1. Introduction

Present-day femtosecond X-ray free-electron lasers (XFELs) [1–4], which are many orders of magnitude brighter than synchrotron sources, have enabled various applications in physics, chemistry, and biology. [5–7] Although they are unparalleled in brightness, the XFELs have a fluctuating spectrum and limited temporal coherence [8] because they are based on self-amplified spontaneous emission (SASE), starting from noise. A longitudinally coherent X-ray source with a narrower and smoother spectrum is still needed for current and future investigations in X-ray science, such as X-ray quantum optics [9–11], nonlinear X-ray scattering [12], and high-resolution X-ray spectroscopy [13–17]. There have been attempts to address these issues, such as the development of fully coherent XFELs through methods like XFEL oscillator (XFELO) [18], Regenerative-Amplifier FEL (RAFEL) [19], and enhanced SASE (ESASE) [20]. However, achieving longitudinal coherence, spectral stability, and high gain in a single pass in the X-ray region remains a challenge. Another potential solution is the utilization of atomic X-ray laser using SASE-based XFEL pulses, which can provide a narrow bandwidth, longitudinal coherence, and spectral stability. This approach complements various direct approaches in XFEL research. [21–26].

The atomic X-ray laser scheme utilizes photoionization from inner-shell states to pump the upper state for lasing action [27–39]. The original proposal by Duguay and Rentzepis [39] was further developed by Stankevich [40], elaborated by Arecchi et al [41] and Elton [42], and detailed calculations were done by Axelrod for K-shell transitions [43]. The realization of this scheme has been a challenge. The technical barrier to overcome for the successful realization of the scheme is the development of a sufficiently fast and strong X-ray pump source whose timescale is of the order of the lifetime of the keV lasing transitions [27,34,43]

The development of XFELs has renewed interest in developing keV X-ray lasers based on inner-shell atomic transitions. In 2012, a neon soft-X-ray laser at 1.46 nm (0.8 keV) was demonstrated using sub-kilovolt XFEL pulses to pump the K-shell electrons of neon atoms [44]. Then, in 2015, lasing action at 0.15 nm (8 keV) pumped by an X-ray free-electron laser was successfully demonstrated [45]

Here, we present numerical simulations based on the atomic kinetics model to investigate the population dynamics of inner-shell vacancy states for an atomic X-ray laser operating around 1.5 keV. When X-rays with slightly higher energy than the absorption edge irradiate an atom, inner-shell photoionizations occur, creating short-lived core-excited ions that subsequently relax via Auger or radiative decay. At current XFEL intensities and time scales, rapid photoionization results in short-lived population inversions that can be exploited for lasing. We study magnesium, aluminum, and silicon and find that a population inversion between K-shell vacancy and L-shell vacancy state is feasible, suggesting another way to develop atomic X-ray lasers based on inner-shell vacancy states. We also discuss their gain characteristics. This atomic X-ray laser, with high monochromaticity, temporal coherence and wavelength stability, may be useful for high-resolution spectroscopy and nonlinear optics in the X-ray region.

2. Result & discussion

This work investigates the population dynamics of atomic inner-shell vacancy states driven by atomic processes such as resonance absorption, radiative transition, and Auger processes. We are particularly interested in the K-shell vacancy dynamics.

Figure. 1 illustrates the atomic inner-shell X-ray laser scheme with the energy level diagram of the most relevant inner-shell states in aluminum. When the photon energy of the pumping X-ray is above the K-edge (1.56 keV threshold in the case of aluminum), the electron in the K-shell is photo-ionized, creating a vacancy in the K-shell (K⁻¹ state). The vacancy state decays by emitting a photon (radiative transition) or an electron (Auger transition, KL⁻¹L⁻¹). These liberated electrons also ionize atoms, creating vacancies in the L and M shells, such as a KL⁻¹ state. The collisional ionization has a larger cross-section for outer shell states than for inner shell states, making the KL⁻¹ state more collisionally populated than the K⁻¹ state. The delicate balance between pumping and depletion of the K⁻¹ and KL⁻¹ states for pumping conditions such as pulse intensity, duration, and photon energy may create the population inversion between the two states. In the discussion below, we show the population dynamics and the creation of inversion.

