1. INTRODUCTION
The development of X-ray sources based on high-peak-power lasers has
been one of the thrust areas of research since the advent of the
table-top terawatt chirped pulse amplified (CPA) laser. This has
stimulated the effort to understand processes involved in intense laser
interaction with matter. Laser-produced plasmas from optically ionized
gases, clusters, and solids have been widely explored as sources of
X-ray and extreme UV over the last decade (McPherson
et al., 1993; Ditmire et
al., 1996; Giulietti & Gizzi,
1998; Rocca, 1999; Hulin et al., 2000; Junkel-Vives et al., 2001; Krainov & Smirnov, 2002). Solids and van der
Waals clusters of rare gases have been found to have excellent X-ray
yields and useful spectral content under intense laser interaction.
Rare gas clusters have solidlike local densities, and therefore absorb
laser radiation very effectively while at the same time producing
negligible debris upon disintegration (Ditmire
et al., 1996). Laser absorption mechanisms in clusters
and plasma hydrodynamics have been subject to intense study since the
discovery of unprecedented X-ray flux from intense laser interaction
with clusters (McPherson et al.,
1993). Monitoring high-energy electrons (Shao
et al., 1996; Chen et al.,
2002), ions (Dobosz et al.,
1999; Nishihara et al.,
2001), neutrons (Zweiback et al.,
2000; Schwoerer et al.,
2001), and X rays (Ter-Avetisyan et
al., 2001; Skobelev et al.,
2002) has helped to unravel some of the processes underlying the
intense X-ray emission. Initial investigations suggested that Mie
resonance of the spherically shaped clusters and resonance heating at
the critical density surface could explain most of the observations
(Milchberg et al., 2001). Further
investigations where the laser pulse duration was varied found the
duration to play a major role in determining particle (ion and
electron) energy spectra and total X-ray flux (Zweiback et al., 1999; Parra et al., 2000; Kumarappan et al., 2002). Plasma expansion
and radiative blast waves, which occur long after the laser pulse
transit, introduce further ionization in the plasma (Edwards et al., 2001). X-ray spectra have
helped to characterize these sources over the years, and so far the
highest observed X-ray line energy from cluster targets has been
identified as 3.3 keV, identified as K-shell emission from
H-like argon (Junkel-Vives et al.,
2001). The presence of a population of hot electrons was found
to be responsible for the excitation of line spectra (Abdallah et al., 2001). However, these
observations were made over a restricted spectral window and therefore
spectral information over a much wider spectral band, particularly at
higher energies, remained unknown. To date, all reports of X-ray
emission from clusters have concentrated on X rays from ions with
L- and M-shell vacancies from atoms of high atomic
numbers (e.g. Kr, Xe) and Kα lines of lighter atoms (e.g., Ar, Ne,
N). In the work presented here, we have observed, for the first time,
X-ray bremsstrahlung over a broad spectral region and intense Kα
radiation at 12.66 keV from krypton irradiated by a focused femtosecond
laser beam with an intensity of less than 3 × 1016
Wcm−2.
2. EXPERIMENTAL PROCEDURE
The experimental chamber used for the laser–cluster interaction
experiments, shown in Figure 1, consists of
an ultrahigh vacuum six-way cross connected to a larger chamber that is
equipped with a high-throughput turbomolecular pump that attains a base
pressure of 10−8 torr and maintains
10−3 torr during experiments. Krypton gas with a
stagnation pressure of 7 bars or 15 bars is pulsed at a repetition rate
of 10 Hz through a cylindrical nozzle with a throat diameter of 2 mm to
form clusters. The high-pressure supersonic gas jet forms a cloud of
atom clusters held together by van der Waals forces when the gas is
cooled by adiabatic expansion through the nozzle. Measurement of the
cluster size distribution is usually difficult to carry out without
partially destroying the clusters; therefore, Rayleigh scattering of
400-nm laser beam from the jet has been monitored to establish the
formation of clusters. Theoretically, the Rayleigh scattered signal
(S) should scale roughly as the cube of the backing pressure,
S ∝ P03 (Ditmire et al., 1996). However, in our own
measurements, as in some other experiments (Ter-Avetisyan et al., 2001), the signal of
the scattered light scales with the square of the stagnation pressure,
S ∝ P02. A semi-empirical
parameter, called the Hagena parameter, determining the average cluster
size for an axisymmetric cylindrical nozzle is Γ* =
kd0.85p0
/T02.29, where d is the
nozzle throat diameter in micrometers, p0 the
backing pressure in millibar, T0 the initial gas
temperature in degrees kelvin (K), and k (= 2890 for Kr) a
constant that depends on the sublimation enthalpy per atom and van der
Waals bond length of the gas (Wormer et
al., 1989). The cluster size scales as
Nc = 32(Γ*/1000)2.25
for Γ* > 1000 (Hagena, 1992).
