1. Introduction

Modulation instability (MI), also known as the Benjamin-Feir instability in water wave physics, is a characteristic phenomenon of a wide range of nonlinear dispersive systems which corresponds to the dynamical energy exchange between a periodic perturbation and a continuous background [1–3]. MI dynamics can be described by the focusing nonlinear Schrödinger equation (NLSE), and have provided a framework for the understanding of numerous important applications in optics, such as pulse train generation, parametric amplification and supercontinuum generation [4–8]. These dynamics are also closely related to fundamental physical phenomena like the Fermi-Pasta-Ulam-Tsingou recurrence [9,10]. More recently, MI has been linked to the generation of rogue waves in optics and hydrodynamics [11,12]. One key result of these studies is that several MI features can be described almost exactly with analytic Akhmediev breather solutions [13–16]. These results laid the foundations for the successful experimental generation of previously unobserved solitary waves such as the Peregrine and Kuznetsov-Ma solitons [17–19]. Usually, only systems with negative second order dispersion ($\beta _2 < 0$) are considered [4,14,17], but a few studies investigated the effects of third [20] and fourth order dispersion [21–24]. Importantly, the latter showed that MI can be induced by negative fourth-order dispersion ($\beta _4 < 0$) in spectral regions with positive $\beta _2$ [22,23].

Recent work demonstrated optical solitons arising from the balance of Kerr nonlinearity with a single, negative high, even order of dispersion $m$, first with $m = 4$ [25,26] before generalizing the results to any arbitrary even order [27]. These novel solitons have been studied in single-pass passive waveguides [28–31], lasers [32,33] and passive resonators [34–37]. Following these results, researchers investigated the effects of high-order dispersion in a broader range of nonlinear systems, such as dark solitons [38], self-similar propagation [39] and MI [40–42].

Most theoretical and numerical studies of MI dynamics can take one of two starting points. As mentioned, the first of these is the NLSE, which is a nonlinear partial differential equation and can be solved numerically [43] or analytically [13,44]. This approach makes the least number of approximations and is necessary, for example, when investigating the field’s evolution in the presence of noisy initial conditions. When the initial condition consists of a continuous wave (CW) background with a seeded sideband, then this sideband grows and, through four-wave mixing (FWM), seeds additional equally spaced sidebands. The second approach considers exclusively the complex amplitudes of each of these sidebands. This leads to a set of coupled ordinary differential equations, one for each of the frequencies that is considered [45]. In the extreme case of the CW background, only the $\pm 1$ side bands are considered. In general, this leads to three coupled complex equations which, through the use of energy conservation and symmetry, can be reduced to two coupled real equations, forming a Hamiltonian system [41,42,46,47].

Although MI has been investigated for individual high dispersion orders, an overall picture of the development of MI as the dispersion order increases is still lacking. Here, we present a theoretical and numerical study of modulation instability dynamics with high-order dispersion $m$. After a general introduction to MI in Section 2, we show that for significantly large $m$ (in practice, $m=6$), the phase matching degrades significantly, which strongly inhibits the generations of additional sidebands outside the spectral region experiencing MI gain. This fully justifies the use of the three-wave truncating method in Section 3, which models the full system by only considering the pump and the first order sidebands [41,42,46,47]. Our results in Section 4 indicate that high-order dispersion may impose an ultimate limit on the MI dynamics and hamper the generation of high-intensity pulses and rogue waves [11,19]. In Section 5, we discuss our results and conclude.

2. Background of modulational instability

We start by recalling that the propagation of optical pulses of envelope $A(z,T)$, in a medium with Kerr nonlinearity and $m^{\textrm {th}}$ order dispersion is described by the modified NLSE [43]

These results are illustrated in Fig. 1(a), which shows the gain spectra as a function of the normalized frequency $\Omega /\Omega _{\textrm {max}}$ for three different dispersion orders $m$. This shows that the frequency range with substantial gain shifts to higher frequencies as the dispersion order increases. Indeed, the gain attains its maximum value when $\Delta \beta =-\gamma P_0$, or for

This can be understood by considering Fig. 1(b), which shows $\Delta \beta /(\gamma P_0)$ versus normalized frequency for the same three dispersion orders. As $m$ increases, the values of $\Delta \beta$ initially are minimal, leading to very low gain, before increasing rapidly and traversing the region with substantial gain very rapidly, leading to an increasingly narrow gain spectrum as previously reported [41,42].

 

Fig. 1. (a) MI intensity gain spectrum for three different orders of dispersion, $m=2$ (blue), $m=6$ (red) and $m=18$ (green) versus normalized frequency $\Omega /\Omega _{\textrm {max}}$. (b) Corresponding wavenumber mismatch $\Delta \beta$. Gain occurs between $\Delta \beta =-2\gamma P_0$ and $\Delta \beta =0$. The gain peaks at $\Delta \beta =-\gamma P_0$, indicated by the horizontal black dashed line.

