Maxwell’s equations are a set of partial differential equations that describe the classic electromagnetic problems, electrodynamics etc. Effective numerical methods are derived to solve the equations in the past decades, and continued to be of great interest to be developed to its completion. In this thesis, we introduce and propose numerical methods using staggered meshes that deal with both two dimensional and three dimensional space problem in polygonal and general curved domains. / Finite difference method, finite volume method, spectral method and staggered discontinuous Galerkin method are discussed in the thesis. A forth order finite difference method using Taylor expansion technic is proposed. The integral form of the original Maxwell’s equations give rise to methods based on more general domain. For the finite volume method, covolume methods both on the cyclic polygon elements and noncyclic polygon elements are derived. To derive a higher order accurate method, staggered discontinuous Galerkin method based on the same domain decomposition present in the finite volume method use Nedelec elements is derived in two dimensional space, and spectral method using nodal high-order method operate on a general domain in 3D with flexible domain geometry is introduced. Numerical results are shown to show the performance oft he above mentioned approximation methods in 2D case. / 麥克斯韋方程組是一組描述經典電磁問題，電磁力學的偏微分方程。在過去數十年，行之有效的偏微分方程數值解已經被推導出並用於求解該方程，該問題現在仍然吸引著學者極大的興趣，並日臻完善。在這篇論文中，我們介紹並提出一些運用曲域交錯網格數值方法在二維和三維的多面體和更一般幾何體處理麥克斯韋方程組問題。 / 本論文對有限差分法，有限體積法，光譜法和交錯間斷有限元方法進行了討論。利用泰勒展開式這一方法推導出一個二維的四階有限差分方法。基於原來的麥克斯韋方程組的積分形式所得到的數值方法更適用於更普遍的域。對於有限體積法，對循環多邊形元素和非環狀多邊形元素的有限體積方法都將被導出。為了得到一個更高階準確的方法，基於有限體積法中使用的域分解方法，使用Nedelec元素，推導了二維空間的高階有限元方法。基於頂點高階數值方法的光譜法對於三維一般定義域的幾何形態更為靈活適用。在二維的定義域中，數值模擬結果驗證上述數值方法的精確性。 / Jian, Fangqiong. / Thesis M.Phil. Chinese University of Hong Kong 2014. / Includes bibliographical references (leaves 62-65). / Abstracts also in Chinese. / Title from PDF title page (viewed on 07, October, 2016). / Detailed summary in vernacular field only. / Detailed summary in vernacular field only.

Zhang Kai. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2005. / Includes bibliographical references (leaves 99-102). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.6 / Chapter 1.1 --- The Generalized PML Theory --- p.6 / Chapter 1.1.1 --- Background --- p.6 / Chapter 1.1.2 --- Derivation --- p.8 / Chapter 1.1.3 --- Reflection Properties --- p.11 / Chapter 1.2 --- Unified Formulation --- p.12 / Chapter 1.2.1 --- "Face-, Edge- and Corner-PMLs" --- p.12 / Chapter 1.2.2 --- Unified PML Equations in 3D --- p.15 / Chapter 1.2.3 --- Unified PML Equations in 2D --- p.16 / Chapter 1.2.4 --- Examples of PML Formulations --- p.16 / Chapter 1.3 --- Inhomogeneous Initial Conditions --- p.23 / Chapter 2 --- Numerical Analysis of PMLs --- p.25 / Chapter 2.1 --- Continuous PMLs --- p.26 / Chapter 2.1.1 --- PMLs for Wave Equations --- p.27 / Chapter 2.1.2 --- Finite PMLs for Wave Equations --- p.31 / Chapter 2.1.3 --- Berenger's PMLs for Maxwell Equations --- p.33 / Chapter 2.1.4 --- Finite Berenger's PMLs for Maxwell Equations --- p.35 / Chapter 2.1.5 --- PMLs for Acoustic Equations --- p.38 / Chapter 2.1.6 --- Berenger's PMLs for Acoustic Equations --- p.39 / Chapter 2.1.7 --- PMLs for 1-D Hyperbolic Systems --- p.42 / Chapter 2.2 --- Discrete PMLs --- p.44 / Chapter 2.2.1 --- Discrete PMLs for Wave Equations --- p.44 / Chapter 2.2.2 --- Finite Discrete PMLs for Wave Equations --- p.51 / Chapter 2.2.3 --- Discrete Berenger's PMLs for Wave Equations --- p.53 / Chapter 2.2.4 --- Finite Discrete Berenger's PMLs for Wave Equations --- p.56 / Chapter 2.2.5 --- Discrete PMLs for 1-D Hyperbolic Systems --- p.58 / Chapter 2.3 --- Modified Yee schemes for PMLs --- p.59 / Chapter 2.3.1 --- Stability of the Yee Scheme for Wave Equation --- p.61 / Chapter 2.3.2 --- Decay of the Yee Scheme Solution to the Berenger's PMLs --- p.62 / Chapter 2.3.3 --- Stability and Convergence of the Yee Scheme for the Berenger's PMLs --- p.67 / Chapter 2.3.4 --- Decay of the Yee Scheme Solution to the Hagstrom's PMLs --- p.70 / Chapter 2.3.5 --- Stability and Convergence of the Yee Scheme for the Hagstrom's PMLs --- p.75 / Chapter 2.4 --- Modified Lax-Wendroff Scheme for PMLs --- p.80 / Chapter 2.4.1 --- Exponential Decays in Parabolic Equations --- p.80 / Chapter 2.4.2 --- Exponential Decays in Hyperbolic Equations --- p.82 / Chapter 2.4.3 --- Exponential Decays of Modified Lax-Wendroff Solutions --- p.86 / Chapter 3 --- Numerical Simulation --- p.93 / Bibliography --- p.99

