# Dispersion tailoring and soliton propagation in silicon waveguides

**Abstract**

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 microm 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 microm. The concept of path-averaged solitons is used to minimize the impact of linear loss and two-photon absorption.

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.

1–3

This Raman gain has been used for

fabricating active optical devices, such as optical

modulators

4

and silicon Raman lasers.

5,6

If

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)

0

typically exceeds 2

m. We also

show that

0

can 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

method

9

to obtain the dispersion relation

共

兲 nu-

merically for the TM

0

and TE

0

waveguide 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.

4–6

The material disper-

sion of Si and SiO

2

is included using the Sellmeier

relations.

8,9

The modal refractive indices are deter-

mined from n

¯

共

兲=

共

兲 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.

Dispersion to the nth order can be calculated from

共

兲 using the relation

n

共

兲=d

n

/d

n

. The wave-

length dependence of the second- and third-order dis-

persion parameters is shown in Fig. 2. The ZDW of

the TM

0

mode occurs near 2.1

m, and that of the

TE

0

mode near 2.3

m. In the wavelength region

near 1.55

m,

2

⬇0.7 ps

2

/m is positive (normal dis-

persion) for both modes. The third-order dispersion is

relatively small with a value of

3

⬇0.002 ps

3

/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 共

2

⬍0兲 near 1.55

m.

This is possible if

0

is reduced to below 1.55

mby

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

0

decreases as W and H are

reduced. Figure 3(c) shows that there is an optimum

value of h for the TM

0

mode for minimizing

0

. The

contours of this optimum value of h are shown in Fig.

3(d) in the W–H plane. Note that

0

is almost always

Fig. 1. Modal refractive indices of the TE

0

(dotted curve)

and TM

0

(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.

May 1, 2006 / Vol. 31, No. 9 / OPTICS LETTERS 1295

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

lower for the TM

0

mode compared with the TE

0

mode. In fact,

0

cannot be reduced to below 1.5

m

for the TE

0

mode because the required width of W

⬍0.3

m becomes impractical. In the case of the TM

0

mode,

0

can 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

0

in 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.

10

According to this theory, a fun-

damental soliton can be excited if

␥

P

0

L

D

=1, where

␥

=2

n

2

/共A

eff

兲 is the nonlinear parameter, P

0

is the

peak power, and L

D

=T

0

2

/兩

2

兩 is the dispersion length

for a pulse of width T

0

. The nonlinear refractive

index of silicon is n

2

⬇4.4⫻ 10

−18

m

2

/W.

11

Using

W=1

m, H =0.6

m, and h=0.3

m, the parameters

of the TM

0

mode at 1.55

m are found to be

2

=−0.56 ps

2

/m,

3

=5.2⫻ 10

−3

ps

3

/m,

0

=1.42

m,

A

eff

=0.38

m

2

, and

␥

=47 W

−1

/m. If we assume L

D

=1 cm, then T

0

=75 fs, corresponding to a full width

at half maximum of 130 fs for the pulse shape gov-

erned by P共t兲= P

0

sech

2

共t / T

0

兲. The required peak

power for

␥

P

0

L

D

=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 共⬇2.4 THz兲 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

A

z

+

␣

2

A +

i

2

2

2

A

t

2

−

3

6

3

A

t

3

= i

兩A兩

2

A −

␣

f

2

A, 共1兲

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

−12

m/W, as ⌫=

TPA

/A

eff

=13 W

−1

/m. FCA is included by

␣

f

=

N, where

=1.45⫻ 10

−21

m

2

for silicon,

2

and N is the density of

carriers produced by TPA. It is obtained by solving

2

N

t

=

TPA

2h

0

P

2

共z兲

A

eff

2

−

N

, 共2兲

where

⬇25 ns is the effective carrier lifetime. For

T

0

Ⰶ

, and at relatively low repetition rates, N can be

approximated as N ⬇2

TPA

P

0

2

T

0

/共3h

0

A

eff

2

兲. For the

device parameters used, N⬇6.1⫻ 10

19

m

−3

. Since

␣

f

⬇8.8⫻ 10

−4

cm

−1

Ⰶ

␣

for this value of N, we can ig-

nore the FCA in Eq. (1).

Fig. 2. Wavelength dependence of

2

and

3

for the TE

0

(dotted curves) and TM

0

(solid curves) modes.

Fig. 3. Dependence of ZDW on (a) W, (b) H, and (c) h for

the TE

0

(dotted curves) and TM

0

(solid curves) modes; (d)

contours of optimum h in the range 0.3–1.1

minW –H

plane for the TM

0

mode.

