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A 3-level Model for Schumann-Runge 2 Laser-Induced Fluorescence 

Glenn S. Diskin* 

NASA Langley Research Center 

Hampton, Virginia 

Walter R. Lempert* and Richard B. Miles* 

Princeton University 

Princeton, New Jersey 



Abstract 

A three level model has been developed for the analysis 
of Schumann-Runge band (B 3 E U <—X 3 E g ) laser- 
induced fluorescence of molecular oxygen, 2 . Such a 
model is required due to the severe lower state depletion 
which can occur when transitions having relatively large 
absorption cross-sections are excited. Such transitions 
are often utilized via ArF* or KrF* excimer or dye-laser 
excitation in high temperature environments. The rapid 
predissociation of the upper state prevents substantial 
repopulation of the lower state by collisional processes, 
and the lower state may be largely depleted, even at 
laser fiuences as low as 10-100 mJ/cm 2 . The resulting 
LIF signal in such cases no longer varies linearly with 
laser pulse energy, and the extent of the sublinear behav- 
ior varies with the particular rovibrational transition of 
interest. Relating the measured signal to the lower state 
population, then, necessitates the use of exceedingly low 
laser fiuences. These low fiuences in turn lead to the 
need to compromise spatial resolution in order to gener- 
ate sufficient signal. 



Introduction 

Schumann-Runge (S-R) 



band system 



The 

(B 3 E~ U <—X 3 E g ) of molecular oxygen, 2 , has been 
the subject of much study, due primarily to its impor- 
tance in atmospheric photochemistry. Measurement of 
flow properties by use of S-R laser-induced fluorescence 
(LIF), has grown in scope since Massey and Lemon 1 



Research Engineer, Hypersonic Airbreathing Propulsion Branch, 
Member AIAA. 

' Research Scientist, Department of Mechanical and Aerospace Engi- 
neering, Member AIAA. 

* Professor, Department of Mechanical and Aerospace Engineering, 
Senior Member AIAA. 

Copyright © 1996 by the American Institute of Aeronautics and Astro- 
nautics, Inc. No copyright is asserted in the United States under Title 
17, U.S. Code. The U.S. Government has a royalty-free license to exer- 
cise all rights under the copyright claimed herein for government pur- 
poses. All other rights are reserved by the copyright owner. 



first proposed the use of a tunable ArF* laser for the 
measurement of temperature and density in air. Since 
that time, S-R LIF has been used to measure single- 
point temperature and density in low-temperature flows 
using an ArF* laser, 2 for imaging and detection of hot 
2 in combustion systems using ArF*, KrF* and dye 
laser systems, 3 " 7 and for measurement of temperature in 
high temperature air using a KrF* laser. 8 S-R LIF has 
also been used in conjunction with stimulated Raman 
scattering to perform velocity measurements in air 
flows. 9 S-R band LIF is often used because (1) oxygen is 
naturally present in many flow situations, and therefore 
doesn't need to be seeded into the flow, and (2) the 
upper, or B-state in the S-R band system is rapidly pre- 
dissociated, eliminating the need for quenching correc- 
tions and providing a signal which is directly 
proportional to lower state number density. An unfortu- 
nate consequence of the rapid predissociation of the 
upper state is that the quantum yield, or fluorescence 
efficiency, of the fluorescence emission is very low, typ- 
ically on the order of 10" 4 . Additionally, numerous 
authors have reported sublinear signal generation at 
moderate laser fiuences, 4 ' 6 ' 8 due primarily to lower state 
depletion, or bleaching. The purpose of this paper is to 
describe a model constructed to simulate the 2 S-R LIF 
signal generation process. 

Analysis of the signal obtained in a laser-induced fluo- 
rescence experiment requires a means to relate the mea- 
sured signal to the population in the state of interest, i.e. 
the lower state involved in the excitation transition. In 
order to relate the collected signal to the lower state pop- 
ulation, one must construct an appropriate model. This 
model should include the processes of laser excitation 
and de-excitation to and from the upper state of the tran- 
sition, de-excitation of the upper state by radiative decay 
(fluorescence), predissociation and collisional quench- 
ing, and redistribution of lower and upper state popula- 
tions by collision-induced energy transfer (rotational 
and/or vibrational). The large predissociation rates asso- 
ciated with transitions in the 2 Schumann-Runge sys- 



tem render a simple steady-state, two-level model, such 
as described by Eckbreth 10 , inappropriate. A model 
more appropriate for 2 Schumann-Runge laser- 
induced fluorescence was described by Laufer, et al. 2 
This quasi-two-level model includes the relevant pro- 
cesses, and uses an analytical integration of the rate 
equations (for top-hat temporal laser excitation) to relate 
the signal to the lower state population. The model 
works well for predissociation-dominated transitions 
with laser intensities low enough not to encounter sig- 
nificant population depletion. Transitions in 2 from the 
ground vibrational level excited by an ArF* laser (the 
(4,0) band) fall into this category. 

