Phonon-Limited-Linewidth of Brillouin Lasers at Cryogenic Temperatures
Myoung-Gyun Suh, Qi-Fan Yang, and Kerry J. Vahala
*
T. J. Watson Laboratory of Applied Physics, California Institute of Technology, Pasadena, California 91125, USA
(Received 12 June 2017; published 2 October 2017)
Laser linewidth is of central importance in spectroscopy, frequency metrology, and all applications of
lasers requiring high coherence. It is also of fundamental importance, because the Schawlow-Townes laser
linewidth limit is of quantum origin. Recently, a theory of stimulated Brillouin laser (SBL) linewidth has
been reported. While the SBL linewidth formula exhibits power and optical
Q
factor dependences that are
identical to the Schawlow-Townes formula, a source of noise not present in conventional lasers, phonon
occupancy of the Brillouin mechanical mode is predicted to be the dominant SBL linewidth contribution.
Moreover, the quantum limit of the SBL linewidth is predicted to be twice the Schawlow-Townes limit on
account of phonon participation. To help confirm this theory the SBL fundamental linewidth is measured at
cryogenic temperatures in a silica microresonator. Its temperature dependence and the SBL linewidth
theory are combined to predict the number of thermomechanical quanta at three temperatures. The result
agrees with the Bose-Einstein phonon occupancy of the microwave-rate Brillouin mode in support of the
SBL linewidth theory prediction.
DOI:
10.1103/PhysRevLett.119.143901
Stimulated Brillouin scattering (SBS) is a third-order
(
χ
3
) optical nonlinearity that results from the interaction
between photons and acoustic phonons in a medium
[1
–
4]
.
SBS has practical importance in optical fiber systems
[5,6]
where it is an important signal impairment mechanism in
long-distance transmission systems
[7]
and makes possible
all-fiber lasers
[8]
as well as tunable, slow-light generation
[9]
. Power fluctuation resulting from thermal phonons has
also been studied in fiber-optic SBS Stokes wave gener-
ation
[10]
, and the intensity and phase noise have been
measured in narrow-linewidth Brillouin lasers
[11]
. More
recently, the SBS process has attracted considerable interest
in microscale and nanoscale devices
[12]
. Brillouin laser
action has been demonstrated in several microcavity reso-
nator systems including silica
[13
–
16]
,CaF
2
[17]
, and
silicon
[18]
, and Brillouin amplification has been demon-
strated in integrated chalcogenide waveguides
[19]
.In
silicon waveguides, the use of confinement to enhance
amplification has been studied
[20]
. SBS is also a powerful
tool for integrated photonics signal processing
[21
–
23]
, and
it has been applied to realize a chip-based optical gyroscope
[24]
. Moreover, at radio-frequency rates, the SBS damping
rate is low enough in certain systems to enable cavity
optomechanical effects
[25]
including optomechanical cool-
ing
[26]
and optomechanical-induced transparency
[27]
.
This work studies a recent prediction concerning the
fundamental linewidth (i.e., nontechnical noise contribu-
tion to linewidth) of the stimulated Brillouin laser (SBL).
The analysis of fundamental fluctuations in Brillouin
devices falls into a more general category of optomechan-
ical oscillators in which phonons participate in the oscil-
lation process
[28,29]
. This participation creates a channel
for fundamental sources of mechanical noise to couple into
the ocillator. For comparison, the SBL fundamental line-
width
[15]
and the conventional laser Schawlow-Townes
(ST) linewidth
[30]
in Hertz are given below,
Δ
ν
SBS
¼
ℏ
ω
3
4
π
PQ
T
Q
E
ð
n
T
þ
N
T
þ
1
Þ
;
ð
1
Þ
Δ
ν
ST
¼
ℏ
ω
3
4
π
PQ
T
Q
E
N
T
þ
1
2
ð
2
Þ
where
n
T
is the number of thermal quanta in the mechanical
field at the Brillouin shift frequency,
N
T
is the number of
thermal quanta in the laser mode (negligible at optical
frequencies and henceforth ignored),
P
is the laser output
power,
Q
T
(
Q
E
) is the total (external)
Q
factor (note:
Q
−
1
T
¼
Q
−
1
0
þ
Q
−
1
E
with
Q
0
the instrinsic
Q
), and
ω
is the laser
frequency. At very low temperatures where
n
T
is negligible,
the quantum-limited SBL linewidth is twice as large as the
Schawlow-Townes linewidth on account of phonon par-
ticipation in the laser process. At finite temperatures
n
T
is
predicted to provide the dominant contribution to the
fundamental SBL linewidth. SBL fundamental linewidth
measurements at room temperature are consistent with this
prediction
[15]
. In this study, the phonon contribution to
Eq.
