A 3-level Model for Schumann-Runge O Laser-Induced Fluorescence 2 Glenn S. Diskin* NASA Langley Research Center Hampton, Virginia Walter R. Lempert† and Richard B. Miles‡ Princeton University Princeton, New Jersey Abstract first proposed the use of a tunable ArF* laser for the A three level model has been developed for the analysis of Schumann-Runge band (B3S - ‹ X3S - ) laser- measurement of temperature and density in air. Since u g that time, S-R LIF has been used to measure single- induced fluorescence of molecular oxygen, O . Such a 2 point temperature and density in low-temperature flows model is required due to the severe lower state depletion using an ArF* laser,2 for imaging and detection of hot which can occur when transitions having relatively large O in combustion systems using ArF*, KrF* and dye absorption cross-sections are excited. Such transitions 2 laser systems,3-7 and for measurement of temperature in are often utilized via ArF* or KrF* excimer or dye-laser high temperature air using a KrF* laser.8 S-R LIF has excitation in high temperature environments. The rapid also been used in conjunction with stimulated Raman predissociation of the upper state prevents substantial scattering to perform velocity measurements in air repopulation of the lower state by collisional processes, flows.9 S-R band LIF is often used because (1) oxygen is and the lower state may be largely depleted, even at naturally present in many flow situations, and therefore laser fluences as low as 10-100 mJ/cm2. The resulting doesn’t need to be seeded into the flow, and (2) the LIF signal in such cases no longer varies linearly with upper, or B-state in the S-R band system is rapidly pre- laser pulse energy, and the extent of the sublinear behav- dissociated, eliminating the need for quenching correc- ior varies with the particular rovibrational transition of tions and providing a signal which is directly interest. Relating the measured signal to the lower state proportional to lower state number density. An unfortu- population, then, necessitates the use of exceedingly low nate consequence of the rapid predissociation of the laser fluences. These low fluences in turn lead to the upper state is that the quantum yield, or fluorescence need to compromise spatial resolution in order to gener- efficiency, of the fluorescence emission is very low, typ- ate sufficient signal. ically on the order of 10-4. Additionally, numerous authors have reported sublinear signal generation at Introduction moderate laser fluences,4,6,8 due primarily to lower state The Schumann-Runge (S-R) band system depletion, or bleaching. The purpose of this paper is to (B3S - ‹ X3S - ) of molecular oxygen, O , has been u g 2 describe a model constructed to simulate the O S-R LIF the subject of much study, due primarily to its impor- 2 signal generation process. tance in atmospheric photochemistry. Measurement of flow properties by use of S-R laser-induced fluorescence Analysis of the signal obtained in a laser-induced fluo- (LIF), has grown in scope since Massey and Lemon1 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 * Research Engineer, Hypersonic Airbreathing Propulsion Branch, order to relate the collected signal to the lower state pop- Member AIAA. ulation, one must construct an appropriate model. This † Research Scientist, Department of Mechanical and Aerospace Engi- model should include the processes of laser excitation neering, Member AIAA. ‡ Professor, Department of Mechanical and Aerospace Engineering, and de-excitation to and from the upper state of the tran- Senior Member AIAA. sition, de-excitation of the upper state by radiative decay (fluorescence), predissociation and collisional quench- Copyright © 1996 by the American Institute of Aeronautics and Astro- ing, and redistribution of lower and upper state popula- nautics, Inc. No copyright is asserted in the United States under Title tions by collision-induced energy transfer (rotational 17, U.S. Code. The U.S. Government has a royalty-free license to exer- and/or vibrational). The large predissociation rates asso- cise all rights under the copyright claimed herein for government pur- poses. All other rights are reserved by the copyright owner. ciated with transitions in the O2 Schumann-Runge sys- 1 tem render a simple steady-state, two-level model, such response