臺灣博碩士論文加值系統

金屬次波長結構具有操控光的應用潛力，因此相關研究已成為奈米光學中熱門領域之一。然而，光與此類結構交互作用之基本機制仍為主要關注的議題。本論文即針對此議題提出理論分析、模型建置與模擬等新穎方法，以揭示其中一些饒富意味且重要的基本物理與其涵義。

我們透過格林理論提出光穿透數種不同金屬結構的理論計算方法。利用虛源法，我們將所需的格林函數整理成三種型式，以對應結構之幾何特性。透過電磁場的邊界條件，可列出聯立方程組，並以數值上常見的線性代數方法解之。其解與先前的理論工作和電磁有限差分時域法得出的結果相當吻合。由於此方法是基於嚴格理論發展而來，因此可以提供豐富的物理內涵，同時也因為能夠進一步分析光學特性與結構間的關係，而拓展了應用的可能性。

我們從模擬中發現光在初期穿透一個次波長狹縫後之振幅有調變的現象。為瞭解此新穎現象及隱藏於其後的物理機制，我們從不同以往的觀點發展出一套新的解析模型。每個光週期產生的電場可視為一個獨立的單元，且根據Fabry-Pérot-like共振，其在狹縫裡將來回振盪，並在傳播到狹縫出口處時有部分的量穿過，以產生一系列的子單元，直到該單元散逸為止；子單元間的時間差與狹縫厚度有關。我們所觀察到的現象即為不同單元產生的子單元互相疊加而成的結果。除了學術重要性外，我們認為此成果可應用到短雷射脈衝的研究上。

我們以點波源模型展示光穿透一個由次波長狹縫與凹槽組合而成之結構後，所產生的繞射現象。根據Fabry-Pérot-like共振，電磁波在狹縫與凹槽裡皆會來回振盪，且在到達出口處有部分會傳播出去。此傳播的波部分將幅射至自由空間中，同時部分流向其他結構。點波源被放置在各出口中心處以建立幅射波模型；其相位可由電磁波模擬裡自由空間中場的分佈及在出口中心處場振盪到峰值的時間決定，而振幅則由結構中的能量流觀察得出。本點波源模型建置的計算結果與模擬相當一致。不同於藉由邊界條件求解的理論方法，本模型提供了基於波動力學與能量守恆的解析解。

最後為應用轉換光學在電磁有限差分時域法中，以產生非均勻網格的模擬方法。我們重新考慮了控制數值穩定之庫倫條件，也分析了相關的數值誤差。為展示本方法的效率，我們模擬一個方型次波長孔洞的穿透頻譜；其中，精細網格被設計在孔洞附近以恰當地解析電磁場的分佈情形，同時在其他地方使用粗糙的網格以降低計算需求。結果顯示計算成本最多可以顯著地降低到原均勻網格的5.31%。因此，本方法應有利於次波長奈米結構之模擬研究。

本論文披載的研究成果可為奈米光學科學研究與工程技術領域提供新方法，以探索更多的未知，也因此對於我們欲進一步瞭解光與次波長結構作用之現象應有所助益。

Study of subwavelength metallic structures is an active research area in nanophotonics because of its capability to tailor light for critical optical applications. The understanding of the light-matter interaction mechanism is still of major concern. This dissertation presents novel methods of theoretical analysis, modeling, and simulation for revealing some intriguing fundamental physics and its implications.

A theoretical analysis via Green's theorem for light transmission through several configurations is proposed. The Green's functions are organized into three types by the method of images. Equation sets are formulated according to boundary conditions for solving the magnetic fields. The results yielded are in excellent agreement with the previous theoretical work and our finite-difference time-domain (FDTD) simulations. Since the method is rigorously developed, it can be helpful for one to gain physical insight and extend the possibility of applications of subwavelength structures from the analysis of their optical properties.

