We study the effect of rotation on the propagation of electromagnetic waves
in slow-light waveguide structures consisting of coupled micro-ring resonators.
We show that such configurations exhibit new a type of the Sagnac effect which
can be used for the realization of highly-compact integrated rotation sensors
and gyroscopes

A. Yariv

Amnon Yariv

H. J. Arditty

J. Scheuer

Jacob Scheuer

M. F. Yanik

M. L. Gorodetsky

S. Ezekiel

English

arXiv

We study the effect of rotation on the propagation of electromagnetic waves in slow-light waveguide structures consisting of coupled microring resonators. We show that such configurations exhibit a new type of Sagnac effect which can be used for the realization of highly compact integrated rotation sensors and gyroscopes

Scheuer, Jacob

Yariv, Amnon

Caltech Authors - Main

PRL 96, 053901 (2006) P H Y S I C A L R E V I E W L E T T E R S
week ending
10 FEBRUARY 2006Sagnac Effect in Coupled-Resonator Slow-Light Waveguide Structures
Jacob Scheuer1,* and Amnon Yariv2
1Center for the Physics of Information, California Institute of Technology, M/C 128-95, Pasadena, California 91125, USA
2Department of Applied Physics, California Institute of Technology, Pasadena, California 91125, USA
(Received 25 April 2005; published 6 February 2006)0031-9007=We study the effect of rotation on the propagation of electromagnetic waves in slow-light waveguide
structures consisting of coupled microring resonators. We show that such configurations exhibit a new
type of Sagnac effect which can be used for the realization of highly compact integrated rotation sensors
and gyroscopes.
DOI: 10.1103/PhysRevLett.96.053901 PACS numbers: 42.60.Da, 42.70.QsFIG. 1. Schematic of the coupled-resonator slow-light rotation
sensor.When an electromagnetic wave propagates in a moving
medium it accumulates additional phase shift, compared to
a wave propagating in a stationary medium, which depends
on the scalar product of the wave propagation direction and
the velocity vector of the medium [1,2]. A particularly
interesting configuration is that of a wave propagating
along a circular path in a rotating medium. In such a
scenario, the additional phase accumulated by the wave
depends on the relation between the propagation directions
of the medium and the wave (codirectional or counter-
directional). This phase difference is often referred to as
the Sagnac effect and in addition to its scientific impor-
tance, it has numerous practical applications such as de-
tection and high-precision measurement of rotation.
In the past few years, much attention has been devoted to
slowing down the propagation speed of light and to coher-
ently stop and store pulses of light [3–6]. There are two
major approaches to achieve significant reduction of the
group velocity of light, which employ either electronic or
optic resonances. Because of the inherent constraints asso-
ciated with the conversion of the optical signals to coherent
electronic states, the electronic resonance approach is less
attractive for practical implementations of slow-light de-
vices. Consequently, significant efforts were focused on
controlling the speed of light using photonic structures
incorporating microcavities and photonic crystals.
Substantial delays and storage of light pulses were pre-
dicted in various coupled-cavities structures such as
coupled-resonator optical waveguides (CROWs) [7] and
side-coupled integrated spaced sequence of resonators
(SCISSORs) [8]. In addition to their practical advantages,
coupled-cavity waveguides (CCW) exhibit similar charac-
teristics to those of periodic atomic lattices, such as energy
(frequency) bands, forbidden gaps, etc., This property
makes optical periodic structures, and, in particular,
CCW, an excellent platform for study and demonstration
of the physical phenomena found and predicted in atomic
systems such as band anticrossing and electromagnetically
induced transparency (EIT) [7].
Recently, Leonhardt and Piwnitski pointed out the ad-
vantages of using the Sagnac effect in slow-light medium
generated by EIT for the realization of an ultrasensitive06=96(5)=053901(4)$23.00 05390optical gyroscope [9]. Subsequently, Steinberg studied the
effect of rotation in coupled photonic crystal defect cav-
ities [10] and Matsko et al. proposed to utilize the disper-
sive characteristics of slow-light propagation in a closed-
loop SCISSOR-like configuration to realize a high-
sensitivity miniaturized optical gyroscope [11]. In that
study, however, the SCISSOR was modeled as a highly-
dispersive conventional waveguide where the slow group
velocity of the light in the SCISSOR stems from the
average interaction of the light with the high-Q resonators.
In this Letter, we study the properties of the Sagnac
effect in a CROW which is wrapped around itself, with
application for a highly compact rotation sensor or an
optical gyroscope. Figure 1 illustrates the geometrical
configuration: light is launched into the input waveguide
and equally divided between the two channels of the 3dB
coupler. The signal in each arm is coupled to a different
end of the circular CROW consisting of directly coupled
ring resonators. Finally, the counterpropagating signals
(marked by the gray and white arrows) are combined by
the 3dB coupler where the output signal in each arm of the
coupler depends on the relative phase difference between
the signals:
jArj2  jEinj2cos2

