115 An Optical Frequency Comb Tied to GPS for Laser ... · 2.2 The Frequency Comb We use a commercial optical frequency comb [8, 9]2 as the basis of our system, with two modifications

1. Introduction

Optical frequency combs are used routinely to gener-ate optical radiation with a frequency uncertainty limit-ed only by the uncertainty of the master clock that setsthe repetition frequency of the comb. The fundamentallimitation is the uncertainty of the primary cesiumclock, currently about 3 × 10–16 [1]. Combs are alsoquite useful in applications that may not require suchhigh accuracy but benefit from the fact that a comb cangenerate radiation over a broad spectrum of opticalwavelengths. In the Precision Engineering Division(PED) of NIST,1 we have a need to calibrate laser

frequency for lasers with wavelengths ranging from530 nm to 1.5 μm, by comparison of the laser frequen-cies to the known frequencies generated by a comb.

Calibration of laser frequency provides equivalentknowledge of the laser vacuum wavelength; we willinterchangeably use the terms “frequency calibration”and “wavelength calibration.” Since 2008, all NISTlaser wavelength calibrations are traceable (eitherdirectly or via a transfer standard) to the comb, and cal-ibrated lasers operating in interferometers form thebackbone of modern length metrology. Thus, comb-based vacuum wavelength measurements effectivelyform the top of the traceability chain for most lengthmeasurements in the United States.

For purposes of laser calibration, a relative uncer-tainty of a few parts in 1012 is more than sufficient for

Volume 115, Number 6, November-December 2010Journal of Research of the National Institute of Standards and Technology

413

[J. Res. Natl. Inst. Stand. Technol. 115, 413-431 (2010)]

An Optical Frequency Comb Tied to GPS forLaser Frequency/Wavelength Calibration

Volume 115 Number 6 November-December 2010

Jack A. Stone andPatrick Egan

National Institute of Standardsand Technology,Gaithersburg, MD 20899-8211

[email protected]@nist.gov

Optical frequency combs can be employedover a broad spectral range to calibratelaser frequency or vacuum wavelength.This article describes procedures andtechniques utilized in the PrecisionEngineering Division of NIST (NationalInstitute of Standards and Technology)for comb-based calibration of laserwavelength, including a discussion ofancillary measurements such asdetermining the mode order. Theunderlying purpose of these calibrationsis to provide traceable standards in supportof length measurement. The relativeuncertainty needed to fulfill this goal istypically 10–8 and never below 10–12,very modest requirements compared to thecapabilities of comb-based frequencymetrology. In this accuracy range theGlobal Positioning System (GPS) servesas an excellent frequency reference thatcan provide the traceable underpinning of

the measurement. This article describestechniques that can be used to completelycharacterize measurement errors in aGPS-based comb system and thus achievefull confidence in measurement results.

Key words: global positioning system;GPSDO; laser frequency calibration; laserwavelength calibration; optical frequencycomb.

Accepted: June 23, 2010

Available online: http://www.nist.gov/jres

1 Not all acronyms will be defined in the text, but definitions can befound in Appendix A. Following reorganization, PED is now theMechanical Metrology Division.

any of our needs. (The relative uncertainty is the samewhether discussing frequency or wavelength.) Themost demanding requirement we face is a need tocalibrate iodine stabilized lasers operating at 633 nm.Under favorable circumstances, these lasers canreproduce their frequency/wavelength at a level of≈4 × 10–12 [2]. There is no benefit in achieving acalibration accuracy much better than the reproducibil-ity. Other calibrations of interest to us include commer-cial lasers operating at 543 nm and at 1.52 μm; forthese lasers the wavelength reproducibility is muchworse, and there is no motivation to achieve a relativeuncertainty better than 10–9.

A comb referenced to a GPS signal is entirely suffi-cient to meet these needs. The GPS signal furthermoreprovides a very clear traceability path to the unit oftime as realized by NIST. In effect, the GPS signal, incombination with a comb, is capable of providing theunit of length (vacuum wavelength) delivered by satel-lite to any point in the world, over a broad range ofwavelengths. As combs become more robust and lesscostly, this “length by satellite” is coming into wide useby National Metrology Institutes (NMIs) and may alsobecome an attractive tool for use in high-level second-ary laboratories. Arguably, the traceability path forthese GPS-based measurements is particularly straight-forward and easy to demonstrate, and a well-function-ing comb is capable of achieving uncertainties manyorders of magnitude better than what is required forlaser calibration. Nevertheless, it is necessary to verifythe reliability of the overall measurement process. Thispaper will describe a suite of tests that we use for thePED comb to assure that the comb indeed deliversorder-of 10–12 accuracy as required for our calibrationneeds. Some further discussions of this topic may befound in Ref. [3]. We would argue that this suite oftests—or similar tests–could be used by anyone toachieve essentially complete confidence in theirmeasurement results for uncertainties as small as a fewparts in 1012. The procedures as described in this articleare directed toward users whose uncertainty require-ments are not more stringent than this level.

2. Implementation of a Comb-BasedSystem for Calibrating LaserFrequency

The operating principles of combs have beendescribed extensively in the literature and will not be

discussed in detail here. Several references [4, 5, 6]give good reviews of frequency combs. Fox et al. [7]explicitly discuss using a GPS in conjunction with acomb for laser calibrations, and some aspects ofthe PED system has been discussed previously byStone et al. [3]. In this paper we cannot review allaspects of comb operation, but in this section we willprovide some needed background information and willdescribe those elements of our measurement systemthat may be somewhat different from standard configu-rations. Figure 1 shows a simplified schematic of thebasic components of the frequency comb and of theset-up for calibration of an iodine stabilized laser.

2.1 Basis of the Measurement

An optical frequency comb is based on a modelocked laser emitting a uniform train of short pulseswith some repetition frequency frep . In terms of frequen-cy, the pulse train consists of a set of uniformly spacedfrequency components, with frequencies fN given by

(1)

where fCEO is the carrier-envelope offset frequency,measured in a manner described later. fCEO can be eitherpositive or negative. N is the mode order, a large inte-ger typically greater than 106. When we calibrate thefrequency of a laser, the output beam of the comb iscombined (via a beamsplitter) with the beam from thelaser under test. The power of the combined beam ismeasured by a fast photodetector, and the output of thedetector then contains beat frequencies fbeat correspon-ding to the difference in frequency between comb com-ponents and the unknown frequency ftest of the laserunder test. Thus, if a beat with one known componentN of the comb output can be isolated, the frequencyftest is determined from a measurement of fbeat :

(2)

In the above equation, fbeat represents a positivequantity—the measured beat frequency—which mustbe added or subtracted depending on whether ftest islarger or smaller than the frequency of the Nth combcomponent. The situation is as shown in Fig. 2, wherethe beat between ftest and the next-lowest comb compo-nent corresponds to a positive sign in Eq. (2), and thebeat with the next-highest comb component corre-sponds to a negative sign.


414

rep ,N CEOf f N f= +

test rep beat .CEOf f N f f= + ±


415

Fig. 1. Basic configuration of comb and optics for calibration of I2 stabilized lasers. Components shown in the diagram are:apd—avalanche photodetector; SHG—second harmonic generator; bs—beamsplitter; m—mirror. Addition optical compo-nents not shown in the diagram include waveplates, focusing lenses, and optical isolators.

Fig. 2. Beat frequencies arising from interference between the test laser and two adjacent comb components. In addition to thetwo beat frequencies indicated in the diagram, there will be an additional RF signal at frequency frep due to interference betweenthe two adjacent comb components, and there will be many higher frequencies due to interference with additional combcomponents not shown in the diagram.

Signs may be determined by making small changesin the frequencies frep and fCEO and observing the corre-sponding change in fbeat . If fbeat is observed to increasewhen frep is increased very slightly, then fbeat must be thebeat with component N + 1 in Fig. 2, and the negativesign must be used in Eq. (2). Conversely, if fbeat

decreases then the positive sign must be used.Similarly, if fCEO is increased slightly, and if this has thesame effect (in terms of increasing or decreasing fbeat )as does an increase in frep , then fCEO is positive. In asetup such as ours, where both the offset frequency andbeat frequency can be servo controlled, the signs aredetermined by the polarity settings of the servo loops;thus it is not necessary to re-determine the signs forevery measurement.

For our system, frep is normally 100 MHz, and weonly measure beat signals between 0 MHz and100 MHz. As shown in Fig. 2, there are always twosuch signals, and the two frequencies sum to 100 MHz.

The accuracy of measurements rests primarily onachieving a low uncertainty in frep . To achieve anuncertainty in ftest of 1 kHz (≈2 parts in 1012), frep mustbe known with a relative uncertainty of 2 parts in 1012,whereas the offset frequency fCEO (typically 20 MHz, or40 MHz after frequency doubling to the visible) andthe beat frequency fbeat (less than 100 MHz) need bemeasured with a relative uncertainty of only 10–5.

