continuous-wave hf optical resonance transfer laser model

Continuous-wave HF optical resonance transfer laser model

Munson A. Kwok and Roger L. Wilkins

A nonreactive kinetics model of the flowing gaseous medium of a cw HF optical resonance transfer laser(ORTL) was developed and a parametric study performed. Cross sections were generated for HF(vi,Jl) +HF(v 2 ,J 2 ) V-V, R - T, and V-R processes by classical trajectory calculations and by surprisal methods.The ORTL medium model was validated by modeling the reported experimental results of past ORTL ex-periments. The collisional processes were found to be important in creating positive gain. The study, ob-serving the effects on small signal gain by varying parameters such as pumping laser radiative flux, flux dis-tribution, flow velocity, flow density, and HF mole fractions, has suggested possible regimes of large gain.

1. Introduction

The cw optical resonance transfer laser (ORTL) hasbeen studied experimentally by Wang and colleagues.1-9

In optical resonance transfer lasers, photon energy froma high flux optical source is resonantly absorbed on vi-brational-rotational transitions by a passive non-reacting molecular gas. A population in the upper stateis rapidly created. Collisional steps follow subsequentlyto access the part of the internal energy manifold in theexcited molecule favorable for population inversionsand for high gain.

The ORTL device is unique in that Wang et al. haveconcentrated on HF or DF as the optical acceptor ofpumping photon flux. They have introduced the ideathat the lasing molecular species need not be the sameas that of the optical acceptor if rapid collisional transferexists. They have consequently invented a largenumber of new laser devices. They have also found thatthe HF (or DF) ORTL, in which the acceptor and lasingspecies are the same, has a potential for high overallefficiency. Namely, a large fraction of the photon fluxfrom a pumping HF laser can be converted efficientlyinto HF ORTL power. This potentially high efficiencycombines with features attractive to chemical lasersystems. Properties conducive to better beam qualityperformance include premixed operation and initiallylaminar flow. The objective of this paper is to investi-

The authors are with Aerospace Corporation, P.O. Box 92957, LosAngeles, California 90009.

Received 27 December 1982.0003/6935/83/172721-11$01.00/0.© 1983 Optical Society of America.

gate the effect of kinetics on various regimes of the zeropower gain medium in an HF ORTL. It is quite clearthat the performance of the ORTL medium depends ona detailed understanding of the individual collisionaltransfer mechanisms involving HF(v1,Jl) + HF(v2,J2)energy transfer on a channel-by-channel basis. Thisis because the pump radiation is quite state-specific onsuch an HF(v,J) basis.

In the early work of the Hughes group, 2 a lasing modelwas constructed to simulate the observed experimentalresults involving the extracavity ORTL. For severalreasons, the model was considerably simplified in itskinetic details. Rate coefficients were not sufficientlywell known on a v,J state-to-state basis, and the intro-duction of lasing to the model led to a complexity andan expense of computer usage which required additionalsimplifications. The crucial major assumption madewas that of rotational equilibration of all the HF(v)states in the ORTL medium. With a multiline HFpumping laser, it is apparent that substantial rotationalnonequilibrium effects could occur. One such effect isthe observed multiline ORTL lasing. We, therefore,set out to examine the ORTL medium with muchgreater detail in the collisional kinetics and kineticpathways by using a model which could depict rota-tional-nonequilibrium effects. To include the neces-sary details in kinetics, we chose to begin with a non-lasing ORTL model.

In one sense, this current study is then an extensionof the previous kinetic studies of Wilkins and Kwok 1 0 11

in which aspects of these classes of laser mechanismshave been closely examined theoretically and experi-mentally and in which code simulation of past experi-mental studies has been conducted using the appro-priate developed kinetic rate data. The objective ofthese code studies has always been the improvement ofvital cross-section data needed for laser modeling.

1 September 1983 / Vol. 22, No. 17 / APPLIED OPTICS 2721

12

~ 10

s w_ -

PUMPING R-R.T K-KJ-5-J-6 i

J-5 -J-7 _J-6 -J-7

PUMPING V-V

DASHED ARROWS:LOSS ORSECONDARYKINETICS

v 3

~~18

_ 15

.D_ 10

HF LASERPUMPING

v 2

23

20

_ 27 in the ground electronic state. HF molecules are raisedfrom the v = 0 state by the resonance absorption of

25 P1-o(J) photons from the pumping laser. The HF(v= 1,J) molecules can then be excited to the v = 2 stateby a second resonance absorption of P2 -1(J) laser

22 photons from the pumping laser or by the following V-Vcollisional energy transfer process:

10:- V-R-_ 15

________ 10

V - 1 V -

Fig. 1. HF energy level diagram showing an optical resonancepumped transfer laser system.

II. Model Development

In Sec. II.A, we discuss some fundamental conceptsneeded for an understanding of an HF (or DF) ORTL.Central to our studies is the collisional energy transferprocesses in the ORTL. The state-to-state rate coef-ficients computed by Wilkins and Kwok12 for use in themodeling are briefly summarized in Sec. II.B. In Sec.II.C we provide a description of the HF ORTLmodel.

A. Fundamental Concepts

A cw HF ORTL system consists of the following ele-ments:

(1) source of pumping laser radiation;(2) premixed medium (He/HF) in transverse flow;(3) mirror structure for coupling the pumping laser

into the medium;(4) ORTL resonator and beam extractor;(5) ORTL gas handling systems.An important feature of an HF ORTL medium with

no chemical reaction is that the same active HF mole-cules can be used more than once (recycled) within themirror structure of the pumped laser. The effectivenumber of cycles has significant influence on the overallORTL efficiency.

This model describes aspects of the first three listedelements of an ORTL system. The model predictions,which will be compared to experimental results, aresmall signal gains of selected spectral lines, HF(v,J)number densities, and medium static temperatures forappropriate spatial locations. The advantage of amodel, of course, is to enable us to make excursions inchoices of parameters which are difficult economicallyto execute experimentally in a short time.

