Near-field integration of a SiN nanobeam and a SiO$_2$ microcavity for Heisenberg-limited displacement sensing

7.8 MB·

by  Ryan Schilling

Near-field integration of a SiN nanobeam and a SiO microcavity
2
for Heisenberg-limited displacement sensing
R. Schilling, H. Schu¨tz, A. Ghadimi, V. Sudhir, D. J. Wilson, and T. J. Kippenberg
Institute of Condensed Matter Physics, E´cole Polytechnique F´ed´erale Lausanne, CH-1015 Lausanne, Switzerland
(Dated: January 26, 2016)
Placing a nanomechanical object in the evanescent near-field of a high-Q optical microcavity
gives access to strong gradient forces and quantum-noise-limited displacement readout, offering an
attractive platform for precision sensing technology and basic quantum optics research. Robustly
implementing this platform is challenging, however, as it requires separating optically smooth sur-
faces by (cid:46) λ/10. Here we describe a fully-integrated evanescent opto-nanomechanical transducer
based on a high-stress Si N nanobeam monolithically suspended above a SiO microdisk cavity.
3 4 2
Employing a novel vertical integration technique based on planarized sacrificial layers, we achieve
beam-diskgapsaslittleas25nmwhilemaintainingmechanicalQ×f >1012Hzandintrinsicoptical
Q∼107. The combined low loss, small gap, and parallel-plane geometry result in exceptionally ef-
6 ficienttransduction,characterizingbyradio-frequencyflexuralmodeswithvacuumoptomechanical
1 couplingratesof100kHz,single-photoncoopera√tivitiesinexcessofunity,andzero-pointfrequency
0 (displacement)noiseamplitudesof10kHz(fm)/ Hz. Inconjunctionwiththehighpowerhandling
2 capacity of SiO2 and low extraneous substrate noise, the transducer operates particularly well as a
sensor. Deployingitina4Kcryostat,werecentlydemonstratedadisplacementimprecision40dB
n below that at the standard quantum limit (SQL) with an imprecision-back-action product <5·(cid:126).
a
In this report we provide a comprehensive description of device design, fabrication, and characteri-
J
zation, with an emphasis on extending Heisenberg-limited readout to room temperature. Towards
5
this end, we describe a room temperature experiment in which a displacement imprecision 30 dB
2
below that at the SQL and an imprecision-back-action product < 75·(cid:126) is achieved. Our results

impacttheoutlookformeasurement-basedquantumcontrolofnanomechanicaloscillatorsandoffer
]
s perspective on the engineering of functionally-integrated (“hybrid”) optomechanical systems.
c
i
t
p
CONTENTS I. INTRODUCTION
o
.
s
c I. Introduction 1 Nanomechanical oscillators [1] are sensitive to weak
i forces and exhibit large zero-point fluctuations, making
s
y II. Device design 3 them an attractive platform for both precision sensing
h technology [2–4] and basic quantum science [5]. Much
A. Nanomechanical beam 3
p effort has been devoted to the development of nanome-
[ B. Optical microdisk 4 chanical transducers in the electrical domain, including
C. Evanescent optomechanical coupling 4 single-electron transistors [6], atomic point contacts [7],
1 and superconducting microwave cavities [8]. Though
v
very successful in recent years [9], these techniques are
5 III. Device fabrication 6
practically limited by the need for cryogenic operation.
4 A. Microdisk fabrication 6 Acomparativelypowerfulapproachistoparametrically
7
B. Planarized sacrificial layer 7 couple a nanomechanical oscillator to an optical cavity.
6
The field of a laser-driven cavity can be quantum-noise-
0 C. Nanobeam fabrication 7
limited at room temperature, and as such represents
. D. Structural release 8
1 a practically ideal form of mechanical transducer, with
0 1. Mesa and sample chip 8 readoutenabledbystandardinterferometrictechniques
6
2. Nanobeam and microdisk 8
1
:
v IV. Device characterization 8 a b
i
X A. Experimental setup 8
r B. Thermal noise measurement 8
a
C. Optical spring effect 9 2 μm
D. g and C versus lateral beam position 9 c
0 0
E. g and C versus beam width and disk
0 0
thickness 10
F. g versus mechanical mode order 10
0
2 μm 1 μm
V. Displacement sensitivity 11
FIG. 1. False-colored scanning electron micrograph
of the device: a high-stress Si N (red) nanomechanical
3 4
VI. Summary and outlook 12 beam integrated into the evanescent mode volume of a
SiO (blue) microdisk. Disk and beam are integrated
2
on a Si (gray) microchip. Subpanel b (c) highlights the
References 13
lateral (vertical) positioning of the beam.
2
andactuationprovidedbyradiationpressure. Moreover, ousdisplacementnoise(typicallythermo-refractivenoise
the finite build-up time of the cavity field allows it to (TRN) in the cavity substrate [35]), optimized systems
do work on the mechanical element, enabling low-noise have achieved room temp√erature displacement impreci-
optical cooling and amplification [10]. Investigation of sions as low as 10−16m/ Hz, sufficient to in principle
these effects has led to two paradigmatic goals in the resolve the zero-point motion in one report [35].
contemporaryfieldofcavityoptomechanics[11]: cooling Despite these advances, the full potential of evanes-
of a solid-state mechanical oscillator to its ground state centcavityoptomechanicshasbeeninhibitedbythedif-
and, concomitantly, read-out of its zero-point motion ficultyofpositioningthenanomechanicalelementwithin
withtheminimaldisturbanceallowedbytheHeisenberg λ/10 ∼ 100 nm of the cavity substrate. Early systems
uncertainty principle (due to radiation pressure shot madeuseofnanopositioningstagesandsufferedfromvi-
noise (RPSN) [12]). The first challenge has been met brational stability [30]. Gavartin et. al. [34] addressed
by several cryogenic optomechanical [13, 14] and elec- this challenge by integrating a Si N nanobeam and a
3 4
tromechanical systems [15] (via resolved-sideband cool- SiO microdisk on a chip; however, due to fabrication
2
ing [16]). The latter, corresponding to a measurement constraints,thebeam-diskseparationwaslimitedto250
atthestandardquantumlimit(SQL)[17], remainsout- nm and the optical Q was reduced by a factor of 10.
