<p>Problems in stochastic homogenization theory typically deal with approximating differential operators with rapidly oscillatory random coefficients by operators with homogenized deterministic coefficients. Even though the convergence of these operators in multiple scales is well-studied in the existing literature in the form of a Law of Large Numbers, very little is known about their rate of convergence or their large deviations.</p><p>In the first part of this thesis, we we establish analytic results for the Gaussian correction in homogenization of an elliptic differential equation with random diffusion in randomly layered media. We also derive a Central Limit Theorem for a diffusion in a weakly random media.</p><p>In the second part of this thesis devise a technique for obtaining large deviation results for homogenization problems in random media. We consider the special cases of an elliptic equation with random potential, the random diffusion problem in randomly layered media and a reaction-diffusion equation with highly oscillatory reaction term.</p> / Dissertation

Mesoscale optical phenomena occur when light interacts with a number of different types of materials, such as biological and chemical systems and fabricated nanostructures. As a framework, mesoscale optics unifies the interpretations of the interaction of light with complex media when the outcome depends significantly upon the scale of the interaction. Most importantly, it guides the process of designing an optical sensing technique by focusing on the nature and amount of information that can be extracted from a measurement. Different aspects of mesoscale optics are addressed in this dissertation which led to the solution of a number of problems in complex media. Dynamical and structural information from complex fluids—such as colloidal suspensions and biological fluids—was obtained by controlling the size of the interaction volume with low coherence interferometry. With this information, material properties such as particle sizes, optical transport coefficients, and viscoelastic characteristics of polymer solutions and blood were determined in natural, realistic conditions that are inaccessible to conventional techniques. The same framework also enabled the development of new, scale-dependent models for several important physical and biological systems. These models were then used to explain the results of some unique measurements. For example, the transport of light in disordered photonic lattices was interpreted as a scale-dependent, diffusive process to explain the anomalous behavior of photon path length distributions through these complex structures. In addition, it was demonstrated how specialized optical measurements and models at the mesoscale enable solutions to fundamental problems in cell biology. Specifically, it was found for the first time that the nature of cell motility changes markedly with the curvature of the substrate that the cells iv move on. This particular work addresses increasingly important questions concerning the nature of cellular responses to external forces and the mechanical properties of their local environment. Besides sensing of properties and modeling behaviors of complex systems, mesoscale optics encompasses the control of material systems as a result of the light-matter interaction. Specific modifications to a material’s structure can occur due to not only an exchange of energy between radiation and a material, but also due to a transfer of momentum. Based on the mechanical action of multiply scattered light on colloidal particles, an optically-controlled active medium that did not require specially tailored particles was demonstrated for the first time. The coupling between the particles and the random electromagnetic field affords new possibilities for controlling mesoscale systems and observing nonequilibrium thermodynamic phenomena.

Most of the physically realizable optical sources are radiating in a random manner given the random nature of the radiation of a large number of atoms that constitute the source. Besides, a lot of natural and synthetic materials are fluctuating randomly. Hence, the optical fields that one encounters, in most of the applications are fluctuating and must be treated using random or stochastic functions. Within the framework of the scalar-coherence theory, one can describe changes of the properties of any stochastic field such as the spectral density and the spectral degree of coherence on propagation in any linear medium, deterministic or random. One of the frequently encountered random media is the atmospheric turbulence, where the fluctuating refractive index of such medium severely degrades any signal propagating through it; especially it causes intensity fades of the signal. The usage of stochastic beams at the transmitter instead of deterministic ones has been suggested sometime ago to suppress the severe effects of intensity fluctuations caused by the atmospheric turbulence. In this dissertation, we study the usage of partially coherent beams in long path propagation schemes through turbulent atmosphere such as one frequently encounters in remote sensing, in the use of communication systems, and in guiding. Also the used detection scheme at the receiver is important to quantify the received signal efficiently, hence we compare the performance of incoherent (direct) detection versus coherent (heterodyne) detection upon the use of either one of them at the receiver of the communication system of beams propagating in turbulent atmosphere and namely we evaluate the signal-to-noise-ratio (SNR) for each case. The scalar-coherence theory ignored the vector nature of stochastic fields, which should be taken into account for some applications such as the ones that depend on the change of the polarization of the field. Recently generalization for the scalar-coherence theory including the vector aspects of the stochastic beams has been formulated and it is well-known as the unified theory of coherence and polarization of stochastic beams. The use of the unified theory of coherence and polarization makes it possible to study both the coherence properties and the polarization properties of stochastic electromagnetic beams on propagation in any linear media. The central quantity in this theory is a 2 × 2 matrix that describes the statistical ensemble of any stochastic electromagnetic beam in the space-frequency domain or its Fourier transform in the space-time domain. In this dissertation we derive the conditions that the cross-spectral density matrix of a so-called planar, secondary, electromagnetic Gaussian Schell-model source has to satisfy in order to generate a beam propagating in vacuum. Also based on the unified-theory of coherence and polarization we investigate the subtle relationship between coherence and polarization under general circumstances. Besides we show the effects of turbulent atmosphere on the degree of polarization and the polarization state of a partially coherent electromagnetic beam, which propagates through it and we compare with the propagation in vacuum. The detection of the optical signals is important; hence it affects the fidelity of the communication system. In this dissertation we present a general analysis for the optical heterodyne detection of stochastic electromagnetic beams. We derive an expression for the SNR when two stochastic electromagnetic beams are mixed coherently on a detector surface in terms of the space-time domain representation of the beams, the beam coherence polarization matrices. We evaluate also the heterodyne efficiency of a heterodyne detection system for stochastic beams propagating in vacuum and we discuss the dependence of the heterodyne efficiency of the detection process on the changes in the beam parameters as the beam propagates in free space.

