On the Asymmetric Spectral Broadening of a Hydrodynamic Modulated Wave Train in the Optical Regime

Abstract

Amplitude modulation of a propagating wave train has been observed in various media including hydrodynamics and optical fibers. The notable difference of the propagating wave trains in these media is the magnitude of the nonlinearity and the associated spectral bandwidth. The nonlinearity and dispersion parameters of optical fibers are two orders of magnitude smaller than the hydrodynamic counterparts, and therefore, considered to better assure the slowly varying envelope approximation (SVEA) of the nonlinear Schrödinger equations (NLSE). While most optics experiment demonstrate an NLSE-like symmetric solutions, experimental studies by Dudley et al. (Optics Express, 2009, 17, 21497-21508) show an asymmetric spectral evolution in the dynamics of unstable electromagnetic waves with high intensities. Motivated by this result, the hydrodynamic Euler equation is numerically solved to study the long-term evolution of a water-wave modulated wave train in the optical regime, i.e., at small steepness and spectral bandwidth. As the initial steepness is increased, retaining the initial spectral bandwidth thereby increasing the Benjamin-Feir Index, the modulation localizes, and the asymmetric and broad spectrum appears. While the deviation of the evolution from the NLSE solution is a result of broadband dynamics of free wave interaction, the resulting asymmetry of the spectrum is a consequence of the violation of the SVEA.

“Extraordinary” modulation instability in optics and hydrodynamics

“…The reason is the asymmetry of the water wave profiles, which is the B A result of significant breather amplification of a factor of three and above. The consequence is the nonlinear Stokes contributions that are always present in water waves at these scales (48,49). Despite these deviations, the patterns in Figs.…”

The Peregrine Breather on the Zero-Background Limit as the Two-Soliton Degenerate Solution: An Experimental Study

“…In fact, when waves become steep, the spectral broadening reduces the ability of the NLSE to accurately describe the wave hydrodynamics. However, this strongly depends on the initial carrier steepness and the bandwidth of the wave train [39].…”

“…In fact, when waves become steep, the spectral broadening applies the natural restriction to the NLSE reducing its ability to accurately describe the wave hydrodynamics. However, this strongly depends on the initial carrier steepness and the bandwidth of the wave train [41]. The next series of tests have been conducted at the University of Sydney wave flume.…”

Phase Evolution of the Time- and Space-Like Peregrine Breather in a Laboratory

“…Moreover, the breather undergoes a gradual asymmetric compression, which affects the NLSE comparison accuracy. It is well-known that these are well-captured in higher-order NLSE-type evolution equations and can be tamed by reducing the wave steepness [22,[37][38][39]. Since it is not possible to significantly reduce the carrier amplitude, the reduction of wave steepness can be achieved by increasing the wave frequency.…”