standing;however,readoutnoisefarbelowthezero-point In this work, we discuss a novel method to monolith-
displacementhasbeenreported[18,19],aswellasRPSN ically integrate a high-stress Si N thin film resonator
3 4
dominatingthethermalforce[20,21]. ReachingtheSQL andaSiO microdiskcavitywithin the evanescent near-
2
ultimately requires a ‘Heisenberg-limited’ displacement field, without deteriorating the intrinsic Q of either el-
sensor for which the product of the read out noise and ement. The critical ingredient is a chemical mechanical
the total force noise is the minimum allowed by the un- polishing technique that allows integration of optically
certaintyprinciple. Thisregimehasbeenapproachedto flat surfaces with sub-100 nm spacing, separated by a
within an order of magnitude by several cryogenic sys- sacrificialfilm. Thisprocedureisusedtocarefullyisolate
tems [15, 18]; it also forms the basis for measurement- Si N and SiO layers during wafer processing, allow-
3 4 2
based quantum feedback protocols such as ground-state inghigh-yieldanddeterministicfabricationofdevicesin
cooling [18, 22] and squeezing [23] of an oscillator. which a nanobeam is monolithically suspended as little
Efficient cavity optomechanical transduction involves as 25 nm above a SiO microdisk — ∼3× smaller than
2
co-localization of optical and mechanical modes with theevanescentdecaylengthofitsWGMs—whilemain-
high Q/(modevolume) and high optical power handling taining mechanical and optical mode qualities in excess
capacity. Moreover, it is desirable that the cavity sup- of 105 and 106, respectively. The process is compatible
portamechanismforefficientinput/outputcoupling. A withe-beamlithography,thusweareabletolocallypat-
diverse zoo (Fig. 2) of micro- and nanoscale cavity op- ternthebeamwithsub-10nmimprecision(openingthe
tomechanical systems (COMS) has risen to meet these door to stress engineering [36]) and laterally position it
challenges,rangingfromcantilevers[24]andmembranes with sub-100 nm imprecision across a full 4” Si wafer.
[25] coupled to Fabry-P´erot cavities to mechanically- A typical device is shown in Fig. 1, corresponding to
compliant whispering-gallery-mode (WGM) microcavi- a 60 × 1 × 0.06µm3 beam positioned 25 nm above a
ties [26] and photonic crystals [27]. They generally em- 0.65-µm-thick, 30-µm-diameter microdisk. By carefully
ploy two types of radiation pressure force coupling: tra- varying the dimensions of the beam, the disk, and their
ditional scattering-type coupling, in which the cavity lateral offset with respect to this nominal geometry, we
field exchanges energy with the mechanical element via achieve optomechanical coupling rates (G) in excess of
momentumtransfer,andgradientforcecoupling[28],in 2π·1GHz/nmwhilemaintainingcavitydecayrates(κ)
whichenergyisexchangedviainduced-dipolecouplingto
a field gradient. The net effect is a parametric coupling 106
G=∂ω /∂x between the cavity resonance frequency ω
atthrnaedAnfstopdhrauecrccemttiaiceopucnphlalairinnelydvicopaplvlreoedrmseignipsrtlieraneacgcionapfvglfiartteayfeod(prdohmimoetlfeooxcnr,t,wroi(cid:126)pcht)·iocGhmmeee[1xcc1hhp]aar.ennsiiscceaacsll erativity, C 0 11110000-2420 1 225239 288 3245 11 561 161730 m31 3KK3060 K
substrate next to the surface of a WGM microcavity, so p 10-4 24 27 9 14
o 22 1815
tdheactayitlseanmgtphleissi∼tseλv/a1n0e,stchenisttfioepldo.loSgiyncoefftehrsetehvaenoepspceonrt- n co 1100--68 8 21202631 74 2
o 19
tunity for strong gradient force coupling to nanoscale ot 10-10 microwave 3 7
mechanical devices. It also has the virtue of naturally ph 10-12 photonic crystals 10
accommodatingopticalandmechanicalsubstratesofdis- e- nano-objects
smimizialtaironmaotferQia/l(manoddegveoolmumeter)y., eMnaobrleionvgers,epWarGatMesopcatin- singl 1100--1146 mmcoiilcrdrro oarrtseosmonsators 12
be input/output coupled with high ideality using ta- 10-18
101 103 105 107 109 1011
pered optical fibers [29], making them well-suited to in-
terferometricdisplacementsensing. Recentworkhasfo- mechanical frequency, W /2p [Hz]
m
cused on coupling of nano-beams [30], -cantilevers [31],
and -membranes [30, 32] to the evanescence of WGM FIG. 2. Survey of single-photon cooperativity C =
0
micro-toroids [30], -spheres [32, 33], and -disks [31, 34], 4g2/κΓ for various cavity optomechanical systems
0 m
with mechanical materials ranging from (ultra low loss) [19, 37–43], adapted with permission from [11]. Non-
high-stress Si3N4 [30] to (ultra low mass) single-layer italicizedreferencesarecitedin[11]. Blueandredpoints
graphene [32], typically using SiO2 as the optical mate- correspond to cryogenic (typically T <10 K) and room
rial. Gradient force coupling as high as G ∼ 2π ·100 temperature experiments, respectively. Diagonal lines
MHz/nm has been achieved [30]. Combined with the indicatetheconditionforC =n ≈k T/(cid:126)Ω ,forvari-
0 th B m
high power handling capacity of SiO2 and low extrane- ousT. Thereportedresultishighlightedwithcrosshairs.
3
aslowas2π·100MHzandradiofrequency(Ω =2π·(1− crodiskresonatorsanddescribesanumericalmodelused
m
10)MHz)flexuralbeammodeswithdampingrates(Γ ) to predict their gradient-force optomechanical coupling.
m
aslowas2π·10Hz. Inconjunctionwiththesmallmass Notably,wefindthatGcanbeimprovedbyanorderof
(m ∼ 10 pg) and large zero-point displacement (x ≡ magnitude by carefully positioning the beam above the
zp
(cid:112)(cid:126)/2mΩ ∼10 fm) of the beam modes, the combined disk. Sec. III describes the fabrication method, particu-
m
low-loss,smallgapandparallel-planegeometryresultin larlytheuseofplanarized(byCMP)sacrificiallayersand
avacuumoptomechanicalcouplingrates(g ≡G·x )as e-beam lithography, which enable precise engineering of
0 zp
highas2π·100kHzandroomtemperaturesingle-photon theverticalandhorizontalbeam-diskseparation,respec-
cooperativitiesashighasC ≡4g2/κΓ =2. Thelatter tively. Sec. IV describes characterization of the device
0 0 m
isnotablyafactorof105timeslargerthanin[34]andon usingthermomechanicalnoisemeasurementsandtheop-
parwiththestate-of-the-artforbothroomtemperature tical spring effect. In Sec. V, we discuss an experiment
and cryogenic COMS (Fig. 2). inwhichthemicrodiskisembeddedintofiber-basedho-
modyne interferometer, enabling displacement read-out
In conjunction with high C , several features of the
0 withanimprecision33dBbelowSzp(Ω )forthefunda-
system make it well-suited for quantum-limited opera- ω m
mentalbeammode. Finally,inSec.VI,weremarkonthe
tion. First, SiO microcavities with the reported di-
2
feasibility of Heisenberg-limited position measurements
mensions and internal loss readily support intracavity
photon numbers of n ∼ 106. This enables quantum andfunctionalizedapplicationswhichtakeadvantageof
c
the heterogeneous integration method.
