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Degenerate laser cavity have been used for second-harmonic generation [1] , [2] and spatially incoherent laser emission using solid-state gain medium and monolithic VCSEL with large area of excitation [3] , [4] . In this work we investigate cw spatial coherent states of a degenerate laser cavity using specially design 1/2-VCSEL based on III-V semiconductor technology, and high numerical aperture optical system, in order to be used in spatio-temporal localization of light also know as Light Bullets [5] . The external cavities implemented are made up two(four) lenses in 4-(-8)f self-imaging configurations. A Zeemax and optical physics study was carried out to design the optimized system of lenses to ensure high numerical aperture and high resolution down to wavelength scale with minimal aberrations for off axis operation. The Semiconductor Gain Mirror (1/2-VCSEL) have been designed based on GaAs materials at 1060nm wavelength and for efficient optical pumping at 800nm using commercial diode laser. The high reflectivity mirror is designed with an GaAs/AlAs bragg mirror and the optical gain is provided through a multi-quantum-well strained-balanced InGaAs/GaAsP active region. The 1/2-VCSEL was designed with the intent to minimize all waveguiding effect such as the thermal lens and strong self-guiding through gain saturation.

We present a generalization of the Haus master equation in which a dynamical boundary condition allows to describe complex pulse trains such as the Q-switched and harmonic transitions of passive mode-locking as well as the weak interactions between localized states. As an example, we investigate the influence of group velocity dispersion on the stability boundaries of the Q-switched regime. We compare our results with that of a time-delayed system., Comment: 4 pages, 5 figures

We analyze both theoretically and experimentally the impact of optical feedback on the dynamics of external-cavity mode-locked semiconductor lasers (VECSELs) operated in the long cavity regime [1]. In particular, by choosing certain ratios between the cavity round-trip time and the feedback delay, we show that feedback acts as a solution discriminator that either reinforces or hinders the appearance of one of the multiple harmonic arrangements of temporal localized structures. For the theoretical modeling, the delayed differential equation model [2] is extended by a term describing the optical feedback. Denoting by A the amplitude of the optical field, G the gain, and Q the saturable losses, the model reads\begin{align*} & \frac{{\dot A}}{\gamma } = \sqrt \kappa \exp \left[ {\frac{{1 - i{\alpha _g}}}{2}G\left( {t - {\tau _c}} \right) - \frac{{1 - i{\alpha _a}}}{2}Q\left( {t - {\tau _c}} \right)} \right]A\left( {t - {\tau _c}} \right) - A(t) + \eta {e^{i{{\Omega }}}}A\left( {t - {\tau _f}} \right),\tag{1} \\ & \dot G = {g_0} - {{\Gamma }}G - {e^{ - Q}}\left( {{e^G} - 1} \right)|A{|^2},\quad \dot Q = {q_0} - Q - s\left( {1 - {e^{ - Q}}} \right)|A{|^2}\tag{2}\end{align*}where time has been normalized to the SA recovery time, α g and α a are the linewidth enhancement factors of the gain and absorber sections, respectively, κ the fraction of the power remaining in the cavity after each round- trip, g 0 the pumping rate, Γ the gain recovery rate, q 0 is the value of the unsaturated losses which determines the modulation depth of the SA, τ c the round-trip time in the cavity, s the ratio of the saturation energy of the SA and of the gain sections and γ is the bandwidth of the spectral filter, η is the feedback rate, Ω is the feedback phase and τ f the round-trip time in the feedback loop. For the experimental realization, the gain medium consisting in 6 quantum wells embedded between a bottom totally reflective Bragg mirror and a top partially reflective Bragg mirror (½VCSEL) was considered; the ½VCSEL was then placed in an external cavity that was closed by a fast semiconductor saturable absorber mirror (SESAM) to operate the laser in the passive mode-locked regime (see fig. 1). In the absence of the optical feedback such a laser can generate so-called harmonic mode-locked solutions denoted by HML n where n is the number of equally spaced pulses per round-trip. The optical feedback induces that each pulse is followed by a train of small copies of itself. The size and position of this echo can be controlled via the feedback rate and the delay time, respectively. When an echo is placed close to the leading edge of another pulse, the pulse experiences less amplification by the gain medium as it is already depleted by the echo. This interaction can lead to the destruction of the main pulse. In consequence, the system settles on a solution where pulses and echos are well-separated and thus, do not interact. This mechanism is visualized in fig. 2 where a HML 4 and HML 2 are exposed to optical feedback and therefor, the output changes to a HML 3 and fundamental mode-locked solution, respectively. This behaviour was also observed in the experiment. In the further steps, a detailed bifurcation analysis is conducted where the influence of the position of the echo is investigated.

