Dispersion tailoring and soliton propagation in
Lianghong Yin, Q. Lin, and Govind P. Agrawal
Institute of Optics, University of Rochester, Rochester, New York, 14627
Received December 16, 2005; accepted January 18, 2006; posted February 8, 2006 (Doc. ID 66720)
The dispersive properties of silicon-on-insulator (SOI) waveguides are studied by using the effective-index
method. Extensive calculations indicate that an SOI waveguide can be designed to have its zero-dispersion
wavelength near 1.5 ?m with reasonable device dimensions. Numerical simulations show that soliton-like
pulse propagation is achievable in such a waveguide in the spectral region at approximately 1.55 ?m. The
concept of path-averaged solitons is used to minimize the impact of linear loss and two-photon absorption.
© 2006 Optical Society of America
OCIS codes: 250.5530, 190.5530, 230.4320, 160.4330.
Silicon-on-insulator (SOI) waveguides have attracted
considerable interest recently, as they can be used for
making inexpensive, monolithically integrated, opti-
cal devices. In particular, stimulated Raman scatter-
ing (SRS) has been used to realize optical gain in SOI
waveguides.1–3This Raman gain has been used for
fabricating active optical devices, such as optical
ultrashort pulses are used with an SOI device, one
can make use of the intensity dependence of the re-
fractive index, provided that the dispersive effects
are properly accounted for. However, the dispersive
properties of SOI waveguides have not been exten-
sively studied so far, although some initial work has
In this Letter we consider dispersion in SOI
waveguides and show that their zero-dispersion
wavelength (ZDW) ?0typically exceeds 2 ?m. We also
show that ?0can be shifted to below 1.5 ?m with rea-
sonable device parameters. Under such conditions,
an ultrashort pulse at 1.55 ?m should form a soliton
as it propagates in the waveguide. This possibility
may lead to new applications of SOI waveguides re-
lated to optical interconnects and high-speed optical
switching. We use a modified nonlinear Schrödinger
equation to study soliton evolution inside SOI
waveguides in the presence of linear loss and two-
photon absorption (TPA).
Our approach makes use of the effective-index
method9to obtain the dispersion relation ???? nu-
merically for the TM0and TE0waveguide modes,
where ? is the modal propagation constant at the fre-
quency ?. For our study, the three important param-
eters are the width W, the height H, and the etch
thickness h for the waveguide geometry shown as the
inset in Fig. 1; the dispersive properties should vary
considerably with these parameters. We first set W
=1.5 ?m, H=1.55 ?m, and h=0.7 ?m, the values
used in recent experiments.4–6The material disper-
sion of Si and SiO2is included using the Sellmeier
relations.8,9The modal refractive indices are deter-
mined from n ¯???=???? c/? and are plotted in Fig. 1
as a function of wavelength. The difference between
the two modal indices is related to the waveguide-
Dispersion to the nth order can be calculated from
???? using the relation ?n???=dn?/d?n. The wave-
length dependence of the second- and third-order dis-
persion parameters is shown in Fig. 2. The ZDW of
the TM0mode occurs near 2.1 ?m, and that of the
TE0mode near 2.3 ?m. In the wavelength region
near 1.55 ?m, ?2?0.7 ps2/m is positive (normal dis-
persion) for both modes. The third-order dispersion is
relatively small with a value of ?3?0.002 ps3/m. We
stress that our results are approximate because of
the use of the effective-index approximation.
The important question from a practical stand-
point is whether SOI waveguides can be designed to
exhibit anomalous dispersion ??2?0? near 1.55 ?m.
This is possible if ?0is reduced to below 1.55 ?m by
choosing the appropriate device parameters. We have
performed extensive numerical calculations to study
how the ZDW depends on W, H, and h and how it can
be controlled by designing the SOI waveguide suit-
ably. The results are shown in Fig. 3. Figures 3(a)
and 3(b) indicate that ?0decreases as W and H are
reduced. Figure 3(c) shows that there is an optimum
value of h for the TM0mode for minimizing ?0. The
contours of this optimum value of h are shown in Fig.
3(d) in the W–H plane. Note that ?0is almost always
and TM0(solid curve) modes for W=1.5 ?m, H=1.55 ?m,
and h=0.7 ?m. The material dispersion of silicon is shown
by a dashed curve. The inset shows the waveguide
Modal refractive indices of the TE0(dotted curve)
May 1, 2006 / Vol. 31, No. 9 / OPTICS LETTERS
0146-9592/06/091295-3/$15.00© 2006 Optical Society of America
lower for the TM0mode compared with the TE0
mode. In fact, ?0cannot be reduced to below 1.5 ?m
for the TE0mode because the required width of W
?0.3 ?m becomes impractical. In the case of the TM0
mode, ?0can be reduced to below 1.55 ?m for W in
the range of 0.5–1.5 ?m provided that H and h are
To find the range of W and H for realizing a specific
value of ?0, we depict in Fig. 4 the contours of con-
stant ?0in the W–H plane with h optimized in each
case. It follows that the dispersion can be made
anomalous at 1.55 ?m for a wide range of device pa-
rameters. As an example, ?2?0 at 1.55 ?m when W
=1 ?m, H=0.6 ?m, and h?0.3 ?m. These device pa-
rameters, although on the low side, are realistic for
One should expect an optical soliton to form inside
an SOI waveguide if ?2?0. A rough estimate of the
pulse parameters can be obtained by using the stan-
dard soliton theory.10According to this theory, a fun-
damental soliton can be excited if ?P0LD=1, where
?=2?n2/??Aeff? is the nonlinear parameter, P0is the
peak power, and LD=T0
for a pulse of width T0. The nonlinear refractive
index of silicon is n2?4.4?10−18m2/W.11
W=1 ?m, H=0.6 ?m, and h=0.3 ?m, the parameters
of the TM0 mode at 1.55 ?m are found to be
?2=−0.56 ps2/m, ?3=5.2?10−3ps3/m, ?0=1.42 ?m,
Aeff=0.38 ?m2, and ?=47 W−1/m. If we assume LD
=1 cm, then T0=75 fs, corresponding to a full width
at half maximum of 130 fs for the pulse shape gov-
erned by P?t?=P0sech2?t/T0?. The required peak
power for ?P0LD=1 is approximately 2.1 W, a rela-
tively low value for 130 fs pulses.