2.2 The Frequency Comb

We use a commercial optical frequency comb [8, 9]2

as the basis of our system, with two modifications tothe commercial comb. The comb system consists of amode-locked erbium fiber laser, an f-2f interferometerfor determining offset frequency, and frequencydoubling optics to reach visible wavelengths. Theerbium laser actually has two outputs that are independ-ently amplified and shaped, and are broadened in twohighly nonlinear fibers. (See Fig. 1.) One output isbroadened so that is covers a wavelength range of oneoctave, between roughly 1 μm and 2 μm, and is usedfor determining the offset frequency fCEO . Initially, theoffset frequency was measured using a two-arminterferometer; frequency-doubled light from a BBOcrystal in one arm of the interferometer (with offset

frequency 2 × fCEO ) interferes with un-doubled light inthe second arm (with offset frequency fCEO ), generatinga beat signal at fCEO for detecting and controlling theoffset frequency. (See Appendix A for definitions ofacronyms such as BBO.) At present we generate thefCEO signal in the single arm of the interferometercontaining the BBO crystal; light emerging from thecrystal contains both doubled and un-doubled compo-nents which interfere with each other to create thesignal. Although walk-off in the BBO crystal reducessignal strength relative to what might be obtained withPPLN [10], we nevertheless achieve S/N ratios usingthis method that are similar to what was achieved withthe two-arm interferometer (roughly 30 dB to 35 dB asmeasured with 300 kHz RBW). The single-armarrangement is advantageous because it is more robustin terms of alignment than is a two-arm interferometer.A feedback loop controlling the pump laser current isused to stabilize fCEO to a 20 MHz reference frequencythat is derived from the GPS signal.

A second output of the erbium laser is independentlyamplified and broadened in a separate nonlinear fiber.This output can be used directly for calibrations in theIR, or it can be doubled in a PPLN crystal to provide avisible output. Three PPLN crystals suffice to cover aspectral range from 530 nm to 700 nm.

Another modification of our comb system is a mech-anism for sliding a glass plate into and out of the free-space region of the comb laser, so as to provide a meansfor rapidly changing the optical length of the resonatorand hence the repetition frequency. This operation isneeded to determine N, as described later. Although thecommercial laser system includes a mechanism basedon a piezo-drive motor for making coarse adjustmentsto the length of the resonator, suitably large shifts inrepetition frequency would require an inordinateamount of time. We can rapidly shift the frequency by208.22 kHz by inserting a 12.7 mm thick piece of BK7into the free space region of the erbium laser resonator.Losses due to reflections from uncoated surfaces of theglass appear to have little effect on laser operation or onmode locking. When the 12.7 mm piece of glass isinserted into the beam, mode lock is usually re-estab-lished immediately and can always be initiated byslightly jiggling the glass.

In passing we note that the change in comb repetitionfrequency when inserting the glass tells us the changein optical length of the laser cavity, arising from thegroup refractive index of the glass. If the group indexcould be determined with low uncertainty, then theabsolute thickness of the glass could be obtained fromthe measured frequency change. The precision of the


416

2 Certain commercial materials and equipment are identified in orderto specify adequately experimental procedures. In no case does suchidentification imply recommendation or endorsement by the NationalInstitute of Standards and Technology, nor does it imply that theitems identified are necessarily the best available for the purpose.

measurement can be very high (sub-nanometer), limit-ed only by the short-term stability of the free-runningcomb, and thus this phenomenon could serve as thebasis of a technique for measuring sub-nanometerthickness variations of transparent materials.

2.3 Reference Frequency (GPSDO)

The repetition frequency (frep) of the mode-lockedlaser is normally phase locked to a multiple of thefrequency of a GPS disciplined rubidium oscillator(GPSDO) [11]. Short-term stability is provided by therubidium oscillator. In a GPSDO, the absolute long-term accuracy is achieved by a servo that adjusts theoutput frequency to agree with GPS signals. The funda-mental accuracy of the system thus rests in the GPSsignal, which is traceable to NIST; it is “calibrated”daily in the sense that NIST continuously monitors thesignal and publishes performance data on the web [12].Some attention should be given to placement of theGPS antenna in a position where it views as manysatellites as possible and where it is not subject toreflections from nearby buildings. Also, the GPSDOunit should be chosen with some care, as certain unitswill provide better short-term stability than others.Short-term performance depends both on hardware(particularly the local oscillator) and on the softwarethat must handle switching between satellite signals,weigh the signals appropriately, and implement asteering algorithm to correct the local oscillatoroutput. These issues have been discussed by Lombardi[13, 14, 15].

2.4 Controlling the Comb Repetition Rate

A photodiode detects comb pulses at the repetitionfrequency frep , and a fast piezo actuator in the erbiumlaser is used to control the laser cavity length and thusset the repetition frequency. Normally, the 10thharmonic of the repetition frequency is phase locked toa 1 GHZ reference derived from the GPSDO signal,thus setting the repetition rate to 100 MHz. Under somecircumstances—particularly for determining the modeorder N—it is useful to operate at other repetitionfrequencies (never far from 100 MHz). This can bedone by replacing the GPSDO signal with a variablereference frequency provided by the output of a synthe-sizer/signal generator. To preserve accuracy, the time-base of the synthesizer/signal generator must be provid-ed by the GPSDO signal. Good performance will not beobtained unless the signal generator has very low phasenoise. When the comb repetition frequency is thus

controlled, measurements of the highest accuracy canbe obtained in a convenient manner, and we use thismode of operation for many of our measurements.

2.5 Controlling the Beat Frequency

We also use a second mode of operation where frep isnot held fixed but is servoed so as to keep the beatfrequency fbeat constant. This alternate approach pro-vides some operational advantages over a signal gener-ator (and is much less expensive to implement). Theprimary advantage of this method is that it can beimplemented in a scheme that will automatically estab-lish servo lock with little or no operator intervention,which is important when it is necessary to rapidly andrepeatedly change the comb repetition frequency. Weuse this approach when measuring the mode order asdescribed in Sec. 2.7. Keeping the beat frequency con-stant also makes it possible to improve signal to noisewith the aid of a narrow filter, as described in Sec. 2.6.A frequency lock that is easy to implement and has awide capture range is described in Ref. [16]. We havetried several variations on this scheme.

There are several possibilities for straightforwardimplementation of a frequency lock with a unique lockpoint, which will provide automatic lock acquisitionwith minimal requirements for operator intervention.An electrical high pass or low pass filter with cutoffnear the desired beat frequency can be used as a dis-criminant for measuring and controlling fbeat . Thepower transmitted through the filter varies rapidly nearthe cutoff frequency. Satisfactory frequency control canbe achieved simply by servoing the comb repetitionfrequency so as to hold the power of the filtered signalconstant at some threshold value; that is, the PI con-troller that normally holds frep constant is instead used tohold the filtered power equal to the threshold value.The control point will change slightly if laser powervaries, but the effect is not severe if the filter rolloff issufficiently sharp.

A system with reasonable performance can be builtsimply by plugging together inexpensive connectorizedstock components. The signal is first pre-filtered toselect the range between 50 MHz and 100 MHz, so thatonly one beat signal is present. Two stock 70 MHz low-pass filters can be cascaded to provide a sharp discrim-inant, suitable for stabilizing the beat frequency at thecenter of a stock 70 MHz bandpass filter. (This willsimultaneously stabilize the signal at frep-fbeat to thecenter of a 30 MHz bandpass filter.) A splitter, mixer,and low pass filter can be used to generate a DC signalproportional to the power transmitted through the


417

filter. If the beat signal has constant amplitude, we seevariations in the lock point over the course of an after-noon of less than ±200 kHz, and the 1-second Allandeviation is 6 kHz. A 10 % change in the amplitude ofthe beat signal shifts the lock point by about 400 kHz.

An order of magnitude better performance can beachieved using a power-insensitive scheme thatdepends on the phase shift of the filtered signal, but thismethod does not result in a unique lock point. Asdescribed in Ref. [16], comparing the filtered power tothe total unfiltered power will give a lock point insen-sitive to power fluctuations, but a unique lock pointwill not be achieved without careful attention as to howsignal levels behave at the edges of the pre-filterpassband. For our application, the potentially tighterlock provided by power-insensitive schemes is of noclear benefit, whereas it is of primary importance toachieve reliable automatic locking, requiring a uniquelock point.