1
2


;
jBrj2  jEinj2sin2

1
2


;
(1)
where Ein and  are correspondingly the input amplitude
and the phase difference between the counterrotating1-1 © 2006 The American Physical Society
PRL 96, 053901 (2006) P H Y S I C A L R E V I E W L E T T E R S
week ending
10 FEBRUARY 2006fields. When the device is stationary, the overall phases
accumulated by both signals are identical, i.e.,   0,
resulting in complete cancellation of Br. On the other hand,
when the device is rotating, the phases accumulated by the
signals differ, resulting in a nonvanishing intensity Br.
To evaluate the phase difference  in a CROW it is
convenient to divide the structure into sections as illus-
trated in Fig. 1: An input section which consists of the
CROW input coupler and part of the first microring (this
section is marked by the dashed white line ‘‘I’’). A recur-
ring section consisting of two halves of a microring
coupled to a complete ring, constituting the main body of
the CROW (this section is defined by two successive
dashed white lines: I ! S1; S1 ! S2, etc.). And an output
section which is similar to the input section (from the line
marked by ‘‘O’’ to the output coupler). Because of the
recurring section, it is convenient to represent each section
by a transfer matrix linking between the input and output
ports of the section. The overall transfer matrix of the
structure is then found simply by multiplying these matri-
ces in the correct order.
The phase accumulated by a wave propagating in non-
stationary waveguide depends primarily on the scalar prod-
uct of the waveguide velocity and the wave vector k. In the
configuration studied here, the contribution of each seg-
ment d~r in each microring is different because the center of
rotation does not necessarily coincide with the center of
any of the microrings. Therefore, in order to construct the
transfer matrix of each section we have to evaluate the
phase accumulated by a wave propagating along a curved
waveguide segment which is rotating around an arbitrary
point.
Figure 2 illustrate the geometry of this problem: a wave
propagating in microring resonator with radius R while the
center of this ring is rotating with angular velocity 
around a fixed point. The distance between the center of
the microring and the center of rotation is ~R0. The phase
accumulated by the wave as it propagates along a segment05390d~r stems from two contributions: The conventional phase
due to the propagation dprop  !=cnjdrj and the
rotation-related phase shift which is given by [1]:
drot 
!n2
c2
1  ~V  d~r; (2)
where! is the optical (angular) frequency, c is the speed of
light in vacuum, n is the refractive index, ~V  ~  ~~R0 
~R is the linear velocity of the segment, and  is the
Fresnel-Fizeau drag coefficient given by   c1 n2)
(for a nondispersive medium). Therefore, the overall
rotation-related phase accumulated by an electromagnetic
wave which propagates in a microring from s to f (see
Fig. 2) is given by:
rot 
Z f
s
drot

!
c2
R2f  s 
!
c2
R ~R0cosf  coss:
(3)
Equation (3) exhibits several interesting properties that
should be noted. First, the rotation-related phase shift is
independent of the waveguide index of refraction n. This is
a well-known property of the Sagnac effect which does not
depend on the refractive index of the medium comprising
the loop. Second, for a complete loop the second term in
(3) vanishes and thus the phase shift is independent of the
center of the rotation. However, in the structure analyzed
here, the propagation section between two couplers does
not form a complete loop, and therefore, the second term
must be included.
For simplicity, we assume that the microrings are iden-
tical and lossless and that the coupling coefficients 
between adjacent microrings are also identical. The trans-
fer matrices for the three types of sections are straightfor-
wardly given byM1 
0 expi=2P D
=2
S 
S;=2
R 
expi=2P D
=2
S 
S;=2
R  0
0@ 1AC
MO  C
expi=2P D
=2
S 
S;=2
R  0
0 expi=2P D
=2
S 
S;=2
R 
0
@
1
A
MR  MCC
expiP D

