Localized Structures in Dispersive Doubly Resonant Optical Parametric Oscillators

Abstract

Recently quadratic dispersive optical resonators have been considered as new sources for frequency comb generation [1–3]. In contrast to Kerr combs, quadratic ones may operate with decreased pump power and can achieve spectral regions non accessible before. These combs in the frequency spectrum correspond to different type of dissipative structures circulating inside the cavity. Therefore the understanding of the dynamics of such structures is of crucial importance. In particular here we focus on the study of localized structures (LSs) since they correspond to broadband, coherent temporal pulses with a fixed repetition rate, which are one of the most important waveforms for frequency comb generation. To do so we consider a dispersive cavity with a quadratic medium phase matched for degenerate optical parametric oscillations (OPOs), and driven by an electric field at frequency 2ω 0 in a doubly resonant configuration. This type of cavities can be described by a mean-field model for the slowly varying envelopes A, and B, of the electric field centered at frequencies ω 0 and 2ω 0 , respectively [4]. These equations are formally equivalent to those describing diffracting cavities [5], although in this case a large walk-off d, related with the difference of group velocities between the fields A and B, is present. Often this walk-off avoids the formation of LSs, and therefore it is desirable to suppress it. This can be done by dispersion engineering as already shown in [6]. In what follows we consider that d = 0 and that we have natural phase-matching such that the phase detuning for A and B relates as θ 2 = 2θ 1 [2]. When bistability between trivial and non-trivial continuous wave (CW) solutions is achieved, LSs of different widths can be formed through the locking of domain walls [4,7]. Figures 1(a) and (b) show two examples of these types of LSs for θ 1 = −2, β 1 = 1, and β 2 = −0.1, where β 1,2 are the group velocity dispersion (GVD) coefficients for A and B repectively. In (c) and (d) the frequency combs corresponding to A and B are plotted for the two types of LSs shown in (a) and (b) respectively. Using numerical continuation techniques we find that these LSs undergo a collapsed snaking bifurcation structure when varying the pump intensity S, as shown in Fig. 1(e). The LSs' solution branches (in blue) oscillate back and forth in S around the Maxwell point of the system, and collapse to that point as increasing the energy of the cavity. Labels (a) and (b) on these branches correspond to the LSs shown in Fig. 1(a) and (b). These types of LSs and its associated collapsed snaking have been studied in detail in the context of Kerr cavities [7]. The region of existence of these structures may be defined by the length of the first branch of solutions, i.e. S 2 –S 1 . We have characterized this region as a function of β 2 for fixed values of θ 1 [see the inset in Fig. 1(e)]. We find that when increasing the value of β 2 these type of solutions tend to disappear. However, when decreasing β 2 , the region of existence broadens, and therefore LSs are easier to find. This type of solutions have not yet been studied in the context of dispersive OPO cavities, and we strongly believe that these LSs may be very relevant for the generation of frequency combs in this type of systems.