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1 DOI: /NPHOTON Supplementary Information Tenfold redution of Brownian noise in high-refletivity optial oatings Garrett D. Cole 1,,*, Wei Zhang 3,*, Mihael J. Martin 3, Jun Ye 3, and Markus Aspelmeyer 1 1 Vienna Center for Quantum Siene and Tehnology (VCQ), Faulty of Physis, University of Vienna, A-1090 Vienna, Austria Crystalline Mirror Solutions GmbH, A-1090 Vienna, Austria 3 JILA, National Institute of Standards and Tehnology and University of Colorado, Boulder, Colorado , USA S1. Finesse Estimation Calulation of the avity finesse requires quantifiation of the various loss mehanisms of the end mirrors. As our rystalline oatings are onstruted from idential multilayers (soured from neighbouring loations on the same epitaxial wafer) we assume that the individual oatings have a mathed refletane. Below we provide a desription of the steps taken in order to haraterize the transmission, satter, and absorption loss of the mirrors. In eah ase measurements are performed both on the original growth wafer and after bonding in order to investigate any potential hanges in the optial properties following the strate-transfer proess. * These authors ontributed equally to this work. NATURE PHOTONICS Mamillan Publishers Limited. All rights reserved.

2 DOI: /NPHOTON In order to determine the mirror transmission, the refletane spetrum of the DBR is reorded via spetrophotometry, as shown in Fig. 1. Fitting of the mirror stopband with a transmission matrix model, using x-ray diffration derived omposition and thikness values as a baseline, it is possible to extrat the individual layer thiknesses for the high and low index omponents of the multilayer. Fortunately, given the long history of appliation in optoeletronis (e.g. Ref. 1), the refrative index and dispersion harateristis are very well known for the AlGaAs materials system. This fitting tehnique is exquisitely sensitive to variations in thikness, with hanges on the order of 1 nm for individual layer thiknesses resulting in shifts of the enter wavelength of the high-refletivity stopband by nearly 10 nm. The fitted transmission matrix model is then used for an aurate estimate of the transmission loss of the mirror. For the 40.5 period DBR employed here, the extrated transmission is 4 ppm. Independent measurement of the transmission of a post-bonded rystalline oating (on a fused silia strate) yields 3.8 ppm, verifying the auray of this approah. One urrent limitation in our proess is the lak of an in situ optial monitor in the MBE system. Thus, it is diffiult to reliably realize a desired transmission value. With this in mind, early mirrors were designed with minimal transmittane in order to avoid the possibility of insuffiient finesse. Figure S1: Post-bond AFM haraterization of the AlGaAs DBR surfae quality. The measured RMS roughness of 1.7 Å (determined by averaging line sans over a 10 µm 10 µm area) leads to a alulated satter loss of 4 ppm for the rystalline oating. The defet in the top orner of the sanned area is exluded from the roughness alulation, but is inluded here to show the potential for imperfetions in the multilayer. NATURE PHOTONICS Mamillan Publishers Limited. All rights reserved.

3 DOI: /NPHOTON SUPPLEMENTARY INFORMATION Atomi fore mirosopy (AFM) is used to haraterize the surfae roughness, and ultimately the satter loss, of both the pre- and post-bonded mirror struture. For our MBEgrown AlGaAs DBR, we reord a root mean square (RMS) roughness value of 1.7 Å over a 10 µm 10 µm san area, as shown in Figure S1. Note that this measurement was made following the transfer proess and remains unhanged when ompared with the on-wafer surfae quality. Assuming normal-inidene illumination of a surfae haraterized by an RMS roughness σ, the satter-limited refletane is determined from the expression derived by Davies. In the long wavelength (for surfae features muh smaller than the wavelength λ) and high refletivity limit, this expression yields a fairly aurate estimate of the refletane and thus the orresponding satter loss value (1-R s ) [3]. Using our measured AFM data and a wavelength of 1064 nm, we alulate a satter loss of 4 ppm for the AlGaAs DBR. The validity of the further simplifiation of taking into aount only the surfae roughness of the multilayer is ultimately verified by the exellent agreement between our theoretial and measured finesse values ( versus 1.5(1) 10 5 respetively). The final omponent neessary to estimate the avity finesse is the optial absorption. This parameter is probed diretly via photothermal ommon path interferometry (PCI) [4]. Absorption data extrated by PCI reveal typial loss levels of ~10 ppm (unhanged for both pre- and postbond onditions) for our MBE-grown AlGaAs mirrors at 1064 nm 5, with the urrent best measurement below 5 ppm. For the sample used in the study, the average absorption value was slightly higher at 1.5±1.19 ppm (with a somewhat large sample-to-sample variation). A series of three linear sans (11.1±0.08 ppm, 1.9 ±0.1 ppm, and 13.5±1.18 ppm) were averaged for this measurement; a typial example is presented in Figure S. We are urrently operating under the assumption that this property is dominated by freearrier absorption (FCA). In ontrast, alulations of the two-photon absorption at 1064 nm yields single digit or -ppm absorption values for realisti intra-avity powers 6. Unfortunately, NATURE PHOTONICS Mamillan Publishers Limited. All rights reserved.

