REVIEW ARTICLE Optical frequency synthesis based on mode ... · fruition due to a number of recent spectacular technological advances, most notably, the use of modelocked lasers for

REVIEW OF SCIENTIFIC INSTRUMENTS VOLUME 72, NUMBER 10 OCTOBER 2001

REVIEW ARTICLE

Optical frequency synthesis based on mode-locked lasersSteven T. Cundiff,a) Jun Ye, and John L. HallJILA, National Institute of Standards and Technology and University of Colorado, Boulder,Colorado 80309-0440

~Received 5 February 2001; accepted for publication 15 July 2001!

The synthesis of optical frequencies from the primary cesium microwave standard has traditionallybeen a difficult problem due to the large disparity in frequency. Recently this field has beendramatically advanced by the introduction and use of mode-locked lasers. This application ofmode-locked lasers has been particularly aided by the ability to generate mode-locked spectra thatspan an octave. This review article describes how mode-locked lasers are used for optical frequencysynthesis and gives recent results obtained using them. ©2001 American Institute of Physics.@DOI: 10.1063/1.1400144#

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I. INTRODUCTION

At the present, measurement of frequencies into thecrowave regime~tens of gigahertz! is straightforward due tothe availability of high frequency counters and synthesizeHistorically, this has not always been the case, with dirmeasurement being restricted to low frequencies. The curcapability arose because an array of techniques was deoped to make measurement of higher frequencies possi1

These techniques typically rely on heterodyning to prodan easily measured frequency difference~zero beating beingthe limit!. The difficulty lay in producing an accuratelknown frequency to beat an unknown frequency against

Measurement of optical frequencies~hundreds of tera-hertz! has been in a similar primitive state until recentdespite the enormous effort of many groups around the wto solve this difficult problem. This is because only fe‘‘known’’ frequencies have been available and it has bedifficult to bridge the gap between a known frequency andarbitrary unknown frequency if the gap exceeds tens ofgahertz~about 0.01% of the optical frequency!. Furthermore,establishing known optical frequencies was itself difficult bcause an absolute measurement of frequency must be bon the time unit ‘‘second,’’ which is defined in terms of thmicrowave frequency of a hyperfine transition of the cesiatom. This requires complex ‘‘clockwork’’ to connect opticfrequencies to those in the microwave region.

Optical frequencies have been used in measurementence since shortly after the invention of lasers. Compariof a laser’s frequency of;531014Hz with its ideal; mil-lihertz linewidth, produced by the fundamental phase difsion of spontaneous emission, reveals a potential dynarange of 1017 in resolution, offering one of the best tools witwhich to discover new physics in ‘‘the next decimal placeNearly 40 years of vigorous research in the many diveaspects of this field by a worldwide community have resul

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in exciting discoveries in fundamental science and develment of enabling technologies. Some of the ambitious loterm goals in optical frequency metrology are just comingfruition due to a number of recent spectacular technologadvances, most notably, the use of modelocked lasersoptical frequency synthesis. Other examples include lafrequency stabilization to 1 Hz and below,2 optical transi-tions observed at a few Hz linewidth~corresponding to aQof 1.531014!,3 and steadily improving accuracy of opticastandards with a potential target of 10218 for cold atom/ionsystems. Indeed, it now seems to be quite feasible to reawith a reasonable degree of simplicity and robustness,optically based atomic clock and an optical frequency sthesizer. Considering that most modern measurement exments use frequency-based metrology, one can foreseemendous growth in these research fields.

The ability to generate wide-bandwidth optical frquency combs has recently provided a truly revolutionadvance in the progress of this field. In this review we dscribe the implementation of a frequency measurement tenique utilizing a series of regularly spaced frequencies tform a ‘‘comb’’ spanning the optical spectrum. The opticfrequency comb is generated by a mode-locked laser.comb spacing is such that any optical frequency can beily measured by heterodyning it with a nearby comb linFurthermore, it is possible to directly reference the cospacing and position to the microwave cesium time standthereby determining the absolute optical frequencies of althe comb lines. This recent advance in optical frequenmeasurement technology has facilitated the realization ofultimate goal of a practical optical frequency synthesizerforms a phase-coherent network linking the entire optispectrum to the microwave standard.

The purpose of this review article is to present the recadvances in optical frequency synthesis and metrologyhave been engendered by using femtosecond mode-lolasers to span vast frequency intervals. This surprising unof two apparently disparate realms likely means that f

9 © 2001 American Institute of Physics

AIP license or copyright, see http://ojps.aip.org/rsio/rsicr.jsp

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3750 Rev. Sci. Instrum., Vol. 72, No. 10, October 2001 Cundiff, Ye, and Hall

readers will be familiar with both. Therefore, in the following we attempt to address either the optical metrologist wing to become knowledgeable about ultrafast science or,versa, an enabling, albeit challenging, undertaking. In Sewe provide background on optical frequency metrology,viewing the state of the art~prior to the introduction ofmode-locked lasers! and the motivation. To put the recenadvances into context and to provide comparison, other teniques, which do not employ mode-locked lasers, areviewed in Sec. III. In sec. IV the basics of mode-locklasers are covered with emphasis on the characteristicsare relevant to optical frequency synthesis. In Sec. Vdescribe how the output comb can be controlled in sucway that is suitable for optical frequency synthesis. Thisfollowed by a description of several recent experiments thave demonstrated the dramatic breakthrough that thtechniques have enabled. In Sec. VI we discuss the impathis work on other fields and expected future developme

II. BACKGROUND: OPTICAL FREQUENCYSYNTHESIS AND METROLOGY

Optical frequency metrology broadly contributes to aprofits from many areas in science and technology. Atcore of this subject is the controlled and stable generatiocoherent optical waves, i.e., optical frequency synthesis. Tpermits high precision and high resolution measuremenmany physical quantities.

In Secs. II A–II C, we will present brief discussions othese aspects of optical frequency metrology, with stablesers and wide bandwidth optical frequency combs makingthe two essential components in stable frequency generaand measurement.

A. Establishment of standards

In 1967, just a few years after the invention of the lasthe international standard of time/frequency was establishbased on theF54, mF50→F53, mF50 transition in thehyperfine structure of the ground state of133Cs.4 The transi-tion frequency is defined to be 9192 631 770 Hz. The renanceQ of ;108 is set by the limited coherent interactiotime between matter and field. Much effort has been invesin extending the coherent atom–field interaction time andcontrolling the first and second order Doppler shifts. Recadvances in laser cooling and trapping technology haveto the practical use of laser-slowed atoms, and a 100-resolution enhancement. With the reduced velocities, Dpler effects have also been greatly reduced. Cs clocks bon atomic fountains are now operational, with reportedcuracy of 3310215 and short-term stability of 1310213 at 1s, limited by the frequency noise of the local rf crystoscillator.5 Through similar technologies, single ions, lascooled and trapped in an electromagnetic field, are nowexcellent candidates for radio frequency/microwave stdards, with a demonstrated frequency stability approach3310213 at 1 s.6 More compact, less expensive~and lessaccurate! atomic clocks use cesium or rubidium atoms inglass cell equipped with all essential clock mechanisms,cluding optical pumping~atom preparation!, microwave cir-cuitry for exciting the clock transition, and optical detectio


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The atomic hydrogen maser is another mature and pracdevice that uses the radiation emitted by atoms directly.7 Al-though it is less accurate than the cesium standard, a hygen maser can realize exceptional short-term stability.

The development of optical frequency standardsbeen an extremely active field since the invention of lasewhich provide the coherent radiation necessary for precisspectroscopy. The coherent interaction time, the determinfactor of the spectral resolution in many cases, is in fcomparable in both optical and rf domains. The optical pof the electromagnetic spectrum provides higher operafrequencies. Therefore the quality factor,Q, of an opticalclock transition is expected to be a few orders higher ththat available in the microwave domain. A superiorQ factorhelps to improve all three essential characteristics of aquency standard, namely, accuracy, reproducibility, andbility. Accuracy refers to the objective property of a standato identify the frequency of a natural quantum transitioidealized to the case that the atoms or the molecules arrest and free of any perturbation. Reproducibility measuthe repeatability of a frequency standard for different realitions, signifying adequate modeling of observed operatparameters and independence from uncontrolled operaconditions. Stability indicates the degree to which the fquency stays constant after operation has started. Idealstabilized laser can achieve a fractional frequency stabili

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where S/N is the recovered signal-to-noise ratio of the renance information, andt is the averaging time. Clearly it isdesirable to enhance both the resolution and sensitivity ofdetected resonance, since they control the time scale nesary for a given measurement precision. The reward is emous: enhancing theQ ~or S/N! by a factor of 10 reduces thaveraging time by a factor of 100.

The nonlinear nature of a quantum absorption procewhile limiting the attainable S/N, permits sub-Doppler reslution. Special optical techniques invented in the 70s andfor sub-Doppler resolution include saturated absorption sptroscopy, two-photon spectroscopy, optical Ramsey fringoptical double resonance, quantum beats, and laser cooand trapping. Cold samples offer a true possibility to obsethe rest frame atomic frequency. Sensitive detection teniques, such as polarization spectroscopy, electron shel~quantum jump!, and frequency modulation optical heterdyne spectroscopy, were also invented during the sameriod, leading to an absorption sensitivity of 131028 and theability to split a MHz scale linewidth typically by a factor o104– 105 at an averaging time of;1 s. All these technologi-cal advances paved the way for subhertz stabilization ofpercoherent optical local oscillators.

To effectively use a laser as a stable and accurate oplocal oscillator, active frequency control is needed due tostrong coupling between the laser frequency and the laparameters. The simultaneous use of quantum absorbersan optical cavity offers an attractive laser stabilization stem. A passive reference cavity brings the benefit of a lin


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3751Rev. Sci. Instrum., Vol. 72, No. 10, October 2001 Optical synthesis with mode-locked lasers

response, allowing the use of sufficient power to achievhigh S/N. On one hand, a laser pre-stabilized by a caoffers a long phase coherence time, reducing the needfrequent interrogations of the quantum absorber. In otwords, the laser linewidth over a short time scale is narrothan the chosen atomic transition width and thus the inmation of the natural resonance can be recovered withoptimal S/N and the long averaging time translates intfiner examination of the true line center. On the other hathe quantum absorber’s resonance basically eliminates intable drifts associated with material standards, such as aity. Frequency stability in the 10216 domain has been measured with a cavity-stabilized laser.8 The use of frequencymodulation for cavity/laser lock has become a standard laratory practice.9 Tunability of such a cavity/laser system cabe obtained by techniques such as the frequency-offsetcal phase-locked loop~PLL!.

A broad spectrum of lasers has been stabilized, frearly experiments with gas lasers~He–Ne, CO2, Ar1, etc.!,to more recent tunable dye lasers, optically pumped sostate lasers~Ti:sapphire, YAG, etc.! and diode lasers. Usuallone or several atomic or molecular transitions are locawithin the tuning range of the laser to be stabilized. Theof molecular ro-vibrational lines for laser stabilization hbeen very successful in the infrared using molecules sucCH4, CO2, and OsO4.

