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Nonlocal Anderson Localization

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.



Scale-Free Instabilities

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.




THz Relativistic Spatial Solitons in Graphene

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.

Last Updated (Thursday, 15 September 2011 14:43)



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 (see the Physical Review Focus, and the related PRL paper).

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.


Last Updated (Thursday, 22 March 2012 06:24)


Mode-Locked Random Lasers in Nature Photonics

The occurrence of the spontaneous synchronization of coupled oscillators is a well known phenomenon.

For example, in standard lasers, by employing specific devices like acousto-optics modulators, it is possible to couple orthogonal electromagnetic modes, make them synchronously oscillating and emitting short pulses.

On the other hand, in ultra-fast femtosecond oscillators, the mode-locking occurs because of the presence of a saturable absorber; as in the former case no disorder is present.

As reported in Nature Photonics (arXiv:1304.3652), Marco Leonetti, Claudio Conti, and Cefe Lopez show that Random Lasers, i.e. lasers with disorder, may be driven through a mode-locking regime when changing the spatial distribution of the excited modes. Indeed, in random lasers, modes are naturally coupled because of the fact that the cavity is open and exhibit a sort of condensation above threshold. When tailoring the pump profile, and exciting modes with a different degree of coupling (from the weakly coupled regime for distant modes, to the strongly coupled one for overlapping modes), random lasers can be made oscillating either in an asynchronous or in a mode-locked regime.

Such an effect has strong influence on the spectral shape of the emitted radiation, showing that random lasers can be made strongly tunable and are engineerable.

This is the first evidence of such a mode-locking transition in the presence of disorder, may have a role in the development of novel kind of laser sources, and it is another step for the assessment of complexity in photonic systems.

The picture below shows an example of a laser cluster pumped by a shaped beam (courtesy of Marco Leonetti)




Two-level laser by Anderson localization and solitons

Two-level lasers are thermodynamically ruled out, because population inversion cannot be achieved at equilibrium.

In a recent paper ArXiv:1102.1207, Viola Folli and Claudio Conti show that, by employing the interaction between a Self-Induced Transparency Soliton and Surface Anderson Localization of Light, it is possible to achieve a two-level laser-like emission in a dynamical regime, when the optical pump pulse has the same carrier frequency of the resonant system, at variance with standard lasers where additional non-radiative (and energetically inefficient) levels are commonly employed.

This shows that the interaction between two apparently un-related forms of light localization, i.e., solitons and Anderson localizations, may provide novel fascinating regimes for light-matter interaction.


The picture above (after parallel FDTD Maxwell-Bloch simulations) shows the comparison between the case when the SIT soliton propagates almost undistorted in the random system (low-index contrast) and the case when it transfers energy to Anderson localizations located in proximity of the surface (high-index contrast).

Related published articles: Optics Letters, Vol. 36, Issue 15, pp. 2830-2832 (2011), and JOSA B, Vol. 29, Issue 8, pp. 2080-2089 (2012).



Scale-Free Optics in Nature Photonics

There is a very limited number of ways for artificially changing the refractive index of a medium. But if one could be able to arbitrarily increase it, unprecedented possibilities would open the road to a novel generation of optical devices.

In a paper in Nature Photonics (January 2011),  E.DelRe, E.Spinozzi, A.J.Agranat and C.Conti demonstrate through theory and experiments that supercooled ferroelectrics (a class of "complex" optical media) display a largely tunable refractive index when acting on their thermal history and resorting to their out-of-equilibrium state.

The so called "polar nano-regions" furnish such a possibility, and lead to the demonstration of a nonliner optical behavior which is intensity independent and may provide the complete suppression of evanescent waves, with a potential impact in whole field of microscopy.

This may constitute the first example of a novel class of non-ergodic metamaterials, where the wavelength of light is effectively cancelled and the propagation is "Scale-Free", that is, beams propagate without distortion at any beam size and intensity.





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)



Quadratic Optical Spatial Solitons in Random Media

Simultons are self-trapped spatial beams, which are sustained by second-harmonic generation. They are optical spatial solitons composed by beams with two frequencies, one fundamental and its second harmonic. So far they have been observed only in media without disorder.

C.Conti, E. D'Asaro, S. Stivala, A. Busacca and G. Assanto, reported on the theory of simultons in quadratic media with random nonlinearity, and on their experimental observation.

This is the first observation of optical solitons in media with a disordered nonlinear susceptibility.


The figure shows the numerical simulation of the two components (a/b and c/d) of a simulton

for two different strengths of disorder.

Details in Opt. Lett. 35, 3760 (2010) .


