1. Introduction

All-optical switching is the phenomenon by which the reflection or transmission of an optical system changes very rapidly upon a small change of input intensity [1]. Since the fast nonlinear effects which are of interest here tend to be weak, leading to high switching intensities, all-optical switching experiments are invariably carried out in guided-wave geometries where the light is transversely confined, enhancing the field strength [2]. This field confinement is particularly strong in high-index semiconductor nanowires, since their cross section is of the order of the square of the wavelength inside the material [3]. Confinement in all three dimensions, such as in cavities (Fabry-Perot, photonic crystal cavities, ring resonators) further enhances the field strength. The enhancement increases with the cavity quality factor [4]. High field intensities inside a cavity enhance the response to input power change [5], however when using cavity resonances to enlarge the optical field, bistability occurs which results in switching hysteresis [6]. It has been argued by Goddard et. al. [7] that the bistable regions are undesirable for all-optical switching and all optical logic, since hysteresis loops in bistable devices can cause ambiguity and increase bit error rates.

Optical fields can also be enhanced using slow light, such as found near a photonic band edge. All-optical switching with slow light, particularly in one-dimensional gratings, has been studied for many years, both theoretically [8, 9] and experimentally [10]. The propagation of high-intensity light near a photonic band edge leads to a nonlinear shift of the band structure so that, at a fixed frequency, previously forbidden states become allowed states, or vice versa, leading to switching [11]. However, such switching is accompanied by bistable regions [12].

Near a regular photonic band edge (RBE) where the frequency ω and the wavenumber k are related by Δω ∝ Δk² [see Fig. 1(a), red], the group velocity is v_g = ∂ω/∂k ∝ Δk, and the local field intensity for a given energy flux scales as $v_g^{-1}$. Despite the ubiquitous nature of slow light near a RBE, in practice no energy buildup occurs. Because of the impedance mismatch between the incoming light and the slow light, the energy coupling at RBE decreases proportionally to v_g, which prevents any field enhancement [13]. Different methods have been developed to overcome the poor coupling at moderately low group velocities around c/40, where c is the speed of light in vacuum. These include apodization [10] and inverse tapers [14]. However these approaches are not scalable for smaller group velocities, since they rely on an intermediate structure with a length which increases proportionally to the slow-down factor.

 

Fig. 1 (a) Dispersion of propagating modes at a RBE (red) and DBE (blue). (c–d) Complex band structure of (c) RBE ( Media 1) and (d) DBE ( Media 2). Blue and red curves correspond to the propagating modes with real wavenumbers as shown in (a), black curves represent evanescent modes with complex wavenumbers. (b) Two-mode waveguide where DBE can be realized.

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Unlike the slow light at RBEs, near degenerate band edges (DBEs) for which Δω ∝ Δk⁴ [see Fig. 1(a), blue], light couples efficiently to low and even zero group velocity modes corresponding to the “Frozen Light” regime [15, 13]. We have previously shown that this favorable coupling is due to the presence of evanescent modes with complex wavenumbers [16, 17]. The complex dispersion relation of a DBE, showing frequency versus the real and imaginary parts of the wavenumber, is shown in Fig. 1(d). Close to a DBE, the dispersion of evanescent and slow propagating modes is found as $\mathrm{\Delta }{k}_j=\sqrt[4]{\xi \mathrm{\Delta }\omega }\text{exp}\left(i\pi j/2\right)$ [17]. Here Δk and Δω are the wavenumber and frequency detuning from the DBE, respectively, j = 0, 2 for the propagating mode and j = 1, 3 for the evanescent modes, and ξ is a real constant characteristic of the DBE dispersion ( Media 2). In Fig. 1(d), the black curves represent the evanescent waves with complex k, and blue — propagating modes with real k. The existence of evanescent modes highlights the difference between DBEs and RBEs: in RBEs [Fig. 1(c) ( Media 1)] evanescent modes exist only inside the photonic band gap, whereas at DBEs propagating and evanescent modes coexist in the transmission band.

In this work, we suggest and demonstrate through numerical modeling that the presence of the evanescent modes near a DBE can strongly enhance nonlinear effects and accordingly lower the threshold for all-optical switching. In Section 2 we formulate the coupled-mode theory used to analyze waveguides with a DBE. In Sec. 3 we analyze power-dependent transmission and reveal that bistability and hysteresis can be suppressed by choosing the excitation conditions. We present conclusions in Sec. 4.

2. Coupled mode description

Degenerate Band Edges can be created in a variety of periodic guided-wave structures [17], such as fiber Bragg gratings [18], nanowires [19] and photonic crystal waveguides [20]. The solutions for light propagation in such structures can be found using full 3D vectorial solvers such as finite difference time domain (FDTD) and transfer matrix methods, however these are computationally intensive and can be ill-suited for nonlinear problems. Perturbative approaches such as coupled mode theory (CMT) and the nonlinear Schrödinger equation (NLSE) are better suited as they are efficient and can provide a physical insight. Here we use CMT, which can account for both the propagating and evanescent modes.

