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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 6 — Mar. 17, 2008
  • pp: 3546–3559
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X-ray lasers from Inner-shell transitions pumped by the Free-electron laser

J. Zhao, Q. L. Dong, S. J. Wang, L. Zhang, and J. Zhang  »View Author Affiliations


Optics Express, Vol. 16, Issue 6, pp. 3546-3559 (2008)
http://dx.doi.org/10.1364/OE.16.003546


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Abstract

We present a approach of generating femtosecond coherent x-ray pulses by using the self-amplified free-electron laser (SASE FEL) to pump the inner-shell x-ray lasers (ISXRL’s). Theoretical simulations are performed. The gain characteristics are analyzed for the two representative schemes of inner-shell x-ray transitions, ie. the self-terminated x-ray lasing (1s)-1→(2p)-1 (λ = 4.5nm) in carbon (Z=6) and the quasi-stationary x-ray lasing (2p)-1→(3s)-1 (λ = 4.1nm) in calcium (Z = 20). When the 10fs x-ray FEL pulses are available at 284eV and 360eV with the pumping intensities of 1.2×1015W/cm2 and 2×1017W/cm2 for C and Ca, respectively, a net gain of 140cm-1 can be predicted. Using a one-dimensional model, the properties of output ISXRL’s are studied. By the Carbon ISXRL scheme, the multi-spiky SASE FEL x-ray pulse with chaotic temporal structure is smoothed to a temporally continuous x-ray pulse with a comparable duration but at a different wavelength. The Calcium scheme, can be used to create one single x-ray laser pulse with a duration as short as 2fs. The spectral bandwidth of the output ISXRL’s is an order of 10-3, which is one order narrower than that of the SASE FEL’s.

© 2008 Optical Society of America

1. Introduction

Free-electron laser (FEL) facilities based on the self-amplified spontaneous emission (SASE) principle have great potential of offering high-brightness, laser-like x-ray radiations at the short wavelength down to 0.085nm and with ultra-short time durations less than 100fs. Recently, the pilot facility free-electron lasers in Hamburg (FLASH) established at the DESY research center has been able to achieve ultra-short (10~50fs), high-power radiations at tunable fundamental wavelengths from 6 to 30nm [1

1. E. N. Ragozin and I. I. Sobel’man, “Laser sources in the soft X-ray spectral region,” Physics-Uspekhi 48(12), 1249–1250 (2005). [CrossRef]

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2. W. Ackermann, G. Asova, V. Ayvazyan, A. Azima, N. Baboi, J. Bahr, V. Balandin, B. Beutner, A. Brandt, A. Bolzmann, R. Brinkmann, O. I. Brovko, M. Castellano, P. Castro, L. Catani, E. Chiadroni, S. Choroba, A. Cianchi, J. T. Costello, D. Cubaynes, J. Dardis, W. Decking, H. Delsim-Hashemi, A. Delserieys, G. Di Pirro, M. Dohlus, S. Dusterer, A. Eckhardt, H. T. Edwards, B. Faatz, J. Feldhaus, K. Flottmann, J. Frisch, L. Frohlich, T. Garvey, U. Gensch, C. Gerth, M. Gorler, N. Golubeva, H. J. Grabosch, M. Grecki, O. Grimm, K. Hacker, U. Hahn, J. H. Han, K. Honkavaara, T. Hott, M. Huning, Y. Ivanisenko, E. Jaeschke, W. Jalmuzna, T. Jezynski, R. Kammering, V. Katalev, K. Kavanagh, E. T. Kennedy, S. Khodyachykh, K. Klose, V. Kocharyan, M. Korfer, M. Kollewe, W. Koprek, S. Korepanov, D. Kostin, M. Krassilnikov, G. Kube, M. Kuhlmann, C. L. S. Lewis, L. Lilje, T. Limberg, D. Lipka, F. Lohl, H. Luna, M. Luong, M. Martins, M. Meyer, P. Michelato, V. Miltchev, W. D. Moller, L. Monaco, W. F. O. Muller, O. Napieralski, O. Napoly, P. Nicolosi, D. Nolle, T. Nunez, A. Oppelt, C. Pagani, R. Paparella, N. Pchalek, J. Pedregosa-Gutierrez, B. Petersen, B. Petrosyan, G. Petrosyan, L. Petrosyan, J. Pfluger, E. Plonjes, L. Poletto, K. Pozniak, E. Prat, D. Proch, P. Pucyk, P. Radcliffe, H. Redlin, K. Rehlich, M. Richter, M. Roehrs, J. Roensch, R. Romaniuk, M. Ross, J. Rossbach, V. Rybnikov, M. Sachwitz, E. L. Saldin, W. Sandner, H. Schlarb, B. Schmidt, M. Schmitz, P. Schmuser, J. R. Schneider, E. A. Schneidmiller, S. Schnepp, S. Schreiber, M. Seidel, D. Sertore, A. V. Shabunov, C. Simon, S. Simrock, E. Sombrowski, A. A. Sorokin, P. Spanknebel, R. Spesyvtsev, L. Staykov, B. Steffen, F. Stephan, F. Stulle, H. Thom, K. Tiedtke, M. Tischer, S. Toleikis, R. Treusch, D. Trines, I. Tsakov, E. Vogel, T. Weiland, H. Weise, M. Wellhoffer, M. Wendt, I. Will, A. Winter, K. Wittenburg, W. Wurth, P. Yeates, M. V. Yurkov, I. Zagorodnov, and K. Zapfe, “Operation of a free-electron laser from the extreme ultraviolet to the water window,” Nat. Photon 1(6), 336–342 (2007). [CrossRef]