The microscopic ORTL cycle can be illustrated withan HF energy level diagram as shown in Fig. 1 describingthe vibrational rotational (v,J) levels of the molecules

HF(vj = 1,Jj) + HF(v 2 = 1,J2) - HF(v = 0,Ji+ HF(v 2 = 2,J2). (1)

Successful ORTL operation seems to require that lasingoccur at wavelengths longer than those of the pumpingtransitions. A partial inversion with a significant gainmust be developed. In the steady state under properconditions, the ORTL transitions being opticallypumped should be nearly saturated without appreciablegain. The collisional deexcitation of molecules to lowerJ levels (or shorter wavelength regimes) leads to otherHF transitions with absorption. These transitionswould tend to have populations in the lower state largerthan those in the upper state due to the inclination ofthe medium toward rotational equilibrium. The col-lisional excitation of molecules to high J states occursdue to V-V processes [Eq. (1)] or (R,T) transfer pro-cesses given in Eq. (2) or combinations of both types.

HF(v1 ,Jl) + M - HF(vJil) + M;

(V, - J1) = +1,+2,+3,+4,+5. (2)

At the higher J states, the partial inversions have thebest chances to develop. The cross sections for collisionchannels represented by Eqs. (1) and (2) should be largefor an efficient ORTL.

After ORTL stimulated emission has occurred, theHF molecule must be recycled to (v,J) states whereoptical or collisional pumping can again occur. Im-portant channels for this behavior are the backwardreactions of Eq. (2) and perhaps the backward reactionsof Eq. (1) if the v = 1 state is involved after the ORTLlasing on the P2(J) transition. The R-R,T processesare favored since they are exothermic. The backwardprocesses of Eqs. (1) and (2) can also be construed to beloss mechanisms affecting the efficiency of the ORTLlasing transitions if they influence the populations ofthe upper states. Other secondary state-scrambling orloss mechanisms might be V-V processes involving v,= 1 and v2 = 2 (or v1 = 2 and V2 = 2) and V-R,T pro-cesses:

HF(vi,Jl) + M - HF(vi,,J) + M, (3)

where M = HF(v 2 ,J2 ), diluents, and v, - v =+1,2, .. ., v1. The V-R,T processes will generally betoo slow to be of great importance in an ORTL cycle. Incontrast, these mechanisms would be quite noticeablein a chemical laser description. Ideally, pumping fluxrates and collisional rates should be sufficiently largeso that one characteristic cycling time is considerablyless than the time for an ORTL fluid element to passthrough the flux fields of the pumping laser. Then in-dividual molecules can be reused and efficiency en-hanced.

2722 APPLIED OPTICS / Vol. 22, No. 17 / 1 September 1983

E (cm 1)

15,000

10,000 _

5000 -

f.

-20

For the Hughes operating condition,2 estimates canbe made of the characteristic time for radiative andcollisional processes. Typically, the pump laser flux 1ph

on a single spectral line is 200 W-cm-2 , and the peakradiative cross section °rad on the absorbing line is 5 X10-17 cm2. The characteristic time rad for radiativepumping is then

Trad = tradph 7 X 10-6 sec.

Single-channel V-V or R-R,T collision processes willhave cross sections ranging from 1 to 5 collisions inprobability. At pressures of HF at 4 Torr, coll =(kcollNHF) 1 - 0.25-1.2 X 10-6 sec. Characteristictimes for ORTL lasing would be of the order of cavitylifetimes-submicroseconds. It seems clear that thelimiting process of an ORTL cycle in this early experi-ment was the radiative pumping process because thephoton fluxes were too low. The entire microscopiccycle might be completed in characteristic times of tensof microseconds.

B. Collisional Energy Transfer State-to-State RateCoefficients for HF + M

The state-to-state rate coefficients used in the modelhave come from two basic calculations: the classicaltrajectory studies by Wilkins13 and recent computationsby Wilkins and Kwok12 using the surprisal analysisapproach. To date, there are no known measured ab-solute cross sections on single-channel HF(v,J) + Mprocesses including some knowledge of the product(v,J) states. This degree of refinement is necessary inthe modeling of an optically pumped medium in whichthe pumping fluxes excite very specific (v,J) states. Tovalidate the computed state-to-state rate coefficients,typical experiments'1 7 were modeled using versionsof the ORTL model code with the same kinetics. Somecomparison with experiments is, of course, necessary todetermine the surprisal parameters when using the in-formation theoretic technique. A detailed kineticslisting is given in the report. 1 8

Some indication of the complexity of the kinetics issummarized in Table I where the types of HF(v1,Jl) +HF(v2,J2) process are described. The number ofchannels possible for each type of process is also given.For the simple V-V energy transfer processes repre-sented by HF(v = 1) + HF(v = 1) - HF(v = 2) + HF(v

= 0), an astonishing number of channels is possiblewhen all possible initial and final J states of <15 areconsidered for near-resonant conditions AE < 500cm-'. To render the ORTL model manageable, welimited the channels listed to I AE I < 100 cm-1. Thesubtleties of the many channels of the V-V type ofprocess would seem to be extremely important in anORTL medium model, since the rate coefficient of eachchannel is typically quite large. There are far fewerchannel types for the R - T processes, but they are alsoextremely important for particular ranges of J becausethe rate coefficients are also quite large (order of 1014cm3 mol- 1 sec-1 ); some 62 HV(v,J) species plus He wasincluded as collision partners giving an effective rep-resentation of over 14,000 pathways. To properly writeequations for V-R,T processes some product J valuesof >15 were considered.

In Fig. 2, the R-T endothermic state-to-state ratecoefficients12 are plotted as a function of initial J for v =1, J' = J = +1 through +5. The wide range of valuesfor the coefficients can be seen. One can see that thereis a limitation to pumping very high J levels in theORTL medium because the rates become too slow.

