We have developed new silicon-on-insulator waveguide designs for simultaneously achieving both low-loss optical confinement and electrical contacts, and we present a design methodology based on calculating the Bloch modes of such segmented waveguides. With this formalism, waveguides are designed in a single thin layer of silicon-on-insulator to achieve both optical confinement and minimal insertion loss. Waveguides were also fabricated and tested, and the measured data were found to closely agree with theoretical predictions, demonstrating input insertion loss and propagation loss better than 0.1 dB and -16 dB/cm, respectively

Baehr-Jones, Tom

Gunn, Lawrence C.

Hochberg, Michael

Scherer, Axel

Walker, Chris

Witzens, Jeremy

English

Caltech Authors - Main

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Hochberg et al. Vol. 22, No. 7 /July 2005 /J. Opt. Soc. Am. B 1493Segmented waveguides in thin silicon-on-insulator
Michael Hochberg, Tom Baehr-Jones, Chris Walker, and Jeremy Witzens
California Institute of Technology, 1200 East California Boulevard, Pasadena, California 91125
Lawrence C. Gunn
Luxtera, Inc., 1819 Aston Avenue, Carlsbad, California 92008
Axel Scherer
California Institute of Technology, 1200 East California Boulevard, Pasadena, California 91125
Received June 4, 2004; revised manuscript received August 30, 2004; accepted October 10, 2004
We have developed new silicon-on-insulator waveguide designs for simultaneously achieving both low-loss op-
tical confinement and electrical contacts, and we present a design methodology based on calculating the Bloch
modes of such segmented waveguides. With this formalism, waveguides are designed in a single thin layer of
silicon-on-insulator to achieve both optical confinement and minimal insertion loss. Waveguides were also fab-
ricated and tested, and the measured data were found to closely agree with theoretical predictions, demon-
strating input insertion loss and propagation loss better than 0.1 dB and −16 dB/cm, respectively. © 2005
Optical Society of America
OCIS codes: 220.3740, 230.7390, 130.1750.T
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g. INTRODUCTION
ecently, low-loss single-mode waveguides in thin silicon-
n-insulator (SOI) have been demonstrated1 with
−7 dB/cm of propagation loss. There are several advan-
ages to working in such a material; in particular, the thin
eometry is helpful in obtaining high field concentrations
n the waveguide cladding, a useful attribute for the con-
truction of optical chemical sensors or silicon waveguides
lad with electro-optic polymers. Typically, the functional-
zation material in such devices resides in the waveguide
ladding, and a large modal overlap with the deposited
ladding layer is desirable. One of the outstanding prob-
ems of this geometry, however, is the difficulty of estab-
ishing an electrical contact with the waveguide without
erturbing the optical mode. This is particularly impor-
ant when a dc or rf electrical field is to be applied directly
nto the waveguiding region. Such a field can be used, for
nstance, to induce modulation through an electro-optic
ndex shift in the cladding.2 As the lateral dimensions of
uch single-mode SOI waveguides are necessarily
500 nm for a 120 nm thin silicon SOI layer to ensure
ingle-mode propagation, it is challenging to generate the
eld amplitudes desired at the surface of the waveguide
ithout electrically contacting the waveguide. Substan-
ial research has been done previously on segmented
aveguides in low-index-contrast, large-mode-volume
eometries.3–5
The SOI geometry that we use in this study was chosen
o have low waveguide loss and relatively large field con-
entrations outside the core of the waveguide. It consists
f 120 nm of silicon (index 3.5) atop a 1.4 mm layer of sili-
on dioxide (index 1.43), which rests on the silicon handle.0740-3224/05/071493-5/$15.00 © 2he structures were clad in polymethyl methacrylate sn
1.43d. The optical mode that such a waveguide supports
s primarily polarized horizontally and is essentially iso-
ated from the substrate by the oxide layer, although it
an be shown that some loss induced by tunneling leak-
ge into the substrate will occur.6 The dispersion plot of
he fundamental mode and modal concentration are de-
ailed in Fig. 1.
The definition of an electrical contact on such a wave-
uide is particularly difficult in that the waveguide is
oth electrically and optically isolated on all sides by sili-
on dioxide cladding. The introduction of an electrical con-
act leads to a significant interruption in the waveguide
ymmetry and therefore a large scattering loss. Here we
how a remedy for this problem by constructing seg-
ented waveguides as shown in Fig. 2.
