of 7
Supplemental Document
Linewidth enhancement factor in a microcavity
Brillouin laser: supplement
Z
HIQUAN
Y
UAN
,
H
EMING
W
ANG
,
L
UE
W
U
,
M
AODONG
G
AO
,
AND
K
ERRY
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 Letter.
This supplement published with The Optical Society on 4 Sepember 2020 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.12730412
Parent Article DOI: https://doi.org/10.1364/OPTICA.394311
Supplementary Material
1
Linewidth
enhancement
factor
in
a
microcavity
Brillouin
laser:
supplementary
material
Z
HIQU
AN
Y
U AN
1,†
,
H
EMING
W
ANG
1,†
,
L
UE
W
U
1,†
,
M
A ODONG
G
AO
1
,
AND
K
ERRY
V
AHALA
1,*
1
T.
J.
Watson
Laboratory
of
Applied
Physics,
California
Institute
of
Technology,
Pasadena,
California
91125,
USA
These
authors
contributed
equally
to
this
work
*
Corresponding
author:
vahala@caltech.edu
Compiled
July
28,
2020
This
document
provides
supplementary
information
to
L
inewidth
enhancement
factor
in
a
microcavity
Brillouin
laser
.
In
section
1,
we
discuss
the
equivalence
of
the
α
factor
and
the
normalized
detuning
of
the
Brillouin
laser,
as
well
as
its
connections
to
the
α
factor
commonly
used
in
semiconductor
lasers.
In
section
2,
we
review
the
frequency
discriminator
measurements
used
in
the
main
text
to
determine
laser
frequency
noise.
In
section
3,
we
present
a
full
analysis
of
the
Brillouin
laser
noise,
including
the
effects
of
pump
phase
noise.
In
section
4,
a
measurement
of
the
α
factor
enhanced
noise
measured
on
a
second
device
is
presented.
In
section
5
measurement
of
the
relative
intensity
noise
of
the
Brillouin
laser
is
presented.
In
section
6,
we
study
the
effect
of
the
anti-Stokes
process
on
the
α
-factor
linewidth
enhancement
in
a
cascaded
Brillouin
laser
system.
1. DERIVATION OF THE ALPHA FACTOR IN BRILLOUIN
LASERS
We derive the
α
factor in stimulated Brillouin laser (SBL) systems
by starting from the Hamiltonian of the system:
H
=
̄
h
(
ω
P
̃
A
̃
A
+
ω
s
̃
a
̃
a
+
̃
b
̃
b
) +
̄
hg
B
(
̃
A
̃
a
̃
b
+
̃
A
̃
a
̃
b
)
(S1)
where
̃
A
,
̃
a
and
̃
b
are the lowering operators of the pump, Stokes
and phonon modes, respectively;
ω
P
,
ω
s
and
are the res-
onance frequencies of the pump, Stokes and phonon modes,
respectively; and
g
B
is the single-particle Brillouin coupling [
1
].
We have ignored terms that are strongly out of phase match
(i.e., energy non-conserving) in the Hamiltonian to simplify the
discussion. The fast time dependencies are removed from the
operators as follows:
A
̃
A
exp
(
i
ω
P,in
t
)
(S2)
a
̃
a
exp
(
i
ω
L
t
)
(S3)
b
̃
b
exp
(
i
L
t
)
(S4)
where
A
,
a
and
b
are the slow-varying lowering operators;
ω
P,in
is the pumping frequency;
ω
L
is the SBL frequency and
L
is the
mechanical vibration frequency. Replacing the operators with
the slow-varying ones results in an effective Hamiltonian:
H
=
̄
h
(
δω
P
A
A
+
δω
a
a
+
δ
b
b
) +
̄
hg
B
(
A
ab
+
Aa
b
)
(S5)
where
δω
P
ω
P
ω
P,in
is the pump mode frequency detuning
compared to the external pump, and
δω
ω
s
ω
L
(
δ
L
) is the detuning of Stokes (phonon) cavity mode compared to
the laser (mechanical vibration) frequency. We note that the slow-
varying amplitudes are directly referenced to the true oscillating
frequencies of each mode instead of the resonance frequencies,
which removes the fast time dependence in the interaction terms.