of the system using a realistic excitation tem- as described by Eckbreth10, inappropriate. A model poral profile is an improvement in the representation of more appropriate for O Schumann-Runge laser- nonlinear effects. This is due to the fact that the system 2 induced fluorescence was described by Laufer, et al.2 responds differently to excitation of high and low inten- This quasi-two-level model includes the relevant pro- sity; a ‘real’ temporal pulse shape incorporates a distri- cesses, and uses an analytical integration of the rate bution of intensities, while a top-hat pulse lumps equations (for top-hat temporal laser excitation) to relate everything into a single, uniform value. What precisely the signal to the lower state population. The model is meant by ‘high’ and ‘low’ intensity is, of course, works well for predissociation-dominated transitions determined by the particular system. with laser intensities low enough not to encounter sig- nificant population depletion. Transitions in O from the Three-level LIF Model 2 ground vibrational level excited by an ArF* laser (the (4,0) band) fall into this category. 2 W W W d Q f 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 WLa WLe W21 v „ v" rotational repopulation rates become large. The reason for this is that the lower ro-vibrational level was 1 (cid:219) Wc (cid:229) N assumed in Ref. [2] to have an infinitely large bath from v'',n 3 n„ N'' 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 Using the notation of Ref. [2], the equations governing vibrational level, i.e. that vibrational re-equilibration the three-level system are: times are much greater than typical laser pulse times and that rotational re-equilibration times are comparable to dN laser pulse times. To keep track of this depletable bath, a 1 = –W (t)(cid:215) N +N (cid:215) (W (t)+W ) (1) dt La 1 2 Le 21 new model was constructed which extends the model of RThefis. [l2e]v einl rtewpore wseanytss. aFlli rsotf, tah eth rirodta lteiovneal lw laesv eilnsc loufd tehde. +WC(cid:215) Ł(cid:230) 1-----–-f---B-f-----(cid:215) N3–Nł(cid:246)1 B ground vibrational level except the one coupled to the upper state by the laser, and provides the bath of mole- dN 2 = W (t)(cid:215) N (2) cules from which the lower level may be refilled. Sec- dt La 1 ond, since for transitions stronger than the (4,0) band the absorption rate may be comparable to the predissocia- –N2(cid:215) (WLe(t)+Wd+W21+Wf +WQ) tion rate, the process of stimulated emission must be included as a mechanism to couple the upper and lower dN (cid:230) f (cid:246) states. The stimulated emission terms, which were dt 3 = –WC(cid:215) Ł -1----–----B-f-----(cid:215) N3–N1ł (3) dropped in the simplification of Ref. [2], are retained in B this model. In this model, described by equations (1)-(3) and depicted schematically above, levels 1 and 2 represent Inclusion of a third level in the model takes away the the ro-vibrational levels in the ground and excited elec- elegance of the solution given in Ref. [2], in that the tronic states, respectively, which are coupled by the solution can no longer be represented in terms of a sim- laser frequency. Level 3 consists of the bath of mole- ple function of two parameters. The analytical solution cules collisionally coupled to level 1, by the rate W . c becomes a very messy algebraic expression which The factor f /(1-f ) which precedes the N term in equa- B B 3 doesn’t allow for easy assessment of the effects of indi- tions (1) and (3) is required to provide detailed balanc- vidual parameters. The system of equations was there- ing of the forward and reverse collisional redistribution fore solved numerically. This solution method has the processes. The model includes the relevant processes of advantage that a more realistic laser pulse shape can be laser-stimulated absorption and emission (W and La employed; Laufer, et al.2 solved the quasi-two-level W ), spontaneous emission (W ), predissociation Le 21 problem for a top-hat excitation pulse shape. For the (W ), collisional quenching of the upper state (W ), and d Q solutions described herein, a Gaussian temporal distri- radiative decay to vibrational levels other than the lower bution was used. A benefit of calculating the temporal level of the transition (the fluorescence of interest, W). f 2 Not included in this model is