物理代写|光学代写Optics代考|PHYSICS 3540

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  • 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|PHYSICS 3540

物理代写|光学代写Optics代考|Cloak for Dielectric Resonator

Many different applications are in pressing need of effectively cloaking resonators (or sensors and detectors), which can efficiently detect signals but has negligible disturbance on the surrounding environment. For example, in physics and engineering experiments, this means that a probe, e.g., the tip of a near-field scanning optical microscope or a microwave antenna, may have a minimal scattering effect on the quantity it is designed to measure $[1,2]$. With the development of transformation optics, the old dream of a device which render an object invisible to the human eye is already within reach $[33,40]$. By transformation optics, the cloak/anticloak interaction has been investigated to realize the sensor cloaking [17]. However, the derived cloak/anticloak has extreme optical properties, permittivity and permeability. And they normally are implemented by exotic metamaterials [9]. The tailored microstructure of such metamaterials has to be much smaller than the wavelength, and this makes it very challenging to realize the desired magnetic properties at optical frequencies. Would it be possible to design a cloaked resonator using conventional

simple isotropic dielectric readily available in nature instead of using metamaterials with extreme optical properties?

The topology optimization based inverse design approach can be adopted to address this question, by finding the geometrical configuration of the conventional nonmagnetic isotropic dielectric cloak for a resonator. Besides, the metasurfacesbased optical illusion or virtual shaping has also been demonstrated to be an alternative approach $[20,38,41]$. Topology optimization is a full-parameter method used to inversely determine the geometrical configuration, which represents distribution of materials [6]. It can be used to implement the structural design for the cases where the scale is large enough to ensure the reasonability for using physical parameters of materials fitting in with statistical hypothesis or continuum hypothesis. In contrast to designing devices by tuning a handful of structural parameters in size and shape optimization, topology optimization method utilizes the full-parameter space to design structures solely based on the user’s desired performance specification. Therefore, topology optimization is more flexible and robust, because of its low dependence on initial structure and implicitly expression of the material distribution in structures.


An infinitely long cylinder domain is illuminated in the free space with monochromatic propagating wave. Due to the invariance of the optical properties along the cylinder axis, the problem can be formulated in a plane perpendicular to the cylinder axis. A first-order absorbing boundary condition is used as an approximation to the Sommerfeld radiation condition in order to truncate the infinite domain. Thus, the computational domain is preset as shown Fig. $3.15$ with one circularly shaped resonator at the center. A time-harmonic optical wave propagates from the left boundary through the computational domain. In the computational domain, the resonator cloak is located in a ring-shaped domain with the same center as the resonator, and it is inversely determined using the topology optimization approach. The rest surrounding medium is set to be vacuum.

For transverse electric polarization, the waves are described by the governing equation as follows:
\nabla \cdot\left[\mu_{r}^{-1} \nabla\left(E_{z s}+E_{z i}\right)\right]+k_{0}^{2} \varepsilon_{r}\left(E_{z s}+E_{z i}\right)=0, \text { in } \Omega \
\mu_{r}^{-1} \nabla E_{z s} \cdot \mathbf{n}+j k_{0} \sqrt{\varepsilon_{r} \mu_{r}^{-1} E_{z s}}=0, \text { on } \partial \Omega
where $E_{z s}$ is the scattering transverse electric field; $E_{z i}$ is the incident transverse electric field; $\varepsilon_{r}$ and $\mu_{r}$ are the relative permittivity and permeability respectively; $k_{0}$ is the free space wave number; $j$ is the imaginary unit; $\Omega$ is the computational domain with trace $\partial \Omega$. This section considers the inverse design case for uniform plane incident waves with the incident transverse electric wave $E_{z i}$ set to be $e^{-j k_{0} \mathbf{k} \cdot \mathbf{x}}$, where $\mathbf{k}$ is the normalized wave vector and $\mathbf{x}$ is the spatial coordinate.

Topology optimization approach is based on the material interpolation between two different materials. And the material interpolation is implemented with the binary distribution defined in the design domain, where the binary distribution with values 0 and 1 respectively represent two material phases. This section considers nonmagnetic materials with unity relative permeability. Then the inverse design for the resonator cloaking is focused on the geometrical configuration corresponding to the spatial distribution of materials with two different relative permittivity. The binary distribution is set to be the design variable, which is relaxed to vary in the interval $[0,1]$ in the gradient information-based topology optimization.

