Letters
https://doi.org/10.1038/s41566-020-0588-y
Earth rotation measured by a chip-scale ring laser
gyroscope
Yu-Hung Lai
1,2,8
, Myoung-Gyun Suh
1,3,8
, Yu-Kun Lu
1,4
, Boqiang Shen
1
, Qi-Fan Yang
1
,
Heming Wang
1
, Jiang Li
1,5
, Seung Hoon Lee
1,6
, Ki Youl Yang
1,7
and Kerry Vahala
1
*
1
T. J. Watson Laboratory of Applied Physics, California Institute of Technology, Pasadena, CA, USA.
2
Present address: OEwaves, Pasadena, CA, USA.
3
Present address: Physics & Informatics Laboratories, NTT Research, Inc., East Palo Alto, CA, USA.
4
Present address: Research Laboratory of Electronics,
MIT-Harvard Center for Ultracold Atoms, Department of Physics, Massachusetts Institute of Technology, Cambridge, MA, USA.
5
Present address:
hQphotonics Inc., Pasadena, CA, USA.
6
Present address: Apple Computer, Cupertino, CA, USA.
7
Present address: Stanford, CA, USA.
8
These authors
contributed equally: Yu-Hung Lai, Myoung-Gyun Suh. *e-mail: vahala@caltech.edu
SUPPLEMENTARY INFORMATION
In the format provided by the authors and unedited.
NaturE PHotoNicS
|
www.nature.com/naturephotonics
Supplement: Earth Rotation Measured by a Chip-Scale Ring Laser Gyroscope
Yu-Hung Lai
1
∗
, Myoung-Gyun Suh
1
∗
, Yu-Kun Lu
1
, Boqiang Shen
1
, Qi-Fan Yang
1
,
Heming Wang
1
, Jiang Li
1
, Seung Hoon Lee
1
, Ki Youl Yang
1
, Kerry Vahala
1
†
1
T. J. Watson Laboratory of Applied Physics, California Institute of Technology, Pasadena, California 91125, USA
∗
These authors contributed equally to this work.
†
vahala@caltech.edu
I. MICRO-OPTICAL GYROSCOPE PERFORMANCE
Table I provides a comparison of performance metrics for recently reported micro-optical gyroscopes including the
gyroscope described in this work.
Reference
Chip-based (Y/N)
ARW (
◦
/
√
h)
Bias instability (
◦
/
h)
Lowest rotation rate measured Ω
pk
(
◦
/
h)
12 (SBL-RLG)
Y
N.R.
N.R.
31
13 (SBL-RLG)
Y
N.R.
N.R.
90000
14 (Spiral)
Y
8.5
45
72
15 (RMOG)
N
0.02
3
360
†
16 (RMOG)
Y
12
15
36
17 (RMOG)
Y
650
21600
432000
18 (RMOG)
N
N.R.
N.R.
1
.
3
×
10
8
This work
Y
0.068
3.6
5
TABLE I. Comparison of the state-of-the-art micro-optical gyroscopes. Reference numbers refer to Main text.
†
Ref
1
. N.R.:
Not reported.
II. EXPERIMENT SETUP
The packaged resonator is shown in Extended Data Figure 1 and the full system setup is shown in the Extended Data
Figure 2. An external-cavity diode laser is used as a pump laser and is amplified by an erbium-doped fiber amplifier
before being split into two arms. In each arm, the optical frequency is shifted by an acoustic-optical modulator
(AOM) and the pump is coupled into the resonator. The pump laser in one arm is phase modulated and locked to the
cavity resonance via Pound-Drever-Hall method, while the frequency of the other laser is freely tuned by adjusting
the AOM drive frequency. The pump powers are monitored and stabilized by a feedback loop. The pumps and SBLs
are combined and photodetected to measure the dual-SBL beating and pump-SBL beating signals. The signals are
analyzed using an electrical spectrum analyzer and two frequency counters. The resonator is packaged in a small brass
box with fiber connectors. The temperature of the resonator is monitored by a thermistor and controlled by a thermal
electric cooler (TEC). Most of the optical components are installed in an isolation chamber to reject acoustic noise,
vibration, and airflow. The system is installed on an automated air-bearing rotation stage for the Earth rotation
measurement.
