arXiv:1901.08217v1 [physics.optics] 24 Jan 2019
Enhanced sensitivity operation of an optical gyroscope nea
r an exceptional point
Yu-Hung Lai
1
∗
, Yu-Kun Lu
1
,
2
∗
, Myoung-Gyun Suh
1
∗
, Kerry Vahala
1
†
1
T. J. Watson Laboratory of Applied Physics, California Institute of
Technology, Pasadena, California 91125, USA
2
State Key Laboratory for Mesoscopic Physics and Collaborative In
novation Center of Quantum
Matter, School of Physics, Peking University, Beijing 100871, Peo
ples Republic of China
∗
These authors contributed equally to this work.
†
vahala@caltech.edu
Exceptional points (EPs) are special spectral de-
generacies of non-Hermitian Hamiltonians gov-
erning the dynamics of open systems. At the
EP two or more eigenvalues and the correspond-
ing eigenstates coalesce
1
–
3
. Recently, it has been
proposed that EPs can enhance the sensitivity of
optical gyroscopes
4
,
5
. Here we report measure-
ment of rotation sensitivity boost by over 4
×
re-
sulting from operation of a chip-based stimulated
Brillouin gyroscope near an exceptional point. A
second-order EP is identified in the gyroscope and
originates from the dissipative coupling between
the clockwise and counterclockwise lasing modes.
The modes experience opposing Sagnac shifts un-
der application of a rotation, but near the excep-
tional point new modal admixtures dramatically
increase the Sagnac shift. Modeling confirms the
measured enhancement. Besides the ability to
operate an optical gyroscope with enhanced sen-
sitivity, this result provides a new platform for
study of non-Hermitian physics and nonlinear op-
tics with precise control.
High-Q optical microresonators have received con-
siderable attention as sensors across a wide range of
applications including biomolecule
6
–
8
and nanoparticle
detection
9
, temperature measurement
10
, and rotation
measurement
11
–
15
. In recent years, a new approach to
enhance the sensitivity of microresonator sensors using
the physics of exceptional points is being studied
4
,
5
,
16
–
20
.
Traditionally, for precise sensing, a perturbation to an
optical microcavity (or to its reference frame as in the
case of a gyroscope) introduces either a linewidth change,
a frequency shift, or a frequency splitting of a reso-
nance that monotonically changes with the strength of
the perturbation. However, operation of these systems
near an exceptional point changes this situation by in-
troduction of a square-root dependence into the trans-
duction that can boost the sensor’s ability to transduce
perturbations
16
.
In this work, we experimentally and theoretically
demonstrate the existence of EPs in a microresonator-
based laser gyroscope, and then measure the enhanced
rotation-rate transduction sensitivity near the EP. The
laser gyroscope is described elsewhere
11
and uses counter-
propagating Brillouin lasers in a high-quality-factor (
Q
≈
10
8
) silica wedge resonator
21
. As shown in Fig.
1
a
pump light at frequencies
ω
p
j
(
j
= 1
,
2) determined by
radio-frequency modulation of a single laser (
∼
1552.5
nm) is coupled into the resonator from both ends of a
fiber taper
22
,
23
. One of the pump frequencies is Pound-
Drever-Hall locked to a resonator mode by feedback con-
trol to the laser. The second pump frequency is then
varied to affect pump detuning change as described be-
low. The two pump powers are stabilized via power feed-
back. Brillouin scattering causes a pump photon with
frequency
ω
p
j
to scatter from a co-propagating acous-
tic phonon with frequency Ω
phonon
into a backward-
propagating Stokes photon with frequency
ω
s
j
. In the
context of a resonator (and as illustrated in Fig.
1
b),
the associated phase matching condition requires that
the Brillouin shift frequency (Ω
phonon
) is close in value
to a multiple of the resonator free-spectral-range (FSR).
This is readily achieved by microfabrication control of
resonator diameter and in effect locates a resonator mode
(the Stokes mode) within the Brillouin gain spectrum for
efficient stimulated Brillouin laser (SBL) action
21
,
24
.
