*
e-mail:
roukes@caltech.edu
Overview
In this document we present a theoretical
exposition of how bifurcation topology can be
controlled for a coupled pair of
parametrically-driven
nonlinear resonators. We then address the
effect of noise on the probability that the system accurately follows the topology of the
bifurcation. We subsequently provi
de further details about our e
xperimental methods and on an
apparatus for realizing a coupled-NEMS BTA. Fi
nally, we assess the e
ffective noise in our
current implementation, and conclude by empl
oying our noise analysis
to make realistic
projections of the ultimate sensitivity limits of this implementation.
Supplementary information
:
Signal Amplification by Sensitive Control of
Bifurcation Topology
R.B. Karabalin, Ron Lifshitz
2*
, M.C. Cross
1
, M.H. Matheny
1
, S.C. Masmanidis
1
,
and M.L. Roukes
1†
Kavli Nanoscience Institute and Condensed Matter Physics 114-36, California Institute of Technology,
Pasadena, CA 91125, USA
2
Raymond and Beverly Sackler School of Physics & Astr
onomy, Tel Aviv University
, Tel Aviv 69978, Israel
*
To whom correspondence on matters of theory should be addressed. E-mail:
ronlif@tau.ac.il
(RL).
†
To whom correspondence on matters of experiment should be addressed. E-mail:
roukes@caltech.edu
(MLR).
Supplementary information:
“Signal amplification by sensitive control of bifurcation topology”
R.B. Karabalin
et al.
-2-
Theoretical Background
Response of Coupled Nonlinear Reso
nators to Parametric Excitation
We begin by providing the calculat
ion of the theoretical response cu
rves that are plotted in Fig. 1
of the main text. We consider two weakly-cou
pled parametrically-driven nonlinear resonators
with slightly different normal frequencies.
Their dynamics are governed
by a pair of coupled
equations of motion (EOM)
()
()
0
~
~
~
~
~
~
~
~
~
~
~
cos
~
~
~
~
~
~
2
3
2
2
2
=
−
+
+
+
+
+
+
±
±
±
±
±
±
±
±
∓
x
x
D
t
d
x
d
x
x
x
t
h
t
d
x
d
t
d
x
d
p
η
α
ω
ω
γ
, (S1)
where all physical parameters—after having divi
ded out the effective ma
ss of the resonators—
are denoted with tildes to distinguish them fr
om the scaled parameters used below. Here
+
x
~
denotes the displacement of the higher-
frequency resonator with frequency
+
ω
from its
equilibrium, and
−
x
~
denotes the displacement from equili
brium of the lower-frequency resonator
with frequency
−
ω
. The Duffing parameter
α
, the linear damping rate
γ
~
, and the coefficient of
nonlinear damping
η
~
are all assumed to be approximately
the same for both resonators, and the
coupling strength between the resonators is denoted by
D
~
. The parameters
h
~
and
p
ω
~
are the
parametric driving amplitude
and driving frequency.
We rescale the units of time and space, to eliminate two additional parameters from the equation
of motion—the average resonance
frequency of the resonators, and the Duffing parameter, which
are both set to 1. Because we
drive the system close to twice
the average resonance frequency,
we express the scaled pump frequency as
2
p
p
ωω
=+Δ
. The EOM then becomes
Supplementary information:
“Signal amplification by sensitive control of bifurcation topology”
R.B. Karabalin
et al.
-3-
()
()
132
1cos2
0
p
xQx
H
txx xxDxx
ωη
−
±±
±±±±±
⎡⎤
++±Δ+ +Δ ++ +−=
⎣⎦
∓
, (S2)
where dots denote derivatives with re
spect to the dimensionless time
t
,
Q
is the quality factor of
the resonators, and
1
<<
Δ
is the scaled frequency difference between the resonators, so that
2
±
ω
are replaced by
Δ
±
1
.
We calculate the response of the coupled resona
tors following the methods of Lifshitz & Cross
(
1,2
). We begin by assuming that the linear
damping is weak or,
equivalently, that
Q
is large,
and define a small expansion parameter
ε
, by expressing the scaled linear damping rate
as
εγ
=
−
1
Q
, with
γ
of order unity. The parametric instabil
ity of the system then occurs for small
driving amplitudes on the order of
ε
near resonance. If, in additi
on, we consider the system near
the onset of the instability, we can assume that th
e effects of nonlinearity
are small as well. Since
the coupling strength is the weak signal to be amp
lified by the BTA, it can also be considered as
a small perturbative correction.
