Experimental results of a two-stage Ni-like Ag soft X-ray laser operated in a seed-amplifier configuration are presented. Both targets were pumped applying the double-pulse grazing incidence technique with intrinsic travelling wave excitation. The injection of the seed X-ray laser into the amplifier target was realized by a spherical mirror. The results show amplification of the seed X-ray laser and allow for a direct measurement of the gain lifetime. The experimental configuration is suitable for providing valuable input for computational simulations.
©2012 Optical Society of America
The generation of soft X-ray lasers (XRL) is often based on amplification of spontaneous emission (ASE). The stochastic nature of this process leads to low spatial coherence and, when not saturated, to significant shot-to-shot fluctuations of the XRL beam profile. It may in addition limit the pointing- and energy stability. These properties limit the use of the XRL in applications, especially for single-shot measurements or if spatial coherence is required. A promising approach circumventing the mentioned disadvantages consists in injection-seeding the medium with a spectrally matched XRL pulse of a secondary source, which is possible due to the high gain provided by the XRL medium. While efficient energy extraction from the XRL medium is still to be demonstrated, several groups succeeded in improving the beam quality of the XRL beams using high-order harmonic radiation (HH) or a second XRL as a seed source. HH offer a high beam quality with full temporal and spatial coherence, but in general suffer from comparably low energy. The number of photons suitable for seeding the XRL medium is further reduced by the spectral mismatch, since the spectral width of typical harmonic lines is significantly larger compared to the XRL transition . Despite these challenges several groups were able to demonstrate high-quality, fully coherent HH-seeded XRL operating at high repetition rates [2, 3]. Utilizing a second XRL as a seed source avoids the problem of spectral overlap and allows for a larger number of seed photons, but of course cannot provide the high beam quality of a HH source. Several experiments have been performed with two XRL media, demonstrating full spatial coherence and diffraction-limited divergence [4–6]. These promising experiments have been performed by utilizing a double-pulse normal-incidence pumping technique where travelling wave (TW) excitation was applied to only one of the two targets. Finally, it should be noted that an increase in spatial coherence could also be obtained by an increase of the gain-column length or simply by free propagation.
We have developed an experimental setup that allows for providing two XRL targets with two counter-propagating beams using the double-pulse grazing incidence pumping (DGRIP) geometry with its intrinsic TW excitation . This technique has also been implemented at the LASERIX facility  and a similar pumping scheme is presented by Imesch et al. . Both targets are efficiently pumped by a double-pulse applied under the optimized GRIP angle . XRL emission is observed independently for both targets, pointing into opposite directions. The injection of the seed XRL into the gain zone of the amplifier medium is accomplished by a spherical extreme ultraviolet (XUV) mirror that images the output plane of the seed medium into the amplifier. The experimental setup is able to provide valuable input for time-dependent Maxwell-Bloch codes such as DeepOne , which is aiming at simulating the temporal evolution of the XRL pulse. In this Letter, we present this experimental configuration and report on results of a first experiment conducted at the PHELIX laser facility.
In the DGRIP scheme, the two pumping pulses are realized as a collinear double-pulse that is applied to the targets under grazing incidence. The double-pulse is created by a Mach-Zehnder interferometer implemented at the front-end of the laser system. The original beam is divided into two arms by a combination of a wave plate and a polarizer, allowing for a continuously adjustable energy ratio of the two resulting beams. The first arm generates the prepulse and includes a compressor, which provides a different duration of the pre- and main pulse after the final compression stage of the laser system. In the second arm, a delay stage is installed to adjust the timing of the two pulses. The two beams are recombined and the resulting collinear double-pulse is injected into the succeeding parts of the Chirped Pulse Amplification (CPA) system.
