## Abstract

Values up to *γ*= 7 × 10^{6}/(Wkm)
for the nonlinear parameter are feasible if silicon-on-insulator based strip and
slot waveguides are properly designed. This is more than three orders of
magnitude larger than for state-of-the-art highly nonlinear fibers, and it
enables ultrafast all-optical signal processing with nonresonant compact
devices. At *λ* = 1.55*μ*m
we provide universal design curves for strip and slot waveguides which are
covered with different linear and nonlinear materials, and we calculate the
resulting maximum *γ*.

©2007 Optical Society of America

## 1. Introduction

Silicon-on-insulator (SOI) is considered a promising material system for dense
on-chip integration of both photonic and electronic devices. Providing low
absorption at infrared telecommunication wavelengths and having a high refractive
index of *n* ≈ 3.48, silicon is well suited for building
compact linear optical devices [1–6]. To efficiently use their inherently large
optical bandwidth, it is desirable to perform all-optical signal processing and
switching on the same chip by exploiting ultrafast
*χ*
^{(3)}-nonlinearities such as
four-wave-mixing (FWM), cross- and self-phase modulation (XPM, SPM) or two-photon
absorption (TPA). Such devices show potential for ultrafast all-optical switching at
low power [7,8].

Third-order nonlinear interaction in SOI-based waveguides can be realized in two ways: First, nonlinear interaction with the silicon waveguide core itself can be used, leading to SPM/XPM overlayed by TPA [9–11]. Second, the silicon core can be embedded in nonlinear cladding material, and interaction with the evanescent part of the guided mode can be exploited. In the latter case, interaction with the nonlinear cladding material can be significantly enhanced when using slot waveguides rather than strips [12,13], whereby the fraction of optical power guided in the low-index region can be maximized by appropriate waveguide dimensions [14].

However, it is not clear from the beginning, which choice leads to more pronounced
non-linearities. The strength of third-order nonlinear interaction in a waveguide is
described by the nonlinear parameter *γ*, the real part of
which depends on the waveguide geometry as well as on the nonlinear-index
coefficient *n*
_{2} of the nonlinear interaction material. To
optimize the waveguide dimensions for maximal nonlinear interaction, a geometrical
measure is needed to rate the spatial confinement of the mode inside the nonlinear
material. For optical fibers or other low index-contrast waveguides, the light
propagates inside a quasi-homogeneous nonlinear material, and an appropriate measure
is the so-called effective core area for nonlinear interaction
*A*
_{eff} [15] which is calculated based on a scalar approximation of the
modal field. The actual cross-sectional power *P* related to the
effective core area *A*
_{eff} accounts then for the nonlinear
deviation
*n*
_{2}
*P*/*A*
_{eff}
from the linear effective refractive index of the waveguide mode.

This widely used notion of an effective area cannot be directly transferred to nonlinear high index-contrast SOI waveguides. In addition, the nonlinearity is usually limited to certain sub-domains of the modal cross section.

In this paper we therefore first extend the standard definition of
*A*
_{eff} to the case of a high index-contrast
*χ*
^{(3)}-nonlinear waveguide and calculate
its effective area *A*
_{eff}. The smaller
*A*
_{eff} becomes, the larger the nonlinear effects will
be for a given *χ*
^{(3)}. We then calculate
universal design parameters for a silicon core and for various cover materials
leading to a minimum *A*
_{eff} for strip and slot waveguides
at the telecommunication wavelength *λ* =
1.55*μ*m. We estimate the nonlinear waveguide
parameter *γ* for optimized waveguide geometries. We find
that *γ* can become more than three orders of magnitude
larger (~7×10^{6}/(Wkm)) than for state-of-the-art
highly nonlinear fibers (~2×10^{3}/(Wkm) [16]), and we infer that ultrafast all-optical switching is
feasible with non-resonant mm-scale SOI-based devices.

The paper is structured as follows: In Section 2, we define the effective area
*A*
_{eff} for nonlinear interaction in high
index-contrast waveguides; mathematical details are given in the Appendix. In
Section 3, we describe the waveguide optimization method, and in Section 4 we
present optimal parameters for different types of SOI-based waveguides. Section 5
deals with different interaction materials; we calculate
*γ* for various waveguides, and we discuss application
examples. Section 6 summarizes the work.

## 2. Effective area for third-order nonlinear interaction

Figure 1 shows cross sections of the waveguides under
consideration. The core domain *D*
_{core} consists of silicon
(*n*
_{core} ≈ 3.48 for
*λ* = 1.55*μ*m), the substrate
domain *D*
_{sub} is made out of silica
(*n*
_{sub} ~ 1-44), and the cover domain
*D*
_{cover} comprises a cladding material with refractive
index *n*
_{cover} <
*n*
_{core}. For the strip waveguide in Fig. 1(a), nonlinear interaction can either occur within the
waveguide core (“core nonlinearity”), or the evanescent part
of the guided light interacts with a nonlinear cover material (“cover
nonlinearity”). The slot waveguide depicted in Fig. 1(b) enables particularly strong nonlinear interaction
of the guided wave with the cover material inside the slot.

For maximum nonlinear interaction in strip or slot waveguides, a set of optimal
geometry parameters *w* and *h* must exist: Given a
nonlinear core and a linear cover, an increase of the waveguide cross section
decreases the intensity inside the core and thus weakens the nonlinear interaction,
while a decrease of the core size pushes the field more into the linear cover
material and again reduces nonlinear effects. If a linear core is embedded into a
nonlinear cover, a very small core produces a mode which penetrates the cover too
deeply thus reducing the optical intensity in the nonlinear material, while for a
large core only a small fraction of light will interact with the nonlinear cover.