cooperativities (C n /n ) approaching unity — a ba-
0 c th
sicrequirementforperformingaHeisenberg-limiteddis-
placementmeasurement—foraroomtemperaturether-
maloccupationofn ≈k T/(cid:126)Ω ∼106,corresponding
th B m II. DEVICE DESIGN
to Ω ∼2π·5 MHz. Another striking feature is the ex-
m
ceptionallylargemagnitudeofthecavityfrequencynoise
producedbyzero-pointmotionoft√hemechanicaloscilla- A. Nanomechanical beam
tor,Szp(Ω )≡4g2/Γ ∼10kHz/ Hz. Thismagnitude
ω m 0 m
ismanyordersofmagnitudelargerthantypicalextrane-
The mechanical resonator we study is a doubly-
ous sources of noise due to laser frequency fluctuations
clamped beam released from a high-stress Si N thin
or TRN [35]. Taking advantage of these strengths, re- 3 4
film [45]. Stressed “nanobeams” are attractive for their
cent deployment of the device in a 4 K Helium cryostat
string-like flexural modes, which possess exceptionally
enabled interferometric measurements with a read-out
high Q/m ratios [46]. Beams with of the dimen-
noise43dBbelowSzp(Ω )(correspondingtoanimpre-
ω m sions studied — {length(l),width(w),thickness(t)} ∼
cision40dBbelowthatnecessarytoreachtheSQL)and
{100,1,0.1} µm — possess effective masses m ∼ 10 pg,
with an imprecision-back-action product of 5·(cid:126), allow-
fundamental frequencies Ω ∼ 2π·10 MHz and room
ing active feedback cooling to near the motional ground m
temperaturequalityfactorsQ >105[45]. Significantly,
state [18]. Below, we demonstrate a measurement with m
Q iswellinexcessofthe“universal”valueof103−104
an imprecision 30 dB below that at the SQL and an m
observed for bulk amorphous glass resonators at tem-
imprecision-back-actionproductof75·(cid:126),usingamoder- peratures above T (cid:38) 1 K [47]. It is also higher than
ate input power of 10 µW. Remarkably, the imprecision
for typical unstressed, single-crystal nanobeams due to
due to microdisk TRN [35] can be 20 dB lower.
surface loss [44]. This exceptional behavior is known to
In the following sections we carefully detail the de- derivefromacombinationoflargeimpedancemismatch
sign,fabrication,andcharacterizationofthedevice,and fromtheanchoringbody[48](suppressingextrinsicloss)
provideademonstrationoflownoisedisplacementmea- andstress-related“dilution”ofintrinsicloss[44,49,50].
surement. Sec.IIgivesanoverviewofnanobeamandmi- From the standpoint of quantum-limited measurement,
an important consequence of their high Q/m is that
high-stress nanobeams exhibit large zero-point fluctua-
tions. Expressedasasingle-sidedspectraldensityevalu-
ated at the mechanical frequency, the above parameters
106 1013 correspondtoapeakzero-pointdisplacemen√tnoiseden-
sity of Szp(Ω ) = 2(cid:126)Q /mΩ2 ∼ 10 fm/ Hz. This
x m m m
value occurs in a radio frequency window, 1-10 MHz,
Q where low noise electronics and laser sources are avail-
m
Ω able; as such, nanobeams were the first solid state me-
Qm105 1012m/2π chanical resonators to be read out electrically (using a
[H metal beam) [51] and optically [35] with an imprecision
z] lower than Sxzp(Ωm).
Measurements of Q for a typical disk-integrated
m
beam with dimensions {l,w,t} = {60,0.6,0.06} µm
are shown in Fig. 3. Despite the complexity of the
104 1011 fabrication procedure (Sec. III), flexural modes exhibit
Ω /2π [MHz] Qm·Ωm/2π ashighas4·1012 Hz,onparwiththestate-
m
of-the-artforhigh-stressSi N nanobeamsofsimilardi-
3 4
mensions [44, 52]. The near-linear eigenfrequency spec-
FIG. 3. Q-factor (red) and Q × frequency product
(blue)ofthefirsteleven, odd-ordered, out-of-planeflex- trum, Ω(mn) ≈2πn·4.3MHz, is consistent with a tensile
ural modes of a nanobeam with dimensions {l,w,t} = stressofT ≈(ρlΩ(0)/π)2 ≈0.8MPaassumingadensity
m
{60,0.6,0.05} µm. Solid red curve is a fit to the Q- of ρ = 2700 kg/m3 [45]. The mechanical-Q spectrum,
dilution model in [44], implying a limiting contribution Q(n) ≈ 3.6·105/(1+0.023·n2), is consistent with the
m
from surface-related intrinsic loss. intrinsiclossmodelof[44,50]. ThedashedlineinFig.3
4
dard lithographic techniques, in conjunction with wet-
etching,canproduceSiO microdiskswithexceptionally
2
rd t highQ(recentlyexceeding108inthetelecommunication
θ d band [55]). This feature is related to the wedged rim of
the disk, which supports WGMs that are spatially iso-
latedfromthesurface,andtherebyfromsurfacescatter-
ing/absorptionloss. Athirdadvantageisthatmicrodisk
WGMs can be evanescently coupled to tapered optical
108 fibers with high ideality [29]. This feature is critical for
sensing applications, in which optical loss produces ele-
κ/2π=10 MHz vated shot-noise imprecision [26].
Microdisk resonators were in this case studied at
λ ≈ 700−800 nm (outside of the telecommunications
107 window), to allow for smaller optical mode volumes. As
Q0 discussed in Sec. IIC, reducing the disk radius (rd) and
100 MHz thickness (td ∼ λ/n) results in smaller mode volumes
with fractionally larger evanescent components, thereby
increasing the optomechanical coupling strength. Fig. 4
photolithography
shows measurements (see Sec. IVA for details) of opti-
106 e-beam lithography
cal Q versus disk radius (r ) for microdisk samples of
d
thickness t = 0.7 µm . Two sets of devices are con-
0 10 20 30 40 50 d
r [µm] sidered. The first set was prepared with photolithog-
d raphy, the second with electron-beam lithography. The
sets differ by their corresponding wedge angle, which is
FIG.4. WGMintrinsicqualityfactorQ asafunctionof
o 30 (11) degrees for wet (e-beam) lithography. For both
disk radius rd for stand-alone SiO2 microdisks of thick- disk preparation methods, intrinsic Q > 106 was mea-
ness t ≈ 700nm. TE and TM modes are not distin-
d sured for radii as low as 10 µm, corresponding to loss
guished. Blue(red)pointscorrespondtodisksprepared
rates of κ∼2π·100 MHz. For shallower wedge angles,
withphotolithography(e-beamlithography),whichpro- Q as high as 4·107 (κ ∼ 2π·10MHz) was obtained —
duce wedge angles of θ ≈30(11)◦. Horizontal lines rep-
notably exceeding (for the same r ) those measured at
resent constant cavity linewidth, κ = 2πc/(λQ ), with d
o telecom wavelengths, where scattering losses are signifi-
λ = 780 nm. Blue (red) dashed line is a guide-to-
cantly lower [55, 56]. Numerical simulations [57] reveal
the-eye for Q ∝ r , corresponding to a fixed finesse of
d that radiation contributes negligibly to the measured
F = 0.6 (1.2)·105. A SEM of a wedged microdisk is
loss. Dotted blue (red) lines in Fig. 4 are guide-to-the-
shown above; blue (gray) indicates SiO (Si).
