The autocorrelation of speckles in deep Fresnel diffraction region and characterizations of random self-affine fractal surfaces

This paper studies the correlation properties of the speckles in the deep Fresnel diffraction region produced by the scattering of rough self-affine fractal surfaces. The autocorrelation function of the speckle intensities is formulated by the combination of the light scattering theory of Kirchhoff approximation and the principles of speckle statistics. We propose a method for extracting the three surface parameters, i.e. the roughness w, the lateral correlation length xi and the roughness exponent alpha, from the autocorrelation functions of speckles. This method is verified by simulating the speckle intensities and calculating the speckle autocorrelation function. We also find the phenomenon that for rough surfaces with alpha = 1, the structure of the speckles resembles that of the surface heights, which results from the effect of the peak and the valley parts of the surface, acting as micro-lenses converging and diverging the light waves.

The accuracy of Kirchhoff's approximation in describing the far field speckles produced by random self-affine fractal surfaces

Based on the rigorous formulation of integral equations for the propagations of light waves at the medium interface, we carry out the numerical solutions of the random light field scattered from self-affine fractal surface samples. The light intensities produced by the same surface samples are also calculated in Kirchhoff's approximation, and their comparisons with the corresponding rigorous results show directly the degree of the accuracy of the approximation. It is indicated that Kirchhoff's approximation is of good accuracy for random surfaces with small roughness value w and large roughness exponent alpha. For random surfaces with larger w and smaller alpha, the approximation results in considerable errors, and detailed calculations show that the inaccuracy comes from the simplification that the transmitted light field is proportional to the incident field and from the neglect of light field derivative at the interface.

The gated integration technique for the accurate measurement of the autocorrelation function of speckle intensities scattered from random phase screens

A new approach based on the gated integration technique is proposed for the accurate measurement of the autocorrelation function of speckle intensities scattered from a random phase screen. The Boxcar used for this technique in the acquisition of the speckle intensity data integrates the photoelectric signal during its sampling gate open, and it repeats the sampling by a preset number, in. The average analog of the in samplings output by the Boxcar enhances the signal-to-noise ratio by root m, because the repeated sampling and the average make the useful speckle signals stable, while the randomly varied photoelectric noise is suppressed by 1/ root m. In the experiment, we use an analog-to-digital converter module to synchronize all the actions such as the stepped movement of the phase screen, the repeated sampling, the readout of the averaged output of the Boxcar, etc. The experimental results show that speckle signals are better recovered from contaminated signals, and the autocorrelation function with the secondary maximum is obtained, indicating that the accuracy of the measurement of the autocorrelation function is greatly improved by the gated integration technique. (C) 2006 Elsevier Ltd. All rights reserved.

Random propagation in complex systems : nonlinear matrix recursions and epidemic spread

This dissertation studies long-term behavior of random Riccati recursions and mathematical epidemic model. Riccati recursions are derived from Kalman filtering. The error covariance matrix of Kalman filtering satisfies Riccati recursions. Convergence condition of time-invariant Riccati recursions are well-studied by researchers. We focus on time-varying case, and assume that regressor matrix is random and identical and independently distributed according to given distribution whose probability distribution function is continuous, supported on whole space, and decaying faster than any polynomial. We study the geometric convergence of the probability distribution. We also study the global dynamics of the epidemic spread over complex networks for various models. For instance, in the discrete-time Markov chain model, each node is either healthy or infected at any given time. In this setting, the number of the state increases exponentially as the size of the network increases. The Markov chain has a unique stationary distribution where all the nodes are healthy with probability 1. Since the probability distribution of Markov chain defined on finite state converges to the stationary distribution, this Markov chain model concludes that epidemic disease dies out after long enough time. To analyze the Markov chain model, we study nonlinear epidemic model whose state at any given time is the vector obtained from the marginal probability of infection of each node in the network at that time. Convergence to the origin in the epidemic map implies the extinction of epidemics. The nonlinear model is upper-bounded by linearizing the model at the origin. As a result, the origin is the globally stable unique fixed point of the nonlinear model if the linear upper bound is stable. The nonlinear model has a second fixed point when the linear upper bound is unstable. We work on stability analysis of the second fixed point for both discrete-time and continuous-time models. Returning back to the Markov chain model, we claim that the stability of linear upper bound for nonlinear model is strongly related with the extinction time of the Markov chain. We show that stable linear upper bound is sufficient condition of fast extinction and the probability of survival is bounded by nonlinear epidemic map.