Considere um meio desordenado formado por $N$ pontos cujas coordenadas são geradas aleatoriamente com probabilidade uniforme ao longo das arestas unitárias de um hipercubo de $d$ dimensões. Um caminhante, partindo de um ponto qualquer desse meio, se desloca seguindo a regra determinista de dirigir-se sempre ao ponto mais próximo que não tenha sido visitado nos últimos $\\mu$ passos. Esta dinâmica de movimentação, denominada caminhada determinista do turista, leva a trajetórias formadas por uma parte inicial transiente de $t$ pontos, e uma parte final cíclica de $p$ pontos. A exploração do meio se limita aos $t+p$ pontos percorridos na trajetória. O sucesso da exploração depende do valor da memória $\\mu$ do viajante. Para valores pequenos de $\\mu$ a exploração é altamente localizada e o sistema não é satisfatoriamente explorado. Já para $\\mu$ da ordem de $N$, aparecem ciclos longos, permitindo a exploração global do meio. O objetivo deste estudo é determinar o valor de memória $\\mu_1$ para o qual ocorre uma transição abrupta no comportamento exploratório do turista em meios unidimensionais. Procuramos também entender a distribuição da posição final do turista após atingir um estado estacionário que é atingido quando o turista fica aprisionado nos ciclos. Os resultados obtidos por simulações numéricas e por um tratamento analítico mostram que $\\mu_1 = \\log_2 N$. O estudo também mostrou a existência de uma região de transição com largura $\\varepsilon = e/ \\ln 2$ constante, caracterizando uma transição aguda de fase no comportamento exploratório do turista em uma dimensão. A análise do estado estacionário da caminhada em função da memória mostrou que, para $\\mu$ distante de $\\mu_1$, a dinâmica de exploração ocorre como um processo difusivo tradicional (distribuição gaussiana). Já para $\\mu$ próximo de $\\mu_1$ (região de transição), essa dinâmica segue um processo superdifusivo não-linear, caracterizado por distribuições $q$-gaussianas e distribuições $\\alpha$-estáveis de Lévy. Neste processo, o parâmetro $q$ funciona como parâmetro de ordem da transição. / Consider a disordered medium formed by $N$ point whose coordinates are randomly generated with uniform probability along the unitary edges of a $d$-dimensional hypercube. A walker, starting to walk from any point of that medium, moves following the deterministic rule of always going to the nearest point that has not been visited in the last $\\mu$ steps. This dynamic of moving, called deterministic tourist walk, leads to trajectories formed by a initial transient part of $t$ points and a final cycle of $p$ points. The exploration of the medium is limited to the $t+p$ points covered. The success of the exploration depends on the traveler\'s memory value $\\mu$. For small values of $\\mu$, the exploration is highly localized and the whole system remains unexplored. For values of $\\mu$ of the order of $N$, however, long cycles appear, allowing global exploration of the medium. The objective of this study is to determine the memory value $\\mu_1$ for which a sharp transition in the exploratory behavior of the tourist in one-dimensional media occurs. We also want to understand the distribution of the final position of the tourist after reaches a steady state in exploring the medium. That steady state is reached when the tourist is trapped in cycles. The results achieved by numerical simulations and analytical treatment has shown that $\\mu_1 = \\log_2 N$. The study has also shown the existence of a transition region, with a constant width of $\\varepsilon = e/ \\ln 2$, characterizing a phase transition in the exploratory behavior of the tourist in one dimension. The analysis of the walk steady state as a function of the memory has shown that for $\\mu$ far from $\\mu_1$, the exploratory dynamic follows a traditional diffusion process (with gaussian distribution). In the other hand, for $\\mu$ near $\\mu_1$ (transition region), the dynamic follows a non-linear superdiffusion process, characterized by $q$-gaussian distributions and Lèvy $\\alpha$-stable distributions. In this process, the parameter $q$ plays the role of a transition order parameter.