Finally, the frequency difference
between the two
resonators can
also be taken to be small, on the order of
ε
. All these perturbative corr
ections can be chosen to
enter the EOM in the same order of the small parameter
ε
by taking the leading order in
±
x
to be
ε
, expressing the scaled para
metric driving amplitude as
h
H
ε
=
, expressing the scaled
frequency difference as
ε
δ
=
Δ
, expressing the scaled coupling constant as
d
D
ε
=
, and driving
the system close to twice the average resonance, taking
p
ωε
Δ=Ω
. The final form of the EOM is
then
[]
()()
0
2
cos
1
2
3
=
−
+
+
+
Ω
+
+
±
+
+
±
±
±
±
±
±
±
∓
x
x
d
x
x
x
x
t
h
x
x
ε
η
ε
ε
εδ
εγ
. (S3)
Supplementary information:
“Signal amplification by sensitive control of bifurcation topology”
R.B. Karabalin
et al.
-4-
Expecting the motion of the resonators away from equilibrium to be on the order of
ε
we try a
solution of the form
()
()
()
()
...
+
+
+
=
±
±
±
t
x
c
c
e
T
A
t
x
it
)
1
(
2
3
.
.
2
ε
ε
. (S4)
The lowest order contribution to this solution is
based on the solution to
the linear equations of
motion of the two simple harmonic oscillators
0
=
+
±
±
x
x
, where
t
T
ε
=
is a slow time
variable, allowing the complex amplitudes
()
T
A
±
to vary slowly in time, due to the effect of all
the perturbative terms in Eq. (S3). Fo
llowing the methods of Lifshitz & Cross
1,2
we obtain a pair
of coupled equations for determining the amplitudes
()
T
A
±
,
()
0
4
3
2
2
2
=
+
+
+
+
±
−
+
±
±
±
Ω
∗
±
±
±
±
A
A
i
A
i
e
A
h
A
A
A
d
dT
dA
i
T
i
η
γ
δ
∓
. (S5)
The explicit time dependence can be rem
oved by taking a solution of the form
()
()
T
i
e
T
a
T
A
2
Ω
±
±
=
, (S6)
yielding a corresponding equation for
()
T
a
±
,
∗
±
±
±
±
−
=
⎟
⎠
⎞
⎜
⎝
⎛
+
+
−
⎟
⎠
⎞
⎜
⎝
⎛
+
Ω
−
±
+
a
h
a
a
i
da
a
i
d
dT
d
i
2
4
3
2
2
η
γ
δ
∓
. (S7)
With the expression (S6) for the slowly varying amplitudes
()
T
A
±
, the steady-state solution to
the scaled equations of motion (S2), for which the complex amplitudes
±
a
are constant in time,
becomes an oscillation at half the drive frequency
1
+
ε
Ω
2
. Note that we are not interested in
Supplementary information:
“Signal amplification by sensitive control of bifurcation topology”
R.B. Karabalin
et al.
-5-
the corrections
x
±
(1)
t
()
of order
2
3
ε
to these oscillations, but rath
er in finding the fixed complex
amplitudes
±
a
of the lowest order terms. These are
obtained by solving the coupled algebraic
equations, obtained from
(S7) by requiring that
0
/
=
±
dT
da
. Setting
0
=
d
decouples the
equations, giving two equations that can be solved in closed from (
1,2
). The solution to a single
equation gives the response curve, shown in Fig.
1a in the main text. The sum of both solutions,
taking into account the relative
π
phase freedom, is plotted in Fig. 1b. For finite coupling
0
≠
d
we can find the roots of the coupled Eqs. (S
7) numerically once values are chosen for the
different parameters. These are shown in Fi
gs. 1c and 1d for positive and negative coupling
respectively. In all plots we use the values
1
=
γ
,
01
.
0
=
η
,
1
.
1
=
δ
, and
05
.
1
=
h
. In 1b we
take
0
=
d
, and in 1c and 1d we take
01
.
0
±
=
d
.
