Theory and modeling of nonlinear light propagation in subwalength structures have attracted significant recent attention, see, e.g., [1–8] and references therein, following remarkable progress with fabrication and a stimulus coming from the need of small footprint photonic devices for information processing applications. Modulational instability (MI) is a well known and important for optical demultiplexing and other applications effect, which has been extensively studied in the chip scale geometries, see, e.g., [8–10]. MI constitutes a process where a constant amplitude pump wave at frequency ω_p is able to create parametric gain for the sidebands at frequencies ω_s (signal) and ω_i = 2ω_p – ω_s (idler).

For the most typical focusing Kerr nonlinearity a well known necessary condition for the MI to exist is determined solely by the dispersion of linear waves. It is expressed as δβ = 2β_p − β_s − β_i > 0, where β_p,s,i = β(ω_p,s,i) are the propagation constants at the respective frequencies, see, e.g. [12] and references therein. If the waveguide dispersion is dominated by the group velocity dispersion (GVD), δβ ≃ −β ₂(ω_p − ω_s)², then MI requires that the GVD coefficient at the pump frequency is negative ${\beta }_2={\partial }_{\omega }^2\beta ({\omega }_p)<0$, i.e. GVD is anomalous. If |β ₂(ω_p)| is relatively small or zero, then the dispersion coefficients of other even orders determine presence or absence of MI [11, 12]. This well established picture is based on the assumption that propagation obeys the nonlinear Schrödinger (NLS) equation with a nondispersive nonlinear parameter. So far the most studied dispersive corrections to the nonlinearity have been the self-steepening term, which introduces dependence of group velocity on light intensity, and Raman nonlinearity, see, e.g., [4, 5, 15, 16]. Group velocity term, ∂_ωβ(ω_p), and hence self-steepening, along with other odd order coefficients in the expansion of β make no impact on the conditions required for MI gain [11, 12, 15]. Raman effect and delayed nonlinear responses of other physical origins [4, 15], while able to influence MI, have different (non-parametric) nature. They do not conserve total energy of the participating photons and the spectral range of the Raman gain is determined by the material properties and not by the wavenumber matching conditions. We disregard any non-parametric effects in what follows. Recently it has been demonstrated that the nonlinearity and magnetic permeability in metamaterials make a strong effective dispersion of nonlinearity, impacting MI [16].

Below we demonstrate, that the frequency dependence of the nonlinear coefficients mediating interaction of the pump, signal and idler photons can generate MI gain when the pump frequency is located in the normal GVD range, and even when the waveguide has only significant normal GVD across the entire range of frequencies. The dispersion of nonlinearity required for this effect can be achieved in subwavelength geometries where degree of light overlap with a nonlinear material strongly varies with wavelength. We also demonstrate, that a simple phenomenological model qualitatively reproducing this effect is the NLS equation with the intensity dependent GVD coefficient.

We consider a waveguide defined by the linear dielectric permittivity ε and nonlinear Kerr susceptibility χ ⁽³⁾, both are functions of transverse coordinates r⃗ _⊥ = (x,y), while the structure is homogeneous along the propagation direction z. The total field is sought as the sum of the pump, signal, and idler: ℰ⃗(r⃗,t) = ∑_{n = p,s,i} E(r⃗,ω_n)e ^{− iω_nt} + c.c.. Each component is described in terms of the slowly varying amplitude A_n(z) of the corresponding linear mode: $\mathbf{\text{E}}(\overrightarrow{r},\hspace{0.17em}{\omega }_n)=I_n^{-1/2}{A}_n(z){\mathbf{\text{e}}}_n({\overrightarrow{r}}_{\perp })e^{i{\beta }_{n}z}$, where $I_n=\int {\int }_{-\infty }^{+\infty }({\mathbf{\text{e}}}_n\times {\mathbf{\text{h}}}_n^{*}+{\mathbf{\text{e}}}_n^{*}\times {\mathbf{\text{h}}}_n){\widehat{e}}_{z}dxdy$, and ê_z is the unit vector along z. e _n(r⃗ _⊥) and h _n(r⃗ _⊥) are the electric and magnetic field profiles of the linear mode, β_n is the corresponding propagation constant. The normalization is such that the power carried by the field n is given by P_n = |A_n|².

