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Nonlinear Dynamics of Mie lasers

Since the pioneering investigations of Gustav Mie in the early days of the last century, the theoretical and experimental study of spherical resonators has deserved much attention from the scientific community. This interest stems from the analysis of fundamental processes such as scattering, energy propagation through disordered media and cavity quantum electrodynamics, and from the large number of applications in photonics, chemistry, meteorology, astronomy and sensing.

Despite the widespread scientific production, the theoretical understanding of Mie lasers is still an open field of research. For what concerns laser emission, existing theoretical approaches either rely on a semi-classical treatment based on light interaction with a two level system or on rate equations, with numerical analysis limited to one and two-dimensional media. The former approach, is not rigorous as the number of physical dimensions grows above one while the latter, accounting only for the contribution of atomic populations, misses important informations on quantum coherence, and cannot be effectively pursued to perform any realistic ab initio computation at ultra-fast time-scales (<ps). Furthermore, although low dimensional models permit a simplified analysis with respect to the fully vectorial set of Maxwell equations they left the general picture unknown. Which is the outcome of a strongly nonlinear ultra-fast and multi-dimensional interaction on scales comparable with the wavelength?

In a recent paper , in collaboration with Andrea Fratalocchi and Giancarlo Ruocco, developed an ab initio rigorous theoretical model of light interaction in the presence of amplifying (or dissipative) materials, deriving a three dimensional vector set ofMaxwell-Bloch (MB) equations within the real representation in terms of the SU(n) algebra. We discretized the resulting equations on a Yee grid and numerically solved them within the Finite-Difference Time-Domain (FDTD) method. The MB-FDTD approach is then applied to investigate the ultrafast dynamics of Mie nanoresonators. Specifically, we perform a series of numerical experiments by investigating the process of laser emission from a single nanosphere, covered by a layer of active material, for different pumping rates.



Solitons (intro)

Spectral theory for nonlinear partial differential equations (PDE) is fascinating. At variance with linear systems the spectrum of a nonlinear wave is composed by a continuos ensemble of plane waves (as for linear wave-packets) and by a set of discrete entities, which are named "solitons". This is true for special classes of PDE that are typically denoted as "integrable".

For integrable PDE the spectrum is conserved during the dynamics;  this means that the number of solitons is conserved, even if there is a strong nonlinear dynamics, and the superposition principle does not hold true.

Solitons typically correspond to bumps of energy (bright solitons) or energy vacancies (dark solitons); these interact, but in integrable systems their interaction is elastic.

Nonlinear systems that are not described by integrable PDE can still support nonlinear waves resembling solitons. In the mathematical community these are denoted as "solitary waves". As it happens for solitons, solitary waves mantain their shape along their evolution or propagation. However solitary waves can be unstable, that is they may decade into radiation or other solitary waves, and the interaction between solitary waves can be strongly unelastic.

In real world systems the subtle distinction between solitary waves and solitons is often difficult to be appreciated. Indeed non integrable PDEs are typically well described by integrable PDE in certain regimes. In the applied physics community the distinction between solitary waves and solitons is often overlooked (even if mathematicians do not like that).


The Phase Diagram of Random Lasers

phasediagramrl.pngIn arXiv:0806.2809 Luca Leuzzi, Claudio Conti, Viola Folli, Luca Angelani and Giancarlo Ruocco proposed a phase diagram for mode-locking processes in lasers displaying a variable degree of disorder. The relevant parameters are the standard deviation of the structural fluctuactions and the pumping rate. Three thermodynamic phases are found, the standard mode-locked phase (ferromagnetic-like), the paramagnetic phase (continuos wave emission) and the glassy mode-locking phases. The complexity is not vanishing in the latter case. This result applies in a huge variety of different physical frameworks, also including Bose condensed gases in random optical lattices and nonlinear optical beam propagation.


See also Summa Complexity


Ab initio Random Laser in Photonic Crystals

Is random lasing (RL) due to the strong localization of light or to delocalized solutions of the photon diffusion equation ?
Probably both of them are true, and probably there is a way for describing the transition from diffusive RL to the localized one. A key difficulty is assessing the observation of a localized state, especially in strongly disordered systems where the Anderson trapping is critical.

Photonic crystals (PhC) play a game apart. As orginally investigated by John in the famous paper in Physical Review Letters of the 1987, in the presence of a complete omnidirectional photonic band gap, even a small amount of disorder nurtures 3D light strong localization at frequencies within the forbidden band. In a recent computational work Claudio Conti and Andrea Fratalocchi have reported on the first ab-initio simulation of RL in a disordered 3D PhC, while showing the transition from de-localized Bloch modes to the Anderdon localization.



Anderson localization in PhC and all of that

The first articles that are often recognized as the starting point in the field of photonic crystals (PhC) are those published in the Physical Review Letters in 1987 by John and Yablonovitch. John pointed out that a particular class of photonic devices, those displaying a 3D periodicity, can foster the observation of 3D strong localization of light, that has often demonstrated to be elusive and is still an intense subject of research.
In our recent computational work we have simulated from first principles the excitation of 3D Anderson localizations in an inverted opal displaying a complete photonic band gap. We reported on what we believe to be the first numerical calculation of the light diffusion constant in these devices, and the calculated diffusion constant is in quantitative agreement with experimental data reported in the literature.

