I
I1
JOURNAL
OF
LIGHTWAVE
TECHNOLOGY,
VOL.
13,
NO.
4,
APRIL
1995
615
Optical Multi-Mode Interference Devices Based
on
Self-Imaging
:
Principles and Applications
Lucas
B.
Soldano and
Erik
C.
M.
Pennings,
Member, IEEE
Invited
Paper
Abstract-This paper presents an overview of integrated optics
outline of the modal propagation analysis (MPA), which
will be used later to describe image formation by general
and restricted multimode interference (Sections
IV
and V,
MMI devices are discussed in Section
VI.
Performances and
compatibility with other components are presented through
examDles of fabricated MMI CouDlers and their aDDlications
routing and coupling devices based on multimode interference.
The underlying self-imaging principle in multimode waveguides
is described using a guided mode propagation analysis. Special
ference devices are discussed, followed by a survey of reported
applications. It is shown that multimode interference couplers
Offer Superior performance, excellent tolerance to polarization
issues concerning the design and operation of multimode inter-
design
and
behavior issues
concerning
I.
and wavelength variations, and relaxed fabrication requirements
when compared
to
alternatives such as directional couplers,
adiabatic
X-
or
Y-junctions, and diffractive
star
couplers.
in elaborate optical circuits ;Section ~11).
we
conclude
by comparing the properties of MMI devices with those
of
more conventional routing and coupling devices.
I.
INTRODUCTION
ODAY’S evolving telecommunication networks are in-
T
creasingly focusing on flexibility and reconfigurability,
which requires enhanced functionality of photonic integrated
circuits (PICs) for optical communications.
In
addition, mod-
em wavelength demultiplexing (WDM) systems will require
signal routing and coupling devices to have large optical
bandwidth and to be polarization insensitive. Also small device
dimensions and improved fabrication tolerances are required
in order to reduce process costs and contribute to large-scale
PIC production.
In recent years, there has been a growing interest in the ap-
plication of multimode interference (MMI) effects in integrated
optics. Optical devices based on MMI effects fulfil all of the
above requirements, and their excellent properties and ease
of fabrication have led to their rapid incorporation in more
complex PICs such as phase diversity networks [l], Mach-
Zehnder switches
[2]
and modulators
[3],
balanced coherent
receivers [4], and ring lasers
[5],
[6].
This paper reviews the principles and properties of MMI
devices and their applications. The operation of optical
MMI
devices is based on the self-imaging principle, presented
in Section 11. Basic properties of multimode waveguides
are introduced early in Section 111, followed by a short
Manuscript received August
8,
1994; revised December 19, 1994. This
work was supported in part by the Netherlands Technology Foundation
(STW)
as part of the programme of the Foundation for Fundamental Research
on
Matter (FOM).
L. B.
Soldano is with Delft University of Technology, Department
of
Electrical Engineering, Laboratory
of
Telecommunication and Remote Sensing
Technology, 2628 CD Delft, The Netherlands.
E. C. M. Pennings is with Philips Research Laboratories, Wideband
Communication Systems, 5656 AA Eindhoven, The Netherlands.
IEEE Log Number 9409613.
11.
THE SELF-IMAGING PRINCIPLE
Self-imaging of periodic objects illuminated by coherent
light was first described more than
150
years ago [7]. Self-
focusing (graded index) waveguides can also produce periodic
real images of
an
object
[8].
However, the possibility of
achieving self-imaging in uniform index slab waveguides was
first suggested by Bryngdahl
[9]
and explained in more detail
by Ulrich [lo], [lll.
The principle can be stated as follows:
Self-imaging is a
property
of
multimode waveguides by which an input field
profile is reproduced
in
single or multiple images at periodic
intervals along the propagation direction
of
the guide.
111.
MULTIMODED
WAVEGUIDES
The central structure of an MMI device is a waveguide
designed to support
a
large number of modes (typically
_>
3).
In order to launch light into and recover light from that
multimode waveguide, a number of access (usually single-
moded) waveguides are placed at its beginning and at its
end. Such devices are generally referred to as
N
x
M
MMI
couplers,
where
N
and
M
are
the number of input and output
waveguides respectively.
