Lim et al. Light: Science & Applications (2019) 8:1 Official journal of the CIOMP 2047-7538https://doi.org/10.1038/s41377-018-0109-7 www.nature.com/lsa

ART ICLE Open Ac ce s s

Probing 10 μK stability and residual drifts inthe cross-polarized dual-mode stabilizationof single-crystal ultrahigh-Q opticalresonatorsJinkang Lim1, Wei Liang2, Anatoliy A. Savchenkov2, Andrey B. Matsko2, Lute Maleki2 and Chee Wei Wong1

AbstractThe thermal stability of monolithic optical microresonators is essential for many mesoscopic photonic applicationssuch as ultrastable laser oscillators, photonic microwave clocks, and precision navigation and sensing. Theirfundamental performance is largely bounded by thermal instability. Sensitive thermal monitoring can be achieved byutilizing cross-polarized dual-mode beat frequency metrology, determined by the polarization-dependentthermorefractivity of a single-crystal microresonator, wherein the heterodyne radio-frequency beat pins down theoptical mode volume temperature for precision stabilization. Here, we investigate the correlation between the dual-mode beat frequency and the resonator temperature with time and the associated spectral noise of the dual-modebeat frequency in a single-crystal ultrahigh-Q MgF2 resonator to illustrate that dual-mode frequency metrology canpotentially be utilized for resonator temperature stabilization reaching the fundamental thermal noise limit in arealistic system. We show a resonator long-term temperature stability of 8.53 μK after stabilization and unveil varioussources that hinder the stability from reaching sub-μK in the current system, an important step towards compactprecision navigation, sensing, and frequency reference architectures.

IntroductionHigh-precision clocks and oscillators are cornerstones

for global navigation to provide accurate timing and dis-tance information, via satellite-to-satellite or satellite-to-ground communications, for the synchronization ofelectronic device communications and for quantuminformation processing using atomic qubits1–3. Likewise,precision time–frequency transfer in timing links ishelpful for synchronization and data exchange in coherentarrays, gravitation sensing, and relativity measurements4.Current remote free-space links relying on radio-

frequency communications have limited data transferrates largely due to the available spectral bands, and oftenrequire large antenna sizes and high power consumptionsdue to diffraction, which naturally demands higher fre-quency carriers. However, the oscillators based on elec-tronics attain their spectral purity from the high-qualityfactor of the resonator circuit, which usually degradeswith increasing frequency. This design, therefore, poten-tially increases not only the size and the power con-sumption but also the complexity of the signal generationsubsystems. As a result, free-space low-noise optical car-rier networks provide an alternative with compact foot-prints for high-precision timing synchronization and highdata rate links4–6.Optical microresonator devices are a potential platform

to provide both high spectral purity and stable clockoperation for future timing networks. High-quality factor

© The Author(s) 2019OpenAccessThis article is licensedunder aCreativeCommonsAttribution 4.0 International License,whichpermits use, sharing, adaptation, distribution and reproductionin any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if

changesweremade. The images or other third partymaterial in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to thematerial. Ifmaterial is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtainpermission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.

Correspondence: Jinkang Lim (jklim001@ucla.edu) orAndrey B. Matsko (andrey.matsko@oewaves.com) orChee Wei Wong (cheewei.wong@ucla.edu)1Fang Lu Mesoscopic Optics and Quantum Electronics Laboratory, Universityof California, Los Angeles, CA 90095, USA2OEwaves Inc., 465 North Halstead Street, Suite 140, Pasadena, CA 91107, USA

1234

5678

90():,;

1234

5678

90():,;

1234567890():,;

1234

5678

90():,;

(Q) solid-state microresonators, especially whispering-gallery resonators7, have demonstrated superior perfor-mances in ultralow-noise laser oscillators and microwavegeneration via optical frequency division8–13. They canalso be disciplined to optical atomic transitions in ana-logous architectures of quartz-crystal microwave oscilla-tors disciplined to radio-frequency atomic hyperfine clocktransitions14. Single-crystal whispering-gallery resonatorsare advantageous when compared to other solid-statemicroresonators since they have Qs more than a billion,significantly higher than those of quartz oscillators (106)and rubidium clock hyperfine atomic transitions (107)used in current time and frequency standards. This high-Q or narrow-linewidth feature provides lower phase noiseand thus potentially lowers the timing jitter of the elec-tromagnetic carrier. Furthermore, the small form factor ofthe resonators also allows for miniaturization towardsspace applications15. However, solid-state micro-resonators experience large thermal noise induced by themedium, potentially hindering the ultimate performanceof the microresonators towards advanced applications inultrastable laser oscillators and photonic microwaveclocks. Temperature-induced aging could also be detri-mental for sensitive optical sensors using the micro-resonators. For instance, the achievable bias stability ofthe microresonator resonant gyroscopic sensor is typicallylarger than 3° h−1 16 and is bounded by the resonatorthermal stability.For sub-millimeter-scale resonators, theoretical models