2 eyemodelsforQ∝r ,consistentwithlossduetosurface
d
absorption/scattering [58], and corresponding to a fixed
is a fit to this model: Qm(n) =Qint/(λ+n2π2λ2), where finesse of F ≡∆ωFSR/κ≈c/(rdκ)=0.6(1.2)·105. As
λ2 =Et2/(12Tl2), E is the elastic modulus of the film, discussed in Sec. IVD, the intrinsic microdisk Q is ulti-
and Q is the intrinsic quality factor of the film when matelyreducedbylossintroducedbythenanobeam,for
int
unstressed. TheinferredvalueofQ ≈6700(usingE = beam-disk separations of less than 100 nm.
int
200 GPa), is roughly an order of magnitude lower than
that for bulk Si N . Interpreted as surface loss, how-
3 4
ever, the inferred coefficient of Qint/t ≈ 1.1·105 µm−1 C. Evanescent optomechanical coupling
iswithinafactoroftwoofthetypicalvalueforLPCVD
SiN thin films [44]. Operating in a 3He cryostat at 0.5
Optomechanical coupling is achieved by placing the
K, we have recently observed Q >104 [53].
int nanobeam near the surface of the microdisk, so that its
In addition to its favorable mechanical properties
mid-sectionoccupiestheevanescentvolumeofoneofthe
when stressed, Si N is an attractive optical material.
3 4 microdiskWGMs. WhentheWGMisexcited,thebeam
It has a relatively large index of refraction, n≈2, and,
experiencesagradientforce,F . Themagnitudeofthis
owingtoits∼3eVbandgap,respectablylowopticalab- opt
force, and likewise the optomechanical coupling factor
sorption at near infrared wavelengths, characterized by
G = ∂ω /∂x, can be derived by computing the work
an imaginary index of n ∼10−5−10−6 [54]. c
im done on the WGM, −δU , by a small displacement of
cav
thebeam,δx: thatis,F =−∂U /∂x≈−GU /ω ,
opt cav cav c
where U is the potential energy stored in the cavity
cav
B. Optical microdisk field [28, 59]. To first order, it can be shown that [30]
ω(0) ∂ (cid:32)(cid:82) ((cid:15)((cid:126)r)−1)|E(cid:126)(0)((cid:126)r)|2d3r(cid:33)
The optical resonator we employ is a SiO microdisk G≈ c beam (1a)
supporting WGMs along its periphery. SiO22microdisks 2 ∂x (cid:82)disk(cid:15)((cid:126)r)|E(cid:126)(0)((cid:126)r)|2d3r
possess several advantages for evanescent sensing. The ω(0) ∂ (cid:32)n2 −1|E(0,beam)|2V (cid:33)
firstadvantageisthatSiO exhibitsawidetransparency ≈ c SiN max beam (1b)
window and a large powe2r handling capacity, enabling 2 ∂x nSiO2 |Em(0a,dxisk)|2 Vdisk
large intracavity photon numbers (n ). The practically
achievable n is typically limited bycKerr and Raman where (cid:15)((cid:126)r) is the local relative permittivity, E(cid:126)(0)((cid:126)r)
c
nonlinearity. At visible and telecommuncation wave- is the unperturbed cavity field amplitude, and
(cid:82)
lengths,othereffectssuchasmulti-photonabsorptiondo indicates an integral over the volume
beam(disk)
not play a significant role in SiO , in contrast to Si and occupied by the beam (disk). The simplified ex-
2
other semiconductors. A second advantage is that stan- pression in Eq. 1b replaces (cid:15) with an index of
5
a w b
t
x g/2π [kHz]
t 200 0 1
d 2
0 5
y m] TM mode 1105
n -200 20
x [ -400 2350
40
-600
-800
-1000
-1000 -500 0 500 1000
r
d y [nm]
w
c
z]
H 10
k out-of-plane
π [ in-plane
l l 2
eff g/0
1
-1000 -500 0 500 1000
y [nm]
r
d y
FIG. 5. (a) Geometry of the nanobeam-microdisk system: x, y represent the vertical (out-of-plane) and lat-
eral (in-plane) position of the beam, respectively, with respect to the inner rim of the disk (thickness t , ra-
d
dius r ). (b) Simulated optomechanical coupling versus beam position for device dimensions {t,w,l,x,r ,t } =
d d d
{0.06,0.4,60,0.025,14.2,0.65}µm. The intensity profile of a TM-like WGM (computed using FEM) is shown in the
background. Solid and dashed white lines denote the disk surface and the boundary within which the beam touches
the disk surface, for the coordinate system defined in (a). Contours indicate lines of constant g for the 4.3 MHz
0
fundamental out-of-plane mode. (c) Measured and simulated g versus y. Black and blue data are for fundamen-
0
tal out-of-plane and in-plane vibrational modes, respectively (see Sec. IVD). Black lines correspond to numerical
solutions to Eq. 1 with a vertical offset of x=25 nm. Gray shading shows the solution space for x=20−30 nm.
refraction n and parameterizes each integral in terms where ρ is the mass density of the beam. In practice
of the intensity-weighted volume of the beam (disk), x ,A ,andξ mustbedeterminednumericallyfora
ev WGM
V ≡ (cid:82) |E |2d3r/|E(0,beam(disk))|2, wedged microdisk. An estimate can be made, however,
beam(disk) beam(disk) 0 max byassumingthemodeshapeofamicrotoroidWGMwith
where E(0,beam(disk)) is the maximum of the unper-
max a minor radius of t /2 [30]. In this case, using n ≈
turbed field within the beam (disk). d (cid:113) SiO2
To gain physical insight into Eq. 1, we consider the 1.4, one has xev ≈λ/(2π n2Si02 −1)≈λ/12, AWGM ≈
configuration shown in Fig. 5. Here the beam is placed 0.15r7/12t1/4λ7/6 and ξ ≈ 1.1(λ/r )1/3 [60]. Using
above the disk, so that it samples the vertical evanes- thesedformdulas, the device geometryd{t,w,l,x,r ,t } =
d d
cence of a WGM. For simplicity, the transverse dimen- {0.06,0.4,60,0.025,14.2,0.65} µm, and assuming λ =
sions of the beam are assumed to be much√smaller than 780 nm, nSiN = 2.0, ρ = 2700 kg/m3, Ωm = 2π ·4.3
that of the evanescent field; that is, w (cid:28) AWGM and MHz, and leff = 10 µm (see Sec. IVF), Eq. 2 predicts
t(cid:28)xev,whereAWGMistheeffectivecross-sectionalarea thatG≈2π·1.0GHz/nm,xzp ≈33fm,andg0 =2π·33
of the WGM and xev is the exponential decay length kHz. AsshowninFig.5d,thisestimateagreeswellwith
of the evanescent field. In this case Vbeam can be ap- numerically and experimentally determined values. No-
proximated as twleff, where leff < l is the intensity- tably, Eq. 2 implies that to achieve large g0, it is nec-
weighted“sampling length” ofthe beam. LikewiseVdisk essary to reduce the vertical gap to x < xev ≈ 100 nm,
can be parameterized as Vdisk ≈ 2πrdAWGM, where and to maximize leff by laterally positioning the beam
rd is the physical disk radius. Assuming the form above the disk.
