Optomechanical coupling in a two-dimensional 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 two-dimensional 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 diffraction-limited 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.

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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

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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-

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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 (#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-

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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
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