10 Their natural linewidths range below1 kHz, limited by molecular fluorescent decay. Usable linwidths are usually>10 kHz due to the transit of moleculethrough the light beam. Transitions to higher levels of thefundamental rovibrational states, usually termed overtbands, extend these rovibrational spectra well into the viswith similar ;kHz potential linewidths. Until recently, therich spectra of the molecular overtone bands have not badopted as suitable frequency references in the visible dutheir small transition strengths.11 However, with one of themost sensitive absorption techniques, which combinesquency modulation with cavity enhancement, an excellS/N for these weak but narrow overtone lines canachieved,12 enabling the use of molecular overtones as stdards in the visible.13,14

Systems based on cold absorber samples potentiallyfer the highest quality optical frequency sources, mainly dto the drastic reductions of linewidth and velocity-relatsystematic errors. For example, a few Hz linewidth onoptical transition of Hg1 was recently observed at the Ntional Institute of Standards and Technology~NIST!.3 Cur-rent activity on single ion systems includes Sr1,15 Yb1,16,17

and In1.18 One of the early NIST proposals of using atomfountains for optical frequency standards19 has resulted ininvestigation of the neutral atoms Mg,20 Ca,21 Sr,22,23 Ba,24

and Ag.25 These systems could offer ultimate frequency stdards free from virtually all of the conventional shifts anbroadenings, to the level of one part in 1016– 1018. Consid-erations of a practical system must always include its csize, and degree of complexity. Compact and low cost stems can be competitive even though their performancebe 10-fold worse compared with the ultimate system. Osuch system is Nd:YAG laser stabilized on HCCD at 10nm or on I2 ~after frequency doubling! at 532 nm, with a


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demonstrated stability level of 4310215 at 300 s averagingtime.26,27 In Fig. 1 we summarize some of the optical frquency standards that are either established or under adevelopment. Also indicated is the spectral width of crently available optical frequency combs generated by molocked lasers.

Accurate knowledge of the center of the resonanceessential for establishing standards. Collisions, electromnetic fringe fields, residual Doppler effects, probe field wavfront curvature, and probe power can all produce undescenter shifts and linewidth broadening. Other physical intactions, and even distortion in the modulation wave forcan produce asymmetry in the recovered signal line shaFor example, in frequency modulation spectroscopy, residamplitude modulation introduces unwanted frequency shand instability and therefore needs to be controlled.28 Theseissues must be addressed carefully before one can befortable talking about accuracy. A more fundamental issrelated to time dilation of the reference system~the secondorder Doppler effect! can be solved in a controlled fashionone simply knows the sample velocity accurately~for ex-ample, by velocity selective Raman process!, or the velocityis brought down to a negligible level using cooling and traping techniques.

B. Application of standards

The technology of laser frequency stabilization has berefined and simplified over the years and has becomeindispensable research tool in many modern laboratoriesvolving optics. Research on laser stabilization has beenstill is pushing the limits of measurement science. Indeednumber of currently active research projects on fundamephysical principles greatly benefit from stable optical sourand need continued progress on laser stabilization. Theyclude laser testing of fundamental principles,29 gravitationalwave detection,30 quantum dynamics,31 searching for drift ofthe fundamental constants,32–34 atomic and molecular structure, and many more. Recent experiments with hydrogenoms have led to the best reported value for the Rydb

FIG. 1. Map of the electromagnetic spectrum showing frequencies oferal atomic and molecular reference transitions and the frequency rangsources. Spanning the frequency difference from Cs to the visible portiothe spectrum is now facilitated by the use of femtosecond lasers. Thequency comb generated by mode-locked femtosecond lasers spans thible and can be transferred to the infrared by difference-frequency gention ~shown by the large arrow!.


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constant and 1S-Lamb shift.35,36 Fundamental physical constants such as the fine-structure constant, ratio of Planconstant to electron mass, and the electron-to-proton mratio are also being determined with increasing precisioning improved precision laser tools.37 Using extremely stablephase-coherent optical sources, we are entering an excera when picometer resolution can be achieved over alion kilometer distance in space. In time keeping, an optifrequency clock is expected to eventually replace the curmicrowave atomic clocks. In length metrology, the realiztion of the basic unit, the ‘‘meter,’’ relies on stable opticfrequencies. In communications, optical frequency metogy provides stable frequency/wavelength reference grid38

A list of just a few examples of stabilized continuouwave~cw! tunable lasers includes millihertz linewidth stablization ~relative to a cavity! for diode-pumped solid statlasers, tens of millihertz linewidth for Ti:sapphire lasers, asubhertz linewidths for diode and dye lasers. Tight phlocking between different laser systems can be achieve39

even for diode lasers that have fast frequency noise.

C. Challenge of optical frequency measurement andsynthesis

Advances in optical frequency standards have resultethe development of absolute and precise frequency measment capability in the visible and near-infrared spectralgions. A frequency reference can be established only afthas been phase coherently compared and linked with ostandards. As mentioned above, until recently opticalquency metrology has been restricted to the limited se‘‘known’’ frequencies, due to the difficulty in bridging thegap between frequencies and the difficulty in establishing‘‘known’’ frequencies themselves.

The traditional frequency measurement takessynthesis-by-harmonics approach. Such a synthesis chaincomplex system that involves several stages of stabilitransfer lasers, high-accuracy frequency references~in bothoptical and rf ranges!, and nonlinear mixing elements. Phascoherent optical frequency synthesis chains linked to thesium primary standard include Cs–HeNe/CH4 ~3.39mm!40,41

and Cs–CO2/OsO4 ~10 mm!.42 Extension to HeNe/I2 ~576nm!43 and HeNe/I2 ~633 nm!44,45 lasers made use of one othese reference lasers~or the CO2/CO2 system40! as an inter-mediate. The first well-stabilized laser to be measured bCs-based frequency chain was the HeNe/CH4 system at 88THz.40 With interferometric determination of the associatwavelength46 in terms of the existing wavelength standabased on krypton discharge, the work led to a definitvalue for the speed of light, soon confirmed by other labotories using many different approaches. Redefinition ofunit of length by adoptingc5299 792 458 m/s became posible with the extension of the direct frequency measuments to 473 THz~HeNe/I2 633 nm system! 10 years laterby a NIST 10-person team, thus creating a direct connecbetween the time and length units. More recently, with iproved optical frequency standards based on cold atoms~Ca!~Ref. 47! and single trapped ions (Sr1),15 these traditional


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frequency chains have demonstrated measurement unceties at the 100 Hz level.

Understandably, these frequency chains are large sresearch efforts that require resources that can be provby only a few national laboratories. Furthermore, the fquency chain can only cover some discrete frequency min the optical spectrum. Difference frequencies of many Tcould still remain between a target frequency and a knoreference. These three issues have represented majostacles to making optical frequency metrology a genelaboratory tool. Several approaches have been proposedtested as simple, reliable solutions for bridging large optifrequency gaps. Some popular schemes include frequeinterval bisection,48 optical-parametric oscillators~OPO!,49

optical comb generators,50,51 sum-and-difference generatioin the near infrared,52 frequency division by three-,53,54 andfour-wave mixing in laser diodes.55 All of these techniquesrely on the principle of difference-frequency synthesis,contrast to the frequency harmonic generation method nmally used in traditional frequency chains. In Sec. III wbriefly summarize these techniques, their operating pciples, and applications. Generation of wide bandwidth ocal frequency combs has provided the most direct and simapproach among these techniques and it is the main topthis review article.

III. TRADITIONAL APPROACHES TO OPTICALFREQUENCY SYNTHESIS

Although the potential for using mode-locked lasersoptical frequency synthesis was recognized early,56 these la-sers did not provide the properties necessary for fulfillithis potential until recently. Consequently, an enormousfort was invested over the last 40 years in ‘‘traditional’’ aproaches, which typically involve phase coherently linksingle frequency lasers. Traditional approaches to opticalquency measurement can be divided into two subcategoone is synthesis by harmonic generation, and the othedifference-frequency synthesis. The former method halong history of success, at the expense of massive resouand system complexity. The latter approach has beenfocus of recent research that has led to systems that areflexible, adaptive, and efficient. Indeed, the main subjectour article, the technique of a wide bandwidth optical frquency comb generator, belongs to the categorydifference-frequency generation. For a historical perspecand to have a direct comparison between the two approacwe first describe the optical frequency chain based on ‘‘clsic’’ harmonic generation.

A. Phase coherent chains: Traditional frequencyharmonic generation

The traditional frequency measurement takessynthesis-by-harmonic approach. Harmonics, i.e., intemultiples, of a standard frequency are generated with a nlinear element and the output signal of a higher-frequeoscillator is phase coherently linked to one of the harmonTracking and counting of the beat note, or the use of a Ppreserves the phase coherence at each stage. Such a pcoherent frequency multiplication process is continued


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higher and higher frequencies until the measurement targthe optical spectrum is reached. In the frequency regionmicrowave to midinfrared, a harmonic mixer can perfofrequency multiplication and frequency mixing/phase coparison all by itself. ‘‘Cat’s whisker’’ W–Si point contacmicrowave diodes, metal–insulator–metal~MIM ! diodes,and Schottky diodes have been used extensively for thispose. In the near infrared to the visible~,1.5 mm!, the effi-ciency of MIM diodes decreases rapidly. Optical nonlinecrystals are better for harmonic generation in these speregions. Fast photodiodes perform frequency mixing~non-harmonic! and phase comparison. Such a synthesis chaincomplex system that involves several stages of stabilitransfer lasers, high-accuracy frequency references~in bothoptical and rf ranges!, and nonlinear mixing elements. Aimportant limitation is that each oscillator stage employs dferent lasing transitions and different laser technologiesthat reliable and cost effective designs are elusive.

1. Local oscillators and phase-locked loops

The most important issue in frequency synthesis isstability and accuracy associated with such frequency trafer processes. Successful implementation of a synthchain requires a set of stable local oscillators at variousquency stages. Maintaining phase coherence across thefrequency gaps covered by the frequency chain demandsphase errors at each synthesis stage be eliminated ortrolled. A more stable local oscillator offers a longer phacoherence time, making frequency/phase comparison mtractable and reducing phase errors accumulated beforeservo can decisively express control. Due to the intrinproperty of the harmonic synthesis process, there aremechanisms responsible for frequency/phase noise entethe loop and limiting the ultimate performance. The firstadditive noise, where a noisy local oscillator compromisthe information from a particular phase comparison step.second, and more fundamental one, is the phase noiseciated with the frequency~really phase! multiplication pro-cess: the phase angle noise increases as the multiplicfactor, hence the phase noise spectral density of the ousignal from a frequency multiplier increases as the squarthe multiplication factor and so becomes progressively woas the frequency increases in each stage of the chain.phase noise microwave and laser local oscillators are thfore important in all PLL frequency synthesis schemes.

The role of the local oscillator in each stage of the fquency synthesis chain is to take the phase information fthe lower frequency regions and pass it on to the next lewith appropriate noise filtering, and to reestablish a staamplitude. The process of frequency/phase transfer typicinvolves PLLs. Sometimes a frequency comparison is carout with a frequency counter that measures the differenccycle numbers between two periodic signals within a pretermined time period. An intrinsic time domain device usto measure zero crossings, e.g., a frequency counter, issitive to signal, and noise, in a large bandwidth and soeasily accumulate counting errors resulting from an insucient signal-to-noise ratio. Even for a PLL, the possibilitycycle slipping is a serious issue. With a specified signa


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noise ratio and control bandwidth, one can estimate theerage time between successive cycle slips and thus knowexpected frequency counting error. For example, a 100 kmeasurement bandwidth requires a signal-to-noise ratio odB to achieve a frequency error of 1 Hz~1 cycle slip per 1s!.57

One function of PLLs is to regenerate a weak signfrom a noisy background, thus providing spectral filteriand amplitude stabilization. This function is described a‘‘tracking filter.’’ Within the correction bandwidth, the tracking filter frequency output follows the perceived rf input sinwave’s frequency. A voltage-controlled oscillator~VCO! pro-vides the PLL’s constant output amplitude, the variable oput frequency is guided by the correction error generafrom the phase comparison with the weak signal inputtracking filter, consisting of a VCO under PLL control,essential for producing reliable frequency counting, with tregenerated signal able to support the unambiguous zcrossing measurement for a frequency counter.