In this figure the essential features of the segmented
aveguide approach are detailed. The electrical contact is
efined as a lateral grating or a planar extension that is
ithographically defined during the same lithographic
tep as the waveguide definition etch. The optical proper-
ies of this geometry are strongly dependent on the peri-
dicity and duty cycle. (We consider the duty cycle to be
he fraction of the period that contains the segment; thus
duty cycle of 0.7 on a period of 1 mm indicates segments
f silicon that are 0.7 mm long.) In theory the lateral sili-
on strips in Fig. 2 continue forever, but, as we will see,
or properly chosen periodicities these can be terminated
fter a relatively short isolation distance. If a low-loss,
ropagating optical mode exists for a particular design, it
s possible to achieve both the desired low-loss optical
uiding and lateral electrical contact to the waveguide.005 Optical Society of America
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1494 J. Opt. Soc. Am. B/Vol. 22, No. 7 /July 2005 Hochberg et al.. BLOCH-MODE ANALYSIS
he analysis of the waveguide layout presented in Fig. 2
an be performed with the aid of Bloch’s theorem. Our
nalysis begins with the conversion of the current-free
nd charge-free Maxwell’s equations into an eigenvalue
roblem, following7
„ 3
1
nsrd2
„ 3 H =
w2
c2
H. s1d
e choose the H field as the field variable, because the ei-
envalue equation is, in such a case, Hermitian and not
eneralized. We shall discuss the practical method of solv-
ng this equation for our structures, but first we will make
ome general observations about the solutions.
In what follows, we will take x and y to be the trans-
erse cross sections of the waveguide, and z is the direc-
ion of propagation. A nonsegmented waveguide, such as
hown in Fig. 1, nsrd has continuous symmetry in z. In the
ase of the segmented waveguide, nsrd has the discrete pe-
iodicity nsr+Dzd=nsrd, where Dz is a translation in the z
irection of the period of the segmentation. Utilizing
loch’s theorem in the propagation direction, we have the
esult that all the eigenvectors of Eq. (1) may be written
n the form
Cswd = fsrdexpsibzd. s2d
ere we are taking Cswd as an eigenvector corresponding
o a particular choice of w, b as the crystal lattice vector,
ig. 1. (a) Diagram of the field pattern of the fundamental op-
ical mode; contours of uEu are plotted, starting at 10% of the
aximum field value and incremented by 10% for each contour.
b) The dispersion diagram consists of the effective index plotted
gainst the free-space wavelength.nd fsrd as the local-field distribution in a unit cell (a
-vector). The propagating modes of a segmented wave-
uide will be among the solutions to Eq. (2).
To solve this problem for complex structures, it is nec-
ssary to discretize it. There are many approaches avail-
ble; we choose the finite-difference time-domain grid as
he basis of the discretization in that such an approach
mplicitly enforces the appropriate continuity and diver-
ence conditions.8 The resulting linear system results in a
arge sparse matrix equation, with approximately 0.6 mil-
ion variables for a typical calculation, with a discretiza-
ion of 0.02 mm. The lowest nonzero eigenvalues can nev-
rtheless quickly be found with the proper choice of
terative methods.9 The modes generated by our direct
olver were found to be identical to the modes generated
y a large finite-difference time-domain simulation with a
ong runway and were substantially faster to generate.
It is important to ensure that Eq. (1) remains Hermit-
an in whatever set of boundary conditions are chosen for
he unit cell. Although the z boundary condition for a
nit-cell calculation is obviously just periodic with the ap-
ropriate Bloch factor, it is less clear how the x and y
oundary conditions should be handled. We wish to deal
ith the infinite-space problem, but for numerical calcu-
ations this is impossible. Instead, it is best to enforce
ero-field boundary conditions on the edge of the unit cell
n x and y, corresponding roughly to having a perfect con-
uctor in this region. Such boundary conditions obviously
o not correspond to the problem at hand, but, for solu-
ions to Eq. (1) that reduce close to zero at these bound-
ries, the introduction of this spurious conductor should
ot disturb the eigenvalue or vector.
One can define an effective index for the Bloch modes
s b /w. With a choice of effective indices in the range of
–4 for cells of periodicity 0.3 mm, the lowest-frequency
igenvalues turn out to be the fundamental propagating
ptical mode that has frequencies in the range of 1–2 mm.
hus solving for the modes of interest simply requires ob-
aining the lowest eigenvalue of the system for varying b
alues. There is no reason this will be true all the time; in
eneral, solving Eq. (1) with functions of the form (2) will
roduce a series of frequencies, only some of which will be
n the range of physically meaningful solutions. It may be
ig. 2. Logical diagram of the planar layout of a segmented
aveguide. For comparison, a nonsegmented waveguide is also
iagrammed, in a configuration suitable for butt coupling into
he segmented waveguide.
