Carrier-Envelope Phase Control of Femtosecond Mode-Locked Lasers

R ESEARCH A RTICLES cavity. Active control of the relative carrier- envelope phase prepares a stable pulse-to- Carrier-Envelope Phase Control pulse phase relation, as presented below, and will dramatically impact extreme nonlinear of Femtosecond Mode-Locked optics. Although it may be natural to think about Lasers and Direct Optical the carrier-envelope phase in the time domain, it is also apparent in a high-resolution measurement of the frequency spectrum. The Frequency Synthesis output spectrum of a mode-locked laser con- David J. Jones,1* Scott A. Diddams,1* Jinendra K. Ranka,2 sists of a comb of optical frequencies separated by the repetition rate. However, the Andrew Stentz,2 Robert S. Windeler,2 1 1 comb frequencies are not necessarily integer John L. Hall, * Steven T. Cundiff *† multiples of the repetition rate; they may also have an offset (Fig. 1B). This offset is due to We stabilized the carrier-envelope phase of the pulses emitted by a femto- the difference between the group and phase second mode-locked laser by using the powerful tools of frequency-domain velocities. Control of the carrier-envelope laser stabilization. We conﬁrmed control of the pulse-to-pulse carrier-envelope phase is equivalent to control of the absolute phase using temporal cross correlation. This phase stabilization locks the ab- optical frequencies of the comb, and vice solute frequencies emitted by the laser, which we used to perform absolute versa. This means that the same control of the optical frequency measurements that were directly referenced to a stable carrier-envelope phase will also result in a microwave clock. revolutionary technique for optical frequency metrology that directly connects the micro- Progress in femtosecond pulse generation has velocities differ inside the laser cavity (Fig. wave cesium frequency standard to the opti- made it possible to generate optical pulses that 1A). To date, techniques of phase control of cal frequency domain with a single laser (14). are only a few cycles in duration (1–4). This femtosecond pulses have employed time do- We used a self-referencing technique to has resulted in rapidly growing interest in con- main methods (5). However, these techniques control the absolute frequencies of the optical trolling the phase of the underlying carrier wave have not used active feedback, and rapid comb generated by a mode-locked laser. with respect to the envelope (1, 5–7). The dephasing occurs because of pulse energy Through the relation between time and fre- absolute carrier phase is normally not important fluctuations and other perturbations inside the quency described below, this method also in optics; however, for such ultrashort pulses, it can have physical consequences (6, 8). Concur- rently, mode-locked lasers, which generate a train of ultrashort pulses, have become an important tool in precision optical frequency measurement (9–14). There is a close connection between these two apparently disparate topics. We exploited this connection to develop a frequency domain technique that stabilizes the carrier phase with respect to the pulse envelope. Using the same technique, we performed absolute optical frequency measurements using a single mode-locked laser with the only input being a stable microwave clock. Mode-locked lasers generate a repetitive train of ultrashort optical pulses by fixing the relative phases of all of the lasing longitudi- nal modes (15). Current mode-locking techniques are effective over such a large bandwidth that the resulting pulses can have a duration of 6 fs or shorter, i.e., approximately two optical cycles (2–4). With such ultrashort pulses, the relative phase between peak of the pulse envelope and the underlying electric-field carrier wave becomes relevant. In general, this phase is not constant from pulse to pulse because the group and phase

Fig. 1. Time-frequency correspondence and relation between ⌬␾ and ␦.(A) In the time domain, the 1JILA, University of Colorado and National Institute of relative phase between the carrier (solid) and the envelope (dotted) evolves from pulse to pulse by Standards and Technology, Boulder, CO 80309–0440, ⌬␾ ␾ϭ⌬␾ ␶ ϩ␾ ␾ the amount . Generally, the absolute phase is given by (t/ ) 0, where 0 is an USA. 2Bell Laboratories, Lucent Technologies, Murray unknown overall constant phase. (B) In the frequency domain, the elements of the frequency comb Hill, NJ 07733, USA. of a mode-locked pulse train are spaced by f . The entire comb (solid) is offset from integer rep␦ϭ⌬␾ ␲ ␦ *These authors contributed equally to this work. multiples (dotted) of frep by an offset frequency f rep/2 . Without active stabilization, is †To whom correspondence should be addressed. E- a dynamic quantity, which is sensitive to perturbation of the laser. Hence, ⌬␾ changes in a mail: [email protected] nondeterministic manner from pulse to pulse in an unstabilized laser.

