A series of seminal works published around 2000 established the physical boundaries of stable spectral broadening of pulses propagating down optical fibers nonlinearly [1–5]. At the heart of this fundamental limitation lies an anomalous group-velocity dispersion (GVD), $\beta _2<0$, since supercontinuum (SC) generation and noise amplification are concomitant features in this regime. As a result, if such a high sensitivity to input pulse fluctuations is to be alleviated, then noisy spectral components need to be coherently seeded, which imposes femtosecond (fs) input-pulse durations on coherent supercontinuum (CSC) generation [4–6]. Having been supported by a large number of theoretical and experimental studies, considering that a CSC requires fs pumping represents a paradigm in nonlinear optics [6,7].

Over the years, some partial solutions have been proposed to circumvent SC instabilities. Longitudinally varying guiding media [1–3,8–11] or modulating long input pulses [12] can mitigate the impact of noise, but their design depend on the initial conditions to a great extent. Tapered [13] and active [14] fibers have also been used to lower the noise of an SC driven by incoherent processes. In contrast, all-normal GVD regimes can allow for CSC generation over a wider range of pump and waveguide parameters [15] because the input noise is not magnified [7]. Notwithstanding, $\beta _2>0$ results in lower spectral broadening efficiencies [1–3] and consequently high input peak powers in the kW range [16–20]. Despite this progress, propagating fs pulses along uniform waveguides with anomalous GVD is still currently the most extended nonlinear dynamics for producing a CSC in integrated waveguides [21–25].

CSC fiber sources have been essential to improve the noise performance of optical coherence tomography [26] and scanning near-field optical microscopy [27]. Unfortunately, the demand for fs pump sources hinders other important CSC applications, in particular, outside laboratories, where compact and affordable technologies are essential. On the one hand, pumped by fs pulses, a fiber CSC often delivers spectral power densities that are too low for coherent nonlinear biomaging so that ultrafast solid-state lasers are generally necessary for these applications [20,28,29]. On the other hand, initiating a CSC from fs pulses binds its realization in fully integrated technologies to the challenging development of on-chip fs mode-locked lasers [30,31] or to ps pulse compression over tens of cm [32].

From a fundamental point of view, if a CSC generated from ps pulses is desired, then the nonlinear gain process leading to the so-called modulation instability (MI) should not exist despite $\beta _2<0$, which is allegedly not possible according to the usual phase-matching condition governing MI [7], hence the difficulties of this goal. In contrast to this widespread belief, it has been shown theoretically that the first-order nonlinear-coefficient dispersion (FOND), $\gamma _1$, corresponding to the self-steepening (or shock) term in the generalized nonlinear Schrödinger equation (GNLSE), can forbid MI even with $\beta _2<0$ [33]. Such a regime, however, would not be practical because it could only be reached, in principle, at sufficiently high powers. Importantly, this requirement implicitly assumes a scarce frequency dependence in the Kerr nonlinear index $n_2$, as is often the case [7]. Notwithstanding, the dispersion of $n_2$ can affect the spectral broadening of pulses pumped close to a two-photon absorption (2PA) resonance in semiconductor waveguides [34]. Motivated by these results and based on recent measurements of the silicon’s $n_2$ above its 2PA threshold at $2.2\,\mathrm{\mu}$m [35], here, a CSC from ps pulses pumped in the anomalous GVD regime is, for the first time to the best of our knowledge, numerically demonstrated. The resonant enhancement of the nonlinear coefficient $\gamma$ in a silicon waveguide close to 2.2 $\mathrm{\mu}$m and its GVD engineering possibilities enable the conditions required for this scenario.

Two reasons confer on this work the potential to open a new avenue in nonlinear guided optics. First, experimental tests of self-steepening effects predicted theoretically, ranging from MI cancellation and a CSC in the long-pulse regime, as studied here, to soliton self-frequency shift deceleration [36], on-chip octave-spanning SC generation [37], or even anomalous self-steepening [38], become now feasible using ps sources and more accessible technologies, as is silicon-on-insulator (SOI), unlike previous approaches [37,38]. Second, these results might also be feasible at telecom [39] or visible wavelengths [40], where chalcogenides [41] and diamond [42] possess their 2PA resonance.