2. Measurements made with phase-coherent chains

As described in Sec. III C, only a few phase-cohereoptical frequency synthesis chains have ever been immented. Typically, some important infrared standards, sas the 3.39mm (HeNe/CH4) system and the 10mm(CO2/OsO4) system, are connected to the Cs standard fiOnce established, these references are then used to mehigher optical frequencies.

One of the first frequency chains was developed atNational Bureau of Standards~NBS!, and it connected thefrequency of a methane-stabilized HeNe laser to thestandard.40 The chain started with a Cs-referenced klystroscillator at 10.6 GHz, with its seventh harmonic linked tosecond klystron oscillator at 74.2 GHz. A HCN laser at 0.THz was linked to the 12th harmonic of the second klystrfrequency. The 12th harmonic of the HCN laser was conected to a H2O laser, whose frequency was tripled to conect to a CO2 laser at 32.13 THz. A second CO2 laser fre-quency, at 29.44 THz, was linked to the difference betwethe 32.13 THz CO2 laser and the third harmonic of the HClaser. The third harmonic of this second CO2 laser finallyreached the HeNe/CH4 frequency at 88.3762 THz. The measured value of HeNe/CH4 frequency was later used in another experiment to determine the frequency of an iodistabilized HeNe laser at 633 nm, bridging the gap betwinfrared and visible radiation.44

The important 10mm spectral region covered by CO2

lasers has been the focus of several different frequechains.41,42,58It is worth noting that in the Whitford chain58 asubstantial number of difference frequencies~generated be-tween various CO2 lasers! were used to bridge the intermediate frequency gaps, although the general principle ofchain itself is still based on harmonic synthesis. CO2 lasersprovided the starting point of most subsequent frequechains that reached the visible frequency spectrum.43,45,59Asnoted above, these frequency chains and measurementsled to accurate knowledge of the speed of light, allowinginternational redefinition of the ‘‘meter,’’ and establishmeof many absolute frequency/wavelength standards throu


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out the infrared~IR!/visible spectrum. More recently, withimproved optical frequency standards based on cold at~Ca! ~Ref. 47! and single trapped ions (Sr1),15 these tradi-tional frequency measurement techniques have demonstmeasurement uncertainties at the 100 Hz level by direlinking the Cs standard to the visible radiation in a singfrequency chain.

3. Shortcomings of this traditional approach

It is obvious that such harmonic synthesis systemsquire a significant investment of human and other resourThe systems need constant maintenance and can be affoonly by national laboratories. Perhaps the most unsatisfyaspect of harmonic chains is that they cover only a few dcrete frequency marks in the optical spectrum. Thereforesystems work on coincidental overlaps in target frequenand are difficult to adapt to different tasks. Another limitatiis the rapid increase of phase noise~asn2! with the harmonicsynthesis order~n!.

B. Difference-frequency synthesis

The difference-frequency generation approach borromany frequency measurement techniques developed foharmonic synthesis chains. Perhaps the biggest advantadifference-frequency synthesis over the traditional harmogeneration is that the system can be more flexible and cpact, and yet have access to more frequencies. We disfive recent approaches, with the frequency interval bisecand the optical comb generator being the most significbreakthroughs. The common theme of these techniques icapability to subdivide a large optical frequency interval insmaller portions with a known relationship to the originfrequency gap. The small frequency difference is then msured to yield the value of the original frequency gap.

1. Frequency-interval bisection

Bisection of frequency intervals is one of the most important concepts in the difference-frequency generatio48

Coherent bisection of optical frequency generates the ametic average of two laser frequenciesf 1 and f 2 by phaselocking the second harmonic of a third laser at frequencyf 3

to the sum frequency off 1 and f 2 . These frequency-intervabisection stages can be cascaded to provide differefrequency division by 2n. Therefore any target frequency capotentially be reached with a sufficient number of bisectstages. Currently the fastest commercial photodetectorsmeasure heterodyne beats of some tens of GHz. Thus, 6cascaded bisection stages are required to connect a fewdred THz wide frequency interval with a measurable micwave frequency. Therefore the capability of measuringlarge beat frequency between two optical signals becoever more important, considering the number of bisectstages that can be saved with a direct measurement. A perful combination is to have an optical comb generatorpable of measuring a few THz optical-frequency differencas the last stage of the interval bisection chain. It is wonoting that in a difference-frequency measurement it is tycal for all participating lasers to have their frequencies innearby frequency interval, thus simplifying system design


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Many optical-frequency measurement schemes hbeen proposed, and some realized, using interval bisecThe most notable achievement so far has been by Ha¨nsch’sgroup at the Max-Planck Institute for Quantum Opti~MPQ! in Garching. They used a phase-locked chain of fifrequency bisection stages to bridge the gap between thedrogen 1S– 2S resonance frequency and the 28th harmoof the HeNe/CH4 standard at 3.39mm, leading to improvedmeasurement of the Rydberg constant and the hydroground state Lamb shift.35 The chain started with a intervadivider between a 486 nm laser~one fourth the frequency othe hydrogen 1S– 2S resonance! and the HeNe/CH4. Therest of the chain successively reduced the gap betweenmidpoint near 848 nm and the fourth harmonicHeNe/CH4, a convenient spectral region where similar diolaser systems can be employed, even though slightly difent wavelengths are required.

2. Optical parametric oscillators

The use of optical parametric oscillators~OPOs! for fre-quency division relies on parametric down conversionconvert an input optical signal into two coherent subhmonic outputs, the signal and the idler. These outputstunable and their linewidths are replicas of the input puexcept for the quantum noise added during the down consion process. The OPO output frequencies, or the origpump frequency, can be precisely determined by phase loing the difference frequency between the signal and the ito a known microwave or infrared frequency.

In Wong’s original proposal, OPO divider stages confiured parallel or series were shown to provide the neemultistep frequency division.49 However, no such cascadesystems have been realized so far, due in part to the difficof finding suitable nonlinear crystals for the OPO operatto work in different spectral regions, especially in the infrred. There is progress on the OPO-based optical-frequemeasurement schemes, most notably optical-frequency dsion by 2 and 3,60,61 that allows rapid reduction of a largfrequency gap. Along with threshold-free differencfrequency generations in nonlinear crystals~discussed next!,the OPO system provides direct access to calibrated tunfrequency sources in the IR region~20–200 THz!.

3. Nonlinear crystal optics

This same principle, i.e., phase locking between theference frequencies while holding the sum frequency cstant, leads to frequency measurement in the near infrausing nonlinear crystals for the sum-and-difference fquency generation. The sum of two frequencies in the ninfrared can be matched to a visible frequency standwhile the difference matches to a stable reference in the minfrared. Another important technique is optical frequendivision by 3. This larger frequency ratio could simplify optical frequency chains while providing a convenient conntion between visible lasers and infrared standards. An ational stage of mixing is needed to ensure the precdivision ratio.53


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FIG. 2. Schematic of the optical-frequency comb geerator based on an intracavity electro-optic modulatHigh finesse~typically a few hundred! of the loadedcavity and a large frequency modulation index~b! areinstrumental to broad bandwidth of the generated co~typically a few THz!.

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4. Four-wave mixing in laser diodes

Another approach to difference-frequency generationlies on four-wave mixing. The idea55 is to use a laser diodeas both a light source and an efficient nonlinear receiveallow a four-wave mixing process to generate phase-cohebisection of a frequency interval of a few THz. The setinvolves two external cavity diode lasers~vLD1 and vLD2!,separated by 1–2 THz, that are optically injected into a thdiode laser for frequency mixing. When the frequency ofthird diode laser (vLD3) is tuned near the interval center ovLD1 and vLD2 , the injection locking mechanism becomeffective and locksvLD3 on the four-wave mixing productvLD11vLD22vLD3 , leading to the interval bisection condtion vLD35(vLD11vLD2)/2. The bandwidth of this procesis limited by phase matching in the mixing diode, and wfound to be only a few THz.55

5. Optical-frequency comb generators

One of the most promising difference-frequency syntsis techniques is the generation of multi-THz optical comby placing a rf electro-optic modulator~EOM! in a low-lossoptical cavity.50 The optical cavity enhances modulation eficiency by resonating with the carrier frequency and all ssequently generated sidebands, leading to a spectral comfrequency-calibrated lines spanning a few THz. A schemof such an optical frequency comb generation processhown in Fig. 2. The single frequency cw laser is locked oone of the resonance modes of the EOM cavity, withfree-spectral-range frequency of the loaded cavity beinginteger multiple of the EOM modulation frequency. The caity output produces a comb spectrum with an intensity proof exp(2ukup/bF),50 where k is the order of the generateside band from the original carrier,b is the EOM frequencymodulation index, andF is the loaded cavity finesse. Thuniformity of the comb frequency spacing was carefuverified.62 These optical frequency comb generato~OFCGs! have produced spectra that extend a few tensTHz,63 nearly 10% of the optical carrier frequency. At JILAwe developed unique OFCGs, one with the capabilitysingle comb line selection64 and the other with efficiencyenhancement via an integrated OPO/EOM system.65

OFCGs had an immediate impact on the field of optifrequency measurement. Kourogi and co-workers66 producedan optical-frequency map~accurate to 1029! in the telecom-munication band near 1.5mm using an OFCG that produce


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a 2 THz wide comb in that wavelength region, connectivarious molecular overtone transition bands of C2H2 andHCN. The absolute frequency of the CsD2 transition at 852nm was measured against the fourth harmonic ofHeNe/CH4 standard, with an OFCG bridging the remaininfrequency gap of 1.78 THz.67 At JILA, we used an OFCG tomeasure the absolute optical frequency of the iodine stlized Nd:YAG frequency near 532 nm.27 The level schemefor the measurement is shown in Fig. 3. The sum frequeof a Ti:sapphire laser stabilized on the Rb two photon trsition at 778 nm and the frequency-doubled Nd:YAG laswas compared against the frequency-doubled output of aode laser near 632 nm. The 660 GHz frequency gap betwthe red diode and the iodine-stabilized HeNe laser at 633was measured using the OFCG.64

An OFCG was also used in the measurement of thesolute frequency of a Ne transition (1S5→2P8) at 633.6 nmrelative to the HeNe/I2 standard at 632.99 nm.68,69The lowerlevel of the transition is a metastable state. Therefore,resonance can only be observed in a discharged neonThe resonance has a natural linewidth of 7.8 MHz. It caneasily broadened~due to unresolved magnetic sublevels! andits center frequency shifted by an external magnetic fieThis line is therefore not a high quality reference standaHowever, it does have the potential of becoming a low c

FIG. 3. Measurement of the iodine stabilized Nd:YAG laser frequency usthe difference-frequency generation schemes, namely, frequency intervsection and OFCG. The sum of the frequency of the doubled Nd:YAG laand a Rb two photon stabilized Ti:S laser at 778 nm is about 1.3 THz higthan the doubled frequency of the iodine-stabilized 633 nm HeNe laser.frequency gap can be bridged with an optical frequency comb generbased on a 632 nm diode laser~with the gap halved at 660 GHz!.


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compact frequency reference that offers frequency calibtion on the order of 100 kHz. A red diode laser probe ainexpensive neon lamp form such a system.

The frequency gap between the HeNe/I2 standard and theneon transition is about 468 GHz, which can easily be msured with an OFCG. The HeNe laser, which is the carriethe comb, is locked to the127I 2R(127)11-5 componenta13.The neon 1S5→2P8 transition frequency was determinedbe 473 143 829.76~0.10! MHz.