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Hochberg et al. Vol. 22, No. 7 /July 2005 /J. Opt. Soc. Am. B 1495ecessary in other circumstances to climb several values
p the eigenspectrum before useful data will be obtained.
t is possible to generate a dispersion diagram for the
loch modes of a given segmented waveguide design, by
se of the effective index defined as above. The fundamen-
al Bloch mode of a segmented waveguide with periodicity
f 0.28 mm and a duty cycle of 0.5 was solved for. We will
efer to this design as series 0, for later comparison with
ther designs. The dispersion diagram is presented in Fig.
, along with a transverse and longitudinal cross-section
lot of the E field intensity.
The dispersion plot in Fig. 3 indicates that the effective
ndex of the Bloch mode closely matches that of a nonseg-
ented waveguide, indicating that there will not be much
nsertion loss from index mismatch if we couple light from
nonsegmented waveguide to a segmented waveguide.
oreover, there is little dispersion in the region of 1550
m for this waveguide. This is not always the case, as
any segmented waveguide periodicities can exhibit
andgaps with high backreflection and prevent the for-
ard propagation of particular frequencies altogether. In
he region of such behavior, the dispersion diagram typi-
ally exhibits a derivative dw /db approaching zero. Fi-
ally, it should be noted that, for our chosen periodicity,
he field amplitudes become small at the edges of the
aveguide domain. Again, unfortunate choices of period-
city may not exhibit this behavior and will exhibit higher
adiative losses, and it is necessary to correctly design the
egmented region to ensure low losses. Because the mode
xhibits low loss in the modeled geometry, the segmenta-
ion need not be extended to infinity, and it is adequate to
nsure that it is longer than several e-folding lengths of
he field to prevent the optical mode from being influ-
nced by the end of the lateral segments. For the devices
abricated here, the segments were extended 2000 nm
rom the center of the waveguide.
. FABRICATION AND TEST
he devices were fabricated with electron-beam lithogra-
hy. Wafers of SOI with approximately 120 nm of silicon
or the top layer and 1.4 mm of oxide were cleaved and
ere spun and cleaned in acetone and isopropanol. Dow
orning’s FOX-12 electron-beam resist was spun onto the
urface at 8000 rpm and baked for 20 min at 180° C. The
afers were exposed at 2250 mC/cm2 at 100 KV in a Leica
BPG-5000+ electron-beam lithography system and de-
eloped in Clariant AZ300 metal-ion-free tetra-methyl
mmonium hydroxide developer. They were rinsed in wa-
er, dried, and then etched in an Oxford Plasmalab 100 in-
uctively coupled plasma, reactive ion etch system for 25
with chlorine flow of 80 SCCM (SCCM denotes cubic
entimeters per minute at STP), forward power of 50 W,
hamber pressure of 12 mTorr, and ICP power of 800 W.
igure 4 shows a scanning electron micrograph of an ex-
mplary device.
Couplers were constructed to couple light into nonseg-
ented waveguides directly from fiber arrays,1 and a tun-
ble laser with a range from 1450 to 1600 nm and power
f 10 mW was used to measure transmission spectra of
hese devices. Many different types of segmented wave-
uide were studied, which will be described further inection 4. For each type of waveguide, we measured the
aveguide loss of the segmented waveguide as well as the
nsertion loss induced by coupling from nonsegmented to
egmented portions of the waveguide. For each design,
00 devices were fabricated with varying numbers of
egmented-to-nonsegmented coupling interfaces and
arying overall lengths of segmented waveguides. Mea-
urement of these resulted in linear fits on the data that
an be used to extract the frequency dependence of cou-
ling and waveguide losses.
. RESULTS
ata were taken for six different segmentation designs.
hree parameters were varied: periodicity, duty cycle, and
enter size (i.e., the size of the unsegmented region of
ig. 3. (a) Dispersion diagram of both the series 0 segmented
aveguide and the normal, unsegmented waveguide. (b) Modal
atterns of the Bloch mode, with contours of uEu plotted, starting
t 10% of the maximum value and with contour increments of
0%. The first plot (a) is on a z plane that intersects the middle of
segment. (c) A plot on a horizontal plane that intersects the sili-
on layer halfway through. For clarity, four periods of the wave-
uide are shown.
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1496 J. Opt. Soc. Am. B/Vol. 22, No. 7 /July 2005 Hochberg et al.aveguide in the center). Table 1 lists the different pa-
ameters studied, as well as the measured waveguide loss
nd insertion losses for nonsegmented-to-segmented butt
oupling at 1485 nm.
A well-contained Bloch mode was found in this series of
ests, and this low-loss optical mode can propagate with
elatively low insertion losses from butt coupling from
onsegmented to segmented waveguide regions. Further-
ore, it should be noted that other designs do not appear
o support light propagation at all. These cases can be
redicted from the Bloch-mode solutions. In the case of se-
ies 3, for instance, a solution to the Bloch mode reveals a
ispersion diagram similar to that of series 0, but that
ode would not be well contained. The E field of such a
ode would stay above 60% of its maximum value at all
oundary conditions, regardless of the distance from the
aveguide core to the boundary. The insertion loss of cou-
ling from nonsegmented to segmented waveguides was
lso studied for series 0 with the finite-difference time do-
ain. The results are plotted along with the frequency-
ependent insertion losses in Fig. 5.
