Multimode Nonlinear Fibre Optics: Theory and Applications

Optical fibres have been developed as an ideal medium for the delivery of optical pulses ever since their inception (Kao & Hockham, 1966). Much of that development has been focused on the transmission of low-energy pulses for communication purposes and thus fibres have been optimised for singlemode guidance with minimum propagation losses only limited by the intrinsic material absorption of silica glass of about 0.2dB/km in the near infrared part of the spectrum (Miya et al., 1979). The corresponding increase in accessible transmission length simultaneously started the interest in nonlinear fibre optics, for example with early work on the stimulated Raman effect (Stolen et al., 1972) and on optical solitons (Hasegawa & Tappert, 1973). Since the advent of fibre amplifiers (Mears et al., 1987), available fibre-coupled laser powers have been increasing dramatically and, in particular, fibre lasers now exceed kW levels in continuous wave (cw) operation (Jeong et al., 2004) and MW peak powers for pulses (Galvanauskas et al., 2007) in all-fibre systems. These developments are pushing the limits of current fibre technology, demanding fibres with larger mode areas and higher damage threshold. However, it is increasingly difficult to meet these requirements with fibres supporting one single optical mode and therefore often multiple modes are guided. Non-fibre-based laser systems are capable of delivering even larger peak powers, for example commercial Ti:sapphire fs lasers now reach the GW regime. Such extreme powers cannot be transmitted in conventional glass fibres at all without destroying them (Gaeta, 2000), but there is a range of applications for such pulses coupled into hollow-core capillaries, such as pulse compression (Sartania et al., 1997) and high-harmonic generation (Rundquist et al., 1998). For typical experimental parameters, these capillaries act as optical waveguides for a large number of spatial modes and modal interactions contribute significantly to the system dynamics. In order to design ever more efficient fibre lasers, to optimise pulse delivery and to control nonlinear applications in the high power regime, a thorough understanding of pulse propagation and nonlinear interactions in multimode fibres and waveguides is required. The conventional tools for modelling and investigating such systems are based on beam propagation methods (Okamoto, 2006). However, these are numerically expensive and provide little insight into the dependence of fundamental nonlinear processes on specific fibre properties, e.g., on transverse mode functions, dispersion and nonlinear mode coupling. For such an interpretation a multimode equivalent of the nonlinear Schrodinger equation, the standard and highly accurate method for describing singlemode nonlinear pulse propagation (Agrawal, 2001; Blow & Wood, 1989), is desirable. In this chapter, we discuss the basics of such a multimode generalised nonlinear Schrodinger equation (Poletti & Horak, 2008), its simplification to experimentally relevant situations and a few select applications. We start by introducing and discussing the theoretical framework for fibres with chi(3) nonlinearity in Sec. 2. The following sections are devoted to multimode nonlinear applications, presented in the order of increasing laser peak powers. A sample application in the multi-kW regime is supercontinuum generation, discussed in Sec. 3. Here we demonstrate how fibre mode symmetries and launching conditions affect intermodal power transfer and spectral broadening. For peak powers in the MW regime, self-focusing effects become significant and lead to strong mode coupling. The spatio-temporal evolution of pulses in this limit is the topic of Sec. 4. Finally, at GW peak power levels, optical pulses can only be delivered by propagation in gases. Still, intensities become so high that nonlinear effects related to ionisation must be taken into account. An extension of the multimode theory to include these extreme high power effects is presented in Sec. 5 and the significance of mode interaction is demonstrated by numerical examples pertaining to a recent experiment. Finally, we end this chapter with conclusions in Sec. 6