在本文中，我們為了三維空間的馬克士威方程組(Maxwell’s equation)制定和分析了一套新種類的交錯間斷伽遼金(discontinuous Galerkin)方法，同時考慮了時間依賴性和時間諧波的馬克士威方程組。我們用了空間離散上交錯笛卡兒網格，這種方法具有許多良好的性質。首先，我們的方法所得出的數值解保留了電磁能量，並自動符合了高斯定律的離散版本。第二，質量矩陣是對角矩陣，從而時間推進是顯式和非常有效的。第三，我們的方法是高階準確，最佳收斂性在這裏會被嚴格地證明。第四，基於笛卡兒網格，它也很容易被執行，並可視為是典型的Yee’s Scheme的以及四邊形的邊有限元的推廣。最後，超收斂結果也會在這裏被證明。 / 在本文中，我們還提供了幾個數值結果驗證了理論的陳述。我們計算了時間依賴性和時間諧波的馬克士威方程組數值收斂結果。此外，我們計算時間諧波馬克士威方程組特徵值問題的數值特徵值，並與理論特徵值比較結果。最後，完美匹配層(Perfect Matching Layer)吸收邊界的問題也有實行其數值結果。 / We develop and analyze a new type of staggered discontinuous Galerkin methods for the three dimensional Maxwell’s equations in this paper. Both time-dependent and time-harmonic Maxwell’s equations are considered. The spatial discretization is based on staggered Cartesian grids which possess many good properties. First of all, our method has the advantages that the numerical solution preserves the electromagnetic energy and automatically fulfills a discrete version of the Gauss law. Second, the mass matrices are diagonal, thus time marching is explicit and is very efficient. Third, our method is high order accurate and the optimal order of convergence is rigorously proved. Fourth, it is also very easy to implement due to its Cartesian structure and can be regarded as a generalization of the classical Yee’s scheme as well as the quadrilateral edge finite elements. Lastly, a superconvergence result, that is the convergence rate is one order higher at interpolation nodes, is proved. / In this paper, we also provide several numerical results to verify the theoretical statements. We compute the numerical convergence order using L2-norm and discrete-norm respectively for both the time-dependent and time-harmonic Maxwell’s equations. Also, we compute the numerical eigenvalues for the time-harmonic eigenvalue problem and compare the result with the theoretical eigenvalues. Lastly, applications to problems in unbounded domains with the use of PML are also presented. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Yu, Tang Fei. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2012. / Includes bibliographical references (leaves 46-49). / Abstracts also in Chinese. / Chapter 1 --- Introduction and Model Problems --- p.1 / Chapter 2 --- Staggered DG Spaces --- p.4 / Chapter 2.1 --- Review on Gauss-Radau and Gaussisan points --- p.5 / Chapter 2.2 --- Basis functions --- p.6 / Chapter 2.3 --- Finite Elements space --- p.7 / Chapter 3 --- Method derivation --- p.14 / Chapter 3.1 --- Method --- p.14 / Chapter 3.2 --- Time discretization --- p.17 / Chapter 4 --- Energy conservation and Discrete Gauss law --- p.19 / Chapter 4.1 --- Energy conservation --- p.19 / Chapter 4.2 --- Discrete Gauss law --- p.22 / Chapter 5 --- Error analysis --- p.24 / Chapter 6 --- Numerical examples --- p.29 / Chapter 6.1 --- Convergence tests --- p.30 / Chapter 6.2 --- Diffraction by a perfectly conducting object --- p.30 / Chapter 6.3 --- Perfectly matched layers --- p.37 / Chapter 7 --- Time Harmonic Maxwell’s equations --- p.40 / Chapter 7.1 --- Model Problems --- p.40 / Chapter 7.2 --- Numerical examples --- p.40 / Chapter 7.2.1 --- Convergence tests --- p.41 / Chapter 7.2.2 --- Eigenvalues tests --- p.41 / Chapter 8 --- Conclusion --- p.45 / Bibliography --- p.46

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The numerical solutions of Maxwell’s equations have been the cornerstone of photonic design for over a century. In recent years, the field of photonics has witnessed a surge in interest in inverse design, driven by the potential to engineer nonintuitive photonic structures with remarkable properties. However, the conventional approach to inverse design, which relies on fully discretized numerical simulations, faces significant challenges in terms of computational efficiency and scalability. This thesis delves into an alternative paradigm for inverse design, leveraging the power of semi-analytical methods. Unlike their fully discretized counterparts, semi-analytical methods hold the promise of enabling simulations that are independent of the computational grid size, potentially revolutionizing the