Our parallel Finite Difference Time Domain code

The Finite Difference Time Domain Algorithm has been introduced in 1966 by Yee. This is currently the most developed numerical technique for solving Maxwell's equations with virtually no approximations.

During the last years, Claudio Conti, in collaboration with Andrea Di Falco and Andrea Fratalocchi, developed from scratch an advanced 3D+1 parallel FDTD code, which is being currently highly optimized on parallel sistems at CINECA. The code is written in F90, including some object oriented features, and it is based on a 3D topology of processors. The code includes material dispersion, nonlinearity, and a variety of different update schemes.

This code is currently being routinely developed in order to deal with some of the fundamental problems in complex photonics, from random lasers to ultra-fast laser propagation in disordered matter.

Time-dependent Nonlinear Optical Susceptibility in Soft-Matter

The time dependent nonlinear optical susceptibility enables to discriminate hydrodynamical and heterogeneous regimes in complex matter. In the former case the system is described by appropriate partial differential equations and can be treated as a continuum. In the latter case the complexity of the material is evidenced by the fact that spatial regions behave differently; the hydrodynamical approach must be replaced by ab-initio models like molecular dynamics. The various heterogenous regions display different time-scales; they can be interpreted as different cooperative self-organized areas of the fundamental constituents (i.e. the colloidal particles).

Notably, hydrodynamics and heterogenous regimes can be discriminated by looking at different time-scales in the nonlinear susceptibility; at fast scales the former have the leading role, at slow scales the different cooperative regions independently evolve and affect the response. Nonlinear susceptibiliy is important because it is related to the multi-point (non local) response and hence directly probes different sample regions. In complex matter the heterogenous regime is also strongly affected by ageing (i.e. out-of-equilibrium dynamics) because the spatial extent of the cooperative regions grows with time since sample preparation.

In our paper we have reported direct experimental evidences of these phenomena by  using nonlinear optics. The following figure shows the measured nonlinear absorption in an out-of-equilibrium colloidal dispersion versus time for different sample ageing times tw and various input powers. At small time-scales the response is hydrodynamical (all the lines collapse), at large time-scales heterogeneous regimes result into different curves that change with the ageing time.


Read more...


Ultrashort pulse propagation and the Anderson localization

Anderson localization of light is the way one can describe the formation of localized resonances of the electromagnetic radiation in a disordered medium.

The mechanism of localization is triggered by an ensemble of photons trapped in the system; this implies that the lifetime of these energy excitations increases well beyond the time of flight of a light beam through the sample.

As a result, if an ultra-short light pulse propagates in the medium, the Anderson states get excited and the transmitted pulse encodes their lifetimes. The shorter the pulse, the wider the spectrum and hence the higher the probability of putting into oscillations the electromagnetic resonances. In addition, the shorter the pulse, the more evident is the photon slowing-down in the tail of the transmitted pulse.

All of this is investigated in our recent paper on the arxiv (http://arxiv.org/abs/0810.1690), where we considered the propagation of an ultra-short light beam in a colloidal medium.

Competion of Bullets and X-waves

Bullets are multidimensional solitary waves that are bell shaped in any direction. Bullets find a role in practically all the fields involving nonlinear waves.
X-waves are algebrically localized solutions of wave equations eventually involving some nonlinearity.

In a recent paper in Physical Review Letters we have found the conditions for the simultaneous generation of X-waves and bullets and numerically investigated their competition.

Competion between bullets and nonlinear X-waves


Nonlinear dressing of X-waves

X-waves exist even in the absence of nonlinearity, for this reason some people refer to this kind of waves as "quasi-localized" waves or "quasi-solitons", while limiting the category of "solitons" or "solitary waves" to non-perturbative solutions of nonlinear equations.

It is interesting to investigate (we did it by numerical experiments) how a linear X-waves is "dressed" (this is a term mutuated by some integrable systems in which exact solutions are built from linear ones, it works well here, but a spectral theory is still missing)  by the nonlinearity.

An example is reported in the picture below which shows an exact linear X-wave solutions and its nonlinear counterpart, as numerically obtained, while increasing the parameter g that measures the strength of the nonlinearity (details in the review chapter below , where other examples can be found)nonlin3


Spooky work : Experiments on ancient human bones !


When one has not acces to nano-technology facilities (or does not like nano-things), it is often possible to look around for some alternative interesting sample for making laser physics. In Italy, we have a lot of ancient stuff, and hence we decided to exploit the richness of our archaeological sites to develop novel, interdisciplinary and (why not?) spooky research. This is case of the recent work [Appl. Phys. Lett. 94,10101(2009)], authored by Marco Leonetti, Silvia Capuani, Marco Peccianti, Giancarlo Ruocco and Claudio Conti, on Enhanced Backscattering of Light from Human Bone Tissue retrieved from the Etruscan site of San Donato (1st-3rd century A.C.). 