The model described in Ref. [2] does not adequately 
account for depletion of the lower vibrational manifold 
when the laser excitation becomes large and collisional 
rotational repopulation rates become large. The reason 
for this is that the lower ro-vibrational level was 
assumed in Ref. [2] to have an infinitely large bath from 
which collisional repopulation can occur. It is more real- 
istic to assume that this lower level can only be refilled 
by those molecules which began the process in the same 
vibrational level, i.e. that vibrational re-equilibration 
times are much greater than typical laser pulse times and 
that rotational re-equilibration times are comparable to 
laser pulse times. To keep track of this depletable bath, a 
new model was constructed which extends the model of 
Ref. [2] in two ways. First, a third level was included. 
This level represents all of the rotational levels of the 
ground vibrational level except the one coupled to the 
upper state by the laser, and provides the bath of mole- 
cules from which the lower level may be refilled. Sec- 
ond, since for transitions stronger than the (4,0) band the 
absorption rate may be comparable to the predissocia- 
tion rate, the process of stimulated emission must be 
included as a mechanism to couple the upper and lower 
states. The stimulated emission terms, which were 
dropped in the simplification of Ref. [2], are retained in 
this model. 

Inclusion of a third level in the model takes away the 
elegance of the solution given in Ref. [2], in that the 
solution can no longer be represented in terms of a sim- 
ple function of two parameters. The analytical solution 
becomes a very messy algebraic expression which 
doesn't allow for easy assessment of the effects of indi- 
vidual parameters. The system of equations was there- 
fore solved numerically. This solution method has the 
advantage that a more realistic laser pulse shape can be 
employed; Laufer, et al. 2 solved the quasi-two-level 
problem for a top-hat excitation pulse shape. For the 
solutions described herein, a Gaussian temporal distri- 
bution was used. A benefit of calculating the temporal 



response of the system using a realistic excitation tem- 
poral profile is an improvement in the representation of 
nonlinear effects. This is due to the fact that the system 
responds differently to excitation of high and low inten- 
sity; a 'real' temporal pulse shape incorporates a distri- 
bution of intensities, while a top-hat pulse lumps 
everything into a single, uniform value. What precisely 
is meant by 'high' and 'low' intensity is, of course, 
determined by the particular system. 

Three-level LIF Model 



w 



La 



w 



Le 



T t 



Wa 



W 



21 



1 


W Q w f 

| V zfi V 


w c 


nUN" 



Using the notation of Ref. [2], the equations governing 
the three-level system are: 



dt 



d_N 2 
dt 



= -W La (t)-N l + N 2 -(W Le (t) + W 2l ) (1) 



+ w c-[jzj--N 3 -N x 



W La (t) ■ Nt 



-N 2 -(W Le (t) + W d+ W 21 + W f+ W Q ) 



(2) 



dN 2 
dt 



= -Wr 



_jj_ 

l-/i 



N 3 -N l 



(3) 



In this model, described by equations (l)-(3) and 
depicted schematically above, levels 1 and 2 represent 
the ro-vibrational levels in the ground and excited elec- 
tronic states, respectively, which are coupled by the 
laser frequency. Level 3 consists of the bath of mole- 
cules collisionally coupled to level 1, by the rate W c . 
The factor f B l{ 1 -f B ) which precedes the N3 term in equa- 
tions (1) and (3) is required to provide detailed balanc- 
ing of the forward and reverse collisional redistribution 
processes. The model includes the relevant processes of 
laser-stimulated absorption and emission (Wi a and 
Wjj,), spontaneous emission (W22), predissociation 
(Wj), collisional quenching of the upper state (Wq), and 
radiative decay to vibrational levels other than the lower 
level of the transition (the fluorescence of interest, Wf). 



Not included in this model is collisional redistribution of 
rotational or vibrational energy in the upper level, as 
these are considered to be slow with respect to the pre- 
dissociation. 

