Page 1

Optomechanical coupling in a two-dimensional photonic crystal defect cavity

E. Gavartin,1R. Braive,2,3I. Sagnes,2O. Arcizet,4A. Beveratos,2T. J. Kippenberg,1,5, ∗and I. Robert-Philip2, †

1Ecole Polytechnique F´ ed´ erale de Lausanne, EPFL, 1015 Lausanne, Switzerland

2Laboratoire de Photonique et de Nanostructures, Route de Nozay, 91460 Marcoussis, France

3Universit´ e Paris Denis Diderot, 75205 Paris, Cedex 13, France

4Institut N´ eel, 25 rue des Martyrs, 38042 Grenoble, France

5Max Planck Institut f¨ ur Quantenoptik, 85748 Garching, Germany

Periodically structured materials can sustain both optical and mechanical modes. Here we inves-

tigate and observe experimentally the optomechanical properties of a conventional two-dimensional

suspended photonic crystal defect cavity with a mode volume of ∼3(λ/n)3. Two families of mechani-

cal modes are observed: flexural modes, associated to the motion of the whole suspended membrane,

and localized modes with frequencies in the GHz regime corresponding to localized phonons in the

optical defect cavity of diffraction-limited size. We demonstrate direct measurements of the optome-

chanical vacuum coupling rate using a frequency calibration technique. The highest measured values

exceed 250 kHz, demonstrating strong coupling of optical and mechanical modes in such structures.

PACS numbers: 42.70.Qs, 43.40.Dx

Cavity optomechanics [1] exploits the coupling of me-

chanical oscillators to the light field via radiation pres-

sure. On the applied side, such coupling may be used to

enable novel radiation pressure driven clocks [2], make

highly sensitive displacement sensors, tunable optical fil-

ters [3], delay lines [4] or enable photon storage on a

chip [5, 6]. On a fundamental level, such systems can

be exploited for demonstrating that nano- and microme-

chanical oscillators can exhibit quantum mechanical be-

havior [7].It has been shown experimentally that it

is possible to cool a mechanical oscillator intrinsically

via radiation pressure dynamical backaction [8]. In or-

der to reach the ground state of mechanical motion and

enable manipulation in the quantum regime, one ap-

proach consists in down-sizing the oscillator, thus shift-

ing the quantum-classical transition towards higher tem-

peratures.Among various sub-micron optomechanical

systems presently investigated [9], suspended membranes

containing a photonic crystal cavity offer strong light con-

finement in diffraction-limited volumes and are therefore

natural candidates for achieving strong optomechanical

coupling. Recently, optomechanical coupling in 1D pho-

tonic crystal systems [10] has been observed in patterned

single [11] and dual nanobeams (zipper cavities) [12, 13].

It would be highly desirable to extend such optomechan-

ical coupling to 2D systems, notably photonic crystal

defect cavities. Such cavities offer the strong light con-

finement possible (i.e. small mode volume), high quality

factor (Q) [14] and have been used for studying cavity

QED using quantum dots [15] or for realizing nanolasers

[16, 17]. Recently, a 2D optomechanical photonic crys-

tal slot cavity has been reported [18]. While mechanical

displacement due to strong radiation force generated by

band-edge modes in bilayer photonic crystal slabs has

been reported as well [19], to date optomechanical cou-

pling of the 2D conventional defect cavity has not been

studied.

In this paper, we demonstrate optomechanical coupling

using a photonic crystal defect (L3) cavity. We provide a

direct and robust experimental determination of the vac-

uum optomechanical coupling rate [30] using frequency

modulation, showing a particularly strong coupling for

the localized mechanical modes, which may also be cou-

pled to quantum dots in future studies [20].

FIG. 1. (a) Scanning Electron Microscope side view of the

cavity.(b) Microphotoluminescence spectrum of the pho-

tonic crystal slab cavity obtained under non-resonant contin-

uous optical excitation at normal incidence with an excitation

power of 100 µW at 532 nm (c) Micrograph (false colors) of a

defect cavity fiber-taper system used to read out mechanical

motion of the cavity. (d) Experimental setup (ECDL: Exter-

nal cavity diode laser, OI: Optical Isolator, EOM: Electro-

optical modulator, FPC: Fiber polarization controller, PhC:

Photonic crystal defect cavity, PD: Photo diode).

