Effetti di non stazionarietà nella dinamica di
ionizzazione di un plasma prodotto da impulsi laser ai ns
L. Labate, C.A. Cecchetti, M. Galimberti, A. Giulietti,
D. Giulietti, L.A. Gizzi, P. Tomassini
IV Congresso italiano di fisica del plasma
Arcetri (FI), 12-14 Gennaio 2004
X-ray spectroscopy of laser-plasmas [1]
Spectrum by an Al
plasma produced by
a 3ns FWHM laser
pulse @ I=1e15
W/cm2
Intensity (A.U.)
X-ray spectroscopy is a powerful diagnostic tool of electron temperature and
density in laser-produced plasmas
Wavelength (Å)
In order to extract information about the plasma parameters, experimental intensity ratios
are compared to numerical simulations accounting for the ion charge state distribution
inside the plasma
IV Congresso italiano di fisica del plasma, Arcetri (FI) 12-14 Gennaio 2004
X-ray spectroscopy of laser-plasmas [2]: the methods
In general, Bragg crystal are used as dispersive element of the spectrometers:
spectral resolution (l/dl)
flat crystals
bent crystals
vertical dispersion
configuration crystals
(both flat and bent)
up to 5x102
up to 103
up to 5x103
Li-like satellites to Heb
Spectrum by an Al
plasma produced by
a 3ns FWHM laser
pulse @ I=1e15
W/cm2,,
obtained using a
spherically bent
mica crystal
Lya doublet
Heb line
IV Congresso italiano di fisica del plasma, Arcetri (FI) 12-14 Gennaio 2004
X-ray spectroscopy of laser-plasmas [3]: spatial resolution
Spatially or temporally resolving the spectra provides informations about
selected regions or times of plasma emitting regions
Spatial resolution can be obtained either by using a
slit between the plasma and the crystal or by using
bent crystal or Vertical Dispersing Crystals
Longitudinal
profile of the
Heb emission
Wavelength
Space
Spatially resolved spectrum by an Al plasma produced by
a 3ns FWHM laser pulse @ I=1e15 W/cm2,,
obtained using a spherically bent mica crystal
IV Congresso italiano di fisica del plasma, Arcetri (FI) 12-14 Gennaio 2004
Modelling X-ray spectra from laser-plasmas [1]
Analysis of X-ray spectra from laser-plasmas
requires the capability to accurately predict the
atomic level populations
Since ionization and recombination processes
depend upon the hydrodynamic variables of the
plasma, in order to do this a full, both hydrodynamic
and atomic, treatment is required:
Hydrodynamic equations
Rate equations between all the ion levels
and different charge state populations
should be solved self-consistenly as a function of space and time
The radiative transfer should be included in some way in the calculations
Without some simplifying assumptions, this task is not affordable
IV Congresso italiano di fisica del plasma, Arcetri (FI) 12-14 Gennaio 2004
Modelling X-ray spectra from laser-plasmas [2]
The general method of X-ray spectra modelling:
separating the plasma simulation into a hydrodynamic part and an atomicphysics/radiative part
Hydrodynamic simulations
with a simplified model of ionization equilibrium
ne(x,t), Te(x,t),
eventually u(x,t)
An estimate of the energy
involved in the ionization
processes, which do not
provide detailed
population informations
useful to interpret X-ray
spectra, is made
Atomic physics/radiative simulations
Examples:
MEDUSA [1] (1D, Lagrangian)
POLLUX [2] (2D, Eulerian)
[1] Christiansen et al., Comput. Phys. Commun. 7, 271 (1974)
[2] Pert G.J., J. Comput. Phys. 43, 111 (1981)
IV Congresso italiano di fisica del plasma, Arcetri (FI) 12-14 Gennaio 2004
Modelling X-ray spectra [3]: post-processing hydrodynamic data
In order to obtain the detailed charge state and level distribution, the rate equations in
collisional-radiative equilibrium must be solved using ne(t) and Te(t) provided by the
hydrodynamic simulations:
NZ : population of charge state Z
ScZ and SRZ : collisional and photo - ionisation rate from charge state Z
Z 1
Z 1
a 3b
and a RR
: three - body and rad. rec. rate from charge state Z  1

In the
ideal case, a set of equations has to be solved for each cell provided by he

hydrodinamic simulations, taking also into account the radiative transfer
Escape factors approximation: the set of rate equations is solved on a single-cell
basis and the emission coefficients are reduced by a factor accounting for the
probability for the photon to be reabsorbed inside the plasma
IV Congresso italiano di fisica del plasma, Arcetri (FI) 12-14 Gennaio 2004
Modelling X-ray spectra [5]: atomic physics codes
A simplification of the rate equations can be made by considering detailed levels
only for the most populated charge states
A set of test cases has been established
in order to sistematically compare nonLTE atomic physics codes
Average ionization of Aluminium as a
function of the electron temperature for an
electron density of 1021 cm-3
(the He-like ion population fraction is also
shown in the inset)[1]
Discrepancies in the results are mainly
due to the principal quantum number
taken into account for the different charge
state
The FLY [2] code has been used for modeling this experiment:
