LASER-INDUCED FLUORESCENCE MEASUREMENT OF PLASMA ION TEMPERATURES: CORRECTIONS FOR POWER SATURATION
M. J. Goeckner and J. Goree
Department of Physics and Astronomy
The University of Iowa
Iowa City, IA 52242-1479
University of Iowa Report number
Journal of Vacuum Science and Technology A7,
LASER-INDUCED FLUORESCENCE MEASUREMENT OF PLASMA ION TEMPERATURES: CORRECTIONS FOR POWER SATURATION
M. J. Goeckner and J. Goree
Department of Physics and Astronomy
The University of Iowa
Iowa City, Iowa 52242
When using high-power pulsed tunable dye lasers to measure plasma ion temperatures, it is important to attenuate the laser intensity. The temperature is found from the Doppler broadening of a spectral line. This may be obscured by saturation broadening, an instrumental effect encountered when too much laser intensity is used. Three useful experimental methods for determining the optimum pulsed laser intensity are found from a semi-classical atomic physics model. As an example, an Ar II transition pumped by a 1 GHz bandwidth laser is examined. The fluorescence line width of room temperature ions broadens from 1.50 GHz to 2.87 GHz when the homogeneous laser intensity is increased from 50 kW/m2 to 5 MW/m2.
The correct measurement of low ion temperatures is an important and difficult task.1 A reliable diagnostic will enable 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
Plasma ion energies are often measured with an in situ electrical device, an energy analyzer. The device has two major drawbacks. The first is the electrical nature of the energy analyzer. This will perturb the local plasma potential. The second drawback is possible contamination of the plasma by an in situ device. Contaminates may produce changes in the plasma characteristics and in the surface conditions.
An alternate method for measurement of ion energies is the use of laser-induced fluorescence (LIF).2-4 LIF measurements do not disturb or contaminate the plasma. When non-perturbative ion measurements are necessary, as with processing plasmas, LIF is the logical diagnostic.
LIF is a relatively new plasma diagnostic technique.5 While LIF can be used for a variety of measurements, we will only consider ion energies. This technique is possible because of the Doppler shift of a moving ion's absorption line. Ion distribution functions are determined by measuring the Doppler broadened spectral line shape.
Some recent studies using this technique have employed pulsed lasers.2,6 Pulsed lasers are characterized by broad frequency bandwidth, 0.5-6 GHz, high intensity, 0.01-1.0 GW/m2, and short pulse duration, 5-20 ns. They introduce instrumental broadening of the spectral line not found when CW lasers are used. Instrumental broadening must be minimized for accurate measurement of ion distribution functions.
One type of instrumental broadening, encountered with the use of high intensity pulsed lasers, is the result of power saturation. Saturation occurs when the stimulated photon emission rate is equivalent to the photon absorption rate and greater then the spontaneous photon emission rate. Increases in laser intensity will not change this balance. When saturation occurs at frequencies in the wings of the laser line, the measured fluorescence line is broadened. This is known as saturation or power broadening.
Ignoring power broadening may result in measurement errors of the ion temperature. These errors can be several orders of magnitude. By reducing the laser intensity, this problem can be avoided. However, reducing the intensity diminishes the fluorescence signal. The balance of a strong signal and minimal instrumental broadening determines the most desirable laser intensity.
To see how laser intensity broadens the fluorescence line width, it is necessary to develop a basic model of LIF. This paper treats LIF as a semi-classical process in a collisionless plasma. Although a completely quantum mechanical description of the photon-ion interaction was described by Glauber7 and Mollow,8-12 we chose to work with the semi-classical model for simplicity. Theoretical results for spatially homogeneous pulses of laser light are presented and discussed.
Three simple experimental methods for selecting the optimum laser intensity are given.
II. SEMI-CLASSICAL MODEL OF LIF
Fluorescence is produced in a simple manner. Laser light, tuned close to a specific transition of the ion, is directed through the plasma. Ions that absorb the laser photons can then decay via spontaneous emission to a lower state. These spontaneously emitted photons, i.e., fluorescence, constitute the LIF signal. The fluorescence line is examined by scanning the laser's frequency through the ion's transition. A number of phenomena, including Doppler broadening, collisional broadening, and saturation broadening, determine the fluorescence line shape.