predict that sub-μK stability is required for achieving thethermodynamically limited noise level corresponding to aresonance frequency stability of 10−13 or less17. Such sub-μK monitoring cannot be achieved with conventionalthermistors due to the thermistor nonlinearity. Thetemperature of the thermistors is generally approximatedby using the Steinhart–Hart thermistor third-orderapproximation. To overcome the detection and controllimit, a cross-polarized dual-mode temperature stabiliza-tion method has been suggested in birefringent crystallineresonators18–20. The approach draws from the frequencystabilization of quartz oscillators using two mechanicalmodes with different thermal sensitivities21. In ourapproach and prior studies, the laser frequency is lockedto an extraordinary polarized resonator mode, and itsradio-frequency modulated sideband, or a second laserfrequency, is subsequently locked to an ordinary polarizedresonator mode. The different thermorefractive coeffi-cients of the orthogonally polarized modes allow thethermal sensitivity of the resonator mode to be detecteddown to sub-μK levels with a precisely measurable cross-polarized dual-mode radio-frequency beat, which facil-itates the precise thermal stabilization of the resonator.Previous experiments implementing this approach have

reported the microresonators that are temperature-

stabilized with 10- to 100-nK precision, which is cali-brated by the in-loop cross-polarized dual-mode beatfrequency measurement. For the sub-μK resonator tem-perature, the laser frequency instability locked to suchresonators should reach less than 10−13 even at 1000 sintegration time based on the theoretical thermodynamicnoise models22. However, the frequency instabilityreported in the experiments shows a substantial dis-crepancy from the theoretical model, implying that thestringent correlation between the resonator temperatureand the cross-polarized dual-mode beat frequency couldno longer hold true in time. This discrepancy raises thequestion of whether the thermodynamically limitedresonator temperature stabilization using the dual-modebeat stabilization method is fundamentally insufficient forthe realistic millimeter-scale ultrahigh-Q resonator or thediscrepancy stems from extraneous noise that needs to beidentified and compensated.First, one can consider that this mismatch arises from

the temperature nonuniformity or temperature gradientcaused by the temperature difference between the reso-nator and its enclosure or between the substrate and theresonator, leading to the imperfect control of the reso-nator thermal expansion. Second, the waveguide effectsand the modal interactions experienced by optical modescan possibly modify the refractive indices and the modevolumes of the resonator. Third, the cross-polarizationmetrology probes predominantly the optical modevolume temperature with the optical mode distribution atthe resonator rim, while the resonator volume tempera-ture—and hence the volumetric size fluctuation—may beless correlated. In addition, the finite heat diffusion time-scale results in a time lag (before thermal equilibrium) inthe resonator stabilization. The first question was studiedby Baumgartel et al.19. Varying the temperature differencebetween the resonator and its enclosure introduces anextraradial deformation caused by the variation in tem-perature gradient along the radial direction, and thedeformation increases with the size of the resonator. Inthis work, we implement a thin cylindrical ultrahigh-Qmagnesium difluoride (MgF2) resonator with a smallerradius by a factor of 2.407 compared to the one used inref. 19 to alleviate the impact of the radial deformationcaused by the resonator temperature gradient, andinvestigate the others. Light guiding effects such asrefractive index modification, mode extinction, and modalarea expansion are further supported by finite-element-method (FEM) numerical simulations on potential effectson the dual-mode temperature stabilization. Moreover,we study the impact of circulating laser-induced heatingand heat diffusion with an associated time delay in theresonator.To interrogate the resonator mode volume temperature,

two low-phase-noise continuous wave (cw) lasers are

Lim et al. Light: Science & Applications (2019) 8:1 Page 2 of 9

locked to a transverse magnetic (TM) mode and atransverse electric (TE) mode of the whispering-galleryresonator, and the dual-mode beat frequency is measuredby heterodyning the two stabilized lasers at a photo-detector. We measure and demonstrate the time-dependent frequency correlation of the dual-mode beatfrequency with the resonator temperature, represented bythe resonant frequency drift due to the thermal expan-sion, by simultaneously counting the dual-mode beatfrequency and a polarization-mode stabilized laser fre-quency. The degree of correlation between the open-loopdual-mode beat frequency and the resonance frequencyapproaches the theoretical bounding limit, where theaveraged resonator volume temperature change is equalto the optical mode volume temperature change, withincreasing integration time. By locking the dual-modebeat frequency to a radio-frequency clock with a passivesuppression of the extraneous noise, we achieve a 51.78×enhancement in long-term frequency instability down to14.55 kHz at 1000 s integration time, corresponding to atemperature instability of 8.53 μK and a linear drift of0.54 kHzmin−1.