The optomechanical device under study consists of

a 262-nm thick InP suspended membrane containing a

two-dimensional photonic crystal defect cavity shown in

Figure 1(a). The cavity, following the design proposed

in [14], contains three missing holes in a line of a perfect

arXiv:1011.6400v1 [physics.optics] 29 Nov 2010

Page 2

2

triangular lattice of holes with a lattice constant of

a = 430 nm and a radius r = 90 nm. At both edges of

the cavity the holes are displaced outwards by d = 0.18a,

in order to obtain high optical quality factors.

cavity is fabricated using electron beam lithography,

inductively coupled plasma etching [21], and wet etching.

The cavity incorporates a single layer of self-assembled

InAsP quantum dots [22] at its vertical center plane for

cavity characterization. The whole structure is grown by

metalo-organic chemical vapor deposition. The quantum

dot density is ∼15×109cm−2, and their spontaneous

emission is centered around 1560 nm at 300 K with

an inhomogeneous broadening of about 150 nm.

presence of the dots inside the cavity allows to identify

the spectral properties of the fundamental optical mode

of the cavity by photoluminescence measurements [23],

as shown in Figure 1(b). The resonance wavelength of

the fundamental mode is centered around 1555 nm and

the cold-cavity quality factor is measured to be ∼104

(cavity linewidth is κ/2π ≈ 20 GHz). The suspended

photonic crystal membrane lies on top of a 10 µm high

mesa structure (see Fig. 1(a)). The mesa structure is

processed to enable positioning of a tapered optical fiber

in the evanescent field of the cavity, while precluding any

interaction with the nearby substrate. The shape of the

membrane resembles a Bezier curve, which was chosen to

increase optomechanical coupling to the flexural modes.

The

The

The setup used in the experiment is depicted in Figure

1(d). An external-cavity diode laser is used for the read-

out of the mechanical motion. Coupling to the optical

modes of the suspended membrane is achieved with the

optical fiber-taper technique [24]. Piezoelectric actuators

enable an accurate positioning allowing to optimize the

gap between the fiber-taper and the defect cavity, and

thus to increase evanescent coupling. Despite careful re-

duction of the taper-cavity gap, only a small fraction of

the light can be coupled into the defect cavity, typically

not exceeding 10 % of the incoming laser power. The

cavity-fiber system is kept in a vacuum chamber with a

pressure below 1 mbar. Laser light coupled inside the

photonic crystal cavity leads to local heating, which in-

duces a thermal effect arising from the temperature de-

pendent refractive index n [25]. As dn/dT > 0 (T is the

temperature) for InP [26], the region detuned to the blue

side of the resonance allows thermal passive locking [25].

In our experiments the laser frequency is chosen to corre-

spond to the blue-detuned side of the fringe of the optical

mode requiring no further locking. Mechanical motion of

the membrane is imprinted on the transmitted optical

intensity through modulation of the internal cavity field.

An electro-optical modulator is used for frequency mod-

ulation to determine the optomechanical coupling rate.

The transmitted signal is detected by a fast receiver and

the electrical signal is analyzed with an oscilloscope as

well as an electronic spectrum analyzer, which is used

for the spectral analysis.

FIG. 2. (a) Detected frequency noise spectrum in the 1 MHz

- 200 MHz range presenting a series of peaks corresponding

to the different mechanical modes labeled by numbers (Black

curve). The red curve represents a spectrum acquired with

the laser being detuned out of resonance. Inset: Calibrated

frequency noise spectrum of the fundamental mode (#1) with

a Lorentzian fit (red line). (b) Spatial displacement pattern

of the first eight mechanical modes and the prominent mode

around 150 MHz, as obtained from finite element modeling.

For a launched laser power of 1.3 mW more than 20

mechanical modes are observed in the frequency range

between 10 MHz and 1 GHz. These modes can be sepa-

rated into two mode families. The first family consists of

flexural modes present in the low-frequency range (below

200 MHz), whereas the second family consists of local-

ized modes. Flexural modes, whose spectrum is shown

in Figure 2(a), correspond to the movement of the whole

membrane. In order to identify the various modes, we

modeled the mechanical properties of the photonic crys-

tal slab structure by finite element modeling (COMSOL

Multiphysics). Realistic geometry parameters were taken

into account, including the under-etching of the mesa

structures between which the membrane is suspended.

