Air trenches for sharp silica waveguide bends

Abstract

Air trench structures for reduced-size bends in low-index contrast waveguides are proposed. To minimize junction loss, the structures are designed to provide adiabatic mode shaping between low- and high-index contrast regions, which is achieved by the introduction of "cladding tapers." Drastic reduction in effective bend radius is predicted. We present two-dimensional (2-D) finite-difference time-domain/effective index method simulations of bends in representative silica index contrasts. We also argue that substrate loss, while present, can be controlled with such air trenches and reduced to arbitrarily low levels limited only by fabrication capabilities. The required trench depth, given an acceptable substrate loss, is calculated in three dimensions using an approximate equivalent current sheet method and also by a numerical solver for full-vector leaky modes. A simple, compact waveguide T-splitter using air trench bends is presented.

Full text

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1762JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 20, NO. 9, SEPTEMBER 2002
Air Trenches for Sharp Silica Waveguide Bends
Miloˇ s Popovic ´, Student Member, IEEE, Kazumi Wada, Shoji Akiyama,
Hermann A. Haus, Life Fellow, IEEE, Fellow, OSA, and Jürgen Michel
Abstract—Air trench structures for reduced-size bends in low-
indexcontrastwaveguidesareproposed.Tominimizejunctionloss,
the structures are designed to provide adiabatic mode shaping be-
tween low- and high-index contrast regions, which is achieved by
the introduction of “cladding tapers.” Drastic reduction in effec-
tive bend radius is predicted. We present two-dimensional (2-D)
finite-difference time-domain/effective index method simulations
of bends in representative silica index contrasts. We also argue
that substrate loss, while present, can be controlled with such air
trenchesandreducedtoarbitrarilylowlevelslimitedonlybyfabri-
cation capabilities. The required trench depth, given an acceptable
substrate loss, is calculated in three dimensions using an approx-
imate equivalent current sheet method and also by a numerical
solver for full-vector leaky modes. A simple, compact waveguide
T-splitter using air trench bends is presented.
Index Terms—Air trench bend (ATB), cladding taper, enhanced
lateral mode confinement, low-index contrast, sharp bend loss,
silica waveguide.
I. INTRODUCTION
L
(SiOB)—has gained widespread use in practice in the fabrica-
tion of passive integrated optical components by virtue of its
useofwell-testedintegrated-circuitindustrymanufacturingpro-
cessesandtechnology[1].Largesilicawaveguidecrosssections
offerlowfiber-to-chipcouplingandpropagationlosses.Amajor
drawback of SiOB technology is the relatively large component
size, where a critical factor is the minimum waveguide bend ra-
dius. This radius is large—normally in the millimeters—in the
low-index contrasts
0.25%−1.5% found in silica [1].
The low density of integration keeps production cost high and
invites yield problems. On the other hand, high-index contrast,
such as silicon-on-insulator (SOI)—while offering dense inte-
gration, poses challenges of fiber-to-chip insertion loss due to
mode shape mismatch and misalignment, scattering loss, and
sensitivity to other fabrication defects and tolerances, as well as
fabrication processing challenges.
OW-INDEXcontrastsilicabenchtechnology—referredto
as planar lightwave circuit (PLC) or silicon optical bench
Manuscript received November 5, 2001; revised June 12, 2002. This work
was supported in part by the Materials Research Science & Engineering Cen-
ters (MRSEC) Program of the National Science Foundation under Award DMR
98-08941, by a Natural Sciences and Engineering Research Council (NSERC)
of Canada PGS-A scholarship, and by a National Partnership for Advanced
Computing Infrastructure (NPACI) supercomputer allocation.
M. Popovic ´ and H. A. Haus are with the Research Laboratory of Electronics,
Massachusetts Institute of Technology, Cambridge, MA 02139 USA (e-mail:
milos@mit.edu).
K. Wada, S. Akiyama, and J. Michel are with the Department of Materials
Science and Engineering, Massachusetts Institute of Technology, Cambridge,
MA 02139 USA.
Digital Object Identifier 10.1109/JLT.2002.802230
Atechnologythatallowsadrasticreductioninthebendingra-
diuswouldovercomeoneofsilica’smajorobstaclestoattaining
truly large-scale optical integration.We propose a schemeusing
air trenches to provide locally enhanced lateral mode confine-
ment. We use adiabatic tapering to avoid abrupt junction-in-
duced mode mismatch and Fresnel reflection in order to minia-
turize optical waveguide bends while preserving low-loss per-
formance [13].
Inthelast30years,muchattentionhasbeendevotedtothede-
sign of waveguide bends and the proper analysis of their losses
[2]–[5], [11],including the useof air trenches. More recent con-
tributions include “deep etching” in InP-ridge waveguide bends
[6], and a proposal of more unconventional, low-
in high-index contrast, such as SOI [7].
Air trenches have been proposed for suppressing bend radia-
tioninseveralways[6],[10].Whentheyareusedtoenhancelat-
eral mode confinement, mode mismatch-induced junction loss
is incurred at points of abrupt change in refractive index and/or
cross-sectional waveguide geometry, limiting the success of the
approach [6], particularly in low-index contrast. To our knowl-
edge, no attempt has been made to use air trenches to improve
bending loss in low-index contrast by properly addressing the
mode-mismatch issue introduced by the present air trench, to
produce small, low-loss bends. We use judiciously placed air
trenches for sharp bending with adiabatic transition to mitigate
junction loss. It is generally recognized that adiabatic mode
shaping results in low-loss tapers and directional couplers (e.g.,
see [8]).
In this paper, we introduce a pair of “cladding tapers” as an
integral part of the etched air trench at the bend (Fig. 1) in order
to provide fast mode transition to and from the high-index con-
trast trench region with low radiation loss and low reflection.
