1150
Vol. 7, No. 9 / September 2020 /
Optica
Letter
Linewidth enhancement factor in a microcavity
Brillouin laser
Zhiquan Yuan,
†
Heming Wang,
†
Lue Wu,
†
Maodong Gao,
AND
Kerry Vahala*
T. J. Watson Laboratory of Applied Physics, California Institute of Technology, Pasadena, California 91125, USA
*Corresponding author: vahala@caltech.edu
Received 2 April 2020; revised 27 July 2020; accepted 28 July 2020 (Doc. ID 394311); published 4 September 2020
The linewidth of regenerative oscillators is enhanced by
amplitude–phase coupling of the oscillator field [Phys. Rev.
160, 290 (1967)]. In laser oscillators, this effect is well known
for its impact on semiconductor laser performance. Here,
this coupling is studied in Brillouin lasers. Because their gain
is parametric, the coupling and linewidth enhancement are
shown to originate from phase mismatch. The theory is con-
firmed by measurement of linewidth in a microcavity Brillouin
laser, and enhancements as large as 50
×
are measured. The
results show that pump wavelength and device tempera-
ture should be carefully selected and controlled to minimize
linewidth. More generally, this work provides a new perspec-
tive on the linewidth enhancement effect.
© 2020 Optical
Society of America under the terms of the OSA Open Access Publishing
Agreement
https://doi.org/10.1364/OPTICA.394311
The linewidth of stimulated Brillouin lasers (SBLs) has received
considerable attention for some time. SBLs based on optical fiber
[1], for example, feature narrow linewidths that are useful for
generation of highly stable microwave sources [2,3]. More recently,
broad interest in microscale and nanoscale Brillouin devices [4]
has focused attention on tiny, often chip-scale, SBLs in several
systems [5–12]. These devices have high power efficiency [13] and
provide flexible operating wavelengths [14], and their fundamen-
tal linewidth can be reduced to less than 1 Hz [12,13]. For these
reasons, they are being applied in a range of applications including
radio-frequency (RF) synthesizers [15,16], ring laser gyroscopes
[12,17,18] and high-coherence reference sources [9].
SBLs derive gain through a process that is parametric in nature
and for which scattering of an optical pump into a Stokes wave
from an acoustic phonon must be phase matched [19,20]. When
the phonon field is strongly damped, the process mimics stimu-
lated emission. Nonetheless, phonon participation introduces
dramatic differences into SBL linewidth behavior compared
with conventional lasers. For example, while the conventional
Schawlow–Townes laser linewidth [21] is insensitive to tempera-
ture, the fundamental SBL linewidth is proportional to the number
of thermomechanical quanta in the phonon mode and therefore
to the Boltzmann energy
k
B
T
[13]. This dependence has been
verified from cryogenic to room temperature [22]. Brillouin lasers
can also oscillate on multiple lines through the process of cascade
[13], in which an initial Stokes wave can serve to pump a second
Stokes wave and so forth. Cascading introduces additional contri-
butions to the SBL linewidth [23]. Finally, the parametric nature
of the process means that pump phase noise couples through to the
laser linewidth, although it is strongly suppressed by the phonon
damping [24].
The fundamental linewidth of lasers is increased by the well-
known linewidth enhancement factor
α
that characterizes
amplitude–phase coupling of the field [25,26]. This quantity
is best known for its impact on the linewidth of semiconductor
lasers [27], and understanding, controlling, and measuring it
have long been subjects of interest [28–30]. Here, the linewidth
enhancement factor is studied in SBLs. The parametric nature
of Brillouin gain is shown to strongly influence this parameter.
Phase mismatch causes a nonzero
α
factor. Measurements of SBL
frequency noise are used to determine
α
versus controlled amounts
of phase mismatch, and the results are in good agreement with
theory. Significant enhancements to the linewidth are predicted
and measured, even when the SBL is operated only modestly away
from perfect phase matching.