]. With the high brilliance output radiation at the soft x-ray regime, major breakthroughs could be realized in fields of many natural sciences [3

3. S. P. Hau-Riege, H. N. Chapman, J. Krzywinski, R. Sobierajski, S. Bajt, R. A. London, M. Bergh, C. Caleman, R. Nietubyc, L. Juha, J. Kuba, E. Spiller, S. Baker, R. Bionta, K. S. Tinten, N. Stojanovic, B. Kjornrattanawanich, E. Gullikson, E. Plonjes, S. Toleikis, and T. Tschentscher, “Subnanometer-scale measurements of the interaction of ultrafast soft X-ray free-electron-laser pulses with matter,” Phys. Rev. Lett 98(14), 145502–145505 (2007). [CrossRef] [PubMed]

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4. H. N. Chapman, A. Barty, M. J. Bogan, S. Boutet, M. Frank, S. P. Hau-Riege, S. Marchesini, B. W. Woods, S. Bajt, H. Benner, R. A. London, E. Plonjes, M. Kuhlmann, R. Treusch, S. Dusterer, T. Tschentscher, J. R. Schneider, E. Spiller, T. Moller, C. Bostedt, M. Hoener, D. A. Shapiro, K. O. Hodgson, D. Van der Spoel, F. Burmeister, M. Bergh, C. Caleman, G. Huldt, M. M. Seibert, F. R. N. C. Maia, R. W. Lee, A. Szoke, N. Timneanu, and J. Hajdu, “Femtosecond diffractive imaging with a soft-X-ray free-electron laser,” Nat. Phys 2(12), 839–843 (2006). [CrossRef]

]. However, the experimental and theoretical investigations have shown a limit temporal coherence of the output fundamental harmonic radiations of SASE FEL’s. The temporal profile of the output FEL pulses is chaotic with many isolated, randomly arriving “spikes”, with typical durations of few femtoseconds to sub-fs [5

5. Y. L. Li, J. Lewellen, Z. R. Huang, V. Sajaev, and S. V. Milton, “Time-resolved phase measurement of a self-amplified free-electron laser,” Phys. Rev. Lett 89(23), 234801–234804 (2002). [CrossRef] [PubMed]

, 6

6. Y. L. Li, S. Krinsky, J. W. Lewellen, K. J. Kim, V. Sajaev, and S. V. Milton, “Characterization of a chaotic optical field using a high-gain, self-amplified free-electron laser,” Phys. Rev. Lett 91(24), 243602–243605 (2003). [CrossRef] [PubMed]

, 7

7. E. L. Saldin, E. A. Schneidmiller, and M. V. Yurkov, “Properties of the third harmonic of the radiation from self-amplified spontaneous emission free electron laser,” Phys. Rev. Spec. Top. Accel. Beams 9(3), 030702–030710 (2006). [CrossRef]

, 8

8. R. Treusch and J. Feldhaus, “SASE free electron lasers as short wavelength coherent sourcesFrom first results at 100 nm to a 1 angstrom X-ray laser,” Eur. Phys. J. D 26(1), 119–122 (2003). [CrossRef]

]. This is due to the intrinsic noisy start-up of the SASE FEL radiations and precludes their straight-forward use in the pump-probe experiments. Approaches were proposed for the generation of single-spike coherent femtosecond or even subfemtosecond (400–600as) x-ray pulses by selectively spoiling the transverse emittance of the electron beam [9

9. P. Emma, K. Bane, M. Cornacchia, Z. Huang, H. Schlarb, G. Stupakov, and D. Walz, “Femtosecond and sub-femtosecond X-ray pulses from a self-amplified spontaneous-emission-based free-electron laser,” Phys. Rev. Lett 92(7), 074801–074804 (2004). [CrossRef] [PubMed]

] or by constructing a multistage high gain high harmonic FEL [10

10. C. B. Schroeder, C. Pellegrini, S. Reiche, J. Arthur, and P. Emma, “Chirped-beam two-stage SASE-FEL for high power femtosecond X-ray pulse generation,” Nucl. Instrum. Meth. A 483(1–2), 89–93 (2002). [CrossRef]

].

Here we show that the femtosecond coherent inner-shell x-ray lasers (ISXRL’s) can be pumped by the SASE x-ray FEL laser pulses through the photoionization processes. Since Duguay and Rentzepis [11

11. M. A. Duguay and G. P. Rentzepis, “Some approaches to vacuum uv and x-ray lasers,” Appl. Phys. Lett 10, 350–352 (1967). [CrossRef]

] first proposed the ISXRL scheme in 1967 for the potential of generating coherent ultra-short (~10fs) x-ray lasing at short wavelengths within the “water window” (2.3–4.4nm), there have been studies of producing possible large population inversion of the inner-shell transitions in various atomic systems pumped by different sources, such as the black-body or Lamor radiations and the electron beams [12

12. E. J. McGuire, “Soft-X-Ray Amplified Spontaneous Emission via the Auger Effect,” Phys. Rev. Lett. 35(13), 844–848 (1975). [CrossRef]

, 13

13. R. C. Elton, “Quasi-stationary populatio inversion on Ka transitions,” Appl. Opt. 14, 2243–2249 (1975). [CrossRef] [PubMed]

