IEEE JOURNAL
OF
QUANTUM ELECTRONICS,
VOL.
QE-9,
NO.
9,
SEPTEMBER
1973 919
Coupled-Mode Theory
for
Guided-Wave
Optics
AMNON YARIV
Absrruct-The problem of propagation and interaction
of
optical radia-
tion
in
dielectric waveguides is cast
in
the coupled-mode formalism. This ap-
proach is useful for treating problems involving energy exchange between
modes.
A
derivation
of
the general theory is followed
by
application to the
specific cases of electrooptic modulation, photoelastic and magnetooptic
modulation, and optical filtering.
Also
treated are nonlinear optical
applications such as second-harmonic generation
in
thin
films and phase
matching.
I. INTRODUCTION
A
GROWING
BODY
of theoretical and experimental
work has been recently building up in the area of
guided-wave optics, which may be defined as the study and
utilization
of
optical phenomena in thin dielectric
waveguides
[l],
[2].
Some of this activity is due to the hopes
for integrated optical circuits in which a number of optical
functions will be performed
on
small solid substrates with
the interconnections provided by thin-film dielectric
waveguides [3],
[4].
Another reason for this interest is the
possibility of new nonlinear optical devices and efficient op-
tical modulators which are promised by this approach
A
variety of theoretical
ad
hoc
formalisms have been
utilized to datein treating thevarious phenomena ofguided-
wave optics. In this paper we present a unified theory cast in
the coupled-mode form to describe a large number of
seemingly diverse phenomena. These include:
1)
nonlinear
optical interactions;
2)
phase matching by periodic pertur-
bations; 3) electrooptic switching and modulation;
4)
photoelastic switching and modulation; and
5)
optical filter-
ing and reflection by a periodic perturbation.
[51-[71.
11.
THE COUPLED-MODE
FORMALISM
We will employ, in what follows, the coupled-mode for-
malism
[X]
to treat the various phenomena listed in Section
I.
Before embarking on a detailed analysis it will prove
beneficial to consider some of the common features
of
this
theory. Consider two electromagnetic modes with, in
general, different frequencies whosecomplex amplitudes are
A
and
B.
These are taken as the eigenmodes of the unper-
turbed medium
so
that they represent propagating distur-
bances
Manuscript received March 9, 1973. This research was supported in
part by the National Science Foundation and in part by the Advanced
Research Projects Agency through the Army Research Office, Durham,
N.C.
The author is with the Department of Electrical Engineering, Califor-
nia Institute
of
Technology, Pasadena, Calif. 91109.
with
A
and B constant.
In the presence
of
a perturbation which, as an example,
can take the place
ofaperiodicelectricfield,
asoundwave,
or
a surface corrugation, power is exchanged between modes
a
and
6.
The complex amplitudes
A
and
B
in this case are no
1ongerconstantbutwillbefoundtodependonz.Theywillbe
shown below to obey relations of the type
where the phase-mismatch constant
A
depends on the
propagation constants
Pa
and
Pb
as well as on the spatial
variation of the coupling perturbation. The coupling
coefficients
K~~
and
Kba
are determined by the physical situa-
tion under consideration and their derivation will take up a
major part of this paper. Before proceeding, however, with
the specific experimental situations, let us consider some
general features of the solutions of the coupled-mode
equations.
A.
Codirectional
Coupling
We take up, first, the case where modes a and
b
carry
(Poynting) power in thesame direction. It is extremely con-
venient to define
A
and
B
in such a way that
IA(z)(
and
I
B(z)l
correspond to the power carried by modes a and
b,
respectively. The conservation of total power is thus ex-
pressed as
t
3)
which, using
(2),
is satisfied when
[9]
If boundary conditions are such that a single mode, say
b,
is
incident at
z
=
0
on the perturbed region
z
>
0,
we have
b(O)=B,,
a(O)=O.
(5)
Subject to these conditions the solutions of
(2)
become
920
IEEE JOURNAL
OF
QUANTUM ELECTRONICS, SEPTEMBER
1973
z=O
2.L
Fig.