The main high-index contrast region in this case contains the
waveguide bend. The result is a reduction in bending radius by
a factor of 10–1000 and in total bend structure edge length by
a factor of 4–60. The theoretical justification for the proposed
idea is presented primarily in terms of two-dimensional (2-D)
finite-difference time-domain (FDTD) simulations of air trench
bends (ATBs) with index contrast between
7%.
In SectionII, we describe the ways inwhich air trencheshave
been used in prior work and justify our approach to the problem
in low-index contrast. In Section III, we first show the perfor-
mance and physical size of a regular waveguide bend, without
air trenches, for a set of chosen index contrasts. We then de-
scribe the method used to produce the ATB structure designs,
and, in Section IV, we show 2-D simulation results, including
dimensions and performance. In Section V, we discuss mode
confinementissuesthatareimportantinproperlyaccountingfor
cavity bends
0.25% and
0733-8724/02$17.00 © 2002 IEEE

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POPOVIC´et al.: AIR TRENCHES FOR SHARP SILICA WAVEGUIDE BENDS1763
Fig. 1.
plan view. (c) Cross-sectional views in the low-index. (d) Air trench regions.
Contour plots representative of the dominant modal electric-field component
are superimposed on the cross sections (not to scale).
ATB structure schematic: (a) Labeled plan view. (b) Dimensioned
Fig. 2.
polarization-dependent components and polarization splitters and rotators.
Optical circuit layout for polarization-independent operation using
the third dimension in the simulations, and we address substrate
leakage and polarization dependence. We calculate the min-
imum air trench depth for tolerable substrate loss. A full-vector
finite-difference numerical mode solver for a 2-D cross-section
leakywaveguideisusedtoverifytheapproximatecomputations
of substrate losses of the proposed waveguides. Finally, in Sec-
tion VI, a waveguide T-splitter using ATBs is shown as an ex-
ample of a simple compound device utilizing the bends.
We state here that the polarization sensitivity of these struc-
tures was not a design consideration because it is recognized
as being inevitably poor, as is generally found in high-index
contrast and especially in a structure with high cross-sectional
asymmetry along the principal polarization directions, such
as with the structures presented here. We used the vertical
(out-of-the-wafer plane, Fig. 1) electric-field polarization for
our designs. We propose that the orders-of-magnitude chip area
savings offered by this approach warrants the making of two
identical optical circuits, one for each polarization. The signal
polarizations may then be split and one rotated such that both
are processed by identical circuits before being recombined in
a similar way at the output, as shown in Fig. 2.
Fig. 3.
junction, (b) an abrupt junction followed by a taper, and (c) the proposed
adiabatic cladding taper. The bend radius is junction loss limited in (a), but
curvature loss limited in (b) and (c). Junction loss is present in (a) and (b),
but is virtually eliminated in (c). The dash-dot line in (c) shows a local Goos–
Hänschen shift and a ray trace of coupling to the bend mode (not to scale).
Incorporating an air trench into a waveguide bend using (a) an abrupt
II. AIR TRENCHES FOR ENHANCED LATERAL CONFINEMENT
The use of air trenches for sharp bending radii, in addition to
being an obvious and integral part of air-clad high-index con-
trast waveguides (e.g., SOI), has been discussed previously in
the context of optical fiber, slab [10], and ridge waveguide bend
loss [6]. In the former case, the trench is placed away from the
core with the intention of extending the reach of the evanescent
part of the leaky-mode (LM) field. The “radiation caustic” is
pushedradiallyoutward,withoutsignificantlyalteringtheshape
of the LM [10]. In the latter case, trenches were “deep etched”
immediatelynexttothecoreofInP/InGaAsP-ridgewaveguides,
providing higher lateral mode confinement at the expense of
junction loss [6].
Intypicalsilica indexcontrasts,theabruptjunctionlossatthe
interface between the standard waveguide and the trench region
[as in Fig. 3(a)] becomes intolerable for any bend radii small
enough to make the air trench useful in reducing size. This de-
mands our use of an adiabatic taper. For clarity, we will briefly
illustrate the argument for the slab (2-D) case. In a straight-to-
straight waveguide junction with a silica input waveguide con-
strained to be single mode, by varying the trench-clad output
waveguide’s core width, mode distributions can be optimally
matched. A surprisingly low loss can be obtained,
even for fiberlike index contrasts in the input waveguide
0.25% . This optimum loss occurs for a wide, greatly over-
moded trench waveguide, with the fundamental mode width es-
sentially determined by the core width [Fig. 3(a)]. When the
trench waveguide is curved, however, for small radii, the mode
width is not determined by the waveguide width, but rather,
by the bend radius (whispering-gallery regime, e.g., see [12]).
The field distribution width of the LM becomes narrower for
smaller bends, setting a lower limit on the bend radius compat-
ible with the above minimized junction loss, e.g., 15 mm for
0.1% (based only on matching the modal width, even
without accounting for qualitative mode shape differences). In
the low-index contrast input waveguide, the field distribution
of the fundamental mode has a minimum width determined by
the index contrast and, thus, a priori cannot be matched well
to the first LM of arbitrarily tight bend radii. To produce small
bend radii, it is necessary to provide a low-loss mechanism to
0.1dB,

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1764JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 20, NO. 9, SEPTEMBER 2002
compress the input mode. We propose the use of an adiabatic
“cladding taper” [Figs. 1 and 3(c)]. Constrained only by the
overall loss of the structure, these tapers allow reduction of the
bendradiusinthetrenchregiontothepointwhereitislimitedby
bending loss, rather than the modal distribution width required
for acceptable junction loss.