The Heisenberg equations of motion for the Stokes mode and
the phonon mode are derived. Then, the quantum operators are
replaced with classical fields as the dominant source of noise in
this system is phonon thermal noise [
1
]. Finally, phenomenolog-
ical damping terms are inserted as follows,
da
dt
=
(
γ
2
+
i
δω
)
a
ig
B
Ab
(S6)
db
dt
=
(
Γ
2
+
i
δ
)
b
ig
B
Aa
(S7)
Supplementary Material
2
where
γ
(
Γ
) is the energy decay rates for the Stokes (phonon)
mode.
We first seek nonzero steady-state solutions to the above
equations that represent SBLs. By writing the equation for
b
using Eq. (S7),
db
dt
=
(
Γ
2
i
δ
)
b
+
ig
B
A
a
(S8)
the equations (S6) and (S8) form a linear system in
a
and
b
. The
requirement for nonzero solutions (i.e., zero determinant of the
coefficient matrix) gives the equation:
(
γ
2
+
i
δω
)
(
Γ
2
i
δ
)
g
2
B
|
A
|
2
0
=
0
(S9)
where the subscript
0 indicates steady state. This complex equa-
tion can be solved as
2
δω
γ
=
2
δ
Γ
(S10)
g
2
B
|
A
|
2
0
=
γ
Γ
4
(
1
+
4
δ
2
Γ
2
)
(S11)
For convenience, we define
α
2
δω
/
γ
=
2
δ
/
Γ
and later
demonstrate that
α
is indeed the linewidth enhancement factor.
With
α
defined, the steady-state pump photon number is,
|
A
|
2
0
=
γ
Γ
4
1
+
α
2
g
2
B
=
γ
2
g
(
1
+
α
2
)
(S12)
where the Brillouin gain coefficient
g
=
2
g
2
B
/
Γ
has been defined.
Since
Γ

γ
in our microcavity system, we can adiabatically
eliminate
b
from Eq. (S6) by setting
db
/
dt
=
0 in Eq. (S8).
da
dt
=
(
γ
2
+
g
|
A
|
2
1
+
α
2
)
(
1
+
i
α
)
a
(S13)
where the definition of
α
has been used. Here,
|
A
|
2
implicitly
depends on
a
through the pump mode dynamics and controls
the gain saturation. Alternatively, Eqn. S13 can be represented
using the amplitude
|
a
|
and phase
φ
a
=
ln
(
a
/
a
)
/
(
2
i
)
variables,
d
|
a
|
dt
=
(
γ
2
+
g
|
A
|
2
1
+
α
2
)
|
a
|
(S14)
d
φ
a
dt
=
(
γ
2
+
g
|
A
|
2
1
+
α
2
)
α
(S15)
which illustrates that
α
=
|
a
|
̇
φ
a
/
̇
|
a
|
represents amplitude-phase
coupling.
