15 June 1998

Ž .Optics Communications 152 1998 182–188

Full length article

Frequency stabilization of a diode laser to Doppler-free spectrumof molecular iodine at 633 nm

H. Talvitie ), M. Merimaa, E. IkonenMetrology Research Institute, Helsinki UniÕersity of Technology, P.O. Box 3000, FIN-02015 Hut, Finland

Received 27 January 1998; accepted 18 March 1998

Abstract

Ž . Ž .We report on frequency stabilization of a diode laser to the hyperfine components of the P 33 6-3 and R 127 11-5transitions of molecular iodine at the He-Ne laser wavelength of 633 nm. Single frequency operation and wavelength controlof the diode laser in a compact form is obtained by employing weak optical feedback from an integrated microlens. Thediode laser driven by an ultra low noise current supply provides nearly shot noise limited detection. A relative frequencystability of 5=10y12 is achieved at an integration time of 100 s. Harmonic distortion of the modulated output of the diodelaser due to spurious optical feedback is considered to be the main effect limiting the day-to-day frequency reproducibility of5=10y11. q 1998 Elsevier Science B.V. All rights reserved.

PACS: 42.62.Fi; 42.55.Px; 42.62.EhKeywords: Frequency stabilization; Laser spectroscopy; Diode laser

1. Introduction

Frequency-stabilized lasers are key devices for the real-ization of the length unit. In practice, one of the most usedradiation sources listed in the mise en pratique of the

w xdefinition of the metre 1 is an iodine-stabilized He-Nelaser at 633 nm. The frequency of the He-Ne laser is

Ž .stabilized to one of the hyperfine components of R 12711-5 transition in the B-X system of 127I . When properly2

operated, these devices offer an absolute optical frequencystandard with a relative uncertainty of 10y11.

As diode lasers have recently become available at 633nm, it has become attractive to take the advantage of theirsimplicity and compactness in the development of aportable frequency standard. The wavelength tunability ofdiode lasers allows also the use of iodine transitions that

Ž .are stronger than the R 127 11-5 transition coincidentwith the He-Ne laser. In addition, diode lasers have very

) E-mail: hannu.talvitie@hut.fi

low intrinsic intensity noise that can provide very sensitivedetection. There are, however, several problems with diodelasers to overcome before they can be used as frequencystandards. The available red diode lasers have generally amulti-mode spectrum with broad lines and a mode-hoplimited wavelength tuning.

Promising results have recently been achieved by stabi-lizing diode lasers to the Doppler-free spectrum of molecu-

w xlar iodine 2–6 . To improve the spectral and tuning prop-erties of the diode laser several methods have been usedincluding resonant optical feedback from a Fabry-Perot

w x w xcavity at 793 nm 2 , injection locking at 657 nm 3 , andexternal cavity configuration with grating feedback at 637

w x w xnm 4 and at 633 nm 5,6 .Another approach to obtain single longitudinal mode

operation and improved wavelength tuning of the diodelaser is the use of weak optical feedback from a closelyŽ . w x;100 mm mounted surface 7–9 . The advantages ofthis method are simple implementation and a very compactstructure. Improved wavelength tuning of the diode lasercan be obtained by adjusting the distance between the

0030-4018r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved.Ž .PII S0030-4018 98 00166-7

( )H. TalÕitie et al.rOptics Communications 152 1998 182–188 183

diode laser and the reflecting surface. A disadvantage isthat the optical feedback is not strong enough to reduce theintrinsic linewidth of the diode laser.

In this paper, we describe a compact and simple diodelaser system that is stabilized to the Doppler-free spectrum

Ž . Ž .of P 33 6-3 and R 127 11-5 transitions of molecularŽ127 .iodine I at the wavelength of 633 nm. The laser2

source is based on a diode laser that employs weak opticalfeedback from an integrated microlens. We also show thatsensitive detection can be achieved even though the laserlinewidth is almost as broad as the detected iodine lines.

