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Dispersion tailoring and soliton propagation in

silicon waveguides

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

modulators4

and silicon

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

been done.7,8

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-

induced birefringence.

Ramanlasers.5,6

If

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

Fig. 1.

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

geometry.

Modal refractive indices of the TE0(dotted curve)

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0146-9592/06/091295-3/$15.00© 2006 Optical Society of America

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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

chosen properly.

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

SOI waveguides.

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

Using

?A

?z+

?

2A +

i?2

2

?2A

?t2−

?3

6

?3A

?t3= i??A?2A −

?f

2A,

?1?

where ?=0.22 dB/cm is the linear loss, and ?=?

+i?/2. The imaginary part of ? is related to the TPA

coefficient,

?TPA=5?10−12m/W,

=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

as

?=?TPA/Aeff

?N

?t

=

?TPA

2h?0

P2?z?

Aeff

2

−

N

?,

?2?

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).

2T0/?3h?0Aeff

2?. For the

Fig. 2.

(dotted curves) and TM0(solid curves) modes.

Wavelength dependence of ?2and ?3for the TE0

Fig. 3.

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

Fig. 4.

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

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OPTICS LETTERS / Vol. 31, No. 9 / May 1, 2006

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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

length, P¯0=?1/L??0

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

gpa@optics.rochester.edu.

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Fig. 5.

(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

suitably.

Output (a) pulse shape and (b) spectrum with

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