Henry [
2
] defined the
α
factor as the ratio of the change in
real part of the refractive index and the change in the imaginary
part. Below we show that this interpretation is consistent with
that derived from the coupled-mode equations. For a system
with Lorentzian gain, the imaginary part of the gain-induced
susceptibility can be written as
χ
I
(
ω
B
) =
χ
B
1
+
4
ω
2
B
/
Γ
2
(S16)
where
Γ
is the gain bandwidth,
χ
B
is a positive constant describ-
ing the strength of the gain at the line center, and the angular
frequency
ω
B
is referenced to the gain center (i.e., detuning
relative to gain center). By the Kramers-Kronig relations,
χ
I
necessarily leads to the real part of the susceptibility
χ
R
through
the relation,
χ
R
(
ω
B
) =
1
π
χ
I
(
ω
B
)
ω
B
ω
B
d
ω
B
=
χ
B
2
ω
B
/
Γ
1
+
4
ω
2
B
/
Γ
2
(S17)
The refractive index can be written as
n
(
ω
B
)
2
=
n
2
+
χ
R
+
i
χ
I
,
where
n
is the material refractive index (dispersion in
n
has been
ignored). Assuming
χ
B

n
2
, we can find the real part
n
and
imaginary part
n
′′
of the refractive index:
n
=
n
+
χ
R
2
n
=
n
+
χ
B
2
n
2
ω
B
/
Γ
1
+
4
ω
2
B
/
Γ
2
(S18)
n
′′
=
χ
I
2
n
=
χ
B
2
n
1
1
+
4
ω
2
B
/
Γ
2
(S19)
The
α
factor can then be obtained as,
α
=
n
/
∂χ
B
n
′′
/
∂χ
B
=
2
ω
B
Γ
(S20)
Setting
ω
B
=
δ
recovers the desired result,
α
=
2
δ
/
Γ
. There
are different conventions regarding the sign of
α
, and here we
choose the negative sign which would be consistent with the
exp
(
i
ω
t
)
phasor used throughout.
To further establish the connection of
α
to linewidth broaden-
ing, the SBL linewidth is derived. We will again assume
Γ

γ
and defer the more general case to Section 3. For this analysis
we add classical noise terms to Eqs. (S6) and (S7),
da
dt
=
γ
2
(
1
+
i
α
)
a
ig
B
Ab
+
f
a
(
t
)
(S21)
db
dt
=
Γ
2
(
1
+
i
α
)
b
ig
B
Aa
+
f
b
(
t
)
(S22)
where
f
a
and
f
b
are classical noise operators for the Stokes and
phonon mode, respectively, satisfying the following correlations:
f
a
(
t
+
τ
)
f
a
(
t
)
=
0
(S23)
f
b
(
t
+
τ
)
f
b
(
t
)
=
n
th
Γ
δ
(
τ
)
(S24)
and
n
th
is the number of thermal quanta in the phonon mode
(thermal quanta in the optical modes are negligible at room
temperature).
Adiabatically eliminating
b
gives
da
dt
=
(
γ
2
+
g
|
A
|
2
1
+
α
2
)
(
1
+
i
α
)
a
+
̃
f
a
(
t
)
(S25)
̃
f
a
f
a
ig
B
A
1
i
α
2
Γ
f
b
(S26)
where we defined a composite fluctuation term
̃
f
a
for the SBL.
Its correlation reads
̃
f
a
(
t
)
̃
f
a
(
0
)
=
f
a
(
t
)
f
a
(
0
)
+
g
2
B
|
A
|
2
0
1
+
α
2
4
Γ
2
f
b
(
t
)
f
b
(
0
)
=
n
th
γδ
(
t
)
(S27)
which is independent of
α
. Applying a standard linewidth anal-
ysis, the SBL linewidth is found as,
ω
SBL
=
γ
2
N
a
n
th
(
1
+
α
2
)
(S28)
where
N
a
=
|
a
|
2
is the steady-state photon number in the Stokes
mode. This is readily shown to agree with Eq. (3) in the main
text in the limit of
Γ
when expressed in terms of output
SBL power.
Supplementary Material
3
10
10
10
-125
-120
-115
-110
-105
-100
-95
-90
-85
Phase Noise (dBc/Hz)
Offset Frequency (Hz)
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Offset Frequency (MHz)
0
5
10
15
20
Frequency Noise (Hz²/Hz)
(a)
(b)
Fig. S1.
SBL noise measurement and fitting.
(a) Blue curve is
the measured phase noise spectrum from the self-heterodyne
output when pump wavelength is 1538 nm and SBL power
is 1.29 mW. Red curve is the fitting according to Eq. (S33) to
obtain the frequency noise
S
. (b) The converted frequency noise
spectrum from panel (a).
