THEORETICAL INVESTIGATION OF X-RAY LASING IN ARGON BY PHOTO-IONIZATION FROM K AND L SHELLS

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1 v.2.1r * #113768ff THEORETICAL INVESTIGATION OF X-RAY LASING IN ARGON BY PHOTO-IONIZATION FROM K AND L SHELLS C. IORGA 1,2,a, V. STANCALIE 1 1 National Institute for Laser, Plasma, and Radiation Physics, Atomistilor 409, P.O.Box MG-36, Magurele-Ilfov, , Romania a : iorga.cristian@inflpr.ro 2 University of Bucharest, Faculty of Physics, Atomistilor 405, P.O.Box MG-11, Magurele-Bucharest, , Romania Received July 17, 2017 Abstract. We theoretically investigate the X-ray lasing in argon gas by means of inner-shell photo-ionization pumping scheme driven by X-ray free electron laser (XFEL) radiation. Atomic data calculations performed within a fully relativistic model potential code reveal the dominant transitions from the K and L shells which may be used in argon lasing experiments. Our study shows that the photo-ionization pumping scheme is most efficient (σ 2p /σ total 94%) for populating the 2p 1 state belonging to Ar 1+, thus creating inversion on the 2p 3s transition, assuming the XFEL pulse is tuned between the two L edges at 270 ev. The generalized one-dimensional Maxwell- Bloch approach is employed in order to model the amplification process at 5.65 nm for this transition. The present simulations show strong amplification resulting in X- ray pulses of up to 100 µj energy with narrow spectral bandwidth. We report X-ray pulse characteristics, such as output energy, time and spectral profile, for the radiation mentioned above as resulted from the simulation. Key words: X-ray laser, Maxwell-Bloch equations, numerical simulation. PACS: Rj, Aa, Fb, Qk, Xx. 1. INTRODUCTION The advent of X-ray free electron lasers (XFELs) has allowed the successful demonstration of X-ray lasing by employing the inner-shell photo-ionization pumping scheme in neon gas [1] at 1.46 nm and in solid copper [2] at 1.54 Å. New atomic X-ray lasers with angstrom wavelengths and femtosecond pulse durations are now reachable through this scheme by making use of the modern XFEL [3 8] pumping capabilities. Previous theoretical studies [9 18] predict high lasing gains for K and L shell transitions in elements with Z 20. It has been shown that a correct approach for modelling X-ray gain is based on Maxwell-Bloch equations [19 23] since it accounts for the phase dynamics between the electric field and the atomic states of the matter. An important aspect in the transient amplification of radiation in high-gain medium is the gain-guiding Romanian Journal of Physics 64, 501 (2019)

2 Article no. 501 C. Iorga, V. Stancalie 2 mechanism [23] which refers to the spatio-temporal interplay between the generated X-ray pulse and the population inversion in the exponential amplification region. Mismatch between the two leads to significant gain reduction. A similar drawback at a higher extend appears in the case of seeded amplification. When using a seeding pulse of short duration compared to the natural timescale of the atomic system, there is a significant delay between the buildup of the polarization and the seeding pulse [24 28], usually only allowing the tail of the seed to be amplified. In order to maximize the amplification one can use a seed pulse with time duration larger than the corresponding atomic timescale. In the case of purely spontaneous emission self-seeded amplification one must carefully choose the parameters for the pumping radiation and the lasing medium such that the non-linear interaction results in a good overlap between the generated X-ray pulse and population inversion. It is also worth mentioning that the rate-equation approach is not able to properly describe the phase dynamic effects such as gain-guiding mechanism and cannot include correct ad-hoc improvements in order to account for a gain-dependent group velocity of the X-ray pulse [23, 29, 30]. A good candidate for investigating inner-shell X-ray lasing is argon gas since it has fully occupied K and L shells with relatively high photo-ionization cross sections. The experimental requirements should be similar to those of the X-ray laser obtained in neon gas [1], apart from the pumping XFEL pulse which must be properly tuned above the ionization potential in each case and needs to be bright enough to overcome the Auger and radiative decay rates of the considered upper lasing level. The sections of this paper are as follows. The second part contains atomic data calculations performed with the aim of finding the most dominant transitions that may be used for X-ray lasing. The next section concerns with the Generalized-Maxwell- Bloch formalism [23, 31, 32] which is used to compute the properties of the generated X-ray pulse. The fourth section presents the results of numerical simulations along with explanations and a discussion. The last part gives the concluding remarks. 2. ATOMIC DATA The atomic data, such as energy levels, dipole moments, oscillator strengths, radiative decay rates, and photo-ionization cross sections, for neutral, singly and doubly ionized argon have been computed within the fully relativistic model-potential Flexible Atomic Code (FAC) [33]. The continuum states necessary for computing the bound-free oscillator strengths and photo-ionization cross sections are determined using the relativistic distorted-wave approximation. The configuration expansions for each ion have been determined by allowing single and double electron promotions from a reference set taken from C. Froese-Fischer et al. [34] while also including the core electron promotions in order to obtain the necessary inner-shell atomic