A good agreement between measurements and modeling

is obtained using a Young modulus of 20 GPa (slightly

smaller than usual values observed in bulk InP materials

[27]attributed to the perforation for the photonic crys-

tal). Figure 2(b) shows the displacement patterns of the

first eight modes as well as a prominent mode around 150

MHz.

Localized modes, shown in Figure 3, correspond to me-

Page 3

3

FIG. 3. (a) Detected frequency noise spectrum in the 200

MHz - 1.1 GHz range presenting a series of peaks correspond-

ing to the different mechanical modes labeled by numbers

(Black curve). The red curve represents a spectrum acquired

with the laser being detuned out of resonance. CP denotes

the calibration peak resulting from the frequency modulation

measurement. Inset: Measured versus simulated frequency

for mechanical modes which could be assigned (#1-15 and

#19-21). (b) Spatial displacement pattern for the first four

orders of localized mechanical modes, as obtained from finite

element modeling. (c) Simulated (blue) and experimentally

determined (red) progression of the resonance frequency of

the localized modes versus mode number (mode #22 was de-

termined in a separate measurement).

chanical displacement of the membrane localized in the

cavity core of the defect. We were able to resolve the

fundamental localized mode at 0.46 GHz as well as the

three higher mode orders (at 0.72 GHz, 0.99 GHz and

1.26 GHz). The inset of Figure 3(a) compares the simu-

lated against the measured resonance frequencies of the

mechanical modes that could be assigned, revealing ex-

cellent agreement both for flexural and localized modes.

The progression of the resonance frequency of the local-

ized modes versus mode number is shown in Figure 3(c)

and follows a linear behavior with mode number.

Importantly, localized mechanical modes coincide spa-

tially fully with the optical defect cavity mode. There-

fore, the photonic crystal not only offers strong optical

confinement, but simultaneously ultra-high phonon con-

finement. We note that the localization occurs in the

absence of a phononic band gap [28]. Due to the co-

localization of the optical and mechanical mode within

the defect cavity, strong optomechanical coupling is ex-

pected. This is visualized in Figure 4(a), where the re-

FIG. 4.

field (left), Scanning Electron Microscope image of the de-

fect (middle) and simulated third order localized mechanical

mode (right). (b) Calibrated frequency noise spectrum of the

third order localized mechanical mode #21 (blue points) with

a Lorentzian fit (red line). A vacuum optomechanical rate of

g0/2π = 268 kHz is determined.

(a) Simulated distribution of the electromagnetic

sults of the mechanical FEM simulation for the third or-

der localized mode are compared with the spatial dis-

tribution of the electromagnetic field of the optical de-

fect mode, which was obtained through a finite differ-

ence time-domain (FDTD) simulation. The high overlap

of the mechanical displacement with the distribution of

electromagnetic energy promises strong optomechanical

coupling. In a second experiment we determined the cou-

pling rate for various flexural modes and the second and

third order localized modes.

Usually, the optomechanical coupling strength is de-

termined by two parameters - the optomechanical cou-

pling parameter G =

dx, with ωcbeing the resonance

frequency of the optical resonator and x denoting the dis-

placement of the mechanical oscillator, and the effective

mass meff[29] of the mechanical mode. The necessity of

introducing the effective mass routinely arises from the

arbitrary definition of x, which often cannot be consis-

tent with the displacement pattern of different mechani-

cal modes of the system. Particularly for photonic crys-

tals it is difficult to define an unambiguous displacement

direction of a mechanical mode. One attempt to circum-

vent this problem is to introduce an effective length Leff

determined by a perturbative expression of the overlap

of mechanical displacement and the electromagnetic field

distribution [12, 13]. Recently, it was suggested that the

vacuum optomechanical coupling rate g0would be a more

proper quantity for optomechanical systems [30, 31]. In

analogy to cavity Quantum Electrodynamics (cQED), g0

is defined as g0= G · xzpf, with xzpf=??/2meffΩmbe-

tor (? indicates the reduced Planck constant and Ωm/2π

the mechanical resonance frequency). As all relevant op-

dωc

ing the zero-point-fluctuations of the mechanical oscilla-

Page 4

4

tomechanical parameters can be derived through knowl-

edge of g0, acquiring its value would make the determi-

nation of G and meff redundant. As described in [30],

the value g0can be determined experimentally via

g0=

?

