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The Optimal Disorder for the Anderson Localization of Light

Many authors investigated the onset of light localization in the regime of strong-disorder.

But how much strong?

Determining how much strong the disorder has to be to give rise to localized states (in three-dimensions) is an extremely difficult enterprise, as it depends on the specific kind of sample that is considered. One has to demonstrate that an optimal region for localization exists, once a parameter that quantifies the strength of the disorder has been identified. This is a very tricky problem, even more tricky if one wants to be far from a perturbative regime as the case of weakly disordered Photonic Crystals, considered here and here.

In a recent article (http://arxiv.org/abs/1003.2555), authored by Silvia Gentilini, Andrea Fratalocchi and Claudio Conti, a specific case has been considered by resorting to parallel large-scale Finite Difference Time-Domain simulations. A molecular dynamics code has been used to generate some disorder realizations of a colloidal solution composed by spherical particles. By varying the filling fraction of the resulting sample, the response of the material to an ultra-wide band (Frequency Comb) optical excitation has been spectrally analyzed.

It has been shown that the maximum spectral delay of the various frequencies in the transmitted spectrum displays a sharp peak in correspondence of a narrow region in the filling fraction for high-index contrast particles (figure below, delay expressed in picoseconds); this also being the first ab-initio numerical investigation of a Frequency Comb in a disordered material. Such a result shows that an optimal disorder for light localization can be indeed identified even for completely randomized samples.

optimal_disorder

 

The Page Rank Equation

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.

pagerank

 

Nonlinear light diffusion, ab-initio results

In this manuscript we report on the numerical simulation of ultra-short intense laser pulse propagating in a random medium.
nonlineardiffusion.png
 

Quantum X-waves

In quant-ph/0409130 and quant-ph/030906 I have reported on a paraxial theory of quantum X-waves and discussed their generation by nonlinear optical processes.

With Alessandro Ciattoni, we developed a general theory of Quantum X-waves and we have shown that classical X-waves may have different quantum counterparts (which share the classical expectation field profile) encoding different degree of entanglement. See arXiv:0704.0442.

quantumxwaves

A proposed experimental setup for generating the "progessive undistorted squeezed vacuum" (see quant-ph/0409130)

 

Matter X-waves

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

nonlin1

 

 

More on Nonlinear X-waves

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.

 

Soliton experimental setup

solitonsetup

 

This picture shows our experimental setup for observing optical spatial solitons. The microscope is used for imaging the light scattered from the top of the sample, which is illuminated by a laser beam through the input inverted microscope.

 

 

The ab-investigation of the 3D Anderson localization

In a recent paper in the arXiv (http://arxiv.org/abs/0810.1690 ), Claudio Conti and Andrea Fratalocchi reported on the first ab-initio investigation in 3D and time-resolved of the Anderson localization of light in a colloid.

 

Femtosecond propagation in disordered matter

The theoretical analysis of ultra-fast laser propagation in disordered matter is enormously complicated by the need to account for a 3D enviroment (Anderson localization and related phenomena are indeed a trivial issue in reduced dimensionality), material dispersion, multiple-scattering processes, polidispersivity and many other phenomena.

Ab initio computation based on Finite Difference Time Domain codes is probably the most extensive approach to the problem.

In a recent article, Silvia Gentilini, Andrea Fratalocchi, and Claudio Conti have reported the first 3D simulations of 100fs laser pulses in a disordered medium by the MD-FDTD approach.

We reported a quantitave calculation of the light diffusion constant (in agreement with experiments from the literature) and also of the dynamics of the diffusion constant.

The figure below shows on the left panel (a) the edge of the trasmitted pulse in log scale (the gray area in the inset is the input pulse, the black line the trasmitted pulse); while on panel (b) the diffusivity constant versus the material filling factor (the inset is the dynamical diffusivity for two different particle diameters)

 

Thermodynamics of soliton gases

In a recent article in Physical Review Letters, in collaboration with Andrea Fratalocchi,  Giancarlo Ruocco and Stefano Trillo, we have shown that within a purely integrable model solitons undergoes non canonical phase transitions.  For the nonlinear Schroedinger equation, many solitons should behave as a gas of free particles. In this respect, phase transitions are not expected. However by employing non canonical phase-space measures, mutuated from the so-called thermostatistics of chaotic systems, it is possible to show that a suitably defined free energy undergoes a methamorphosis as the number of solitons grows.
A notable issue is that this mathematical circumstance has a clear physical signature, i.e. the development of a dispersive shock wave, which is the result of the cooperative evolution of the solitons. This theoretical result fixes a link between chaotic and integrable systems, which is fairly not trivial and can be experimentally tested. The experiments are now being performed in our group.
 