Doppler broadening is a simple process. It arises from the Doppler shift, 2[[pi]][[Delta]][[nu]] = k*v, of the absorption spectrum. If this were the only line broadening mechanism, a Maxwellian ion distribution would result in a line width characterized by the uncorrected frequency full width at half maximum (FWHMunc) of
where mi is the ion mass, c is the speed of light, Ti is the ion temperature in units of energy, and [[nu]]01 is the central frequency of the transition. While this equation offers an easy method for determining the ion temperature, we will show it can be incorrect.
Collisional broadening of the fluorescence line has been well studied.13 Our equations will at first include source terms in which collisional transitions could be accounted for, but these terms are assumed to be small and will be dropped from the final analysis.
Saturation or power broadening is caused by multiple processes. They are absorption, stimulated emission, and spontaneous emission of photons. This broadening can be understood by examining how these processes change the population densities of three states of an ion, denoted 0, 1, and 2. The laser is tuned near the 0 -> 1 transition while an ion decaying via the 1 -> 2 transition produces a fluorescence photon. This is shown in Fig. 1.
Absorption occurs when an ion in state i interacts with a photon near a transition frequency [[nu]]ij. This causes an upward transition from state i to state j. Single ions, with a velocity component v along the laser beam, have a normalized probability Bij/ 4[[pi]] L([[nu]],[[nu]]ij,v) d[[nu]] dt of absorbing a single photon with frequency from [[nu]] to [[nu]] + d[[nu]] in time dt. The proportionality constant, Bij, is the Einstein absorption coefficient defined for light of isotropic intensity. Multiplying by the intensity frequency spectrum, I(x,[[nu]],t), and the density of ions of velocity v, fi(x,v,t), gives the number of these transitions occurring in time dt. Integrating over frequency gives the rates of change, produced by absorption, for the densities of states i and j,
Stimulated emission occurs when an ion in state j interacts with a photon near the transition frequency [[nu]]ij. This causes a downward transition from state j to state i. In the semi-classical model the interaction probability is the same as that of absorption, L([[nu]],[[nu]]ij,v) d[[nu]] dt.14 This process changes the densities of states i and j at the rates
where Bji is the Einstein stimulated emission coefficient defined for light of isotropic intensity.
Spontaneous emission occurs without the presence of external photons. This decay is a downward transition from state j to state i. The density of state i changes at the rate
where Aji is the Einstein spontaneous emission coefficient. The sum of all these spontaneous decay rates is the decay rate for state j
where the 1/e time of state j is denoted by [[tau]]j.
Other processes, unrelated to the laser light, can change the density of a state. They include ion-ion collisions, electron-ion collisions, and spontaneous emission from other upper states. These will be included in a single source term, Si(x,v,t), for each state.
Combining these processes we develop rate equations for the ion distribution functions of each of the ion states. The set of rate equations14 for states 0, 1, and 2 are
is an effective isotropic laser intensity. This is the complete set of empirical rate equations governing the LIF process. To use these equations the laser intensity spectrum, I(x,[[nu]],t), the ion absorption spectrum, L([[nu]],[[nu]]ij,v), and the ion distribution functions, fi(x,v,t), must be specified.
Lasers do not have a monochromatic intensity spectrum. They emit a frequency spectrum dependent upon the physical characteristics of the laser cavity and lasing medium. Because these characteristics vary from laser to laser, the power spectrum varies from laser to laser. A simple model of the laser beam employs a Gaussian frequency distribution
with [[theta]](t) a dimensionless temporal dependence, [[xi]](x) a dimensionless spatial dependence, [[nu]] the frequency, [[nu]]l the central frequency, and d[[nu]]l the bandwidth. Experimenters should characterize their laser. If the laser has a different power spectrum, the following equations should be re-evaluated.
Spatial dependence will be treated in a future paper. Presently the beam is assumed to be uniform. Temporal dependence may vary from shot to shot. To keep the model as simple as possible, [[theta]](t) is assumed to be unity during the laser pulse, 0 <= t <= T, and zero otherwise. Thus
is our model of the laser intensity.
