Modulation instability in the sub-cycle regime

Abstract

Summary form only given. Modulation instability (MI) in Xe-filled kagomé-style hollow core photonic crystal fibre (PCF) has recently been demonstrated experimentally using 500 fs laser pulses centred at λ = 800 nm with energies of a few μJ [1]. It is known that MI leads to the formation of fundamental solitons [2] with average amplitude and temporal width given by [3]: τ<;sub>0<;/sub> = √2T<;sub>0<;/sub>(ΠN), where T<;sub>0<;/sub> is the duration of the input pulse, N = √(γP<;sub>0<;/sub>T<;sub>0<;/sub>/|β2|) the soliton order, γ the nonlinear coefficient, P0 the pump peak power and β<;sub>2<;/sub> the group velocity dispersion. This means that τ<;sub>0<;/sub> depends on N. The average temporal width of the solitons emerging from the instability can be arbitrarily set by suitable choice of gas pressure and pump pulse energy and duration. This theory is however only strictly valid within the approximations of the nonlinear Schrödinger equation; for few-cycle pulses a more complete model based on the full field equation must be used [4]. Using a statistical approach we explore numerically the differences between multi-cycle MI with a predicted soliton duration τ<;sub>0<;/sub> > c/ λ ~2.7 fs, and sub-cycle MI τ<;sub>0<;/sub> <; c/ λ ~2.7 fs. After running many simulations (~1000) for both cases we retrieve, at a given fibre position, the temporal location, duration and peak power of certain discrete peaks in the intensity envelope. For a predicted average soliton duration of 7 fs (multi-cycle regime) the simple analysis above works as expected (Fig. 1); N = 1 solitons are predominantly generated (dashed curve), the peak in the distribution of pulse durations being at 7 fs. Even after propagation over 3 dispersion lengths the distribution is maintained, as expected for solitons. The small deviations are likely to be due to higher order effects in these extremely short and broadband pulses.For a predicted average soliton duration of 1 fs, i.e., in the sub-cycle regime, we see that instead of an <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> = 1 distribution, the majority of pulses condense to a single state with duration 1.5 fs, two small distributions appearing along the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> = 1 and <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> = 2 lines, corresponding respectively to half-cycle and single-cycle solitons (Fig. 2a). Upon propagation this state evolves , eventually converging to that in Fig. 2b. Remarkably, although the NLSE-based MI theory is not valid, MI-like break-up still occurs, leading to an incoherent train of pulses of duration 0.5 to 1.5 fs, with peak powers greater than 50 MW. A very broad supercontinuum is generated (Fig. 2c) and the numerical and experimental spectra agree extremely well, supporting the validity of the numerical model in this extreme case.