Impact of phase modulation on the dynamics of temporal localized structures in injected Kerr microcavities

We theoretically investigate how phase modulation alters the dynamics of temporal localized structures (TLSs) in vertically emitting Kerr micro-cavities under detuned optical injection operating in the normal dispersion regime. We show that the emergence of TLSs in general is governed by a synchronization between the imposed modulation and the intrinsic pulse dynamics. We perform a multi-parameter bifurcation analysis of the underlying delay-algebraic equation model in the uniform field limit and demonstrate that weakly nonlinear and dissipative Hermite-Gauss modes shape the dynamics of dark TLSs, leading to a complex hybrid bifurcation structure. Beyond the uniform field limit, both bright and dark modulated TLSs are shown to exist and to occupy distinct equilibrium positions within the cavity. An effective equation of motion for the TLS positions is derived, showing good agreement with the full model.