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Brillouin Backaction Thermometry for Modal Temperature Control
Yu-Hung Lai
1
, Zhiquan Yuan
1
, Myoung-Gyun Suh
1
, Yu-Kun Lu
1
, Heming Wang
1
, Kerry J. Vahala
1
1
T. J. Watson Laboratory of Applied Physics, California Institute of Technology, Pasadena, California 91125, USA
These authors contributed equally to this work.
vahala@caltech.edu
Stimulated Brillouin scattering provides optical gain for efficient and narrow-linewidth lasers in
high-Q microresonator systems. However, the thermal dependence of the Brillouin process, as well as
the microresonator, impose strict temperature control requirements for long-term frequency-stable
operation. Here, we study Brillouin back action and use it to both measure and phase-sensitively lock
modal temperature to a reference temperature defined by the Brillouin phase-matching condition.
At a specific lasing wavelength, the reference temperature can be precisely set by adjusting resonator
free spectral range. This backaction control method is demonstrated in a chip-based Brillouin laser,
but can be applied in all Brillouin laser platforms. It offers a new approach for frequency-stable
operation of Brillouin lasers in atomic clock, frequency metrology, and gyroscope applications.
The realization of microresonator-based Brillouin
lasers [1–8] has generated interest in their potential ap-
plication to compact and potentially integrated Brillouin
systems[9]. Moreover, high-coherence Brillouin lasers,
featuring short-term linewidths below 1 Hz [3, 4, 8, 10],
have been used for precision measurement and signal gen-
eration. This includes microwave synthesis [11, 12], in-
terrogation of atomic clocks [13], and rotation measure-
ment [6, 8, 14]; all of which require excellent temper-
ature stability of the laser mode volume[15]. Here, we
demonstrate a new method for temperature stabiliza-
tion of Brillouin lasers based on backaction produced by
the Brillouin anti-Stokes process. This process is shown
to provide for phase sensitive locking to a temperature
set point
T
0
given by the following condition, which ex-
presses the Brillouin phase matching condition (see Fig.
1) [3, 10]:
B
(
T
0
) =
m
×
FSR(
T
0
)
(1)
where Ω
B
(
T
) is the Brillouin shift at temperature
T
(see
Supplementary Information) and
m
×
FSR(
T
) is an in-
teger multiple (
m
) of the resonator free-spectral-range
(FSR(
T
)) at temperature
T
. The actual temperature
T
0
can be set by micro-fabrication control of
FSR
at a spec-
ified operating wavelength.
The short term frequency stability of a stimulated Bril-
louin laser (SBL) is set by fundamental noise associ-
ated with the thermal occupancy of photons involved in
the Brillouin process [10]. In high-Q microresonators,
this adds a white noise contribution to frequency noise
with an equivalent short time linewidth less than 1 Hz
[6, 8, 10].
However, on longer time scales the fre-
quency stability is most often set by temperature vari-
ations. Here, thermo-optical and thermo-mechanical ef-
fects change the cavity resonant frequency [16–18], while
the temperature dependence of the sound velocity causes
drifts in the Brillouin frequency shift [19]. The latter cou-
ples temperature to the lasing frequency through mode
pulling [10] and can also induce short term linewidth
variation through the Brillouin
α
parameter [20]. To
compensate temperature drift, measurement of cavity
temperature using modes belonging to different polar-
izations was demonstrated [21, 22] and has been re-
cently employed in fiber optic Brillouin laser systems
and silicon-nitride chip-resonator systems to stabilize fre-
quency [13, 15, 23]. These dual-polarization modes fea-
ture different frequency tuning rates versus temperature,
thereby providing a way to convert change in modal
temperature to measurement of a frequency change. In
contrast to this method which relies upon an exter-
nal frequency reference to establish locking, the back-
action method described here features an intrinsic refer-
ence temperature
T
0
given by Eq. (1). It’s sensitivity
limit is also determined by the fundamental white fre-
quency noise of the laser as opposed to the integrated
laser linewidth.