|Em(0a,bxeam)|/|Em(0a,dxisk)| = ξe−xx+etv/2, neglecting the weak Anumericalmodelforg (x,y)isshowninFig.5b. In-
position dependence of V , and assuming the effec- 0
tive mass of a point probbeea,mm = ρtwl/2, the vacuum trinsic WGM mode shapes, E(cid:126)0)((cid:126)r), were computed us-
optomechanical coupling rate can be approximated as ing an axially-symmetric finite element model (COM-
SOL FEM axial symmetric package [57]). The energy
(cid:115)
g0 ≈ 21ωxc(0)n2SniN−12πrtwAleff ξe−xx+etv/2 · ρtw(cid:126)lΩ astnodretdheinentehregyWsGhiMft,dUuce(a0v)to≈the21(cid:82)bdeisakm(cid:15),((cid:126)r∆)|EU(cid:126)(0)(((cid:126)rx),|y2d)3≈r,
ev SiO2 d WGM m cav
(2) 1(cid:82) ((cid:15)((cid:126)r)−1)|E(cid:126)(0)((cid:126)r)|2d3r,werecomputedbynumeri-
4 beam
6
calintegrationinMatlab. Differentiatingthe2Denergy
landscape gives G(x,y) = ω ∂ (∆U (x,y)/U(0)) for
c∂x cav cav
out-plane-motion. Fig. 5b shows g (x,y)=G(x,y)·x
0 zp
for a beam and disk with the dimensions given above,
for a TM-like WGM mode. Contours indicate that the
optimalpositionofthebeamisaboveandinsidethein-
ner rim of the disk, and that the magnitude of g scales
0
exponentially with vertical displacement from the disk
surface, with a decay length of ∼ 100 nm. A horizon-
tal cut through the contours for x=25 nm is shown in
Fig. 5c. Upper and lower curves show models for fun-
damental in-plane (IP) and out-of-plane (OP) flexural
modes. Significantly, maximizing g(OP) also minimizes
0
g(IP); this opens a wide spectral window, ∆Ω∼Ω , for
0 m
measurement of the out-of-plane mode. Experimental
measurements (see Sec. IVB) of g (25 nm,y) are also
0
shown in Fig. 5c . The model agrees well with experi-
ment assuming a vertical offset of 25±5 nm.
III. DEVICE FABRICATION
ThefabricationprocessisoutlinedinFig.6. Fourkey
elements of the process, detailed in the following sub-
sections,are: (A)fabricationoftheSiO microdisk,(B)
2
formation of a planarized sacrificial layer using chemi-
cal mechanical polishing (CMP), (C) fabrication of the
Si N nanobeam,and(D)releaseofthesacrificiallayer.
3 4
Ofparticularimportanceisthesacrificiallayer,whichal-
lowsthemechanical(Si N )andoptical(SiO )elements
3 4 2
tobedesignedindependentlywhilemaintainingthehigh
optical quality and achieving a vertical beam-disk sepa-
ration of less than 100 nm. Also important is the use of
e-beamlithographytopatterntheSi N ,asthisenables
3 4
fine tuning of the lateral beam position.
A. Microdisk fabrication
The process begins with an undoped, float-zone (FZ)
Siwafer,onwhicha750nmfilmofSiO isgrownbydry
2
oxidation. Three structures are patterned into the SiO
2
film: the microdisk, rectangular pads that later serve
as a support for the nanobeam and a reference plane
for CMP polishing, and markers that are later used for
e-beam alignment. As illustrated in Fig. 7, the SiO
2
pattern is processed in two stages. In the first stage all
structuresaredefined. Inthesecondstagethemicrodisk
is etched preferentially, recessing it from the pads and
defining the vertical gap between disk and the beam.
Details of the SiO patterning process are as follows:
2
The first mask, containing all structures, is exposed in
1.1 µm of Microchemicals AZ 1512 photoresist using a
Karl Su¨ss MA 150 mask aligner and a broadband Hg
lamp. A subsequent reflow step is used to smooth the
pattern boundaries and minimize standing wave pat-
terns. Afterwards, the pattern is transfered to SiO by
2 FIG. 6. Fabrication process flow: blue, red, green, and
etching in a room-temperature bath of BHF. The pho-
(light) gray indicate SiO , Si N , Al O , and (poly-)Si,
2 3 4 2 3
toresist is then stripped and a second mask is applied.
respectively.
The second mask covers all structures on the wafer ex-
cept for the microdisk, leaving it exposed for etching
(Fig.7a). ThemicrodiskispreferentiallyetchedinBHF pads due to its reduced thickness. Also seen in Fig. 7 is
until it is 10-100 nm thinner than the surrounding pads a matrix of sacrificial pads surrounding the disk. This
(laterdefiningthebeam-diskgap). Theresult,afterthe matrix extends across the entire wafer and is only bro-
photoresistisstripped,isshowninFig.7b. Notethatthe ken where microdisks or alignment marks (not shown)
microdiskreflectsadifferentcolorthanthesurrounding are placed. As discussed in Sec. IIIB, a uniform ma-
7
trix of pads is necessary to achieve a flat surface when where features are sparse experience a higher pressure
performing CMP the sacrificial layer. and thus a higher polishing rate than areas where fea-
The final result of microdisk fabrication is illustrated turesaredense. Inordertoreducethepoly-Sithickness
inFig.6b. Blueindicates(inprofile)thepatternedSiO to less than 100 nm over the entire 100 mm wafer, a
2
film, with the microdisk in the center and nanobeam uniform polishing rate is critical. This is the reason for
support pads on either side. Not shown are sacrificial patterning a matrix of sacrificial pads (Fig. 7).