ResultsTheoretical model of dual-mode temperature stabilizationThe resonator is made from a z-cut magnesium

difluoride (MgF2) single crystal. It is a thin disk-shapedresonator with a radius (R) of 1.35 mm, which is sizablysmaller than that of our prior study to mitigate the tem-perature gradient and shorten the heat diffusion time toachieve fast thermal equilibrium. A ring-down time of3.63 μs is measured, corresponding to Q of 2.1 × 109, asillustrated in Fig. 1a. MgF2 is a birefringent crystal withne= 1.382 and no= 1.37, where (o) represents theordinary polarized light or polarized with a TM mode and(e) represents the extraordinary polarized light or polar-ized with a TE mode, for the z-cut resonator. The reso-nance frequency in the whispering-gallery resonator has alinear function of temperature dependence that can bequantified using the thermal expansion coefficient,describing the resonator expansion perpendicular to thecrystal axis (αl,(o)), and thermorefractive coefficients (αn,(o,e)).The expansion along the symmetry axis has a much weakerimpact on the mode frequency and can be neglected. Atroom temperature, these coefficients are αl,(o)= (1/R)(dR/dT)= 8.9 × 10−6, αn,(e)= (1/ne)(dne/dT)= 0.64 × 10−6, andαn,(o)= (1/no)(dno/dT)= 0.23 × 10−6 23.The thermal dependence of the TM and TE mode fre-

quency difference, fTM−TE= fo− fe, (i.e., heterodyne beatnote), can be determined by ΔfTM–TE/ΔT= –ν0(αn,(o)− αn,(e)), where ν0 is the optical carrier frequency. To derive thisexpression, we assume that the modes occupy the samemode volume and are nearly degenerate so that an increasein resonator size does not change their relative frequency.

For an optical carrier wavelength of 1565.5 nm, we char-acterize the dual-mode beat frequency thermal fluctuation(i.e., ΔfTM–TE) in our MgF2 resonator, which is 78.71ΔTm

MHz based on the difference between the TM and TEmode thermorefractive coefficients, where ΔTm is thetemperature change in the whispering-gallery mode chan-nel. On the other hand, the frequency shift induced bythermal expansion is described by ν0αl,(o) for the ordinarypolarized mode (or the electric field of the mode is per-pendicular to the crystal axis) in the resonator. The coeffi-cient αl,(o) is typically greater than αn,(o,e). The value of thefrequency fluctuation of the ordinary polarized mode isestimated to be 1.705ΔTR GHz based on the thermalexpansion coefficient, where ΔTR is the resonatorvolume temperature change. We note that the temperaturefluctuations are quantified with precision frequencymetrology. The ratio of the dual-mode beat frequencyvariation and a single mode frequency variation due to thethermal expansion becomes a factor of 21.66 whenthe mode volume temperature variation (ΔTm) is equal tothe resonator volume temperature variation (ΔTR). Thiscorrelation can be utilized for stabilizing the resonatortemperature by locking the radio-frequency dual-modebeat to a low-noise radio-frequency clock with properfeedback actuation.

Integration of waveguide effects on the resonatortemperature stabilizationTo detail the impacts of waveguide modal effects on the

resonator temperature stabilization, we perform the FEMmodeling of the MgF2 resonator with a 12.5 μm radius ofcurvature and a 1.35mm bending radius. The simulationsclearly show the well-confined TM and TE modes at ourexperimental laser frequencies. The purities of these twomodes are high enough, and their extinction ratios from theorthogonal modes are 35 dB (Supplementary Informa-tion S1). The eigenfrequencies of the TE and TM modesare determined to be fTE= 191.44842724346256 THz andfTM= 191.45602707701247 THz, respectively. The separa-tion of the two frequencies (fTM–TE) is 7.599 GHz, which isclose to the experimentally observed value of 5.953 GHz.The deviation is attributed to slight differences in thewaveguide geometry between the numerical model and thefabricated device, as well as in minute material constantvariations.We calculate the effective refractive indices including the

waveguide modal effect for both TM and TE resonantmodes in the resonator. The FEM simulation shows that themodal effect modifies the refractive indices and the result-ing refractive indices decrease by a factor of 5 × 10−3 forboth modes, as illustrated in Fig. 1c. This is small becausethe modal areas for both confined modes are fairly large(modal area of 75.52 μm2 for the TM mode and 74.56 μm2

for the TE mode) and the bending radius is also large.

Lim et al. Light: Science & Applications (2019) 8:1 Page 3 of 9

The refractive index change at the different temperature(T+ΔT) is computed by using temperature-dependentrefractive index equations: ΔneffiΔT [0.09797–5.57293 ×10−4T] × 10−5 and ΔnoffiΔT [0.04183–5.63233 × 10−4T] ×10−5 from ref. 17. The thermorefractive index changes dueto the temperature change are small—we use a ΔT= 10 Kto visualize the modification. Regardless of the temperature(even with ΔT= 10 K), the refractive index modificationdue to the waveguide modal effect is equally a factor of 5 ×10−3 for each mode, implying that the difference inrefractive indices [αn,(o)− αn,(e)] can thus be safelyconsidered the same for our dual-mode temperaturestabilization. We also simulate the possible anisotropicthermal expansion of the mode cross-sections. For 1 Kof the temperature change, the resonator deformationis small enough, as illustrated in Supplementary Informa-tion S2, and the distinct mode area change is not obser-ved. If we consider that our resonator mode volumetemperature can be stabilized to sub-μK, the modal sizefluctuations can therefore be effectively neglected forstabilization.