The search for gravitational waves from the coalescence of black hole binary systems in data from the LIGO and Virgo detectors. Or: A dark walk through a random forest

The LIGO and Virgo gravitational-wave observatories are complex and extremely sensitive strain detectors that can be used to search for a wide variety of gravitational waves from astrophysical and cosmological sources. In this thesis, I motivate the search for the gravitational wave signals from coalescing black hole binary systems with total mass between 25 and 100 solar masses. The mechanisms for formation of such systems are not well-understood, and we do not have many observational constraints on the parameters that guide the formation scenarios. Detection of gravitational waves from such systems — or, in the absence of detection, the tightening of upper limits on the rate of such coalescences — will provide valuable information that can inform the astrophysics of the formation of these systems. I review the search for these systems and place upper limits on the rate of black hole binary coalescences with total mass between 25 and 100 solar masses. I then show how the sensitivity of this search can be improved by up to 40% by the the application of the multivariate statistical classifier known as a random forest of bagged decision trees to more effectively discriminate between signal and non-Gaussian instrumental noise. I also discuss the use of this classifier in the search for the ringdown signal from the merger of two black holes with total mass between 50 and 450 solar masses and present upper limits. I also apply multivariate statistical classifiers to the problem of quantifying the non-Gaussianity of LIGO data. Despite these improvements, no gravitational-wave signals have been detected in LIGO data so far. However, the use of multivariate statistical classification can significantly improve the sensitivity of the Advanced LIGO detectors to such signals.

Stationary random response of multidegree-of-freedom systems

An approximate approach is presented for determining the stationary random response of a general multidegree-of-freedom nonlinear system under stationary Gaussian excitation. This approach relies on defining an equivalent linear system for the nonlinear system. Two particular systems which possess exact solutions have been solved by this approach, and it is concluded that this approach can generate reasonable solutions even for systems with fairly large nonlinearities. The approximate approach has also been applied to two examples for which no exact or approximate solutions were previously available.

Also presented is a matrix algebra approach for determining the stationary random response of a general multidegree-of-freedom linear system. Its derivation involves only matrix algebra and some properties of the instantaneous correlation matricies of a stationary process. It is therefore very direct and straightforward. The application of this matrix algebra approach is in general simpler than that of commonly used approaches.

Measuring temporal fluctuation on femtosecond scale in a random medium

We report an observation of femtosecond optical fluctuations of transmitted light when a coherent femtosecond pulse propagates through a random medium. They are a result of random interference among scattered waves coming from different trajectories in the time domain. Temporal fluctuations are measured by using cross-correlated frequency optical gating. It is shown that a femtosecond pulse will be broadened and distorted in pulse shape while it is propagating in random medium. The real and imaginary components of transmitted electric field are also distorted severely. The average of the fluctuated transmission pulses yields a smooth profile, probability functions show good agreement with Gaussian distribution. (c) 2007 Elsevier B.V. All rights reserved.

Topology optimization of silicon anode structures for lithium-ion battery applications

This thesis presents a topology optimization methodology for the systematic design of optimal multifunctional silicon anode structures in lithium-ion batteries. In order to develop next generation high performance lithium-ion batteries, key design challenges relating to the silicon anode structure must be addressed, namely the lithiation-induced mechanical degradation and the low intrinsic electrical conductivity of silicon. As such, this work considers two design objectives of minimum compliance under design dependent volume expansion, and maximum electrical conduction through the structure, both of which are subject to a constraint on material volume. Density-based topology optimization methods are employed in conjunction with regularization techniques, a continuation scheme, and mathematical programming methods. The objectives are first considered individually, during which the iteration history, mesh independence, and influence of prescribed volume fraction and minimum length scale are investigated. The methodology is subsequently extended to a bi-objective formulation to simultaneously address both the compliance and conduction design criteria. A weighting method is used to derive the Pareto fronts, which demonstrate a clear trade-off between the competing design objectives. Furthermore, a systematic parameter study is undertaken to determine the influence of the prescribed volume fraction and minimum length scale on the optimal combined topologies. The developments presented in this work provide a foundation for the informed design and development of silicon anode structures for high performance lithium-ion batteries.

Nonperturbative dynamical decoupling with random control

Parametric fluctuations or stochastic signals are introduced into the rectangular pulse sequence to investigate the feasibility of random dynamical decoupling. In a large parameter region, we find that the out-of-order control pulses work as well as the regular pulses for dynamical decoupling and dissipation suppression. Calculations and analysis are enabled by and based on a nonperturbative dynamical decoupling approach allowed by an exact quantum-state-diffusion equation. When the average frequency and duration of the pulse sequence take proper values, the random control sequence is robust, fault-tolerant, and insensitive to pulse strength deviations and interpulse temporal separation in the quasi-periodic sequence. This relaxes the operational requirements placed on quantum control devices to a great deal.