(1)
is verified by determination of
n
T
over a wide range
of temperatures followed by comparison to the Bose-
Einstein phonon occupancy.
It is important to note that in addition to the fundamental
phase noise [Eq.
(1)
] there is also an important technical
noise contribution to the Brillouin linewidth. Specifically,
because the Brillouin process is fundamentally a parametric
process, pump-phase-noise leaks into the phase of the
Stokes wave
[11,15,29,31]
. This pump noise is theoretically
predicted and experimentally observed to be strongly
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© 2017 American Physical Society
suppressed by the stronger damping of phonons relative to
the optical Stokes wave
[11,31]
, but can nonetheless
dominate the Brillouin linewidth if the pump linewidth is
large enough. In prior work, low noise optical pumping has
been shown to enable observation of the fundamental
Brillouin noise in resonators like those studied here
[15,22]
. These pumping conditions are used here to observe
the fundamental noise.
Figure
1(a)
shows the measurement setup. Pump and
signal light are conveyed using fiber optic cables. After
passing through an optical circulator, the pump laser passes
into the cryostat using a fiber vacuum feedthrough. Inside
the cryostat the pump laser power is evanescently coupled
to a silica disk microresonator using a fiber taper that is
positioned piezoelectrically [Fig.
1(b)
]. Pumping power to
the resonator as high as 20 mW was possible. The silica
microresonator, shown in Fig.
1(c)
, is a wedge design
[14]
.
The cryostat is an open-loop continuous-flow unit and
was cooled to 77 K using liquid nitrogen and to 8 K using
liquid helium.
Brillouin laser action proceeds as diagrammed in
Fig.
1(d)
where cascaded lasing is illustrated. Pump light
coupled to a resonator mode induces Brillouin gain over a
narrow band of frequencies that are downshifted by the
Brillouin shift frequency
Ω
=
2
π
¼
2
nV
s
=
λ
P
where
V
s
is
the sound velocity,
n
is the refractive index, and
λ
P
is the
pumping wavelength
[15]
. At room temperature in the
silica devices tested here, the Brillouin-shift frequency is
10.8 GHz for optical pumping near
1
.
55
μ
m. When the
cavity free-spectral range approximately equals
Ω
, stimu-
lated Brillouin lasing is possible creating a first-Stokes
wave (FSR-
Ω
matching occurs for resonator diameters near
6 mm in the silica devices tested here). This Stokes wave
propagates backward relative to the pump wave on account
(a)
(b)
(d)
(e)
ECDL
EDFA
PDH Feedback
Cryostat
μ - resonator
PD
PD
Electrical signal
Optical signal
(c)
Oscilloscope
L(f) Analyzer
/ ESA
OSA
6 mm
PC
Fast
PD
Frequency
Pump 1
1
st
Stokes
Forward
FSR
Backward
2
nd
Stokes
3
rd
Stokes
cavity mode
SBS gain
Stokes wave
Pump Laser
1566.0
1566.5
1567.0
1567.5
1568.0
1568.5
-100
-80
-60
-40
-20
0
Optical Power (dBm)
Wavelength (nm)
1
st
Stokes
3
rd
Stokes
5
th
Stokes
-250
0
250
-100
-50
0
RF Power (dBm)
Frequency (kHz + 21.53 GHz)
RBW = 1 kHz
FIG. 1. Experimental setup and Brillouin laser action. (a) Experimental setup showing external cavity diode laser (ECDL) pump,
erbium-doped fiber amplifier (EDFA), polarization control (PC), and circulator coupling to the cryostat. Green lines indicate optical
fiber. A fiber taper is used to couple to the microresonator. Pump and even-ordered stimulated Brillouin laser (SBL) waves propagate in
the forward direction while odd-ordered SBL waves propagate in the backward direction and are coupled using the circulator.
Photodetectors (PD) and an oscilloscope monitor the waves propagating in both directions. A fast photodetector measures the first and
third beatnote which is measured using an electrical spectrum analyzer (ESA) and phase noise [L(f)] analyzer. An optical spectrum
analyzer (OSA) also measures the backward propagating waves. The pump laser is locked to the microresonator optical resonance using
a Pound-Drever-Hall (PDH) feedback loop. (b) Schematic of the optical fiber taper coupling setup inside the cryostat. Optical fiber (red)
is glued to an aluminum holder which is fixed on a 3-axis piezoelectric stage. The microresonator is mounted on a copper plate. (c) Top
view of the 6 mm wedge disk resonator. (d) Illustration of cascaded Brillouin laser action. Pump and even Stokes orders propagate in the
forward direction while odd orders propagate in the backward direction. Green curves represent the Brillouin gain spectra. Brillouin shift
frequency (
Ω
) and free-spectral range (FSR) are indicated. (e) Optical spectrum measured using the OSA and showing cascaded
Brillouin laser action to fifth order. Inset: typical electrical beatnote spectrum produced by the first- and third-order SBL signals.