The discrepancy between the simulated and observed
esults is actually quite small, differing by approximately
.15 dB. This translates into approximately a 1% differ-
nce in field amplitudes, which indicates excellent agree-
ent between theory and experiment.
Table 1. Listing of Segmented Waveguide Ser
eries
Periodicity
smmd
Duty
Cycle
0.28 0.5
0.28 0.5
0.28 0.65
0.28 0.8
0.28 0.35
0.34 0.5
0.38 0.5
aDevices that did not appear to support a mode sno light transmission observedd
ig. 4. Scanning electron micrograph of the series 0 segmented
aveguide.The full frequency dependence of the waveguide loss is
hown in Fig. 6. It is important to note that it is relatively
at in frequency. The loss that is observed is comparable
ith the waveguide loss that has been measured1 in such
structure for a nonsegmented waveguide, or approxi-
ately −7 dB/cm. The additional loss can be attributed to
abrication errors within the more complex periodic seg-
ented waveguide, as well as to more substrate leakage
redicted for the specified geometry. As substrate leakage
s heavily dependent on the particular mode pattern, it is
ot surprising that there would be more such loss from a
egmented waveguide compared with a normal wave-
uide. Moreover, this is the most probable explanation for
he differences in losses between different segmented
aveguide designs.
udied and Essential Parameters at 1.48 mma
r
d
Insertion Loss
(dB)
Waveguide Loss
sdB/cmd
−0.303+ /−0.007 −25.5+ /−0.7
−0.569+ /−0.004 −40.4+ /−0.6
−1.04+ /−0.03 −39.2+ /−1.1
X X
−0.161+ /−0.002 −16.3+ /−0.4
X X
X X
s marked for their measurements.
ig. 5. (a) and (b) Measured coupling insertion loss for the seg-
ented waveguide designs studied, in decibels, plotted against
he free-space wavelength in micrometers. Series 0, in (a), shows
imulated data, which are presented for comparison.ies St
Cente
smm
0.5
0.4
0.5
0.5
0.5
0.5
0.5
have X’
5
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Hochberg et al. Vol. 22, No. 7 /July 2005 /J. Opt. Soc. Am. B 1497. CONCLUSION
e have demonstrated that some segmented waveguide
eometries can support relatively low-loss optical modes
ithin thin SOI films. Moreover, these modes are quite
asy to couple into from nonsegmented waveguides, ex-
ibiting surprisingly low loss for simple butt coupling. We
elieve that this will permit segmented waveguides to of-
er viable options for electrically contacting optical
aveguides in a single-layer structure. Also note that seg-
ented waveguides offer broader applications beyond
imply allowing electrical contact. For instance, one could
mploy segmented waveguides as low-loss frequency fil-
ers or as mode expanders by varying the periodicity of
he grates.10 Because the optical modes in these
aveguides were supported in a broadband pattern, such
configuration might be useful in isolating a signal band
ig. 6. Measured waveguide loss in decibels per centimeter as a
unction of the free-space wavelength for various segmented
aveguide designs.rom a pump wavelength in an optical system.CKNOWLEDGMENTS
he authors gratefully acknowledge the support of the
efense Advanced Research Projects Agency under the
hip Scale-Wavelength Division Multiplexing program,
ontract N00421-02-D-3223. Luxtera, Inc., is gratefully
cknowledged for its donation of computer time.
T. Baehr-Jones, the corresponding author, can be
eached by e-mail at thorolf@caltech.edu.
EFERENCES
1. T. Baehr-Jones, M. Hochberg, C. Walker, and A. Scherer,
“High Q resonators in thin silicon-on-insulator,” Appl.
Phys. Lett. 85, 3346–3347 (2004).
2. B. Maune, R. Lawson, C. Gunn, A. Scherer, and L. Dalton,
“Electrically tunable ring resonators incorporating nematic
liquid crystals as cladding layers,” Appl. Phys. Lett. 83,
4689–4691 (2003).
3. Z. Weissman and A. Hardy, “Modes of periodically
segmented waveguides,” J. Lightwave Technol. 11,
1831–1838 (1993).
4. D. Nir, Z. Weissman, S. Ruschin, and A. Hardy,
“Periodically segmented wave-guides in Ti:LiNbO3,” Opt.
Lett. 19, 1732–1734 (1994).
5. Z. Weissman, “Evanescent field sensors with periodically
segmented waveguides,” Appl. Opt. 36, 1218–1222 (1997).
