The evolution of stratified shear flows with multilayer density distributions is discussed briefly from a theoretical perspective, generalizing the results of Caulfield [J. Fluid Mech. 258, 255 (1994)] to allow for asymmetry. Three distinct types of instability are predicted to occur according to linear theory. In the laboratory, we measure the density profile and the velocity profile continuously, and so are able to identify the flow characteristics that are applicable when each of the different instabilities grow. Knowledge of the bulk Richardson number is insufficient to predict the observed properties of the instabilities of the flow. The parameter that is most determinant of the selection of a particular type of instability is found to be the ratio R of the depth of the intermediate density layer to the depth over which the velocity varies, though any asymmetry in the flow (either in the velocity or density fields) also plays a role. If R is close to 1, and hence the layer of intermediate density occupies a significant portion of the shear layer, overturnings appear in the intermediate layer, which are long lived, and strongly two dimensional. These overturnings are the three layer stratified generalization of the Kelvin–Helmholtz instability first discussed by Taylor [Proc. R. Soc. London Ser. A 132, 499 (1931)]. Such modes inefficiently mix the background flow, and the major mixing mechanisms are found to consist of overturnings in the lower fluid layer (and, to a lesser extent, the upper layer). These overturnings are clearly manifestations of an (asymmetric) three layer generalization of the Holmboe [Geophys. Publ. 24, 67 (1962)] instability. In general, all three instabilities can be observed simultaneously at markedly different wavelengths and phase speeds for extended periods of time, even though linear theory may predict significantly different growth rates.
Abstract-In order to both experimentally and numerically investigate nonlinear femtosecond ultrabroadband-pulse propagation in a silica fiber, we have extended the finite-difference time-domain (FDTD) calculation of Maxwell's equations with nonlinear terms to that including all exact Sellmeier-fitting values. We have compared results of this extended FDTD method with experimental results, as well as with the solution of the generalized nonlinear Schrödinger equation by the split-step Fourier method with a slowly varying-envelope approximation. To the best of our knowledge, this is the first comparison between FDTD calculation and experimental results for nonlinear propagation of a very short (12 fs) laser pulse in a silica fiber.Index Terms-FDTD, femtosecond, GNLSE, monocycle optical pulse, nonlinear chirp, nonlinear fiber optics, nonlinear propagation, Raman, self-phase modulation, self-steepening, Sellmeier, silica fiber, SVEA, ultrabroad-band spectrum.T HERE WAS RECENTLY significant interest in the generation of single-cycle optical pulses by optical pulse compression of ultrabroad-band light produced in fibers. We reported some experiments on the ultrabroad-band pulse generation using a silica fiber [1], [2] and an Ar-gas filled hollow fiber [3], and the optical pulse compression by nonlinear chirp compensation [1], [3]. For these experiments on generating few-optical-cycle pulses, characterizing the spectral phase of ultrabroad-band pulses analytically as well as experimentally is highly significant. We performed an experiment of 12-fs optical pulse propagation [1] [2]. In order to compare an FDTD calculation results with the experimentally measured ultrabroad spectra of such an ultrashort laser pulse, we extend the GJTH algorithm to that considering all exact Sellmeier's fitting values for ultrabroad spectra. Owing to broad spectrum of pulse propagating in a fiber, it becomes much more important to take accurate linear dispersion into account. It is well known that at least two resonant frequencies must be required for the linear dispersion to fit accurately to a refractive index data. Recent report by Kalosha and his coworker considers the linear dispersion with two resonant frequencies and the nonlinear terms without Raman effect [6]. For the single-cycle pulse generation experiment, we must use at least sub-5 fs [3], [7] or commercially available 12-fs pulses. Such a time regime is comparable to the Raman characteristic time of 5 fs [4] in a silica fiber. Therefore, it is very important to consider not only the accurate linear dispersion of silica but also the Raman effect in a silica fiber in the few-optical-cycles regime. In addition, owing to the high repetition rate and pulse intensity stability, in particular, ultrabroad-band supercontinuum light generation and few-optical-cycles pulse generation by nonlinear pulse propagation in photonic crystal fibers and tapered fibers [8], [9], which both are made of silica, have attracted much attention. In this work, we have extended the FDTD method with nonl...
A numerical approach called Fourier direct method ͑FDM͒ is applied to nonlinear propagation of optical pulses with the central wavelength 800 nm, the width 2.67-12.00 fs, and the peak power 25-6870 kW in a fused-silica fiber. Bidirectional propagation, delayed Raman response, nonlinear dispersion ͑self-steepening, core dispersion͒, as well as correct linear dispersion are incorporated into "bidirectional propagation equations" which are derived directly from Maxwell's equations. These equations are solved for forward and backward waves, instead of the electric-field envelope as in the nonlinear Schrödinger equation ͑NLSE͒. They are integrated as multidimensional simultaneous evolution equations evolved in space. We investigate, both theoretically and numerically, the validity and the limitation of assumptions and approximations used for deriving the NLSE. Also, the accuracy and the efficiency of the FDM are compared quantitatively with those of the finite-difference time-domain numerical approach. The time-domain size 500 fs and the number of grid points in time 2048 are chosen to investigate numerically intensity spectra, spectral phases, and temporal electric-field profiles up to the propagation distance 1.0 mm. On the intensity spectrum of a few-optical-cycle pulses, the self-steepening, core dispersion, and the delayed Raman response appear as dominant, middle, and slight effects, respectively. The delayed Raman response and the core dispersion reduce the effective nonlinearity. Correct linear dispersion is important since it affects the intensity spectrum sensitively. For the compression of femtosecond optical pulses by the complete phase compensation, the shortness and the pulse quality of compressed pulses are remarkably improved by the intense initial peak power rather than by the short initial pulse width or by the propagation distance longer than 0.1 mm. They will be compressed as short as 0.3 fs below the damage threshold of fused-silica fiber 6 MW. It is demonstrated that the carrier envelope phase ͑CEP͒ causes the difference on the temporal electric-field profile and the intensity spectrum for the initial peak power of the order of megawatts. At the propagation distance longer than the coherence length for third-order harmonics, the difference grows in the spectral components around the third-order and higher-order harmonics. The CEP can be a sensitive marker to monitor the evolution of nonlinear optical process by a few-optical-cycle electric-field wave-packet source.
Change in the refractive index at around 630 nm in the femtosecond time region has been measured for the first time for a 75-wt% PbO-doped silica single-mode fiber with a highly sensitive, time-resolved interferometer using a heterodyne pump-probe technique. The result shows that the instantaneous response, nonlinear refractive index is 9.8 times as large as that of a conventional fused-silica fiber. This suggests that the PbO fiber is useful for efficient optical pulse compression and ultrafast optical switching in the visible and near-infrared wavelength region.
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