4 DOI: /NPHOTON with these thin films and at these low arrier onentrations, diret haraterization of the arrier type and onentration via Hall measurement is extremely diffiult. Assuming that the orresponding absorption oeffiient of 0.38 m -1 (alulated from the 1.5 ppm absorption value over a round-trip penetration depth of 163 nm) is dominated by a p-type impurity bakground and extrapolating from the data presented in Kudman and Seidl 7, the estimated arrier onentration for this struture is m 3. Figure S: Typial PCI line-san for haraterizing the optial absorption of the bonded rystalline oating. The average absorption in this example is 11.1±0.08 ppm at 1064 nm. Averaging of three line sans yields a mean absorption value of 1.5 ppm for the 40.5-period AlGaAs DBR. With FCA in the high index layers being the ultimate limit, it should be possible to generate high refletivity AlGaAs-based mirrors out to approximately 3 µm 8, while moving to wavelengths shorter than the urrent implementation at 1064 nm will neessitate a signifiant inrease in the average Al ontent in order to stave off interband absorption. Previous work with visible VCSELs has demonstrated high quality AlGaAs DBRs with enter wavelengths as short as 650 nm 9, though further researh will be neessary to quantify the minimum ahievable absorption levels in this wavelength window. Note that operation at 698 nm will be diretly relevant for Sr optial atomi lok appliations. 4 NATURE PHOTONICS Mamillan Publishers Limited. All rights reserved.

5 DOI: /NPHOTON SUPPLEMENTARY INFORMATION S. Birefringene Calulations As disussed in the manusript, the mode struture of the fixed spaer avity onstruted with the AlGaAs-based rystalline oatings reveals two orthogonal polarization eigenmodes with a splitting of 4(0.4) MHz. Interestingly, the refrative index of unstrained (100)-oriented ternary Al x Ga 1-x As alloys is in-plane isotropi, meaning that, from the point of view of a refleting surfae-normal beam, there is no radial variation in index. Thus, for an ideal avity onstruted from perfet rystalline oatings, there will be no birefringene. However, anisotropy an easily be introdued through the appliation of strain in the system, leading to a orresponding diretional dependene on index via the elasto-opti effet. For the bonded mirrors there are two possible paths for strain being imparted on the mirror: 1) bending-indued strain generated when fabriating the urved mirror and ) internal strains resulting from the epitaxial growth proess. Path 1 requires a finite element model in order to alulate the imparted strain on the deformed mirror. In this ase the mirror dis (8 mm in diameter by 6.83 µm in thikness) is fored into a presribed displaement desribed by m (with x and y lying within the plane of the originally flat mirror and z being the vertial diretion normal to the DBR surfae), foring the mirror to onform to a sphere of 1-m radius with a maximum defletion of 8 µm at the mirror enter. Inorporating the average density and elasti onstants of the single-rystal multilayer (ρ=4551 kg/m 3 ; 11 =119.94, 1 =55.38, and 44 =59.15 GPa), the mean in-plane stress,, is found to be approximately 500 kpa (ompressive) on the front surfae of the mirror and is essentially uniform aross the mirror fae given the onstant radius of urvature of the deformed multilayer. Note that this is an extremely small stress value ompared with the hundreds of MPa or even GPa levels of internal stress found in typial ion NATURE PHOTONICS Mamillan Publishers Limited. All rights reserved.

