A general design algorithm for low
optical loss adiabatic connections in
waveguides
Tong Chen, Hansuek Lee, Jiang Li, and Kerry J. Vahala
∗
T. J. Watson Laboratory of Applied Physics, California Institute of Technology,
Pasadena, California, 91125, USA
*vahala@caltech.edu
Abstract:
Single-mode waveguide designs frequently support higher
order transverse modes, usually as a consequence of process limitations
such as lithography. In these systems, it is important to minimize coupling
to higher-order modes so that the system nonetheless behaves single mode.
We propose a variational approach to design adiabatic waveguide connec-
tions with minimal intermodal coupling. An application of this algorithm
in designing the “S-bend” of a whispering-gallery spiral waveguide is
demonstrated with approximately 0
.
05 dB insertion loss. Compared to other
approaches, our algorithm requires less fabrication resolution and is able to
minimize the transition loss over a broadband spectrum. The method can be
applied to a wide range of turns and connections and has the advantage of
handling connections with arbitrary boundary conditions.
© 2012 Optical Society of America
OCIS codes:
(230.7370) Waveguide; (230.7390) Waveguide, planar; (220.0220) Optical design
and fabrication.
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1. Introduction
Progress in optical communications has motivated much research in optical circuits composed
of basic elements such as modulators, switches and splitters. In designing any type of inte-
grated optical circuit, linking two given points or elements with minimal loss is a universal
problem [1]. It is becoming even more important in the light of the recent development of
ultra-low-loss optical waveguides [2–6]. With propagation loss less than 0
.
1 dB/m [5, 6], the
additional loss imposed by connections in these structures must be addressed carefully. The
current design strategy is normally based on piece-wise construction from a certain family of
curves, such as ellipses and sinusoidal curves [1,7,8]. However, it is not clear that these families
will always provide the best possible solution to the general problem of linking two points in
an arbitrary optical circuit. Herein, we propose a novel algorithm to the optical connection de-
sign. The design algorithm minimizes the transition loss by avoiding excitation of higher-order
optical modes. It is physically based and applicable to any waveguide connection (single mode
or multimode) between two points in a photonics circuit. The approach is flexible enough to
handle arbitrary boundary conditions.
An ideal design algorithm will minimize two types of loss: bending loss and transition loss.
The first mechanism is a result of radiative loss present in any bent waveguide. The second
mechanism, which is more critical in connection design, comes from the abrupt change or
discontinuity of curvature in the connection. It includes not only mismatch of overlap in the
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propagating mode at the joint points but also the intermode power exchange (cross talk) caused
by the variation of curvature along the connection waveguide path. This cross talk can cause
distortion via multimode interference. Because bending loss is most readily reduced by making
the bend radius larger or by increasing the mode confinement, it is the connection loss that is the
primary focus here. However, the proposed algorithm, by minimizing all intermode coupling,
actually minimizes both bending loss and transition loss [9, 10].
A generalized waveguide connection is illustrated in Fig. 1 with different curvatures at the
terminals of waveguides A and B. In prior work, several different methods have been applied
to reduce the insertion loss of this connection. In the offset approach, a small lateral offset in
waveguides of different curvature (but having the same tangent) is applied to improve mode
overlap [11]. However, this method always results in some mismatch of the modes and also
requires high resolution fabrication. In the matched bent approach, the connection is designed
to minimize the leaky-mode excitation at the end of the bend [12]. In particular, if the length
of the bent waveguide is a multiple of the beat length of the fundamental and a higher-order
mode, then only the fundamental mode is present at the output of the bend. This approach has
the side effect of introducing a wavelength dependence into the design. In another method, the
curvature is smoothly transformed between the endpoints using specialized curves [13]. One
such curve is a clothoid curve [14], which is widely used in highway design [15, 16], robot
path planning [17] and computer graphics [18]. However, this method is not general enough
to minimize loss in problems containing other constraints such as footprint minimization and
general boundary conditions. Our algorithm, in contrast, is based on loss minimization and con-
nection designs satisfying arbitrary boundary conditions. A variational approach is introduced
to achieve the optimal curve connecting two waveguide endpoints having specified curvature
and tangent vectors. Section 2 introduces our general design algorithm and section 3 describes
the experimental result to validate our algorithm.
Fig. 1: An illustration of the generic connection design problem. Waveguides A and B are
shown linked by the connection waveguide (dashed). To create a low insertion loss, coupling
to higher-order modes must be reduced at the connecting points as well as through out the
transition. The inset shows the curvature (
κ
) versus path length,
z
.
