of 6
The 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.
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 two-level 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 thermo-mechanical 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.
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 mecha-
nism 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 generation [10]. More recently, the SBS process has
attracted considerable interest in micro and nanoscale de-
vices [11]. Brillouin laser action has been demonstrated
in several microcavity resonator systems including silica
[12–15], CaF
2
[16] and silicon [17], and Brillouin am-
plification has been demonstrated in integrated chalco-
genide waveguides [18]. In silicon waveguides, the use of
confinement to enhance amplification has been studied
[19]. SBS is also a powerful tool for integrated photon-
ics signal processing [20–22], and it has been applied to
realize a chip-based optical gyroscope [23]. Moreover,
at radio-frequency rates, the SBS damping rate is low
enough in certain systems to enable cavity optomechani-
cal effects [24] including optomechanical cooling [25] and
optomechanical-induced transparency [26].
This work studies a recent prediction concerning the
fundamental linewidth (i.e., non technical noise contribu-
tion to linewidth) of the stimulated Brillouin laser (SBL).
The formula for SBL linewidth in Hertz (full-width half
maximum) [14] and the conventional laser Schawlow-
Townes linewidth (2-level laser system) [27] are given be-
low,
ν
SBS
=
~
ω
3
4
πPQ
T
Q
E
(
n
T
+
N
T
+ 1)
,
(1)
ν
2
Level
=
~
ω
3
4
πPQ
T
Q
E
(
N
T
+
1
2
)
(2)
where
n
T
is the number of thermal quanta in the me-
chanical field at the Brillouin shift frequency,
N
T
is the
number of thermal quanta in the Stokes optical field (neg-
ligible at optical frequencies and henceforth ignored),
P
is
the SBL output power,
Q
T
(
Q
E
) is the total (external) Q-
factor, and
ω
is the laser frequency. Eq. (1) is valid when
the Brillouin gain bandwidth is much broader than the
optical cavity linewidth. At very low temperatures where
n
T
is negligible, the quantum limited SBL linewidth is
twice as large as the Schawlow-Townes linewidth on ac-
count of phonon participation in the laser process. At
finite temperatures
n
T
is predicted to provide the dom-
inant contribution to the fundamental SBL linewidth.
SBL linewidth measurements at room temperature are
consistent with this prediction [14]. In this study, the
phonon contribution to eq. (1) is verified by determina-
tion of
n
T
over a wide range of temperatures using eq.
(1) followed by comparison to the Bose-Einstein phonon
occupancy at these temperatures.
Figure 1(a) shows the measurement setup. Pump and
signal light are conveyed using fiber optic cable (green
lines in figure 1(a)). After passing through an opti-
cal 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 (Figure 1(b)). Pumping power to the
resonator as high as 20 mW was possible. The silica
microresonator, shown in figure 1(c), is a wedge design
[13]. The resonator diameter was approximately 6 mm to
phase match the Brillouin process at cryogenic tempera-
tures (see discussion below). The cryostat is an open-loop
continuous-flow unit made by Janis and was cooled to 77
K using liquid nitrogen and to 8 K using liquid helium.
Brillouin laser action proceeds as diagrammed in fig-
ure 1(d) where cascaded lasing is illustrated. Pump light
coupled to a resonator mode induces Brillouin gain over
a narrow band of frequencies shown in green (typically
arXiv:1706.03359v1 [physics.optics] 11 Jun 2017
2
(a)
(b)
(d)
(e)
ECD
L
EDFA
PDH Feedbac k
Re
fl
ected pump,
1
st
S BL, 3
rd
SBL, ...
circulat or
Cr yostat
μ
- resonator
PD
PD
Ele ctric al s ig nal
Optic al sig nal
Pu
mp,
2
nd
S BL, 4
th
SBL, ...
Pu
mp
(c)
Oscilloscope
L(f) Anal yze r
/ ESA
OSA
6 mm
PC
Fast
PD
F
requenc
y
Pump 1
1
st
S toke s
F
orw
ard
FSR
Bac
k
w
ard
2
nd
S toke s
3
rd
S toke s
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 P
ower (dBm)
Wavelength (nm)
1
st
Stokes
3
rd
Stokes
5
th
Stokes
-250
0
250
-100
-50
0
RF P
ower (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 oscillopscope monitor the waves propagating in both directions.
A fast photodetector measures the 1
st
/ 3
rd
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 5th order. Inset: typical electrical beatnote spectrum produced by the 1
st
and 3
rd
order Stokes laser signals.
