We study the formation of three-dimensional structures in type-I intracavity second harmonic generation model where polarization degrees of freedom due to the birefringence of the χ<sup>(2)</sup> crystal is not considered. The device consists of an optical cavity ﬁlled with a quadratic nonlinear material and driven by an external beam at the fundamental frequency. The transmitted part of this ﬁeld is coupled into the cavity where it undergoes second-harmonic conversion. These dissipative structures consist of regular or localized 3D lattices of bright spots travelling at the group velocity of light in the material. We show evidence of stable three dimensional structures such as stripes and cylinders.
We consider an optical ring cavity filled with a metamaterial and with a Kerr medium. The cavity is driven by a coherent radiation beam. The modelling of this device leads to the well known Lugiato-Lefever equation with high order diffraction term. We assume that both left-handed and right-handed materials possess a Kerr focusing type of nonlinearity. We show that close to the zero-diffraction regime, high-order diffraction effect allows us to stabilise dark localised structures in this device. These structures consist of dips or holes in the transverse profile of the intracavity field and do not exist without high-order diffraction effects. We show that high order diffraction effects alter in depth the space-time dynamics of this device. A weakly nonlinear analysis in the vicinity of the first threshold associated with the Turing instability is performed. This analysis allows us to determine the parameter regime where the transition from super- to sub-critical bifurcation occurs. When the modulational instability appears subcritically, we show that bright localised structures of light may be generated in two-dimensional setting. Close to the second threshold associated with the Turing instability, dark localised structures are generated.
Broad area Vertical-Cavity Surface-Emitting Lasers (VCSELs) have peculiar polarization properties which are a field of study by itself.<sup>1-3</sup> These properties have already been used for localized structure generation, in a simple configuration, where only one polarization component was used.<sup>4</sup> Here, we present new experimental and theoretical results on the complex polarization behavior of localized structures generated in an optically-injected broad area VCSEL. A linear stability analysis of the spin-flip VCSEL model is performed for the case of broad area devices, in a restrained and experimentally relevant parameter set. Numerical simulations are performed, in one and two dimensions. They reveal existence of vector localized structures. These structures have a complex polarization state, which is not simply a linear polarization following the one of the optical injection. Experimental results confirm theoretical predictions. Applications of this work can lead to the encoding of small color images in the polarization state of an ensemble of localized structures at the surface of a broad area VCSEL.
We are interested in spatio-temporal dynamics of cavity solitons (CSs) in a transverse section of a broad area vertical cavity surface emitting laser (VCSEL) with saturable absorbtion subjected to time-delayed optical feedback. In the absence of delayed feedback, a single branch of localized solutions appears in the parameter space. However, in the presence of the delayed feedback, multistability of CS solutions emerges; The branches of CSs fill the surface of the "solution tube" in the parameter space, which is filled densely with increasing delay time. Further, our study reveals that the multistability of stationary solutions is caused by a delayed-induced phase bifurcation of CSs. Furthermore, it was shown that stability properties of CSs strongly depend on the delayed feedback parameters. In particular, the thresholds of the drift and phase bifurcations as well as corresponding bifurcation diagrams are obtained by a combination of analytical and numerical continuation methods. It turns out that both thresholds tend to zero in the limit of large delay times. In addition, we demonstrate that the presence of the delayed optical feedback can induce Andronov-Hopf bifurcation and a period doubling route to chaos. Moreover, a coupling between this bifurcation scenario with aforementioned delay-induced multistability leads to a complex spatio-temporal behavior of the system in question. The results of analytical bifurcation analysis are in agreement with those obtained by direct numerical integration of the model equation.
We consider a broad area vertical-cavity surface-emitting laser(VCSEL) subject to injection and to time-delayed feedback. We present analytical and numerical analysis of the dependence of the drift instability threshold and on the feedback strength, feedback phase, and carrier relaxation time. we demonstrate that due to finite carrier relaxation rate the delay induced drift instability can be suppressed to a certain extent. We give analytical estimation of the soliton velocity near the drift instability point which is in a good agreement with numerical results obtained using the full model equations.
We investigate a control technique, known as rocking, for the formation of phase bistable patterns of light by means of spatially periodic injection in a broad area nonlinear optical system. More precisely, we consider a vertical-cavity surface-emitting laser (VCSEL). The spatial periodic injection or spatial rocking is found to convert the initially phase-invariant oscillatory system into a phase-bistable pattern forming one. We investigate the role of carrier lifetime on the efficiency of rocking in a broad-area VCSEL structure. This simple and robust device received a special attention owing to advances in semiconductor technology. We show that the regime where rocking works depends strongly on the ratio between the time scales associated with the electric field and the carrier density. The size of the rocking region depends on the ratio between the time scales.
We consider a broad area vertical-cavity surface-emitting laser (VCSEL) subject to optical injection. We experimentally investigate the spontaneous formation of a Cavity Soliton (CS) in a medium size (80μm diameter) VSCEL. CSs are generated and erased when sweeping optical injection power and proper frequency detuning between the master laser and the VCSEL is set.
