Testing ultrafast mode-locking at microhertz relative optical linewidth
aa r X i v : . [ phy s i c s . op ti c s ] J a n Testing ultrafast mode-locking atmicrohertz relative optical linewidth
Michael J. Martin , ∗ , Seth M. Foreman , , T. R. Schibli , and Jun Ye JILA, National Institute of Standards and Technology and University of ColoradoDepartment of Physics, University of Colorado, Boulder, CO 80309–0440 Current address: Varian Physics, Room 246, 382 Via Pueblo Mall, Stanford, CA 94305-4060 ∗ Corresponding author:
Abstract:
We report new limits on the phase coherence of the ultrafastmode-locking process in an octave-spanning Ti:sapphire comb. We find thatthe mode-locking mechanism correlates optical phase across a full opticaloctave with less than 2.5 m Hz relative linewidth. This result is at least twoorders of magnitude below recent predictions for quantum-limited individ-ual comb-mode linewidths, verifying that the mode-locking mechanismstrongly correlates quantum noise across the comb spectrum. © 2018 Optical Society of America
OCIS codes: (320.7090) Ultrafast lasers; (120.3940) Metrology; (140.4050) Mode-lockedlasers.
References and links
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1. Introduction
Optical frequency combs have become ubiquitous tools for precision optical measure-ment [1, 2]. They have enabled a new generation of optical frequency references basedon narrow transitions in single trapped ions [3, 4, 5, 6] and cold, neutral, atomic ensem-bles [7, 8, 9, 10, 11, 12, 13]. In addition, the comb’s ability to phase coherently transfer op-tical references across large spectral gaps allows for direct optical atomic clock comparison,placing new constraints on the evolution of fundamental constants [6] as well as driving atomicclock technology [7]. Recently, this broadband phase coherence has facilitated the productionof ground-state ultracold polar molecules via stimulated Raman adiabatic passage [14, 15].Previous evaluations have proven the frequency comb’s suitability for precision opticalmetrology. The fractional frequency uncertainty of Ti:sapphire-based frequency combs hasbeen evaluated at the 10 − level at 1000 s [16, 17]. Experiments testing the phase coherenceof Ti:sapphire combs have been able to place upper limits on the relative linewidth of differ-ent spectral regions for both locked and free running combs at 20 mHz [18] and 9 mHz [19],respectively, and were ultimately limited by by differential-path technical noise caused by aircurrents and mirror vibrations. The frequency comb used in this work was previously comparedo a 10 W average power Yb fiber comb locked to a common optical reference, with a resulting1 mHz resolution bandwidth-limited relative linewidth [20]. This indicates that the Ti:Sapphirecomb should be capable of supporting narrower relative linewidths. Here we report a new lowerlimit to the intrinsic phase coherence of a mode-locked Ti:sapphire laser phase locked to an op-tical reference, by both linewidth and phase noise measurements. This work demonstrates thatthe mode-locking process correlates the phase noise of individual frequency comb modes at alevel far below the quantum noise limit of individual free-running comb modes.A complete description of the frequency of a given output mode of a comb is given by n n ( t ) = n f rep ( t ) + f ( t ) + e n ( t ) . (1)Here n is the index labeling the harmonic of the repetition rate f rep ( t ) and f ( t ) is the carrierenvelope offset frequency. Both f rep ( t ) and f ( t ) are radio frequency (RF) signals that can bemeasured by directly observing the output pulse on a photodetector, and by employing a self-referencing technique ( e.g. [21]), respectively. The term e n ( t ) represents frequency noise in thevicinity of mode n that is not described by fluctuations in f rep ( t ) or f ( t ) . In other words, e n ( t ) accounts for mode-dependent noise terms that are at least quadratic in order with respect to n as a result of pulse-to-pulse fluctuations, which could be in part caused by time-dependent fluc-tuations of higher-order intracavity dispersion. Additionally, and most importantly, e n ( t ) is alsoassumed to include fluctuations not related to any other comb mode—fluctuations that are com-pletely uncorrelated across the comb. In this way, e n ( t ) also accounts for possible spontaneousemission-induced frequency noise that does