3 3 Theoretical investigation of X-ray lasing in Argon by photo-ionization Article no. 501 data. A limited number of strongly interacting configurations has been included in the overall model as it is shown in Table 1. Assessment of the current data is pro- Table 1 Configuration expansions used for calculating the atomic data corresponding to Ar 0, Ar 1+, and Ar 2+. Ion Reference set Single e pr. Double e pr. Core e pr. (Fixed core: 1s 2 2s 2 2p 6 ) (Ref. set) (Ref. set) (1s 2 2s 2 2p 6 ) Ar 0 Even: 3s 2 3p 6, 3s3p 6 3d, 3s 2 3p 5 4p ; 3 n 7, 3 n 4, 3 n 4, Odd: 3s 2 3p 5 3d, 3s 2 3p 5 4s ; 0 l 4, 0 l 3, 0 l 3, +8s,8p; +5s,5p,6s,6p; +5s,5p,6s,6p; Ar 1+ Even: 3s3p 6, 3s3p 5 4p,3s 2 3p 4 3d, 3 n 7, 3 n 4, 3 n 5, 3p 6 3d, 3s 2 3p 4 4s, 3p 6 4s, 3s 2 3p 4 4d ; 0 l 4; 0 l 3, 0 l 4, Odd: 3s 2 3p 5, 3s 2 3p 4 4p, 3p 6 4p +5s,5p; +5s,5p; Ar 2+ Even: 3s 2 3p 4, 3p 6, 3s 2 3p 3 4p, 3p 5 4p ; 3 n 7, 3 n 4, 3 n 4, Odd: 3s3p 5, 3s3p 4 4p, 3s 2 3p 3 3d, 0 l 4; 0 l 3, 0 l 4, 3p 5 3d, 3s 2 3p 3 4s, 3p 5 4s, 3s 2 3p 3 4d; +5s,5p; +5s,5p; vided by comparison with NIST [35]. In the case of low energy levels, the relative error is 3%, while for the radiative decay rates the agreement is about 10 20%. Agreement with measured results [36, 37] is improved up to 1% and 0.1% relative error for L and K shell transition energies, respectively. The transitions proposed to be investigated for efficient X-ray lasing are presented in Table 2. Table 2 K and L shell transitions in Ar 1+ relevant for X-ray lasing along with corresponding transition energy E(eV), wavelength λ(nm), radiative transition probability A(s 1 ), and dipole moment µ(a. u.) Upper level Lower level E(eV) λ (nm) A (s 1 ) µ (a.u.) 1s 1 ( 2 S 1/2 ) 2p 1 ( 2 P1/2 o ) p 1 ( 2 P o 3/2 ) p 1 ( 2 P o 1/2 ) p 1 ( 2 P o 3/2 ) s 1 ( 2 S 1/2 ) 3p 1 ( 2 P o 1/2 ) p 1 ( 2 P o 3/2 ) p 1 ( 2 P o 1/2 ) 3s 1 ( 2 S 1/2 ) p 1 ( 2 P o 3/2 ) 3s 1 ( 2 S 1/2 ) By computing the photo-ionization cross section at photon energies slightly above the ionization potential for each state, the highest values have been obtained for 2p 1 final state with σ 2p1/2 (270 ev) = 1.49 Mb and σ 2p3/2 (270 ev) = 3 Mb amounting to roughly 31% and 63% of the total photo-ionization cross, σ total (270 ev) = 4.76 Mb. For photon energies above the two L edges, for example at 350 ev, the photo-ionization cross section for the 2s 1 final state, σ 2s (350 ev) = 0.24 Mb, is significantly lower than the ones corresponding to the 2p 1 doublet, σ 2p1/2 (350 ev) =