S 
C;
R  0
0 expiP D

S 
C;
R 
0
@
1
ACMC
MC 
expi=2P D
=2
S 
C;=2
R  0
0 expi=2P D
=2
S 
C;=2
R 
0
@
1
A
C 
i

p

1 
p
1
1 

1 
p
 !
P  !nR=c; 

S  !R
2=c2; S;R  !R ~R0 sin=c
2; ;C;R  !R ~R0 cos=c
2;
(4)
where MI, MO, and MR are, respectively, the transfer matrices for the input section, the output section, and the recurring1-2
θs
θf
R
0
~
R
θ
Ω
Propagating wave
Center of 
rotation
FIG. 2. Phase accumulation in a rotating microring resonator.
PRL 96, 053901 (2006) P H Y S I C A L R E V I E W L E T T E R S
week ending
10 FEBRUARY 2006section, D  1 for the signal propagating with the device
rotation and 1 for the signal counter propagating to the
device rotation.  is the angle between adjacent microrings
(see Fig. 1). The overall transfer matrix connecting the
input and output of the CROW is, therefore, given by
MCCW  MOMR
N1=2MI; (5)
where N is the number of microrings comprising the
CROW which must be odd for the configuration illustrated
in Fig. 1.
Equations (5), (4), and (1) allow us to calculate the
output signal Br for various parameters. A closer inspec-
tion of (4) allows us to eliminate some of the terms because
we are not interested in the complete transmission function
of the CROW but rather in the phase difference between
the two paths. The phase terms proportional to ~R0 can be
rewritten as unit matrices multiplied by a common phase0 5 10
0
20
40
60
80
100
120
140
160
180
Relat ive slope
quadratic fit
Numb
S N
/S
1
0 100 200 300 400 500 600 700 800 900 1000
0
1
2
3
4
5
6
|B
r|2
[a
.u
.]
Ω [Hz]
Nring=1
Nring=6
Nring=11
Nring=16
Nring=21(a)
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
0
1
2
3
4
5
6
7
8
x108
numeric
analytic
κ
R
es
p o
ns
i v
ity
d|
B
r|2
/d
Ω
[1
/H
z ]
Ring r
0 10 20 30 40
0
0.5
1
1.5
2
2.5
3
3.5
x105
R
es
po
ns
iv
ity
d|
B
r|2
/d
Ω
[ 1
/H
z]( )
κ
πω 22
2
2
1
R
N
cd
Bd r +=
Ω
S N
/S
1
S N
/S
1
(b)
(c) (d)
|B
r|2
[a
.u
.]
|B
r|2
[a
.u
.]
R
es
p o
ns
i v
ity
d|
B
r|2
/d
Ω
[1
/H
z ]
i v
ity
d|
B
r|2
/d
Ω
[1
/H
z ]
R
e s
po
ns
i v
ity
d|
B
r|2
/d
Ω
[ 1
/H
z]
05390factor. Since this phase factor is identical for both paths, it
has no effect on the outcome of (1), and therefore, the
output signal Br is independent of ~R0. This is an important
conclusion because ~R0 defines the area of the effective
CROW ring. The Sagnac sensitivity in conventional wave-
guide loops is directly related to the area of the loop, thus
we cannot analyze this effect in the CROW loop simply by
assuming an effective ring waveguide with the dispersion
relation of a CROW. It should be noted that the rotation
induced phase due to the waveguide sections connecting
the 3dB coupler to the CROW is neglected because it can
be made significantly smaller than that of the CROW.
Figure 3(a) depicts the output intensity jBrj2 as a func-
tion of  for CROWs with different number of resonators.
The parameters of these CROWs are defined in the figure
caption. As can be expected, the output intensity increases
with  and the responsivity (slope) increases with the
number of microrings. For the rotation-sensing application,
a steeper slope is advantageous because it corresponds to
higher sensitivity, i.e., the ability to detect slower rotation
rates. Figs. 3(b)–3(d) show the relative responsivity of the
CROW loop for a varying number of rings (3b), coupling
coefficient (3c), and the microrings radius (3d). The re-
sponsivity increases with increasing number of microrings
N, smaller coupling coefficient k, and larger microrings’
radius R.
It is worthy to quantify some of these trends because
they reveal the inherent differences between the Sagnac ef-
fect in CROWs and in conventional waveguides.
Figure 3(b) shows a quadratic fit to the dependence of
the responsivity on the number of rings comprising the
CROW. The fit indicates that the responsivity of a closed-
loop CROW, SN , consisting of N microrings is related to
that of a single ring according to SN  N  12S1=4. It is
well known that the slope of the output signal of a single-15 20 25
er of rings
adius [µm]
50 60 70 80 90
FIG. 3. (a) Output signal intensity as a
function of the structure angular velocity
for various numbers of rings, R 
25 m,   0:01. (b) Dependence of
