Temporal Localized Turing Patterns in Mode-Locked Semiconductor Lasers

: We show that large aspect-ratio Vertical External-Cavity Surface-Emitting Lasers 13 (VECSELs) with a saturable absorber can be operated in the regime of spatio-temporal mode- 14 locking. The emitted pulses exhibit a spatial profile resulting from the phase locking between an 15 axial plane-wave with a set of tilted waves having a hexagonal arrangement in the Fourier space. 16 We show that these pulsating patterns are temporally localized, i.e. they can be individually 17 addressed by modulating the optical pump.The theoretical analysis shows that the emergence of 18 these pulsating patterns is a signature of a Turing instability whose critical wave vector depends on 19 spherical aberrations of the optical elements. Our result reveals that large aspect-ratio VECSELs 20 offer unique opportunities for studying fully developed spatio-temporal dynamics.


23
Large aspect-ratio (Large Fresnel number) lasers [1,2] are a playground for studying pattern 24 formation ruled by paradigmatic partial differential equations [«-8] where light is structured both in time and space, has recently appeared [«6]. 48 In this paper we realize a spatio-temporal mode-locked VECSEL and we operate it in the regime 49 of temporal LS. The mode-locked pulses exhibit a spatial profile consisting of a combination of 50 an axial plane-wave with a set of tilted waves having a hexagonal arrangement in the Fourier 51 space. These plane waves are phase locked and their interference gives birth to an hexagonal 52 pattern in the near-field emission profile. We show that these spatio-temporal mode-locked 53 pulses can be individually addressed by shining short pump pulses, hence we call them temporal 54 localized patterns. Our theoretical analysis reveals that they arise from a Turing instability whose 55 critical wave vector is determined by spherical aberrations of the optical elements. near-field profiles are imaged on two CCD cameras. The near-field is also imaged on an array of 70 optical fibers for spatially resolved detection at 10 GHz bandwidth. Finally, the total emission is 71 monitored by a «« GHz bandwidth detection system and by an optical spectrum analyzer. 72 Fig. 1. a) Experimental set-up showing the L-shape VECSEL. 1 ȷ distance between the gain section and lens 1 , 2 ȷ distance between 1 and lens 2 , 3 ȷ distance between 2 and lens 3 , 4 ȷ distance between 3 and lens 4 , 5 ȷ distance between 4 and the SESAM, HRM= high reflectivity beam splitter (>99.5% at 1.060 nm). b) Calculated waist size of the fundamental Gaussian mode on the gain mirror (see Supplemental 1-2B, Eq. S5) as a function of the position of the SESAM ( = 5 − ) for ℎ =»0 mm and for two positions of ȷ = 2.5 mm (blue curve) and = −3.5 mm (red curve). For ℎ = 40 mm, SI condition condition is given byȷ 0 = −0.8 mm, 0 = −1.3 m, hence, in terms of Δ = − 0 , Δ = +3.3 mm (blue curve) and Δ = −2.7 mm (red curve). These numerical curves fit with good agreement the experimentally measured values of when the VECSEL is pumped at 2«0 mW. At this power thermal lens exhibits a focal length of ℎ ≈ »0mm [«9].
The coefficients of the matrix defined in Eq.
(2) can be used to calculate the stability of the Close to SI condition, stability of the cavity requires Δ > 0 when Δ > 0 and Δ < 0 when 118 Δ < 0, while the analysis of is shown in Fig. 1 b). We can notice that the waist of the  Finally, it is worth noting that, close to SI condition, the ABCD roundtrip matrix can be 125 approximated to (see Supplemental 1-2C)

127
As shown in Fig. 1 b), the analysis of ( ) from Eq.
(2) together with experimental measurements 128 of ( ) allow to determine the cavity parameters with rspect to SI condition. As | Δ |→ 0  The time behavior of the pattern of Fig. 2 is shown in Fig. « a). It features multistability where is the round-trip number and we defined the effective nonlinearity as The nonlinear response of the active material to a pulse is ℎ ( ) = (1 − − ) / . We define as where we define the following dimensionless parametersȷ the effective diffraction parameter 225˜= /(4 ) + 2 1,⊥ + 2 2,⊥ , the wavefront curvature˜= / and the aberration parameter pumping. It reveals that a homogeneous emission of temporal LS appear subcritically below the 255 lasing threshold. The corresponding C-shape is represented by the blue line in Fig. 5(b). When