Displacement vector 𝒟⃗(r⃗,t) = ∑_{n = p,s,i} D(r⃗_,ω_n)e ^{− iω_nt} + c.c. is assumed to have the form: D(ω_n) = ε ₀ ε(ω_n)E(ω_n) + ε ₀ χ ⁽³⁾(–ω _n;ω_k, −ω_l, ω_m)⋮E(ω_k)E ^*(ω_l)E(ω_m), where the third-order succeptibility tensor is given by ${\chi }_{iprs}^{(3)}(-{\omega }_n;\hspace{0.17em}{\omega }_k,\hspace{0.17em}-{\omega }_l,\hspace{0.17em}{\omega }_m)={\chi }_{nklm}[\rho ({\delta }_{ip}{\delta }_{rs}+{\delta }_{ir}{\delta }_{ps}+{\delta }_{is}{\delta }_{pr})/3+(1-\rho ){\delta }_{iprs}]$ [3], ρ is the nonlinear anisotropy parameter (ρ = 1 in the isotropic approximation). Using an approach identical to the one developed in [3, 8] we derive the following system of equations for the amplitudes of interacting waves:

The condition g > 0 is expressed as:

Dispersion of nonlinearity is defined by three distinct contributions. First, is the material dispersion of the χ ⁽³⁾ tensor; second, is the geometrical dispersion induced by dependencies of the modal profiles and of the overlap integrals on values of ω_p,s,i; third, is that each γ_nklm is trivially proportional to ω_n. The latter factor acting on its own makes ${\Gamma }_{-}\sim {\omega }_p-\sqrt{{\omega }_p^2-{({\omega }_p-{\omega }_s)}^2}>0$ and hence can not create MI gain with negative δβ. Thus to achieve Γ_– < 0, one has to rely on material and geometrical contributions to the dispersion of nonlinearity. The material dispersion is usually weak and also poorly characterized in terms of its variations with multiple frequencies. Contrary, geometrical dispersion is expected to be strong and also controllable with the waveguide geometry and choice of the operating wavelength. Below we illustrate possibility of MI with δβ < 0 in some ordinary waveguide geometries. Hereafter we take χ = (4/3)ε ₀ εcn ₂, where n ₂ is the constant Kerr coefficient and ε = ε(ω_p). The geometrical dispersion is accounted for by computing guided modes at the required frequencies with the help of the Comsol’s Maxwell solver.

As our first example we consider a suspended Al_0.25Ga_0.75As waveguide with the geometry and profile of one of the guided modes (dominant electric field component is oriented horizontally) shown in Fig. 1(a). The GVD of this mode is normal for λ > 1.66μm, see white area in Fig. 1(b). The full line in Fig. 2(a) shows the plot of δβ as function of the signal and idler wavelengths for λ_p = 1.7μm. One can see that δβ is negative everywhere. Hence MI can be provided only through the mechanism related to the dispersion of nonlinearity and is possible only if Γ₋ < 0. We have found that Γ₋ < 0 in the broad range of the signal and idler frequencies, see the dashed line in Fig. 2(a), while Γ₊ is always positive. Thus, by increasing the pump power P, one can always satisfy the MI condition (8). The calculated gain together with the threshold pump power P_th(λ_p) and typical MI gain profile are shown in Figs. 2(b) and 2(c), respectively. For P = 800W, the maximum gain is achieved for λ_s ≈ 1.65μm (λ_i ≈ 1.75μm), and the characteristic MI length is L_MI = 1/g ∼ 0.03mm. Assuming the excitation with picosecond pulses, T ₀ = 1ps, we estimate the dispersion length, $L_D=T_0^2/\left|{\beta }_2\right|\sim 100\text{mm}$, and the walk-off length between the signal and idler, L_w = T ₀/|1/v_gs − 1/v_gi| ∼ 1mm, v_gs,gi = 1/∂_ω β(ω_s,i) to be much longer than the MI length. This justifies our CW based approach to analyze MI, Eqs. (4)–(6). In Fig. 3 we illustrate the MI development along the waveguide length for the cases of δβ > 0 (λ_p = 1.6μm) and δβ < 0 (λ_p = 1.7μm). The results were obtained by numerical integration of Eq. (1) for the case of three waves. During evolution, the energy from the pump is almost entirely transferred to the signal and idler, and then back to the pump again, the whole process repeats periodically with propagation distance. Such recurrence is typical e.g. for MI processes in optical fibres and is known to persist even with the full account of higher order harmonics [13]. Studies of possible modifications of the recurrence process due to the dispersion of nonlinearity in subwavelength structures are beyond the scope of present work.

 

Fig. 1 Guided mode of Al_0.25Ga_0.75As waveguide suspended in air: (a) profile of the dominant electric field component (e_x) at λ_p = 1.7μm for 500nm×300nm. Waveguide is indicated by solid lines; (b) calculated GVD, D = −2πcβ ₂ /λ ², white area indicates the range of normal GVD.

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Fig. 2 MI in Al_0.25Ga_0.75As waveguide: (a) δβ and Γ₋ at λ _p = 2πc/ω_p = 1.7μm as functions of the signal/idler wavelengths. n ₂ = 1.5 · 10⁻¹⁷m²/W, D = −0.07ps/nm/mm; (b) gain as function of pump power and signal/idler wavelengths, g > 0 within shaded areas with darker colours corresponding to larger values of g. Dashed lines show the threshold power P_th = δβ/(4 Γ_–); (c) gain as function of signal/idler wavelength for P = 800W.

