IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 14, NO. 4, APRIL 2002 483
Critical Coupling and Its Control in Optical
Waveguide-Ring Resonator Systems
Amnon Yariv
Abstract—The coupling of optical waveguides to ring resonators
holds the promise of a new generation of switches (modulators)
which employ orders of magnitude smaller switching (modulation)
voltages (or control intensities). This requires a means for voltage
(or intensity) control of the coupling between the waveguide and
the resonator. Schemes for achieving such control are discussed.
Index Terms—Electrooptic modulation, integrated optics,
optical resonators, optical waveguides, optoelectronic devices.
R
ECENTLY, we have been witnessing an ever-increasing
pace of activity in the area of coupling between optical
waveguides and ring or micro resonators [1]–[3]. Devices based
on this coupling hold the promise of a new modality of light
switching, amplification, and modulation.
Before this field can proceed to the realm of scientific and
technological applications, we must develop methods, prefer-
ably electrical or optical, to precisely control the coupling. This
is the main concern of this note.
The generic geometry is illustrated in Fig. 1. A waveguide
and a ring resonator, both enter and emerge from a coupling
region (the dashed box) where power exchange takes place. This
exchange is describable in terms of universalrelations which are
independent of the specific embodiment [4]. Some key results
of that analysis needed here are stated in what follows.
If the coupling is limited only to waves traveling in one sense
i.e., no reflectiontakesplace, and if the total powersentering and
leaving the coupling region (“box”) are equal (lossless case),
then the coupling can be described by means of two constants
and and a unitary scattering matrix
(1)
(2)
Equations (1) and (2) are supplemented by the circulation con-
dition in the ring
(3)
Manuscript received August 13, 2001; revised October 26, 2001. This work
was supported by the Office of Naval Research and the Defense Advanced Re-
search Project Agency.
The author is with the Department of Applied Physics, California Institute of
Technology, Pasadena, CA 91125 USA.
Publisher Item Identifier S 1041-1135(02)00875-3.
Fig. 1. The generic geometry for waveguide ring resonator coupling.
where and , real numbers, give respectively, the loss (or
gain) and the phase shift per circulation. The above equations
are solved to yield the transmission factor in the input wave-
guide
(4)
In the above, we use power normalization so that
, are
the respective traveling wave powers. We will, without loss of
generality, take the incident power
to be unity. At reso-
nance
, an integer, and
(5)
This simple universal relation, plotted in Fig. 2, has two very
important features which are the key for most of the proposed
applications. 1) The transmitted power
is zero at a value
of coupling
, “critical coupling,” and 2) For high Q res-
onators,
near unity, the portion of the curve to the right of the
critical coupling point is extremely steep. “Small” changes in
for a given , or vice versa, can control the transmitted power,
, between unity and zero. If we can learn how to control
and/or , we have a basis for a switching technology. If we can
do it sufficiently rapid, we have the basis of a new type of an
optical modulator.
The first of the proposed coupling control schemes is illus-
trated in Fig. 3. It incorporates a Mach–Zehnder interferometer
(MZI) sandwiched between two 3-dB couplers (the “composite”
interferometer, CI) into the ring resonator. The MZI introduces
a differential phase shift
between its two arms. A light cir-
culating, say in a clockwise sense, in the ring need enter one arm
1041-1135/02$17.00 © 2002 IEEE
484 IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 14, NO. 4, APRIL 2002
Fig. 2. The universal transmission plot for the configuration of Fig. 1. ——
=
0
:
999
,——
=0
:
98
.
Fig. 3. A composite interferometer for achieving voltage (or light) control of
the coupling between a waveguide and a ring resonator.
of the CI on the left and exit it on the right. Using the same wave
designation
, ( ), as in Fig. 1, the CI is described by
[4]
(6)
so that
(7)
We note, for example, that if
is zero , and
, i.e., unity transmission. When , ,
. Critical coupling and in general coupling control are,
thus, achieved by controlling
. Using (7) in (4) leads directly
to the transmission expression
(8)
If the two arms of the MZI consistof an electrooptic material the
differential phase shift
is proportional to the applied voltage
.
where is the voltage causing a differential phase shift
in the MZI. Using the last relation and (7) in (5) leads to the
following expression for the transmission at resonance (
, )
(9)
The power transmission
in the “through” fiber can, thus, be
controlled via the applied voltage. When
the transmis-
sion is unity. When
critical coupling
occurs and the transmission is zero. For
(high ring res-
onator) the voltage
needed to turn the transmission off (from
a transmission of unity at
)is
(10)
For a value of
, . Since in conven-
tional electrooptic modulators the “on” to “off” voltage is ap-
proximately
, this reduction by nearly two orders of magni-
tude is, potentially, one of the most important features of this
approach since, using present day materials, it points the way to
modulation voltages measured in tenths of millivolts.
A simple directional coupler can be used instead of the CI
proposed above. The geometry is the same as that depicted in
Fig. 3 except that the MZI section is eliminated and
is now
equal to
i.e., the electrooptically induced phase mismatch
between the two waveguides. This results in a value of
.
The control of the phase shift
, and thus, switching can
be achieved, instead of electrooptically, by injecting an optical
signal into one arm of the MZI of Fig. 3 and utilizing the Kerr
effect.The reductions in the ratio
will be reflected here in
similar reductions in the switching light intensity compared to
nonresonant geometries. Related resonant reductions have been
predicted by Heebner and Boyd [5].
A plot of the transmission as given by (4) as a function of
optical frequency
(or equivalently , where is the
optical path) is shown in Fig. 4. We note the critical coupling at
in the lower solid trace. We also note net transmission
gain results if gain is provided [6]
when
(11)
We note from Fig. 4 that coupling control can also be achieved
by controlling the internal loss parameter
. An experiment
demonstrating such control is described in [6].
YARIV: CRITICAL COUPLING AND ITS CONTROL IN OPTICAL WAVEGUIDE-RING RESONATOR SYSTEMS 485
Fig. 4. Waveguide power transmission against frequency (
=
!L=c
) for the
geometry of Fig. 1 with internal loss factor
as a parameter.
j
t
j
=
0
:
9998
.
——
=0
:
9998
critical coupling,
lb
j
=0
,
1
-
1
-
1
-
=1
transparency,
lb
j
=1
,----
=1
:
00006 1
<<
1
=t
transmission gain,
lb
j
>
1
,
111111
=1
:
20002
=1
=t
laser oscillation.
The modulation bandwidth is limited to , where
is the optical frequency and Q the (loaded) figure of merit of
the resonator
(12)
in the case of a ring resonator of radius R near critical coupling.
The bandwidth can also be estimated from Fig. 4. In the case
of a ring resonator with
cm, ,
m, , we obtain from (12) and
Hz.
C
ONCLUSION
The control of coupling between optical resonators and
waveguides opens up new possibilities of modulating and
switching light. Some generic schemes for achieving this con-
trol by means of applied voltages or injected optical (control)
power have been proposed and discussed.
A
CKNOWLEDGMENT
The author would like to thank Y. S. Park of the Office of
Naval Research and D. Honey of the Defense Advanced Re-
search Project Agency for their generous support of this work.
Productive discussions with Prof. K. Vahala, Dr. R. Lee, and J.
Choi are gratefully acknowledged.
R
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