Sωω(Ωm)Γm

4¯ n,

(1)

with ¯ n being the average phonon occupancy of the

mechanical mode, Γm/2π being the mechanical damping

rate and Sωω(Ωm) being the frequency spectral density

of cavity frequency noise evaluated at the mechanical res-

onance frequency. For high phonon occupancy one can

approximate ¯ n ≈ kBT/?Ωm? 1, with kBas Boltzmann’s

constant. Sωω(Ωm) can be experimentally determined by

a frequency modulation technique [32].

We performed a measurement of the cavity frequency

noise by applying a known phase modulation to the laser

using an LiNbO3 electro-optical modulator (cf. Figure

1). The incoming laser power was reduced to 0.9 mW.

To make sure that the mechanical modes were not driven

thermally, the laser frequency was slightly detuned from

the side of the fringe in both directions just before the

experiment. With no change in the resonance frequency

of the mechanical mode occurring, we inferred that the

mechanical modes were only driven by thermal Brown-

ian motion (T=300 K). The modulation frequency was

chosen to be close to the resonance frequency of the me-

chanical mode to be calibrated. The measurement also

allows calibration of the frequency noise produced by the

mechanical mode. A detailed account of the calibration

method is given in [33]. A calibrated frequency noise

spectrum for the fundamental flexural mode is shown in

the inset of Figure 2(a), and the optomechanical vacuum

coupling rate was determined to be g0/2π = 234 Hz.

The mechanical quality factor for this mode is Qm= 890

being the highest for all flexural modes. The coupling

rate for the flexural modes increases with the mode num-

ber up to several kHz. The optomechanical coupling for

the localized mechanical modes was determined to be

g0/2π = 199 kHz (Qm = 160) and g0/2π = 268 kHz

(Qm= 180) for the second and third order, respectively.

A calibrated frequency noise spectrum for third order lo-

calized mode is shown in Figure 4(b). The high values of

g0/2π give a definite experimental proof of the high op-

tomechanical coupling between a photonic crystal defect

cavity and a localized mechanical mode. These coupling

values are two orders of magnitudes higher than mea-

sured in whispering gallery mode toroidal resonators [34]

and doubly-clamped strained silicon nitride beams in the

near-field of a silica toroidal resonator [9], both of which

are ca. g0/2π ≈ 1 kHz. Moreover, the measured values

are as high as the recently reported coupling of a flexural

mode to a photonic crystal slot cavity [4]. The values of

g0/2π which could be unambiguously obtained are spec-

ified in Table I.

TABLE I. Optomechanical vacuum coupling rates for various

modes determined experimentally via the frequency modula-

tion technique.

Mode

index

1

2

5

8

20

21

Measured frequency

Ωm/2π (MHz)

9.54

24.45

46.87

59.83

716

991

Measured vacuum

coupling rate g0/2π (kHz)

0.23

0.67

2.26

7.26

199

268

The calibration technique used has the clear advantage

that it allows to determine g0for any mechanical mode

from the mere knowledge of its mechanical linewidth and

the mechanical mode occupancy. The method does not

require knowledge about the optical linewidth, the spe-

cific transduction mechanism of the signal through the

optical cavity, the spatial distribution of electromagnetic

energy or the displacement pattern of the mechanical

mode.Thus the method is particularly suitable for

optomechanical calibration in photonic crystals due

to the frequently complex spatial distribution of both

mechanical and optical modes. Moreover, the present

planar architecture can be used for photon-phonon

conversion experiments [4].

In conclusion, we demonstrated optomechanical cou-

pling in a two-dimensional III-V photonic crystal defect

cavity. We observed both flexural as well as localized

mechanical modes. Furthermore, we provide direct mea-

surements of the vacuum optomechanical coupling rate

in a photonic crystal and measure for the first time the

coupling of a localized photonic crystal defect mode with

a localized 2D mechanical mode. Coupling rates for the

localized modes exceed 250 kHz and are two orders of

magnitude larger than in conventional optomechanical

systems. By integrating a single quantum dot in the de-

fect cavity, a variety of experiments can be envisioned

such as laser cooling [20] and coupling of a quantum me-

chanical oscillator to an artificial atom, once the ground

state of the mechanical oscillator is reached.

Funding for this work was provided by European

NanoSci-ERA project NanoEPR, through QNEMS and

MINOS by the FP7, the NCCR Quantum Photonics,

the SNF and through an ERC Starting Grant SiMP.

