Wiener chaos expansion and simulation of electromagnetic wave propagation excited by a spatially incoherent source

© 2010 Society for Industrial and Applied MathematicsThe definitive version of this paper is available at: http://dx.doi.org/10.1137/090749219DOI: 10.1137/090749219First, we propose a new stochastic model for a spatially incoherent source in optical phenomena. The model naturally incorporates the incoherent property into the electromagnetic wave equation through a random source term. Then we propose a new numerical method based on Wiener chaos expansion (WCE) and apply it to solve the resulting stochastic wave equation. The main advantage of the WCE method is that it separates random and deterministic effects and allows the random effects to be factored out of the primary partial differential equation (PDE) very effectively. Therefore, the stochastic PDE is reduced to a set of deterministic PDEs for the coefficients of the WCE method which can be solved by conventional numerical algorithms. We solve these secondary deterministic PDEs by a finite-difference time domain (FDTD) method and demonstrate that the numerical computations based on the WCE method are considerably more efficient than the brute-force simulations. Moreover, the WCE approach does not require generation of random numbers and results in less computational errors compared to Monte Carlo simulations

Design and implementation of ultra-high resolution, large bandwidth, and compact diffuse light spectrometers

My research on the new concepts for spectrometer has been focused on the development of true multi-dimensional spectrometers, which use a multi-dimensional [two-dimensional (2D) or 3D] mapping of the spectral information into space. I showed that by combining a simple dispersive element (a volume hologram) formed in very inexpensive polymers with a basic Fabry-Perot interferometer, we can form a spectrometer with ultra-high resolution over a large spectral bandwidth, which surpasses all conventional spectrometers. I strongly believe that the extension of this mapping into three dimensions by using synthetic nanophotonic structures with engineered dispersion can further improve the performance and reduce the overall spectrometer size into the micron regime. The need for efficient modeling and simulation tools comes from the sophisticated nature of the new 3D nanophotonic structures, which makes their simple analysis using well-known simple formulas for the propagation of the electromagnetic fields in bulk materials impossible. In my Ph.D. research, I developed new approximate modeling tools for both the modeling of incoherent sources in nanophotonics, and for the propagation of such optical beams inside the 3D nanophotonic structures of interest with several orders of magnitude improvement in the simulation speed for practical size devices without sacrificing accuracy. To enable new dispersive properties using a single nanophotonic structure, I have focused in my Ph.D. research into polymer-based 3D photonic crystals, which can be engineered using their geometrical features to demonstrate unique dispersive properties in three dimensions that cannot be matched by any bulk material even with orders of magnitude larger sizes. I have demonstrated the possibilities of using a very compact structure for wavelength demultiplexing and also for spectroscopy without adding any other device.Ph.D.Committee Chair: Adibi, Ali; Committee Member: Bhatti, Pamela; Committee Member: Callen, William; Committee Member: Gaylord, Thomas; Committee Member: Zhou, Hao-Mi

Wave Propagation

A wave is one of the basic physics phenomena observed by mankind since ancient time. The wave is also one of the most-studied physics phenomena that can be well described by mathematics. The study may be the best illustration of what is “science”, which approximates the laws of nature by using human defined symbols, operators, and languages. Having a good understanding of waves and wave propagation can help us to improve the quality of life and provide a pathway for future explorations of the nature and universe. This book introduces some exciting applications and theories to those who have general interests in waves and wave propagations, and provides insights and references to those who are specialized in the areas presented in the book

Doctor of Philosophy

dissertationModeling techniques are provided for accurate and efficient solution of near-field radiative heat transfer in complex, three-dimensional and multiscale geometries. These techniques are applied to investigate the physics of near-field thermal radiation in several configurations. A closed-form expression based on fluctuational electrodynamics is derived and applied for modeling size effect on the emissivity of metallic and dielectric thin films. The emissivity of dielectric films increases with increasing film thickness, while metallic films show the inverse behavior. The critical thickness, above which no size effect is observed, is about a hundred nanometers for metals and a few centimeters for dielectrics. A novel computational method, called the thermal discrete dipole approximation (T-DDA), for modeling near-field radiative heat transfer in arbitrary geometries is proposed and verified. The T-DDA is based on discretizing objects into cubical subvolumes behaving as electric point dipoles. The objects are submerged in an infinite lossless medium and can interact with an infinite surface. An extensive convergence analysis of the method is performed using the exact results for two spheres. The convergence of the T-DDA mostly depends on the dielectric function of the objects and the object size to gap ratio. An error less than 5% was achievable in the T-DDA using the available computational resources. The T-DDA is applied to model near-field thermal radiation between a silica probe and a silica surface separated by a gap of size d. When d --> 0, the probe-surface heat rate is dominated by the contribution of surface phonon-polaritons and approaches a d^-2 power law. In this limit, the total heat rate and the resonance location can be predicted using the proximity approximation. When the probe tip size is comparable to the gap thickness, localized surface phonons also contribute to heat transfer and induce a resonance splitting in the thermal spectrum. In this regime, the spheroidal dipole approximation predicts the resonant frequencies accurately, and it provides a rough estimate of the heat rate. Finally, the T-DDA analysis of probe-sample interactions demonstrates that the resonance redshift observed in near-field thermal spectroscopy is caused by the reflection interactions between the probe and the sample

Synchronization of Coupled and Periodically Forced Chemical Oscillators

Physiological rhythms are essential in all living organisms. Such rhythms are regulated through the interactions of many cells. Deviation of a biological system from its normal rhythms can lead to physiological maladies. The tremor and symptoms associated with Parkinson\u27s disease are thought to emerge from abnormal synchrony of neuronal activity within the neural network of the brain. Deep brain stimulation is a therapeutic technique that can remove this pathological synchronization by the application of a periodic desynchronizing signal. Herein, we used the photosensitive Belousov--Zhabotinsky (BZ) chemical reaction to test the mechanism of deep brain stimulation. A collection of oscillators are initially synchronized using a regular light signal. Desynchronization is then attempted using an appropriately chosen desynchronizing signal based on information found in the phase response curve.;Coupled oscillators in various network topologies form the most common prototypical systems for studying networks of dynamical elements. In the present study, we couple discrete BZ photochemical oscillators in a network configuration. Different behaviors are observed on varying the coupling strength and the frequency heterogeneity, including incoherent oscillations to partial and full frequency entrainment. Phase clusters are organized symmetrically or non-symmetrically in phase-lag synchronization structures, a novel phase wave entrainment behavior in non-continuous media. The behavior is observed over a range of moderate coupling strengths and a broad frequency distribution of the oscillators