1. Università degli studi dell’Insubria Como, 22 settembre, 2005 The hunt for 3D global or localized structures in a semiconductor resonator Ph.D student: Lorenzo Columbo Supervisor: Prof. Luigi Lugiato External supervisor: Prof. Massimo Brambilla (Politecnico di Bari) The hunt for 3D global or localized structures in a semiconductor resonator Ph.D student: Lorenzo Columbo Supervisor: Prof. Luigi Lugiato External supervisor: Prof. Massimo Brambilla (Politecnico di Bari)

2. OutlineOutline Short introduction. 2D and 3D structures localization in a dissipative optical system: Cavity Solitons in a VCSEL below lasing threshold and Cavity Light Bullets in a nonlinear resonator filled with a two level system. Future agenda and Conclusions Beyond the Single Longitudinal Mode Approximation: The dynamical model, the Linear Stability Analysis and the first numerical results. Fully localized structure in a self-focusing passive regime. 3D Pattern formation in a semiconductor resonator driven by a coherent injected field.

3. The spontaneous formation in the transverse profile of the field emitted by a Vertical Cavity Surface Emitting Laser (VCSEL) driven by a coherent injected field and slightly below lasing threshold of highly spatial correlated structures (global structures) or that of independent, isolated intensity peaks (localized structures or Cavity Solitons (CSs)) represents a valid example of Pattern formation in Optics. P attern formation in a semiconductor resonator E R (reflected field) Active layer (MQWs GaAs-GaAlAs) n-contact Bragg reflector E In E In ) (injected field plane wave) p-contact n-contact VCSEL device (Bottom Emitter) GaAs Substrate ~150  m ~ m~ m Rolls Numerical simulations: Global and localized structures in |E R | 2 transverse profile Transverse intensity field profile on the exit window Honeycombs Cavity Solitons

4. Even if a complete physical interpretation is still missing, from a fundamental point of view these phenomena result from the competition\balance between linear and nonlinear effects in the radiation-matter interaction combined with the resonator’s feedback\dissipation action. A pplications of Cavity Solitons to the optical information technology Nonlinear effects: self-focusing, saturable absorption.. Linear effects: diffraction Resonator’s action: feedback, dissipation  - -  - -  |ER|2|ER|2 x y We don’t have it in the Spatial Solitons case Intensity field profile of a single CS x y Ideal scheme of a binary optical memory. Es: CSs based parallel optical memory From a mere applicative point of view, since CSs can be externally excited, erased and drifted by means of suitable addressing beams (as it has been predicted and very recently observed ( Nature 419 699, 2002 )), these micro pixels are candidate to realize all optical devices for parallel information storage and processing. Nonlinear medium Gaussian pulse Plane wave E I ERER

5. The previous results are valid in the Single Longitudinal Mode Approximation (SLMA) according to which the field profile is uniform in the propagation direction in all the system’s configurations. Although this condition is well verified in a VCSEL for example, we could ask what would happen in the longitudinal field profile when it is not fulfilled (long cavities, high values of mirrors’ transmissivity etc.) B eyondSingle Longitudinal Mode Approximation!! B eyond Single Longitudinal Mode Approximation!! x y z Plane wave z ? A. Is it possible to observe spontaneous 3D confinement? B. In this case could we externally control these new fully localized structures like what happens with CSs? ? ? CS: Transverse localization.

6. Starting from 2002 we tried to answer to the previous questions by considering first a optical prototype: 1. B eyond SLMA: two level system EIEI ERER ETET 4 3 1 T=0 00 aa nn mm  cc  a = atomic transition frequency  0 = input field frequency  n = generic empty cavity mode Unidirectional ring resonator filled with a vapour of two level atoms and driven by a coherent injected beam 2 Nonlinear medium E I = injected field (Plane wave) E T = transmitted field E R = reflected field Nonlinear medium EE

7. 1.1. M axwell-Bloch equation Maxwell-Bloch equation describing system dynamics in the slowly varying envelope approximation (SVEA), paraxial approximation and after adiabatic elimination of the atomic variables, but without introducing any hypothesis on the longitudinal field profile: Boundary condition: E = normalized envelope of the intracavity field Y = normalized envelope of the injected field L A = resonator length=nonlinear medium length  a = normalized absorption coefficient at resonance T = transmission coefficient (R=1-T)  =(  a -  0 )/    a0 = (  c -  0 ) L A /c z = normalized propagation coordinate x, y = normalized transverse coordinate t = normalized time coordinate

8. 1.2. 3D global and 3D localized structures We predicted in this case in more than one parametric regime the formation of 3D global structures and 3D self-confinement phenomena ( M. Brambilla et al., PRL 93 2042, 2004 ). We named Cavity Light Bullets (CLBs) the fully localized structures travelling along the resonator with a constant spatial dimensions and a constant period. Isosurface plot of the intracavity intensity field profile For particular value of the injected field Y some filaments contract into stable fully localized structures. We then answered to question A a) 3D filaments b) Cavity Light Bullets