Analytical descriptions of third-order nonlinear interaction in optical fibers are
given in textbooks [15,17]. The derivations are adapted to low index-contrast material
systems, and it is assumed that the nonlinear susceptibility is constant over the
whole cross section. These approximations are excellent for optical fibers and other
low-index-contrast systems, but they do not hold for high index-contrast (HIC)
waveguides. For example, in the analysis of low index-contrast systems, it is
usually assumed that ∇ ∙ **E** =
*ε* ∇ ∙ **D**, which
requires ∇*ε* ≈ 0 in the entire
cross section of the waveguide (see for example Eq. (2.1.18) in [15]). This approximation is not valid for HIC material systems,
and the accuracy of standard equations for fibers is questionable when applied to
SOI waveguides. We therefore derive a relation for the nonlinear waveguide parameter
*γ* which is adapted to high index-contrast
waveguides, where in addition only parts of the cross section are nonlinear. The
result is similar to the relations presented in [18]. The mathematical details of the derivation are given in
the Appendix.

In the following, the total domain *D*
_{tot} =
*D*
_{core} ∪ *D*
_{sub}
∪ *D*
_{cover} denotes the total cross section of
the waveguide. *D*
_{tot} includes a domain which is filled
with the nonlinear interaction material and which is referred to as
*D*
_{inter}. The quantity
*n*
_{inter} denotes the linear refractive index of the
nonlinear material in this interaction domain *D*
_{inter}.
For the case of core nonlinearity we have *D*
_{inter} =
*D*
_{core}, *D*
_{inter} =
*n*
_{core}, and for cover nonlinearity
*D*
_{inter} = *D*
_{cover},
*n*
_{inter} = *n*
_{cover} has to
be used, see Fig. 1. We further approximate the third-order nonlinear
susceptibility tensor
*χ*̲̃^{(3)} by a scalar
*χ*̃^{(3)} which is constant within
*D*
_{inter}. A simple relationship of the form
*γ* ∝
*χ*̃^{(3)}/(*n*
^{2}
_{inter}
*A*
_{eff})
can then be derived for the nonlinear waveguide parameter
*γ*, see Eq. (16). Denoting the electric and magnetic field vectors of
waveguide mode *μ* by
E*μ* (*x*,*y*)
and H*μ*
(*x*,*y*), respectively, the effective area
*A*
_{eff} for third-order nonlinear interaction is given
by (see Eq. (15) in the Appendix)

${Z}_{0}=\sqrt{\frac{{\mu}_{0}}{{\epsilon}_{0}}}=377\Omega $ is the free-space wave impedance, and **e**
_{z} is the unit vector pointing in positive *z*-direction. For
low-index contrast material systems with homogeneous nonlinearity, Eq. (1), (15) reduces to the usual definition of an effective area [15, Eq. (2.3.29)] as is shown in Eq. (17) of the Appendix.

The modal fields E_{μ} (*x*,*y*) and H_{μ} (*x*,*y*) are classified by the terms TE and
TM. TE refers to a waveguide mode with a dominant electric field component in
*x*-direction (parallel to the substrate plane), whereas the
dominant electric field component of a TM mode is directed parallel to the
*y*-axis (perpendicular to the substrate plane).

## 3. Waveguide optimization method

To evaluate the integrals in Eq. (1), both the electric and the magnetic fields of the
fundamental waveguide modes are calculated using a commercially available vectorial
finite-element mode solver [19]. For core (cover) nonlinearity, the computational domain
extends from -1.5 *μ*m to
+1.5*μ*m (-2*μ*m
to +2*μ*m) in the *x*-direction,
and from - 1*μ*m to
+2*μ*m (-1.5*μ*m to
+2.5 *μ*m) in the *y*-direction,
terminated by perfectly matched layers of 0.4*μ*m
thickness in all directions. To improve accuracy, second-order finite elements are
used. The size of the finite elements outside the core region is
Δ*x* ≈ Δ*y*
≈ 40 nm, whereas the silicon strips and the gaps are each divided into at
least 10 elements both in the *x*- and in the
*y*-direction. To better resolve the discontinuities of the normal
electric field components, two layers of 2 nm wide finite elements are placed on
each side of each dielectric interface. For the structures operated in TM
polarization, the fields are evaluated and stored on a rectangular grid with step
size Δ*x*
_{store} ≈ 5 nm in the
*x*-direction and Δ*y*
_{store}
≈ 2 nm in the *y*-direction. For TE polarization, the
values Δ*x*
_{store} ≈ 2nm in
*x*-direction and Δ*y*
_{store}
≈ 5nm in *y*-directionare chosen. The exact step sizes of
the grids are matched to hit the dielectric boundaries.

For optimization, the waveguide parameters *w* and *h*
are alternately scanned in a certain range. The resulting values for
*A*
_{eff} are slightly scattered due to numerical
inaccuracies. Therefore, a fourth-order polynomial is fitted to the data points, and
the local minimum of the polynomial is taken as a starting point for the next scan.
The iteration is stopped when the geometrical parameters repeatedly change by less
than 0.5 nm between subsequent iterations.

## 4. Optimal strip and slot waveguides

Third-order nonlinear interaction is maximized for five different cases: Core
nonlinearities in strip waveguides for both TE- and TM-polarization, cover
nonlinearities in strip waveguides for both polarizations, and cover nonlinearities
in TE-operated slot waveguides. For the exploitation of core (cover) nonlinearities,
different values of *n*
_{cover} ∈
{1.0,1.1,…2.5} (*n*
_{cover} ∈
{10,1.1,…3.0}) are considered.