III. KERR-INDUCED CONTRIBUTIONS TO THE STIMULATED BRILLIOUN LASER LINEWIDTH
Here we study contributions to the stimulated Brillioun laser (SBL) linewidth that result from the Kerr nonlinearity.
The primary noise input is assumed to be thermo-mechanical noise that drives the SBL mode and is responsible for
the Schawlow-Townes-like linewidth of the Brillouin laser mode
2
. At room temperature, this noise contribution is
nearly 3 orders larger than that from quantum contributions to the linewidth. The dynamics of the SBL can be
approximated as follows:
2,3
da
dt
=
−
γ
2
a
+
g
|
A
|
2
a
+
f
(
t
)
(S1)
2
where
a
is the slow-varying amplitude of the SBL in the rotating frame determined by the laser. The square amplitude
|
a
|
2
is normalized to photon number,
γ
is the decay rate,
g
is the Brillioun gain parameter,
A
is the amplitude of the
pump mode, and
f
(
t
) is a Langevin fluctuation term with correlation given by,
〈
f
(
t
+
τ
)
f
∗
(
t
)
〉≈
γN
T
δ
(
τ
)
(S2)
where
N
T
are the number of thermal quanta in the acoustic field and a quantum contribution in the normalization is
ignored
2
. Lasing occurs when the pump reaches the lasing threshold
|
A
|
2
=
γ
/(2g), and the linewidth of the laser is
given by (ignoring quantum contributions),
∆
ω
=
γ
2
N
s
N
T
(S3)
where
N
s
is the steady-state coherent photon number in the optical mode.
The Kerr effect becomes important when the laser power is increased. To consider this effect we add a Kerr term
to the dynamical equation:
da
dt
= (
ig
K
N
s
−
γ
2
)
a
+
g
|
A
|
2
a
+
ig
K
(
N
−
N
s
)
a
+
f
(
t
)
(S4)
where
g
K
is the single-photon Kerr shift of the SBL mode and
N
≡|
a
|
2
is the laser mode photon number. To estimate
the linewidth correction from the Kerr effect, we first linearize the gain saturation near the laser operating point,
g
|
A
|
2
≈
γ
2
+
g
′
(
N
−
N
s
)
(S5)
where
g
′
<
0 is the gain derivative with respective to laser photon number
N
. This expression assumes that the rate
of change of the system is slow enough so that the photons in the pump mode (
|
A
|
2
) adiabatically follow changes in
the lasing mode photons (see later discussion relating to eq. S17 below). We can then write down the equations for
photon number
N
and the instantaneous laser frequency fluctuation
̇
φ
≡
( ̇
a/a
−
̇
a
∗
/a
∗
)
/
(2
i
)
−
g
K
N
s
separately (Note:
in defining
̇
φ
we have removed a constant frequency term associated with the Kerr effect):
̇
N
= 2
g
′
(
N
−
N
s
)
N
+ (
fa
∗
+
f
∗
a
)
(S6)
̇
φ
=
g
K
(
N
−
N
s
) +
1
2
i
(
f
a
−
f
∗
a
∗
)
(S7)
where the frequency fluctuation equation contains a Kerr effect term that couples the photon number fluctuation
N
−
N
s
to the phase noise. For processes slower than the relaxation time scale 1
/
(
g
′
N
s
),
̇
N
can be neglected. As a
result, fluctuations in the photon number are given by,
N
−
N
s
=
−
1
2
g
′
N
(
fa
∗
+
f
∗
a
) =
−
1
2
g
′
(
f
a
+
f
∗
a
∗
)
(S8)
Substituting this result into the equation of
̇
φ
gives,
̇
φ
=
−
g
K
2
g
′
(
f
a
+
f
∗
a
∗
)
+
1
2
i
(
f
a
−
f
∗
a
∗
)
(S9)
From which the following correlation is readily computed,
〈
̇
φ
(
t
+
τ
)
̇
φ
(
t
)
〉
=
〈
f
(
t
+
τ
)
f
∗
(
t
)
〉
2
N
(
1 +
g
2
K
g
′
2
)
= (1 +
α
2
)∆
ωδ
(
τ
)
(S10)
where
α
≡
g
K
/g
′
is the amplitude-phase coupling factor and where we replaced
N
with
N
s
for steady state operation.