pillarsandalignmentmarks. Inthenextprocessingstep, TheobjectiveoftheCMPprocessistoremovepoly-Si
all structures are buried in a sacrificial material, onto untilthepadsareexposed,whilemaintainingathinlayer
which a Si N film will be grown. above the recessed microdisk (Fig. 6d). This procedure
3 4
is complicated by the fact that the polishing rate varies
acrossthewaferand,moreimportantly,thatthepolish-
ingrateabovethemicrodiskisfasterthantherateabove
B. Planarized sacrificial layer
theadjacentnanobeamsupportpads. Thelatterresults
in a poly-Si layer which is thinner above the microdisk
Afterpatterning,theSiO2 filmiscoveredwithalayer than at the nanobeam supports. To reduce this “dish-
of sacrificial material. The sacrificial layer is used as ing” effect, the support pads are brought as close the
a substrate for deposition and patterning of the Si3N4 microdiskaspossible(limitedto7µmbyphotolithogra-
film, meanwhile protecting the underlying microdisk. A phyandBHFbiasing). Tofurtherreducedishing,atwo
crucial consideration is the thickness and flatness of the step polishing technique is used. First a slurry designed
sacrifical layer, which is initially uneven because of its to etch poly-Si is used to remove the bulk of the mate-
conformity to the underlying SiO2 pattern. To thin rial,leavingapproximately100nmabovethepads. The
and planarize the sacrificial layer, a delicate chemical- remainingmaterialisremovedwithaslurrydesignedto
mechanical polishing (CMP) procedure is followed. etch SiO faster than poly-Si. When the surface of the
2
Poly-Si is chosen as the sacrificial material because it SiO padsisreached,thedishingeffectbeginstoreverse,
2
can be isotropicallyetchedwith high selectivity toSiO2 resulting in an overall flat surface.
andSi3N4,iswell-suitedtoCMP,canwithstandthehigh The gap between the microdisk and nanobeam is not
temperatures required for LPCVD Si3N4 (> 800◦C), determined by the thickness of the sacrificial layer, but
andcanbeusedtoundercutthenanobeamandthemi- rather by the predefined difference in thickness between
crodiskinasinglestep. A1.5µmthicklayerisdeposited the microdisk and the pads (Fig. 6b). During the final
by LPCVD at 600◦C using silane and disilane as reac- steps of CMP, however, the support pads are etched.
tants. In addition, immediately before poly-Si deposi- The final gap is therefore smaller than originally de-
tion, a 5 nm aluminum oxide (Al2O3) film is deposited finedbythinningofthemicrodisk. Inordertoprecisely
atop the SiO2 using atomic layer deposition. This film tunethegap,thethicknessoftheclampingpadsisitera-
laterservesasanetch-stoptoprotectthemicrodiskwhen tivelymeasuredbyreflectometryuntiladesiredvalueis
releasingtheSi3N4 nanobeam. (Al2O3 etchesover100× reached. ThesampleisthenreadyforSi3N4 deposition.
slower than Si N in flourine-based RIE used, and thus
3 4
afewnanometersissufficienttoprotectthemicrodisk.)
Aprofileoftheper-polishedsacrificiallayerissketched
in Fig. 6c. The Al O etch-stop film is indicated by C. Nanobeam fabrication
2 3
green. Immediately above the etch-stop is the layer of
poly-Si (gray). Because of the underlying SiO2 struc- To form the nanobeam, a 50-100 nm film of high-
tures, the surface of the poly-Si is uneven. This surface stress Si N is deposited onto the planarized poly-Si
3 4
is planarized by CMP before Si3N4 is deposited. layer. LPCVD is performed at 800◦C using dichlorosi-
CMP involves pressing the wafer against a rotating lane and ammonia, producing a nearly stoichiometric
polishing pad in the presence of an abrasive and cor- Si N . High stoichiometry is important for reducing
3 4
rosive chemical slurry. Abrasion is provided by SiO2 absorption caused by hydrogen and oxygen impurities
particles 30-50 nm in diameter. The slurry PH is ad- [54]. Moreover,thestress(800MPa)resultingfromhigh
justed to achieve the desired polishing rate. In practice temperature deposition is important for achieving high
the polishing rate is also a function of applied force, ro- mechanical quality factors [45].
tation speed, and wafer topography. Areas of the wafer Tomaximizeoptomechanicalcoupling,itisnecessary
to fine tune the lateral beam-disk separation with 100
nm precision (Fig. 5c). This is accomplished using e-
beam lithography to define the beams, in conjunction
withthealignmentmarkersdefinedduringSiO pattern-
2
ing. Importantly, after Si N deposition, the markers
3 4
are buriedunder Si N and poly-Si, andcannot beseen
3 4
by the electron-beam. A series of etch steps are used
to locally uncover the markers; in addition, to improve
contrast, the exposed markers are used as a hard mask
to etch 2 µm into the underlying Si, using a highly se-
lective flourine-based etch. The resulting high-contrast
FIG.7. Definingtheverticalgapbetweenthediskand markers permit alignment of the Si N mask with sub-
3 4
the nanobeam: (a) Top view of patterned SiO2 prior to 100 nm precision.
selective etch of the microdisk. Photoresist protects the
The nanobeams, support pads, and sample labels are
sacrificial structures, while a window is exposed around
patterned in a 180 nm-thick hydrogen silsesquioxane
the microdisk. (b) Top view after selective etch of the
(HSQ)negativephotoresist. Toreducewritingtime,the
microdisk and removal of the photoresist. The altered
pattern is separated into two parts, one containing the
color of the microdisk indicates thinning.
nanobeamsandonecontainingthepadsandlabels. The
8
former is written with a high resolution of 5 nm, while IV. DEVICE CHARACTERIZATION
thelatteriswrittenwitha50nmresolution. Proximity
effectcorrectionisusedtoensureahighfidelitypattern.
A. Experimental setup
Thee-beampatternistransferredtoSi N usinganSF
3 4 6
RIE etch. The resulting structure is shown in Fig. 8a.
Samples are characterized using the experimental
setup shown in Fig. 9. Light from a 765−785 nm tun-
able diode laser (New Focus Velocity 6312) is coupled
D. Structural release intothemicrodiskusingataperedopticalfiber(780HP)
[29]. Theforward-scattered(“transmitted”)fieldismon-
itoredusingoneoftwotechniques: directdetectionwith
1. Mesa and sample chip
an avalanche photodiode (Thorlabs APD110) and bal-
anced homodyne detection with a pair of fast Si pho-
Beforethenanobeamandmicrodiskarereleased,they todiodes (FEMTO HCA-S-100). DC- and AC-filtered
are elevated from the surrounding wafer on a rectangu- photosignals are split between an oscilloscope (Tek-
lar “mesa”. This later enables alignment of a straight tronix DPO4034) and a spectrum analyzer (Tektronix
tapered optical fiber to the microdisk [61]. Figure 8b RSA5106A). To calibrate laser-cavity detuning, a frac-
shows the mesa defined in a 5µm mask of Microchemi- tion of the input field is simultaneously passed through
calsAZ9260photoresist. Flourine-basedRIEisusedto a 20-cm-long (FSR ∼350 MHz) fiber loop cavity. Opti-
remove the surrounding poly-Si. The underlying sacrifi- caldecayratesareinferredfrommeasurementsoftrans-
cial SiO2 pads are removed by a subsequent BHF etch, mitted power versus laser detuning (Fig. 9b). Mechan-
exposing the Si substrate. To create the elevated mesa, ical properties, including the optomechanical coupling
exposed Si is recessed an additional 50µm by DRIE. rates,areinferredfrommeasurementsofthermomechan-
Afterreleasingthemesa,thesamplechipsaredefined. ical cavity frequency noise [62] (Fig. 9c). To calibrate
To define the sample chips, the wafer is coated with a this noise, the input field is frequency modulated using
protective photoresist layer and partially diced (300µm an electro-optic modulator (EOSpace). Residual ampli-
deep) with a high precision Si dicing saw. Partial dic- tude modulation — an important source of calibration
ing is important as it leaves the wafer intact, enabling error — is actively suppressed by stabilizing the phase
further processing using wafer-scale equipment. After ofanout-of-loopheterodynebeat[63]. Toeliminategas
partial dicing the photoresist is stripped, so that final dampingofthenanobeam,thesamplechipandthefiber
release steps can be carried out. couplingsetup(basedonanAttocubestack)areembed-
ded in a vacuum chamber operating at <10−5 mbar.