To provide both the seed- and the amplifier XRL target with an individual pump beam, the PHELIX double-beam option has been used. The beam containing the double-pulse structure is divided by a wave plate and a polarizer. To recombine the two beams, a half-mirror is installed in a way that only the lower half of the first beam and the upper half of the second beam are injected into the next section of the laser system. The respective other halves of the beams are disposed of. At the end of the laser chain each of the two independent beams is 90 mm x 90 mm in size while being vertically separated by 25 mm. Each beam provides up to 50 J of energy that can be compressed down to 300 fs in duration. The timing between the two beams can be adjusted by a motorized delay stage with sub-picosecond resolution.
Inside the target chamber, the DGRIP scheme allowed for a very simple and compact setup, relying on one target and two beams (Fig. 1 ). Each of these beams was brought to its designated height by a periscope before being focused by a spherical mirror positioned in GRIP geometry. The angle of the incident laser beam with respect to the target surface was 26.5° for both targets. Taking the wavelength of the PHELIX laser of 1054 nm into account, the electron density in which the laser energy was predominantly deposited amounts to 2 × 1020 /cm3, following , where denotes the critical electron density, = 1 × 1021 /cm3.
Both line foci were aligned onto the same target, vertically separated by 6 mm (Fig. 1, lower inset). According to ray tracing simulations, the different optical path lengths between each spherical mirror and the respective target added up to a TW velocity of 1.1c. Due to the symmetry of the setup, this value applied to both targets, with the travelling waves propagating in opposite directions. The line focus of the seed target (amplifier target) was 50 µm (120 µm, defocused) in width and 11 mm (16 mm) in length. The targets themselves were only 6 mm or 8 mm long and thus shorter than the line foci. This design was chosen to provide a homogeneous intensity distribution along the target. Especially in the case of the defocused line focus providing the amplification medium, inhomogeneity issues had been of concern prior to the experiment. The TW mismatch adds up to 1.8 ps (2.4 ps) at the end of the 6 mm (8 mm) long target, which is in the same order as the duration of the IR heating pulse (2 ps). The IR diagnostic system consisted of an f = 200 mm achromatic lens that imaged both line foci onto a CCD camera positioned outside the target chamber. A pinhole CCD camera was installed to monitor the homogeneity of the plasma columns on-shot. The distance between target, pinhole and the chip of the CCD camera was chosen to image both line foci onto the 26 mm x 26 mm chip and resulted in a magnification of 1.5.
The pulse duration of the pre- and the main pulse were 200 ps and 2 ps, respectively, and their energy ratio was kept constant at 25% - 75%. The resulting irradiances on the seed target (amplifier target) were in the order of 3 × 1012 W/cm2 (2 × 1012 W/cm2) for the prepulse and 9 × 1014 W/cm2 (7 × 1014 W/cm2) for the main pulse.
To inject the seed XRL, the corresponding line focus was tilted by 3°, according to the vertical distance of the two targets (6 mm) and the distance to the spherical XUV mirror (117 mm). The latter was positioned at the same height as the amplifier and is tilted by 1.5°. This way, the height difference between the two XRL media is compensated and the propagation of the seed beam is parallel to the amplifier medium. The reflectivity of the XUV mirror was measured to be 30% at the XRL wavelength and its focal length was 100 mm. The spatial coupling between the seed XRL and the amplifier medium was in the order of 0.24% in this geometry. In more detail, the coupling efficiency takes into account the spatial mismatch between the seed XRL beam size (2.4 mm x 1.2 mm) and the estimated gain zone (~50 µm x 100 µm) of the amplifier as well as the reflectivity of the spherical XUV mirror (30%). This design was chosen since it is comparably insensitive to pointing instabilities and divergence fluctuations of the seed XRL, which is advantageous when working with a laser system of low repetition rate (1 shot / 90min). To detect the XRL signal, a flat 45° XUV mirror was used, deflecting both XRL beams onto a 16-bit back-illuminated CCD camera positioned 20 cm away from the target. In addition, several exchangeable Zr filters between 1 µm and 3 µm thickness were installed to optimize the XRL signal within the dynamic range of the camera.