A progression from an abrupt junction to a cladding taper
is illustrated in Fig. 3. As argued above, an abrupt junction
[Fig. 3(a)] requires a large bending radius. A taper after the
abrupt junction can be used to compress the mode so that a
tighter bend radius can be used [Fig. 3(b)]. However, a more
natural way to build the transition is by introducing the taper as
a slow perturbation of the waveguide’s index profile, as shown
in Fig. 3(c). This eliminates the reduced but present Fresnel re-
flection at the junction of Fig. 3(b) and requires a less abrupt
modification of the propagating mode while providing equally
fasttapering.Theproposedairtrenchisintroducedgradually(in
the form of the aforementioned cladding taper) and away from
the core, where the field is small, in order to allow an adiabatic
evolution of the mode to a shape better matched to the leaky
bend mode used in the high-index bend region.
In the presented examples, we use a square cross-section,
low-index contrast buried waveguide with a good match to the
fiber mode for inherently low-loss fiber-chip coupling. We em-
ploy air trenches surrounding (and etched through part of) the
core region to provide high lateral confinement in the plane,
while leaving the mode weakly confined in the vertical direc-
tion [see mode contour plots in Fig. 1(c) and (d)]. By having
air in place of much of the cladding material near the core, the
fundamental mode has a lower modal index than the cladding
index. At first glance, this might suggest inevitable leakage to
the cladding in the trench region, but this is in fact not a nec-
essary consequence. In order to curb leakage to the substrate
and also prevent the evanescent tails of the field in the vertical
direction from radiating sideways in a bend, the air trench is
etched well below the core [Fig. 1(d)]. As discussed in Sec-
tion V, etching below the core allows reduction of substrate loss
to arbitrarily low levels, as limited only by concerns with fabri-
cation of a high-aspect ratio structure.
Weanalyzethebend inthetrenchregionusing somestandard
semianalytic tools [2]–[5], [11]. The trench tapers are designed
using some initial ray optics intuition and iteratively optimized
throughsimulations(aloneoraspartofthecompletebendstruc-
ture) using FDTD.
III. AIR TRENCH BEND: BEND AND TAPER DESIGN METHOD
Optimizationforminimumsizegivenarequiredtransmission
efficiency of a device, such as the present ATB—consisting of a
bend and two tapers—is a complex problem where variation in
the parameters of one subcomponent in general influences all
others. For a simpler approach, we choose a bend using some
simple criteria and then optimize the taper parameters with re-
spect to the chosen bend design without recursion. In our higher
indexexamples,wherethebendaccountsforasignificantpartof
the structure size, we choose the minimum bend radius meeting
curvature-loss requirements. In lower index examples, where
the bend radius contributes little to overall structure size, we
oversize the bend radius in order to lower the taper-bend junc-
Fig. 4.
ATBs, showing improvement in size for the case of 0.1-dB loss. It is also shown
that,onthisscaleofsizevariation,therelativechangeinradiusissmallfor0.01-
and 0.001-dB loss.
Bend (radius and “total box”) size for regular waveguide bends and
tion loss [Fig. 1(a)]. For loss analysis and design, we separate
the structure into its two constituent tapers and the high-index
bend between them.
A. Regular and ATB Loss
In a conventional curved waveguide, bending loss is mini-
mized by coupling to the lowest order LM [2]–[5], [11]. The
total loss in a corner bend comes from the bending loss of the
chosen LM and from junction loss at the interfaces between the
straight and bent waveguides. In the ATB, additional loss is in-
curred by propagation through each cladding taper and at the
junction where each taper meets the low-index contrast wave-
guide (Fig. 1). The latter is primarily related to the change of
phase front curvature.
Furthermore, in regular waveguide bends, the radius and
width of the bend and its axial offset with respect to the straight
waveguides are free parameters, but where single-mode waveg-
uides are used, the width of the straight waveguide is generally
fixed. (Tapering out to a larger width at the junction is also
possible, but not desirable, since mode-matching can instead
be accomplished by reducing the bend radius. Reduction of the
bend radius is limited at some point by bend loss.) In the ATB
case, the waveguide width at the taper side of the taper-bend
junction [
, Fig. 1(b)] is also a free parameter and need not
be constrained by single-modedness in the local normal-mode
sense. This is because the input single-mode waveguide is
much wider than that in the high-index contrast bend region
[
, Fig. 1(b)]; a wider taper output
allows for a less steep (and thus, more efficient) taper.
First, we consider bending loss, namely, in regular wave-
guide bends without an air trench, or in case of the ATB, we
consider the curvature loss in the bend-containing trench re-
gion. Radiation loss in regular bends and ATBs is evaluated nu-
merically according to the approaches in [4] and [5] after the
customary conformal transformation for bent waveguides [3].
Bendsare designed for an overall98% transmission or
loss per right-angle bend, and the bend parameters are shown
in Fig. 4 and Tables I and II for three chosen index contrasts,
0.1-dB

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POPOVIC´et al.: AIR TRENCHES FOR SHARP SILICA WAVEGUIDE BENDS1765
TABLE I
REGULAR AND ATB RADII AND TOTAL
SIZES FOR 98% TRANSMISSION
TABLE II
ATB LOSS BUDGET
including (for ATBs) separately calculated proportions of loss
attributed to junction mode mismatch and bending radiation.
The regular bend is constrained to have single-mode input and
output waveguides, while the ATB bend region is only con-
strained to have them smaller than the low-index waveguide
—which effectively presents no constraint. The
bending loss is evaluated in the usual way by solving for the
complex propagation constant of the lowest order LM using ei-
ther the linearized, conformally transformed dielectric constant
profile, where the modal field is expressed in terms of Airy
functions [4], [5], or the Wentzel–Kramers–Brillouin (WKB)
method for very low loss bends [12]. Where both can be used,
the two approaches yield comparable numbers.