Correlation measurements between the dual-modefrequency and the resonator temperatureWe encapsulate the microresonator in a compact alumi-

num oven with temperature control and photodetector unitsin a small form factor, as shown in Fig. 1b. The resonatortemperature is prestabilized at 295.7 K with mK precisionover hours via the proportional–integral–differential (PID)control of a thermoelectric cooler (TEC) in a resonatormount. The resonator is then placed in a compact vacuumchamber, and the chamber is evacuated to maintain a highvacuum level (8 × 10−6 torr) to eliminate convective heattransfer and environmental perturbations such as pressureand humidity. For instance, strong acoustic noise peaks areeliminated from the dual-mode beat phase noise, as illu-strated in Supplementary Information S3. Figure 2 shows theschematic experimental measurement set-up. We utilize twocw lasers to interrogate a TM mode and a TE mode,respectively. The lasers that we used for stabilization to theresonator have a linewidth of 3 kHz at 100ms integrationtime. The laser helps to interrogate ultrahigh-Q resonanceswithout unwanted heating of the resonator so that the

0.1a

b

c

0.0

Inte

nsity

(a.

u.)

–0.10 5 10

1 mm

Time (μs)

Modal area: 75.52 μm2

TETM

T = 295.7 K

T + 10 K

Refractive index Modified index

Refractive index Modified index

f TE =

191

.448

4272

4346

256

TH

z

f TM

= 1

91.4

5602

7077

0124

7 T

Hz

Modal area: 74.56 μm2

Input PM fiber

Oven

PD

out

put

15

Q = 2.1×109

1.382 1.374416

1.374424

1.362363

1.362366

1.370

1.382008225

1.370002595

ne

no

no + Δno

ne + Δne

Fig. 1 Integrated thin-disk MgF2 ultrahigh-Q whispering-gallery resonator and FEM simulations of the MgF2 whispering-gallery resonatormodes. a Ring-down measurement of the resonator shows a cavity lifetime of 3.63 μs corresponding to Q= 2.1 × 109. Inset: image of the thin-diskwhispering-gallery resonator. b Integrated package. The resonator is placed in a compact aluminum oven with a PID controlled TEC and the laserlight is delivered into the resonator via an optical fiber. Both TM and TE laser signals are measured at a single photodetector. Optical resonance modefrequencies for TM and TE are located at fTM= 191.45602707701247 THz and fTE= 191.44842724346256 THz. c The frequency difference between thetwo orthogonal modes (fTM–TE) is evaluated to be 7.599 GHz, close to the experimentally observed value. The FEM simulation shows modal areas andeffective index changes due to the whispering-gallery resonator guiding effect for individual modes. The waveguide effect lowers the refractiveindices by a factor of 5 × 10−3 for both polarization modes, numerically examined across a ΔT of 10 K

Lim et al. Light: Science & Applications (2019) 8:1 Page 4 of 9

short-term scale temperature stability is not limited by thelaser. The orthogonally polarized laser beams are combinedwith a fiber-optic polarization beam combiner andevanescently coupled into the whispering-gallery resonatorvia a prism24.Each laser frequency is locked to the resonance mode

via the Pound–Drever–Hall (PDH) technique25. Theoptical power used to interrogate the resonator is 20 μWfor each resonance mode. We first find the resonancefrequency of a TM mode and then interrogate a TE modenear the TM mode, offering the best PDH error signals. Asingle photodetector is implemented, and the two PDHerror signals are separated by two different electro-opticmodulation frequencies. The frequency difference in TMand TE modes is deterministically measured at5.953 GHz. This dual-mode beat frequency is divided by64 with a prescaler for stabilization to a 93.015MHzradio-frequency clock. We then examine the Allandeviation (AD) of the TM laser frequency and the dual-mode beat frequency to measure the degree of correlationin time. The stability of the TM laser frequency is mea-sured by downconverting the optical frequency via theheterodyne beating of the TM laser against a commercialfiber laser frequency comb locked to an ultrastable laserpossessing 1 Hz linewidth and ≈1 Hz s−1 drift26. Figure 3ashows both the TM laser frequency (fTM) measurementdata and the dual-mode beat frequency (fTM–TE) mea-surement data with 1 s counter gate time. Figure 3b plotsthe calculated AD from the frequency measurementdatasets. For the open-loop measurement, both ADs show

frequency drift over time. The long-term stability of750 kHz at 1000 s integration time implies less than mKresonator temperature stability dominated by the thermalexpansion coefficient. We extract the correlation betweenΔfTM and ΔfTM–TE along the integration time (τ) bymeasuring their ratio as expressed by: β(τ)=ΔfTM(τ)/ΔfTM–TE(τ).Figure 3c shows that β(τ) approaches the theoretically