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We now consider phase matching for the opposite case in which $\omega$ is large. We then find from Eqs. (2) and (5) that

To assess the validity of this argument, we numerically solve Eq. (1) for different dispersion orders $m$, by using the split-step Fourier method [43]. For the results discussed below, we used the following parameters. The nonlinear parameter is $\gamma = 1.2\,\textrm {W}^{-1}\textrm {km}^{-1}$, while the dispersion coefficients are defined as $\beta _m = -m!T_0^m/L_m$, where $L_m = 1\,\textrm {m}$ and $T_0 = 100\,\textrm {fs}$, following Ref. [49]. These values correspond to $\beta _2 = -20\,\textrm {ps}^2\textrm {km}^{-1}$, which is close to the value of second-order dispersion in standard single-mode fibers [18]. The input field is a weakly modulated CW field of the form

The evolution of the spectral intensity against normalized longitudinal coordinates $z/L_{NL}$ for $m = 2$, $m = 6$ and $m = 18$, are shown in Fig. 2(a)-(c), respectively. For $m = 2$, following the exponential growth of the $\pm 1$ sidebands at $\Omega _{\textrm {mod}}$ due to MI gain, several spectral sidebands are generated at $n\times \Omega _{\textrm {mod}}$ through cascaded FWM [43]. This is highlighted in Fig. 2(d), which shows the calculated spectrum at the distance of maximum temporal compression (i.e. highest temporal peak power) [14]. At this particular propagation length, the spectrum consists of a set of discrete frequencies separated by $\Omega _{\textrm {mod}}$, the relative intensities of which follow an exponential decay, resulting in a triangular shape on a log scale, typical of parametrically-driven systems [14,50]. For $m = 6$, the spectral evolution follows similar dynamics, as seen in Fig. 2(b). However, the second and third-order sidebands generated through FWM have a significantly smaller intensity and appear at $35.6$ and $67.2\,\textrm {dB}$ below the first-order sidebands, as seen in Fig. 2(e), due to the much larger phase mismatch induced by higher-order dispersion, consistent with Eq. (9). Finally, for $m = 18$, only the pump and the $\pm 1$ order sidebands can be observed due to a large phase mismatch. These results confirm that for sufficiently large dispersion order $m$, the large phase mismatch between the first frequencies at $\Omega _{\textrm {mod}}$ and $2\Omega _{\textrm {mod}}$ inhibits the generation of higher-order sidebands through FWM. Therefore, the system can be accurately described by only three waves, namely the pump and the first order sidebands [46].

 

Fig. 2. (a)-(c) Numerically calculated spectral evolution of an input field defined as Eq. (10), for $m=2$, $m=6$, and $m =18$, respectively. $L_{NL} = (\gamma P_0)^{-1}$ is the nonlinear length. The white dashed line indicates the length of maximum compression. (d)-(f) Corresponding spectral intensity at propagation distance of maximum compression.

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3. Three-wave truncating model

As discussed in Sec. 1, one way of analyzing parametric amplification is through a set of coupled ordinary differential equations, one for each frequency. Such an approach may be applied when $\Omega _{\textrm {mod}}>\Omega _{\textrm {max}}/2$, a condition we take to be satisfied in Secs. 3 and 4. Trillo and Wabnitz showed that two coupled differential equations can describe such system in the case of quadratic dispersion [46]. Yao et al. considered quartic dispersion using the same approach [41], before Wang et al. generalized this to arbitrary dispersion order [42]. Assuming the spectrum to be symmetric, it is found that

Reducing the system to only three spectral components also restricts the dynamics that can arise from the interaction between these components–the small number of Fourier components limits the formation of high-intensity peaks on an otherwise low-intensity background. Considering the Hamiltonian in Eq. (12) and the initial condition $\eta \rightarrow 0$ we find that ${\cal H}=0$. From Eq. (12) we can then find that the largest value that $\eta$ can reach for a given ratio $\Omega _{\textrm {mod}}^m/\Omega _{\textrm {max}}^m$ is

For the particular case $\Omega _{\textrm {mod}}=\Omega _{\textrm {peak}}= \Omega _{\textrm {max}}/2^{1/m}$ (Eq. (8)), we find that $\eta _{\textrm {max}}={4}\left/{7}\right.$, independent of $m$. This means that a fraction ${2}\left/{7}\right.$ of the total energy is in each of the sidebands, and a fraction ${3}\left/{7}\right.$ on the central frequency. According to Eq. (14), the peak intensity of the signal then takes the value $(11+4\sqrt 6)/7\approx 2.971$, again, independent of $m$. The deepest modulation of 3 is obtained when $\eta _{\textrm {max}}={2}\left/{3}\right.$, corresponding to the modulation frequency $\Omega _{\textrm {mod}}/\Omega _{\textrm {max}}=({5}\left/{12}\right.)^{1/m}$. This means that upon evolution, the peak intensity reaches a maximum value that is $3\times$ higher than that of the CW background.