Amplitude Equation for an Imp
erfect Pitchfork Bifurcation
We wish to consider in more detail what happe
ns at the bifurcation
upon an upward frequency
sweep, as the second mode starts oscillating. In
the presence of weak c
oupling the normal modes
are slightly modified from pure
motion of the individua
l resonators. Diagonalization of the linear
terms in (S7), keeping only corrections of order
δ
/
d
, yields the modified modes
()
∑
=
−
+
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎝
⎛
−
=
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎝
⎛
=
2
1
)
(
)
2
(
)
1
(
with
,
2
1
,
1
2
n
n
n
T
a
a
a
d
d
v
v
v
δ
δ
, (S8)
where the mode frequencies are
unchanged to first order in
δ
/
d
. Substitution into Eq. (S7)
yields a set of nonlinearly-coup
led equations of motion for th
e modified mode amplitudes
()
T
a
n
,
Supplementary information:
“Signal amplification by sensitive control of bifurcation topology”
R.B. Karabalin
et al.
-6-
∗
∗
−
=
⎥
⎦
⎤
⎢
⎣
⎡
⎟
⎠
⎞
⎜
⎝
⎛
−
−
+
⎟
⎠
⎞
⎜
⎝
⎛
+
+
⎟
⎠
⎞
⎜
⎝
⎛
+
Ω
−
−
1
2
2
1
2
2
1
2
2
2
1
2
1
1
2
2
2
1
2
1
4
3
2
2
a
h
a
a
a
a
a
a
d
a
a
i
a
i
dT
d
i
δ
η
γ
δ
, (S9)
∗
∗
−
=
⎥
⎦
⎤
⎢
⎣
⎡
⎟
⎠
⎞
⎜
⎝
⎛
+
+
−
+
⎟
⎠
⎞
⎜
⎝
⎛
+
+
⎟
⎠
⎞
⎜
⎝
⎛
+
Ω
−
2
1
2
2
1
2
2
1
2
1
2
2
2
2
2
2
2
1
2
1
4
3
2
a
h
a
a
a
a
a
a
d
a
a
i
a
i
dT
d
i
δ
η
γ
, (S10)
where we now measure the drive frequency with
respect to the second mode frequency by
defining
()
δ
+
−
Ω
=
Ω
d
2
.
Our aim is to obtain an equation for the growth
of the amplitude of the second mode at the
bifurcation, as the frequency is swept upward. At that point the first mode will have already
reached a certain non-zero amplitude, which can be determined analytically
(
2
) by solving (S9)
after setting
0
2
=
a
. To find the initial growth of th
e second mode we linearize (S10) in
2
a
.
Taking the coupling
δ
/
d
to be weak, and assuming an initial growth of the form
ℜ
∈
=
⎟
⎠
⎞
⎜
⎝
⎛
+
σ
φ
σ
φ
π
,
,
with
,
4
2
a
e
ae
a
T
i
, (S11)
we find that the phase
φ
and the growth rate
σ
satisfy the relations
.
2
sin
2
0
,
2
cos
2
2
2
φ
φ
γ
σ
h
h
+
Ω
=
+
−
=
(S12)
Thus, the bifurcation occurs
as the drive frequency
2
Ω
is increased and reac
hes a critical value
of
()
2
2
2
/
γ
−
−
=
Ω
h
C
.
Supplementary information:
“Signal amplification by sensitive control of bifurcation topology”
R.B. Karabalin
et al.
-7-
Next, we wish to include nonlinearity to saturate
the growth of the second
mode, and to include
the coupling to the first mode,
which is already oscillating,
to affect the topology of the
bifurcation. Performing a calculati
on similar to that found in sect
ion 1.3.3 of Lifshitz & Cross
(
2
), we find that
,
)
(
4
2
⎟
⎠
⎞
⎜
⎝
⎛
+
=
φ
π
i
e
T
B
a
(S13)
where the real-valued saturated amplitude
)
(
T
B
satisfies the equation
()
⎟
⎠
⎞
⎜
⎝
⎛
−
+
+
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
Ω
−
Ω
Ω
−
Ω
=
4
ˆ
sin
32
9
3
8
1
2
1
3
1
2
3
2
π
φ
φ
γ
η
δ
η
γ
γ
a
h
d
B
B
dT
dB
C
C
C
, (S14)
where
1
a
is the amplitude of the first mode determined earlier, and
1
ˆ
φ
is the phase
1
φ
of the
first mode plus
()
3
/
arctan
η
. It can be shown that for weak nonlinear damping (
1
<<
η
),
4
/
ˆ
1
π
φ
φ
+
≅
.