1015

4 K \ ~~~T-=30 K

10

10 0I 2 3 4 5 6 7 8 9 10 11 12 13 14

Fig. 2. R-R,T endothermic state-to-state rate coefficients for T =

300 K predicted using surprisal analysis. HF(v = 1,J) + HF.

Table 1. HF + M Collision Processes v < 2, J 15

k,Channel cm 3

mol1Type Equation type types sec-

R-T HF(v1,J1 ) + M - HF(v1,J,) + M

M = , 2,j 2 HF(v 2 ,J2 ) + HeJSi- = 1,+2,+3,+4,5 225 1014

V-V HF(v = 1,J1) + HF(v2 1,Jl)HF(v = 0,Jl') + HF(v2 = 2,JS), JAE I < 500 cm-1 30,000 1013-1014

V-V AEI < 100 cm'1 426V-R,T HF(v1,J1 ) + M-HF(vl,J1) +, 340 1012


C. HF ORTL Model

The HF ORTL model is built upon the NEST code,19

which handles nonequilibrium chemistry problems withone dependent variable, time t, or distance x. Theoriginal code provided for photolytic excitation as afunction of time by selected species in a fixed opticalvolume. The code accounted for the energy input bythe photon flux into the gas volume, but it did not pro-vide information on the spatial variation of the fluxwithin that volume due to optical thickness. The needto study individual HF(v,J) states required a significantexpansion of the NEST code and the photolytic excita-tion or flashlamp option. Crucial to the ORTL simu-lation is a consistent expression of the interaction oflaser radiation with the gaseous HF species within thecontext of the code's flashlamp option.

For this work an expanded version of NEST,19 desig-nated NESTE, has been developed. The version in-cludes the capability to handle 1100 kinetic equations,125 species, and the appearance of a species in 250equations. It also can handle 8 flashlamps (or laserlines). Recently the code has been upgraded furtherto 2000 equations and 16 flashlamps. This capabilitywould provide an opportunity to study more complexORTL systems such as the cw DF ORTL. The flash-lamp option has also been modified to accept analyticalfunctions as a function of t or x. These include sinus-oidal functions, square or rectangular functions, theGaussian function, and step functions.

The ORTL model describes the changes in the ORTLmedium as a function of x, the direction of subsonicflow. For a simple 1-D flow the transformation betweenx and t involves U, an average velocity of the flow,

t = x/U. (4)

The model is schematically illustrated in Fig. 3. It hasbeen assumed that the ORTL medium is a constantarea, constant velocity flow. Necessarily in a 1-D flow,this medium has a constant density as a function of x.Changes in the gasdynamic parameters of this flow aremoderated because of the large amount of diluent em-ployed, but gas heating will be seen to occur.

The initial conditions of the medium are staticpressure p, static temperature T, average velocity U,

P, T, all UI

PHOTONbFLUX I I

X ORTL MEDIUM

X

TOP VIEW

M

PHOTON FRONT VIEWFLUX _ _ _

Fig. 3. ORTL model schematic. Parameters for initial conditionsare shown.

gas composition of the ith species (He and HF) caj, anddimensions d X W of the flow cross section. The initialconditions for photon excitation include the spatial andspectral line distributions of the pump laser fluxes aswell as the absolute values of that power distribution.

For the current work, the rectangular flux profile withdimensions a and b will be used. The flux profile ap-proximates that which is observed. The coupling of thepump laser into the ORTL medium includes, in thesecases, a double-pass configuration as shown in Fig. 3.The one-pass interaction length is L.

The important ORTL model outputs as a functionof x include

(1) gain coefficient at the spectral line center of HF2 - 1 and 1 - 0 P-branch lines;

(2) HF(vJ) number densities;(3) static temperature;(4) static pressure;(5) Voigt line shape and parameters.The conventional formulation of the gain coefficient

and line-shape expressions are summarized in the re-port.18 At the line center, the small signal gain coeffi-cient go (cm- 1 ) is given by

go(vo) = Crad(vO)gu N- _ Ns~gu g9o

(5)

where 0-rad(VO) is the radiative cross section for stimu-lated emission at line center with frequency vo, gu andgj are statistical weights, and Nu and N1 are the upperand lower state densities in no.-cm-3.

Some discussion dealing with resonance absorptionwithin the spectral line is necessary in entering theproper initial conditions for the photolytic excitation.It will be assumed that the pump laser is homogeneouslybroadened. Then the lasing mode of the pump laserhas a high probability of being near the spectral linecenter. It will be assumed that, on a given spectral line,the lasing occurs on one mode at line center. LargeFresnel number medium pressure flowing lasers withunstable resonators, such as cw HF lasers, generally willfit this assumption quite well.20

With these assumptions in mind, one can formulatethe photon interaction expression for the NESTE codeas

d = x 11 - exp[-k(vua;x)L]} (6)dx hVuINAL

for an optical volume of area a X [0rad(v)]/a (cm2 ) andlength L (cm) as defined in Fig. 3. The rate of changeof upper state density Nu (mole cm-3), an upperHF(v,J) state, due to the photons is given by this ex-pression. The flux of pumping laser photons for thetransition is I(x) in W-cm-2 at ORTL spectral linecenter frequency Puj, while h is Planck's constant andNA is Avogadro's number. The absorption coefficientat line center h (vl) (cm-') for an isolated line is relatedto the radiative cross section by

k(.1) = Crad(Vul)91NA N -_ (7)\g1 guj

where rad(Vul) is the radiative cross section for ab-sorption at line center and where gu and gj are, respec-


Table II. ORTL Input Parameters

TotalFlux % influx V A. 1 as 10

Transition input (W/CM2 ) (1014 sec- 1) (sec-1) (10-16 Cm2) M (X 10-3)

P1 (5) 0.07 46.2 1.122 114.0 7.9 1.49 1.94P 1 (6) 0.21 138.6 1.108 113.0 8.3 1.66 6.67P 1 (7) 0.21 138.6 1.093 113.0 8.75 1.84 7.48P 1 (8) 0.05 29.7 1.078 112.0 9.04 1.93 1.71P 2 (5) 0.11 72.6 1.073 195.0 14.9 1.56 3.39P 2 (6) 0.22 145.2 1.059 193.0 15.6 1.72 7.58P2 (7) 0.10 66.0 1.045 192.0 16.4 1.82 3.69P2 (8) 0.04 25.1 1.030 192.0 17.0 1.94 1.52

PT = 41 Torr, XHF = 0.03, I(X) = 660 W/Cm2.

tively, the rotational statistical weights of upper andlower states of the transition. The derivation has as-sumed no strong spatial dependences of N or N, lowerand upper state densities, in the propagation or L di-rection and that the laser mode widths are very narrowcompared to ORTL medium linewidths.