The second contributor to loss in a finite-angle bend is junc-
tion loss at the start and end points. Junction loss between two
straight slab waveguides is determined by matching guided and
radiation modes at the junction, using a scattering matrix ap-
proach (e.g., see [12]). For a straight-to-bend junction, and ne-
glecting scattering into reflected (backward) radiation modes,
we obtain a simple expression that includes the overlap inte-
gral of the leaky bend mode field computed previously with the
mode of the straight waveguide. The appropriate expression for
power coupling efficiency at a straight-to-bent slab junction is
given by
(1)
resulting in a junction loss of
are propagation constants,
plitudes of the two waveguides at the junction,
radius,and istheradialcoordinatethatisidenticaltothetrans-
verse
coordinate of the straight waveguide along the junc-
tion. The one-dimensional (1-D) overlap integral in this TE slab
waveguide case is, for example,
(dB). Here,
are the modal electric-field am-
is the bend
(2)
and simplifies to the case for straight waveguides for
, whereover the range of
tegrand (modal field) significantly contributes to the integral.
The integral is terminated at the radiation caustic
the straight slab junction case, FDTD simulations were done
to check on the validity of ignoring reflected radiation modes,
which confirmed a high accuracy for low junction losses of
1dB.Whentheinputwaveguideisoflow-indexcontrast,such
as the case considered here, loss in regular waveguide bends is
dominated by bending loss, and junction loss can be ignored.
For trench-clad bends (the bend region of ATBs), junction loss
must be incorporated into the loss figure.
values where the in-
. For
B. Cladding Tapers
In the case of the ATB, the purpose of our cladding taper is
to adiabatically shape the fundamental mode of the low-index
contrast input waveguide [at the input of Section I, Fig. 1(a)] to
the shape of the fundamental mode, in the local normal mode
sense, of the high-index contrast (air trench-clad) output [at the
outputofSectionII,Fig.1(a)].Inlow-indexcontrast,thisoutput
mode can be much smaller than the minimum possible width of
the input waveguidemode. The output waveguidewidth is fixed
by the bend design from optimization of the trench region’s
taper-bendjunctionloss.Forsimplicity,thecladdingtaperscon-
sidered are piecewise linear, but it is recognized that because of
the need for adiabatic shaping of phase front curvature in addi-
tion to mode shape, the optimal taper of this kind will have a
more complex shape. For our analysis, a cladding taper with a
constantwidth core is broken uplengthwiseinto two distinctre-
gions with piecewise linear tapers: one where air surrounds the
core and cladding [the outer, “cladding” taper—I in Fig. 1(a)]
and one where air surrounds the core directly [the inner, “core”
taper—II in Fig. 1(a), which transitions to the bend, III].
Optimization of the taper is dealt with in a less analytic
manner than the bend due to its geometric complexity. The
initial estimate for the tapering angle of each of the inner
and outer tapers is obtained from considering the ray-optical
description of the input and output fundamental modes. Ray
angles from the plane- wave description of input and output
(local normal) modes are used in conjunction with an ex-
trapolated Goos–Hänschen shift to determine a taper angle
that would guide the input ray directly into the output ray
with minimal loss (i.e., in a “single bounce”), as illustrated in
Fig. 3(c). While this is a highly simplistic picture, it is useful in
providing an initial estimate. The refining of the taper param-
eters requires few steps and is done here by hand, via iterated

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1766 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 20, NO. 9, SEPTEMBER 2002
Fig. 5.
and the total size box (dash-dot square). Inset shows field plot of simulation
used for throughput efficiency lower bound (180 bend) due to simulation size
constraints. Key in lower right corner refers to transmission data in Fig. 8.
Example A. Dominant electric-field plot from FDTD simulation,
computer simulation. The tapers are finally characterized by
full 2-D FDTD simulations, which, while computationally
costly, should provide an accurate estimate of performance.
Results of individual taper efficiency are listed in Table II, with
equal throughput efficiency in either direction, as required by
reciprocity.
We split the total loss between the cladding tapers and the
trench bend in such a way that the tapers (which dominate the
sizeoflowerindexcontraststructures—Figs.5–7)carryalarger
part of the losses in order to minimize total structuresize. Bend-
region loss can be much improved with small radius increases
(see Fig. 4). In lower index contrast, where bends constitute a
smallpartoftheATBstructure,radius-sensitivebendlosscanbe
practically removed and taper-bend junction loss reduced with
a small radius increase, then redistributed among the tapers to
reduce overall structure size.
IV. ATB EXAMPLES WITH 2-D SIMULATION RESULTS
We present 2-D simulation results for several example ATBs,
designed in the manner described above, chosen to demonstrate
the proposed idea and illustrate its effectiveness in reducing
bend size in various index contrasts. Examples A–C range in
indexcontrast from 0.25% to 7%, where thelast example is out-
side the range of typical index contrasts found in silica waveg-
uides
1.5% [1]. It was used as our first structure be-
cause of its small size in terms of wavelengths and, thus, short
simulation time. Refractive indexes and dimensions for the ex-
ample structures are given in Table III. The refractive indexes
that are chosen for all examples are ones that could be produced
in SiO N -core/SiO -cladding waveguides. SiON has an index
tunability over the range of 1.45–1.96 [15].
Fig. 6.
size box is outlined (dash-dot square) for ATB and for a regular bend (inset) for
reference. Key in lower right corner refers to transmission data in Fig. 8.
Example B. Dominant electric-field plot from FDTD simulation. Total
Fig. 7.
sizeboxshownforreference(dash-dotsquare).Thekeyinthelowerrightcorner
refers to transmission data in Fig. 8.
Example C. Dominant electric-field plot from FDTD simulation. Total
A. FDTD Simulation Setup
The ATB structure is reduced to two dimensions for
simulation by using the effective index method (EIM) with
perturbative correction [14], which works well in spite of the
square cross-sectional geometry of the waveguides and high
aspect ratio of the trench (see Section V). In applying the
EIM, it is assumed that the mode is always well confined if

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POPOVIC´et al.: AIR TRENCHES FOR SHARP SILICA WAVEGUIDE BENDS 1767
TABLE III
ATB STRUCTURE DIMENSIONS
the waveguide is straight, and that substrate (or bulk cladding)
leakage beneath the air trench is negligible. Thus, we concern
ourselves only with loss due to bending in the plane. The core
effective index is obtained by the traditional EIM, while for the
cladding, it was obtained by perturbation of the propagation
constant, assuming the same shape of mode field as in the core
[14].