not affect the global timing and phase parametersof the comb as a result of imperfect mode-locking. This is in contrast to the case of an idealcomb, where the mode n is perfectly defined with respect to mode m in the limit where both e m and e n are zero, due to the mode locking mechanism enforcing a fixed phase relation across thecomb. While the relative coherence of comb teeth is only limited by the quality of the mode-locking process, a free-running frequency comb, when compared to an external reference, hasnoise properties dominated by vibrational noise in the mirror mounts and thermal drifts in thelaser cavity coupling to both f rep and f .Actively phase locking a Ti:sapphire comb to an optical reference requires control of both f rep and f via control of the laser cavity length and pulse group delay ( e.g. [22, 23]). Combin-ing the comb and a continuous wave (CW) laser results in a time-dependent heterodyne beat,formed when a given comb tooth n interferes with the CW laser. This RF signal is denoted by f b , n ( t ) . By phase locking f b , n ( t ) to an RF source, the comb ideally acquires the optical phaseinformation of the reference laser and the RF source. This is due to the fixed phase relationshipbetween the comb’s output modes, enforced by the mode-locking process. When locked viacontrol of f rep , f b , n ( t ) is related to the RF reference frequency, f RF , and the comb degrees offreedom by f b , n ( t ) = f RF + d f b , n ( t ) = n f rep ( t ) + f ( t ) + e n ( t ) − n CW ( t ) . (2)Here, d f b , n ( t ) is the locking error due to the finite gain of the servo, and n CW ( t ) is the frequencyof the CW optical reference. We defer consideration of shot noise, which adds a white phasenoise term to the right side of Eq. (2), to Section 3. Additionally locking f to an RF reference,with locking error d f ( t ) , constrains both comb degrees of freedom. Solving for f rep ( t ) yields f rep ( t ) = n (cid:2) f RF + d f b , n ( t ) − e n ( t ) + n CW ( t ) − f − d f ( t ) (cid:3) . (3)It is important to note that the locking errors and noise term e n ( t ) write noise onto f rep ( t ) , thusglobally affecting the comb. Using Eq. (1) and f rep ( t ) given in Eq. (3), the optical frequency ofa comb mode numbered m is given by n m = mn (cid:2) f RF + d f b , n ( t ) − e n ( t ) + n CW ( t ) (cid:3) + (cid:16) − mn (cid:17) [ f + d f ( t )] + e m ( t ) . (4)gain the added noise term, e m ( t ) , represents extra frequency noise added to mode m by bothcorrelated and uncorrelated laser dynamics. Thus, measuring the comb mode m relative to mode n in a precise way allows upper limits on the intrinsic noise properties of the comb to be deter-mined. One way to accomplish this, as we report in this work, is to use the second harmonicof the optical reference to which mode n is locked to compare the phase coherence of combmodes n and m = n in a direct way. In this case, the expected heterodyne beat between thesecond harmonic light and the comb mode 2 n , which represents an out-of-loop measurementof the relative coherence of modes n and 2 n , is given by f b , n ( t ) = n n ( t ) − n CW ( t ) = f RF − f − d f ( t ) + d f b , n ( t ) + e n ( t ) − e n ( t ) . (5)Here, the term e n ( t ) − e n ( t ) represents the time-fluctuating out-of-loop frequency noise addedby the comb dynamics.The Wiener–Khinchin theorem relates time-domain frequency fluctuations to single-sidedfrequency noise power spectral density by S n ( f ) = Z ¥ cos ( pt f ) R xx ( t ) d t . (6)Here, S n ( f ) is the power spectral density associated with the time-fluctuating frequency x ( t ) .The autocorrelation term R xx ( t ) is defined by R xx ( t ) = h x ( t ) x ( t + t ) i = lim T → ¥ T Z T / − T / x ( t ) x ( t + t ) dt . (7)Additionally, S n ( f ) is related to phase power spectral density by S f ( f ) = S n ( f ) / f . (8)In the case that x ( t ) represents the noise e n ( t ) − e n ( t ) , then Eq. (6) provides a descriptionof the frequency noise power spectral density induced by this term on the out-of-loop beat.Equation (6) will include the possible effects of correlated dynamics between e n ( t ) and e n ( t ) ,which add in quadrature with the completely uncorrelated components of e n ( t ) and e n ( t ) .Thus, S n ( f ) of Eq. (6), with x ( t ) = e n ( t ) − e n ( t ) , represents an upper limit to the completelyuncorrelated noise between modes n and 2 n .