2. FREQUENCY DISCRIMINATOR MEASUREMENTS
In this section, the frequency noise measurement is studied to
arrive at the transfer function that relates the measured phase
noise spectrum (see Fig. S1 (a)) to the frequency noise spec-
tral density
S
plotted in Fig. 3 of the Main Text. As shown in
the Supplement Information of our previous work [
3
], pump
noise conversion is believed to be the dominant noise source at
low offset frequency, while white Schawlow-Townes-like noise
dominants at high offset frequency (usually over 100 kHz).
For white frequency noise, the correlation of the time deriva-
tive of the phase satisfies,
̇
φ
(
t
+
τ
)
̇
φ
(
t
)
=
ω
N
δ
(
τ
)
(S29)
where
ω
N
is the Lorentzian full-width-at-half-maximum
linewidth in rad/s, including both fundamental (
ω
SBL
) and
technical contributions. The two-sided spectral density function
for the instantaneous frequency
ν
̇
φ
/
(
2
π
)
is given by the
Fourier transform of the correlation function:
S
w
=
1
4
π
2
̇
φ
(
t
+
τ
)
̇
φ
(
t
)
e
2
π
i f
τ
d
τ
=
ω
N
4
π
2
(S30)
where
S
w
is the white frequency noise spectral density as in the
main text.
On account of the time delayed path in the frequency dis-
crimination system, the detected output returns a signal with
a noisy phase
φ
(
t
+
τ
)
φ
(
t
)
, where
τ
is the interferome-
ter delay. We are thus interested in the frequency noise of
ν
(
τ
)
(
̇
φ
(
t
+
τ
)
̇
φ
(
t
))
/
(
2
π
)
. By the time-shifting property of
the Fourier transform,
S
ν
(
τ
)
(
f
) =
S
w
(
2
e
2
π
i f
τ
e
2
π
i f
τ
) =
4 sin
2
(
π
f
τ
)
S
w
(S31)
The detected output from the self-heterodyne interferometer is
analyzed by a phase noise analyzer. Therefore, converting to
phase noise gives,
S
φ
(
τ
)
(
f
) =
1
f
2
S
ν
(
τ
)
(
f
) =
4
sin
2
(
π
f
τ
)
f
2
S
w
(S32)
A typical measured phase-noise spectrum is shown in Fig. S1.
In fitting the spectrum, there is both the
sinc
2
-shaped noise spec-
trum contributed by the SBL laser, and a noise floor contributed
by the photodetector noise equivalent power (NEP). Thus, the
following equation is used to describe the total phase noise,
S
Total,
φ
(
f
) =
S
NEP
+
4
π
2
τ
2
sinc
2
(
π
f
τ
)
S
w
(S33)
where
sinc
(
z
)
sin
z
/
z
,
S
Total,
φ
(
f
)
is the total measured phase
noise, and
S
NEP
is the NEP contributed phase noise (determined
by averaging the measured phase noise between 8 MHz to 10
MHz).
S
w
and the time delay
τ
are fitting parameters in the mea-
surement (the fiber delay has around 1 km length and therefore
provides an approximate delay of
τ
4.67
μ
s). The fitting is
performed within the frequency range between 0.1 MHz and 3
MHz, since technical noise becomes significant below 0.1 MHz,
while the fringe contrast is reduced for frequencies higher than
3 MHz on account of reduced resolution.
To explicitly illustrate the measured noise is approximately
white over this frequency range, we convert the phase noise
from discriminator measurement to frequency noise by dividing
out the response function,
4
π
2
τ
2
sinc
2
(
π
f
τ
)
. As shown in Fig.
S1 (b), the overall frequency noise is nearly white except for
some spikes resulting from zeros in the response function in
combination with the NEP noise contributions.