In order to model a periodic waveguide with a DBE, we start with a longitudinally uniform waveguide with two types of modes: the fundamental and a higher-order mode with propagation constants K̃₁ and K̃₂, respectively, both at a given frequency Ω̃. The total field is thus characterized by four amplitudes $E_{1,2}^{\pm }$, where the signs correspond to forward (+) and backward (−) propagation. We take the modulus of group velocity v and the refractive index n of these modes to be the same. In the presence of the periodic modulation (“grating”), the modes can be coupled and the amplitudes become envelopes $E_{1,2}^{\pm }\left(z,t\right)$ that vary in space and time. Two gratings are sufficient to realize a DBE: one grating, with coupling coefficient ρ₁ and period 2π/(2K̃₁), couples Mode 1 to its counter-propagating counterpart. The other grating, with the coupling coefficient ρ₂ and period 2π/( K̃₁ + K̃₂ +δ), couples Mode 1 to the counter propagating Mode 2 [17]. The grating coupling and Kerr nonlinear interactions between the modes can be described by the coupled-mode equations [18],

At low intensities the nonlinear terms in Eqs. (1) can be ignored. We can then find the linear dispersion relation by substituting plane-wave solution of the form $E_{1,2}^{\pm }\left(z,t\right)=C_{1,2}^{\pm }{e}^{i\left(kz-\omega t\right)}$, where ω and k are the frequency and wavenumber detuning from Ω̃ and the associated wavenumbers K̃_1,2, respectively. We choose ω to be real and find the associated k as eigenvalues which are either real, or come in complex conjugate pairs. We then adjust the linear parameters ρ_1,2 and δ so that the dispersion relation has a DBE at its upper band edge. We find that for most ρ’s it is sufficient to adjust only δ in order to eliminate the quadratic dependence of the band edge and turn it into a quartic DBE. In this paper we take these parameters to have the values δ = −1.7166, ρ₁ = 1 and ρ₂ = 0.5, and we adjust the length scale such that v = 1. The eigenvectors, the grating modes, are superpositions of the uniform waveguide modes, and can be either propagating (k real) or evanescent (k complex). The complex dispersion of propagating and evanescent modes for these parameters is shown in Fig. 1(d).

3. Results

We now consider a DBE waveguide of a finite length of ρ₁L = 12, in normalized units [9]. This normalized length corresponds approximately to 8 cm in a silica fiber grating and 20 μm in a perforated silicon nanowire, where the refractive index modulation depths are in order of 10⁻⁴ and 1, respectively. We note that the fundamental fiber mode has a nonlinearity which is much larger than that of the other modes, i.e., Γ₁₁ ≫ Γ₁₂ ≫ Γ₂₂. Therefore, in our analysis below we only include Γ₁₁ ≡ Γ, and put Γ₁₂ = Γ₂₂ = 0. We have ascertained that this does not qualitatively affect our results, which can therefore predict the general switching features in nanowire waveguides, for which Γ₁₁ also dominates as well.

We first consider the low-intensity limit and solve for CW light tuned to the band edge frequency ω_D, incident from a uniform section of the waveguide. Since the waveguide has two different modes, the complex amplitudes i_1,2 of which can be independently adjusted, the input can be parameterized, by two real parameters α(0 < α < π/2) and β(−π < β < π), where i₁ = cos(α)exp(iβ), and i₂ = sin(α). Figure 2(a) shows the reflection from the structure versus α and β. In Fig. 2(b) we plot the field inside the waveguide at the two points marked A and B in Fig. 2(a), where, respectively, the reflection is lowest and highest.

 

Fig. 2 (a) Reflection versus the input parameters (α, β) at the DBE frequency for a wave-guide of length ρ₁L = 12. (b) Field intensity inside the waveguide for the condition of minimum (blue) and maximum (red) reflection, corresponding to points A and B in (a), respectively. Dashed black line shows asymptotic for the blue line at small z. (c) Field enhancement inside the waveguide versus waveguide length (loglog scale) at input state A, showing that the enhancement is ∝ L².

