(1)

(2)

--. - /' -1;

B

manner with a fractional frequency uncer­tainty approaching I part in 1019.

The use ofa mode-locked laser for opticalfrequency metrology was first demonstratedwith picosecond lasers (12, 13), with thecomb of frequencies emitted from the laserserving as an "optical frequency ruler" or"synthesizer." The spacing of the comb teethis given by the repetition rate !rep at whichpulses are emitted; the overall frequency off­set of the comb teeth is given by fceo' Recentexperiments (14-19) have demonstrated thatthe frequency comb associated with a femto­second mode-locked laser can be readily con­trolled and is more versatile and precise thanexisting technologies (20).

We search for potential limitations inthe femtosecond laser synthesizer approachby rigorously comparing four such synthe­sizers that use two different types of con­struction (20). The basic scheme of ourmeasurements is to compare pairs of fem­tosecond laser synthesizers (labeled by in­dices I and 2) and verify with optical het­erodyne techniques that the output modeshave their expected frequency positions rel­ative to a continuous-wave reference laserwith frequency le = 456 THz (Fig. 1). Themode spacing (i.e., repetition rate) of thetwo synthesizers can be written as

h ep1 = (fL- loeo1 - fb1)/N1

whereh,1 b2 are the beat frequencies betweenhand modes NI 2 of the respective femtosecond

Femtosecond'------i~ Laser f-J!....~.l'--.y..l-.l-~

Synthesizer 2

Femtosecondr----~ Laser

Synthesizer 1

measurements indicate that et could have beensmaller than its present value by a factor of10-5in the early universe some 10 billion yearsago (11). The next generation of optical fre­quency standards should enable searches at thelevel of 10-18per year. However, these search­es require a low-noise, broadband (> I00 THz)frequency synthesizer that is able to phase­coherently generate an arbitrary optical (ormicrowave) frequency at its output given anoptical frequency reference input. Within thiscontext, we demonstrate that an optical fre­quency synthesizer using a mode-lockedfemtosecond laser can operate in such a

JLFig. 1. Two approaches to .hI1compare femtosecond laser ~ I J;synthesizers that are refer- A j ----tool ~Ix

enced to the same optical iJLL'frequency t; (AJ The posi-tion of the (N + 1)th modefrom one synthesizer is ---U-LN ,:1 N+lcompared against the corre-sponding mode from thesecond synthesizer. (B) The I i<- I

mode spacings of the two K: ~synthesizers are set to be I.r...equal thereby enabling the b2:<

comparison ofgroups of modes (~1000) from eachsynthesizer in anarbitrary region ofthe opticalspectrum.

Long-Sheng Ma,'·2*t Zhiyi Bi!* Albrecht Bartels,'*Lennart Robertsson," Massimo Zucco.' Robert S. Windeler.4

Guido Wilpers,' Chris Oates,' Leo Hollberg,'Scott A. Diddams3*t

Optical Frequency Synthesis andComparison with Uncertainty at

the 10-1 9 Level

A femtosecond Laser-based opticaL frequency synthesizer is referenced to anopticaL standard, and we use it to demonstrate the generation and controL ofthe frequency of eLectromagnetic fieLds over 100 terahertz of bandwidth withfractionaL uncertainties approaching 1 part in 101 9

. The reproducibility of thisperformance is verified by comparison of different types of femtosecond Laser­based frequency synthesizers from three Laboratories.

"Bureau International des Poids et Mesures (BIPM),Pavillon de Breteuil, 92312 Sevres, France. "PhysicsDepartment, East China Normal University (ECNU),Shanghai 200062, China. "Nationel Institute of Stan­dards and Technology (NIST), 325 Broadway, Boulder,CO 80305, USA. 40FS Laboratories, 700 MountainAvenue, Murray Hill, NJ 07974, USA.

*These authors contributed equally to this work.'[To whom correspondence should be addressed. E­mail: lsma@phy.ecnu.edu.cn (L.S.M.), sdiddams@boulder.nist.gov (SAD.)

At present, microwave frequency standardsbased on the ground-state hyperfine transitionin cesium atoms have fractional frequency un­certainties at the level of I part in 1015 (1).However, new optical frequency standards (1)and clocks (2, 3) using ultrastable lasers (4, 5)locked to optical transitions in laser-cooled ionsand atoms have the potential to be orders ofmagnitude more stable and accurate. For exam­ple, the stability of neutral calcium and mercuryion optical frequency standards already exceedsthat of the best microwave standards by one ortwo orders of magnitude (2), and the fractionalfrequency uncertainty of optical standardsbased on single ions is anticipated to approach10-18 (6). However, such an extremely sta­ble and accurate standard is of little value ifits frequency cannot be readily distributedto users and compared to those of otherstandards based on various atomic speciesoperating at different frequencies.