estimated value of 21.66 when ΔTR= ΔTm withincreasing integration time, which indicates that theresonator volume temperature equilibrates to the modevolume temperature via heat diffusion and time aver-aging. This is supported by a heat transfer simulationwith circulating laser power as a heat source (details inSupplementary Information S4). The simulation shows aheat gradient along the radius due to the light absorp-tion by the resonator in Fig. 3d. The average tempera-tures near the mode volume and the entire resonatorvolume are probed individually and their difference iscalculated in time, as illustrated in Fig. 3e. The differ-ence between the mode volume temperature and theresonator volume temperature becomes constant after30 s, which could be the characteristic time required toreach heat equilibrium. The longer time averaging helpsβ(τ) to be closer to the theoretical value. The deviationof β(τ) from the theoretically estimated value of 21.66 isattributed to the atmospheric temperature variationtriggering the TEC control to maintain the enclosuretemperature, which can be an extra heat source for theresonator.

Ref. laser

cw laser 1

cw laser 2

FBS

FBS EOM

AOM

SERVO

SERVO

SERVO

TM

TE

7 MHz

12 MHz

Optical path

Electronic path

EOMFBS

PD

PD outVacuum

FB

C

FBC

PBC

TM-ref beat

Dual-mode beat

PD

TETM

Fig. 2 Orthogonally polarized dual-mode temperature stabilization set-up. The two cw lasers are coupled into the ultrahigh-Q MgF2 resonatorin a vacuum chamber with two orthogonally polarized modes. Each laser is locked to the resonance via a Pound–Drever–Hall (PDH) lock scheme. Theinset (right bottom) shows the PDH error signals for the TM and TE modes interrogated by the two lasers, respectively. The dual-mode beatfrequency is measured at a single photodetector and the TM laser frequency stability is measured by the heterodyne beating of it against anultrastable Fabry–Pérot mirror cavity laser reference possessing 1 Hz linewidth and ≈1 Hz s−1 drift. The dual-mode beat frequency is locked to a radio-frequency clock via intensity modulation with an acousto-optic modulator (AOM). Both the TM laser frequency and the dual-mode beat frequencyare simultaneously counted and their phase noises are recorded. FBC fiber-optic beam combiner, FBS fiber-optic beam splitter, EOM electro-opticmodulator, PBC polarization beam combiner, PD photodetector

Lim et al. Light: Science & Applications (2019) 8:1 Page 5 of 9

Dual-mode temperature stabilization and residual driftsWe stabilize the dual-mode beat frequency by control-

ling the intensity of the TE laser into the resonator with

an acousto-optic modulator (AOM). The inset in Fig. 4ashows the frequency measurement of the in-loop dual-mode beat carrier locked to a radio-frequency reference

1.5

1.0

0.5

0 20 40 60

40201.00

1.050

1.10

80 100

22ΔTR = ΔTm

Tm

– T

R (a

. u.)

� (�

)

20

18

16

Time (s)

A heat source(circulating laser power)in a mode volume

100Integration time (s)

100010

a

d ec

5.0TM laser frequencyDual-mode beat frequency

TM laser frequencyDual-mode beat frequency

2.5

0.0

–2.5

0.2106

105

104

103

102

0.0

–0.2

–5.00 20 40 60

Time (min)80 100 100

Integration time (s)1000101

bFr

eque

ncy

(MH

z)

Freq

uenc

y (M

Hz)

Alla

n de

viat

ion

(Hz)

Fig. 3 Correlation between a resonator-stabilized TM laser frequency (fTM) and the dual-mode beat frequency (fTM–TE). a The stabilized TMlaser frequency and the dual-mode beat frequency are simultaneously counted for 2 h, with a correlation in the frequency changes. b The calculatedAllan deviations without the dual-mode temperature stabilization show the frequency drift in time. The long-term stability of 0.75 MHz at 1000 sintegration time shows sub-mK resonator temperature stability dominated by the thermal expansion coefficient. c The measured β(τ), i.e., ΔfTM/ΔfTM–TE, shows that the frequency correlation approaches the theoretically estimated value of 21.66 at ΔTR= ΔTm with increasing integration time,likely because the resonator volume temperature equilibrates to the mode volume temperature via heat diffusion and averaging. d The simulatedresonator volume temperature distribution at 100 s due to a circulating laser heat source in the optical mode volume. e The simulation result showsthe time-dependent difference between the mode volume temperature (Tm) and the averaged resonator volume temperature (TR). The differencebecomes constant after 30 s, which can be the characteristic heat diffusion time required to reach an equilibrium point

a106

105

104

103

b

AD

TM

-unl

ocke

d /A

DT

M-lo

cked

102

101

1 10

–20 10 20 30

Time (min)40 50

0

Dua

l-mod

e fr

eq. (

Hz)

TM

lase

r A

llan

devi

atio

n (H

z)

2

Locked

Unlocked

In-loop singal

Integration time (s)

100 1000 11

10

10

Integration time (s)