FDTD resultsfor transmission lossare obtained bylaunching
a short pulse (on the order of 50 fs) into the input waveguide
with enough spectral width to span the wavelength spectrum of
interest. A discretization of
20 points per wavelength is used,
with the perfectly matched layer (PML) absorbing boundary
condition imposed on the edges of the 2-D computational do-
main. The total power flux through the output waveguide cross-
sectional plane is monitored, and an overlap integral with the
fundamental mode of the waveguide is carried out to obtain
transmissionefficiencyatthecentralwavelength(and,similarly,
for reflection back to the input waveguide). A simulation is ter-
minated once the pulse has left the computational domain and
all fields have sufficiently died out.
B. Regular Bend Results
Radiation losses of regular waveguide bends (without air
trenches) are evaluated using standard tools for 2-D (bent slab)
waveguides mentioned previously, making use of the EIM to
“collapse” the third dimension. Since their loss per wavelength
along the propagation direction is very low, FDTD would be
too computationally costly (and perhaps inaccurate) for such
structures. The values obtained for the regular bend radius
without an air trench—as required for a transmission of 98% or
loss of 0.1 dB/90 —are shown in Table I (using the effective
indexes from Table IV).
C. ATB Results
ATB structures corresponding to each of these regular bends
are thendesigned andsimulated using2-DFDTD(verticalelec-
TABLE IV
COMPARISON OF EFFECTIVE INDEX METHOD AND VECTOR MODE SOLVER
tric or TE-like polarization). Contrary to convention, we refer
to the polarization of the mode with a vertical dominant elec-
tric-field component
as TE-like or quasi-TE. We do so
because this mode in the trench region looks TE-like, and for
approaches the slab waveguide TE mode. In the square
cross-section low-index input waveguide, the modes are degen-
erate, and the labeling does not matter. Figs. 5–7 are plan view
plotsoftheelectric-fieldamplitudesuperimposedontheoutline
of the structure. The absence of obvious radiation gives quali-
tative evidence that these bends exhibit low loss. The new bend
radii and total or “box” edge lengths, including tapers (i.e., the
effective radii), are listed in Table I alongside regular bend data
and show drastic size reduction for the same performance. As
a metric for the size of bends, we use a box placed around the
bend that encompasses the bend structure as well as
the input and output waveguide mode power (Figs. 5–7, inset in
Fig. 6 for regular bends).
For
7%,athroughputefficiencyof98%isachievedwith
an ATB size of 20 m (a trench region bend radius of 7 m), as
compared with a regular bend of 80 m. For
duction from 2.5 mm to an ATB of 96
trench radius of 9 m. For the lowest
(Fig. 5 inset) was simulated for loss to reduce computational re-
quirements and is representative of the 90 ATB because bend
loss is a small part of the total loss (Table II). The 90 ATB field
plot in Fig. 5 is given from a simulation with coarse discretiza-
tion. The lowest index contrast bend is reduced from a 15-mm
radius to 250
m.
Fig. 8 shows the transmission and backscattering power
ratio spectra over the
-band communications window of
1530–1570 nm, exhibiting little wavelength dependence as
expected, because single-mode coupling is being used through
the entire device, and no use is made of either resonance or
multimode/path interference. The exception is backscattering
in the 180 turn (A), where weak Fabry–Pérot resonances are
seen due to the proximity of the input and output waveguides.
Markers placed in Fig. 8 at the central wavelength (1550 nm)
show the values of transmission and reflection into the fun-
damental waveguide mode (obtained using overlap integrals).
These represent the bend transmission loss values of interest,
where 98% was targeted in order to compare the bend sizes.
Notably, reflection in all cases is well below −30 dB and, thus,
does not present a problem, at least in theoretical design.
A final example is given to illustrate the impact of the
cladding taper by removing the portion of the taper in the
cladding region in Example B with
throughput of 69% or a loss of 1.5 dB results in this case.
99.9% of
0.7%, a re-
m is achieved, with a
of 0.25%, a 180 bend
0.7% (Fig. 9). A

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1768 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 20, NO. 9, SEPTEMBER 2002
Fig. 8.
B (dashed), and C (solid) through the output and input reference planes,
respectively. True transmission and reflection into the fundamental mode is
shown at the central wavelength (A-triangle, B-circle, C-square).
Transmitted and backscattered power for cases A (dashed–dotted),
Fig. 9.
performance. The core taper is virtually lossless, and the loss here is caused by
the abrupt junction. No gain results if the core taper is lengthened.
The ATB from Example B without cladding tapers shows very poor
In all of the example cases (A–C), bend size is significantly
reduced, most drastically in lower index contrasts, which also
have better fiber-chip coupling (i.e., matching to the fiber
mode). However, in the vertical (out-of-plane) direction, the
aspect ratio of the air trenches for low
problems.
’s may pose fabrication
V. CONFINEMENT IN THE 2-D CROSS SECTION: EFFECTIVE
INDEXES, SUBSTRATE LOSS, POLARIZATION
Here, we present some justification for the use of the EIM
in the full structure simulations, and we consider substrate loss
and polarization issues, the proper discussion of which requires
considerationofthefull2-Dcrosssectionoftheproposedwave-
guide structure.
A. Effective Index Versus Exact Modes of the Ideal Air Trench
All2-DFDTDsimulationsshowninthispaperarecarriedout
using effective indexes for the core, cladding, and air regions
Fig. 10.
trench,freeofsubstrateloss.Fundamentalquasi-TEand-TMmodesoftheideal
trench for Examples A–C are in (c)–(e), respectively. Mode plots are to scale
individually, but not in proportion to each other. For an idea of relative scale,
note that the waveguide width in each of (c)–(e) is the same.