 

Fig. 1. Atomic inner-shell X-ray laser scheme. When a K-shell electron is photo-ionized, it creates the K⁻¹ state (a vacancy in the K-shell). The state decays via radiative transition or Auger transition. The lower state KL⁻¹ is populated by the collisional processes of liberated electrons. Under a proper condition, a population inversion is created between the K⁻¹ and KL⁻¹ states. The related inner shell-vacant states are shown. In this simulation, atomic processes involving with 277 configurations and ionization charges up to +12 ions are included in the case of Al.

Download Full Size | PDF

A typical XFEL pulse generates around 10¹² photons at a few keV energy level within a duration of a few tens of femtoseconds [46,47]. When the XFEL interacts with a solid density target, it preferentially liberates inner-shell electrons, leaving atoms in inner-shell vacant states. These states are highly excited transient states and lie above the first ionization limit. The atoms stabilize to lower energy states via either Auger decay or radiative decay within a timescale comparable to the XFEL pulse duration. The photoelectrons and Auger electrons produced in the process also ionize the solid target, resulting in the production of secondary electrons with much lower energies than primary electrons. The substantial amount of ionization enables the secondary electrons to establish a Maxwellian distribution rapidly through collisions.

To study atomics processes initiated by XFEL pulses, it is necessary to solve rate equations that describe the atomic states and their transitions. Additionally, the time-dependent temperature and density conditions of the XFEL-heated dense matter must be estimated. To achieve this, population distributions, photon absorption rates, internal energies, and average charge states should be considered in a time-dependent and self-consistent manner. To address these challenges, collisional-radiative models have been extended to a specific atomic kinetics model called SCFLY [48]. SCFLY includes an extensive set of configurations required for solid density plasmas created by XFEL pulses; in the case of Al, 277 configurations are considered to Al ^+ 12, 284 configurations are considered to Mg ^+ 11 in the case of Mg and 346 configurations are considered to Si ^+ 13. At the intensity of ∼ 10¹⁰ W/cm², where the simulations have been done in this investigation, most of the processes arise from low charge ions, and their population magnitudes dominate. Using the temporal conditions of an XFEL pulse as input parameters, SCFLY simulates self-consistently the time-dependent population and ionization distributions, assuming that electron equilibration is instantaneous.

To study the population dynamics of relevant states and determine the optimal conditions, we used the SCFLY code to solve the rate equations numerically for the populations of the relevant states of interest when neutral atoms are ionized by a short, intense X-ray pulse. The SCFLY code establishes a grid of atomic states in an atom or ion and calculates the population distribution by considering the rates of population and depopulation resulting from various atomic processes. A set of rate equations is solved in matrix form,

As an ion is surrounded by electrons, photons and other ions, the highly-lying bound states become effectively inaccessible due to interactions with the surrounding electrons and ions, which reduces the ionization potential. This phenomenon is known as ionization potential depression (IPD) and plays a significant role in determining the relevant states. Therefore, the number of the states require to determine the population distributions is finite. The cut-off limit is often determined using the Stewart and Pyatt model [49,50]. The SCFLY code utilizes a screened-hydrogenic model to simulate plasmas driven by XFELs, where the statistical distribution among fine-structure or term states is used, and an atomic state is described by the principal quantum number of each bound electron. The SCFLY code employs on calculated and averaged data in a Dirac-Hartree-Slater approximation [51], with a focus on the precision of atomic data [52,53,54].

The recent work on the effect of nonthermalized electron [55] on the population and emission spectra indicates that the contribution from non-thermalization of electron distribution becomes significant from the laser intensity of ∼ 10¹⁷ W/cm² or higher. Since the laser intensity in this case is smaller by 7 orders of magnitude, the contribution from non-thermal electron will be negligible, if any. The assumption of Maxwellian distribution is still applicable to the simulation conditions in this study.

Since the realization of attosecond XFEL pulse, we have done simulations for short pulses in the ranging from 0.01 fs to 5 fs to investigate how high gain can be achieved and determine the required XFEL intensities within the attosecond XFEL conditions.

