1. Spring School on Solitons in Optical Cavities Cargèse, May 8-13, 2006 Introduction to Cavity Solitons and Experiments in Semiconductor Microcavities Luigi A. Lugiato Dipartimento di Fisica e Matematica, Università dell’Insubria, Como (Italy) Collaborators: - F. Prati, G. Tissoni, L. Columbo (Como) - M. Brambilla, T. Maggipinto, I.M. Perrini (Bari) - X. Hachair, F. Pedaci, E. Caboche, S. Barland, M. Giudici, J.R. Tredicce, INLN (Nice) - R. Jaeger (Ulm) - R. Kheradmand (Tabriz) - M. Bache (Lingby) - I Protsenko (Moscow)

2. Program - Science behind Cavity Solitons: Pattern Formation (Maestoso) - Cavity Solitons and their properties (Andante con moto) - Experiments on Cavity Solitons in VCSELs (Allegro) Future: the Cavity Soliton Laser (Allegro vivace) Future: the Cavity Soliton Laser (Allegro vivace) - My lecture will be “continued” by that of Willie Firth -The lectures of Paul Mandel and Pierre Coullet will elaborate the basics and the connections with the general field of the basics and the connections with the general field of nonlinear dynamical systems nonlinear dynamical systems - The other lectures will develop several closely related topics

3. Optical Pattern Formation x y z

4. Optical pattern formation: old history - J. V. Moloney - A huge and relevant Russian literature (A.F. Sukhov, N.N. Rosanov, I. Rabinovich, S.A. Akhmanov, (A.F. Sukhov, N.N. Rosanov, I. Rabinovich, S.A. Akhmanov, M.A. Vorontsov etc.) M.A. Vorontsov etc.) In particular, N.N. Rosanov introduced and studied “Diffractive Autosolitons”, In particular, N.N. Rosanov introduced and studied “Diffractive Autosolitons”, precursors of Cavity Solitons precursors of Cavity Solitons A recent review: LL, Brambilla, Gatti, Optical Pattern Formation in Advances in Atomic, molecular and optical physics, Vol. 40, p 229, Academic Press, 1999

5.  The mechanism for spontaneous optical pattern formation from a homogeneous state is a, exactly as e.g. in hydrodynamics, state is a modulational instability, exactly as e.g. in hydrodynamics, nonlinear chemical reactions etc nonlinear chemical reactions etc  : a random initial spatial modulation, on top of  Modulational instability: a random initial spatial modulation, on top of a homogeneous background, grows and gives rise to the formation of a a homogeneous background, grows and gives rise to the formation of a pattern pattern  In optical systems the modulational instability is produced by the combination of nonlinearity and diffraction. combination of nonlinearity and diffraction. In the paraxial approximation diffraction is described by the transverse Laplacian: In the paraxial approximation diffraction is described by the transverse Laplacian: Nonlinear Optical Patterns 1

6. Nonlinear Optical Patterns 2  Optical patterns may arise  in propagation  in propagation  in systems with feedback, as e.g.  in systems with feedback, as e.g. optical resonators or single feedback mirrors optical resonators or single feedback mirrors  Optical patterns arise for many kinds of nonlinearities (  (2),  (3), semiconductors, photorefractives..) photorefractives..)  There are stationary patterns and time-dependent patterns of all kinds

7. Input Nonlinear Medium  nl Cavity Output (Plane Wave ) ( Pattern ) Nonlinear Medium  nl Nonlinear media in cavities HexagonsHoneycombRolls Optical Pattern Formation

8. MEAN FIELD MODELS normalized slowly varying envelope of the electric field cavity damping rate (inverse of lifetime of photons in the cavity) input field of frequency  0 cavity detuning parameter,  c = longitudinal cavity frequency nearest to  0 cubic, purely dispersive, Kerr nonlinearity diffraction parameter Mean field limit  thin sample, high cavity finesse The purely dispersive case (L.L., Lefever PRL 58, 2209 (1987)) The purely dispersive case (L.L., Lefever PRL 58, 2209 (1987))

10. The purely absorptive case (LL, Oldano PRA 37, 96 (1988) ; The purely absorptive case (LL, Oldano PRA 37, 96 (1988) ; Firth, Scroggie PRL 76, 1623 (1996)) Firth, Scroggie PRL 76, 1623 (1996)) saturable absorption, C = bistability parameter MEAN FIELD MODELS as “simple” as pattern formation models in nonlinear chemical reactions, hydrodynamics, etc. The “ideal” configuration for mean field models (mean field limit, plane mirrors) has been met in broad area VCSELs (Vertical Cavity Surface Emitting Lasers).