One of the most compelling motivations forthe development of advanced atomic frequencystandards is that their intercomparison wouldallow one to search for possible time variationsof fundamental constants (7, 8). For example,laboratory-based tests of the stability of thefine-structure constant et show no time depen­dence at the present measurement limit of~ 10-15 per year (9, 10), whereas astronomical

1843

la'

,

L_~

I~--- -~---t---

-L---- ~ _! i

i I

l..- --t-. ... ... i (11) I

t-~ -1~---d----III ,-

]_ ······-r·· -

10'

B

~ 10.13

j 10.14

10.15

i 10.16

I 10.17

.li 10.18«

10-19

--t;=r'iTmrt=r'ITffii1=i'IT;;;;;f=i'TiiiiiFni'iffiP

10' 10' la' 10'Averagingtime (s)

frequency synthesizers BIPM-C2 and NIST-BB1 at 456 THz, (ii) the opticaltransfer oscillatorof (18) comparing a continuous-wave laser and its secondharmonic,and (iii) the high-quality microwave synthesizerof (24). Curves ivand v are the upper limits of instabilities of femtosecond laser-basedsynthesizers connecting microwave and optical domains as reported in (25)and (16), respectively.

-Measured differencesfrom expected values- Weightedmean:-1.1x 10-20:t 1.4X10.19

- I . t rti"_1 " -"W .rt

- Aug. 15 "",,.22 Aug.29 Aug.30 Sept. 30 0cI.16ECNU·C1 BIPM-C2 BIPM-C2 BIPM-C2 NIST·BB2 NIST·BB2458Ttu 458_ 458_ 375 THz "'Ttu ....-1.01 1.01 L,7000 WOOO WOO, L.-13000T=4333. T=7412s T...... To6010s T...,38OQs T.14568 I

Gy=3.4x10·17 aye2.4x10·17ay=2.3X10·17 CJy= 1.4x10·111 ay= 2.9x10·16 cry= 3.4x10·111

-

A

~- 2";-

0-.!:!.s" 0e";E;Cl1ii" -2

~11.

-4Fig.2. (A) Summaryof comparisonsof three femtosecond lasersynthesizers(BIPM-CZ, ECNU-Cl, andNIST-BB2) with a fourth synthesizer (NIST-BB 1) onsix different days in 2003. Alsogiven are the number of comb lines (L) usedin the comparison,the total averagingtime (T) inseconds, and the 1-s Allandeviation "x for each comparison. (B) Relative fractional frequency instabil­ity as given bythe Allan deviation for (i) our multiline comparisonof optical

from each of the two combs with k1 = k2 =k to generate the frequency difference signal.In this case, the expected beat frequency be­tween the two combs can be written as

fi(k) - h(k) =L :«>f,w' (7)

"When the relative phase between the opticalpulsetrains from the two synthesizersis set to zero (i.e.,the pulses from each synthesizer reach the detec­tor at the same time), all the modes that areappropriatelyphased generate a strong beat signalwith signal-to-noise ratio as high as 60 dB withina 300-kHz bandwidth (21).

Thee ferrrtosecond laser synthesizers(BIPM-C2, ECNU-Cl, and NlST-BB2) werecompared with a fourth synthesizer (NIST­BB 1) on 6 days over a period of severalmonths. A summary of these frequency com­parisons (Fig. 2A) is plotted as the differencebetween the measured frequencies and the val­ues predicted by Eqs. 5 to 7. On each day, theheterodyne beat of interest was recorded with afrequency counter for total integration times ofseveral thousand seconds. For each date in Fig.2A, the data points are the results obtained withdifferent gate times and experimental values ofL; and A. The statistical uncertainties shownon the points are used as weights in computingthe weighted mean (22). These uncertainties aredetermined from the calculation of the Allandeviation (23). Such a weighting accounts forthe presence of flicker-like noise in the data atlonger averaging times and provides a realistic(larger) estimate of the uncertainty. Using stan­dard statistical methods (22), we combined allthe data from the 6 days to calculate the weight­ed mean. The result is equal to -1.1 X 10-20,with an uncertainty of 1.4 X 10-19, corre­sponding to a 95% confidence level determinedfrom a X2 analysis (22). Both the weighted meanand the uncertainty are normalized by the com­pared optical frequency to give the fractionalval­ues presented. It can thus be concludedthat withinthis confidence level, no systematic effects aredetectable at a level of 1.4 X 10-19 .