www.sciencemag.org SCIENCE VOL 288 28 APRIL 2000 635 R ESEARCH A RTICLES serves to stabilize the carrier phase with re- the frequency domain offset ␦ enabled us to use frequency doubling a single-frequency laser spect to the envelope. This phase stabilization the powerful techniques developed for stabili- and then measuring the difference between was verified with temporal cross correlation. zation of single-frequency lasers. It is possible the fundamental and second harmonic by The utility of this method for absolute optical to lock the position of the frequency comb to a subdividing it. The subdivision may be done frequency metrology was demonstrated by known optical frequency, such as that of a by optical bisection (20, 21), comb genera- measuring the frequency of a stable single- single-frequency laser (12). However, employ- tion (22), or, most typically, a combination of frequency laser directly from a microwave ing this approach to determine and lock ␦ is the two (9, 10, 17, 23). The goal of the clock. problematic because the optical frequencies are subdivision is finally to obtain a frequency Time versus frequency. The connection Ͼ106 times the repetition rate. In addition, it interval small enough that it can be directly between the pulse-to-pulse carrier-envelope introduces the complication of a highly stabi- compared to the cesium frequency. A recent phase shift and the absolute frequency spec- lized single-frequency laser, not to mention the series of landmark experiments by Ha¨nsch trum can be understood by considering how uncertainty in the single-frequency continuous and co-workers demonstrated that mode- the spectrum is built up by a temporal train of wave (CW) laser itself. locked lasers are the preferred implementa- pulses (7, 9, 14). It is easily shown with A more elegant approach is to use a self- tion of optical comb generators (9–11). We Fourier transforms that a shift in time corre- referencing technique, which is based on recently performed absolute frequency mea- sponds to a phase shift that is linear with comparing the frequency of comb lines on the surements using only the comb generated by frequency; that is, the phase at angular fre- low-frequency side of the optical spectrum to a mode-locked laser (13). The laser and mi- quency ␻ is ␻t for a time shift of t. Within a those on the high-frequency side that have crostructure fiber were similar to those used narrow spectral bandwidth, successive pulses approximately twice the frequency. Let the here, although the technique is quite different interfere, but a signal will be observed only at frequency of comb line n, which is on the red in that the laser comb was not stabilized in ϭ frequencies where they add constructively, side of the spectrum, have a frequency fn position but rather only used to divide down ␲ ϩ␦ i.e., have a phase shift of 2n . For a pulse nfrep . The comb line corresponding to 2n, an optical frequency interval. train with time ␶ between pulses, these fre- which will be the blue side of the spectrum, The self-referencing technique described ϭ ␶ϭ ϭ ϩ␦ quencies are fn n/ nfrep, where n is an will have frequency f2n 2nfrep .We here is a dramatic step beyond these previous ϭ ␶ ␦ integer and frep 1/ is the repetition fre- obtain by frequency doubling fn and then techniques because it uses only a single ϭ␦ quency of the pulse train. Thus, we recover taking the difference 2fn – f2n . To imple- mode-locked laser and does not need any the fact that the frequency spectrum consists ment the technique in a simple way, we need stabilized single-frequency lasers. We think of a comb of frequencies spaced by the rep- an optical spectrum that spans a factor of 2 in that it will make precision absolute optical etition rate of the pulse train. If we include frequency, known as an optical octave. This frequency metrology into an easily accessible the pulse-to-pulse phase shift ⌬␾, then the is obtained by spectrally broadening the laser laboratory tool. phase difference between successive pulses at pulse in air-silica microstructure fiber (16). Laser and stabilization. The heart of the angular frequency ␻ is now ␻␶Ϫ⌬␾. Again, Full experimental detail is given below. experiment is a titanium-doped sapphire for constructive interference, this phase dif- Optical frequency metrology tech- (Ti:S) laser (shown in Fig. 2) that is pumped ference is set equal to 2n␲, which shows that niques. Optical frequencies are preferred for with a single-frequency, frequency-doubled ϭ ϩ␦ now the frequencies are fn nfrep , where the precision measurements that test funda- Nd:YVO4 laser operating at 532 nm. The Ti:S ␲␦ϭ⌬␾ 2 frep. Hence, a pulse-to-pulse phase mental physical theories (17, 18). In part, this laser