100

100

�0.8

1000

Fig. 4 Resonator temperature stabilization and stability enhancement. a TM laser frequency stability when the dual-mode beat frequency isunlocked (red) and locked (navy). Inset: in-loop dual-mode beat frequency measurement showing less than ±1 Hz deviation. b At the characteristicequilibrium time, the largest stability enhancement is achieved. For τ < 30 s, the stability enhancement is likely to be limited by the response of theresonator temperature to the feedback control. For τ > 30 s, we measure the roll-off of the stability enhancement due to the frequency drift

Lim et al. Light: Science & Applications (2019) 8:1 Page 6 of 9

with 1 s gate time. While the dual-mode beat frequency islocked, we count the TM laser frequency and plot themeasured AD, as shown in Fig. 4a (navy scatters). At thecharacteristic heat equilibrium time, the largest stabilityenhancement is accomplished, as shown in Fig. 4b. In thisregime, further noise suppression is limited by the feed-back gain for the dual-mode beat control to avoid theelectrical-line harmonic noise peaks on top of theuncompensated frequency noise. For τ < 30 s, the stabilityenhancement could be limited by the response of theresonator temperature to the feedback control so that themagnitude of the enhancement decreases. For τ > 30 s, weobserve a linear frequency drift in AD, leading to the roll-off of the stability enhancement in Fig. 4b.We investigate this frequency drift by measuring and

analyzing the frequency noise of the dual-mode beat fre-quency and find that the drift could be attributed to thenoise of the radio-frequency clock to which the dual-mode beat frequency is locked. Figure 5a illustrates thestabilized dual-mode beat frequency spectrum (inset) andits single-side-band (SSB) phase noise. The dual-modebeat frequency stabilization loop suppresses noise up to300 Hz offset frequency. When the higher feedback gainwas applied, we found strong 60 Hz harmonic peaks in the

error signal, which limits the available feedback gain.However, we found that the feedback bandwidth and gaincould not be the fundamental limiting factors for thelong-term temperature stability of the resonator. Inprinciple, time averaging with more than the character-istic time, which is required to reach the resonator heatequilibrium (≈30 s), can help to rule out fast frequencyfluctuations. The dual-mode beat frequency noise starts toconverge to the reference clock noise below 1 Hz offsetfrequency. The locked dual-mode beat frequency SSBphase noise has a decaying slope of f−1 while the radio-frequency reference SSB phase noise has a decaying slopeof f−4.6 near the carrier frequency, which implies that thetwo curves could have a crossover so that the achievabledual-mode beat frequency stability can be bounded by theradio-frequency clock noise beyond the crossover point,which potentially introduces the frequency drift in ourcurrent system. To verify this assumption, we deliberatelyreduce proportional feedback gain without losing fre-quency locking and measure the stability again. Althoughthe stabilities at τ < 100 s are rather worse, the frequencydrift is significantly suppressed, as illustrated in Fig. 5b.The drifts estimated with a linear function fitting show11.8 and 0.54 kHzmin−1 at the two different gains. The

105

40

0

–40

–80

–120105

TM

lase

r A

llan

Dev

. (H

z)

Cav

ity te

mpe

ratu

re (

μK)

TM

lase

r fr

eque

ncy

(MH

z)

SS

B p

hase

noi

se (

dBc/

Hz)

104

104

103

103

101

101

101

102

102

Fourier frequency (Hz)

102

103

100

100

Integration time (s)

10

Integration time (s)

100 1000

180012006000

AD

TM

-unl

ocke

d /A

DT

M-lo

cked

1

10

100

–1

0

1

Time (s)

Unlocked Unlocked0

–30

–60

–250 2500Frequency (kHz)

RBW = 1 kHz

w/ prop. gain = P/10

w/ prop. gain = Pw/ prop. gain = P/10

prop. gain = p/10

w/ prop. gain = P

Locked Locked

prop. gain = P

Locked

f –4.6

f –1

Inte

nsity

(dB

)

rf clock

f TM-TE: 5.953 GHz carrier

a b

c d�0.8

�0.52

Fig. 5 Single-sideband (SSB) phase noise of the dual-mode beat frequency, Allan deviation, and their enhancement at differentproportional feedback gains. a SSB phase noise measurements of the dual-mode beat frequency at 5.953 GHz. Inset: the stabilized dual-mode beatfrequency spectrum at 5.953 GHz. b Measured TM laser frequency in time with the two different feedback gains. The frequency drifts estimated witha linear function fitting show two examples at 11.8 kHz min−1 (navy) and 0.54 kHz min−1 (green), respectively. The red line is the unlocked (open-loop) TM laser frequency for comparison. c TM laser frequency Allan deviations with different feedback gains. By lowering the proportional gain, thelong-term stability is improved, illustrating that the radio-frequency reference noise could be responsible for the frequency drift. d The lowerproportional gain makes the enhancement slower along the integration time, but the enhancement roll-off is avoided with the higher gain. Theenhancement factor at 1000 s integration time is 51.78

Lim et al. Light: Science & Applications (2019) 8:1 Page 7 of 9

AD at 1000 s integration time is improved by an order-of-magnitude compared to that with the higher gain and theresonator temperature remains to be approximately 10 μKfrom 1 to 1000 s integration time. The TM laser frequencyinstability is improved approximately by a factor of51.78–14.55 kHz at 1000 s integration time compared tothe instability (753.8 kHz) before stabilizing the dual-mode mode beat frequency as illustrated in Fig. 5d.