450

quencies of single lines adjacent to mode Nfrom each of the two combs (i.e., k1 = Js = 1).This method does not require time synchroni­zation between the optical pulse trains from thetwo synthesizers. When fceol' l ce02' l repI' andl rep2 are controlled as described in Eqs. 1 to 4,the expected difference frequency between thek = 1 lines of two combs can be written as

In a second case (Fig. lB), we require/repl =

l rep2' This allows the use of groups of lines

fi(l) - h(l) = (fb2 - fb,) +

[(f,w2 + fb2) - (Iow' + fb,)]/N (5)

This difference can be determined experi­mentally by subtracting the two measuredbeats Alx andA2x:

300

.. 1 s Uncertainty • Averaged Uncertainty

('1" ..." ...

," >+1

(ii)

" ..(1111 ,

..--.....JP

"(Ivl ~

I I I

NI

"5 10.1

B-C>

0iD 10.2N

.U;Q)s:C>-(j) 10.3'0z-c.0;t loAQ)oc::J

350 400Optical Frequency (THz)

Fig. 3. Uncertaintyof the optical frequencyoutput of a femtosecond laserfrequencysynthesizer.Shownare uncertainties at averagingtimes of 1s (...) and several hours (.). The frequency error bars on thesepoints indicatethe spectral width over which the measurements were made. The various lines indicatethe uncertainty limits imposed by (i) the Doppler-induced fluctuations at 1 s, (ii) the 1-s uncertaintyintroduced by the phase locks controllingthe femtosecond synthesizer, (iii) the 1-s fundamental shotnoise limitation for the control of the femtosecond synthesizer, and (iv) the estimated limit due todifferential Doppler shifts occurring in the control and measurement systems on a 1-hour time scale.

(4)

synthesizer. The frequencies of the outputmodes are then given by

wherek1,2 = 0, -r l , ±2, .... In our experiments,we require NI = N2 = N, such that 4frep =

frep, - fre", and /:;f= J;(k,) - fzCJ0 are indepen­dent of the frequencyh' Therefore, using Eqs. 1to 4, 8/rep and 4f can be determined from/ceoI'l ce02' AI' A2' kI' Js, and N This enables high­precision tests of the spectral purity and intrinsicnoise of the two combs themselves.

In the case of Fig. lA, with/repl ~ lrep2butnot necessarily equal, we can compare the fre-

(3)

h(k2) =A -fb2 + k2 X (A -L:« -fb2)1N2

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Figure2B (curve i) shows an example of thecomputed Allan deviation for the frequencycomparison (method of Fig. lB) betweenBIPM-C2 andNIST-BBl at456THz. The verylow Allan deviation of ~2.3 X 10-17 at 1 s isachieved when the beams from the two femto­second lasers are made to be mostly collinearwith the 456-THz laser that controls each syn­thesizer. This implies that the path length fluc­tuations from the two synthesizers to the het­erodyne photodetector are common mode at anapproximate level of <10 nm in 1 s of averag­ing. This low instability enables us to reachstatistical uncertainties as low as a few parts in1019 with less than 1000 s of averaging (Fig.2B). Also shown in Fig. 2B are Allan devia­tions from other optical and microwave fre­quency synthesizers.

Figure 3 shows the short-term Cl s) andaveraged uncertainties obtained from our mea­surements, along with measured and projecteduncertainties associated with the laser controlsystems, Doppler shifts, and fimdamental shotnoise. The short-term (1 s) uncertainty arisesfrom differential mechanical vibrations andvariations in air pressure and temperature. Theestimated level of these fluctuations (Fig. 3, linei) is in agreement with the measurements acrossthe optical spectrum, except near the frequencyA = 456 THz where we could arrange theoptical paths collinearly for maximumcommon-mode suppression. The limit set bythe performance of the laser control (Fig. 3, lineii) is only a factor of 1.5 below the l-s uncer­tainty near 456 THz The estimated shot noise­limited uncertainty at l-s averaging for thecontrol of the femtosecond laser synthesizerrelative toI L is shown in Fig. 3, line in.