, 14

14. H. C. Kapteyn, “Photoionization-pumped x-ray lasers using ultrashort-pulse excitation,” Appl. Opt 31(24), 4931–4939 (1992). [CrossRef] [PubMed]

, 15

15. D. C. Eder, “Tabletop x-ray laser,” Phys. Plasmas 1, 1744–1752 (1994). [CrossRef]

, 16

16. S. J. Moon and D. C. Eder, “Theoretical investigation of an ultrashort-pulse coherent x-ray source at 45 angstrom,” Phys. Rev. A 57(2), 1391–1394 (1998). [CrossRef]

, 17

17. K. Moribayashi, A. Sasaki, and T. Tajima, “X-ray emission by ultrafast inner-shell ionization from vapors of Na, Mg, and Al,” Phys. Rev. A 59(4), 2732–2737 (1999). [CrossRef]

, 18

18. D. Kim, C. Tóth, and C. P. J. Barty, “Population inversion between atomic inner-shell vacancy states created by electron-impact ionization and Coster-Kronig decay,” Phys. Rev. A 59(6), R4129–R4132 (1999). [CrossRef]

, 19

19. E. Fill, D. Eder, K. Eidmann, J. Meyer-ter Vehn, G. Pretzler, A. Pukhov, and A. Saemann, “Relativistic plasma pumping of x-ray lasers,” in Proceeding of 6th international conference on X-ray lasersY. Kato, H. Takuma, and H. Daido, ed., (Institute of Physics, London, 1998) pp. 301–308.

, 20

20. S. M. Hooker, “Inner-shell soft X-ray lasers in Ne-like ions driven by optical field ionization,” Opt. Commu 182(1-3), 209–219 (2000). [CrossRef]

, 21

21. J. S. Liu, R. X. Li, Z. Z. Xu, and J. G. Liu, “Approximately analytical model for inner-shell photoionization x-ray lasers in low-Z elements,” Phys. Rev. A 6303(3), 033809–033815 (2001). [CrossRef]

, 22

22. D. Kim, S. H. Son, J. H. Kim, C. Toth, and C. P. J. Barty, “Gain characteristics of inner-shell photonization-pumped L23M1 transition in Ca,” Phys. Rev. A 63, 023806–023811 (2001). [CrossRef]

]. Complex inner-shell atomic processes involved in the ISXRL’s were studied integrally. The Kα inner-shell lasing has higher fluorescence efficiency than the ML or higher shell transitions. The K-line shift due to multiple-L vacancies in ions of elements with z≤20 helps reduce the resonance reabsorption of the KL laser radiations [13

13. R. C. Elton, “Quasi-stationary populatio inversion on Ka transitions,” Appl. Opt. 14, 2243–2249 (1975). [CrossRef] [PubMed]

]. A quasi-steady population inversion can be expected due to the faster Coster-Kronig decay rate of the outer-shell vacancies than the Auger decay rate of the inner-shell vacancies. However, the ISXRL has not been demonstrated experimentally, because of the huge reabsorption of the stimulated line emission in propagation and the critical requirement for the high intense pumping source with short rising time due to the fast decay of inner-shell vacancies. The high brilliant SASE x-ray FEL pulses with ultra-short durations make an ideal photoionization pumping source for the high gain ISXRL’s. The FEL photo-pumped XUV lasers of Helium and Lithium at (λ = 58.4nm) and (λ = 19.9nm) have been theoretically investigated by Lan et al, using the photoionization with subsequent three-body recombination scheme and the direct inner-shell ionization of the K-shell electron with transient inversion scheme [23

23. K. Lan, E. E. Fill, and J. Meyer-Ter-Vehn, “Simulation of He-alpha and Ly-alpha soft X-ray lasers in helium pumped by DESY/XFEL-radiation,” Europhys. Lett 64(4), 454–460 (2003). [CrossRef]

, 24

24. K. Lan, E. E. Fill, and J. Meyer-Ter-Vehn, “Photopumping of XUV lasers by XFEL radiation,” Laser. Part. Beams 22, 261–266 (2004). [CrossRef]

]. However, in helium or lithium with only a single outer electron, the fast atomic processes like the Auger decay were not included. The results can not be deduced to other ISXRL systems with complex inner-shell atomic processes and shorter wavelengths.

Fig. 1. (Color online) Schematic diagram of the main inner-shell vacancy levels and atomic processes included in the pumping of inner-shell x-ray lasers.

In this work, we studied theoretically two characteristic FEL photo-ionization pumped ISXRL systems, carbon and calcium, with representative inner-shell x-ray transitions of Kα (1s)-1→(2p)-1 (λ = 4.5nm) and L 23 M 1 (2p)-1→(3s)-1 (λ = 4.1nm), respectively [18

18. D. Kim, C. Tóth, and C. P. J. Barty, “Population inversion between atomic inner-shell vacancy states created by electron-impact ionization and Coster-Kronig decay,” Phys. Rev. A 59(6), R4129–R4132 (1999). [CrossRef]

, 21

21. J. S. Liu, R. X. Li, Z. Z. Xu, and J. G. Liu, “Approximately analytical model for inner-shell photoionization x-ray lasers in low-Z elements,” Phys. Rev. A 6303(3), 033809–033815 (2001). [CrossRef]

, 22

22. D. Kim, S. H. Son, J. H. Kim, C. Toth, and C. P. J. Barty, “Gain characteristics of inner-shell photonization-pumped L23M1 transition in Ca,” Phys. Rev. A 63, 023806–023811 (2001). [CrossRef]

]. Numerical simulations are performed using a model developed in the paper. Temporal evolutions of the population of relevant states and the gain coefficients of inner-shell transitions are investigated, pumped by FEL pulses with different durations and intensities. Moreover, the gain performances of ISXRL’s pumped by multi-spiky FEL pulses are presented. With the assumption of a travelling wave pump, we predict that single-spike femtosecond innershell x-ray pulses can be resulted, with a coherence time of 10fs for the self-terminated carbon ISXRL system, and 3fs for the quasi-stationary calcium system.