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2. Design Algorithm
2.1. Overview
In a straight waveguide, the fundamental mode is centered in the waveguide, while it is slightly
shifted from center in a curved waveguide [19]. This mismatch results in loss when the light
transitions between two segments with discontinuous curvature. To this end, we require the
junction of the connection region to the waveguides A and B to feature a continuous curvature.
Likewise, the curvature is required to be continuous along the connection waveguide. Even with
the continuous curvature, the evolution of curvature itself along the connection waveguide can
introduce inter-mode power transfer [20, 21]. Accordingly, the slow or adiabatic evolution of
curvature along the connector is desirable.
Conformal mapping provides a way to both simplify the discussion and provide a more intu-
itive understanding of the light-wave evolution [22]. If
n
(
x
,
y
)
denotes the transverse refractive
index of the waveguide,
κ
(
z
)
is the curvature,
(
x
,
y
)
are the transverse coordinates (
x
=
R
1
and
x
=
R
2
at the inner and outer boundary) and z is the coordinate along the direction of propaga-
tion, then using the conformal transformation (see Fig. 2),
−
Conformal
Mapping
−
Fig. 2: Conformal mapping between a bent waveguide and a straight waveguide. The
transverse refractive index,
n
(
x
,
y
)
, of the curved waveguide is mapped to
n
2
eq
(
u
,
v
,
z
)=
n
2
(
u
,
v
)
e
2
u
κ
(
z
)
for a straight guide.
(
x
,
y
)
→
(
u
,
v
)
:
u
=
R
2
log
(
x
/
R
2
)
,
v
=
y
(1)
one can show that a bent waveguide with curvature
κ
(
z
)
behaves like a straight waveguide with
a refractive index profile [22] given by,
n
2
eq
(
u
,
v
,
z
)=
n
2
(
u
,
v
)
e
2
u
κ
(
z
)
(2)
With this mapping, the original problem can be transformed to the equivalent problem of de-
signing the refractive index profile in a non-uniform straight waveguide.
The transition loss can be calculated from coupled-mode theory. We consider a medium with
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general dielectric constant
ε
(
r
)
and constant permeability
μ
. The fields are expressed by the
superposition of local normal modes,
{
e
p
(
x
,
y
)
,
h
p
(
x
,
y
)
}
, having propagation constants
{
β
p
}
E
=
∑
p
A
(
β
p
,
z
)
e
p
(
x
,
y
)
(3)
H
=
∑
p
A
(
β
p
,
z
)
h
p
(
x
,
y
)
(4)
with orthonormal condition
S
ˆ
z
·
(
e
p
×
h
∗
q
)
dS
=
δ
pq
(5)
where ˆ
z
is the unit vector along the axis of propagation and “
S
” represents the local surface
area that is normal to the axis of propagation. For slow variation in
ε
, the coupling coefficient
C
(
β
p
,
β
q
)
between two modes
p
and
q
is given by (
q
=
p
)
C
(
β
p
,
β
q
)=
1
4
S
ˆ
z
·
e
q
×
∂
h
∗
p
∂
z
−
e
∗
p
×
∂
h
q
∂
z
dS
=
ω
4
(
β
p
−
β
q
)
S
e
q
·
e
∗
p
∂ε
∂
z
dS
(6)
where
ω
is the optical frequency in radians/sec. We see that the coupling is directly proportional
to
∂ε
∂
z
and only exists between the modes with the same polarizations. Now consider a region
from
z
0
to
z
1
and suppose that at
z
=
z
0
the only non-zero modal amplitude is
A
(
β
p
,
z
0
)
.To
leading order, the propagation solution is given by:
A
(
β
q
,
z
1
)
∼
A
(
β
p
,
z
0
)
z
1
z
0
C
(
β
p
,
β
q
)
exp
i
(
β
p
−
β
q
)
z
dz
×
exp
−
i
z
1
z
0
β
q
(
z
)
dz
(7)
Thus, power transfer from mode
p
to
q
is found to be proportional to
(
∂ε
∂
z
)
2
/
(
β
p
−
β
q
)
2
. Namely,
|
A
(
β
q
,
z
1
)
A
(
β
p
,
z
0
)
|
2
∝
z
1
z
0
1
(
β
p
−
β
q
)
2
S
(
1
ε
∂ε
∂
z
)
dS
2
dz
(8)
Based on the estimation in Eq. (8), it is possible to design a curve that will mitigate the power
transfer during the variation of curvature. In particular, with the equivalent index profile of
curved waveguide
ε
eq
(
x
,
y
,
z
)=
ε
(
x
,
y
)
e
2
u
κ
(
z
)
and the width of waveguide much smaller than its
curvature radius (
i.e. w
=
R
2
−
R
1
R
1
), we have
1
ε
∂ε
∂
z
≈
2
(
R
2
−
x
)
∂κ
(
z
)
∂
z
(9)
Herein, it is apparent that a narrower waveguide is desirable to minimize the power transfer.