2. Experiment
The basic approach we take for our measurement of the out-of-loop coherence between modes n and 2 n is to stabilize the comb to a CW laser, and then use the second harmonic of the sameCW laser as a reference. When compared against the comb, this second harmonic referenceenables the phase coherence of the comb across a full optical octave to be tested. The opticalphase lock to the CW laser is implemented by servo control of the laser cavity length andpump power, while an f –2 f interferometer provides the additional signal used to stabilize f .The heterodyne beat between the comb and second harmonic CW reference represents the out-of-loop signal, which includes contributions from both the f rep and f phase locked loops asdescribed by Eq. (5). Any differential-path effects limit the sensitivity of this measurement, andwe have taken care to limit their effect by careful design.The specific octave spanning Ti:sapphire frequency comb used in this experiment is similarto the system described in [24]. The relatively low repetition rate of 95 MHz leads to high pulseenergy. This facilitates self phase modulation in the laser crystal, which broadens the spectrumto a full octave, the wings of which are not resonant with the cavity and are immediately trans-mitted by the output coupler. As detailed in Fig. 1, approximately 3 nm wide spectral regions at elf-referenced octave-spanning comb PCF NPRO1064 nm
NPRO and2 nd harmonic Broadened comb lock CW n CW n nb f nb f , CW n PPLN X2 lock fnf rep + fnf rep +
575 nm f Fig. 1. After passing through an f − f interferometer for self-referencing, the remaining spec-trum (600–1100 nm) from the octave-spanning frequency comb is broadened in photonic crystalfiber (PCF) and overlapped with the output of an NPRO Nd:YAG at 1064 nm. The resultingheterodyne beat, f b , n , and the carrier envelope offset frequency, f , are locked to RF references.Polarization-selective doubling of the overlapped beams ensures that only the CW light is fre-quency doubled, while collinear beam propagation reduces technical noise. The out-of-loop beatat 532 nm, f b , n , is analyzed as detailed in the text.
575 nm and 1150 nm are used to measure f with a standard f –2 f interferometer. This RF sig-nal is used to lock f , via group delay actuation (using the same method as described in [22]), toan RF source that shares a common timebase with all the RF sources in the experiment. The re-maining optical spectrum that is not used to measure f consists of 600–1100 nm light, which isrebroadened to an optical octave centered near 750 nm using photonic crystal fiber (PCF). Theoutput of the PCF is combined with a Nd:YAG non-planar ring oscillator (NPRO) CW opticalreference at 1064 nm with polarization orthogonal to the comb. After passing through opticalband-reject filters to remove the majority of the comb power in the unused central portion of thecomb spectrum, the co-propagating comb and 500 mW of 1064 nm light pass through a tem-perature stabilized periodically poled lithium niobate (PPLN) crystal, doubling the 1064 nmCW light while overlapped with the comb, reducing technical noise due to differential patheffects. The 1064 nm comb light is not doubled due to its orthogonal polarization, althoughif it were the resulting RF beat would be distinguishable from the true out-of-loop signal byits central frequency. A l / l at 532 nm wave plate is placed before the PPLNcrystal in order to ensure that the the second harmonic CW light has the same polarization asthe 532 nm comb light, since the PPLN outputs parallel polarizations of second harmonic andfundamental CW light. A single beam exits the PPLN crystal, with the 1064-nm comb lightpolarized orthogonally to the other three components of interest.A Glan-Thompson polarizer separates the majority of the 1064 nm comb light and ∼ n n ( n ≃ × )by phase locking the heterodyne beat, via cavity length and pump power control, to an