2. Model Analysis

The population of the upper state is mostly depleted through the subsequent Auger decay, while the energetic Auger electrons can collisionally ionize the outer-shell electrons of the neutral atoms, 2p for C or 3s for Ca, increasing the population of the ions at the lower state. This destroys the inversion for the (1s)-1→(2p)-1 (λ = 4.5nm) transition in carbon and makes its Kα radiation a self-terminated process. However, the lower state (3s)-1 of calcium undergoes the Coster-Kronig decay process faster than the decay of the deeper vacancy state (2p)-1. A quasi-stationary population inversion can be expected for the L 23 M 1 transition in calcium.

In our study, the inner-shell-vacancy levels (IVL’s) of three ionic stages (initial neutral, single ionized, double ionized) are considered [18

18. D. Kim, C. Tóth, and C. P. J. Barty, “Population inversion between atomic inner-shell vacancy states created by electron-impact ionization and Coster-Kronig decay,” Phys. Rev. A 59(6), R4129–R4132 (1999). [CrossRef]

, 22

22. D. Kim, S. H. Son, J. H. Kim, C. Toth, and C. P. J. Barty, “Gain characteristics of inner-shell photonization-pumped L23M1 transition in Ca,” Phys. Rev. A 63, 023806–023811 (2001). [CrossRef]

, 21

21. J. S. Liu, R. X. Li, Z. Z. Xu, and J. G. Liu, “Approximately analytical model for inner-shell photoionization x-ray lasers in low-Z elements,” Phys. Rev. A 6303(3), 033809–033815 (2001). [CrossRef]

]. Single IVL’s are considered for the depletion of the neutral atoms through the photoionization of inner-shell electrons and the electron collisional ionization of outer-shell electrons. Double IVL’s are considered to include the decay channels from the single IVL’s, such as photoionization, electron collisional ionization, and Auger or Coster-Kronig decay processes. The gain of the inner-shell x-ray lasers is calculated through the rate equations concerning the relevant transitions:

N0˙=N0(Rtotale+Rtotalp)
(1)
N2˙=N0(R02e+R02p)D2N2
(2)
N1˙=N0(R01e+R01p)D1N1
(3)

The ground state of the neutral atom is represented by the state 0, and the upper and the lower states of the single ionic stage by the states 2 and 1, respectively. Nk is the population for each state k; R0ke=neσe0kve is the electron impact ionization rate of the monoenergetic Auger electrons, R0kp=Iσ(v)/hv the photoionization rate with the spectral bandwidth of the FEL Δν/ν≪1 considered; Rtotal is the total rates summed over all the single ion levels; Dk the decay rate including radiative, Auger and Coster-Kronig processes, photo-ionization and electron collisional ionization from single ions to all the possible double ions. The gain of x-ray lasing between the inner-shell vacancies is given by Ginv = σg(N 2-gN 1). g = g 2/g 1 is the ratio of the statistical weights and σg=Aulλ38πcΔλλ is the stimulated emission cross section. Because of the short life time of the inner-shell vacancies, natural broadening is rather significant for the calculation of Δλ/λ [14

14. H. C. Kapteyn, “Photoionization-pumped x-ray lasers using ultrashort-pulse excitation,” Appl. Opt 31(24), 4931–4939 (1992). [CrossRef] [PubMed]

].

The effective gain Geff is the difference between Ginv and αabs the absorption of the lasing line by neutrals and ions: Geff = Ginv-αabs. The photoabsorption loss of the lasing line includes all the photoionization and photoexcitation processes in ions and neutral atoms. The absorption coefficient is theoretically given by αabs()=∑i(∑tt Nitσ itt()+∑tNitαit()) [25

25. J. L. Zeng, F. T. Jin, J. M. Yuan, Q. S. Lu, and Y. S. Sun, “Detailed-term-accounting-approximation simulation of x-ray transmission through laser-produced Al plasmas,” Phys. Rev. E 62(5), 7251–7257 (2000). [CrossRef]

]. Nit is the population density for term t of ion stage i, σ itt is the cross section for the bound-bound photoexcitation transition from term t to t′, and σit is the photoionization cross section from term t, is the photon energy of the inner-shell x-ray lasing.