Understanding the phenomenon of wave scattering by random media is a ubiquitous problem that has instigated extensive research in the field. This thesis focuses on wave scattering by quasiperiodic media as an alternative approach to provide insight into the effects of structural aperiodicity on the propagation of the waves. Quasiperiodic structures are aperiodic yet ordered so have attributes that make them beneficial to explore. Quasiperiodic lattices are also used to model the atomic structures of quasicrystals; materials that have been found to have a multitude of applications due to their unusual characteristics. The research in this thesis is motivated by both the mathematical and physical benefits of quasiperiodic structures and aims to bring together the two important and distinct fields of research: waves in heterogeneous media and quasiperiodic lattices. A review of the past literature in the area has highlighted research that would be beneficial to the applied mathematics community. Thus, particular attention is paid towards developing rigorous mathematical algorithms for the construction of several quasiperiodic lattices of interest and further investigation is made into the development of periodic structures that can be used to model quasiperiodic media. By employing established methods in multiple scattering new techniques are developed to predict and approximate wave propagation through finite and infinite arrays of isotropic scatterers with quasiperiodic distributions. Recursive formulae are derived that can be used to calculate rapidly the propagation through one- and two-dimensional arrays with a one-dimensional Fibonacci chain distribution. These formulae are applied, in addition to existing tools for two-dimensional multiple scattering, to form comparisons between the propagation in one- and two-dimensional quasiperiodic structures and their periodic approximations. The quasiperiodic distributions under consideration are governed by the Fibonacci, the square Fibonacci and the Penrose lattices. Finally, novel formulae are derived that allow the calculation of Bloch-type waves, and their properties, in infinite periodic structures that can approximate the properties of waves in large, or infinite, quasiperiodic media.

Next generation photonics-based technologies will ultimately rely on novel materials and devices. For this purpose, phenomena at subwavelength scales are being studied to advance both fundamental knowledge and experimental capabilities. In this dissertation, concepts specific to near-field optics and experimental capabilities specific to near-field microscopy are used to investigate various aspects of the statistical properties of random electromagnetic fields in the vicinity of optically inhomogeneous media which emit or scatter radiation. The properties of such fields are being characterized within the frame of the coherence theory. While successful in describing the far-field properties of optical fields, the fundamental results of the conventional coherence theory disregard the contribution of short-range evanescent waves. Nonetheless, the specific features of random fields at subwavelength distances from interfaces of real media are influenced by the presence of evanescent waves because, in this case, both propagating and nonpropagating components contribute to the detectable properties of the radiation. In our studies, we have fully accounted for both contributions and, as a result, different surface and subsurface characteristics of inhomogeneous media could be explored. We investigated different properties of random optical near-fields which exhibit either Gaussian or non-Gaussian statistics. We have demonstrated that characteristics of optical radiation such as first- and second-order statistics of intensity and the spectral density in the vicinity of random media are all determined by both evanescent waves contribution and the statistical properties of the physical interface. For instance, we quantified the subtle differences which exist between the near- and far-field spectra of radiation and we brought the first experimental evidence that, contrary to the predictions of the conventional coherence theory, the values of coherence length in the near field depend on the distance from the interface and, moreover, they can be smaller than the wavelength of light. The results included in this dissertation demonstrate that the statistical properties of the electromagnetic fields which exist in the close proximity of inhomogeneous media can be used to extract structural information. They also suggest the possibility to adjust the coherence properties of the emitted radiation by modifying the statistical properties of the interfaces. Understanding the random interference phenomena in the near-field could also lead to new possibilities for surface and subsurface diagnostics of inhomogeneous media. In addition, controlling the statistical properties of radiation at subwavelength scales should be of paramount importance in the design of miniaturized optical sources, detectors and sensors.