collisional redistribution of B are the Einstein coefficients for absorption and 21 rotational or vibrational energy in the upper level, as emission, respectively, and j is the overlap integral of these are considered to be slow with respect to the pre- the transition and laser lineshapes. dissociation. j = (cid:242) g(n )h(n )dn , The primary parameter of interest for the user of this Dn model is the fluorescence signal, denoted S . The fluo- where (cid:242) g(n )dn = 1 and (cid:242) h(n )dn = 1 f rescence signal at any time, t, per unit volume, emitted Dn Dn into 4p steradians, Sˆ (t), is equal to W * (cid:215) N (t) and and g(n ) and h(n ) are the transition and laser line- f f 2 , the total signal generated during the laser pulse is: shapes, respectively. The symbol n is defined as n ⁄c. Sˆ = (cid:242) W * (cid:215) N (t)dt, (4) The Einstein coefficient B12 used in equation (5) applies f f 2 to the ro-vibrational transition of interest, and is given t p by where t is the laser pulse duration and W* is the por- p f A SR g tion of the fluorescence signal which is detectable due to B = ----------2--1------(cid:215) ------------J------(cid:215) ----u-, spectral filtering. In order to obtain the quantity Sˆf , 12 8p hcn 3 2J''+1 gl equations (1)-(3) must be integrated to find N2(t), which where SRJ is the rotational line strength and is given in is then integrated according to equation (4). To perform Ref. [13] for each of the branches of 3S -3S transitions. the required integration, appropriate values for the The coefficient B is given as B · g/g . The A used 21 12 l u ij parameters Wc, Wd, Wf, W21 and fB must be found, as for the calculations presented herein were taken from well as an appropriate functional form of WL(t). With Ref. [14]; W21 and Wf are found from the same Aij data. these values, discussed subsequently, equations (1)-(4) can be integrated numerically using one of the standard The remaining component of W (t) is I (t). The tempo- L L techniques. For this work, the public domain software ral pulse shape of the ArF* excimer laser used in this package Octave [11], version 1.1.1, was employed. work can be closely approximated by a Gaussian distri- Octave provides a front-end for the Lawrence Livermore bution with a 15 ns FWHM (t ); the KrF* of Ref. [8] by ordinary differential equation solver, LSODE [12], writ- a Gaussian with t =20 ns. Noting that the laser pulse ten by Alan C. Hindmarsh. energy, E , is equal to the integral of the product of the L beam area, A, and its intensity, the laser temporal inten- Parameters in the Rate Equations sity is given by, In order to solve the equations governing the model sys- tfeomun, dth. eW pca raism ethteer sc Wolcli, sWiodn,a Wl Qre, pWofp, uWla2t1io ann dr afBte m iuns tt hbee I (t) = F----(cid:215) 4----l--n---2--(cid:215) e–4ln2(cid:215) Ł(cid:230)t----–-t---t--0ł(cid:246)- 2, (7) lower vibronic level (N ), and is a function of the fluid L t p 1 density and temperature. W is the upper rovibronic pre- d where F =E /A is the laser fluence. dissociation rate, and is only a function of the upper L level quantum state, as are W, the fluorescence rate, and f The predissociation rates were taken from references W , the rate of spontaneous emission at the laser fre- 21 [15] and [16], converting from linewidths (FWHM) by quency. W is the collisional quenching rate, and is a Q W = 2p c·Dn . function, in general, of the temperature and densities of d pre all collision partners. The rotational Boltzmann fraction The electronic quenching rate for the B-state of O is not associated with the lower level, f , is a function of tem- 2 B known, due to the fact that it typically competes poorly perature, the lower vibrational level, and the lower rota- with that state’s rapid predissociation. For upper vibra- tional level. The laser excitation rate parameters, W (t) La tional levels v'=0 and v' > 12, though, the quenching rate and W (t), should be decomposed into their various Le may be comparable to the low predissociation rates of pieces so that they can be better understood. As those levels. Based on data provided in Ref. [6], in described in Ref. [10], which no effects of quenching were seen in an atmo- spheric pressure flame for v'=0, N'=18, we assume a W (t) = I (t)(cid:215) B (cid:215) j ⁄c (5) La L 12 maximum quenching rate at those conditions