However, the natural energy widths of the monochromatic ISXRL’s are only 0.064eV for Kα transition in CII and 0.17eV for L 23 M 1 transition in CaII. The shifted-energy in CIII and CaIII of the same electron transitions is 5eV and 2eV, respectively. So the absorption due to the photoexcitation in other ions can be ignored and the absorption coefficient in our calculation can be approximated as αabs(cm -1)=µ(cm 2/gm) *ρ(gm/cm 3). µ(cm 2/gm) is the photoabsorption cross section taken from a synthesis of the currently available experimental data and the theoretical calculations in Ref. [26

26. B. L. Henke, E. M. Gullikson, and J. C. Davis, “X-Ray Interactions: Photoabsorption, Scattering, Transmission, and Reflection at E=50-30,000eV,Z=1-92,” Atomic Data and Nucl. Data Tables 54, 181–342 (1993). [CrossRef]

]. The 2s (for C) and 3p (for Ca) electrons have 4~5 times larger photoionization cross sections than that of the outer (2p in carbon and 4s in calcium) electrons. As long as there are 2s or 3p electrons in the lasant ions, the approximation of the photoabsorption by the photoionization of the inner-shell x-ray lasers is appropriate.

In the numerical simulations, we use the Flexible Atomic Code (FAC) [27

27. M. F. Gu, “Indirect X-Ray Line-Formation Processes in Iron L-Shell Ions,” Astrophys. J 582, 1241–1250 (2003). [CrossRef]

] to generate the atomic data like the level energies (j-j coupling), and the rates of the photoionization, the electron collisional ionization and the radiative decay. This code is based on the relativistic configuration interaction with independent particle basis wavefunctions. Other inner-shell atomic processes Auger or Coster-Kronig rates and total decay rates (line widths) are taken from the calculations carried out by McGuire [28

28. E. J. McGuire, “K-Shell Auger Transition Rates and Fluorescence Yields for Elements Be-Ar,” Phys. Rev. 185(1), 1–6 (1969). [CrossRef]

, 29

29. E. J. McGuire, “Atomic L-Shell Coster-Kronig, Auger, and Radiative Rates and Flourescence Yields for Na-Th,” Phys. Rev. A 3(2), 587–594 (1971). [CrossRef]

, 30

30. E. J. McGuire, “M-Shell Auger and Coster-Kronig Electron Spectra,” Phys. Rev. A 5(3), 1052–1059 (1972). [CrossRef]

]. Notice that we are talking about inner-shell atomic processes with a time scale of only tens of femtoseconds, other slow effects like the ion-ion collision and the electron-ions 3-body recombination on picosecond or even longer time scale are not important and not included in this model.

3. ISXRL’s pumped by ideally smooth FEL x-ray pulses

Capable of providing the high intense, ultra-short (10~50fs) x-ray radiations at wavelength from 6 to 30nm, the free-electron laser FLASH (or XFEL in proposal) makes the ideal pumping source for the generation of coherent ultra-short (~10fs) inner-shell x-ray laser pulses. Third harmonics of the fundamental frequency have photon energies of 284 and 360eV, right above the photoionization threshold of the inner-shell 1s, 2p electrons of carbon and calcium, which produces 5 to 10 times larger inner-shell photoionization cross sections than those of the outer shell electrons.

To test the simulation model, the inner-shell x-ray lasers are first pumped by the x-ray FEL pulses without random spikes. The temporal intensity profile is assumed as [23

23. K. Lan, E. E. Fill, and J. Meyer-Ter-Vehn, “Simulation of He-alpha and Ly-alpha soft X-ray lasers in helium pumped by DESY/XFEL-radiation,” Europhys. Lett 64(4), 454–460 (2003). [CrossRef]

]

I(t)={Ipsin2(πt2τ)0t2τ0t>2τ
(4)

where t is the time, τ is the full width at half-maximum intensity (FWHM) and Ip is the peak value at t = τ. Simulations are performed under different pumping conditions.

3.1. Self-terminated inner-shell lasing in Carbon

For the calculation of the self-terminated (1s)-1→(2p)-1 (λ = 4.5nm) inner-shell x-ray transition in carbon, the energy level diagram includes one neutral level, three single IVL’s and eight double IVL’s. When energy differences for levels are small, they are treated as one level. Figure. 2 shows the ion population and the gain evolution for the FEL radiation at 284eV with a peak intensity of 3×1014W/cm2 and FWHM of τ = 20fs. The initial neutral density of carbon is set as 2.28×1021cm-3. The pumping pulse starts at time t = 0. The time evolution of the populations of single IVL’s (1s)-1, (2s)-1, (2p)-1 is shown in Fig. 2(a). Fig. 2(b) shows that Ginv is obtained from the beginning of the third harmonic FEL pulses because the photoionization cross section of the inner-shell upper state (1s)-1 is larger than that of the out-shell lower state (2p)-1. The upper state with one 1s inner-shell hole decays mainly through the Auger process (10fs)-1 with no radiation. The generated Auger electrons with an energy of 280eV ionize the neutral carbon atoms to the lower state (2p)-1, which is deleterious to the population inversion. The value of Ginv reaches its maximum (140cm-1) at the peak time (t = 20fs) of the pumping radiation and rapidly decreases when excessive Auger electrons are produced.

We also investigated the influence of the FEL pulse width on the gain performance of the self-terminated (1s)-1→(2p)-1 (λ = 4.5nm) transition in carbon. Figure. 3 shows the temporal evolution of Ginv along with FEL pumping lasers profiles of different pulse durations. The gain reaches its maximum before (for 2τ = 60fs) or after (for 2τ = 10, 20fs) the peak of the x-ray pulses, while the gain maximum happens at the same time when the pumping radiation reaches its peak for 2τ = 40fs, shown in Fig. 2(b). It can be understood by the comparison between the (10fs)-1 Auger decay rate of the upper (1s)-1 state and the τ/2 rising time of the x-ray FEL pumping pulses. They are equal for the pulse 2τ = 40fs. At the ascending half of the 2τ = 40fs pulse, the population of the upper state is accumulated. While, at the descending half the upper state starts to be depleted through the Auger decay. The x-ray photons of the second half pulse have no contribution to the gain coefficient of the inner-shell x-ray transitions. So for the shorter x-ray pulses 2τ = 10, 20fs more proportion of x-ray photons in the pulses can participate in establishing the population inversion for the ISXRL’s. The higher pumping efficiency can be reached for the x-ray FEL pulses with shorter durations, as shown in Fig. 3(b).

