Discrete solitons in coupled nonlinear active cavities

1. Discrete solitons in coupled nonlinear active cavities JAROSLAW E. PRILEPSKY Nonlinearity and Complexity Research Group Aston University, Birmingham, UK LENCOS, Sevilla, July 2012

2. In collaboration with: • Alexey Yulin • Magnus Johansson • Stanislav Derevyanko Centro de Fisica Teorica e Computacional, Universidade de Lisboa, Portugal Dep. of Physics, Chemistry and Biology (IFM), Linkoping University, Sweden NCRG, Aston University, Birmingham, UK

3. Light manipulation in optical cavities • Localized bright spots in driven optical cavities have received a great deal of attention because of their potential applications in information processing Ackemann, Firth, Oppo, 2009. • A relatively new area is the study of collective excitations in coupled nonlinear cavities (resonators): coupled waveguides with the facet mirrors.

4. Coupled-mode equations e.g. ,Peschel,Egorov, Lederer 2004

5. Discrete nonlinear coupled cavity models Discrete mean field equations for cavity arrays (a passive case!): A discrete Lugiato-Lefever model Peschel,Egorov, Lederer 2004 An effective model for quadratic cavity solitons Egorov, Peschel, Lederer 2005 A model with a saturable conservative nonlinearity Yulin, Champneys, Skryabin 2008 A model with saturable non-conservative nonlinearity Yulin, Champneys, 2010

6. Our model: active nonlinear media Active media: gain exceeds damping in the linear limit: δ<γ dissipative terms amplitude

7. Analysis: bistability as a starting point • Strategy: Set An=A (or set C=0, anticontimuum limit), and study the response curve P=P(|A|). • When we have a multivalued curve (bistability), we can find solitons as homoclinic connections between stable states. • Set An(T)=An+an(T) and linearize F(A+a(T)) with respect to an(T). Set an(T)=a exp(λT+iqn) + b* exp(λ*T-iqn) and study the resulting eigenvalues λn=λn(q,parameters). If there are any Re[λn]>0 – unstable, otherwise - stable. UNSTABLE H2 H1 BISTABILITY STABLE Seek for stable solitons at C≠0 in the bistable region starting from decoupled stable states H1 and H2

8. Response curve and spectrum. I

9. Bright solitons. I: C-snakes

10. Bright solitons. II: H1→H2→H1 M

11. Grey solitons. I: H2→H1→H2 Snaking diagrams C(|Amin|) and P(|Amin|) (for C=0.15)for the grey DCS corresponding to the H2-H1-H2 connection.

12. Inhomogeneous (periodic) background. I C=0.15

13. Inhomogeneous (periodic) background. II: Comparison Snaking diagrams P(|A|) for homogeneous H statesand {P(|Amax|), P(|Amin|)} for the periodic I-state. Bistability region for H-states is highlighted. C=0.15

14. Bright solitons. III: H1→I→H1 Snaking diagrams P(|Amax|) and M(P) for H1→I→H1 (bright) solitons, C=0.15; profiles of stable DCSs.

15. Grey solitons. III: H2→I→H2 Snaking diagrams P(|Amax|) and M(P) for H2→I→H2 (grey) solitons, C=0.15. Inset shows a stable solution profile.

16. Conclusions • We have found a zoo of stable DCS in coupled active lasing cavities. Aside from `usual' DCS, corresponding to the connections between homogeneous states H1 and H2, we have found a new type of DCS involving a periodic inhomogeneous I-state, which has not identified before in optical cavities. The existence of the great variety of stable DCS paves the way to the more versatile and sophisticated patterning and manipulation of transverse light distribution. Notably, the family of grey H2-I-H2 solutions marked as ■‘9' can be stable when the bistability of H-states is absent. • Further challenges: solitons in the absence of bistability, quasilinear solitons (no conservative nonlinearity), compare properties of new DCS with the usual ones (dynamics etc.) Other models (Lugiato-Lefever etc.) • Ref: http://arxiv.org/abs/1202.4660 (submitted to Optics Letters) THANK YOU!