The primary parameter of interest for the user of this 
model is the fluorescence signal, denoted Sf . The fluo- 
rescence signal at any time, t, per unit volume, emitted 
into 4it steradians, Sf(t) , is equal to W** ■ N 2 (t) and 
the total signal generated during the laser pulse is: 



B 21 are the Einstein coefficients for absorption and 
emission, respectively, and <p is the overlap integral of 
the transition and laser lineshapes. 

<p = \ g(v)h(\/)dv , 

Av 

where J g(v)dv = 1 and J h(v)dv = 1 

Av Av 

and g(v) and h(v) are the transition and laser line- 
shapes, respectively. The symbol v is defined as v/c . 



S f = jW f * ■N 2 (t)dt. 



(4) 



where f„ is the laser pulse duration and W** is the por- 
tion of the fluorescence signal which is detectable due to 
spectral filtering. In order to obtain the quantity Sf , 
equations (l)-(3) must be integrated to find N 2 (t), which 
is then integrated according to equation (4). To perform 
the required integration, appropriate values for the 
parameters W c , W d , Wf, W 2] and f B must be found, as 
well as an appropriate functional form of Wjjt). With 
these values, discussed subsequently, equations (l)-(4) 
can be integrated numerically using one of the standard 
techniques. For this work, the public domain software 
package Octave [11], version 1.1.1, was employed. 
Octave provides a front-end for the Lawrence Livermore 
ordinary differential equation solver, LSODE [12], writ- 
ten by Alan C. Hindmarsh. 

Parameters in the Rate Equations 

In order to solve the equations governing the model sys- 
tem, the parameters W c , W d , Wq, Wp W 2 j and /# must be 
found. W c is the collisional repopulation rate in the 
lower vibronic level (Nj), and is a function of the fluid 
density and temperature. W d is the upper rovibronic pre- 
dissociation rate, and is only a function of the upper 
level quantum state, as are Wp the fluorescence rate, and 
W 2 j, the rate of spontaneous emission at the laser fre- 
quency. Wq is the collisional quenching rate, and is a 
function, in general, of the temperature and densities of 
all collision partners. The rotational Boltzmann fraction 
associated with the lower level, /g, is a function of tem- 
perature, the lower vibrational level, and the lower rota- 
tional level. The laser excitation rate parameters, W^Jt) 
and WiJt), should be decomposed into their various 
pieces so that they can be better understood. As 
described in Ref. [10], 

W La {t) = / L (f)-B 12 -<p/c (5) 

W Le (t) = I L (t) • B 21 • (p/c (6) 

where I L (t) is the time-dependent laser intensity, B ]2 and 



The Einstein coefficient B 12 used in equation (5) applies 
to the ro-vibrational transition of interest, and is given 
by 



S R j 
2/' + 1 



a n ~ 15 

%-KhcV 

where S j is the rotational line strength and is given in 
Ref. [13] for each of the branches of E- T, transitions. 
The coefficient B 21 is given as B 12 ■ g/g u - The A,-.- used 
for the calculations presented herein were taken from 
Ref. [14]; W 21 and Wt are found from the same A r; data. 

The remaining component of W L (t) is Ijjt). The tempo- 
ral pulse shape of the ArF* excimer laser used in this 
work can be closely approximated by a Gaussian distri- 
bution with a 15 ns FWHM (x); the KrF* of Ref. [8] by 
a Gaussian with x=20 ns. Noting that the laser pulse 
energy, E L , is equal to the integral of the product of the 
beam area, A, and its intensity, the laser temporal inten- 
sity is given by, 



L i 4 % 



where <£>=E L /A is the laser fluence. 



(7) 



The predissociation rates were taken from references 
[15] and [16], converting from linewidths (FWHM) by 

W d = 2nc-Av„ 



"pre- 



The electronic quenching rate for the B-state of O2 is not 
known, due to the fact that it typically competes poorly 
with that state's rapid predissociation. For upper vibra- 
tional levels v'=0 and v' > 12, though, the quenching rate 
may be comparable to the low predissociation rates of 
those levels. Based on data provided in Ref. [6], in 
which no effects of quenching were seen in an atmo- 
spheric pressure flame for v'=0, N'=18, we assume a 
maximum quenching rate at those conditions of 1/10 of 
that state's predissociation rate. Also assuming no varia- 
tion in quenching cross-section with temperature, we 
have: W Q = 7.8-10 9 p-(300/T) 1/2 sec -1 . This rate is com- 



parable to quenching rates for other electronically 
excited species. This rate is comparable at atmospheric 
pressure to the predissociation rates for v'=0 and v' > 12, 
and therefore must be included in the model. 