The effective gain Geff can only be positive when the upper state is populated sufficiently. The photoabsorption loss αabs(cm -1) increases linearly with matter density. As a result, the absorption of the lasing line in carbon is approximately 90cm-1. Figure. 3(b) shows sufficient high net gain ≃120cm-1 can be reached for the inner-shell Kα transitions of carbon, pumped by an ideal FEL radiation pulse with a peak intensity of 6×1014W/cm2 and a pulse duration 2τ = 10 fs.

Fig. 2. (Color online) Temporal change of (a) the population density (PD) of CI(neutral atom) and Single IVL’s in CΠ. The initial neutral density of carbon is 2.28×1021 cm-3; (b) Ginv and Geff of the (1s)-1→(2p)-1 (λ = 4.5nm) transition in carbon for the FEL pumping radiation at 284eV with a 3×1014W/cm2 peak intensity and a FWHMτ = 20fs.

3.2. Quasi-stationary inner-shell lasing in Calcium

As one characteristic example of the quasi-stationary population inversion scheme, the inner-shell x-ray transition (2p)-1→(3s)-1 (λ = 4.1nm) of calcium (Z = 20), pumped by the third harmonics of FEL at photo energy of 360eV, is also studied. We consider one neutral level, six single IVL’s and forty double IVL’s. The initial neutral density of Ca is set as 1.2×1021cm-3. The absorption coefficient of the lasing line is approximately 480cm-1 [26

26. B. L. Henke, E. M. Gullikson, and J. C. Davis, “X-Ray Interactions: Photoabsorption, Scattering, Transmission, and Reflection at E=50-30,000eV,Z=1-92,” Atomic Data and Nucl. Data Tables 54, 181–342 (1993). [CrossRef]

]. Figure. 4 gives the simulation results of (a) the temporal evolution of the population of CaI and IVL’s of CaII, and (b) (c) the variation of Ginv and Geff of inner-shell L 23 M 1 x-ray transitions in calcium for two different pumping conditions. The population inversion is achieved from the beginning of the pumping pulse, through direct photoionization (larger cross section of (2p)-1 than that of (3s)-1) of the neutral calcium atoms. The Auger decay rate of the upper state is ~(3.9fs)-1, which is 2.5 times faster than the (10fs)-1 Auger decay in carbon, resulting the secondary Auger electrons with an energy of 300eV. The Auger electrons are energetic enough to collisionally ionize the neutral atoms to the lower state (3s)-1, which is harmful for the population inversion. However, the lower state undergoes the Coster-Kronig Auger process with a fast decay rate of (0.8fs)-1, which gives the L 23 M 1 transitions a quasi-stationary population inversion.

Fig. 3. (Color online) Temporal change of Ginv and Geff of the (1s)-1→(2p)-1 (λ = 4.5nm) transition in carbon for pumping radiation with different pulse durations: (a) 2τ = 20, 60fs with the same pump intensity Ip = 3×1014W/cm2; (b) 2τ = 10, 20fs with the same radiation energy.
Fig. 4. (Color online) (a) Temporal evolution of the population of CaI and IVL’s of single ionized ions CaII, pumped by an XFEL pulse at 360eV with a peak intensity of 3×1017W/cm2, pulse duration 2τ = 30fs. (b) Temporal change of Ginv and Geff of inner-shell x-ray transitions (2p)-1→(3s)-1 (λ = 4.1nm) in calcium for the same pumping condition in (a). (c) Temporal change of Ginv and Geff for a smaller peak intensity of 1×1017W/cm2 with a shorter pulse duration 2τ = 10fs.

The quasi-stationary population inversion is depleted because of the rapid depletion of the neutral atom CaI through the photoionizaiton by the high power FEL radiation and the electron ionization by the hot Auger electrons. From Fig. 4(c), a shorter (10fs) and smaller pumping power (Ip = 1×1017W/cm2) can generate a higher net peak gain ≃120cm-1. But the duration of net gain is quite short about 2fs.

Compared to 6×1014W/cm2 pump power required by the self-terminated Kα transitions in carbon (Z=6), more pump intensity is required to get the same effective gain. It can be explained by the more complex electron structure, faster Auger decay rate of the upper-state, higher Aul rate, and larger reabsorption of the (2p)-1→(3s)-1 line transition in calcium.

