Research Article
Vol. 9, No. 7 / July 2022 /
Optica
701
Brillouin backaction thermometry for modal
temperature control
Yu-Hung Lai,
†
Zhiquan Yuan,
†
Myoung-Gyun Suh,
Yu-Kun Lu,
Heming Wang,
AND
Kerry J. Vahala*
T. J. Watson Laboratory of Applied Physics, California Institute of Technology, Pasadena, California 91125, USA
*Corresponding author: vahala@caltech.edu
Received 21 March 2022; revised 31 May 2022; accepted 1 June 2022; published 24 June 2022
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 frequencies impose
strict temperature control requirements for long term frequency-stable operation. Here, we study Brillouin backac-
tion 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 the resonator’s 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.
© 2022 Optica Publishing Group under
the terms of the Optica Open Access Publishing Agreement
https://doi.org/10.1364/OPTICA.459082
The realization of microresonator-based Brillouin lasers [1–8]
has generated interest in their potential application 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 generation. This includes microwave synthesis [11,12],
interrogation of atomic clocks [13], and rotation measurement
[6,8,14], all of which require excellent temperature stability of the
laser mode volume [15]. Here, we demonstrate a new method for
temperature stabilization 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 expresses the
Brillouin phase matching condition (see Fig. 1) [3,10]:
B
(
T
0
)
=
m
×
FSR
(
T
0
),
(1)
where
B
(
T
)
is the Brillouin shift at temperature
T
(see
Supplement 1), and
m
×
FSR
(
T
)
is an integer multiple (
m
) of
the resonator’s free spectral range [FSR
(
T
)
] at temperature
T
. The
actual temperature
T
0
can be set by micro-fabrication control of
FSR at a specified operating wavelength.
The short term frequency stability of a stimulated Brillouin
laser (SBL) is set by fundamental noise associated 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 frequency
stability is most often set by temperature variations. Here, thermo-
optical and thermo-mechanical effects 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 couples temperature to the lasing frequency through
mode pulling [10] and can also induce short term linewidth varia-
tion through the Brillouin
α
parameter [20]. To compensate for
temperature drift, measurement of cavity temperature using modes
belonging to different polarizations was demonstrated [21,22] and
has been recently employed in fiber optic Brillouin laser systems
and silicon-nitride chip-resonator systems to stabilize frequency
[13,15,23]. These dual-polarization modes feature different fre-
quency 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 that relies upon an
external frequency reference to establish locking, the backaction
method described here features an intrinsic reference temperature
T
0
given by Eq. (1). Its sensitivity limit is also determined by the
fundamental white frequency noise of the laser as opposed to the
integrated laser linewidth.
We consider a Brillouin ring laser geometry, shown in Fig. 1(a),
wherein two pumping waves (light and dark gray arrows) are
coupled from a waveguide into counterclockwise and clockwise
directions of the resonator. The frequencies of the pumping waves
are close to resonance, but are not necessarily on resonance. Exact
details of this geometry and explanations of the Brillouin process
are provided in Ref. [24]. Briefly, each pump wave provides power
that is sufficient to excite a corresponding Stokes laser wave [SBL1
or SBL2, shown as green and blue arrows in Fig. 1(a), respectively]
that propagates opposite to the pumping wave and with a lower
(Brillouin-shifted) frequency as a result of the phase matching con-
dition. Power transfer from the pumping waves occurs by way of
2334-2536/22/070701-05 Journal © 2022 Optica Publishing Group
Research Article
Vol. 9, No. 7 / July 2022 /
Optica
702
AOM2
AOM1
PD
F-V circ
FC
Temperature feedback
LED
TEC
EDF
A
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
SB
L2
Brillo
ui n Mode
Cascade
d Mo
de
Gain
Abs orpt
ion
Pushin g
SB
L1
Cas cad
ed Las er
SB
L2
Gain
Pushin g
Back
Act io n
(b)
Abs orpt
ion
ω
ω
∆ω
s
ω
s
ω
r
ω
c
Ω
B
(T
H
)
Ω
B
(T
L
)
m
× FSR
Fig. 1.
Experimental setup and illustrations of the cascaded Brillouin laser induced backaction. (a) Pump 1 (counterclockwise, gray) and pump 2 (clock-
wise, black) waves are generated from an external cavity diode laser (ECDL) amplified by an erbium doped fiber amplifier (EDFA). Both 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 the Brillouin laser wave SBL1 (SBL2) discussed in
(b). Pump 1 has a slightly higher power so that the cascaded SBL is generated in the counterclockwise 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 track-
ing 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
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 (
1ω
s
/
2
π
) when the cas-
caded 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
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 (
1ω
s
/
2
π
) when the cascaded laser power increases.
the Stokes scattering process. After laser action occurs, however, a
strong anti-Stokes process occurs that is driven by the lasing fields.