of 1/10 of that state’s predissociation rate. Also assuming no varia- W (t) = I (t)(cid:215) B (cid:215) j ⁄c (6) Le L 21 tion in quenching cross-section with temperature, we where I (t) is the time-dependent laser intensity, B and have: WQ = 7.8·109 p·(300/T)1/2 sec-1. This rate is com- L 12 3 parable to quenching rates for other electronically For O S-R LIF, the upper and lower electronic states 2 excited species. This rate is comparable at atmospheric are both triplet states. Due to the coupling of the mole- pressure to the predissociation rates for v'=0 and v' > 12, cule’s nuclear angular momentum and electron spin and therefore must be included in the model. angular momentum, the energies of the three spin com- ponents of each level are slightly different. That these Estimation of the collisional refilling rate, W differences are not identical in the upper and lower rovi- c bronic levels is the source of observable triplet splitting. The final parameter required is the collisional redistribu- As is customary, we denote N the quantum number for tion rate, W . That this is written as a singular parameter c angular momentum excluding electron spin, and J the represents a great simplification in the dynamic rota- quantum number including spin. For each N, the possi- tional energy transfer processes which occur in the ble J values are N-1, N, and N+1. Although the spin probed medium during the period of laser-induced components are usually unresolved or only partially removal of molecules from a single rotational level, or resolved in absorption, this is not always the case. For even sublevel. A more complete model would include the cases where only one or two components may be the summation of a number of rotational energy transfer excited, we need to take into account any collisional reactions of the form, coupling between molecules of differing electron spin. N +N (cid:219) N +N (8) The microwave absorption data provided in Ref. [17] i,v j–1,v i–1,v j,v include half-linewidths for (N", J") levels for v"=0. with appropriate forward and reverse rate coefficients. These lifetimes are related to the collisional rates by Similar expressions could be written for reactions Wc = 2p c·Dn m wave. Ref [17] also provides information involving the transfer of multiple rotational quanta. The about the spin re-equilibration, by invoking a propensity result would be a set of ordinary differential equations, rule. By this rule, collisions which change electron spin one for each of the rotational levels in each of the vibra- are unlikely, due to the weak coupling in O2 between the tional levels, possibly including each of the electronic electron spin angular momentum and the nuclear angu- levels, for all of the chemical species present. Clearly, lar momentum. the complete model would be algebraically cumbersome and would necessitate evaluation of each of the transfer Using this information, the bath of molecules which rates. If one compares the expressions derived from may collisionally replace those molecules removed from equations of the form of equation (8) to equation (3), the lower state of the transition consists, for the purpose one may interpret the rate W as the product of the popu- of this model, of those molecules in the same vibrational c lation of the local bath of molecules and a Boltzmann level and having the same electron spin as those of the fraction-weighted rotational transfer rate. This interpre- lower state of the transition. In other words, each elec- tation of the collisional repopulation rate causes diffi- tron spin group acts as if it is independent of the others, culty in assigning a value to it for the purpose of using and its bath consists of the molecules of like spin in the the model. A simpler interpretation and assessment of remaining rotational levels in the lower vibronic state. the rate, W , follows. c Using an average value of the half-widths from Whenever a molecule in a level 1 is removed by absorp- Ref. [17], and assuming that the collision cross-section tion of a photon, the equilibrium of the rovibrational dis- is independent of temperature, we estimate tribution is disturbed. The restoration of the local Wc = 7.78·109 p·(300/T)1/2 sec-1. equilibrium occurs through the effects of collisional redistribution of energy among the remaining mole- With the understanding