However, the cw pump laser used in the early re-ported work at Hughes may not belong in this categoryof homogeneous broadening. The cross section of thelasing zone in the direction of the resonator is -(1.2 X2.3) cm2, and the stable resonator is far from confocal(300-cm radius of curvature mirror and flat mirror 70cm apart). The Hughes chemical laser is operated ona number of high order transverse modes, thus provid-ing a large number of modes within the spectral line-width of a lasing transition. It is also assumed that theORTL medium is pressure broadened or homoge-neously broadened. The entire population of a specificHF(v,J) state is then available to participate in theresonance absorption or stimulated emission process.There are no so-called hole burning effects. Fromcomputations using linewidth formulas, the ORTLmedium pressure broadening width is found, for theHughes studies, to be indeed about equal to the line-width for Doppler broadening. This calculation forpressure broadening is a lower bound because the effectsof elastic or inelastic long range or grazing collisionshave not been included.

In this case, an identical expression to that of Eq. (7)for k(vu1 ) arises if I(x) vs frequency is essentially con-tinuous and nonzero across part of the absorption line,and k (v) is taken as the average k within this range.Then I as in Eq. (6) represents the total photon flux inthis range instead of the flux at line center. Within thelinewidth, the error is not great since k is 0.79k (vi) fora Gaussian distribution. Thus for either of the two setsof above defined broadening conditions of pump laserand ORTL medium, Eqs. (6) and (7) are applicable withproper definitions of k(vu1 ) and I(vUI;x).

Table II presents a typical set of inputs for modelingthe Hughes' ORTL. The double-pass coupling of pumplaser radiation into the ORTL medium is estimated bymultiplying the incoming flux by a factor M. Thefactor is determined by estimating the return flux afterthe first passage through the length L of the mediumand taking an approximate spatially averaged lower

state density determined iteratively. Although spatialdependences in the flux direction are weak, there arenoticeable adjustments so M is not quite equal to 2.

Ill. Simulation of Hughes ORTL. Model Validationand Kinetic Variations

The experimental studies on the HF ORTL reportedby Baily et al. 2 of Hughes Aircraft form the basis for oursimulation validation. In that report a careful para-metric study was made of a typical ORTL medium.Number densities, static temperatures, and small signalgains were reported. The principal variation was in thepartial fraction of HF. The objective of our validationstudy is to simulate as many details of the experimentalstudies as possible.

Comparison of the ORTL model can be made withthe Hughes experimental number densities. The modelpump laser conditions are summarized in Table II. Interms of Fig. 3, the dimensions of the flux beam aregiven by a = 2.3 cm and b = 0.38 cm, while the dimen-sions of the ORTL flow are w = 0.3 cm and d = 6 cm.This geometry yields an effective L = 0.78 cm. Theinitial temperature is 300 K, and the centerline velocityis 7500 cm sec-1 for a parabolic profile with an averagevelocity of 5000 cm sec-1. The model densities of theORTL medium are determined at the center of thepumping laser beam in the direction of flow (i.e., x =0.19 cm). Individual HF(v,J) populations weresummed over J for each level. The correspondingmodel results are shown in Fig. 4 along with experi-mental results reported by Baily et al. The indepen-dent variable is the HF initial mole fraction. It is ap-parent that the agreement is very good. For the higherpressure (and higher HF density) ORTL medium con-dition, the fit is less good. It would appear that onereason for the disagreement might be an HF loss (i.e.,a smaller than nominal HF flow) in the experiments at78 Torr. In agreement with experimental results, ro-tational nonequilibrium effects are not evident at thesetime scales on the centerline of the pumping beam.

The model results are shown in Fig. 5 for static tem-perature in the 41-Torr cases. The model static tem-perature data are likewise extracted from the x =0.19-cm position. The comparison with experimentalresults is also given. Generally, the model temperaturesare 5-10% lower than observed rotational temperatures.


0.05,

Rev-v

o EXPERIMENT- MODEL

S

0.05 0.06 0.07

(a)

FLUX * 660 WIcm2

PI . 41 Torr

XHF . 0.03

TI 30'K

20 30I t Ipsec

40 50

0 0.15 - 0.30X cm) U 5200 cm-sec

-HFO

7

E

4d 101

(b)

HFI)

I / HF(2)

o EXPERIMENT

I / - MODELI I I I I I

LU 0 0.01 0.02 0.03 0.04 0.05 0.06

HF MOLE FRACTION, XHF

Fig. 4. Comparison of ORTL model and experiments of ORTLmedium HF(v) number densities as a function of HF mole fraction:

(a) p = 41 Torr; (b) p = 78 Torr. Flux = 660 W/cm 2 .

700

650

0N 10(b) FLUX: ON

o

P1(9)

20 30 40I t seI

0.15 0.30X (cm) U - 5200 cm-sec

tlI

Fig. 6. Small signal gain coefficients as a function of flow distanceor time t. Pump initial conditions as in Table II; 41 Torr, 660 W/cm2 .

(a) P 2- 1(J) lines; (b) P1 . 0 (J) lines.