20 - 60 MHz wide) that are down-shifted by the Brillouin
shift frequency Ω
/
2
π
= 2
nV
s
P
where
V
s
is the sound
velocity,
n
is the refractive index and
λ
P
is the pump-
ing wavelength [14]. 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 (FSR) approximately equals Ω, stim-
ulated Brillouin lasing is possible creating a 1
st
-Stokes
laser wave. This wave propagates backward relative to
the pump wave on account of the Brillouin phase match-
ing condition, and emerges from the cryostat at the fiber
input (figure 1(a)). With increasing pump power, the
1
st
-Stokes laser wave will grow in power and ultimately
induce laser action on a 2
nd
-Stokes laser wave, which,
by phase matching, propagates in the forward direction.
This cascaded laser process is illustrated in figure 1(d)
to 3
rd
order. Phase matching ensures that odd (even)
orders propagate backward (forward), and follow fiber-
optic paths in figure 1(a) to measurement instruments.
Figure 1(e) is a spectrum of cascaded laser action to 5
th
order measured using the OSA in figure 1(a). Odd orders
appear stronger than the even orders (and the pump sig-
nal) because the OSA is arranged to detect odd orders
(see figure 1(a)). Even-order detection occurs because of
weak back-scattering 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 laser wave to the circulating photon number,
p
n
1
,
3
(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)
Freq uen cy
SBS Gain
Pump
Cavity Resonance
On gain peak
Off gain peak
Off gain peak
FSR
FIG. 2.
Aligning the 3
rd
Stokes wave to the Brillouin
gain spectrum maximum
(a) Illustration showing spectral
placement of Stokes wave with respect to Brillouin gain spec-
trum 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 3
rd
-order Stokes wave power (indicated
by the black circle in the plots). The three panels show mea-
surements performed at T = 300 K, 77 K and 8 K. Each color
corresponds to a distinct pumping wavelength. The 1
st
-order
Stokes wave also appears in the spectral map as the stronger
peak near the 3
rd
-order Stokes wave. The pump wave is not
observable in the linear-scale spectrum as it propagates in
the direction opposite to the 1
st
-order and 3
rd
-order Stokes
waves.
P
min
clamp
is determined (with corresponding pumping
wavelength) from the fitted red curve as the minimum power
point.
of its preceding (n-1)
st
-pump wave [14].
̇
p
n
=
g
n
p
n
1
p
n
ω
n
Q
T
p
n
,
(3)
where
g
n
=
~
ω
n
v
2
g
Γ
g
B
V
eff
~
ω
n
v
g
Γ
2
π
(
g
B
A
eff
)
(4)
is the Brillouin gain coefficient for the n
th
-Stokes laser
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 [28]),
g
B
is the bulk Brillouin gain
coefficient of silica, and
V
eff
is the effective optical mode
volume of the n
th
-Stokes laser wave.
g
B
/A
eff
is the nor-
malized Brillouin gain coefficient in
W
1
m
1
unit that
is typically measured in optical fibers.
The gain coefficient has the spectral profile of the Bril-
louin gain spectrum (i.e., green curves in figure 1(d)). 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
=
ω
n
1
Q
E
~
ω
n
1
p
n
1
1
g
~
ω
3
Q
T
Q
E
(5)
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 linewidth of the
Stokes laser mode follows by substitution of eq. (5) into
eq. (1),
ν
clamp
=
g
4
π
(
n
T
+ 1)
.
(6)
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 de-
termine
g
. If combined with eq. (6) and measurement
of ∆
ν
clamp
then
n
T
can be determined at each operating
temperature.
g
depends on the placement of the Stokes wave within
the Brillouin gain spectrum (see green spectral curve in
figure 1(d)), and the value of
g
at the spectral maxi-
mum (defined as
g
0
) was also determined for comparison
with theory. To determine this maximum, the pump was
tuned while recording
P
clamp
. This causes the Brillouin
shift frequency Ω to also tune, and therefore to vary the
spectral location of the Stokes wave within the Brillouin
gain band (see figure 2(a)).
P
clamp
will be minimum
(denote as
P
min
clamp
) when the Stokes wave is spectrally
aligned to the maximum value of
g
, thereby allowing
determination of the pumping wavelength corresponding
4
to maximum
g
. At this pumping wavelength, the value
P
min
clamp
can be used to determine
g
o
from eq. (5) when
Q
T
and
Q
E
are measured.
Spectra showing multiple measurements of clamped
power for the 3
rd
-Stokes wave at different pumping wave-
lengths are presented in figure 2(b). The three panels
show spectra at T= 300, 77, and 8 K. The 3
rd
-Stokes
wave spectral peak at each pumping wavelength is identi-
fied by a black circle. At each temperature, the minimum
clamped power and corresponding wavelength are deter-
mined from the quadratic fit (red curve in figure 2(b)).
At this pumping wavelength
g
=
g
0
(center case in green
box in figure 2(a)). As an aside, the power clamping con-
dition for the 3
rd
-Stokes wave was determined by mon-
itoring the onset of laser action in the 4
th
-Stokes wave.