2. Experimental setup

2.1. Laser source

The laser source used in this experiment is a commer-Žcially available 15-mW diode laser BlueSky Research

.PS010 operating near 635 nm. The diode laser packagew xcontains a virtual point source microlens 10 that is glued

Ž .closely to the laser chip distance 30 mm . A schematiclayout of the microlens-coupled diode laser is shown inFig. 1. The primary function of the microlens is to providea circular, diffraction-limited output beam. In addition,even though the 250-mm thick microlens is anti-reflection

Ž .coated R;0.8% , the outer surface of the lens reflects asmall fraction of the beam back to the laser. This weakoptical feedback produces additional wavelength depen-

Ž .dent loss of the laser cavity period of 0.5 nm and forcesthe laser to oscillate mainly in a single longitudinal modew x11 . The obtained side-mode suppression ratio is typically20–30 dB.

The optical feedback from the microlens is too weak toincrease the photon lifetime in the laser cavity and thus theintrinsic linewidth of the diode laser is not narrowed.

Ž .Fortunately, a relatively long cavity optical length 3 mmŽ .and high output power 15 mW of the laser yield a

linewidth that is suitable for this application. The linewidthis determined from the measured frequency modulationŽ .FM noise of the laser shown in Fig. 2. The white FMnoise level corresponds to a Lorentzian linewidth of about2 MHz. This value is in good agreement with the calcu-lated value of 1.8 MHz that can be derived using the

Fig. 1. Schematic drawing of a 635-nm diode laser with anintegrated microlens.

Fig. 2. Upper curve: Spectral density of the frequency modulationŽ .FM noise of the diode laser. Lower curve: Spectral density ofthe relative intensity noise. The shot noise level of detection iscalculated for the DC-photocurrent of the detector.

w xmodified Schawlow-Townes linewidth formula 12,13 withthe values shown in Fig. 1 and a typical value for thelinewidth enhancement factor of as5. In practice, thelinewidth is slightly broadened due to 1rf-type FM noiseto a value of about 3 MHz. The laser linewidth is narrowenough to enable detection of hyperfine lines of iodine, thewidths of which are typically 4 MHz.

The intrinsic intensity noise of diode lasers is very low.Ž .To avoid excess intensity noise and FM noise due to

w xinjection current noise, a current supply 14 with ex-tremely low noise of about 50 pArHz1r2 is used. It shouldbe noted that this value is well below the shot noise levelof the used injection current of 70 mA. The measuredrelative intensity noise of the laser is shown in Fig. 2. Theintensity noise reaches the white noise floor at 10 kHz andis typically only about 1 dB above the shot noise level ofthe detection.

Wavelength tuning of the diode laser can be obtainedby changing the temperature or injection current of thelaser. Within one longitudinal mode the dependences areabout y45 GHzrK and y2.4 GHzrmA. On a larger

Žscale, the temperature tuning rate is y150 GHzrK 0.2.nmrK , which provides a wavelength tuning range of

about 8 nm by changing the temperature from 0 to 408C.To reach the wavelength of 633 nm the diode laser iscooled to about 168C. The gaps in the wavelength tuningcaused by mode hops can be covered by adjusting the

w xdistance between the diode laser and the microlens 8 . Inpractice, the distance adjustment is obtained by applyingmechanical stress to the heatsink where the diode laserchip and the microlens are mounted. It is possible to selectabout ten different longitudinal modes within "0.4 nm ata time. The maximum required displacement of the mi-crolens is about half of the wavelength, and thus theproperties of the output beam remain practically un-

w xchanged 15 . By changing the injection current or thetemperature of the laser it is possible to achieve a continu-ous frequency tuning of more than 30 GHz.

( )H. TalÕitie et al.rOptics Communications 152 1998 182–188184

Fig. 3. Experimental setup to stabilize the diode laser frequency toa Doppler-free hyperfine component of molecular iodine. Abbre-viations: LD – laser diode, mL – microlens, L – lens, PD –photodetector, PBS – polarizing beamsplitter, BS – beamsplitter,lr4 – quarter-wave plate, and lr2 – half-wave plate.

2.2. Doppler-free detection

The Doppler-free hyperfine spectra of molecular iodineare detected using a collinear saturated absorption configu-ration. The experimental setup is shown in Fig. 3. Thecircular output beam of the microlens laser is collimatedand directed through a Faraday optical isolator. The beampasses a 10-cm iodine cell as the saturating beam with anoptical power of about 5 mW. The 1re2-diameter and thepeak intensity of the saturating beam are 0.8 mm and 1.6

2 Ž .Wrcm , respectively. A fraction 4% of the beam is thenreflected back as a probe beam. A polarizing beamsplitterand a quarter-wave plate are used to direct the probe beamto a photodetector and to improve optical isolation of thediode laser. The cold finger temperature of the iodine cell