of level 3 as the bath described cules. This redistribution process, then, is in some way above, it is clear that the Boltzmann fraction, fB, used in related to the bimolecular collision rate. The collisions equations (1) and (3) is the rotational portion of the which are expected to be important are those involving complete Boltzmann fraction in the lower vibronic state, molecules in the energetic neighborhood of level 1, and and must be computed for the initial rotational tempera- an unspecified partner. In order for collisions to be ture of the gas. effective in the context of this model, their effect must be felt in the time-frame of the laser pulse. Due to the f = 2----J---'-'---+----1---(cid:215) e–Fv''(N'',J'')(cid:215) hc⁄kT, (9) relative slowness (with respect to the laser pulse length, B Q r t ) of vibrational re-equilibration, only the vibrational level containing level 1 will be considered to contribute where Qr is the rotational partition function and to the replenishment of level 1, for the purpose of this Fv"(N",J") is the rotational energy associated with v", model. N" and J". 4 One final comment is in order, regarding the use of the only » 0.17 of its initial value. Hence, approximately microwave-derived collisional transfer rate. The micro- 83% of the initial population in the lower vibronic state wave absorption is a purely spin-changing process, i.e. has been lost, to predissociation and via both radiative the nuclear angular momentum is unchanged. For this and nonradiative decay to other vibrational levels in the reason, the disturbance to rotational equilibrium associ- lower electronic state. The majority of this loss is to pre- ated with the absorption of microwave radiation is dissociation, for the conditions described. expected to be small. Under intense excitation, however, Schumann-Runge electronic absorption may signifi- Comparisons with Data cantly perturb the local rotational energy distribution, In order to assess the performance of the model, a com- removing molecules preferentially from the energetic parison was made between calculations such as these neighborhood of level 1. The relationship between the and experimental laser-induced fluorescence data. A microwave linewidths and the rotational refilling rate, sequence of calculations was performed for each of sev- Wc, under such conditions may no longer be valid. eral transitions, for a range of laser fluence, F, from 10-1to 10+3 mJ/cm2. Two sets of experimental data were Calculations using the Model used for comparison. The first set was obtained from Calculations were made of the laser-induced fluores- Ref. [8], which contains data for air at 1800K, for the cence signal, Sˆ , as a function of the laser fluence, F , (2,7) P(9) and (0,6) R(17) transitions excited by an f for relevant rotational states in several of the vibrational injection-locked KrF* excimer laser. The bandwidth and bands accessible using the ArF* or KrF* excimer lasers. pulse duration of this laser were reported to be 0.8 cm-1 Prior to integration, variables were normalized by and 20 ns, respectively. Comparisons of these data with appropriate constants, as follows: model calculations are shown in Figures 2(a) and(b). The agreement is seen to be excellent for the (2,7) tran- Nˆ = N ⁄N , Nˆ = N ⁄N , sition, and although the model somewhat overpredicts 1 1 1,0 2 2 1,0 the depletion observed for the (0,6) excitation, the agreement is still good enough to provide a reasonable Nˆ3 = N3⁄N3,0 = (N3⁄N1,0)(cid:215) fB⁄(1– fB). assessment of the fluence at which nonlinear behavior becomes a concern. For each set of parameter values, equations (1) - (4) were integrated forward in time from the initial condi- The second data set was obtained in a manner identical tions, {t = 0.,Nˆ = 1.,Nˆ = 0.,Nˆ = 1.}, past the to that described in Ref.