Guided-wave configurations are of great interest in nonlinear optics because they allow controlling the dynamics of light pulses geometrically [1]. However, the relation between features of nonlinearly propagating pulses, such as resonant frequencies, and the geometrical parameters of the guiding medium, e.g., the core diameter of an optical fiber, is not trivial at all. Governing key nonlinear processes, the group velocity dispersion (GVD) curve usually acts as a proxy of pulse nonlinear dynamics. For instance, the spectral resonances called dispersive waves (DWs) may appear at a detuning from the pumping frequency ${\omega _0}$ given by ${-}3{\beta _2}/{\beta _3}$ [2] in the anomalous dispersion regime, i.e., with ${\beta _2} \lt 0$, or by ${-}2{\beta _2}/{\beta _3}$ [3] in the normal dispersion regime, i.e., with ${\beta _2} \gt 0$, where ${\beta _n}$ denotes the ${n^{{\rm th}}}$-order dispersion. These results can reduce the search of a waveguide for a given nonlinear function to the design of a cross section providing the corresponding GVD curve [4]. Nonetheless, these expressions are not valid, e.g.,  for any value of ${\beta _3}$ or input pulse peak power, as shown in Ref. [5] for cases with ${\beta _2} \lt 0$. Therefore, the waveguide design procedure only simplifies provided that the nonlinear regime where the above formulas were derived still holds, which is not always obvious a priori.

From this point of view, finding a waveguide with a targeted nonlinear performance may be seen as two consecutive problems. First, for a given input pulse, values of parameters of the nonlinear Schrödinger equation (NLSE), say $({\beta _2},{\beta _3})$, allowing the target behavior should be determined. This step will hereinafter be referred to as the inverse nonlinear design. Second, a cross section of the platform of choice providing the values $({\beta _2},{\beta _3})$ obtained from the inverse nonlinear design should be found. A common approach to tackle this second problem is to create maps of the space defined by the waveguide design to visualize how $({\beta _2},{\beta _3})$ vary with the geometrical parameters [6].

Mapping the space of geometrical parameters of a given guided-wave device notably benefits from invariances that Maxwell’s equations exhibit [7]. The resulting relations allow calculating analytically how the GVD curve changes owing to geometrical and material scalings, thus avoiding superfluous numerical calculations [7]. Analogously, here, a novel approach based on an invariance of the NLSE will be developed to delimit different pulse nonlinear dynamics in the parameter space associated to the NLSE, including not only ${\beta _n}$ and the waveguide nonlinear coefficient $\gamma$, but also the duration ${T_0}$ and the peak power ${P_0}$ of the incident pulse.

The rich phenomenology that the NLSE encompasses makes it difficult to map its parameter space based on accurate solutions of the NLSE [8,9]. Instead, the scaling transformations undergone by a pulse during nonlinear propagation will be sufficient to distinguish nonlinear regimes. This approach will be illustrated here, deriving a threshold in a NLSE parameter space with ${\beta _2} \gt 0$, from which DWs experience self-frequency shift, and, as a result, resonant frequencies evolve along the propagation. This mechanism is in contrast to the scenarios previously explored in the normal dispersion regime, where DWs are locked at frequencies determined by the GVD curve [3,4,8–11]. This Letter is organized in three sections. Given the key role of the dynamical length scales in this work, their definition will first be reviewed before deriving the equation governing their evolution. An analytical study of this equation will subsequently lead to the threshold that will eventually separate the regimes of locked and shifting DW emission. Second, the NLSE will be numerically solved for different values of the NLSE parameters around the boundary between the two different dynamical regimes, which will evince the feature of DW self-frequency shift. Finally, the physical mechanisms underlying this effect will be discussed.

The nonlinear length ${L_{{\rm NL}}} = 1/(\gamma {P_0})$ and the dispersive length ${L_{\rm D}} = T_0^2/|{\beta _2}|$ along with the sign of ${\beta _2}$ are used to characterize different regimes such as linear (${L_{\rm D}} \ll {L_{{\rm NL}}}$), solitonic (${L_{{\rm NL}}} = {L_{\rm D}}$ with ${\beta _2} \lt 0$), or wave-breaking (${L_{{\rm NL}}} \ll {L_{\rm D}}$ with ${\beta _2} \gt 0$) [1]. Nevertheless, as the pulse propagates and its duration and peak power change, ${L_{{\rm NL}}}$ and ${L_{\rm D}}$ no longer represent the length scales corresponding to nonlinearities and dispersion. In Ref. [12], the $z$-dependent functions

On the experimental side, this conservation law of the NLSE removes a fundamental source of error in the measurement of $\gamma$ [13]. On the theoretical side, this property allows for looking at the nonlinear regimes described by the NLSE in the different manners of how ${\cal L}_{{\rm NL}}^{- 1}$ and ${\cal L}_{\rm D}^{- 1}$ compete with each other [11]. The present work constitutes the backbone of such a framework. It will show how this approach provides, in a comparatively simple way, relevant insights into complex dynamics of the pulse.