Žcan be controlled and it is normally stabilized to 15.0".0.1 8C. The iodine cell is provided by Opthos Instruments,

Inc.The Doppler-free hyperfine components are detected

using the third harmonic lock-in technique. This methodprovides a negligible Doppler background and the possibil-ity to modulate the laser frequency via the injection cur-

Ž .rent. A modulation frequency 1 f of 10 kHz and amodulation amplitude of about 7 MHz are used. Thechosen modulation amplitude is relatively large becausethe hyperfine lines are broadened by the broad laser line.The modulation frequency is chosen so that the intensitynoise at the 3 f detection frequency of 30 kHz has already

Ž .reached the noise floor to obtain high signal-to-noise SrNratio. The third harmonic signal is detected using a lock-inamplifier. In order to lock the laser frequency, the outputsignal of the lock-in amplifier is fed back to the lasercurrent via a PI-controller. A 1-kHz unity-gain bandwidthof the feedback loop is achieved.

3. Results and discussion

3.1. Linear absorption

ŽA linear absorption spectrum of molecular iodine Fig.. Ž .4 near the R 127 11-5 transition was recorded by detect-

ing the transmitted intensity through the iodine cell. Thefrequency scan over 30 GHz was obtained by sweeping thetemperature of the diode laser. The strongest lines are

w xidentified according to Ref. 6 . The region consists ofseveral strong and well separated transitions that could beused for the purpose of frequency stabilization. An attrac-

Ž .tive transition is P 33 6-3 which is about 37 times strongerŽ .than R 127 11-5 and is only 1 GHz apart from the He-Ne

w xfrequency 1,5,6,16 . Part of the spectrum containing bothŽ . Ž .P 33 6-3 and R 127 11-5 transitions is shown in Fig.Ž .4 b . The transitions consist of 21 hyperfine componentsŽ .vertical lines assigned as b –b and a –a , respec-1 21 1 21

tively.

Fig. 4. Linear absorption spectrum of iodine at 633.0 nm with aŽ . Ž .frequency span of a 35 GHz and b 2.5 GHz. The strongest

Ž .transitions are identified by vertical lines in a . The vertical linesŽ . Ž .in b indicate the positions of the hyperfine components of P 33

Ž .6-3 and R 127 11-5 transitions. The frequency is given relative toŽ .the a component of R 127 transition. The cold finger tempera-15

ture was 20.08C and the continuous absorption background ofiodine is subtracted.

( )H. TalÕitie et al.rOptics Communications 152 1998 182–188 185

3.2. Hyperfine spectrum

The third harmonic signal of the hyperfine spectrum ofŽ .the P 33 6-3 transition is shown in Fig. 5. The numbers

from 1 to 21 refer to the assigned hyperfine components ofb –b . Due to the relatively large modulation amplitude1 21

Ž .of the laser 14 MHz peak-to-peak , some of the hyperfineŽ .components b , b , b , b , b are only partially re-5 6 11 19 20

Ž .solved and a doublet b –b remains unresolved. The9 10Ž .observed linewidth FWHM of the hyperfine components

is broadened by the laser linewidth and is about 8 MHz.The SrN ratio of the third harmonic signal was deter-

Ž .mined by measuring the root-mean-square rms noisedensity around the 3 f signal with a FFT spectrum ana-lyzer. A SrN ratio of 200 for a 1-Hz bandwidth is

Ž .typically obtained for the P 33 6-3 transition. It should benoted that the signal shown in Fig. 5 consists of rms valuesof the third harmonic signal but the visual appearance of

Žthe noise is determined by the peak-to-peak values for a.10-Hz bandwidth . We are also able to detect the hyperfineŽ .components of R 127 11-5 transition with a SrN ratio of

about 20. The ratio of 10 between the SrN ratios differsfrom the value of 37 given by the line strengths, becausethe intensity noise is increased by a factor of about 3 whenthe laser frequency is tuned to the center of the Doppler-

Ž .broadened P 33 transition. The main reason for this isattenuation of the main longitudinal mode of the laser thatincreases the intensity noise due to mismatch of the anti-correlation between the main laser mode and very weak

w xside modes 17 .