Assuming the width of the waveguide to be unchanged, a tempting objective functional to be
minimized is given by:
E
κ
(
s
)
=
z
1
z
0
∂κ
(
s
)
∂
s
2
ds
(10)
where, for mathematical simplicity, we have replaced the
z
with the arc-length parameter
s
.
This equation is the working equation used here to minimize the power transfer in a connection
waveguide having evolving curvature along the connection path.
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2.2. S-Bend Design
In this subsection, we apply our algorithm to design a low insertion loss S-bend connection
waveguide [5]. S-bend connections waveguides occur in spiral waveguides wherein two inter-
laced Archimedean spirals (one clockwise and one counter clockwise) must be joined near the
center of either spiral. An application of recent interest has been the creation of ultra-low opti-
cal loss waveguides for true time delay applications [23–27]. Significant efforts have been put
into realization of these structures [2–4]. Recently, we have demonstrated a silica-on-silicon
waveguide having optical attenuation of 0
.
08 dB/m over path lengths as long as 27 meters [5].
These delay-line designs feature two embedded Archimedean-shaped spirals, coupled using an
S-bend waveguide [5]. In previous research [3, 4], the S-bend waveguide paths are normally
constructed from sinusoidal curves, ellipses or other families of curves that are not necessarily
the optimal choice. In contrast, using the above described algorithm, we can design an optimal
S-bend so as to minimize excitation of higher-order modes.
We note that the functional
E
is the
L
2
norm of the variation of curvature along the arc.
This problem is therefore similar to the “minimization of variation of curvature (MVC)” prob-
lem in computer graphics and free-way design [15]. The variation of
E
leads to the following
corresponding Euler-Lagrange equation,
κ
(
s
)=
0
(11)
The solution family is
κ
=
k
0
+
k
1
s
, which are similarity transformations of the basic Euler spi-
ral (
i.e.
clothoid)
κ
=
s
[14], where the curvature varies linearly along the length of the curve.
Indeed, the Euler spiral has been employed as the transition curve in connecting straight and
bent waveguides [13]. This solution is, however, unsatisfying for a variety of reasons. First,
the variational equation is expressed in terms of
κ
rather than curve
(
x
,
y
)
; also, the bound-
ary conditions are expressed in terms of curvature rather than
z
. Similarly, positional endpoint
constraints are missing. The most general endpoint constraints are the specification of curve
length, end point positions, and end point tangents. Of these, the end point position constraints
require Lagrange multipliers. The end point constraints
(
x
0
,
y
0
)
and
(
x
1
,
y
1
)
for connection to
waveguides A and B are expressed as the integral of the unit tangent vector of direction
θ
(
s
)
:
x
0
+
s
0
cos
(
θ
(
s
))
ds
=
x
1
x
0
+
s
0
sin
(
θ
(
s
))
ds
=
y
1
(12)
Therefore, by adding the Lagrange multipliers (
λ
1
and
λ
2
) and eliminating
κ
in favor of
θ
, the
following the functional must be minimized over the length
l
[28]
E
=
l
0
(
θ
)
2
+
λ
1
sin
θ
+
λ
2
cos
θ
ds
(13)
The corresponding Euler-Poisson equation is then given by:
2
θ
+
λ
1
cos
θ
−
λ
2
sin
θ
=
0
.
(14)
Without loss of generality we may assume
λ
1
=
0 and observe that
dy
ds
=
sin
θ
.
θ
−
λ
2
y
=
0
.
(15)
Upon integration, the resulting equation is
κ
=
λ
2
y
(16)
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where
κ
(
s
)=
θ
(
s
)
and the constant of integration has been set at zero by a translation of the
curve. As an aside, it is interesting to note that for
λ
2
=
0, this equation is equivalent to that
of the Euler spiral. To approximately solve this ODE (Eq. (16)), we may consider a family of
curves with curvature given in terms of a cubic polynomial of arc length “
s
” [28, 29].
κ
(
s
)=
a
0
+
a
1
s
+
a
2
s
2
+
a
3
s
3
.