Counter-pumping is performed on the same resonant
mode number (
m
) so that laser action on two counter-
propagating Stokes waves also occurs on one mode num-
ber (set to
m
-6 in this measurement). The device then
functions as a ring laser gyroscope in which rotation mea-
surement is possible by comparing (via heterodyne detec-
tion) the relative frequencies of the Stokes waves
25
. Line
narrowing to sub-Hertz values results from laser action
within the high-Q system
24
thereby boosting the rota-
tional sensitivity. To control the frequency of the Stokes
waves for study of EP physics, a small frequency mis-
match between the cavity mode and the gain spectral
center is used to vary modal dispersion and thereby pull
the Stokes lasing modes toward the gain spectral center
(Fig.
1
b). As described below this pulling mechanism
is arranged to be different for the two Stokes waves and
it is controlled by varying the pump frequency detun-
ing ∆
ω
p
=
ω
p
2
−
ω
p
1
. Finally, because of the taper-
induced dissipative coupling (rate
κ
) between CW and
CCW modes, the EPs emerge for a critical pump de-
tuning ∆
ω
p
= ∆
ω
c
. When ∆
ω
p
is close to ∆
ω
c
the gyro
sensitivity is boosted due to the steep slope of the square-
root response curve (Fig.
1
c).
The above features of the gyro system are now studied
quantitatively. For simplicity, the case without rotation
is considered first. The equation of motion of the system
reads
id
Ψ
/dt
=
H
0
Ψ where Ψ = (
α
1
, α
2
)
T
is the column
vector for the two SBL modes.
H
0
is the non-Hermitian
2
b
c
Microresonator Brillouin Gyroscope
Brillouin Process
Exceptional Points
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Normalized Pump Detuning Angular Frequency
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
SBL Beating Angular Frequency
(a.u.)
Without EP
With EP
0.8
1
1.2
1.4
0
2
4
6
8
Sensitivity
pump 1
SBL 1
Angular frequency
6 x FSR
Brillouin gain
Cavity mode
a
Pump 1
Pump 2
SBL 1
SBL 2
utput
(without EP)
utput
(EP enhanced)
otation
ε
input
Δ
/Δ
Δ
/Δ
Δ
Δ
Δ
Figure
1.
EP-enhanced microresonator-based Brillouin gyroscope. a
, The dual stimulated Brillouin laser process in
a microresonator. Center: The green (blue) solid curve repr
esents pump 1 (pump 2) with angular frequency
ω
p
1
(
ω
p
2
) and
the red (yellow) solid curve represents SBL 1 (SBL 2) with ang
ular frequency
ω
s
1
(
ω
s
2
). The orange wavy line represents the
acoustic phonons with angular frequency Ω
phonon
. Left: The Brillouin energy and the momentum conservation c
onstraints
(phase matching) are illustrated for scattering of a pump wa
ve into a Stokes wave. Right: Because of resonator imperfect
ions
including the fiber taper, the CW and CCW modes experience dis
sipative coupling at rate
κ
. This coupling creates eigenmodes
that map to a Bloch sphere containing dual EPs (black dots). T
he trajectories on the Bloch sphere show the evolution of two
eigenmodes (red for SBL1 and yellow for SBL2) when the pump de
tuning decreases from +
∞
to
−∞
. The low-loss and high-
loss eigenmodes inside the locking zone are plotted in solid
and dashed black curves, respectively (see Methods for addi
tional
discussion).
b
, Efficient laser action requires that each Stokes mode (black
with linewidth
γ
and separated from the pump by
a multiple of the cavity FSR) lies within the Brillouin gain b
and (orange with linewidth Γ) which, through the phase match
ing
condition, is shifted relative to the pump by Ω
phonon
= 4
πnc
s
/λ
p
(refractive index
n
, speed of sound in silica
c
s
and pump
wavelength
λ
p
). In this work, the FSR is
∼
1.8 GHz so that 6
×
FSR approximately matched the Brillouin shift. Dispersion
from the Brillouin gain pulls the Stokes lasing modes by diffe
rent amounts towards the gain center on account of the differe
nce
∆
ω
p
in pump angular frequencies.
c
, The blue solid curve (red dashed curve) shows the dependenc
e of the dual-SBL beating
angular frequency ∆
ω
s
versus the normalized pump detuning frequency ∆
ω
p
/
∆
ω
c
for
κ
6
= 0 (
κ
= 0) as per Eq.
4
. The yellow
wavy arrow represents the input rotation signal, while the r
ed dotted and blue solid wavy arrows represent the output sig
nal
with and without EP, respectively. The inset shows the
κ
6
= 0 sensitivity normalized to the
κ
= 0 sensitivity, indicating the
enhancement near the EP.