We see that the Kerr effect modifies the laser linewidth by a 1 +
α
2
factor similar to the well-known Henry linewidth
enhancement factor in semiconductor lasers.
This analysis can be readily extended to the beating of two SBLs by introducing cross-phase modulation terms:
da
1
dt
= (
ig
K
N
s
1
+ 2
ig
K
N
s
2
−
γ
2
)
a
1
+
g
|
A
1
|
2
a
1
+
ig
K
[(
N
1
−
N
s
1
) + 2(
N
2
−
N
s
2
)]
a
1
+
f
1
(
t
)
da
2
dt
= (
ig
K
N
s
2
+ 2
ig
K
N
s
1
−
γ
2
)
a
2
+
g
|
A
2
|
2
a
2
+
ig
K
[(
N
2
−
N
s
2
) + 2(
N
1
−
N
s
1
)]
a
2
+
f
2
(
t
)
(S11)
3
where subscripts 1 and 2 indicate quantities associated with CW and CCW SBL modes, and we have assumed the
same gain and loss for the two SBLs. The equations for
N
1
and
̇
φ
1
are now given by,
̇
N
1
= 2
g
′
(
N
1
−
N
s
1
)
N
1
+ (
f
1
a
∗
1
+
f
∗
1
a
1
)
(S12)
̇
φ
1
=
g
K
(
N
1
−
N
s
1
) + 2
g
K
(
N
2
−
N
s
2
) +
1
2
i
(
f
1
a
1
−
f
∗
1
a
∗
1
)
(S13)
and similar equations exist for
N
2
and
̇
φ
2
. We are interested in the phase difference
θ
≡
φ
2
−
φ
1
. For rates within
the relaxation time 1
/g
′
N
s
1
,
2
, it is given by,
̇
θ
=
g
K
2
g
′
[(
f
1
a
1
+
f
∗
1
a
∗
1
)
−
(
f
2
a
2
+
f
∗
2
a
∗
2
)]
−
1
2
i
[(
f
1
a
1
−
f
∗
1
a
∗
1
)
−
(
f
2
a
2
−
f
∗
2
a
∗
2
)]
(S14)
Its correlation reads
〈
̇
θ
(
t
+
τ
)
̇
θ
(
t
)
〉
= (1 +
α
2
)(∆
ω
1
+ ∆
ω
2
)
δ
(
τ
)
(S15)
where ∆
ω
1
and ∆
ω
2
are the individual SBL linewidths when the Kerr effect is absent. Again, it can be seen that the
noise of the beatnote is corrected by a 1 +
α
2
factor.
Now we proceed to find the gain saturation
g
′
parameter, which requires study of the dynamics of the pump:
dA
dt
=
−
γ
2
A
−
g
|
a
|
2
A
+
√
γ
ex
P
ex
(S16)
where
γ
ex
is the external coupling factor and
P
ex
is the input power to the pump mode. For processes slower than
the cavity decay rate (
dA/dt
γA/
2), the derivative can be ignored yielding,
|
A
|
2
=
γ
ex
P
ex
(
γ/
2 +
g
|
a
|
2
)
2
(S17)
To eliminate the input term
γ
ex
P
ex
, consider the following steady state form of Eq. (S16):
0 =
−
γ
2
A
0
−
gN
s
A
0
+
√
γ
ex
P
ex
, A
0
=
√
γ
2
g
(S18)
where the pump has its steady-state clamped threshold value (
A
0
) and
N
s
is, as above, the steady-state laser photon
number. The input term can be solved as
γ
ex
P
ex
=
(
γ
2
+
gN
s
)
2
γ
2
g
(S19)
which, when plugged into the equation for
|
A
|
2
, gives
|
A
|
2
=
γ
2
g
(
γ/
2 +
gN
s
)
2
(
γ/
2 +
g
|
a
|
2
)
2
(S20)
The gain saturation parameter can then be found as,
g
′
=
∂
(
g
|
A
|
2
)
∂
|
a
|
2
∣
∣
∣
∣
|
a
|
2
=
N
s
=
−
2
g
γ/
2
γ/
2 +
gN
s
(S21)
Using this expression, the overall dependence of the linewidth on the SBL power can be made explicit:
(1 +
α
2
)∆
ω
=
[
1 +
1
4
(
g
K
g
+
2
g
K
N
s
γ
)
2
]
γ
2
N
s
N
T
(S22)
For the silica resonator used here, the SBL gain per photon is
g
= 3
×
10
−
3
Hz, and the Kerr shift per photon is
g
K
= 1
×
10
−
5
Hz. Therefore
α
is on the order of 10
−
1
.
