Spectral Element Method Simulation of Linear and Nonlinear Electromagnetic Field in Semiconductor Nanostructures

<p>In this dissertation, the spectral element method is developed to simulate electromagnetic field in nano-structure consisting of dielectric, metal or semiconductor. The spectral element method is a special kind of high order finite element method, which has spectral accuracy. When the order of the basis function increases, the accuracy increases exponentially. The goal of this dissertation is to implement the spectral element method to calculate the electromagnetic properties of various semiconductor nano-structures, including photonic crystal, photonic crystal slab, finite size photonic crystal block, nano dielectric sphere. The linear electromagnetic characteristics, such as band structure and scattering properties, can be calculated by this method with high accuracy. In addition, I have explored the application of the spectral element method in nonlinear and quantum optics. The effort will focus on second harmonic generation and quantum dot nonlinear dynamics. </p><p>The electromagnetic field can be simulated in both frequency domain and time domain. Each method has different application for research and engineering. In this dissertation, the first half of the dissertation discusses the frequency domain solver, and the second half of the dissertation discusses the time domain solver.</p><p>For frequency domain simulation, the basic equation is the second order vector Helmholtz equation of the electric field. This method is implemented to calculate the band structure of photonic crystals consisting of dielectric material as well as metallic materials. Because the photonic crystal is periodic, only one unit cell need to be simulated in the computational domain, and a periodic boundary condition is applied. The spectral accuracy is inspected. Adding the radiation boundary condition at top and bottom of the computational region, the scattering properties of photonic crystal slab can be calculated. For multiple layers photonic crystal slab, the block-Thomas algorithm is used to increase the efficiency of the calculation. When the simulated photonic crystals are finite size, unlike an infinitely periodic system, the periodic boundary condition does not apply. In order to increase the efficiency of the simulation, the domain decomposition method is implemented. </p><p>The second harmonic generation, which is a kind of nonlinear optical effect, is simulated by the spectral element method. The vector Helmholtz equations of multiple frequencies are solved in parallel and the consistence solution with nonlinear effect is obtained by iterative solver. The sensitivity of the second harmonic generation to the thickness of each layer can be calculated by taking the analytical differential of the equation to the thickness of each element. </p><p>The quantum dot dynamics in semiconductor are described by the Maxwell-Bloch equations. The frequency domain Maxwell-Bloch equations are deduced. The spectral element method is used to solve these equations to inspect the steady state quantum dot dynamic behaviors under the continuous wave electromagnetic excitation.</p><p>For time domain simulation, the first order curl equations in Maxwell equations are the basic equations. A spectral element method based on brick element is implemented to simulate a nano-structure consisting of woodpile photonic crystal. The resonance of a micro-cavity consisting of a point defect in the woodpile photonic crystal block is simulated. In addition, the time domain Maxwell-Bloch equations are implemented in the solver. The spontaneous emission process of quantum dot in the micro-cavity is inspected. </p><p>Another effort is to implement the Maxwell-Bloch equations in a previously implemented domain decomposition spectral element/finite element time domain solver. The solver can handle unstructured mesh, which can simulate complicated structure. The time dependent dynamics of a quantum dot in the middle of a nano-sphere are investigated by this implementation. The population inversion under continuous and pulse excitation is investigated. </p><p>In conclusion, the spectral element method is implemented for frequency domain and time domain solvers. High efficient and accurate solutions for multiple layers nano-structures are obtained. The solvers can be applied to design nano-structures, such as photonic crystal slab resonators, and nano-scale semiconductor lasers.</p>Dissertatio

On the 3D electromagnetic quantitative inverse scattering problem: algorithms and regularization