4. Gain performances pumped by multi-spiky SASE FEL pulses

The ideal x-ray FEL pulses with an ultra-short duration of 10fs have been proved to be efficient for generating high gain photo-ionization pumped ISXRL’s. However, the actual temporal structure of the radiation pulses from the SASE FEL is a composition of a few isolated and random arriving spikes, with durations of femtosecond to sub-fs, as shown in Fig. 5(a), we assumed here 2fs[6

6. Y. L. Li, S. Krinsky, J. W. Lewellen, K. J. Kim, V. Sajaev, and S. V. Milton, “Characterization of a chaotic optical field using a high-gain, self-amplified free-electron laser,” Phys. Rev. Lett 91(24), 243602–243605 (2003). [CrossRef] [PubMed]

, 7

7. E. L. Saldin, E. A. Schneidmiller, and M. V. Yurkov, “Properties of the third harmonic of the radiation from self-amplified spontaneous emission free electron laser,” Phys. Rev. Spec. Top. Accel. Beams 9(3), 030702–030710 (2006). [CrossRef]

]. The intensity envelop has the same temporal profile as in Eq. 4. Gain performances of the ISXRL’s in carbon and calcium pumped by such x-ray SASE FEL pulses are shown in Fig. 5. We also used the same pump energy as in the ideal x-ray FEL pumping scheme. The time duration is 2τ = 10fs.

We found that, by guiding the chaotic temporal multi-spiky SASE FEL pulses into the carbon system, the high gain (1s)-1→(2p)-1 (λ = 4.5nm) ISXRL with a peak net gain of 140cm-1 is generated. Furthermore, the spiky temporal structure from the SASE FEL pulses is smoothed by removing the gaps between those spikes. This results a 10fs continuous temporal profile of the gain coefficients Ginv and Geff. This is due to the accumulation effect of the individual 2fs x-ray spikes on the inner-shell Kα transition in carbon. The life time of the upper state (1s)-1 is about 10fs. The population of the upper state is accumulated during the photon-pump of five isolated x-ray spikes.

However, for the calcium L 23 M 1 inner-shell x-ray transitions (λ = 4.1nm), the life time of the upper state (2p)-1 is about 3.9fs. The population can only be sustained for the time of almost two spikes. As a result, one single peak of effective gain is produced, and the time duration is 2fs as shown in Fig. 5(c).

5. Pump geometry and output inner-shell x-ray pulses

Considering the photoionization pumped ISXRL as an amplified spontaneous emission (ASE) system, we give brief discussions and estimations about the pump geometry and properties of the output inner-shell x-ray lasers. For carbon and calcium ISXRL system with an initial density of 2.28×1021cm-3 and 1.2×1021cm-3, the absorption length of the pump radiation at 284eV and 360eV is about latt = 4.5µm and 3µm, respectively. Very small amplification can be got along the pump direction inside the target. Furthermore, the gain duration is quite short, only 10fs and 2fs, respectively. In this situation, a traveling wave pump scheme should be used. The grazing-incidence pump scheme proposed for outer-shell soft x-ray lasers could realize an inherent traveling pump [32

32. 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 18.9nm,” Phys. Rev. Lett 94(10), 103901–103904 (2005). [CrossRef] [PubMed]

], is also adopted here for the pump of ISXRL’s. So a pump geometry of irradiating a grazing-incidence focused-line on a foil is utilized, as shown in Fig. 6, a 5µm×100µm or a 3µm×100µm FEL radiation line focused on a 5µm thick carbon or 3µm thick calcium foil with a grazing-incidence angle θ from the surface. The traveling pump speed depends on the grazing-incidence angle, and is very close to c, the velocity of light.

Fig. 5. (Color online) (a) Temporal structure of the third harmonic radiation pulses from the SASE FLASH at DESY. The pulses envelop has a peak intensity Ip and a time duration of 2τ = 10fs. The duration of each spikes is 2fs. Temporal change of Ginv and Geff of the inner-shell x-ray lasers in carbon (b) and calcium (c), pumped by the third harmonic SASE FEL pulses in (a) with a peak intensity of 1.2×1015W/cm2 and 2×1017W/cm2, respectively.
Fig. 6. (Color online) Pumping geometry of ISXRL’s pumped by FEL radiation pulse with a grazing-incidence angle θ.
Fig. 7. The temporal profile of the output ISXRL pulse of carbon and calcium at 4.5nm and 4.1nm, from the end of a 100µm focused-line, pumped by the 10fs multi-spiky FEL radiation with the same intensity as in Fig. 5.

The output intensity Iout of ISXRL’s is predicted using a one-dimensional amplifier model [33

33. R. C. Elton, X-Ray Lasers (Academic, San Diego, 1990).

]:

Iout=J(expGeffL1)Geff
(5)

where Geff (t) is the effective gain resulted from the FEL pumping, L is the amplification length and J = Nu 0ΩsAul is the spontaneous emission per unit length, Ωs = Sout/4πL 2 is the solid angle of the x-ray lasing emission, Sout = latt×lwidth is the output area, lwidth is line width and normally is set to be equal to latt. ISXRL is amplified in a long cylindrical plasma. Small signal amplification, far from saturation is considered. No spatial effect is included in this simple model. The gain is assumed constant along the pump line, which is rational for the grazingincidence traveling pump scheme. Spatial effects in the other two transverse directions, perpendicular and parallel to the foil surface, depend on the grazing-incidence angle and the focus of the FEL pulses ((i.e. laser intensity spatial distribution) on the target foil, respectively. Detailed study of these effects needs 3D simulations of the grazing-incidence FEL radiations transfer into the gain medium, the kinetics of atomic processes and the spatial distributions of secondary electrons, which are not included in this 1D amplification model and would be observed in our future work.