物理代写|光学代写Optics代考|Results and Discussion

In this section, the resonator cloaking performance is investigated, with including the sensitivity to the incident angle. The inverse design method is further applied to the cases with dielectric materials $\mathrm{SU} 8, \mathrm{Si}$ and $\mathrm{SiO}_{2}$ to reveal the origin of inversely designed resonator cloaking.

The dielectric material with relative permittivity $\varepsilon_{r}=2$ is chosen for both the resonator and cloak. The incident wavelength is set to be $600 \mathrm{~nm}$. The radius of the resonator and exterior radius of the ring-shaped design domain are set to be $0.5$ – and 2 -fold of the incident wavelength respectively. Then, the resonator cloak is derived as shown in Fig. 3.16, where the inversely designed resonator is shown in Fig. $3.16 \mathrm{a}$, and the total fields for the cloaked and uncloaked resonator are plotted respectively in Fig. $3.16 \mathrm{~b}$ and c. With the inversely designed resonator cloak shown in Fig. $3.16 \mathrm{a}$, the scattering induced by the resonator is reduced to be $0.08$-fold compared with that of the uncloaked case; and the filed is kept to resonate in the central domain with $1.30$-fold enhancement. From the total field in Fig. 3.16b, one can conclude that the inversely designed resonator cloak achieves the phase matching by effectively weakening the scattering field in the outside surrounding and the total field is enhanced in the resonator by guiding and focusing the field in the cloak.
The resonator cloak in Fig. $3.16 \mathrm{a}$ is inversely designed for incident wave with fixed incident angle. Its performance has a strong dependence on the incident angle. Therefore, the incident angle-insensitive inverse design is implemented to extend the incident angle bandwidth. The inverse design procedure is implemented by setting the design objective to be the sum of equally weighted quotients corresponding to different incident angles valued in a specified incident bandwidth. By specifying the incident bandwidth to be $-5^{\circ} \sim 5^{\circ}$, the incident angle-insensitive inverse design of resonator cloak is derived as shown in Fig.3.17a with total field distribution corresponding to different incident angles respectively shown in Fig. $3.17 \mathrm{~b} \sim \mathrm{g}$. In Fig. $3.17 \mathrm{~h}$, the incident angle spectra of the inversely designed resonator cloak is plotted. These results demonstrate that reasonably good cloaking effect is achieved within the moderate angle range.

物理代写|光学代写Optics代考|PHYSICS 3540


物理代写|光学代写Optics代考|Cloak for Dielectric Resonator

许多不同的应用都迫切需要有效地隐蔽谐振器(或传感器和检测器),它可以有效地检测信号,但对周围环境的干扰可以忽略不计。例如,在物理和工程实验中,这意味着探头(例如近场扫描光学显微镜或微波天线的尖端)可能对其设计测量的量具有最小的散射效应[1,2]. 随着变换光学的发展,使人眼看不到物体的设备的古老梦想已经触手可及[33,40]. 通过变换光学,已经研究了斗篷/反斗篷相互作用以实现传感器隐形[17]。然而,衍生的斗篷/反斗篷具有极端的光学特性、介电常数和磁导率。它们通常由外来超材料实现 [9]。这种超材料的定制微结构必须比波长小得多,这使得在光学频率下实现所需的磁性非常具有挑战性。是否有可能使用传统方法设计一个隐形谐振器


可以采用基于拓扑优化的逆向设计方法来解决这个问题,方法是找到用于谐振器的传统非磁性各向同性介电斗篷的几何结构。此外,基于超表面的光学错觉或虚拟成形也已被证明是一种替代方法[20,38,41]. 拓扑优化是一种全参数方法,用于反向确定几何配置,代表材料的分布[6]。可用于在规模足够大的情况下进行结构设计,以保证使用符合统计假设或连续假设的材料物理参数的合理性。与通过在尺寸和形状优化中调整少量结构参数来设计设备相比,拓扑优化方法利用全参数空间来设计结构,仅基于用户所需的性能规格。因此,拓扑优化更加灵活和稳健,因为它对初始结构的依赖性低,并且隐含地表达了结构中的材料分布。


一个无限长的圆柱域在自由空间中被单色传播波照亮。由于沿圆柱轴的光学特性的不变性,该问题可以在垂直于圆柱轴的平面中表述。一阶吸收边界条件用作索末菲辐射条件的近似值,以截断无限域。因此,计算域被预先设定,如图 1 所示。3.15在中心有一个圆形谐振器。时谐光波从左边界通过计算域传播。在计算域中,谐振器斗篷位于与谐振器同心的环形域中,采用拓扑优化方法逆确定。其余的周围介质设置为真空。