12. Kerr slice with feedback mirror (Firth, J.Mod.Opt.37, 151 ( 1990)) F thin Kerr slice B | F Plane Mirror - Crossing the Kerr slice, the radiation undergoes phase modulation. - In the propagation from the slice to the mirror and back, phase modulation is converted into an amplitude modulation is converted into an amplitude modulation - Beautiful separation between the effect of the nonlinearity and that of diffraction, only one forward-backward propagation  Simplicity diffraction, only one forward-backward propagation  Simplicity - Strong impact on experiments

13. 11 11 1 0 00 0 Encoding a binary number in a 2D pattern?? Problem: different peaks of the pattern are strongly correlated

14. The solution to this problem lies in the concept of Localised Structure The concept of Localised Structure is general in the field of pattern formation: - it has been described in Ginzburg-Landau models (Fauve Thual 1988) and Swift-Hohenberg models (Glebsky Lerman 1995), and Swift-Hohenberg models (Glebsky Lerman 1995), - it has been observed in fluids (Gashkov et al., 1994), nonlinear chemical reactions (Dewel et al., 1995), in vibrated granular layers (Tsimring reactions (Dewel et al., 1995), in vibrated granular layers (Tsimring Aranson 1997; Swinney et al, Science) Aranson 1997; Swinney et al, Science)

15. 1D case Spatial structures concentrated in a relatively small region of an extended system, created by stable fronts connecting two spatial structures coexisting in the system Solution: Localised Structures Theory 1D: P. Coullet, C. Riera, and C. Tresser, Phys. Rev. Lett. 84, 3069-3072 (2000)

16. 1D case Spatial structures concentrated in a relatively small region of an extended system, created by stable fronts connecting two spatial structures coexisting in the system Solution: Localised Structures Theory 1D: P. Coullet, C. Riera, and C. Tresser, Phys. Rev. Lett. 84, 3069-3072 (2000)

17. Localised Structures Tlidi, Mandel, Lefever

18. - Localised structure = a piece of a pattern - The scenario of localised structures corresponds to a pattern “broken in pieces” “broken in pieces” E.g. a Cavity Soliton corresponds to a single peak of a hexagonal pattern ( Firth, Scroggie PRL 76, 1623 (1996)) - -WARNING: there is a smooth continuous transition from a pattern (in the rigid sense of complete pattern or nothing at all) to a scenario of independent localised structures (see e.g. Firth’s lecture)

19. Program - Science behind Cavity Solitons: Pattern Formation (Maestoso) - Cavity Solitons and their properties (Andante con moto) - Experiments on Cavity Solitons in VCSELs (Allegro) Future: the Cavity Soliton Laser (Allegro vivace) Future: the Cavity Soliton Laser (Allegro vivace) - My lecture will be “continued” by that of Willie Firth -The lectures of Paul Mandel and Pierre Coullet will elaborate the basics and the connections with the general field of the basics and the connections with the general field of nonlinear dynamical systems nonlinear dynamical systems - The other lectures will develop several closely related topics

20. Intensity xy CAVITY SOLITONS The cavity soliton persists after the passage of the pulse. Each cavity soliton can be erased by re-injecting the writing pulse. Intensity profile Nonlinear medium  nl Holding beam Output field Writingpulses - Cavity solitons are independent of one another (provided they are not too close to one another) and of the boundary. close to one another) and of the boundary. - Cavity solitons can be switched on and off independently of one another. - What is the connection with standard solitons?