According to this scaling law at room temperature
(T0 = 293 K) and stagnation pressure 7 bar, we
estimate the mean number of atoms per cluster as
〈Nc〉 ∼ 6.5 ×
104, which gives an estimate of the radius of clusters as
Rc ≈
r0〈Nc〉1/3
∼ 10 nm and the corresponding quantities for 15 bar is 3.6 ×
105 and ∼17.5 nm, respectively. The density is
∼1023 cm−3, of the order of solid
density.
Experimental system used for X-ray observations from clusters. L:
focusing lens; P: Laser power meter; D: X-ray detector (XR-100CR,
Amptek); AMP: spectroscopic amplifier; MCA: Multichannel analyzer; OSC:
Oscilloscope; VC: vacuum chamber; A: variable aperture.
Part of the energy from a 5-TW CPA Ti:sapphire laser (λ = 800 nm,
τp = 50 fs) beam is used to irradiate the
cluster cloud using a f = 100 mm UV-grade fused silica lens
with a measured focal spot radius of 8 μm. The laser pulse duration
is adjusted by varying the grating separation in the pulse compressor
of the CPA laser system. Single photon counting is used to measure
X-ray spectra in the range 2–20 keV using a Peltier cooled Si:PIN
detector, which gives a measured energy resolution of 350 eV at 14.4
keV and excellent detection efficiency in the spectral range. Single
X-ray events are accumulated using a gated multichannel analyzer
calibrated with a Co57 source. The spectra are corrected for
the efficiency of the entire detection and measurement system (e.g.,
filter and window transmission) prior to analysis (X-ray transmission
data tables obtained from www.cxro.lbl.gov). The count rate is
kept below 0.1/pulse to avoid distortion of the spectra due to pile
up by limiting the detection solid angle to less than 5 ×
10−4 sr using apertures.
3. RESULTS AND DISCUSSION
Kα and Kβ emission from krypton at 12.66 keV and 14.1 keV,
respectively, has been observed, as shown in Figure
2, for a cluster radius 10 nm and a laser intensity of 2 ×
1016 Wcm−2. The laser intensity is an order
of magnitude lower than that used in many similar experiments (Dobosz et al., 1997) with substantially
larger clusters. The low energy continuum part of the spectrum is
filtered using a 350-μm-thick aluminum filter to isolate the
K-shell radiation having ∼28% transmission at 12.66 keV.
The measured X-ray spectra corrected for filter transmission factors,
shown in Figure 3 as a function of laser
intensity, have broad features corresponding to a bremsstrahlung
continuum in the 3–15 keV region. These spectra are qualitatively
similar to those found in condensed phase experiments (Korn et al., 2002). The strong resemblance
of the spectra to that emitted by solid density matter and the fact
that unclustered gas targets cannot be heated efficiently to extreme
high temperatures indicate that absorption and emission happens while
the high density phase is sustained. After the clusters have expanded
and form uniform plasma, the density has fallen drastically so that
collisional effects are minimal. The continuum emission originates from
either free–free or free–bound transitions during
interactions of electrons with ions and can be effectively used as a
diagnostic of the temperature of the plasma (Yaakobi
et al., 1996). For both free–free and
free–bound transitions, the spectrum depends exponentially on
photon energy, as I(hν) ∝
exp(−hν/kBTe),
which indicates a Maxwellian distribution of electrons, where
h is Planck's constant, kB
the Boltzmann constant, and Te the
electron temperature. Thus the slope of the semi-logarithmic plot
directly gives the electron temperature of the plasma. The thick solid
lines in Figure 3 are exponential fits to the
measured spectra. A minimum of 6.8 × 1015
Wcm−2 is required to produce measurable Kα
radiation. Furthermore, the line to continuum ratio grows as a function
of laser intensity due to more effective heating to higher
temperatures. The onset of Kα radiation indicates that hollow Kr
atoms with K-shell vacancies are being formed by collisions
with high-energy electrons. Since it is not possible that these inner
shell electron excitations responsible for Kα, β emissions are
produced through direct field ionization by the relatively low laser
intensities used here, the dominant mechanism responsible must be
excitation due to high-energy inelastic electron–ion collisions.