Amplitude–phase coupling occurs at a specified optical
frequency when the real and imaginary parts of the optical sus-
ceptibility (equivalently, refractive index and gain) experience
correlated variations subject to a third parameter. The ratio of
the real to imaginary variation is the
α
parameter [27]. With a
nonzero
α
parameter, noise that normally couples only into the
laser field amplitude can also couple into the phase. And because
phase fluctuations are responsible for the finite laser linewidth [25],
the nonzero
α
factor thereby causes linewidth enhancement. For a
physical understanding of how a nonzero
α
parameter arises within
the SBL system, consider Fig. 1(a) (a detailed analysis is provided
in Section 1 of Supplement 1). Optical pumping at frequency
ω
P
on a cavity mode causes a Lorentzian-shaped gain spectrum
through the Brillouin process. The Brillouin gain spectrum is
frequency downshifted by the phonon frequency
(Brillouin
shift frequency) relative to the pumping frequency. Laser action
at frequency
ω
L
is possible when a second cavity mode lies within
the gain spectrum, which requires that
1ω
≡
ω
P
−
ω
L
is close in
value to
. Perfect phase matching corresponds to laser oscillation
at the peak of the gain (i.e.,
1ω
=
). Also shown in Fig. 1(a) is
the refractive index spectrum associated with the gain spectrum
according to the Kramers–Kronig relations. It is apparent that
α
(the ratio of the variation of real to imaginary susceptibility) will be
2334-2536/20/091150-04 Journal © 2020 Optical Society of America
Letter
Vol. 7, No. 9 / September 2020 /
Optica
1151
(a)
(b)
Fig. 1.
SBL phase mismatch illustration and experimental setup.
(a) Brillouin gain process in the frequency domain. Purple (brown) curve
refers to the pump (Stokes) cavity mode at frequency
ω
P
(
ω
S
). Blue curve
refers to the SBL laser signal at frequency
ω
L
. Orange and red curves
correspond to gain (g) spectrum and refractive index (
1
n), respectively.
Brillouin shift frequency (
), gain spectrum linewidth (
0
), and cavity
linewidth (
γ
) are also indicated. Frequency detunings
δω
and
δ
are
defined in the text. (b) Experimental setup for
α
and linewidth measure-
ment. An external cavity diode laser (ECDL) (Newport, TLB-6728) near
1550 nm passes through an erbium-doped fiber amplifier (EDFA) and is
coupled to the microcavity (a silica wedge resonator [8]) using a tapered
fiber [31,32]. Its frequency is Pound–Drever–Hall locked (not shown)
to the center of the cavity resonance. Pump power is controlled using an
acousto-optic modulator (AOM) as an attenuator in combination with
a feedback loop (not shown). The resonator diameter is around 7.1 mm,
corresponding to an FSR of 10.8 GHz, which is selected to closely match
the Brillouin shift frequency in silica at 1550 nm. The resonator chip tem-
perature is actively stabilized to 26.5000
±
0.0005
◦
C using a temperature
controller. The SBL emission propagates in the opposite direction of the
pumping due to the phase-matching condition. The emission is coupled
to a series of measurement instruments through a circulator. An optical
spectrum analyzer (OSA) is used to record the laser and pump spectra as
well as to measure SBL power. Pump and SBL signals are mixed on a fast
photodetector (PD) (Thorlabs, DXM30AF) to measure their frequency
difference. Another PD monitors the pumping power. An interferometer
is used to measure the laser frequency noise. Therein, the laser signal is
sent into an AOM that is split into frequency-shifted (first order) and
unshifted (zeroth order) signals. The latter is delayed in a 1-km-long fiber,
and then the two signals are mixed on a PD (Newport, 1811-FC). The
delay sets up a frequency-to-amplitude discriminator with a discrimina-
tion gain that is proportional to the amount of interferometer delay. To
measure the frequency noise spectral density, the detected current is mea-
sured using an electrical phase noise analyzer (PNA), and the spectrum is
fit to obtain the two-sided spectral density of the SBL laser (Section 2 of
Supplement 1).
zero for phase-matched operation, while it increases with increased
frequency detuning relative to perfect phase matching.