A
full-modal propagation analysis is probably the most
comprehensive theoretical tool to describe self-imaging phe-
nomena in multimode waveguides. It not only supplies the
basis
for
numerical modelling and design, but it also provides
insight into the mechanism of multimode interference. Other
approaches make use of ray optics [12], hybrid methods [13],
or BPM type simulations. We follow here the guided-mode
propagation analysis (MPA), proposed first in [ll] for the
formulation of the periodic imaging.
0733-8724/95$04.00
0
1995 IEEE
I
-.‘
616
JOURNAL
OF
LIGHTWAVE
TECHNOLOGY.
VOL.
13,
NO.
4,
APRIL
1995
Self-imaging may exist in three-dimensional multimode
structures, for which MPA combined with two-dimensional
(finite-element or finite-difference methods) cross-section cal-
culations can provide
a
useful simulation tool
[14].
How-
ever, the current trend of etch-patterning produces step-index
waveguides, which are, in general, single-moded in the trans-
verse direction.
As
the lateral dimensions are much larger
than the transverse dimensions, it is justified to assume that
the modes have the same transverse behavior everywhere in
the waveguide. The problem can thus be analyzed using a
two-dimensional (lateral and longitudinal) structure, such as
the one depicted in Fig.
1,
without losing generality. The
analysis hereafter is based on such a
2-D
representation
of
the
multimode waveguide, which can be obtained from the actual
3-D
physical multimode waveguide by several techniques,
such
as
the effective index method (EIM) [15] or the spectral
index method (SIM) [16].
A. Propagation Constants
Fig.
1
shows
a
step-index multimode waveguide of width
Whl.
ridge (effective) refractive index
nr
and cladding (effec-
tive) refractive index
nc.
The waveguide supports
m
lateral
modes
(as
shown in Fig.
2)
with mode numbers
v
=
0,
1,
...
(m
-
1)
at
a
free-space wavelength
Xo.
The lateral
wavenumber
k,,
and the propagation constant
8,
are related
to the ridge index
n,
by the dispersion equation
(1)
kp,
+
P;
=
IC&?
with
27r
k”
=
-
XO
(U
+
1)7r
k,,
=
___
WW
where the “effective” width
We,
takes into account the
(polarization-dependent) lateral penetration depth of each
mode field, associated with the Goos-Hahnchen shifts at
the ridge boundaries. For high-contrast waveguides, the
penetration depth is very small
so
that
We,
N
WM.
In
general, the effective widths
W,,
can be approximated by the
effective width
We,
corresponding to the fundamental mode
[17],
(which shall be noted
We
for simplicity):
where
o
=
0
for TE and
o
=
1
for TM. By using the binomial
expansion with
k&
<<
kgn:,
the propagation constants
8,
can
be deduced from (1)-(3)
(5)
Therefore, the propagation constants in
a
step-index mul-
timode waveguide show a nearly quadratic
respect to the mode number
v.
By defining
L,
as the beat length of the
modes
7r
4n,
W,“
L
2-w-
-
,-
PO -PI
3x0
dependance with
two lowest-order
ItC
nr
I
ID
i
II
2
D
Fig.
1.
Two-dimensional representation of
a
step-index multimode wave-
guide; (effective) index lateral profile (left), and top view of ridge boundaries
and coordinate system (right).
v=O
1
2
3
4
5
6
7
8...
Fig.
2.
Example
of
amplitude-normalized lateral field profiles
I.’~
(
y).
cor-
responding to the first
9
guided modes in a step-Index multimode waveguide
the propagation constants spacing can be written
as
Y(Y
+
2)7r
3L,
(Po
-
Pu)
=
(7)
B.
Guided-Mode Propagation Analysis
An input field profile
Q(y,
0)
imposed at
z
=
0
and totally
contained within
We
(Fig.
3),
will be decomposed into the
modal field distributions
?I,,(y)
of
all
modes:
where the summation should be understood as including
guided as well as radiative modes. The field excitation co-
efficients
c,
can be estimated using overlap integrals
e,
=
/m
(9)
based on the field-orthogonality relations.
If the “spatial spectrum” of the input field
q(y,
0)
is narrow
enough not to excite unguided modes, (a condition satisfied
for all practical applications), it may be decomposed into the
1--
--
I
I
I1
SOLDANO
AND
PENNINGS:
OPTICAL
MULTI-MODE
INTERFERENCE
DEVICES
BASED
ON
SELF-IMAGING
617
the latter being a consequence of the structural symmetry with
respect to the plane
y
=
0.