• ion stages from neutral down to Be-like: only ground state
• Li-like: n=1-10 (6-10 hydrogenic)
• He-like: n=1-10 (6-10 hydrogenic)
• H-like: n=1-10
[1] Lee R.W. et al., J. Quant. Spectr. Radiat. Transfer 58, 737 (1997)
[2] Lee R.W., Larsen J.T., J. Quant. Spectr. Radiat. Transfer 56, 535 (1996)
IV Congresso italiano di fisica del plasma, Arcetri (FI) 12-14 Gennaio 2004
Modelling X-ray spectra [4]: SS vs TD solution
When tat << thydr a steady-state (SS) solution can be considered, by solving the rate
equations at each time of interest with dNZ/dt = 0
For an indication of this solution to be acceptable, a simplified system with two charge
states can be solved, in order to calculate the relaxation times
Relaxation Time (s)
10
Relaxation times for an Al
plasma at a density
ne=1e21xm-3
10
-8
n e=1.0 10 21 cm -3
Be-li ke to Li l ike Al
Li-l ike to He l ike Al
He-li ke to H l ike Al
-9
10
-10
10
-11
10
-12
0
200
400
600
800
1000
Electron Temperature (eV)
These calculations show that relaxation time from He-like to H-like Al at temperatures
greater than 100 eV is comparable to the rise-time of nanosecond pulse
In this case a full time-dependent treatment of the atomic physics may be necessary
IV Congresso italiano di fisica del plasma, Arcetri (FI) 12-14 Gennaio 2004
The experimental technique
Tight-focus irradiation of a solid Al target in order to have a spherical expansion
Focusing lens: f/4
Flat Bragg TlAP crystals have been used as
dispersive elements both for time-integrated
and time-resolved spectra
CCD
l
Filter
Shield
0.7 mm Pb
x
Crystal
Lens
Laser
Target
Spectral resolution: l/dl~102
X-ray spectra has been resolved in time to obtain the
temporal evolution of line intensity ratios
IV Congresso italiano di fisica del plasma, Arcetri (FI) 12-14 Gennaio 2004
The laser system
YLF oscillator, 1053 nm
Phosphate amplifiers
Equivalent plane monitor of the laser beam
3 ns (FWHM) gaussian temporal profile
Single transverse and longitudinal mode
High-quality, near diffraction-limited
focal spots (beam waist ~ 8µm)
IV Congresso italiano di fisica del plasma, Arcetri (FI) 12-14 Gennaio 2004
Temporal resolution of the spectra
The time evolution of the spectral emission in the spectral range from 6.0 to 8.0 Å has
been recorded by means of an X-ray streak-camera coupled to the flat crystal
spectrometer
In particular, X-ray emission has been recorded during the rise-time of the laser-pulse
Image provided by the time resolving diagnostics,
corrected for the imaging system aberrations,
during the rise-time of the laser pulse
Temporal resolution has been
estimated to be ~20 ps
IV Congresso italiano di fisica del plasma, Arcetri (FI) 12-14 Gennaio 2004
Streak-camera intensity calibration
Streak-camera images have been calibrated by using time-integrated spectra provided
by a high-dynamics, linear spectrometer viewing the plasma at the same angle
Streak-camera
response at different
sweep speeds
.
Cross-calibration
5
3
2
Log(Intensity(A.U.))
4
1
IV Congresso italiano di fisica del plasma, Arcetri (FI) 12-14 Gennaio 2004
Experimental ratio vs. time
Different time-intagrated line intensity ratios have been used to check numerical results
Due to experimental reasons, the time-resolved Lya-to-Heb line intensity ratio has been
used to study the ionization dynamics during the rise-time of the laser pulse:
Lya: 2p -> 1s
Heb: 1s2 1S -> 1s3p 1P
The use of the resonance line Lya requires a careful consideration of the reabsorption
effects
Experimental Lya-to-Heb ratio during the rise-time of the laser-pulse
IV Congresso italiano di fisica del plasma, Arcetri (FI) 12-14 Gennaio 2004
Hydrodynamic simulations: 2D density map
Hydrodynamic simulation shows the expansion, due to the tight-focusing, has a 2D
behaviour
2D map of the mass density at the pulse peak
history of electron density profile obtained
by considering a full ionization
IV Congresso italiano di fisica del plasma, Arcetri (FI) 12-14 Gennaio 2004
Hydrodynamic simulations: 2D temperature map
Temperature also rapidly decreases 50 µm far from the original target surface
2D map of the electron temperature at the pulse peak
history of electron temperature profile
IV Congresso italiano di fisica del plasma, Arcetri (FI) 12-14 Gennaio 2004
Post-processing hydrodynamic data
A numerical predictions about the temporal behaviour of the Lya-to-Heb intensity
ratio can be retrieved by post-processing hydrodynamic data using FLY
Since the plasma is strongly inhomogeneous, an estimate of the most emitting regions
for each line should be done, in order to get a temperature and density history
This has been done by post-processing the hydrodynamic 2D data cell-by-cell by the
code FLY
IV Congresso italiano di fisica del plasma, Arcetri (FI) 12-14 Gennaio 2004
2D maps of the line emission
+20
+10
0
Lya line
-10
-20
history of line emission in a
3ns interval centered at the
pulse peak
µm
+20
history of the maximum
emission for each line
+10
0
Heb line
-10
-20
-25
0
+25
+50
+75 µm
IV Congresso italiano di fisica del plasma, Arcetri (FI) 12-14 Gennaio 2004
