圓柱結構上的波導邊值問題之解析研究

本論文研究的起源其脈絡係近年來產學界對於在兆赫頻段(10¹² Hz)導波問題所興起的蓬勃興趣，其中單一裸露金屬線此一結構業已成為在此波段的導波最適候選之一。在此結構上所傳播的表面波乃是所謂的表面電漿子之一例。本論文的主要結果乃呈現於第二章之倒數第二節，在該節中我們對於關乎單一裸露金屬線結構的「主波」概念進行了認真的考察，此概念亦延伸定義為（在無限多的可行解中）具有最低損耗的模式解，而我們也指出在這樣定義下所會產生的找根困難。另外，文獻上可見的許多對於同一色散關係式的形式殊異之表達，經考察已歸結出三種數學任意性之來源：exp(-iωt)與exp(iωt)慣習之選擇問題、對於複數平面上開根號函數的分支選擇、Bessel與modified Bessel方程之擇一以用。而文獻上看似形式相同的表達式往往同表而殊裡，其通常包含了定義各異、或甚至未被妥善定義的參數。我們在論文中亦提供了一個精簡而直截的對於一重要行列式展開的計算法，該行列式會生出色散關係所對應的超越方程。此計算中的其一關鍵乃在為看似冗雜的方程係數引入適當反映其間結構的符號。本論文的第一章是全部的理論基礎：Maxwell方程。此一看似熟悉的題材實則隱含一些在其公設化過程中的隱微議題，而我們在該章中也詳細而謹慎的分析如何適當的依該方程形成與求解電磁波的邊值問題。

關鍵字

單一裸露金屬線；表面電漿子； Maxwell 方程；邊值問題

並列摘要

This thesis was initiated in the context of recent interest in THz waveguiding, for which the so-called "Sommerfeld wire," a metallic single wire, has become one of the most promising candidates to propagate waves in this frequency range. The propagated surface wave on this structure is an instance of surface plasmon polaritons. The main result of this thesis is presented in the penultimate section of Chapter 2. The idea of "principal waves" for a single wire is discussed at length and this idea is extended to be solutions having the lowest attenuation among the allowed infinitely many solutions, and difficulties in tracing such solutions are then noted. Also, the origin of the many form discrepancies found in the literature is addressed for a very general, and widely applied, transcendental equation that implicitly defines the dispersion relations for modes propagating on a circular cylinder. It is found that there are 3 elements contributing to the freedom of a final form of the reached dispersion relation: the choice between the exp(-iωt) and exp(iωt) convention, the choice of the principal branch for the complex square root function, and the use of the Bessel versus modified Bessel equations. The seemingly identical forms sometimes found in the literature are usually ostensible, involving parameters that are defined differently or that have not been properly defined. A succinct way of expanding the determinant giving the transcendental equation is also presented, the gist of such calculation lying in part in introducing symbols that properly reflect the structure of the seemingly messy coefficients. Chapter 1 of this thesis is the whole of the theoretical foundation: Maxwell's equations. This familiar topic actually involves certain subtleties in its postulation and the proper implementation scheme in solving an electromagnetic boundary value problem is very carefully analyzed.

並列關鍵字

Sommerfeld wire ； Surface Plasmon Polariton ； Maxwell’s Equations ； Boundary Value Problems

參考文獻

[21] Freeman J. Dyson (1999): “Why is Maxwell’s theory so hard to understand?” in James Clerk Maxwell Commemorative Booklet. Edinburgh: James Clerk Maxwell Foundation

[66] Edward J. Rothwell, Michael J. Cloud (2009): Electromagnetic (2nd ed.). Boca Raton, FL: CRC Press

[2] Max Abraham, Richard Becker (1937): The Classical Theory of Electricity and Magnetism. London: Blackie

[6] J. C. Ashley, L. C. Emerson (1974): “Dispersion relations for non-radiative surface plasmons on cylinders,” Surf. Sci., 41(2),pp. 615–618

[7] Shaghik Atakaramians, Shahraam Afshar V., Tanya M. Monro, Derek Abbott (2013): “Terahertz dielectric waveguides,” Adv. Opt. Photon., 5, pp. 169-215

國際替代計量

圓柱結構上的波導邊值問題之解析研究

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