Scaling back to the parameters of
Eq. (S2), and dropping terms of order
2
η
we find that at the
bifurcation the modes [with eigenvector
s given by (S8)] are oscillating as
()
()
()
.
4
2
1
cos
,
2
1
cos
2
2
1
1
1
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
+
⎥
⎦
⎤
⎢
⎣
⎡
Δ
+
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
⎥
⎦
⎤
⎢
⎣
⎡
Δ
+
=
φ
π
ω
φ
ω
t
t
X
t
x
t
X
t
x
p
p
(S15)
The amplitude of the second mode
()
t
X
2
satisfies the equation
Supplementary information:
“Signal amplification by sensitive control of bifurcation topology”
R.B. Karabalin
et al.
-8-
[]
()
()
3
1
3
2
2
2
16
3
3
8
1
2
X
Q
D
X
Q
X
Q
D
dt
dX
C
C
C
C
p
ω
η
ω
ω
ω
ω
Δ
+
+
−
−
Δ
+
−
Δ
=
, (S16)
which has the typical form of an im
perfect pitchfork bifurcation, with
p
ω
Δ
acting as the control
parameter, and where
1
2
2
−
⎟
⎠
⎞
⎜
⎝
⎛
=
QH
Q
C
ω
.
Analysis of the Effects of Noise
Noise in the system will have its largest ef
fect on the measurement process as the pump
frequency passes through the pitchfork bifurca
tion of the second mode. The dynamics in this
vicinity can be analyzed usi
ng Eq. (S16) for the amplitude of
the second mode, supplemented
with a noise term. For small signal and noise the important time range is when
X
2
is small, so
that the nonlinear term in Eq. (S16) is not invo
lved. We write the linearized Eq. (S16) in the
form
)
(
2
2
t
f
s
rtX
dt
dX
+
+
=
(S17)
where
rt
is the linear ramp of control para
meter (the parameters in front of
X
2
in the first term on
the right hand side of Eq. (S
16)) choosing to measure time
t
from the bifurcation point,
s
is the
term leading to the imperfect bifurcation proportional to the coupling
D
(the last term in Eq.
(S16)), and
f(t)
is the noise force term, assumed to
be Gaussian white noise of strength
F
defined
by
)
'
(
2
)
'
(
)
(
t
t
F
t
f
t
f
−
=
δ
. (S18)
Supplementary information:
“Signal amplification by sensitive control of bifurcation topology”
R.B. Karabalin
et al.
-9-
The conventional force spectral density
F
S
is related to this noise strength by
F
S
F
4
=
. The
Fokker-Planck equation for th
e probability distribution
P(X
2
)
of
X
2
at time
t
corresponding to Eq.
(S17) can be solved to give the Gaussian distribution
)
(
2
/
)]
(
[
)
(
2
1
2
2
2
2
)
(
t
t
X
X
t
D
e
X
P
σ
σ
π
−
−
=
, (S19)
where
)
(
t
X
D
is the deterministic solu
tion, given by equation (S17)
without the noise term, and
)
(
t
σ
is the time dependent width.
)
(
t
X
D
grows away from zer
o due to the signal
s
. The explicit
expressions, assuming the control parameter ramp
starts at a value far below the bifurcation
point, are
'.
2
)
(
,
'
)
(
)
'
(
2
2
/
)
'
(
2
2
2
2
dt
e
F
t
dt
e
s
t
X
t
t
t
r
t
t
t
r
D
∫
∫
∞
−
−
∞
−
−
=
=
σ
(S20)
For long times
2
/
1
−
>>
r
t
these expressions give
.