Hamiltonian governing the time evolution:
H
0
=
(
ω
0
+
i
(
g
1
|
A
1
|
2
−
γ/
2
)
iκ
iκ
ω
0
+
i
(
g
2
|
A
2
|
2
−
γ/
2
)
)
(1)
and
α
1
(
A
1
) and
α
2
(
A
2
) represent the photon-number-
normalized amplitudes of the CW and CCW SBL (pump)
modes, respectively.
ω
0
is the unpumped frequency of
the Stokes’ cavity mode and
γ
is the cavity damping
rate.
g
j
=
g
0
/
(1 + 2
i
∆Ω
j
/
Γ) (
j
= 1
,
2) represents the
Brillouin gain factor where
g
0
is the gain coefficient, Γ
is the gain bandwidth, and ∆Ω
j
=
ω
p
j
−
ω
s
−
Ω
phonon
is the frequency mismatch with
ω
s
the Stokes frequency
and Ω
phonon
the Brillouin shift
24
. The real part of the
Brillouin gain factor leads to amplification of the Stokes
mode, while the imaginary part is responsible for the
3
-200 -150 -100
-50
0
50
100
150
200
Pump Detuning Frequency
(kHz)
-15
-10
-5
0
5
10
15
Shifted SBL Frequency
SBL1 (Theory)
SBL2 (Theory)
SBL1 (Experiment)
SBL2 (Experiment)
-200
-100
0
100
200
(kHz)
0
20
40
I
p1s2
/I
p1s1
(dB)
Theory
Data
ȟ
/
(kHz)
(
±
െ
)/
ȟ
/
ȟ
/
ȟ
/
ȟ
/
(
ȟ
െȟ
)/
S1, S2
P1, S2
P1, S1
P2, S2
P2, S1
Slope = 1/2
Slope = 1
Figure
2.
Measurement of the eigenmode properties. a
, Typical measured dual-SBL beating spectrum.
b
, Typical
pump-SBL beating spectrum with frequency axis shifted appr
oximately 10
.
845 GHz to center the pump1-SBL1 beating peak.
The individual pump-SBL beating peaks are identified.
c
, Measured dual-SBL beating frequency versus pump detuning
frequency (blue circles). Red solid curve is fitting (
γ/
Γ = 0
.
073 and
κ
= 1
.
80kHz) and black dotted line corresponds to
κ
= 0
(
γ/
Γ = 0
.
073). The data have a slope 1/2 (slope 1) near (away from) the E
P in the log-log plot provided in the inset. This
data used another mode with larger
κ
compared to panel d.
d
, Measured shifted frequencies of the two SBLs (
ω
s
±
−
ω
r
)
/
2
π
versus pump detuning frequency. Theoretical values of (
ω
s
+
−
ω
r
)
/
2
π
and (
ω
s
−
−
ω
r
)
/
2
π
with
γ/
Γ = 0
.
076 and
κ
= 1
.
23kHz
are plotted as red and dashed-yellow lines, respectively. T
he experimental data of the shifted SBL1 (SBL2) frequency is
shown
as blue (purple) circles. The inset shows the measured power
ratio of CCW components of the lasing modes (blue circles)
obtained by analysis of spectral components in (b) and agree
s reasonably well with the theoretical prediction (red soli
d curve).
mode pulling effect.
κ
is the dissipative coupling rate
between two SBL modes, which originates from the fiber
taper and imperfections in the resonator (see Method).
In the absence of backscattering (
κ
= 0), the CW
and CCW SBL processes are independent because the
Brillouin gain is intrinsically directional as a result of
the phase matching condition (Fig.
1
a). The steady-
state lasing condition requires the power loss rate
γ
to be balanced by the Brillouin gain, which leads to
the clamping condition of the pump powers
|
A
j
|
2
=
γ
(1 + 4∆Ω
2
j
/
Γ
2
)
/
2
g
0
24
. As shown in the Methods, these
conditions remain valid for nonzero dissipative backscat-
tering (
κ
6
= 0) within the regime where EP-enhanced
rotation measurement is performed (the unlocked regime
defined below). As a result, Eq. (
1
) simplifies above laser
threshold to the following form:
H
0
=
(
ω
0
+
γ
Γ
∆Ω
1
iκ
iκ
ω
0
+
γ
Γ
∆Ω
2
)
(2)
With the introduction of
κ
the lasing system exhibits a
frequency locking-unlocking transition when varying the
pump detuning frequency. The locking regime is known
in ring laser gyroscopes to create a sensing dead band for
rotations
25
. In the frequency unlocked regime, the two
lasing modes oscillate with distinct angular frequencies
ω
s
+
and
ω
s
−
, which are the eigenvalues of the Hamilto-
nian (Eq. (
2
)).