This anti-Stokes process creates absorption near the pumps; the
absorption strength is proportional to laser power, and this absorp-
tive 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 frequency pulling of the
laser frequencies such that about 1 MHz of pump frequency tuning
induces 10s of kHz of Brillouin laser frequency pulling (
1ω
s
/
2
π
).
This configuration (non-degenerate, counterpropagating SBLs) is
the starting point for implementation of the backaction tempera-
ture control method, and a schematic of the key spectral features is
provided in Fig. 1(b).
Now suppose that the power of SBL1 is increased so that
it begins to function as a pumping wave for a new laser wave
[cascaded laser wave shown in red in Fig. 1(b)]. 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
action, but now in the cascaded wave, induces absorptive back-
action on the pump, SBL1, that is proportional to the cascaded
laser power. It is important to note that on account of the phase
matching condition, this absorption acts only on SBL1 (i.e., it
is directional and does not affect SBL2). The spectrum of the
absorption experienced by SBL1 is shown in purple in Fig. 1(b)
and compensates for SBL1 optical gain (not shown in the figure)
provided by the original pump 1. As a result, SBL1 power is held
constant, which is equivalent to the gain of the cascaded laser field
[shown in orange in Fig. 1(b)] being clamped (even if pump 1
power is increased). The backaction absorption spectrum has a
spectral profile similar to the Brillouin gain, and the location of its
maximum relative to the frequency of SBL1 is determined by the
phase matching condition for the backaction process. Specifically,
maximum 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 Supplement 1.
Associated with the backaction absorption is a contribution to
the refractive index that pushes the frequency of SBL1 away from
the absorption center. At
T
=
T
0
, the pushing is zero since this
temperature corresponds to perfect phase matching for which the
SBL1 frequency is at the absorption maximum, where the refrac-
tive index contribution is zero. However, cases where
T
<
T
0
and
T
>
T
0
[upper and lower panels in Fig. 1(b), respectively] result
in a slight phase mismatch. Defining a phase mismatch parameter
as
1ω
≡
m
×
FSR
−
B
(
T
)
, for
1ω >
0 (upper panel) and
1ω <
0 (lower panel), the SBL1 frequency is pushed higher and
lower, respectively. Importantly, since SBL1 and SBL2 are coun-
terpropagating waves, the frequency of SBL2 is not affected by
the cascade laser backaction due to the phase matching condition.
Thus measurement of the frequency difference between 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 fre-
quency tuning with respect to temperature is a common mode for
SBL1 and SBL2, as they share the same cavity longitudinal mode.
To implement this measurement, it is convenient to use the
power dependence of the frequency pushing. Specifically, a weak
modulation of pump 1 power (
P
) will induce a correspond-
ing frequency modulation of SBL1 through modulation of the
backaction-induced refractive index. The resulting pump power
(
P
) dependence of the SBL2-SBL1 beat frequency (
1ω
s
) is given
by the following equation (Supplement 1):
Research Article
Vol. 9, No. 7 / July 2022 /
Optica
703
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
s
Slope = 27.6 MHz
/
(K*W)
(a)
(b)
(c)
T
0
= 26.561
°C
Fig. 2.
Measurement of Brillouin backaction. (a) The beating frequency of the counterpropagating 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
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
o
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.
∂1ω
s
∂
P
≈
4
g
0
γ
ex
~
ω
p
γ
2
0
d
B
d T
(
T
−
T
0
),
(2)
where
1ω
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
0
is the Brillouin gain bandwidth. This result shows
that frequency discrimination of
1ω
s
combined with subsequent
phase-sensitive detection will provide an error signal whose
magnitude and sign vary as
T
−
T
0
.
In the experiment, we used a 36 mm diameter silica wedge res-
onator on silicon [3] with 8
μ
m thickness and wedge angle of 30
◦
.
The resonator is packaged (similar to Ref. [14]) with a thermal elec-
trical cooler (TEC), a light emitting diode, and a thermistor. The
TEC and thermistor are used for coarse temperature control and
monitoring. The ultrahigh-quality factor of the microcavity and
the precisely controlled resonator size enable efficient generation
of SBL action along 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 Fig. 1(a) caption.
To set up the phase-sensitive servo temperature control, 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 demod-
ulates 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 TEC (coarse
control).
