Quantum theory of femtosecond solitons in optical fibres

We use the time-dependent Hartree approximation to obtain the solution IO the quantum higher-order non-linear Schrodinger equation. This equation describes femtosecond pulses propagating in non-linear optical fibres and can have soliton solutions. These solitons travel at velocities that differ from the picosecond solitons obtained from the standard quantum non-linear Schrodinger equation. We find that these femtosecond solitons cannot propagate in graded-index fibres; rather, they require quadruple-clad fibres. This is the first investigation of quantum effects in femtosecond solitons to our knowledge. There is considerable interest in the non-linear Schrodinger equation (NU) in terms of both classical and quantum phenomena [l-61. In particular it has been used extensively to model the propagation of pulses in non-linear optical fibres; however, the NLS is generally not valid for pulses with durations shorter than the picosecond time scale. Yet the recent development of optical sources that generate pulses in the femtosecond domain makes possible the exploration of many new phenomena. Therefore the investigation of solitons arising from the higher-order NLS (HNLS), which can be used in the femtosecond time domain, is of interest. One of the simpletst HNLS is [7] where C, d and p are constants. We follow the conventional notation in the mathematical literature, which uses t and x to represent normalized space and time, respectively. This equation gives rise to soliton solutions when p = 6d [7]. Equation (1) reduces to the NLS forp=d=O. In certain circumstances the HNLS can be used to describe femtosecond pulses propagating in optical fibres; these are outlined in [8] and described in detail by us in [9]. Using experimental fibre parameters to evaluate the physical parameters in equation (1) we find that the pulse width must be below 200 fs for wavelengths in the 1.48-1.57um region in order for d and p to become significant. In addition, the dispersion parameters, pz and b3, given by the second and third derivatives of the propagation constant respectively, evaluated at the carrier frequency wo, must be negative, This necessitates a quadruple-clad fibre rather than the typical graded-index fibres used in calculations and experiments to date. This is a significant feature of our results [9]. The soliton self-frequency shift (SSFS) [lo, 111 may be an important effect 0954-899R/92/03U157+05 $4.50