Random Lasers (intro)

In a nutshell, a random laser  is the coherent emission from active stochastic resonators.

In a series of articles around 1966, a Russian scientist V. S. Letokhov, of the Lebedev Physics Institute in Dubna considered the generation of light in the interstellar medium. In the presence of scatterers, as for example dust particles, photons diffuse like neutrons and, if some mechanism (following Letokhov a “negative absorption”) is able to increase their number, a sort of photonic reactor can be realized. At a threshold value for the amplification, a quantum generator of light, not properly a laser, is realized. Nowadays the process is known as RANDOM LASER.

 

The Letokhov’s basic idea is that of a stochastic resonator: a scattering medium with a large number of modes (roughly, waves with different directions) which are strongly coupled and display radiation losses. If the number of modes is sufficiently large their overlapping resonances merge and the result is a nonresonant feedback sustaining a laser-like action; its properties are today largely investigated, while, in many respects, they are still unknown. There are several analogies with neutrons in a reactor.

 

Letokhov’s analysis was not limited to the interstellar medium, but also included particles dispersed in a host medium able to amplify light, like a liquid dye. Indeed, the first demonstration of a random laser, in 1994, was done in the laboratory using such a kind of medium; since then the amount of related literature has grown steadily.

 

Examples of Random Lasers and Stochastic Resonators include granular systems, bio-logical tissue, nano-structured devices and many others.

 

Random Laser Experimental Setup

This is our experimental setup on random lasers.

 setuprl

The following picture (after arXiv:0809.1547 ) displays the random laser emission from a TiO2 colloidal dispersion averaged over 100 pump laser shots and versus the pump energy.

 figurerl


Enhanced Backscattering of Light

 

Fundamental thermodynamics at a mesoscopic level

If you have studied thermodynamics in your life, you probably had to face with adiabatic pistons and their applications, and you probably know that one driving force for this topic has been the development of steam engines. But what happens when the dimensions of an adiabatic piston shrink down to a mesoscopic level ?

Here mesoscopic does not mean quantum mechanics, but the regime in which the mass of the piston is comparable with that of the gas, which, incidentally, seems to correspond to the typical scales of biophysics and bridges classical mechanics and quantum mechanical length scales. Crosignani and Di Porto (and little help by me) have pointed out that in this regime the fluctuations of the piston position are comparable to the piston size. As Capek and Sheehan have outlined in their book "Challenges to the Second Law Thermodynamics," this result raises serious doubts about the validity of the second law in the mesoscopic regime. Indeed, if on one hand we are not discovering a perpetuum mobile  (because we are dealing with fluctuations), on the other hand "the entropy variation associated with these displacements are at odd with a standard corollary of the second law, namely" [Capek, Sheean, ibid.]

A closed system in an equilibrium state, once an internal constraint is removed, eventually reaches a new equilibrium state characterized by a larger value of the entropy

As a result, one can imagine a thermodynamic cycle which has the net effect of extracting work for its thermal surroundings ().

The adiabatic piston problem is continuosly attracting the interest of various researchers involved in thermodynamics, since 1965, and its extension to the mesoscopic realm is likely to play a role in thermodynamics of life and modern nano- and micro-devices as MEMS and NEMS technologies.


In this paper, we have deepened our analysis of the piston problem by comparing the cases of a diathermal piston and that of an adiabatic piston. Notably enough there is a remarkable difference between them: the adiabatic piston undergoes much larger fluctuactions (by a factor given by the square root of the ratio between the piston mass and the gas molecule mass). See cond-mat/0611323.
 

The MD-FDTD technique (intro)

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) 

 

Random lasers and spin-glasses

There are two basic ingredients for complexity: randomness and nonlinearity. Random Lasers display both of them.

In order to assess the complexity of a given physical systems, one has to make reference to some paradimatic approaches; these typically furnish the basic ideas in a clear and well defined way. In the science of complexity, Spin Glass Theory is the most fundamental paradigm.

However Spin Glass Theory is an abstract and largely idealized approach to the thermodynamics and the dynamics of complex systems; a major difficulty is finding a specific physical system that is described by Spin Glass Theory within a reasonable accuracy. The common situation is that Spin Glass Theory only furnishes a qualitative description of the physical process.

Random Lasers seem to be an important exception. As detailed in two articles in Physical Review Letters and Physical Review B, we derived a model for multimode Random Lasers that is based on Spin Glass Theory. This approach enables to predict mode-locking transitions of these devices and the fact that these are glassy transitions.

Ab-initio numerical simulations and experiments are now being performed to substantiate these theoretical predictions.

See also The phase diagram of random lasers