Figure 2(a) shows the temporal evolution of the population densities of the upper state (K⁻¹) and lower state (KL⁻¹), as well as the inversion density for aluminum in response to an XFEL pulse. The simulation considers a 1 um-thick aluminum foil irradiated by an XFEL pulse with a photon energy of 1580 keV, a pulse intensity and duration of 5 × 10¹⁰ W/cm² and 1 fs at full width at half maximum (FWHM), respectively, and a beam diameter is 200 µm. The photon energy is slightly higher than the ionization energy of the K shell. The time origin is set to be the time when the intensity of the incident XFEL pulse reaches its maximum. The population of the K-shell vacancy state increases rapidly following the pumping X-ray pulse due to the high photo-ionization cross-section of the K-edge at 1580 keV and the high intensity of the pump pulse. The K-shell vacancy state has fast decay channels, such as radiative transition to the lower state (KL⁻¹), Auger transition to the next-higher ionization state (KL⁻¹L⁻¹), and collisional ionization into the L shell, resulting in a short lifetime of 2 fs. The rapid increase in the population of the upper state during the pumping phase indicates that the pumping to the state via inner-shell photoionization is much stronger than the loss caused by the various decay channels. After the pulse, the population of the upper state begins to drop at almost the same rate as in the pumping phase.

 

Fig. 2. The temporal changes of (a) population density and inversion density and (b) G_inv is the gain from population inversion and G_eff is the absorption subtracted value from G_inv in Aluminum. Photon energy is 1580 eV which is over threshold of the K-edge, 1560 eV. Pulse intensity is 5 × 10¹⁰ W/cm² and pulse duration is 1 fs. Maximum value of gain is 6.9×10⁴cm⁻¹

Download Full Size | PDF

On the other hand, the population of the lower state builds up more gradually, reaching 2.5 times lower than that of the upper state. In addition to the large photoionization cross-section, the fast Coster-Kronig decay from the lower state is also allowed, contributing to the decrease in population. This makes the lifetime of the lower state short, and its population begins to decline around 200 as before the peak of the pumping pulse. The competition between the populations of the upper and the lower state results in the population inversion. The inversion density (${N_{inv}}$) is define by ${N_{inv}} = {N_{up}} - \left( {\frac{{{g_{up}}}}{{{g_{low}}}}} \right){N_{low}}$, where ${N_{up}}$ and ${N_{low}}$ are the population of the upper and lower state, and ${g_{up}}$ and ${g_{low}}$ are the statistical weight of the upper and lower state, respectively. The inversion density is plotted as a blue-dotted line. The inversion is successfully created and lasts for 3 fs. Due to the strong pumping condition, the inversion density mostly follows the population of the upper state. G_inv is the gain coefficient from the population inversion defined $G = \frac{1}{{8\pi }}{\left( {\frac{M}{{2\pi K{T_{ion}}}}} \right)^{\frac{1}{2}}}{A_{ul}}\lambda _{ul}^3{g_u}\left( {\frac{{{N_u}}}{{{g_u}}} - \frac{{{N_l}}}{{{g_l}}}} \right)$, where M is ion mass, K is Boltzmann constant, T_ion is ion temperature in room temperature, A_ul is the spontaneous decay rate from upper level to lower level, λ_ul is the transition wavelength, N_u and N_l is the populations of the upper level and lower level respectively. And g_u and g_l is statistical weights of lower and upper level, respectively. G_eff is effective gain which is the absorption subtracted gain value from G_inv which is proportional to the inversion density, reaches a maximum value of 6.9×10⁴cm⁻¹ at −260 as in Fig. 2(b).

