Vol. 4, No. 11/November 1987/J. Opt. Soc. Am. B 1829

Constants of the 5s 12g state of Na2 from two-photonspectroscopy

G.-Y. Yan, T. P. Duffey, W.-M. Du,* and A. L. Schawlow

Department of Physics, Stanford University, Stanford, California 94305-4060

Received February 6, 1987; accepted June 16, 1987

Fluorescence spectroscopy is used as an aid in the identification of two-photon transitions in Na2 . Several methodsare introduced and used to identify the transitions and analyze the spectra. By using these methods, the upper-level rotational quantum number, energy, and electronic character may be determined without any assumptionsabout the state to which the upper level belongs. Twenty-nine two-photon transitions were observed and analyzed.A Dunham expansion for the energy levels allowed 18 of the 29 to be associated with the 12' (3s + 5s) state. Theywere used to calculate new values of the Dunham coefficients for that state. These were in close agreement withvalues obtained in polarization-labeling experiments.

INTRODUCTION

A number of cw laser-induced two-photon transitions inmolecular sodium have been reported' since Harvey2 andWoerdman3 first explored these transitions. Identificationof those two-photon transitions, produced with modest laserpower, is not an easy matter. Study of the peculiar lineshapes in the molecular two-photon transitions has shownthat this kind of two-photon absorption will not occur unlessa nearly resonant enhancement level exists.",4 The enhanc-ing level must be such that transitions are allowed from it toboth the initial and the final levels of the two-photon transi-tion. Thus, for diatomic molecules, it must differ from eachof these end levels by one unit of angular momentum, andthere must be reasonably large Franck-Condon factors forthe transitions. In Na2, for laser wavelengths in the 570-600-nm range, these resonant-enhancement levels belong tothe A ',, state.

The presence of such a level can be used to identify thetwo-photon transition. In this paper we report the identifi-cation of 29 two-photon transitions in Na2 through a newmethod using analysis of fluorescence spectra originating onthe upper and intermediate levels of the transitions.

One earlier method is to record the two-photon absorptionand the one-photon excitation spectra simultaneously,searching for a one-photon A-X transition within whoseDoppler width the two-photon transition occurs. Such aone-photon transition is tentatively identified -as the firststage in the two-photon process. Polarization dependenceof the two-photon absorption then provides informationabout the J value of the upper level and the electronic stateto which it belongs.5 In practice, several transitions fromthe X state to the A state occur within the Doppler width.This makes it difficult to identify unambiguously the A-Xtransition associated with the two-photon transition. An-other method' uses Dunham expansions for the states' ener-gy levels to identify two-photon transitions whose two stageshave nearly equal frequencies and appropriate rotationalquantum numbers. Unfortunately, lack of knowledge aboutthe Rydberg states, compared with the X and A states,affects the reliability of assignments by this method.

Fluorescence spectroscopy is a useful technique in assign-ing molecular levels. From each upper or intermediate lev-el, pairs of fluorescence lines (AJ = +1 and -1) are observedfor a number of lower-state v values. As the constants forthe X and A states are well known and differ by a factor ofroughly 3/2 (because v" 0 - 2 and v' 20), the spacings ofthese fluorescence lines can be used to identify the lower andintermediate levels involved and their rotational quantumnumbers. However, difficulties arise in applying this meth-od to the case of a two-photon transition with photons ofnearly equal frequencies. Fluorescence lines originating inthe upper level will be obscured by several considerablystronger A-X transition series and their satellite lines. Inthe present work, various methods were used to discriminatethese lines from the lines of other fluorescence series. Anal-ysis of the pertinent lines was performed using Dunhamexpansions for the energy levels of the X and A states. Thelower and intermediate levels of the transition can thus beidentified, as can the J value, energy, and electronic charac-teristic of the upper level, without the need to rely on anycoefficients of the Rydberg states.