21. Solitons in propagation problems Temporal Solitons: no dispersion broadening z “Temporal” NLSE: dispersion propagation Solitons are localized waves that propagate (in nonlinear media) without change of form Spatial Solitons: no diffraction broadening “Spatial” NLSE: 1D 2D x y z diffraction

22. Cavity Solitons are dissipative ! E.g. they arise in the LL model, which is equivalent to a “dissipative NLSE” diffraction dissipation Dissipative solitons are “rigid”, in the sense that, once the values of the parameters have been fixed, they have fixed characteristics (height, radius, etc)

23. Typical scenario: spatial patterns and Cavity Solitons Honeycomb pattern Roll pattern Cavity Solitons

24. On/off switching of Cavity Solitons - Coherent switching: the switch-on is obtained by injecting a writing beam in phase with the holding beam; the switch-off by injecting a writing beam in phase with the holding beam; the switch-off by injecting a writing beam in opposition of phase with respect to the writing beam in opposition of phase with respect to the writing beam - Incoherent switching: the switch-on and the switch-off are obtained independently of the phase of the holding beam. independently of the phase of the holding beam. E.g. in semiconductors, the injection of an address beam with a frequency E.g. in semiconductors, the injection of an address beam with a frequency strongly different from that of the holding beam has the effect strongly different from that of the holding beam has the effect of creating carriers, and this can write and erase CSs. of creating carriers, and this can write and erase CSs. (See Kuszelewicz’s lecture) The incoherent switching is more convenient, because it does not require control of the phase of the writing beam ~2ns CS on CS off CS off CS off ~5 ns CS on

25. Motion of Cavity Solitons KEY PROPERTY: Cavity Solitons move in presence of external gradients, e.g. 1)Phase Gradient in the holding beam, 2)Intensity gradient in the holding beam, 3)temperature gradient in the sample, In the case of 1) and 2) usually the motion is counter-gradient, e.g. in the case of a modulated phase profile in the holding beam, each cavity soliton tends to move to the nearest local maximum of the phase A complete description of CS motion, interaction, clustering etc. will be given in Firth’s lecture. Phase profile Possible applications: realisation of reconfigurable soliton matrices, serial/parallel converters, etc

26. Experiments on Cavity Solitons - - in macroscopic cavities containing e.g. liquid crystals, photorefractives, saturable absorbers - in single feedback mirror configuration (Lange et al.) - in semiconductors The semiconductor case is most interesting because of: - - miniaturization of the device - - fast response of the system Review articles on Cavity Solitons - - L.A.L., IEEE J. Quant. Electron. 39, 193 (2003). - - W.J. Firth and Th. Ackemann, in Dissipative solitons, Springer Verlag (2005), p. 55-101.

27. Program - Science behind Cavity Solitons: Pattern Formation (Maestoso) - Cavity Solitons and their properties (Andante con moto) - Experiments on Cavity Solitons in VCSELs (Allegro) Future: the Cavity Soliton Laser (Allegro vivace) Future: the Cavity Soliton Laser (Allegro vivace) - My lecture will be “continued” by that of Willie Firth -The lectures of Paul Mandel and Pierre Coullet will elaborate the basics and the connections with the general field of the basics and the connections with the general field of nonlinear dynamical systems nonlinear dynamical systems - The other lectures will develop several closely related topics

28. Nature 419, 699 (2002) The experiment at INLN (Nice) and its theoretical interpretation was published in

29. Tunable Laser CCD Holding beam Writing beam Detector linear array VCSEL BS aom C L C MM Experimental Set-up S. Barland, M. Giudici and J. Tredicce, Institut Non-lineaire de Nice (INLN) BS: beam splitter, C: collimator, L: lens, aom: acousto-optic modulator

30. Active layer (MQW) E R Bottom Emitter (150  m) Features 1) Current crowding at borders (not critical for CS) 2) Cavity resonance detuning  (x,y) 3) Cavity resonance roughness (layer jumps) See R.Kuszelewicz et al. "Optical self-organisation in bulk and MQW GaAlAs Microresonators", Phys.Rev.Lett. 84, 6006 (2000) n-contact The VCSEL Th. Knoedl, M. Miller and R. Jaeger, University of Ulm Bragg reflector GaAs Substrate E In p-contact