6 DOI: /NPHOTON beam sputtered films. There are two take-home messages from this exerise: i) for a bonded mirror with a 1-m ROC, we are imparting very little strain in the multilayer, so there is no fear of damaging the struture, and ii) given the minisule stress (and orresponding strain value), this is likely not the ause of the birefringene. For a quik estimate, path (growth-indued strain) an be alulated using simple analytial expressions. As we employ a well-known epitaxial materials system, it is quite easy to extrat the internal strain arising from slight differenes in the inter-atomi spaing between the onstituent omponents of the multilayer, or in other words, due to the differential thermal ontration of the layers upon ooling, from the growth, to room temperature, T. Notably, for Al x Ga 1-x As ompounds, the lattie parameter is nearly exatly mathed for typial deposition temperatures 10. At room temperature, the high index film, onsisting of binary GaAs, is haraterized by a lattie parameter, a, of Å, while for Al 0.9 Ga 0.08 As this value is Å. This slight differene in lattie onstant leads to a mismath strain, e m, of when ooling from ~900 K to 300 K, in general agreement with previous investigations showing CTEdriven mismath values slightly over 1 millistrain for films of similar average aluminum ontent Assuming an average isotropi Young s modulus, E, of 100 GPa for the multilayer, the resulting mismath-driven stress, X, is then 17 MPa (simply determined from σ = ee). Obviously, this value is signifiantly larger (>50 ) than the bending-indued stress desribed above and is thus the more likely soure of the observed polarization splitting. Realizing a first-order estimate of the index hange with this internal strain is fairly straightforward. The prinipal axes of birefringene in zinblende rystals under a uniform applied strain is the [110] family of diretions; in that ase the resulting index differene 1 6 NATURE PHOTONICS Mamillan Publishers Limited. All rights reserved.

7 DOI: /NPHOTON SUPPLEMENTARY INFORMATION between the parallel and perpendiular axes is desribed by, where ε is the real part of the dieletri onstant at optial frequenies, α pe is the photoelasti oeffiient, and X is the internal stress in the multilayer. Plugging in the average photoelasti oeffiient of Pa -1 and high frequeny dieletri onstant of 1, both values from Ref. [13], along with the previously alulated stress value (17 MPa), yields an index differene of roughly Calulation of the frequeny splitting requires knowledge of the optial path length inluding the vauum avity (L av = 35 mm), plus the additional field penetration depth into the mirrors 15. For the 40.5-period AlGaAs DBR employed in this experiment, the penetration depth, L pen, at 1064 nm is alulated to be 163 nm. The hange in length from the strain-driven index perturbation, δ, is determined to be L pen Δn = 3.3 Å. Converting this physial length to a frequeny differene, Δf, results in a maximum polarization splitting of 5.3 MHz for a enter wavelength of 1064 nm (f 0 = Hz) using. This value ompares well with our measurement of 4(0.4) MHz for a randomly oriented pair of rystalline oatings. For future runs, the mirror diss should inlude features indiating the rystallographi orientation of the oatings in order to maintain the desired alignment. NATURE PHOTONICS Mamillan Publishers Limited. All rights reserved.

8 DOI: /NPHOTON S3. Thermal Noise Measurement and Calulations The Methods Summary provides basi information as required for the alulation of the thermal noise results presented in the Letter. Below we present a detailed disussion of the thermal noise model and introdue the relevant parameters used in the present work. Thermal noise theory: The single-sideband power spetral density of the thermal noise displaement (G total ), onsisting of Brownian motion of the spaer (G spaer ), strate (G ), and oating (G ), strate thermoelasti noise (G TE ) and oating thermo-opti noise (G TO ), is desribed by the following 5,16-3 (symbols and orresponding values defined in Table 1): G total k BT = π f ( f ) = G ( R 4k BT 1 σ + π f w Y + G TE m ( f ) + G spaer r TO L ( f ) + G ) Y D w m ( f ) spaer Y φ ( f ) + G spaer B 3 / ( f ) + G T 1 σ w Y (1 σ )(1 σ 4k + π φ f m Y ) TE ( f ) + G ( f ) [ Y (1 + σ ) (1 σ ) + Y (1 + σ ) (1 σ )] φ TO (1) Here, G TE ( f ) = 8 (1 + σ π ) α kbt w κ m 0 + du + dv 3 3 / π u e u / /{( u + v )[( u + v ) + Ω TE ( f )]} Ω TE ( f ) = w m C π f / κ and G 4k T B TO ( f ) = Γ ( )( / ) 3 / TO f α D β λ α DC C ( w mc f / κ >> 1 π wm fκ C π ) G 4k T B TO ( f ) = ( α D β λ α DC / C ) ( wmc f / κ << 1 π w κ m π ), where ΓTO ( f ) is the thikness orretion fator of the oating layer 5,. The expression of G TO at high and low frequenies are based on the treatments shown in Ref. 1 and. 8 NATURE PHOTONICS Mamillan Publishers Limited. All rights reserved.