1.
The variation of the mode power in the case
of
codirectional Fig.
2.
The transfer
of
power from an incident forward wave
B(z)
to a
coupling
for
phase-matched and unmatched operation. reflected wave
A(z)
in the case of contradirectional coupling.
B(z)
=
BoeiA2/'{
cos
[9(4~'
+
A2)'''zI
the space betweenz
=
0
andz
=
L.
Sincemodeaisgenerated
by the perturbation we have a(L)
=
0.
With these boundary
A
-
2
(4K
+
A
)
sin
[4(4,('
+
~')"~~z]j
(6)
conditions the solution of
(1
1)
is
given by
where
K~
=
I
K=,,I
2.
Under phase-matched condition
A
=
0,
a
A(Z)
=
B(O)
SL SL
complete spatially periodic Ijower transfer between modesa
-A
sinh
-
f
is
cosh
-
and
b
takes place with a period
ir/2K.
2
2
2iK,be-i(4.2/2)
sinh
[f
(z
-
L)]
-i(Az/2)
B(z)
=
B(0)
-
c
-A
sinh
--
+
is
cosh
-
SL
SL
2
2
b(=,
t)
=
~~~~(~b~-@b~)
cos
(KZ).
(7)
-{A
sinh
[$
(z
-
L)]
+
is
cosh
A
plot of the mode intensities
1
a1
and
1
bJ
is shown in Fig.
1.
This figure demonstrates the fact that
for
phase mismatch
A
>>
I
K~,,[
the power exchange between the modes is negligi-
s
E
44K2
-
A',
K
E
]K,bI.
(1
3)
ble. Specific physical situations which are describable in
terms of this picture will be discussed further below. Under phase-matching conditions
A
=
0
we have
B.
Contradirectional Coupling
In
this case the propagation in the unperturbed medium
is
described by
A(z)
=
B(0)
tf)
-
sinh
[x(z
-
L)]
cosh
(KL)
cosh
[K(Z
-
L)]
cash
(KL)
B(z)
=
B(0)
-
(1
4)
A
plot
of
the mode powers
1
B(z)J and
1
A(z)J for this case
(')
is
shown in Fig.
2.
For sufficiently large arguments of the
where
A
and
B
are constant. Mode a corresponds to a left
(-z)
traveling wave whileb travels to the right.
A
time-space
periodic perturbation can lead to power exchange between
the modes. Conservation
of
total power can be expressed as
cosh and sinh functions
in
(14),
the incident-mode power
decays exponentially along the perturbation region. This
decay, however, is due not to absorption but to reflection of
power into the backward traveling mode
a.
This case will be
considered in detail in following sections, where acoustoop-
d
tic, electrooptic, and spatial index perturbation will be
-
(1
AI2
-
IBI')
=
0
dz
(9)
treated. The exponential-decay behavior
of
Fig.
2
will be
shown in Section
VI11
to correspond to the stopband region
which is satisfied by
(2)
if we take of periodic optical media.
(10)
111.
ELECTROMAGNETIC DERIVATIONS
OF
THE
COUPLED-
MODE EQUATIONS
so
that
A.
TE
Modes
-
dA
=
K,bBe-iAz
dB
=
K,b*Ae"z
dz
dz
(11)
Consider the dielectric waveguide sketched in Fig.
3.
It
consists of a film of thickness
t
and index
of
refraction
nz
In this case we take the mode
b
with an amplitude
B(0)
to be sandwiched between media with indices
n,
and
n,.
Taking
incident at
z
=
0
on the perturbation region which occupies
(a/ay)
=
0,
this guide can, in the general case, support a
YARIV:
COUPLED-MODE THEORY
92
1
power flow of
1
A
I
W/m. The normalization condition is
nl
thus
n2
-
propagation
n3
x=-t
where the symbol m denotes themth confined TE mode cor-
Fig.
3.
The basic configuration of
a
slab dielectric waveguide.
responding to mth eigenvalue
of
(19).