(17)
This curve family provides a very good approximation to the original variational problem and
provides an analytical expression of the connection path [28]. To see this, assuming small turn-
ing angles,
y
is almost proportional to
s
plus a constant offset, so substituting into Eq. (16),
κ
≈
c
1
s
+
c
2
, and then integrating twice yields Eq. (17).
The coefficients of the polynomial (
a
i
) are determined by matching the endpoint positions,
−600
−400
−200
0
200
400
600
−300
−200
−100
0
100
200
300
X (
μ
m)
Y (
μ
m)
Physical Space
0
500
1000
1500
2000
−0.5
0
0.5
Arc length (
μ
m)
π
Tangent Vector
0
500
1000
1500
2000
0
0.002
0.004
0.006
0.008
0.01
Arc length (
μ
m)
Curvature (
μ
m
−
1
)
Curvature
0
500
1000
1500
2000
−5
0
5
10
x 10
−5
Derivative of Curvature
Arc length (
μ
m)
Derivative of curvature (
μ
−2
m )
Fig. 3: The optimal S-bend design that minimizes mode coupling between clockwise and
counter-clockwise Archimedean spiral waveguides. A jump from 0
.
5
π
to
−
0
.
5
π
of the tan-
gent vector in the upper right panel is due to the convention that the tangent vector is defined
as
[
−
0
.
5
π
,
0
.
5
π
]
(for example, a tangent vector of 0
.
6
π
is considered as 0
.
6
π
−
π
=
−
0
.
4
π
).
The geometry property is still continuous.
endpoint tangents and the curvature between the S-bend and Archimedean spiral. By symme-
try, we only need to design the incoming arc of the S-bend. This curve starts at the origin
(
x
=
0
,
y
=
0
)
and has curvature
κ
=
0at
x
=
0
,
y
=
0, which leads to the following formulation
⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
θ
1
=
θ
0
+
s
1
0
κ
(
s
)
ds
=
θ
0
+
s
1
0
a
1
s
+
a
2
s
2
+
a
3
s
3
ds
κ
1
=
a
1
s
+
a
2
s
2
+
a
3
s
3
|
s
=
s
1
κ
1
=
d
κ
(
s
)
ds
s
=
s
1
=
d
ds
a
0
+
a
1
s
+
a
2
s
2
+
a
3
s
3
s
=
s
1
(
x
1
,
iy
1
)=
s
1
0
exp
(
i
θ
(
s
))
ds
(18)
The additional constraint of curvature at the starting point has been enforced and gives
a
0
=
0. By solving a set of unknowns (
a
1
,
a
2
,
a
3
,
θ
0
,
s
1
) to match the input parameters
#171778 - $15.00 USD
Received 3 Jul 2012; revised 6 Sep 2012; accepted 7 Sep 2012; published 20 Sep 2012
(C)
2012
OSA
24
September
2012
/ Vol.
20,
No.
20
/ OPTICS
EXPRESS
22825
(
θ
1
,
κ
1
,
κ
1
,
(
x
1
,
iy
1
))
from the end point of the Archimedean spiral, a curve for the adiabatic
coupler is successfully defined as shown is Fig. 3.
3. Experimental Verification
Measurement of connector loss in waveguides is complicated by the insertion loss associated
with coupling light into the waveguide. To avoid this problem entirely we use optical back-
scatter reflectometry [30]. The spiral waveguides and S connector waveguide are fabricated via
procedures outlined in Ref. [5]. This process begins with lithography and etching with buffered
hydrofluoric acid of silica on silicon. The oxide layer then functions as an etch mask for an
isotropic dry etch of the silicon using XeF
2
. Further details on the processing are given in
Ref. [5]. Figure. 4(a) shows a 7 m (physical length) spiral waveguide that is approximately
4
.
5 cm in diameter. The angle-cleaved input facet (7 degree cleave angle) is at the upper left
corner of the chip. Optical fiber and index-matching oil are used for end-fire coupling. The
waveguides have a width of 170 microns. A magnification of the S-shaped connection is shown
in Fig. 4(b). The connection width was tapered to 10 microns at its narrowest point to relax the
condition in Eq. (9). The spiral waveguides were characterized using a Luna OBR 4400 back-
scatter reflectometer [30]. A measured backscatter trace is given in Fig. 4(c). As can be seen
in the trace, the adiabatic connector creates a singularity in the backscattering signal. Higher-
resolution measurement of this singularity reveals that this region rises and plateaus over a
length of several mm at the center of the S-bend connection (see inset to Fig. 4(c)). The appar-
ent lack of any decrease in backscatter signal within this region required that another method
be applied to analyze the coupler loss.