#### 4.1. Strip waveguides and core nonlinearity

For the case of *core nonlinearity*, silicon is used as nonlinear
interaction material. Silicon is of point group
*m*3*m*. If Kleinman symmetry is assumed, the
susceptibility tensor has two independent elements,
*χ*̃^{(3)}
_{1111} =
*χ*̃^{(3)}
_{2222} =
*χ*̃^{(3)}
_{3333} and
*χ*̃^{(3)}
_{1122} =
*χ*̃^{(3)}
_{1212} =
*χ*̃^{(3)}
_{1221} =
*χ*̃^{(3)}
_{2211} =
⋯ =
*χ*̃^{(3)}
_{1133} =
⋯ =
*χ*̃^{(3)}
_{2233} =
…, where the indices 1, 2 and 3 refer to the crystallographic [100],
[010] and [001] directions. For an isotropic nonlinearity,
*χ*̃^{(3)}
_{1122}/*χ*̃^{(3)}
_{1111}
= 1/3, but for silicon a larger ratio
*χ*̃^{(3)}
_{1122}/*χ*̃^{(3)}
_{1111}
= 0.48±0.03 has been measured [20]. The assumption of an anisotropic nonlinearity is thus
not valid in the strict sense and implies that the components of the nonlinear
polarization vector that are not oriented parallel to the exciting electric
field vector are neglected. However, the error in calculating the nonlinear
waveguide parameter *γ* is negligible: The TM (TE)
mode fields have a dominant E_{μy}-component (E_{μx}-component), resulting, e.g., in an inaccurate
*x*-component (*y*-component) of the nonlinear
polarization. To calculate the overlap integral in Eq. (14) these components are weighted with the weak E_{μx}-component (E_{μy}-component) for TM (TE). The overall error is thus very small compared to
the contributions of the nonlinear polarization’s
*y*-component (*x*-component). The errorin
*γ* would increase, if the interaction between
modes of orthogonal polarizations was of interest: The nonlinear polarization
generated by a TM (TE) mode is then projected onto a TE (TM) mode field. A
small, but inaccurate *x*-component
(*y*-component) of the nonlinear polarization is thus weighted
with the dominant component E_{μx}-component (E_{μy}-component), whereas the large *y*-component
(*x*-component) of the nonlinear polarization is weighted by the
weak E_{μx}-component (E_{μy}-component). However, from a practical point of view, theses inaccuracies
are small compared to the uncertainties in measured nonlinearities of silicon, Table 1.

Figure 2 shows the results for *core
nonlinearity* in a TM-operated strip waveguide. The dominant electric
field component (E_{μy}) is discontinuous at the horizontal dielectric interfaces with a strong
field enhancement in the low-index material. Therefore the optimal cross
sectional shape of the waveguide core must be narrow and high. This is confirmed
by the results of the optimization. It can further be seen that a high index
contrast between the core and the cover material always allows for higher field
confinement and stronger nonlinear interaction within the core. Effective
nonlinear interaction areas as small as *A*
_{eff} =
0.054*μ*m^{2} can be obtained for
*n*
_{cover} = 1.0.

Figure 3 shows the results for *core
nonlinearity* in a TE-operated strip waveguide. Using analogous
arguments as for the TM case, the optimal cross section of the waveguide core
must now be wide and flat. Again, a high index contrast between the core and the
cover material always allows for higher field confinement and stronger nonlinear
interaction within the core. For low values of
*n*
_{cover}, the minimal effective area of nonlinear
interaction is slightly smaller for TE polarization than it was TM - for
*n*
_{cover} = 1.0 we now find
*A*
_{eff} =
0.050*μ*m^{2}. TE-operated strip waveguides
with silica cover (*n*
_{cover} = 1.44) and with nearly
optimal width *w* = 400nm and height *h* = 200nm
have previously been used in experiments [4,21].

#### 4.2. Strip waveguides and cover nonlinearity

The results for *cover nonlinearity* in TM-operated strip
waveguides are shown in Fig. 4. The dominant electric field component (E_{μy}) is discontinuous at horizontal dielectric interfaces with a strong field
enhancement in the nonlinear low-index material. Under these circumstances, the
optimal cross sectional shape of the waveguide is rather wide and flat except
for very low refractive indices of the cladding material. It is further found
that there is an optimal refractive index *n*
_{cover}
≈ 1.7 for which *A*
_{eff} assumes a minimal
value of 0.33*μ*m^{2}. For lower indices, too
big a fraction of the electromagnetic field has to be guided within the
waveguide core to prevent leakage into the substrate. This part of the field
does not contribute to the nonlinear interaction, which makes the effective area
bigger. For higher refractive indices, the field enhancement at the dielectric
interface decreases, which reduces the nonlinear interaction with the cover
material.

In the case of a TE-operated strip waveguide with *cover
nonlinearity*, discontinuous field enhancement can be exploited at both
sidewalls. This results in smaller effective nonlinear interaction areas as can
be seen from Fig. 5. The minimum of *A*
_{eff}
now shifts to *n*
_{cover} ≈ 1.3 and amounts to
roughly 0.24*μ*m^{2}.

#### 4.3. Slot waveguides and cover nonlinearity

For a slot waveguide, most of the light is confined to the slot area, and
reducing the slot width *w*
_{slot} increases the
intensity in the nonlinear material. Within the range of technologically
feasible slot widths, the effective nonlinear interaction area
*A*
_{eff} therefore always decreases with
*w*
_{slot} and no optimal value for
*w*
_{slot} can be found. For the design of slot
waveguides, the minimum slot width will be dictated by technological issues,
e.g. the maximum aspect ratio that the fabrication process can achieve, or the
difficulty of filling a narrow slot with nonlinear interaction material.
Therefore *w*
_{slot} ∈ {60 nm, 80 nm
… 200 nm} is fixed during the optimization procedure.

Figure 6 shows the optimal parameters as a function of
the refractive index *n*
_{cover} of the nonlinear cover
material with the slot width *w*
_{slot} as a parameter.
The width *w* of the individual strips mainly depends on
*n*
_{cover}, whereas the optimal height
*h* shows substantial variations with both
*n*
_{cover} and *w*
_{slot}.
For *w*
_{slot} ≥ 100nm, there is again an
optimal refractive index *n*
_{cover} for which
*A*
_{eff} is minimum. The existence of this minimum
can be explained physically: For larger refractive indices, the
discontinuity-induced field enhancement at the dielectric interfaces decreases.
For lower refractive indices, the increase in field enhancement is
over-compensated by the fact that a minimum fraction of the electromagnetic
field has to be guided in the high-index core material to prevent leakage into
the substrate. This fraction of the field does not contribute to the nonlinear
interaction and thus increases *A*
_{eff}. For
*w*
_{slot} < 100 nm, the guidance of the
fundamental mode is always strong enough to prevent it from leaking into the
substrate, and *A*
_{eff} decreases monotonically as
*n*
_{cover} decreases.