The main drawback of stabilizing the beat frequency,rather than the repetition rate, is that this approachrequires very careful measurement of the repetitionfrequency, which is no longer controlled by the repeti-tion rate servo. This will require long averaging timesand will severely test the performance of a frequencycounter. Actually, the counter does not need anextremely accurate internal timebase, because achiev-ing suitable accuracy of the timebase is only possiblevia using the GPSDO as an external timebase for thecounter. But good timing resolution is required in orderto achieve accurate measurements in short periods oftime. For example, with 1 ns timing resolution, a 1 sfrequency measurement has a resolution of only 1 partin 109, which may not be sufficient for purposes ofdetermining mode order. Thus, it is necessary to pur-chase a counter that has good timing resolution and it isnecessary to use long measurement samples to achievehigh accuracy. This point is discussed further inSec. 4.2. In any event, if measurements with the beatfrequency locked agree well with results obtained withthe repetition frequency locked, this can serve as across check to give confidence in both the frequencycounter and in the frep servo.

2.6 Beat Frequency Measurement

To measure the beat frequency, the laser beams fromthe comb and test laser are combined at a beamsplitterand the time-varying power in the superimposed beams(at frequency fbeat ) is detected with an avalanchephotodetector. The primary difficulties with measuringfbeat arises from the fact that only a very small fraction

of the comb power is in the mode that gives rise to abeat signal. The problem is particularly a concern for usbecause of the relatively low repetition frequency ofour comb laser (100 MHz) and the corresponding highdensity of comb lines. After frequency doubling to thered, the output power of the comb is a few milliwattsspread over several nanometers spectral width. Thisspread corresponds to 1.5 × 104 comb lines, only one ofwhich will contribute to the signal fbeat that we wish tomeasure. The rest of the comb power degrades the sig-nal through its contribution to shot noise and throughsaturation of the detector. Depending on the detectorand its built-in amplification, saturation may occur dueto either the DC comb power or due to RF signals at frep

(and multiples of frep) arising from interference betweenadjacent comb lines, because these signals are muchlarger than is the signal at fbeat . If too much amplifica-tion is built into the detector, it will be difficult to avoidsaturation. Saturation can be avoided if the detector haslower amplification and its output is filtered prior toadditional stages of amplification.

Several strategies are needed to maximize signal tonoise and reduce detector saturation effects. Probablythe most important single step is simply to take greatcare in aligning the beam from the test laser withthe beam from the comb. To be more precise, what isideally needed is perfect matching of the wave-fronts–two coaxial Gaussian beams with the same waistposition and same size, traveling in the same direction.The ultimate method for assuring matching of thewavefronts—and rejecting unmatched portions of thebeams that do not contribute to signal but contribute tosaturation and noise—is to couple both beams into asingle mode fiber prior to the detector. If a fiber is notused, it must be assured that the beams overlap verywell, travel accurately in the same direction, and havesimilar waist size (or equivalently, similar spot sizes inthe far-field region). A factor of two mismatch in thewaist size will result in a 1.9 dB reduction in signalstrength (calculated from the overlap integral assumingthat the beams are Gaussian and both can be focusedcompletely onto the detector). For beams of equal size,there will also be >1.9 dB reduction in signal if thebeams are misaligned by an angle greater than 2/3 ofthe beam divergence (half angle), or if one beam is off-set laterally from the other by more than 2/3 of thewaist radius, or if the positions of the waists are sepa-rated along the direction of propagation by a distancegreater than the Rayleigh range.

Other things can be done that will slightly improvesignal to noise and will reduce the potential problem ofdetector saturation by the comb power. First, a narrow-


418

band optical filter or grating can be used to rejectunwanted spectral components of the comb signal. Agrating in Littrow configuration provides potentiallybetter performance than do optical filters and, unlikethe narrow band filter, works over a range of wave-lengths. The grating alleviates the problem of detectorsaturation and slightly improves signal to noise.

Similarly, a non-symmetric beamsplitter or polariz-ing optics can be used to combine the comb and testlaser beams in proportions other than 50 %, which isslightly advantageous when the test laser has much lesspower than the comb (an iodine stabilized laser, forexample). If the test laser has 1/10 of the power of thecomb prior to mixing, then S/N is optimized (assumingshot noise) by a splitting ratio of 75 % to 25 % in favorof the test laser. The improvement in S/N relative to a50 % beamsplitter is almost negligible—only 1 dB—but the total power on the detector is reduced by 40 %and thus helps avoid saturation.

Perhaps the simplest and most effective way ofimproving S/N is to limit the detection bandwidth forthe electronic signal going to the frequency counter. Wetypically stabilize fbeat at a value that is centered on abandpass filter with a width of about 12 MHz. Thisfilter provides about 6 dB better S/N than would beobtained by a filter of 50 MHz width (the maximumwidth that would assure separation of the signal at fbeat

from the signal at frep – fbeat ). This provides satisfactoryS/N for any of our current measurement needs. A yetnarrower filter would provide better S/N but might notbe broad enough to accept the signal of a modulatedlaser; iodine stabilized lasers at 633 nm have a modula-tion width of 6 MHz, and one commercial 1.52 μmlaser has a modulation width of 10 MHz.

If it is necessary to further reduce noise, this can bedone by removing the modulation from the beat signal,allowing use of a narrower filter. A suitably fast beatfrequency servo should narrow the modulation width,but the effect is not profound for our system. Betterresults are obtained by adding an AC signal into thefeedback system so as to modulate the comb frequencyin a manner that cancels the modulation of the test laser.Cancellation is possible by adjusting the amplitude andphase of a sinusoidal voltage output of a phase lockedloop. The PLL must be locked to a reference signalwhich has a fixed phase relationship to the laser modu-lation. Some modulated lasers provide a reference out-put that can be used for this purpose. If no such outputexists, it is also possible to derive a reference from thebeat frequency servo error signal (but performance willnot be as good).

We can narrow the linewidth of the beat signal from6 MHz to about 400 kHz using a PLL. If future meas-urements require improved S/N, this narrowing of themodulation width will allow us to replace our 12 MHzbandpass filter with a filter of ≤ 1 MHz width—sufficiently large to comfortably accept the beat signalwhile providing an additional 11 dB improvementin S/N.

2.7 Determining the Mode Order

The mode order N must be determined. One way todo this would be to employ a wavemeter, but a suitablyaccurate wavemeter is expensive, requires periodiccalibration, and would require more than one set ofoptics to cover the visible and IR wavelengths of inter-est to us. Alternatives based on changing the combrepetition rate [17, 18] are more attractive as a methodof determining order, as they avoid the expense andmaintenance associated with a separate instrument. Ourmethod is most similar to that of Ref. [17] and isclosely analogous to multicolor interferometry [19].Implementation of these methods is very straightfor-ward when measuring very stable sources. It is some-what more difficult to implement when measuring thewavelength of typical commercial stabilized lasers, dueto the frequency fluctuations of these lasers.

Usually, the frequency of the laser under test isknown approximately before beginning the measure-ment. In fact, the mode order for commercial stabilizedlasers is almost always known, since the tolerance onthe output frequency is much less than the comb spac-ing. However, certain assembly errors occasionallycause a commercial laser to have a frequency that is notat the expected order. Even if the exact order is notknown, only a narrow range of orders will be realisti-cally possible for a gas laser. For example, the frequen-cy of a 633 nm He-Ne laser will always lay withina known range of 2.8 GHz [20], corresponding to28 possible values for N. Within this range, the ordercan easily be determined by changing the repetition rateby some amount Δfrep . For our system, Δfrep is normallyset at 208.22 kHz, as explained previously. In all thefollowing analysis we assume that Δfrep << frep .

To determine the mode order, we first use Eq. (2) tofind N0 , the value of N that gives a result ftest as close aspossible to the expected frequency of the test laserwhen the repetition rate is frep. We next find the order N0′which, for repetition rate frep + Δfrep , gives a value forftest as close as possible to the value obtained with frep . Ifthe initial guess N0 is correct, then the two values


419

obtained for ftest must be the same (within experimentaluncertainty). If the first estimate N0 is in error byΔN, and if ΔN is not too large [more precisely,|ΔN | < frep / (2Δfrep ) = 240 orders], then N0′ will be inerror by the same amount ΔN. Thus one result for ftest

will be in error by ΔN frep and the second will be in errorby ΔN (frep + Δfrep ), so the two results of Eq. (2) willdisagree with each other by

(3)

Therefore the true order must be

(4)

where “Round” represents rounding to the nearestinteger. If there is an error in measuring δftest such thatthe error in δftest /Δ frep exceeds 0.5, then N will be inerror by 1. This is never a problem when measuring alaser stabilized by saturated absorption, but it canbecome a significant problem when measuring certaincommercial stabilized lasers, which have much greaterfrequency fluctuation and drift. Unless the variations offtest are somehow measured and corrected, it will benecessary to assume that ftest is constant, and the varia-tions of ftest between measurements will appear as errorsin δftest . To avoid an error in determining N, it is neces-sary to tune the comb repetition rate by amounts thatare at least twice as large as these frequency fluctua-tions. Zhang et al. [17] recommend that the repetitionfrequency be shifted by more than ten times thefrequency fluctuations in the laser under test. Our shiftof 208 kHz is not as large as recommended, but we cancompensate for the relatively small shift as describedbelow.

