發展應用於電磁與金屬奈米微粒電漿子問題之三維時域擬譜方法

Abstract

本研究中我們針對馬克思威爾方程式發展三維時域樂勤得(Legendre)擬譜方法來分析單顆銀奈米球和銀奈米棒陣列的光學性質。藉由三套商用軟體並搭配自行開發的平行化程式，我們建構了一個從網格建立到計算完成後資料處理的完整解決方案。 經由適當的理論分析，我們先找出馬克思威爾方程式的適定性(well-posed)邊界運算子，然後導出對應於原方程式物理邊界條件的特徵邊界條件。運用這些理論結果，我們建構在半離散階段呈現漸近穩定的含有同步近似項之數值架構。接著透過檢查數個含有解析解例子的收斂性，我們得到理論所預測的結果。對於銀的光學特性，由於古典Drude模型不適用於光波段，我們使用一個頻段從可見光到近紅外線更準確的模型來描述其色散特性。同樣經由嚴格Mie理論的驗證，不論近場或遠場都符合數值理論的要求。我們更進一步系統性的比較近場和遠場的特性，發現隨著奈米粒子尺寸增大或使用介電常數大於空氣之背景材料，都會使兩者頻譜峰值的差距拉大並呈現紅移，同時也使頻帶變寬。這些差異性可提供設計時參考。 奠基於本數值架構，我們也建構一等效於實際光學實驗量測過程的模擬程序，來驗證嵌於氧化鋁基板上具近似六角晶格排列的銀奈米棒陣列之近場和遠場光學特性，模擬結果顯示相當吻合量測所得。模擬結果也顯示近場中電場在相鄰銀奈米棒之間有增強的現象。透過等效原理，我們也發現這種銀奈米棒陣列結構的遠場主要由表面磁場貢獻，反而不是一般所關注的電場。此一結果可進一步提供設計更有效表面增強拉曼散射基板。

In this study, a three-dimensional (3-D) Legendre pseudospectral time-domain (PSTD) algorithm for solving the Maxwell equations is developed to analyze optical properties of single silver nanoparticles and two-dimensional (2-D) silver nanorod arrays. An in-house PSTD parallel program along with three commercial softwares provides a thorough solution from the construction of mesh grids to data post-processing. Our approach starts by conducting an analysis for finding well-posed boundary operators for the Maxwell equations. We then derive equivalent characteristic boundary conditions for common physical boundary constraints. These theoretical results are then employed to construct a pseudospectral penalty scheme which is asymptotically stable at the semidiscrete level. Through verified by a number of cases with exact solutions, the expected convergence patterns of the proposed scheme are observed. Due to inadequacy of the classical Drude model in the visible spectra, a more precise Drude-Lorentz model with carefully chosen parameters is used to characterize the linear dispersive nature of silver form visible to near infrared regime. Numerical validations are conducted based on solving both the near-field and far-field of Mie scattering problem, and expected convergence is observed. With a systematic comparison of near- and far-field behaviors of single silver nanoparticles, the significant differences in peak wavelengths increase and represent red-shifted, and their bandwidths become broader, as particle size increases and the relative permittivity of the surrounding medium has a larger value than the vacuum does. Taking into account such differences provides useful suggestions in designing plasmonic structures. Based on this numerical scheme, a program equivalent to the experimental procedure is constructed for investigating both near- and far-field electromagnetic characteristics of 2-D silver-nanorod quasi-hexagonal arrays embedded in a substrate of anodic aluminum oxide, and the simulated far-field scattering spectra agree with the experimental observations. We show that enhanced electric field is created between adjacent nanorods and, most importantly, far-field scattered light wave is mainly contributed from surface magnetic field, instead of the surface enhanced electric field. The identified near-field to far-field connection produces an important implication in the development of more efficient surface-enhanced Raman scattering substrates.