The timing between the two IR pump beams was verified by using a fast-rise-time photodiode placed at the target position and a 15 GHz oscilloscope, resulting in a temporal resolution of 15 ps. The distance of 117 mm between the target and the spherical XUV mirror used for injection was known better than the uncertainty of the photodiode measurement. Within that precision, the delay between the two IR beams was set to 780 ps according to 2 x 117 mm. In the following, this will be referred to as a delay of 0 ps. Thus a negative delay corresponds to the seed XRL arriving at the amplifier medium earlier with respect to the IR pumping pulse and vice versa.
As described above, the footprint camera was installed in a way that allowed for recording both the almost-collimated seed XRL as well as the ASE signal emerging from the amplifier target. The upper inset of Fig. 1 shows an example of a recorded footprint. The larger XRL signal corresponds to the ASE of the amplifier medium. Its large divergence of 60 mrad exceeds the value expected from the aspect ratio of the line focus width (120 µm) and target length (8 mm). The smaller seed target was operated with a line focus of 50 µm width and 8 mm target length. Assuming that the seed XRL was collimated after the spherical XUV mirror, its divergence can be estimated to 8 mrad. In the direction of the plasma expansion both signals show the same divergence of 3 mrad, which is to be expected since both targets were irradiated with an identical double-pulse structure at comparable intensities. It should be noted that the seed XRL signal is clipped by the target surface indicated by the dashed line. Since the Zr filters on this particular shot were chosen to favour the signal-to-noise ratio of the amplifier ASE, the almost-collimated seed XRL signal exceeds the dynamic range of the CCD camera. The XRL pulse energies of the seed and amplifier signal were estimated via the quantum efficiency of the CCD camera, the reflectivity of the flat mirror and the transmission of the Zr filters .
The shadow that is clearly visible in the footprint of the seed signal is caused by absorption from the amplifier plasma. When changing the delay between the IR pump beams we were able to synchronize the seed XRL beam with the gain lifetime of the amplifier medium. This method to study the XRL pulse duration and the temporal evolution of the gain has already been demonstrated using HH pulses as a seed source by Mocek et al. for OFI XRL  and Hasegawa et al. for solid target XRL . The results are illustrated in Fig. 2 , which show the footprint patterns of the seed XRL for different delays between −7.5 ps and + 7.0 ps with their corresponding line-outs. The latter have been obtained by vertical binning along the seed XRL signal.
Before −3 ps and after + 3 ps the amplifier medium attenuates the seed XRL, causing the shadow inside of the footprint. At −1.5 ps the shadow disappears, indicating the onset of the amplification process. The data at + 1.5 ps indicates that the part of the seed XRL beam passing through the amplifier medium is more intense than the surrounding part, which becomes even more obvious in the shot at 0 ps, where the amplified part of the seed XRL beam saturates the CCD camera. Interpreting the shots at ± 1.5 ps as the very onset of the amplification process one can expect a FWHM XRL pulse duration shorter than 3 ps. This fits well with the results obtained by Klisnick et al., who directly measured a Ag XRL pulse duration of 2 ps using a streak camera . In that experiment the Ag target was pumped with two separately focused pulses of similar intensities and duration and the same delay of 200 ps. Oshi et al. reported an Ag XRL pulse duration of 8 ps , directly measured with a streak camera. There the XRL target was also pumped by two pulses, but with a larger delay of 500 ps and more importantly with longer pulse durations of 400 ps and 4.8 ps for the pre- and the main pulse, respectively. Especially the longer main pulse duration contributes to a longer XRL pulse duration, as it was demonstrated by Dunn et al. . To our knowledge, the shortest XRL pulse duration measured up to today of 1 ps has been reported by Wang et al.  for a HH-seeded, Ne-like titanium XRL.