Also, optical losses between the resonator and the OSA
were calibrated to determine the clamped power. Table
I summarizes the measured minimum clamped powers,
P
min
clamp
, and their corresponding pumping wavelengths.
Q
T
and
Q
E
are also given and were determined by fit-
ting both the linewidth and the transmission minimum
of the Stokes mode. Finally,
g
0
, calculated using eq. (5),
is compiled in the table I.
As an aside, the Brillouin gain bandwidth, ∆
ν
B
, is also
extracted from a quadratic fit of the curves in fig. 2(b)
[14]. We measure 20 MHz at 300 K, 25 MHz at 77 K,
and 35 MHz at 8 K. These linewidths reflect the damp-
ing rate of the Brillouin process and have been the focus
of theory and experiment in silica optical fiber [29, 30].
The measured temperature dependence is not consistent
with theory and is believed to result from different op-
tical and acoustical mode families participating in the
Brillouin process at different temperatures. To partially
test this hypothesis, Brillouin linewidths were measured
at room temperature by inducing Brillouin laser action
on a range of different cavity modes. Linewidths in the
range 15 MHz - 45 MHz were measured suggesting that
damping of the Brillouin process is strongly affected by
the spatial structure of the mode. This could, for ex-
ample, result from differences in the surface interactions
of the various spatial acoustical modes with the wedge
resonator dielectric-air interface. It is important to note
that this behavior in no way affects the measurement of
n
T
since it is the measured value of
g
and not the theo-
retical value of
g
that matters.
TABLE I.
Experimental parameters for Brillouin gain
(
g
0
) calculation
T
λ
Q
T
Q
E
P
min
clamp
g
0
(
K
) ( nm ) ( x 10
6
) ( x 10
6
) ( mW ) ( Hz )
300 1567.0
40
50.5
0.4
0.2272
77 1531.1
82.5
91
0.2423 0.1082
8
1530.8
94
103
0.0935 0.2174
To measure the laser linewidth, the beat of the 1
st
and 3
rd
Stokes laser waves is detected using a fast pho-
todetector. An electrical spectrum analyzer trace of this
beat is provided as the inset in figure 1(e). As described
elsewhere [21] the phase noise of this beat signal pro-
vides spectral components associated with the fundamen-
tal phase noise of the Stokes waves and can be used to in-
fer the linewidth. Moreover, because the 1
st
-Stokes wave
has more power than the 3
rd
-Stokes wave (see figure 2(b))
the fundamental linewidth of the 1
st
-Stokes wave is nar-
rower. Accordingly, fundamental phase noise in the beat
signal is dominated by the 3
rd
-Stokes wave. Also, micro-
cavity technical frequency noise, while present, is reduced
in this measurement because the two Stokes laser waves
lase within a single cavity.
The phase noise of 1
st
/3
rd
-order SBL beatnote is mea-
sured at the
P
min
clamp
wavelength determined in figure
2(b). A Rhode-Schwarz FSUP26 phase noise analyzer
was used. The measured phase noise spectra at 300 K,
77 K and 8 K are shown in figure 3(a). The theoretical
phase noise spectrum for an SBL limited by fundamental
noise is given by the expression,
L
(
ν
) =
ν
min
2
ν
2
=
g
0
8
πν
2
(
n
T
+ 1)
(7)
where ∆
ν
min
is the fundamental linewidth given by eq.
(6) with
g
=
g
0
and the second equality in eq. (7) uses
eq. (6). The black dashed lines in figure 3(a) give mini-
mum noise-level fits to the measured phase noise spectra
using eq. (7). Even with the common-mode noise sup-
pression noted above, there is considerable technical noise
coupling to the phase noise spectrum from the cryogenic
system; and fitting is not possible over the entire spectral
range. The corresponding ∆
ν
min
is plotted in figure 3(b).
By using
g
0
from the table I and the ∆
ν
min
data in fig-
ure 3(b), eq. (7) provides values for
n
T
at the three tem-
peratures. These inferred
n
T
values are plotted versus
temperature in figure 3(c). The Bose-Einstein thermal
occupancy is also provided for comparison. The discrep-
ancy between the lowest temperature
n
T
value and the
Bose-Einstein value could result from parasitic optical
heating or temperature difference between the tempera-
ture sensor and the resonator. A calibrated temperature
of 22 K is estimated using the Bose-Einstein curve.
In summary, stimulated Brillouin lasers are unusual
because their fundamental linewidth is predicted to be
limited by thermo-mechanical quanta of the Brillouin
mode. We have confirmed this prediction by determin-
ing the thermal phonon occupancy versus temperature
using the SBL phase noise. Measurements at 300 K, 77
K and 8 K are in good agreement with the expected Bose-
Einstein occupancy. This work provides a possible way to
reduce the SBL linewidth for precision measurements. It
also lends support to the theoretical prediction that the
quantum-limited linewidth of an SBL is strongly influ-