RFsynthesizer. The remaining CW light at 1064 is transmitted by the polarizer, along with themajority of the comb and CW component at 532 nm.The transmitted light contains approximately 500 m W of second harmonic CW light andcomb near 532 nm, which is filtered to reject a large component of residual fundamental CWlight. We measure the resulting heterodyne beat between the comb mode 2 n and the second har-monic CW light. The expected RF frequency is given by Eq. (5), where n CW is now specificallyreferring to the frequency of the NPRO. In principle, the time-domain frequency error due tofinite servo gain, given by d f lock ( t ) = − d f ( t ) + d f b , n ( t ) , (9) -7 -6 -5 -4 -3 -2 M ea n s qu a r e vo lt a g e ( V ) -40 -20 0 20 40 Fourier frequency – f beat (MHz)
125 Hz (a) -7 -6 -5 -4 -3 -2 M ea n s qu a r e vo lt a g e ( V ) -40 -20 0 20 40 Fourier frequency – f beat (mHz) m Hz (b) Fig. 2. (a)
Out-of-loop beat mixed to 50 kHz at 100 kHz span with 125 Hz resolution bandwidth.Servo bumps are evident due to the contribution of finite locking gain and bandwidth to d f lock ( t ) . (b) Out-of-loop beat mixed to 1 Hz at 100 mHz span and 244 m Hz resolution bandwidth. Astrong coherent carrier is evident, even in this narrow resolution bandwidth. can be estimated from the in-loop phase error signals and is indistinguishable from fundamentalnoise described by the term e n ( t ) − e n ( t ) . The phase noise power spectral density associatedwith this term can be expressed via Eqs. (6–8) as S f ( f ) = f Z ¥ cos ( p f t ) h R d f d f ( t ) + R d f b , n d f b , n ( t ) − R d f d f b , n ( t ) i d t . (10)If the cross-correlation term, R d f d f b , n ( t ) , is zero, the expected contribution of the locking errorto the out-of-loop noise can thus be estimated by a weighted sum of the in-loop phase noisepower spectral densities. This will only occur if there are no common noise sources for theservos locking f and the comb to the NPRO reference.While Eq. (10) gives a prediction for the out-of-loop frequency noise from the in-loop lockingerror, there are two additional important sources of out-of-loop noise that are indistinguishablefrom fundamental comb noise. Shot noise propagates through the optical phase locks for both f b , n and f , and adds in quadrature with the shot noise on the detector for f b , n , as discussedin Section 3. Out-of-loop technical noise such as differential path Doppler noise or amplitudeto phase conversion in the PCF [25] also contributes to the measured out-of-loop phase noisespectrum.One important feature of Eq. (5) is that it does not depend on the frequency of the NPRO,which, despite being quite stable due to its monolithic construction, has drifts on the order of1 MHz/min. A shift of 1 MHz will show up on the 1 Hz level in f b , m ( t ) if m = n + i.e. theconjugate beat is used. This is clearly unacceptable for measuring comb linewidth on the m Hzlevel. Thus, only f b , n ( t ) is considered.Figure 2 shows fast Fourier transforms (FFTs) of f b , n ( t ) , after it has been mixed down tonear DC. Figure 2(b) represents the narrowest resolution bandwidth obtainable by the FFTinstrument used in this experiment, due to an instrument-limited measurement time of ∼ -5 -4 -3 -2 -1 P h a s e no i s e s p ec t r a l d e n s it y (r a d / H z / ) -2 -1 Fourier Frequency (Hz) I n t e g r a t e d R M S ph a s e (r a d ) Fig. 3. Left axis: Estimate of the out-of-loop phase noise contribution of d f lock based on aweighted incoherent sum of in-loop phase noise power spectral densities of both servo loops(blue) compared to the measured overall out-of-loop phase noise spectral density (red). Rightaxis: Root mean square (RMS) phase error integrated down from 100 kHz for both the contribu-tion of the servo errors on out-of-loop noise (dashed blue) and the measured out-of-loop phasenoise spectral density (dashed red). Fig. 3, is complimentary to the linewidth measurement. The out-of-loop phase noise is directlyvisible in the sidebands of Fig. 2(a). We note that the integrated root-mean-square (RMS) phaseis 0.35 rad when integrated down from 100 kHz to 10 mHz. This result, when combined withFig. 