3.3. Frequency stability

To investigate the long term frequency stability of thediode laser system a beat frequency measurement wasperformed. An iodine-stabilized He-Ne laser locked to the

Ž .a component of R 127 11-5 was used as a reference.15

The beat signal was detected using an avalanche photode-

Ž .Fig. 5. Hyperfine spectrum of the P 33 6-3 transition obtained bythe third harmonic detection. Numbers 1–21 refer to componentsb –b , respectively. The frequency is given relative to the b -1 21 21

component. The detection bandwidth is 10 Hz.

wFig. 6. Relative stabilities square root of the Allan varianceŽ .xs 2,t of the beat frequency between the diode laser and any

iodine-stabilized He-Ne laser, when the diode laser is free runningŽ . Ž . Ž .A , locked to the a component of R 127 11-5 B , and locked18

Ž . Ž .to the b component of P 33 6-3 C . The dashed lines are21

estimates of frequency stability based on the signal-to-noise ratioof detection.

tector and analyzed with a RF-spectrum analyzer and afrequency counter. The frequency stability is shown in Fig.

Ž .6 as the square root of the Allan variance s 2,t of they

relative frequency fluctuations. The relative frequency sta-bility slope of the reference laser is approximately 1=

10y11 ty1r2 up to an integration time of ts3000 s andthus its contribution to the measurement results is small.

Ž .Plot A in Fig. 6 shows the frequency stability of thediode laser for free-running operation. The stability isnearly independent on the integration time correspondingto 1rf-type frequency fluctuation of the laser. The twoother plots represent results when the diode laser was

Ž . Ž .locked to the a component of R 127 11-5 B and to the18Ž . Ž . y1r2 Ž .b component of P 33 6-3 C . The t -slopes in B21

Ž .and C are due to white frequency noise. The Allanvariance slopes can be estimated from

kDny1r2s 2,t f t , 1Ž . Ž .y

n SrN0

where Dn is the linewidth of the hyperfine transitionŽ .FWHM , n is the laser frequency, SrN is the signal-to-0

noise ratio of the error signal measured in a 1-Hz band-width, and t is the integration time. The factor k takesinto account the steepness of the slope of the error signalwhich depends on the used harmonic detection method andthe ratio of the peak-to-peak modulation amplitude and thetransition linewidth. For a ratio of 1.7 and third harmonicdetection, the factor is ks0.4, whereas for first harmonicdetection it would be ks1. The values of these factors arebased on numerical simulations. The dashed lines in Fig. 6are deduced using the experimental values of Dns8

Ž . Ž .MHz, SrNs200 for P 33 and SrNs20 for R 127transitions. The calculated slopes are in good agreementwith the measured values. The measured frequency stabil-

Ž . y11 y1r2ity slope for the P 33 transition is 4.0=10 t . A

( )H. TalÕitie et al.rOptics Communications 152 1998 182–188186

y12 Ž .relative stability of 4.8=10 2 kHz is reached at anintegration time of 100 s. The frequency stability at long

Ž .integration times t)100 s is limited by slow changes ofthe operation parameters of the laser. For different daysand reproduced parameter settings, the mean laser fre-quency varied up to "50 kHz. The observed day-to-day

Ž .reproducibility 1s of the laser frequency is 25 kHz,y11 Žcorresponding to a relative value of 5=10 see Section

.3.5 .

3.4. Frequency interÕals

The frequency intervals of the hyperfine componentswere measured by locking the diode laser to the resolved

Ž .components of P 33 6-3 transition while keeping theŽ .reference laser locked to the a component of R 12715

11-5 transition. The measurement was restricted to thecomponents b –b due to the limited bandwidth of the4 21

frequency counter. The frequency measurements were re-peated several times and the average frequency values n LD

of each component relative to the b component were21

derived. The values are tabulated in Table 1 with theirŽ .experimental standard deviations 1s . The frequency val-

w xues n 1,16 recommended by the Comite Interna-´CIPMŽ .tional des Poids et Mesures CIPM are given for compari-

son and their standard uncertainty is 20 kHz. During the

interval measurement the average frequency of the b21

component relative to the a component was y519 21315Ž .kHz ss3 kHz which differs from the CIPM value of

y519 224 kHz by Dn s9 kHz. Taking this into ac-corrŽcount, the frequency differences are given as n ynLD CIPM