9. We also demonstrated the possibility to excite or erase one or more independent CLBs by means of suitable addressing beams in both “parallel” or “serial” configurations. We also managed to drift transversely a single CLB. x z CLB external control (2+1) dim Switching on of one or more CLBs We then answered to question B a) Switching on of a single CLBb) Switching on of two parallel CLBsc) Switching on of a CLB train

10. OutlineOutline Short introduction. 2D and 3D structures localization in a dissipative optical system: Cavity Solitons in a VCSEL below lasing threshold and Cavity Light Bullets in a nonlinear resonator filled with a two level system. Future agenda and Conclusions Beyond the Single Longitudinal Mode Approximation: The dynamical model, the Linear Stability Analysis and the first numerical results. Fully localized structure in a self-focusing passive regime. 3D Pattern formation in a semiconductor resonator driven by a coherent injected field.

11. Why are semiconductors devices relevant? They have a very fast dynamics Their growth and hence their energy spectrum can be controlled with high precision degree They can be miniaturized They already have broad applications in telecommunications and optoelectronics etc. ? ? 2. B eyond SLMA: semiconductor resonators

12. Unidirectional ring resonator filled with a Bulk or a Multi Quantum Wells (MQWs) semiconductor sample Phenomenological model used to describe radiation matter-interaction by means of a complex susceptibility: where in the passive configuration: while in the active configuration: with, N= carrier density, N 0 = transparency carrier density, A = absorption\gain coefficient, n = background refractive index,  e = half width of the excitonic absorption line,  e = central frequency of the excitonic absorption line,  = linewidth enhancement factor Fast carrier dynamics → we cannot adiabatically eliminate carrier dynamics  ETET 4 3 1 T=0 EIEI 2 Nonlinear medium EE EE ERER

13. 2.1. M axwell-Bloch equations D = normalized difference between N and N 0   = normalized cavity detuning d = diffusion coefficient  = nonradiative decay constant  photon life time  = pump parameter (  1→laser) (1a) (1b) Maxwell-Bloch equations describing system dynamics within the rate equation, SVEA and paraxial approximations but without introducing any hypothesis on the longitudinal field profile: Boundary condition:

14. Intensity field profile for a fixed (x,y) value In the general case, the nonlinear character of eq. (1a)-(1b) prevents us to solve them analytically  Equating to zero the time derivatives and the terms with the laplacian operators we can get numerically their stationary and transversely homogeneous solutions X s, where X stands for the generic variable; it turns out these solutions are associated to a non uniform field profile in the propagation direction. Linear Stability Analysis We study the stability of X s against spatially modulated perturbations by applying a well known approximate method: the Linear Stability Analysis (LSA).

15. Contrary to what happens in the Single Longitudinal Mode Approximation, the a priori unknown z-dependence of X s introduces an high degree of complexity in LSA. In particular, looking for solution of Maxwell-Bloch equations in the form: with  X<<X s we cannot derive for each modal  amplitude an equation for describing its the temporal evolution. Then, extending the results obtained in the two level system, we adopt an alternative approach: we expand  X on the transverse Fourier basis keeping implicit its z-dependence : Thus we get for each (k x, k y ) a system of two linear ordinary differential equations for, that we rename, and its c.c. Step1 Fourier expansion

16. The easiest way to proceed at this point is to introduce the polar  representation  of  E s  and  E 0 where  s,  s, ,  are real quantities. After some simple algebra, we then get: where k  =(k x 2 + k y 2 ) 1/2,  (z)  2 s (z) and r and u are auxiliary variables linked to  and  trough the linear transformation: Step 2 (2a) (2b)

17. Combining eq. (2a) and (2b) we derive the following 2 nd order linear differential equation for r  where the coefficients A, B, H i, i=1..5 depend on the physical parameters, X s, k  and also on. We then reduce the initial problem to that of solving the previous equation. Since the complicated expressions of the polynomial coefficients it is not easy (possible?) to find an analytical general solution of this equation; on the other hand we can approximate it around the regular singular point  as superposition of the two linearly independent series solutions r 1 and r 2  where c 1, c 2  are arbitrary complex constants. We also get for u from (2a) and (2b): Step 3

18. Keeping fixed the other quantities, it represents a nonlinear implicit equation for the which, solving our LSA problem, tells us how evolves in time the generic transverse mode amplitude of the perturbation: given a stationary transversely homogeneous state it is unstable if exists at least one “zero” of the function C with Re  0. Step 4 NOTE: We checked the validity of this LSA by reproducing the results obtained in the SLMA framework for a parametric regime which fulfils the SLMA conditions. Finally, taking into account the boundary conditions for r and u, we get an algebraic homogeneous system for c1 and c2 which admits non trivial solution if and only if the following condition is fulfilled: Observation: From a computational point of view the implicit character of the equation C( )=0 represents and additional CPU time consuming factor. This forced us to implement a parallel numerical code for LSA.  