the relative sensitivity on the number of
microrings, R  25 m,   0:01.
Dependence of the sensitivity on:
(c) the coupling coefficient (R 
25 m, N  9) and on (d) the micror-
ings’ radius (N  9,   0:01).
1-3
105 106 107 108 109 1010 1011 1012
1000
2000
3000
4000
5000
6000
7000
8000
9000
Q
R
es
po
ns
iv
ity
d|
B
r|2
/d
Ω
[1
/H
z]
R
es
po
ns
iv
ity
d|
B
r|2
/d
Ω
[1
/H
z]
FIG. 4. The impact of the quality factor on the sensitivity. R 
25 m, N  9,   0:03.
PRL 96, 053901 (2006) P H Y S I C A L R E V I E W L E T T E R S
week ending
10 FEBRUARY 2006ring device is proportional to the ring area [12], and there-
fore, the responsivity of the CROW-based device is pro-
portional to the total area of the microrings composing it.
Alternatively, it can be shown that the responsivity of the
CROW gyro is proportional to the square effective length
of the CROW, Leff  RN=jj [7,8]. In the interesting
limit of   1 where the CROW loop is reduced to a
‘‘conventional’’ waveguide loop, this result coincides
with the conventional expression for the responsivity be-
cause L2  4Aloop. It should be emphasized that, unlike
the conventional Sagnac effect, the overall area circum-
scribed by the CROW does not affect the output signal.
This result, which clearly demonstrates the difference be-
tween the Sagnac effect in conventional and in CROWs, is
interesting and, to some extent, counterintuitive because
one might expect the Sagnac effect contributions from
adjacent microrings to cancel each other. Figure 3(c) also
compares the numerically calculated responsivity accord-
ing to (5) and an analytic expression based on the derived
responsivity of a single microring, and the quadratic de-
pendence of the responsivity on the number of microrings
in the CROW.
For practical applications, the CROW-based gyroscope
exhibits several inherent advantages compared to conven-
tional Sagnac loops: (1) The dependence of the gyro output
signal on the interring coupling allows the enhancement of
the device sensitivity without requiring larger area; (2) the
independence of the responsivity of the CROW gyro on ~R0
indicates that the arrangement of microrings comprising
the CROW is insignificant, and thus, enables a more effi-
cient utilization of the chip area.
In order to take advantage of the complete capacity of
the CROW gyro it is desired that all cavities have the same
resonance frequency. In practice, when such a structure is05390fabricated, there could be some deviations between the
resonances of the cavities due to fabrication errors. How-
ever, this problem can be overcome by post-tuning the
cavities after the fabrication using, e.g., the electro-optic
or thermo-optic effects.
The ultimate limiting factor of the ability of a rotation
sensor to detect slow angular velocity is the minimal
detectable output power jBrj2 which is shot noise limited.
While ideal microring resonators are lossless, when light
propagates in real resonators it experiences propagation
loss that can be included in our analysis by introducing an
imaginary part to the index of refraction in (4). The propa-
gation loss decreases the output signal and reduces the
responsivity djBrj2=d of the rotation sensor. Figure 4
shows the responsivity of a CROW rotation sensor as a
function of the resonators’ quality factor (Q). As shown in
the figure, for resonators withQ> 107, the influence of the
propagation loss is negligible and has small effect on the
device response. This transition occurs when the intrinsic
losses of the cavities become smaller than the ‘‘coupling’’
losses (i.e., Qint  Qloaded). Since high-Q (>107) single-
mode, planar-technology-based microring resonators are
being fabricated by many research groups [13], the propa-
gation losses in the cavities do not limit significantly the
sensitivity of the CROW rotation sensor.
J. S. is grateful to Beny Steinberg (Tel-Aviv University)
for stimulating discussions and useful comments. The
support of DARPA is gratefully acknowledged.1-4*Electronic address: koby@caltech.edu
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Sagnac Effect in Coupled-Resonator Slow-Light Waveguide Structures

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