### 数学代写|拓扑学代写Topology代考|MATH3531

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (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

## 数学代写|拓扑学代写Topology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.

## 数学代写|拓扑学代写Topology代考|Modelling

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:
$$\left{\begin{array}{l} \nabla \cdot\left[\mu_{r}^{-1} \nabla\left(E_{z s}+E_{z i}\right)\right]+k_{0}^{2} \varepsilon_{r}\left(E_{z s}+E_{z i}\right)=0, \text { in } \Omega \ \mu_{r}^{-1} \nabla E_{z s} \cdot \mathbf{n}+j k_{0} \sqrt{\varepsilon_{r} \mu_{r}^{-1}} E_{z s}=0, \text { on } \partial \Omega \end{array}\right.$$
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 $e^{-j k n \cdot \mathbf{x}}$, 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.

## 数学代写|拓扑学代写Topology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 $\mathrm{SU}, \mathrm{Si}$ 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. $3.16 \mathrm{~b}$, 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. 3.16a 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.

## 数学代写|拓扑学代写Topology代考|Modelling

$$\left{ ∇⋅[μr−1∇(和和s+和和一世)]+ķ02er(和和s+和和一世)=0, 在 Ω μr−1∇和和s⋅n+jķ0erμr−1和和s=0, 上 ∂Ω\正确的。$$

## 有限元方法代写

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## MATLAB代写

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