M. J. Goeckner, J. Goree, and T.E. Sheridan

Department of Physics and Astromomy

The University of Iowa

Iowa City, IA 52242-1479

**Abstract**

When using laser-induced fluorescence to measure plasma ion velocity
distribution functions, high power inhomogeneous laser intensities produce
important undesirable changes in the Doppler broadened fluorescence line width.
A basic model of laser-induced fluorescence is discussed. As typical examples,
fluorescence from the ArII 3d'^{2}G9/2 <->
4p'^{2}F^{0}7/2 -> 4s'^{2}D5/2 transition with three
spatial laser intensity modes are examined. The ion velocity distribution
function can be measured correctly only if the maximum intensity for all
regions of the laser beam is below the saturation broadening threshold. A
procedure to determine if this is achieved is described.

**I. INTRODUCTION**

A correct measurement of low plasma ion energies is an important and difficult
task.^{1} A reliable diagnostic enables the accurate study of ion
heating, ion acoustic waves, and related phenomena. These processes can be
important in gas discharge, fusion, and other plasmas.^{2} One
possible diagnostic uses laser-induced fluorescence from the ions.

Laser-induced fluorescence,^{3} LIF, may be used for a variety of
measurements; here we consider measuring only ion velocity distributions. Ion
velocity distribution functions are determined by measuring the Doppler
broadened spectral line shape. Measurements of velocity distributions will be
in error if other other processes broaden the spectral line.

Power broadening of the spectral line, an instrumental broadening encountered with the use of high intensity pulsed lasers, is the result of saturation of the probed transition for ions in a velocity range v to v + dv. Saturation occurs when the stimulated photon emission rate is balanced by the photon absorption rate and is much larger than the spontaneous photon emission rate. Increasing the laser intensity will increase the saturated velocity range but will not change this balance for velocities already saturated.

Power broadening can be reduced by decreasing the laser intensity. However,
decreasing the laser intensity will lower the detected fluorescence signal. An
optimum laser intensity will balance between a strong signal and minimal
broadening. We have discussed three methods of optimizing the intensity for a
spatially homogeneous laser beam in an earlier paper.^{4}
Inhomogeneous laser beams complicate these optimization methods.

When saturation broadening occurs in some areas of the laser beam and not others, the Doppler broadened fluorescence line is distorted. By examining how inhomogeneities in the laser intensity distort the fluorescence signal, this undesirable distortion can be avoided.

We examined three spatial intensity cases: a homogeneous laser beam, the sum of two homogeneous components, and a Gaussian spatial distribution. We show that the distortions take the form of false tails in the ion velocity distribution functions, which we call pseudo-temperatures.

**II. Theory**

The LIF process is dominated by transitions between three states of an ion.
These states, labeled 0, 1, and 2, along with the laser photon, [[nu]]01, and
the fluorescence photon, [[nu]]12, are shown in Fig. 1. The laser photons are
used to probe the Doppler broadened 0 - 1 transition. The relative signal from
the detected fluorescence photons is plotted as a function of the laser's tuned
frequency. Doppler shift, 2[[pi]][[Delta]][[nu]] = **k *** **v**, gives
the velocity distribution function.

We consider only three spatial intensity modes, homogeneous, two component, and Gaussian. However, the fluorescence line shapes from two modes will be the same if there is a one to one and on to spatial mapping of the laser intensities.

The number of fluorescence photons detected in a solid angle, d[[Omega]],
is^{4}

, 1

where

,

,

and

.

Here [[Phi]](**x**,**v**,t) is an effective isotropic laser intensity,
L([[nu]],[[nu]]10,**v**) d[[nu]] d**v** is the absorption probability,
I(**x**,[[nu]],t) is the laser intensity, f0(**x**,**v**,t) is the
distribution function for state 0, B01 is the Einstein coefficient of
absorption for isotropic intensity light, B10 is the Einstein coefficient of
stimulated emission from state 1 to state 0 for isotropic intensity light, A1i
is the Einstein coefficient of spontaneous emission from state 1 to state i,
[[tau]]i is the lifetime of state i, and [[nu]]10 is the transition frequency
from state 1 to state 0.

The theoretical prediction of the fluorescence line shape is found by plotting
Nobs as a function of the laser frequency. This theoretical line shape may be
plotted as a function of energy by using the Doppler shift of the ion
transition, 2[[pi]] [[Delta]][[nu]] = **v** * **k**, and the energy
equation, E = 0.5 mv^{2}.

An undistorted Maxwellian ion velocity distribution, f(v) [[proportional]]
exp(-v^{2}), will give a straight line when the log of the fluorescence
signal is plotted as a function of energy. The ion temperature, in units of
energy, may be found from the slope of this line using the formula, Ti =
-(slope)^{-1}. A distortion of a Maxwellian distribution will manifest
as a second line added to the correct distribution. The slope of the false
secondary line gives a pseudo-temperature.

Not all ion velocity distributions are Maxwellian. This precludes ignoring a measured tail of a velocity distribution. The tail may be real and an important part of the distribution.