Instabilities on longer time scales or fre­quency offsets that result in systematic errorsare of greater concern. In our comparisons, thevarious synthesizers were separated by ~2 mon a steel table. With temperature variations of~0.1QC per hour, the thermal expansion of thesteel results in a fractional Doppler shift on theorder ofa few parts in 1018

. We have attemptedto cancel this effect by arranging the opticalpaths for our experiments in a symmetric fash­ion with as much common-mode rejection aspossible. In addition, the relatively long dataacquisition times of several hours provide someimmunity by averaging over temperature fluc­tuations on the 100- to 1000-s time scale. Inprinciple, the control system compensates forall Doppler shifts inside the control loop path atfrequencyI L = 456 THz; however, because ofdispersion (or physically different paths) this isnot the case for the emitted frequencies far fromIL' For example, if the nonlinear optical fiber inBIPM-C2 and ECNU-Cl expands at the saanerate given above, the resulting Doppler shift forthe frequencies near 333 THz is ~200 J.1Hz,which corresponds to a fractional shift of 5 X10-19 (Fig. 3, line iv), This indicates that longeraveraging times (::::::100,000 s) or direct mea-

surement and compensation of all Dopplershifts would be required to reduce the uncer­tainty below the level of 10-19 .

Considering the very different designs ofthese synthesizers (broadband operation versusnonlinear microstructure fiber), it is notable thatour data do not point to the existence of anyfimdamentallimitations to the uncertainty. Ourresults appear to be limited mainly by noise ofa technical nature (thermal and mechanicalfluctuations) and total integration time. The re­producibility demonstrated in our experimentsfmnly establishesthe femtosecond laser synthe­sizer as a reliable tool for optical frequencycomparisons with uncertainties approaching10-19, and demonstrates its value for precisionmeasurements in experimental physics.

References and Notes1. P. Gill. Ed.. Proceedings of the 6th Symposium on

Frequency Standards and Metmlogy (World soenuttc.Singapore. 2002).

2. S.A Diddams et al .. Science 293. 825 (2001).3. J Ye,L.-S. Ne, J Hall. Pfrys. REV. Lett. 87. 270001 (2001).4. B. C. Young. F. C. Cruz.W. M. ltano. J. c. Bergqutst.

Phys. Rev. Lett. 82. 3799 (1999).5. C. W. Oates. E. A Curtis. L. Hollberg. Opt. Lett. 25.

1003 (2000).6. H. G. pelmet. IEEE Trans. InstJUm. Meas. 31. 83 (1982).7. S. G. Karshenboim. Can. }. Phys. 78. 639 (2CXXl).8. J.D. Presrage. R.J.Tjoelker. L. Maleki, Phys. Rev. Lett.

74.3511 (1995).9. H. Marion et al .. Phys. Rev. Lett. 90. 150801 (2003) .

10. S. Blze et al .. Phys. Rev. Lett. 90. 150802 (2003).11. J.K. Webb et al .. Phys. Rev. Lett. 87.091301 (2001).12. J.N. Eckstein. AI. rerguson. T. W. Hansrh, Phys. Rev.

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13. Th. udem R. Holzwarth, T. W. Hansrh, Nature 416.233 (2002).

14. Th. udem. J. Reichert. R. Holzwarth, T. W. Hansrh,Phys. Rev. Lett. 82. 3568 (1999).

15. Th. udem. J. Reichert. R. Holzwarth, T. W. Hansrh,Opt. Lett. 24. 881 (1999).

16. R. Holzwarth et al .. Phys. Rev. Lett. 85. 2264 (2CXXl).17. S. A Diddams et al .. Opt. Lett. 27. 58 (2002).18. J. Stenger et al .. Phys. Rev. Lett. 88. 073601 (2002).19. M. Zimmermann et al .. Opt. Lett. 29. 310 (2004).20. See supporting data on Science Online.21. R. K. Shelton et al .. SCience 293. 1286 (2001).22. I. Lira. Evaluating the Measurement Uncertainty (In­

stitute of Physics. Bristol. UK. 2002).23. D. W. Allan. IEEE Trans. Ultrason. Ferroeiect. Freq.

Control 34. 647 (1987).24. A. Senccpta. D. Popovir;F.L.Walls, in Procooiings of the

1999 joint Meeting of the EuropeanFmquencyand TlI1JeForum (EFTF) and the IEEE International FmquencycomaSymposium (FCS). Besarxon France. 13 to 16 April 1999(IEEE. Ptscateway Nj, 1999). pp. 615-619.

25. A Bartels. S. A Dlddarns, T. M. Rarnond, L. Hollberg.Opt. Lett. 28. 663 (2002).

26. We thank 1. Remond, R. Fox. and J Bergquist fa" theirmntributims to this work, andJ.Hall. D.Wine/and.andJYe for their thoJghtful canments m this manuscript Thework at NISTwas funded in part by NASA The project atECNUwas funded in part by the Sdenceand TethnologyCornmissim of Shanghai Munidpality (01DJGK014.022261033). Shanghai Munidpal Educatim Cornrnission,National SCience Foundation of China (10274020). andMinistry of Education China(02106).

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