9 DOI: /NPHOTON SUPPLEMENTARY INFORMATION Table 1. Symbols and parameters used in the thermal noise alulations. Here, spaer refers to the zerodur avity spaer, the index is for the fused silia strates, and represents the AlGaAs rystalline oating. Parameter Desription Value λ Wavelength µm f Fourier frequeny L Cavity length 35 mm R Spaer radius 15 mm r Central bore radius 5 mm Y spaer// Young s modulus 91 / 7 / 100 GPa spaer/strate/oating φ spaer// Loss angle / / spaer/strate/oating σ / Poisson ratio 0.17/0.3 strate/oating w m Beam radius 50 µm D Coating thikness 6.83 µm T Temperature 300 K k B Boltzmann s onstant J/K C / Heat apaity per volume / J/ Km 3 strate/oating κ / Thermal ondutivity strate/oating 1.38 / 6.9 W/Km α / Effetive oeffiient of / /K thermal expansion strate/oating β Effetive thermorefrative oeffiient (oating) The orresponding frequeny noise G υ is G ν ν ( f ) = Gx( f ), () L Where G x is the thermal displaement noise orresponding to eah ontribution shown in Eq. 1 and υ is the optial frequeny. Eqs. 1 and allow us to determine the theoretial thermal-noiselimited frequeny noise, whih is shown as the violet line in Figure 4b. For the type of frequeny ounter employed in this measurement, the theoretial thermal noise-limited frational frequeny stability, σ y, is equal to 4 G f f f y 4 ( ) (sin( π τ / )) sin( π τ ) ν σ ( τ ) = 3 df. (3) 0 4 ν ( πfτ ) NATURE PHOTONICS Mamillan Publishers Limited. All rights reserved.

10 DOI: /NPHOTON Employing Eqs. 1 to 3 and the parameters of the Fabry-Pérot avity listed in Table 1, the thermal noise limited values of G x and σ y (at 1 Hz and 1 s, respetively) an be alulated. These results are summarized in Table. To failitate measurement of the ontribution from the mirror oatings, a relatively short spaer length of 35 mm is hosen for the experiment. Table. Theoretial alulation of thermal noise in the rystalline-oating avity. Displaement values G x are shown at 1 Hz and the Allan deviation σ y at 1 s. Assumed loss angles of zerodur, fused silia, and the rystalline oating are , , and , respetively. Brownian motion Noise type G x ( ) σ y m /Hz ( ) Spaer % 86% 1.1 Substrate % Coating % Substrate thermoelasti % Coating thermo-opti % Contribution to total (σ y ) Total σ y ( ) The results in Table learly demonstrate that Brownian noise makes the main ontribution to the predited total thermal noise. The frational frequeny stability as predited by the total theoretial thermal noise is at 1 s, where the ontribution from the rystalline oating is approximately 11% when assuming a loss angle of as measured in free-standing resonators. This an be ompared with onventional dieletri end mirrors where 70% of the loss is attributed to the amorphous SiO /Ta O 5 layers. Though admittedly small in terms of the overall frational ontribution, the oating Brownian noise ontribution is large enough to be isolated from other noise soures, allowing us to extrat the oating loss angle by measuring the stability of the 1064 nm avity-stabilized laser. The rystalline oating loss angle is evaluated to be 0(4) 10-5,orresponding to the measured σ y of f b1064 at 1 s averaging time, using a loss angle of for the zerodur spaer and for the fused silia strate 16. The 10 NATURE PHOTONICS Mamillan Publishers Limited. All rights reserved.