ω
s
±
−
ω
r
=
γ/
2Γ
1 +
γ/
Γ
(
∆
ω
p
±
√
∆
ω
2
p
−
∆
ω
2
c
)
(3)
where
ω
r
≡
ω
0
+
γ
(
ω
p
1
−
Ω
phonon
)
/
Γ and ∆
ω
c
≡
2Γ
κ/γ
is the critical frequency. In deriving this result it is im-
portant to note that the Hamiltonian (Eq. (
2
)) depends
weakly upon its own eigenvalues through the appearance
of ∆Ω
1
and ∆Ω
2
(see derivation in Methods). The SBL
beating frequency is readily extracted by taking the dif-
4
°
°
ȟ
/
(
ȟ
െȟ
)/
/
ȳ
/
ȳ
Figure
3.
Measured Sagnac transduction factor
S
(∆
ω
p
)
compared with model.
The blue dots are data
(each point is an average of four measurements) while the
red curve is the theoretical prediction using Eq. (
6
). The
mode pulling factor 1
/
(1 +
γ/
Γ) slightly reduces the Sagnac
factor at large pump detuning. The black dashed line gives
the conventional (non EP-enhanced) Sagnac factor. The inse
t
shows a log-log plot of 5 data points near the EP with a slope
of -1/2, further verifying that the sensitivity enhancemen
t is
proportional to (∆
ω
p
−
∆
ω
c
)
−
1
/
2
.
ference of the above eigenfrequencies, ∆
ω
s
≡ |
ω
s
+
−
ω
s
−
|
:
∆
ω
s
=
γ/
Γ
1 +
γ/
Γ
√
∆
ω
2
p
−
∆
ω
2
c
(4)
This equation is plotted in Fig.
1
c. The dissipative
coupling between the clockwise (CW) and counterclock-
wise (CCW) lasing modes induces second-order EPs at
critical pump-detuning frequencies
|
∆
ω
p
|
= ∆
ω
c
where
the eigenfrequencies as well as the eigenmodes coalesce.
For pump detuning
|
∆
ω
p
|
>
∆
ω
c
the eigenfrequencies
bifurcate and the eigenmodes are an unbalanced hy-
bridization of CW and CCW modes. For pump detuning
|
∆
ω
p
|
<
∆
ω
c
the eigenfrequencies (real part of the eigen-
values) are equal, but have different loss rates.
An electrical spectrum analyzer was used to measure
the photo-detected dual-SBL beating frequency ∆
ω
s
/
2
π
(Fig.
2
a) and the SBL-pump beating frequency (Fig.
2
b). Plots of these frequencies versus the pump frequency
detuning are given in Fig.
2
c and Fig.
2
d. Comparisons
with Eq. (
3
) and Eq. (
4
) are provided and are in good
agreement with measurement. Moreover, the ratio of the
CCW components in the eigenmodes was measured from
the intensity of the CCW-pump beating with the SBL
signals (see Method for analysis) and is plotted as the
inset of Fig.
2
d. There is a reasonable agreement between
the model and measurement. Within the locked regime,
only one Stokes mode is lasing so this measurement is
no longer possible. Further discussion is provided in the
Methods.
When the resonator experiences an angular rotation
rate Ω (positive for CW direction), the Sagnac effect fur-
ther lifts the degeneracy of the CW and CCW modes
by shifting the CW and CCW mode frequencies by
∆
ω
Sagnac
=
∓
2
πD
Ω
/n
g
λ
where
D
is the resonator di-
ameter,
n
g
is the group index of the passive cavity mode,
and
λ
is the laser wavelength
11
. This modifies the SBL
beating frequency as follows:
∆
ω
s
=
γ/
Γ
1 +
γ/
Γ
√
(∆
ω
p
−
Γ∆
ω
Sagnac
/γ
)
2
−
∆
ω
2
c
(5)
Accordingly, the counter-pumped Brillouin system can
serve as a gyroscope for measuring the rotation signal Ω
by monitoring the dual-SBL beating ∆
ω
s
. For compar-
ison with measurements below, the Sagnac transduction
factor
S
is calculated as the derivative of the SBL split-
ting frequency with respect to the applied rotation rate
amplitude Ω:
S
=
∂
∆
ω
s
∂
Ω
∣
∣
∣
∣
Ω=0
=
2
π
1 +
γ/
Γ
∆
ω
p
√
∆
ω
2
p
−
∆
ω
2
c
D
n
g
λ
(6)
where a linear response requires Γ∆
ω
Sagnac
/γ
≪
∆
ω
p
.