We consider a Brillouin ring laser geometry shown in
Fig. 1a wherein two pumping waves (dark and light grey
arrows) are coupled from a waveguide into clockwise and
counter-clockwise directions of the resonator. The fre-
quencies of the pumping waves are close to a resonance,
but are not necessarily on resonance. Exact details of
this geometry and explanations of the Brillouin process
are provided in reference [24]. Briefly, each pump wave
provides power that is sufficient to excite corresponding
Stokes laser waves (SBL1 and SBL2 shown as green and
blue arrows) that propagate opposite to their pumping
wave and with a lower (Brillouin-shifted) frequency as
a result of the phase matching condition. Power trans-
fer from the pumping waves occurs by way of the Stokes
scattering process. After laser action occurs, however, a
strong anti-Stokes process occurs that is driven by the
lasing fields. This creates absorption near the pumps
which is proportional to laser power, and this absorptive
backaction clamps their circulating powers and hence the
laser gain. The exact lasing frequencies can be controlled
by tuning of the pumping frequencies, which causes fre-
quency pulling of the laser frequencies such that about
1 MHz of pump frequency tuning induces 10s of kHz of
Brillouin laser frequency pulling (∆
ω
s
/
2
π
). This config-
uration (non-degenerate, counter-propagating SBLs) is
arXiv:2205.07181v1 [physics.optics] 15 May 2022
2
AOM2
AOM1
PD
F-V circ
FC
Temperature feedback
LED
TEC
EDFA
ECDL
Ref
Power
Dithering
Optical signal
Cascade SBL
SBL1
SBL2
Pump1
Pump2
Electrical signal
LIA
PM
Servo
PD
PDH
Detection
PDH
Error
Frequency
Modulation
a
FG
SB
L1
Cas cad
ed Las er
SBL2
~
~
Brilloui
n Mode
Casc
ade
d Mo
de
Gain
Absorption
Pushin g
SB
L1
Cas cad
ed Las er
SBL2
~
~
Gain
Pushin g
Back
Act io n
b
Absorption
ω
ω
∆ω
s
ω
s
ω
r
ω
c
B
(T
H
)
B
(T
L
)
m × FSR
Figure
1.
Experimental setup and illustrations of the cascaded Brillouin laser induced backaction. a,
Pump
1 (counterclockwise, grey) and Pump 2 (clockwise, black) waves are generated from an external cavity diode laser (ECDL)
amplified by an Erbium doped fiber amplifier (EDFA). Both the pump 1 and pump 2 frequencies are shifted using acousto-optic
modulators (AOM) to create a relative frequency offset. The frequency of Pump 1 is further Pound-Drever-Hall (PDH) locked
to the cavity resonance using a phase modulator (PM). Green (blue) arrow refers to Brillouin laser waves SBL1 (SBL2) discussed
in panel
b
. Pump 1 has a slightly higher power so that the cascaded SBL is generated in the counter-clockwise direction (red).
The beat signal of SBL1 and SBL2 is generated by a photodetector (PD) and its frequency is measured by a frequency counter
(FC) and then frequency discriminated using a frequency tracking circuit (F-V circ) for subsequent phase sensitive detection by
a lock-in amplifier (LIA), and temperature control using a light emitting diode (LED) and thermal electrical cooler (TEC).
b,
Upper panel (low temperature case, T
L
): the Brillouin shift Ω
B
(T
L
) is smaller than the mode frequency difference (
m
×
FSR,
where
m
the mode number difference and FSR is the free spectral range in angular frequency). Backaction on SBL1 pushes
its frequency away from the backaction absorption (purple) maximum, thereby increasing its frequency. This decreases the
SBL1-SBL2 beating frequency (∆
ω
s
/
2
π
), when the cascaded laser power increases. Note: SBL2’s frequency is not affected by
the mode-pushing effect because the backaction absorption is directional according to the phase-matching condition. Lower
panel (high temperature case, T
H
): the Brillouin shift Ω
B
(T
H
) is larger than the mode frequency difference so that SBL1
has its frequency pushed lower. This increases the SBL1-SBL2 beating frequency (∆
ω
s
/
2
π
), when the cascaded laser power
increases.
the starting point for implementation of the back-action
temperature control method and a schematic of the key
spectral features is provided in Fig. 1b.
Now suppose that the power of SBL1 is increased so
that it begins to function as a pumping wave for a new
laser wave (the cascaded laser wave shown in red in Fig.