All the two-photon transitions identified had upper levelsin 12; states, located in the region 33 700-35 000 cm-1 . Itwas then possible to assign 18 of the 29 transitions to 1 (3s+ 5s) by using a set of Dunham coefficients obtained by aleast-squares-fit technique. The coefficients obtained forthe 12g (3s + 5s) state are close to those of Taylor et al.6 Wewill discuss possible reasons for our inability to fit the re-maining 11 lines.

EXPERIMENT

The experimental setup is shown in Fig. 1. A Coherent 599-21 dye laser was pumped by an argon laser. Typical single-mode output power of the dye laser was 100 mW. Thecoherent radiation was focused into a cross oven through alens with a focal length of 10 cm. The beam was reflectedback upon itself by a spherical mirror with R = 20 cm. Inorder to minimize the effects of feedback into the dye laser, amirror mounted on a speaker driven at audio frequency was

0740-3224/87/111829-06$02.00 © 1987 Optical Society of America

Yan et al.

1830 J. Opt. Soc. Am. B/Vol. 4, No. 11/November 1987

Fig. 1. Experimental setup.

used. The oven operated in the temperature440°C and was filled with argon acting as a bipressure of roughly 1 Torr. The temperature w;by a thermocouple meter. A P28 photomuwith a UV filter (Corning 2X7-59 and Cornihplaced at the end of one arm of the oven to mo:fluorescence. On the other arm, the fluorescEaged vertically onto a Spex-170 spectrometer band a lens. The resolution of the spectromet0.6 A at 6000 A. A cooled RCA-8850 photomi(PMT) placed at the exit slit of the spectrometea photon counter. The electronics of the pho:system consisted of a discriminator, a preamlratemeter. A dark-count rate of 20/sec could bEcareful adjustment of the electronics.

RESULTS

A simple and strong fluorescence spectrum excituned to 16 704.169 cm-' was recorded as shoThe vibrational and rotational constants for Istates are quite different. This leads to vibrotational spacings of R-A transition lines beinjthe spacings in A-X lines. Such a difference cJreject all A-X transition lines except those lrelated to the first step in the two-photon translines that come from the enhancing level in theupper level are marked in Fig. 2. A small sispectrum in Fig. 2 was rerecorded at a slower sceis shown in the expanded section of Fig. 2. Tspacings of 17.9 and 27.7 cm-' were measuredA-X transitions, respectively, within an accuraThe error is a result of uncertainties in the ca]resolution of the spectrometer as well as in thethe spectrum. Nevertheless, we were able to dtwo-photon transition to be J + 2 - J + 1 - J.we established that v" _ 1 and J" is in theUsing the Dunham coefficients for the A andthen found the R-A-X transition to be (v,(1, 44).7 Finally, we compared the wavelengthlines in both series with values calculated assumassignment. We are confident that this two-ption in Na2 takes place between a II+ state witthe ground state (1, 44) through the A state (

intermediate enhancement level. The term energy of theupper level is 33 946.818 cm-l.

This transition (16 704.169 cm-') is rather easily assignedIETER because the one-photon fluorescence ending on the A state is

very strong. Other two-photon excitations produce consid-PMT erably weaker R-A fluorescence lines. Assignment of these

transitions is a more complicated matter. As in the methodIHO ON described above, the key is to determine which lines origi-

COUNTER nate in the upper level of the individual two-photon transi-tion. Such lines will have strengths that vary as the squareof the intensity of the incident laser, whereas lines originat-ing in the A state vary like a saturated one-photon transi-tion. Thus one may distinguish between two series of linesby blocking one of the two counterpropagating beams and

x-y RECORDER noting how the line strengths change. The intensity-squared dependence is followed only approximately for two-photon transitions in Na2 because of distortion in the line