In this equation, the coefficient 1
/
(1 +
γ/
Γ) is a cor-
rection from the mode pulling effect and the factor
∆
ω
p
/
√
∆
ω
2
p
−
∆
ω
2
c
is the transduction enhancement
caused by the square-root dependence at the EP. This
enhancement originates from the steep slope of the re-
sponse curve near the EP (Fig.
1
c) so that the Sagnac
transduction factor surpasses the conventional value.
To measure rotations and verify the EP enhancement,
the resonator was packaged in a small metal box with
one edge hinged and the opposing end attached to a PZT
stage in a manner similar to that used in ref.
11
. A sinu-
soidal oscillation was generated by the PZT to create a
sinusoidal rotation of the resonator at a 1 Hz rate hav-
ing a fixed amplitude (410 deg/h). The resulting time-
varying dual-SBL beating frequency was recorded using
a frequency counter and the amplitude of the modulated
frequency was extracted by applying a fast-Fourier trans-
form to the counter signal. Frequency modulation ampli-
tudes were recorded at a series of pump frequency detun-
ings. The resulting Sagnac transduction factor (i.e., the
SBL difference frequency modulation amplitude divided
by the applied rotation-rate amplitude) is plotted in Fig.
3
. A boosted transduction factor by up to 4
×
compared
to the non-EP-enhanced case is observed when operat-
ing close to the EP (i.e., near the critical detuning fre-
quency). There is good agreement between Eq. (
6
) and
the measurement as shown in Fig.
3
. As an aside, ther-
mal and pumping power fluctuations were the primary
source of noise affecting the measurement. Specifically,
as the system operates closer to the EP, these mecha-
nisms exert a greater impact on the measurement leading
to relatively larger error bars
26
.
In summary, exceptional-point-enhancement of a laser
gyroscope’s sensitivity has been experimentally demon-
5
strated. The gyroscope implementation used a microres-
onator Brillouin laser and verified a boost of the Sagnac
rotation transduction factor up to 4
×
near the EP. The
microresonator-based counter-pumped Brillouin system
makes it possible to engineer the gain/loss rate of CW
and CCW modes independently and can facilitate the
study of EP physics. This work therefore provides a new
platform for studying EPs in a nonlinear optical system
while also demonstrating sensitivity improvement of on-
chip sensors.
METHODS
Origin of the dissipative coupling
In a standing-wave mode
basis, the optical loss induced by the fiber taper or any other
spa-
tially localized absorption or dissipative scattering ele
ment will be
different for each mode and can be captured by the following co
n-
tribution to the Hamiltonian:
H
taper
=
−
iγ
1
0
0
−
iγ
2
.
(7)
Changing to a traveling wave basis (CW and CCW) by using the
relation
|
Φ
±
i
= (
|
CW
i ± |
CCW
i
/
√
2 gives the following Hamilto-
nian in the new basis,
H
taper
=
−
iγ
common
0
0
−
iγ
common
+
0
iκ
iκ
0
(8)
where
γ
common
= (
γ
1
+
γ
2
)
/
2 and
κ
= (
γ
1
−
γ
2
)
/
2. The first term is
the common loss (out-coupling loss of the taper) while the se
cond
term is the dissipative backscattering in Eq. (
1
).
Validity of clamping condition
Note that the Hamiltonian in
Eq.
1
depends on its eigenvalues
ω
s
j
through the Brillouin gain
factor
g
j
=
g
0
/
[1 + 2
i
(
ω
p
j
−
Ω
phonon
−
ω
s
j
)
/
Γ]. However, by sepa-
rating the Brillouin gain factor into real part and imaginar
y parts
as follows:
Re(
g
j
) =
g
0
1 + 4
(
ω
p
j
−
Ω
phonon
−
ω
s
j
)
2
Γ
2
(9)
Im(
g
j
) = Re(
g
j
)
1
−
2
i