From our experimental data, it is worth mentioning that the optimum injection condition is confined to a very small time window. This demonstrates the necessity to synchronize a double-XRL seed- amplifier setup with sub-picosecond resolution, despite the pulse duration of the seed XRL pulse being significantly larger compared to HH seed sources. At the optimum timing condition, comparison of the power density of the amplified part of the seed yields a gain factor of 2. In this case, the energy of the amplified part of the seed XRL adds up to 1 µJ in a 0.3 mm x 0.4 mm beam. However, the energy content of the ASE emerging from the amplifier medium (5 mm x 0.8 mm) is of the same order of magnitude, indicating that the seeding process still needs be improved. One effect that could have reduced the seeding efficiency is that the seed XRL was refracted out of the gain zone before reaching the end of the amplification medium. Comparison of the position of the amplified seed to the target surface yields that the seed has been refracted by 6 mrad. This angle would correspond to a displacement of 36 µm after the 6 mm long target, which is comparable to the size of the gain zone. Therefore, one can estimate that refraction has limited the seeding efficiency. Another consequence of this effect would be that the medium length available for the amplifier ASE is reduced. This could explain the observed large divergence that exceeded the expected value given by the aspect ratio of the line focus.
Finally, it should be noted that the time-scale of the measurement series is not absolutely calibrated and the 0 ps delay does not coincide with the simultaneous arrival of the seed XRL and the heating pulse at the amplifier target.
Time-dependent Maxwell-Bloch simulations
For qualitative comparison and general conclusions on the relationship between the seed XRL pulse and the temporal development of the gain in the amplifier target we performed simulations with the 1D Maxwell-Bloch code DeepOne. The simulations rely on a Ne-like Zn medium, for which reliable atomic data is available from the Flexible Atomic Code FAC . In this simulation the seed- and the amplifier-target were irradiated with a double-pulse structure as it was used in the experiment described above. According to ARWEN simulations this will correspond to a mean temperature of 531 eV and a mean electron density of 1.1 × 1020 /cm3 in the gain zone of the seed and amplifier medium.
The DeepOne code distinguishes between two different scenarios. In the first case, any ASE effect is excluded and the population inversion is depleted only by the seed XRL pulse. This situation yields the amplified seed pulse of maximum energy, since the complete energy stored in the gain medium is transferred without any additional losses. In the second case, losses due to ASE are implemented. Although the one-dimensional code cannot directly distinguish between the energy stored in the seed and the ASE beam, the comparison of the temporal evolution in both cases allows for conclusions on the seeding efficiency. If there are substantial differences, the losses due to ASE are dominant and the seeding efficiency was low. If the pulse profiles are similar, losses due to ASE can be neglected. The energy stored in the gain medium has efficiently been transferred to the seed pulse. Thus, the deviation in the temporal evolution of the XRL output can serve as a measure of the seeding efficiency.
We performed several simulations with different delays between the arrival of the seed XRL and the IR pulse heating the amplifier target, both considering and excluding the losses due to ASE. In these simulations, the zero-position of the timescale was chosen so that the arrival of the 2 ps FWHM IR heating pulse of the amplifier target takes place at 5 ps. The arrival time of the 3 ps long seed pulse was varied and its intensity was matched to the experimental conditions. Figures 3(a) and 3(b) show the temporal evolution of the output XRL pulse for seeding 0 ps and 2 ps before the IR pulse arrives, taking into account only the seed pulse (dotted red lines) and the seed pulse together with losses due to ASE (blue lines). In Fig. 3(a), the two temporal profiles are substantially different. A strong ASE mixed with the seed XRL is present, dominating the 1.5 µJ output. Due to the low energy (360 nJ) of the seed pulse, the ASE is mixed with a low intensity XRL seed (1.5 × 106 W/cm2 at maximum) and they are amplified together. Seeding earlier allows the seed XRL to be amplified before the ASE starts to develop. The seed pulse reaches higher intensity and dominates the amplification process. This can be seen in Fig. 3(b), where the seed pulse arrives at the amplifier medium 2 ps before the maximum of the heating pulse. In this case, the simulation yields a pulse energy of 1.6 µJ for the amplified seed XRL. The predicted output energy of 1.6 µJ is larger than the energy measured in the experiment. Refraction of the amplified beam can explain this difference. With the electron density gradient computed in , 2.4 × 1022 /cm4, it is possible to estimate that 14% of the energy, i.e. 0.2 µJ, will be lost.