The propagation symmetry of electromagnetic fields is affected by encounters with material systems. The effects of such interactions, for example, modifications of intensity, phase, polarization, angular spectrum, frequency, etc. can be used to obtain information about the material system. However, the propagation of electromagnetic waves imposes a fundamental limit to the length scales over which the material properties can be observed. In the realm of near-field optics, this limitation is overcome only through a secondary interaction that couples the high-spatial-frequency (but non-propagating) field components to propagating waves that can be detected. The available information depends intrinsically on this secondary interaction, which constitutes the topic of this study. Quantitative measurements of material properties can be performed only by controlling the subtle characteristics of these processes. This dissertation discusses situations where the effects of near-field interactions can be (i) neglected in certain passive testing techniques, (ii) exploited for active probing of static or dynamic systems, or (iii) statistically isolated when considering optically inhomogeneous materials. This dissertation presents novel theoretical developments, experimental measurements, and numerical results that elucidate the vectorial aspects of the interaction between light and nano-structured material for use in sensing applications.

In an optically random system, such as naturally occurring and man-made media, light undergoes pronounced multiple scattering. This phenomenon has shown a remarkable potential in characterizing complex materials. In this regime, scattering occurs from each individual center of the scattering and independent scattering events lead to multiple light scattering. This phenomenon is often described as a random walk of photons and can be modeled in terms of a diffusion equation based on the radiative transfer theory. In this thesis, we used optical path-length spectroscopy (OPS), which is an experimental method to obtain the path-length probability density of the propagating light in multiple scattering media, with a low-coherence optical field to investigate the distribution of photon path lengths in a skin cell model system. This method is capable of measuring the transport mean free path of light in a highly scattering medium and depth-resolved profiles of the backscattered light. Our OPS experimental configuration is based on a fiber-optic Michelson interferometer geometry using single mode optical fibers. We performed OPS based on low-coherence interferometry (LCI) on three-dimensional organotypic models of esophageal cell invasion by measuring the optical path-length distribution of backscattered light in normal and invasive conditions. The optical path-length distribution of light waves inside the cell samples provides information on how a change in the extracellular matrix affects invasiveness of the esophageal cells and induction of signaling pathways. Also, we demonstrated the compatibility to study the structural changes during a two-week period for in vitro cell samples.

No description available.

The presence of gas hydrates in permafrost regions has been confirmed by core samples recovered from the Mallik gas hydrate research wells located within Mackenzie Delta in the Northwest Territories of Canada. Strong vertical variations of compressional and shear velocities and weak surface seismic expressions of gas hydrates indicate that lithological heterogeneities control the lateral distribution of gas hydrates. Seismic scattering studies predict that typical horizontal scales and strong velocity contrasts due to gas hydrate concentration will generate strong forward scattering, leaving only weak energy to be captured by surface receivers. In order to understand the distribution of gas hydrates and the scattering effects on seismic waves, heterogeneous petrophysical reservoir models were constructed based on the P-wave and S-wave velocity logs. Random models with pre-determined heterogeneity scales can also be used to simulate permafrost interval as well as sediments without hydrates. Using the established relationship between hydrate concentration and P-wave velocity, we found that gas hydrate volume content can be determined by correlation length and Hurst number. Using the Hurst number obtained from Mallik 2L-38, and the correlation length estimated from acoustic impedance inversion, gas hydrate volume fraction in Mallik area was estimated to be 17%, approximately 7x108 m3 free gas stored in a hydrate bearing interval with 250,000 m2 lateral extension and 100 m depth. Simulations of seismic wave propagation in randomly heterogeneous models demonstrate energy loss due to scattering. With the available modeling algorithm, the impact of heterogeneity scales on seismic scattering and optimum acquisition geometries will be investigated in future studies.