To test these predictions, we numerically solved Eq. (1) for $\Omega _{\textrm {mod}} = \Omega _{\textrm {peak}}$ and for $m$ ranging from 2 to 20, to calculate the fractional spectral power in the pump (blue dots) and the positive first sideband order (red diamonds) at the distance of maximum compression (see Fig. 2) with the same initial condition as in Sec. 2. These results are shown in Fig. 3(a), and are in very good agreement with our predictions for sufficiently large $m$ ($m \geq 8$), i.e. when the three wave truncating model is a very good approximation. The corresponding calculated normalized temporal peak powers $P/P_0$ (green squares), shown in Fig. 3(b), are again in excellent agreement with our prediction (green dashed line) for sufficiently large $m$. The maximum normalized power monotonically decreases for increasing $m$ until reaching the $P/P_0=2.971$ limit at this modulation frequency. As the dispersion order increases, only three spectral components contribute due to poor phase matching, leading to a smaller peak power in the temporal domain. This is illustrated in Fig. 3(c), which shows the calculated temporal intensity for $m=2$ (blue), $m=6$ (red) and $m=18$ (dashed green). As shown in Fig. 3(b), the highest intensity is $2.971$ for $m = 18$, about half of the highest intensity of the $m = 2$ case, for the same initial conditions. This suggests again that high-order dispersion inhibits the generation of high-intensity waves, previously observed in nonlinear systems with low dispersion orders [11,17,19].

 

Fig. 3. (a) Numerically calculated fractional spectral power in the pump (blue dots) and positive first-order sideband (red diamonds) at the maximum compression length, against dispersion order $m$. The blue and red-dashed lines indicate the predicted minimal fractional power in the pump and maximum power in the first sideband, corresponding to ${3}\left/{7}\right.$ and ${2}\left/{7}\right.$, respectively. (b) Corresponding normalized temporal peak power $P/P_0$ versus dispersion order. (c) Examples of simulated temporal intensity profiles versus normalized time $t\times f_{\textrm {mod}}$ at the distance of maximum compression for $m=2$ (blue), $m=6$ (red) and $m=18$, (dashed green). The horizontal dashed line indicates $P/P_0 = 2.971$.

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Figures 4(a) and (b) show the evolution of $\eta _{\textrm {max}}$ and the highest intensity versus normalized frequency for $m = 14$, respectively. The values obtained from numerically solving Eq. (1) (red dots) are in excellent agreement with the analytical values calculated from Eqs. (13) and (14) (solid blue curves). This highlights again that for sufficiently large dispersion order $m$, the system dynamics are well described by Eqs. (11)–(14).

 

Fig. 4. (a) Energy fraction $\eta$ calculated from Eq. (13) (solid blue line) and numerical simulations (red dots) against normalized frequency for $m = 14$. (b) Corresponding calculated from Eq. (14) and numerical simulation ratio of the highest intensity that can be reached relatively to the CW background. The green diamonds indicate $\Omega _{\textrm {peak}}$.

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4. Effects of the MI gain profile

As discussed in Sec. 2, the frequency range experiencing gain shifts to higher frequencies as the dispersion order increases, but also narrows. Therefore, for frequencies $\Omega \neq \Omega _{\textrm {peak}}$ the MI gain drops significantly.

We numerically calculate the field evolution for the same input as in Sec. 2 with white intensity noise, for $m = 18$, for three different modulation frequencies. These modulation frequencies and their corresponding MI gain are shown in Fig. 5(a), with $\Omega _{\textrm {mod,1}} = 0.6\times \Omega _{\textrm {peak}}$ (black diamonds), $\Omega _{\textrm {mod,2}} = 0.75\times \Omega _{\textrm {peak}}$ (green squares), and $\Omega _{\textrm {mod,3}} = 0.95\times \Omega _{\textrm {peak}}$ (red dots). For $\Omega _{\textrm {mod,1}}$, the spectral dynamics are dominated by sidebands growing from noise at the frequency of $\Omega _{\textrm {peak}}$, as seen in Fig. 5(b). This is highlighted in Fig. 5(c), which shows the intensity spectrum at $z/\textrm {L}_{\textrm{NL}} = 34.8$. The spectrum displays an incoherent pump and sidebands centered around $\pm \Omega _{\textrm {peak}}$. This leads to a small intensity in the time domain (see inset). A similar case is observed for $\Omega _{\textrm {mod,2}}$ at the same propagation distance, as seen in Fig. 5(d). In this case, the sidebands appear again around $\pm \Omega _{\textrm {peak}}$ as the evolution dynamics are also dominated by an exponential growth of noise. This is again confirmed by the temporal intensity profile. Finally, for $\Omega _{\textrm {mod,3}}$, the MI gain at this frequency is significantly larger, which leads to evolution dynamics dominated by the seeded pump modulation. The intensity spectrum profile at $z/\textrm{L}_{\textrm{NL}} = 19.1$, shown in Fig. 5(e), displays two coherent sidebands centered at the modulation frequency $\Omega _{\textrm {mod,3}}$, while the corresponding temporal intensity profile displays a train of Akhmediev breathers [13]. We note that the calculations for Fig. 5 were carefully carried out to have noise with the same statistics for each of the three frequencies.