The results obtained using OFCGs made the advantof larger bandwidth very clear. However the bandwidachievable by a traditional OFCG is limited by cavity dispesion and modulation efficiency. To achieve even larger bawidth, mode-locked lasers were introduced, thus triggerintrue revolution in optical frequency measurement. This issubject of the remainder of this review article.

IV. MODE-LOCKED LASERS

The OFCGs described above actually generate a traishort pulses. This is simply due to interference among mowith a fixed phase relationship and is depicted in Fig. 4. Tfirst OFCG was built to generate short optical pulses ratthan for optical-frequency synthesis or metrology.70 Laterwork provided even shorter pulses from an OFCG.71

A laser that can sustain simultaneous oscillation on mtiple longitudinal modes can emit short pulses; it justquires a mechanism to lock the phases of all the modwhich occurs automatically in an OFCG due to the actionthe EOM. Lasers that include such a mechanism are refeto as ‘‘mode locked’’~ML !. While the term mode lockingcomes from this frequency-domain description, the actprocesses that cause mode locking are typically describethe time domain.

The inclusion of gain65,72and dispersion compensation73

in OFCGs brings them even closer to ML lasers. The useML lasers as optical comb generators has been developeparallel with the OFCG, starting with the realization that

FIG. 4. Schematic of the pulse train generated by locking the phassimultaneously oscillating modes. The upper panel show the output inteas the number of modes is increased from 1 to 2 to 3. The lower panel sthe result for 30 modes, both phase locked and with random phasesmode spacing is 1 GHz.


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regularly spaced train of pulses could excite narrow renances because of the correspondence with a comb infrequency domain.74–77 Attention was quickly focused onML lasers as the source of a train of short pulses.56,78–93Therecent explosion of measurements based on ML lasersbeen largely due to development of the Kerr-lens-molocked ~KLM ! Ti:sapphire laser94–96 and its capability togenerate pulses sufficiently short so that the spectral wapproaches an optical octave. In many recent results hasobtained a spectral width exceeding an octave by specbroadening external to the laser cavity.89,91,97

ML lasers have succeeded in generating a much labandwidth than OFCGs, which is very attractive. In additiothey tend to be ‘‘self-adjusting’’ in the sense that they do nrequire the active matching between cavity length and molator frequency that an OFCG does. Although the spacbetween the longitudinal modes is easily measured~it is justthe repetition rate! and controlled, the absolute frequencpositions of the modes is a more troublesome issue andquires some method of active control and stabilization. Tincredible advantage of having spectral width in excess ooctave is that it allows the absolute optical frequencies todetermined directly from a cesium clock,83,89,91without theneed for intermediate local oscillators. Synthesis of optifrequencies directly from a cesium clock can also be accoplished with less than a full optical octave, albeit at the prof a somewhat more complicated apparatus.

In addition to the large bandwidth, ML lasers also haan important advantage over OFCGs with respect tophase coherence of the modes. In an OFCG, only adjamodes are phase coherently coupled by the EOM. In alaser, the ultrashort pulse results from phase locking ofall ofthe lasing modes. This means that there is very strong muphase coherence among all of the modes, which is keythe remarkable results obtained using ML lasers for opticfrequency metrology.

A. Introduction to mode-locked lasers

ML lasers generate short optical pulses by establishinfixed phase relationship among all of the lasing longitudimodes~see Fig. 4!.98 Mode locking requires a mechanismthat results in higher net gain~gain minus loss! for a train ofshort pulses compared to cw operation. This can be donan active element, such as an acousto-optic modulatorpassively by saturable absorption~real or effective!. PassiveML yields the shortest pulses because, up to a limit,self-adjusting mechanism becomes more effective thantive mode locking, which can no longer keep pace with tultrashort time scale associated with shorter pulses.99 Realsaturable absorption occurs in a material with a finite numof absorbers, for example, a dye or semiconductor. Real srable absorption usually has a finite response time assocwith relaxation of the excited state. This typically limits thshortest pulse widths that can be obtained. Effective srable absorption typically utilizes the nonlinear index of rfraction of some material together with spatial effects orterference to produce a higher net gain for more intepulses. The ultimate limit on minimum pulse duration inML laser is due to an interplay among the ML mechanis

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~saturable absorption!, group velocity dispersion~GVD!, andnet gain bandwidth. Strong coupling of the cavity modesnecessary for successfully using mode-locked lasersoptical-frequency synthesis. Mode-locking techniques tutilize essentially instantaneous response, such as theeffect, provide the strongest coupling, and therefore arepreferred technique. The Kerr lens mode-locking techniqdescribed below currently dominates the field and provithe characteristics required.

Because of its excellent performance and relative splicity, the KLM Ti:S laser has become the dominant lasfor generating ultrashort optical pulses. A diagram of a tycal KLM Ti:S laser is shown in Fig. 5. The Ti:S crystalpumped by green light from either an Ar1-ion laser~all linesor 514 nm! or a diode-pumped solid state~DPSS! laser emit-ting 532 nm. Ti:S absorbs 532 more efficiently, so 4–5 Wpump light is typically used from a DPSS laser, while 6–8of light from an Ar1-ion laser is usually required. The Ti:crystal provides gain and serves as the nonlinear materiamode locking. The prisms compensate for the group velodispersion~GVD! in the gain crystal.100 Since the discoveryof KLM, 94,95 the pulse width obtained directly from the Mlaser has been shortened by approximately an order of mnitude by first optimizing the intracavity dispersion96 andthen using dispersion compensating mirrors.101,102 yieldingpulses that are less than 6 fs in duration, i.e., less thanoptical cycles. Here we will briefly review how a KLM laseworks. While there are other ML lasers and mode-locktechniques, we will not discuss them because of the ubiqof KLM lasers at the present time. We also note that pulof similar duration were achieved earlier,103 however thisresult relied on external amplification at a low repetitiowith pulse broadening and compression, which does notserve a suitable comb structure for optical-frequency synsis.

The primary reason for using Ti:S is its enormous gbandwidth, which is necessary for supporting ultrashpulses by the Fourier relationship. The gain band is typicaquoted as extending from 700 to 1000 nm, although lascan be achieved well beyond 1000 nm. If this entire bawidth could be mode locked as a hyperbolic secant or Gaian pulse, the resulting pulse width would be 2.5–3 fs. Whthis much bandwidth has been mode locked, the spectrufar from smooth, leading to longer pulses.

The Ti:S crystal also provides the mode-locking mecnism in these lasers. This is due to the nonlinear indexrefraction~Kerr effect!, which is manifested as an increasethe index of refraction as the optical intensity increases.cause the intracavity beam’s transverse intensity profile

FIG. 5. Schematic of a typical Kerr-lens mode-locked Ti:sapphire lase


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Gaussian, a Gaussian index profile is created in the Ti:S ctal. A Gaussian index profile is equivalent to a lens, hencebeam slightly focuses, with the focusing increasing withcreasing optical intensity. Together with a correctly potioned effective aperture, the nonlinear~Kerr! lens can act asa saturable absorber, i.e., high intensities are focusedhence transmit fully through the aperture while low intenties experience losses~see Fig. 6!. Since short pulses produchigher peak powers, they experience lower loss, makmode-locked operation favorable. While some KLM laseinclude an explicit aperture, the small size of the gain regcan act as one. This mode-locking mechanism has the adtage of being essentially instantaneous; no real excitatiocreated that needs to relax. It has the disadvantages obeing self-starting and of requiring a critical misalignmefrom optimum cw operation.

Spectral dispersion in the Ti:S crystal due to the vartion of the index of refraction with wavelength will result itemporal spreading of the pulse each time it traversescrystal. At these wavelengths, sapphire displays ‘‘normdispersion, where longer wavelengths travel faster thshorter ones. To counteract this, a prism sequence is usewhich the first prism spatially disperses the pulse, causthe long wavelength components to travel through mglass in the second prism than the shorter wavelencomponents.100 The net effect is to generate ‘‘anomalousdispersion to counteract the normal dispersion in the Tcrystal. The spatial dispersion is undone by placing the prpair at one end of the cavity so that the pulse retraces itsthrough the prisms. With the optimum choice of materialis possible to minimize both GVD and third order dispersioyielding operation that is limited by the fourth ordedispersion.104 It is also possible to generate anomalous dpersion with dielectric mirrors;105 these are typically called‘‘chirped mirrors.’’ They have the advantage of allowinshorter cavity lengths but the disadvantage of less adjustaity if used alone. Also, at present, they are only availafrom a small number of suppliers. Chirped mirrors also alloadditional control over higher order dispersion and habeen used in combination with prisms to produce pulses eshorter than those achieved using prisms alone.101,102

FIG. 6. Schematic of the Kerr-lens mechanism. The upper diagram shthat, for low intensity, much of the intensity does not pass throughaperture. At high intensity, the Gaussian index profile, which is due tononlinear index of refraction, acts as a lens and focuses the beam ancreases net transmission.


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B. Frequency spectrum of mode-locked lasers

To successfully exploit ML lasers for the generationfrequency combs with known absolute frequencies, it is nessary to understand the spectrum emitted by a mode-lolaser, how it arises, and how it can be controlled. While ialways possible to describe the operation of these laseeither the time or frequency domain, details of connectthe two are rarely presented.106 Unless sufficient care istaken, it is easy for misunderstandings to arise when atteming to convert understanding in one domain into the othe

We will start with a time domain description of thpulses emitted by a mode-locked laser. This is shown in7~a!. The laser emits a pulse every time the pulse circulatinside the cavity impinges on the output coupler. This resin a train of pulses separated by timet5 l c /vg where l c isthe length of the cavity andvg is the net group velocity. Dueto dispersion in the cavity, the group and phase velocitiesnot equal. This results in a phase shift of the ‘‘carrier’’ wawith respect to the peak of the envelope for each roundtWe designate the shift between successive pulses asDf. It isgiven by

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1

vpD l cvc mod 2p,

wherevp is the intracavity phase velocity andvc is the car-rier frequency. This pulse-to-pulse shift is shown in Fig. 7~a!.The overall carrier-envelope phase of a given pulse, whobviously changes from pulse to pulse ifDfÞ0, includes anoffset which does not affect the frequency spectrum. Wetherefore not concerned with this offset phase here, althoit is a subject of current interest in the ultrafast optics comunity.

FIG. 7. Time–frequency correspondence betweenDf andd. ~a! In the timedomain, the relative phase between the carrier~solid line! and the envelope~dotted line! evolves from pulse to pulse byDf. Generally, the absolutephase is given byf5Df(t/t)1f0 , wheref0 is an unknown overall con-stant phase.~b! In the frequency domain, the elements of the frequencomb of the mode-locked pulse train are spaced byf rep. The entire comb~solid line! is offset from integer multiples~dotted line! of f rep by an offsetfrequencyd5Df f rep/2p. Without active stabilization,d is a dynamic quan-tity that is sensitive to perturbation of the laser. HenceDf changes in anondeterministic manner from pulse to pulse in an unstabilized laser.


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1. Comb spacing

The frequency spectrum of the pulse train emitted bML laser consists of a comb of frequencies. The spacingthe comb lines@Fig. 7~b!# is simply determined by the repetition rate of the laser. This is easily obtained by Fourtransforming a series ofd-function-like pulses over time. Therepetition rate is in turn determined by the group velocand the length of the cavity.