For our system, we will have an error in N unlessfrequency fluctuations and drift can be kept belowΔfrep /2 = 108 kHz. This condition can usually be ful-filled but it is not guaranteed. Frequency fluctuationscan be reduced by using long sampling times, an effi-cient solution to the problem if the fluctuations arecharacterized by a white-noise spectrum. Simplyincreasing the averaging time will not completely solvethe problem for some commercial lasers, which may besubject to frequency variations that are quasi-linear orquasi-periodic on long time scales (ranging from a fewminutes up to more than an hour). A more efficientmethod of averaging out long-term drift is to switch

back and fourth several times between two values offrep on a time scale short relative to the time scale of thedrifts. For most commercial lasers, reasonablesampling times probably lie between 20 s and 2 min.

As mentioned previously, we shift the repetitionfrequency by sliding a piece of glass into or out of thefree space region of the comb resonator. Whenperforming this test, we servo-control the beat frequen-cy so as to keep fbeat constant (rather than directlystabilizing the repetition frequency). Mode locking willusually self-initiate when the glass is inserted. On occa-sion operator intervention is required, but, on average,the process of switching the repetition rate requires lessthan 20 s.

Figure 3 shows repeated measurements of ftest for atypical commercial laser (633 nm). If fin is a frequencymeasurement with the glass inserted and fout is a meas-urement without the glass, then the data is taken repeat-ing a pattern fin fout fout fin . Each measurement shown inFig. 3 is actually a 10 s sample, with two measurementsare taken in succession (except for the very first andlast point). The sequence fin fout fout fin yields two meas-urements of δftest whose average is insensitive to lineardrift.

A long-term drift is apparent in the frequency of thetest laser. Over the period of the first 10 measurementsthe average drift between successive readings is about25 kHz, which will have a non-negligible effect on theresults (but is still smaller than typical random fluctua-tions from one reading to the next, suggesting that alonger averaging time would be beneficial for testingthis laser). In Fig. 3, the data points connected by asolid line show the results when the correct order isused in the analysis. The dashed line shows what theresults would look like if the order were misidentifiedby 1. For the dashed line, the frequency clearly changesas we switch between glass-in and glass-out; the clearsystematic difference between the readings fin andthe readings fout indicate that the order has beenmisidentified.

A typical measurement sequence could consist ofthree repetitions of the pattern fin fout fout fin —a total of12 frequency measurements (6 measurements of δftest).The data of Fig. 3 represents 4 such measurementsequences. The results are unambiguous in the sensethat, for each of these 4 measurement sequences, theaverage value for δftest is consistent with 0 and is incon-sistent with ± 208 kHz (a change of N by ± 1) at a 99.7 %confidence level or better, giving us good assurance


420

test rep .f N fδ = Δ × Δ

0 test rep[ / ]N N Round f fδ= − Δ

that the order has been identified correctly. Thus weconclude that, for this particular laser, a set of threerepetitions of the fin fout fout fin pattern, with 10 s sampletime for each data point, suffices to provide a reliablevalue for the order N that can be obtained in a reason-able amount of time (about 5 minutes).

If the average value of δftest is m and if the standarddeviation of the mean of all measurements is σ, theneither neighboring order is excluded with 99 %confidence if

(5)

where k is given by the Student’s t-distribution(k = 3.36 for 6 samples and a one-sided confidenceinterval of 99 %). This provides strong evidence thatthe order is identified correctly. Equation (5) willalways provide a conservative criterion for claimingthat the order has been identified correctly, although itmay be unnecessarily conservative in some situationswhere slightly different analyses may be moreappropriate.

If all of the measurements shown in Fig. 3 are takentogether, the average value of δftest is 17 kHz with anuncertainty of 13 kHz, where these values are quite

small relative to the fluctuations of the test laser. Thegood agreement between glass-in and glass-out fre-quency values provides real-time assurance that themeasurements are being done correctly, at a level sig-nificantly smaller than uncertainties in the test laserassociated with its drift. This is one of several tests wecan do to assure that measurements are being donecorrectly. Of course, this good level of agreement willnot verify everything in our measurement process, butit would catch subtle errors in the measurement of frep orgross errors in measurement of fbeat .

In most cases, the procedure above is all that wouldbe needed to distinguish the order. One exceptionwould be if there was very little prior knowledge as tothe order, such as when measuring a tunable diodelaser. If the true mode order does not lie within therange N0 ± frep / (2Δfrep ), then the analysis of Eq. (4)breaks down, and ambiguities in possible solutions canarise. (An analogous problem occurs in two-color inter-ferometry when prior knowledge of the measuredlength is not sufficiently accurate.) If frep /Δfrep wereexactly 480, so that 481 orders of frep exactly matched480 orders of ( frep +Δfrep ), then the same beat frequencywould be predicted for possible solutions differing by480 orders, and it would be impossible to distinguishthe ambiguity. When frep is not exactly an integral


421

Fig. 3. Repeated measurement results for ftest (deviations from the average value) while switching between twovalues of the repetition frequency. The solid line shows the correct answer. The dashed line shows what resultswould look like if the order were misidentified by 1. Uncertainties of the individual frequency measurementsare too small to be visible on the graph.

208 kHz m kσ− >

multiple of Δfrep , the predicted beat frequencies do notrepeat exactly at intervals of frep /Δfrep orders, and inprinciple the unambiguous region can be extended. Infact, the sum total of all the data shown in Fig. 3 is suf-ficiently accurate to relax the requirements on priorknowledge of the wavelength. However, a much small-er set of measurements could efficiently distinguish theorder unambiguously, if a series of small frequencyshifts is used to perturb the 208 kHz shifts. Because thenon-ambiguity interval is given by ( frep /Δfrep ), it is clearthat we could use a very small shift Δfrep to expand thenon-ambiguity interval. A geometric sequence of ever-decreasing shifts can be used to expand the non-ambi-guity interval to cover the entire visible spectrum orbeyond. In practice, these small frequency shifts do notneed to be done independently of the large shifts(208 kHz via inserting the glass) that are used todistinguish N from N + 1; if the 208 kHz shifts areperturbed by additional smaller shifts made in someother manner, it is possible to completely eliminateambiguity without increasing the total number ofrequired measurements.

3. Traceability and EvaluatingMeasurement Uncertainty

An important aspect of a GPS-based comb system isthat it is based on a traceable frequency standard,provided by the GPS system. The traceability of GPSand of GPS-based combs has been discussed in variousarticles [3, 13-15]. In this section we discuss trace-ability from the standpoint of the VIM definition [21],“…the property of the result of a measurement or thevalue of a standard whereby it can be related to statedreferences, usually national or international standards,through an unbroken chain of comparisons, all havingstated uncertainties.” For laser calibrations via a GPS-comb, the unbroken chain of comparisons to nationalstandards can be remarkably short, and the uncertaintybudget can be remarkably simple. We will not discusslegal or documentary aspects of traceability.

In the simplest possible measurement system, thefrequency of the test laser is determined from Eq. (2):

(2)

This equation is valid if no additional frequency off-sets are introduced by devices such as acousto-opticmodulators (AOMs). In essence, determining ftest thusdepends on determining three frequencies, and the

argument for traceability of the measurement rests ontwo points:

• All frequency determinations are traceable toNIST and to the SI second via a unbroken chainof measurements, with the GPS satellite systemplaying the central role: (1) a frequency is eithermeasured by a counter, with its timebase provid-ed by the GPSDO, or it is effectively determinedby servolocking the frequency to a reference sig-nal derived from the GPSDO (2) the frequencyreference for the GPSDO is provided by the sig-nal from GPS satellites and (3) this GPS signal islinked to NIST primary standards (“calibration”)by measurements that are recorded in the NISTGPS Archives [12].

• Our uncertainty budget for laser calibrationsusing the GPS-comb is entirely dominated byeffects that can be quantified by measuring short-term fluctuations of fbeat when the comb iscompared to a stable laser.

For our application, we need not be concerned withsmall uncertainties such as the uncertainty of the NISTprimary cesium standard; the only uncertainties of anysignificance for us are those exceeding a few parts in1013. The argument we wish to make is that all of thesesources of uncertainty in the GPS-comb system areerrors that vary on short time scales, typically less thanone or two days. Therefore, the combined magnitude ofthese effects can be quantified by measuring the appar-ent variations in a stable test laser for a period of timesomewhat longer than 1 day. Because the test laser isnot perfectly stable, this test will overestimate measure-ment errors, but the resulting upper bound on measure-ment uncertainty is nevertheless sufficient for mostneeds.