Similar arguments hold for explaining the behaviour of the optimal height: For
decreasing refractive indices, the height increases in the case of
*w*
_{slot} ≥ 120 nm to prevent leakage
into the substrate. For *w*
_{slot} < 120nm
this does not seem to be crucial, and the optimal height even decreases slightly
for small values of *n*
_{cover}. Using slot waveguides
with technologically feasible gap widths of 100 nm results in effective
nonlinear interaction areas as small as *A*
_{eff} = 0.086
*μ*m^{2} or
*A*
_{eff} =
0.105*μ*m^{2} for
*n*
_{cover} = 1.2 or *n*
_{cover} =
1.5, respectively.

## 5. Nonlinear parameters for different materials

The previous analysis shows that outstandingly small effective areas
*A*
_{eff} can be obtained in SOI-based waveguides, and it
can be expected that, depending on the properties of the employed materials, highly
nonlinear integrated waveguides can be realized. We will now estimate the nonlinear
parameter Re {*γ*} for different interaction materials.

Nonlinear properties of optical materials are commonly described by a nonlinear
refractive index which depends on the intensity *I* of an optical
wave, *n* = *n*
_{0} +
*n*
_{2}
*I*, and by a corresponding
intensity-dependent power absorption coefficient *α* =
*α*
_{0} +
*α*
_{2}
*I*. The nonlinear
refractive index *n*
_{2} and the TPA coefficient
*α*
_{2} are linked to the scalar third-order
nonlinear optical susceptibility
*χ*̃^{(3)} by [15, Eq. (2.3.13)]

TPA leads to a strong decay of optical power along the direction of propagation and
can therefore severely impair nonlinear parametric effects such as SPM, XPM andFWM [22]. A measure of this impairment is the TPA figure of merit
FOM_{TPA}, which is the nonlinear phase shift related to the associated
intensity change and may be expressed through the nonlinear parameter
*γ*, see Eq. (16),

An optical power *P*
_{0} launched into a waveguide of length
*L* would account for a nonlinear phase shift of
Δϕ_{nl} = Re
{*γ*}*P*
_{0}
*L*
in the absence of loss. TPA reduces the power along the propagation length, $P\left(L\right)=\frac{{P}_{0}}{\left(1+\frac{\Delta {\varphi}_{\mathrm{nl}}}{2\pi}{\mathrm{FOM}}_{\mathrm{TPA}}\right)}$, thereby reducing the nonlinear phase shift. To achieve
SPM-induced nonlinear phase-shifts Δϕ_{nl}
> 2*π* (Δϕ_{nl}
> *π*), the interaction material should satisfy
FOM_{TPA} > 1 (FOM_{TPA} > 0.5) [23].

Tables 1 and 2 list the calculated optimum nonlinear parameters
Re{*γ*} as defined in Eq. (16) for various nonlinear core and cover materials,
polarizations and structures. In both tables these calculations are based on
material data at the specified wavelengths, namely on the linear refractive index
*n*
_{0} and on the nonlinearity coefficient
*n*
_{2}. In addition, the TPA figure of merit
FOM_{TPA} is specified. All material data were taken from the references
listed in the last column of both tables. For some materials, no FOM_{TPA}
data at 1550 nm could be found. Some nonlinearity data were only available from
third-harmonic generation experiments, which is indicated in Table 2 by an asterisk^{*}) after the wavelength. In
these cases the calculated maximum nonlinear parameter Re
{*γ*} might be inaccurate, but should still reflect the
correct order of magnitude.

Table 1 refers to the case of *core
nonlinearity* with silicon as the nonlinear core material. Reported
nonlinearity coefficients *n*
_{2} for silicon range from 4.3
× 10^{-18}m^{2}/W to 14.5 ×
10^{-18}m^{2}/W. The nonlinear parameters
Re{*γ*} have been calculated for optimized strip
waveguides with air as a cover material (*n*
_{cover} = 10).
Optimal strip widths and heights for TM-polarization,
(“TM_{strip}”, *A*
_{eff} =
0.054 *μ*m^{2}) and for TE-polarization
(“TE_{strip}”, *A*
_{eff} =
0.050 *μ*m^{2}) are obtained from Figs. 3 and 2. Depending on the value of *n*
_{2},
the resulting nonlinear waveguide parameters range from 322/ (Wm) to 1180/ (Wm). TPA
figures of merit around 1 indicate that parametric effects such as SPM, XPM and FWM
will usually be impaired by TPA.

Table 2 refers to the case of *cover
nonlinearity*. The interaction material must have a linear refractive index
*n*
_{inter} = *n*
_{0} smaller than
the index of silicon and provide low linear and nonlinear absorption in the desired
wavelength range. There is a vast choice of such materials, and we have concentrated
on the most prominent ones for which reliable data on nonlinear parameters could be
obtained. These materials are subdivided into three groups: Inorganic materials
(glasses), organic materials (polymers) and nanocomposites (e.g. artificial
nanocrystals).

For each material, we have estimated the nonlinear parameter Re
{*γ*} for three different cases: A TM-operated strip
waveguide (“TM_{strip}”), a TE-operated strip
waveguide (“TE_{strip}”), and a TE-operated slot
waveguide with *w*
_{slot} = 100nm
(“TE_{slot}”). All these waveguides have geometries
optimized for the respective cover material, see Figs. 4, 5 and 6. The nonlinear parameter Re {*γ*}
denotes the contribution of the nonlinear cover material only - the contribution of
the silicon core is not taken into account, and the values for Re
{*γ*} as listed in Tab. 2 are to be understood as lower bounds for the nonlinear
parameter. While the waveguides discussed in Tab. 1 are designed with a nonlinear core material, the
structures in Tab. 2 have been optimized for cover nonlinearity; the
contribution of the silicon core is in this case significantly smaller than could be
inferred from Tab. 1.