2. Nanobeam and microdisk
B. Thermal noise measurement
Toreleasethenanobeamandundercutthemicrodisk,
Mechanical mode frequencies Ω , damping rates Γ ,
m m
the partially diced wafer is immersed in 40% KOH at
and optomechanical coupling rates g , were determined
45◦C, selectively removing poly-Si but also etching Si. 0
by analyzing the cavity resonance frequency noise pro-
The etch time is fine-tuned with two opposing criteria
ducedbythermalmotionofthenanobeam. Anin-depth
in mind: (1) to ensure that the microdisk is undercut
description of this method is given in [62]. Important
sufficiently far from its rim to avoid optical losses and
details are recounted below for clarity.
(2) to ensure that Si underneath the nanobeam clamp-
Thermalmotionofthenanobeamx(t)iswrittenonto
ing point is not etched away. After KOH etching, the
the cavity resonance frequency ω (t) via their optome-
c
wafer is rinsed in water and any remaining potassium
chanical coupling, G = dω /dx. To measure ω (t), we
c c
is neutralized in a bath of hydrochloric acid. Organic
monitor the power of the transmitted field while oper-
cleaningisthenperformedusinganexothermicmixture
ating at a fixed detuning of |∆|≈κ/2. Referred to the
ofthreepartssulfuricacidtoonepart30%hydrogenper-
outputvoltage(V)ofthephotodetectortransimpedance
oxide (a “piranha etch”). After rinsing again, the wafer
amplifier, the uncalibrated noise spectrum can be ex-
is transfered directly to the ethanol bath of a critical pressed as S (Ω) = |G (Ω)|2S (Ω), where G (Ω) is
V Vω ω Vω
point drying (CPD) machine. After CPD, the wafer is
themeasurementtransferfunctionandS (Ω)istheap-
ω
broken into sample chips along the partially diced lines,
parent cavity frequency noise. G (Ω) is calibrated by
Vω
concluding the fabrication process.
applyingaphasemodulationtoneofknowndepth(β )
cal
and frequency (Ω ) to the input, resulting in a narrow
cal
spectral peak with area |G (Ω )|2β2 Ω2 /2 [62].
Vω cal cal cal
A representative measurement is shown in Fig. 9c.
Red,blue,andblackcomponentscorrespondtothermal
noise, Sth(Ω), the calibration tone, Scal(Ω), and mea-
ω ω
surement imprecision, Simp(Ω), respectively. The full
ω
signal can be modeled as
S (Ω)=Sth(Ω)+Scal(Ω)+Simp(Ω) (3a)
ω ω ω ω
≈2g2n ·L(Ω−Ω ) (3b)
0 th m
FIG. 8. Defining the nanobeam and the ”mesa”. (a) + βc2alΩ2cal ·G(Ω−Ω )+Simp(Ω), (3c)
TopviewofsampleafteretchingofSi N (pinkandpur- 2 cal ω
3 4
ple). Surrounding SiO2 structures, including microdisk, where L(Ω) = 4Γ /(Γ2 + 4Ω2) is a normalized
appear green. (b) Image of the “mesa” photomask. m m
Lorentzian(characterizingthemechanicalsusceptibility)
9
Ω
a cal vacuum chamber
~ nanobeam
spectrum
~ analyzer
µ-disk
AOM
-
λ EOM oscilloscope
taper
laser
fiber loop cavity
b c
1.0 Γ
n m
o κ 2π
si
mis 0.9 2π Hz]
ns γ 2z/
d tra 0.8 2π 2) [H (∆FM2/2π)2
e π
z 2
ali /(ω
m 0.7 S 2n (g/2π)2
th 0
r
o
N
0.6
1000 500 0 500 1000
Laser-cavity detuning [MHz] Frequency [MHz]
FIG. 9. (a) Overview of the experimental apparatus, described in Sec. IVA. (b) Representative optical Q measure-
ment. WGM loss rates (κ) and mode splitting (γ) are inferred from the cavity transmission profile (red), generated
by sweeping the diode laser frequency while monitoring the transmitted power. The sweep is calibrated by simul-
taneously monitoring transmission through a fiber loop cavity (blue). (c) Representative thermomechanical noise
measurement. Ω ,Γ ,andg areinferredfromthecenterfrequency,linewidth,andareabeneaththethermalnoise
m th 0
peak (pink), respectively. The latter is calibrated by normalizing to the area beneath a FM tone (blue).
√
and G(Ω) = e−Ω2/(2B2)/ 2πB2 is a normalized Gaus- A measurement of the optical spring effect is shown
sian (characterizing the window function of the spec- in Fig. 10, corresponding to the sample also character-
trum analyzer, which is assumed to have a resolution izedinFig.9c. Theinjectedpowersused—P =60,120
bandwidth B (cid:28) Γ ). To calibrate the vertical axis in nW — were chosen to avoid instabilities due to pho-
m
Fig. 9c, it is assumed that |G (Ω )| ≈ |G (Ω )|. tothermal/radiation pressure damping. The cavity was
Vω m Vω cal
Fitting the calibrated spectrum to Eq. 3 gives Ω , Γ , critically coupled (κ ≈ κ/2 ≈ 2π · 550 MHz) and
m m ex
and g . The last inference requires knowledge of n . laser detuning was estimated from the mean transmit-
0 th
Byusinginputpowerslowenoughtoneglectphotother- ted power. Overlaid models correspond to Eq. 4 with
mal/radiation pressure damping (∼10 nW), we assume thevalueg =2π·60kHz,inferredfromaleast-squared
0
that n ≈k ·295K/((cid:126)Ω ). fit to the low power measurement. This value is within
th B m
10% of that inferred from thermal noise in Fig. 9c.
C. Optical spring effect
D. g and C versus lateral beam position
0 0
As a cross-check of the thermal noise measurement,
g0 was independently estimated from the optical spring As discussed in Sec. IIC, g0 depends sensitively on
effect [11]. In the experimentally relevant bad cavity the lateral positioning of the nanobeam, and assumes a
limit(Ωm (cid:28)κ),themechanicalfrequencyshiftproduced maximum (minimum) value for out-of-plane (in-plane)
by a radiation pressure optical spring is flexuralmodeswhencenteredabovetheWGM.Thisbe-
8g2 ∆/κ havior was studied by sweeping the lateral position of
∆Ωm(∆)≈ κ0 ·nc(∆)· 1+4(∆/κ)2 (4) thebeamusinganappropriatee-beammask(Sec.IIIC).