Estimating measurement uncertainty via studyingthe repeatability of the measurement is a well-estab-lished technique often used in dimensional metrologyat NIST. However, it often requires years of data totruly sample all sources of error in a typical dimension-al measurement system. For the GPS-comb system, wewill argue that the required time is much shorter, some-where between 1 day and 1 week. There are severalnatural timescales of the GPS system that come intoplay. One natural time scale is the sidereal day, atimescale that includes effects due to diurnal variationsof the ionosphere and diurnal temperature variations.GPS satellites pass overhead twice per day, so thateffects associated with individual satellites or with


422

test rep beat .CEOf f N f f= + ±

multipath reflections into the antenna will repeat withthis period. Another important scale is set by timeconstants in algorithms that steer the GPSDO localoscillator. Usually these time constants are no morethan a few hours, but some exceptional GPSDOsystems might update steering only infrequently [15]. Itis desirable that the GPSDO manufacturer provideinformation as to what is the longest timescale associat-ed with their steering algorithms. The natural timescaleof the system is either 1 day or, if the steering algo-rithms employ longer timescales, then it is the longesttimescale associated with the steering algorithms. Inthis article, we will define “short-term” or “long-term”in reference to the longest timescale thus defined. Atesting period that samples all errors must be longerthan the longest timescale associated with the GPSsystem and must be longer than 1 day; probably twodays of testing would be minimally required to assurethat all errors are sampled. We believe that all plausibleGPSDO errors are short-term and that there are nolong-term systematic offsets at levels of interest to us(above 1 part in 1013). Although it is true that someGPSDO units have much better performance thanothers in terms of Allan deviation for a particularsampling interval, and many commercial units aresimply not well designed for use as a frequencystandard, the authors are aware of no instances wherecommercial units give systematic long-term frequencyoffsets. Consequently a study of short-term repeatabili-ty will suffice to quantify all uncertainties associatedwith the GPSDO.

The argument presented above regarding time scalesis not strictly true; some relevant time scales are muchlonger. For example, the 11-year sunspot cycle couldalso have some bearing on the measurements; increasedsolar activity may reduce short-term stability of theGPS signal and will thus increase measurement uncer-tainty. Any such degradation of the GPS signal will beseen as increased Allan deviations as recorded in theNIST archives. If the Allan deviation significantlyincreases at some point in time, it will be necessary toincrease estimated measurement uncertainty and maybecome necessary to re-quantify the short-termstability of the system by comparison to a stable laser.Significant changes in the environment of theantenna—including seasonal changes in nearbyfoliage—might also slightly degrade performance on along time scale, requiring re-quantification of theshort-term stability.

With the basic link to the SI unit provided by GPS,the primary traceability issue is in evaluating uncertain-ty due to short-term fluctuations. For measurements at

the 10–12 level, it is the fluctuations in frep that are of pri-mary importance. These fluctuations arise primarilyfrom fluctuations in the received GPS signal (includingeffects such as fluctuations in the clocks of the GPSsatellites, varying atmospheric delays, and multipathreflections), as modified by the smoothing provided bythe GPSDO. If frep is phase locked to a referencefrequency derived from the GPSDO (our normal modeof measurement), then there are additional fluctuationsin frep due to imperfect performance of the servo. If fbeat

rather than frep is stabilized, then frep must be measured,and there are additional fluctuations due to the timingresolution of the frequency counter. Regardless ofwhich scheme is employed, it is possible to put anupper limit on the uncertainty associated with short-term stability. Ideally, this would be done by com-paring the GPSDO/comb system to perfect localoscillator—an ultra-stable laser. But useful informationregarding GPSDO/comb stability can often be obtainedby comparing it to a less-than perfect laser, such as aniodine stabilized laser or possibly even a polarizationstabilized laser (although a polarization stabilized laseris not sufficiently stable for implementation of somediagnostics to be described later). When comparing thecomb to an imperfect laser, the measurements quantifythe combined instabilities of the two systems and thusprovide an upper bound on the stability of the GPS-comb. There is no inherent reason that any laser cannotbe used for the measurement, as long as the upperbound is low enough to serve the needs of the calibra-tion lab. The laser employed for the stability measure-ment need not be calibrated prior to the measurement.In principle, if a laboratory does not own an iodinestabilized laser but iodine stabilized lasers are sent tothe lab for calibration, then a customer laser could beused to carry out the stability measurements. (Ofcourse, if a customer needs precise knowledge of theshort-term Allan deviation of his/her iodine stabilizedlaser, this information cannot be obtained withoutbetter quantification of the GPS-comb short-termstability.)

We have measured comb stability by comparing it toone of our iodine stabilized lasers. To achieve the bestpossible results, the laser sits undisturbed in a tempera-ture-controlled laboratory and is warmed up for manyhours prior to beginning the measurement. Also, theGPSDO should be fully warmed up and finished withany self-survey that may be performed to determine thelocation of the antenna. Even under these circum-stances, drift of the test laser frequency may not beinsignificant; if the drift is incorrectly attributed to thecomb, it can make the comb appear to be less stable


423

than it actually is. Nevertheless, the amount of drift issmall enough to satisfy our needs.

Figure 4 shows the Allan deviation of fbeat mea-sured over a period of 6 days. The measurementquantifies the combined effect of short-termfluctuations in the GPSDO, the comb, the iodinestabilized laser, and all ancillary equipment suchas frequency counters. The shape of the curve isvery similar to the manufacturer’s published resultsfor this GPSDO, although the magnitude of theobserved Allan deviation is smaller than shown bythe manufacturer for sample times longer than 10 s.The results of Fig. 4 provide a useful upper boundon fluctuations of the comb system, demonstratingthat we can achieve good accuracy for measuringtimes in excess of 100 s, where the Allan deviationfalls below 1 part in 1012.

During the period of time when these data werecollected (July 27 to August 1, 2006), there was littlesunspot activity and the NIST archives do not indicateany unusual errors in the GPS signal. The results ofFig. 4 are indicative of what can be expected undersimilar near-ideal conditions, but measurement uncer-tainty should be re-evaluated when the NIST archivesshow decreased stability of the GPS signal.

The data shows that for measurement times longerthan approximately 1000 s the Allan deviation fallsbelow 5 × 10–13 and remains below this value up to thelongest time measured. If we can argue that the long-term average of these measurements has an uncertaintymuch less than 5 parts in 1013, then we can concludethat the standard uncertainty of measurements in excessof 1000 s duration is on the order of 5 × 10–13. Moredirect evidence of stability is given in Fig. 5, whichshows results for beat frequency measurementsbetween the comb and laser with 22 min (1340 s) aver-aging time. The data is graphed as a function of time ofday, and was collected over a period of six days. Thedata suggests that there may be small correlationsbetween the measured frequency and time of day, butany such systematic fluctuations are unlikely to exceed1 kHz peak-to-valley.

Because of these systematic variations on a time scaleof many hours, the 22 min Allan deviation may slightlyunderestimate the true uncertainty of a 22 min measure-ment. The long-term variations are reflected in the factthat the Allan deviation does not decrease as the aver-aging time is increased from about 1000 s to10 000 s (but decreases again for sample times in excessof six hours, dropping to 1.5 × 10–13 at 12 hours).


424

Fig. 4. Allan deviation of fbeat , comparing the comb to an iodine stabilized laser. Error bars corresponding to thestandard uncertainty are shown for longer averaging times, where the error is large enough to be significant.

The sample standard deviation of the data in Fig. 5 is370 Hz (7.8 × 10–13), and 95 % of the points are within730 Hz of the mean. This suggests that a reasonablek = 1 estimate of the uncertainty for a measurementaveraged over 22 min is 370 Hz (7.8 × 10–13), wherethis value might be slightly inflated by (possible) driftof the iodine stabilized laser. To this could be added theexpected uncertainty (relative to the SI unit of time) ofthe six-day average, but in the absence of serious satel-lite failure, six-day averages are almost always in errorby less than 1 part in 1013 and thus have no effect on theresult. Indeed, the NIST archives show that the six-dayaverage during this test was in error by only 1 × 10–14,and the additional error added by our GPSDO over aperiod of 6 days should be well less than the 1.5 × 10–13

Allan deviation observed at the longest sampling times.Even with the most pessimistic possible assumptions,these additional uncertainties would have almost noeffect on the overall uncertainty for 22 min averages,increasing the uncertainty from 7.8 × 10–13 to 7.9 × 10–13.

In a similar manner, any laboratory with access to aniodine stabilized laser can estimate short-term uncer-tainties of their system, without appeal to an outside

agency such as an NMI. Once again, we reiterate thatthese short-term uncertainties capture all the importantsources of uncertainty in the measurement process andthat no additional measurements are needed to evaluatethe uncertainty of the system. Thus, documentation ofthe short-term performance, combined with the trace-ability of the GPS signal, might form a basis forclaiming traceable comb-based measurements with arigorously evaluated uncertainty.