2(b)—which shows no significant features 50 mHz away from the carrier—indicates thatthe 244 m Hz instrument-limited coherent linewidth in Fig. 2(b) is a robust upper limit to thebeat linewidth. The estimated locking error contribution to the phase noise from typical errorsignal spectral densities is additionally shown in Fig. 3. Here, Eq. (10) has been used with theassumption that R d f d f b , n ( t ) →
0. The discrepancy between the predicted out-of-loop phasenoise near 50 kHz is due to the servo bump of the RF tracking filter used in the f phase lock.The f servo does not have the bandwidth to track this noise, so it appears only on the in-loopspectrum. Further discrepancies at Fourier frequencies in the 1 kHz range show the inadequacyof the assumption that R d f d f b , n ( t ) →
0, reflecting the fact that the cavity length and groupdelay servos are coupled when used to lock the comb to an optical reference and additionallymay share common technical noise sources. When integrated from 100 kHz to 10 mHz, theexpected integrated RMS phase predicted by the in-loop locking error is 70 mrad below theobserved out-of-loop integrated phase error.To overcome the measurement time-limited resolution bandwidth of 244 m Hz and examinethe out-of-loop beat at significantly improved phase noise sensitivity, we take the approachof studying the phase noise of a harmonic of the out-of-loop beat. A step recovery diodeimpedance matched at 100 MHz generates over 30 harmonics, amplifying the phase noise.The diode generates a ∼
100 ps pulse for every high-to-low voltage zero crossing, and thisoutput can be represented in the time domain as V ( t ) = V ¥ (cid:229) k = − ¥ L [ t − k / f in − D f ( t ) / p ] ≃ V ¥ (cid:229) n = − ¥ a n exp [ i p n f in t − in D f ( t )] . (11)Here, L ( t ) is a temporally narrow function compared to the inverse input frequency, f in , and Hz Synth . FFT80 MHz20 MHzSRD nb f (a) P o w e r ( a r b . ) -100 -50 0 50 100 f – f center (Hz) n = 1 n = 5 n = 10 (b) -10 -9 -8 -7 -6 -5 M ea n s qu a r e vo lt a g e ( V ) -80 -60 -40 -20 0 20 40 60 80 Fourier Frequency – f beat (mHz) m Hz, n = 10 (c) Fig. 4. (a)
RF electronics for analyzing the tenth harmonic of the out-of-loop beat. The 20 MHzsignal is filtered with a 2 kHz passband crystal filter and mixed to 100 MHz before being sentto the step-recovery diode (SRD). The tenth harmonic is selected for analysis. (b)
Schematicdepiction of a Lorentzian lineshape (green) broadened by frequency multiplication factors of n = n =
10 (red), yielding linewidths enhanced by a factor of 25 and 100, respec-tively. (c)
Out-of-loop signal, 190 mHz span with a factor of 10 frequency multiplication. Thiscorresponds to a 100-fold increase of the phase noise power spectral density in the vicinity of thecarrier, resulting in a 100-fold increase in linewidth, yet a strong coherent peak is still observedin the 244 m Hz resolution bandwidth. a k are the Fourier series coefficient for L ( t ) . The above approximation only holds if the phasecan be approximated as stationary on the timescale of the envelope width. In the limit where L ( t ) is a perfect Dirac delta function, Eq. (11) is exact. The final sum in Eq. (11) shows that forharmonic n the phase noise power spectral density, S n f ( f ) , is related to that of the fundamental, S f ( f ) , by S n f ( f ) = n S f ( f ) . (12)In order to focus on the phase noise nearest the carrier, a narrow crystal filter centered at20 MHz with a 2 kHz passband and 3 dB maximum ripple rejects phase and amplitude noisegreater than 1 kHz away from the carrier. This further enforces the assumption of stationaryphase noise on the time scale of 1 / f in and prevents broadband phase noise from causing carriercollapse