3. FULL ANALYSIS OF THE BRILLOUIN LASER NOISE
In this section, a more complete analysis of the SBL frequency
noise is presented that includes both the effect of the pumping
noise and also does not make the adiabatic approximation (i.e.,
Γ

γ
). The equations of motion for the Stokes, phonon and
pump mode amplitudes, with damping and pumping terms,
are:
da
dt
=
γ
2
(
1
+
i
α
)
a
ig
B
Ab
(S34)
db
dt
=
Γ
2
(
1
+
i
α
)
b
ig
B
Aa
(S35)
dA
dt
=
(
γ
2
+
i
δω
P
)
A
ig
B
ab
+
κ
A
in
(S36)
where the pump and Stokes mode have the same decay rate
γ
,
κ
is the external coupling rate,
A
in
>
0 is the external pumping
amplitude (normalized to photon rate), and the other symbols
have the same meaning as in Section 1.
It is convenient to work with amplitude (
|
a
|
,
|
b
|
,
|
A
|
) and
phase (
φ
a
=
ln
(
a
/
a
)
/
(
2
i
)
, similar definitions for
φ
b
and
φ
A
)
Supplementary Material
4
variables. Their equations can be rewritten as,
d
|
a
|
|
a
|
dt
=
γ
2
+
g
B
|
A
||
b
|
|
a
|
sin
θ
(S37)
d
|
b
|
|
b
|
dt
=
Γ
2
+
g
B
|
A
||
a
|
|
b
|
sin
θ
(S38)
d
|
A
|
|
A
|
dt
=
γ
2
g
B
|
a
||
b
|
|
A
|
sin
θ
+
κ
A
in
|
A
|
cos
φ
A
(S39)
d
φ
a
dt
=
γ
2
α
g
B
|
A
||
b
|
|
a
|
cos
θ
(S40)
d
φ
b
dt
=
Γ
2
α
g
B
|
A
||
a
|
|
b
|
cos
θ
(S41)
d
φ
A
dt
=
δω
P
g
B
|
a
||
b
|
|
A
|
cos
θ
κ
A
in
|
A
|
sin
φ
A
(S42)
where we defined the phase difference
θ
=
φ
A
φ
a
φ
b
. The
steady-state solutions (indicated by a subscript 0) are given by,
cos
θ
0
=
α
1
+
α
2
(S43)
sin
θ
0
=
1
1
+
α
2
(S44)
|
A
|
2
0
=
γ
2
g
(
1
+
α
2
)
(S45)
|
b
|
2
0
=
γ
Γ
N
a
(S46)
κ
A
in
cos
φ
A
,0
=
|
A
|
0
(
γ
2
+
gN
a
1
+
α
2
)
(S47)
δω
P
=
α
1
+
α
2
gN
a
κ
A
in
|
A
|
0
sin
φ
A
,0
(S48)
where we used the definition
g
=
2
g
2
B
/
Γ
. Also, although we
expressed everything in terms of SBL photon numbers
N
a
≡|
a
|
2
0
,
it is the input amplitude
A
in
that determines
N
a
.
Because the pump mode is Pound-Drever-Hall (PDH) locked
to the cavity resonance
φ
A
,0
=
0. Thus, the input amplitude and
detuning can be further simplified as
κ
A
in,0
=
(
γ
2
+
gN
a
1
+
α
2
)
γ
2
g
1
+
α
2
(S49)
δω
P,0
=
α
1
+
α
2
gN
a
(S50)
We note that the
δω
P,0
obtained here is, up to zeroth order of
γ
/
Γ
,
equal to the negative of beatnote change between the pump and
SBL signals induced by amplitude-phase coupling, as measured
in Fig. 2a in the main text.