The 370 Hz standard deviation of the data in Fig. 5should be a true measure of the uncertainty for 22 minsampling. Essentially the same sample standard devia-tion is obtained for 22 min averages whether data isaveraged from a single day or for the full six dayperiod. If the comb could be compared to a perfectstabilized laser, and if sunspot activity were constant,there would be every reason to believe that a similarstandard deviation would be calculated from data takenover a period of six years as was obtained for just sixdays. Our estimated expanded uncertainty (1.8 × 10–12

with k = 2) should be reliable as long as the NISTarchives don’t indicate any unusual problems.


425

Fig. 5. Beat frequency vs. time of day (hours past midnight). Measurements were taken over a period of six days. The differentsymbols in the graph correspond to data from different days. Uncertainties of the individual beat frequency measurements are too smallto be visible on the graph.

Finally, note that there is no uncertainty componentassigned to the measurement result (ftest ) arising fromuncertainty in the determination of N in Eq. (2). If theprocedure for determining N as given in Sec. 2.7 wereapplied in precisely the manner that was described,then about 1 measurement in 100 would be in error byan amount exceeding 105 u, where u is the standarduncertainty we have assigned to the measurement. (Inreality, a misidentified order would normally be anunexpected result that would undoubtedly prompt addi-tional measurements for verification.) The same situa-tion arises in multicolor interferometry, and it has longbeen recognized that this has no bearing on an uncer-tainty budget. Misidentification of N is classified as a“blunder,” outside the uncertainty budget. There aremany other blunders that can also occur in a combmeasurement, where the term “blunder,” within thecontext of metrology, refers to an avoidable mistake butnot necessarily a large mistake [22]. For example, theGPS signal might be lost, or the beat frequency can bemiscounted if the S/N is low. These blunders will oftengive rise to small errors but, as in the case of mis-identifying N, can potentially cause errors many ordersof magnitude larger than the standard uncertainty.Regardless of the size of the error, they have no bearingon an uncertainty budget, and they do not relate totraceability as defined by the VIM. Blunders arediscussed in the next section.

4. Blunders

Blunders are not explicitly discussed in the VIMdefinition of traceability. Nevertheless, for the GPS-comb system, blunders represent the primary impedi-ment to achieving confidence in measurement results atour claimed uncertainty, where “blunders” include bothmisuse of equipment (blunders by an operator) andblunders by the designers of equipment. (Beyondsimply being “out of spec,” poorly designed equipmentmight not even function in accord with its intendedoperating principles.) Therefore it is desirable to verify,as far as possible, that blunders have not occurred.

4.1 Blunders in the Design or Use of the GPSDO

If a laboratory were to use its own cesium clock as astandard, not tied to GPS, it might be necessary to sendit to an NMI for calibration, but if we use the clocks ofthe GPS satellites, calibrations are performed automat-ically. Thus, the system does not need to be calibratedto avoid long-term offsets. Short-term performance of

the GPSDO can be verified as described already. Thus,it is probably not necessary to send a unit to an NMI forcalibration. Furthermore, although there can be severalbenefits from obtaining an NMI calibration, it may stillbe necessary to locally verify short-term performanceof the entire GPS-comb system, including effects suchas antenna placement which can only be evaluated byin situ testing.

As stated previously, we are not aware of a GPS unitthat has ever been produced that has a long-term fre-quency offset, if “long-term” is defined as describedpreviously. (By contrast, a systematic time offset caneasily occur in a GPS system.) There is, nevertheless, apractical problem in defining “long-term,” because thiscan only be done if the manufacturer provides reliableinformation regarding operation of their unit. To someextent, the manufacturer’s information can be supple-mented by looking at results such as shown in Fig. 4,where the structure of the plateau between 1000 s and10 000 s suggests that “long-term” must be longer than10 000 s. Certainly if a similar plateau or an increase inthe Allan deviation occurred at the longest samplingtimes, this might indicate that the test of short-termstability is not long enough to capture all sources oferror. Of course, the possibility that the software makessome steering corrections once every two weeks oreven once per year can not be ruled out by dataobtained over a six-day period; it is still of interest toverify from the manufacturer that such infrequentsteering corrections do not occur.

The plateau in Fig. 4 also conveys another importantpiece of information. The shape of the curve is charac-teristic of the steering algorithm for this particularmodel GPSDO (that is, the shape of the curve corre-sponds to published data for this model) and bears noresemblance to what would occur if the rubidium clockwere not controlled by GPS. The characteristic shapeseen in the diagram proves that the critical GPSDOsubsystems are performing correctly. The shape wouldnot be seen if GPS steering were disabled by an opera-tor blunder or by a broken line to the antenna. It wouldnot be seen if the GPS steering servo or the rubidiumcell had failed. One would expect that a GPSDO willinclude diagnostics that warn a user of such catastroph-ic failures, but even in the absence of these diagnosticsthe shape of the curve proves that major failures (thatcould give rise to long-term offsets) have not occurred.

It is difficult to imagine other plausible scenarios thatwould give a long-term error that is small enough to goundiscovered but large enough to affect our measure-ments at the 10–12 level. Nevertheless, in principle adesign blunder or firmware bug might produce such a


426

small long-term offset. The best defense against sucherrors is for the manufacturer to carefully compare eachnew model (or new firmware update) to a model whoseoperation has been previously verified. The new modelmight also be sent to a NMI for testing, and this has agood chance of uncovering design blunders. In-housecomparisons by the manufacturer have a slight advan-tage because they can be carried out over longer peri-ods of time, but in some manner the manufacturershould also directly or indirectly compare his model tonational standards. If these modest and reasonablemeasures are carried out to assure the integrity of a newmodel, it is difficult to see how a hidden systematic off-set would ever occur, even at the 10–14 level. It is prob-ably more logical for a manufacturer to thus verifyoperation of his product line than for every individualunit to be sent for calibration to an NMI.

An alternate method of testing for subtle GPSDOerrors might be to compare to another well-character-ized frequency standard. One possibility is to compareto a second GPSDO from a different manufacturer,using an independent antenna, and show that both unitsgive the same frequency. We have tried this approachwith limited success for two units that share the sameantenna (thus not a perfect test). The two units arevery different in operation—one uses a rubidium localoscillator and the second, from a different manufactur-er, used quartz—and consequently we can argue thatthere should be very little correlation in the errors ofthe two units (other than errors associated with thecommon antenna). Unfortunately, the quartz unit doesnot have sufficiently good stability to provide a usefulupper bound on short-term performance, but for aver-aging times approaching 1 day, where the fractionaldifference in frequency of the two units falls below3 × 10–13, this test provides further confidence that ourGPSDO is working properly. However, the test isneither as comprehensive nor as sensitive as is the com-parison to an iodine stabilized laser.

4.2 Blunders in Determining Comb Frequenciesfrep and fCEO

The short-term stability of the comb is evaluatedalong with the stability of the GPSDO as describedpreviously. A remaining question is whether the combcould be subject to systematic offsets even if theGPSDO is operating correctly. For example, therecould be errors in the locking of frep and fCEO or in themeasurement of these frequencies.

One of the more likely sources of trouble would bepoor S/N levels in the f – 2f interferometer, leading to

an incorrectly controlled value for fCEO . The accuracyof fCEO can be checked by using an independentfrequency counter to measure fCEO , although there issome danger that noise might generate the same error inthe counter as in the electronics controlling fCEO . Thispossibility can be eliminated if the counter measure-ments are unchanged when the signal is filtered withfilters of differing bandwidths. Confidence in properoperation will also be increased if it is observed that themeasured value for ftest does not change when the f – 2finterferometer is slightly misaligned, thus reducing theS/N. Finally, a good test of overall system performance,particularly sensitive to errors in fCEO , is to demonstratethat the measured value of ftest remains unchanged if thesign of fCEO is reversed by changing the polarity of theservo. By switching the polarity several times, it ispossible to distinguish a systematic offset frominstability of the test laser. This is a worthwhile testbecause consistent results will not be obtained unlessboth fCEO and fbeat are being measured correctlyand the signs of the two frequencies are understoodcorrectly.

It should also be verified that the repetition frequen-cy frep is being controlled correctly (or, if fbeat is con-trolled, that frep is measured correctly.) Actual miscount-ing of frep is highly unlikely, because S/N is very large.Furthermore, a miscount of 1 in a sampling time under10 s would be a large error that would be immediatelyobvious. One can imagine slow thermal-dependentphase shifts that would cause more subtleerrors—hence more difficult to detect—but even theseerrors should be detected by the multi-day stabilitymeasurements described previously. Nevertheless, it isworthwhile to verify proper operation by measuring frep

directly using an independent frequency counter, usingthe GPSDO as an external timebase for the counter.(An error in frep due to an error in the GPSDOfrequency would not be revealed by this test, butverification of GPSDO performance has already beendiscussed.) As mentioned previously, the frequencycounter used for this test must have good timingresolution and accuracy, and long measurement inter-vals will be required if the measurement is to achieve arelative uncertainty of 10–12 (which is an uncertainty of10–4 Hz in frep ). For example, a 100 ps timing resolutionwith a sample time of 100 s is only nominally sufficientto achieve the desired accuracy. Averaging shorter sam-ples may not achieve comparable accuracy if there aresystematic timing offsets in the counter. For suitablylong samples, verifying that the average result is inde-pendent of sampling time can be used to argue that anysuch offsets are not affecting results.


