Self-pulsing dynamical solitons in cw-pumped backward optical parametric oscillators

We present a model of degenerate and non-degenerate counter-propagating Optical Parametric Oscillators (OPOs) in the quasi-phase matching interaction, so that both the signal and idler fields propagate backward with respect to the direction of the continuous input pump field. For the degenerate backward OPO, we perform, for the first time to our knowledge, the stability analysis of the non-linear inhomogeneous stationary solutions where we take into account the time-dependent feedback boundary conditions. It proves that inhomogeneous stationary solutions are always unstable, whatever the cavity length and the pump power are. Numerical simulations confirm this unconditional instability and reveal the appearance of self-pulsing of the backward signal. The latter is amplified and compressed as it propagates in the cavity. Above a critical pulse steepening chromatic dispersion saturates this singular behaviour leading to a new type of stable dynamical solitary structures. The velocity and the modulational period of the selected solitary wave have been determined by means of the KPP approach.
Considering the non-degenerate case, the most striking feature is that the amplification of the solitary pulses is saturated by the temporal walk-off between the signal and idler beams. In this case, the dispersion allows the stability domains to be enlarged.
It results from this study that dispersion and temporal walk-off are the key ingredients for the spontaneous structuration of solitary waves in the backward quadratic interaction.