Fig. 4. Contours of constant ZDW for the TM

0

mode as a

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.

1296 OPTICS LETTERS / Vol. 31, No. 9 / May 1, 2006

We solve Eq. (1) with the split-step Fourier

method

10

and 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

3

is 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.

10

In this

approach, the input peak power is increased by aver-

aging the pulse peak power over the waveguide

length, P

¯

0

=共1/L兲兰

0

L

P

0

共z兲dz, and requiring

␥

P

¯

0

L

D

=1.

This amounts to enhancing the input peak power by

a factor of F

e

=

␣

t

L/关1 − exp共−

␣

t

L兲兴, where

␣

t

=

␣

+⌫P

¯

0

is the total effective loss. For the parameters used, F

e

equals 1.58. As shown in Fig. 5, the broadening in-

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

TM

0

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

(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.

May 1, 2006 / Vol. 31, No. 9 / OPTICS LETTERS 1297

- CitationsCitations91
- ReferencesReferences14

- "1) Geometry: The dispersion of nanowires can be manipulated most directly via control of their geometrical dispersion, which is accomplished by changes in the dimensions or aspect ratio of the waveguide core [29], [31]–[33], [59]. Of course this approach also, via a change in waveguide cross-sectional area, can alter drastically the effective waveguide nonlinearity.Fig. "

[Show abstract] [Hide abstract]**ABSTRACT:**Single-mode Si-wire waveguides, fabricated in the Si-on-insulator (SOI) platform, are the basis for a growing number of potential applications in linear and nonlinear integrated optical devices and systems. This paper reviews the fundamental optical physics and behavior of these waveguides and demonstrates how their reduced transverse dimensions and index contrast lead to a series of unique and distinct modal properties. These properties include readily tunable waveguide dispersion (including dispersion flattening), large longitudinal fields, a decrease in group velocity over those of the bulk materials, and anisotropic nonlinear optical properties. In turn, new devices and device structures are achievable as a result of these features and examples of new device structures are provided.- "where E, β2, β3, and γ represent the amplitude field, group velocity dispersion parameter, third-order dispersion, and nonlinear parameter. The third-order dispersion in equation (2) is considered negligible because this dispersion only cause a minor effect to the pulse [11, 12]. The terms αlin, αTPA, αFCA, and αG are the absorptions occur within silicon; linear propagation loss, free carrier absorption (FCA), two photon absorption (TPA), and glucose absorption (GA), respectively. "

[Show abstract] [Hide abstract]**ABSTRACT:**The split-step Fourier technique is used to study the effect of temperature in triple stage microring resonating sensor (TSMRRS). The optical bright soliton beam is used as the probe pulse into the TSMRRS and the effect of temperature variations on various concentrations of glucose in deionize water is investigated. The detection of glucose is measured by intensity variations of output signals from through and drop ports of TSMRRS. The temperature variations cause an exact intensity reduction of glucose concentration in deionize water when the temperature increased by 1 °C. The performance of TSMRRS glucose sensor is improved for temperature range similar with standard room temperature which shows that TSMRRS is suitable candidate for chemical sensing application.- "μm close to the bandgap wavelength. Moreover , even if the waveguide size is increased to move the ZDW to longer wavelength, the dispersion slope near the ZDW is not small, as shown in187188189, causing a limited low-dispersion bandwidth. Recently, a dispersion engineering technique for integrated high-index-contrast waveguides has been proposed , in which an off-center nano-scale slot controls modal distribution at different wavelengths [59, 60]. "

[Show abstract] [Hide abstract]**ABSTRACT:**Group IV photonics hold great potential for nonlinear applications in the near- and mid-infrared (IR) wavelength ranges, exhibiting strong nonlinearities in bulk materials, high index contrast, CMOS compatibility, and cost-effectiveness. In this paper, we review our recent numerical work on various types of silicon and germanium waveguides for octave-spanning ultrafast nonlinear applications. We discuss the material properties of silicon, silicon nitride, silicon nano-crystals, silica, germanium, and chalcogenide glasses including arsenic sulfide and arsenic selenide to use them for waveguide core, cladding and slot layer. The waveguides are analyzed and improved for four spectrum ranges from visible, near-IR to mid-IR, with material dispersion given by Sellmeier equations and wavelength-dependent nonlinear Kerr index taken into account. Broadband dispersion engineering is emphasized as a critical approach to achieving on-chip octavespanning nonlinear functions. These include octave-wide supercontinuum generation, ultrashort pulse compression to sub-cycle level, and mode-locked Kerr frequency comb generation based on few-cycle cavity solitons, which are potentially useful for next-generation optical communications, signal processing, imaging and sensing applications.

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