design and optimization of photonic structures. To achieve this goal, we put forth a more generalized formalism for semi-analytical methods and have developed a comprehensive differential theory to underpin their operation. This theoretical foundation not only enhances our understanding of these methods but also paves the way for their broader application in the field of photonics. In the final stages of our investigation, we illustrate how the semi-analytical simulation framework can be effectively employed in practical photonic design scenarios. We demonstrate the synergy of semi-analytical methods with ray tracing techniques, showcasing their combined potential in the creation of large-scale optical lens systems and other complex optical devices.

Traditional light microscopy suffers from the diffraction limit, which limits the spatial resolution to λ/2. The current trend in optical microscopy is the development of techniques to bypass the diffraction limit. Resolutions below 40 nm will make it possible to probe biological systems by imaging the interactions between single molecules and cell membranes. These resolutions will allow for the development of improved drug delivery mechanisms by increasing our understanding of how chemical communication within a cell occurs. The materials sciences would also benefit from these high resolutions. Nanomaterials can be analyzed with Raman spectroscopy for molecular and atomic bond information, or with fluorescence response to determine bulk optical properties with tens of nanometer resolution. Near-field optical microscopy is one of the current techniques, which allows for imaging at resolutions beyond the diffraction limit. Using a combination of a shear force microscope (SFM) and an inverted optical microscope, spectroscopic resolutions below 20 nm have been demonstrated. One technique, in particular, has been named tip enhanced near-field optical microscopy (TENOM). The key to this technique is the use of solid metal probes, which are illuminated in the far field by the excitation wavelength of interest. These probes are custom-designed using finite difference time domain (FDTD) modeling techniques, then fabricated with the use of a focused ion beam (FIB) microscope. The measure of the quality of probe design is based directly on the field enhancement obtainable. The greater the field enhancement of the probe, the more the ratio of near-field to far-field background contribution will increase. The elimination of the far-field signal by a decrease of illumination power will provide the best signal-to-noise ratio in the near-field images. Furthermore, a design that facilitates the delocalization of the near-field imaging from the far-field will be beneficial. Developed is a novel microscope design that employs two-photon non-linear excitation to allow the imaging of the fluorescence from almost any visible fluorophore at resolutions below 30 nm without changing filters or excitation wavelength. The ability of the microscope to image samples at atmospheric pressure, room temperature, and in solution makes it a very promising tool for the biological and materials science communities. The microscope demonstrates the ability to image topographical, optical, and electronic state information for single-molecule identification. A single computer, simple custom control circuits, field programmable gate array (FPGA) data acquisition, and a simplified custom optical system controls the microscope are thoroughly outlined and documented. This versatility enables the end user to custom-design experiments from confocal far-field single molecule imaging to high resolution scanning probe microscopy imaging. Presented are the current capabilities of the microscope, most importantly, high-resolution near-field images of J-aggregates with PIC dye. Single molecules of Rhodamine 6G dye and quantum dots imaged in the far-field are presented to demonstrate the sensitivity of the microscope. A comparison is made with the use of a mode-locked 50 fs pulsed laser source verses a continuous wave laser source on single molecules and J-aggregates in the near-field and far-field. Integration of an intensified CCD camera with a high-resolution monochromator allows for spectral information about the sample. The system will be disseminated as an open system design.