The first group of nonlinear cover materials comprises different glasses. Silica
glass (SiO_{2}) is not a typical nonlinear material, but for comparison, we
have calculated the corresponding nonlinear parameters. We note that the resulting
values Re {*γ*} ≤ 1.0/ (Wm) are in the same
order of magnitude as the nonlinear parameters obtained for modern highly-nonlinear
fibers based on lead silicate glasses, Re {*γ*} = 1.86/
(Wm) [16]. Lead silicate glasses, bismite glasses, tellurite glasses
and chalcogenide glasses feature high linear and high nonlinear refractive indices
*n*
_{0} and *n*
_{2}. The high
linear indices considerably reduce the discontinuity-induced field enhancement at
the dielectric interfaces, so that *A*
_{eff} increases and
the nonlinear parameter decreases. For the slot waveguide, we find
*A*
_{eff} = 0.62
*μ*m^{2} given *n*
_{cover}
= 2.81, which is roughly a factor of 6 bigger than the value of
*A*
_{eff} =
0.104*μ*m^{2} for
*n*
_{cover} = 1.5. Still, the nonlinear parameters Re
{*γ*} are nearly two orders of magnitude larger than
for state-of-the-art highly nonlinear fibers [16].

The second group of nonlinear materials comprises nonlinear organic materials.
Nonlinearities in these materials can either arise from the polymer backbone, or
from chromophore units embedded in the host matrix or laterally attached to the
backbone. For the conjugated polymers PDA (polydiacetylene), PTA (polytriactelyene)
and TEE (tetraethynylethene), nonlinearities are roughly two orders of magnitude
stronger than for SiO_{2}. Please note that the nonlinear refractive indices
for PTA and TEE have been measured via third-harmonic generation (THG) at a pump
wavelength of 1900 nm, and the results cannot offhand be applied to SPM at 1550 nm.
The order of magnitude might be correct, though. The organic dye functionalized
main-chain polymer PSTF66 exhibits large nonlinear losses, whereas the side chain
polymer DANS (4-dialkyamino-4’nitro-stilbene) exhibits TPA figures of
merit that are suitable for devices based on nonlinear phase shifts. For
single-crystalline poly(p-toluene sulphonate) (PTS) polydiacetylene, nonlinear
refraction is even four orders of magnitude stronger than for SiO_{2}, and
nonlinear parameters Re {*γ*} in the order of 6950/ (Wm)
can be expected for slot waveguides without severe impairment by TPA. For strip
waveguides, Re {*γ*} reduces by roughly 50%, but is still
about 3 820/ (Wm). Using single-crystal PTS as a nonlinear interaction material
around a pre-structured silicon waveguide core might also solve the problem of poor
processability of single crystal PTS.

Lastly, we consider the case where the slot waveguide is filled with artificial
silicon nanocrystals. At *λ* = 813 nm this nanocomposite
material exhibits huge nonlinearities (about five order of magnitudes stronger than
in SiO_{2}) without impairment by TPA. It is questionable which
nonlinearities can be obtained at 1550nm, but even if only values of
*n*
_{0} = 1.50 and *n*
_{2} =
10^{-16} m^{2}/ W are assumed, as has been done by other authors [7], large nonlinear parameters Re
{*γ*} up to 4000/ (Wm) can be expected.

## 6. Discussion

For state-of-the-art highly nonlinear fibers, the highest nonlinear parameters Re
{*γ*} are in the order of 2/ (Wm) [16]. According to our estimations, a nonlinear parameter more
than three orders of magnitude larger can be expected for SOI-based strip and slot
waveguides covered with appropriate nonlinear interaction materials. Approximately
one order of magnitude is gained from the strong confinement of the electromagnetic
field. Because waveguides with cover non-linearities allow to choose from a broad
spectrum of interaction materials, the extremely nonlinear PTS-system can be chosen,
which leads to an additional improvement of approximately two orders of magnitude
compared to lead silicate glass.

Highly-nonlinear integrated strip and slot waveguides are viable for on-chip all-optical signal processing as shall be illustrated by estimating the lengths required for a passive SPM/XPM-based switch and a passive wavelength converter based on FWM.

The nonlinear phase shift Δϕ_{nl} experienced by an
optical signal through SPM or XPM in a lossless waveguide is proportional to the
optical power *P* and the interaction length *L*,
Δϕ_{nl} = Re
{*γ*}*PL* or
Δϕ_{nl} = 2Re
{*γ*}*PL*, respectively. For many
nonlinear signal processing schemes, a nonlinear phase shift of
Δϕ_{nl} = *π* is
required. If an optical peak power of *P* = 100 mW and a slot
waveguide with a nonlinear waveguide parameter of Re {*γ*}
= 6 950/ (Wm) are assumed, a nonlinear phase shift of *π*
requires a slot waveguide with a length of *L* = 4.5mm or
*L* = 2.3mm, respectively. For Re {*γ*} = 3
820/ (Wm) as calculated for a TE-operated strip waveguide, the length increases to
*L* = 8.2mm or *L* = 4.1mm, again for SPM or XPM,
respectively.

Neglecting waveguide loss and pump depletion, and assuming phase matching, the
conversion efficiency for degenerate FWM is given by
*η*
_{FWM} = (Re
{*γ*}
*P*
_{pmp}
*L*)^{2} , where
*P*
_{pmp} denotes the pump power [15]. Assuming again a slot waveguide with Re
{*γ*} = 6950/(Wm) and *P*
_{pmp}
= 100mW, a conversion efficiency of 100% can be obtained for an estimated waveguide
length of *L* = 1.4mm. For a TE-operated strip waveguide with
Re{*γ*} = 3820/(Wm), this length increases to
*L* = 2.6mm.

These results indicate that broadband, i. e., nonresonant ultrafast all-optical signal processing is feasible with compact mm-long integrated devices based on highly nonlinear slot and strip waveguides. We note that in all cases the assumed power levels are far too low to induce saturation of the nonlinear phase shift due to a Kerr-induced decrease of the discontinuity-induced field enhancement [39]. As with all nonlinear switching processes, the switching power and/or the interaction length can be considerably reduced at the expense of bandwidth by using resonant structures [7]. Compared to signal processing schemes based on active integrated devices, e.g., semiconductor optical amplifiers, passive schemes need higher power levels. However, passive Kerr-based devices are ultra-fast, do not exhibit pattern effects, and do not require active cooling.