1b). The power of SBL2 is intentionally kept below
threshold so that no cascading occurs. As noted above
for the original pumping waves, the onset of laser ac-
tion, now in the cascaded wave, induces absorptive back-
action on its pump, SBL1, that is proportional to the
cascaded laser power. It is important to note that on
account of the phase matching condition, this backaction
acts only SBL1 (i.e., it is directional and does not affect
SBL2). The absorptive backaction experienced by SBL1
is shown in purple in Fig. 1b and compensates SBL1
optical gain (not shown in the figure) provided by the
original Pump 1. As a result, SBL1 power is held con-
stant, which is equivalent to the gain of the cascaded laser
field (shown in orange in Fig. 1b) being clamped (even if
Pump 1 power is increased). The backaction absorption
spectrum has a similar spectral profile to the Brillouin
gain and the location of its maximum relative to the fre-
quency of SBL 1 is determined by the phase matching
condition for the backaction process. Specifically, max-
imum nonlinear absorption occurs for the condition of
perfect phase matching given by Eq. (1).
The use of this backaction for temperature control
is now briefly summarized, but a detailed analysis is
given in Supplementary Information. Associated with
the backaction absorption is a dispersion contribution
that pushes the frequency of SBL1 away from the ab-
sorption center. At
T
=
T
0
the pushing is zero since
this temperature corresponds to perfect phase match-
ing for which the SBL 1 frequency is at the absorption
maximum. However, cases where
T < T
0
and
T > T
0
(upper and lower panels in Fig. 1b) result in slight
phase mismatch. Defining a phase mismatch parameter
as ∆
ω
m
×
FSR
B
(
T
), for ∆
ω >
0 (upper panel)
and ∆
ω <
0 (lower panel) the SBL 1 frequency is pushed
higher and lower, respectively. Since the frequency of
SBL2 is not affected by this process, measurement of the
frequency difference of SBL1 and SBL2 gives a direct
way to measure
T
T
0
. As an aside, in reaching this
conclusion, it is important to note that cavity resonant
frequency tuning with respect to temperature is common
mode for SBL1 and SBL2 as they share the same cavity
longitudinal mode.
3
SBL1 pump power P (mW)
∆ω
s
/2
π
(kHz)
∆ω
s
/2
π
(kHz)
P = 6.5 mW
T (°C)
Data
Linear Fit
3
3.5
4
4.5
5
5.5
6
6.5
7
7.5
0
10
20
30
40
50
60
Non-cascaded Region
Cascaded Region
27.369
27.158
27.002
26.794
26.685
26.589
26.573
26.549
26.492
26.395
26.299
26.206
26
26.5
27
27.5
-15
0
15
30
26
26.5
27
27.5
0
20
40
60
Temperature (°C)
Temperature (°C)
(MHz
/
W)
∆ω
/2
π
∂P
s
Slope = 27.6 MHz
/
(K*W)
a
b
c
T
0
= 26.561
°C
Figure
2.
Measurement of Brillouin backaction. a,
The beating frequency of the counter-propagating SBLs is plotted
versus pump 1 power at a series of resonator temperatures as indicated. Frequencies at all temperatures closely track one
another in the noncascaded regime with a slight power dependence is induced by the Kerr effect. In the cascaded regime,
Brillouin backaction resolves the temperature differences with beating frequency showing a distinct dependence upon pump
power. The slope of this dependence changes sign at T
0
(approximately 26.561
C).
b,
Measured SBL beating frequency
change per unit pump power plotted versus temperature tuning. The temperature at perfect phase matching condition (
T
0
) is
indicated.
c,
Measured SBL beating frequency change plotted versus temperature. Slope is 41 kHz/K at 6.5 mW pump power.
To implement this measurement it is convenient to use
the power dependence of the frequency pushing. Specif-
ically, a weak modulation of the pump 1 power (
P
) will
induce a corresponding frequency modulation of SBL1
through modulation of the backaction dispersion. The
resulting pump power (
P
) dependence of the SBL2-SBL1
beat frequency (∆
ω
s
) is given by the following equation
(see SI):
ω
s
∂P
4
g
0
γ
ex
~
ω
p
γ
2
Γ
d
B
dT
(
T
T
0
)
(2)
where ∆
ω
s
ω
r
ω
s
,
ω
s
(
ω
r
) is the absolute angular
frequency of SBL1 (SBL2),
g
0
is the Brillouin gain,
ω
p
is
the pump angular frequency,
γ
ex
is the external coupling
rate,
γ
is the total cavity loss rate, and Γ is the Brillouin
gain bandwidth. This result shows that frequency dis-
crimination of ∆
ω
s
combined with subsequent phase sen-
sitive detection will provide an error signal whose mag-
nitude and sign vary as
T
T
0
.
