In this work a theoretical treatment of self-amplification of extreme ultraviolet light in a gas medium is presented. The process is initiated by focusing short light pulses produced by a free-electron laser onto a helium gas target, to transfer a part of the population from the ground state of the atom to the selected doubly excited state below the second ionization threshold. The main decay channel of the upper state is autoionization, but it can also decay via fluorescence to singly excited states by emitting $\sim\!30\,$nm light belonging to the extreme ultraviolet region of the spectrum. If the target pressure and pump pulse intensity are high, the intensity of spontaneously emitted light in the direction of the pump pulse propagation increases enough to cause stimulated emission. This leads to an exponential amplification of the light intensity of the emitted pulse, which is directional and coherent.
The treatment of the problem with Maxwell-Bloch equations is presented, which describe the propagation of electromagnetic fields in the target together with the temporal and spatial evolution of the state populations and coherences between them. Spontaneous emission is modeled by a source term, which injects energy into the system and produces random fluctuations of the atomic polarization. In addition to the basic three-level scheme, further ionization with the pump and emitted field is considered by means of the rate equations. Results for a wide range of pump intensities and target pressures are presented, showing clearly the regions of spontaneous emission, exponential amplification, and saturation of the emitted light. The treatment includes both the coherent pump pulses, produced by a seeded free-electron laser, as well as partially coherent pump pulses, generated by amplification of light emitted randomly by the free-electron laser.
To our knowledge, this is the first time the full set of Maxwell-Bloch equations for a three-level system, where one of the states decays by autoionization, is solved, including the spontaneous decay of the autoionizing state. In our case, the upper state is chosen to be the $3a\,^1\!P\,^o$ doubly excited state in helium, a prototype of strongly autoionizing resonances in atomic physics, featuring a lifetime similar to the pulse length of the free-electron laser light in the extreme ultraviolet region.
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