∗tobias.kippenberg@epfl.ch

†isabelle.robert@lpn.cnrs.fr

[1] T. J. Kippenberg and K. J. Vahala,

(2008); F. Marquardt and S. M. Girvin, Physics 2, 40

Science 321, 1172

Page 5

5

(2009); I. Favero and K. Karrai, Nature Photon. 3, 201

(2009).

[2] T. J. Kippenberg, et al., Phys. Rev. Lett. 95, 033901

(2005); T. Carmon, et al., ibid. 94, 223902 (2005).

[3] J. Rosenberg, et al., Nature Photon. 3, 478 (2009); G. S.

Wiederhecker, et al., Nature (London) 462, 633 (2009).

[4] A. H. Safavi-Naeini and O. Painter, arXiv:1009.3529

(2010).

[5] S. Weis, et al., Science doi:10.1126/science.1195596.

[6] D. Chang, et al., arXiv:1006.3529 (2010).

[7] K. Schwab and M. Roukes, Phys. Today 58, 36 (2005).

[8] O. Arcizet, et al., Nature (London) 444, 71 (2006); S. Gi-

gan, et al., ibid. 444, 67 (2006); A. Schliesser, et al., Phys.

Rev. Lett. 97, 243905 (2006).

[9] J. D. Thompson, et al., Nature (London) 452 (2008);

G. Anetsberger, et al., Nature Phys. 5, 909 (2009); M. Li,

et al., ibid. 456, 480 (2008).

[10] J. Foresi, et al., Nature (London) 390, 143 (1997).

[11] M. Eichenfield, et al., Nature (London) 462, 78 (2009).

[12] M. Eichenfield, et al., Nature (London) 459, 550 (2009).

[13] M. Eichenfield, et al., Opt. Express 17, 20078 (2009).

[14] Y. Akahane, et al., Nature (London) 425, 944 (2003).

[15] T. Yoshie, et al., Nature (London) 432, 200 (2004).

[16] O. Painter, et al., Science 284, 1819 (1999).

[17] S. Strauf, et al., Phys. Rev. Lett. 96, 127404 (2006).

[18] A. H. Safavi-Naeini, et al., Appl. Phys. Lett. 97, 181106

(2010).

[19] Y.-G. Roh, et al., Phys. Rev. B 81, 121101(R) (2010).

[20] I. Wilson-Rae, P. Zoller, and A. Imamoglu, Phys. Rev.

Lett. 92, 075507 (2004).

[21] A. Talneau, et al., Appl. Phys. Lett. 92, 061105 (2008).

[22] A. Michon, et al., J. Appl. Phys. 104, 043505 (2008).

[23] R. Hostein, et al., Opt. Lett. 35, 1154 (2010).

[24] M. Cai, O. Painter, and K. J. Vahala, Phys. Rev. Lett.

85, 74 (2000).

[25] T. Carmon, L. Yang, and K. Vahala, Opt. Express 12,

4742 (2004).

[26] P. Martin, et al., Appl. Phys. Lett. 67, 881 (1995).

[27] http://www.ioffe.ru/SVA/NSM/Semicond/InP/mechanic.html.

[28] M. Maldovan and E. L. Thomas, Applied Physics Letters

88, 251907 (2006).

[29] A. Gillespie and F. Raab, Phys. Rev. D 52, 577 (1995);

M. Pinard, Y. Hadjar, and A. Heidmann, Eur. Phys. J.

D 7, 107 (1999).

[30] M. L. Gorodetsky, et al., Opt. Express 18, 23236 (2010).

[31] A. H. Safavi-Naeini and O. Painter, Opt. Express 18,

14926 (2010).

[32] A. Schliesser, et al., New J. Phys. 10, 095015 (2008).

[33] See SI at [URL by APS] for detailed description.

[34] A. Schliesser, et al., Nature Phys. 4, 415 (2008).