Fig. 5. ORTL medium static temperature as a function of HF molefraction. p = 41 Torr. Flux = 660 W/cm2. Note expanded

ordinate.

o EXPERIMENT- MODEL

I I I I I I

Unlike the vibrational number densities, which rise toa nearly uniform condition, the static temperaturesteadily rises in the flow direction throughout thepumping laser flux field due obviously to the absorptionof laser power. The positioning of observations forcomparison with model calculations is, therefore, cru-cial. It is possible that this is the basis for the slightdisagreement.

Model predictions are in general qualitative agree-ment with reported observations of zero power gaincoefficients. In Fig. 6(a), plots are shown of the smallsignal gain coefficients go at spectral line center of the2 - I band of positive gain HF transitions as a functionof flow direction through the pump laser beam. Thisplot is for the flux of 660 W cm- 2 of the pump laser withthe pump line distribution tabulated in Table II.Similar results are computed for the P1 0 lines. The

0.05 0.06


.2-)

10 1

(a) 10140.01 0.02 40 0.04

HF MOLE FRACTION, XHF

I l I I I I I

7060OFF

0.38

OFF

0.38

6001-

550 -

500 -

W_450

I_-,

400 -

350-

3001-

0 0.01 0.02 0.03 0.04HF MOLE FRACTION.XHF

In'' ' * ' '. Al) J

10 17

Table Ill. Observed Lasing Compared to Model Gain Coefficient

Normalized Normalizedlasing model

Pump lasers Pressure (From Fig. 4 gainCase (W-cm- 2 ) (Torr) XHF of Ref. 2) coefficient

1 440 78 0.03 1.0 1.02 440 78 0.01 1.0 0.93 440 78 0.06 0.0 0.04 660 78 0.01 0.5 0.25 660 78 0.03 0.8 0.66 660 78 0.06 0.0 0.07 660 41 0.01 <0.05 0.28 660 41 0.03 0.5 0.99 660 41 0.06 0.75 2.2

initial total pressure is 41 Torr, and the initial statictemperature is 300 K with 0.03 HF in He. The largestgain coefficients for our model are in general agreementwith the Hughes observations between 0.012 and 0.06cm-1 at the centerline of the pump beam. The modelgain coefficients are seen to rise virtually linearly withdistance from a threshold. This behavior suggests thatthe radiative pump flux levels are too low in these earlyexperiments. On this basis, one might expect ineffi-cient recycling of HF molecules and extremely ineffi-cient ORTL lasing in this experiment. Indeed theoverall efficiency was observed to be -2%.

A further comparison of experimental and modelresults is given in Table III where the lasing results arecompared with model small signal gain results. The.normalization of results is made to one of the betterdocumented experimental cases, i.e., 440 W/cm 2 casewith P(4) and P2(4) pumping at 78-Torr total pressureand 3% HF. The model small signal gain coefficientsfor this table are taken at the x = 0.38-cm position. Thelargest gain coefficient at this position is used. This isbecause some gain plots exhibit the triangular structureas in Fig. 6 while others do not, depending on conditions.In reality, the axis for the experimental lasing may beat some position equivalent to smaller x. The corre-lation is fairly good. Case 7 must describe a case tooclose to experimental threshold where lasing output isquite sensitive to slight variations. In Case 9 the modelgain coefficient at P2(10) is somewhat overpredicted.Since most relevant ORTL studies will concentrate onthe XHF regime of <0.03, this deviation is not consid-ered serious for present applications.

The simplest global assessment of the relative im-portance of the R-T and V-V processes is the directremoval of those particular cross sections in the mod-eling code and running a particular case. If one choosesthe case described in Fig. 6, 8 in Table III, and removesthe R-T cross sections, one produces no positive gainexcept around <0.001 cm-1 at x = 0.38 cm in P1 (6) andP1(7). This illustrates that the R-T collisional pro-cesses are important in v = 1 or v = 2 in producing anORTL gain at higher J states than those that arepumped by the laser. This observation is further sup-portd by the modeling results for the same ORTL case(660 W-cm-2 , 41 Torr, XHF = 0.03) when the V-Vprocesses are removed. Gain coefficients on the

Pi-o(J) branch are unaffected. In fact, they are largerbecause the HF(v = 1) states have not been depleted bycollisional V-V pumping to HF(v = 2). However, gaincoefficients on the P2 1(J) branch are greatly reducedby the absence of the 1 + 1 - 0 + 2 V - V collisionalmechanisms. Since one potential benefit of ORTL isthe conversion of multiline chemical laser operation tonear single-line ORTL lasing at an elevated v level, suchas v = 2, the V-V processes appear crucial in pro-ducing the relatively large zero power gains on impor-tant lines of the P2 -1 (J) branch of HF.

In recent work,7 an ORTL lasing model, which in-cludes rotational nonequilibrium effects, was developed.This model, however, does not include by far the fullkinetic details present in our model. The R-R,T ratecoefficients used by Hughes were determined fromempirical rotational relaxation rate measurements' 4

fitted to an exponential gap representation. The ex-ponential gap representation tends to give R-R,T ratecoefficients, for intermediate J HF(v,J) states J > 5,substantially smaller than our informational theoreticapproach based on an inverse power law representation.Our approach has fitted other experiments14- 7 suc-cessfully as well as the system studied here. Resolutionof such differences such as the R-R,T cross sections willbe quite important for understanding the behavior ofHF ORTL lasers at intermediate J values, the principaloperational region for the device.

IV. ORTL Model Parametric StudiesWe now explore regimes of ORTL operation in which

good overall efficiencies might be expected and shorteffective-cyle times might be realized. Accordingly weexamined the effects of

(A) flow velocity variation (variation of the time thata fluid element spends in pump beam);

(B) changes in the J distributions of pump beam;(C) pump flux variation;(D) initial static temperature variation and tem-

perature control.Since our comparison will deal with small signal gain

coefficients, we establish as a base line the reportedHughes case: 20-W ORTL output at 78 Torr, XHF =0.019, for a pumping laser input of 300 W or 440 W/cm 2

and a P(4)-P 1(7), P2 (4)-P2(7) pattern. The reportedoverall efficiency was 0.053. For that case our modeling


0.40,-

E

.F

FLUX: ONt /pisecl OFF

oXHF ' 006

* 0.01o ~003

0.30-

S

Iy

n:

0.20 .-.