IV.
GENERAL INTERFERENCE
This section investigates the interference mechanisms which
are independent of the modal excitation, that is, we pose no
I
restriction on the coefficients
c,
and explore the periodicity
A.
Single Images
Fig.
3.
Multimode waveguide showing the input field
*(y,o),
a
mirrored
single image at
(3L,),
a direct single image at
2(3L,).
and two-fold images
at
;(3L,)
and
%(3L,).
By inspecting (13), it can be seen that 6(y,
L)
will be an
image
of
6(y,
0)
if
guided
modes alone
m-1
@(Y,O)
=
CV+U(Y).
(10)
u=o
The field profile at a distance
z
can then be written as a
superposition of all the guided mode field distributions
m-1
Q(Y,
.)
=
CU+U(Y) exp[j(wt
-
PU.)].
(11)
u=o
Taking the phase of the fundamental mode as a common
factor out of the sum, dropping it and assuming the time de-
pendence exp(jwt) implicit hereafter, the field profile 6(y,
z)
becomes
m-1
The first condition means that the phase changes of all the
modes along
L
must differ by integer multiples of 27r. In this
case, all guided modes interfere with the same relative phases
as in
z
=
0;
the image is thus a
direct
replica of the input field.
The second condition means that the phase changes must be
alternatively even and odd multiples of
7r.
In this case, the
even modes will be in phase and the odd modes in antiphase.
Because of the odd symmetry stated in (1
6),
the interference
produces an image
mirrored
with respect to the plane y
=
0.
Taking into account (15), it is evident that the first and
second condition of (17) will be fulfilled at
L
=
p(3L,)
with
p
=
0,1,2,.
.
.
(18)
q(Y,
2)
=
c~?l~(Y)exP[j(Po
-
b).].
(12)
A useful expression for the field at a distance
2
=
L
is then
for
p
even and
p
odd, respectively. The factor
p
denotes
the periodic nature of the imaging along the multimode
waveguide. Direct and mirrored single images of the input
field 6(y,
0)
will therefore be formed by general interference
at distances
z
that are, respectively, even and odd multiples
of the length
(3L,),
as shown in Fig.
3.
It should be clear at
this point that the direct and mirrored single images can be
exploited in bar- and cross-couplers, respectively.
u=o
found by substituting
(7)
into (12)
m-1
(13)
u=o
The
Of
'(Y,
L),
and
the
Of
images
and
Next, we investigate multiple imaging phenomena, which
be
determined
by
the
excitation
provide the basis
for
a
broader range of
MMI
couplers.
the properties of the mode phase factor
(14)
exp
[j
v(v
+
3LiT
It will be seen that, under certain circumstances, the field
@(y,L)
will be a reproduction (self-imaging) of the input
field
6
(y
,
0).
We call
General Interference
to the self-imaging
mechanisms which
are
independent of the modal excitation;
and
Restricted Interference
to those which
are
obtained by
exciting certain modes alone.
The following properties will prove useful in later deriva-
tions:
B.
Multiple Images
In addition to the
single
images at distances given by (18),
multiple
images can be found as well. Let us first consider
the images obtained half-way between the direct and mirrored
image positions, i.e., at distances
P
2
(19)
L=
-(3L,)
with
p=
1,3,5,...
.
The total field at these lengths is found by substituting (19)
into (13)
even for
v
even
odd
for
U
odd
v(v
+
2)
=
{
and,
+,,(y) for
v
even
(16)
with
p
an odd integer. Taking into account the property of (15)
and the mode field symmetry conditions of (16),
(20)
can be
=
{
-?,hu(y) for
v
odd
618
122
prn
offset
Fig.
4.
Schematic layout of a
2
x
2
MMI
coupler based on the
general interference mechanism
[19].
The multimode waveguide length is
LMMI
cz
250
pm.
Offsets
are
used to minimize losses at the transitions
between waveguides of different curvature. Note the widely spaced access
branches, which decrease coupling between access waveguides and obviates
blunting due to poor photolithography resolution.
written as
U
even
uodd
The last equation represents a pair of images of
Q(y,
0),
in quadrature and with amplitudes 1/
a,
at distances
z
=
(3L,),
;(3L,),
.
. .
as shown in Fig.
3.
This two-fold imaging
can be used to realize 2
x
2 3-dB couplers.