generates a 90-MHz pulse train with pulse shift between the carrier and envelope corre- preference is associated with the very narrow widths as short as 10 fs using Kerr lens mode- sponds to an offset of the frequency comb fractional linewidths displayed by optical res- locking (2). The output pulse spectrum is typi- from simple integer multiples of the repeti- onances, allowing precise measurements to cally centered at 830 nm with a width of 70 nm. tion rate. be made in relatively short times. For similar For the generation of a 10-fs pulse, the normal In a mode-locked laser, the carrier slips reasons, optical resonances will probably be dispersion of the Ti:S crystal is compensated by through the envelope as the pulse circulates used in future atomic clocks. However, any incorporating a pair of fused silica prisms inside in the cavity. Because an output pulse is only absolute frequency measurement must be de- the cavity (24). After the second prism, the produced once per cavity round trip, what rived from the 9.193-GHz cesium hyperfine optical frequencies of the pulse are spatially matters is the accumulated carrier phase, with transition, which defines the second as one of resolved across the high-reflector mirror; this respect to the envelope, per round trip. If this the SI base units. The frequencies involved property will be used to stabilize the absolute is an integer multiple of 2␲, then there is no and the large ratio between the cesium tran- frequency of the laser. pulse-to-pulse phase shift for the emitted sition frequency and optical frequencies (a We previously noted that the relative car- pulses, and the frequencies are all integer factor of ϳ5 ϫ 104 times the transition fre- rier-envelope phase ⌬␾ in successive pulses multiples of the repetition rate, without an quency) represent serious obstacles. generated by mode-locked lasers is not con- offset. If the accumulated carrier phase is an Before this work, two techniques have stant because of a difference between the integer multiple of 2␲ plus a rational fraction been used to make absolute optical frequency group and phase velocities inside the cavity. of 2␲, then the phase will periodically shift, measurements. The first technique is a phase- As shown in Fig. 1, this is represented by the and the frequency comb will be offset by the coherent frequency chain of oscillators that frequency offset ␦ of the frequency comb ϭ repetition rate times the rational fraction. For spans from the cesium reference transition to from fnϭ0 0. With the pulse repetition rate example, if the pulse-to-pulse phase shift is optical frequencies (19). The complexity and frep, the relative phase is related to the ␲ ␲␦ϭ⌬␾ 2 /8, then every eighth pulse will have the difficulty of these chains requires substantial offset frequency by 2 frep. Thus, by ␦ ⌬␾ same phase, and the frequency offset will be investment and is typically undertaken only stabilizing both frep and , can be con- frep/8. In the most general case, the phase and at national research facilities. Furthermore, trolled. Toward this end, as shown in Fig. frequency offsets are arbitrary. Furthermore, because of their complexity, the chains may 2, the high-reflector mirror (behind the in a free-running laser (even one with its not run on a daily basis but rather are typi- prism) is mounted on a piezoelectric trans- repetition rate locked), the phase-group ve- cally used to calibrate intermediate standards ducer tube that allows both tilt and transla-

locity difference drifts with time. such as the HeNe/I2 stabilized laser and CO2 tion. By comparing a high harmonic of the Self-referencing technique. Choosing to lasers stabilized to OsO4 resonances. pulse repetition rate with the output of a control the carrier-envelope phase by locking A newer measurement technique involves high-stability radio frequency (RF) synthe-

636 28 APRIL 2000 VOL 288 SCIENCE www.sciencemag.org R ESEARCH A RTICLES sizer, a feedback loop can lock the repeti- the continuum (500 to 900 nm, containing tor to the beat to enhance the signal-to-noise

tion rate frep by translating the mirror. Be- f2n) is directed through one arm that contains ratio by substantially reducing the noise cause the pulse spectrum is spatially dis- an acousto-optic modulator (AOM). The bandwidth. From the tracking oscillator out- persed across the mirror, tilting of this mir- near-infrared portion of the continuum (900 put, we generate an error signal that is pro-