Have you ever been thinking to the fact that your bones could be used by a Ph.D. student in a far away future for his/her thesis? Take care of your bones !



The Enlightened Game of Life

The link between light and the development of complex behavior is as much subtle as evident. Examples include the moonlight triggered mass spawning of hard corals in the Great Barrier, or the light-switch hypothesis in evolutionary biology, which ascribes the Cambrian explosion of bio-diversity to the development of vision. Electromagnetic (EM) radiation drastically alters complex systems, from physics (e.g., climate changes) to biology (e.g., structural colors or bioluminescence). So far emphasis has been given to bio-physical, or digital, models of the evolution of the eye with the aim of understanding the environmental influence on highly specialized organs. In this manuscript, we consider the way the appearance of photosensivity affects the dynamics, the emergent properties and the self-organization of a community of interacting agents, specifically, of cellular automata (CA).


Nonlinear optics in the X-ray regime

Understanding nonlinear processes at the smallest accessible spatio-temporal scale is a frontier of modern research. In this respect, the new generation of X-ray Free-Electron Lasers (X-ray FEL) opens unprecedented possibilities: an example is nonlinear optics in the X-ray region. The new X-ray FEL are expected to deliver femtosecond pulses in the wavelength range 0.05nm - 1nm with peak power greater than 100 GW. This corresponds to intensities up to 1023Wcm-2 for focused beams with 10nm spot-size.
Even not taking into account relativistic effects and particle production (expected at intensities of 1026Wcm-2), the underlying fundamental physical processes are in many respects unknown. Photons at the atomic-scale wavelength (0.1 nm) have macroscopic propagation lengths in condensed materials (even considering photoelectric absorption) and are sensitive to the granular structure of matter . Therefore, the physics of intense X-ray beams have to take into account nonlinear effects accumulating along large propagation distances in largely inhomogeneous environments.
In two papers published in Optics Express and Physical Review B , within a collaboration with Andrea Fratalocchi, Francesco Sette and Giancarlo Ruocco, we quantify the nonlinear contribution to the linear polarization expected by employing the future X-ray FEL. Within a purely classical formulation, we show that the nonlinear effects are comparable to the linear response for powers of the order of 10GW (a basically wavelength-independent result, even including photo-absorption).

Nonlinear X-ray dynamics is expected to become a relevant field for several fundamental studies employing ultrafast X-ray FELs.

Enhanced Backscattering Cone Experimental Setup

Light propagation in random media with low absorption and high scattering is characterized by weak localization phenomena. Their signature is an increase of the diffuse light intensity in the backward direction with respect to the background. This sharp peak is detectable by an experiment in which angular dependence of intensity is monitored around the backward direction. The Enhanced Backscattering Cone (EBS) setup, is an ad hoc system to study light propagation in solid and liquid multiple scattering materials.


EBS enables to estimate the transport mean free path l. This parameter takes in account the average distance between two scattering events and the angular deviation from the propagation direction at any single scattering event.

The FWHM W of the backscattering cone and l are related by the following relation: l=0.7 λ ε/2 π W

where λ is the incoming radiation wavelength and ε is a correction factor to be used if sample is enclosed in a couvette (it depends on refraction index of the couvette).

The enhancement factor EF is the ratio between peak and diffuse light intensity and is connected to the sample length and its absorption; theoretically EF=2.

The following figure shows a measure of the EBS cone from TiO2 in methanol (filling fraction 20%).


The following figure shows a measure of the EBS cone from TiO2 in methanol at higher filling fraction (36%). The EBS cone is larger and, correspondingly, the mean free path is reduced.



Meint P. Van Albada and Ad Lagendijk "Observation of Weack Localization of Light in a Random Medium" Phys. Rev. Lett. 55 2692-2695 (1985)
J.X. Zhu, D.J. Pine, and D.A. Weitz  "Internal reflection of diffusive light in random media " Phys. Rev. A 44 3948-3959  (1991)

Solitonic shocking catastrophe

Solitons are known as non-perturbative solutions of nonlinear wave equations. However solitons are not the most prounounced nonlinear effect. When incresing the input power (or equivalently the amount of nonlinearity), one first discovers perturbative effects over linear propagation (as beam self-phase modulation); then one finds out that solitons can be generated; and finally one has to do with shock waves.
But what is the link between solitons and shock waves? Our experimental results, in collaboration with Andrea Fratalocchi, Marco Peccianti, Giancarlo Ruocco and Stefano Trillo,   (the setup is shown here), reported in our recent paper on the arXiv  Observation of a Gradient Catastrophe Generating Solitons, display the way a shock front generates solitons.

This is the experimental demonstration of our previous theoretical paper, in which we investigate phase-transitions of an ensemble of solitons. To tell the whole truth, we have to add that it is not correct to think to the shock wave as a way to generate solitons. According to the inverse scattering theory , the solitons are already present in the initial beam, and they are indeed the origin of the shock. Such a result is so intricated that a variety of further investigations are needed, and this will be the subject of our future work.