6. T. Baehr-Jones, M. Hochberg, and A. Scherer, “An analytic
method for calculating substrate leakage in dielectric
waveguides,” J. Lightwave Technol., submitted for
publication.
7. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic
Crystals (Princeton U. Press, 1995).
8. A. Taflove and S. C. Hagness, Computational
Electrodynamis (Artech, 2000).
9. G. H. Golub and C. F. Van Loan, Matrix Computations
(Johns Hopkins U. Press, 1996).
0. Z. Weissman and A. Hardy, “2-D Mode tapering via tapered
channel waveguide segmentation,” Electron. Lett. 28,
1514–1516 (1992).


Segmented waveguides in thin silicon-on-insulator

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Hochberg et al. Vol. 22, No. 7 /July 2005 /J. Opt. Soc. Am. B 1493Segmented waveguides in thin silicon-on-insulator
Michael Hochberg, Tom Baehr-Jones, Chris Walker, and Jeremy Witzens
California Institute of Technology, 1200 East California Boulevard, Pasadena, California 91125
Lawrence C. Gunn
Luxtera, Inc., 1819 Aston Avenue, Carlsbad, California 92008
Axel Scherer
California Institute of Technology, 1200 East California Boulevard, Pasadena, California 91125
Received June 4, 2004; revised manuscript received August 30, 2004; accepted October 10, 2004
We have developed new silicon-on-insulator waveguide designs for simultaneously achieving both low-loss op-
tical confinement and electrical contacts, and we present a design methodology based on calculating the Bloch
modes of such segmented waveguides. With this formalism, waveguides are designed in a single thin layer of
silicon-on-insulator to achieve both optical confinement and minimal insertion loss. Waveguides were also fab-
ricated and tested, and the measured data were found to closely agree with theoretical predictions, demon-
strating input insertion loss and propagation loss better than 0.1 dB and −16 dB/cm, respectively. © 2005
Optical Society of America
OCIS codes: 220.3740, 230.7390, 130.1750.T
=
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. INTRODUCTION
ecently, low-loss single-mode waveguides in thin silicon-
n-insulator (SOI) have been demonstrated1 with
−7 dB/cm of propagation loss. There are several advan-
ages to working in such a material; in particular, the thin
eometry is helpful in obtaining high field concentrations
n the waveguide cladding, a useful attribute for the con-
truction of optical chemical sensors or silicon waveguides
lad with electro-optic polymers. Typically, the functional-
zation material in such devices resides in the waveguide
ladding, and a large modal overlap with the deposited
ladding layer is desirable. One of the outstanding prob-
ems of this geometry, however, is the difficulty of estab-
ishing an electrical contact with the waveguide without
erturbing the optical mode. This is particularly impor-
ant when a dc or rf electrical field is to be applied directly
nto the waveguiding region. Such a field can be used, for
nstance, to induce modulation through an electro-optic
ndex shift in the cladding.2 As the lateral dimensions of
uch single-mode SOI waveguides are necessarily
500 nm for a 120 nm thin silicon SOI layer to ensure
ingle-mode propagation, it is challenging to generate the
eld amplitudes desired at the surface of the waveguide
ithout electrically contacting the waveguide. Substan-
ial research has been done previously on segmented
aveguides in low-index-contrast, large-mode-volume
eometries.3–5
The SOI geometry that we use in this study was chosen
o have low waveguide loss and relatively large field con-
entrations outside the core of the waveguide. It consists
f 120 nm of silicon (index 3.5) atop a 1.4 mm layer of sili-
on dioxide (index 1.43), which rests on the silicon handle.0740-3224/05/071493-5/$15.00 © 2he structures were clad in polymethyl methacrylate sn
1.43d. The optical mode that such a waveguide supports
s primarily polarized horizontally and is essentially iso-
ated from the substrate by the oxide layer, although it
an be shown that some loss induced by tunneling leak-
ge into the substrate will occur.6 The dispersion plot of
he fundamental mode and modal concentration are de-
ailed in Fig. 1.
The definition of an electrical contact on such a wave-
uide is particularly difficult in that the waveguide is
oth electrically and optically isolated on all sides by sili-
on dioxide cladding. The introduction of an electrical con-
act leads to a significant interruption in the waveguide
ymmetry and therefore a large scattering loss. Here we
how a remedy for this problem by constructing seg-
ented waveguides as shown in Fig. 2.