To study the gain characteristics, we vary the conditions of pumping photon energy, pulse intensity, and pulse duration. After each simulation of the populations, we calculated the gain, and traced its maximum value. First, we investigate the gain variation with respect to the pumping photon energy. Figure 3(a) shows the gain and absorption coefficient of the Al for various photon energies around the K shell absorption edge at 1560 eV, as shown in the absorption curve. The gain saturates at this energy and reaches its local maximum at 1580 eV before decreasing. The pumping pulse at 1550 eV still has spectral components around 1560 eV, even though they are weak, because of its energy bandwidth of 11 eV FWHM. This weak spectral component is still capable of creating population inversion; thus, we observe the gain, although small, at the pumping photon energy of 1550 eV. Around the absorption edge, the absorption varies, which is called the x-ray absorption near-edge structure (XANES). While there are variations of the absorption near the near edge, the effective gain is about 16 times larger than the absorption, as shown in Fig. 3(a). The simulations shown in Fig. 3(b), and c was done at 1580 eV, at which XANES effect is already small. The XANES effect will not change the main conclusions of this study. In Fig. 3(b), the gain variation for pumping intensities is shown. In these simulations, the photon energy is fixed at 1580 eV where is higher than the K-edge threshold, and the simulation was conducted for pulse durations of 0.01 fs, 0.1 fs and 1 fs FWHM. The results demonstrate that the attosecond pumping pulse is more effective than the femtosecond pumping pulse. There is a significant jump in gain value between the 1 fs and 0.1 fs pumping pulse duration, whereas there is much less increase between 0.01 fs and 0.1 fs pumping pulse durations. The gain increases with pulse intensity and saturates around 3-5 × 10¹⁰ W/cm² in 0.01 fs and 0.1 fs pulses. This suggests that the intensity is already sufficient to deplete the neutral states and reaches saturation. Moreover, the intensity of 5 × 10¹⁰ W/cm² in 0.01 fs and 0.1 fs duration is strong enough to deplete the K shell in neutral Al. On the other hand, in the case of 1 fs pulse, the gain keeps increasing even beyond the 5 × 10¹⁰ W/cm² intensity.

 

Fig. 3. Gain characteristics. (a) G_eff vs photon energy at an intensity of 5 × 10¹⁰ W/cm² and a pulse duration of 1 fs FWHM. The absorption cross-section is also displayed. (b) G_eff vs pulse intensity at a photon energy of 1580 eV for pulse durations of 0.01, 0.1, 1 fs (c) G_eff vs pulse duration at photon energy of 1580 eV for intensities of 0.5 × 10¹⁰ W/cm² to 3.5 × 10¹⁰ W/cm²

Download Full Size | PDF

The gain variation with respect to the pulse duration for a given pulse intensity is shown in Fig. 3(c). First, we note that the gain increases non-linearly with the variation of pulse duration and, as expected, is lower for longer pulse durations. Upon closer inspection, we observe that the gain increase for the variation of pulse intensity from 0.5 × 10¹⁰ W/cm² to 3.5 × 10¹⁰ W/cm² differs for different pulse durations. While the gain decrease from 0.1fs to 5 fs FWHM, the slope of the gain decrease is much steeper at higher pulse intensities. For example, the gain decreases 64 times from 0.01 fs to 5 fs at 3.5 × 10¹⁰ W/cm², but it decreases 9 times at 0.5 × 10¹⁰ W/cm², in case of G_inv. In the case of longer pulse durations, the upper state is slowly populated, and at the same time, it has a greater chance of being depopulated via various decay processes. Thus, the population of the upper state cannot be higher than that in the case of shorter pulse duration. Therefore, even though the pulse intensity is increased 7 times, the gain is enhanced only 1.84 times in the case of a pulse duration of 5 fs.

We also investigate the gain characteristics of neighboring atoms such as Mg and Si, as shown in Fig. 4. For comparison purposes, the simulation results for aluminum, which were shown in Fig. 3(c), are also plotted. Simulations were carried out to obtain the gain characteristics around the photon energies at the K-shell absorption edge of each material. The pumping photon energies are set at 1315 eV and 1870eV for Mg and Si, respectively. It is notable that, in general, the gain variation with respect to pulse duration is similar to that obtained for Al, where shorter pumping pulses lead to higher gains. However, since the relative ratios between atomic process rates are different from each material, the rate of gain increase differs between material. For example, in the case of a pumping of 0.5 × 10¹⁰ W/cm², the gain increases 6.7 times for the change of pump pulse duration from 2 fs to 0.01 fs FWHM in case of Al, but 8 times in the case of Si although the gain value is lower than Al’s. Additionally, note that the slope of gain increase in the case of Si is much smaller, so gain crossing points exist between Mg, Al, and Si. For a detailed understanding of gain crossing, we look into the temporal changes of populations before and after the crossing points.