Page 6

6

Supplementary information - Optomechanical coupling in a two-dimensional photonic

crystal defect cavity

DETERMINATION OF THE OPTOMECHANICAL VACUUM COUPLING RATE

The optomechanical vacuum coupling rate g0/2π is determined via a frequency modulation technique. An electro-

optical modulator (EOM) is used to phase modulate the laser carrier before its coupling into the cavity. The modulated

phase Φ of the signal can be written as

Φ(t) = ω0t + β cos(Ωmodt), (S1)

with ω0as the radial laser frequency, t as the time, β as the phase shift factor and Ωmod/2π as the modulation

frequency. The transduced signal from the photonic crystal cavity is detected with a fast receiver and the spectrum is

resolved with an electrical spectrum analyzer (ESA). The spectrum exhibits the Lorentzian spectra of the mechanical

modes and a calibration peak at Ωmod/2π resulting from the modulation [1, 2]. If the modulation frequency is chosen

to be close (a couple of mechanical linewidths) to the resonance frequency Ωm/2π of the mechanical mode to be

calibrated, g0can be determined as [3]

g2

0≈

1

2¯ n

β2Ω2

mod

2

Γm

4 · RBW

Sωω(Ωm)

Sωω(Ωmod), (S2)

with ¯ n as the average phonon occupancy, Γm/2π as the dissipation rate of the mechanical oscillator, RBW as the

resolution bandwidth of the ESA, Sωω(Ωm) as the double-sided frequency spectral noise density evaluated at the

mechanical resonance frequency and Sωω(Ωmod) as the double-sided frequency noise spectral density evaluated at the

modulation frequency. Only the proportion between Sωω(Ωm) and Sωω(Ωmod) is required to be known, and it can

be readily obtained from the measured peak power spectral density values of the mechanical and calibration signals.

The power spectral density is usually given in dBm in the spectrum of the ESA. The proportion can be found with

the following relationship

Sωω(Ωm) = Sωω(Ωmod) · 10−(Pmod−Pm)/10,(S3)

with Pmodbeing the power at the peak of the calibration signal expressed in dBm and Pmbeing the power at the

peak of the mechanical mode, which is expressed in dBm as well.

The frequency modulation technique can be also used to calibrate the spectrum in absolute units. This is done by

directly determining Sωω(Ωmod) via [3]

Sωω(Ωmod) =(βΩmod)2

4 · RBW. (S4)

By using an equivalent scheme depicted in Equation S3, the spectrum can be calibrated. The calibration holds true

as long as the frequencies of the values to be calibrated are close enough to the resonance frequency of the calibration

peak. Otherwise, a generalized transduction coefficient needs to be included [3].

CALCULATION OF CORRECTION FACTORS FOR THE ESA SPECTRUM

When measuring electrical signals with an ESA, one usually expresses the power detected at a certain frequency in

the logarithmic units of dBm. By averaging one obtains an average of the logarithmic power, which is not equal to the

logarithmic expression of the averaged power - the value that is of significance in the measurements. The discrepancy

is different for random signal, such as background noise and the mechanical spectrum (resulting from thermal motion),

and for coherent signals, such as a calibration peak with a large enough signal-to-noise ratio. For random signals it

is known [4] that the measured value is about 2.5 dB below the actual one, whereas for truly coherent sources the

measured signal corresponds to the real one. For signals that have contributions both from a coherent source and

Page 7

7

random signals, such as a calibration peak with a rather low signal-to-noise ratio, the correction factor depends on

the magnitude of both contributions.

Modern ESAs can often compensate this discrepancy internally, however as this function was not applied during

our measurements, we need to use the correction described above.

expression for the correction of the measured signal.

In the following we will derive the general

A signal arriving on the ESA can be decomposed in its in-phase (I) and out-of-phase (Q) part. Spectrum analyzers

respond to the magnitude of the signal within their RBW passband [4]. The magnitude of a signal (voltage) v

represented by I and Q is given by

?

The average power arriving on the detector is given by

?v2

v =I2+ Q2.(S5)

¯P =

50

?

=

?I2+ Q2

50

?

,(S6)

where an input impedance of 50 Ω is considered (equations are generally dimensionless in this section) and the

brackets ?? imply that the mean value is taken of the expression inside them. In the average mode the ESA averages

the logarithmic input signal, thus leading to the following expression for the measured power

?

¯Pmeas= 10 · log

?I2+ Q2

50

??

. (S7)

The logarithmic value of the actual power of the signal is given by

¯Plog= 10 · log?¯P?

(S8)

It should be intuitive that the values for¯Plogand¯Pmeasare not equal, just as the log of the average is not the same

as the average of the log.