when it is multiplied by the diode. Using the system whose key components are shownschematically in Fig. 4(a), we select the 10th harmonic of the step recovery diode and mix itdown to near DC. Figure 4(c) shows the beat note is still resolution bandwidth limited, evenwith the 100-fold increase in phase noise power spectral density. -3 -2 -1 P h a s e no i s e s p ec t r a l d e n s it y (r a d / H z / ) -2 -1 Fourier Frequency (Hz) I n t e g r a t e d R M S ph a s e (r a d ) Fig. 5. Left axis: Estimated shot noise floor (blue) compared to the measured out-of-loop phasenoise spectral density (red). Phase noise at Fourier frequencies above 12 kHz is limited by broad-band light noise on the out-of-loop detector. Right axis: integrated root-mean-square (RMS)phase error from 100 kHz to 10 mHz for the estimated shot noise contribution (dashed blue).
3. Discussion
From Fig. 4(c), it is clear that there is no significant phase noise near the carrier. In order to usethis measurement to extrapolate an upper limit to the actual non-instrument limited linewidth,we choose a functional form of the phase spectral density corresponding to a random walk inphase, S f ( f ) = Cf . (13)This type of phase diffusion is found due to spontaneous emission in laser systems, resultingin the famous Schawlow-Townes limit [26, 27] and would arise if the comb were limited by an“intrinsic linewidth” due to random-walk relative phase diffusion amongst comb modes. Theoptical power spectral density corresponding to this form of phase noise is a Lorentzian withfull width at half maximum (FWHM) given by D n
FWHM = p C . Using Eq. (12), the FWHM ofthe n th harmonic of the step recovery diode, D n n FWHM , will thus be related to that of the firstharmonic by
D n n FWHM = n D n
FWHM . (14)This broadening effect is illustrated in Fig. 4(b). By applying this relationship to the Fourier-limited linewidth measurement of 244 m Hz with multiplication factor n =
10, we can extrapo-late an upper limit to the comb intrinsic linewidth of 2.44 m Hz.Figure 3 indicates that there is approximately a 70 mrad difference between the extrapolatedintegrated phase error and the measured integrated phase error. The extra ∼
100 mrad in-loopintegrated phase error between 100 kHz and 10 kHz that is due only to the RF tracking filterservo bump indicates that the true difference, which is an estimate of the total out-of-loop noise,is at least twice as large. The extra phase noise could come from technical sources, such asamplitude to phase conversion in the PCF [25] or differential-path Doppler noise. Fundamentalnoise sources that cause out-of-loop noise are shot noise and the noise term e n ( t ) − e n ( t ) .In order to estimate the effect of shot-noise limited detection on the out-of-loop beat, weconsider the signal to noise ratio at each of the relevant detectors assuming shot-noise-limiteddetection. By modeling each servo using simple proportional-integral (PI) transfer functionsnd using empirically determined gain and bandwidth coefficients for both the cavity lengthand group delay servos, we show in Fig. 5 the effect of shot noise on the out-of-loop beat. Thisnoise floor is compared with the measured out-of-loop spectrum, indicating that below 30 kHz,the measurement is technical noise limited. This justifies the choice of a 100 kHz upper boundon the phase noise integration shown in Fig. 3. Under a sufficiently large servo bandwidth, thenoise contribution due to Eq. (9) would ideally be eliminated, leaving the shot noise floor asthe ultimate limit to the measurement’s sensitivity to other noise sources, such as out-of-looptechnical noise or intrinsic comb noise. As seen in Fig. 5, the total estimated integrated phasedue to shot noise is of order 100 mrad.While it is surprising that out-of-loop technical noise sources do not dominate the measuredout-of-loop spectrum, it is important to note that many of the typical noise sources, such