 (2010) [11] Vrba, P., et al.: Analysis of Laser Pumping by Capillary Pinching Discharge in Argon and Nitrogen. Springer proceedings in Physics 115 (2007) [12] Vrba, P., Vrbova, M.:, Optimization of Nitrogen Filled Capillary Pinch for Soft X-ray Laser Recombination Pumping. Proc. SPIE 6702, 67020W (2007) [13] P. Vrba, M. Vrbova, N.A. Bobrova and P.V. Sasorov, Cent. Eur. J. Phys. 3, 564 (2005) 87 Acta Polytechnica 53(2):79--87, 2013 1 Introduction 2 Plasma modeling 3 An analysis of the modified Shin experiment 4 The role of capillary parameters 4.1 Initial density 4.2 Electric current 4.3 Capillary radius 4.4 Optimal set-up for Carbon filling 4.5 Optimal set-up for Nitrogen filling 4.6 Initial density and the border between inner and wall ablated material 4.7 Exponential decay parameter 5 Discussion and conclusion Acknowledgements References