Light and Complexity

In Graphene, the optical nonlinearity arises from the interaction of light with quasi-electrons moving in the peculiar band structure. Specific regimes, as massless relativistic propagation, which can be electromagnetically driven, allow a strong enhancement of the nonlinearity. This happens in certain spectral ranges, and particularly in the TeraHertz bandwidth.

Haiming Dong, Claudio Conti and Fabio Biancalana in http://arxiv.org/abs/1107.5803 investigate a novel class of self-trapped beams in the THz bandwidth, sustained by relativistically moving particles (Dirac fermions) in Graphene, a novel frontier between nonlinear optics and field-theory driven condensed-matter physics.

In some respects, the nonlinear optical processes that can be observed in soft-matter are still unknown.

The reason is to be ascribed to the fact that, so far, only "simple" colloidal materials have been considered, as nematic liquid crystals or weakly interacting colloidal solutions. The realm of "complex" regimes, as dynamic phase-transitions or non-ergodic phases, is still to be extensively explored.

Novel effects may arise from the competition between different kinds of nonlinear optical processes, which, in soft-matter, can be many, as  thermodiffusion, electrostriction and thermal phenomena.

An example is the paper on Optical Supercavitation in Soft Matter, authored by C. Conti and E. DelRe and published in Phys. Rev. Lett. 105, 118301 (2010) [arXiv:1008.2616].

The figure below shows the experimental evidence of the supercavitation process: an optical beams enters inside the highly absorbing colloidal solution by inducing a dynamic phase-transition region, thus stronly resembling what happens in hydrodynamic supercavitation, where an high-speed bullet induces a liquid-gas transition in water and can propagate experiencing a reduced friction. Details in Phys. Rev. Lett. 105, 118301 (2010) [arXiv:1008.2616].

In an extended version of the Enlightened Game of Life (http://arxiv.org/abs/0810.3179), we investigate a model for light-driven species selection. A population of interacting agents is placed in a electromagnetic  (EM) cavity and different species, with different degrees of photosensitive response, are included in the game. The model shows that the interaction with the EM field favors photosensitive species; in addition they may self-organize in order to localize the EM field and extract more efficiently the life-sustaining wave-energy. The figure below shows the trend of the number of photosensitive cells and the degree of localization of the field versus time (details in the paper), when a nonlinear interaction between the Cellular Automata and the EM wave is included.

This is a toy model for investigating various field-driven complex systems and placing on a physical ground evolutionary schemes, which ascribe a leading role to the development of the eye. See also here.

In a recent article in Europhysics Letters, authored by N. Perra, V. Zlatic, A. Chessa, C. Conti, D. Donato and G. Caldarelli an equation for the page-rank in the WWW and in several other kinds of networks has been derived. The equation strongly resembles other famous ones in physics, like the Schroedinger or the Helmotz equations, and it enables to make fascinating connections between theoretical physics and the science of scale-free networks. The arXiv version of this paper has been formerly commented in the New Scientist issue of August 2008.

The picture shows a localized solution of the Page Rank equation, corresponding to the formation of an HUB node.

People coming from soliton theory (especially nonlinear optics) are used to associate self-localization to a nonlinear self-action. Actually, self-localized waves do even exist for linear systems, and, surprisingly, they have been mostly studied in recent years, AFTER solitons and bullets.

The typical argument against localized waves (to be intended as the linear waves) is that they carry infinite enegy. However, this holds true for many nonlinear waves, as for example dark solitons and vortices. Finite energy realizations of localized waves have been observed, as it has been for darks and vortices.

The situation is even more intricate when localized waves propagate in the presence of a nonlinear self-action. The result are the NONLINEAR X-WAVES. They aresolutions to 3D+1 nonlinear evolution equations carrying infinite energy and have been investigated both experimentally and theoretically. [see Conti C., Phys. Rev. E 70, 046613 (2004) and the book chapter  NonlinearXwaves (.pdf, 1.2MB) in the book Localized Waves (ISBN 0-470-10885-1) for an introduction]

In other words:

1D nonlinear waves with infinite energy = dark solitons

2D nonlinear waves with infinite energy = vortex solitons

3D nonlinear waves with infinite energy = nonlinear X waves (or X solitons)

The world of nonlinear X waves is still mainly unexplored; not only it is fascinating because it deals with 3D waves, which have a counterpart in Bose Einstein condensates, but also because they are expected to play a role in ultrafast laser propagation. Indeed, contrary to light bullets, X-waves are sustained by normal dispersion, which is always the case in the relevant experiments.