31. Experimental results In the homogeneous region: formation of a single spot of about 10  m diameter Observation of different structures (symmetry and spatial wavelength) in different spatial regions Interaction disappears on the right side of the device due to cavity resonance gradient (400 GHz/150  m, imposed by construction) Intensity (a.u.) x (  m) Frequency (GHz) x Above threshold, no injection (FRL) Intensity (a.u.) x (  m) Frequency (GHz) x Below threshold, injected field

32. Experimental demonstration of independent writing and erasing of 2 Cavity Solitons in VCSELS below threshold, obtained at INLN Nice S. Barland et al, Nature 419, 699 (2002)

33. E = normalized S.V.E. of the intracavity field E I = normalized S.V.E. of the input field N = carrier density scaled to transp. value  = cavity detuning parameter  = linewidth enhancement factor 2C = bistability parameter Where Choice of a simple model: it describes the basic physics and more refined models showed no qualitatively different behaviours.  (x,y) = (  C -  0 ) /  +  (x,y) Broad Gaussian (twice the VCSEL) The Model M. Brambilla, L. A. L., F. Prati, L. Spinelli, and W. J. Firth, Phys. Rev. Lett. 79, 2042 (1997). L.Spinelli, G.Tissoni, M. Brambilla, F. Prati and L. A. L., Phys.Rev.A 58, 2542 (1998)

34. Theoretical interpretation x (  m) 0 37.5 75 112.5 150 -2.25 -2.00 -1.75 -1.50 -1.25  Patterns (rolls, filaments) Cavity Solitons The vertical line corresponds to the MI boundary CS form close to the MI boundary, on the red side

35. Pinning by inhomogeneities Broad beam only Experiment Add local perturbation Broad beam only Cavity Solitons appear close to the MI boundary, Final Position is imposed by roughness of the cavity resonance frequencyNumerics  (x,y)

36. 7 Solitons: a more recent achievement X. Hachair, et al., Phys. Rev. A 69, 043817 (2004).

37. CS can also appear spontaneously........... In this animation we reduce the injection level of the holding beam starting from values where patterns are stable and ending to homogeneous solutions which is the only stable solution for low holding beam levels. During this excursion we cross the region where CSs exist. It is interesting to see how pattern evolves into CS decreasing the parameters. Qualitatively this animation confirms the interpretation of CS as “elements or remains of bifurcating patterns”. Experiment Numerics

38. Depending on current injection level two different scenarios are possible (Hachair et al. IEEE Journ. Sel. Topics Quant. Electron., in press) 5% above threshold 20% above threshold VCSEL above threshold

39. Despite the background oscillations, it is perfectly possible to create and erase solitons by means of the usual techniques of WB injection

40. Program - Science behind Cavity Solitons: Pattern Formation (Maestoso) - Cavity Solitons and their properties (Andante con moto) - Experiments on Cavity Solitons in VCSELs (Allegro) Future: the Cavity Soliton Laser (Allegro vivace) Future: the Cavity Soliton Laser (Allegro vivace) - My lecture will be “continued” by that of Willie Firth -The lectures of Paul Mandel and Pierre Coullet will elaborate the basics and the connections with the general field of the basics and the connections with the general field of nonlinear dynamical systems nonlinear dynamical systems - The other lectures will develop several closely related topics

41. Cavity Soliton Laser - A cavity soliton laser is a laser which may support cavity solitons (CS) even without a holding beam : simpler and more compact device! even without a holding beam : simpler and more compact device! CS are embedded in a dark background: maximum visibility. - In a cavity soliton laser the on/off switching must be incoherent - A cavity soliton emits a set of narrow be18ams (CSs), the number and position of which can be controlled position of which can be controlled CSL

42. The realization of Cavity Soliton Lasers is the main goal of the FET Open project FunFACS. - CW Cavity Soliton Laser - Pulsed Cavity Soliton Laser (Cavity Light Bullets) Approaches: - Laser with saturable absorber - Laser with external cavity or external grating LPN Marcoussis INLN Nice INFM Como, Bari USTRAT Glasgow ULM Photonics LAAS Toulouse

43. Conclusion Cavity Solitons are interesting !

44. Control of two independent spots Spots can be interpreted interpreted as CS 50  W writing beam (WB) in b,d. WB-phase changed by  in h,k All the circled states coexist when only the broad beam is present