Estimation of the collisional refilling rate, W c 

The final parameter required is the collisional redistribu- 
tion rate, W c , That this is written as a singular parameter 
represents a great simplification in the dynamic rota- 
tional energy transfer processes which occur in the 
probed medium during the period of laser-induced 
removal of molecules from a single rotational level, or 
even sublevel. A more complete model would include 
the summation of a number of rotational energy transfer 
reactions of the form, 



W,>+^-l,V^^-l,V +iV J,V 



(8) 



with appropriate forward and reverse rate coefficients. 
Similar expressions could be written for reactions 
involving the transfer of multiple rotational quanta. The 
result would be a set of ordinary differential equations, 
one for each of the rotational levels in each of the vibra- 
tional levels, possibly including each of the electronic 
levels, for all of the chemical species present. Clearly, 
the complete model would be algebraically cumbersome 
and would necessitate evaluation of each of the transfer 
rates. If one compares the expressions derived from 
equations of the form of equation (8) to equation (3), 
one may interpret the rate W c as the product of the popu- 
lation of the local bath of molecules and a Boltzmann 
fraction-weighted rotational transfer rate. This interpre- 
tation of the collisional repopulation rate causes diffi- 
culty in assigning a value to it for the purpose of using 
the model. A simpler interpretation and assessment of 
the rate, W c , follows. 

Whenever a molecule in a level 1 is removed by absorp- 
tion of a photon, the equilibrium of the rovibrational dis- 
tribution is disturbed. The restoration of the local 
equilibrium occurs through the effects of collisional 
redistribution of energy among the remaining mole- 
cules. This redistribution process, then, is in some way 
related to the bimolecular collision rate. The collisions 
which are expected to be important are those involving 
molecules in the energetic neighborhood of level 1 , and 
an unspecified partner. In order for collisions to be 
effective in the context of this model, their effect must 
be felt in the time-frame of the laser pulse. Due to the 
relative slowness (with respect to the laser pulse length, 
x) of vibrational re-equilibration, only the vibrational 
level containing level 1 will be considered to contribute 
to the replenishment of level 1, for the purpose of this 
model. 



For 2 S-R LIF, the upper and lower electronic states 
are both triplet states. Due to the coupling of the mole- 
cule's nuclear angular momentum and electron spin 
angular momentum, the energies of the three spin com- 
ponents of each level are slightly different. That these 
differences are not identical in the upper and lower rovi- 
bronic levels is the source of observable triplet splitting. 
As is customary, we denote N the quantum number for 
angular momentum excluding electron spin, and J the 
quantum number including spin. For each N, the possi- 
ble J values are N-l, N, and N+l. Although the spin 
components are usually unresolved or only partially 
resolved in absorption, this is not always the case. For 
the cases where only one or two components may be 
excited, we need to take into account any collisional 
coupling between molecules of differing electron spin. 
The microwave absorption data provided in Ref. [17] 
include half-linewidths for (N", J") levels for v"=0. 
These lifetimes are related to the collisional rates by 
W c = 27tcAv^ wave . Ref [17] also provides information 
about the spin re-equilibration, by invoking a propensity 
rule. By this rule, collisions which change electron spin 
are unlikely, due to the weak coupling in 2 between the 
electron spin angular momentum and the nuclear angu- 
lar momentum. 

Using this information, the bath of molecules which 
may collisionally replace those molecules removed from 
the lower state of the transition consists, for the purpose 
of this model, of those molecules in the same vibrational 
level and having the same electron spin as those of the 
lower state of the transition. In other words, each elec- 
tron spin group acts as if it is independent of the others, 
and its bath consists of the molecules of like spin in the 
remaining rotational levels in the lower vibronic state. 

Using an average value of the half-widths from 
Ref. [17], and assuming that the collision cross-section 
is independent of temperature, we estimate 
W c = 7.78- 10 9 p-(300/T) 1/2 sec" 1 . 

With the understanding of level 3 as the bath described 
above, it is clear that the Boltzmann fraction, f B , used in 
equations (1) and (3) is the rotational portion of the 
complete Boltzmann fraction in the lower vibronic state, 
and must be computed for the initial rotational tempera- 
ture of the gas. 



ft 



27" + 1 -F V .{N", J") ■ hc/kT 



(9) 



where Q r is the rotational partition function and 
F V >{N",J") is the rotational energy associated with v", 
N" and J". 



