Pumping a photoionization atomic inner-shell x-ray laser with x-ray free-electron laser radiation Page: 4 of 10
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of small-signal gain calculations at different photon pumping energies in section 3. In section 4, we present a
one-dimensional self-consistent gain model to determine the X-ray laser output and show results for a typical
SASE sample pump pulse.
2. DESCRIPTION OF THE INNER-SHELL LASING SCHEME
The basic idea is to use an XFEL to create a population inversion by inner-shell photoionization. The concept
of a photoionization-pumped x-ray laser is not new,21,22 but due to the lack of appropriate x-ray pumping
sources, it could not be experimentally realized so far. Focusing an XFEL into a gas target, an elongated plasma
column of core-excited ions is created by inner-shell photoionization within the first few femtoseconds of the
XFEL pulse.13 The core-excited ions relax by either Auger decay or radiative decay. In case of neon, the
Auger lifetime of a K-shell hole is 2.7 femtoseconds, therefore allowing a transient population inversion of only
femtosecond duration. This ultra-short lived population inversion forms the basis of the x-ray transition. Since
the inner-shell photoionization and build-up of population inversion happens on an ultrafast time scale, i.e. on a
time scale short to typical ion-electron collisions in a gas target of moderate density, the ion temperature in the
plasma column is expected to remain cold. The line width of the x-ray lasing transition is then dominated by the
Auger width of the core-hole, which opens the pathway to very narrow high-gain lasing transitions. The proposed
lasing scheme is self-terminating, i.e. each atom can at most contribute one photon of a given transition during
the amplification process. The lower lasing state in our pumping scheme is efficiently depleted by another inner-
shell photoionization event. This is in stark contrast to recently proposed XUV FEL pumped lasing schemes
using inner valence electrons,18 where depletion of the lower lasing level is caused by Coster-Kronig transitions,
i.e. lasing occurs between two autoionizing states. Our scheme is widely applicable, by tuning the pumping
photon energy above the core-ionization edge of the lower lying lasing state and can be extended to any other
atomic species.
The geometry of the XRL is determined by the focus of the XFEL. Focusing an XFEL pulse of 1 keV photon
energy with a Gaussian spatial beam profile to a focal radius of r 1 pm results in a focal depth of approximately
5 mm, which defines the length L of the amplifying plasma column of extremely low aspect ratio. The spatial
coherence properties of the XRL can be characterized by the geometry of the x-ray amplifying plasma column3:
The transverse coherence length is given by LT = +. For a lasing transition around A 1 nm, we therefore
find a transverse coherence length of LT 1 pm, implying that the XRL will have nearly full spatial coherence.
3. SMALL SIGNAL GAIN CALCULATIONS
In a simple one-dimensional gain model, amplification of radiation in the exponential gain regime is governed by
j(z, t) j(O, t) ( en)z 9 t(
where j(z, t) denotes the flux at point z as a function of time t (to describe the temporal pulse structure), j(O,t)
is the seed (in our case, spontaneous emission), n is the density of the gain material and g(t) is the so-called
small-signal gain cross section, which is given by
g(t) nu (t)ustm -nL(t)abs " (2)
The small-signal gain cross section is determined by the occupancies of the upper and lower lasing states nu and
nL, respectively and by the cross-sections stm and cabs for stimulated emission and absorption,
2rc2 9lU
Qstim = AU-L 2Aw , %abs =stim -, (3)
where AUy-L is the Einstein A coefficient for the radiative transition and gu and gL are the statistical weights of
the upper and lower lasing levels. Eq. (3) gives the cross sections at the peak of the line, supposing a Lorentzian
line shape. The line width of the transition Ao is dominated by the total lifetime of the upper and lower
states (Auger-life time and radiative lifetime). This is in contrast to traditional optical-laser pumped atomic
x-ray lasers, where the line width is determined by Doppler and Stark broadening for high density plasmas. In
our proposed x-ray laser scheme the amplifying plasma column is cold (close to room temperature) since the2
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Rohringer, N & London, R. Pumping a photoionization atomic inner-shell x-ray laser with x-ray free-electron laser radiation, article, July 17, 2009; Livermore, California. (https://digital.library.unt.edu/ark:/67531/metadc866244/m1/4/: accessed April 18, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.