History of the line emission profiles: Lya
The instantaneous profile of the Lya emission exhibits a narrow peak, sitting on a
long tail due to the plume of the plasma
The location of the peak moves inward by approximately 10µm during the entire pulse
history of the Lya
emission profile
IV Congresso italiano di fisica del plasma, Arcetri (FI) 12-14 Gennaio 2004
History of the line emission profiles: Heb
The instantaneous profile of the Heb emission exhibits a narrow peak, sitting on a
long tail due to the plume of the plasma
The location of the peak moves inward by approximately 10µm during the entire pulse
history of the Heb
emission profile
IV Congresso italiano di fisica del plasma, Arcetri (FI) 12-14 Gennaio 2004
Atomic physics simulations: electron density and temp. history
Since the emission regions are very peaked, the electron density and temperature
corresponding to their peak can be used for the atomic physics simulation
During the rise-time of the laser pulse density and temperature conditions are obtained
to which atomic relaxation times of the order of some hundredths of ps are expected
IV Congresso italiano di fisica del plasma, Arcetri (FI) 12-14 Gennaio 2004
Atomic physics simulations: SS vs TD solution
Simulated history of the Lya-to-Heb line intensity ratio both in SS and TD
conditions (assuming a thin plasma)
Early during the emission, time dependent and steady-state model show different
results. Later on both models give identical values for the Lya-to-Heb ratio
IV Congresso italiano di fisica del plasma, Arcetri (FI) 12-14 Gennaio 2004
Atomic physics simulations: plasma opacity [1]
Line reabsorption effects has been taken into account by allowing for plasma sizes
given by the widths of the emission regions
The widths at different height with respect to the maximum have been considered
100% (optically thin line)
90%
50%
Heb emission profile
at the peak of the
laser pulse
10%
IV Congresso italiano di fisica del plasma, Arcetri (FI) 12-14 Gennaio 2004
Atomic physics simulations: plasma opacity [2]
Atomic physics simulations corresponding to these different values for the thickness
of the reabsorption region have been performed, both SS and TD
comparison between
exprimental line ratio
and simulated one for
different thickness of the
reabsorption region
in TD conditions (similar
results hold for the SS
case)
The plasma can be considered as optically thin for the two lines
IV Congresso italiano di fisica del plasma, Arcetri (FI) 12-14 Gennaio 2004
Line thicknesses: Doppler decoupling [1]
The small optical depth of the two lines can be explained by considering the
decoupling to the Doppler effect
longitudinal
profile of the
radial
component of the
fluid velocity at
the peak of the
pulse
Hydrodynamic simulations show a strong velocity gradient
near to the regions of maximum emissivity for the two lines
IV Congresso italiano di fisica del plasma, Arcetri (FI) 12-14 Gennaio 2004
Line thicknesses: Doppler decoupling [2]
By taking into account the Doppler emission and absorption profiles in two different
points, the probability for a photon to be reabsorbed can be calculated
reabsorption
probability due
to the Doppler
effect for a
photn emitted
in the point of
maximum
emissivity
Lya
Heb
The width of these curves allows to consider the plasma as optically
thin for both the lines
IV Congresso italiano di fisica del plasma, Arcetri (FI) 12-14 Gennaio 2004
Evidence of transient ionization
The history of the experimental line ratio can now be compared to the atomic physics
predictions both in the SS and TD case
Early during the emission, experimental data agrees well with TD calculations only
L.A.Gizzi et al., Letter on Phys. Plasmas 10,
4601 (2003).
IV Congresso italiano di fisica del plasma, Arcetri (FI) 12-14 Gennaio 2004
Summary and conclusions
An experiment has been performed devoted to detect transient effects in the
ionization dynamics during the rising of a ns laser pulse
A single longitudinal mode laser beam has been tight-focused on a thick solid Al target in
order to obtain a nearly spherical expansion of the produced plasma
2D maps of the lines emissivity has preliminary been obtained by post-processing
hydrodynamic data in order to evaluate the density and temperature of the emitting
regions as well as its dimensions
By comparing with the experimental ratio, the plasma has been found to be optically thin
for the two lines. An evaluation of the effects of the Doppler decoupling due to the
velocity gradient has been performed
Finally the atomic physics code has then been used in order to simulate the time history of
Lya-to-Heb line intensity ratio both in SS and TD conditions
The comparison of these histories with the experimental one has shown the existence of
transient effects during the rising of the laser pulse, when ionization relaxation times have
been found to be comparable with hydrodynamic times
IV Congresso italiano di fisica del plasma, Arcetri (FI) 12-14 Gennaio 2004