2. Comb position

If all of the pulses had the same phase relative toenvelope, i.e.,Df50, then the spectrum is simply a seriescomb lines with frequencies that are integer multiples ofrepetition rate. However this is not in general true, due todifference between the group and phase velocities insidecavity. To calculate the effect of a pulse-to-pulse phase son the spectrum, we write the electric field,E(t), of a pulsetrain. At a fixed spatial location, let the field of a single pulbeE1(t)5E(t)ei (vct1f0). Then the field for a train of pulseis

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5(n

E~ t2nt!ei ~vct1n~Df2vct!1f0!,

whereE(t) is the envelope,vc is the carrier frequency,f0 isthe overall phase offset, andt is the time between pulses~forpulses emitted by a mode-locked lasert5tg , wheretg is thegroup roundtrip delay time of the laser cavity!. If we take theFourier transform, we obtain

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letting E(v)5*E(t)e2 ivtdt and recalling the identity* f (x2a)e2 iaxdx5e2 iaa* f (x)e2 iaxdx we obtain

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The significant components in the spectrum are the oneswhich the exponential in the sum add coherently becausephase shift between pulsen and n11 is a multiple of 2p.EquivalentlyDf2vt52mp. This yields a comb spectrumwith frequencies

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or, converting from angular frequency,f m5m frep1d whered5Df f rep/2p and f rep51/t is the repetition frequencyHence we see that the position of the comb is offset frinteger multiples of the repetition rate by a frequencyd,


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which is determined by the pulse-to-pulse phase shift. Thshown schematically in Fig. 7~b!. Thus the frequency spectrum of a mode-locked laser is a series of comb lines wfrequencies given by

f n5n f rep1d,

where n is an integer that indexes the comb lines. Thsimple equation is the basis for much of what follows.

C. Frequency control of the spectrum

For the comb generated by a ML laser to be usefulsynthesizing optical frequencies, control of its spectrum, ithe absolute position and spacing of the comb lines, is nessary. In terms of the above description of the output putrain, this means control of the repetition rate,f rep, and thepulse-to-pulse phase shift,Df. Once the pulses have beeemitted by the laser,f rep cannot be controlled.Df can becontrolled by shifting the frequency of the comb, for eample, with an acoustooptic modulator.107 However it is gen-erally preferable to control bothf rep andDf by making ap-propriate adjustments to the operating parameters of theitself. Some experiments only require control of the repetion rate, f rep. This can easily be obtained by adjusting tcavity length using a piezoelectric transducer~PZT! to trans-late one of the end mirrors. Temperature stabilizing the bplate for the laser reduces cavity length drift and allowsordinary PZT to achieve sufficient range. PZT stacks ctwist as they expand or contract, thereby requiring a mcomplex arrangement.108

Many experiments are simplified by locking bothf rep

and d. To do so, both the roundtrip group delay and troundtrip phase delay must be controlled. Adjusting the city length changes both. If we rewritef rep andd in terms ofroundtrip delays, we find

f rep51

tg; d5

vc

2ptg~ tg2tp!,

wheretg5 l c /vg is the roundtrip group delay andtp5 l c /vp

is the roundtrip phase delay. Bothtg and tp depend onl c ,therefore another parameter must be used to control tindependently. Methods for doing this will be discussed laThe equation ford may seem unphysical because it depenon vc , which is arbitrary, however there is an implicit dpendence onvc that arises due to the dispersion invp ~andhencetp! that cancels the explicit dependence. Note thatvp

must have dispersion forvpÞvg and thatvg must be con-stant~dispersionless! for stable mode-locked operation.

D. Spectral broadening

It is generally desirable to have a comb that spansgreatest possible bandwidth. At the most basic level, thisimply because it allows measurement of the largest possfrequency intervals. Ultimately, when the output spectrfrom a comb generator is sufficiently wide, it is possibledetermine the absolute optical frequencies of the comb ldirectly from a microwave clock, i.e., without relying ointermediate phase-locked oscillators. The simplest of thtechniques requires a spectrum that spans an octave, i.e


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high frequency components have twice the frequency oflow frequency components. In the transform limit and forspectrum with a single peak, this bandwidth would corspond to a single cycle pulse, which has yet to be achieveoptical frequencies. Thus it has been necessary to relyexternal broadening of the spectrum. Fortunately, the optheterodyne technique employed for detection of single cocomponents is very sensitive, therefore it is not necessarhave a 3 dBbandwidth of an octave. Detection is typicalfeasible even when the power at the octave points is 10dB below the peak.

Self-phase modulation~SPM! occurs in a medium with anonlinear index of refraction, i.e., a third order optical nolinearity. It generates new frequencies, thereby broadenthe spectrum of a pulse. This process occurs in the gcrystal of a ML laser and can result in output spectra texceed the gain bandwidth. In the frequency domain it cbe viewed as four-wave mixing between the comb lines. Tamount of broadening increases with the peak power percross-sectional area of the pulse in the nonlinear mediConsequently, an optical fiber is often used as a nonlinmedium because it confines the optical power into a smarea and results in an interaction length that is longer tcould be obtained in a simple focus. While the small crosection can be maintained for very long distances, the hpeak power in fact is typically only maintained for a rathshort distance because of group velocity dispersion infiber, which stretches the pulse over time and reducespeak power. Nevertheless, it has been possible to generaoctave of usable bandwidth with an ordinary single spamode optical fiber by starting with very short and intenpulses from a low repetition rate laser.109 The low repetitionrate increases the energy per pulse despite limited avepower, thereby increasing the broadening. The fiber was o3 mm long. The pulses were pre-chirped before belaunched into the fiber so that the dispersion in the fiwould recompress them.

The recent development of microstructured fiber hmade it possible to easily achieve well in excess of an octbandwidth using the output from an ordinary KLM Ti:laser.97,110 Microstructured fibers consist of a fused siliccore surrounded by air holes. This design yields a waveguwith a very high contrast of the effective index of refractiobetween the core and the cladding. The resultant waveging provides a long interaction length with a minimum beacross section. In addition, the waveguiding permits designof the zero point of the group velocity dispersion to be withthe operating spectrum of a Ti:S laser~for ordinary fiber thegroup velocity dispersion zero can only occur for wavlengths longer than 1.3mm!. Figure 8 shows the dispersiocurves for several different core diameters.110 This dispersionproperty means that the pulse does not disperse and thelinear interaction occurs over a long distance~centimeters tometers, rather than millimeters in ordinary fiber!. Typical in-put and output spectra are shown in Fig. 9. The output sptrum is very sensitive to the launched power and polarition. It is also sensitive to the spectral position relative tozero-GVD point and the chirp of the incident pulse. Becauthe fiber displays anomalous dispersion, pre-compensatio


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FIG. 8. Properties of the microstructure fiber.~a! Calculated group velocity dispersion for a silica fiber with a core-cladding index difference of 0.1 and adiameter of 1mm ~solid curve! and a fiber with a core-cladding index difference of 0.3 and a core diameter of 2mm ~dotted curve!. The GVD of bulk silicais shown by the dashed curve.~b! Measured GVD of the microstructure fiber~squares! and standard single mode fiber~circles!. ~c! Calculated waveguideGVD contribution~dashed line!, material dispersion of bulk silica~dotted curve!, and resulting net GVD~solid! curve of the microstructure fiber. Experimentally measured values from~b! are shown by circles.~d! Calculated net GVD of the microstructure fiber as the fiber dimensions are scaled for 1.4and 4mm core diameters.~Reproduced from Refs. 97 and 110 with permission of the authors.!

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the dispersion is not required if the laser spectrum is centetoward the long wavelength side of the zero-GVD point~i.e.,in the anomalous dispersion region!. Because the pulse ituned close to a zero-GVD point, the output phase profildominated by third order dispersion.111

Nonlinear processes in the microstructure fiber also pduce broadband noise in the radio-frequency spectrumphotodiode that detects the pulse train. This has deletereffects on the optical-frequency measurements describedlow because the noise can mask the heterodyne beats@seeFig. 10~c!#. The noise increases with increasing input puenergy and appears to display threshold behavior.112 Thismakes it preferable to use short input pulses since less brening, and hence less input power, is required. The eorigin of the noise is currently uncertain, but it may be dueguided acoustic-wave Brillouin scattering,113–115 Ramanscattering, or modulation instabilities. This problem canovercome by increasing the resolution of the detection stem.

FIG. 9. Continuum generation by and air–silica microstructured fiber. Sphase modulation in the microstructured fiber broadens the output olaser so that is spans more than one octave. The spectra are offset verfor clarity.


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V. OPTICAL-FREQUENCY MEASUREMENTS USINGMODE-LOCKED LASERS

Optical frequency measurements are typically madedetermine the absolute optical frequency of an atomic, mlecular, or ionic transition. Typically, a single frequency lasis locked to an isolated transition, which may be amongrich manifold of transitions, and then the frequency of tsingle frequency laser is measured. Mode-locked lasers

f-heally

FIG. 10. Correspondence between optical frequencies and heterodynein the rf spectrum.~a! Optical spectrum of a mode-locked laser with modspacingf rep plus a single frequency laser at frequencyf l . ~b! Detection bya fast photodiode yields a rf spectrum with equally spaced modes due tmode-locked laser~long lines! plus a pair of intervening beats signals dueheterodyning~short lines!. ~c! Typical experimental rf spectrum showinboth repetition rate beats and heterodyne beats.


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typically employed in measurement of the frequency ofsingle frequency laser. This is performed by heterodyningsingle frequency laser against nearby optical comb linesthe mode-locked laser. The resulting heterodyne rf sig~see Fig. 10! contains beats at frequenciesf b5u f l2n f rep

2du where f l is the frequency of the single frequency lasf rep the repetition rate of the ML laser,n an integer, andd theoffset frequency of the ML laser~see Sec. I B 2!. This yields,in the rf output of a photodetector, a pair of beats withevery rf frequency interval betweenm frep and (m11) f rep.One member of the pair arises from comb lines withn f rep

. f l and the other from lines withn f rep, f l . Both the beatfrequencies andf rep can be easily measured with standardequipment. Typically,f rep itself is not measured, but ratheone of its harmonics~10th to;100th harmonic!, to yield amore accurate measurement in a given measurementsince the uncertainty is divided by the harmonic numbMore sophisticated techniques can yield even greater prsion in the measurement off rep and can, in addition, allowcomparison of the stability of two optical frequencies indpendent off rep.116 To map from these beat frequency mesurements to the optical frequencyf

l, we need to known and

d. Sincen is an integer, we can estimate it if we have preous knowledge aboutf l to within 6 f rep/4. Typically f rep isgreater than 80 MHz, putting this requirement easily inrange of commercially available wave meters, which haveaccuracy of;25 MHz. Alternatively,n can be determined bydithering f rep and analyzing the resulting changes in the otical beat note. Thus, measurement ofd is the remainingproblem. This can be done by comparison of the comb tointermediate optical-frequency standard, which is discusin Sec. V B, or directly from the microwave cesium standaas will be described below. Several such techniques for dilinks between microwave and optical frequencies arescribed in Sec. V C.

A. Locking techniques

Although measurement off rep andd are in principle suf-ficient to determine an absolute optical frequency, it is gerally preferable to use the measurements in a feedbackto actively stabilize or lock one or both of them to suitabvalues. If this is not done, then they must be measuredmultaneously with each other and withf b to obtain meaning-ful results. In Sec. III, we described the physical quantitiesthe laser that determinef rep and d. Here we describe thetechnical details of adjusting these physical quantities.