0.10

0 0.19 -l 0.38X (cml U 1950 cm-sec 1

Fig. 7. Small signal gain coefficents as a function of t or X at fourfold

slower flow velocity. Only P 2 (J) branch lines are presented. p = 78

Torr, XHF = 0.03, T = 300 K, and Jp = 4 pump flux distribution.

N.__

1 I 1 1 ( -v0 1 2 3 4 5

Jp, PUMP LASER LOWEST P p) LINE

Fig. 8. Dependence

yielded a maximum gain coefficient [on P2 (7) or P2 (8)]of 0.05 cm-l on the downstream edge of the beam. Atthe centerline of the beam, x = 0.19 cm, the gain is-0.020 cm- 1 .

A. Flow Velocity

One principal control parameter is the velocity of theflow. Basically this variation adjusts the time that anyfluid element spends within the pump laser beam. InFig. 7, the P2 (J) gain coefficients are shown as a func-tion of x for a case in which the velocity is a factor of 4slower than cases previously discussed, i.e., 1300 cmsec1. The pump flux level is 440 W cm-2 with aP(4)-P(7) pattern, and the other flow parameters arean initial pressure of 78 Torr, initial temperature 300K, and XHF = 0.03. A concurrent set of P1(J) gaincoefficients at half of the values shown is also generated.It is quite clear that there is considerable improvementin peak gains around 0.095 cm-l or a total gain Go 1.2.This is a fourfold to fivefold improvement on the orig-inal Hughes reported results on gain. One might expectthat conversion efficiency and, therefore, overall effi-ciency should improve dramatically. Conversion effi-ciency is the ratio of total ORTL laser power to pumpinglaser power absorbed. In fact, this advance has beenexperimentally reported by the Hughes group in whichoverall efficiencies between 25 and 30% were observed.The dominant P2 (J) spectra, according to the model,would shift to higher J states than P2(8) under theseconditions.

The results in Fig. 7 also serve to reinforce the ideathat for a given pump flux level and HF density, thereis a minimum characteristic time for the fluid elementto remain in the beam. In the depicted case, this min-imum time appears to be 30 ,usec and the number ofpossible HF cycles of reuse might be 5 or 6 in 150,lsec.

A maximum characteristic time for a fluid elementin the pump beam appears to be indicated. As the flow

of peak gain coefficientdistribution.

on pump flux J

moves through the pump beam, collisional relaxationprocesses (R-R,T and V-R,T) tend to become moredominant and lead to heating effects. Direct relaxationprocesses and heating tend to degrade the gain fartherdownstream in the flow direction.

B. Pump J Distribution

The pattern of pump HF chemical laser lines is cru-cial to the potential success of the ORTL for high powerapplications. Accordingly, a modeling study has beenconducted in which the flux input pattern has beenvaried in J in the following manner. The input fluxintensity pattern in Table II is maintained; however, theJ pattern on both the P2 - 1 and P1_0 branches isshifted. The lowest J in the P(J) branch is designatedJp as an identification of the pattern, i.e., J = 2 repre-sents a P 1(2), P 1(3), P1(4), P 1(5), P2 (2), P2 (3), P2 (4),P2 (5) pattern. For a total flux of 660 W cm- 2 at 41Torr, 300 K, the results are exhibited in Fig. 8. Themaximum achievable gain coefficient is plotted as afunction of Jp. It can be seen, unsurprisingly, that anORTL medium can work very well at low Jp pumpingpatterns and would probably approach very high overallefficiencies. This is the result of high densities forresonance absorption at the nearly Boltzmannized lowerJ HF states and also the result of very fast R-R,T ratesat these low J values. At higher middle Jp around 5 or6, the peak gain coefficient appears to settle above 0.08cm-' for XHF = 0.03 and 0.06. The peak gain has beendriven down mainly by the steady large increases instatic temperature. These temperatures at the down-stream side are large. With Jp = 4, the downstreamtemperature rises from 300 to nearly 1000 K, with Jp =6, to over 2000 K. Therefore, with the given positivegain, the inversions (ratios of upper state density tolower state density) must be quite high. These great


6 l6 7

0.2

E

I 0.1aw

_0

ip 5

1000 10,000

FLUX W-cm )

100,000

Fig. 9. Dependence of peak gain coefficient on total pumping radiative flux.

inversions must be partly due to the matchup of pumpspectra with the near-Boltzmann distribution in HFpopulation at the elevated temperatures above 1000 K.It can be shown that the P2(J) line of peak gain movesto higher J as Jp, the lowest pump line, increases. Thispeak gain line always occurs at J greater than Jp. Itreflects the fact that the inversions are found at higherJ where the lower state densities are initially droppingoff. In this sense, the peak P2(J) lasing line in ORTLdepends on the pump laser design. Non-negligible gaincoefficients are generated on lines that are simulta-neously pumped by the chemical laser at all patterns ofJp. This behavior would tend to lower the input cou-pling efficiency (fraction of pump laser power absorbed)as certain strong pump lines are effectively enhancedin passage through the ORTL medium and not ab-sorbed. Too thick an ORTL medium in the directionof the pump laser beam propagation will obviouslyconvert the ORTL medium to an inefficient amplifierfor some lines. At low Jp, where the gain coefficientsare large, this problem will have a great influence inlowering the coupling efficiency.