Residual intensity noise contributionsWe investigate the impact of the coupled laser intensity

fluctuation resulting in the fluctuation of the absorbed cir-culating optical power via the amplitude fluctuation of theintracavity laser field. Since we implement the TE laserintensity modulation feedback, the intensity fluctuation ofthe TE laser into the resonator is under control, but theintrinsic relative intensity noise (RIN) or the coupled powerchange of the TM laser could trigger the intracavity laserfield fluctuation. To quantify the impact of RIN, we measurethe resonance frequency shift induced by the laser intensitymodulation as illustrated in Supplementary Information S5.The coupled laser power is modulated by an AOM with a1Hz top-hat function and the modulated laser power(0.75%) into the resonator and the corresponding TM laserfrequency shift are measured. We determine the resonancefrequency shift on the coupling power modulation—2.66kHz for 1% coupled power change corresponding to 200 nW.The integrated RIN of the lasers used in this measurement is≈10 ppm from 10Hz to 10MHz offset frequencies27. Byextrapolating RIN to the carrier frequency with 20 dB/dec-ade, we estimate that the impact of RIN could be one part in103 up to 1Hz offset frequency. Hence, the RIN-associatedpower fluctuation can introduce a TM laser frequencyfluctuation of 266Hz, which is small compared to our cur-rently measured frequency instability, but the laser RIN orthe coupling power fluctuation could be a contributingbound when operating at the thermodynamic noise limit.

DiscussionWe implement the cross-polarized dual-mode tempera-

ture stabilization for a birefringent high-Q whispering-gallery resonator and improve the long-term stability by≈51× at 1000 s integration time. We achieve 10 μK resonatortemperature instabilities even up to 1000 s integration time,enabling this compact optical resonator module to serve as ahigh-performance frequency reference in potential metrol-ogy, timing, and frequency transfer applications. Thenumerical simulations reveal the possible thermorefractivecoefficient and mode volume modification due to thewaveguide effect and the heat diffusion process in the MgF2resonator. The characteristic time for the resonator tem-perature equilibrium is deduced and matched with theexperimental result. Further, more accurate models considerthe impact of TEC control because the ambient temperature

variations can trigger the TEC to maintain the enclosuretemperature, leading to a change in the resonator tem-perature. Improved system thermal isolation from the sur-rounding environment may be needed. Although the currentlong-term frequency or temperature stability seems to berestricted by the low Fourier frequency noise of the radio-frequency clock used for the dual-mode beat frequencystabilization, further investigations and understandings onnoise are desirable for operating the resonator stability atfundamental thermodynamic noise limits.

Materials and methodsFEM optical and thermal modeling and simulations for theMgF2 resonatorThe thin cylindrical MgF2 resonator is modeled using 2D-

axis symmetry in COMSOL Multiphysics with a 1.35mmbending radius and a 12.5 μm radius of curvature. Theguided modes in the resonator are calculated by solvingMaxwell’s electromagnetic wave equations in the frequencydomain, which provides the eigenfrequencies of the modes.By performing mode analysis near the optical carrier fre-quency, the family of TE and TMmodes associated with theeigenfrequencies are found. We probe the effective modalareas and the effective mode indices, and study their reso-nator temperature dependence by performing simulationsat different resonator temperatures. For the resonator heatdiffusion modeling and simulation, the time-dependentheat transfer equation solver in COMSOL is implemented,where a user-defined heat source, the resonator absorptionof circulating laser power in the modal area, is used. Thesize of the heat source is estimated by the modal area cal-culated by the mode analysis study. The physical constantsand the optical parameters used in the simulation are listedin Supplementary Information S4.

AcknowledgementsThe authors appreciate the helpful discussions with Shu-Wei Huang, WentingWang, Yoo Seung Lee, and Abhinav K. Vinod. The authors acknowledge supportfrom DARPA and Air Force Research Laboratory under contract FA9453-14-M-0090.

Author contributionsJ.L., C.W.W., A.B.M., and L.M. designed the experiment, and J.L. developedthe experimental set-up, performed the experiment, and analyzed themeasurements. W.L. and A.A.S. developed the optical resonator along withthe package assembly. J.L. performed the FEM numerical simulations andanalyzed them with A.B.M. and C.W.W. All authors contributed to themanuscript preparation.

Conflict of interestThe authors declare that they have no conflict of interest.

Supplementary information is available for this paper at https://doi.org/10.1038/s41377-018-0109-7.

Received: 4 August 2018 Revised: 28 October 2018 Accepted: 18 November2018

Lim et al. Light: Science & Applications (2019) 8:1 Page 8 of 9

References1. Lewandowski, W. & Arias, E. F. GNSS times and UTC.Metrologia 48, S219–S224

(2011).2. International Telecommunication Union. ITU-T Recommendation ITU-T

G.8271.1/Y.1366.1: Network Limits for Time Synchronization (ITU, 2013).3. Legaie, R., Pichen, C. J. & Pritchard, J. D. Sub-kilohertz excitation lasers for

quantum information processing with Rydberg atoms. J. Opt. Soc. Am. B 35,892–898 (2018).