Optical 2
x
2 MMI couplers based on the single and two-
fold imaging by general interference have been realized in
111-V semiconductor waveguides
[
181,
[
191, in silica-based
dielectric waveguides [20], and in non-lattice matched 111-V
quantum wells [21], 131.
Fig. 4 shows the schematic layout of the InGaAsP 2
x
2
multimode interference coupler reported in [18], [19]. The
8-
pm
wide multimode section supports 4 guided modes. Excess
losses of
0.4-0.7
dB, with extinction ratios of -28 dB at the
cross state (3L,
=
500
pm) and imbalances well below 0.1
dB for the 3-dB state
($3L,
=
250
pm)
were measured for
TE
and TM polarizations at
A0
=
1.52 pm. The imbalance of
an
N
x
M
coupler is defined as the maximum to minimum
output power ratio for all
M
outputs, expressed in dB. This
definition will be used throughout the paper.
In general, multi-fold images are formed at intermediate
z-
positions
[
121. Analytical expressions for the positions and
phases of the N-fold images have been obtained 1221 by
using Fourier analysis and properties of generalized Gaussian
sums. A very brief summary of the bases and results of that
derivation is given here. The starting point is to introduce a
field
Qin(y)
as the periodic extension of the input field q(y,
0);
antisymmetric with respect to the plane
y
=
0
(which, for
this analysis, is chosen to coincide with one guide’s lateral
JOURNAL
OF
LIGHTWAVE TECHNOLOGY,
VOL.
13,
NO.
4,
APRIL
1995
boundary), and with periodicity
2W,
03
Si,(Y)
[*(y-
W2We,0)
-
*(-y+v2We;0)]
1)=--03
(22)
and to approximate the mode field amplitudes by sine-like
functions
$Ju(Y)
2
sin(k,vy). (23)
Based on these considerations,
(10)
can be interpreted as
a (spatial) Fourier expansion, and it is shown 1221 that, at
distances
P
N
L
=
-(3L,)
where
p
2
0
and
N
2
1
are
integers with no common divisor,
the field will be of the form
with
where
C
is a complex normalization constant with
IC1
=
fi,p
indicates the imaging periodicity along
z,
and
q
refers
to each of the
N
images along
y.
The above equations show that, at distances
z
=
L,
N
images are formed of the
extended
field
qin(y),
located at the
positions
yn,
each with amplitude
1/m
and phase
pn.
This
leads to
N
images (generally not equally spaced) of the
input
field
Q(y,
0)
being formed inside the physical guide (within the
guide’s lateral boundaries), as shown in Fig. 5. The multiple
self-imaging mechanism allows for the realization of
N
x
N
or
N
x
M
optical couplers. Shortest devices are obtained for
p
=
1. In this case, the optical phases of the signals in a
N
x
N
MMI coupler are, (apart from a constant phase), given by
r
prs
=
-(s
-
l)(2N
+
T
-
s)
+
7r
for
T
+
s
even
(28)
4N
and
7l
cprS
=
-(r
+
s
-
l)(2N
-
T
-
s
+
1)
for
T
+
s
odd
4N
(29)
where
T
=
1,
2,
...
N
is the (bottom-up) numbering of
the input waveguides and
s
=
1,
2,...N
is the (top-down)
numbering of the output waveguides.
It is important to note that the phase relationships given
by (28) and (29)
are
inherent to the imaging properties of
multimode waveguides. It appears that the output phases of the
4
x
4 coupler satisfy the phase quadrature relationship, and
that this MMI device can be used as a 90O-hybrid which is a
key component in phase-diversity or image rejection receivers
and which can be used to avoid the quadrature problem in
interferometric sensors.
7-
r
I
SOLDANO
AND
PENNINGS:
OPTICAL
MULTI-MODE INTERFERENCE
DEVICES BASED
ON
SELF-IMAGING
(b)
Fig
5
Theoretical light intensity patterns corresponding to general
or
pared
interference mechanisms in two multimode waveguides, leading to
a
mrrored
single image
(a),
and
a
4-fold image
(b)
Note
also
the multi-fold images at
intermediate distances, non-equally spaced along the lateral axis Reproduced
by kind pernussion
of
J
M
Heaton, @British Crown Copynght DRA 1992
Several 4
x
4
MMI
optical hybrids have been demonstrated
in different technologies and sizes, such as 10-25 mm long
semi-bulk constructions of sandwiched glass sheets [23], and
ion-exchanged waveguides on glass substrates [24], [25].