The classical photon absorption spectrum is derived by modeling a motionless ion as a driven harmonic oscillator.14 For light of frequency [[nu]], near the transition [[nu]]ij, this model gives a Lorentzian absorption probability. Thus
where Aji is the decay rate from state j to state i. When the Doppler shift is included, the absorption curve for an ion with velocity v in the laboratory rest frame becomes
where v is the component of the ion velocity parallel to the laser beam.
The ion distribution functions are assumed to be homogeneous Maxwellians,
where k represents the atomic state and nk the state's spatial density. Before the laser is turned on at time t = 0, each state has the same form:
To obtain a large signal the experimenter must choose a state 0 with a long lifetime and a state 1 with a short lifetime. This gives state 0 a large initial density and state 1 a small initial density. Thus
n0 >> n1. 5
With this semi-classical model of LIF, theoretical calculations of the fluorescence signal can be performed. By examining how the laser intensity distorts the signal, methods are found to determine the optimum intensity for use of LIF as a diagnostic tool.
The LIF signal comes from detecting the photon emitted as an ion decays from state 1 to state 2. The number of fluorescence photons collected is
where d[[Omega]] is the detector's solid angle. Solving the coupled rate equations, Eqs. 2-4, using the assumption that the external source terms Si are small, and integrating over time yields
where we have defined
When state 0 and state 2 are identical, A10 will replace A12 in Eq. (6).
To determine the line width, Eq. (6) is calculated numerically. By plotting Nobs versus the laser frequency, we arrive at our theoretical prediction of the line shape. As an example, we examined the Ar II transition 3d' 2G9/2 <-> 4p' 2F07/2 -> 4s' 2D5/2. This transition was used by Anderegg et al.2 to observe ion heating. They employed a laser with a bandwidth, d[[nu]]l, of 1 GHz and a pulse duration, T, of 17 ns. These parameters were used in our computation; a list is provided in Table 1.
Fig. 2 shows the fluorescence FWHM as a function of ion temperature. The correct line widths are found by computing Eq. (6) for a set of laser frequencies near the transition frequency of a motionless ion. They are compared to the uncorrected line width found from Eq. (1). The fluorescence line is broadened by high laser intensities. When this broadening occurs, Eq. (1) will not provide the correct ion temperature.
Fig. 3 shows the fluorescence FWHM as a function of laser intensity. If the laser intensity is increased above an certain level, saturation broadening occurs. Below this level, the fluorescence line width has little variation.
Fig. 4 shows the fraction of ions, as a function of laser intensity, that make the 0 -> 1 -> 2 transition when the laser is tuned to [[nu]]01. This is a measure of the relative signal level. The signal diminishes proportional to the intensity when the laser power is weak. Above a certain laser intensity, the signal strength reaches an asymptotic limit. This asymptotic limit is caused by the saturation of the 0 -> 1 transition.
Optimizing the laser intensity requires the balance of a strong fluorescence signal with minimal saturation broadening. Three arbitrary methods for choosing the optimum intensity are suggested by an examination of Figs. 2 - 4.
The first experimental method for optimizing the laser intensity is inferred from Fig. 2. As the laser intensity is lowered, the FWHM of the fluorescence curve will approach the value found by Eq. (1). The experimenter measures the fluorescence FWHM as a function of the intensity, while the plasma parameters remain constant. Plotting the measured data on log-log graphs will result in curves as shown in Fig. 3. The tangent of the experimental curve will be of the form
log10(FWHM) = [[sigma]] log10(I) + constant
where [[sigma]] is the slope. The laser intensity is chosen to be optimum when [[sigma]] = 0.015. Curve fitting algorithms, such as a cubic spline fit, can determine slopes. Figure 4 shows that decreasing the intensity below the optimum value will decrease the signal strength, and Fig. 3 shows that increasing the intensity broadens the fluorescence line. In Fig. 3 the optimum intensity for Ti = 1.0 eV is 500 kW/m2.
The second experimental method for optimizing the laser intensity is inferred from Fig. 4. The relative LIF signal, when the laser is tuned to [[nu]]10, becomes asymptotic at high and at low laser intensities. The optimum laser intensity is half the intensity at which the the asymptotes intersect. Intensities below this value will decrease signal strength and intensities above it will broaden the fluorescence line. In Fig. 4 the asymptotes for Ti = 1.0 eV intersect at I = 500 kW/m2. Thus the optimum laser intensity is 250 kW/m2. This compares well with the optimum found by the first method.