Analysis (Sections 1 and 3 of Supplement 1) shows that the
α
factor enhancement of the fundamental SBL linewidth
1ν
SBL
is
1ν
SBL
=
1ν
0
(
1
+
α
2
),
(1)
where
1ν
0
is the non-enhanced (
α
=
0) SBL linewidth [given
below in Eq. (3)] and the linewidth enhancement factor can be
expressed using two equivalent frequency-detuning quantities
relative to perfect phase matching:
α
=
2
δ
0
=
2
δω
γ
.
(2)
In the first quantity, phonon mode detuning
δ
≡
−
1ω
is
normalized by
0
, the Brillouin gain bandwidth (i.e., phonon decay
rate constant). In the second quantity, optical mode detuning
δω
≡
1ω
−
FSR (where FSR is the unpumped cavity free spectral
range) is normalized by
γ
, the photon decay rate constant. Note
that the sign of
α
changes to either side of perfect phase matching.
Also, as an aside,
δω
is the mode pulling induced by the Brillouin
gain spectrum [13].
1ν
0
is given by
1ν
0
=
(
0
γ
+
0
)
2
~
ω
3
L
n
th
4
π
Q
T
Q
ex
P
SBL
,
(3)
where
~
is the reduced Planck constant,
P
SBL
is the SBL output
power, and
n
th
is the number of thermal quanta in the phonon
mode. This expression is the same as that derived in Ref. [13],
except for the omission of the zero-point energy terms, and also
the inclusion of the near-unity correction factor
[
0/(γ
+
0)
]
2
relating to the finite damping rate of the phonons (derivation is
given in Section 3 of Supplement 1).
As a first step toward verification of Eqs. (1) and (2), it is nec-
essary to measure the phase mismatch detuning at each point
where the linewidth will be measured. The experimental setup and
information on the high-
Q
silica whispering gallery microcavity
used to generate Brillouin laser action are provided in Fig. 1(b) and
its caption. To vary the phase mismatch detuning, the pump laser
wavelength
λ
P
is tuned, which is achieved by selecting different
longitudinal modes within the same transverse mode family as
pump and Stokes modes. This has the effect of varying
through
the relationship
=
4
π
nc
s
/λ
P
(where
n
is the refractive index and
c
s
is the speed of sound in the microcavity). Since
is not directly
measurable in the experiment, we instead obtained information on
the phase mismatch using
δω
, which requires measurement of
1ω
and FSR.
The frequency
1ω
is determined by first measuring the beating
frequency of the pump and the SBL using a fast photodetector,
followed by measurement of the detected current on an electrical
spectrum analyzer. Beyond being influenced by mode pulling
as noted above, this beating frequency is also slightly shifted via
backaction of the amplitude–phase coupling (see Section 3 of
Supplement 1) and the optical Kerr effect [33], both of which are
proportional to the SBL powers. Therefore, to account for these
effects, the beat note frequencies were measured at five different
SBL power levels. Representative measurements performed at
three pump wavelengths are shown in Fig. 2(a). The
y
-intercept of
these plots provides the required beating frequency in the absence
of the above effects, and a summary plot of a series of such mea-
surements is provided as the blue square data points in Fig. 2(b). As
an aside, the data point near 1559 nm is missing because of strong
mode crossings at this wavelength in the SBL microcavity (i.e.,
higher-order mode families become degenerate with the SBL mode
family).
To determine the FSR at each pumping wavelength, the mode
spectrum of the resonator is measured by scanning a tunable laser
whose frequency is measured using an RF calibrated interferometer
[34]. The measured FSR is plotted versus wavelength as the dotted
line in Fig. 2(b). Measurement of the FSR this way also ensured
that pumping was performed on the same transverse mode family
with which the pumping wavelength was tuned. This is important
Letter
Vol. 7, No. 9 / September 2020 /
Optica
1152
(a)(b)(c)
Fig. 2.
