Determination of the Rydberg Constant by Doppler-Free Two ...

F. BIRABEN et al.: DETERMINATION OF THE RYDBERG CONSTANT ETC. 929where L is the cavity length, N an integer number, @ the Fresnel phase shift1@ = - cos-'(1 - L/R)mxand Y the reflective phase shift for light of frequency v.The contribution of @ is measured by comparison of the modes TEMOO and TEMOlfrequencies for each radiation in the cavity. The contribution of Y is eliminated in threesuccessive steps, where the etalon spacings are alternately 10 em, 50 cm and 10 em. A driftof the silver coating has been observed during the time of our measurement (approximately4 months). This aging effect results in an increase of about 0.8 A of the apparent etalonlength at 633 nm with respect to the same length at 778 nm. This effect can easily be takeninto account in our results. The numbers N are deduced from etalon transmission recordingsfor various wave-lengths.For each transition, the beat frequency between the two He-Ne lasers has to beextrapolated at null light power to eliminate systematic effects due to two-photon lightshift. Such an extrapolation is shown in fig. 3, where the transition involved is the2Sl12(F=1)-8D512 transition in hydrogen. Each dot is obtained 2s the average of tenmeasurements of the beat frequency at the centre of the atomic resonance. With a 40 Wlight power, the light shift is about 330 kHz. Taking into account the imprecision of the lightpower scale, this experimental value is in good agreement with the theoretical one(300 kHz). Figure 3 clearly shows that this extrapolation quite eliminates the two-photonlight shift.tf 0itwo-photon lineposit ion01TT-it-'100 kHz(atomic frequency)0 10 20 30 40 50power(W1Fig. 3. - Extrapolation of the two-photon line position vs. the light power. The dashed line denotes alinear fit and the extrapolation result is indicated on the y-axis.4. Results.The previous experimental method has been applied to three different transitions:2SIl2+ 8D5k in H and D and 2SlI2+ 10D5,2 in H. The respective wave-lengths in air are777.8 nm, 777.6 nm and 759.6 nm. Table I gives the experimental frequency measurementsafter extrapolation to null light power and the corrections due to the second-order Dopplereffect and to the hyperfine structure [18,19]. Using the theoretical work of Erickson [20],and taking into account the recently measured value of m,lme[211, we can deduce the

930 EUROPHYSICS LETTERSTABLE I. - Experimental results.Hydrogen 8Dslz Hydrogen 10DsE DeuteriumExperimental result (MHz) 385324758.54(20) 394 572 421.06(19) 385429 619.56(23)x2 770 649 517.08(40) 789 144842.12(38) 770 859239.12(46)Second-order Doppler effect (MHz) + 0.044 + 0.045 + 0.022hyperfine splitting (MHz) 44.389 44.389 13.64125'1~nDan hyperfine splitting (MHz) - 0.028 - 0.014 - 0.0082Sl,z-nDm energy splitting (MHz) 770649561.49(40) 789 144 sS6.54(38) 770859252.78(46)R, - 109737 (em-') 0.315682(57) 0.315 711(53) 0.315682(65)Final resultR, = 109737.315692(60) cm-'Rydberg constant. The three values of R, are in good agreement. Our final result isR, = 109737.315692(60) cm-'.The various experimental errors are evaluated in table I1 in the typical example of the2S1,2-8D5n transition in H. In table I11 our result is compared with other recentmeasurements of the Rydberg constant. Our measurement improves the precision on R, bya factor of about 2 and gives a value a little larger than the preceding ones: a slightdisagreement with the most precise one[3] can be noticed.Moreover, our experiment also provides a measurement of the isotopic shift between Hand D in the 8D5,2 level. We obtaindexp = 209691.29(7) MHz .TABLE 11. - Errm Budget for the Rydberg constant measurement.ComponentExtrapolatioon of the two-photon line positionReflection phase shift measurement and aging of coatingsFresnel phase shift measurementZ2 stabilized He-Ne laserStark effectr.m.s. sumPrecision (parts in lo1')2.62.53.02.01.05.2TABLE 111. - Comparison with recent measurements.(Rm- 109737) cm-'GOLDSMITH et al. [11PETLEY et al. [ZIAMIN et d. [3]HILDUM et al. 151BARR et al. [6]Present result0.31500(32)0.31521(64)0.31544(11)0.314 92(21)0.31500(110)0.315 69(6)These values are all corrected using the new definition of c and the experimental ratio m,/mp [Zl].

F. BIRABEN et al.: DETERMINATION OF THE RYDBERG CONSTANT ETC. 931The above precision is much smaller than the precisions on each 2S1/2-8D5/2 measurement,since various systematic errors cancel. This result is in good agreement with the theoreticalvalue of Erickson [20] when modified to reflect the most precise determination of the protonto-electronmass ratio %/me [21]:Atheor = 209 691.32 MHz .If this measurement is considered as a way to measure mplme, the Rydberg constantvalue allows us to deduce-_ mp - 1836.15272(64) ,mewhich agrees with the last result of Van Dyck et al. [211-_mPme- 1836.152 701(37) .We notice that, up to now, this method cannot give a precision better than on themplm, ratio because of uncertainties due to nuclear-size effects.In conclusion, the measurements reported here have allowed us to improve theexperimental precision on the Rydberg constant, because of the very small line width of the2S112-nD5i2 lines observed in hydrogen. Elimination of residual stray electric field and othercauses of broadening will allow us to obtain optical relative line widths smaller than lo-'. Ina near future, precision of the order of lo-'' on the Rydberg constant will thus be achieved.***The authors are indebted to Prof. B. CAGNAC for stimulating discussions. We thank P.JUNCAR, R. GOEBEL and Y. MILLERIOUX from the

932 EUROPHYSICS LETTERS[11] BIRABEN F. and LA3ASTIE f., @t. COWZ~UYZ., 41 (1982) 49.[12] GIACOBINO E., BIRABEN F., GRYNBERG G. and CAGNAC B., J. Phys. (Pa?%), 38 (1977) 623.[131 HANSCH T. W. and COUILLAUD B., Opt. Commun., 35 (1980) 441.[141 BORDE C., C. R. Acd. Sci. B, 282 (1976) 341.[15] BIRABEN F., BASSINI M. and CAGNAC B., J. Phys. (Paris), 40 (1979) 445.[161 JENNINGS D. A., POLLOCK C. R., PETERSEN F. R., DRULLINGER R. E., EVENSON K. M., WELLSJ. S., HALL J. L. and LAYER H. P., Opt. Lett., 8 (1983) 136.[17] LAYER H. P., DESLATTES R. D. and SCHWEITZER W. G. jr., AppI. Opt., 15 (1976) 734.[18] HEBERLE J. W., REICH H. A. and KUSCH P., Phys. Rev., 101 (1952) 612.[19] REICH H. A., HEBERLE J. W. and KUSCH P., Phys. Rev., 104 (1956) 1586.[20] ERICKSON G. W., J. Phys. Chem. Ref. Data, 6 (1977) 831.[21] VAN DICK jr. R. S., MOORE F. L., FARNHAM D. L. and SCHWINBERG P. B., Bull. Am. Phys.Soc., 31 (1986) 244.