e range 400- shape due to beam focusing and the presence of enhancingiffer gas at a intermediate levels. Nevertheless, comparison of fluores-as monitored cence spectra for one and two beams can be used to distin-ltiplier tube guish between R-A transition lines and the A-X ones, pro-ng 7-60) was vided that the offset is not less than the relevant naturalnitor the UV linewidth.ence was im- An example of this type of line is shown in Fig. 3(a). Ay two prisms section of the spectrum in Fig. 3(b) was taken with bothr was about beams, while Fig. 3(c) was taken with one. The vertical

iltiplier tube scale in Fig. 3(c) is one half that of Fig. 3(b) to facilitater operated as comparison; this was accomplished by doubling the x-y re-ton-counting corder sensitivity while obtaining the second spectrum.)lifier, and a Compare Figs. 3(b) and 3(c); those lines whose height is lessobtained by in Fig. 3(c) than in Fig. 3(b) may be inferred to be R-A

a LASER LINE

ted by a laser R-Awn in Fig. 2. V 15 16 17 18 19 20 21n n Mthe A and Xrational andg about 2/3 of A-X

anbeusedto v" 0 1 2 3

ines that areition. ThoseA state or the cr z3ction of thetn rate. Thishe rotational Ifor R-A and " k'

icy of 1 cm~l. 5800 5900 1 6000 6100 A[ibration andrecording of

.etermine theIn addition, 27.7cm

range 43-45.X states, we46)-(21, 45)-position of alling the above Fig. 2. Fluorescence spectrum produced by two-photon excitation

at 16 704.169 cm-'. Series produced for Rydberg-state to A-statehoton transi- (R-A) and A-state to X-state (A-X) transitions are indicated. Ex-th J = 46 and panded section depicts splittings between AJ = 1 transitions:2.-45) the 27.7 cm-' for A-X and 17.9 cm-' for R-A.

Yanet al.

i _ w _

Vol. 4, No. 11/November 1987/J. Opt. Soc. Am. B 1831

R-AV = 22

n

1 2 3

23n

4 5

ILASER LINE

I

6 7 8

5800 5900 6000

5800 0 5900A

A

(a)

6100 6200

n

1

5800 5900

(b) (c)Fig.3. (a) Fluorescence spectrum produced by two-photon excitation at 16 672.645 cm-1 . (b) Portion of spectrum in (a). Two counterpropa-gating beams were used. (c) Same portion of spectrum as in (b) with single-beam excitation. The vertical scale here is one half that of (b).

transitions since the two-photon absorption that resultedfrom the presence of both beams was rejected. According tothe relation between the rotational spacing of R-A and A-Xtransition lines, we presumed that one of those series thatappear in Fig. 3(a) was related to the two-photon transitionat 16 672.645 cm-'. The procedure detailed in the previousparagraphs was then used to identify the two-photon transi-tion as (v, 15)-(26, 14)-(5, 15). In addition to checking thewavelengths of lines in the relevant series with calculatedvalues, comparison of the line strengths in the series with

their calculated Franck-Condon factors enabled the correctassignment to be established securely.

A few of the two-photon transitions studied have verysmall offsets4; the intensity-dependence technique de-scribed above cannot be utilized to discriminate R-A fromA-X transitions in these spectra. Figure 4 depicts excita-tion spectra of one such two-photon transition at an excita-tion of 16 737.980 cm-'. The lower and upper curves in Fig.4 were taken with one and two beams, respectively. There isno Lorentzian peak that is due to saturation.8 The intensity

A- XV =

-

Yan et al.

1832 J. Opt. Soc. Am. B/Vol. 4, No. 11/November 1987

ONE BEAM

-2 0DETUNING (GHz)

Fig. 4. Comparison of excitation spectra for two counterpropagat-ing beams and single beam. The excitation is at 16 737.980 cm- 1 .