11 DOI: /NPHOTON SUPPLEMENTARY INFORMATION error in the AlGaAs loss angle is based on a 10% error in the measured σ y as well as a 0% unertainty in the loss angles of the avity omponents, inluding the zerodur spaer and fused silia strates *. Details on the Yb frequeny omb: The arrier-envelop-offset, f eo, is generated by a f-f self-referening interferometer and stabilized on a RF referene via feedbak to an AOM, ating as the fast atuator, and the temperature of a fiber Bragg grating as the slow atuator 5. The bandwidth of the f eo servo loop is approximately 160 khz. To phase lok the Yb omb to the 698 nm lok laser, the optial heterodyne beat, f b698, between a omb tooth and the lok laser is stabilized using the intraavity EOM as a fast atuator and a avity end-mirror mounted on a piezoeletri transduer (PZT), as well as a PZT-based fiber strether, as slow atuators. The bandwidth of the f b698 servo loop is about 150 khz. Stabilization of the 1064 nm laser: The avity-stabilized laser at 1064 nm is a ontinuous wave all solid-state, diode-pumped non-planar ring osillator (NPRO). Sine the free-running laser linewidth is only a few khz muh narrower than the Fabry-Pérot avity linewidth an additional pre-stabilization stage an be avoided. The referene avity and supporting struture are installed in a vauum hamber with a residual pressure of 10-6 torr (1 torr = Pa) provided by an ion pump ( liter/s pumping speed). The temperature of the vauum hamber is 300 K with variations smaller than 1 mk over a 4 hour period. The vauum hamber and optial omponents for the RF sideband lok are * If we perform an idential alulation with a fused silia loss angle of 10-7, we obtain a loss angle of 4(4) 10-5 for the stratetransferred AlGaAs multilayer. NATURE PHOTONICS Mamillan Publishers Limited. All rights reserved.

12 DOI: /NPHOTON loated on a passive vibration isolation platform (Minus K, Hz resonane frequeny). The laser output is divided into two omponents: the majority of the optial power is sent to the Yb omb system for interomparison, while the remaining 10% of the output power is fiber-oupled to the platform for RF sideband loking to the referene avity. A fiber-based EOM provides phase modulation for loking, with its residual amplitude modulation (RAM) atively stabilized to well below the predited thermal noise floor 6-8. The single frequeny laser is stabilized to the Fabry-Pérot avity via feedbak to the laser avity PZT with a servo bandwidth of approximately 40 khz. S4. Redution of exess thermo-opti with an optimized AlGaAs oating As disussed in the Artile, our AlGaAs multilayer displays signifiant thermo-opti noise at frequenies exeeding 10 Hz. This exess noise arises from the relatively large values of the effetive thermorefrative and thermal expansion oeffiients for this design and materials system, as an be seen in Table 1. However, it is important to note that this limitation should be fairly straightforward to overome. Aording to the thermo-opti noise expression given in Eq. 1 (from Refs. [5] and []), the addition of a half-wavelength ap to our urrent multilayer design an redue this noise ontribution by a fator of roughly, as shown in Figure S3. Consequently, employing suh an optimized design, the oating Brownian noise is then predited to dominate the total oating thermal noise. Compared with the theoretial thermal noise of a traditional SiO /Ta O 5 multilayer as shown in Figure 4, the orresponding improvement of the AlGaAs multilayer with the additional ap layer will be at least a fator of 10 over the entire range from 1 Hz to 100 Hz. Future implementations of AlGaAs-based rystalline oatings will 1 NATURE PHOTONICS Mamillan Publishers Limited. All rights reserved.

13 DOI: /NPHOTON SUPPLEMENTARY INFORMATION thus inorporate this additional material in order to signifiantly redue the thermo-opti noise ontribution. Figure S3: Comparison of the thermal noise performane of strate-transferred AlGaAs oatings with and without a half-wavelength ap. The green dot-dashed line is the AlGaAs Brownian noise, the yan dotted line is the thermoopti noise for an AlGaAs multilayer using a standard quarter-wave (QW) design, while the blue solid line is the total thermal noise for suh a struture. When a half-wavelength ap is added to the QW AlGaAs multilayer, the thermo-opti noise is redued, partiularly at low frequenies, as indiated by the red dashed line. In this ase the total thermal noise for the optimized struture would be limited primarily by the Brownian noise urve. Referenes: [1] Jewell, J. L., Sherer, A., MCall, S. L., Gossard, A. C. & English, J. H. GaAs-AlAs monolithi miroresonator arrays. Appl. Phys. Lett. 51, (1987). [] Davies, H. The refletion of eletromagneti waves from a rough surfae. Pro. IEE 101, (1954). [3] Bennett, H. E. & Porteus, J. O. Relation between surfae roughness and speular refletane at normal inidene. J. Opt. So. Am. 51, (1961). [4] Alexandrovski, A., Fejer, M., Markosian, A. & Route, R. Photothermal ommon-path interferometry (PCI): new developments. In Pro. SPIE 7193, Solid State Lasers XVIII: Tehnology and Devies. 7193D (009). [5] Harry, G., Bodiya, T. & DeSalvo, R. Optial Coatings and Thermal Noise in Preision Measurement. (Cambridge University Press, 01). pp. 45, 136, 161, 95, 99. [6] Krishnamurthy, S., Yu, Z. G., Gonzalez, L. P., & Guha, S. Temperature and wavelength-dependent twophoton and free-arrier absorption in GaAs, InP, GaInAs, and InAsP. J. Appl. Phys. 109, (011). [7] Kudman, I. & Seidel, T. Absorption edge in degenerate p-type GaAs. J. Appl. Phys. 33, (196). [8] Spitzer, W. G. & Whelan, J. M. Infrared absorption and eletron effetive mass in n-type gallium arsenide. Phys. Rev. 114, (1959). [9] Knigge, A., Zorn, M., Wenzel, H., Weyers, M. & Trankle, C. High effiieny AlGaInP-based 650 nm vertialavity surfae-emitting lasers. Eletron. Lett 37, 1 13 (001). [10] Ettenberg, M. & Paff, R. J. Thermal expansion of AlAs. J. Appl. Phys. 41, (1970). NATURE PHOTONICS Mamillan Publishers Limited. All rights reserved.

