以散射矩陣法研究二維奈米金屬圓柱之表面電漿共振

A Study of Surface Plasmon Resonance on 2-D Metal Cylindrical Nanoparticles by Scattering Matrix Method

蘇英杰(Ying-Jie Su)

指導教授 ： 張建成

共同指導教授 ： 陳瑞琳

國立臺灣大學/工學院/應用力學研究所/碩士(2006年)

https://doi.org/10.6342/NTU.2006.10432

摘要

表面電漿共振行為是存在於介電質與金屬間的界面現象，本研究主要是從麥克斯威爾電磁場理論出發，以赫姆霍茲方程式為電磁波動的統御方程式，使用散射矩陣法分析多顆圓柱散射體的散射場問題，藉此討論奈米尺度下的二維圓柱之表面電漿現象及散射場行為。
單顆圓柱散射體的散射場解析解很早就被解出，而多個圓柱陣列的散射場問題因為涉及到入射光在圓柱與圓柱間的多重散射，故散射行為較單顆圓柱的散射複雜，因此圓柱陣列的多重散射問題需要利用加法定理來處理；散射矩陣法的主要精神即是先用圓柱座標下的特殊函數（如貝索函數與漢克函數）對平面波和圓柱散射體的內外域電磁場做無窮級數展開，再藉由特殊函數的加法定理將所有圓柱散射體的展開中心移到同一個展開中心，最後可以得到一組連結整個散射系統的入射電磁場係數及散射電磁場係數的線性方程組，將該組線性方程配合電磁場在散射體邊界的連續條件，便可分別求出圓柱陣列中各個圓柱體的內部電磁場與外部散射場，再利用線性疊加原理即可求得整個圓柱系統的全域電磁場分佈。
散射矩陣法可以解決二維圓柱陣列彼此間的多重散射問題，但必須限制各圓柱彼此間必須是平行排列且互不相交，而且系統的背景介質必須是均勻的。本文主要探討多顆奈米柱在耦合作用的影響下所產生的散射場行為，並討論多顆圓柱彼此間距大小、入射波入射圓柱散射體系統的不同方向、及圓柱的幾何尺寸大小對於表面電漿共振行為的影響，由散射截面積去討論上述各種不同條件下的共振波長。在模擬圓柱陣列的散射問題時，文中主要使用銀光學參數之實驗結果為圓柱的材料參數，並以可見光波長範圍做討論。

關鍵字

表面電漿共振 ； 散射矩陣法 ； 金屬圓柱

並列摘要

The scattering matrix method was applied to the analysis of surface plasmon resonance of infinite two-dimensional metal nanoparticles. The object of this thesis is to discuss the surface plasmon resonance on nanoparticles in different illumination direction. We also analysis the electric fields and magnetic fields with different particle size when the surface plasmon resonance occurs.

Since the multiple scattering should be considered, the scattering problem of many-cylinders is more complicated than single cylinder. By using scattering matrix method to solve the scattering problem of many-cylinders, first we have to express the incident field（plane wave）and scattered field by special function（for example, Bessel function and Hankel function）under cylindrical coordinate, then use the addition theorem of special function to get a linear system of equations to relate the incident field coefficients and scattered field coefficients. The incident and scattered field coefficients for every cylinder can be solved from the linear equations by matching electromagnetic boundary condition pointwisely.

並列關鍵字

surface plasmon resonance ； scattering matrix method ； metal cylinder

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