In this thesis, 3D quantitative microwave imaging algorithms are developed with emphasis on efficiency of the algorithms and quality of the reconstruction. First, a fast simulation tool has been implemented which makes use of a volume integral equation (VIE) to solve the forward scattering problem. The solution of the resulting linear system is done iteratively. To do this efficiently, two strategies are combined. First, the matrix-vector multiplications needed in every step of the iterative solution are accelerated using a combination of the Fast Fourier Transform (FFT) method and the Multilevel Fast Multipole Algorithm (MLFMA). It is shown that this hybridMLFMA-FFT method is most suited for large, sparse scattering problems. Secondly, the number of iterations is reduced by using an extrapolation technique to determine suitable initial guesses, which are already close to the solution. This technique combines a marching-on-in-source-position scheme with a linear extrapolation over the permittivity under the form of a Born approximation. It is shown that this forward simulator indeed exhibits a better efficiency. The fast forward simulator is incorporated in an optimization technique which minimizes the discrepancy between measured data and simulated data by adjusting the permittivity profile. A Gauss-Newton optimization method with line search is employed in this dissertation to minimize a least squares data fit cost function with additional regularization. Two different regularization methods were developed in this research. The first regularization method penalizes strong fluctuations in the permittivity by imposing a smoothing constraint, which is a widely used approach in inverse scattering. However, in this thesis, this constraint is incorporated in a multiplicative way instead of in the usual additive way, i.e. its weight in the cost function is reduced with an improving data fit. The second regularization method is Value Picking regularization, which is a new method proposed in this dissertation. This regularization is designed to reconstruct piecewise homogeneous permittivity profiles. Such profiles are hard to reconstruct since sharp interfaces between different permittivity regions have to be preserved, while other strong fluctuations need to be suppressed. Instead of operating on the spatial distribution of the permittivity, as certain existing methods for edge preservation do, it imposes the restriction that only a few different permittivity values should appear in the reconstruction. The permittivity values just mentioned do not have to be known in advance, however, and their number is also updated in a stepwise relaxed VP (SRVP) regularization scheme. Both regularization techniques have been incorporated in the Gauss-Newton optimization framework and yield significantly improved reconstruction quality. The efficiency of the minimization algorithm can also be improved. In every step of the iterative optimization, a linear Gauss-Newton update system has to be solved. This typically is a large system and therefore is solved iteratively. However, these systems are ill-conditioned as a result of the ill-posedness of the inverse scattering problem. Fortunately, the aforementioned regularization techniques allow for the use of a subspace preconditioned LSQR method to solve these systems efficiently, as is shown in this thesis. Finally, the incorporation of constraints on the permittivity through a modified line search path, helps to keep the forward problem well-posed and thus the number of forward iterations low. Another contribution of this thesis is the proposal of a new Consistency Inversion (CI) algorithm. It is based on the same principles as another well known reconstruction algorithm, the Contrast Source Inversion (CSI) method, which considers the contrast currents – equivalent currents that generate a field identical to the scattered field – as fundamental unknowns together with the permittivity. In the CI method, however, the permittivity variables are eliminated from the optimization and are only reconstructed in a final step. This avoids alternating updates of permittivity and contrast currents, which may result in a faster convergence. The CI method has also been supplemented with VP regularization, yielding the VPCI method. The quantitative electromagnetic imaging methods developed in this work have been validated on both synthetic and measured data, for both homogeneous and inhomogeneous objects and yield a high reconstruction quality in all these cases. The successful, completely blind reconstruction of an unknown target from measured data, provided by the Institut Fresnel in Marseille, France, demonstrates at once the validity of the forward scattering code, the performance of the reconstruction algorithm and the quality of the measurements. The reconstruction of a numerical MRI based breast phantom is encouraging for the further development of biomedical microwave imaging and of microwave breast cancer screening in particular

Multiple Volume Scattering in Random Media and Periodic Structures with Applications in Microwave Remote Sensing and Wave Functional Materials

