10
3
10
4
10
5
10
6
10
7
10
8
10
9
−180
−160
−140
−120
−100
−80
−60
−40
−20
o!set frequency (Hz)
L(f) (dBc/Hz)
−500
0
500
−90
−80
−70
−60
−50
−40
−30
−20
−10
0
kHz +21.721GHz
RF Power (dBm)
Supplementary Figure S1:
High RF power, low white phase noise.
Main panel: Single-sideband phase noise of a high-
power open-loop Brillouin oscillator. A record low white ph
ase noise floor (-160 dBc/Hz) for a microcavity-based microw
ave
source is achieved for offset frequencies
>
200 MHz. Inset: RF spectrum of an open-loop Brillouin oscill
ator measured directly
from a highspeed photodetector without any amplification.
2
Supplementary Note 1: Cascade and phase noise in a Brillouin
microwave oscillator
For cascaded Stimulated Brillouin lasers, the N-th Stokes w
ave is excited once the intra-cavity circulating power
of the (N-1)-th Stokes wave reaches the threshold excitatio
n level. It has been shown that the excitation of the
N-th Stokes wave will cause the intracavity power of the (N-1
)-th Stoke wave to be clamped [44] on account of gain
clamping. For microwave generation by heterodyne mixing of
the first and third Stokes waves, the operation described
in the main text is to achieve maximum output power of the thir
d Stokes wave just before threshold excitation of
the fourth Stokes wave. It can be shown that the Schawlow-Tow
nes frequency noise of a third Stokes wave at the
threshold of excitation of the fourth Stokes wave is:
S
ν
(
f
) =
~
ωg
c
[
n
T
+
N
T
+ 1]
4
π
2
≈
g
c
kT ω
4
π
2
Ω
B
=
g
c
kT
4
π
2
c
2
nV
A
(S1)
where
n
T
(
N
T
) is the number of thermal quanta in the mechanical (optical)
field. At room temperature,
N
T
is
negligible and
n
T
1, which gives the approximate form in the last two expressio
ns in equation (S1).
ω
(Ω
B
) is
the angular frequency for the optical (mechanical) field, an
d
ω
Ω
B
=
c
2
nV
A
.
g
c
is the microcavity Brillouin gain and is
related to the bulk Brillouin gain parameter
g
0
by
g
c
=
c
2
g
0
n
2
V
eff
(1+
4∆Ω
2
Γ
2
)
(Lorentzian Brillouin gain spectrum).
V
eff
is
the effective mode volume, Γ is the Brillouin phonon damping r
ate, and ∆Ω is the offset of the cavity FSR relative
to the Brillouin phonon frequency Ω
B
. It can be seen from equation (S1) that the noise limit is inde
pendent of
both the pump frequency and the phonon frequency. Assuming t
hat cascade is not inhibited, equation (S1) sets a
limit of the 1
/f
2
phase noise of the Brillouin oscillator upon photomixing th
e first and the third Stokes lines. From
equation (S1) and the volume dependence of
g
c
, it can be seen that a larger effective mode volume will help re
duce
the microcavity Brillouin gain parameter
g
c
and also reduce the Schawlow-Townes limit of the Brillouin o
scillator.
Moreover, operation at cryogenic temperatures would lower
this noise limit as well through reduction of the phonon
thermal quanta. The incident pump power at which the cascade
transitions from the third to fourth Stokes oscillation
is given by the following form:
P
th
=
27
π
2
n
2
V
eff
̃
g
0
Q
2
T
λ
2
(S2)
where ̃
g
0
≡
g
0
1+
4∆Ω
2
Γ
2
, and
Q
T
is the loaded Q factor of the cavity at critical coupling. The
refore, in systems wherein
the phase noise is limited by cascade the high optical Q enabl
es attainment of this noise limit at low pump power
levels.
3
Supplementary Note 2: High-RF-power, low white-phase-noi
se of the Brillouin Microwave Oscillator
High incident power to the fast photodetector means high RF p
ower without amplification, and also a lower shot-
noise-limited white phase noise floor. Higher RF power direc
tly from the photodetecter also makes the interconnection
of the microwave Brillouin oscillator to the other RF compon
ents (e.g. the frequency divider) more convenient since it
avoids the requirement of a microwave amplifier. Considerin
g the heterodyne beating of two lasers with power
P
1
and
P
2
, onto a photodetector with responsivity
R
s
and load impedance
R
L
, the RF power is given as
P
RF
= 2
R
2
s
P
1
P
2
R
L
.
The shot-noise-limited phase noise floor is given as
L
(
f
) = 10
log
(
qR
s
(
P
1
+
P
2)
R
L
P
RF
)
. In the experiment, power levels as
high as 11.8 mW and 4.4 mW for the first and third Stoke lines wer
e delivered to the fast photodetector (PD). The
generated average photocurrent was 10.66 mA, with a photode
tector responsivity of 0.65 A/W. The microwave power
measured at the RF spectrum analyzer was -1.38 dBm as shown in
the inset of figure S1. Taking into account about
4.3 dB loss of the RF coax cable and DC block, the microwave pow
er generated directly at the PD is 2.9 dBm (1.9
mW). It is important to note that this number is over 20 dB larg
er than what has been possible using microcombs [31,
32, 35]. For comparison, the calculated RF power is 3.4 dBm an
d the calculated shot-noise limited white phase noise
floor is -163 dBc/Hz. In the measurement, the white phase nois
e floor shown in figure S1 is -160 dBc/Hz for offsets
above 200 MHz, which is close to the calculated value. (Note:
the instrument white phase noise floor is lower than
-164 dBc/Hz.) In summary, a record-high, RF power and low pha
se noise floor for a microresonator-based microwave
source has been demonstrated.