ror provides a linear phase change with to 1100 nm, containing fn) traverses the other grammable to be (m/16)frep, thus allowing us frequency (i.e., a group delay for the pulse), arm of the apparatus, passing through a to lock the relative carrier-envelope phase thereby controlling both the repetition rate 4-mm-thick ␤-barium-borate frequency-dou- from0to2␲ in 16 steps of ␲/8. and the offset frequency (11). The maxi- bling crystal. The crystal is angle-tuned to Temporal cross correlation. Verification mum required tilt angle is 10Ϫ4 rad, sub- efficiently double at 1040 nm. The beams of control of ⌬␾ in the time domain is ob- stantially less than the beam divergence, so from the two arms are then mode-matched tained by interferometric cross correlation be- cavity misalignment is negligible. and recombined. The combined beam is fil- tween two different, not necessarily adjacent, To stabilize the offset frequency of a sin- tered with a 10-nm bandwidth interference pulses in the pulse train (5). In fact, we gle mode-locked laser, without external in- filter centered at 520 nm and focused onto an performed a time-averaged cross correlation formation, it is useful to generate a full opti- avalanche photo diode (APD). Approximate- between pulses i and i ϩ 2 using the corre- cal octave. The typical spectral output gener- ly 5 ␮W are incident on the APD from the lator shown in Fig. 4. A multipass cell in one ated by the Ti:S laser used in these experi- arm containing the AOM, whereas the fre- arm of the correlator is used to generate the ments spans 70 nm or 30 THz, whereas the quency-doubling arm provides ϳ1 ␮W. The required 20-ns delay. For the purpose of min- ϳ Ϯ ␦Ϫ center frequency is 350 THz; that is, the resulting RF beats are equal to ( fAOM), imizing dispersion, the beam splitter is a ␮ spectrum spans much less than a full octave. where fAOM is the drive frequency of the 2- m-thick polymer pellicle with a thin gold Propagation through optical fiber is common- AOM and is generated to be 7/8frep. The RF coating. To obtain a well-formed interfero- ly used to broaden the spectrum of mode- beats are then fed into a tracking oscillator gram, we chose the mirror curvatures and locked lasers through the nonlinear process of that phase-locks a voltage-controlled oscilla- their separations to mode-match the output self-phase modulation, based on the intrinsic intensity dependence of the refractive index (the Kerr effect). Optical fiber offers a small mode size and a relatively long interaction length, both of which enhance the width of the generated spectrum. However, chromatic dispersion in the optical fiber rapidly stretches the pulse duration, thereby lowering the peak power and limiting the amount of generated spectra. Although zero dispersion optical fiber at 1300 and 1550 nm has existed for years, optical fiber that supports a stable, fundamental spatial mode and has zero dispersion near 800 nm has been available only in the past year. In this work, we employed a recently developed air-silica microstructure fiber that has zero group velocity dispersion at 780 nm (16). The sustained high intensity (hundreds of GW/cm2) in the fiber generates a stable, single-mode, phase-coherent continuum that stretches from 510 to 1125 nm (at Ϫ20 dB) as shown in Fig. 3. Through four- wave mixing processes, the original spectral comb in the mode-locked pulse is transferred to the generated continuum. As described above, the offset frequency ␦ is obtained by Fig. 2. Experimental setup for locking the carrier-envelope relative phase. The femtosecond laser is taking the difference between 2fn and f2n. Figure 2 details this process of frequency located inside the shaded box. Solid lines represent optical paths, and dashed lines show electrical paths. The high-reflector mirror is mounted on a transducer to provide both tilt and translation. doubling fn in a nonlinear crystal and com- bining the doubled signal with f2n on a pho- todetector. The resulting RF heterodyne beat Fig. 3. Continuum gener- is equal to ␦. In actuality, the beat arises from ated by air-silica micro- a large family of comb lines, which greatly structure fiber. Self-phase enhances the signal-to-noise ratio. After suit- modulation in the micro- structured fiber broadens able processing (described below), this beat the output of the laser so is used to actively tilt the high-reflector mir- that it spans more than ror, allowing us to stabilize ␦ to a rational one octave. Approximate- fraction of the pulse repetition rate. ly 25 mW are coupled The experimental implementation of the into the fiber to generate f-2f heterodyne system is shown in Fig. 2. the displayed continuum. The wavelengths/frequen- The continuum output by the microstructure cies denoted by fn and f2n fiber is spectrally separated into two arms by are used to lock the offset frequency as described in the text. The spectra are offset from each other a dichroic beamsplitter. The visible portion of for clarity.