.qDn . The frequency differences are less than "20corr

kHz for most components, but are larger for the partiallyŽ .resolved components b , b , b , b , b . This is mainly5 6 11 19 20

caused by the frequency pushing effect of the neighboringcomponents due to our relatively large modulation ampli-tude and broad hyperfine lines. It is assumed that thiseffect is negligible for the values given by CIPM. Tocalculate the frequency shifts due to the pushing effect the

w xequations described in Refs. 18,19 were used. A pureLorentzian line shape is assumed for the hyperfine compo-nents. The experimental values of 8 and 7 MHz are used

Ž .for the linewidth FWHM of the hyperfine componentsand the frequency modulation amplitude, respectively. Thecalculated frequency shifts dn given in Table 1 are incalc

good agreement with the measured frequency differencesand well within the frequency standard uncertainty of 0.02MHz given by CIPM. It should be noted that according tothe calculation the frequency of the b component is15

shifted by about q20 kHz due to the a component of1Ž .R 127 11-5 transition existing only 1 MHz apart from the

center of b .15

Table 1Ž .Measured frequency values n of the hyperfine components b –b of P 33 6-3 transitionLD 4 21

Ž . Ž . Ž . Ž . Ž .Component Measured n MHz CIPM n MHz n yn qDn kHz Calculated shift dn kHzLD CIPM LD CIPM corr calc

Ž .b y660.528 5 y660.52 1 04Ža . Ž .b y610.760 5 y610.71 y41 y495Ža . Ž .b y593.974 9 y594.01 45 496

Ž .b y547.420 5 y547.42 9 07Ž .b y487.116 7 y487.08 y27 y78

Ža .b y y461.27 y y5039Ža .b y y453.23 y 35910Ža . Ž .b y438.911 22 y439.02 118 11411

Ž .b y347.384 3 y347.36 y15 y112Ž .b y310.320 4 y310.28 y31 113Ž .b y263.602 5 y263.59 y3 014Ž .b y214.550 7 y214.56 19 y115Ž .b y179.320 9 y179.30 y11 y516Ž .b y153.945 11 y153.94 4 517Ž .b y118.239 11 y118.22 y10 118

Ža . Ž .b y36.823 4 y36.72 y94 y8819Ža . Ž .b y21.911 5 y21.98 78 7420

b 0 0 9 1421Žb. Ž .b y519.213 3 y519.224 021

The frequencies are given relative to the b component. The digits in parentheses represent the standard deviation of the measured values21Ž .1s . The frequency values n recommended by CIPM and their differences to the measured values are also given. The last columnCIPM

shows the calculated frequency shifts dn caused by the frequency pushing effect of neighboring components due to the wide modulationcalc

amplitude used.Ža.Partially resolved or unresolved component.Žb. Ž .Frequency is given relative to the a component of R 127 .15

( )H. TalÕitie et al.rOptics Communications 152 1998 182–188 187

3.5. Frequency shifts

To estimate the reasons for the day-to-day variation ofthe laser frequency, we determined the sensitivity of thefrequency to the iodine cell pressure and the modulationamplitude. The laser frequency as a function of the iodinepressure was measured by changing the cold finger tem-perature in the range from 9.2 to 20.08C which corre-sponds to a change of iodine pressure from 10.1 to 26.9Pa. The frequency shifts of the laser locked to threedifferent hyperfine components are shown in Fig. 7. Theobserved linear pressure shift rates are b : y11.2 kHzrPa,17

b : y11.6 kHzrPa, and b : y9.1 kHzrPa. The mea-18 21

sured shift rates are consistent with those reported earlierfor an external-cavity diode laser applying an external

w xiodine cell 5 and for a He-Ne laser with an intra-cavityw xcell 20 .

The frequency shift as a function of the modulationamplitude for the b component is shown in Fig. 8. The21

observed shift is less than 10 kHz for modulation ampli-tudes up to 10 MHz, but for larger amplitudes the shiftincreases rapidly. This can partly be explained by thefrequency pushing effect of the neighboring components.The dashed line in Fig. 8 shows the frequency shift due tothe pushing effect calculated using the equations and pa-rameters described earlier.

