頻域譜方法的發展與應用

在此篇論文中，基於勒詹德多項式與補償機制的高精確度多區域之頻域譜方法主要被發展來在數值上解電磁問題。提出此些頻域譜方法的主要目的在於能夠更精確地分析光波導的模態，尤其是漏波模態，以及解決因光波和奈米尺度金屬結構產生強烈交互作用的電漿子現象的散射問題。我們在這會研究一些重要的波導結構，並且將會展現此頻域譜方法結合延伸坐標型完美匹配吸收層在解決無奇異場型尖角之漏波模態波導結構時，包括M型、W型、及更複雜的六氣孔光纖，可提供指數收斂的高精確度計算，其精確度可達e-15等級。其他一些帶有尖角的直線形狀波導，如光子線波導與脊型波導，也會被拿來驗證其計算結果，並可得知在和其他演算法比較下可算出很相似的精確結果。另一方面，在解決圓形銀圓柱的波散射問題上，這裡提出的頻域譜方法將會被證明其在計算近場時，如果入射波的場強度為1時，此頻域譜方法可達到e-9等級的數值計算精確度，同時亦帶有傑出的指數收斂特性。我們接著會把此頻域譜方法應用於分析更複雜的金屬耦合結構，包括多根圓柱、多根方柱、以及多根介質包裹柱。除此之外，為了能更有效率地減少計算時間和記憶體使用量，我們使用了透過圖形處理器達成之平行計算來加速反覆解線性或特徵系統過程中所需的矩陣乘法。

關鍵字

頻域譜方法

並列摘要

The high-accuracy multidomain pseudospectral frequency-domain (PSFD) methods based on the Legendre polynomials with penalty scheme are developed in this dissertation mainly to numerically solve electromagnetic problems. The primary aim of the proposed PSFD methods is to more accurately analyze optical waveguide modes, especially the leaky ones, and solve scattering problems in plasmonics, in which optical waves would strongly interact with nanometer-sized metallic structures. Some important waveguide structures will be studied, and we’ll show that the proposed PSFD method incorporated with the stretched-coordinate perfectly matched layers (PMLs) will be able to provide high computational accuracy on the order of e-15 with exponential convergence for solving leaky-mode waveguide structures without singular-field corners, including the standard W-type, M-type, and the more complicated six-air-hole fibers. Several rectilinear waveguides with sharp corners, like the photonic wire and the Rib waveguides, will be examined as well, and their computed accuracies also show well agreements comparing with other algorithms. On the other hand, in solving wave scattering of a silver circular cylinder, the formulated PSFD method will be demonstrated to achieve numerical accuracy in near-field calculations on the order of e-9 with respect to a unity field strength of the incident wave with excellent exponentially convergent behavior in numerical accuracy, which will then be applied to analyze more complicated coupled metallic structures involving circular cylinders, square cylinders, and dielectric coated cylinders. In addition, for efficiently reducing computation time and memory usage, parallel calculations through graphic processing unit (GPU) will be utilized to accelerate matrix multiplications in the processes of iteratively solving linear or eigen-systems.

並列關鍵字

Pseudospectral Frequency-Domain Method

參考文獻

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頻域譜方法的發展與應用

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