In this figure the essential features of the segmented
aveguide approach are detailed. The electrical contact is
efined as a lateral grating or a planar extension that is
ithographically defined during the same lithographic
tep as the waveguide definition etch. The optical proper-
ies of this geometry are strongly dependent on the peri-
dicity and duty cycle. (We consider the duty cycle to be
he fraction of the period that contains the segment; thus
duty cycle of 0.7 on a period of 1 mm indicates segments
f silicon that are 0.7 mm long.) In theory the lateral sili-
on strips in Fig. 2 continue forever, but, as we will see,
or properly chosen periodicities these can be terminated
fter a relatively short isolation distance. If a low-loss,
ropagating optical mode exists for a particular design, it
s possible to achieve both the desired low-loss optical
uiding and lateral electrical contact to the waveguide.005 Optical Society of America
2
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1494 J. Opt. Soc. Am. B/Vol. 22, No. 7 /July 2005 Hochberg et al.. BLOCH-MODE ANALYSIS
he analysis of the waveguide layout presented in Fig. 2
an be performed with the aid of Bloch’s theorem. Our
nalysis begins with the conversion of the current-free
nd charge-free Maxwell’s equations into an eigenvalue
roblem, following7
¹ 3
1
nsrd2
¹ 3 H =
w2
c2
H. s1d
e choose the H field as the field variable, because the ei-
envalue equation is, in such a case, Hermitian and not
eneralized. We shall discuss the practical method of solv-
ng this equation for our structures, but first we will make
ome general observations about the solutions.
In what follows, we will take x and y to be the trans-
erse cross sections of the waveguide, and z is the direc-
ion of propagation. A nonsegmented waveguide, such as
hown in Fig. 1, nsrd has continuous symmetry in z. In the
ase of the segmented waveguide, nsrd has the discrete pe-
iodicity nsr+Dzd=nsrd, where Dz is a translation in the z
irection of the period of the segmentation. Utilizing
loch’s theorem in the propagation direction, we have the
esult that all the eigenvectors of Eq. (1) may be written
n the form
Cswd = fsrdexpsibzd. s2d
ere we are taking Cswd as an eigenvector corresponding
o a particular choice of w, b as the crystal lattice vector,
ig. 1. (a) Diagram of the field pattern of the fundamental op-
ical mode; contours of uEu are plotted, starting at 10% of the
aximum field value and incremented by 10% for each contour.
b) The dispersion diagram consists of the effective index plotted
gainst the free-space wavelength.nd fsrd as the local-field distribution in a unit cell (a
-vector). The propagating modes of a segmented wave-
uide will be among the solutions to Eq. (2).
To solve this problem for complex structures, it is nec-
ssary to discretize it. There are many approaches avail-
ble; we choose the finite-difference time-domain grid as
he basis of the discretization in that such an approach
mplicitly enforces the appropriate continuity and diver-
ence conditions.8 The resulting linear system results in a
arge sparse matrix equation, with approximately 0.6 mil-
ion variables for a typical calculation, with a discretiza-
ion of 0.02 mm. The lowest nonzero eigenvalues can nev-
rtheless quickly be found with the proper choice of
terative methods.9 The modes generated by our direct
olver were found to be identical to the modes generated
y a large finite-difference time-domain simulation with a
ong runway and were substantially faster to generate.
It is important to ensure that Eq. (1) remains Hermit-
an in whatever set of boundary conditions are chosen for
he unit cell. Although the z boundary condition for a
nit-cell calculation is obviously just periodic with the ap-
ropriate Bloch factor, it is less clear how the x and y
oundary conditions should be handled. We wish to deal
ith the infinite-space problem, but for numerical calcu-
ations this is impossible. Instead, it is best to enforce
ero-field boundary conditions on the edge of the unit cell
n x and y, corresponding roughly to having a perfect con-
uctor in this region. Such boundary conditions obviously
o not correspond to the problem at hand, but, for solu-
ions to Eq. (1) that reduce close to zero at these bound-
ries, the introduction of this spurious conductor should
ot disturb the eigenvalue or vector.
One can define an effective index for the Bloch modes
s b /w. With a choice of effective indices in the range of
–4 for cells of periodicity 0.3 mm, the lowest-frequency
igenvalues turn out to be the fundamental propagating
ptical mode that has frequencies in the range of 1–2 mm.
hus solving for the modes of interest simply requires ob-
aining the lowest eigenvalue of the system for varying b
alues. There is no reason this will be true all the time; in
eneral, solving Eq. (1) with functions of the form (2) will
roduce a series of frequencies, only some of which will be
n the range of physically meaningful solutions. It may be
ig. 2. Logical diagram of the planar layout of a segmented
aveguide. For comparison, a nonsegmented waveguide is also
iagrammed, in a configuration suitable for butt coupling into
he segmented waveguide.