In the experiment, we used a 36 mm-diameter silica
wedge resonator on silicon [3] with 8
μ
m thickness and
wedge angle of 30 degrees. The resonator is packaged
(similar to ref. [14]) with a thermal electrical cooler
(TEC), a light emitting diode, and a thermistor. The
TEC and thermistor are used for coarse temperature con-
trol and monitoring. The ultra-high-quality factor of the
microcavity and the precisely controlled resonator size
enable efficient generation of stimulated Brillouin laser
action in the opposite propagation directions (operating
wavelengths close to
λ
1553
.
3 nm). The intrinsic Q
factor is 300 million and the SBL threshold is 0.9 mW.
Details on the optical pumping as well as generation of
SBL1, SBL2 and the cascaded laser wave are provided in
the Figure 1a caption.
To setup the phase-sensitive servo temperature con-
trol, the beat frequency of SBL1 and SBL2 is dithered
by modulating pump 1 (and in turn the cascaded SBL
power) using a power modulator (orange dashed line).
The frequency of the beat signal is monitored using a
frequency counter. A frequency tracking circuit demodu-
lates the dithered signal and phase sensitive measurement
is performed using a lock-in amplifier (Stanford Research
SR830) to generate the error signal. Temperature control
applies the feedback signal to a light emitting diode (fine
control) and a thermal electrical cooler (coarse control).
Figure 2 shows the measured SBL1/SBL2 beating fre-
quency (∆
ω
s
/
2
π
) when sweeping the pump power 1 from
below to above cascaded laser operation. Sweeps are per-
formed at a series of temperatures as indicated. As ex-
pected, when the pump power is low in the non-cascaded
regime, all temperatures provide identical traces. The
observed slope on all of these traces is the result of fre-
quency shift provided by the Kerr effect [25]. On the
other hand, for higher pump powers in the cascaded
regime, the back-action effect is apparent with each tem-
perature showing a distinctly different linear dependence
on power. Significantly, the slope of this dependence is
observed to change sign as discussed above, correspond-
ing to temperatures above and below
T
0
in Eq. (2).
This sign change is essential for implementation of the
phase sensitive detection servo control. An experimen-
4
Figure
3.
Thermal tuning of the backaction with power dithering. a,
The open-loop SBL beat frequency (backaction
regime of Fig. 2) is measured versus time at a series of chip temperatures. A weak sawtooth power modulation is applied to
Pump 1.
b,
Zoom-ins of the plot in
a
showing how the polarity and amplitude of the sawtooth beat frequency modulation
depends on temperature.
Inset,
Temperature stabilization of the resonator with servo control activated.
tal plot showing the sign change in slope is provided
in Fig. 2b, where the absolute temperature reference
T
0
=26.561
C is also measured. The beating frequency
is observed to show a linear power dependence on tem-
perature over the narrow range measured (see Fig. 2c
measured at 6.5 mW pump power). The temperature
tuning rate is smaller than the corresponding value pro-
vided by the dual-polarization approach [15]. Nonethe-
less, it should be possible to substantially increase this
rate in resonators designed for forward-Brillouin scat-
tering, wherein the reduced Brillouin-shift is accompa-
nied by much narrower linewidths (see SI). By fitting
these data sets, the experimental back-action strength
2
ω
s
/
(
∂P∂T
) is measured to be 2
π
×
27
.
6 MHz/(W
·
K),
and is consistent with the theory from Eq. (2) (2
π
×
22
.
5
MHz/(W
·
K), see SI).
The linear power and temperature dependence of the
backaction are further illustrated in Figure 3a where the
beating frequency is measured versus time at a series
of temperatures. To illustrate the change in slope with
power at each temperature a weak and slowly-varying
saw-tooth power modulation is applied. Zoom-in views
of the corresponding saw-tooth modulation in the beat
frequency are presented in Fig. 3b. The change in polar-
ity and amplitude of the backaction-induced modulation
are apparent as the temperature is set to values above
and below T
0
.