Steady temperature rises are due to the overall ab-sorption of large amounts of pump radiation. Needlessto say, the overall efficiency of ORTL is not unity, andthat pump power which is not extractable as ORTLlasing will eventually heat the gas. As the size of themedium grows in larger devices, so that the character-istic times for a fluid element within a pump beam be-come longer, heating mechanisms become more im-portant. On these time scales, this gas heating is dueto the R-T rates and may be a sensitive indicator for thecorrect rate scheme here. The gas heating problem isaggravated at intermediate J values since the energygaps in transferring from J to J - 1 on a given vibra-tional level are increasingly proportional to J. Fromthis model study, it seems clear that gas temperaturecontrol is an important consideration in ORTL. At theelevated temperatures above 1000 K, it must be notedthat the kinetics scheme is no longer considered accu-rate because literally tens of thousands of V-V channelswith energy defects less than the average thermal energyhave not been included. Extensions on J on R-T and

V-R collisions should also have been made. Never-theless, the model still reflects some of the generalqualitative behavior of the ORTL medium.

C. Pump Flux VariationsAs shown in Fig. 7, it is quite clear that the total pump

fluxes of the early Hughes studies were too low to beefficient. If the flux is raised to 50,000 W cm- 2 for thecase in Fig. 7, the peak gain coefficient is attained in 40,usec rather than 150 sec. This improvement shouldalso improve the effective cycle time on a microscopicbasis since radiative pumping times are much faster. InFig. 9, the effect of flux increases on peak gain coeffi-cient is shown. For an optimum ORTL condition, itappears that the pumping flux should exceed 5000 or600 W cm-2 per pump lasing spectral line. At suffi-ciently elevated fluxes, there is the additional advantagethat the peak gain is attained before gas heating be-comes a serious problem in the flow direction. Typi-cally the static temperature does not exceed 750 K at thepeak gain in these cases.

D. Temperature

Temperature emerges as an important variable in anycw HF ORTL device. Two aspects of temperature ef-fects have been studied with this ORTL model. Theeffect of initial temperature of the flow has been ex-amined. Then the possible control of gas heating by thepump laser beam is briefly discussed.

Setting the initial temperature can have two effectsin an ORTL. The temperature can be raised so that theinitial distribution of HF(v = 0,J) densities bettermatches the pump laser spectral distribution. Theobvious practical limit is reached quickly, however. ForJp = 6, an initial ORTL medium temperature around2000 K would have to be produced for good matching.Even if the practical limit were not a factor, the gaincoefficients would generally be lower because of theiradverse dependence on temperature. Furthermore, asJp increases so that the pump laser spectral pattern


I1 I , , I I . I In1

o I t $ ~~~~~~ ~~~~~~P2 (7) 0.02 - __ _ _ _ _ _ _ _ _ _ _ _ _I ~~~~~~~~~~P )

0.01 f..ok I' I l l l l

0 10 20 30 40 50 60 70

FLUX ON t se OFFI I I I 0 0.15 0.30 0.38

X cm} (U 5200 cm-sec 1)

Fig. 10. Effect of elevated initial static temperature on P 2 _ 1(J) small signal gain coefficients.The case illustrated is similar to that in Fig. 8.

moves to higher J values, one encounters slower R-Rcollisional cross sections, as given in Fig. 2.

The second effect of initial temperature is moresubtle. In Fig. 10, small signal gain coefficients areplotted as a function of x or t for a case in which theinitial temperature is slightly elevated. The case isidentical to that shown in Fig. 6 except for the initialtemperature. The temperature dependence of the R-Rand V-V rate coefficients is such that the coefficientsare faster at elevated temperatures. Steady state isapparently reached quite quickly, although the peakgain is about half of that reported with the initial tem-perature at 300 K. The temperature rise is under 100K. The reaching of a peak gain condition in 1-2 ,usecas such low fluxes is important in establishing an ef-fective cycle time for the kinetics, and this characteristictime will determine the ultimate size of ORTL flowsrelative pump chemical laser flows. Thus the slightheating of the ORTL medium may be beneficial at thecost of some of the gain.

Control of gas heating by the pump beam can occurin two ways. If the overall efficiency of the ORTL isunity, there will be no gas heating. Thus power ex-traction from ORTL lasing is important in moderatingthe effects of gas heating. The current model examinesthe worst case, in which no laser power is extracted. Amanipulation of the specific heat Cp of the gas can beeffected by adding a high Cp species, but poor HFquencher, such as SF6 in small quantities. For a 78-Torr case, such as case 5 in Table III, the addition of 15Torr of SF6 into the model initial conditions results in70% of the peak gain while the temperature is nearlyhalved.

V. Conclusion

A model has been developed to study the gas flow ofan optically resonance pumped transfer laser. Themodel is validated by simulating the Hughes groupexperiments. Collision energy transfer processes of theR-T and V-V types are shown to be instrumental increated ORTL gain inversions and the experimentally

observed levels of small signal gain coefficients. Inparticular, V-V processes are found to be important inthe HF 2 - 1 band gain coefficients. A successfuldepiction of the gain coefficients is found to be quitesensitive to the details of individual V-V pathways.

The model has been used to search for regimes of fa-vorable small signal gain. Parametric variations in flowvelocity, pump J distribution, pump flux, and gastemperature control have simulated experimentallyobserved results or eventually were verified by experi-ments. The agreement is in the qualitative sense sincetrends in small signal gain followed trends in efficiencypower outputs or spectral distributions2 7 quite well.

The model also predicts regimes of large peak smallsignal gain coefficients. For peak gains above 0.20cm-', this regime dictates that the pump laser pro-duce

(1) >5000-W-cm- 2 flux (1000 W-cm- 2 in one spectralline);

(2) a pumping pattern with Jp 4 where the lasinglines are P1 (Jp = 4), P1(5), P1(6), P1(7) and P2 (Jp = 4),P 2 (5), P2 (6), P2 (7), and over 80% of the power is in theP(5) and P(6) lines of both branches.

For the ORTL He and HF, conditions for this regimemight be

(1) total pressures up to 100 Torr;(2) elevated initial temperatures up to 600 K for

proper flow cross-section sizes;(3) HF mole fractions between 0.01 and 0.03 for a

double-pass coupling mirror system for the pumplaser.

For a multiple-pass coupling mirror system the molefraction would be proportionally reduced (10 passes,one-fifth of the mole fraction listed). The optical depthfor the pump laser beams' total passage through themedium should be unity.