[18], in a variation of the 0 1 2 3 completion of the laser pulse, until the fluorescence sig- RELIEF technique. These data were collected by con- nal had reached its final value. A representative plot of ducting an excitation scan for each of three nominal val- the time histories, for the (15,3) R (11)transition, using ues of ArF* laser pulse energy, after preparing prior to 1 fB=0.127, F =25 mJ/cm2 is shown in Figure 1. The value each laser pulse a sample of vibrationally excited O2 by of f is consistent with a rotational temperature of stimulated Raman scattering (SRS) and allowing the B approximately 300K. In the Figure, the strong laser vibrational distribution to evolve for a fixed period of excitation causes N to drop rapidly, while the large time (1.0 m s). During this time interval, vibrational-to- 1 value of W at atmospheric pressure and room tempera- vibration energy transfer creates a substantial population c ture causes N to follow closely behind N . This rela- in v">1, while leaving the rotational temperature sub- 3 1 tively rapid refilling allows significantly more signal to stantially unchanged. These vibrationally excited states be generated than would have been in the absence of may be probed using S-R LIF. The variation in ArF* collisional refilling. The strong excitation nevertheless laser fluence was achieved by inserting zero, one or two causes a reduction in the lower state population avail- thicknesses of an absorbing glass into the beam path. able for pumping, and hence a reduction in signal from The glass chosen absorbed approximately 30% of the that which would have been generated if N were able to light, and so the pulse energy was varied by an order of 1 remain essentially constant. Near the peak of the laser magnitude over this sequence of excitation scans. For pulse, N is approximately 2% of N , and hence stimu- each excitation scan, the ArF* laser wavelength was 2 1 lated emission does not cause a significant reduction in incremented by 1 step (equivalent to 0.0493 cm-1) for signal in this case. The resultant signal is only 44% of each data point, and the scan encompassed 700 grating that which would be achieved with infinitely fast refill- steps. Due to absorption of the laser beam by atmo- ing from an inexhaustible bath, i.e. N1=N1,0=constant. spheric O2, the energy arriving at the measurement loca- As the pulse finishes, rotational re-equilibration between tion varied over the course of the excitation scan; a PIN N and N occurs, and the final lower state population is photodiode provided a monitor for the energy arriving at 1 3 the measurement location. The LIF data from the excita- 5 tion scans were least-squares fit to a sum of Voight pro- 3-level model. With this maximum fluence and appro- files, and the transition-specific peak values were priate signal collection and detection efficiencies, the extracted. The data from these three excitation scans maximum measurable linear signal can be calculated. thus consisted of three line-center signals for each of the The equation for signal photons collected per pixel, as transitions excited in the scan, each at a different ArF* derived in Ref. [10], is laser fluence. Due to the congested nature of the S-R spectrum and the broadband spectral signal collection, B (cid:215) j only one unambiguous transition could be isolated, NPPmax = F max(cid:215) Vh coll(cid:215) n(cid:215) -----1--2c-------(cid:215)-- h f, (10) namely the (15,3) R (11) line. The range of laser fluence 1 utilized in this test was not sufficient to provide data in where h is the fluorescence efficiency, or Stern-Volmer f the linear (non-depleting) regime, but the lowest fluence factor, V is the fluid volume element and h is the coll is predicted by the model to be within 10% of linear. A combined collection / detection efficiency. The quantity plot of line-center signal versus laser fluence for the n is the number density of molecules in level 1; (15,3) R (11) transition at a rotational temperature of n=N X a f k /g , where N X is the O number 1 T O2 v" B elec T O2 2 300K is shown in Figure 3. The experimental data and density, a is the vibrational Boltzmann fraction and k / v" model calculation have been forced to the same value at g is the fraction of spin components excited elec a fluence of 6 mJ/cm2, and the model reasonably pre- (g =3). If we assume a cubic volume element and elec dicts the signal levels for the higher fluences. It should square pixels of size h˜, then V = h˜3⁄M3, where M is be pointed out that the vibrational temperature associ- the magnification, and the sheet thickness, t, is h˜ ⁄M. ated with these data is undefined, as the vibrational Using the notation of Ref. [10], the quantity Vh is coll energy distribution is evolving in time following the equal to h˜3/M/[4f (M+1)]2. # SRS event. As an example, consider the case of a