Nonlinear silicon-on-insulator waveguides for all-optical signal processing

Values up to γ = 7× 106/(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 μm 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 OCIS codes: (130.4310) Nonlinear; (130.2790) Guided Waves; (160.4330) Nonlinear optical materials; (190.3270) Kerr effect; (190.4710) Optical nonlinearities in organic materials; (190.5970) Semiconductor nonlinear optics. 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. 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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 telecom-munication wavelengths and having a high refractive index of n ≈ 3.48, silicon is well suited for building compact linear optical devices [1][2][3][4][5][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), crossand 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][10][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 nonlinearities. 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 subdomains 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 μm. 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-ofthe-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. 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 μm), 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.

Effective area for third-order nonlinear interaction
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 indexcontrast 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 , n 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) (1) Z 0 = μ 0 /ε 0 = 377 Ω 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).

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 μm to +1.5 μm (−2 μm to +2 μm) in the x-direction, and from −1 μm to +2 μm (−1.5 μm to +2.5 μm) in the y-direction, terminated by perfectly matched layers of 0.4 μm thickness in all directions. To improve accuracy, second-order finite elements are used. The size of the finite elements outside the core region is Δx ≈ Δy ≈ 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 Δx store ≈ 5nm in the x-direction and Δy store ≈ 2nm in the y-direction. For TE polarization, the values Δx store ≈ 2nm in x-direction and Δy store ≈ 5nm in y-direction are 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.

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 ∈ {1.0, 1.1,...3.0}) are considered.

Strip waveguides and core nonlinearity
For the case of core nonlinearity, silicon is used as nonlinear interaction material. Silicon is of point group m3m. If Kleinman symmetry is assumed, the susceptibility tensor has two independent elements, χ  Table 1. Figure 2 shows the results for core nonlinearity in a TM-operated strip waveguide. The dominant electric field component (E μ y ) 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 μm 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 μm 2 . TEoperated strip waveguides with silica cover (n cover = 1.44) and with nearly optimal width w = 400 nm and height h = 200 nm have previously been used in experiments [4,21].

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 μ y ) 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 μm 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 μm 2 .

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 inter- 1 -0 1 Fig. 6. TE-operated slot waveguide with cover nonlinearity. Optimized geometrical parameters for a minimum effective area A eff (a) Optimal strip width w as a function of the linear refractive index n cover of the nonlinear cover material for various slot widths w slot ∈ {60 nm, 80 nm,...,200 nm} (b) Optimal strip height h (c) Minimized effective area A eff for nonlinear interaction (d) Dominant component (E μ x ) of the electric modal field for n cover = 1.5 and w slot = 100 nm. Click for an animation of E μ x for w slot = 100 nm and increasing n cover (file size 700kB).  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 ≥ 100 nm, 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 < 120 nm 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 μm 2 or A eff = 0.105 μm 2 for n cover = 1.2 or n cover = 1.5, respectively.

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 (2) 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 and FWM [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(L) = P 0 /(1 + Δφ nl 2π FOM TPA ), thereby reducing the nonlinear phase shift. To achieve  Table 1. Core nonlinearity. Calculated maximum nonlinearity parameters Re{γ} ∝ 1/A eff for optimized strip waveguides with a nonlinear silicon core and a linear air cladding n cover = 1, operated in TM or TE polarization. The calculation is based on data for silicon at the specified wavelengths: Linear refractive index n 0 , nonlinearity coefficient n 2 and TPA figure of merit FOM TPA were taken from the references listed in the last column. -The resulting nonlinear parameters Re{γ} ≈ 400/(Wm) are remarkably large. However, the material suffers from non-negligible two-photon absorption leading to a figure of merit FOM TPA ≈ 0.3 ... 0.9.
SPM-induced nonlinear phase-shifts Δφ nl > 2π (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 1 550 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 = 1.0). Optimal strip widths and heights for TM-polarization, ("TM strip ", A eff = 0.054 μm 2 ) and for TE-polarization ("TE strip ", A eff = 0.050 μm 2 ) are obtained from Figs. 3 and 2. Depending on the value of n 2 , the resulting nonlinear waveguide parameters range from 322/ (W m) to 1180/ (W m). 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 = 100 nm ("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 [7] ) Third-order nonlinearity obtained by third-harmonic generation Table 2. Cover nonlinearity. Calculated maximum nonlinearity parameters Re{γ} ∝ 1/A eff for optimized strip and slot waveguides with a linear silicon core and various nonlinear cover materials, operated in TM or TE polarization. The calculation is based on cover material data at the specified wavelengths: Linear refractive index n 0 , nonlinearity coefficient n 2 and TPA figure of merit FOM TPA were taken from the references listed in the last column. Three material groups are considered: Inorganic materials like glasses, organic substances, and nanocomposites. -Most remarkable are the large nonlinear parameters Re{γ} ≈ (70 ... 150)/(Wm) and Re{γ} ≈ 300/(Wm) for chalcogenide glasses and for the side-chain polymer DANS, respectively, and the record value of Re{γ} ≈ 7 000/(Wm) for the single-crystalline organic material PTS, a number which is 1 000 times larger than for a higly nonlinear bismite glass. These material groups have also very good TPA figures of merit in the order of FOM TPA ≈ 4 ...27. 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/ (W m) 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/ (W m) [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 μm 2 given n cover = 2.81, which is roughly a factor of 6 bigger than the value of A eff = 0.104 μm 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 1 900 nm, and the results cannot offhand be applied to SPM at 1 550 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 (4dialkyamino-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/ (W m) 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/ (W m). Using single-crystal PTS as a nonlinear interaction material around a prestructured 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 1 550 nm, 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 4 000/ (Wm) can be expected.

Discussion
For state-of-the-art highly nonlinear fibers, the highest nonlinear parameters Re {γ} are in the order of 2/ (W m) [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 nonlinearities 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/XPMbased 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 = 4.1 mm, 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} = 6 950/ (W m) and P pmp = 100 mW, a conversion efficiency of 100 % can be obtained for an estimated waveguide length of L = 1.4 mm. For a TE-operated strip waveguide with Re {γ} = 3 820/ (W m), 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.

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 nonresonant ultrafast nonlinear signal processing is possible with mm-scale integrated SOI-based devices.

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.

Appendix: Third-order nonlinear interaction in high index-contrast waveguides
In this Appendix we derive the basic nonlinear propagation equation for a nonlinear high-indexcontrast 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 thirdorder 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 μth 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 c ) and H μ (x, y, ω c ) denote the vectorial electric and magnetic mode profiles in a transverse plane of the waveguide, β μ (ω c ) is the associated propagation constant of the carrier wave, and P μ is used for power normalization of the numerically computed mode fields, 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 β (n) μ = d n β μ d ω n ω=ω 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[ω c t−β (ω c )z] represents a scalar function, and F = E μ (x, y, ω c )/ P μ (F = H μ (x, y, ω c )/ P μ ) is a vector field. The amplitudes associated with the μth mode on the right-hand side are then projected out by taking the scalar product of both sides with H μ (x, y, ω c ) (E μ (x, y, ω c )) 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, 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 γ = 3ω c ε 0 Z 2 0 4A eff n 2 inter χ (3) .
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 , H μ (x, y) ≈ n inter Z 0 F(x, y) 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)].