4 Article no. 501 C. Iorga, V. Stancalie Mb and σ 2p3/2 (350 ev) = 1.44 Mb, respectively. The σ total : σ 2p : σ 2s = 1 : : ratio for hν = 350 ev shows that two-color X-ray lasing may be developed by simultaneously inverting the 2s 3p and 2p 3s transitions [38]. Lasing is also possible for K-shell transition, but the photo-ionization cross section for populating the 1s 1 upper lasing level is rather small compared to the other values obtained in this paper, σ 1s (3.5 kev) 0.07 Mb. Considering all of the above, the highest output X-ray pulse energy can be obtained by lasing on the 2p 3s transition due to the overwhelming probability for populating the upper-lasing level through photo-ionization and due to the fact that the XFELs can provide up to photons per pumping pulse at 270 ev. 3. THEORETICAL MODEL In order to simulate the amplification of the radiation arising from 2p 3s transition via inner-shell photo-ionization pumping scheme, one must numerically solve the semiclassical Maxwell-Bloch system. To simplify the calculations it is useful to work within the time frame moving with the speed of light by using the retarded time τ = t z/c, where c is the speed of light. Prior to solving these equations it is necessary to compute the XFEL pulse propagation in argon medium described by the following non-linear coupled equations: ρ 0,0 (τ,z) τ = σ total J(τ,z)ρ 0,0 (τ,z) (1) J(τ,z) z = nσ total J(τ,z)ρ 0,0 (τ,z) (2) Here, the pumping flux is given by J(τ,z) and since the pulse is assumed to be tuned to 270 ev, which is well above the first L edge of 248 ev, the phase effects can be neglected for the driving field. The argon gas is assumed to be under normal pressure and room temperature at density n cm 3. The occupation probability for the neutral argon atoms is denoted as ρ 0,0 and its initial value is 1 along the whole medium. The propagation of the XFEL radiation in gas results in strong attenuation of the pumping pulse along a 3 cm path and rapid photo-ionization of the argon atoms. After performing the macroscopic calculations for the pumping field and neutral argon density, we employed the Generalized-Maxwell-Bloch approach [23, 31, 32] in order to describe the phase dynamics between the newly formed X-ray pulse and the atomic level densities. The electric field is treated classically by using the approximation of the slowly varying envelope, E(τ,z) = 1/2(ɛ(t,z)e iωτ + ɛ (τ,z)e iωτ ). In the case of lasing on the 2p 3s transition, the equations for the

5 5 Theoretical investigation of X-ray lasing in Argon by photo-ionization Article no. 501 atomic densities are: ρ 2pj,2p j (τ,z) = Γ 2pj ρ 2pj,2p τ j (τ,z) + σ 2pj ρ 0,0 (τ,z)j(τ,z) Im(ɛ(τ,z)z 3s,2pj ρ 3s,2pj ), with j = 1/2, 3/2 ρ 3s,3s (τ,z) = A 3s ρ 2pj,2p τ j (τ,z) + σ 3s ρ 0,0 (τ,z)j(τ,z)+ + Im(ɛ(τ,z)z 3s,2pj ρ 3s,2pj ) j=1/2, 3/2 (3) (4) ρ 3s,2pj (τ,z) τ = Γ 2p j + A 3s ρ 2pj,3s(τ,z)+ 2 + iɛ 2 z 2p j,3s(ρ 2pj,2p j ρ 3s,3s ) + S, with j = 1/2, 3/2 The diagonal terms of the reduced density matrix operator are the occupation probability of the levels and the non-diagonal terms represent the coherences between the levels. The terms Γ and A stand for Auger and radiative decay rates, respectively, and z ij is the dipole moment computed between bound states i and j. The S term represents the stochastic spontaneous emission which is modelled as a Gaussian white noise described by the following function : < S(t)S (t ) >= c2 ρ 2p,2p A 2p Γ 2 2p 4πωz 2p,3s δ(t t ) (6) The correlation function is properly normalized in order to inject energy in the system with the corresponding spontaneous emission rate and to account for the Lorentzian line shape of this process [19, 23]. The polarization is given by the contributions of the non-diagonal terms of the reduced density matrix: (5) P (τ,z) = 2n i j z ij ρ ij (τ,z)e i(ω ω ij)τ (7) The electric envelope is directly computed from the polarization as: ɛ(τ, z) z = i 2πω P (τ,z) (8) c The number of photons per pulse is computed by time integrating the photon flux over the duration of the time frame T and within an area S given by: T N hν (z) = cs ε 0 ɛ(τ,z) 2 dτ (9) hν 0 2