Before concluding that a soliton would form when
such pulses are launched into the waveguide, we
should consider the impact of linear loss, TPA, and
free-carrier absorption (FCA). SRS can be ignored for
130 fs pulses because their bandwidth ??2.4 THz? is
much less than the Raman shift of 15.6 THz for Si.
We modify the standard nonlinear Schrödinger equa-
tion to include TPA and FCA, and obtain
2/??2? is the dispersion length
?t3= i??A?2A −
where ?=0.22 dB/cm is the linear loss, and ?=?
+i?/2. The imaginary part of ? is related to the TPA
=13 W−1/m. FCA is included by ?f=?N, where ?
=1.45?10−21m2for silicon,2and N is the density of
carriers produced by TPA. It is obtained by solving2
where ??25 ns is the effective carrier lifetime. For
T0??, and at relatively low repetition rates, N can be
approximated as N?2?TPAP0
device parameters used, N?6.1?1019m−3. Since ?f
?8.8?10−4cm−1?? for this value of N, we can ig-
nore the FCA in Eq. (1).
2?. For the
(dotted curves) and TM0(solid curves) modes.
Wavelength dependence of ?2and ?3for the TE0
the TE0(dotted curves) and TM0(solid curves) modes; (d)
contours of optimum h in the range 0.3–1.1 ?m in W–H
plane for the TM0mode.
Dependence of ZDW on (a) W, (b) H, and (c) h for
function of W and H in the range of ?0=1.4–2 ?m; etch
thickness h is optimized for each set of W and H and is in
the range h/H=0.4–0.8.
Contours of constant ZDW for the TM0mode as a
OPTICS LETTERS / Vol. 31, No. 9 / May 1, 2006
We solve Eq. (1) with the split-step Fourier
method10and study soliton propagation inside a 5 cm
long SOI waveguide. The input and output pulse
shapes and the corresponding spectra are plotted in
Fig. 5 under several different conditions. The pulse
does not maintain its width because of ?3, ?, and
?TPA. The impact of ?3is found to be relatively minor.
Even without TPA, linear loss leads to pulse broad-
ening, and TPA enhances this broadening further.
However, we should stress that the pulse would
broaden by a factor of 4 in the absence of nonlinear
effects. Clearly, soliton effects help because the pulse
broadens by a factor of less than 2.
We can reduce pulse broadening even further by
using the concept of a path-averaged soliton.10In this
approach, the input peak power is increased by aver-
aging the pulse peak power over the waveguide
This amounts to enhancing the input peak power by
a factor of Fe=?tL/?1−exp?−?tL??, where ?t=?+?P¯0
is the total effective loss. For the parameters used, Fe
equals 1.58. As shown in Fig. 5, the broadening in-
LP0?z?dz, and requiring ?P¯0LD=1.
duced by linear loss and TPA can be reduced consid-
erably when the input power is increased by this fac-
tor. Moreover, the pulse spectrum becomes almost
identical to that of the input pulse.
In summary, we studied the dispersive properties
of SOI waveguides and found that the ZDW of the
TM0mode is approximately 2.1 ?m for the device
used in Ref. 4. We used numerical calculations to re-
veal the dependence of the ZDW on the three design
parameters of the device. The results show that ?0
can be reduced to below 1.5 ?m with reasonable
waveguide dimensions. Propagation of a 130 fs pulse
in the spectral region near 1.55 ?m reveals that such
a pulse can nearly maintain its shape and spectrum
over a 5 cm long waveguide because of the solitonlike
effects in the anomalous-dispersion regime.
We acknowledge Da Zhang, Fatih Yaman, and Nick
Usechak for helpful discussions. This work was sup-
ported by the National Science Foundation under
grant ECS-0320816. G. Agrawal’s e-mail address is
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(solid curves) and without (dashed curves), TPA effects
with third-order dispersion and linear loss included in both
cases; dotted curves show input profiles. The curve marked
path-averaged shows that loss-induced pulse broadening
can be reduced by increasing the input peak power
Output (a) pulse shape and (b) spectrum with
May 1, 2006 / Vol. 31, No. 9 / OPTICS LETTERS