Final Report Summary - INDIGO (Interaction of Nonlinearity and Disorder: Gateways to Optics) Project background: One of the features which makes bulk and planar optical waveguides as well as the discrete waveguide lattices such an attractive contemporary area of research is that it is a rare example of a physical system where the fundamental concepts of seemingly very diverse areas of physics (such as quantum mechanics, statistical physics, Bose-Einstein Condensation) suddenly come together and literally “shine through”. Anderson localization, Berezinski-Kosterlitz-Thouless transition (BKT), quantum wave-particle duality, Klein particle tunnelling, bound states in continuum - these are just a few examples of the long list of the diverse phenomena that have direct analogue in optical waveguide systems. Some of these optical analogues have already been observed experimentally, others await implementation. However coupled optical waveguides are not just an exciting playground for theoretical and experimental physicists – they have direct applications in various areas of modern optoelectronics including all-optical routing, switching, time gating, field couplers etc. Yet another important example of the interaction of the nonlinearity and disorder is given by continuous optical beam propagation in nonlinear disordered waveguides and its interaction with disordered defects (singular or forming a lattice) or nonlinear focusing using the intensity feedback and spatial light modulators. The INDIGO project resulted in significant broadening of our understanding of all these challenging areas. Scientific results and impact: One of the main results obtained in the project is a full and self-consistent theoretical description of the equilibrium thermal state of a system of coupled optical waveguides with different states of polarization in the presence of the cross-phase modulation (XPM) and the four wave mixing (FWM). A universal analytical form of the equilibrium probability density functions for the Stokes parameters was obtained and it was studied how this form depends on the material parameters and the initial conditions. It was found that irrespective of the initial conditions the discrete waveguide system tends to evolve to circularly polarization state provided that the nonlinearity is strong and the waveguide is large enough. This research was the first attempt at studying polarization effects on thermalization in discrete optical waveguide systems which significantly increases its research impact. Another result established for the first time the existence of strongly localized moving discrete dissipative breather-solitons in Kerr nonlinear media and studied their properties. Additionally it was studied how a broad optical wavepacket scatters on a randomly moving narrow defect. Using a well-known mathematical analogy between the CW optical beam dynamics and time-dependent quantum mechanics the problem was reformulated as that of a nonstationary 1D quantum scattering on a randomly moving point defect (quantum dot) which immediately increased the outreach of the results to include the field of condensed matter physics (especially people studying non-adiabatic quantum pumping). It was demonstrated that for a sufficiently long waveguide (long time dynamics in quantum problem) each component of a linear wavepacket (wavefunction) is naturally separated into the Berry-phase contribution (resulting from the singular part of the wave amplitude in the co-moving frame) and the non-adiabatic correction (arising from fast oscillating, slow decaying tails of the same amplitude). In the special limit of a delta-correlated continuous Gaussian random walk there was obtained a closed analytical expressions for the ensemble averaged amplitude in the co-moving frame and the exact expressions for the average propagator of the light beam (quantum wave packet) was derived.Finally the project studied theoretically the problem of light imaging through a turbid media where in addition to the wavefront shaping and optimization by means of spatial light modulator a nonlinear focusing crystal was employed which increased the maximum achievable enhancement at least by a factor of 3 compared to a purely linear medium. The resulting focus was shown to be sharper than the corresponding linear one by approximately the same factor.Impact on the career development: During the research and training period at Weizmann Institute of Science the research fellow, Dr Stanislav Derevyanko:(a) acquired the knowledge of state-of-the-art techniques of numerical simulations of disordered pulse propagation in optical waveguides. (b) was exposed to experimental techniques in nonlinear optics, in particular: systems of coupled optical waveguides, SLM phase mask optimization, speckle optics.(c) acquired competence in project management, teaching and research commercialization methods;(d) increased the number of inter-European collaboration links at the level of Principal Investigator. In particular within the lifetime of the project new collaboration links with Dept. of Physics at the University of Duisburg-Essen, Germany and University of Athens, Greece were established. (e) Organized and managed a minisymposium entitled ``The Interplay Between Nonlinearity and Disorder’’ within the framework of the ICCMMSE -2014 conference that took place on 4-7 April, 2014 in Athens, Greece.As a result of the project the host group has offered Dr Derevyanko a 6 month extension of his visiting fellowship in order to finalize and carry out the paper submission on enhanced focusing in the nonlinear disordered medium.Socio-economic impact of the project: The innovative nature of the project, interdisciplinary approach and complementary competence and skills of Dr. Derevyanko and the Ultrafast Optics Group along with the undertaken training programme helped enhancing the EU scientific excellence with the findings of the project being of benefit to the European telecommunications and computing industry (via opening possibilities of building all-optical processing components and advanced optoelectronic devices). At the same time the enhanced focusing of light in a turbid and nonlinear media demonstrated in the project has wide potential applications in the field of biological and medical imaging (both invasive and non-invasive) thus increasing the quality of life and fulfilling one of the EU research targets of developing better diagnostics and more effective therapies.