Brillouin gain phase mismatch and
α
factor. (a) Beating frequency between the pump laser and the SBL is plotted as a function of SBL power.
Linear fitting is applied to eliminate the influence of the Kerr effect and
α
factor backaction, and the
y
-axis intercept is plotted as
1ω
in (b). Blue, red, and
yellow traces correspond to measurements at 1545 nm, 1538 nm, and 1532 nm, respectively. (b) The extrapolated beating frequency (squares) and FSR
(triangles) are plotted versus wavelength. The calculated
α
factor (red circles) is plotted versus wavelength using Eq. (2). The Brillouin gain center occurs
at around 1548 nm, where FSR
=
1ω
. (c) Total (
Q
T
), intrinsic (
Q
0
), and external (
Q
ex
) quality factors are plotted versus wavelength. The values are
measured in the same transverse mode family.
because the mode volume would change strongly if the mode
family were to change. In Fig. 2(b), the phase-matching condition
(gain center) occurs when FSR equals
1ω
(
δω
=
0) at a pump
wavelength around 1548 nm. We can also use the Brillouin shift
at the gain center to infer that
c
s
=
5845 m
/
s, which is consistent
with the material properties of silica [35].
Finally,
γ
is determined via measurement of the cavity
linewidth at each wavelength (equivalently, the total
Q
-factor
Q
T
of the resonator). By measuring both the linewidth and
transmission on cavity resonance it is possible to extract both the
intrinsic
Q
-factor
Q
0
and external coupling
Q
factor
Q
ex
at each
wavelength (1
/
Q
T
=
1
/
Q
0
+
1
/
Q
ex
). A plot of the results is
provided in Fig. 2(c). The
Q
0
values inferred this way are relatively
constant across the measured modes, while
Q
ex
exhibits variation
that reflects the wavelength dependency of the coupling condition.
The
Q
factors are significantly lower than those of state-of-the-art
resonators of the same kind [8], which is intentional and increases
the sensitivity of the noise measurement that follows. Using
Eq. (2), the theoretical
α
factor as a function of wavelength from
1532 nm to 1563 nm is plotted in Fig. 2(b) (red circles). Deviations
of beating frequency and the
α
factor from a linear trend are the
result of variations in the total
Q
factor across the measured wave-
lengths. The largest
α
factor is greater than 7, so a fundamental
linewidth enhancement of more than 1
+
7
2
=
50 is expected at
the largest detuning values.
A frequency discriminator method [36,37] is used to measure
the noise spectrum of the two-sided white frequency noise spectral
density
S
w
of the SBL, as described in the Fig. 1(b) caption. The
fundamental noise component in
S
w
, defined as
S
F
, is related to the
fundamental SBL linewidth through 2
π
S
F
=
1ν
SBL
[33] [
1ν
SBL
is given in Eq. (1)]. And the inverse power dependence contained
in
1ν
SBL
is used to extract
S
F
from the measurement of
S
w
. Data
plots of
S
w
versus inverse power at three pumping wavelengths are
given in the inset of Fig. 3 and reveal this power dependence. Of
importance to this measurement is that optical pumping power
was controlled by attenuation of the pump so that its phase noise
was constant throughout the measurement. Therefore, only the
Fig. 3.
SBL frequency noise enhancement. Measured SBL frequency
noise
S
F
(blue), theoretical
S
F
[Eq. (3)] prediction (green) with
α
obtained
from Fig. 2(b), and non-enhanced
S
0
formula (
α
=
0) [13] prediction
(yellow); all are plotted versus pump wavelength normalized to 1 mW
output power. Error bars on the
S
F
noise correspond to the error in deter-
mining slope (see inset). Error bars on the Eq. (3) prediction mainly arise
from
1ω
and Q measurement errors. Variations of the
α
=
0 prediction
mainly arise from
Q
ex
differences. Inset: SBL frequency noise
S
w
is plotted
versus the reciprocal of SBL output power. A linear fitting is applied to
determine
S
F
from the slope, and then plotted in the main panel. Blue,
red, and yellow data correspond to measurements at 1545 nm, 1538 nm,
and 1532 nm, respectively.