As an alternate measurement, we plotted the ratio of out-going to in-going backscatter
strength at equidistant points from the spiral center. By symmetry, these pairs of points will
have the same curvature. The ratio of backscatter strength can be written as
log
(
P
backscatter
(
z
2
)
P
backscatter
(
−
z
2
)
)=
−
α
1
·
z
−
α
0
(19)
where the center of the spiral is the origin,
P
backscatter
(
z
2
)
and
P
backscatter
(
−
z
2
)
are out-going and
in-coming waveguide backscattering strength at the equidistant points from the spiral center;
the waveguide length between these two points is
z
;
α
1
is the waveguide loss per unit length
and
α
0
is the insertion loss of the S-bend connection. Such a plot is shown in Fig. 4(d). The
slope of the linear fit gives approximately 0
.
35 dB/m loss for the waveguide, while the intercept
gives an estimated insertion loss for the adiabatic coupler of 0
.
05 dB. This insertion loss and
the indicated confidence interval result from linear regression on all of the points. Data points
within 0
.
25 meters of the S-bend have been omitted in this estimate as there is a large increase
in the variance on account of the steep slope associated with the backscatter singularity (see
Fig. 4(c)). As an aside, the spiral device of this measurement was fabricated using a contact
aligner and therefore features a higher waveguide loss as compared to that reported in Ref. [5],
wherein devices were fabricated using a Canon stepper lithography tool.
To evaluate the spectral performance of the connection, backscattering measurements were
performed over 1536
−
1598 nm with an instrument measurement window set to 10 nm. Figure.
5 shows the spectral dependence of the intercept (see Fig. 4(d)) measured in a 1 meter spiral.
There is, overall, a weak variation across the spectrum, however, the variation is larger than that
inferred from the confidence interval in Fig. 4(d). We attribute this to the reduced length of this
spiral and hence the smaller number of points used in the regression. Reducing the measure-
ment window to 10 nm to study the spectral variation is also believed to have contributed to a
larger variation.
#171778 - $15.00 USD
Received 3 Jul 2012; revised 6 Sep 2012; accepted 7 Sep 2012; published 20 Sep 2012
(C)
2012
OSA
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September
2012
/ Vol.
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/ OPTICS
EXPRESS
22826
0
2
4
6
8
−140
−130
−120
−110
−100
−90
−80
−70
Length(m)
Amplitude(dB/mm)
Input facet
Output facet
Connection
4.6715
4.672
4.6725
4.673
4.6735
4.674
−100
−80
−60
−40
Length (m)
Amplitude (dB/mm)
0
1
2
3
4
5
6
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
Length (m)
Loss (dB)
Original OBR Data
Linear Fitting
95% Confidence Interval
~ 0.05 dB Loss
0.5mm
1cm
(a)
(b)
(c)
(d)
Fig. 4:
(a)
Optical micrograph of a spiral waveguide having a physical path length of 7 me-
ters. The input port is in the upper left of the image, and there are two small spirals at the
input and output ports (not resolved in the backscatter trace of panel (c)). The entire chip is
4
.
5cm
×
4
.
5 cm.
(b)
A magnified view of the adiabatic coupling section (approximately 1 mm
in diameter). Light brown regions are silicon (under oxide or exposed) while darker brown
regions along the border of the light brown are silica that has been undercut by dry etching.
Very-dark-brown border regions are also silica but having a wedge profile. For further details
see Ref. [5].
(c)
Optical backscatter reflectometer measurement of the spiral waveguide. Be-
sides occasional random noise spikes that we believe are associated with small dust particles
on the surface of the waveguide, the major singularities in the backscatter signal correspond
to the input facet, the optical wave transiting the inner adiabatic coupling region of the spiral
and the output facet. The inset shows a close-in view of the adiabatic coupler region (
i.e.
, peak
of the singularity). There is no apparent drop in signal within this region.
(d)
Analysis of the
adiabatic coupler insertion loss using backscatter data. Data points are generated by taking
the ratio of backscatter signals at symmetrically offset distances away from the adiabatic cou-
pler in (a). The intercept reveals the insertion loss of the S connection as given by a range of
possible values falling within a confidence interval determined by linear regression.
#171778 - $15.00 USD
Received 3 Jul 2012; revised 6 Sep 2012; accepted 7 Sep 2012; published 20 Sep 2012
(C)
2012
OSA
24
September
2012
/ Vol.
20,
No.
20
/ OPTICS
EXPRESS
22827