This Letter is organized into three sections. First, the concept underpinning the conclusion of this work will be justified on physical grounds, and values of $\gamma _1$ and $\beta _2<0$ guaranteeing coherent spectral broadening will be analytically derived. Simulations of the GNLSE will confirm this result. Second, a silicon waveguide with $\gamma _1$ and $\beta _2$ values suiting the stability condition derived initially will be designed and employed to demonstrate numerically coherent spectral broadening of ps pulses in the anomalous dispersion regime. Finally, using ps pulses with tens of W of peak power propagated over 2 cm, a CSC spanning 70 THz will be obtained numerically. It is worth emphasizing that this result arouses additional interest because it narrows notably the existing gap between fully integrated pulsed lasers, which tend to deliver ps pulses with tens of mW of peak power [30], and the typical conditions enabling a CSC, namely fs pulses with hundreds of W of peak power [22–25].

Let us consider, on the one side, the degenerate four-wave mixing (FWM) process $2\,\omega _p \to \omega _s + \omega _i$ triggering MI, where $\omega _p$ denotes the pump wave and $\omega _i\,(\omega _s)$ indicates the idler (signal) wave that maximizes the MI gain at a given pump power $P_p$. By considering a well-known result [7], in the low-power regime (or, equivalently, neglecting FOND), the power of $\omega _i\,(\omega _s)$ will initially experience an exponential gain proportional to $P_p$. On the other side, and according to Ref. [33], such a gain must vanish at high powers owing to the FOND. Consequently, MI cancellation might be possible in a high-power regime. Nevertheless, the number of pump photons decreases as the pulse broadens spectrally and so does the power corresponding to the pump wave, which might eventually give rise to MI. Despite the fact that MI would then be unavoidable, the MI gain, in the presence of FOND, must be an upper bounded function of the pump power, as follows from its behavior at low and high powers. Therefore, if some values of the GNLSE parameters led to a $\mathrm {max}(g_\mathrm {MI})$ below the linear propagation loss of a waveguide, $\alpha$, then MI should not impact the pulse nonlinear dynamics, and coherent spectral broadening would then be feasible regardless of the initial conditions and $\beta _2<0$. Accordingly, the following calculation aims at finding the region of the parameter space $(\beta _2, \beta _3, \gamma _0, \gamma _1)$ where $\mathrm {max}(g_\mathrm {MI})<\alpha$.

Solving the GNLSE [7],

 

Fig. 1. Output spectra (gray traces) and their mean (black line) and modulus of the degree of coherence (magenta line) of ps pulses propagating with $\beta _2<0$ and (a) FOND above the threshold value in Eq. (4) and (b) conventional FOND.

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By considering the measurements of silicon’s $n_2$ carried out in Ref. [35], the FOND of a silicon waveguide can be expected to be significantly enhanced at wavelengths around $2.2\,\mathrm{\mu}$m. Furthermore, pumping above $2.2\,\mathrm{\mu}$m, say $2.3\,\mathrm{\mu}$m, can additionally avoid 2PA losses. On this basis, a silicon channel waveguide with a cross section of $340\,\mathrm {nm}$ height $\times$ $1600\,\mathrm {nm}$ width embedded in a silica cladding is engineered to comply with Eq. (4). The input pulses are those used in Fig. 1, and the spectra and coherence of the pulses after propagating 2 cm are shown in Fig. 2(a). Analogous to Fig. 1, the numerical results with $\gamma _1=\gamma _0/\omega _0$ are plotted in Fig. 2(b). The $\beta _2$ and $\gamma$ profiles (modeled to all orders unlike in Fig. 1) employed in these simulations correspond to the fundamental quasi-transverse electric (TE) mode and are provided in Fig. 3(a). The loss coefficient is also $50$ m$^{-1}$ [43,44]. The comparison between Fig. 2(a) and Fig. 2(b), where the coherence reduces to $\langle |g_{1,2}|\rangle = 0.559$, further supports Eq. (4) and its experimental demonstration in a foundry-compatible SOI waveguide.

 

Fig. 2. Output spectra (gray traces) and their mean (black line), and modulus of the degree of coherence (magenta line) of ps pulses propagating through (a) a silicon waveguide with $\beta _2<0$ and (b) fixing the FOND in panel (a) to its conventional value.

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Fig. 3. (a) GVD (magenta curve), and real (black solid line) and imaginary (black dashed line) parts of the nonlinear coefficient of the silicon waveguide used in the simulations. (b) Spectra (gray traces) and their mean (black line), and modulus of the complex degree of coherence (magenta line) at 2 cm. (c) Pulse profile at a propagation distance of 1.2 cm. (d) Spectra at 1.2 cm (black line) and 1.5 cm (green line), and FWM gain curves (dashed lines) matching the spectral sidelobes.