The temporal profile of the output intensity of ISXRL at the end of the 100µm line is simulated, as shown in Fig. 7. Pumped by a multi-spiky FEL radiation pulse at 284eV, a continuous 5fs (FWHM) ISXRL pulse of carbon at 4.5nm is resulted, with a peak output intensity of 1.2×108W/cm2. Whereas, one single 1fs (FWHM) ISXRL pulse of calcium at 4.1nm with a peak intensity of 7.1×108W/cm2 is generated. In another word, the SASE FEL x-ray pulse with chaotic temporal structure is now converted into one single femtosecond ISXRL pulse.

The total pump energy and the energy conversion efficiency (CE) are also estimated, 15µJ with CE about 1×10-8 for carbon Kα inner-shell lasing, and 1.5mJ with CE 5×10-11 for calcium L 23 M 1. The calculated CE is very small, which is because of the small attenuation length latt of the pump radiation and the small solid emission angle ΩsS (out) = latt×lwidth in Eq. 5. If the initial density is decreased, say 2.28×1020cm-3 for gaseous carbon, the latt increases inversely to 45µm. With the same FEL pump intensity 1.2×1015W/cm2, the peak gain coefficient is reduced to 16cm-1. The pump line is set to be 45µm×100µm. The energy CE arises to 1×10-6. The total pump energy is now 135µJ. However, the divergence angle latt/L of the output ISXRL also becomes large.

At last, the output ISXRL’s are shown to have better temporal coherence with a spectral bandwidth Δλ/λ of 1.5×10-3 and 4.2×10-3 for carbon and calcium, which are one order narrower than the harmonic FEL radiations with a bandwidth of 10-2 [31

31. E. L. Saldin, E. A. Schneidmiller, and M. V. Yurkov, “Self-amplified spontaneous emission FEL with energy-chirped electron beam and its application for generation of attosecond x-ray pulses,” Phys. Rev. Spec. Top. Accel. Beams 9(5), 050702–050707 (2006). [CrossRef]

]. The coherence time τc = λ 2/cΔλ is 10fs and 3fs, respectively.

6. Conclusion and Discussion

Theoretical simulations are performed to investigate the feasibility of producing the high gain ISXRL’s pumped by the high brilliance SASE FEL. The gain performance of the inner-shell x-ray transitions is studied with different FEL pumping conditions. For the self-terminated (2p)-1→(3s)-1 (λ = 4.1nm) transitions in carbon (Z=6) and the quasi-stationary population inversion scheme transitions (2p)-1→(3s)-1 (λ = 4.1 nm) in calcium (Z = 20), the net gain of 140cm-1 can be pumped by the 10fs SASE FEL at 284eV and 360eV with the pumping intensities of 1.2×1015W/cm2 and 2×1017W/cm2, respectively. Using a one-dimensional model, the temporal profile of the output intensity of ISXRL’s is simulated. We find that, the multi-spiky FEL radiations can be converted to a 5fs (FWHM) continuous carbon ISXRL pulse at 4.5nm, due to the accumulation effect of the upper state (1s)-1. One single spike of 1fs (FWHM) calcium ISXRL pulse at 4.1nm can be resulted. The output ISXRL’s have the spectral bandwidth of an order of 10-3, which is one order narrower than that of the SASE FEL’s. The ISXRL’s have better temporal coherence property than the FEL’s.

The energy conversion efficiency is also estimated. By changing the initial density, the gain coefficient of ISXRL’s varies, also does the attenuation length and the divergence of the output ISXRL’s. A higher conversion efficiency can be expected with a gaseous carbon target. Experiments need to be done to test this prediction.

For other ISXRL’s with different x-ray wavelengths, depending on the Kα or L 23 M 1 transitions of the elements, the FEL pumped gain performance and the required pump intensity for sufficient population inversions are determined by many parameters, mainly including the complexity of the atomic levels, the fast inner-shell atomic processes like the Auger decay, photoionization and the secondary electron collisional ionizations, and the huge reabsorption of the inner-shell lasing by the ions. Schemes, like the shifted line model for the Kα transition in z≤20 elements and the closed outer shell elements or ions, can help realize the FEL pumped ISXRL’s with shorter wavelengths.

We propose that by simply adding an ISXRL system at the output of the high brilliance SASE FEL, femtosecond temporal coherent inner-shell x-ray pulses could be generated. It has many applications in the pump-probe experiments. The experimental sketch is simple with no change required inside the SASE FEL system. The FEL’s can keep its high brilliance characteristic for other important applications.

Acknowledgments

This work is jointly supported by the National Natural Science Foundation of China (under Grant Nos. 60678007, 60621063), National Basic Research Program of China (973 Program) (Grant No.2007CB815101) and the National Hi-tech ICF Program.

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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 18.9nm,” Phys. Rev. Lett 94(10), 103901–103904 (2005). [CrossRef] [PubMed]

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OCIS Codes
(140.2600) Lasers and laser optics : Free-electron lasers (FELs)
(140.7240) Lasers and laser optics : UV, EUV, and X-ray lasers
(260.5210) Physical optics : Photoionization
(260.6048) Physical optics : Soft x-rays

ToC Category:
Lasers and Laser Optics

History
Original Manuscript: November 6, 2007
Revised Manuscript: January 31, 2008
Manuscript Accepted: February 3, 2008
Published: March 3, 2008

Citation
J. Zhao, Q. L. Dong, S. J. Wang, L. Zhang, and J. Zhang, "X-ray lasers from Inner-shell transitions pumped by the Free-electron laser," Opt. Express 16, 3546-3559 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-6-3546


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References

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