The mean values given in Equations S6 and S7 can be determined by multiplying the variable being measured

with its probability density function (PDF) and integrating over the possible values of the variable. The PDFs of the

quadratures I and Q can be assumed to be Gaussian with a certain variance σI for I and σQfor Q, and a certain

offset of the distribution from 0 being µIfor I and µQfor Q. In general, one can assume the variance to be the same

for both quadratures, so that σI= σQ= σ. For random signals it holds that µI= µQ= 0, whereas for a coherent

signal one has µQ= 0, but µI= µ > 0.

Using the assumptions stated above, one can write

PDF(I) =

1

√2πσexp

?

−(I − µ)2

2σ2

?

(S9)

and

PDF(Q) =

1

√2πσexp

?

−Q2

2σ2

?

.(S10)

From the Equations S6 and S7 it is obvious that we are interested in the mean values of v2. This requires one

to find the PDFs for this variable from the PDFs of I and Q. This is accomplished by using a general relationship,

which links the PDFs fXi(xi) of independent random variables Xi,i = 1,2,...n to the PDF fY(y) of some variable

Y = G(X1,X2,...Xn). This relationship is given by

?∞

fY(y) =

−∞

?∞

−∞

...

?∞

−∞

dx1dx2...dxnfX1(x1)fX2(x2)...fXn(xn)δ (t − G(x1,x2,...xn)), (S11)

where δ denotes Dirac’s delta function.

Page 8

8

Using Equation S11 one can determine the PDF of v2to be

PDF?v2?=

1

2σ2exp

?

−v2+ µ2

2σ2

?

· J0

?vµ

σ2

?

, (S12)

with J0

Using this expressions for the PDF of v2, one can calculate¯P and¯Pmeasby using Equations S6 and S7. For¯P

one obtains

?

v2µ

σ2

?

as a Bessel function of the first kind.

¯P (µ,σ) =

?v2

50

?

=

?∞

,

0

dv2v2

50· PDF?v2?

=µ2+ 2σ2

50

(S13)

and for¯Pmeasone obtains

¯Pmeas(µ,σ) =

?

?∞

10 ·

10 · log

?v2

50

??

?v2

µ2

2σ2

ln(10)

=

0

dv 10 · log

?

50

?

· PDF?v2?

µ2

50

=

Γ

?

0,

?

+ ln

?

??

, (S14)

where Γ

?

0,

µ2

2σ2

?

is an incomplete Gamma function and ln denotes the natural logarithm.

For random signals we set µ to 0 obtaining

¯Pmeas(µ = 0,σ) = lim

µ→0

¯Pmeas(µ,σ) = lim

= −10?ΓE+ ln?25

ln(10)

µ→0

??

10 ·

?

Γ

?

0,

µ2

2σ2

ln(10)

?

+ ln

?

µ2

50

??

σ2

, (S15)

with ΓEas the Euler-Mascheroni constant (ΓE≈ 0.577216).

Finally, we obtain a general expression for the discrepancy between the actual and the measured signal

∆(µ,σ) =¯Pmeas− 10 · log?¯P?=

10 ·

?

Γ

?

0,

µ2

2σ2

?

+ ln

?

µ2

µ2+2σ2

??

ln(10)

.(S16)

For the special case of a random signal the final expression reads

∆(µ = 0,σ) = lim

µ→0∆(µ,σ) = −10 · ΓE

ln(10),(S17)

The discrepancy in this case is a constant value being approximately ∆(µ = 0,σ) = −2.50682 and independent of

σ as expected.

For a strongly coherent signal with µ ? σ one obtains ∆(µ ? σ,σ) = 0. This means that for a strong coherent

drive there is no discrepancy between the measured and the real one.

Page 9

9

FIG. S1. Reflection coefficient ΓEOM of the power being received by the combined EOM+cable system versus frequency

measured with a network analyzer.

FIG. S2. Transmission coefficient TBNC of the power being transmitted through the BNC cable linking the signal generator

and the EOM versus frequency. The measurement was performed with a network analyzer.

EXPERIMENTAL IMPLEMENTATION

The phase shift factor β can be determined by

β = V/Vπ· π =?2PrfZπ/Vπ,(S18)

with V as the applied voltage, Vπ as the voltage needed to induce a phase shift of π, Prf as the input power of

the signal generator providing the modulation signal and Z as the impedance between the signal generator and the

EOM. To have a quantitavie understanding of the actual voltage applied on the EOM, we directly measured the

reflectance of the system consisting of the EOM and the BNC cable linking it to the signal generator. Moreover, we

measured transmittance through the BNC cable to account for potential losses. The measurements were performed

using a network analyzer. Figure S1 shows the fraction of the reflected power ΓEOM of the combined EOM+cable

system as a function of the output frequency of the signal generator. Figure S2 shows the fraction of transmitted

power TBNCthrough cable as a function of frequency. It is obvious that for high frequencies the power loss becomes

fairly strong, which is due to radiation losses in the BNC cable. This fact is undermined by Figure S2 showing the

reflection coefficient ΓBNCthrough the cable, which is well below the one of the total EOM+cable system.