One final comment is in order, regarding the use of the 
microwave-derived collisional transfer rate. The micro- 
wave absorption is a purely spin-changing process, i.e. 
the nuclear angular momentum is unchanged. For this 
reason, the disturbance to rotational equilibrium associ- 
ated with the absorption of microwave radiation is 
expected to be small. Under intense excitation, however, 
Schumann-Runge electronic absorption may signifi- 
cantly perturb the local rotational energy distribution, 
removing molecules preferentially from the energetic 
neighborhood of level 1. The relationship between the 
microwave linewidths and the rotational refilling rate, 
W c , under such conditions may no longer be valid. 

Calculations using the Model 

Calculations were made of the laser-induced fluores- 
cence signal, Sf , as a function of the laser fiuence, <I>, 
for relevant rotational states in several of the vibrational 
bands accessible using the ArF* or KrF* excimer lasers. 
Prior to integration, variables were normalized by 
appropriate constants, as follows: 

#1 = N,/N ia ,N 2 = N 2 /N lA) , 

N 3 = N 3 /N xo = (N 3 /N U0 )-f B /(l-f B ). 

For each set of parameter values, equations (1) - (4) 
were integrated forward in time from the initial condi- 
tions, {t - 0., JVi = 1., N 2 - 0., N3 - 1.}, past the 
completion of the laser pulse, until the fluorescence sig- 
nal had reached its final value. A representative plot of 
the time histories, for the (15,3) R^ll^ransition, using 
/g=0.127, <I>=25 mJ/cm is shown in Figure 1. The value 
of f B is consistent with a rotational temperature of 
approximately 300K. In the Figure, the strong laser 
excitation causes Nj to drop rapidly, while the large 
value of W c at atmospheric pressure and room tempera- 
ture causes iVj to follow closely behind Nj, This rela- 
tively rapid refilling allows significantly more signal to 
be generated than would have been in the absence of 
collisional refilling. The strong excitation nevertheless 
causes a reduction in the lower state population avail- 
able for pumping, and hence a reduction in signal from 
that which would have been generated if Nj were able to 
remain essentially constant. Near the peak of the laser 
pulse, N2 is approximately 2% of Nj, and hence stimu- 
lated emission does not cause a significant reduction in 
signal in this case. The resultant signal is only 44% of 
that which would be achieved with infinitely fast refill- 
ing from an inexhaustible bath, i.e. Nj=Nj =constant. 
As the pulse finishes, rotational re-equilibration between 
Nj and N 3 occurs, and the final lower state population is 



only =0.17 of its initial value. Hence, approximately 
83% of the initial population in the lower vibronic state 
has been lost, to predissociation and via both radiative 
and nonradiative decay to other vibrational levels in the 
lower electronic state. The majority of this loss is to pre- 
dissociation, for the conditions described. 

Comparisons with Data 

In order to assess the performance of the model, a com- 
parison was made between calculations such as these 
and experimental laser-induced fluorescence data. A 
sequence of calculations was performed for each of sev- 
eral transitions, for a range of laser fiuence, <I>, from 

1 1 o n 

10 to 10 mJ/cm . Two sets of experimental data were 
used for comparison. The first set was obtained from 
Ref. [8], which contains data for air at 1800K, for the 
(2,7) P(9) and (0,6) R(17) transitions excited by an 
injection-locked KrF* excimer laser. The bandwidth and 
pulse duration of this laser were reported to be 0.8 cm 
and 20 ns, respectively. Comparisons of these data with 
model calculations are shown in Figures 2(a) and(b). 
The agreement is seen to be excellent for the (2,7) tran- 
sition, and although the model somewhat overpredicts 
the depletion observed for the (0,6) excitation, the 
agreement is still good enough to provide a reasonable 
assessment of the fiuence at which nonlinear behavior 
becomes a concern. 