Nevertheless, since the laser line width is Gaussian, it is expected that this gradient will be higher , increasing the energy loss by refraction. If the electron density gradient was 2.5 times higher (e.g. 6.0 × 1022 /cm4), the experimental result of 1 µJ would be reproduced. This time (−2 ps) is the optimum time for an efficient amplification of the seed XRL in the following sense. As explained earlier, the difference in the temporal evolution in the two scenarios can be interpreted as a measure of the seeding efficiency. Thus, the optimum timing between the arrival of the seed XRL and the IR heating pulse will be given when the two curves are identical. In Fig. 3(c) the calculated standard deviation of the temporal profiles is shown for different delays, illustrating that the minimum standard deviation (maximum efficiency of the seeding process) is obtained when the 3 ps long seed pulse arrives 2 ps before the maximum of the heating pulse. Furthermore, it can be seen that the timing window for optimum injection is confined to about 2 ps.
The code predicts a longer gain duration compared to the experimental data. This can be expected since the code uses a simplified atomic model that assumes that the lasing ions are always present in the plasma, which is not the case in the experiment. Thus, the experimental data obtained can be used to calibrate and improve the code, including some ad hoc reduction of the gain. We adjusted our code to reduce progressively the population inversion to mimic the recombination of the ions inside the plasma and then predict the gain duration observed experimentally. We found that starting to reduce the population inversion 1 ps after the arrival of the main pulse gives us a maximum gain duration of less than 3 ps, in good agreement with the experimental results.
As explained, these experimental results are a valuable tool for testing and calibrating the DeepOne code. Therefore, results from future experiments utilizing the Butterfly configuration can provide a valuable input for its further development. The absolute calibration of the time scale used in the measurements would be highly interesting for a comparison with the simulated results. For this purpose, techniques as demonstrated by Mocek et al.  or Staub et al.  could be considered for future experiments.
Conclusion and outlook
In conclusion, a new experimental design to pump two independent XRL in the transient collisionally excited (TCE) scheme has been developed. Both XRL media benefit from the DGRIP technique and travelling wave excitation. Using the two XRL targets in a seed-amplifier configuration demonstrated signs of amplification and allowed for determining the lifetime of the transient gain in a Ni-like silver medium within the range of 3 ps. For qualitative comparison, simulations performed by the Maxwell-Bloch code DeepOne have been carried out. The results confirm that the amplification process should be clearly measureable, even when taking into account the large injection losses due to the spatial mismatch. As a further result, it is predicted that the most efficient energy extraction of the amplifier medium will take place when the several ps long seed XRL pulse arrives at the amplifier target shortly before the maximum of the IR pumping pulse is reached.
Further investigations with this Butterfly geometry will be carried out using molybdenum as target material. The reduced pumping energy necessary for the Mo XRL operation allows us to run the experiment at a higher repetition rate and thus to use an injection model with an increased spatial coupling efficiency. In addition, the increase in repetition rate will allow for a measurement of the spatial coherence of the amplified seed. Due to the limited number of shots this has not been performed in the experiment described in this Letter. Temporally resolved measurements of the energy ratio of the seed XRL and the ASE of the amplifier will allow for drawing more detailed conclusions on the temporal evolution of the gain, which can be compared with the results of the DeepOne code. Finally, the setup is suitable for performing seed-/ amplifier experiments as proposed in , where simulations with gain media pumped by exceptionally large line foci yielded promising results.