5
when the SBL power is low, and the correction of Kerr effect
to the linewidth is on the order of 10
−
3
.
4
However, since the magnitude of
g
′
decreases as the SBL power increases,
α
will increase with SBL power. And
eventually, the Kerr-induced noise scales with power and dominates the linewidth. This scaling holds even if we take
the quantum noise sources into account. An optimal SBL operating power can be found by minimizing the overall
linewidth,
N
s
=
γ
g
K
√
1 +
g
2
K
4
g
2
≈
γ
g
K
(S23)
The corresponding linewidth is
(1 +
α
2
)∆
ω
=
g
K
N
T
(S24)
which sets a limit to the gyroscope sensitivity. At room temperature
N
T
≈
570 and the dual-SBL linewidth limit is
estimated to be 0
.
011 Hz (equivalently 0
.
01
◦
/
√
h angle random walk). This linewidth is substantially smaller than
the one currently measured. Also, this ARW is about 7
×
smaller than the one currently measured.
We note that the nature of the noise limitation described here is similar to a nonlinear passive resonator as outlined in
Matsko
et. al.
4
, where the noise in the nonlinear Kerr frequency shift dominates the frequency noise as the intracavity
power increases. The main difference is that the SBL system is thermal-limited, with the vacuum fluctuations replaced
by thermal quanta. The single photon Kerr shift
g
K
can be expressed as
g
K
=
n
2
~
ω
2
c
n
2
0
V
(S25)
where
n
2
is the Kerr-nonlinear refractive index of the resonator material,
V
is the mode volume,
n
0
is the linear
refractive index,
ω
is the angular frequency of the SBL, and
c
is the speed of light in vacuum. Increasing mode volume
and choosing a material with lower
n
2
can reduce the Kerr nonlinear effect and subsequently improve the performance
limit of the SBL gyro.
IV. OTHER NOISE
Concerning other sources of noise, Kerr-induced cross-phase modulation from pump power fluctuations introduces
frequency fluctuations into the two SBL waves that are common mode and hence cancel out when their difference
frequency is measured. In addition, power fluctuations from the pump waves will also cause power fluctuations in the
SBL waves. These will in turn cause self and cross phase modulation that shifts the difference frequency of the two
SBL waves. However, it has been shown that difference frequency shift caused by changes in the SBL powers varies
like the difference in the SBL powers
3
. Therefore, because technical power noise from the pump is largely common
mode (i.e., both the clockwise and counter-clockwise pump waves are derived from the same laser) it is expected that
this noise does not contribute significantly to the beat frequency. Finally, phase noise from the pump can couple into
the SBL waves, however, this component of noise can be shown to be weak because the cavity damping rate is much
weaker than the phonon damping rate
2
.
1
Liang, W., Ilchenko, V., Eliyahu, D., Dale, E., Savchenkov, A., Matsko, A. & Maleki, L. Whispering Gallery Mode Optical
Gyroscope.
Proc. 2016 IEEE International Symposium on Inertial Sensors and Systems
, 89–92 (2016).
2
Li, J., Lee, H., Chen, T. & Vahala, K. J. Characterization of a high coherence, Brillouin microcavity laser on silicon.
Opt.
Express
20
, 20170–20180 (2012).
3
Wang, H., Lai, Y.-H., Yuan, Z., Suh, M.-G. & Vahala, K. J. Petermann-factor limited sensing near an exceptional point.
arXiv:1911.05191
(2019).
4
Matsko, A. B., Liang, W., Savchenkov, A. A., Ilchenko, V. S. & Maleki, L. Fundamental limitations of sensitivity of whispering
gallery mode gyroscopes.
Phys. Lett. A
382
, 2289–2295 (2018).