Figure 2 shows the measured SBL1/SBL2 beating frequency
(
1ω
s
/
2
π
) when sweeping pump power 1 from below to above
cascaded laser operation. Sweeps are performed at a series of tem-
peratures as indicated. As expected, 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 backaction
effect is apparent with each temperature showing a distinctly dif-
ferent linear dependence on power. Significantly, the slope of this
dependence is observed to change sign as discussed above, corre-
sponding to temperatures above and below
T
0
in Eq. (2). This sign
change is essential for implementation of the phase-sensitive detec-
tion servo control. An experimental plot showing the sign change
in slope is provided in Fig. 2(b), 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 temperature
over the narrow range measured [see Fig. 2(c) measured at 6.5 mW
pump power]. The temperature tuning rate is smaller than the
corresponding value provided by the dual-polarization approach
[15]. Nonetheless, it should be possible to substantially increase
this rate in resonators designed for forward-Brillouin scattering,
wherein the reduced Brillouin shift is accompanied by much nar-
rower linewidths (see Supplement 1). By fitting these data sets, the
experimental backaction strength
∂
2
1ω
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 Supplement 1].
The linear power and temperature dependence of the backac-
tion are further illustrated in Fig. 3(a) 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. 3(b). The change in polarity 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 system is
demonstrated by closing the servo control loop. Here, a faster
sinusoidal power modulation (200–500 Hz) is used to generate a
Research Article
Vol. 9, No. 7 / July 2022 /
Optica
704
Fig. 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 saw-tooth 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.
small frequency dither on the SBL beating frequency. It is demodu-
lated by a frequency to voltage conversion circuit, and the error
signal is generated by phase-sensitive detection with a lock-in
amplifier as before. The output of a proportional-integral (PI)
servo drives a 1 W white LED [see Fig. 1(a)] 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. 3(a). After an initial relaxation oscillation,
the long term temperature drift (
>
10 s) is stabilized.
With the servo-control loop disconnected, but with the power
dither active, the beating frequency exhibits a continuous drift as
large as 2.3 kHz in 1 h, corresponding to around 0.13
o
C temper-
ature change per hour. This is apparent in both the measured SBL
beat frequency (see Fig. 4 inset) and its Allan deviation (ADEV)
measurement presented in the main panel in Fig. 4. However, with
the servo-control loop connected, there is no observable drift in the
beating frequency over 1 h of measurement (see Fig. 4 inset). Also,
over 1500 s averaging time (limited by data size of 1 h) the ADEV
remains around 2 Hz, corresponding to about 0.1 mK temperature
variation. In the short term, the drift suppression is believed to be
limited by the thermal response of the cavity to the LED. In the
future, a faster form of temperature feedback (e.g., an integrated
resistive heater placed near the resonator) should further reduce
this response 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 res-
onator to an absolute temperature defined by the phase matching
condition. The backaction has been shown to induce both linear
power and temperature dependences in a readily measured optical
beat frequency. The polarity of the power dependence depends
upon operation above or below the phase matching temperature.
Fig.4.
Allan deviation of SBL beating frequency
1ω
s
. Allan deviations
of SBL beating frequency with (blue) and without (red) servo control. The
error bar gives the standard deviation. Inset: SBL beating frequency versus
time with (blue) and without (red) servo control.
This feature and the high stability of the beat frequency were used
to servo control the optical mode temperature to 0.1 mK stability
levels. While not yet at the temperature stability level of the cross-
polarization 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 aspect of the phase-sensitive control is
that the noise limit is determined by SBL frequency noise at large
offset frequencies set by backaction modulation. Here, the rate is
500 Hz, but could be set even higher if necessary to avoid laser fre-
quency noise at low-offset frequencies. Moreover, characterization
of the absolute optical frequency stability that is achievable by this
method is a possible area of investigation. The range of material
Research Article
Vol. 9, No. 7 / July 2022 /
Optica
705
systems on which Brillouin laser action has been demonstrated
(including integrable platforms) suggests that this method can find
wide use.
Funding.
Defense Advanced Research Projects Agency (N66001-16-1-4046);
Air Force Office of Scientific Research (FA9550-18-1-0353); Kavli Nanoscience
Institute.
Acknowledgment.
Yu-Kun Lu thanks the Caltech SURF program for
financial support.
Disclosures.
The authors declare no conflicts of interest.
Data availability.
Data underlying the results presented in this paper are not
publicly available at this time but may be obtained from the authors upon reason-
able request.
Supplemental document.
See Supplement 1 for supporting content.
†
These authors contributed equally to this paper.
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