www.sciencemag.org SCIENCE VOL 288 28 APRIL 2000 637 R ESEARCH A RTICLES from both arms. The entire correlator is in a parameters, we determined the center of the the phase offset between experiment and theory vacuum chamber held below 300 mTorr to envelope and compared it with the phase of the to a phase imbalance in the correlator. The minimize the effect of the dispersion of air. underlying fringes to find ⌬␾. A fit of the fringe number of mirror bounces in each arm is the The second-order cross correlation was mea- peaks assuming a hyperbolic secant envelope same, and mirrors with the same coatings were sured with a two-photon technique (25)by produced nearly identical results. A plot of the used for 22 of the 23 bounces in each arm. focusing the recombined beam with a spher- experimentally determined relative phases for Nevertheless, because of availability issues, ical mirror onto a windowless GaAsP photo- various offset frequencies, along with a linear there is a single bounce that is not matched. diode. The band gap of GaAsP is large fit of the averaged data, is given in Fig. 5B. Furthermore, the large number of bounces nec- enough and the material purity is high enough These results show a small offset of 0.7 Ϯ 0.35 essary to generate the delay means that a very so that appreciable single-photon absorption rad from the theoretically expected relation small phase difference per bounce can accumu- ⌬␾ ϭ ␲␦ does not occur. This yields a pure quadratic 4 /frep (the extra factor of 2 results late and become significant. In addition, the intensity response with a very short effective because the cross correlator compares pulses i pellicle beam splitter will introduce a small temporal resolution. and i ϩ 2). The experimental slope is within 5% phase error because of the different reflection A typical cross correlation is shown in Fig. of the theoretically predicted value and demon- interface for the two arms. Together, these ef- 5A. To determine ⌬␾, we fit the fringe peaks of strates our control of the relative carrier-enve- fects can easily account for the observed offset. the interferogram to a correlation function as- lope phase. Despite our extensive efforts to The group-phase dispersion due to the residual suming a Gaussian pulse envelope. From the fit match the arms of the correlator, we attribute air only accounts for a phase error of ϳ␲/100. We think that this correlation approach represents the best measurement strategy that can be Fig. 4. Cross-correlator. made short of demonstration of a physical pro- The second-order cross cess that is sensitive to the phase. correlation between pulse We think that the uncertainty in the individ- ϩ i and pulse i 2 is mea- ual phase measurements shown in Fig. 5B aris- sured to determine the pulse-to-pulse carrier-en- es both from the cross-correlation measurement velope phase shift. The itself and from environmental perturbations of components inside the the laser cavity that are presently beyond the shaded box are enclosed bandwidth of our stabilizing servo loops. In- in a vacuum chamber at deed, a measurement in the frequency domain 300 mTorr to minimize made by counting a locked offset frequency the dispersive effects of ␦ϭ air. The number of mirror 19 MHz with 1-s gate time revealed a bounces in both arms are standard deviation of 143 Hz, corresponding to matched, also to mini- a relative phase uncertainty of 10 rad. The mize phase errors. The correlator uses a shorter effective gate time, second-order cross corre- which decreases the averaging and hence in- lation is measured with creases the standard deviation. Nevertheless, two-photon absorption in 3 4 a GaAsP photodiode. the uncertainty in the time domain is 10 to 10 times that in the frequency domain (see below), indicating that the correlator itself contributes to the measurement uncertainty. With pulses generated by mode-locked lasers now approaching the single-cycle re- gime (3, 4), the control of the carrier-enve- Fig. 5. Correlation re- lope relative phase that we have demonstrat- sults. (A) Typical cross ed is expected to dramatically impact the correlation (solid line) between the pulse i and field of extreme nonlinear optics. This in- pulse i ϩ 2, along with cludes above-threshold ionization and high a ﬁt of the correlation harmonic generation/x-ray generation