Passively mode-locked semiconductor VECSELs realized in a V-shaped cavity configuration ( Fig. 1a ) have proven to provide strong performance figures and are thus promising candidates for photonic applications. Furthermore, these lasers can also be a source of interesting pulsed dynamics, as the double gain pass within one round-trip in the external cavity favors the emergence of pulse clusters over harmonic mode-locking [1] . Here, we investigate this cavity effect in the regime of temporally localized states where the gain relaxation time has to be much shorter than the cavity length.

Spatial and temporal localized structures in optical cavities, also called Dissipative Solitons, have attracted a great deal of attention in the last two decades [1] , [2] . Beyond their fundamental interest, these structures can be used to code and process information bits in optical resonators. In general, two conditions appear to be necessary for a system to exhibit LSs: i) to have a large aspect-ratio and ii) to feature a multistable response. We have recently shown that an optically pumped passively mode-locked Vertical External Cavity Surface Emitting Laser (VECSEL) can be operated in a regime where mode-locked pulses becomes localized, i.e. they are individually addressable [3] , [4] . This regime requires long external cavity, i.e. cavity roundtrip time be significantly larger than gain recovery time, τ c >> τ g . Both the gain section and the saturable absorber mirrors needs to be properly engineered to provide multistability between different PML solutions with different number of pulses per roundtrip. Moreover, VECSEL geometry is very interesting because the external cavity can be designed in a self-imaging configuration which allow for reaching high-Fresnel number, hence paving the way also to spatial LSs [5] .

Passive-mode locking (PML) is a fascinating phenomenon allowing for the generation of ultrashort optical pulses. Semiconductor lasers are widely used as PML sources and Vertical External-Cavity Surface-Emitting Semiconductor Lasers (VECSELs) are explored in various applications that require stable lasing with high output power and good beam quality. Here, PML can be obtained by closing the external cavity with a semiconductor saturable absorber (SA) mirror (see Fig. 1 (a) ). Operating in the long cavity regime, where the round-trip time is much longer than the semiconductor gain recovery time, the PML pulses become independent and addressable temporal localized states (TLSs) below the lasing threshold [1] (see Fig. 1 (b) ).

Time-delayed dynamical systems materialize in situations where distant, point-wise, nonlinear nodes exchange information that propagates at a finite speed. However, they are considered devoid of dispersive effects, which are known to play a leading role in pattern formation and wave dynamics. We show how dispersion may appear naturally in delayed systems and we exemplify our result by studying theoretically and experimentally the influence of third order dispersion in a system composed of coupled optical micro-cavities. Dispersion induced pulse satellites emerge asymmetrically and destabilize the mode-locking regime.

In this work we show that a high finesse Fabry-Perot micro-cavity containing a Kerr medium, coupled to a external cavity and in presence of optical injection can give rise to phase-locked temporal dissipative solitons. These Localized Structures (LSs) may possess a strongly asymmetrical oscillating tail which is the result of Third Order Dispersion induced by the cavity. Dispersion is found to induce soliton molecules and tunable Frequency Combs. Because of the external locking to the injection field, the LSs are phase coherent with each other, and could be used for interferometric applications.

