Giovanna Tissoni, Franco Prati, Lorenzo Columbo, Luigi A. Lugiato, Reza Kheradmand INFM, Dipartimento di Fisica e Matematica, Università dell’ Insubria, Como, Italy Massimo Brambilla, INFM, Dip. Di Fisica Interateneo, Università e Politecnico di Bari, Italy Igor Protsenko Lebedev Physics Institute, Moscow, Russia Xavier Hachair, Massimo Giudici, Emilie Caloche, Francesco Pedaci, Stephane Barland, Jorge Tredicce Institut Non-Linéaire de Nice, France Cavity solitons in broad-area driven VCSELs above threshold

CAVITY SOLITONS Cavity solitons persist after the passage of the pulse, and their position can be controlled by appropriate phase and amplitude gradients in the holding field Phase profile Intensity xy Intensity profile In a semiconductor microcavity: Brambilla, Lugiato, Prati, Spinelli, Firth, Phys. Rev. Lett. 79, 2042 (1997). Nonlinear medium  nl Holding beamOutput field Writing pulses Possible applications: realisation of reconfigurable soliton matrices, serial/parallel converters, etc

The experiment at INLN First solid experimental demonstration of CS in semiconductor microresonators. S. Barland et al, Nature 419, 699 (2002)

Outline CS in driven VCSELs above threshold: existence, switching on/off (theory and numerical simulations) Model for CS in VCSELs above threshold, beyond the rate-equation approximation CS in driven VCSELs above threshold: existence, switching on/off (experiment)

In presence of diffraction, for a free running laser (FRL), the simple adiabatic elimination (AE) of material polarization is not a good approximation see Jakobsen, Moloney, Newell and Indik, Phys. Rev. A 45, 8129 (1992) (case of two-level laser, no injected signal) Necessity of a more accurate model Center Manifold AE see Oppo, D'Alessandro and Firth, Phys. Rev. A 44, 4712 (1991) (it solves the problem of short- wavelength instability for negative detunings) Complete model, able to describe also the material polarization dynamics.

The Model Equation for P: same structure as in Agrawal’s model: a complex parameter multiplies all the r.h.s. BUT the nonlinear term has the form (1+i  )ED, where D=N/N 0 -1, while in Agrawal’s model it was kEN Starting point: Yao, Agrawal et al., Opt. Comm. 119, 246 (1995) The rate equations (PRA model) are readily recovered with a standard adiabatic elimination of P:

Presence of “effective” damping  and detuning  in the macroscopic polarization equation. They depend on N and then on D.  (D) and  (D) can be derived: - from direct microscopic calculations (Agrawal’s model) - from fitting the calculated gain curves

Linear fit:

Linear stability analysis of the homogeneous steady-state: the lower intensity branch is Hopf unstable for K = 0 up to Depending on the injection frequency, the Hopf instability has different on the left non-homogeneous (K  0) characteristics: For      i.e.  input frequencies on the left (right) of the gain maximum, a non-homogeneous (K  0) (homogeneous (K = 0)) emission is favoured. (stationary state of the free-running laser above threshold)

Depending on current injection level two different scenarios are possible: 1. Only a portion of the lower intensity branch of the homogeneous steady-state curve is unstable 2. The whole lower intensity branch of the homogeneous steady-state curve is unstable In case 1, numerical simulations demonstrate that usual stable CS can be found, exactly as below the lasing threshold. The background is stable, and they can be written and erased in the usual way In case 2, numerical simulations demonstrate that stable CS can be found, but they are sitting on unstable oscillating background. Nevertheless, they can be written and erased in the usual way

1% above threshold 10% above threshold

20% above threshold 100% above threshold

Temporal behavior field phase field amplitude The background is rapidly oscillating, while the CS peak is almost constant, both in amplitude and phase Transverse profile of the field amplitude

CSs can be written and erased Erasing 5 CSsInjection of one CS

Experimental setup VCSEL: Broad-area (150  m VCSEL); SL: High power tunable laser ( nm); IC: Power supply stabilized better than 1 0 / 00 ; TC: Thermal stabilization better than 0,01°C; CCD: Ccd camera for near-field detection of the output intensity profile; DA: Photodiode for fast detection of local output intensity; PZT: piezo electric translator; FP: Fabry Perot interferometer (FSR 2.5 THz, Finesse 140); BC: Beam expander; AOM : Acousto Optical Modulator; OD: Optical diode

Spontaneous formation of 7 CS Region of Existence CS no HB (FRL) CS with HB Patterns Hom.