Finally, long term temperature stabilization of the sys-
tem is demonstrated by closing the servo control loop.
Here, a faster sinusoidal power modulation (200 to 500
Hz) is used to generate a small frequency dither on the
SBL beating frequency. It is demodulated by a frequency
to voltage conversion circuit and the error signal is gen-
erated by phase-sensitive detection with a lock-in am-
plifier as before. The output of a proportional-integral
(PI) servo drives a 1 W white LED (see Figure 1a) for
faster fine-control of temperature. The temperature is
also controlled by a TEC that provides slower feedback.
The temperature feedback result is shown in the inset of
Fig. 3a. After an initial relaxation oscillation, the long
term temperature drift (
>
10 seconds) is stabilized.
With the servo-control loop disconnected, but with the
power dither active, the beating frequency exhibits a con-
tinuous drift as large as 2.3 kHz in an hour, corresponding
to around 0.13
C temperature change per hour. This is
apparent in both the measured SBL beat frequency (see
Fig. 4 inset) and its Allan Deviation (ADEV) measure-
ment presented in the main panel in Fig. 4). However,
with the servo-control loop connected there is no observ-
able drift in the beating frequency over an hour of mea-
surement (see Fig. 4 inset). Also, over 1500 s averaging
time (limited by data size of 1 hour) the Allan Deviation
remains around 2 Hz, corresponding to about 0.1 mK
temperature variation. In the short term, the drift sup-
pression is believed to be limited by the thermal response
of the cavity to the LED. In the future, a faster form of
5
10
τ
(s)
With Feedback
Power Dithered Only
0
1000
2000
3000
Time (s)
-2
-1
0
With Feedback
Power Dithered Only
10¹
10²
10³
10
10¹
10²
10³
10
10¹
10
-
¹
σ
(
τ
) (Hz)
∆ω
s
/2
π
∆ω
s
/2
π (
kHz
)
T
(
mK
)
Figure
4.
Allan Deviation of SBL beating frequency
ω
s
.
Allan deviations of SBL beating frequency with (blue)
and without (red) servo control. The error bar gives the stan-
dard deviation.
Inset,
SBL beating frequency versus time
with (blue) and without (red) servo control.
temperature feedback (e.g., an integrated resistive heater
placed near the resonator) should further reduce this re-
sponse time. Feed forward frequency correction could
also be employed.
In summary, we have investigated Brillouin backaction
and shown that it provides a way to phase sensitively
lock an optical resonator to an absolute temperature de-
fined by the phase matching condition. The back-action
has been shown to induce both linear power and tem-
perature dependences in a readily measured optical beat
frequency. The polarity of the power dependence de-
pends upon operation above or below the phase match-
ing temperature. This feature and the high stability of
the beat frequency were used to servo control the opti-
cal mode temperature to 0.1 mK stability levels. While
not yet at the temperature stability level of the cross po-
larization method when applied to microcavities (0.008
mK) [26], cross-polarization stabilization has existed for
a decade. And we believe the initial results presented
here can be substantially improved. An important as-
pect of the phase sensitive control is that the noise limit
is determined by SBL frequency noise at large offset fre-
quencies set by backaction modulation. Here, the rate is
500 Hz, but could be set even higher if necessary to avoid
laser frequency noise. Moreover, characterization of the
absolute optical frequency stability that is achievable by
this method is a possible area of investigation. The range
of material systems on which Brillouin laser action has
been demonstrated (including integrable platforms) sug-
gests that this method can find wide use.
Acknowledgments
This project was supported by the Defense Advanced
Research Projects Agency (DARPA) through SPAWAR
(grant no. N66001-16-1-4046), the Air Force Office of
Scientific Research (FA9550-18-1-0353) and the Kavli
Nanoscience Institute. Yu-Kun Lu would like to thank
the Caltech SURF program for financial support.
Author Contributions
Y.H.L. and K.V. conceived the microresonator cascaded
Brillouin thermometer. Y.H.L., Y.K.L., Z.Y., H. W. and
K.V. constructed the theoretical model. M.G.S. fabri-
cated the ultra-high-Q silica microresonator. Y.H.L.,
Z.Y., and M.G.S. performed the experiment and ana-
lyzed the data. All authors contributed to writing the
manuscript. K.V. supervised the project.