lean H /air flame 2 Imaging of S-R LIF at 1 atm. and 2300K, with the mole fraction of O , X , 2 O2 The results presented in Figures 2 and 3 provide justifi- equal to 2%. The quantity NPPmax/Vh coll is calculated cation for using this 3-level model in a predictive mode. for several LIF transition options, and these are pre- In the course of designing a LIF experiment, one must sented in Table 1. Also shown in Table 1 are required estimate signal levels, in order to determine the spatial Table 1: Calculated LIF Signal Maxima, resolution achievable for a particular laser fluence, at per collection volume 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 F -N----P----P----m----a--x- Vh coll using O S-R LIF. At relatively low fluences, the signal Transition max Vh for NPP=4000 2 mJ/cm2 coll no longer varies linearly with laser fluence; one must cm-3 cm3 therefore operate below this level. It is not possible to operate in a fully saturated regime, since in the limit of (7,1) P(17) 200 4.9·1010 8.2·10-8 very large laser fluence and complete depletion of N , 1 the signal is still dependent on the collisional refilling (10,2) P(11) 5 1.3·1010 3.1·10-7 rate. This rate, as discussed earlier, is a function of the local thermodynamic conditions, and the signal (15,3) R1(11) 1 3.8·109 1.1·10-6 obtained, then, would also be a function of those condi- tions. (0,6) R(17) 100 8.0·109 5.0·10-7 The quantity that is needed, in order to design an LIF (2,7) P(9) 4 7.3·108 5.5·10-6 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 values of Vh coll to achieve NPP=4000. This value was population to be deduced from the signal without need- chosen as it provides 400 photoelectrons per pixel (and a ing to understand the temporal dynamics of the signal noise-to-signal ratio of 1/20, or 5%) for a combined fil- generation process. Operation in this regime also allows ter and detector efficiency of 10%. In order to convert correction of signal variations which are due to laser these values to signal levels, a collection geometry is energy fluctiuations. If the departure from linear behav- required. Table 2 presents several representative values ˜ ior is limited to, say, 5%, the maximum laser fluence for the collection parameters, M, f#, and h, and the allowable can be predicted for any transition, using this resulting Vh coll and spatial resolution, t. In order to 6 Table 2: Typical Collection Parameters References 1 Massey, G.A.; and Lemon, C.J.:“Feasibility of Measuring M f# h˜ , m m Vh coll , cm3 t=h˜ /M , m m TInedmupceerda Otur eF alunodr Desecnesnictye ,”F lIuEcEtuEa tJioounrsn ianl A oifr Q Uusainngtu Lma Eselre-c- 2 tronics, QE-20(5), 454 (1984). 2 1.4 50 2.21·10-10 25 2 Laufer, Gabriel; Robert L. McKenzie and Douglas G. Fletcher: “Method for measuring temperatures and densities in 2 1.4 200 1.42·10-8 100 hypersonic wind tunnel air flows using laser-induced O2 fluo- rescence,” Applied Optics, 29(33), 4873 (1990). 1 2 50 4.88·10-10 50 3 Lee, Michael P.; Paul, Phillip H.; and Ronald K. Hanson, “Laser-fluorescence imaging of O in combustion flows using 2 1 2 200 3.12·10-8 200 an ArF laser,” Optics Letters, 11(1), 7 (1986). 4 Goldsmith, J.E.M; and R.J.M.Anderson, “Laser-induced flu- 0.5 4.5 50 3.43·10-10 100 orescence spectroscopy and imaging of molecular oxygen in flames,” Optics Letters, 11(2), 67 (1986). 0.5 4.5 200 2.19·10-8 400 5 Copeland, Richard A; et al., “Vibrationally excited O in 2 flames: Measurements on v"=9-11 by laser-induced fluores- ˜ cence,” Journal of Chemical Physics, 86(5), 2500 (1987). achieve the different values of h, binning of pixels will 6 Andresen, Peter; et al., “Laser-induced fluorescence with probably be required. Note that, in order to offset the tunable excimer lasers as a possible method for instantaneous low levels of NPP /Vh , and allow for spectral fil- max coll temperature field measurements at high pressures: checks with tering and detector quantum efficiency, both of which an atmospheric flame,” Applied Optics, 27(2), 365 (1988). further degrade the signal, it is necessary to operate with 7 Kim, G.