Browsing by Author "Luo, Ma"

To study periods of fundamental groups of algebraic varieties, one requires an explicit algebraic de Rham theory for completions of fundamental groups. This thesis develops such a theory in two cases. In the first case, we develop an algebraic de Rham theory for unipotent fundamental groups of once punctured elliptic curves over a field of characteristic zero using the universal elliptic KZB connection of Calaque-Enriquez-Etingof and Levin-Racinet. We use it to give an explicit version of Tannaka duality for unipotent connections over an elliptic curve with a regular singular point at the identity. In the second case, we develop an algebriac de Rham theory for relative completion of the fundamental group of the moduli space of elliptic curves with one marked point. This allows the construction of iterated integrals involving modular forms of the second kind, whereas previously Brown and Manin only studied iterated integrals of holomorphic modular forms.

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.

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.

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.

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.

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.

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.

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.

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.