Competing interests
The authors declare no competing interests.
Data Availability
The data that support the plots within this paper and
other findings of this study are available from the corre-
sponding authors upon reasonable request.
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7
SUPPLEMENTARY INFORMATION
Below we derive the temperature dependence of the frequency shift in the power-dithered cascaded SBL. In the
cavity-mode rotating frame, we write the pump, SBL, and cascaded SBL in the following form,
̇
A
=
[
i
(
ω
p
ω
0
)
γ
2
]
A
g
s
|
α
|
2
A
+
γ
ex
P
~
ω
p
,
(S1)
̇
α
=
[
i
(
ω
s
ω
1
)
γ
2
]
α
+
g
s
|
A
|
2
α
g
c
|
β
|
2
α,
(S2)
̇
β
=
[
i
(
ω
c
ω
2
)
γ
2
]
β
+
g
c
|
α
|
2
β,
(S3)
where
A
,
α
,
β
are the normalized photon number amplitudes of the pump, SBL, and cascaded SBL, respectively. Here
we have adiabatically eliminated the phonon fields as described elsewhere [10]. We also note that the term involving
g
c
in Eq. S2 results from anti-Stokes scattering of the cascaded laser field
β
. The
ω
p
,
ω
s
, and
ω
c
are the lasing angular
frequencies, and the
ω
0
,
ω
1
,
ω
2
are the cavity angular frequencies.
P
is the input pump power (i.e., pump 1 in Fig. 1
in the main text).
g
s,c
are defined by,
g
s,c
=
g
0
1 +
2
i
∆Ω
s,c
Γ
,
(S4)
∆Ω
s
=
ω
p
ω
s
s
,
(S5)
∆Ω
c
=
ω
s
ω
c
c
,
(S6)
B
s
c
,
(S7)
where Ω
s
and Ω
c
are the phonon angular frequencies associated with the SBL and the cascaded SBL, respectively.
B
is the Brillouin shift, which is equal to 4
πnc
s
p
(
n
is the refractive index,
c
s
is the speed of sound in silica, and
λ
p
is the pump wavelength).
g
0
is the Brillouin nonlinear coefficient [10],
γ
ex
is the external coupling coefficient, and
P
is the input pump power.
In steady-state, if we assume good phase matching (∆Ω
s,c

Γ), then the real parts and imaginary parts of Eq.(S2)
and Eq.(S3) give,
γ
2
=
g
0
(
|
A
|
2
−|
β
|
2
)
=
g
0
|
α
|
2
,
(S8)
ω
s
ω
1
=
2
g
0
Γ
(
|
A
|
2
∆Ω
s
+
|
β
|
2
∆Ω
c
)
(S9)
ω
c
ω
2
=
2
g
0
Γ
|
α
|
2
∆Ω
c
.
(S10)
From Eq. (S8) we get a photon number relation
|
A
|
2
=
|
α
|
2
+
|
β
|
2
as well as the clamping condition for the Stokes
wave,
|
α
|
2
= 2
g
0
. These can be used to eliminate
|
α
|
2
and
|
β
|
2
in Eq. (S9) and Eq. (S10), which yields,
ω
s
ω
1
=
2
g
0
Γ
|
A
|
2
(∆Ω
s
+ ∆Ω
c
)
γ
Γ
∆Ω
c
,
(S11)
ω
c
ω
2
=
γ
Γ
∆Ω
c
.
(S12)
To study the dependence of
ω
s
on the input power, we take the partial derivative with respect to
|
A
|
2
. Since
ω
1
,
ω
2
,
ω
p
, Ω
B
are independent of power the following hold,
∂ω
c
|
A
|
2
=
γ
Γ
∆Ω
c
|
A
|
2
=
γ/
Γ
1 +
γ/
Γ
∂ω
s
|
A
|
2
(S13)
∂ω
s
|
A
|
2
=
2
g
0
Γ
(
ω
p
ω
c
2Ω
B
)
+
2
g
0
Γ
|
A
|
2
(∆Ω
s
+ ∆Ω
c
)
|
A
|
2
γ
Γ
∆Ω
c
|
A
|
2
=
2
g
0
Γ
ω
p
ω
c
2Ω
B
1 +
γ/
Γ
1+
γ/
Γ
(
1 +
2
g
0
|
A
|
2
Γ
)
.