Page 10

10

FIG. S3. Reflection coefficient ΓBNC of the power being transmitted through the BNC cable linking the signal generator and

the EOM versus frequency. The measurement was performed with a network analyzer.

The factor β can be thus determined as

β =

?

2Prf·?1 − 10ΓEOM/10?· 10TBNC/10· Zoutπ/Vπ. (S19)

Measurements gave a value of Vπ ≈ 7 V which is consistent with the value provided of the manufacturer of the

EOM.

A spectrum of the fundamental flexural mode is shown in Figure S4 together with the calibration peak. As discussed

in the previous section, the measured spectrum needs to be readjusted. This is done by first fitting the mechanical

with a Lorentzian and the calibration peak with a Gaussian. The noise level is obtained from the baseline of the

Lorentzian, and this value is subsequently used to determine the variance σ as defined in Equations S9 and S10

through use of Equation S15. From the Gaussian fit we obtain the maximum value of the calibration peak. Using

this value together with the determined value of σ, one can numerically solve Equation S14 to obtain µ. The value

of µ2/50 is the corrected maximum value of the calibration peak. The spectrum of the random signals, consisting

of the background noise and the mechanical response, is corrected with the constant value given in Equation S17.

The corrected values for the maxima of the mechanical spectrum and the calibration peak are used together with

Γm, inferred from the Lorentzian fit, to determine g0as described in Equation S2. By using Equation S4 one can

calibrate the spectrum in absolute frequency units.

The values of g0/2π, that could be unambiguously determined, are summarized in Table S1. The value of g0/2π

without the correction for logarithmic averaging is also stated together with the calculated correction for the calibration

peak. The required correction for the calibration peak is fairly small for most measurements due to the large enough

signal-to-noise ratio. In those cases the main correction stemmed from the adjustment of the random signals by a

constant factor of around 2.5 dBm.

Mode Ωm/2πg0/2πg0/2π (without correction) Correction factor

for the calibration

peak

0.000 dBm

0.004 dBm

0.007 dBm

0.000 dBm

0.045 dBm

0.322 dBm

1

2

3

4

5

6

9.54 MHz

24.48 MHz

46.87 MHz

59.83 MHz

716.39 MHz 199.1 kHz 148.4 kHz

990.88 MHz 267.8 kHz 193.4 kHz

234 Hz

668 Hz

2.26 kHz

7.26 kHz

175 Hz

500 Hz

1.69 kHz

5.44 kHz

TABLE S1. Table with g0 for different mechanical modes as well as different calculation conditions. The last column specifies

the correction factor for the peak height of the calibration peak.

Page 11

11

FIG. S4. Mechanical mode (blue) with a calibration peak (green) including the respective fits.

The values of g0/2π, that could be unambiguously determined, are summarized in Table S1. The value of g0/2π

without the correction for logarithmic averaging is also stated together with the calculated correction for the calibration

peak. The required correction for the calibration peak is fairly small for most measurements due to the large enough

signal-to-noise ratio. In those cases the main correction stemmed from the adjustment of the random signals by a

constant factor of around 2.5 dBm.

∗tobias.kippenberg@epfl.ch

†isabelle.robert@lpn.cnrs.fr

[1] A. Schliesser, G. Anetsberger, R. Riv` ere, O. Arcizet, and T. J. Kippenberg, New J. Phys. 10, 095015 (2008).

[2] A. Schliesser, R. Riv` ere, G. Anetsberger, O. Arcizet, and T. J. Kippenberg, Nature Phys. 4, 415 (2008).

[3] M. L. Gorodetsky, A. Schliesser, G. Anetsberger, S. Deleglise, and T. J. Kippenberg, Opt. Express 18, 23236 (2010).

[4] Agilent Application Note 1303: Spectrum and Signal Analyzer Measurements and Noise, Agilent Technologies.