Let us consider the NLSE,

According to Eq. (5), to the first order in $h$, the pulse duration changes linearly, while its phase already evolves nonlinearly. Therefore, the equation of motion derived for ${\cal L}_{\rm D}^{- 1}$ should contain more information than that corresponding to ${\cal L}_{{\rm NL}}^{- 1}$. If a Gaussian pulse with a linear chirp is considered to evaluate ${\cal L}_{\rm D}^{- 1}$ at any distance $z$, then Eq. (5) leads —with no further approximation— to the following nonlinear length-scale propagation equation (see details in Appendix A):

Equation (7) will be numerically tested using parameters values from Ref. [10]: ${\beta _2} = 7.5\;{{\rm ps}^2}/{\rm km}$, $\gamma = 2.5\; {{\rm W}^{- 1}} {{\rm km}^{- 1}}$, ${T_{{\rm FWHM}}} = 1\;{\rm ps} $, and ${P_0} = 0.6\;{\rm kW} $, with a Gaussian input pulse. These values will be employed along this work unless otherwise indicated. In Fig. 1(a), ${\cal L}_{\rm D}^{- 1}$ predicted by Eq. (7) is compared with simulations of Eq. (4) over the propagation distance. The excellent agreement between these results validates Eq. (7), meaning that the strong distortion experienced by the pulse in this regime [8] does not impact the evolution of ${\cal L}_{\rm D}^{- 1}$ (and thus ${\cal L}_{{\rm NL}}^{- 1}$) notably. It is worth mentioning that ${\cal L}_{\rm D}^{- 1}$ approaches the values obtained for $\kappa ({\beta _2} = 0)$ initially and tends to the values corresponding to $\kappa (\gamma = 0)$ at the output. This further supports the use of Eqs. (1) and (2) as dynamical nonlinear and dispersive length scales [3,11,12] and suggests that the width of the gray strip plotted in Fig. 1(a) could be used to evaluate the error associated with Eq. (7).

 

Fig. 1. (a) Dispersive length obtained from simulations of Eq. (4) for ${\beta _{n \gt 2}} = 0$ (solid line). The gray strip depicts the results predicted by Eq. (7) (see details in the text). (b) Results as in (a) but for ${\beta _4} \lt {\beta _{4,{\rm th}}}$. The simulated dispersive length for ${\beta _4} \gt {\beta _{4,{\rm th}}}$ is also included.

Download Full Size | PDF

In this work, Eq. (6) will be tested in regimes with ${\beta _2} \gt 0$ and ${\beta _4} \lt 0$. According to Eq. (6), ${\cal L}_{\rm D}^{- 1}$ reaches a maximum when $\Theta = 0$. It is worth indicating that Eq. (7) does not allow an extreme for ${\beta _2} \gt 0$. Since ${\cal L}_{{\rm NL}}^{- 1}(z) \gt 0$, and thus ${\cal L}_{\rm D}^{- 1}(z) \lt {\cal L}_{{\rm NL}}^{- 1}(0) + {\cal L}_{\rm D}^{- 1}(0)$, $\Theta = 0$ is possible if

 

Fig. 2. (a) Spectra at different propagation distances and theoretical position of the resonance (dotted pink line) for ${\beta _4} \gt {\beta _{4,{\rm th}}}$. The input spectrum is also included (long-dashed line). (b) Spectra at different propagation lengths for ${\beta _4} \lt {\beta _{4,{\rm th}}}$. In both figures, $\nu _{{\rm DW}}^{{\rm th}}$ has been calculated using the theoretical result derived in the text for a locked DW.

Download Full Size | PDF

For the sake of completeness, it is worth mentioning that ${\cal L}_{\rm D}^{- 1}$ experiences a minimum when ${\beta _2} \lt 0$ [11] due to nonlinear pulse compression [14], unlike the case studied here, where the maximum in ${\cal L}_{\rm D}^{- 1}$ is due to higher-order dispersion. Interestingly, Eq. (7) also predicts a minimum for ${\cal L}_{\rm D}^{- 1}$ when ${\beta _2} \lt 0$. This result suggests that Eqs. (3) and (6) could also provide new insights into nonlinear regimes where the incident pulse is pumped at wavelengths with anomalous dispersion [15,16], including the impact of higher-order dispersion.