The objective of my research is two-fold: to study wave scattering phenomena in dense volumetric random media and in periodic wave functional materials. For the first part, the goal is to use the microwave remote sensing technique to monitor water resources and global climate change. Towards this goal, I study the microwave scattering behavior of snow and ice sheet. For snowpack scattering, I have extended the traditional dense media radiative transfer (DMRT) approach to include cyclical corrections that give rise to backscattering enhancements, enabling the theory to model combined active and passive observations of snowpack using the same set of physical parameters. Besides DMRT, a fully coherent approach is also developed by solving Maxwell’s equations directly over the entire snowpack including a bottom half space. This revolutionary new approach produces consistent scattering and emission results, and demonstrates backscattering enhancements and coherent layer effects. The birefringence in anisotropic snow layers is also analyzed by numerically solving Maxwell’s equation directly. The effects of rapid density fluctuations in polar ice sheet emission in the 0.5~2.0 GHz spectrum are examined using both fully coherent and partially coherent layered media emission theories that agree with each other and distinct from incoherent approaches. For the second part, the goal is to develop integral equation based methods to solve wave scattering in periodic structures such as photonic crystals and metamaterials that can be used for broadband simulations. Set upon the concept of modal expansion of the periodic Green’s function, we have developed the method of broadband Green’s function with low wavenumber extraction (BBGFL), where a low wavenumber component is extracted and results a non-singular and fast-converging remaining part with simple wavenumber dependence. We’ve applied the technique to simulate band diagrams and modal solutions of periodic structures, and to construct broadband Green’s functions including periodic scatterers.PHDElectrical EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/135885/1/srtan_1.pd

Methods in wave propagation and scattering

Supervised by Jin A. Kong.Also issued as Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2001.Includes bibliographical references (p. 195-213).by Henning Braunisch

Methods in wave propagation and scattering

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2001.Vita.Includes bibliographical references (p. 195-213).This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Aspects of wave propagation and scattering with an emphasis on specific applications in engineering and physics are examined. Frequency-domain methods prevail. Both forward and inverse problems are considered. Typical applications of the method of moments to rough surface three-dimensional (3-D) electromagnetic scattering require a truncation of the surface considered and call for a tapered incident wave. It is shown how such wave can be constructed as a superposition of plane waves, avoiding problems near both normal and grazing incidence and providing clean footprints and clear polarization at all angles of incidence. The proposed special choice of polarization vectors removes an irregularity at the origin of the wavenumber space and leads to a wave that is optimal in a least squared error sense. Issues in the application to 3-D scattering from an object over a rough surface are discussed. Approximate 3-D scalar and vector tapered waves are derived which can be evaluated without resorting to any numerical integrations. Important limitations to the accuracy and applicability of these approximations are pointed out. An analytical solution is presented for the electromagnetic induction problem of magnetic diffusion into and scattering from a permeable, highly but not perfectly conducting prolate spheroid under axial excitation, expressed in terms of an infinite matrix equation. The spheroid is assumed to be embedded in a homogeneous non-conducting medium as appropriate for low-frequency, high-contrast scattering governed by magnetoquasistatics. The solution is based on separation of variables and matching boundary conditions where the prolate spheroidal wavefunctions with complex wavenumber parameter are expanded in terms of spherical harmonics. For small skin depths, an approximate solution is constructed, which avoids any reference to the spheroidal wavefunctions. The problem of long spheroids and long circular cylinders is solved by using an infinite cylinder approximation. In some cases, our ability to evaluate the spheroidal wavefunctions breaks down at intermediate frequencies. To deal with this, a general broadband rational function approximation technique is developed and demonstrated. We treat special cases and provide numerical reference data for the induced magnetic dipole moment or, equivalently, the magnetic polarizability factor. The magnetoquasistatic response of a distribution of an arbitrary number of interacting small conducting and permeable objects is also investigated. Useful formulations are provided for expressing the magnetic dipole moment of conducting and permeable objects of general shape. An alternative to Tikhonov regularization for deblurring and inverse diffraction, based on a local extrapolation scheme, is described, analyzed, and illustrated numerically for the cases of continuation of fields obeying Laplace and Helmholtz equations. At the outset of the development, a special deconvolution problem, where a parameter describes the degree of additive blurring, is considered. No a priori knowledge on the unblurred data is assumed. A standard solution based on an output least squares formulation includes a regularization parameter into a linear, shift-invariant filter. The proposed alternative approach takes advantage of the analyticity of the smoothing process with respect to the blurring parameter. Here a simple local extrapolation scheme is employed. The problem is encountered in applications involving potential theories dealing with magnetostatics, electrostatics, and gravity data. As a generalization to the dynamic case, inverse diffraction of scalar waves is considered. Examples are presented and the two methods compared numerically. The problem of inferring unknown geometry and material parameters of a waveguide model from noisy samples of the associated modal dispersion curves is considered. In a significant reduction of the complexity of a common inversion methodology, the inner of two nested iterations is eliminated: The approach described does not employ explicit fitting of the data to computed dispersion curves. Instead, the unknown parameters are adjusted to minimize a cost function derived directly from the determinant of the boundary condition system matrix. This results in a very efficient inversion scheme that, in the case of noise-free data, yields exact results. Multi-mode data can be simultaneously processed without extra complications. Furthermore, the inversion scheme can accommodate an arbitrary number of unknown parameters, provided that the data have sufficient sensitivity to these parameters. As an important application, the sonic guidance condition for a fluid-filled borehole in an elastic, homogeneous, and isotropic rock formation is considered for numerical forward and inverse dispersion analysis. The parametric inversion with uncertain model parameters and the influence of bandwidth and noise are investigated numerically. The cases of multi-frequency and multi-mode data are examined. Finally, the borehole leaky-wave modes are classified according to the location of the roots of the characteristic equation on a multi-sheeted Riemann surface. A comprehensive set of dipole leaky-wave modal dispersions is computed. In an independent numerical experiment the excitation of some of these modes is demonstrated. The utilization of leaky-wave dispersion data for inversion is discussed.by Henning Braunisch.Ph.D