A considerable effect that can produce frequency shiftis harmonic distortion of the modulated output of the laserw x18 , which can be produced by nonlinear modulationresponse. A nonlinear response appears if the diode laser is

w xcoupled with an external cavity 21 or even if weakw xoptical feedback is present 22 . In the case of an external-

cavity diode laser, coupled cavities can be avoided using adiode laser with a high-quality anti-reflection coating.Since the degree of nonlinearity increases with the strengthand distance of the optical feedback, the modulation re-sponse of the microlens-coupled diode laser should also benearly linear. However, additional optical feedback fromthe optical setup can still cause nonlinear behavior. We

Fig. 7. Frequency shift of the diode laser locked to the compo-nents b , b , and b as a function of iodine pressure. The17 18 21

dashed lines are linear fits to the data. The error bars indicate thestandard deviation of the measurement.

Fig. 8. Frequency shift of the diode laser locked to the componentb as a function of modulation amplitude. The error bars indicate21

the standard deviation of the measurement. The dashed line is thecalculated shift due to the frequency pushing effect.

noticed that even a very weak optical feedback can pro-duce harmonic distortion of the modulated laser output.

Since the diode laser is modulated through the injectionŽ .current, both frequency modulation FM and intensity

Ž . Žmodulation IM are produced. Harmonic components at.3 f of the IM signal at the laser output produce a back-

ground to the detected third harmonic signal and thuscause frequency shift of the laser. The measured ampli-

Ž .tudes rms values, relative to the dc-level of the IMharmonic components for the 7 MHz frequency modula-tion amplitude are typically about 1 f : 1=10y4, 2 f : 3=

y7 y7 Ž .10 , and 3 f : -1=10 measurement is noise limited .Under optical feedback the amplitude of the 3 f componentcan be increased or decreased by a factor of 10 dependingon the operation point of the laser. In our detection system,the frequency shift due to the 3 f IM component is esti-mated to be less than 25 kHz for a relative amplitude of

y7 Ž .10 . The harmonic components mainly 2 f and 3 f ofthe FM signal can also produce frequency shift as they areconverted to 3 f intensity modulation at the slopes of the

w xabsorption spectrum 18 . Preliminary calculations showthat the frequency shifts due to the 2 f and 3 f FM compo-nents can be even "70 kHz and "300 kHz for 1%

Ž .amplitudes relative to the 1 f component , respectively.The frequency shift due to the 3 f FM component is largestfor the hyperfine components at the slopes of the Dopplerbackground. Our present measurement accuracy of theamplitudes of FM components is limited to 1%. Accordingto the observed frequency shifts of the laser, however, weexpect the harmonic FM components to be much smaller.

Other parameters that can cause frequency variation ofthe diode laser system are optical power, beam alignmentin the saturated absorption configuration, weak side modes

Ž .of the laser separated by 50 GHz , broad linewidth andasymmetric lineshape of the laser, and offset drift ofelectronics. However, we believe that our present day-to-day reproducibility of 25 kHz is mainly limited by nonlin-ear modulation response of the diode laser caused byspurious optical feedback from the optical setup. We ex-

( )H. TalÕitie et al.rOptics Communications 152 1998 182–188188

pect better reproducibility by improving the optical isola-tion of the system and by stabilizing the laser to one of thehyperfine components of b –b located on the top of the12 14

Doppler-broadened background.

4. Conclusion

In this work we have developed a simple and compactdiode laser system, the frequency of which is stabilized to

Ž .the Doppler-free spectrum of the P 33 6-3 transition ofmolecular iodine at 633 nm. The system utilizes weakoptical feedback from an integrated microlens to obtainsingle frequency operation and improved wavelength tun-ing of the diode laser. Even though the laser linewidth isalmost as broad as the Doppler-free absorption lines ofiodine, a third harmonic locking signal with a signal-to-

Ž .noise ratio of 200 bandwidth 1 Hz is achieved. This ismainly due to the nearly shot noise limited intensity noiseof the diode laser that is obtained using an ultra low noisecurrent supply. A relative frequency stability slope of4=10y11 ty1r2 up to an integration time of 100 s and aday-to-day frequency reproducibility of 5=10y11 areachieved. The measured frequency intervals of the hyper-

Ž .fine components of the P 33 6-3 transition are in verygood agreement with values recommended by the CIPMwhen the frequency pushing effect of neighboring compo-nents is taken into account. Nonlinear modulation responseof the diode laser caused by unwanted optical feedback isconsidered to be the main limiting factor of the presentfrequency reproducibility.

Acknowledgements

The authors would like to thank J. Hu for usefuldiscussions and assistance in beat frequency measure-

ments. This work is a collaboration project with the Centerfor Metrology and Accreditation.

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