Third-order three-wave mixing in single-mode fibers: exact solutions and spatial instability effects

Abstract

Exact solutions are presented to the steady-state coupled-mode equations that govern the nonlinear parametric interaction of a central-frequency wave with a pair of upshifted and downshifted sidebands in isotropic single mode optical fibers. This solution accounts for pump depletion as well for as a possible phase mismatch among the waves. The existence is predicted of eigensolutions propagating unchanged along the fiber, which may be either spatially stable or spatially unstable, depending on the total power and the propagation-constant mismatch. The presence of spatially unstable eigensolutions dramatically affects the power exchange among the three waves. The physical implications of this instability for the frequency-conversion process, as well as its potential application to all-optical switching, are discussed

Experimental study of the reversible behavior of modulational instability in optical fibers

“…Square-pulse spectrum on a logarithmic scale at an input pump power P 0 ϭ 2.35 W after 1 km. 10. As many as eight detectable modes are involved in this energy exchange process.…”

“…12 From the application point of view, some authors have suggested exploitating the reversibility of MI for switching purposes. 10,24 An all-optical switch based on longterm MI dynamics must be operated in a range of parameters where Ϫ1 Ͻ Ͻ 0 so, the output pump (or sideband) power strongly depends on the initial relative phase ⌽( ϭ 0), yielding either a periodic power exchange between the waves or aperiodic pump depletion, which leads asymptotically to ϭ 0 according to the launching conditions. However, in the case of scalar MI considered here, the truncated three-wave model used to describe such dynamics is no longer valid for Ϫ1 Ͻ Ͻ 0 because higher-order sideband harmonics fall inside the MI gain band and therefore can not be neglected.…”

“…[8][9][10][11] In this approach the phenomenology of MI is limited to the interaction between the pump wave and the pair of Stokes and antiStokes waves of frequencies s ϭ 0 Ϫ ⍀ and a ϭ 0 ϩ ⍀, respectively. Provided that the sideband detuning from the cw pump ⍀ satisfies ⍀ c /2 Ͻ ͉⍀͉ Ͻ ⍀ c , the higher-order Fourier modes may be neglected, as they fall outside the MI gain band and are therefore not linearly unstable.…”

Analytical model and experimental verification of the critical power for modulation instability in optical fibers

“…The theoretical approach to modulation instability can be based on either analyzing a fourwave mixing (FWM) framework or solving the nonlinear Schrödinger equation (NLSE). In the FWM method the pump depletion is taken into account; however, the analytic solution for most cases is not available [8,9]. The general approach to the NLSE that provides exact analytical solutions for the temporal evolution of optical pulses propagating in a dispersive Kerr medium is the Akhmediev breather formalism [10,11].…”

Full and semi-analytic analyses of two-pump parametric amplification with pump depletion

“…In addition, we extend the approach to any phase mismatch by neglecting the periodic behavior of the power flow. Depletion and the periodic behavior of the flow of power have previously been described analytically by the use of Jacobi elliptic functions [12][13][14]. In this work we derive a simpler analytical model of saturated dual-pumped FOPA.…”