(S14)
8
Eq. (S13) shows that the cascaded Brillouin laser frequency is also affected by the mode pulling effect[24], such that
the cascaded laser frequency moves toward to the cavity mode center and is less sensitive to the original Brillouin laser
drift. Assuming the operation is only slightly above the cascaded threshold, such that Eq. (S8) gives 2
g
0
|
A
|
2
γ
,
then simplifies Eq. (S14) into
∂ω
s
|
A
|
2
=
1
1 +
γ/
Γ
2
g
0
Γ
(
ω
p
ω
c
2Ω
B
)
.
(S15)
Then, by assuming
γ/
Γ

1, we can drop the correction factor in Eq. (S15). Next, the
|
A
|
2
can be further replaced
by
P
=
~
ω
p
γ
2
|
A
|
2
ex
, which is the input pump power above cascade threshold (obtained by solving the steady state
of Eq. (S1) with clamping conditions for
|
α
|
2
). Then, Eq. (S15) simplifies to the following,
∂ω
s
∂P
=
2
g
0
γ
ex
~
ω
p
γ
2
Γ
(
ω
p
ω
c
2Ω
B
)
.
(S16)
The derivative with respect to temperature is now taken in this expression to arrive at the following,
2
ω
s
∂P∂T
≈−
4
g
0
γ
ex
~
ω
p
γ
2
Γ
d
B
dT
(S17)
In writing this expression the temperature dependence of
ω
p
ω
c
ω
0
ω
2
2
m
×
FSR is neglected. This term’s tem-
perature dependence is dominated by the thermorefractive and thermoexpansion effects, which are much weaker than
the temperature dependence of the Brillouin frequency. For example, in silica glass the thermoexpansion coefficient is
α
L
= 0
.
51
×
10
6
/
K and the thermorefractive coefficient is
dn/dT
= 11
.
6
×
10
6
/
K, so that
d
(
ω
p
ω
c
)
/dT
2
π
×
180
kHz/K. This compares to
d
B
/dT
2
π
×
1
.
16 MHz/K at 1550 nm (estimated from 1
.
36 MHz/K at 1320 nm [18]),
and justifies the simplification involved in Eq. (S17).
Now, if we introduce an independent backward propagating SBL as a reference with higher angular frequency
ω
r
,
the result is,
2
ω
s
∂P∂T
=
2
(
ω
r
ω
s
)
∂P∂T
=
4
g
0
γ
ex
~
ω
p
γ
2
Γ
d
B
dT
(S18)
which is the temperature derivative of Eq. 2 in the main text. The silica resonator has
g
0
/
2
π
= 0
.
61 mHz, Γ
/
2
π
= 30
MHz,
γ
ex
/
2
π
= 110 kHz,
γ/
2
π
= 860 kHz,
ω
p
/
2
π
= 193 THz, giving a theoretical estimation of
2
ω
s
/∂P∂T
=
2
π
×
22
.
5 MHz/(W.K), which compares favorably with the experimental value in Fig. 2) of 2
π
×
27
.
6 MHz/(W.K).
The difference here mainly originates from the uncertainty of certain parameters.
As an aside, Eqs. (S17) and (S18) show that frequency tuning response with respect to temperature depends
inversely upon the Brillouin damping rate Γ. Along these lines, forward Brillouin scattering in high-Q silica mi-
croresonators has produced damping rates much smaller than for back scattering on account of the smaller required
Brillouin shift for backscattering [2, 27]. We intend to investigate this approach as a means to increase the backaction
response.
To estimate
g
0
above (in the unit of rad/s), we use the following equations[28]:
g
0
~
ω
3
2
P
clamp
Q
T
Q
E
(S19)
2
π
ν
clamp
n
th
,
(S20)
where
P
clamp
is the clamped power of SBL at the cascading threshold,
Q
T
(
Q
E
) is the total (external) quality factor,
ν
clamp
the full-width-half-maximum of the fundamental SBL linewidth under cascaded clamping conditions,
n
th
is the number of thermal phonon quanta at the operating temperature. For the 36 mm silica resonator at room
temperature, we measured ∆
ν
clamp
= 0
.
35 Hz and used the theoretical
n
T
577.