References and links
1. D. C. Hochhaus, J. Seres, B. Aurand, B. Ecker, B. Zielbauer, D. Zimmer, C. Spielmann, and T. Kühl, “Tuning the high-order harmonic lines of a NdGlass laser for soft X-ray laser seeding,” Appl. Phys. B 100(4), 711–716 (2010). [CrossRef]
2. Ph. Zeitoun, G. Faivre, S. Sebban, T. Mocek, A. Hallou, M. Fajardo, D. Aubert, P. Balcou, F. Burgy, D. Douillet, S. Kazamias, G. De Lachèze-Murel, T. Lefrou, S. Le Pape, P. Mercère, H. Merdji, A. S. Morlens, J. P. Rousseau, and C. Valentin, “A high-intensity highly coherent soft X-ray femtosecond laser seeded by a high harmonic beam,” Nature 431(7007), 426–429 (2004). [CrossRef] [PubMed]
3. Y. Wang, E. Granados, F. Pedaci, D. Alessi, B. Luther, M. Berrill, and J. J. Rocca, “Phase-coherent, injection-seeded, table-top soft-X-ray lasers at 18.9 nm and 13.9 nm,” Nat. Photonics 2(2), 94–98 (2008). [CrossRef]
4. S. Sebban, H. Daido, N. Sakaya, Y. Kato, K. Murai, H. Tang, Y. Gu, G. Huang, S. Wang, A. Klisnick, Ph. Zeitoun, F. Koike, and H. Takenaka, “Full characterization of a high-gain saturated x-ray laser at 13.9 nm,” Phys. Rev. A 61(4), 043810 (2000). [CrossRef]
5. M. Nishikino, M. Tanaka, K. Nagashima, M. Kishimoto, M. Kado, T. Kawachi, K. Sukegawa, Y. Ochi, N. Hasegawa, and Y. Kato, “Demonstration of a soft-x-ray laser at 13.9 nm with full spatial coherence,” Phys. Rev. A 68(6), 061802 (2003). [CrossRef]
6. M. Tanaka, M. Nishikino, T. Kawachi, N. Hasegawa, M. Kado, M. Kishimoto, K. Nagashima, and Y. Kato, “X-ray laser beam with diffraction-limited divergence generated with two gain media,” Opt. Lett. 28(18), 1680–1682 (2003). [CrossRef] [PubMed]
7. D. Zimmer, B. Zielbauer, V. Bagnoud, U. Eisenbarth, D. Javorkova, and T. Kuehl, “An improved double-pulse non-normal incidence pumping geometry for transient collisionally excited soft X-ray lasers,” Opt. Express 16(14), 10398–10404 (2008). [CrossRef] [PubMed]
8. D. Ros, K. Cassou, B. Cros, S. Daboussi, J. Demailly, O. Guilbaud, S. Kazamias, J.-C. Lagron, G. Maynard, O. Neveu, M. Pittman, B. Zielbauer, D. Zimmer, T. Kuhl, S. Lacombe, E. Porcel, M.-A. du Penhoat, P. Zeitoun, and G. Mourou, “LASERIX: An open facility for developments of EUV and soft X-ray lasers and applications - Developments of XUV sources using high power laser facilities: ILE, ELI,” Nucl. Instrum. Methods Phys. Res., Sect. A 653(1), 76–79 (2011). [CrossRef]
9. C. Imesch, F. Staub, and J. E. Balmer, “Gain-saturated Ni-like antimony laser at 11.4 nm in grazing-incidence pumping geometry,” Opt. Commun. 283(1), 66–70 (2010). [CrossRef]
10. R. Keenan, J. Dunn, P. K. Patel, D. F. Price, R. F. Smith, and V. N. Shlyaptsev, “High-repetition-rate grazing-incidence pumped X-Ray laser operating at 18.9 nm,” Phys. Rev. Lett. 94(10), 103901 (2005). [CrossRef] [PubMed]
11. E. Oliva, P. Zeitoun, M. Fajardo, G. Lambert, D. Ros, S. Sebban, and P. Velarde, “Comparison of natural and forced amplification regimes in plasma-based soft-x-ray lasers seeded by high-order harmonics,” Phys. Rev. A 84(1), 013811 (2011). [CrossRef]
12. B. L. Henke, E. M. Gullikson, and J. C. Davis, “X-Ray Interactions: Photoabsorption, scattering, transmission, and reflection at E = 50-30,000 eV, Z = 1-92,” At. Data Nucl. Data Tables 54(2), 181–342 (1993). [CrossRef]
13. T. Mocek, S. Sebban, G. Maynard, Ph. Zeitoun, G. Faivre, a. Hallou, M. Fajardo, S. Kazamias, B. Cros, D. Aubert, G. de Lachèze-Murel, J. Rousseau, and J. Dubau, “Absolute time-resolved X-Ray laser gain measurement,” Phys. Rev. Lett. 95, 19–22 (2005).