## 7. Summary

SOI-based nonlinear strip and slot waveguides are well suited for ultrafast
all-optical signal processing if an appropriate cover material is applied. A newly
introduced effective area A eff for third-order nonlinear interaction in high
index-contrast waveguides with nonlinear constituents serves as a basis for the
optimization of different SOI-based waveguide structures with respect to a maximum
nonlinearity parameter *γ* We provide universal optimal
design parameters for strip and slot waveguides covered with different nonlinear
interaction materials, and we calculate the resulting maximum nonlinear parameter
*γ*. It is found that *γ*
can be more than three orders of magnitude larger compared with state-of-the-art
highly nonlinear fibers. Estimating the waveguide lengths for different nonlinear
signal processing schemes, we infer that non-resonant ultrafast nonlinear signal
processing is possible with mm-scale integrated SOI-based devices.

## Appendix: Third-order nonlinear interaction in high index-contrast waveguides

In this Appendix we derive the basic nonlinear propagation equation for a nonlinear high-index-contrast waveguide. We start from Maxwell’s curl equations for the electric and the magnetic field,

where **B** = *μ*
_{0}
**H** and where the electrical displacement **D** =
*ε*
_{0}
*n*
^{2}
**E**+**P**
^{(nl)}
contains the third-order nonlinear polarization
**P**
^{(nl)}. Assuming a medium response that is local in
space, **P**
^{(nl)} can be written in tensor notation,

where ⋮ denotes the tensor product; the spatial argument
**r** was omitted. The optical signal propagating in the
*μ*th mode of the waveguide is described in
slowly-varying envelope approximation (SVEA) of a carrier signal at
frequency *ω _{c}*,

Here,
*A _{μ}*(

*z*,

*t*) is the complex envelope,

*E*(

_{μ}*x*,

*y*,

*ω*) and

_{c}*H*(

_{μ}*x*,

*y*,

*ω*) denote the vectorial electric and magnetic mode profiles in a transverse plane of the waveguide,

_{c}*β*(

_{μ}*ω*) is the associated propagation constant of the carrier wave, and

_{c}*P*is used for power normalization of the numerically computed mode fields,

_{μ}In this definition,
*A _{μ}*(

*z*,

*t*) has the dimension √W, and the power of the signal averaged over some optical periods is given by |

*A*(

_{μ}*z*,

*t*)|

^{2}We further need the orthogonality of the transverse mode fields [40],

where we have omitted the arguments (*x*,*y*,
*ω _{c}*).

Three approximations are involved in the following analysis: First, we assume
that the nonlinear polarization is weak compared to the linear contribution
and can therefore be treated as a small perturbation that changes the
complex amplitude
*A _{μ}*(

*z*,

*t*) during propagation. Second, the SVEA is used, and we assume that the nonlinear response of the medium is instantaneous on the time-scale of the pulse envelope

*A*(

_{μ}*z*,

*t*), which allows us to simplify the triple convolution integral in Eq. (7) into a normal tensor product for the mode fields. Third, the dispersion relation of the waveguide is approximated by a second-order Taylor expansion about the carrier frequency

*ω*,

_{c}where ${\beta}_{\mu}^{\left(n\right)}=\frac{{d}^{n}{\beta}_{\mu}}{d{\omega}^{n}}{|}_{\omega ={\omega}_{c}}$. We note that there are no restrictions for the shape of
the mode fields, for the refractive index profile of the waveguide or for
the spatial distribution of
*χ*̲^{(3)}.

The derivation of the nonlinear propagation equation for a single
monochromatic signal involves several algebraic modifications which will be
described only briefly. We first insert the nonlinear polarization according
to Eq. (7) into the right-hand side of Eq. (5). We then use a mode expansion according to Eq. (8) (Eq. (9)) on the left-hand side of Eq. (6) (Eq. (5)) and apply the identity ∇ ×
(ΦF) = Φ(∇ ×
F)+ (∇Φ) ×
F, where Φ =
*A _{μ}*(

*z*,

*t*)e

^{j[∇ct-β(ωc)z]}represents a scalar function, and F = E

_{μ}(

*x*,

*y*,

*ω*)/P

_{c}_{μ}(F = H

_{μ}(

*x*,

*y*,

*ω*)/P

_{c}_{μ}) is a vector field. The amplitudes associated with the

*μ*th mode on the right-hand side are then projected out by taking the scalar product of both sides with H

^{*}

_{μ}(

*x*,

*y*,

*ω*)(E

_{c}^{*}

_{μ}(

*x*,

*y*,

*ω*)) followed by an integration over the entire cross section. The resulting equations are then added and Eq. (11) is applied. We finally obtain the nonlinear Schrödinger equation,

_{c}where the nonlinear parameter *γ* is given by

The spatial arguments (*x*,*y*) have been again
omitted. The quantity
*χ*̲̃^{(3)} is
the frequency-domain representation of the nonlinear susceptibility tensor.

For many cases of practical interest, only the core or the cover material
have a *χ*
^{(3)} nonlinearity, which is
usually isotropic. The third-order nonlinear susceptibility tensor
*χ*̲^{(3)} can then assumed
to be zero outside a nonlinear interaction domain
*D*
_{inter} (refractive index
*n*
_{inter}), and it is nonzero and constant
inside *D*
_{inter}. Further,
*χ*̲̃^{(3)} may
be approximated by a scalar
*χ*̃^{(3)}so that
*χ*̲̃^{(3)}. E_{μ}E_{μ}E^{*}
_{μ}= *χ*̃^{(3)}|E_{μ}|^{2}E_{μ} holds. To evaluate only the effects of the waveguide geometry, the
strength of the nonlinear interaction of the guided modes can then be
compared to a hypothetical plane wave in bulk nonlinear material with the
same nonlinear susceptibility *χ*
^{(3)}
and the same refractive index as *D*
_{inter}.