The second data set was obtained in a manner identical 
to that described in Ref. [18], in a variation of the 
RELIEF technique. These data were collected by con- 
ducting an excitation scan for each of three nominal val- 
ues of ArF* laser pulse energy, after preparing prior to 
each laser pulse a sample of vibrationally excited O2 by 
stimulated Raman scattering (SRS) and allowing the 
vibrational distribution to evolve for a fixed period of 
time (1.0 (as). During this time interval, vibrational -to- 
vibration energy transfer creates a substantial population 
in v">l, while leaving the rotational temperature sub- 
stantially unchanged. These vibrationally excited states 
may be probed using S-R LIE The variation in ArF* 
laser fiuence was achieved by inserting zero, one or two 
thicknesses of an absorbing glass into the beam path. 
The glass chosen absorbed approximately 30% of the 
light, and so the pulse energy was varied by an order of 
magnitude over this sequence of excitation scans. For 
each excitation scan, the ArF* laser wavelength was 
incremented by 1 step (equivalent to 0.0493 cm ) for 
each data point, and the scan encompassed 700 grating 
steps. Due to absorption of the laser beam by atmo- 
spheric O2, the energy arriving at the measurement loca- 
tion varied over the course of the excitation scan; a PIN 
photodiode provided a monitor for the energy arriving at 
the measurement location. The LIF data from the excita- 



tion scans were least-squares fit to a sum of Voight pro- 
files, and the transition-specific peak values were 
extracted. The data from these three excitation scans 
thus consisted of three line-center signals for each of the 
transitions excited in the scan, each at a different ArF* 
laser fluence. Due to the congested nature of the S-R 
spectrum and the broadband spectral signal collection, 
only one unambiguous transition could be isolated, 
namely the (15,3) Rj(l 1) line. The range of laser fluence 
utilized in this test was not sufficient to provide data in 
the linear (non-depleting) regime, but the lowest fluence 
is predicted by the model to be within 10% of linear. A 
plot of line-center signal versus laser fluence for the 
(15,3) Rj(ll) transition at a rotational temperature of 
300K is shown in Figure 3. The experimental data and 
model calculation have been forced to the same value at 
a fluence of 6 mJ/cm , and the model reasonably pre- 
dicts the signal levels for the higher fluences. It should 
be pointed out that the vibrational temperature associ- 
ated with these data is undefined, as the vibrational 
energy distribution is evolving in time following the 
SRS event. 

Imaging of S-R LIF 

The results presented in Figures 2 and 3 provide justifi- 
cation for using this 3-level model in a predictive mode. 
In the course of designing a LIF experiment, one must 
estimate signal levels, in order to determine the spatial 
resolution achievable for a particular laser fluence, at 
some nominal thermodynamic conditions. The data pre- 
sented have shown that simply increasing the laser flu- 
ence in order to increase signal is not an option when 
using O2 S-R LIF. At relatively low fluences, the signal 
no longer varies linearly with laser fluence; one must 
therefore operate below this level. It is not possible to 
operate in a fully saturated regime, since in the limit of 
very large laser fluence and complete depletion of Nj, 
the signal is still dependent on the collisional refilling 
rate. This rate, as discussed earlier, is a function of the 
local thermodynamic conditions, and the signal 
obtained, then, would also be a function of those condi- 
tions. 

The quantity that is needed, in order to design an LIF 
experiment, then, is the maximum laser fluence one may 
use, while remaining nominally in the linear regime. 
Operation in the linear regime allows the lower state 
population to be deduced from the signal without need- 
ing to understand the temporal dynamics of the signal 
generation process. Operation in this regime also allows 
correction of signal variations which are due to laser 
energy fluctiuations. If the departure from linear behav- 
ior is limited to, say, 5%, the maximum laser fluence 
allowable can be predicted for any transition, using this 



3-level model. With this maximum fluence and appro- 
priate signal collection and detection efficiencies, the 
maximum measurable linear signal can be calculated. 
The equation for signal photons collected per pixel, as 
derived in Ref. [10], is 



NPP = <$> 

max max 



Vr U/r n 



B n 



■*!/• 



(10) 



where % is the fluorescence efficiency, or Stern- Volmer 
factor, V is the fluid volume element and T| coll is the 
combined collection / detection efficiency. The quantity 
n is the number density of molecules in level 1; 

n=N T X 2CV'fB K /gelec' wnere Nt^02 * s th e ®2 number 
density, oc v " is the vibrational Boltzmann fraction and k/ 
g e l ec is the fraction of spin components excited 
(g e l ec =3). If we assume a cubic volume element and 
square pixels of size h , then V = h /M 3 , where M is 
the magnification, and the sheet thickness, t, is h/M . 
Using the notation of Ref. [10], the quantity Vr| coll is 
equal to h 3 /M/[4f # (M+l)] 2 . 