with envelope (dashed line). intense femtosecond pulses. Above-threshold (B) Plot of the relative ionization with circularly polarized light has phase versus the offset recently been proposed as a technique for frequency (normalized to the pulse repetition determining the absolute phase (6). Measure- rate). As indicated, a ments of x-ray generation efficiency also linear ﬁt of our aver- show effects that are attributed to the evolu- aged data produces the tion of the pulse-to-pulse phase (8). ␲ expected slope of 4 Absolute optical frequency metrology. In with a 5% uncertainty. addition to applications in the time domain, the The origin of the phase offset is discussed in stabilized mode-locked laser shown in Fig. 2 the text. has an immediate and revolutionary impact also in optical frequency metrology. As shown sche-

matically in Fig. 1B, when both the frep (comb spacing) and the offset frequency ␦ (comb position) are stabilized, lying underneath the broadband continuum envelope is a frequency comb with precisely defined intervals and

known absolute frequencies. By stabilizing frep

638 28 APRIL 2000 VOL 288 SCIENCE www.sciencemag.org R ESEARCH A RTICLES in terms of the primary 9.193-GHz cesium stan- ue. The width of the Gaussian distribution leads optical input. The phase can either be locked so dard, we can then use this frequency comb as a us to suspect that the measurement scatter is that every pulse has the identical phase or made self-referenced “frequency ruler” to measure most likely due to phase noise in the RF rubid- to vary so that every ith pulse has the same any optical frequency that falls within the band- ium atomic clock used to stabilize the repetition phase. In the frequency domain, this means that width of the comb. With this technique, a direct rate of the laser. Only minimal environmental the broad spectral comb of optical lines has link between the microwave and optical do- stabilization of the laser cavity was performed. known frequencies, namely a simple (large) mains is now possible with a single stabilized With a higher quality reference clock, improved multiple of the pulse-repetition frequency plus a femtosecond laser. environmental isolation of the mode-locked la- user-defined offset. This is particularly conve- To demonstrate this application, we ser cavity and higher bandwidth servo loops, nient if the repetition rate of the laser is locked present results using this procedure to mea- we expect lower amounts of scatter, not only to an accurate microwave or RF clock because sure a CW Ti:S laser operating at 778 nm and for data such as those presented in Fig. 6B, but then the absolute optical frequencies of the 5 ϭ 3 5 ϭ locked to the S1/2 (F 3) D5/2 (F 5) also for time domain results analogous to those entire comb of lines are known. These results two-photon transition in 85Rb. The experi- in Fig. 5B. will impact extreme nonlinear optics (8, 27), mental setup is shown in Fig. 6A. A portion The average of our frequency measure- which is expected to display exquisite sensitiv- of the stabilized frequency comb is combined ments over several days, giving Ϫ4.2 Ϯ 1.3 ity to electric field of the pulse. The self-refer- with the 778-nm stabilized Ti:S laser and kHz from the CIPM (1997) value, agrees encing method also represents a dramatic ad- spectrally resolved with a 1200 lines per mil- quite well with a previous measurement of vance in optical frequency metrology, making limeter grating. The heterodyne beat between the JILA rubidium two-photon stabilized ref- measurement of absolute optical frequencies the frequency comb and the CW stabilized erence laser, in which we measured an offset possible with a single laser. Ti:S laser is measured with a photodiode of Ϫ3.2 Ϯ 3.0 kHz (13). In this previous ϳ positioned behind a slit that passes 1nmof work, the position of the broadened femto- References and Notes bandwidth about 778 nm. By counting both second comb was not locked but rather the 1. G. Steinmeyer, D. H. Sutter, L. Gallmann, N. Matus- the offset frequency ␦ and the heterodyne comb position was calibrated with the funda- chek, U. Keller, Science 286, 1507 (1999). 