A normal form for frequency combs and localized states in Kerr–Gires–Tournois interferometers

Abstract

We elucidate the mechanisms that underlay the formation of temporal localized states and frequency combs in vertical external-cavity Kerr–Gires–Tournois interferometers. We reduce our first-principles model based upon delay algebraic equations to a minimal pattern formation scenario. It consists in a real cubic Ginzburg–Landau equation modified by high-order effects such as third-order dispersion and nonlinear drift, which are responsible for generating localized states via the locking of domain walls connecting the high and low intensity levels of the injected micro-cavity. We interpret the effective parameters of the normal form in relation with the configuration of the optical setup. Comparing the two models, we observe an excellent agreement close to the onset of bistability.

Temporal localized states and square-waves in semiconductor micro-resonators with strong time-delayed feedback

“…One can see that for increasing γ values, first bright TLSs start to exist and the bistability between dark and bright TLSs is established only above some finite value of γ. Moreover, one can see that the width of the stability regions remains constant for large γ values, where the adiabatic elimination of the carriers can be performed, connecting our results with past studies based upon an instantaneous nonlinearity [11][12][13].…”

“…To understand the origin and the behaviour of the periodic pulse solution presented in Fig. 1 (b), we perform a bifurcation analysis of the system (1)-( 3) using a recently developed extension of DDE-BIFTOOL [19] that allows for the bifurcation analysis of algebraic and neutral delayed equations [12,13,[20][21][22]. We start the analysis with the situation where the injection is set to be resonant with an external cavity mode, i.e., ϕ = 0.…”

“…From the CW branch, one can see that the branch of the TLSs emerges in a subcritical Adronov-Hopf bifurcation. Indeed, in the bistable CW region, similar to the case of the injected microcavity subjected to the Kerr nonlinearity [11,12], the coexistence of two stable CW solutions can lead to the formation of domain walls between them that exhibit oscillatory tails, for certain sets of parameters. In particular, two opposed fronts may lock at discrete distances forming stable bound states which results in the formation of TLSs.…”

“…3 is difficult since the role and impact of e.g., GDD and TOD on the TLS formation is hidden. Hence, to obtain a better understanding of the underlying physical mechanisms, we derive a partial differential equation (PDE) that approximates the dynamics of the full DAE model ( 1)-(3) in the limit of a good cavity (η → 1) and small detunings δ using a rigorous multiple time scale analysis as discussed in [12,14,21,25]. We start our analysis by eliminating the field Y (t) and transforming the DAE to a neutral delay differential equation (NDDE) that reads…”

“…* gurevics@uni-muenster. de Recently, the promising potential of injected Kerr microcavities enclosed into a long external feedback cavity as sources of high-power tunable OFCs, temporal localized structures (TLSs) and square-waves (SWs) was demonstrated [11][12][13]. There, it was shown that the natural modeling approach consists in using singularly perturbed timedelayed systems, in which the algebraic constraints simply corresponds to the boundary conditions linking the optical field in the various parts of the compound cavity system.…”