6 Article no. 501 C. Iorga, V. Stancalie 6 Fig. 1 XFEL attenuation and X-ray laser amplification along the 3 cm length argon gas cell for pump pulse duration of τ F W HM = 20 fs and different focusing areas, π µm 2, 2π µm 2, 5π µm 2. The (1)-(2) coupled equations for the propagation of the XFEL pulse in the neutral argon gas are self-consistently solved first. This allows the calculation of the pumping rate for the lasing levels. Next, the Gaussian White Noise distribution is generated. At this point, equations (3)-(8) are solved in a self-consistent manner. The propagation of the XFEL pulse and X-ray lasing field have been performed using the second order Adams-Bashforth method while the evolution for the neutral density was done using the Crank-Nicholson scheme. The time evolution of the level occupation probabilities has been solved by employing a split time step method [23, 32, 39]. The computations have been performed using discretized time steps of t = s and corresponding spatial propagation steps of z = c t similar to the calculations done by C. Weninger and N. Rohringer [23]. 4. NUMERICAL RESULTS Assuming an incoming XFEL pulse tuned to 270 ev containing photons acting on an argon gas medium of density cm 3 kept within a cell of 3 cm length, we simulated lasing on the 2p 3s transition. The fine splitting of the upper state has been considered, but the 2p( 2 P o 3/2 ) 3s(2 S 1/2 ) transition has been proved to be clearly dominant and is the one presented in this paper. Different focusing areas and pump pulse FWHM durations have been considered in order to determine the optimum parameters for maximum gain. The number of photons belonging to the attenuated XFEL and amplified X-ray pulses, computed with eq. (2) and (9), respectively, are presented in Fig. 1, for a pump pulse duration of 20 fs and π µm 2, 2π µm 2, and 5π µm 2 focusing areas. These parameters are chosen to explicitly show the spatial limiting factor for increasing the output energy by tightening the focus. This limitation is given by the large percentage of transmitted XFEL pulse after propagation in the argon gas. For example, in the case of 20 fs FWHM pump pulse duration and at π µm 2 and 2π µm 2 focusing areas, the transmitted pump pulse is 71% and 43%, respectively, so the energy stored in the system becomes much lower compared to the case of 5π µm 2 focus where

7 7 Theoretical investigation of X-ray lasing in Argon by photo-ionization Article no. 501 Fig. 2 Evolution of population inversion (up) and normalized XRL (ɛ(t, z)) temporal profile (down) for 5 (left), 10 (middle), and 20 fs (right) pump pulse duration. The black dotted line follows the maximum of the XRL profile while the green dotted line follows the peak of the population inversion. there is 0%(10 55 ) pump pulse transmission, most of it being absorbed by the medium. After the exponential amplification ends at 0.7 cm, in the case of 5π µm 2 focus, the radiation is linearly amplified up until 2.1 cm taking full advantage of the energy stored in the system as opposed to the π µm 2 and 2π µm 2 cases. Similar features are presented for the 50 fs pump pulse duration when tightening the focus from 2π µm 2 to π µm 2, reducing the pumping energy stored in the system which is reflected in the output energy of the X-ray pulses in Table 3. There is also a temporal limiting factor which prevents increasing the laser gain when shortening the pump pulse duration and it is related to the rise time of population inversion. This parameter depends on both the duration of the pump pulse and the non-linear interaction between the pump radiation and the medium. The simulations reveal that these rise times generally become much shorter than the corresponding pump pulse duration along the propagation path, due to strong interaction with the medium (σ Mb). This only results in further mismatch between the generated X-ray radiation and the population inversion for shorter pump durations as can be seen in Fig. 2. One would usually expect that by halving the time duration of the pump pulse and thus doubling the intensity, the output energy would be higher. However, due the gain guiding mechanism this is not always the case, as one can see by comparing, for example, the output energies in Table 3 for the case of 5π µm 2 focus and pump pulse