LASER LINE

A-X

ON RESONANCER-A

V, - 16 1 7

-1.4 GHz -

+ 2.4 GHz

+2.6 GHz A A

ATOMIC SODIUM0 LINE -

1 2

I I I5900 6000 6100 A

Fig. 5. Fluorescence spectrum for two-photon excitation at16 737.980 cm-'. Four series are indicated; the frequency offsetsbetween the two-photon excitation and the relevant one-photonexcitation of each series are also given. There is a factor-of-10reduction in the vertical scale beyond X = 5970 A.

ter was set to transmit one of the lines in a series, and the dyelaser was scanned. The difference between the frequency ofthe two-photon transition and that of a one-photon excita-tion could be found. Figure 5 shows four series and thefrequency deviations of their excitation transitions from thetwo-photon transition. Only one (J" around 14-16, J' = J'- 1, and v" - 1, and v" 0) is on resonance with the two-photon transition. The transition (19-14)-(0, 15) was theninferred to be the first step of the two-photon transition at16 737.980 cm-1, using the values of v" and J" shown in thespectrum. We next attempted to identify the one-photonfluorescence lines that originated in the upper level of thetwo-photon transition by comparing rotational spacings ofthe series. Finally, the transition was assigned as (v, 15)-(19, 14)-(v, 15).

After we identified the J values of the upper levels for 29two-photon transitions, we tried to establish their v valuesand relevant electronic states. All levels measured are ar-rayed between 33 700 and 35 000 cm-'; these belong to the12; state (or states). Taylor et al. have found more than sixgerade Rydberg V states in Na2 (Ref. 6) by using polariza-tion-labeling spectroscopy. Among them, only three elec-tronic states [, (3s + 5s), II+ (3s + 3d), and 'Z+ (3s +4d)] have levels with energies in the proper region to beupper states of the two-photon transitions that we studied(see Fig. 6). For each state, the molecular (Dunham) coeffi-cients were used to assign a tentative v value to each ob-served transition by matching energies of calculated andobserved transitions. Plots (one for each electronic state) ofthe deviation between the observed and calculated energiesversus v were then made (Figs. 7 and 8). Figure 7 indicates asystematic deviation of the actual data from the values cal-culated for the 12g (3s + 5s) state, implying that the major-ity of lines originate in that state. Figure 8 shows randomdeviations of the data from theoretical values for 1Z± (3s +4d). The plot for 1Z7 (3s + 3d) is similarly random. Basedon the data given in Fig. 7, 18 of the observed transitions

50 .000rof the two-photon transition with both beams was almosttwice that with one beam. It should be noted that this two-photon transition is much weaker than the previous twodiscussed. Its intensity is only a few percent of the transi-tion at 16 704.169 cm 1 . The corresponding one-photonfluorescence spectrum is shown in Fig. 5. For such a spec-trum, it is difficult to isolate the lines that come from theupper level of the two-photon transition. However, the pe-culiar line shape of the two-photon transition indicates thatthe enhancing level should be located almost midway be-tween the upper and lower levels of the two-photon transi-tion. Several series of fluorescence lines are observed in thespectrum. This implies that several one-photon transitionsexist with frequencies within the Doppler linewidth of thetwo-photon transition. Among these, that transition whosefrequency lies within a few tens of megahertz of the two-photon transition frequency is the one associated with thetwo-photon transition. For this type of two-photon transi-ton, we first determined the A-X transition relevant to thetwo-photon transition. The related R-A fluorescence serieswas then found by using the relation between rotationalspacings for the R-A and A-X transitions. The spectrome-

E.-

I-

x0

002

0o

(.2a

zLi

45 000

40000

35,000

30,00

3s+ns 3s+nd-iT 1 n=7

n-5

n-5 6 { IONIZATION

////////t n= 3 * ////////LIMIT

n= 7

25 000

Fig. 6. Energy regions spanned by various states of Na2. Thehatched region depicts the regime studied in these experiments.

Yan et al.

Vol. 4, No. 11/November 1987/J. Opt. Soc. Am. B 1833

E 120-

080

Z 40-

> o ;*i 5 20 25 30

Fig. 7. Plot of energy deviations versus (tentative) v values for the12+ (3s + 5s) state.