14. N. Hasegawa, T. Kawachi, A. Sasaki, M. Kishimoto, K. Sukegawa, M. Tanaka, R. Z. Tai, Y. Ochi, M. Nishikino, K. Nagashima, and Y. Kato, “Direct measurement of the temporal profile of the amplification gain of the transient collisional excitation neonlike manganese x-ray laser medium,” Phys. Rev. A 76(4), 043805 (2007). [CrossRef]
15. A. Klisnick, J. Kuba, D. Ros, R. Smith, G. Jamelot, C. Chenais-Popovics, R. Keenan, S. Topping, C. Lewis, F. Strati, G. Tallents, D. Neely, R. Clarke, J. Collier, A. MacPhee, F. Bortolotto, P. Nickles, and K. Janulewicz, “Demonstration of a 2-ps transient X-ray laser,” Phys. Rev. A 65(3), 033810 (2002). [CrossRef]
16. M. Kado, A. Sasaki, K. Nagashima, K. Sukegawa, M. Kishimoto, M. Tanaka, M. Nishikino, Y. Ochi, T. Kawachi, and N. Hasegawa, “Measurement of temporal durations of transient collisional excitation X-ray lasers,” Appl. Phys. B 78(7-8), 961–963 (2004). [CrossRef]
17. J. Dunn, R. F. Smith, R. Shepherd, and R. Booth, “Pulse duration measurements of a picosecond laser-pumped 14.7 nm X-ray laser,” 9th International Conference on X-ray Lasers (2004).
18. Y. Wang, M. Berrill, F. Pedaci, M. Shakya, S. Gilbertson, Z. Chang, E. Granados, B. Luther, M. Larotonda, and J. Rocca, “Measurement of 1-ps soft-X-ray laser pulses from an injection-seeded plasma amplifier,” Phys. Rev. A 79(2), 023810 (2009). [CrossRef]
19. M. Gu, “The Flexible Atomic Code,” AIP Conf. Proc. 730, 127–136 (2004), doi:. [CrossRef]
20. F. Ogando and P. Velarde, “Development of a radiation transport fluid dynamic code under AMR scheme,” J. Quant. Spectrosc. Radiat. Transfer. 71(2-6), 541 (2001). [CrossRef]
21. K. Cassou, Ph. Zeitoun, P. Velarde, F. Roy, F. Ogando, M. Fajardo, G. Faivre, and D. Ros, “Transverse spatial improvement of a transiently pumped soft-X-ray amplifier,” Phys. Rev. A 74(4), 045802 (2006). [CrossRef]
22. F. Staub, M. Braud, J. Balmer, and J. Nilsen, “Absolute timing measurements of the Ni-like Pd and Sn soft-x-ray lasers,” Phys. Rev. A 72(4), 043813 (2005). [CrossRef]
23. E. Oliva, P. Zeitoun, P. Velarde, M. Fajardo, K. Cassou, D. Ros, S. Sebban, D. Portillo, and S. le Pape, “Hydrodynamic study of plasma amplifiers for soft-x-ray lasers: a transition in hydrodynamic behavior for plasma columns with widths ranging from 20 μm to 2 mm,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 82(5), 056408 (2010). [CrossRef] [PubMed]