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Fig. 3 Dynamics of the pump and signal waves for δβ > 0 (a) and δβ < 0 (b). Initial conditions: P(z = 0) = 800W, P_s(z = 0) = 0.8mW, P_i(z = 0) = 0, λ_p = 1.6μm, λ_s = 1.55μm (a), λ_p = 1.7μm, λ_s = 1.65μm (b). Solid/dashed line corresponds to pump/signal.

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To qualitatively estimate the impact of material dispersion of χ ⁽³⁾, we also calculated P_th(λ) by using χ_nklm = (4/3)ε ₀ ε($\overline{\omega }$_nklm)cn ₂, where $\overline{\omega }$ = (ω_n +ω_k + ω_l + ω_m)/4. This gave only negligible deviations from the results shown in Fig. 2(b), thus confirming the dominant role of the geometrical dispersion of nonlinearity.

As our second example we consider a silicon-on-insulator dielectric slot waveguide with the nonlinear polymer cladding [14], see Fig. 4(a). Localization of the fundamental mode is sensitive to wavelength, and therefore the geometrical dispersion of nonlinearity is expected to be significant. GVD of a typical structure is shown in Fig. 4(b) and it is normal for any wavelength. Hence δβ < 0, see Fig. 5(a), and without the dispersion of nonlinearity, no MI is expected for any pump wavelength. However, Γ₋ has been found negative, see an example of the Γ₋ dependence on the signal and idler wavelength for λ_p = 1.7μm shown in Fig. 5(a). Therefore, the MI condition (8) is satisfied for P > P_th, see Fig. 5(b). The ratio of the characteristic lengths (see AlGaAs discussion) has also been found favorable for MI observation over typical sub-milimeter propagation distances, where walk-off due to GVD is negligible for picosecond pulses. Thus, while the frequency conversion in the optical processing experiment with a similar slot waveguide [14] has relied on the two frequency pumping (classical four wave mixing setup), our MI mechanism allows to obtain the necessary gain only with the single pump wave. Note also that two photon absorbtion and free carrier generation in AlGaAs are negligible for λ ≥ 1.5μm. They are significant in silicon, but reduced in the slot geometry with the polymer cladding [14].

 

Fig. 4 Fundamental mode of the silicon-polymer slot waveguide: a) profile of the dominant electric field component (e_x) at λ_p = 1.7μm for 500nm×220nm silicon waveguides, wall-to-wall separation 50nm, silica glass substrate and nonlinear polymer cladding. Geometry is indicated by solid lines; (b) calculated GVD.

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Fig. 5 The same as Figs. 2(a) and (b) but for the slot geometry, λ_p = 1.7μm (D = −0.0015), n _{2, Polymer} = 16.9 · 10⁻¹⁸ m ²/W, n _{2, Silicon} = 4 · 10⁻¹⁸ m²/W, n _{2, SiO2} = 2.5 · 10⁻²⁰ m ²/W.

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While the results presented above demonstrate the existence of MI induced by the dispersion of nonlinearity, they provide us with little intuition on why and where such behavior can be expected. Indeed, all the information needed to analyze MI condition in Eq. (8) is hidden inside the complex overlap integrals determining the values of γ_nklm. To formulate a more transparent qualitative approach to the problem, we introduce a generalized NLS equation where all the dispersion coefficients include nonlinear contributions $i{\partial }_{z}A=-{\sum }_{n=0}^N\frac{i^n}{n!}({\beta }_n+{\gamma }_n|A{|}^2){\partial }_t^{n}A$. Monochromatic solution of this equation is given by A = ae^{iκz − iδt}, where κ = ∑_n(β_n + γ_n|a|²)δⁿ/n! and δ = ω – ω_p. Thus the linear propagation constant is β = ∑_{n =0} β_nδⁿ/n!, while the frequency dependence of the nonlinear waveguide parameter γ is given by γ = ∑_{n =0} γ_nδⁿ/n!. Through the appropriate choice of the phase shift and of the reference velocity one can always fix β ₀ = β ₁ = 0. The minimal model capturing the nonlinearity induced MI is then

 

Fig. 6 Nonlinear parameters γ ₀ (a) and γ ₂ (b) for the waveguides as in Fig. 1 (solid lines) and Fig. 4 (dashed lines).

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In summary: Using modal expansion we have demonstrated that the confinement induced dispersion of nonlinearity creates conditions for observation of MI in subwavelength AlGaAs and silicon waveguides with large normal GVDs. We calculated the pump power thresholds required for this type of MI to be around 1kW and presented a generalization of the NLS equation accounting for this effect. To reduce the threshold power to levels more favorable for applications, further understanding of the relevant physics and design work are necessary.