19. 2.2. Numerical simulations Using the indications of LSA we study system dynamics by numerical integration of eq. (1a)-(1b) with the relative boundary condition. For this highly demanding computational task we developed a parallel code. First stage of investigation: close to the atomic system In this first stage of investigation we take advantage of the results obtained in the atomic system; in fact from eq. (1a)-(1b) neglecting diffusion (d=0) and after adiabatic elimination of the carrier density variable in the limit  >>1, we get in the passive case: which is formally equivalent to the equation describing system dynamics in the atomic case. Following this analogy, the idea is to look for fully confined structures still using eq. (1a)-(1b) with d=0 and  >>1 in parametric regimes linked to those in which we observed CLBs through relations: Two level system Do you remember?

20. Self-defocusing passive parametric regimes  e =2,  0 =-0.3,  T=0.1,d=0 and  →300.0) Longitudinal filaments……and fully localized structures When, as happens in this case, d=0 and Im  <<1<<  the instability domains are independent from . In spite of this,  still plays a role in influencing system’s dynamical evolution. We observe in the general case highly correlated longitudinal filaments at regime. Stationary transversely homogeneous states curves (independent from  ) x z Two fully localized structures for  =50.0 Two stable fully localized structures obtained by cutting two longitudinal filaments and letting the system evolve. They are not independent from each other. Y=22.975

21. Self-focusing passive parametric regimes  e =-2,  0 =-0.4,  T=0.1,d=0 and  →500.0) Longitudinal filaments …… ((2+1) dim) When, as happens in this case, d=0 and Im  <<1<<  the instability domains are independent from . In spite of this,  still plays a role in influencing system dynamical evolution. Longitudinal filaments  ~300  Stationary transversely homogeneous states curves (independent from  ) 17.011.0 Y x z ((2+1) dim)

22. …….. and fully confined structures …….. and fully confined structures Although we still don’t observe phenomena of spontaneous structures localization in the propagation direction, we proved that a longitudinal confined portion of a longer filament represents a stable system’s solution for a sizable interval of Y values. Fully localized structure (  ~300) x z Intensity field profile on the exit window ! The localized structure disappears when we decrease  under a certain threshold

23. ObservationObservation Since we have:  =  nr /k p where  nr is the nonradiative carrier density decay constant, while k p =cT/nL A is the inverse of the photon life time, we can think to get large value of  by increasing L A. a) We could consider for example Edge Emitter configurations l~250  b) Moreover, we can get the same result by considering the case: medium length ≠ cavity length and increasing the latter Semiconductor sample Input mirrorOutput mirror l LALA EiEi ETET L A <<l a)b) We put  >>1

24. OutlineOutline Short introduction. 2D and 3D structures localization in a dissipative optical system: Cavity Solitons in a VCSEL below lasing threshold and Cavity Light Bullets in a nonlinear resonator filled with a two level system. Future agenda and Conclusions Beyond the Single Longitudinal Mode Approximation: The dynamical model, the Linear Stability Analysis and the first numerical results. Fully localized structure in a self-focusing passive regime. 3D Pattern formation in a semiconductor resonator driven by a coherent injected field.

25. Future Agenda 9.65 t.u.1.3 t.u. 0.35 t.u. x z Switching on process of a single localized structure by using an external addressing beam Passive case Looking for fully localized structures in less critical parametric domains and\or configurations. Systematic study of the proprieties of these localized structures in analogy to what we did for CLBs.

26. Future Agenda Active case We can also consider the active configurations below or above lasing threshold. In the laser configuration should we remove the rate equation approximation? (We already did some calculations about this! ) ? [ ] + = The Vertical External Cavity Surface Emitting Laser (VECSEL) configuration is for example already used to produce mode locking laser operation.→ (longitudinal localization) The VCSEL configuration is already used to observe CSs even above lasing threshold.→ (transverse localization) ?

27. ConclusionsConclusions Funfacs European Project Even in this case, the first numerical investigations show the existence of both global (longitudinal filaments) and fully localized structures; the latter are candidate to be the semiconductor analogous of CLBs. We extended the model describing system dynamics in SLMA to include a generic intracavity field longitudinal profile; we applied to the LSA of the stationary and transversely homogeneous field configurations a semianalytical approach developed in a prototype. We looked for 3D pattern formation and 3D self-confinement in a semiconductor resonator driven by a coherent injected field. This work is supported by the Funfacs ( Fundamentals, Functionalities and Applications of Cavity Solitons)- F.E.T. VI P.Q. UE. In the framework of this European collaboration with many other theoretical and experimental research units, I am going to join the Computational Nonlinear and Quantum Optics group at the University of Strathclyde (Scotland) for a visiting period of six months.

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