-S.; et al., “Identification and Imaging of Hot O Vh coll on the order of 10-8 to 10-7 cm3, even for the (v"=2, 3, or 4) in Hydrogen Flames Using 193 nm- and 2210 most efficient transitions. These levels of Vh are nm-range Light,” Applied Physics B, 53, 180 (1991). coll achievable only with spatial resolution on the order of 8 Grinstead, J.H.; Laufer, G.; and J.C. McDaniel, Jr.: “Single- 200-400 m m. The less efficient transitions may require pulse, two-line temperature-measurement technique using KrF that spatial resolution be limited to near 1 mm. The laser-induced O2 fluorescence,” Applied Optics, 34(24), 5501 (1995). result of this calculation indicates that, if O S-R LIF is 2 to be used, the spatial resolution will need to be limited, 9 Miles, R.; et al., “Velocity measurements by vibrational tag- ging and fluorescent probing of oxygen,” Optics Letters, possibly severely, in order to remain in the non-deplet- 12(11) 861 (1987). ing, linear signal regime. 10 Eckbreth, Alan C.: Laser Diagnostics for Combustion Tem- perature and Species, Energy and Engineering Science Series, Summary 7, A.K.Gupta and D. G. Lilley (eds.), Abacus Press, Cam- A 3-level model has been developed to simulate the O bridge, Mass., 1988. 2 Schumann-Runge laser-induced fluorescence process. 11 Octave, A high-level interactive language for numerical Calculations using the model compare favorably with computations, Copyright © 1993, 1994, 1995 John W. Eaton; available experimental data, and these calculations indi- available via public ftp from bevo.che.wisc.edu/pub/octave. cate that population depletion causes the signal in many 12 Radhakrishnan, Krishnan; and Alan C. Hindmarsh: cases to respond sublinearly to excitation laser fluence, “Description and Use of LSODE, the Livermore Solver for Ordinary Differential Equations,” NASA Reference Publica- even for low to moderate values of the fluence. The tion 1327, 1993. implication of this sublinear response is that, in order to 13 Tatum, J.B.: “Hönl-London Factors for 3S –-3S – Transi- remain in the linear regime, and hence in order to be tions,” Canadian Journal of Physics, 44, 2944 (1966). able to relate the signal to lower state population, very 14 Allison, A.C; Dalgarno, A.; and N.W.Pasachoff, “Absorp- low laser fluences are required. These low fluences, tion by Vibrationally Excited Molecular Oxygen in the Schu- combined with the fact that predissociation of the O 2 mann-Runge Continuum,” Planet. Space Sci., 19, 1463 (1971). B-state results in very low fluorescence yields, require 15 Lewis, B.R.; Gibson, S.T; and P.M.Dooley, “Fine-Structure that the spatial resolution in an O Schumann-Runge 2 dependence of predissociation linewidth in the Schumann- LIF be severely limited. This model provides a guide for Runge bands of molecular oxygen,” Journal of Chemical design of such an O LIF experiment, and may be used Physics, 100(10), 7012 (1994). 2 to ascertain, a priori, whether the achievable spatial res- 16 Lewis, B.R.: personal communication. olution is sufficient for resolution of the spatial scales of 17 Smith, Earl W.: “Absorption and dispersion in the O micro- 2 interest. wave spectrum at atmospheric pressures,” Journal of Chemical Physics, 74(12), 6658 (1981). 18 Diskin, Glenn S.; Lempert, Walter R.; and Richard B. Miles: 7 “Observation of Vibrational Relaxation Dynamics in X3S g- Oxygen following Stimulated Raman Excitation to the Signal, linear Signal, calculated using 3-level model v=1 Level: Implications for the RELIEF Flow Tagging Tech- Signal, from Ref. [8] nique,” AIAA paper 96-0301, presented at the 34th Aerospace Sciences Meeting and Exhibit, Reno, Nevada, January, 1996. 3 2.5 1 0.25 s N1 NSi2gnal y unit 2 0.8 NI3 0.2 bitrar 1.5 L N ar malized 0.6 0.15 2 , Sign Signal, 1 , norL al, no , I30.4 0.1 rma 0.5 N , N1 lized 0 0.2 0.05 0 200 400 600 800 1000 1200 F , mJ/cm2 Figure 2(b). Variation of LIF Signal with laser fluence, F , for (0,6) R(17) transition at 1800K, 1 atm. 0 0 0 10 20 30 40 50 60 time, ns Signal, linear Figure 1. Calculated time histories for (15,3) R (11) Signal, calculated using 3-level model 1 transition. F = 25 mJ/cm2; Trot = 300K; p = 1 atm. Signal, this work 0.25 Signal, linear Signal, calculated using 3-level model Signal from Ref. [8] 0.2 0.2 s nit y u 0.15 ar 0.15 bitr s ar y unit gnal, 0.1 ar Si bitr 0.1 ar 0.05 al, n g Si 0.05 0 0 10 20 30 40 50 60 F , mJ/cm2 Figure 3. Variation of LIF Signal with laser fluence, F , 0 for (15,3) R1(11) transition at 300K, 1 atm. 0 5 10 15 20 25 30 35 40 F , mJ/cm2 Figure 2(a). Variation of LIF Signal with laser fluence, F , for (2,7) P(9) transition at 1800K, 1 atm. 8