Using
(1
7) in
(20)
we determine
finite number of confined TE modes with field components
E,,
H,,
and
Hz,
andTMmodeswithcomponents
H,,
E,, and
E,.
The "radiation" modes of this structure which are not
cM
=
2hm
y2.
(21)
and
will
be ignored. The field component
E,
of the TE
modes, as an example, obeys the wave equation Since the modes are orthogonal we have
confined to the inner layer are not considered in this paper
[P.,
(t
+
-
11
+
--)(hm~
+
qmz),
4m
Pm
We take
E,(x,
z,
t)
in
the form
B.
TM
Modes
Ey(x,z,l)
=&y(x)e"wt-flz'.
(16)
I
The field components are
The transverse function
&,,(x)
is taken as
H,(x,
z,
t)
=
Xy(x)ei(Wt-iBZ)
COS
(hx)
-
(q/h)
sin
(hx)],
&"(X)
=
-t<xIO
which, applying
(15)
to regions
1,
2,
3,
yields
The continuity of
H,
and E, at the interfaces requires that
From
the
requirement
that
and
Hz
becontinuous
at
x
=
thevariouspropagationconstantsobeytheeigenvalueequa-
and
x
=
-t,
we obtain'
tion
tan
(ht)
=
4+P
.
(1
9)
tan
(hi)
=
htP
+
4)
h(l
-
y)
ha
-
(25)
where
This equation in conjunction with (18)
is
used to obtain the
eigenvalues
p
of the confined TE modes.
2
2
Theconstant Cappearingin (17) isarbitrary. Wechooseit
ji
G
-sp,
n2 n
q
-4j
4.
in
such a way that the field
&,(x)
in (17) corresponds to a
n3
nl
power flow
of
1
W
(per unit width in they direction)
in
the The normalization constant C is chosen
so
that the field
mode.
A
mode whose
EN
=
A&
.(x)
will thus correspond to a represented by
(23)
and
(24)
carries
1
W per unit width in the
y
direction.
The assumed
form
of
E,
in
(17)
is such that
E,
and
X,
=
(i/wp)
a
o,/ax
are continuous at
x
=
0
and that
E,
is continuous at
x
=
-I.
All
that is left is to require continuity of
aE,/ax
at
x
=
-I.
This leads
to
(1
9).
1
HUEx*
dx
=
!.-
/rn
x,"o
dx
=
1
2
-m
2u
-m
E
922
IEEE
JOURNAL
OF
QUANTUM
ELECTRONICS, SEPTEMBER
1973
Multiplying
(31)
by
&y(m)(x),
and integrating and making
use of the orthogonality relation
(22)
yields
2W€o
dx
=
-
Pn
This condition determines the value of
C,
as
[lo]
I
(27)
C. The Coupling Equation
The wave equation obeyed by the unperturbed modes is
a2E
at
V2E(r,
t)
=
pe
-3
.
We will show below that in most of the experiments of in-
terest to
us
we can represent the perturbation as a distributed
polarization source
Ppert(r,t),
which accounts for the devia-
tion of
the medium polarization from that which accompanies
the unperturbed mode.
The wave equation for the perturbed
case follows directly from Maxwell's equations
if
we take
D
=
coE
+
P.
with similar equations for the remaining Cartesian com-
ponents of
E.
We may take theeigenmodes of
(28)
as an orthonormal set
in
which to expand
E,
and write
where
1
extends over the discrete set of confined modes and
includes both positive and negative traveling waves. The in-
tegration over
/3
takes in the continuum
of
radiation modes,
and C.C. denotes complex conjugation. Our chiefinterest lies
in perturbations which couple only discrete modes
so
that, in
what follows, we will neglect the second term
on
the right side
of
(30).
Problems of coupling to the radiation modes arise in
connection with waveguide losses
[
1
11
and grating couplers
Substituting
(30)
into
(29),
assuming "slow" variation
so
that
d2Am/dz2
<<
Dm
dAm/dz,
and recalling that
&ycml
(x)
P21.
ei(wt
-
Omz)
obeys the unperturbed wave equation
(28),
gives
where
A
m(-j
is the complex normal mode amplitude
of
the
negative traveling
TE
mode while
A
m(+)
is that
of
the positive
one. Equation
(32)
is the main starting point for the follow-
ing discussion in which we will consider a number of special
cases.