asdifferential path effects, were designed to be common-mode in our measurement. The results of[20] show that even without such careful design, the total effect of measurement noise does notinhibit sub-millihertz measurement precision. Beyond common mode cancelation, additionalout-of-loop noise sources may be obscured due to other higher-order correlations amongst theseprocesses.In analogy to the Schawlow-Townes limit, the effect of spontaneous emission noise on fre-quency combs has been explored [28, 29, 30, 31]. With spontaneous emission as the onlyquantum noise source, Paschotta et al. find quantum-induced timing jitter causes extra phasenoise in the spectral wings [29]. Wahlstrand et al. take a more complete approach, consider-ing all quantum noise drivers and empirically determined coupling coefficients to extrapolatethe quantum-limited linewidth of individual comb modes as a function of frequency [31]. Bothof these results are predictions of the spectral width of a comb mode compared to an outsidereference, not between individual lines of the same comb. However, they are useful concep-tual tools to understand the scale of the quantum noise. While a locked in-loop error signallinewidth can be measured to be arbitrarily small, it is the passive mode-locking mechanismthat keeps the relative coherence of comb modes an octave away from being differentially af-fected by spontaneous emission noise. We obtain a conservative lower-limit for the free-runningquantum-limited linewidth of our system from the result given by Paschotta et al. D n = D n ST h + ( pdnt p ) i . (15)Here, D n ST is the result obtained by directly applying the Schawlow-Townes limit to the comb, dn is the distance from the central frequency, and t p is the output pulse width. When we insertthe relevant parameters, we obtain D n ≃ m Hz for comb wavelengths near 1064 nm and532 nm. The analysis of Wahlstrand et al. indicates that full consideration of noise couplingprocesses can result result in linewidths orders of magnitude larger in these spectral regions.By observing a relative linewidth between comb lines an octave apart that is at least twoorders of magnitude lower than that predicted for a free-running quantum-limited comb, wehave shown that the mode-locking mechanism does an excellent job of correlating quantum-driven phase noise between two comb modes that are one octave apart. Even though only twodegrees of freedom are controlled, the passive mode-locking process leads to a well-definedphase relationship across the visible and near-IR spectrum.
4. Conclusion
By carefully controlling sources of technical noise, we have placed a new limit on the phase co-herence of an optical frequency comb by using second harmonic generation to compare modes n and 2 n . Phase noise measurements show a total RMS integrated optical phase error from100 kHz to 10 mHz of 0.35 rad, and that the majority of accumulated phase error is due to finiteervo gains, not technical noise from the PCF or intrinsic noise from the comb. We have addi-tionally placed limits on the fundamental phase-coherence of second harmonic generation, anextension of the results of Stenger et al. [19]. The relative linewidth of comb modes an octaveapart is less than 2.5 m Hz, with no significant phase noise features near the carrier, indicatingthat the mode-locking process strongly correlates quantum noise due to spontaneous emissionacross the comb. This result is at least two orders of magnitude below the predicted individualcomb modes’ quantum-limited linewidths. Thus, local phase perturbation due to spontaneousemission at a given wavelength is converted into a global phase perturbation, affecting all modesequally within the mode-locking bandwidth. The robust broadband phase coherence shown heredemonstrates that there is essentially no practical limit to comb-facilitated coherent distributionof optical clock signals to arbitrary visible and near-IR wavelengths.