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of the Brillouin phase matching condition, and emerges
from the cryostat at the fiber input [Fig.
1(a)
]. With
increasing pump power, the first-Stokes wave will grow
in power and ultimately induce laser action on a second-
Stokes wave, which, by phase matching, propagates in the
forward direction. Phase matching ensures that odd (even)
orders propagate backward (forward), and follow fiber-optic
paths in Fig.
1(a)
to measurement instruments. Figure
1(e)
is
a spectrum of cascaded laser action to fifth order measured
using the OSA. Odd orders appear stronger than the even
orders (and the pump signal) because the OSA is arranged to
detect odd orders. Even-order detection occurs because of
weak backscattering in the optical system.
The SBL cascade obeys a system of rate equations
relating the circulating photon number,
p
n
, of the
n
th-
Stokes wave to the circulating photon number,
p
n
−
1
, of its
preceding
ð
n
−
1
Þ
st-pump wave
[15]
.
_
p
n
¼
g
n
p
n
−
1
p
n
−
ω
n
Q
T
p
n
;
ð
3
Þ
where
g
n
¼
ℏ
ω
n
v
2
g
Γ
g
B
V
eff
≈
ℏ
ω
n
v
g
Γ
Ω
2
π
g
B
A
eff
ð
4
Þ
is the Brillouin gain coefficient for the
n
th-Stokes wave in
Hertz units. Here,
ω
n
is the optical frequency of the
n
th
Stokes wave,
v
g
is the group velocity,
Γ
is the phonon-
photon mode overlap factor (defined as the optical mode
area,
A
eff
, divided by the acousto-optic effective mode area
[32]
),
g
B
is the bulk Brillouin gain coefficient of silica,
V
eff
is the effective optical mode volume of the
n
th-Stokes
wave, and
Ω
¼
FSR is assumed.
g
B
=A
eff
is the normalized
Brillouin gain coefficient in W
−
1
m
−
1
units.
As pumping to the resonator is increased, a Stokes wave
will begin to lase and increase in power until it clamps
when the threshold condition for the next Stokes wave in
the cascade is reached. This clamped power,
P
clamp
, follows
directly from the steady-state form of Eq.
(3)
,
P
clamp
¼
ω
n
−
1
Q
E
ℏ
ω
n
−
1
p
n
−
1
≈
1
g
ℏ
ω
3
Q
T
Q
E
ð
5
Þ
where the approximation results from letting
ω
n
−
1
≈
ω
n
and in the final result the Stokes order,
n
, is suppressed. At
this clamped power, the fundamental SBL linewidth
follows by substitution of Eq.
(5)
into Eq.
(1)
,
Δ
ν
clamp
¼
g
4
π
ð
n
T
þ
1
Þ
:
ð
6
Þ
It is useful to note that the SBL linewidth in the clamped
condition is independent of
Q
T
and
Q
E
. From Eq.
(5)
,
measurement of
P
clamp
,
Q
T
, and
Q
E
are sufficient to
determine
g
. If combined with Eq.
(6)
and measurement
of
Δ
ν
clamp
then
n
T
can be determined at each operating
temperature; see the Supplemental Material
[33]
.
(a)
1564
1565
1566
1567
1568
1569
Power (mW)
0
0.5
1
1.5
2
2.5
(b)
1529
1530
1531
1532
1533
0
0.2
0.4
0.6
0.8
1
Wavelength (nm)
1528
1529
1530
1531
1532
1533
0
0.1
0.2
0.3
0.4
0.5
T = 300 K
T = 77 K
T = 8 K
Power (mW)
Power (mW)
Frequency
SBS Gain
Pump
Cavity Resonance
On gain peak
k
a
e
p
n
i
a
g
f
f
O
k
a
e
p
n
i
a
g
f
f
O
FSR
FIG. 2. Aligning the third-Stokes wave to the Brillouin gain
spectrum maximum. (a) Illustration showing spectral place-
ment of Stokes wave with respect to Brillouin gain spectrum
maximum. Variation of the pumping wavelength causes the
Brillouin shift frequency (
Ω
) to vary and thereby scans the
Stokes wave across the Brillouin gain peak. (b) Measured
spectra of
P
clamp
, the clamped third-order Stokes wave power
(black circle). The three panels show measurements per-
formed at
T
¼
300
, 77, and 8 K. Each color corresponds
to a distinct pumping wavelength. The first-order Stokes wave
also appears in the spectral map as the stronger peak near the
third-order Stokes wave. The pump wave is not observable in
the linear-scale spectrum as i
t propagates in the direction
opposite to the first-order and third-order Stokes waves.
P
min
clamp
is determined from the fitted red curve as the minimum
power point.
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