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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. 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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.