4
)
(
,
2
)
(
2
/
2
/
1
4
/
1
2
/
2
/
1
2
2
rt
rt
D
e
F
r
t
e
s
r
t
X
⎟
⎠
⎞
⎜
⎝
⎛
=
⎟
⎠
⎞
⎜
⎝
⎛
=
π
σ
π
(S21)
Note the super-exponential
growth of both the center and widt
h of the distribution due to the
increasing amplification rate, proporti
onal to the bifurcation parameter
rt
. We now calculate the
probability
↑
P
that at long times
X
2
falls in the basin of attraction of the
↑
branch (and
↑
↓
−
=
P
P
1
). The shift
rt
s
/
−
in the basin boundary away from
X
2
=0
is small compared with
X
D
for
2
/
1
−
>>
r
t
, and so
∫
∞
↑
≈
0
2
2
)
(
dX
X
P
P
giving
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎟
⎠
⎞
⎜
⎝
⎛
=
⎟
⎠
⎞
⎜
⎝
⎛
≈
−
↓
↑
2
/
1
4
/
1
4
/
1
4
erf
2
erf
F
r
s
X
P
P
D
π
σ
. (S22)
Supplementary information:
“Signal amplification by sensitive control of bifurcation topology”
R.B. Karabalin
et al.
-10-
Note that the expression for
↓
↑
−
P
P
is independent of time, and
we may choose any time for its
evaluation that is long
enough compared with
2
/
1
−
r
so that the approximations we have made are
good, but short enough so that the
nonlinear term in the evolutio
n equation is not yet important.
Such a time always exists for the limit of small
signal and noise of intere
st. Also note that the
effective bandwidth for the noise appearing in the
“signal to noise” ratio in the argument to the
error function in Eq. (S22) is
2
/
1
r
and is determined by the frequency ramp-rate.
As we show below in Eq. (S27), th
e BTA output signal is proportional to
↓
↑
−
P
P
. For the limit of
small signals Eq. (S22) reduces to
2
/
1
4
/
1
4
/
1
4
F
r
s
P
P
⎟
⎠
⎞
⎜
⎝
⎛
≈
−
↓
↑
π
. (S23)
For M sweeps through the bifurcation, the di
stribution of the fraction of up traces follows
Poisson statistics, which for large M
reduces to a Gaussian with mean
↑
P
, and standard deviation
M
2
/
1
for
↑
P
close to 1/2. For a sweep rate
π
ω
ν
2
/
sw
sw
=
, this leads to the error estimate for
the measurement of the signal
s
of
()
Hz
/
4
/
2
/
1
sw
F
S
r
ν
π
.
The noise term
f(t)
in Eq. (S17) ultimately derives from physical noise forces on the beams
)
~
(
~
t
f
±
with
)
~
~
(
~
2
)
~
(
~
)
~
(
~
t
t
F
t
f
t
f
′
−
=
′
±
±
±
δ
δ
leading to terms
m
t
f
/
)
~
(
~
±
on the right hand side of
Eq. (S1). Proceeding through the transformations as
in Eqs. (S2)-(S16) l
eading to the evolution
equation for
X
2
, but now including this noise
term, relates the force strength in Eq. (S17) to these
fundamental forces. For small beam coupling, the
dominant noise source is
just from the higher
frequency beam, and then we find
F
QH
m
F
~
2
)
(
2
1
2
3
2
0
⎟
⎠
⎞
⎜
⎝
⎛
=
ω
α
. (S24)
Supplementary information:
“Signal amplification by sensitive control of bifurcation topology”
R.B. Karabalin
et al.
-11-
Fabrication and Methods
Device Fabrication: Coupled-NEMS BTA
Fabrication is based on a GaAs-based multilayer grown by molecular beam epitaxy (MBE) upon
a GaAs substrate. The device structural la
yer itself is a 200nm thick multilayer structure,
comprising a stack of th
ree layers forming a vertically-oriented (
i.e.
out of the wafer plane)
p-i-n
diode: a topmost 100 nm
n
-GaAs (10
19
cm
-3
) layer, a 50 nm
i
-GaAs layer, and a 50 nm
p
-GaAs
(10
18
cm
-3
). The
i
-GaAs layer has a
p
-type background concentration of ~5x10
15
cm
-3
arising
from natural impurities, which is negligible comp
ared to the intentionally doped regions. These
structural
p-i-n
diode layers are grown on a sacrificial
p
-Al
0.8
Ga
0.2
As (10
18
cm
-3
) layer, which in
turn is grown on a
p
-doped (10
18
cm
-3
) GaAs (001) substrate by MBE.