After the steady-state solutions are obtained, the dynamical
equations are linearized by defining relative amplitude change
variables (e.g.,
δ
a
=
|
a
|
/
|
a
|
0
1) and phase change variables
(e.g.,
δφ
a
=
φ
a
φ
a
,0
). Also, Langevin terms are added to the
right side of the equations. These are, as before, classical and
include only the thermal noise contributions. The linearized
equations with noise terms are:
d
δ
a
dt
=
γ
2
(
δ
A
+
δ
b
δ
a
αδθ
) +
f
δ
a
(S51)
d
δ
b
dt
=
Γ
2
(
δ
A
+
δ
a
δ
b
αδθ
) +
f
δ
b
(S52)
d
δ
A
dt
=
γ
2
δ
A
gN
a
1
+
α
2
(
δ
a
+
δ
b
αδθ
)
(S53)
d
δφ
a
dt
=
γ
2
(
αδ
A
+
αδ
b
αδ
a
+
δθ
) +
f
δφ
,
a
(S54)
d
δφ
b
dt
=
Γ
2
(
αδ
A
+
αδ
a
αδ
b
+
δθ
) +
f
δφ
,
b
(S55)
d
δφ
A
dt
=
gN
a
1
+
α
2
(
αδ
a
+
αδ
b
αδ
A
+
δθ
)
(
γ
2
+
gN
a
1
+
α
2
)
(
δφ
A
+
f
δφ
,
A
)
(S56)
where
f
z
represents noise input to the variable
z
. It is convenient
to switch to the frequency domain using
d
/
dt
i
ω
. The power
spectral density of each noise term can be written as,
S
f
,
δ
a
=
S
f
,
δφ
,
a
=
0
(S57)
S
f
,
δ
b
=
S
f
,
δφ
,
b
=
n
th
2
Γ
|
b
|
2
0
(S58)
S
f
,
δφ
,
A
=
S
φ
,Pump
(S59)
where
S
φ
,Pump
is the input phase noise contributed by the pump,
and each noise term is independent of others. We have ignored
the relative intensity noise of the pump, but it can also be ana-
lyzed similarly.
The above linear equations can be directly inverted, and the
solution for
δφ
a
is, to the lowest order in
ω
,
δφ
a
=
i
(
γ
+
Γ
)
ω
(
α
Γ
f
δ
a
αγ
f
δ
b
Γ
f
δφ
,
a
+
γ
f
δφ
,
b
)
γ
γ
+
Γ
f
δφ
,
A
(S60)
where the lowest order of
ω
approximation remains valid when
ω

γ
. From here we obtain the phase noise of the SBL,
S
φ
,SBL
=
α
2
Γ
2
S
f
,
δ
a
+
α
2
γ
2
S
f
,
δ
b
Γ
2
S
f
,
δφ
,
a
+
γ
2
S
f
,
δφ
,
b
(
γ
+
Γ
)
2
ω
2
+
(
γ
γ
+
Γ
)
2
S
f
,
δφ
,
A
(S61)
S
φ
,SBL
=
Γ
2
(
1
+
α
2
)
(
γ
+
Γ
)
2
ω
2
γ
2
N
a
n
th
+
(
γ
γ
+
Γ
)
2
S
φ
,Pump
(S62)
Converting to frequency noise gives,
S
ν
,SBL
=
Γ
2
(
1
+
α
2
)
4
π
2
(
γ
+
Γ
)
2
γ
2
N
a
n
th
+
(
γ
γ
+
Γ
)
2
S
ν
,Pump
(S63)
Thus, the fundamental linewidth of the SBL is given by
ω
SBL
=
(
Γ
γ
+
Γ
)
2
(
1
+
α
2
)
γ
2
N
a
n
th
(S64)
Note that the above derivation automatically incorporates non-
adiabaticity and the linewidth enhancement factor. Also, the
transduction of the pump phase noise is, when the pump mode
is PDH locked,
S
ν
,SBL
=
(
γ
γ
+
Γ
)
2
S
ν
,P
(S65)
and is independent of the
α
factor.
We briefly comment on the noise behavior when
δω
P
is tuned
away from its PDH-locked value, which happens because the
PDH locking can reduce, but not totally eliminate, the drifting in
δω
P
. Repeating the previous analyses, we arrive at the following