Schematic of (a) actual air trench cross section and (b) the idealized
derived for an equivalent 2-D problem using the EIM with per-
turbation correction [14]. The latter was necessary because air
trenches (like rectangular waveguides) cannot be treated using
the standard EIM, as is possible for rib waveguides (except
to adopt the cladding index as the effective index value in the
cladding).
For our simulations, effective indexes are obtained assuming
a guided mode and, thus, a waveguide cross- section in which
the air trench and cladding regions extend infinitely above and
below the core [Fig. 10(b)]. Because the index contrast is much
higherinthelateral( axisinFig.10)thaninthevertical( axis)
direction, the behavior is very much slablike—confinement is
weak in the vertical direction—and the maximum waveguide
widthfor “singlemodedness” isroughlythatoftheslab.Having
goodlateralandpoorverticalconfinementisacceptablebecause
we are only concerned with making sharp bends in the plane.
In fact, allowing poor vertical confinement improves the lateral
confinement, although it does present other undesirable proper-
ties,suchasenhancedsubstrateloss.Theairtrenchandcladding
are in reality of finite vertical extent [Fig. 10(a)], and there will
be substrate loss if the modal index is lower than the index of
the bulk cladding below the trench. In the case of an ideal air
trench of infinite extent, the mode is guided [Fig. 10(b)–(e)].
A vectorial finite-difference mode solver (with metallic-wall
boundary conditions placed far from the waveguide) is used to
verify the appropriateness of the values obtained through the
EIM. Table IV shows a comparison of the modal indexes in the
2-D EIM approximation and from the three-dimensional (3-D)
full-vector solver, and Fig. 11 shows the lateral field distribu-
tions of the 2-D and 3-D solutions along the
axis for case C,

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POPOVIC´et al.: AIR TRENCHES FOR SHARP SILICA WAVEGUIDE BENDS1769
Fig. 11.
the horizontal plane at the waveguide axis, as obtained from a vectorial mode
solver (dash-dot) and as approximated using the EIM (solid). The EIM field is
representative of the 2-D FDTD simulations.
Comparison of the dominant electric-field component distribution in
theworstapproximation,accordingtoTableIV.Thegoodmatch
in Fig. 11 provides a justification for the use of the EIM in our
2-D simulations.
B. Polarization Dependence and Loss
The quasi-TE and quasi-TM mode propagation constants in
Table IV show that the TM-like mode is more weakly con-
fined and that this structure will not be polarization-indepen-
dent, especially for tight bend radii [Fig. 10(c)–(e)]. Instead of
attempting to compensate for polarization dependence, it is pro-
posed that the real estate savings would justify building two
identical circuits (Fig. 2).
Polarization-mode mismatch can hamper the operation of a
single polarization device as well by introducing loss, via mode
scatteringatinterfaces.However,intheATBstructurewithideal
(vertically infinite, guiding) air trench waveguides, polarization
mixing at the interfaces is forbidden by the symmetry of the
fields. In an actual ATB, this symmetry is broken at the inter-
faces by the combination of a substrate layer present below and
the lateral asymmetry of the leaky bend mode of one of the
waveguides. However, in our application, the substrate is kept
far enough away to provide low loss, and with it, this coupling
term will also be small.
C. Substrate Loss and Minimum Air Trench Depth
In order to curb the substrate loss in the actual ATB
[Fig. 10(a)], the air trench must be etched sufficiently deep
beneath the core for the evanescent tail of the field to be small
where it reaches the bulk cladding. With
modal field will be oscillatory in the bulk substrate below the
trench, and the mode will exhibit leakage through a tunneling
process from the core, through the “cladding ridge” beneath it,
into the bulk cladding (the field is evanescent in the vertical
direction in the “ridge”). A deep trench, as demanded here,
requires high aspect ratio etching. The mode is weakly confined
in the vertical direction, so the air trenches must be several
, the
Fig. 12.
sheet method (ECSM). (b) Mode solver LM solution example: Real part of the
? field component for the waveguide of Example B with the bulk cladding 0.6
core heights from the axis.
Calculating substrate loss: (a) Setup of the 2-D equivalent current
core heights [
loss. Since the aspect ratio increases with lower index contrast,
fabrication issues will place a lower limit on the index contrast
range for which this technique is practical. Making use of the
whispering-gallery regime in bending and removing the inner
wall of the trench bend completely may ease this difficulty.
It is of interest to evaluate the required depth of the air trench
[Fig. 10(a)] given an acceptable substrate loss (much less than
total ATB loss). We proceed along two parallel routes to eval-
uate substrate loss: a semianalytic perturbation method using
the mode solution of the ideal (guiding trench) and a numerical,
full-vector mode solver for LMs, both taking into consideration
the 2-D cross section of the waveguide.
1) 2-D Equivalent Current Sheet Method: We define an
equivalent current sheet and use a cylindrical-vector Green’s
function (e.g., see [9]) approach to evaluate the far-field
radiation and, thus, loss per unit length. Since we are interested
in small loss, we consider a “perturbative” calculation. That is,
ending the air trench at a prescribed depth [Fig. 10(a)] implies
a perturbation of the refractive index from the ideal trench
[Fig. 10(b)] below that depth. We assume that silica cladding
continues below the trench. We also assume that the mode
shape remains unperturbed above the air trench-bulk cladding
interface and use this field to define an equivalent current
sheet just above the bottom of the air trench to represent the
fields below [Fig. 12(a), region of interest]. Assuming that
the field does not change much in crossing the interface, the
current sheet can be moved just below the trench, where we
now have a region of uniform index (the bulk cladding or
substrate). For a more accurate treatment, we can account for
Fresnel reflection due to the interface when translating across
it fields that generate the current sheet. A far-field solution of
the radiation pattern into the bulk cladding can then easily be
calculated integrating a cylindrical-vector Green’s function
over this source (factoring out first the propagation direction
). A field attenuation coefficient results
, Fig. 1(b)] deep for acceptable substrate
(3)

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1770 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 20, NO. 9, SEPTEMBER 2002
Fig. 13.
of Examples A–C (Figs. 5–7) computed by approximate ECSM (lines) and
numerical vector-field mode solver for leaky waveguides (symbols). The
discrepancy is due to reflection at the interface with the bulk cladding, not
accounted in the ECSM calculation.