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The nonlinear dynamics typically underpinning a CSC can be divided into two stages [49]. First, self-phase modulation (SPM) induces nonlinear chirping and thus pulse spectral broadening. Second, certain frequencies generated by SPM overlap temporally owing to GVD and additional coherent FWM processes occur. If some of these interactions fulfill phase matching, then new spectral components, so-called dispersive waves (DWs), are produced. According to this picture, the results reported until now are associated with the first CSC evolution stage. Note, however, that SPM-based spectral broadening has now been extended to longer pulse regimes, in sharp contrast to the state of the art. As such, Fig. 2 also paves the way for a CSC driven by ps pulses.

To explore beyond the SPM stage in the waveguide used in Fig. 2, 1-ps-long sech-pulses (also with 50 W of peak power and pumped at 2.3$\,\mathrm{\mu}$m) will now be considered. These simulations provide a fully coherent output spectrum spanning 70 THz from $1.76$ $\mathrm{\mu}$m up to $3\,\mathrm{\mu}$m, as shown in Fig. 3(b). Let us discuss the reasons making these results foreseeable from the conclusions based on Fig. 2. Qualitatively speaking, the FOND should not drastically alter the main mechanisms governing CSC generation. First, the initial CSC stage should still rely on nonlinear chirping. Although FOND will introduce an asymmetric spectral broadening, canceling MI will be definitely its crucial role. At this point, it is worth noting that in the absence of MI, short- and long-pulse regimes should not be fundamentally different. [2PA has also been included in the simulations in Fig. 3(b) using the measurements reported in Ref. [46], see Fig. 3(a). Nevertheless, this process only reduces the SPM broadening efficiency [50] and similar results to Fig. 3(b) are obtained over shorter distances when 2PA is neglected. Raman scattering has been numerically checked to have a minor impact and therefore it has been neglected in these simulations. Neither higher-order absorption nor free-carrier effects have been considered [21,45]. Second, frequency overlapping and efficient FWM interactions can also occur in this case during the second CSC stage. The FOND will certainly act as an additional source of frequency dispersion [7] and will also impact phase matching conditions [51], but these effects, yet important, should not modify these processes substantially. For the sake of completeness, it is worth mentioning that when $\gamma _1=\gamma _0/\omega _0$ is fixed in this case, instabilities also prevent the CSC dynamics, as expected in view of Fig. 2(b). The second CSC stage is analyzed in Figs. 3(c) and 3(d), where the pulse at the beginning of the DW formation and the evolution of the DWs are plotted, respectively. This temporal profile results from self-steepening [7], i.e., the FOND, and resembles the wave-breaking phenomenon typically due to the interplay between SPM and normal GVD [7,15,50]. In the presence of a negative third-order dispersion, wave-breaking may induce a phase-matched FWM process where a blue-shifted frequency generated through SPM acts as a pump and interacts with a signal wave with frequency $\omega _0$ [50]. Its gain curve is also represented in Fig. 3(d) and it is in accordance with the high-frequency lobe generated during the propagation. After this spectral lobe is formed, a low-frequency broad resonance also appears. Similarly, the interaction between a different pump wave produced by SPM and the blue-shifted DW gives rise to a gain band accounting for the red-shifted DW. Although this picture provides an interesting insight on the possible origin of the spectral sidelobes, the emission of DWs is here a more dynamic process due to the power dependence of the phase-matching conditions. This feature contrasts with the common scenario of locked DWs and resembles the emission of shifting DWs in wave-breaking regimes with large higher-order dispersion [52].

The fact that current CSC generation schemes often employ pulses one order of magnitude shorter and with peak powers one order of magnitude higher than those used in Fig. 3(b) states that there is still room for improvement of CSC dynamics provided that the dispersion of the nonlinear coefficient comes into play. Indeed, if, e.g., 300-fs-long pulses are considered, then a CSC similar to that in Fig. 3(b) is obtained using 10 W of peak power. It is also worth mentioning that, according to preliminary simulations based on Ref. [53], high coherence is still preserved when technical noise is taken into account in Fig. 3(b). Accordingly, this work may disrupt actual approaches for integrating CSC on-chip, even at telecom or visible wavelengths where chalcogenides and diamond exhibit their 2PA resonance.

In conclusion, the dispersion of the nonlinear coefficient allows crossing fundamental frontiers in nonlinear optics. In this Letter, conditions for canceling allegedly intrinsic instabilities during SC generation have been analytically derived. This novel nonlinear dynamics has been achieved in a foundry-compatible silicon waveguide relying on a resonance of the Kerr nonlinear index. Based on this finding, a CSC in the long pulse regime, considered until now unfeasible, has been proven numerically.