Four-wave mixing of linear waves ans solitons in fibers with higher-order dispersion

The use of any optical system, including fibers, for frequency conversion relies on the ability to satisfy so-called phase-matching conditions, which depend critically on the dispersive properties of the system, and on strong enough nonlinearity to allow a reduction in the threshold pump power. The recent surge of interest in theoretical and experimental studies of optical parametric processes in photonic crystal f ibers – 8 (PCFs) is related to their high nonlinearities, achieved by reduction in core size, and to the possibility of dispersion control by suitable design of the f iber core and photonic crystal cladding. Here we study parametric generation of new frequencies resulting from four-wave mixing (FWM) of solitons and continuous wave (cw’s) in optical f ibers under conditions in which the effects of higher-order dispersion are important, i.e., when pulses are short and (or) the frequency dependence of the groupvelocity dispersion (GVD) is steep. Interest in this problem arises from the fact that the Fourier components of a soliton are dispersionless, while freely propagating cw’s are strongly dispersive. Therefore the phase-matching conditions are expected to be satisfied at frequencies different from those generated by the mixing of cw’s. Addressing this issue is timely because of the availability of strongly nonlinear small-core PCFs, which decrease the threshold for observing parametric processes by 1 to 2 orders of magnitude compared with that for conventional fibers: gconven 1023 W21 m21, gpcf 1022 1021 W21 m21, where g is the nonlinear fiber parameter. Strongly nonlinear PCFs have already been used to demonstrate the coupling of solitons and cw radiation in supercontinuum generation, strong red and blue resonant, or Cherenkov, radiation from solitons – 8 and cancellation of the soliton self-frequency shift by the spectral recoil. Note that it is natural to expect that FWM of solitons and cw radiation is one of the many nonlinear processes contributing to the shape of the supercontinuum spectra. The problem of mixing of solitons and cw’s using the idealized nonlinear Schrödinger equation, i.e., with higher-order dispersion disregarded, was analyzed in a number of papers in the past. – 16 In these cases several exact analytical solutions for a soliton sitting on the cw background were found and different perturbation techniques suggested. – 14 However, none of these studies addressed the issue of generation of new frequencies by mixing of solitons and cw light, which is the central focus of this Letter. We assume that the dynamics of the dimensionless amplitude A t, z of the fundamental f iber mode is governed by the generalized nonlinear Schrödinger equation