Successive switch-on and off of two CS Sequence Movie

CS motion The presence of a phase or amplitude gradient induces a motion of the CS. Here a phase gradient is used to displace the CS from its original position

Numerical simulations demonstrate that CSs persist above the laser threshold, possibly sitting on unstable background, and can be switched on and off in the usual way The usual AE of material polarization (rate equation approximation) is not a good approximation above the lasing threshold => Model describing the material polarization Conclusions Experimental results show the spontaneous formation of CS, the possibility of switching them on and off, and controlling their position, exactly as below threshold This Research is performed in the framework of the european project FunFAcs CB8-6-THU Room 2, Xavier Xachair, Physics of cavity solitons in semiconductors

The VCSEL  Realised by ULM University (PIANOS project) (R. Jaeger, T. Knoedl, M. Miller) Grabherr Jaeger, Miller Thalmaier, Heerlein, Michalzik, Ebeling, Phot. Tech.Lett. 10, 1061 (98)  Frequency: 980 nm  Diameter 150  m  Configuration: “Bottom emitting” Bragg Mirror GaAs Substrate Definition of threshold Bragg Mirror

Profil d’émission Sans Injection I=300 mAInjection I=300 mA  La répartition du courant fait que les bords du VCSEL « lasent » avant la partie centrale  Sous injection la partie centrale peut s’accrocher à la fréquence du laser maître. Profil transverse d’émission, résolu spectralement Étude expérimentale Introduction théorie expérience conclusion

Gradient de fréquence de résonance de la cavité En-dessous du seuil nominal Puissance d’injection constante Pas de 50 GHz in Étude expérimentale Introduction théorie expérience conclusion Le VCSEL présente un gradient de fréquence de résonance le long de sa section transverse (40 GHz/150 µm). Le long de cet axe le désaccord en fréquence entre la résonance de la cavité et le FM change. Formation de structures de fréquences spatiales différentes le long de cet axe. La ligne verticale séparant la région homogène de celle possédant des structures spatiales correspondant à la frontière d’instabilité modulationnelle

Adaptation du modèle aux conditions expérimentales Prise en compte de différents éléments suivants Le désaccord en fréquence entre le champ injecté et la résonance de la cavité est fonction de l’espace Les inhomogénéités introduites lors la fabrication du VCSEL (générées de manière aléatoire) Profil spatial du courant injecté : I I(x,y) Expérience  (x,y)  (x,y) = (  C -  in ) (x,y)/  +  (x,y) Distribution stochastique gaussienne de moyenne nulle Simulation numérique Étude expérimentale Introduction théorie expérience conclusion

Interpretation théorique  x (  m) structures spatiales Solitons de cavité En fixant la fréquence d’injection, on peut associer une valeur de désaccord en fréquence pour chaque point de l’axe horizontal. Domaine d’existence des SC La ligne verticale: séparation entre la région homogène et celle présentant des structures spatiales ensemble des points où  remplit les conditions de l’instabilité modulationnelle (IM). Les SC se développent au voisinage de la frontière de IM et de plus leurs existence est limitée à une bande verticale prés de cette ligne. Étude expérimentale Introduction théorie expérience conclusion  structures spatiales solitons ce cavité

Observation expérimentale Gradient de fréquence de résonance : 40 GHz/ 150 mm Faisceau de maintien (FM) : ~ 3 mW Faisceau d’écriture (FE) : ~ 1 µW Points d’application du FE Champ proche avant l’application du FE Champ proche après l’application du FE Résultats expérimentaux Introduction théorie expérience conclusion