For Maxwellian electron distributions, the cross section for electron
impact ionisation is (Brunner, 1997)
where nj is the number of electrons in
the shell j, ηj is the ionization
energy in electron volts, η is the kinetic energy of the
electrons, and g(η/ηj) has
the form
From the above, the required electron energy to enable inner shell
vacancies must be equal to or greater than the ionization energy, which
for the krypton 1s electron is 14.3 keV. Observation of
Kα, β lines is therefore evidence of the existence of electrons
with energies higher than 14.3 keV. The electron temperature evaluated
from the continuum spectrum is shown in Figure
4. The solid line is a least squares fit of the data to
kTe ∝ I1/2,
which is more slowly varying than the ponderomotive energy scaling,
Up = 9.33 ×
10−14I (Wcm−2), which rules
out predominance of direct ponderomotive heating mechanism.
X-ray spectrum transmitted through 350-μm aluminum filter showing
the K-shell radiations from Kr cluster plasma. Integrated
X-ray yield across Kα line profile after correcting for filter
transmission is ∼105
photons/pulse/4πsr.
X-ray emission from Rc = 4 nm krypton
clusters irradiated with a laser of intensity (a) 2.5 ×
1015 (b) 6.8 × 1015, (c) 1.2 ×
1016, and (d) 2 × 1016
Wcm−2. Spectra are corrected for filter
transmissions.
Plasma temperature deduced from the continuum spectra measured as a
function of laser intensity.
The high local density within clusters, which is of the order of
solid density, and the low average density of the medium, demands that
the initial absorption and heating occurs within the cluster prior to
expansion. Even though anomalous flux of X rays from clusters was
observed almost a decade ago, none of the existing models fully
explains laser–cluster interactions. Ditmire et al.
(1996) proposed a nanoplasma model that still
remains the most successful model for predicting anomalous heating of
clusters. In this model, a nanoplasma is formed by rapid heating of the
cluster by collisional processes followed by hydrodynamic expansion.
The heating process encounters a strong resonance when the electron
density is equal to 3ncr, where
ncr is defined for the excitation
wavelength, ncr = ε0
meω2/e2,
where me and e are the electron
rest mass and charge, respectively, and ω is the angular frequency
of the laser. The enhanced absorption happens because of the resonance
in the dipole moment of the plasma sphere of radius
Rc given by P =
Rc3[(ε −
1)/(ε + 2)]E where ε is the plasma
dielectric function, ε = 1 −
ωp2/[ω(ω +
iνei /2)],
νei is the electron–ion collision
frequency, ωp2 =
ne
e2/ε0
me defines the plasma frequency,
ne, the electron density, and E
represents the laser electric field. P undergoes resonance
when ε + 2 = 0 or ωp2 =
3ω2 while ω >> νei.
However, such a resonance has a short lifetime and can happen during
the buildup and decay of the electron density according to simulations
(Milchberg et al., 2001). It has
also been suggested that, in addition to the
3ncr resonance heating, resonance
absorption at the critical density surface will contribute to heating
for large clusters (Milchberg et al.,
2001). Even though for clusters with dimensions much smaller
than the wavelength of light defining the angle of incidence at poles
of the plasma sphere is questionable and therefore resonance absorption
at the critical density surface resembling laser–solid
interactions is not fully justifiable when the electric field inside
the cluster is uniform. Rhodes and colleagues (McPherson et
al. 1993) formulated a model for small
clusters that relies primarily on coherent intracluster collisions. The
basic assumption of this model is that individual atoms inside the
cluster are ionized through direct field ionization. It successfully
explains the threshold for the appearance of L-, M-,
and N-shell emission from xenon for clusters of
Nc < 103 atoms. However, our
observations of harder X rays at low laser intensities suggest that
this scaling may not directly apply for K-shell emissions.
X-ray yield from 3 to 15 keV as a function of pulse duration for
larger clusters (Rc ∼ 17.5 nm) is
measured and given in Figure 5. Total X-ray
yield is more than 107 photons/pulse/4πsr.
Quantitatively X-ray yield in our measurements is significantly higher
than that observed for krypton clusters of comparable dimension and for
∼20 times the laser intensity (Dobosz et al. 1997). The highest observed X-ray energy from
krypton by Dobosz et al. (1997) was
1.75 keV, and any indication of K-shell emission at higher
X-ray energies was absent. Figure 5 shows
that there exists an optimum pulse duration at which the X-ray yield is
maximum. For fixed cluster size and pulse energy, the integrated X-ray
yield, which is a measure of absorption and electron–ion
collisions, first increases significantly with increasing pulse
duration toward a maximum at 300 fs and decreases for longer pulses.