We used the ArII transition 3d'^{2}G9/2 <->
4p'^{2}F^{0}7/2 -> 4s'^{2}D5/2 to examine distortion
of the velocity distribution. This transition has been used by Anderegg et
al.^{2} to observe ion heating. They employed a laser with a bandwidth,
d[[nu]]1, of 1 GHz and a pulse duration, T, of 17 ns. The parameters used in
our computations are provided in Table 1.

Pseudo-temperatures may result from two mechanisms. The first mechanism is
weak saturation broadening in all regions of the laser beam. This shown for
homogeneous laser beams in Fig. 2. Saturation broadening, an undesirable
effect, can be controlled by reducing the laser intensity.^{4}

The second mechanism, an enhancement of the first mechanism, is caused by a small bright portion of an inhomogeneous laser beam saturating the ion transition while the rest of the laser mode is less intense and does not saturate the transition. We call this inhomogeneous saturation broadening. Figure 3 shows inhomogeneous saturation broadening for a beam composed of two homogeneous components, a high intensity area and a low intensity area. The ratio of the high intensity area to the low intensity area is 1:10.

Figure 4 is the LIF signal for a Gaussian spatial laser intensity distribution. The pseudo-temperature seen in this figure is produced by a combination of the two mechanisms. Because spatial changes in intensity are not steplike, this distribution most realistically models actual experimental conditions.

Pseudo-temperatures are caused by saturation of the ion transition in any part of the laser beam. Elimination of distortion in the measured ion velocity distribution requires that no portion of the laser beam saturates the ion transition. This requirement allows the experimenter a simple procedure to determine if any distortion of the measured distribution occurred.

The procedure involves repeating the measurement at a lower laser intensity. The ion velocity distribution is measured. The laser intensity is lowered, by at least a factor of ten. The velocity distribution function is measured again. If the measured velocity distribution does not change, that distribution is correct. If the measured distribution does change, the laser intensity must be reduced again. This procedure is repeated until the correct distribution is measured.

**III. CONCLUSIONS**

Distortions of the ion velocity distribution function determined using LIF will occur if a portion of the laser beam saturates the ion transition. These distortions may be eliminated by attenuating the laser intensity.

To avoid these distortions, an experimenter can make repeated measurements of a velocity distribution, lowering the laser intensity between each measurement. If lowering the laser intensity does not change the measured velocity distribution, this type of distortion has not occurred.

ACKNOWLEDGEMENT

This work is supported by The Iowa Department of Economic Development.

FIGURE CAPTIONS

Fig. 1 Ion states in the LIF process. Laser photons, [[nu]]01, are used to probe the 0 - 1 state transition. The collected fluorescence photons, [[nu]]12, reveal the number of ions in state 1.

Fig. 2 The LIF signal for a homogeneous spatial intensity mode. These graphs
show the same normalized fluorescence signal. The signal is a function of
frequency in the left graph and energy in the right graph. They are related by
the Doppler shift, 2[[pi]] [[Delta]][[nu]] = **v** * **k**, and the
energy equation, E = 0.5 mv^{2}.

Fig. 3 The LIF signal for a two component spatial intensity distribution.

Fig. 4 The LIF signal for a Gaussian spatial intensity distribution.

Table 1 - The parameters used in Figs. 2 - 4. They correspond to probing the
ArII transition, 3d'^{2}G9/2 <-> 4p'^{2}F^{0}7/2
-> 4s'^{2}D5/2, with a typical broad bandwidth high intensity pulsed
laser. The laser intensity, I([[nu]]), is assumed to be a Gaussian with a FWHM,
d[[nu]]1, of 1.0 GHz.

^{_____________________________________________________________________________________________________________________________________________________________}

^{}PARAMETER VALUE UNITS REFERENCE

^{______________________________________________________________________________}

^{}d[[nu]]1 1.0 GHz

[[lambda]]0 611.492 nm 5

A10 0.212 10^{8} s^{-1} 6

A12 0.759 10^{8}s^{-1} 6

B10 121.9 10^{11} m^{2}(Js)^{-1}

B01 97.55 10^{11}m^{2}(Js)^{-1}

[[tau]]1 8.51 ns

[[tau]]0 >= 1.0 ms

T 17.0 ns

mi 39.962 amu

^{}

REFERENCES

^{1}N. D'Angelo and M. J. Alport, Plasma Phys. **24**, 1291
(1982).

^{2} F. Anderegg, R. A. Stern, B. A. Hammel, M. Q. Tran, P. J. Paris,
and P. Kohler, Phys. Rev. Lett. **57** 329, (1986) and references therein.

^{3} R. A. Stern and J. A. Johnson, Phys. Rev. Lett. **24**, 1548
(1975).

^{4} M. J. Goeckner and J. Goree, J. Vac. Sci. Technol. A **7**,
(May/June 1989).}

^{5} Göran Nolén, Physica Scripta **8**, 249 (1973).}

^{6} Gustavo García and José Campos, J. Quant. Spectrosc.
Radiat. Transfer **34**, 85 (1985).