The cavity length is a key parameter for locking tcomb spectrum. It is useful to examine a specific examplethe sensitivity to the length. Consider a laser with a repetitrate of 100 MHz and a center wavelength of 790 nm~;380THz!. The repetition rate corresponds to a cavity length1.5 m which is 3.8 million wavelengths long~roundtrip!. Alength decrease ofl/2 gives an optical-frequency shift oexactly one order,1100 MHz, for all of the optical-frequency components. The corresponding shift of theMHz repetition rate is (100 MHz)/(380 THz)512.6331027 fractionally or126.31 Hz. Since the cavity length imost sensitive to environmental perturbations on both~vibrations! and slow~temperature! time scales, good contro


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of it warrants close attention. Furthermore, we note tlength variations have a much larger effect on the optifrequency of a given comb line than on the repetition rat

Since the adjustment and locking of two quantities,f rep

and d, are desired, a second parameter in addition tocavity length must be controlled. We will present detailsusing the swivel angle of the back mirror to obtain the dsired degree of freedom.85 This is the mostly widely imple-mented and the one with which we are most familiar. We wshow that the swivel angle changes the group delay, whwould lead one to believe that it should be used to conf rep. However, we will also show thatd does not depend onl c . Thus, it is necessary to controlf rep by adjustingl c anddby the swivel angle. This is counterintuitive, and occurs bcause in order for a change in the swivel angle to leavefrequency of a given optical mode constant~i.e., as a fixedpoint in the comb spectrum! it must changed to compensatefor the change inf rep.

We note that the language of ‘‘fixed points’’ in the comspectrum, introduced by Telle, can also be very usefuldiscussing the combined action of various control parameon the comb spectrum.117 This yields insight into optimumcontrol strategies and noise contributions.

1. Comb spacing

The comb spacing is given byf rep51/tg5vg / l c , wherevg is the roundtrip group velocity andl c is the cavity length.The simplest way of lockingf rep is by adjustingl c . Mount-ing either end mirror in the laser cavity on a translatipiezo-electric actuator easily does this~see Fig. 5!. The ac-tuator is typically driven by a phase-locked loop that coparesf rep or one of its harmonics to an external clock. Forenvironmentally isolated laser, the short time jitter inf rep islower than in most electronic oscillators, althoughf rep driftsover long times. Thus the locking circuit needs to be cafully designed for a sufficiently small bandwidth so thatf rep

does not have fast noise added while its slow drift is beeliminated.

Locking the comb spacing alone is sufficient for mesurements that are not sensitive to the comb position sucmeasurement of the frequency difference between two la~f L1 and f L2!. For example, suppose the frequency off L1 ishigher than its beating comb line, whilef L2 is lower than itscorresponding comb line. The sum of these two beatstaken using a double-balanced mixer, the resulting sumquency is f s5 f L12n f rep2d2( f L22m frep2d)5( f L12 f L2)2(m2n) f rep. Where n f rep1d, f L1 and m frep1d. f L2 , aminus sign arises due to the absolute value in determinthe beat frequencies~see the expression forf b above!. If f L1

and f L2 are known with accuracy better thanf rep/2, then(m2n) can be determined and hencef L12 f L2 from f s .

2. Comb position

The frequency of a given comb line is given byf n

5n f rep1d, wheren is a large integer. This means that simply changing the cavity length can control the frequencycomb lines. However, this also changes the comb spacwhich is undesirable if a measurement spans a large numof comb lines. Controllingd instead of f rep allows a rigid


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shift of the comb positions, i.e., the frequency of all the lincan be changed without changing the spacing.

The comb position depends on the phase delay,tp , andthe group delay,tg . Each delay in turn depends on the cavlength. In addition, the comb position depends on the carfrequency, which is determined by the lasing spectrum.obtain independent control of both the comb spacingposition, an additional parameter, besides the cavity lenmust be adjusted.

A controllable group delay is produced by a small rotion about a vertical axis~swivel! of the end mirror of thelaser in the arm that contains the prisms~see Fig. 5!. This isbecause the different spectral components are spread outially across the mirror. The dispersion in the prisms resuin a linear relationship between the spatial coordinate andwavelength. Hence the mirror swivel provides a linear phwith frequency, which is equivalent to a group delay.118 Thegroup delay depends linearly on the angle for small anglethe pivot point for the mirror corresponds to the carrier fquency, then the effective cavity length does not change.angle by which the mirror is swiveled is very small, appromately 1024 rad. If we assume that swiveling the mirror onchanges the group delay by an amountau, whereu is theangle of the mirror anda is a constant that depends on tspatial dispersion on the mirror and has units of s/rad, twe rewrite

f rep51

l c /vg1au; d5

vc

2p S 12l c /vp

~ l c /vg!1au D .

From these equations we can derive how the comb frequcies depend on the control parameters,l c andu. To do so wewill need the total differentials

d f rep5] f rep

]udu1

] f rep

] l cdlc

52a

~ l c /vg1au!2 du21/vg

~ l c /vg1au!2 dlc

>2avg

2

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l c2 dlc ,

dd5vc

2p

l ca

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au/vp

@~ l c /vg!1au#2 dlc

>vc

2p

vg2a

vpl cdu,

where the final expressions are in the approximationau! l c /vg . From this, we see thatd is controlled solely byu. The concomitant change inf rep can be compensated for bchanges in the cavity length.

Physically, we can understand these relationshipsconsidering how the optical frequencies of individual moddepend on the length and swivel angle. This is shown scmatically in Fig. 11 for an example of a 1 cm long cavity.From the upper part it is clear that the dominant effect


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changing the cavity length is the position change of eaoptical mode, the repetition rate~spacing between modes!changes much less, by a ratio of the repetition rate/optfrequency. Thus the change in the repetition rate is onlyparent for larger changes in the length~cf. the left and rightupper panels in Fig. 11!. This is because the repetition ratemultiplied by the mode number to reach the optical frquency. Swiveling the mirror does not change the frequeof the mode at the pivot point~mode 10 005 in the lowerpanel of Fig. 11!, but causes the adjacent modes to moveopposite directions. This can be understood in terms ofrequency dependent cavity length, in this example itcreases for decreasing frequencies.

3. Comb position and spacing

It is often desirable to simultaneously control/lock bothe comb position and spacing, or an equivalent set oframeters, say, the position of two comb lines. In an idsituation, orthogonal control off rep andd may be desirable toallow the servo loops to operate independently. If this canbe achieved, one servo loop will have to correct chanmade by the other. This is not a problem if they have vedifferent response speeds. If their responses are similarteraction between loops can lead to problems, includingcillation. If necessary, orthogonalization can be achievedeither mechanical design in some cases, or by electromeans in all cases.

Note that we earlier assumed that the pivot point oftilting mirror corresponds to the carrier frequency. Thisoverly restrictive, because moving the pivot point will givan additional parameter that can be adjusted to help orthonalize the parameters of interest. We can generalize our trment to include an adjustable pivot point by allowingl c todepend onu. The resulting analysis shows that it is imposible to orthogonalized and f rep by a simple choice of the

FIG. 11. Schematic showing how the modes of a cavity depend oncavity length ~upper panels! and swivel angle of the mirror behind thprisms. For illustrative purposes, a 1 cmcavity was used. Comparison of thleft and right upper panels shows the weak dependence of the mode spon cavity length.


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FIG. 12. Schematic of a circuit thaactuates the length-swivel PZT baseon two error signal inputs~V1 andV2!.The circuit allows an appropriate admixture to be generated by adjustinthe potentiometers, labeled bya andb.

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pivot point. Further examination of the lower panel in Fig.makes it clear why this must be so. The comb line at~ornear! the pivot point does not change its frequency whenswivel angle changes. This is inconsistent with being ablecontrol d while holding f rep constant, which is equivalent ta rigid shift of all of the comb lines~none are constanttherefore there can be no pivot point!.

Although we cannot orthogonalized and f rep by choos-ing a pivot, the analysis does show how to do so, we meneed to make the length change of the cavity proportionathe swivel angle. This can be implemented electronically aamounts to inverting the linear matrix equation connectdd andd f rep to du anddlc . An electronic remedy also addresses the practical issue that experimental error siggenerated to control the comb often contain a mixture of tdegrees of freedom. For example, the two error signalscorrespond to the position of a single comb line~usually withrespect to a nearby single frequency laser! and the combspacing. A more interesting situation is when the error snals correspond to the position of two comb lines. Typicathis will be obtained by beating a comb line on the lofrequency side of the spectrum with a single frequency laand a comb line on the high frequency side with the secharmonic of the single frequency laser. In both of thecases, the error signals contain a mixture ofd and f rep,which in turn are determined by a mixture of the contparametersl c andu.

As will be evident below, having a pair of error signathat correspond to the positions of two comb lines is the minteresting.119 This is obtained by beating the two comb linagainst two single frequency lasers with frequenciesf L1 andf L2 . The beat frequencies are given byf bi5 f Li2(ni f rep

1d) with i 51, 2 ~for clarity we have assumed thatf Li isabove the nearest comb line with indexni!. Taking the dif-ferentials of these equations and inverting the result wetain

d f rep5d fb12d fb2

n22n1→ 1

n~d fb12d fb2!,

dd5n1 /n2d fb22d fb1

12n1 /n2→d fb222d fb1 ,

where the expressions after the arrows are forf b252 f b1 ,i.e., we are using the fundamental and second harmonics


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single laser withn being the index of the laser comb modjust below the fundamental frequency. These equationsbe combined with those connectingdd andd f rep to du anddlc to obtain

du52pvpl c

vcvg2a

~d fb222d fb1!,

dlc52l c2

vgn@~122An!d fb11~An21!d fb2#,

A52pvp

vcl c.

These equations directly connect the observables withcontrol parameters for this configuration~note that the prod-uct An is of the order of 1!. Equations of similar form havebeen derived without determining the values of tcoefficients.119

It is desirable to design an electronic orthogonalizatcircuit that is completely general. Although we have exprsions telling us what the coefficients connecting the obseables and control parameters should be, we typically doknow the values of all of the parameters that appear incoefficients. Furthermore, there may be technical factorscause unwanted mixing of the control parameters or esignals that must be compensated for. Such coupling arfor example, because the two transducers have differingquency response bandwidths. Finally, a general circuitreadily be adapted to experimental configurations other tthe one discussed in detail above.

The electronic implementation depends on the actuamechanisms. We employed a piezoelectric transducer tubwhich application of a transverse~between the inside andoutside of the tube! voltage results in a change of the tublength. By utilizing a split outer electrode, the PZT tube cbe made to bend in proportion to the voltage differencetween the two outer electrodes. The common mode voltaor the voltage applied to the inner electrode, causes theto change its length, which we will designate ‘‘piston’’ modMounting the end mirror of the laser cavity on one end of tPZT tube allows it to be both translated and swiveled, whcorresponds to changing the cavity length (l c) and the mirrorangle~u!.

The circuit shown in Fig. 12 allows a mixture of tw


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error inputs to be applied to both the piston and swimodes of the mirror. The coefficientsa andb are adjusted viathe potentiometers where the neutral midpoints correspona5b50 and more generally

a,b}Rup2Rlow

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whereRi represents the resistance above or below the fepoint. The gains of the amplifiers,g1 andg2 , provide addi-tional degrees of freedom. The length and swivel anglerelated to the input voltages by

l c}g1V11g2V12g2aV2

u}2g2~bV11V2!21,a,b,1.

Thus if a5b50, then l c only responds toV1 and u onlyresponds toV2 . By adjustinga,l c can be made to also respond toV2 , with either sign, this is also the case forb andV1 .

4. Other control mechanisms

In addition to tilting the mirror after the prism sequences, the difference between the group and phase dcan also be adjusted by changing the amount of glass incavity120 or adjusting the pump power.120,121

The amount of glass can be changed by moving a pror by introducing glass wedges. Changing the amountglass changes the difference between the group delayphase delay due to dispersion in the glass. It has the divantage of also changing the effective cavity length. Furthmore, rapid response time in a servo loop cannot be achiebecause of the limitation of mechanical action on the retively high mass of the glass prism or wedge.