4. Deschenes, J. D. et al. Synchronization of distant optical clocks at the fem-tosecond level. Phys. Rev. X 6, 021016 (2016).

5. Boroson, D. M. et al. Overview and results of the lunar laser communicationdemonstration. Proceedings of SPIE 8971, Free-Space Laser Communication andAtmospheric Propagation XXVI 89710S (SPIE, San Francisco, California, UnitedStates, 2014).

6. Riehle, F. Optical clock networks. Nat. Photonics 11, 25–31 (2017).7. Grudinin, I. S. et al. Ultrahigh Q crystalline microcavities. Opt. Commun. 265,

33–38 (2006).8. Lim, J. et al. Chasing the thermodynamical noise limit in whispering-gallery-

mode resonators for ultrastable laser frequency stabilization. Nat. Commun. 8,8 (2017).

9. Alnis, J. et al. Thermal-noise-limited crystalline whispering-gallery-mode reso-nator for laser stabilization. Phys. Rev. A 84, 011804 (2011).

10. Xie, Z. D. et al. Extended ultrahigh-Q-cavity diode laser. Opt. Lett. 40,2596–2599 (2015).

11. Lim, J. et al. A stabilized chip-scale Kerr frequency comb via a high-Q referencephotonic microresonator. Opt. Lett. 41, 3706–3709 (2016).

12. Liang, W. et al. High spectral purity Kerr frequency comb radio frequencyphotonic oscillator. Nat. Commun. 6, 7957 (2015).

13. Huang, S. W. et al. A low- phase-noise 18 GHz Kerr frequency microcombphase-locked over 65 THz. Sci. Rep. 5, 13355 (2015).

14. Savchenkov, A. A. et al. Stabilization of a Kerr frequency comb oscillator. Opt.Lett. 38, 2636–2639 (2013).

15. Maleki, L. et al. Miniature atomic clock for space applications. Proceedings ofSPIE 9616 Nanophotonics and Macrophotonics for Space Environments IX96160L (SPIE, San Diego, California, United States, 2015).

16. Liang, W. et al. Resonant microphotonic gyroscope. Optica 4, 114–117(2017).

17. Savchenkov, A. A., Matsko, A. B., Ilchenko, V. S., Yu, N. & Maleki, L. Whispering-gallery-mode resonators as frequency references. II. Stabilization. J. Opt. Soc.Am. B 24, 2988–2997 (2007).

18. Fescenko, I. et al. Dual-mode temperature compensation technique for laserstabilization to a crystalline whispering gallery mode resonator. Opt. Express 20,19185–19193 (2012).

19. Baumgartel, L. M., Thompson, R. J. & Yu, N. Frequency stability of a dual-modewhispering gallery mode optical reference cavity. Opt. Express 20,29798–29806 (2012).

20. Strekalov, D. V., Thompson, R. J., Baumgartel, L. M., Grudinin, I. S. & Yu, N.Temperature measurement and stabilization in a birefringent whisperinggallery mode resonator. Opt. Express 19, 14495–14501 (2011).

21. Vig, J. R. (ed). Dual-mode oscillators for clocks and sensors. Proceedings of the1999 IEEE International Ultrasonics Symposium, 17–20 October (IEEE, CaesarsTahoe, NV, USA, 1999).

22. Matsko, A. B., Savchenkov, A. A., Yu, N. & Maleki, L. Whispering-gallery-moderesonators as frequency references. I. Fundamental limitations. J. Opt. Soc. Am.B 24, 1324–1335 (2007).

23. Ghosh, G. Handbook of Thermo-Optic Coefficients of Optical Materials withApplications. (Academic, San Diego, CA, 1998).

24. Lim, J., Liang, W., Matsko, A. B., Maleki, L. & Wong, C. W. Time-dependentcorrelation of cross-polarization mode for microcavity temperature sensingand stabilization. CLEO: Appl. Technol. https://doi.org/10.1364/CLEO_AT.2017.JF2D.4 (2017).

25. Drever, R. W. P. et al. Laser phase and frequency stabilization using an opticalresonator. Appl. Phys. B 31, 97–105 (1983).

26. Notcutt, M., Ma, L. S., Ye, J. & Hall, J. L. Simple and compact 1-Hz laser systemvia an improved mounting configuration of a reference cavity. Opt. Lett. 30,1815–1817 (2005).

27. Dale, E. et al. On phase noise of self-injection locked semiconductor lasers.Proceedings of SPIE Laser Resonators, Microresonators, and Beam Control XVI89600X (SPIE, San Francisco, California, United States) https://doi.org/10.1117/12.2044824 (2014).

Lim et al. Light: Science & Applications (2019) 8:1 Page 9 of 9