Localization in a nonlocal medium sounds like a contradiction. However, it is interesting trying to figure out the interplay between processes of localization and nonlocal effects. This is even more complicated if one takes into account localizations induced by structural disorder. The interplay between the characteristic length scales of localization and nonlocality was not considered before.

In the manuscript Opt. Lett. 37, 332 (2012), [arXiv:1201.3923], Viola Folli and Claudio Conti report on a theoretical analysis of Anderson localization in a nonlocal nonlinear medium; it is shown that nonlocality stabilizes localizations with respect to the action of nonlinearity, and analytical expressions for the power needed to destabilize the Anderson states are obtained. The nonlocal model allows indeed a simple treatment of the nonlinear Anderson localizations.

The picture below shows the evolutions of the states in the presence of nonlocal nonlinearity.

Optomechanics deals with optically induced deformations. In a microstructured fiber, the high field intensity may lead to novel kind of optomechanical effects. Among these, there is the Kerr nonlocal optical response due to an optically induced deformation. A specific geometry has been considered by Anna Butsch, Claudio Conti, Fabio Biancalana and Philip St. J. Russell in arXiv:1108.5190.

In a dual slab waveguide geometry, embedded in a fiber, a novel form of optomechanical optical self-challenging, i.e., of non diffracting light beams, has been theoretically investigated, while identifying a novel and notable geometry for the generation of spatial solitons: a fiber!

Indeed, fibers were only known to host optical TEMPORAL solitons, but in this novel geometry, thanks to the impressive developments in the design and fabrication of microstructured fibers, SPATIAL solitons can propagate, potentially extending for several kilometers; this is a new frontier for self-trapped beams, in a word : OPTOMECHANICONS.

The investigation of the interaction between a nonlocal nonlinear response and disorder is still at its infancy. In the manuscript arXiv:1005.0578 [Phys.Rev.Lett.104,193901(2010)], we consider the interplay between nonlocality and disorder in solitary wave propagation. Such a topic has relevance in practically all the problems concerning nonlinear wave propagation in soft-matter, as the material density fluctuactions always introduce a certain degree of disorder and, at the same time, long range correlations in the material response function turn out in a nonlocal nonlinear response. In addition nonlocality is also relevant in Bose-condensed gases, plasma physics and many other fields.

The outcome of our analysis is that, for any kind of nonlocality, as the correlation length increases (i.e., the degree of nonlocality increases), the Brownian motion of the solitons is frustrated, and nonlocality filters out fluctuactions. This happens even if nonlocality involves large material regions, if compared with the local case, and hence, in principle, the solitons feel an enhanced disorder. In some sense, nonlocality acts as a cooling mechanism of soliton dynamics.

The figure below shows trajectories of the solitons in a nonlocal disorder medium (details in arXiv:1005.0578).

In Phys. Rev. Lett. 92, 120404 (2004), Conti and Trillo investigate X-wave solutions for Bose Condensed gases.

Theoretical modelling of light driven self-organization and nonlinear processes in soft-matter

 
A key in soft-matter is self-organization and supra-molecular structuring, controlling these mechanisms will provide a variety of possibilities in terms of applications and fundamental studies. In this respect nonlinear optical processes in soft-matter seem to open novel roads, on one hand for investigating novel physical phenomena, on the other to fine tune and control soft-matter.

Laser-driven self-organization strongly affects laser propagation, and this feedback effect has been only recently considered in a couple of papers by us, also including complex structured soft-materials. However, if these approaches fill the gap while considering electrostrictive mechanisms and the corresponding nonlinear optical response in structured soft-matter, on the other hand they are based on a continuous description of SM, which is not expected to be valid when the dimension of the beam is comparable to that of the particles. This is exactly the opposite limit with respect to the longitudinal optical binding of an array of particles, for which various theoretical models have been reported.

In this respect the current theoretical situations is as follows: there is the theory of solitons and modulational instability that describes trapping of many particles within a continuous model, then there is the theory of optical binding that describes trapping of an array of particles (eventually including some particle-particle interaction).

There is hence the need for a comprehensive theoretical approach that accounts for both of these two limits and is able to take into account the landscape of soft-matter due to the particle-particle interaction and the way it is affected by external fields. However, the numerical simulations of the various self-organization processes in soft-matter is enormously complicated by the need of taking into account a full 3D environment and nonlinear effects. In addition soft-materials are typically characterized by complicated shapes that role out standard multiple scattering approaches, like Mie theory.