13th Annual Review of Progress in Applied Computational Electromagnetics at the Naval Postgraduate School, Monterey, CA, March 17-21, 1997, Conference Proceedings Volumes I & II

Includes Volumes 1 &

Annual Review of Progress in Applied Computational Electromagnetics

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Accurate and efficient full-wave modelling for indoor radio wave propagation

The transition towards next-generation communication technologies has increased the need for accurate knowledge about the wireless channel. Knowledge of radio wave propagation is vital to the continued development of efficient wireless communications systems capable of providing a high data throughput and reliable connection. Thus, there is an increased need for accurate propagation models that can rapidly predict and describe the propagation channel. This is extremely challenging for indoor environments given the large variety of materials encountered and very complex and widely varying geometries.Currently, empirical or ray optical models are the most common for indoor propagation. Empirical models based on measurement campaigns provide limited accuracy, are very costly and time-consuming but provide rapid predictions. Deterministic models are applied to the geometrical representation of the environment and are based on Maxwell’s equations. They can produce more accurate predictions than empirical models. Ray tracing, an approximate model, is the most popular deterministic model for indoor propagation. The current trend of research is focused on improving its accuracy. Full-wave propagation models are based on the numerical solution of Maxwell’s equations. They are able to produce accurate predictions about the wireless channel. However, they are very computationally expensive. Thus, there has been limited attempts at developing indoor propagation models based on full-wave techniques. In this work, the Volume Electric Field Integral Equation (VEFIE) is used as the basis of a full-wave indoor propagation model. The 2D and 3D formulations of the VEFIE are applied to model the propagation of radio waves indoors. An enhancement to the 2D VEFIE, called 2D to 3D models, is developed to improve its accuracy and utilise its efficiency. It is primarily used for the prediction of time domain characteristics due to its high efficiency whereas the 3D VEFIE is shown to be suitable for frequency domain predictions