n
u
I
B
u
t
o
r
o
3
p
i
m
i
a
M
n
m
b
w
t
c
n
t
w
i
r
s
e
t
e
t
e
f
f
3
T
p
f
w
C
s
w
E
v
a
t
d
s
c
c
F
e
m
a
o
t
g
S
w
i
s
3
s
v
s
c
p
4
D
T
c
F
w
p
a
1
a
c
g
Hochberg et al. Vol. 22, No. 7 /July 2005 /J. Opt. Soc. Am. B 1495ecessary in other circumstances to climb several values
p the eigenspectrum before useful data will be obtained.
t is possible to generate a dispersion diagram for the
loch modes of a given segmented waveguide design, by
se of the effective index defined as above. The fundamen-
al Bloch mode of a segmented waveguide with periodicity
f 0.28 mm and a duty cycle of 0.5 was solved for. We will
efer to this design as series 0, for later comparison with
ther designs. The dispersion diagram is presented in Fig.
, along with a transverse and longitudinal cross-section
lot of the E field intensity.
The dispersion plot in Fig. 3 indicates that the effective
ndex of the Bloch mode closely matches that of a nonseg-
ented waveguide, indicating that there will not be much
nsertion loss from index mismatch if we couple light from
nonsegmented waveguide to a segmented waveguide.
oreover, there is little dispersion in the region of 1550
m for this waveguide. This is not always the case, as
any segmented waveguide periodicities can exhibit
andgaps with high backreflection and prevent the for-
ard propagation of particular frequencies altogether. In
he region of such behavior, the dispersion diagram typi-
ally exhibits a derivative dw /db approaching zero. Fi-
ally, it should be noted that, for our chosen periodicity,
he field amplitudes become small at the edges of the
aveguide domain. Again, unfortunate choices of period-
city may not exhibit this behavior and will exhibit higher
adiative losses, and it is necessary to correctly design the
egmented region to ensure low losses. Because the mode
xhibits low loss in the modeled geometry, the segmenta-
ion need not be extended to infinity, and it is adequate to
nsure that it is longer than several e-folding lengths of
he field to prevent the optical mode from being influ-
nced by the end of the lateral segments. For the devices
abricated here, the segments were extended 2000 nm
rom the center of the waveguide.
. FABRICATION AND TEST
he devices were fabricated with electron-beam lithogra-
hy. Wafers of SOI with approximately 120 nm of silicon
or the top layer and 1.4 mm of oxide were cleaved and
ere spun and cleaned in acetone and isopropanol. Dow
orning’s FOX-12 electron-beam resist was spun onto the
urface at 8000 rpm and baked for 20 min at 180° C. The
afers were exposed at 2250 mC/cm2 at 100 KV in a Leica
BPG-5000+ electron-beam lithography system and de-
eloped in Clariant AZ300 metal-ion-free tetra-methyl
mmonium hydroxide developer. They were rinsed in wa-
er, dried, and then etched in an Oxford Plasmalab 100 in-
uctively coupled plasma, reactive ion etch system for 25
with chlorine flow of 80 SCCM (SCCM denotes cubic
entimeters per minute at STP), forward power of 50 W,
hamber pressure of 12 mTorr, and ICP power of 800 W.
igure 4 shows a scanning electron micrograph of an ex-
mplary device.
Couplers were constructed to couple light into nonseg-
ented waveguides directly from fiber arrays,1 and a tun-
ble laser with a range from 1450 to 1600 nm and power
f 10 mW was used to measure transmission spectra of
hese devices. Many different types of segmented wave-
uide were studied, which will be described further inection 4. For each type of waveguide, we measured the
aveguide loss of the segmented waveguide as well as the
nsertion loss induced by coupling from nonsegmented to
egmented portions of the waveguide. For each design,
00 devices were fabricated with varying numbers of
egmented-to-nonsegmented coupling interfaces and
arying overall lengths of segmented waveguides. Mea-
urement of these resulted in linear fits on the data that
an be used to extract the frequency dependence of cou-
ling and waveguide losses.
. RESULTS
ata were taken for six different segmentation designs.
hree parameters were varied: periodicity, duty cycle, and
enter size (i.e., the size of the unsegmented region of
ig. 3. (a) Dispersion diagram of both the series 0 segmented
aveguide and the normal, unsegmented waveguide. (b) Modal
atterns of the Bloch mode, with contours of uEu plotted, starting
t 10% of the maximum value and with contour increments of
0%. The first plot (a) is on a z plane that intersects the middle of
segment. (c) A plot on a horizontal plane that intersects the sili-
on layer halfway through. For clarity, four periods of the wave-
uide are shown.
w
r
a
c
t
r
n
m
t
p
r
d
m
m
b
w
p
a
m
d
r
0
e
m
s
fl
w
a
m
f
m
p
i
n
s
g
t
w
S
0
1
2
3
4
5
6
F
w
F
m
t
s
1496 J. Opt. Soc. Am. B/Vol. 22, No. 7 /July 2005 Hochberg et al.aveguide in the center). Table 1 lists the different pa-
ameters studied, as well as the measured waveguide loss
nd insertion losses for nonsegmented-to-segmented butt
oupling at 1485 nm.
