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 = 1.55μm. We estimate the nonlinear waveguide parameter γ for optimized waveguide geometries. We find that γ can become more than three orders of magnitude larger (~7×10⁶/(Wkm)) than for state-of-the-art highly nonlinear fibers (~2×10³/(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 = 1.55μm), 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.

 

Fig. 1. Waveguide cross-sections (a) Strip waveguide. For core or cover nonlinearity, the nonlinear interaction domain D _inter is limited to the core domain (D _inter = D _core) or to the cover domain (D _inter = D _cover), respectively. (b) Slot waveguide. The nonlinear interaction domain is limited to the cover domain (D _inter = D _cover).

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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 χ̲̃⁽³⁾ by a scalar χ̃⁽³⁾ which is constant within D _inter. A simple relationship of the form γ ∝ χ̃⁽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 Eμ (x,y) and Hμ (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 μm to +1.5μm (-2μm to +2μm) in the x-direction, and from - 1μm to +2μm (-1.5μm to +2.5 μm) in the y-direction, terminated by perfectly matched layers of 0.4μm 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 ≈ 5 nm in the x-direction and Δy _store ≈ 2 nm in the y-direction. For TE polarization, the values Δx _store ≈ 2nm in x-direction and Δy _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 m3m. If Kleinman symmetry is assumed, the susceptibility tensor has two independent elements, χ̃⁽³⁾ ₁₁₁₁ = χ̃⁽³⁾ ₂₂₂₂ = χ̃⁽³⁾ ₃₃₃₃ and χ̃⁽³⁾ ₁₁₂₂ = χ̃⁽³⁾ ₁₂₁₂ = χ̃⁽³⁾ ₁₂₂₁ = χ̃⁽³⁾ ₂₂₁₁ = ⋯ = χ̃⁽³⁾ ₁₁₃₃ = ⋯ = χ̃⁽³⁾ ₂₂₃₃ = …, where the indices 1, 2 and 3 refer to the crystallographic [100], [010] and [001] directions. For an isotropic nonlinearity, χ̃⁽³⁾ ₁₁₂₂/χ̃⁽³⁾ ₁₁₁₁ = 1/3, but for silicon a larger ratio χ̃⁽³⁾ ₁₁₂₂/χ̃⁽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.

 

Fig. 2. TM-operated strip waveguide with core nonlinearity. Optimized geometrical parameters for a minimum effective area A _eff (a) Optimal strip width w and height h as a function of the refractive index n _cover of the linear cover material (b) Minimized effective area A _eff of nonlinear interaction. (c) Dominant component (E_μy) of the electric modal field for n _cover = 1.5

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Fig. 3. TE-operated strip waveguide with core nonlinearity. Optimized geometrical parameters for a minimum effective area A _eff (a) Optimal strip width w and height h as a function of the refractive index n _cover of the linear cover material (b) Optimized effective area A _eff of nonlinear interaction (c) Dominant component (E_μx) of the electric modal field for n _cover = 1.5

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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 = 0.054μm² 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 = 0.050μm². 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 0.33μm². 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.

 

Fig. 4. TM-operated strip waveguide with cover nonlinearity. Optimized geometrical parameters for a minimum effective area A _eff (a) Optimal strip width w and height h as a function of the linear refractive index n _cover of the nonlinear cover material (b) Minimized effective area A _eff of nonlinear interaction (c) Dominant component (E_μy) of the electric modal field for n _cover = 1.5

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Fig. 5. TE-operated strip waveguide with cover nonlinearity. Optimized geometrical parameters for a minimum effective area A _eff (a) Optimal strip width w and height h as a function of the linear refractive index n _cover of the nonlinear cover material (b) Minimized effective area A _eff of nonlinear interaction (c) Dominant component (E_μx) of the electric modal field for n _cover = 1.5

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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 0.24μm².

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.

 

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 ∈_μx {60nm, 80nm,…,200nm} (b) Optimal strip height h (c) Minimized effective area A _eff for nonlinear interaction (d) Dominant component (E_μx) of the electric modal field for n _cover = 1.5 and w _slot = 100nm. Click for an animation of E_μx for w _slot = 100nm and increasing n _cover (file size 700kB). [Media 1]

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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 μm² or A _eff = 0.105μm² 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 χ̃⁽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 > 2π (Δϕ_nl > π), the interaction material should satisfy FOM_TPA > 1 (FOM_TPA > 0.5) [23].

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.

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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^-18m²/W to 14.5 × 10^-18m²/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 μm²) and for TE-polarization (“TE_strip”, A _eff = 0.050 μm²) 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.

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} ≈ 7000/(Wm) for the single-crystalline organic material PTS, a number which is 1000 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.

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The first group of nonlinear cover materials comprises different glasses. Silica glass (SiO₂) 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 μm² given n _cover = 2.81, which is roughly a factor of 6 bigger than the value of A _eff = 0.104μm² 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 SiO₂. 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 SiO₂, 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 SiO₂) 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 m²/ 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.