of 4
Supplemental Document
Brillouin backaction thermometry for modal
temperature control: supplement
Y
U
-H
UNG
L
AI
,
Z
HIQUAN
Y
UAN
,
M
YOUNG
-G
YUN
S
UH
,
Y
U
-K
UN
L
U
,
H
EMING
W
ANG
,
AND
K
ERRY
J. V
AHALA
T. J. Watson Laboratory of Applied Physics, California Institute of Technology, Pasadena, California 91125,
USA
Corresponding author:
vahala@caltech.edu
These authors contributed equally to this paper.
This supplement published with Optica Publishing Group on 24 June 2022 by The Authors under
the terms of the Creative Commons Attribution 4.0 License in the format provided by the authors
and unedited. Further distribution of this work must maintain attribution to the author(s) and the
published article’s title, journal citation, and DOI.
Supplement DOI: https://doi.org/10.6084/m9.figshare.19967975
Parent Article DOI: https://doi.org/10.1364/OPTICA.459082
Brillouin Backaction Thermometry for
Modal Temperature Control:
supplemental document
This supplement provides details about the theoretical analyses presented in the main text.
Below we derive the temperature dependence of the frequency shift in SBL1 (see main text)
induced by power-dithering of the cascaded SBL. In the cavity-mode rotating frame, we write the
pump, SBL, and cascaded SBL in the following form,
̇
A
=
h
i
ω
p
ω
0

γ
2
i
A
g
s
|
α
|
2
A
+
s
γ
ex
P
̄
h
ω
p
(S1)
̇
α
=
h
i
(
ω
s
ω
1
)
γ
2
i
α
+
g
s
|
A
|
2
α
g
c
|
β
|
2
α
(S2)
̇
β
=
h
i
(
ω
c
ω
2
)
γ
2
i
β
+
g
c
|
α
|
2
β
(S3)
where
A
,
α
,
β
are the normalized photon number amplitudes of the pump, SBL1, and cascaded
SBL, respectively. Here we have adiabatically eliminated the phonon fields as described elsewhere
[
1
]. 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
s
and
c
are the phonon angular frequencies associated with SBL1 and the cascaded
SBL, respectively.
B
is the Brillouin shift, which is equal to
4
π
nc
s
/
λ
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 [1],
γ
ex
is the external coupling coefficient,
Γ
is the phonon damping rate.
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
=
γ
/
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
,
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
2
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
2
B
1
+
γ
/
Γ
1
+
γ
/
Γ

1
+
2
g
0
|
A
|
2
Γ

(S14)
Eq. (S13) gives the mode pulling induced in the cascaded Brillouin laser frequency [
2
] caused
by changes in the SBL1 frequency. This effect moves the cascaded wave’s frequency towards its
cavity mode frequency. Next, we assume for simplicity that operation is only slightly above the
cascaded threshold, such that Eq. (S8) gives 2
g
0
|
A
|
2
γ
. In this case Eq. (S14) simplifies into,
∂ω
s
|
A
|
2
=
1
1
+
γ
/
Γ
2
g
0
Γ
ω
p
ω
c
2
B

(S15)
The correction factor
γ
/
Γ
is small (
γ
/
Γ
1) in this system since the cavity linewidth is typically
much smaller than the phonon damping rate (i.e., justification for adiabtic approximation above).
Therefore, we drop the correction factor in Eq. (S15). Next, the
|
A
|
2
can be further replaced by
P
=
̄
h
ω
p
γ
2
|
A
|
2
/
γ
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
̄
h
ω
p
γ
2
Γ
ω
p
ω
c
2
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
̄
h
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 temperature dependence is dominated by the thermorefractive and ther-
moexpansion 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 [3]), 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
̄
h
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 tempera-
ture depends inversely upon the Brillouin damping rate
Γ
. Along these lines, forward Brillouin
scattering in high-Q silica microresonators has produced damping rates much smaller than for
back scattering on account of the smaller required Brillouin shift for backscattering [
4
,
5
]. We
intend to investigate this approach as a means to increase the backaction response.
2
To estimate
g
0
above (in the unit of rad/s), we use the following equations [6]:
g
0
̄
h
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 SBL1 at the cascading threshold,
Q
T
(
Q
E
) is the total (exter-
nal) quality factor,
ν
clamp
is the full-width-half-maximum of the fundamental SBL1 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.
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