607- 40

20

<, O 8 1 ,o 1 2 I | 4 , | 6 , 18 0 . 8 0 2 1 1 1

0 -20

0 40 -

2 -60 L

Fig. 8. Plot of energy deviations versus (tentative) v values for the1+ (3s + 4d) state.

were assigned to 12g+ (3s + 5s) (see Table 1). Also, newvalues for the Dunham coefficients of the (3s + 5s) state werecalculated by a least-squares-fit technique (see Table 2).These closely match the data found in polarization-labelingexperiments. 6

An attempt to fit the remaining 11 lines to the (3s + 5s)

potential well leads to large deviations (>2 cm-'). Fittingthe same lines to 'I+ (3s + 3d) or 12g (3s + 4d) yields large,random deviations. No line can be fitted to 12; (3s + 3d)with a deviation of less than several wave numbers. Notethat all levels populated by a two-photon excitation withfrequency of roughly 16 000 cm-1 are located close to the topof the 'I+ (3s + 3d) potential well with v values as high as 60.It is likely that for such high v values the Dunham expansionfor the energy levels is no longer accurate, owing to uncer-tainties in the values of the coefficients. Thus the assign-ment of the remaining lines to transitions between A 'Z, and'Z+ (3s + 3d), with a large difference in vibrational quantumnumbers, is not completely ruled out.

Among the eleven unfitted lines, there are three that canbe assigned to 12T (3s + 4d) with deviations of less than a fewwave numbers. For this assignment, the v values of theupper levels [2' (3s + 4d)] differ from those of the interme-diate level (A 1;U) by approximately 10. We have calculat-ed the Franck-Condon factors in transitions between the Astate and 1g (3s + 4d) state and have found that theFranck-Condon factors for Av of approximately 10 are lessthan that for Av = 0, 1 by approximately 4 orders ofmagnitude. Furthermore, if the lines are inferred to belongto 12g (3s + 4d), then a strong fluorescence, associated withtransitions from the Rydberg state to the A state with Av =+1, should exist at roughly 100 nm to the blue side of the

excitation frequencies. This strong fluorescence was notobserved. Finally, we note that the polarization-labelingexperiments of Taylor et al. studied states of the (3s + 4d)well with vibrational quantum numbers as high as 20 (10 < J< 39). Our assignments involve vibrational states with v ofapproximately 12-17 in the (3s + 4d) state. Thus a good fit

Table 1. Assigned Two-Photon Transitions

1+ (3s + 5s)-A l+u-Xlg Vobs vcalc (A-X; Vcalc - obs Elower Eupper Ecalc Ecalc - Eupper(v, J)-(v', J')-(v", J") (cm-') cm'1) (cm'1) (cm-,) (cm-') (cm-') (cm-')