IV.
NONLINEAR INTERACTIONS
In this section we consider the exchange of power between
three modes of different frequencies brought about through
the nonlinear optical properties of the guiding
or
bounding
layers. The relevant experimental situations involve second-
narmonic generation, frequency up-conversion, and optical
parametric oscillation.
To
be specific we consider first the
case of second-harmonic generation from an input mode at
w/2
to an output mode at
w.
The perturbation polarization is
taken as
The complex amplitude of the polarization is
where
dijk("')
is an element
of
the nonlinear optical tensor and
summation over repeated indices is understood. We have
allowed, in
(34),
for apossible dependence ofdij,ontheposi-
tion
r.
A.
Case
I:
TEinpUt-TEoutpUt
Without going, at this point, into considerations in-
volving crystalline orientation, let
us
assume that an optical
field parallel to the waveguide
y
direction will generate a
second-harmonic polarization along the same direction
where
P
and
E
represent complex amplitudes, and
d
corresponds to a linear combination of
dijk
which depends
on
the crystal orientation.
In
this special case
an
input TE
mode at
w/2
will generate an output TE mode at
w.
Using
(30)
in
(35)
gives
We consider a case of a single mode input, say
n.
In that case
the double summation of
(36)
collapses to asingle term
n
=
p.
923
YARIV: COUPLED-MODE THEORY
If we then use
Py(r,t)
as
[Ppert(r,
t)Jy
in (32) we get
with
where we took
d(r)
=
d(z)Ax),
In the interest of conciseness let
us
consider the case where
the inner layer
2
is nonlinear and where both the input and
output modes are well confined. We thus have
qm,pm
>>
h,
and
h,d
=
T.
From
(17)
and
(21)
we get
8-
The overlap integral
S(n,n,m)
is maximum for
n
=
m
=
1, i.e.,
fundamental mode operation both at
w
and 0/2. For this
case the overlap integral becomes
(39)
and where the, now-superfluous, .mode-number subscripts
have been dropped. Integrating
(40)
over the interaction dis-
tance
1
gives
The normalization condition
(20)
was chosen
so
that
I
A
I
is the power per unit width in the mode.
We
can thus rewrite
(42)
as
where we used
,P
adz,
E/€,,
=
n2.
Note that
(P12/wt)
is
the intensity (watts/square meter) of the input mode. Except
for a numerical factor of
1.44,
this expression is similar to
that'derived for the bulk-crystal case'[ 131. Efficient conver-
Fig.
4.
The orientation
of
a 43m crystal
for
converting a
TM
input at
w/2
to
a
TE
wave at
w.
x,
y,
z
are thedielectric-waveguide coordinates,
while
I,
2,
and 3 are the crystalline axes.
Top
surface is (100):
sion results when the phase-matching condition
is satisfied. In this case the factor sin2 (Al/2)(A1/2l2 is
unity. Phase-matching techniques will be discussed later.
B.
Case
Ii:
TMingut-TEoutput
The anisotropy of the nonlinear optical properties can be
used in such a way that the output at
w
is polarized
orthogonally to the field of the input mode at w/2. To be
specific, we co'nsider thecase of an input TM mode and an
output TE mode. If, as an example, theguiding layer
(or
one
of the bounding layers) belongs to the 43m crystal class
(GaAs, CdTe, InAs), it is possible
to
have a guide geometry
as shown in Fig.
4.
x,y,z
is the waveguide coordinate system
as defined in Fig.
4,
while 1,'2, and 3 are the conventional
crystalline axes. For input TM mode with
E
I
I
x
we have
The nonlinear optical properties of a3m crystals are
described by
[13]
so
that
Taking
and using
(8Hy/8z)
=
-iwt
E,
gives