Measurements of g versus lateral beam position are
0
where ∆ is the laser-cavity detuning, n (∆) = shown in Fig. 11a. (In-plane modes exhibit typically
c
(4P/((cid:126)ω κ))(κ /κ)/(1 + 4(∆/κ)2) is the intracavity 10× lower g , and were not considered.) for beam and
0 ex 0
photon number, and P is the power injected into the disk dimensions of {l,w,t} = {60,0.4,0.06} µm and
cavity. (We note that radiation pressure damping also {r,t ,θ} = {15µm,0.60µm,30deg.}, respectively, and
d
occursforadetunedinputfield; however,inthedevices for a vertical gap of 25 nm. In agreement with numer-
studied,forwhichΩ /κ∼0.01,thiseffectwasfoundto ical modeling (dashed line), g assumes a maximum of
m 0
be overwhelmed by photothermal damping [64].) 2π·40 kHz as the outer edge of the beam eclipses the
10
gests that κ is ultimately dominated by beam-induced
1012 ∆Ωm scattering/absorption loss, rather than deterioration of
1010 ∆ > 0 intrinsic disk loss (Fig. 4), implying that an additional
u.] 108 10-fold reduction in κ may yet be realized with appro-
[a.V 106 priate beam shaping/positioning.
S 104 ∆ < 0
102
100 E. g0 and C0 versus beam width and disk
thickness
-600 -400 -200 0 -200 400 600
(Ω−Ω )/2π [Hz]
m
Widerbeams(w∼λ)andthinnerdisks(t <λ)were
d
600
fabricatedinanattempttoincreaseg andC (seeEq.2).
0 0
400 Measurements of {g0,C0} vs w for two microdisk thick-
Hz] 200 Fneixsseeds,dtidme≈ns0io.4n3s oafndth0e.6n3anµomb,eaamreasnhdowmnicirnodFiisgk. a1r1e.
[m 0 {t,l} ≈ {0.06,60} µm and {rd,θ} ≈ {15 µm,30 deg.},
Ω
∆ 200 respectively. The lateral beam position was chosen to
maximizeg forthe0.4µm-widebeam(seeFig.11). For
P= 120 nW 0
-400 the TE optical modes studied, a roughly 2× increase in
P= 60 nW
g wasobservedforthe30%thinnerdisk. Inbothcases,
-600 0
g scaled roughly linearly for widths w ∈ [0.4,1] µm.
2 -1 0 1 2 0
C also increased with w, roughly in proportion to g2,
∆/κ 0 0
for both t . This is due to the fact that κ (not shown)
d
FIG.10. Opticalspringmeasurement. (a)Thermalnoise was roughly independent of w for both disk thicknesses
spectrum of the fundamental beam mode as a function and a factor of four larger for the thinner disk. The
of laser detuning. Blue and red spectra indicated blue highestoptomechanicalcouplingratewehavemeasured,
(∆>0)andred(∆<0)detuning,respectively. Lighter g0 ≈ 2π·150 kHz, was for a 1 µm-wide beam coupled
shadesindicatesmallerdetuning. Bluespectraareverti- toa0.43µm-thickdisk. Thehighestcooperativities ob-
callyoffset. (b)Plotofopticalspringshift,∆Ωm,versus served, C0 > 2.5, were for 1 µm-wide beams coupled to
normalized detuning, ∆/κ. Dashed black lines are a fit disks of both thicknesses.
to Eq. 4 using g as a free parameter.
0
rim of the disk.
Also shown in Fig. 11b are measurements of κ versus F. g0 versus mechanical mode order
lateral beam position (y). When the beam is displaced
far from the disk, κ converges to the intrinsic value of
g wasalsostudiedforhigherordermechanicalmodes.
0
∼2π·100MHzobservedinFig.4,suggestingthatCMP
As shown in Fig. 11d , g decreases as the vibrational
0
didnotsignificantlyaffectmicrodisksurfacequality. As
node spacing approaches the dimensions of the effective
the beam is brought within 100 nm of the disk, κ is
sampling length l . In this case the model in Sec. IIC
eff
observed to increase sharply. The observed exponential
— which assumes rigid displacement of a beam with ef-
dependence κ on y is independent of mode polarization
fective mass m = ρtwl/2 — breaks down. A simple
and similar to the scaling observed in [35] with a beam
extensionofthemodelisshownasaredlineinFig.11d.
coupledtoamicrotoroid. Theabsolutemagnitudeofthe
Here m is computed with respect to optical-intensity-
loss is also inconsistent with bulk Si N optical absorp-
3 4 weighted displacement of the mechanical mode:
tion — specifically, accounting for the relatively small
fraction of energy stored in the beam, the observed loss
would require an imaginary index of ∼ 10−4, which is (cid:82) ρ|u(r)|2d3r
m= (cid:82) beam (cid:82) (5a)
1-2 orders of magnitude larger than conventionally ob- | |E(r)|2u(r)d3r/ |E(r)|2d3r|2
beam beam
served for Si N at NIR wavelengths [54, 65]. We thus
3 4 ρtwl 1
cwoanvjeegcutuidreetchoautpltihnigslionstsoitshdeubeetaoms.catteringfromand/or ≈ 1−(−1)nsinc2(cid:16)n2πlelff(cid:17) (5b)
Combining measurements of g and κ with typical
0
room temperature mechanical damping rate of Γ =
m where (cid:126)u(x,y,z) ≈ sin(nπx/l)zˆ is the displacement pro-
2π·15Hz(weobservednochangeinΓ forsmallbeam-
m file of the nth-order out-of-plane flexural mode. The
disk seperation, suggesting that squeeze-film gas damp-
latter expression is appropriate when the transverse di-
ing [52] was not a factor), the single-photon cooperativ-
mensions of the beam are much smaller than that of
ityisobservedtoapproachC ∼1. Thisvalueislimited
0 the WGM, and assumes that the intensity distribution
by the unfavorable scaling of g2/κ as g begins to satu-
0 0 sampled by the beam is uniform along the beam axis
rate. Despite this limitation, the inferred C represents
0 with an effective sampling length l . Using Ω ∝ n
a nearly 50 dB increase over our prior chip-scale imple- (cid:16) (cid:17) √ eff m
mentation [34], owing to the combined 100-fold increase givesg0(n)/g0(0) ≈|sinc n2πlelff |/ nforoddnand0for
ofg and10-foldreductioninκ. Increaseg isduetothe even n. The model shown in Fig. 11d agrees quantita-
0 0
precise vertical and lateral positioning of the beam af- tively with experiment assuming an effective length of
fordedbyCMPande-beamprocessing. Reducedκisdue l =9.6µmastheonlyfreeparameter. Asimpleroute
eff
to greater isolation of the disk during beam patterning, toincreasingg wouldbetoremovemassfromthebeam
0
making use of the poly-Si sacrificial layer. Fig. 11b sug- outside of the effective sampling length (see Fig. 13).