The use of modern parallel computational resources has opened the possibility to novel multidisciplinary approaches committing together molecular dynamics, for simulating soft-matter complex behaviour, and Time Domain Finite Difference Parallel Techniques, for solving the electromagnetic field dynamics.

 
Methodology

In a recent article some of us report on the first results on a fully-comprehensive ab-initio numerical analysis of laser propagation in a disordered medium.

A colloidal dispersion structure is obtained by a Molecular-Dynamics (MD) code, and a Maxwell's equations solver (FDTD) is used for the laser propagation. This approach has lead to the first ab-initio computation of the light diffusion constant, also including nonlinear effects, in quantitative agreement with experiments. By MD-FDTD it is possible to unveil, within a first-principle formulation, the interplay between the 3D structure of a complex material, its nonlinear response (eventually including light amplification) and the features of the transmitted laser pulses. By using this approach we are developing a comprehensive approach to the self-organization in the presence of an external field, while also including different particle-particle interaction potential and the feedback of the re-organization in the field dynamics.

  C. Conti, G. Ruocco, S. Trillo "Optical Spatial Solitons in Soft Matter" Phys. Rev. Lett. 95, 183902 (2005); C. Conti, N. Ghofraniha, G. Ruocco, S. Trillo "Laser beam filamentation in soft matter" Phys. Rev. Lett.  97, 123903 (2006) References added by Claudio Conti 

C. Conti, L. Angelani and G. Ruocco "Light diffusion and localization in three-dimensional nonlinear disordered media" Phys. Rev. A 75, 033812 (2007) 

Within the fundamentals of nonlinear optics there are the processes of instability of a beam, either temporal or spatial. In the presence of nonlinearity,  a beam can break up because of a variety of physical effects, in primis, the so-called modulational instability (MI). MI appears in focusing media, and destroy (in the spatial case) a wide beam into fragments, which are characterized by a dominant spatial period. In other words, through MI a periodical pattern arises from a plane wave, the period being determined by the wave intensity; this intensity dependent period is the dominant spatial scale.

In scale-free optics, beam evolution is not affected by the spatial scales: non-diffracting beams can be generated at any beam size and power independently. Thus one could expect that MI is not present for scale-free nonlinearities, because MI has a dominant spatial case. And this is indeed the case: if one makes the standard analysis, no MI is retrieved in the scale-free model.

However, as shown by Viola Folli, Eugenio Del Re, and Claudio Conti in Physical Review Letters 108, 0339012 (2012), [http://arxiv.org/abs/1201.3865], other kinds of instabilities arise in the peculiar nonlocal nonlinearity of the out-of-equilibrium crystals, which are used to activate the scale-free response. At variance with MI, no spatial scale arises in these instabilities and the beam breaks into many spots with a distributions of sizes and powers. Preliminary evidence of the process was reported in the first observartion of the scale-free solitons, the figure below shows the numerical evidence of the scale-free instabilities and the corresponding spectrum.

The reported analysis is just a first step in of several theoretical open roads of the scale-free model.

Solitons are solutions of nonlinear partial differential equations. In many cases the nonlinearity is "local", meaning that the nonlinear part of the equation is only a function of the relevant field in a specific point of the coordinates.

If, in the nonlinear part, fields at different points are involved, the nonlinearity is "nonlocal".

One could think that, roughly speaking, a nonlocal nonlinearity is "more nonlinear" than a local one, as it combines, in a complicated way, fields at different positions, instead, e.g., of being a simple power of the field in a single point.

In this respect, it looks really un-expected and, a bit, disorienting, the fact that when one considers a "strongly nonlocal model", the relevant solitons are described by a sort of linear equation.

For these reasons, the word "linearons" was used by C. Conti, M. Schmidt, P.St.J.Russell and F. Biancalana, to dub a specific class of highly-nonlocal (highly non-instantaneous, indeed) solitary waves in a novel kind of photonic device: a photonic crystal fiber filled by a strongly nonlinear re-orientational liquid.

The LINEARONS have remarkable and un-expected properties and applications, many to be investigated, and the quest for their observation in experiments is just started...

Details in arXiv:1010.0331 (http://prl.aps.org/abstract/PRL/v105/i26/e263902)