Recurrent nonlinear modulational instability via down-conversion in quadratic media

TL;DR

This paper reveals a genuine recurrence in modulational instability for quadratic media, driven by non-degenerate downconversion during second-harmonic generation rather than cascading Kerr-like dynamics. By developing reduced three-wave and four-wave truncations, along with Lie-transform averaging, the authors describe and predict quasi-periodic energy exchange between the SH pump and fundamental-sidebands for both pure SH and phase-locked mixed FF-SH pumping. The three-wave model captures the recurrence accurately at large SHG mismatch and remains valid up to the bifurcation point; near phase-matching, the four-wave approach and averaging explain regularity and spectral features despite added frequencies. These findings provide a practical framework for experimental observation and control of MI recurrence in quadratic media and point to new routes for short-scale pulse sources in nonlinear optics.

Abstract

We investigate the induced modulation instability in second-harmonic generation beyond the early stage of the linearized growth of the modulation. We find a regime of recurrence (quasi-periodic conversion and back-conversion between the pump and the modulation) which is genuine of the parametric conversion process in quadratic media. Such recurrence is mainly driven by a process of non-degenerate downconversion, showing no analogy to the cascading regime which mimics the cubic (Kerr) nonlinearities. We consider two different steady states, i.e., a pure second-harmonic and a mixed fundamental/second-harmonic state. Both exhibit this dynamics, which we show to be amenable to a description in terms of reduced frequency-truncated models. The comparison with full numerical simulations of the starting model prove the validity and robustness of the reduced models in characterizing in a simple and elegant way a wide range of modulationally-unstable steady states.

Figures (12)

• Figure 1: Bifurcation structure of background (stationary) eigenmodes: SH intensity fraction $\eta_\mathrm{e}=|u_{20}|^2$ of equilibria vs. SHG mismatch $\delta k$. Solid and dashed lines indicate stable (center-type) and unstable (saddle-type) equilibria, respectively. The (saddle) branches with $\eta_\mathrm{e}>1$ are physically inaccessible. Stable mixed FF-SH eigenmodes branches with different locked phases $\phi_e = 0$ and $\phi_e=\pi$ exist for $\delta k <2$ and $\delta k >-2$, respectively. The tails of these branches, i.e., for $|\delta k| \gtrsim 4$ and $\eta_\mathrm{e} <0.05$ (blue color traits), correspond to the cascading or Kerr-like regime responsible for a recurrent MI of different type compared with that discussed in this paper Trillo23. Such Kerr-like MI is ruled by an effective focusing NLSE obtained for FF normal GVD ($\beta_1=1$) over the right tail ($\delta k \gtrsim 4$) or FF anomalous GVD ($\beta_1=-1$) over the left tail ($\delta k \lesssim -4$).
• Figure 2: (a,b) False-color plots of MI gain $g$ of the SH eigenmode ($u_{20}=1$) vs. frequency $\Omega$ and SHG mismatch $\delta k$, in the (a) normal ($\beta_1=1$) and (b) anomalous ($\beta_1=-1$) GVD regime. (c,d) Gain spectral profile $g(\Omega)$ sampled at different values of $\delta k$ in the (c) normal; (d) anomalous GVD regime. In (a,b) the dashed white curves gives the optimum (peak gain) frequency $\Omega_p=\sqrt{\delta k/\beta_1}$ corresponding to phase-matching of 3WM downconversion.
• Figure 3: (a) MI gain $g$ of the mixed FF-SH phase-locked eigenmode with $\phi_e=0$ vs. frequency $\Omega$ and SHG mismatch $\delta k$, in the normal GVD regime ($\beta_1=\beta_2=1$). (b) Gain spectral profile for $dk=1.6, 1.8, 2$ (bifurcation point in Fig. 1).
• Figure 4: Evolution of FF (left) and SH (right): (a,b) false-color plots of the intensity dynamics in $(t,z)$ plane. (c,d) Intensity patterns vs. time $t$ and (e,f) relative Fourier spectra vs. dimensionless detunings $\Omega=(\omega-\omega_0)T_0$ at FF and $\Omega=(\omega-2\omega_0)T_0$ at SH, sampled at first peak conversion distance $z=3.35$. Here, $a_1=0.1$, $\phi_1=0$, and $\beta_1=\beta_2=1$, $\delta k=3$, and the input modulation is at $\Omega=\Omega_p=\sqrt{3}$ such that $\delta k_3=0$.
• Figure 5: Recurrent evolution in space variable $z$ of intensity fractions of the SH (green) and sideband pair (orange), comparing those obtained from the PDEs Eqs. \ref{['eq:SHGsystem']} (thick solid and dashed lines) with those from the 3WM truncation [Eqs. \ref{['eq:Hred']}-\ref{['eq:period']}] (thin solid and dashed lines with crosses). Parameters and input are as in Fig. \ref{['fig4']}.