As an example, consider the case of a lean H 2 /air flame 
at 1 atm. and 2300K, with the mole fraction of Oj, X Q 2, 
equal to 2%. The quantity AfPP max /Vr| co ji * s calculated 
for several LIF transition options, and these are pre- 
sented in Table 1. Also shown in Table 1 are required 

Table 1: Calculated LIF Signal Maxima, 
per collection volume 



Transition 


^max 

mJ/cm 


NPP 

T max 


VTlcoll 

forNPP=4000 

3 
cm 


VTlcon 

-3 

cm 


(7,1)P(17) 


200 


4.9-10 10 


8.2-10" 8 


(10,2) P(ll) 


5 


1.3-10 10 


3.1-10" 7 


(15,3)Ri(H) 


1 


3.8-10 9 


1.1-10" 6 


(0,6)R(17) 


100 


8.0-10 9 


5.0-10" 7 


(2,7) P(9) 


4 


7.3-10 8 


5.5-10" 6 



values of Vr| co [| to achieve NPP=4000. This value was 
chosen as it provides 400 photoelectrons per pixel (and a 
noise-to-signal ratio of 1/20, or 5%) for a combined fil- 
ter and detector efficiency of 10%. In order to convert 
these values to signal levels, a collection geometry is 
required. Table 2 presents several representative values 
for the collection parameters, M, f # , and h , and the 
resulting Vr) co u and spatial resolution, t. In order to 



Table 2: Typical Collection Parameters 



M 


f# 


h, jim 


Vr lcoll . cm3 


t= h M , (xm 


2 


1.4 


50 


2.21-10" 10 


25 


2 


1.4 


200 


1.42-10" 8 


100 


1 


2 


50 


4.88-10' 10 


50 


1 


2 


200 


3.12-10" 8 


200 


0.5 


4.5 


50 


3.43- 10" 10 


100 


0.5 


4.5 


200 


2.19-10" 8 


400 



achieve the different values of h , binning of pixels will 
probably be required. Note that, in order to offset the 
low levels of NPP mia /'Vr\ co ii, an d allow for spectral fil- 
tering and detector quantum efficiency, both of which 
further degrade the signal, it is necessary to operate with 
Vr| co || on the order of 10 to 10 cm , even for the 
most efficient transitions. These levels of Vr| co || are 
achievable only with spatial resolution on the order of 
200-400 tim. The less efficient transitions may require 
that spatial resolution be limited to near 1 mm. The 
result of this calculation indicates that, if O2 S-R LIF is 
to be used, the spatial resolution will need to be limited, 
possibly severely, in order to remain in the non-deplet- 
ing, linear signal regime. 

Summary 

A 3-level model has been developed to simulate the 2 
Schumann-Runge laser-induced fluorescence process. 
Calculations using the model compare favorably with 
available experimental data, and these calculations indi- 
cate that population depletion causes the signal in many 
cases to respond sublinearly to excitation laser fluence, 
even for low to moderate values of the fluence. The 
implication of this sublinear response is that, in order to 
remain in the linear regime, and hence in order to be 
able to relate the signal to lower state population, very 
low laser fluences are required. These low fluences, 
combined with the fact that predissociation of the 2 
B-state results in very low fluorescence yields, require 
that the spatial resolution in an 2 Schumann-Runge 
LIF be severely limited. This model provides a guide for 
design of such an 2 LIF experiment, and may be used 
to ascertain, a priori, whether the achievable spatial res- 
olution is sufficient for resolution of the spatial scales of 
interest. 



References 

Massey, G.A.; and Lemon, C.J.:"Feasibility of Measuring 
Temperature and Density Fluctuations in Air Using Laser- 
Induced 2 Fluorescence," IEEE Journal of Quantum Elec- 
tronics, QE-20(5), 454 (1984). 

Laufer, Gabriel; Robert L. McKenzie and Douglas G. 
Fletcher: "Method for measuring temperatures and densities in 
hypersonic wind tunnel air flows using laser-induced 2 fluo- 
rescence," Applied Optics, 29(33), 4873 (1990). 

3 Lee, Michael P.; Paul, Phillip H.; and Ronald K. Hanson, 
"Laser-fluorescence imaging of 2 in combustion flows using 
an ArF laser," Optics Letters, 11(1), 7 (1986). 

Goldsmith, J.E.M; and R.J. M.Anderson, "Laser-induced flu- 
orescence spectroscopy and imaging of molecular oxygen in 
flames," Optics Letters, 11(2), 67 (1986). 