 

Fig. 4. The gain of magnesium, aluminum, and silicon as a function of pulse duration for pumping intensity of (a) 0.5 × 10¹⁰ W/cm² and (b) 3.5 × 10¹⁰ W/cm². The photon energy is where the maximum G_eff is achieved for each material.

Download Full Size | PDF

For each material, we analyzed the population dynamics for different pulse durations 0.01 and 2 fs for clarify the crossing point around 1.4 fs . In the case of pulse intensity of 0.5 × 10¹⁰ W/cm², there is a gain crossing between the Mg and Al, while Si consistently exhibits the lowest gain in every pulse duration, as shown as Fig. 4(a). In Fig. 5(a), (b), (c), representing a pulse duration of 0.01 fs, the population of the upper state in each material is on the same order of magnitude as the gain value, while the population of the lower state is comparable between materials. The gain values follow the population of the upper states because the upper state population is dominant in every material. For a pulse duration of 2 fs, as depicted in Fig. 5(d), (e), (f), the population of the upper state in Si is the lowest, resulting in the lowest gain value. The population of the upper state between Mg and Al is comparable, and the population of the lower state is higher in Al. However, the rise in the population of the lower state in Al is slower compared to Mg, allowing for a higher inversion density to be achieved. Consequently, the gain values between Mg and Al reverse while Si maintains the lowest gain values.

 

Fig. 5. Temporal changes of population density and inversion density in each material and conditions. Pulse intensity is 0.5 × 10¹⁰ W/cm² for all condition. (a), (d) are Mg and (b), (e) are Al and (c), (f) are Si. And pulse duration is 0.01 fs for (a), (b), (c) and 2 fs for (d), (e), (f).

Download Full Size | PDF

In the case of a pulse intensity of 3.5 × 10¹⁰ W/cm², there is crossing point between Al and Si around 1.4 fs, while Mg exhibits the lowest gain value across all pulse duration, as shown in Fig 4(b). For the pulse duration of 0.01 fs, as depicted in Fig. 6(a), (b), (c), the populations of the lower states are comparable among the materials and significantly lower than the populations of the upper states, rendering them negligible. However, there is a clear distinction in the population of the upper states among the materials, and the gain value follows the order of the upper states in each material. In Fig. 6(d), (e), (f), representing a pulse duration of 2 fs, Mg displays a lower inversion density and consistently maintains the lowest gain value across all pulse durations. The population of the upper state is higher in Al compared to Si; however, the rate of change in the lower state populations is higher and faster in Al. As a result, the gain value between Al and Si becomes similar and undergoes a reversal in longer pulse duration.

 

Fig. 6. Temporal changes of population density and inversion density in each material and conditions. Pulse intensity is 3.5 × 10¹⁰ W/cm² for all condition. (a), (d) are Mg and (b), (e) are Al and (c), (f) are Si. And pulse duration is 0.01 fs for (a), (b), (c) and 2 fs for (d), (e), (f).

Download Full Size | PDF

The SASE-XFELs has spikes in the temporal profile but they don’t make a critical difference to the gain dynamics in this simulation. We define the temporal profile as Gaussian.

In conclusion, we have studied the population dynamics between the K⁻¹ and KL⁻¹ states in the ultrafast timescale, considering attosecond XFEL pulses. The photon energy around the K-shell absorption edge of materials is used to creating the K shell vacancy. For pulse durations longer than 1 fs FWHM, the gain increases as the pump pulse intensity increases even beyond 1 × 10¹¹ W/cm². In the case of a short pulse such as 0.1 fs FWHM, the gain starts to saturate even at 5 × 10¹⁰ W/cm². The comparison of the gain characteristics between Mg, Al, and Si shows interesting points: due to the different ratios between the rates of atomic processes such as photoionization rate from the K-shell, radiative transition rate, and Auger transition rate, (1) the degree of variation of the gain coefficient for the pumping conditions is different, and (2) the saturated gains are different for strong pumping conditions. However, there is a crossing point in the gain between materials due to the different rates of atomic processes. With the arrival of the attosecond coherent XFEL, we can explore the inner-shell dynamics in the attosecond timescale for the development of atomic X-ray lasers. Atomic X-ray lasers have applications in high-resolution spectroscopy and nonlinear optics in the X-ray region.