Bulk cladding loss versus trench etching depth for waveguides
wherethenumeratorrepresentstheradialcomponentoffar-field
radiated power per unit waveguide length, the denominator rep-
resents power guided in the mode, and substrate loss is
(dB/m).
For air trench waveguide substrate loss calculations, we use
the original equivalent current sheet and ignore reflection from
the interface, as proper compensation for the reflection on a
2-D interface would complicate the calculation. Normalized
2-D substrate loss results obtained for ATB Examples A–C
presented in this paper are shown in lines in Fig. 13, where
guided-mode solutions of the ideal air trench waveguide from
a vector-field mode solver were used to define the current
sheet. From these plots, we choose a trench depth for which the
substrate loss is much smaller than the ATB loss of 0.1 dB.
2) Numerical Mode Solver for 2-D Vector Modes: A second
setofsubstratelossresultscomeswithlessphysicalintuitionbut
with (numerically) exact solutions and is obtained directly from
avectorialmodesolverdesignedforleakywaveguides.Weused
a staggered vector-field discretization like that in FDTD [17],
[18] and implemented PML absorbing boundary conditions in
the frequency domain to allow for LMs (e.g., the 1-D example
in [19]). The resulting complex eigenvalue problem was solved
using a sparse matrix iterative eigenvalue solver (e.g., eigs in
MATLABorARPACK).Thesubstratelossresultsdirectlyfrom
the imaginary part of the mode propagation constant. The com-
puted substrate loss values are plotted in Fig. 13 along with the
current sheet results. The discrepancy is attributable to ignored
reflection in the the current sheet method. An example modal
solution is shown in Fig. 12(b).
For our examples A–C, the mode resembles an air trench
mode only within the “air square” [Table IV and Fig. 1(b)],
so we take a propagation distance of two “air square” edges
for total substrate loss. For a chosen 0.01-dB loss, this gives
loss per unit length requirements of 0.00005, 0.00014, and
0.00039 dB/ m for cases A–C, respectively. Total trench depth
Fig. 14.
?-field component plot from FDTD simulation. Transmission and reflection
into source mode given in lower right corner.
Simple waveguide tee using ATBs (0.7% index contrast): Dominant
[as in Fig. 10(a)] is twice the distance of the trench from
the waveguide axis in Fig. 13, if we assume a trench that is
symmetric about the core axis but leaks only to the bottom.
The loss-per-unit-length values above map onto required
normalized trench depth in Fig. 13 (using the mode solver data
set). Unnormalizing by the core height
obtain required total trench depths of 24, 15, and 5
Examples A–C. Clearly, a lower index contrast poses greater
challenges for fabricating the air trenches but also offers greater
reduction in bend size.
Another concern that emerges with the introduction of air
trenches is scattering loss, which is common to all high-index
contrast structures. We argue that our bends are small enough to
have an acceptably low scattering loss. For example, a 20- m
ATB structure (as in case C), assuming a scattering loss of
5 dB/cm, would incur a 0.01-dB total loss due to scattering per
90 bend, as compared with the bending loss of 0.1 dB/bend.
from Table IV, we
m for
VI. WAVEGUIDE TEE INCORPORATING ATBS
We show as an application a primitive compound structure
that makes use of the proposed ATBs, a waveguide T-splitter
(90
junction). In this structure, additional waveguide off-
sets at the input bend junction are required for optimal trans-
mission. The field plot and performance are given in Fig. 14.
An evenly split, total transmission of 95% is achieved with less
than
40-dB reflection to the source. In this example, care was
not taken to design for simple fabrication, and the structure’s
sharp corner at the
junction would present a problem. An al-
ternative geometry for the power-splitting section, such as a
rectangularmultimode-interference(MMI)sectionwithoutputs
wider apart, could be used at the
herewastoshowcompactwide-anglesplittingusingATBs.The
structureis180 m
100 m insizeand intherefractiveindex
system of Example B.
junction instead. The intent
VII. CONCLUSION
By introducing air trenches gradually, away from the core
first, in a configuration that allows for adiabatic mode transition

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POPOVIC´et al.: AIR TRENCHES FOR SHARP SILICA WAVEGUIDE BENDS1771
from low to high (trench) index contrast regions, a dramatic re-
duction in the bending radius of otherwise low-index contrast
waveguides is possible without incurring large junction losses
through mode mismatch and Fresnel reflection. The “total box
size”(effectiveradius)ofabendwasreducedbyafactorof4–60
in our silica examples. Because bending radius is one of the pri-
mary factors limiting the density of integration in silica, the use
of ATBs, such as the ones presented, may allow dense integra-
tion leading to reduced cost and better yield, while preserving
the good fiber coupling and propagation loss properties of silica
PLCs. We have provided 2-D simulations of example structures
with support for this treatment. Arbitrarily low substrate loss is
possibletowithinthelimitationsonfabricatinghighaspectratio
trenches, ranging in depth between 5 and 25
ples. In addition, we suggest that complete removal of the inner
wall of the ATB (with necessary modifications to the design)
could further improve not only bend loss but also the manufac-
turability by reducing the vertical structure’s aspect ratio. We
showed a T-splitter, as a simple compound device using these
ATBs.
m in our exam-
ACKNOWLEDGMENT
The authors gratefully acknowledge the use of the NPACI
Cray T-90 supercomputer at the San Diego Supercomputer
Center (SDSC) for all FDTD simulations. Author M. Popovic ´
also wishes to thank C. Manolatou for use of her well-written
FDTD simulation code and for invaluable and frequent advice
and M. J. Khan and M. Watts for fruitful discussions and useful
suggestions.