Delayed dynamical systems (DDSs) describe a large number of phenomena in nature and they exhibit a wealth of dynamical regimes such as localized structures (LSs) [1], fronts and chimera states. A fertile perspective lies in their interpretation as spatially extended diffusive systems which holds in the limit of long delays [2]. A strong limitation of DDSs for this aim comes from the difficulty of taking into consideration chromatic dispersion. As such, DDSs description of wave propagation is limited to low dispersive media and/or narrow-spectrum signals. Nevertheless, chromatic dispersion plays a leading role in many phenomena occurring during wave evolution [3]. Beyond second order dispersion (SOD), third order dispersion (TOD) is the lowest order non-trivial parity symmetry breaking effect, which leads to convective instabilities [4] and drifts.

The possibility of using light bullets (LBs), i.e. pulses simultaneously confined in the transverse and the propagation directions, has attracted a lot of interest in the last twenty years, both for fundamental and practical reasons. In this work, we review the recent theoretical, numerical, and experimental advances regarding the realization of LBs in semiconductor passively mode-locked lasers. It was recently shown [1] that phase invariant Temporal Localized Structures (LSs) can evolve from passive mode-locking (PML). The PML regime leads to the emission of temporal pulses much shorter than the cavity round-trip and it is achieved by combining two elements, a laser amplifier providing a gain and a nonlinear loss element, usually a saturable absorber (SA). The different dynamical properties of the SA and of the gain create a window for regeneration only around the pulse. For cavities with a large aspect ratio, as defined by the ratio of the gain recovery time and of the cavity length r = τ/ τ g , the PML pulses may become individually addressable and coexist with the off solution [1]. It was later established in which conditions the transverse profile of these temporal LSs can self-organize, see Fig. 1, yielding a new robust LB formation scenario [2]. We present an intuitive theory [2,3] that allows to understand the LBs as hybrids between the temporal localized structures and spatial diffractive autosolitons. Using the path following techniques within pde2path framework, we performed the bifurcation analysis of the Haus master equation. We were able to unveil a new mechanism of instability for the both temporal [4], and spatial LSs which also appears as an instability for the full three-dimensional problem.

Driven optical cavities widely used for generation of optical frequency combs are known to support the so-called temporal cavity solitons (CSs) manifesting themselves in the form of short optical pulses propagating along the cavity. A formation of temporal CSs in a standard silica fiber cavity [1] and in a continuous-wave driven nonlinear optical microresonators [2] were recently studied experimentally. In particular, the experimental investigation of the temporal CSs interaction performed in [1] indicates that due to a very fast decay of the CS tail stable soliton bound states (BS) are hardly observable. Here we demonstrate theoretically that the interaction of bright and dark temporal CSs is strongly modified in the presence of even very small high order chromatic dispersion terms. We show that the appearance of weakly decaying dispersive waves (sometimes referred to as a Cherenkov radiation, see Fig. 1) emitted by CSs leads to a strong enhancement of their interaction at large distances and, hence, strongly facilitates experimental observation of the BS. We present an analytical theory of the CSs interaction [3] and show that for the CSs interaction the Newton's 3rd Law is violated in the presence of third order dispersion. We consider dynamics of CSs in a driven optical cavity described by the well-known Lugiato-Lefever equation with high order dispersion terms. We derive the equations governing the slow time evolution of the coordinates of two well separated solitons interacting weakly via their exponentially decaying tails. We show that the interference between solitons and the Cherenkov radiation produces an oscillating pattern responsible for the formation of a large number of stable BS with large CSs separations (see Fig. 2). Furthermore, we demonstrate that the soliton interaction is very asymmetric: while one of the two solitons remains almost unaffected by the interaction force, another one is strongly modified by the interaction. Finally, we discuss the effect of forth and higher order dispersions on the CSs interaction and the influence of the high order dispersion on the interaction of dark CSs.

In this paper we report on the drift bifurcation of single dissipative soliton in a three-component reaction-diffusion system due to a change of its shape. We derive analytical expression for the velocity of the dissipative soliton as a function of an appropriate control parameters. To prove the correctness of the calculations, numerical simulations are carried out showing good agreement with the analytical predictions.