Recently, ultra-compact (sub-millimeter length) 4
x
4
deeply etched waveguide couplers were fabricated by reactive-
ion etching in
111-V
semiconductor material [26], [27]. These
devices (shown in Fig. 6) attained excess losses below 1 dB,
imbalances from 0.3-0.9 dB and phase deviations of the order
of
5".
V.
RESTRICTED INTERFERENCE
Thus far, no restrictions have been placed on the modal
excitation. This section investigates the possibilities and real-
izations of
MMI
couplers in which only some of the guided
modes in the multimode waveguide are excited by the input
field(s). This selective excitation reveals interesting multiplic-
ities of
v(v+
2), which allow new interference mechanisms
through shorter periodicities of the mode phase factor of (14).
A.
Paired Integerence
By noting that
mods[v(v
+
2)]
=
0
for
v
#
2,5,8,.
. .
(30)
it is clear that the length periodicity of the mode phase factor
of (14) will be reduced three times if
Therefore, as shown in [28], [29], single (direct and inverted)
images of the input field
q(y,
0)
are now obtained at (cf. (18))
L
=
p(L,)
with
p
=
0,1,2,.
. .
(32)
provided that the modes
v
=
2,
5,
8,
. . .
are not excited in the
multimode waveguide. By the same token, two-fold images are
found at (p/2)L, with
p
odd. Based on numerical simulations,
619
400pm
,
Lm,,=3Lrd4
,
400pm
Fig. 6. Schematic layout
of
the 4
x
4 90O-hybrid and modal propagation
analysis within the multimode waveguide [27]. The length and the width
of
the multimode waveguide are
L,,,
N
945 pm and
W,,,
N
21.6 pm,
respectively.
N-fold images will be formed at distances (cf. (24))
P
N
L
=
-(Lr)
(33)
where
p
2
0
and
N
2
1
are
integers having no common
divisor.
One possible way of attaining the selective excitation of
(31) is by launching an even symmetric input field
@(y,
0)
(for
example, a Gaussian beam) at
y
=
kWe/6. At these positions,
the modes
v
=
2,
5,
8,
. . .
present a zero with odd symmetry,
as shown in Fig. 2. The overlap integrals of (9) between
the
(symmetric) input field and the (antisymmetric) mode fields
will vanish and therefore
c,
=
0
for
v
=
2,5,8,
.
. .
Obviously,
the number of input waveguides is in this case limited to two.
When the selective excitation of (31) is fulfilled, the modes
contributing to the imaging are paired, i.e. the mode pairs
0-
1,
3-4, 6-7,
.
. .
have similar relative properties. (For example,
each even mode leads its odd partner by a phase difference of
7r/2 at
z
=
L,/2-the 3-dB length-, by a phase difference
of
7r
at
z
=
L,-the cross-coupler length-, etc). This
mechanism will be therefore called
paired
interference. Two-
mode interference
(TMI)
can be regarded in this context as a
particular case of paired interference.
2
x
2
MMI
couplers based on the paired interference
mechanism have been demonstrated in silica-based dielectric
rib-type waveguides with multimode section lengths of 240
pm (cross state) and
150
pm (3-dB state) [30], [31]. Insertion
loss
lower than
0.4
dB, imbalance below 0.2 dB, extinction
ratio of
-
18 dB, and polarization-sensitivity loss penalty of
0.2 dB were reported for structures supporting 7-9 modes.
Calculations predict that power excitation coefficients as low
as -40 dB for the modes
v
=
2,5,8 can be achieved through a
correct positioning of the access waveguides, remaining below
-30
dB for a 0.1-pm misalignment [29].
Recently, extremely small paired-interference
MMI
devices
were reported [32]. The 3-dB (cross) couplers, realized in a
raised-strip InP-based waveguide,
are
107-pm (216-pm) long,
and show 0.9-dB (2-dB) excess loss and -28 dB crosstalk.
B.
Symmetric Inte$erence
Optical N-way splitters can in principle be realized on the
basis of the general N-fold imaging at lengths given by (24).
However, by exciting only the even symmetric modes, l-to-
N
beam splitters can be realized with multimode waveguides
four times shorter [33].