We suggest a technique to amplify a train of attosecond pulses, produced by high-harmonic generation (HHG) of an infrared (IR) laser field, in an active medium of a plasma-based X-ray laser. This technique is based on modulation of transition frequency of the X-ray laser by the same IR field, as used to generate the harmonics, via linear Stark effect, which results in redistribution of the resonant gain and simultaneous amplification of a wide set of harmonics in the incident field. We propose an experimental implementation of the suggested technique in active medium of C 5+ ions at wavelength 3.4 nm in the "water window" range and show the possibility to amplify by two orders of magnitude a train of attosecond pulses with pulse duration down to 100 as. We show also a possibility to isolate a single attosecond pulse from the incident attosecond pulse train during its amplification in optically deep modulated medium. Abstract:In this supplemental material we derive an analytical solution describing an amplification of the highharmonic field in an active medium of a plasma-based X-ray laser in the linear regime within the three-level medium model and constant population inversion approximation. We provide also more detailed numerical study of an amplification process in both linear and nonlinear regimes taking into account a saturation effect within the fivelevel medium model.

Coherent intense attosecond X-ray pulses could lead to a fast dynamical imaging of the biological macromolecules and other material nanostructures with a unique combination of a record high temporal and spatial resolution. Plasma based X-rays laser sources are capable to produce high energy X-ray pulses but with relatively long picosecond duration. The sources based on high-harmonic generation (HHG) of a laser field allow to produce much shorter pulses but of lower energy. We suggest two different paths towards intense sub-femtosecond X-ray sources, namely i) via efficient transformation of the picosecond radiation of the X-ray plasma lasers into the trains of sub-femtosecond pulses in a resonantly absorbing medium, and ii) via amplification of HHG radiation in the active medium of the X-ray plasma lasers. We show that essentially the same technique can be used for realization of both paths. This technique is a modulation of the parameters of the resonant transition (accordingly in absorbing or amplifying medium) produced under the action of sufficiently strong infrared or optical field. We propose experimental realization of the suggested technique in the passive/active media of i) Li III ions modulated by the mid-IR laser field and ii) C VI ions modulated by the optical laser radiation. I.

We derive an analytical solution uncovering the origin of few-cycle attosecond pulse formation from vacuumultraviolet (VUV) radiation in an atomic gas simultaneously irradiated by a moderately strong infrared (IR) laser field, which does not perturb atoms in the ground state, but induces rapid quasistatic ionization from the excited states [Opt. Lett. 36, 2296 (2011)]. The derived solution shows that the pulses are produced due to periodic switching of the resonant interaction between the incident VUV radiation and the atoms: turning it off near the crests of the IR-field strength and switching it back on near the IR field zero-crossings. We extend the method originally proposed in [Opt. Lett. 36, 2296 (2011)] to non-hydrogenlike media and show that the pulses can be produced from resonant VUV radiation in a variety of atomic gases. The pulses are nearly bandwidth-limited without external adjustment of phases of the generated sidebands. Proximity of the carrier frequency of the produced pulses to intra-atomic resonances may allow their utilization for nondestructive steering of ultrafast dynamics of the bound electrons. The experimental possibilities for attosecond pulse formation from 58.4 nm VUV radiation in helium and from 73.6 nm VUV radiation in neon dressed by the 3.9 μm laser field, as well as from 122 nm VUV radiation in atomic hydrogen dressed by CO 2-laser field are discussed.

We establish a close physical analogy between coherent forward scattering of γ-ray radiation in the vibrating quasi-resonant nuclear absorber and the XUV field propagation in the quasi-resonant atomic medium in the presence of the moderately strong IR field. We prove that both processes, under certain conditions, are described by similar Maxwell–Bloch equations for a two-level medium with modulated parameters of the resonant transition. It results in similar transformation of both γ-ray and XUV fields at the exit from the medium, fully determined by the characteristics of applied modulation and spectral content of the incident fields. We find an appropriate analytical solution describing transformation of the electromagnetic field as a result of its propagation in the modulated medium. We show, in particular, that recently observed effects of (i) suppressed resonant absorption in coherent γ-ray scattering of vibrating absorber and (ii) ionization rate modulation in IR pump–XUV probe experiments, present themselves as different manifestations of the same general physical phenomenon of modulation induced transparency (MIT). That transparency is induced by modulating the parameters of the resonant transition. While only partial MIT was observed so far, we suggest certain conditions for conducting some realistic experiments, which should demonstrate nearly 100% transparency in both processes.

We perform an ab initio study of the ultimate capabilities and limits of applicability of the method for few-cycle pulse formation via the resonant interaction of an extreme ultraviolet (XUV) radiation with atoms dressed by moderately strong infrared (IR) laser field proposed in ([Phys. Rev. Lett. 105, 183902 (2010)] and [Opt. Lett. 36, 2296). Taking into account all the multiphoton processes in the systems under consideration on the basis of numerical solution of the three-dimensional time-dependent Schrödinger equation (TDSE) in the single-active-electron approximation, we show the possibilities to produce 1.1 fs pulses from 124.6 nm XUV radiation via linear Stark effect in atomic hydrogen, as well as 500 as pulses from 58.4 nm XUV radiation via excited-state ionization in helium. We derive a generalized analytical solution, which takes into account the interplay between sub-laser-cycle Stark effect and excited-state ionization and allows to analyze the results of TDSE calculations. We found that the ultimate intensity of the IR field suitable for few-cycle pulse formation via the linear Stark effect or excited-state-ionization is limited by the threshold for atomic ionization from the resonant excited state or the ground state, respectively. We show that the pulses with shorter duration can be produced in the medium of ions with higher values of the ionization potential.( ) XUV d τ is calculated by filtering out the low-frequency components of the total dipole moment, ( ) d τ , at the frequencies of low-order harmonics of the IR field and below. An additional filtering is applied for the spectral components with photon energies exceeding the atomic ionization potential, which accounts for a strong photoabsorption of XUV radiation just above the ionization potential.

We suggest a highly efficient method of ultrashort pulse formation from the resonant XUV radiation due to sub-laser-cycle modulation of the excited state of non-hydrogen-like atoms by a nonionizing IR laser field. This modulation results in formation of the Raman Stokes and anti-Stokes sidebands in coherently forward scattered radiation, which, in turn, leads to formation of short pulses, when the phases of the sidebands are matched. This method is a generalization of recently suggested technique [Phys. Rev. A 88, 053849 (2013)] to non-hydrogen-like medium. Possibility to form 2 fs XUV pulses in the gas of helium atoms and 990 as XUV pulses in the plasma of Li+ ions with efficiencies over 80% is shown.

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