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[4] D. Rowland, “Nonperturbative calculation of bend loss for a pulse in a
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[5] I. C. Goyal, R. L. Gallawa, and A. K. Ghatak, “Bent planar waveguides
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[6] L. H. Spiekman, Y. S. Oei, E. G. Metaal, F. H. Groen, P. Demeester,
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[7] C. Manolatou, S. G. Johnson, S. Fan, P. R. Villeneuve, H. A. Haus,
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[8] D.Marcuse,TheoryofDielectricOpticalWaveguides,2nded.
MA: Academic, 1991.
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[12] E. C. M. Pennings, “Bends in optical ridge waveguides: Modeling and
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[13] K. Wada, M. Popovic ´, S. Akiyama, H. A. Haus, and J. Michel,
“Micron-size bending radii in silica-based waveguides,” in Proc.
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13–14.
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Miloˇ s Popovic ´ (S’98) was born in Zajeˇ car, Yu-
goslavia, in 1977. He received the B.Sc. degree
in electrical engineering from Queen’s University,
ON, Canada, in 1999, and the M.S. degree from the
Massachusetts Institute of Technology, Cambridge,
in 2002. He is currently working toward the Ph.D.
degree, pursuing integrated optics as his research
topic.
Kazumi Wada received the B.S. and M.S. degrees
and the Ph.D. degree in instrumentation engineering
from Keio University, Yokohama, Japan, in 1973,
1975, and 1982, respectively.
Since joining NTT Laboratories, Tokyo, Japan,
in 1975, he has been engaged in research on defects
in silicon and III–V semiconductor materials and
devices, including oxygen precipitation kinetics in
Si. From 1981 to 1982, he was a Visiting Scientist
at the Massachusetts Institute of Technology (MIT),
Cambridge, conducting research on defects in
GaAs. From 1982 to 1983, he was a Visiting Specialist at the Science and
Technology Agency of the Japanese Government working on establishing a
guideline on semiconductor materials research in Japan. Since 1998, he has
been with the MIT Microphotonics Center, Department of Materials Science
and Engineering, conducting research on silicon-based photonic integrated
circuits. His current interest is on-chip optical interconnection for high-speed
Si-LSIs. He is author and coauthor of more than 100 refereed journal papers
and has edited 13 books. He is an Associate Editor of the Japanese Journal of
Applied Physics.
Dr. Wada has been serving as Associate Editor of the Institute of Electrical
and Electronics Engineers (IEEE)/The Minerals, Metals & Materials Society
(TMS) JOURNAL OF ELECTRONIC MATERIALS. He is an active Member of the
Materials Research Society, the Electrochemical Society, and the Japan Society
of Applied Physics.
Shoji Akiyama received the B.S. and M.S. degrees
in chemistry from Kyoto University, Kyoto, Japan, in
1995 and 1997, respectively.
In 1997, he joined Shin-Etsu Handotai (SEH)
Company, Ltd. In 2000, he entered Professor L.C.
Kimerling’s Group at the Department of Materials
Science and Engineering, Massachusetts Institute of
Technology (MIT), Cambridge.

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1772 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 20, NO. 9, SEPTEMBER 2002
Hermann A. Haus
LF’91) was born in Ljubljana, Slovenia, in 1925.
He attended the Technische Hochschule, Graz, and
the Technische Hochschule, Vienna, Austria. He
received the B.Sc. degree from Union College,
Schenectady, NY, in 1949, the M.S. degree from
the Rensselaer Polytechnic Institute, Troy, NY, in
1951, and the Sc.D. degree from the Massachusetts
Institute of Technology, Cambridge, in 1954.
He joined the Faculty of Electrical Engineering at
MIT in 1954, where he is an Institute Professor. He
(S’50–A’55–SM’58–F’62–
is engaged in research in electromagnetic theory and lasers. He is the author or
coauthor of six books and more than 360 journal articles.
Dr.HausisaFellowoftheAmericanPhysicalSocietyandtheOpticalSociety
of America (OSA) and a Member of the National Academy of Engineering, the
National Academy of Sciences, and the American Academy of Arts and Sci-
ences. He is the recipient of the 1984 Award of the Institute of Electrical and
Electronics Engineers (IEEE) Quantum Electronics and Applications Society,
the1987CharlesHardTownesPrizeoftheOpticalSocietyofAmerica,the1990
Outstanding Paper Award of the IEEE TRANSACTIONS ON EDUCATION, the 1991
IEEE Education Medal, the 1994 Frederic Ives Medal of the Optical Society of
America, the President’s 1995 National Science Medal, the Ludwig Wittgen-
stein-Preis Award of the Austrian Government 1997, and the Willis E. Lamb
Medal, 2001. He holds honorary doctorate degrees from Union College, the
Technical University of Vienna, Vienna, Austria, and the University of Ghent,
Ghent, Belgium.
Jürgen Michel was educated in Germany and
received the degree in physics from the University
of Cologne, Cologne, Germany, in 1983, and the
Ph.D. degree in applied physics from the University
of Paderborn, Paderborn, Germany, in 1987.
He was a Research Scientist at the University
of Paderborn. Prior to joining MIT in 1990, he
was a Postdoctoral Member of Technical Staff at
AT&T Bell Laboratories, studying defect reactions
and defect properties in semiconductor materials.
Currently, he is a Senior Research Associate in the
Microphotonics Center at the Massachusetts Institute of Technology (MIT),
Cambridge. He manages and conducts research projects in silicon-based
photonic materials and devices and contamination issues in silicon processing.
His main focus is currently on optical-clock distribution and coupling and
packaging issues in microphotonics and on-chip WDM devices. He is author
or coauthor of more than 80 technical articles.