 

Fig. 5. (a) MI gain gain spectrum for $m = 18$ (solid blue line). The markers indicate the three example modulation frequencies. $\Omega _{\textrm {mod,1}} = 0.6\times \Omega _{\textrm {peak}}$ (black diamonds), $\Omega _{\textrm {mod,2}} = 0.75\times \Omega _{\textrm {peak}}$ (green squares) and $\Omega _{\textrm {mod,3}} = 0.95\times \Omega _{\textrm {peak}}$ (red dots). (b) Spectral intensity evolution of CW input modulated at $\Omega _{\textrm {mod,1}} = 0.6\times \Omega _{\textrm {peak}}$. The white dash lines indicate the modulation frequency. (c)-(e) Spectral intensities at propagation distance of emerging sidebands. The markers indicate the input modulation frequencies. Insets, corresponding temporal intensity.

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5. Discussion and conclusion

We note that our results in this work and previous studies [41,42,46] assume that $\Omega _{\textrm {mod}} > \Omega _{\textrm {max}}/2$, so that only the $\pm 1$ order sidebands experience MI gain. We performed additional numerical simulation for $\Omega _{\textrm {mod}} = 0.45\times \Omega _{\textrm {max}}$. We use similar input field as defined in Eq. (10), but with a larger modulation modulus $|\varepsilon | = 10^{-2}$ so that these sidebands may be observable in spite of their low gain (as seen in Fig. 5). The intensity spectra at the distance of maximum compression for $m = 6$ and $m=18$ are shown in Fig. 6(a) and (b), respectively. In both cases, the intensity of $\pm 1$ order sidebands at $\Omega _{\textrm {mod}}$ is significantly smaller than that at $2\Omega _{\textrm {mod}}$ due to the significantly larger MI gain around $\Omega _{\textrm {peak}}$ as $m$ increases. For $m = 6$, several spectral sidebands are generated outside the MI gain range, through FWM, however for $m = 18$ only the $\pm 1$ and $\pm 2$ order sidebands are generated, due to poor phase matching, similar to Fig. 2(c) and (f).

 

Fig. 6. Spectral intensity at propagation distance of maximum compression, for modulation frequency $\Omega _{\textrm {mod}} = 0.45\times \Omega _{\textrm {max}}$ for $m = 6$ (a), and $m = 18$ (b). The red arrows indicate the modulation frequency. (c)-(d) Corresponding temporal intensity. The red dashed line indicates the input temporal intensity.

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Equation (12), which does not include the effects of noise, shows that when $\Omega _{\textrm {mod}} > \Omega _{\textrm {max}}/2$ and $m$ is sufficiently large so the three-wave approximation can be applied, the evolution only depends on the dispersion via $\Omega ^m/\Omega ^m_{\textrm {max}}$. This implies that every order of dispersion exhibits the same range of behavior and that the same evolution is observed at different orders of dispersion provided, $\Omega ^m/\Omega ^m_{\textrm {max}}$ is the same. A similar observation, namely that when $m$ is sufficiently large, the dynamics starts become universal, was earlier observed in the investigation of the linear evolution of short pulses [51].

In conclusion, we have studied the dynamics of MI in the presence of even, high-order dispersion. Our theoretical analysis shows that high-order dispersion leads to increasingly poor phase matching, which strongly inhibits the generation of spectral sidebands through FWM and, therefore, limits pulse formation in the time domain. For sufficiently large dispersion order $m$, the system can be described by a three-wave model, while its dynamics and maximum achievable intensity can be easily derived from the Hamiltonian [41]. These predictions are confirmed by full numerical simulations based on the NLSE.

We expect these results to trigger future experimental studies of MI with high-order dispersion. Some of the dynamics described in this work may be observed in waveguides with dominant high-order dispersion [28], or by using a recirculating loop configuration [9,52,53] and recent progresses in cavity dispersion engineering [32,54,55].