8 Article no. 501 C. Iorga, V. Stancalie 8 durations of 5 fs, 10 fs and 20 fs, respectively. Table 3 Output energy of the X-ray pulse at the end of the medium in units of µj. S F \ τ F W HM 5 fs 10 fs 20 fs 50 fs π µm µj 21.1 µj 32.8 µj 30.2 µj 2π µm µj 51.8 µj 69.9 µj 54.7 µj 5π µm µj 84.9 µj µj 27.5 µj 10π µm µj 34.9 µj 51.4 µj 0.68 µj 20π µm µj 0.11 µj 1.95 µj µj 50π µm µj µj µj µj The generated pulses are initialized with a FWHM duration of about 20 fs as given by the non-linear interaction between the pumping radiation and the gas. They become shorter up until saturation sets in, reaching 3 4 fs per peak. Multipeak structures are formed following the Rabi flopping at saturation [23]. The spectral bandwidth of the pulses is initially 0.17 ev corresponding to the inverse lifetime of the 2p 1 ( 2 P3/2 o ) state. Figure 3 shows the evolution of the spectral bandwidth of the X-ray laser for XFEL pulse durations of 5 fs, 10 fs, and 20 fs and focusing area of 10πµ m 2. The spectrum is rapidly narrowed in the exponential amplification regime down to a value around 0.05 ev, 0.06 ev, and 0.08 ev, which is maintained up until saturation when the spectral bandwidth is broadened up to 0.13 ev, 0.22 ev, and 0.45 ev for the XFEL pump pulse durations of 5 fs, 10 fs, and 20 fs respectively. After saturation, the final values of 0.13 ev, 0.12 ev, and 0.09 ev are reached and maintained until the end of the medium. The pump pulse parameters have a rather Fig. 3 Evolution of XRL spectrum profile for a 10π µm 2 focusing area and pump pulse FWHM durations of 5 (left), 10 (center), and 20 fs (right), respectively. low influence on the final values for the FWHM time duration and spectral bandwidth but they strongly affect the profile shapes of the output X-ray pulses in the process of amplification, resulting in multipeak structures in the time domain (see Fig. 2)

9 9 Theoretical investigation of X-ray lasing in Argon by photo-ionization Article no. 501 or sidebands in the frequency domain (see Fig. 3). These features are generally maintained along the parameter range considered in this work. 5. CONCLUSION This paper theoretically investigates the possibility of X-ray lasing by innershell photo-ionization scheme in argon gas kept within a 3 cm long cell at atmospheric pressure and room temperature. Atomic calculations revealed that lasing on the 2p 3s transition at 5.65 nm is most efficient via this pumping scheme, due to the high photo-ionization cross section and pumping possibilities of the XFEL at 270 ev. The Maxwell-Bloch simulations predict X-ray pulses of up to 100 µj output energy with temporal duration of about 3 4 fs for the peaks, and spectral bandwidth of 0.1 ev, with small variations depending on the pump pulse parameters. Higher XFEL intensity combined with good overlap between the X-ray field and population inversion induce saturation at shorter propagation distances leading to strong Rabi oscillations modulating the population inversion giving rise to multipeak structure in the time domain and further broadening of the spectrum. A discussion regarding the limitations derived from the gain-guiding mechanism is presented. These results can be useful for further experimental demonstration of X-ray lasing in argon gas at XFEL and high-power-laser facilities. Acknowledgements. This work has been financed by the National Authority for Research and Innovation in the frame of Nucleus programme-contract 4N/2016. REFERENCES 1. N. Rohringer, D. Ryan, R. A. London, M. Purvis, F. Albert, J. Dunn, J. D. Bozek, C. Bostedt, A. Graf, R. Hill, S. P. Hau-Riege, and J. J. Rocca, Nature (London), vol. 481, p. 488, H. Yoneda, Y. Inubushi, K. Nagamine, Y. Michine, H. Ohashi, H. Yumoto, K. Yamauchi, H. Mimura, H. Kitamura, T. Katayama, T. Ishikawa, and M. Yabashi, Nature, vol. 524, p. 446, W. Ackermann, G. Asova, V. Ayvazyan, et al., Nature Photon., vol. 1, pp , P. Emma, R. Akre, J. Arthur, et al., Nature Photon., vol. 4, pp , T. Ishikawa, H. Aoyagi, T. Asaka, et al., Nature Photon., vol. 6, pp , M. Altarelli et al., Technical Design Report of the European XFEL, H. S. Kang, K. W. Kim, and I. S. Ko International Particle Accelerator Conference, Shanghai, China, p. 2074, R. Ganter et al., SwissFEL-Conceptual design report, Technical Report, Paul Scherrer Institute (PSI), Villigen, Switzerland, T. S. Axelrod, Phys. Rev. A, vol. 13, pp , H. C. Kapteyn, Appl. Opt., vol. 31, pp , G. L. Strobel, D. C. Eder, R. A. London, M. D. Rosen, R. W. Falcone, and S. P. Gordon, Proc. SPIE, vol. 1860, pp , 1993.

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