intrinsic contribution to linewidth could cause the observed power
dependence. The slope is equal to
S
F
normalized to an output
power of 1 mW. Linear fitting provides the slopes that are plotted
versus wavelength in the main panel of Fig. 3. The corresponding
minimum measured fundamental noise is about
S
F
=
0.2 Hz
2
/
Hz
Letter
Vol. 7, No. 9 / September 2020 /
Optica
1153
(
1ν
SBL
=
1.25 Hz) near the phase-matching condition (gain
center), and the maximum fundamental noise is more than
S
F
=
10 Hz
2
/
Hz (
1ν
SBL
=
63 Hz), corresponding to 50
×
noise
enhancement, at the largest mismatch detunings. Comparison to
Eq. (1) is provided via the green curve in Fig. 3. In this plot,
Q
T
,
Q
ex
, and
α
[Figs. 2(b) and 2(c)] measurements at each wavelength
are used with no free parameters.
γ
can be obtained from
Q
T
, and
we can infer
0/
2
π
to be 34.7 MHz, assuming it is constant over the
wavelength. Also,
n
th
=
572 is used (corresponding to the operat-
ing temperature of 26.5
◦
C). There is overall good agreement with
the measured linewidth values. The conventional
S
0
=
1ν
0
/(
2
π)
(with
α
=
0) is also plotted for comparison.
The nonzero intercept on the
y
axis of the inset in Fig. 3 is
believed to be related to transferred pump phase noise associated
with imperfect Pound–Drever–Hall locking. This contribution
will increase with increasing
α
. Both it and the linewidth-
enhancement-factor contribution to the pump phase noise are
discussed in Section 3 of Supplement 1. We have also verified the
α
measurement results in another SBL resonator. Details can be
found in Section 4 of Supplement 1.
As an aside, the relative intensity noise of the SBLs is another
important characteristic of laser operation, and a typical mea-
sured spectrum is shown in Section 5 of Supplement 1. Also, as
noted in the introduction, Brillouin lasers can provide multiline
oscillation via cascade [13], and under these conditions additional
terms appear in the linewidth expression [23]. In the context of
the present discussion, it is therefore of interest to consider the
impact of the
α
factor on linewidth under conditions of cascaded
operation. This is done in Section 6 of Supplement 1.
We have studied the linewidth enhancement factor
α
in a
Brillouin laser. A modification to the fundamental linewidth for-
mula that incorporates the
α
factor was theoretically derived and
then tested experimentally in a high-
Q
silica whispering gallery
resonator. Phase matching of the Brillouin process determines
the sign and magnitude of
α
. Under perfect phase-matching con-
ditions, corresponding to laser oscillation at the Brillouin gain
maximum,
α
=
0. However, measurement and theory show that
the mismatch (induced here by tuning of the pumping wavelength)
leads to
α
factors greater than 7, yielding frequency noise and fun-
damental linewidth enhancement as large as 50
×
. The sign of
α
can also be controlled through the sign of the frequency mismatch
detuning. Although the phase-matching condition was controlled
here via tuning of the pumping wavelength, it should also be pos-
sible to vary phase matching and therefore
α
through control of
the temperature. This would vary the Brillouin shift frequency
by way of the temperature dependence of the sound velocity. The
results presented here stress the importance of proper pumping
wavelength selection and observance of temperature control for
narrow-linewidth operation of SBLs. These considerations will be
important in all applications of these devices that are sensitive to
frequency noise and linewidth.
Funding.
Air Force Office of Scientific Research (FA9550-18-
1-0353); Caltech Kavli Nanoscience Institute.
Acknowledgment.
The authors thank Y. Lai, Q. Yang, and
C. Bao for helpful discussions.
Disclosures.
The authors declare no conflicts of interest.
See Supplement 1 for supporting content.
†
These authors contributed equally to this Letter.
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