The lead frame structure
and wire bond pads are patterned
by photolithography. This is
followed by deposition of a thin
~5nm Ti adhesion layer and a 50nm Au layer, subs
equently standard liftoff is employed. The
backside of the
p
+
-doped wafer is coated with Ti/Au in
order to provide a bottom electrical
contact. The NEMS devices
themselves are defined by elec
tron beam lithogra
phy, which is
followed by deposition of a 60nm Ti layer and liftoff.
This Ti mask layer protects the desired
structural regions during a dry etch using ar
gon ion-beam milling to a depth of 250 nm.
Subsequently, the patterned devices are
suspended by removing the sacrificial Al
0.8
Ga
0.2
As layer
using a selective, wet chemical etch in dilute hydr
ofluoric acid. This st
ep also removes the Ti
masks, exposing the Au
electrodes and pads.
Coupling
Two different physical phenomena contribute to
the coupling between beams: elastic mechanical
coupling mediated through the substrate, and an
electrostatic dipole-di
pole interaction between
Supplementary information:
“Signal amplification by sensitive control of bifurcation topology”
R.B. Karabalin
et al.
-12-
adjacent
p-i-n
diode
structures in the
voltage-biased beams.
We discuss each
mechanism in turn;
they are pictorially
depicted in Figure S1.
Electrostatic Coupling
. When a DC bias voltage is appl
ied to both beams, charges of opposite
sign accumulate on their top and bottom surfaces forming dipole moments. Two such identical
dipoles interact electrostatically with a force
(
3
)
)
(
)
(
4
9
2
1
2
0
5
2
0
0
x
x
t
t
V
d
A
F
i
m
bias
beam
dipoles
−
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
−
=
ψ
ε
ε
πε
, (S25)
where
d
is the distance between the beams,
A
beam
is a beam’s surface area
L
×
w
,
x
1
and
x
2
are
the out-of-plane displacements at the center of the beams,
ψ
0
≈
1.2V is a built-in potential,
t
i
= 50nm is the thickness of
the insulating layer, and
t
m
≈
78nm is the depletion width of the
p-i-n
diode (
4
).
This coupling has a simple linear form with
a negative coefficient, meaning a repulsive
interaction ensues. This arises from the fact th
at, with the same voltage
applied to both beams,
dipole moments of the same orientation develop in
the beams. For the geometry of the devices
used in these experiments (
d=
400nm,
A
beam
= L
×
w =
6
μ
m
×
0.5
μ
m, and
V
offset
≈
1.1V ), a force
of approximately
−
1pN is generated from a 1nm
difference in displacements.
Figure S1.
We model the interactions be
tween beams as arising from
two mechanisms: elastic coupling through the substrate (right) and
electrostatic dipole-dipole interaction (left). Device geometry is
designed so that the coupling forces are of the same order.
Supplementary information:
“Signal amplification by sensitive control of bifurcation topology”
R.B. Karabalin
et al.
-13-
Mechanical Coupling
. Without any applied voltage ther
e is residual attraction between the
beams due to elastic coupling
through their shared elastic support, even though no ledge is
shared by the beams. Finite element numerical si
mulations are used to estimate the magnitude of
elastic coupling mediated through th
e substrate, the displacement
colormap of the mechanically
interacting beams is shown in Fi
gure S1. We find the interaction to be attractive and linearly
dependent on displacement difference. We design
the geometry of the system so that at the
difference in displacements of ~1nm, the effec
tive mechanical coupling force is approximately
~1pN, as a result the dipole–dipole interact
ion compensates the elastic coupling within
experimentally accessible voltage range.
Summary of Measurement Apparatus and Methods
Samples are mounted in a room temperature
vacuum chamber, which is pumped down to a
typical pressure of 5 mTorr for experiments. This chamber is fitted with a transparent sapphire
optical window to enable optical interferometry
, however to minimize the impact of spurious
light on device performance (for example, due
to heating and inadvertent generation of
photocarriers) we place a neutral
density filter with a 10-fold ex
tinction factor in front of the
optical port. Illumination is provided by an infrared laser diode emitting 2 mW at 904 nm. The
laser is focused to a spot of ~10
μ
m in diameter upon the device. Th
e reflected signal is detected
by a low-noise, high-bandwidth photoreceiver (N
ew Focus 1801, bandwidth=125 MHz, optical
noise power spectral density = 30 pW/
√
Hz, referred to input).