The results presented until now have shown that the evolution of ${\cal L}_{\rm D}^{- 1}$ can unveil signatures of distinct regimes. Once the corresponding parameter region has been derived, further information about the pulse and its spectrum needs to be obtained by solving the NLSE in such a parameter region. Figure 2(a) includes spectra at different distances obtained solving Eq. (4) numerically for ${\beta _4} = - 0.04\;{{\rm ps}^4}\,\,{{\rm km}^{- 1}}$, thus corresponding to the monotonic evolution of ${\cal L}_{\rm D}^{- 1}$ as seen in Fig. 1(b). A locked DW is observed in this case. Such a resonant frequency agrees with the idler wave of the four-wave mixing (FWM) process $2 {\omega _p} \to {\omega _0} + {\omega _{{\rm DW}}}$, where ${\omega _p}$ fulfills the phase-matching condition ${\beta _p}({\omega _{{\rm DW}}}) + \beta ({\omega _0}) - 2 \beta ({\omega _p}) = 0$, and hence ${\beta _2} + (7/12){\beta _4}{({\omega _p} - {\omega _0})^2} = 0$ [1], which leads to the resonant frequency ${\omega _{{\rm DW}}} - {\omega _0} = \pm {(48/7)^{1/2}}{({\beta _2}/|{\beta _4}|)^{1/2}}$, as plotted in Fig. 2(a). (The upper DW is not shown for the sake of simplicity.) This is fully in line with the mechanism underlying DW emission proposed in Refs. [3,11]. In contrast, when ${\beta _4} = - 0.06\;{{\rm ps}^4}/{\rm km}$, and thus ${\cal L}_{\rm D}^{- 1}$ is no longer monotonic, see Fig. 1(b), the DW shifts continuously as the pulse propagates, as Fig. 2(b) shows. Besides illustrating an example of inverse nonlinear design based on Eqs. (3) and (6), this DW self-frequency shift might be exploited for realizing tunable light sources, which brings us to the final part of this work.

 

Fig. 3. (a) Evolution of the dispersive length scale. (b) Spectral resonances at several propagation distances. (c) Dips in the pulse spectra corresponding to the pump waves leading to the resonances plotted in (b) and GVD dispersion curve. Note that frequencies beyond the ZDF (dotted pink line) also serve as pump waves. (d) Spectral resonances at a fixed distance for several input peak powers.

Download Full Size | PDF

Since DW emission only demands one zero-dispersion frequency (ZDF), i.e., ${\beta _3}$ [3,8,10], DW self-frequency shift can also be expected when higher-order dispersion is governed by ${\beta _3}$ instead of ${\beta _4}$. [As indicated before, ${\beta _3}$ effects do not appear in Eq. (6) owing to the pulse symmetries assumed here.] In addition, from an experimental point of view, such a GVD curve would be more feasible and would avoid modulation instability [1,9]. (Although the dynamical length scales are valid concepts as far as the NLSE holds, incoherent regimes [17] might be out of the scope of this formalism.) These reasons motivate simulations of the NLSE in the parameter space $({\beta _2},{\beta _3},\gamma)$, with ${\beta _3}$ preserving one of the ZDFs produced by ${\beta _4} \lt {\beta _{4,{\rm th}}}$ (remember that ${\beta _{4,{\rm th}}} \lt 0$). This requirement leads to ${\beta _{3,{\rm th}}} = (|{\beta _{4,{\rm th}}}|{\beta _2}{/2)^{1/2}}$, and hence ${\beta _{3,{\rm th}}} = 0.45\;{{\rm ps}^3}/{\rm km}$ here. It is worth remarking that in Ref. [10], where locked DW emission was observed pumping at wavelengths with normal dispersion, ${\beta _3} = 0.2\;{{\rm ps}^3}/{\rm km} \lt {\beta _{3,{\rm th}}}$, which is also in line with the findings reported here.

In Fig. 3(a), the evolution of the length scales for ${\beta _3} = 0.5\;{{\rm ps}^3}/{\rm km} \gt {\beta _{3,{\rm th}}}$ is plotted. Note that ${\cal L}_{\rm D}^{- 1}$ reaches a maximum, similarly to Fig. 1(b) for ${\beta _4} \lt {\beta _{4,{\rm th}}}$. DW self-frequency shift is also produced in this case, as seen in Fig. 3(b). According to Refs. [3,11], locked DW emission can be understood as a two-step mechanism where self-phase modulation (SPM) creates a pump frequency that is subsequently transformed into a DW via a phase-matched FWM process. If ${\beta _3}$ is considered, then $2 {\omega _p} \to {\omega _0} + {\omega _{{\rm DW}}}$ occurs in phase matching when ${\omega _p}$ corresponds to the ZDF and, thus, ${\omega _{{\rm DW}}} - {\omega _0} = - 2 {\beta _2}/{\beta _3}$ [3,11]. Remember that the resonance observed in Fig. 2(a) has also been successfully explained on this basis. However, this mechanism becomes more intricate when the spectral broadening induced by SPM goes beyond the ZDF, as occurs here in view of Fig. 3(c). On the one hand, SPM will compete with DW emission, reducing the available pump photons. On the other hand, photons beyond the ZDF will enable additional (non-phase-matched) FWM processes leading to new DWs located at frequencies beyond ${-}2{\beta _2}/{\beta _3}$, see Fig. 3(b). This complex interplay gives rise to a continuous shift of the spectral resonance, as shown in Fig. 3(b). Finally, attending to Fig. 3(d), the DW self-frequency shift also allows tuning DWs through the input power at a fixed propagation distance [5,18].

In conclusion, a novel framework to analyze the NLSE by means of the evolution of the length scales accounting for dispersion and nonlinearities has been derived and validated numerically. This approach has been applied in nonlinear regimes where the incident pulses are pumped in normal dispersion and has led to unexplored scenarios where DWs experience self-frequency shift as the pulse propagates. This formalism may also provide useful insights into nonlinear regimes with anomalous dispersion.