A well-contained Bloch mode was found in this series of
ests, and this low-loss optical mode can propagate with
elatively low insertion losses from butt coupling from
onsegmented to segmented waveguide regions. Further-
ore, it should be noted that other designs do not appear
o support light propagation at all. These cases can be
redicted from the Bloch-mode solutions. In the case of se-
ies 3, for instance, a solution to the Bloch mode reveals a
ispersion diagram similar to that of series 0, but that
ode would not be well contained. The E field of such a
ode would stay above 60% of its maximum value at all
oundary conditions, regardless of the distance from the
aveguide core to the boundary. The insertion loss of cou-
ling from nonsegmented to segmented waveguides was
lso studied for series 0 with the finite-difference time do-
ain. The results are plotted along with the frequency-
ependent insertion losses in Fig. 5.
The discrepancy between the simulated and observed
esults is actually quite small, differing by approximately
.15 dB. This translates into approximately a 1% differ-
nce in field amplitudes, which indicates excellent agree-
ent between theory and experiment.
Table 1. Listing of Segmented Waveguide Ser
eries
Periodicity
smmd
Duty
Cycle
0.28 0.5
0.28 0.5
0.28 0.65
0.28 0.8
0.28 0.35
0.34 0.5
0.38 0.5
aDevices that did not appear to support a modesno light transmission observed
ig. 4. Scanning electron micrograph of the series 0 segmented
aveguide.The full frequency dependence of the waveguide loss is
hown in Fig. 6. It is important to note that it is relatively
at in frequency. The loss that is observed is comparable
ith the waveguide loss that has been measured1 in such
structure for a nonsegmented waveguide, or approxi-
ately −7 dB/cm. The additional loss can be attributed to
abrication errors within the more complex periodic seg-
ented waveguide, as well as to more substrate leakage
redicted for the specified geometry. As substrate leakage
s heavily dependent on the particular mode pattern, it is
ot surprising that there would be more such loss from a
egmented waveguide compared with a normal wave-
uide. Moreover, this is the most probable explanation for
he differences in losses between different segmented
aveguide designs.
udied and Essential Parameters at 1.48 mma
r
d
Insertion Loss
(dB)
Waveguide Loss
sdB/cmd
−0.303+ /−0.007 −25.5+ /−0.7
−0.569+ /−0.004 −40.4+ /−0.6
−1.04+ /−0.03 −39.2+ /−1.1
X X
−0.161+ /−0.002 −16.3+ /−0.4
X X
X X
’s marked for their measurements.
ig. 5. (a) and (b) Measured coupling insertion loss for the seg-
ented waveguide designs studied, in decibels, plotted against
he free-space wavelength in micrometers. Series 0, in (a), shows
imulated data, which are presented for comparison.ies St
Cente
smm
0.5
0.4
0.5
0.5
0.5
0.5
0.5
dhave X
5
W
g
w
e
h
b
f
w
m
s
e
t
t
w
a
f
A
T
D
C
c
a
r
R
1
F
f
w
Hochberg et al. Vol. 22, No. 7 /July 2005 /J. Opt. Soc. Am. B 1497. CONCLUSION
e have demonstrated that some segmented waveguide
eometries can support relatively low-loss optical modes
ithin thin SOI films. Moreover, these modes are quite
asy to couple into from nonsegmented waveguides, ex-
ibiting surprisingly low loss for simple butt coupling. We
elieve that this will permit segmented waveguides to of-
er viable options for electrically contacting optical
aveguides in a single-layer structure. Also note that seg-
ented waveguides offer broader applications beyond
imply allowing electrical contact. For instance, one could
mploy segmented waveguides as low-loss frequency fil-
ers or as mode expanders by varying the periodicity of
he grates.10 Because the optical modes in these
aveguides were supported in a broadband pattern, such
configuration might be useful in isolating a signal band
ig. 6. Measured waveguide loss in decibels per centimeter as a
unction of the free-space wavelength for various segmented
aveguide designs.rom a pump wavelength in an optical system.CKNOWLEDGMENTS
he authors gratefully acknowledge the support of the
efense Advanced Research Projects Agency under the
hip Scale-Wavelength Division Multiplexing program,
ontract N00421-02-D-3223. Luxtera, Inc., is gratefully
cknowledged for its donation of computer time.
T. Baehr-Jones, the corresponding author, can be
eached by e-mail at thorolf@caltech.edu.
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Segmented waveguides in thin silicon-on-insulator

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