Changing the pump power changes the power of thetracavity pulse. This has empirically been shown to chathe pulse-to-pulse phase.120 Intuitively, it might be expectedthat this would lead to a change in the nonlinear phase sexperienced by the pulses as they traverse the crystal. Hever, a careful derivation shows that the group velocity adepends on the pulse intensity in such a way as to mocancel out the phase shift.122 Thus it is not too surprising thathe correlation between power and phase shift is the oppoof what is expected from this simple picture.120 The phaseshift is attributed to the shifting of the spectrum that accopanies changing the power. This will yield a changing grodelay due to group velocity dispersion in the cavity. Sucheffect is similar to the control ofu. Nevertheless, it is possible to achieve very tight locking due to the much largservo bandwidth afforded by an optical modulator compato physically moving an optical element.121 Although theservo loop stabilizes the frequency spectrum of the molocked laser, it also reduces amplitude noise, presumablycause amplitude noise is converted into phase noise bylinear processes in the laser. More work is needed to funderstand the mechanism and to exploit it for control ofcomb.

Frequency shifts of the comb have also been obtainedfocusing light from a diode laser into the Ti:sapphicrystal.123 The diode laser light depletes the gain, there


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altering the intracavity intensity and thus changing the nlinear phase shift. High modulation bandwidth can alsoobtained using this technique.

The offset frequency of the comb can also be controlexternally to the laser cavity by using and acouto-opmodulator. This has been exploited to lock the comb toreference cavity.107

B. Measurements using an intermediate reference

The simplest use of the frequency comb from a molocked laser is to measure the frequency difference betwtwo optical sources, typically single frequency lasers. If tabsolute optical frequency of one of the two sourcesknown, this yields an absolute measurement of the otThis strategy had been demonstrated by Kourogi andworkers using the OFCG described in Sec. III B 5.66 In Secs.V B1, V B2, and VC, we present two representative recexamples of such measurements based on wider combserated by ML lasers.

1. Cesium D 1 line

One of the first measurements to utilize the broadbacomb from a KLM Ti:S laser was performed by Ha¨nsch’sgroup at MPQ.85 This experiment measured the absolute fquency of the cesiumD1 line and was, we believe, the firsthat utilized phase coherent control of the position of a fetosecond laser comb. The KLM Ti:S laser generated 73pulses and was used as part of a complex frequency c~see Fig. 13! that connected a 3.39mm CH4 stabilized HeNe

FIG. 13. Schematic of the frequency chain used to measure the frequenthe cesiumD1 line using a mode-locked laser. The methane-stabilized Helaser served as an intermediate.~Reproduced from Ref. 85 with permissioof the authors.!


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laser at 88.4 THz to a laser locked on the cesiumD1 line viaa saturation spectrometer. The KLM Ti:S laser spanned18.4 THz gap between the fourth harmonic of the HeNe la~353.5 THz! to the frequency of the cesiumD1 line of 335.1THz. The HeNe laser was transportable and its absolutequency had been measured earlier at the PhysikaliTechnische Bundesanstalt~PTB! in Braunschweig using theitraditional frequency chain starting from the microwave csium clock. The MPQ frequency chain involved a color ceter laser operating at 1696 nm and two diode lasers at 8and 894.1 nm. Four phase-locked loops were required tothe lasers and the femtosecond comb. Only the positionthe comb was controlled, not the spacing~repetition rate!,which was simply measured to determine the frequencyterval. The experiment yields results85 ~see Fig. 14! with anaccuracy that was three orders of magnitude better thanvious measurements. When combined with other measments, it provided a new determination of the fine structconstant.

2. Measurement across 104 THz

The measurements described above stimulated intinterest in the optical frequency metrology communiClearly the goal was to increase the frequency gap that cbe spanned using a ML laser. This led to work at JILA thdemonstrated measurement across 104 THz.88 This was

FIG. 14. Fg54→Fe54 component of the CsD1 transition. The absolutefrequency is determined from the fitted Lorentzian~solid line!. ~Reproducedfrom Ref. 85 with permission of the authors.!

FIG. 15. Experimental setup used to bridge the 104 THz gap betwI2-stabilized Nd:YAG and Rb-stabilized Ti:sapphire single frequency las


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achieved by starting with a laser that generated pulsesproximately 10 fs in duration~see Fig. 15!. The alreadybroad spectrum of these pulses was further increased byphase modulation in standard single mode optical fi~SMF!. The usable width of the resulting spectrum was aproximately 165 THz. One side of the 104 THz gap that wmeasured was the frequency of the 5S1/2(F53)→5D5/2(F55) two photon transition of85Rb at 778 nm. A single fre-quency Ti:S laser was locked to this transition. The othside of the gap was a single frequency Nd:YAG laser at 10nm that had its second harmonic locked to thea10 compo-nent of theR(56)32-0 transition in127I2. The 778 nm laserserved as a known reference, with a Comite´ International desPoids et Mesures~CIPM! recommended uncertainty of onl65 kHz. The measurement yielded an improved measument of the frequency of the 1064 nm laser~Fig. 16! whichhad a CIPM recommended uncertainty of620 kHz. Theresults yielded an offset of 23.1 kHz from the CIPM recommended value with a measurement uncertainty of62.8 kHz,with additional65 kHz uncertainty due to the 778 nm stadard. This experiment began the present epoch in whichmeasurement inaccuracy is less than that of the availstandards, due to femtosecond lasers.

C. Direct optical to microwave synthesis

While the measurements described above showedpromise of the large bandwidth comb generated by a femsecond ML laser, it quickly became clear that it was possito use them to measure optical frequencies directly frommicrowave primary standard for frequency. There are sevtechniques for doing this,83 depending on how much bandwidth is generated by the ML laser. The simplest of therequires a spectrum that spans a full octave, i.e., becausblue end of the spectrum is twice the frequency of theend. Although a ML laser that directly generates an octavespectrum has very recently been reported,124 all of the fre-n

s.

FIG. 16. Measured Nd:YAG frequency with respect to the CIPM recomended value of 281 630 111 740 kHz. The dashed line is the averageplotted points~23.2 kHz!. The inset is a histogram of the;800 individualmeasurements on the third day. Note that direct measurement of oustandard~Fig. 18! reveals it to be23.765 kHz relative to the CIPM rec-ommended value.


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quency measurements so far have used external broadeof the spectrum to achieve the full octave~see Sec. III D!. Inthe following we describe three measurements that utilizfull octave. We also describe an important measurementused a more complex chain with several bisection stagethat a full octave was not necessary.

1. Methane stabilized HeNe and hydrogen 1S – 2Stransition with bisection

The first experiment to utilize a mode-locked lasermake absolute optical frequency measurements referedirectly to a microwave clock was performed at MPQ.92 Thisexperiment simultaneously measured the frequency omethane stabilized HeNe laser and the 1S– 2S interval inhydrogen. Without an octave spanning mode-locked laserequired a more complex frequency chain than the teniques described in the following. The chain included fiphase-locked intermediate oscillators and a single bisecstage~see Fig. 17!. It used the mode-locked laser combspan the frequency intervals between 4f and 4f –D f andbetween 4f –D f and 3.5f –D f , where f ;88.4 THz is thefrequency of the HeNe laser andD f is chosen so tha28f – 8D f is the frequency of the H 1S– 2S transition. Therepetition rate of the mode-locked laser was locked to asium atomic clock. Note that measurement of the inter4 f –D f to 3.5f –D f yields 0.5f and hencef itself.

Later work by the same group compared the resultsthis chain with andf – 2 f chains~described below!.121 Theresults put an upper limit of 5.1310216 on the uncertainty ofthe newf – 2 f chains.

FIG. 17. Frequency chain used to simultaneously measure the frequenthe hydrogen 1S– 2S transition and that of a methane stabilized HeNe las~Reproduced from Ref. 92 with permission of the authors.!


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2. I2 stabilized Nd:YAG by f – 2 f

Given an octave spanning frequency comb and the oput of a stabilized single frequency laser, the frequencythe single frequency laser can be determined by measuthe frequency interval between the laser and its secondmonic, i.e., 2f 2 f 5 f , wheref is the frequency of the singlefrequency laser. This technique requires the least amouneffort to control the comb, only the comb spacing needs tocontrolled. In principle, the comb spacing does not havebe locked, it only needs to be measured. Of course, thatadd uncertainty to the final frequency measured if the msurement intervals are not exactly coincident due to gatingthe counters. As described in Sec. V A1, if the beat frequcies are chosen so that the frequency difference betwefand a lower frequency comb line is measured at the satime the frequency difference between 2f and a higher fre-quency comb line is measured and the appropriate differeor sum taken, the absolute frequency position of the comcanceled and does not enter into the final measurement@seeFig. 18~a!#. This means that the comb position does not neto be controlled and any change in it does not matter as las it occurs on a time scale that is slow compared tospeed with which the differencing occurs. This is easachieved by heterodyning the two beat signals@see Fig.18~b!#.

This technique was utilized at JILA to make one of tfirst direct microwave to optical measurements using a sinmode-locked laser.91 In this experiment the octave spanninfrequency spectrum was obtained by spectral broadeningpiece of microstructured optical fiber.97,110 The resultsyielded a frequency for thea10 component of theR(56)32-0transition of 127I234.4 kHz higher than the recommende

of.

FIG. 18. Direct measurement of the absolute frequency of an I2-stabilizedNd:YAG laser. The only input is a microwave clock.~a! Relationship be-tween the single frequency laser (f 1064), its second harmonic (2f 1064), thecomb from the mode-locked laser~dashed lines! and the measured heterodyne beats,d1 and d2 . Also shown are frequencies of the Rb-stabilizeTi:sapphire laser (f 778) and I2-stabilized HeNe (f 633) measured in a secondexperiment.~b! Schematic of the setup.


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CIPM value ~see Fig. 19!. The standard deviation was 1kHz. Including uncertainty in the realization of thea10 fre-quency, the final value reported was 563 260 223 565 kHz.

3. Rb stabilized Ti:sapphire and I2 stabilized HeNe byNd:YAG f – 2 f

Once the absolute frequency of a stabilized singlequency laser is known by using an octave spanning comcan be used as a ‘‘reference’’ and the frequency of any ocomb line determined. This allows it to be used to measthe frequency of any other source that lies within the fquency span of the comb. This concept was utilized to msure the frequency of a Ti:sapphire laser stabilized to5S1/2(F53)→5D5/2(F55) two photon transition in85Rband that of a HeNe laser stabilized to thea13 component oftransition 11-5,R(127) of 127I2 ~peak i!.91 Both measure-ments yield results with uncertainties of one part in 1011 orbetter. The measurement of the metrologically important 6nm HeNe laser was carried one step further when an innational intercomparison was made between the absomeasurement using the JILA femtosecond comb and a trtional frequency chain maintained at the National ReseaCouncil of Canada.125 This time the 633 nm laser was locketo the f peak of the I2 spectrum. The agreement between tJILA and NRC measurements was 2206770 Hz, yielding ameasurement accuracy of 1.6310212. Such frequency inter-comparison between different laboratories using differtechnologies often reveals and permits studies of systemerrors of either locking or frequency measurement teniques.

Further work showed that the dominant source of insbility in these measurements was actually the microwaveerence oscillator used to lock the comb spacing.119 The factthat optical frequencies are;33106 times the repetition ratemeans that the frequency instability in the microwave sougets amplified by a similar factor. This was ascertainedlocking the frequency between the fundamental ofNd:YAG laser and the nearest comb line with a phase-locloop. The uncertainty in the apparent position of the otcomb lines was shown to increase with the frequency seration from the locked line. In general these results showfor short times the mode-locked laser is very stable, bu

FIG. 19. Summary of direct rf to optical measurement off 1064 plotted withrespect to the CIPM recommended value. The117.2 kHz average of allmeasurements is shown by the dashed line.


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drifts over longer times, necessitating the use of an extereference.