Because the laser energy per pulse is kept constant, at the highest
peak intensity (shortest pulse duration) the X-ray yield is
approximately 2.5 times smaller than that at the optimum pulse
duration. The broad peak of the curve is between about 250 and 800 fs,
which correspond to an intensity of (3–0.93) ×
1016 Wcm−2. The electron oscillation
amplitude in the laser field of magnitude E0 at
this intensity is x0 = eE0
/(mω2) ∼ 14–8.5 nm (neglecting
the dielectric response of the cluster). This is comparable to the
cluster dimensions within predicted cluster size distribution
Nn ∼ Rn
exp(−Rn2/2Rc2)
(Parks et al., 2001; Krainov & Smirnov, 2002).
Dependence of total X-ray yield for krypton clusters of radii 10 nm
in the 3–15-keV region, as a function of laser pulse duration at
fixed pulse energy. Laser intensity corresponding to pulse duration 250
fs is 2.4 × 1016 Wcm−2.
In almost all models developed so far on cluster heating there is a
general agreement that a substantial contribution originates from
electron–ion inverse bremsstrahlung, though various resonances
enhance the absorption when the conditions are right. During
collisions, part of the energy can also be radiated as X-ray
bremsstrahlung. When the oscillation amplitude is comparable to the
cluster dimensions, there is significant overlap between the electron
and ion clouds to facilitate effective collisional absorption. For the
shortest pulses used here, the oscillation amplitude is ∼25 nm and
hence the electron cloud overlap with the ions is minimal during the
laser pulse. Moreover, the ponderomotive energy for the electrons,
Up, is higher than the measured thermal
energy with short pulses, and therefore the collision frequency scales
with laser electric field as (Silin, 1965),
νei = Zene
meω3 ln
Λ/rπε02E03,
and hence rapidly decreases with increasing laser field amplitude. Here
ln Λ is the Coulomb logarithm. This reduction in collision
frequency in conjunction with the larger oscillation amplitude
contributes to less effective heating and X-ray yield with shorter
pulses. For longer pulses than the optimum, the hydrodynamic expansion
of the cluster reduces the density and thus the number of collisions.
The efficiency of laser coupling to the cluster nanoplasma can
therefore be enhanced by altering the electron dynamics in the laser
field using variable pulse width irradiation as indicated by our
results.
4. CONCLUSION
We have measured X-ray spectra in the region of 3 to 15 keV from
laser-heated krypton clusters. The direct observation of broadband
bremsstrahlung radiation demonstrates the role of collisions during
cluster heating and evolution. Dependence of X-ray yield on pulse
duration is arguably an indication of collisional effects during laser
cluster interaction through better overlap between electron cloud and
the ion background in a single ionizing cluster. For longer pulse width
(a few hundred femtoseconds) and moderate intensities, the oscillating
electrons spend most of their time inside the cluster, enabling more
energy to be absorbed through collisions with ions due to relatively
small amplitude oscillations over a larger number of laser cycles. In
the case of shorter pulses with same laser energy density, electrons
are driven with higher oscillation amplitude and hence result in less
effective electron–ion overlap and therefore less efficient
collisional heating. A reduced total X-ray flux is observed for pulses
shorter than 300 fs. Therefore, the results shows that careful
selection of laser and cluster parameters enhances the X-ray spectral
yield even at moderate laser intensities. Detailed and quantitative
theoretical investigations to establish the effect of collisions in
cluster heating are underway and will be published elsewhere in the
near future.
Hard X-ray K-shell emission lines can potentially be used as
a powerful source of ultrashort pulses of incoherent X-ray radiation
for time-resolved X-ray diffraction and imaging applications in
ultraclean environments. For large clusters,
Rc ∼ 17.5 nm, we have measured X-ray
photon yield >107/pulse, useful for imaging,
diffraction, and medical applications. Laboratory X-ray sources in the
10–20-keV region are particularly useful for medical applications
like mammography. The required laser intensities are readily available
in medium-scale laboratories and, furthermore, the source is debris
free and can be operated at a reasonably high repetition frequency.
Previous observations under similar experimental conditions have been
limited to line emission with energies of a few kiloelectron volts,
seriously limiting the applicability of the source.
ACKNOWLEDGMENTS
The authors acknowledge the SHEFC and the EPSRC for financial support.