Cavity solitons are localized light peaks in the transverse section of nonlinear resonators. These structures are usually formed under a coexistence condition between a homogeneous background of radiation and a self- organized patterns resulting from a Turing type of instabilities. In this issue, most of studies have been realized ignoring the nonlocal eﬀects. Non-local eﬀects can play an important role in the formation of cavity solitons in optics, population dynamics and plant ecology. Depending on the choice of the nonlocal interaction function, the nonlocal coupling can be strong or weak. When the nonlocal coupling is strong, the interaction between fronts is controlled by the whole non-local interaction function. Recently it has shown that this type of nonlocal coupling strongly aﬀects the dynamics of fronts connecting two homogeneous steady states and leads to the stabilization of cavity solitons with a varying size plateau. Here, we consider a ring passive cavity filled with a Kerr medium like a liquid crystal or left-handed materials and driven by a coherent injected beam. We show that cavity solitons resulting for strong front interaction are stable in one and two-dimensional setting out of any type of Turing instability. Their spatial profile is characterized by a varying size plateau. Our results can apply to large class of spatially extended systems with strong nonlocal coupling.
We consider a broad area vertical-cavity surface-emitting laser (VCSEL) subject to injection and to time-delayed feedback. We show that near the nascent optical bistability regime, the space-time dynamics of this device is described by a generalized Swift-Hohenberg equation with delay. We classify different regime of stability of the homogeneous steady states in terms of dynamical parameters. We show that the delay modifies strongly the stability domain of both periodic and localized structures solutions. Finally, we show that the delay feedback induces a spontaneous motion of bright peaks in one and in two-dimenional transverse plane. Bifurcation diagram associated with these localized structures is constructed.
Cavity solitons are controllable two-dimensional transverse Localized Structures (LS) in dissipative optical cavities.
Such LS have been suggested for use in optical data storage and information processing. Typically,
diffraction constrains the size of these light spots to be of the order of the square root of the diffraction coefficient of the system. Due to recent advances in the development of metamaterials, the diffraction strength in a
cavity could be controlled by adding a left-handed material layer in a Fabry-Perot resonator together with a traditional
nonlinear material. This system thus potentially allows for LS beyond the size limit imposed by natural
diffraction. However, when the diffraction strength becomes smaller, the non-local response of the left-handed
metamaterial starts to dominate the nonlinear spatiotemporal dynamics. Considering a typical linear non-local
response, we develop a mean-field model describing the spatiotemporal evolution of LS. First, the influence of this
non-local response on the minimal attainable width of the LS is studied [Gelens et al., Phys. Rev. A <b>75</b>, 063812
(2007)]. Secondly, we elaborate on the different possible mechanisms that can destabilize the LS, leading to
stable oscillations, expanding patterns, or making the LS disappear. Furthermore, we also show multiple routes
towards excitability present in the system. We demonstrate that these different regions admitting stationary,
oscillating or excitable LS unfold from two Takens-Bogdanov codimension-2 points [Gelens et al., Phys. Rev. A
Recent advances in the field of left-handed materials (LHM) have renewed the interest for the study of the propagation properties in these materials. In particular, many works have been devoted to the study of the Veselago plate, which constitutes the simplest linear system based on LHM. Recently, Zharov and co-workers introduced a description of the nonlinear properties of LHM. In particular, they suggested that LHM could be used as the basis of power limiters, all optical switches, etc. In this communication, we consider a Fabry-Perot cavity filled with two materials having refractive indexes of opposite signs, and driven by an external coherent field. Using the well-known mean-field approximation, we derive a partial differential equation describing the bidirectional propagation of light in this optical cavity. In the absence of reflection at the interface between the LHM and the RHM, our model is similar to the well-known LL-model. However, an important difference that appears when considering LHM is that the sign of the diffraction term can be reversed by varying the geometry. Besides the bistable behaviour of the homogeneous response curve of the system, our study reveals the existence of modulational instability. The wavelength of the spatial dissipative structure emerging from that instability can be tuned continuously by varying the diffraction coefficient.
Left-handed materials have attracted growing attention in recent years. Although many efforts have been concentrated on the study of linear devices filled with left-handed materials, it was only recently that the nonlinear optical properties and optical response of such materials have been described. However, no analytical description of light propagation in these media has been proposed. In this communication, we develop an analytical model describing the interaction of coherent electromagnetic radiation in structures containing both right- and left-handed materials with a Kerr type nonlinearity. We study the paraxial propagation of electromagnetic radiation in such structures. Using a multiple time and space scales analysis, we derive a system of two coupled nonlinear Schrödinger equations for the amplitudes of the fields, allowing for both forward and reflected waves to be present in the structure. Diffraction, group velocity dispersion and Kerr nonlinear effects are taken into account. Appropriate boundary conditions are derived. Finally, we point out the differences between propagation in right- and left-handed materials.