427

Another useful operational test is to verify that thesame value for ftest is obtained if frep is servolocked to areference or if fbeat is servolocked while frep is measured.(Similarly, if frep is controlled but a synthesizer is usedto generate different repetition frequencies, consistencyof measurement results obtained with different repeti-tion frequencies will provide good evidence that noblunders are causing bad results.) Self-consistency ofthe measurements of ftest also provides evidence that thebeat frequency is being measured correctly. The beatfrequency measurement can also be checked moredirectly, as described below.

4.3 Blunders in Beat Frequency Measurement

There is some danger that poor S/N ratio, electricalinterference, amplifier oscillation, or similar effects candegrade the measurement of fbeat . A simple methodsometimes used to verify proper counting is to misalignthe comb and laser beams so as to decrease the S/Nratio by more than 5 dB; if this does not change themeasured beat frequency then it is apparent that thereare no difficulties in the measurement due to poor S/N.(Possible electrical interference problems must bechecked separately.) As an alternative to this approach,we have implemented a real-time method for verifyingproper frequency counting. We simultaneously measuretwo beat signals, one at some frequency f and thesecond, arising from interference of the next comborder, at frequency frep – f. When the sum of the twosignals is exactly frep , this provides an excellent real-time indicator that there are no problems with the beatfrequency measurement.

If the two frequencies are sampled for 1 s and thetwo counters disagree by 1 Hz, this is indicative of1 miscounted cycle. We use external arming of thecounters to guarantee simultaneous measurements ofthe beat signals, and under these conditions the coun-ters should add exactly to frep . (Note that advancedcounters may employ noise-reduction techniques [23,24] that, if not disabled, can complicate simultaneousmeasurements—as is particularly evident when meas-uring a frequency modulated laser.) Actually, a fewmiscounts could be tolerated, as this would not affectresults at the 10–12 level, but one should be aware thatthe actual number of miscounts might be significantlylarger than indicated by the failure of the summation tomatch frep , because errors in the summation due tomiscounting associated with white noise tend to cancel.(The errors never cancel completely, but they willcancel on average if the bandpass widths of the two

filters are identical and the center frequencies of thetwo filters add to frep .)

4.4 Summary—How to Guarantee ThatMeasurement Blunders Do Not Occur

In summary, we believe that a laboratory can avoidalmost any imaginable measurement blunder if certainsteps are taken.

(a) The NIST GPS archives should be consulted toassure that the GPS system is operating within normal-ly expected uncertainty levels.

(b) It must be verified that the GPSDO faithfullyreproduces the GPS signal without long-term offsets.As discussed previously, this can be assured if (1) themanufacturer has directly or indirectly compared thespecific model used for the comb measurements (andthe specific firmware) to national standards and (2) theoperator monitors GPSDO diagnostics to assure thatthere is not a system failure (such as failure to lock tothe GPS signal due to RF interference [15]). Also, itwill build confidence if the Allan deviation, such asshown in Fig. 4, is consistent with expectations (basedon manufacturer’s specifications and data) for the par-ticular model in use.

(c) After verifying that the GPS signal and the GPSDOare operating correctly as described above, then theprimary remaining concerns are that frep , fbeat , and fCEO

might not be controlled and measured correctly. Thetimebase of these measurements is supplied by theGPSDO and thus has been verified, and a number ofstrategies have been discussed that will uncover errorsin the determination of frep , fbeat , and fCEO due to mis-counting or faulty servo performance.

(d) Probably the best way to demonstrate that noblunder has been made in a particular calibration is tolook for internal consistency between various ways ofperforming the measurement, including measurementswith the offset frequency polarity reversed and withdifferent repetition frequencies. Good consistencydemonstrates that the order number is known,shows that there are no significant errors in measuringfbeat or fCEO , and provides additional evidence thatimperfect frequency synthesis does not introduceerrors in frep that vary nonlinearly with frequency. It alsoverifies that the operator is not making calculationerrors.


428

(e) An additional source of error can be very impor-tant, even when the tests above demonstrate properoperation of the comb/GPSDO and demonstrate thatthe frequency counting is error-free. Even when thecurrent frequency of the test laser is being measuredabsolutely correctly, it must be further verified that thecurrent frequency has not been perturbed away fromthe normal operating frequency. The frequency ofthe test laser can potentially be shifted due to beamreflections re-entering the laser (optical feedback).Reflections will reduce the apparent stability of the testlaser and might shift the average value of ftest .Reflections are particularly likely from the surface ofthe photodetector; this surface should always be tiltedso that the lens focusing the laser beams onto the detec-tor does not form a cat’s eye reflector. A Faraday isolatoror an AOM can be used to greatly reduce reflections.

Optical feedback is, in fact, a common source ofproblems when calibrating a laser against the comb. Ifthere is any concern that feedback might be affectingresults, this can be tested by inserting a 0.3 opticaldensity filter into the beam from the test laser. Thefilter will substantially change the feedback, attenuat-ing any possible reflected amplitude by a factor of two(power by a factor of 4). The filter will also shift thephase of the reflected light. If the filter reduces the sizeof frequency fluctuations or causes a statistically signif-icant shift in the average frequency, then feedback isclearly a problem. Otherwise, we can conclude thatfeedback is not causing significant errors. The filtermust be tilted slightly so that a specular reflection fromthe surface does not cause feedback.

A final note: if a laser has a two-frequency output(such as Zeeman-stabilized lasers intended for use indisplacement interferometers) there is a danger that themeasured frequency is not what is needed for theintended application. The customer must specify whichfrequency component is relevant to his/her needs.(For a heterodyne displacement interferometer, it isthe component in the variable-length arm of theinterferometer.)

5. Concluding Remarks

We have described the comb system used in PED,which is our most accurate realization of the unit oflength as embedded in laser vacuum wavelength. ThePED comb has never been directly compared tostandards from other NMIs. An indirect comparisoncan be inferred through previous international compar-isons of our iodine stabilized lasers, but the uncertainty

of this indirect comparison is an order of magnitudeabove our claimed uncertainty for comb measurements.Furthermore, the central pieces of equipment used inthe experiment, the GPSDO and the comb, have neverbeen calibrated. In spite of these facts, we can be fullyconfident of our uncertainty claim. Although it is neverpossible to rule out every conceivable error of a meas-urement system, we can realistically establish a degreeof confidence in our comb measurements that is greaterthan our confidence in many other types of measure-ments carried out within PED, even including measure-ments whose uncertainties have been confirmed bydirect international intercomparisons. Confidence incomb measurements can be significantly higher thanfor many other types of measurements because of thegreat strength of the internal consistency tests, whichprovide continuous assurance that the measurement isworking properly.

In principle, a portable comb referenced to a portablecesium frequency standard can be used to establishinternational equivalence of comb systems; it canprovide high-accuracy, realistic, in situ testing of acomplete GPSDO-based comb system, including theeffect of GPS antenna performance as mounted in itsparticular environment. At the present time, however,there are no portable systems in use for direct verifica-tion of comb performance. The expense associated withmaintaining such systems and carrying out the compar-isons is difficult to justify. If it is indeed possible (asclaimed here) to develop full confidence in a system viainternal consistency testing, while relying on GPS tomaintain the link to international standards, then thereis little justification for deployment of a portablesystem to prove international equivalence of combsystems with modest uncertainty claims.

Our confidence in our link to international standardsis based on the traceability provided by the NIST GPSarchives, and also based partly on the fact that GPSDOunits similar to ours have been compared to NISTstandards (even though our particular unit has not).Given the validity of this link, we have argued thatshort-term errors in the system are the only errors ofsignificance, and we can establish an upper limit on ourmeasurement uncertainty via comparison to an iodinestabilized laser. Tests have been described to rule outthe possibility of blunders or system failures, leavingfew remaining plausible sources of undiscoveredproblems. Short of exotic errors such as deliberatesabotage, blunders that are not completely implausiblemight include firmware bugs in the GPSDO or analysiserrors. If firmware problems in our system have goneundetected, they must occur on time scales exceeding


429

our six-day test of short-term stability. Almost anymeasurement system could in principle have firmwareerrors that would appear only infrequently, makingthem very difficult to detect by a test (or by an inter-laboratory measurement comparison) of finite duration,and we have little choice but to accept this small risk.Likely analysis blunders, such as sign errors or an errortyping data into a spreadsheet, will be uncovered ifinternal consistency is verified in the manner that hasbeen described. More subtle analysis blunders thatwould not be immediately evident are largely confinedto rounding errors, not a major concern for moderncomputers.

