Optomechanical coupling in a twodimensional photonic crystal defect cavity.
ABSTRACT Periodically structured materials can sustain both optical and mechanical modes. Here we investigate and observe experimentally the optomechanical properties of a conventional twodimensional suspended photonic crystal defect cavity with a mode volume of ~3(λ/n)³. Two families of mechanical 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 diffractionlimited size. We demonstrate direct measurements of the optomechanical vacuum coupling rate using a frequency calibration technique. The highest measured values exceed 80 kHz, demonstrating high coupling of optical and mechanical modes in such structures.

Conference Paper: Optomechanics with photonic crystals slab mirrors and cavities
R. Braive, I. RobertPhilip, I. Sagnes, I. Abram, A. Beveratos, T. Antoni, K. Makles, A. Kuhn, T. Briant, P.F. Cohadon, A. Heidmann, E. Gavartin, T.J. Kippenberg[Show abstract] [Hide abstract]
ABSTRACT: We investigate optomechanical effects in photonic crystal slab membranes, either including a cavity or acting as an endmirror in a FabryPerot cavity. We in particular demonstrate the nonlinear behavior of the membranes fundamental mode.Lasers and ElectroOptics Pacific Rim (CLEOPR), 2013 Conference on; 01/2013  SourceAvailable from: David A HoweX. Luan, Y. Huang, Y. Li, J. F. McMillan, J. Zheng, S. W. Huang, P. C. Hsieh, T. Gu, D. Wang, A. Hati, D. A. Howe, G. Wen, M. Yu, G. Lo, D. L. Kwong, C. W. Wong[Show abstract] [Hide abstract]
ABSTRACT: Highquality frequency references are the cornerstones in position, navigation and timing applications of both scientific and commercial domains. Optomechanical oscillators, with direct coupling to continuouswave light and nonmateriallimited f Q product, are long regarded as a potential platform for frequency reference in radiofrequencyphotonic architectures. However, one major challenge is the compatibility with standard CMOS fabrication processes while maintaining optomechanical high quality performance. Here we demonstrate the monolithic integration of photonic crystal optomechanical oscillators and onchip high speed Ge detectors based on the silicon CMOS platform. With the generation of both high harmonics (up to 59th order) and subharmonics (down to 1/4), our chipset provides multiple frequency tones for applications in both frequency multipliers and dividers. The phase noise is measured down to 125 dBc/Hz at 10 kHz offset at ~ 400 {\mu}W droppedin powers, one of the lowest noise optomechanical oscillators to date and in roomtemperature and atmospheric nonvacuum operating conditions. These characteristics enable optomechanical oscillators as a frequency reference platform for radiofrequencyphotonic information processing.Scientific reports. 10/2014; 4.  SourceAvailable from: de.arxiv.orgAlejandro W. Rodriguez, PuiChuen Hui, David N. Woolf, Steven G. Johnson, Marko Loncar, Federico Capasso[Show abstract] [Hide abstract]
ABSTRACT: Whether intentionally introduced to exert control over particles and macroscopic objects, such as for trapping or cooling, or whether arising from the quantum and thermal fluctuations of charges in otherwise neutral bodies, leading to unwanted stiction between nearby mechanical parts, electromagnetic interactions play a fundamental role in many naturally occurring processes and technologies. In this review, we survey recent progress in the understanding and experimental observation of optomechanical and quantumfluctuation forces. Although both of these effects arise from exchange of electromagnetic momentum, their dramatically different origins, involving either real or virtual photons, lead to different physical manifestations and design principles. Specifically, we describe recent predictions and measurements of attractive and repulsive optomechanical forces, based on the bonding and antibonding interactions of evanescent waves, as well as predictions of modified and even repulsive Casimir forces between nanostructured bodies. Finally, we discuss the potential impact and interplay of these forces in emerging experimental regimes of micromechanical devices.Annalen der Physik 09/2014; · 1.51 Impact Factor
Page 1
Optomechanical coupling in a twodimensional photonic crystal defect cavity
E. Gavartin,1R. Braive,2,3I. Sagnes,2O. Arcizet,4A. Beveratos,2T. J. Kippenberg,1,5, ∗and I. RobertPhilip2, †
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 twodimensional
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 diffractionlimited 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 downsizing the oscillator, thus shift
ing the quantumclassical transition towards higher tem
peratures.Among various submicron optomechanical
systems presently investigated [9], suspended membranes
containing a photonic crystal cavity offer strong light con
finement in diffractionlimited 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
bandedge 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 nonresonant 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 fibertaper 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 262nm thick InP suspended membrane containing a
twodimensional 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
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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 selfassembled
InAsP quantum dots [22] at its vertical center plane for
cavity characterization. The whole structure is grown by
metaloorganic 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 coldcavity 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 externalcavity 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 fibertaper technique [24]. Piezoelectric actuators
enable an accurate positioning allowing to optimize the
gap between the fibertaper and the defect cavity, and
thus to increase evanescent coupling. Despite careful re
duction of the tapercavity 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
cavityfiber 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 bluedetuned 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 electrooptical 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 lowfrequency 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 underetching 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
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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 (#115 and
#1921). (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 ultrahigh 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 timedomain (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 zeropointfluctuations 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 electrooptical 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 doublyclamped strained silicon nitride beams in the
nearfield 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 photonphonon
conversion experiments [4].
In conclusion, we demonstrated optomechanical cou
pling in a twodimensional IIIV 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
NanoSciERA 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
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Supplementary information  Optomechanical coupling in a twodimensional 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 doublesided frequency spectral noise density evaluated at the
mechanical resonance frequency and Sωω(Ωmod) as the doublesided 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 signaltonoise 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
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7
random signals, such as a calibration peak with a rather low signaltonoise 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 inphase (I) and outofphase (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 EulerMascheroni 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.
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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
signaltonoise 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
signaltonoise 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.
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 Available from Emanuel Gavartin · May 22, 2014
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