### 物理代写|光学代写Optics代考|PHYSICS3540

statistics-lab™ 为您的留学生涯保驾护航 在代写光学Optics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写光学Optics代写方面经验极为丰富，各种代写光学Optics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 物理代写|光学代写Optics代考|Self-consistency of Adjoint Analysis for Topology

In frequency domain, the field variables of the optical waves are complex, comprising the amplitude and phase of the field. Cost functions of the topology optimization problems are usually included the conjugate of the field variables, e.g., the energy functionals, which are the product of the field variables and their conjugates. The energy functionals are popularly used, because of the well-posedness of their least square forms. For the sensitivity analysis of a topology optimization problem, adjoint method is popularly used, where the first-order variational is implemented for the corresponding augmented Lagrangian [10].

During the adjoint analysis procedure, it is necessary to implement the variational of the conjugate operation to the field variables. Mathematically, the conjugate operator is Gâteaux differential instead of Fréchet differential, to the field variables [130]. This was ignored in several previous researches, such as the literatures in the Refs. $[77,87]$. The Gâteaux differentiability of the conjugate operator can cause the incompleteness of the adjoint sensitivity, i.e., the adjoint sensitivity of the real-valued cost

functions is complex instead of real for the design variable, which is real-valued distribution defined on the computational domain. If this incomplete sensitivity is used directly, the design variable will be evolved to be complex during the iterative procedure, where the initial of the design variable is set to be real. This adjoint sensitivity is then self-inconsistent from the pointview of keeping the real-value property of the design variable. Therefore, the real part extraction operator is used to extract the real part of the derived adjoint sensitivity, and to artificially enforce the self-consistency of the adjoint sensitivity.

The consequence of such enforced self-inconsistency is that the derived structural topology has dependence on the phase of the incident wave. This phase-dependence is unreasonable, because the incident waves can not be inherently distinguished by only altering their phases. To solve the problem on the self-inconsistency of the adjoint sensitivity, Fréchet differentiability should be ensured for the cost function. The conjugate operator in the cost function can be removed by splitting the complex field variables into the corresponding real and imaginary parts respectively defined on real functional spaces instead of complex functional spaces. Then the Gâteaux differentiability induced by the conjugate operator is avoided. The splitting of the complex variables brings about the splitting of the wave equations, which are complex partial differential equations. The method of splitting complex partial differential equations or variational problems into their corresponding coupled systems for the real and imaginary parts of the field variables has been systematically discussed in the Refs. $[3,65]$.

## 物理代写|光学代写Optics代考|Dielectric Material Based Topology Optimization

It has been mentioned that the control of optical waves is realized by structures with complex spacial configurations using pre-selected materials and the incident waves can have complicated polarizations. Most of those situations cannot be reduced into two dimensional, except for a minority of cases involving linear polarized waves. Most of the reports on topology optimization in optics have focused on applications, including beamsplitters $[80,86]$, photonic crystals $[37,89]$, cloaks $[7,8,38]$, sensors and resonators $[104,105]$, metamaterials $[28,77,137]$, excitation of surface plasmons [10], and electromagnetic and optical antennas [35, 36, 48, 134], without presenting the systemical topology optimization methodology for optical waves propagating in three-dimensional space. Therefore, it is necessary to develop a unified and systematic topology optimization approach that sufficiently considers the physical complexity of three-dimensional optics.

It is not straightforward to develop the finite element-based topology optimization method for optical waves in three-dimensional space, because the divergence-free condition needs to be enforced. In the two-dimensional transverse electric or magnetic wave cases, the divergence-free conditions are automatically satisfied during the reducing procedure of the Maxwell’s equations with deriving the Helmholtz equations, and the node element-based Galerkin finite element method can be naturally used to directly discretize the Helmholtz equations $[53,69]$. Being different from the two-dimensional cases, the divergence-free conditions can not be automatically satisfied in solving the three-dimensional optical waves with the node element-based Galerkin finite element method, and this results in the spurious solutions.

For this problem, two dominant approaches have been developed to enforce the divergence-free conditions and eliminate the spurious solutions. The first approach is to add a penalty term with the least square form of the divergence-free condition to the weak form of the wave equation, and then discretize the weak form with node elements. However, the use of penalty term can not eliminate the divergence of the solution completely and it affects the solution accuracy. Therefore, the divergencefree condition can not be satisfied accurately by the penalty approach [53]. The second approach is the use of edge elements that assign degrees of freedom to the edges rather than to the nodes of the elements, where the vector basis with inherent satisfaction of divergence-free condition is used to implement interpolation $[53,69,72,116]$. The edge elements have also solved the problems on the inconvenience of imposing boundary conditions at material interfaces and the difficulty in treating conducting and dielectric edges and corners due to field singularities $[53,69]$. Therefore, the edge element-based finite element method is the more reasonable choice for discretizing the three-dimensional wave equations and developing the topology optimization method for three-dimensional optical waves.

## 物理代写|光学代写Optics代考|Metal Material-Based Topology Optimization

A metal surface with a negative real part of the permittivity can trap optical waves with achieving surface plasmon polaritons $[31,81]$. The metals used for surface plasmon polaritons are usually noble metals, e.g., silver (Ag), gold (Au) and Aluminum (Al). At optical frequencies, the metal’s free electrons can sustain, under certain conditions, oscillations with distinct resonance frequencies $[66,127,140]$. The existence of surface plasmons is characteristic for the interaction between metal and light, where the Kretschmann-Raether and Otto configurations are commonly used for plasmon excitation.

Many innovative concepts of surface plasmon polaritons have been developed over the past few years, e.g., localized surface plasmon resonances [100], extraordinary optical transmission [30] and transformational plasmon optics $[52,60]$. Correspondingly, many related applications have also been proposed for surface plasmon polaritons, e.g., biomolecular manipulation and labeling [23], surface enhanced Raman spectroscopy [70], chemical and biological sensors [33], photo-voltaics [22], nearfield lithography and imaging [84], optical trapping [67, 76], nano optic circuits [1], opto-electronic devices, wavelength-tunable filters, optical modulators [16,34,41, $42]$, plasmonic Luneburg lens and surface plasmonic cloaking $[52,60]$.

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。