14 DOI: /NPHOTON [11] Reinhart, F. K. & Logan, R. A. Interfae stress of Al x Ga 1-x As-GaAs layer strutures. J. Appl. Phys. 44, (1973). [1] van der Ziel, J. P. & Gossard, A. C. Absorption, refrative index, and birefringene of AlAs-GaAs monolayers. J. Appl. Phys. 48, (1977). [13] Adahi, S. & Oe, K. Internal strain and photoelasti effets in Ga 1-x Al x As/GaAs and In 1-x Ga x As y P 1-y /InP rystals. J. Appl. Phys. 54, (1983). [14] Adahi, S. Physial Properties of III-V Semiondutor Compounds: InP, InAs, GaAs, GaP, InGaAs, and InGaAsP (John Wiley & Sons, New York, 199). Chapter 9. [15] Babi, D. & Corzine, S. Analyti expressions for the refletion delay, penetration depth, and absorptane of quarter-wave dieletri mirrors. IEEE J. Quantum Eletron. 8, (199). [16] Harry, G. M. et al. Thermal noise in interferometri gravitational wave detetors due to dieletri optial oatings. Classial Quant. Grav. 19, 897 (00). [17] Numata, K., Kemery, A. & Camp, J. Thermal-noise limit in the frequeny stabilization of lasers with rigid avities. Phys. Rev. Lett. 93, 5060 (004). [18] Kessler, T., Legero, T. & Sterr, U. Thermal noise in optial avities revisited. J. Opt. So. Am. B 9, (01). [19] Cerdonio, M., Conti, L., Heidmann, A. & Pinard, M. Thermoelasti effets at low temperatures and quantum limits in displaement measurements. Phys. Rev. D 63, (001). [0] Numata, K., Ando, M., Yamamoto, K., Otsuka, S. & Tono, K. Wide-band diret measurement of thermal flutuations in an interferometer. Phys. Rev. Lett. 91, 6060 (003). [1] Braginsky, V. B., Gorodetsky, M. L. & Vyathanin, S. P. Thermo-refrative noise in gravitational wave antennae. Phys. Lett. A 71, (000). [] Evans, M. et al. Thermo-opti noise in oated mirrors for high-preision optial measurements. Phys. Rev. D 78, (008). [3] Gorodetsky, M. L. Thermal noises and noise ompensation in high-refletion multilayer oating. Phys. Lett. A 37, (008). [4] Dawkins, S. T., MFerran, J. J. & Luiten, A. N. Considerations on the measurement of the stability of osillators with frequeny ounters. IEEE Trans. Ultrason. Ferroeletr. Freq. Control 54, (007). [5] Benko, C. et al. Full phase stabilization of a Yb:fiber femtoseond frequeny omb via high-bandwidth transduers. Opt. Lett. 37, (01). [6] Wong, N. C. & Hall, J. L. Servo ontrol of amplitude modulation in frequeny-modulation spetrosopy: demonstration of shot-noise-limited detetion. J. Opt. So. Am. B, (1985). [7] Li, L., Liu, F., Wang, C. & Chen, L. Measurement and ontrol of residual amplitude modulation in optial phase modulation. Review of Sientifi Instruments 83, (01). [8] Kessler, T. et al. A -40-mHz-linewidth laser based on a silion single-rystal optial avity. Nature Photonis 6, (01). 14 NATURE PHOTONICS Mamillan Publishers Limited. All rights reserved.