(20, 49)-(23, 48)-(3, 47) 16 583.591 16 583.493 -0.098 887.013 34 054.195 34 054.178 -0.017(19, 50)-(22, 49)-(2, 50) 16 601.803 16 601.780 -0.023 778.337 33 981.943 33 982.060 0.117(18, 27)-(21, 28)-(2, 27) 16 611.073 16 611.141 0.068 508.198 33 730.344 33 729.908 -0.436(18, 20)-(21, 21)-(2, 22) 16 614.564 16 614.569 0.005 470.254 33 699.382 33 698.365 -1.017(17, 38)-(20, 37)-(1, 36) 16 635.586 16 635.618 0.032 440.325 33 711.497 33 711.108 -0.389(16, 48)-(19, 47)-(0, 46) 16 647.220 16 647.235 0.015 410.210 33 704.651 33 705.168 0.517(25, 15)-(26, 14)-(5, 15) 16 672.645 16 672.651 0.006 888.895 34 225.511 34 226.467 0.956(19, 46)-(21, 45)-(1, 44) 16 704.169 16 704.135 -0.034 538.480 33 946.818 33 945.960 -0.852(18, 48)-(20, 47)-(0, 48) 16 720.465 16 720.409 -0.056 439.029 33 879.959 33 879.739 -0.220(17, 15)-(19, 14)-(0, 15) 16 737.980 16 738.056 0.076 116.365 33 592.325 33 593.422 1.097(20, 19)-(21, 20)-(1, 19) 16 785.420 16 785.435 0.015 295.239 33 866.079 33 864.341 -1.738(24, 11)-(24, 12)-(3, 11) 16 789.656 16 789.737 0.082 567.965 34 147.275 34 147.339 0.064(32, 57)-(28, 56)-(4, 57) 16 844.667 16 844.857 0.190 1192.985 34 882.319 34 880.289 -2.030(21, 47)-(21, 48)-(0, 47) 16 845.549 16 845.608 0.059 424.473 34 115.571 34 115.369 -0.202(31, 24)-(27, 23)-(4, 22) 16 914.976 16 915.180 0.204 777.325 34 607.277 34 607.019 -0.258(34, 41)-(28, 40)-(4, 41) 16 933.615 16 933.756 0.141 958.980 34 826.210 34 827.669 1.459(25, 34)-(23, 35)-(1, 34) 16 946.066 16 946.225 0.159 418.751 34 310.847 34 310.408 -0.442(28, 63)-924,62)-(0, 63) 17 025.481 17 025.384 -0.097 692.060 34 743.022 34 743.724 0.722

Table 2. Dunham Coefficients for 12, (3s + 5s) (cm-')

Y00 Y01 Y,0 Y20 X 102 y30 X 102 Y40 X 105 Yll X 103 y2l X 105

This work 31 772.43 0.11363 109.412 3.3524 -2.5374 9.877 -1.2701 1.0942

Taylor's experimentsa 31 770.01 0.11138 111.764 -33.3030 -1.0100 - 0.8474 -

aRef 6.

Yan et al.

1834 J. Opt. Soc. Am. B/Vol. 4, No. 11/November 1987

Table 3. Unassigned Two-Photon Transitions

R'2g-A V+-X'2;g Vobs "calc Vobs - Vcalc Elower Eupper

(J)-(v', J')-(v", J" (cm') (cm') (cm-') (cm-') (cm-')

(83)-(25, 82)-(1, 83) 16 806.204 16 805.700 -0.504 1278.634 34 891.042(26)-(23, 27)-(2,26) 16 815.871 16 815.950 0.079 500.008 34 131.750(29)-(22, 28)-(1, 29) 16 850.653 16 850.674 0.021 370.048 34 071.354(23)-(24, 24)-(2, 23) 16 923.617 16 923.622 0.005 477.241 34 324.475(16)-(21, 17)-(0, 16) 16 948.120 16 948.115 -0.005 121.293 34 017.533(15)-(21, 16)-(0,15) 16 949.700 16 949.680 -0.020 116.365 34 015.765(61)-(25, 62)-(1, 61) 17 006.295 17 006.204 -0.091 808.899 34 821.489(63) (24, 64)-(0, 63) 17 049.480 17 049.358 -0.122 692.060 34 791.020(62)-(27, 61)-(2, 62) 17 053.178 17 053.232 0.054 943.613 35 049.969(45)-(23, 44)-(0, 43) 17 065.486 17 065.472 0.014 369.188 34 500.16(61)-(28, 62)-(1, 61) 17 103.224 17 103.179 -0.045 808.899 - 35 015.747

is to be expected; yet deviations for these lines are all greaterthan 2 cm-'.

The poor fit, small Franck-Condon factors, and lack ofstrong blueward fluorescence prevent assigning any lines to'Z+ (3s + 4d). The 11 unassigned lines are shown in Table3. In the regime reached by the two-photon transitions thatwe have studied, several singlet and triplet electronic statesoverlap.9 Perturbation of 1Z' (3s + 5s) resulting from thepresence of these states may be responsible for the largedeviation in fitting these lines. Further experiments areneeded to explore the higher vibrational levels of the 12; (3s+ 5s) state and refine the values of the molecular constantsfor this state.