This leads to the concept of an effective nonlinear interaction area
*A*
_{eff}: In a waveguide with a nonlinear
interaction region *D*
_{inter} the cross-sectional
power *P* is transported. Relating *P* to the
effective area *A*
_{eff} leads to an effective
intensity *I* =
*P*/*A*
_{eff}. This intensity
*I* should be attributed to a plane wave which propagates
in a homogeneous medium with the same optical properties as seen in
*D*
_{inter}. For this effective area we find

The nonlinear waveguide parameter *γ* then
simplifies to the expression

For complex values of *χ*̃^{(3)}
the nonlinear parameter *γ* will be also complex,
and parametric *χ*
^{(3)}-processes (e.g.
SPM, XPM, FWM) will be impaired by nonparametric processes (e.g. TPA).

For low index-contrast material systems, the approximation
*n*
_{core} ≈
*n*
_{cover} ≈
*n*
_{inter} holds, and the longitudinal field
components become negligible. The transverse components of the mode fields E_{μ}(*x*,*y*) and H_{μ}(*x*,*y*) may then be approximated by a
scalar function *F*(*x*,*y*), E_{μ}(*x*,*y*) ≈
*F*(*x*,*y*) **e**
_{x}, ${\mathscr{H}}_{\mu}\left(x,y\right)\approx \frac{{n}_{\mathrm{inter}}}{{z}_{0}}F\left(x,y\right){\mathbf{e}}_{y}.$. If we further assume a homogeneous nonlinearity, then
*D*
_{inter} = *D*
_{tot},
and Eq. (15) can be simplified to

This relation is identical with the usual definition of an effective area
*A*
_{eff} [15, Eq. (2.3.29)].

## Acknowledgement

This work was supported by the Center for Functional Nanostructures (CFN) of the Deutsche Forschungsgemeinschaft (DFG) within projects A3.1 and A4.4, and by the European project TRIUMPH (grant IST-027638 STP). We acknowledge fruitful discussions with U. Gubler.

## References and links

**1. **B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Toen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene. “Ultra-compact Si-SiO_{2}
microring resonator optical channel dropping
filters,” IEEE Photon. Tech-nol. Lett. **10**:549–551,
1998. [CrossRef]

**2. **T. Fukazawa, F. Ohno, and T. Baba. “Very compact arrayed waveguide
grating using Si photonic wire waveguides,”
Japan. Journ. of Appl. Phys. **43**:L673–L675,
2004. [CrossRef]

**3. **W. Bogaerts, R. Baets, P. Dumon, V. Wiaux, S. Beckx, D. Taillaert, B. Luyssaert, J. Van Campenhout, P. Bienstmann, and D. Van Thourhout. “Nanophotonic waveguides in
silicon-on-insulator fabricated with CMOS
technology,” J. Lightw. Technol. **23**:401–412,
2005. [CrossRef]

**4. **T. Tsuchizawa, K. Yamada, H. Fukuda, T. Watanabe, J. Takahashi, M. Takahashi, T. Shoji, E. Tamechika, S. Itabashi, and H. Morita. “Microphotonics devices based on
silicon microfabrication technology,”
IEEE J. Sel. Topics Quantum Electron. **11**(1):232, 2005. [CrossRef]

**5. **Y. A. Vlasov, M. O’Bolye, H. F. Hamann, and S. J. McNab. “Active control of slow light on a
chip with photonic crystal waveguides,”
Nature **438**:65–69, November
2005. [CrossRef] [PubMed]

**6. **M. Lipson. “Guiding, modulating, and emitting
light on silicon — challenges and
opportunities,” J. Lightw. Tech-nol. **23**:4222–4238,
2005. [CrossRef]

**7. **C. A. Barrios. “High-performance all-optical silicon
microswitch,” Electron. Lett.40, 2004. [CrossRef]

**8. **T. Fujisawa and M. Koshiba. “All-optical logic gates based on
nonlinear slot-waveguide couplers,” J.
Opt. Soc. Am. B **23**:684–691,
2006. [CrossRef]

**9. **H. K. Tsang, C. S. Wong, T. K. Liang, I. E. Day, S. W. Roberts, A. Harpin, J. Drake, and M. Asghari. “Optical dispersion, two-photon
absorption and self-phase modulation in silicon waveguides at
1.5*μ*m
wavelength,” Appl. Phys. Lett. **80**:416–418,
2002. [CrossRef]

**10. **H. Yamada, M. Shirane, T. Chu, H. Yokoyama, S. Ishida, and Y. Arakawa. “Nonlinear-optic silicon-nanowire
waveguides,” Japan. Journ. of Appl. Phys. **44**:6541–6545,
2005. [CrossRef]

**11. **E. Dulkeith, Y. A. Vlasov, X. Chen, N. C. Panoiu, and R. M. Osgood. “Self-phase-modulation in submicron
silicon-on-insulator photonic wires,”
Opt. Express **14**:5524–5534,
2006. [CrossRef] [PubMed]

**12. **V. R. Almeida, Q. Xu, C. A. Barrios, and M. Lipson. “Guiding and confining light in void
nanostructure,” Opt. Lett. **29**:1209, 2004. [CrossRef] [PubMed]

**13. **Q. Xu, V. R. Almeida, R. R. Panepucci, and M. Lipson. “Experimental demonstration of
guiding and confining light in nanometer-size low-refractive-index
material,” Opt. Lett. **29**:1626, 2004. [CrossRef] [PubMed]

**14. **P. Müllner and R. Hainberger. “Structural optimization of
silicon-on-insulator slot waveguides,”
IEEE Photon. Technol. Lett. **18**:2557–2559,
2006. [CrossRef]

**15. **G. P. Agrawal. Nonlinear Fiber Optics. Academic Press, San
Diego, third edition, 2001.

**16. **J. Y. Y. Leong, P. Petropoulos, S. Asimakis, H. Ebendorff-Heideprim, R. C. Moore, Ken. Frampton, V. Finazzi, X. Feng, J. H. V. Price, T. M. Monro, and D. J. Richardson. “A lead silicate holey fiber with ?=
1860(Wkm)-1 at 1550nm,” In Optical Fiber
Communication (OFC) Conference Anaheim (CA), USA, March 2005.
PDP22.