∇⋅[μr−1∇(和和s+和和一世)]+ķ02er(和和s+和和一世)=0, 在 Ω μr−1∇和和s⋅n+jķ0erμr−1和和s=0, 上 ∂Ω\正确的。
在哪里和和s是散射横向电场;和和一世是入射横向电场;er和μr分别是相对介电常数和磁导率;ķ0是自由空间波数;j是虚数单位;Ω是有迹的计算域∂Ω. 本节考虑均匀平面入射波与入射横向电波的逆设计情况和和一世设置为和−jķ0ķ⋅X, 在哪里ķ是归一化的波向量和X是空间坐标。

拓扑优化方法是基于两种不同材料之间的材料插值。并且材料插值是使用设计域中定义的二进制分布实现的,其中值为 0 和 1 的二进制分布分别代表两个材料相。本节考虑具有统一相对磁导率的非磁性材料。然后,谐振器遮蔽的逆向设计侧重于与具有两种不同相对介电常数的材料的空间分布相对应的几何构型。将二元分布设为设计变量,在区间内放宽变化[0,1]基于梯度信息的拓扑优化。

物理代写|光学代写Optics代考|Results and Discussion


具有相对介电常数的介电材料er=2被选择用于谐振器和斗篷。入射波长设为600 n米. 谐振腔的半径和环形设计域的外半径设置为0.5– 和 2 倍的入射波长。然后,得到谐振器斗篷,如图 3.16 所示,其中反向设计的谐振器如图 3.16 所示。3.16一个, 隐形和非隐形谐振器的总场分别绘制在图 2 中。3.16 b和 c。使用如图所示的反向设计的谐振器斗篷。3.16一个,由谐振器引起的散射减少为0.08- 与未隐藏的情况相比的倍数;并且该领域被保持在中心领域与1.30-倍增。从图 3.16b 中的总场可以得出结论,反向设计的谐振器斗篷通过有效削弱外部周围的散射场来实现相位匹配,并通过引导和聚焦场中的场来增强谐振器中的总场。披风。
图 1 中的谐振器斗篷3.16一个是针对具有固定入射角的入射波逆向设计的。其性能对入射角有很强的依赖性。因此,采用入射角不敏感逆向设计来扩展入射角带宽。逆向设计过程是通过将设计目标设置为对应于在指定入射带宽中的不同入射角的等权重商的总和来实现的。通过指定事件带宽为−5∘∼5∘, 谐振器斗篷的入射角不敏感逆设计如图 3.17a 所示,不同入射角对应的总场分布分别如图 3.17a 所示。3.17 b∼G. 在图。3.17 H,绘制了反向设计的谐振器斗篷的入射角光谱。这些结果表明,在中等角度范围内可以实现相当好的隐身效果。

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术语 广义线性模型(GLM)通常是指给定连续和/或分类预测因素的连续响应变量的常规线性回归模型。它包括多元线性回归,以及方差分析和方差分析(仅含固定效应)。



有限元是一种通用的数值方法,用于解决两个或三个空间变量的偏微分方程(即一些边界值问题)。为了解决一个问题,有限元将一个大系统细分为更小、更简单的部分,称为有限元。这是通过在空间维度上的特定空间离散化来实现的,它是通过构建对象的网格来实现的:用于求解的数值域,它有有限数量的点。边界值问题的有限元方法表述最终导致一个代数方程组。该方法在域上对未知函数进行逼近。[1] 然后将模拟这些有限元的简单方程组合成一个更大的方程系统,以模拟整个问题。然后,有限元通过变化微积分使相关的误差函数最小化来逼近一个解决方案。





随机过程,是依赖于参数的一组随机变量的全体,参数通常是时间。 随机变量是随机现象的数量表现,其时间序列是一组按照时间发生先后顺序进行排列的数据点序列。通常一组时间序列的时间间隔为一恒定值(如1秒,5分钟,12小时,7天,1年),因此时间序列可以作为离散时间数据进行分析处理。研究时间序列数据的意义在于现实中,往往需要研究某个事物其随时间发展变化的规律。这就需要通过研究该事物过去发展的历史记录,以得到其自身发展的规律。


多元回归分析渐进(Multiple Regression Analysis Asymptotics)属于计量经济学领域,主要是一种数学上的统计分析方法,可以分析复杂情况下各影响因素的数学关系,在自然科学、社会和经济学等多个领域内应用广泛。


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