In our study of two-photon transitions with the samefrequencies, the Franck-Condon effect plays an importantrole. Although the A state is located midway between the Xstate and the Rydberg states that have been reported, thedual requirement of existence of an enhancing level andfavorable Franck-Condon factors for the one-photon A-Xand R-A transitions is not easily met. Thus the likelihoodof strong two (equal-frequency) photon transitions betweenX and Rydberg states is reduced; this eliminates the possi-bility of obtaining more information about the Rydbergstate through two-photon absorption with equal frequen-cies. Cw polarization labeling and two-step spectroscopycombined with the fluorescence spectroscopy method thatwe have used are useful tools to solve this problem.

SUMMARY

In summary, fluorescence spectroscopy following two-pho-ton excitation of Rydberg levels in molecular sodium hasbeen performed. Two kinds of fluorescence spectra (linesoriginating from the upper level and those originating fromthe intermediate level) were obtained for 29 two-photontransitions; the two types of spectra were distinguished byseveral methods. The values of rotational quantum num-bers, J, the energy positions, and the electronic characteris-tics were determined unambiguously for the upper levels.The majority of upper levels were assigned to the II+ (3s +5s) state, with deviations from expected values of about 1cm-'. Eleven lines cannot be assigned with any confidence.Possibly these lines are associated with high-lying levels (v

60) of the IT, (3s + 3d) state, which must be in this energyrange. For such high v values, the tabulated Dunham coeffi-cients for this state are quite inadequate. Indeed, Jeung10

has calculated that the potential curve for this state isstrongly anharmonic. An alternative is to assign the lines tol2; (3s + 5s), under the assumption that the presence ofother electronic states perturbs some of the 12g (3s + 5s)levels. Further experiments are needed to investigate eitherof these possibilities.

ACKNOWLEDGMENT

This research supported in part by the National ScienceFoundation under grant NSF PHY-86-04441 and in part bythe U.S. Office of Naval Research under contract ONRN00014-C-78-0403.

* Present address, Department of Physics, DartmouthUniversity, Hanover, New Hampshire 03755.

REFERENCES AND NOTES

1. G. P. Morgan, H.-R. Xia, and A. L. Schawlow, J. Opt. Soc. Am.72, 315-320 (1982).

2. K. C. Harvey, thesis, G. L. Rep. No. 2442 (Stanford University,Stanford, Calif., 1975) (unpublished).

3. J. P. Woerdman, Chem. Phys. Lett. 43, 279-282 (1976).4. H.-R. Xia, G.-Y. Yan, and A. L. Schawlow, Opt. Commun. 39,

153-159 (1981).5. R. G. Bray and R. M. Hochstrasser, Mol. Phys. 31, 1199-1211

(1976).6. A. J. Taylor, K. M. Jones, and A. L. Schawlow, J. Opt. Soc. Am.

73, 994-998 (1983).7. Throughout this paper, we have abandoned the conventional

notation for molecular transitions, e.g., v'-v" PQR(J"), believ-ing that it lacks clarity in the case of two-photon transitions.Instead we have used (v, J)-(v' J')-(v", J'), where, as usual,unprimed v and J represent the upper level, singly primedvariables are for the intermediate level, and doubly primedvariables are for the lower level in the transition. To clarify, thetransition that we denote (v 46)-(21, 45)-(1, 44) would moreconventionally be described as R-A v-21 R(45) A- X21-I R(44).

8. G.-Y. Yan and H.-R. Xia, Sci. Sin. A, 18, 504-515 (1985).9. L. Li, S. F. Rice, and R. W. Field, J. Mol. Spectrosc. 105,344-350

(1984).10. G.-H. Jeung, Phys. Rev. A. 35, 26-35 (1987).

Yan et al.