Supplementary information:
“Signal amplification by sensitive control of bifurcation topology”
R.B. Karabalin
et al.
-14-
We calibrate the displacement response of
the interferometer using the known
amplitude at the onset of nonlinear response
for doubly-clamped beams, which arises
from the Duffing instability. Direct
frequency response measurements for the
coupled resonators ar
e made using an RF
vector network analyzer (Hewlett Packard
3577A) as shown in Figure S2. The
amplitudes of the two peaks are different
due to slight variations in their
actuation efficiencies. The family
of curves displayed represents
drive amplitudes from 50mV
rms
to 1.2V
rms
. The onset of nonlinea
rity occurs for a ~600mV
rms
drive level, which yields an op
tical signal of approximately 70
μ
V
rms
. The estimated accuracy of
this calibration is of order 10%.
We have configured the sample geometry and e
xperimental apparatus so
that the laser spot
illuminates both NEMS resonators simultaneously. In this case the output from the
photodetector represents the summed contribution fr
om the coupled beams. As described in the
main text, the response to a parametric pump si
gnal in the 26MHz range is for both beams to
become excited when the pump is roughly twice th
eir natural resonance fr
equency. Depending
on the sign of
V
in
t
()
the pump induces coupled vibrations
that are either in
-phase (yielding
strong optical reflection) or out-o
f-phase (giving a weak
optical response).
Parametric frequency
sweeps are measured with
a spectrum analyzer (Agilent 4395A).
13.08
13.12
13.16
13.20
13.24
0
5
10
15
20
|a
1
+a
2
| (nm)
Frequency (MHz)
Figure S2.
Duffing nonlinearities dominate at
high excitation levels. The onset of nonlinear
behavior depends only on quality factor and
geometry of the beam, hence it provides a effective
way of calibrating the optical interferometer
responsivity.
Supplementary information:
“Signal amplification by sensitive control of bifurcation topology”
R.B. Karabalin
et al.
-15-
BTA Measurement Protocol
To evaluate the BTA’s performance we first nul
l the beam-beam coupling
with a DC input signal
set to
)
(
DC
in
V
=
0V. Then, to simulate a small-signal input
to be amplified, we
add to this static
input a simple single-ton
e, square-wave periodic waveform of the form
V
in
(
AC
)
(
t
)
=
v
in
sq(
ω
in
t
)
,
where
sq(
ω
in
t
)
is a square-wave functio
n that changes from -1
to +1 with a period of
T
in
=
1/
ω
in
. We use typical
input frequencies
ω
in
/(2
π
)~
170Hz. This can be increased
without significant
change to the ou
tput signal if
ω
in
remains a factor of 2 lower than the sweep
rate discussed below. A function generato
r (Agilent 33250A) is used to source both the
(summed) DC and AC input vo
ltages simultaneously; this summed signal is applied to both
NEMS actuation electrodes via a DC/RF bias tee.
The function generator (Agile
nt 33250A) used to provide
the aforementioned ~26MHz
parametric pump signal, provides is own inte
rnal frequency modulation in the form of a
triangular sweep signal (ramp waveform) to provi
de an output with in
stantaneous frequency
ω
(
t
)
=
ω
bif
+−
ω
offset
+Δ
ω
sw
rmp(
ω
sw
t
)
⎡
⎣
⎤
⎦
.
(S26)
Here,
rmp(
ω
sw
t
)
is a triangle function that changes from
0 to 1 and back again with period
T
sw
=
1/
ω
sw
. This frequency modulation serves
to sweep the pump
signal through the
bifurcation point,
ω
bif
, starting from a frequency
ω
bif
−
ω
offset
below it, to a frequency
ω
bif
+Δ
ω
offset
−Δ
ω
sw
⎡
⎣
⎤
⎦
above it. In most of our measurements an FM rate
ω
sw
/(2
π
)
=
557Hz is
used. However, for our study of BTA amplifica
tion bandwidth, as described in the main text,
this was varied between 70Hz and 3kHz. As men
tioned, a 3dB decrease in gain was observed for
ω
sw
/(2
π
)~
2kHz.