Copeland, Richard A; et al., "Vibrationally excited 2 in 
flames: Measurements on v"=9-ll by laser-induced fluores- 
cence," Journal of Chemical Physics, 86(5), 2500 (1987). 

Andresen, Peter; et al., "Laser-induced fluorescence with 
tunable excimer lasers as a possible method for instantaneous 
temperature field measurements at high pressures: checks with 
an atmospheric flame," Applied Optics, 27(2), 365 (1988). 

Kim, G.-S.; et al., "Identification and Imaging of Hot 2 
(v"=2, 3, or 4) in Hydrogen Flames Using 193 nm- and 210 
nm-range Light," Applied Physics B, 53, 180 (1991). 

8 Grinstead, J.H.; Laufer, G.; and J.C. McDaniel, Jr.: "Single- 
pulse, two-line temperature-measurement technique using KrF 
laser-induced 2 fluorescence," Applied Optics, 34(24), 5501 
(1995). 

Miles, R.; et al., "Velocity measurements by vibrational tag- 
ging and fluorescent probing of oxygen," Optics Letters, 
12(11)861(1987). 

Eckbreth, Alan C: Laser Diagnostics for Combustion Tem- 
perature and Species . Energy and Engineering Science Series, 
7, A.K.Gupta and D. G. Lilley (eds.), Abacus Press, Cam- 
bridge, Mass., 1988. 

Octave, A high-level interactive language for numerical 
computations, Copyright © 1993, 1994, 1995 John W. Eaton; 
available via public ftp from bevo.che.wisc.edu/pub/octave. 

Radhakrishnan, Krishnan; and Alan C. Hindmarsh: 
"Description and Use of LSODE, the Livermore Solver for 
Ordinary Differential Equations," NASA Reference Publica- 
tion 1327, 1993. 

Tatum, J.B.: "Honl-London Factors for Z~- E~ Transi- 
tions," Canadian Journal of Physics, 44, 2944 (1966). 

14 Allison, A.C; Dalgamo, A.; and N.W.Pasachoff, "Absorp- 
tion by Vibrationally Excited Molecular Oxygen in the Schu- 
mann-Runge Continuum," Planet. Space Set, 19, 1463 (1971). 

15 Lewis, B.R.; Gibson, S.T; and P.M.Dooley, "Fine-Structure 
dependence of predissociation linewidth in the Schumann- 
Runge bands of molecular oxygen," Journal of Chemical 
Physics, 100(10), 7012 (1994). 
16 



Lewis, B.R.: personal communication. 



17 



Smith, Earl W.: "Absorption and dispersion in the 2 micro- 
wave spectrum at atmospheric pressures," Journal of Chemical 
Physics, 74(12), 6658(1981). 

18 Diskin, Glenn S.; Lempert, Walter R.; and Richard B. Miles: 



"Observation of Vibrational Relaxation Dynamics in 
X Eg Oxygen following Stimulated Raman Excitation to the 
v=l Level: Implications for the RELIEF Flow Tagging Tech- 
nique," AIAA paper 96-0301, presented at the 34th Aerospace 
Sciences Meeting and Exhibit, Reno, Nevada, January, 1996. 



0.25 



Signal, linear 

Signal, calculated using 3-level model 

• Signal, from Ref. [8] 



0.15 




0.05 



30 40 
time, ns 
Figure 1. Calculated time histories for (15,3) Rj(l 1) 
transition. * = 25 mj/cm 2 ; Trot = 300K; p = 1 atm. 



Signal, linear 

-Signal, calculated using 3-level model 
Signal from Ref. [8] 



0.15 - 



fa 
S3 

p 




0.05 



<E>, mJ/cm 

Figure 2(a). Variation of LIF Signal with laser fluence, <I>, 
for (2,7) P(9) transition at 1800K, 1 atm. 




400 600 800 
<t> mJ/cm 2 



1000 1200 



Figure 2(b). Variation of LIF Signal with laser fluence, <I>, 
for (0,6) R(17) transition at 1800K, 1 atm. 






0.25 



0.2 



0.15 



0.1 - 



0.05 



Signal, linear 

Signal, calculated using 3-level model 

• Signal, this work 



' / * 



10 



20 30 40 
<E>, mJ/cm 2 



50 



60 



Figure 3. Variation of LIF Signal with laser fluence, <J>, 
for (15,3) Ri(ll) transition at 300K, 1 atm.