For our measurements, the greatest danger is that ablunder will affect results at the 10–12 level but will notbe large enough to be apparent from measurements ofiodine stabilized lasers, where the answer is knownwith an uncertainty of a few parts in 1011 if the laser isoperating properly. Thus, the blunders must fall withina very specific range if the system gives a plausible butincorrect result for the calibration of an iodine stabi-lized laser. For example, a sign error can easily occurwhen accounting for the frequency shift of an AOMused as an optical isolator, but the resulting error wouldbe obvious if an iodine stabilized laser were measured.Smaller, subtler errors would be more likely in a moresophisticated system. A system that uses a cesium clockwill have more possibilities for errors or blunders thanwill a GPSDO, even though it is potentially capable ofhigher accuracy. When the measurement can be embod-ied with the simplicity of Eq. (2) and all frequencies aretied to the GPSDO, the list of plausible blunders isshort, and tests can be devised for any blunder that canbe anticipated.

Just as we have confidence in our measurements, wecan have good confidence in the results from other lab-oratories if they follow procedures similar to what hasbeen described here. Arguably, anyone in the world canreceive a GPS signal and use it to calibrate lasers overa wide range of wavelengths with good confidence inthe results. The basic traceability is delivered by theGPS system, and additional internal testing can verifyperformance of a particular piece of equipment, quanti-fy short-term uncertainty, and demonstrate competenceof personnel. The laboratory can carry out tests toverify operation at a level appropriate to those lasersthat they are calibrating—on the order of 10–12 for lasersstabilized by saturated absorption or ≤ 10–9 for polariza-tion or Zeeman stabilized lasers. There is no clear needfor additional testing or interaction with an NMI toprove competence. An interlaboratory comparison of

some form is always helpful to instill confidence, but itis not clearly necessary. Also, it may be possible toachieve very good confidence in the measurementusing a comparison whose uncertainty is not sufficient-ly low to fully test the uncertainty of the comb-GPSsystem [3]. The VIM definition of traceability can besatisfied simply by quantifying and documenting short-term errors while relying on NIST calibrations of theGPS system to provide a link to the SI second.

6. Appendix A: Acronyms

Below are definitions of acronyms used in this paper(including several acronyms that are in such wide-spread use that they might not need definition but arenevertheless included for completeness).

AOM: acousto-optic modulatorBBO: beta barium borateCEO: carrier-envelope offsetGPS: global positioning systemGPSDO: GPS disciplined oscillatorIR: infraredNIST: National Institute of Standards and

TechnologyNMI: National Metrology InstitutePED: Precision Engineering Division, NISTPI: proportional-integralPLL: phase locked loopPPLN: periodically poled lithium niobateRBW: Resolution BandwidthRF: radio frequencyRMS: Root mean squareSI: Système international d‘unités (the inter-

national metric system)S/N: signal to noise, or signal to noise ratioVIM: Vocabulaire International de Métrologie

(International Vocabulary of Metrology.)

7. References

[1] http://www.nist.gov/pml/div688/timekeeping.cfm (accessed10/5/2010).

[2] R. Hamid, E. Sahin, M. Celik, G. Özen, M. Zucco,L. Robertsson, and L.-S. Ma, 10–12 level reproducibility of aniodine-stabilized He-Ne laser endorsed by absolute frequencymeasurements in the BIPM and UME, Metrologia 43, pp.106-108 (2006).

[3] J. Stone, L. Lu, and P. Egan, Calibrating Laser VacuumWavelength With a GPS-based Optical Frequency Comb,Measure: The Journal of Measurement Science 2, pp. 28-38(2007).

[4] J. Ye and S.T. Cundiff, eds., Optical frequency combs and theirapplications, Springer, New York (2005).


430

[5] T. Udem, R. Holzwarth, and. T. W. Hänsch, Optical frequencymetrology, Nature (London) 416, pp. 233-237 (2002).

[6] P. Maddaloni, P. Cancio, and P. De Natale, Optical combgenerators for laser frequency measurement, Meas. Sci.Technol. 20, 052001 (19 pp.) (2009).

[7] R. W. Fox , S. A. Diddams, A. Bartels, and L. Hollberg, Opticalfrequency measurements with the global positioning system:tests with an iodine-stabilized He-Ne laser, Appl. Opt. 44,pp.113-120 (2005).

[8] F. Adler, K. Moutzouris, A. Leitenstorfer, H. Schnatz,B. Lipphardt , G. Grosche, and F. Tauser, Phase-locked two-branch erbium-doped fiber laser system for long-term precisionmeasurements of optical frequencies, Optics Express 12,pp. 5880 (2004).

[9] The commercial system, based on the work of reference [8], issold by Toptica Photonics AG. Some components of the Topticasystem, including the f – 2f interferometer and servo electron-ics, were designed and manufactured by Menlo SystemsGmbH.

[10] Y. Jiang, Z. Bi, L. Robertsson, and L-S Ma, A collinear self-referencing set-up for control of the carrier-envelope offsetfrequency in Ti : sapphire femtosecond laser frequency combs,Metrologia 42, pp. 304-307 (2005).

[11] The GPSDO is Timing Solutions model TSC 4410A.[12] http://nist.gov/pml/div688/grp40/gpsarchive.cfm (accessed

10/5/2010).[13] M. A. Lombardi, The Use of GPS Disciplined Oscillators as

Primary Frequency Standards for Calibration and MetrologyLaboratories, Measure: The Journal of Measurement Sciencevol. 3, pp 56-65 (2008).

[14] A. Lombardi, L. M. Nelson, A. N. Novick, and V. S. Zhang,Time and frequency measurements using the global positioningsystem, Cal Lab Magazine, Jul/Aug/Sept 2001, pp. 29-33(2001).

[15] M. A. Lombardi, A. N. Novick, and V .S. Zhang,Characterizing the performance of GPS disciplined oscillatorswith respect to UTC(NIST), Proc. of the 2005 IEEE Freq. Cont.Symp., pp. 677-684 (2005).

[16] G. Ritt, G. Cennini, C. Geckeler, and M. Weitz, Laser frequen-cy offset locking using a side of filter technique, Appl.Phys. B 79, 363-365 (2004).

[17] J. Zhang, Z. H. Lu, Y. H. Wang, T. Liu, A. Stejskal, Y. N. Zhao,R. Dumke, Q. H. Gong, and L. J. Wang, Exact FrequencyComb Mode Number Determination in Precision OpticalFrequency Measurements, Laser Physics 17, pp. 1025-1028.(2007.)

[18] L.-S. Ma, M. Zucco, S. Picard, L. Robertsson, and R. S.Windeler, A New Method to Determine the Absolute ModeNumber of a Mode-Locked Femtosecond-Laser Comb Used forAbsolute Optical Frequency Measurements, IEEE J. Sel. Top.Quantum Electron. 9, 1066-1070 (2003).

[19] C. R. Tilford, Analytical Procedure for determining lengthsfrom fractional fringes, Applied Optics 16, pp. 1857-1860(1977).

[20] J. A. Stone, J. E. Decker, P. Gill, P. Juncar, A. Lewis, G. D.Rovera, and M. Viliesid, Advice from the CCL on the use ofunstabilized lasers as standards of wavelength: the helium-neonlaser at 633 nm, Metrologia 46, pp. 4611-4618 (2009)

[21] International Vocabulary of Basic and General Terms inMetrology (VIM), BIPM, IEC, IFCC, ISO, IUPAC, IUPAP,OIML (1993).

[22] http://www.physics.unc.edu/~deardorf/uncertainty/defini-tions.html (accessed 5/19/2010).

[23] S. T. Dawkins, J. J. McFerran, and A. N. Luiten,Considerations on the Measurement of the Stability ofOscillators with Frequency Counters, IEEE Transactionson Ultrasonics, Ferroelectrics, and Frequency Control,pp. 918-925 (2007).

[24] S. Johansson, New frequency counting principle improvesresolution, Proc. IEEE Int. Freq. Contr. Symp. Expo., pp.628-635 (2005).

About the authors: Jack Stone and Patrick Egan arephysicists in the Mechanical Metrology Division of theNIST Physical Measurement Laboratory. The NationalInstitute of Standards and Technology is an agency ofthe U.S. Department of Commerce.


431

Documents

115 An Optical Frequency Comb Tied to GPS for Laser ... · 2.2 The Frequency Comb We use a commercial optical frequency comb [8, 9]2 as the basis of our system, with two modifications