Summa Complexity

The interplay between disorder and nonlinearity in wave propagation is a very challenging topic for theoretical research. The reason lies in the fact that the two effects are strongly competing, and treating both of them at the same level is really a difficult enterprise. At the moment, most of the approaches either tackle with nonlinearity as a perturbation to disorder, or with disorder as a perturbation to nonlinearity. In both of the cases, the results of the analysis do not seem to contain a substantially innovative picture, as it appears to be the case in practically all the related fields of research (as nonlinear optics or Bose-Einstein condensation).

The application of spin glass theory seems to open new perspectives, if methods from the statistical mechanics of disordered systems can be applied to nonlinear waves. Such an interdisciplinary approach is developed and extensively described in a recent paper ,authored by L. Leuzzi and C. Conti, (arXiv:1009.3290) which extends previous theoretical investigations (see, e.g., these articles in this website).

The main result of this approach is the predicition of the existence of a complex landscape (i.e., many energetically equivalent states) for nonlinear waves in disordered media, and, specifically, the explicit calculation of the  complexity in terms of the strength of nonlinearity for various degrees of disorder. Specific dynamic and thermodynamic phases, organized in a universal phase-diagram and with different complexity, can be identified.

All of this finds application in random photonics, nonlinear waves in random systems and Bose-Einstein condensation, including finite temperature effects. Details in arXiv:1009.3290.


Optical Supercavitation in Soft-matter

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].

In this work, the way the onset of a dynamic phase transtion (namely, the passage from a liquid to a gelified state) leads to a novel  kind of beam self-tunnelling  effect , in an otherwise highly absorbing soft-colloidal material, is demonstrated by means of theory, numerical computation and experiments.

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].



The Equation of Random Lasers

Haus, Gross and Pitaevskii, the fellowship of random lasers. Our proposed equation for describing the linewidth of random lasers is based on the work of these three scientists. In arXiv:1005.2082 (Josa B 27, 1446 (2010)), the derivation of our model (originally introduced herePhys. Rev. Lett. 101, 143901 (2008)) is detailed and it is shown how a single equation quantitatively predicts the observed random laser linewidth.

The model is not based on the diffusive approximation for light; conversely it starts from the general Haus model for passively mode-locked lasers and arrives to an equation that is formally identical to that describing one-dimensional zero-temperature Bose-Einstein condensation in an external potential (the Gross-Pitaevskii equation).

The basic idea follows that of a stochastic resonator, originally introduced by Lethokov in 1968, i.e., an open system sustaining a large number of electromagnetic resonances with overlapping linewidths, as sketched in the following figure


By extending the ideas of Haus for multi-mode lasers to such a kind of resonator we derive the following equation:



The Fourier transform of the solution gives the spectrum of the Random Laser.

The equation contains only one parameter: the "nonlinear eigenvalue" E, which depends on the gain bandwidth and the coefficients that measure the spread of the mode decay-time distribution. When the nonlinear eigenvalue E is greater than 1, a bell-shaped linewidth is attained.

This allows to define rigorously a threshold for random lasers, and is in quantitative agreement with our experiments.

This also shows how random lasing can be understood as a classical condensation process of electromagnetic resonances in open disordered resonators.


Waterlight in Photonic Crystal Fibers

The fact that the fundamental models for nonlinear optics (and Bose-Einstein condensation) , under specific conditions, reduce to an hydrodynamic form is well known.

However, in the temporal domain, it is very difficult to experimentally observe hydrodynamic-like phenomena, like shock formation, as many other competing "optical" effects intervene (like modulational instability or higher order dispersion).

In a recent paper, written in collaboration with F. Biancalana, P.J. Russel and S. Stark of the Max Planck Institute for the Science of Light, we found that an engineered Photonic Crystal Fiber (PCF) may allow the observation of temporal shocks and related wave-breaking phenomena. The driving process is the Raman effect that induces a spectral drift in the soliton spectrum, which, hydrodynamically, corresponds to water falling along a slope under the action of gravity. This circumstance opens the roads to optimize the fiber for an hydrodynamic regime of propagation of ultra-short optical pulses, which can be exploited for novel pulse compression and supercontinuum generation schemes, and for theoretically assessing the main paradigms of extreme nonlinear optics.

Figure below shows the wave breaking regime on an optical pulse, as predicted by the hydrodynamic equations (details in  arXiv:1005.0995, Phys. Rev. A, 82: 013838)



Solitons and nonlocality, disorder and frustration

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).

Brownian dynamics of a solitary wave


Light-driven genetic selection and localization in the Enlightened Game of Life

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.