The amplitude of light transmitted through a subwavelength slit at early stage, as revealed in an FDTD simulation, is found to be modulated. To understand this novel phenomenon and underlying physics, a new model is developed. The field of one light period is considered as an individual unit. It is partially transmitted through the slit to produce a subunit train based on Fabry-Pérot-like resonance. The superposition of the trains produces the observed light. Besides academic importance, this study may be applicable to photonics with short laser pulses.

A point source model for light diffraction from a subwavelength slit surrounded by grooves is demonstrated. Due to Fabry-Pérot-like resonance, the wave in each indentation (slit or groove) travels back and forth and propagates out at their exits. The propagating wave partially radiates into free space and partially flows into other indentations simultaneously. The point sources are placed at the exit centers to model the radiation; the phases are obtained from the spatial peak shift and the peak time of the simulated field while the amplitudes are obtained from the energy fluxes inside the indentations. The results of superposed waves show excellent agreement with the simulation. Different from the solution yielded according to the boundary conditions, this analytical solution is based on the wave dynamics and the conservation of energy.

An FDTD simulation method that applies the methodology of transformation optics for generation of non-uniform grids is developed. The numerical stability is re-considered. Also, an error analysis is performed. To demonstrate the efficiency, we obtain the light transmission spectrum through a rectangular subwavelength aperture, where fine cells are designed to be around the aperture. It is shown that the computational cost can be remarkably reduced down to 5.31% of the uniform case at the most. Therefore, the method should be useful to the simulation study of subwavelength nanostructures.

These findings revealed in this dissertation provide new means to reveal more discoveries in the science and engineering of nanophotonics, and thus should be helpful to advance our understanding of the interaction between light and subwavelength structures.

口試合格證明 i
摘要 iii
Abstract v
Acknowledgments vii
Contents ix
List of Figures xiii
List of Tables xxi

1 Introduction p.1
1.1 Development of theoretical models p.2
1.2 FDTD simulation and other numerical schemes p.7
1.3 The aim and organization of this dissertation p.11
2 Theoretical Analysis for Light Transmission through Subwavelength Structures via Green's Theorem p.13
2.1 Formulation of time-harmonic fields p.14
2.2 Application of Green's theorem p.17
2.3 Single slit p.20
2.4 A slit surrounded by grooves p.24
2.5 Double slit p.27
2.6 Indented double slit p.30
2.7 Discussions p.34
2.8 Summary p.37
3 Modeling of Modulated Light Transmission through a Subwavelength Slit at Early Stage p.39
3.1 Fabry-Pérot-like resonance and modulated light transmission at early stage p.40
3.2 Analytical model based on Fabry-Pérot-like resonance p.43
3.3 Model parameters obtained from FDTD simulation p.46
3.4 Modeling results and comparison with simulation p.48
3.5 Discussions p.50
3.6 Summary p.51
4 Method of Modeling Light Diffracted from a Subwavelength Slit Surrounded by Grooves with Point Sources p.53
4.1 Simulation of light diffraction from a subwavelength slit surrounded by grooves p.54
4.2 Point source modeling with the coefficients yielded from FDTD simulation p.59
4.3 Modeling results p.65
4.4 Summary p.67
5 FDTD Simulation with Transformation Optics p.69
5.1 Concept of transformation optics: generation of non-uniform grids p.70
5.2 Courant condition with the derived numerical anisotropic medium p.72
5.3 Error analysis p.76
5.4 Light transmission through a subwavelength aperture p.80
5.5 System setup p.83
5.6 Convergence of transmittance spectrum with uniform grids p.85
5.7 Results from 1D and 2D non-uniform grids and their efficiency p.87
5.8 Summary p.89
6 Conclusion p.91
A Solutions to the equation sets formulated from Green's theorem p.95
A.1 The high-order terms of the Green's function in the integral p.95
A.2 Single slit p.96
A.3 A slit surrounded by grooves p.99
A.4 Double slit p.104
A.5 Indented double slit p.107
Bibliography p.111

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