The principal problems and limitations of the ORTLmedium have also been detected with this simple model.The pumping spectral line distribution required tendstoward lower J-P-branch lines at or below P(5) andP(6). This distribution is not typical of most efficientcw HF lasers, which tend toward higher J lines. Ele-


vated initial temperature operation appears to be notimportant in achieving 0.2-cm-' peak gain operation;however, this condition is important in high efficiency,and it is an additional operational complication.

The ORTL medium is possibly limited in two of threedimensions by optical depth requirements and by gasheating. In the direction of propagation of the pumplaser, the model has shown that in a multiple linepumping pattern in rotational nonequilibrium, gain canbe generated on transitions on which pumping laserlines exist. For too large a dimension (or optical depth),significant amplification will occur. This effectivelylowers the input coupling efficiency (and, therefore, theoverall efficiency). Significant gas heating by pumplaser radiation is generated by the small signal gainmodel at longer time scales in the flow direction.Practically, this effect could limit the extent of theORTL medium in the flow direction (within the pumpbeam) to characteristic times below 300 ,usec. ORTLlasing would moderate this effect. Additives such asSF6 augmenting the specific heat of the flow are alsohelpful in controlling gas heating.

Our current model of an ORTL device can undergoimprovements. A major advance would be the con-version of the current code to a lasing model, at least atthe conventional level of using Fabry-Perot mirrors.Then the defined efficiencies of an actual ORTL candirectly be estimated. Since parts of the ORTL me-dium experience large elevated temperatures, a reex-amination of R - T mechanisms, and a further en-largement of the V-V manifold would seem useful.

Finally, we note that an ORTL device seems an ex-cellent relevant empirical test for HF(v,J) + M kineticsand laser modeling of cw HF lasers because of its com-parative simplicity. This case is unique compared withothers we have studied because the radiative pumpingflux is on during the interesting collisional kinetic pro-cesses. Also elements of simple premixed gasdynamicsare included.

References1. J. H. S. Wang, J. Finzi, and F. N. Mastrup, App. Phys. Lett. 31,

35 (1977).2. P. K. Baily, J. Finzi, G. W. Holleman, K. K. Hui, and J. H. S.

Wang, "High Energy Optical Resonance Transfer Study," HughesAircraft Co. Technical Report FR-79-73-538 (Jan. 1979),65 pp.,and FR-79-73-999 (Aug. 1979).

3. K. K. Hui, P. K. Baily, J. Finzi, J. H. S. Wang, and F. N. Mastrup,Appl. Opt. 19, 831 (1980).

4. J. Finzi, J. H. S. Wang, K. K. Hui, P. K. Baily, G. W. Holleman,and F. N. Mastrup, IEEE J. Quantum Electron. QE-16, 912(1980).

5. J. H. S. Wang, J. Finzi, P. K. Baily, G. W. Holleman, K. K. Hui,and F. N. Mastrup, Appl. Phys. Lett. 36, 24 (1980).

6. G. W. Holleman and H. Injeyan, "Multiwavelength 2-5 Mi-crometer Laser," Hughes Aircraft Corp. Technical Report FR80-72-653 (June 1980), 90 pp.

7. P. K. Baily, J. H. S. Wang, J. Finzi, R. C. Smith, J. Paranto, andD. Bruns, "Optical Resonance Transfer Laser (ORTL) Devel-opment," Hughes Aircraft Co. Technical Report FR-82-72-814(July 1982), 121 pp.

8. J. H. S. Wang, J. Finzi, P. K. Baily, K. K. Hui, and G. W. Holle-man, J. Phys. Paris C9, 463 (1981).

9. J. H. S. Wang, J. Finzi, P. K. Baily, K. K. Hui, and G. W. Holle-man, Proc. Soc. Photo-Opt. Instrum. Eng. 270, 106 (1981).

10. R. L. Wilkins and M. A. Kwok, J. Chem. Phys. 73, 3198 (1980).11. R. L. Wilkins and M. A. Kwok, J. Chem. Phys. 70, 1705 (1979).12. R. L. Wilkins and M. A. Kwok, J. Chem. Phys. 78,7153 (1983).13. R. L. Wilkins, J. Chem. Phys. 67, 5838 (1977).14. J. J. Hinchen and R. H. Hobbs, J. Chem. Phys. 65, 2732 (1976).15. J. J. Hinchen and R. H. Hobbs, J. Appl. Phys. 50, 628 (1979).16. R. M. Osgood, Jr., P. B. Sackett, and A. Javan, J. Chem. Phys. 60,

1464 (1974).17. M. J. Bina and C. R. Jonas, Appl. Phys. 22, 44 (1973).18. R. L. Wilkins and M. A. Kwok, "Modeling Study of a CW HF

Optical Resonance Transfer Laser Medium," The AerospaceCorp. Technical Report 0082(2603)-3 (1982).

19. E. B. Turner, G. Emanuel, and R. L. Wilkins, "The NEST

Chemistry Computer Program," Aerospace Corp. TechnicalReport TR-0059(6240-40)-1, The Aerospace Corp., El Segundo,Calif. (30 June 1970), 150 pp.

20. H. Mirels, AIAA J. 17, 478 (1979).

This study could not have been performed withoutthe major assistance of Karen L. Foster in programmingand computations. We thank George Judd for adapt-ing the NEST program to our problem needs. We ap-preciate the continuing discussions with L. Wilson andD. Drummond of AFWL/ARAC. We thank K. K. Hui,J. H. S. Wang, and P. Baily of Hughes Aircraft Co. fordiscussing their modeling results3 including ORTL os-cillation processes but with fewer kinetic processes. Wethank Lydia Hammond for typing the manuscript.

This work was supported in part by the Air ForceWeapons Laboratory and conducted under U.S. AirForce Space Division (ASFD) contract F04701-81-C-0082.


continuous-wave hf optical resonance transfer laser model

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