4. Self-referenced synthesizer

Direct optical synthesis from a microwave clock is posible using only a single mode-locked laser without an ailiary single frequency laser. This is done by directly frquency doubling the long wavelength portion~nearfrequencyf ! of the octave-spanning spectrum and comparit to the short wavelength side~near frequency 2f !. Thus itrequires more power in the wings of the spectrum produby the femtosecond laser than if an auxiliary single fquency cw laser were used. However, the fact that mcomb lines contribute to the heterodyne signal means thstrong beat signal can be obtained even if the doubled lighweak.

This technique was first demonstrated at JILA by Jonet al.89 A diagram of the experiment is shown in Fig. 20. Thoctave-spanning spectrum is again obtained by extebroadening in microstructure fiber. The output is spectraseparated using a dichroic mirror. The long wavelength ption is frequency doubled using ab-barium-borate crystal.Phase matching selects a portion of the spectrum near 1nm for doubling. The short wavelength portion of the spetrum is passed through an acoustooptic modulator~AOM!.The AOM shifts the frequencies of all of the comb lines. Thallows the offset frequency,d, to be locked to zero, whichwould not otherwise be possible due to degeneracy in thspectrum between thef – 2 f heterodyne beat signal and threpetition rate comb. The resulting heterodyne beat diremeasuresd and is used in a servo loop to fix the value ofd.The repetition rate is also locked to an atomic clock. The eresult is that the absolute frequencies of all of the comb liare known to within an accuracy limited only by the rf stadard.

The resulting comb was then used to measure thequency of a 778 nm single frequency Ti:sapphire laser twas locked to the 5S1/2(F53)→5D5/2(F55) two photon

FIG. 20. Schematic of the self-referenced synthesizer. Solid lines repreoptical paths and dashed lines are electrical connections.


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transition in 85Rb. This was done by combining the comwith the single frequency laser using a 50–50 beamspl@see Fig. 21~a!#. A small portion of the spectrum near 778 nis selected and the beat between the comb and the sfrequency laser measured. A histogram of the measuredquencies is shown in Fig. 21. Averaging over several dyielded a value of24.261.3 kHz compared to the CIPMrecommended value. This agrees quite well with the msurement described in section VC3 for the same Rb sys

5. Recent measurements

As this review article is being written, many more mesurements are being performed and reported. The rapidof advancement guarantees that any list will be out of datethe time of publication. A very important set of measurments, which warrants notice, is being made of candidafor use as optical clocks. These include a single Hg1 ion andcalcium in a magneto-optic trap. A single trapped Hg1 ionrecently yielded3 the narrowest optical resonance ever oserved, with a width of only 6.7 Hz, equivalent toQ;1.431014. This incredibly narrow line greatly reduces the ucertainty in the measurement of the frequency, which wrecently done using a mode-locked laser.126 The resultsyielded an uncertainty of;3 Hz, making it the most precisoptical measurement to date. A mode-locked laser hasbeen used to measure the frequency interval betweenfrequencies of the Hg1 ion and trapped Ca atoms.127 Thefrequency of Ca has also been measured at PTB usinfemtosecond laser directly.128 Other high precision measurements published during the last year include those of sinYb ~Ref. 116! and In ~Ref. 129! ions.

VI. OUTLOOK

In the last 2 years we have seen remarkable advancethe field of optical frequency synthesis and metrology, wfemtosecond mode-locked lasers becoming a truly powetool for optical frequency synthesis and metrology. The vsimplification that results from using femtosecond lasers

FIG. 21. Frequency measurement using the self-referenced comb. Thegram shows the experimental setup for measuring the two photon transin 85Rb using the self-referenced comb.


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making optical frequency metrology a routine laboratotechnique, as opposed to an enterprise that requires emous effort and expense. This paradigm shift is demstrated by the large number of results using femtoseclasers that have been already been reported in the lastmonths.

Here in Sec. VI we will briefly discuss some issues thappear to be important in the future.

A. Time domain implications

As described in Sec. III, frequency offset arises from tpulse-to-pulse shift of the carrier-envelope phase. Thus ctrolling the frequency offset is equivalent to controlling thpulse-to-pulse phase shift. This has been shown experimtally by cross correlation~see Fig. 22!. It is the first steptowards controlling the carrier-envelope phase of ultrafpulses~not just the pulse-to-pulse phase change!. The carrier-envelope phase is expected to affect extreme nonlineartics, for example, x-ray generation.130 As discussed belowthe overall carrier-envelope phase is also manifest in coent control experiments where interference between diffeorders occurs.

If an experimental technique can be developed thasensitive to the carrier-envelope phase and works withdirect output of a mode-locked oscillator~i.e., without thatamplification that will be necessary for extreme nonlineoptics!, it may in turn benefit optical-frequency synthesis bcause it may create a simpler technique for determinicontrolling the comb offset frequency.

B. Higher repetition rate

Increasing the repetition rate of the mode-locked lasebeneficial because it increases the comb spacing. Thiscreases the power in each individual comb line, which ma

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FIG. 22. Time–domain cross correlations. The upper panel shows a tycross correlation between pulsei and pulsei 12 emitted from the laseralong with the fit of the envelope. The relative phase is extracted by msuring the difference between the peak of the envelope and the nefringe. This is plotted in the lower panel as a function of locking frequen~relative to the repetition rate!. The linear fit produces the expect slope of 4pwith a small overall phase shift that is attributed to the correlator.


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the heterodyne signals with a single frequency laser stronIt also makes it easier to separate out the beat signal amothe forest of comb lines and it reduces the requirements oapriori knowledge of the frequency of a source. A Kerr-lemode-locked ring laser was demonstrated that achievesetition rates of 2 GHz.131 This laser uses dispersioncompensating mirrors, rather than a prism sequence.necessitates the use of an alternate technique for contrthe comb, such as ‘‘servoing’’ the pump power. Versionsthis laser have been used in measurements.121 Note, for agiven average power, high repetition rates correspondlower pulse energy, which reduces the external nonlinspectral broadening. Thus we expect that there is a maximuseful repetition rate.

C. Direct generation of an octave

Most rf to optical-frequency synthesizers basedmode-locked lasers reported to date use microstructuredto broaden the output spectrum of the laser so that it spafull octave. While this has proven to be an effective tecnique, it would clearly be simpler if an octave could be geerated directly from a mode-locked laser. Obtaining antave directly from the laser would also mitigate conceabout phase noise generated in the fiber used to broadespectrum. It might seem impossible for a Ti:sapphire lasegenerate such a broad spectrum due to the limited gain bwidth of Ti:sapphire of;300 nm. However lasers have oftebeen observed to generate spectral components outside ogain region. This is due to self-phase modulation in the gcrystal. Recently Ellet al.124 reported the generation of aoctave directly from a laser by including a second waistthe laser cavity. A glass plate is positioned at the secwaist, thereby yielding additional self-phase modulatioThis laser employed special pairs of double-chirped mirrso that a time focus occurred at the second waist.132

D. High precision atomic and molecular spectroscopyand coherent control

A phase stable femtosecond comb represents a mstep towards ultimate control of light fields as a general laratory tool. Many dramatic possibilities are ahead. For hresolution laser spectroscopy, the precision-frequency coprovides a tremendous opportunity for improved measument accuracy. For example, in molecular spectroscopy,ferent electronic, vibrational, and rotational transitions cbe studied simultaneously with phase coherent light of vous wavelengths, leading to determination of molecustructure and dynamics with unprecedented precision.sensitive absorption spectroscopy, multiple absorption orpersion features can be mapped efficiently with cohermultiwavelength light sources. A phase-coherent widbandwidth optical comb can also induce the desired mupath quantum interference effect for a resonantly enhantwo photon transition rate.93 This effect can be understooequally well from the frequency domain analysis and timdomain Ramsey-type interference. The multipulse inter


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ence in the time domain gives an interesting variation ageneralization of the two-pulse-based temporal coherent ctrol of the excited state wave packet.

Stabilization of the relative phase between the pulsevelope and the optical carrier should lead to more preccontrol of the pulse shape and timing, and open the doormany interesting experiments in the areas of extreme nonear optics and quantum coherent control. The first stepwards coherent control of a chemical reaction is selecexcitation of atomic or molecular excited state populatioThis is usually achieved by controlling therelative phase ofpairs or multiplets of pulses. If the phase of the pulse~s! thatarrive later in time is the same as the excited state–grostate atomic phase, the excited state population is increawhereas if the pulse is antiphased, the population is returto the ground state. With ultrashort pulses, it is possibleachievesingle pulsecoherent control by interference between pathways of different nonlinear order, for exampbetween three photon and four photon absorption. Suchterference between pathways is sensitive to the absophase. This has been discussed for multiphoton ionization133

For a pulse with a bandwidth that is close to spanningoctave, interference between one and two photon pathwwill be possible, thereby lessening the power requirementsimulation of this is shown in Fig. 23~a!, where the upperstate population is plotted as a function of absolute phasethe ideal three-level system, shown in the inset. In Fig. 23~b!,the dependence on pulse width is shown. The effect is msurable for pulses that are far from spanning an octave. Hever real atoms do not have perfectly spaced levels, whwill reduce the effect, and this simple simulation ignorselection rules.

FIG. 23. Results of a simple simulation showing that the upper state polation of the three-level system is sensitive to the absolute pulse phaseupper state can be reached by both single photon and two photon pathThe phase dependence is shown in~a! and the dependence on the pulswidth is shown in~b!.


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E. Optical clocks

Measurement of absolute optical frequencies has sdenly become a rather simple and straightforward task.tablished standards can now be easily recalibrated125 andmeasurement precision has reached an unprecedelevel.134 What is the next step? With the stability of thoptical-frequency comb currently limited by the microwareference used for phase lockingf rep, direct stabilization ofcomb components based on ultrastable optical refereholds great promise. The initial demonstration of precisphase control of the comb shows that a single cw laser~alongwith its frequency-doubled companion output! can stabilizeall comb lines~covering one octave of the optical frequenspectrum! to a level of 1–100 Hz at 1 s.119 With controlorthogonalization, we expect the system will be improvedthat every comb line is phase locked to the cw referebelow the 1 Hz level. Then we could generate a stablecrowave frequency directly from a laser stabilized to antical transition, essentially realizing an optical atomic clocAt the same time, an optical-frequency network spanningentire optical octave~.300 THz! is established, with mil-lions of regularly spaced frequency marks stable at thelevel every 100 MHz, forming basically an optical-frequensynthesizer. The future looks very bright, considering theperior stability ~10215 at 1 s! offered by optical oscillatorsbased on a single mercury ion,126 a single ytterbiuim ion,116 asingle indium ion,129 or cold calcium atoms.126–128 Indeed,prototype optical clocks have recently been demonstrabased on single Hg1 ~Ref. 135! and I2.

136 Within the nextfew years it will be amusing to witness friendly competitiobetween the Cs and Rb fountain clocks and various optclocks based on Hg1, Ca, or some other suitable system.

ACKNOWLEDGMENTS

The authors wish to acknowledge the contributionsthe many individuals who have contributed to this work.particular they acknowledge S. A. Diddams and D. J. Jofor their substantial contributions in the lab, R. S. WindelJ. K. Ranka, and A. J. Stentz~Bell Labs, Lucent Technolo-gies! for providing the microstructure fiber, T. W. Ha¨nschand colleagues~Max-Planck Institute for Quantum Optics!for stimulating the work in this field and discussing theongoing work with the authors. The authors thank L. Hoberg, J. Bergquist, T. Udem, L. S. Ma, T. H. Yoon, andFortier for many useful discussions. This work at JILA wsupported by NIST and NSF. The authors are staff membin the NIST Quantum Physics Division.

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