**17. **Y. R. Shen. Nonlinear Optics. John Wiley and Sons, New
York, 1984.

**18. **X. Chen, N. C. Panoiu, and R. M. Osgood. “Theory of raman-mediated pulsed
amplification in silicon-wire waveguides,”
IEEE J. Quantum Electron. **42**:160–170,
2006. [CrossRef]

**19. ** RSoft Design Group, Inc.,
http://www.rsoftdesign.com. FemSIM 2.0 User Guide
2005.

**20. **J. J. Wynne. “Optical third-order mixing in GaAs,
Ge, Si and InAs,” Phys. Rev. **178**:1295–1301,
February 1969. [CrossRef]

**21. **H. Fukuda, K. Yamada, T. Shoji, M. Takahashi, T. Tsuchizawa, T. Watanabe, J. Takahasi, and S. Itabashi. “Four-wave mixing in silicon wire
waveguides,” Opt. Express **13**:4629–4637,
2005. [CrossRef] [PubMed]

**22. **V. Mizrahi, K. W. DeLong, G. I. Stegeman, M. A. Saifi, and M. J. Andrejco. “Two-photon absorption as a
limitation to all-optical switching,”
Opt. Lett. **14**:1140–1142, December
1989. [CrossRef] [PubMed]

**23. **U. Gubler and C. Bosshard. Molecular design for third-order nonlinear optics.
Advances in Polymer Science158:123–191,
2002. [CrossRef]

**24. **M. Dinu, F. Quochi, and H. Garcia. “Third-order nonlinearities in
silicon at telecom wavelengths,” Appl.
Phys. Lett. **82**:2954–2956,
2003. [CrossRef]

**25. **R. S. Friberg and P. W. Smith. “Nonlinear optical glasses for
ultrafast optical switches,” IEEE J.
Quantum Electron. **23**:2089, 1987. [CrossRef]

**26. **K. Kikuchi, K. Taira, and N. Sugimoto. “Highly nonlinear bismuth oxide-based
glass fibers for all-optical signal processing,”
Electron. Lett. **38**:166, 2002. [CrossRef]

**27. **H. Nasu, O. Matsushita, K. Kamiya, H. Kobayashi, and K. Kubodera. “Third harmonic generation from
Li_{2}O- TiO_{2}-TeO_{2}
glasses,” J. Non-Cryst. Solids. **124**:275–277,
1990. [CrossRef]

**28. **T. Cardinal, K. A. Richardson, H. Shim, A. Schulte, R. Beatty, K. Le Foulgoc, C. Meneghini, J. F. Viens, and A. Villeneuve. “Non-linear optical properties of
chalcogenide glasses in the system As-S-Se,”
J. Non-Cryst. Solids. **256&257**:353–360,
1999. [CrossRef]

**29. **J. M. Harbold, F. Ö. Ilday, F. W. Wise, J. S. Sanghera, V. Q. Nguyen, L. B. Shaw, and I. D. Aggarwal. “Higly nonlinear As-S-Se glasses for
all-optical switching,” Opt. Lett. **27**:119–121,
2002. [CrossRef]

**30. **P. D. Townsend, G. L. Baker, N. E. Schlotter, C. F. Klausner, and S. Eternad. “Waveguiding in spun films of soluble
polydiacetylenes,” Appl. Phys. Lett. **53**:1782–1784,
1988. [CrossRef]

**31. **K. Rochford, R. Zanoni, G. I. Stegeman, W. Krug, E. Miao, and M. W. Beranek. “Measurement of nonlinear refractive
index and transmission in polydiacetlyene waveguides at
1.319*μ*m,”
Appl. Phys. Lett. **58**:13–15,
1991. [CrossRef]

**32. **U. Gubler. Third-order nonlinear optical effects in organic materials. PhD
thesis, Swiss Federal Institute of Technology Zürich,
2000.

**33. **T. Kaino. “Waveguide fabrication using organic
nonlinear optical materials,” J. Opt. A:
Pure Appl. Opt. **2**:R1–R7,
2000. [CrossRef]

**34. **M. Asobe, I. Yokohama, T. Kaino, S. Tomaru, and T. Kurihara.Nonlinear absorption and refraction in an organic dye
functionalized main chain polymer waveguide in the
1.5*μ*m wavelength region. Appl. Phys. Lett. **67**:891–893,
1995. [CrossRef]

**35. **D. Y. Kim, M. Sundheimer, A. Otomo, G. Stegeman, W. H. G. Horsthuis, and G. R. Möhlmann. “Third order nonlinearity of
4-dialkyamino-4’nitro-stilbene waveguides at
1319nm,” Appl. Phys. Lett. **63**:290–292,
1993. [CrossRef]

**36. **A. K. Bhowmik and M. Thakur. “Self-phase modulation in
polydiacetylene single crystal measured at
720-1064nm,” Opt. Lett. **26**:902–904,
2000. [CrossRef]

**37. **B. L. Lawrence, M. Cha, J. U. Kang, W. Toruellas, G. Stegemann, G. Baker, J. Meth, and S. Etemad. “Large purely refractive nonlinear
index of single-crystal P-toluene sulphonate (PTS) at
1600nm,” Electron. Lett. **30**:447–448,
1994. [CrossRef]

**38. **G. Vijaya Prakash, M. Cazzanelli, Z. Gaburro, L. Pavesi, F. Iacona, G. Franzo, and F. Priolo. “Linear and nonlinear optical
properties of plasma-enhanced chemical-vapour deposition grown silicon
nanocrystals,” J. Mod. Opt. **49**:719–730,
2002. [CrossRef]

**39. **T. Fujisawa and M. Koshiba. “Guided modes of nonlinear slot
waveguides,” IEEE Photon. Technol. Lett. **18**:1530–1532,
2006. [CrossRef]

**40. **D. Marcuse. Light Transmission Optics. Van Nostrand
Reinhold, New York, 1972.