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For the band edge frequency at the condition B, almost all the light is reflected back and the field decays into the grating, similar to the behavior of a RBE. In contrast, at condition A we get frozen light: the reflection is only slightly less, but the energy builds up inside the waveguide. The total field initially grows as z² before it decays, where z is the distance from the interface. This quadratic dependence is shown with a black dashed line in Fig. 2(b). For an infinitely long frozen light waveguide, at the DBE frequency all the light is reflected back, but remarkably the field diverges as z². This was originally predicted by Figotin and Vitebskiy [13] for a different geometry, and we confirm that it occurs at a DBE in the framework of the model in Sec. 2. Indeed, Fig. 2(c) shows that at the condition A the maximum field is proportional to (ρ₁L)². The reason for the divergence of the field is due to the degeneracy between the propagating and the evanescent grating modes at the DBE frequency. To satisfy the boundary condition, they must span the input profile, V_i = [i₁, i₂]^T = span(V_pr, V_ev). An input profile which is orthogonal to V_i, which corresponds to condition A in Fig. 2(a), requires V_pr and V_ev to have large amplitudes. At the condition B, V_i is parallel to V_er and V_pr and each has a small amplitude.

We now consider the high-intensity regime with positive nonlinearity. This has the effect of increasing the refractive index with intensity, thus shifting the band structure to lower frequencies, leading to an increase in transmission for frequencies near the top of the band edge. To find the nonlinear stationary solutions, we solve Eqs (1) by backward integration of a given output field, $E\left(0\right)={\left[\sqrt{S_t}\left(t_1,t_2\right),\left(0,0\right)\right]}^T$, where S_t is the total transmitted flux and $\sqrt{{\left|t_1\right|}^2+{\left|t_2\right|}^2}=1$. After integration, we find $\mathbf{E}\left(-L\right)={\left[\sqrt{S_i}\left(i_1,i_2\right),\sqrt{S_r}\left(r_1,r_2\right)\right]}^T$, where S_i and S_r are the incident and reflected flux respectively and $\sqrt{{\left|i_1,r_1\right|}^2+{\left|i_1,r_2\right|}^2}=1$. Because in experiment the inputs (i₁ and i₂) are directly controlled, the output (t₁ and t₂) were searched numerically in order to match the inputs after integration. In Fig. 3(a) we show the total transmission versus the input flux (S_i) at the band edge frequency for the monomode cases: α = β = 0 (blue) and α = π/2, β = 0 (red), corresponding to the conditions C and D [Fig. 2(a)] respectively. We have determined the stability of stationary solutions with respect to small perturbations, and show the linearly stable branches with the continuous curves and unstable states with plus signs.

 

Fig. 3 (a–b) Transmission vs. the input flux at the band edge frequency: (a) the DBE with input amplitudes i₁ = 1 (red) and i₂ = 1 (blue), corresponding to points C and D in Fig. 2(a), respectively; (b) the DBE with input amplitudes corresponding to point A in Fig. 2(a) (blue) and a RBE (black). Solid lines indicate stable while and plus (+) signs — unstable stationary solutions. (c) Threshold power for switching (100% transmission) versus the normalized waveguide length. (d) Field profile inside the waveguide at the nonlinear resonance.

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Figure 3(b) compares the transmission versus the input flux between DBE at the condition A and at a RBE with the same length, ρ₁L = 12. We see that in both cases the transmission is a strongly nonlinear function of the input power, which can be used for optical switching. The key difference is that the transmission curve for a RBE is bistable, whereas for a DBE with the input condition A there is no bistability. Another key difference between a DBE and RBE is that while for a RBE the threshold slightly increases with length, for an DBE it decreases as L⁻² (Fig. 3(c)). This happens due to the strong field enhancement inside the waveguide with DBE at both low and high intensities, see Figs. 2(b) and 3(d) respectively. We also point out that unstable solutions [(+)’s in Figs. 3(a–b)] give rise to self-pulsations associated with gap-soliton excitations at much lower powers in DBE than in RBE.

Finally, we compare the switching threshold between the DBE and RBE in a fiber grating made of silica. For a RBE there is only a single mode and one grating, and for comparison with DBE we take the grating coefficient as ρ = ρ₁ = 1. In an additional scale the Fig. 3(b) we give the input flux rescaled using the effective cross section of 20 μm² and n⁽²⁾ = 2.86 × 10⁻²⁰ W/m² for silica fiber. Note that the threshold exceeds 10 kW for an RBE, whereas for the same length fiber with a quartic DBE the threshold is 800 W.

4. Conclusions

We have predicted strong slow-light enhanced all-optical switching near the degenerate band edge of nonlinear waveguides. Such waveguides support frozen light at the band edge frequency and this facilitates strong field increase inside the waveguide. As a consequence, nonlinear effects are strongly enhanced at DBEs, which enables low-threshold all-optical switching. While in conventional cavity structures the field is enhanced due to multiple reflections in the cavity, here the enhancement is based on a different physical mechanism involving the interference of near-degenerate slowly propagating and evanescent modes. The threshold for the all-optical switching between the low and full transmission can be flexibly designed, as it scales inversely proportional to the squared length of the waveguide. Moreover, in the frozen light regime the bistability and hysteresis can be avoided. Our estimates indicate that such switching can be realized in silica fibers and nanowire waveguides.