The third experimental method for optimizing the laser intensity is also inferred from Fig. 4. The experimenter tunes the laser to find the maximum signal strength. The laser intensity is then diminished until the fluorescence signal is reduced by a factor of five. For Ti = 1.0 eV this optimum intensity is 190 kW/m2. This intensity is close to those found by the other methods.
Note that the results presented above are for a laser frequency bandwidth of 1 GHz. By selecting a laser with a narrower bandwidth, which produces less instrumental broadening, lower temperatures can be measured more accurately. Likewise, using a laser with a wider bandwidth will result in more instrumental broadening and less accuracy.
Large errors in the measured ion temperatures result from neglecting instrumental broadening, including saturation broadening. Saturation occurs when the laser intensity is high enough that the 0 -> 1 photon absorption and 1 -> 0 stimulated emission rates are approximately equal and large compared to the 1 -> 0 spontaneous emission rate, f0(x,v,t) B01[[Phi]](x,v,t) ~ f1(x,v,t) B10[[Phi]](x,v,t) > A10 f1(x,v,t). If the laser intensity is lower than this, saturation and saturation broadening will be minimized. Saturation broadening can be seen in Figs. 2 and 3. Saturation can be seen in Fig. 4.
Optimal laser intensities, which balance a strong signal with minimal saturation broadening, can be determined experimentally. They are chosen so that very weak saturation occurs. This assures a relatively strong fluorescence signal but minimal power broadening. The three arbitrary methods, discussed above, involve either measuring the fluorescence frequency bandwidth or the signal strength as a function of the laser intensity.
We wish to thank P. Kleiber and T. Whelan for helpful discussions. This work is supported by The Iowa Department of Economic Development. J. Goree also receives partial support from the NORAND Corporation Applied Academics Program.
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Table 1 The parameters used for Figs. 2 - 4. This corresponds to probing the Ar II transition, 3d' 2G9/2 <-> 4p' 2F07/2 -> 4s' 2D5/2, with a typical broad bandwidth high intensity pulsed laser.
PARAMETER VALUE UNITS REFERENCE
d[[nu]]l 1.0 GHz
[[lambda]]0 611.492 nm 15
A10 2.12 x 107 s-1 16
A12 7.59 x 107 s-1 16
B10 12.19 x 1012 m2(Js)-1
B01 9.755 x 1012 m2(Js)-1
[[tau]]1 8.51 ns
[[tau]]0 >= 1.0 ms
T 17.0 ns
mi 39.962 amu
Fig. 1 The laser-induced fluorescence process. Ions are pumped from state 0 to state 1. The ions in state 1 then have a probability of spontaneously decaying into state 2. Photons produced by this spontaneous transition comprise the fluorescence signal. The strength of this signal is determined by density of ions in state 1. The mechanisms changing the density of state 1 are: stimulated and spontaneous photon emission by ions in state 1, and photon absorption by ions in state 0.
Fig. 2 The LIF frequency line width as a function of ion temperature for various laser intensities. The solid line represents a fluorescence line width that is uncorrected for all broadening mechanisms except Doppler broadening. As the homogeneous laser intensity is lowered, this uncorrected line width becomes a better approximation. Lowering the laser intensity below a certain level produces little change in the fluorescence line width.
Fig. 3 The LIF frequency line width as a function of laser intensity for various ion temperatures. Saturation broadening occurs for large intensities. The laser intensity can be optimized by producing this curve experimentally and finding the slope of the tangent, [[sigma]]. The intensity is chosen to be optimum when [[sigma]] = 0.015. Laser intensities above the optimum will broaden the fluorescence line width.
Fig. 4 The fraction of ions that produce fluorescence photons when the laser frequency is tuned to maximize the signal. Saturation broadening occurs when a large fraction of all ions produce fluorescence. The laser power levels can be optimized by determining the asymptotes of the signal strength for high and low intensities. The optimum is half the intensity at which these lines cross.
An alternate method of optimizing the laser is to decrease the intensity until the fluorescence signal is a fifth of the maximum. Laser intensities below the optimum will simply reduce the signal strength.