2. M. T. Asaki et al., Opt. Lett. 18, 977 (1993). beat signal between the CW Ti:S and the mental and second harmonic of a secondary, 3. U. Morgner et al., Opt. Lett. 24, 411 (1999). comb ( f beat), the unknown frequency is de- stabilized CW laser, which itself was mea- 4. D. H. Sutter et al., Opt. Lett. 24, 631 (1999). ϭϮ␦ϩ Ϯ 5. L. Xu et al., Opt. Lett. 21, 2008 (1996). termined by funknown nfrep fbeat. sured with the octave-spanning comb. The sign ambiguity of f arises because it These results demonstrate absolute optical 6. P. Dietrich, F. Krausz, P. B. Corkum, Opt. Lett. 25,16 beat (2000). is not known a priori whether the individual frequency measurements with a single mode- 7. R. J. Jones, J.-C. Diels, J. Jasapara, W. Rudolph, Opt. frequency comb member closest to the 778- locked laser. This technique represents an Commun. 175, 409 (2000). nm laser is at a higher or lower frequency. enormous simplification over conventional 8. C. G. Durfee et al., Phys. Rev. Lett. 83, 2187 (1999). ␦ 9. T. Udem, J. Reichert, R. Holzwarth, T. W. Ha¨nsch, A similar ambiguity exists for .Asthe frequency metrology techniques, including Phys. Rev. Lett. 82, 3568 (1999). 778-nm frequency is already known within multiplier chains, and even other femtosec- 10. ࿜࿜࿜࿜ , Opt. Lett. 24, 881 (1999). ϭ 11. J. Reichert, R. Holzwarth, Th. Udem, T. W. Ha¨nsch, much better than frep/2 45 MHz, funknown ond methods, described earlier. The tools de- is found by simply incrementing or decre- scribed here should make absolute optical Opt. Commun. 172, 59 (1999). 12. S. A. Diddams, D. J. Jones, L.-S. Ma, S. T. Cundiff, J. L. menting n and using the appropriate sign of frequency synthesis and measurement a com- Hall, Opt. Lett. 25, 186 (2000). ␦ 13. S. A. Diddams et al., Phys. Rev. Lett., in press. fbeat and . Figure 6B displays one set of mea- mon laboratory practice instead of the heroic surement results relative to the Comite´ Interna- effort it has been heretofore. 14. Various schemes for using mode-locked lasers in optical frequency metrology were recently discussed in work tional des Poids et Mesures (CIPM) (1997) Conclusion. We have demonstrated stabili- by H. R. Telle et al. [Appl. Phys. B 69, 327 (1999)]. recommended value of 385,285,142,378 Ϯ 5.0 zation of the carrier phase with respect to the 15. See, for example, A. E. Siegman, Lasers (University kHz for the optical rubidium transition (26). pulse envelope of ultrashort pulses produced by Science Books, Mill Valley, CA, 1986). 16. J. Ranka, R. Windeler, A. Stentz, Opt. Lett. 25,25 Our first demonstration of this technique is a mode-locked laser using a self-referencing (2000). within the uncertainty of the CIPM (1997) val- technique that does not require any external 17. J. D. Prestage, R. L. Tjoelker, L. Maleki, Phys. Rev. Lett. 74, 3511 (1995). 18. A. Huber et al., Phys. Rev. Lett. 80, 468 (1998). Fig. 6. Frequency me- 19. See, for example, H. Schnatz, B. Lipphardt, J. Helmcke, trology with the self- F. Riehle, G. Zinner, Phys. Rev. Lett. 76, 18 (1996). 20. H. R. Telle, D. Meschede, T. W. Ha¨nsch, Opt. Lett. 15, referenced frequency 532 (1990). comb. (A) Experimental 21. P. A. Junger et al., IEEE Trans. Instrum. Meas. 44, 151 setup to measure the (1995). 5 ϭ 3 5 S1/2 (F 3) D5/2 22. M. Kourogi, K. Nakagawa, M. Ohtsu, IEEE J. Quantum (F ϭ 5) two-photon Electron. 29, 2692 (1993). transition in 85Rb with a 23. J. L. Hall et al., IEEE Trans. Instrum. Meas. 48, 583 self-stabilized frequency (1999). comb. (B) Histogram of 24. R. L. Fork, O. E. Martinez, J. P. Gordon, Opt. Lett. 5, one set of measure- 150 (1984). 25. J. K. Ranka, A. L. Gaeta, A. Baltuska, M. S. Pschenich- ments in relation to nikov, D. A. Wiersma, Opt. Lett. 22, 1344 (1997). the recommended 26. T. Quinn, Metrologia 36, 211 (1999). CIPM (1997) value of 27. Ch. Spielmann et al., Science 278, 661 (1997). 385,285,142,378 Ϯ 5.0 28. The authors gratefully acknowledge J. Ye for help kHz for the rubidium with developing and running the rubidium stabilized transition. The standard Ti:S laser, J. Levine for assistance in establishing deviation at the 1-s gate common-view global positioning system–based time is 5 kHz, which is time comparison, and T. W. Ha¨nsch for discussions. an absolute uncertainty Research support from the NIST competence pro- 11 gram and the NSF are acknowledged. D.J.J. and of 1 part in 10 . S.A.D. are supported by the U.S. National Academy of Sciences/National Research Council postdoc- toral fellows program.

13 March 2000; accepted 10 April 2000