### 物理代写|光学代写Optics代考|PHS2062

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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

## 物理代写|光学代写Optics代考|Cloak for Perfect Conductor

The cloaks for perfect conductor are inversely designed using the developed method. This is a typical min-type optimization problem. Topology optimization-based inverse design of two-dimensional optical cloaks have been investigated for transverse magnetic and transverse electric incident waves, where two-dimensional is the reduced case with an infinite extension assumed in the third dimension $[3,4$, 16]. Three-dimensional design is more flexible and practical for the consideration of realistic situations.

In the following, optical cloaks are designed for a spherical perfect conductor. To cloak the sphere, the scattering field should be minimized to achieve phase matching of the total field around the conductor. The inverse domain of the cloak is set to be a cube with side length equal to 7 times the incident wavelength, as shown in Fig. 3.3, where the cloak domain is set to be a spherical shell with external and internal radii equal to $2.5$ and $0.75$ times the incident wavelength, and the cloaked conductor is enclosed in a central spherical domain with a radius equal to $0.75$ times the incident wavelength. The computational domain is discretized by $63 \times 63 \times 63$ brick elements.

For a magnetic field described optical cloak, the objective in Eq.3.8 is set to minimize the normalized square norm of the scattered magnetic field
$$J=\frac{1}{J_{0}} \int_{\Omega_{a}} \mathbf{H}{s} \cdot \overline{\mathbf{H}}{s} \mathrm{~d} \Omega$$
where $\Omega_{o}$ is the domain outside the spherical shell-shaped design domain; $J_{0}$ is the square norm of the uncloaked scattered magnetic field in the outside domain of the cloak. The obtained cloak topology, found by solving the corresponding topology optimization problem, is shown in Fig. 3.4a, with incident wave, uncloaked field, and cloaked field shown in Fig. $3.4 \mathrm{c}, \mathrm{d}$, and $\mathrm{e}$, where the incident wave is set to be the

uniform plane wave $\mathbf{H}{i}=\left(0,0, e^{-j k{0} x}\right)$ with $k_{0}=20 \pi \mathrm{rad} / \mathrm{m}$. For an electric field described optical cloak, the objective in Eq. $3.24$ is set to minimize the normalized square norm of the scattered electric field
$$J=\frac{1}{J_{0}} \int_{\Omega_{o}} \mathbf{E}{s} \cdot \overline{\mathbf{E}}{s} \mathrm{~d} \Omega$$
where $J_{0}$ is the square norm of the uncloaked scattered electric field in the outside domain of the cloak. The obtained cloak topology, found by solving the corresponding topology optimization problem, is shown in Fig. 3.5a, with incident wave, uncloaked field, and cloaked field respectively shown in Fig. $3.5 \mathrm{c}, \mathrm{d}$, and e, where the incident wave is set to be the uniform plane wave $\mathbf{E}{i}=\left(0,0, e^{-j k{0} x}\right)$ with $k_{0}=20 \pi \mathrm{rad} / \mathrm{m}$. Objective convergent histories for both these two cases were respectively plotted in Fig. 3.4b and b, which has demonstrated the robustness of the convergent process of the solving procedure.

The inversely designed cloaks have effectively reduced the scattering energy in the outside domain of the cloaks, and this can be confirmed by comparing the uncloaked and cloaked fields shown in Figs. $3.4 \mathrm{~d}$ and e, $3.5 \mathrm{~d}$ and e.

## 物理代写|光学代写Optics代考|Dielectric Resonator

This section considers max-type optimization problems, for which dielectric-based optical resonator design is a typical task. Optical resonators are designed to concentrate the optical energy in a specified spherical domain, where the total field should be maximized, hence achieving a resonance of the total field in this domain. The computational domain of the resonator is set to be a cube with side length equal to $2.5$ times the incident wavelength, as shown in Fig. 3.7, where the design domain is set to be a spherical shell with external and internal radii respectively equal to 1 and

$0.3$ times the incident wavelength, and the resonating domain is the central spherical domain with radius equal to $0.3$ times the incident wavelength. The computational domain is discretized by $50 \times 50 \times 50$ brick elements.

For the magnetic field described case, the objective in Eq. $3.8$ is set to maximize the normalized square norm of the total magnetic field in the resonating domain
$$J=\frac{1}{J_{0}} \int_{\Omega_{r}} \mathbf{H} \cdot \overline{\mathbf{H}} \mathrm{d} \Omega$$
where $\Omega_{r}$ is the resonating domain; $J_{0}$ is the square norm of the total magnetic field in the resonating domain, with dielectric filled in the design domain. After implementing the solution procedure introduced in Sect. 3.1.5, the obtained resonator topology is shown in Fig. 3.8a, with incident wave and resonating field shown in Fig. 3.8c and d, and in which the incident field is set to be the uniform plane wave $\mathbf{H}{i}=\left(0,0, e^{-j k{0} x}\right)$ with $k_{0}=20 \pi \mathrm{rad} / \mathrm{m}$. For the electric field described case, the objective in Eq. $3.24$ is set to maximize the normalized square norm of the total electric field in the resonating domain
$$J=\frac{1}{J_{0}} \int_{\Omega_{r}} \mathbf{E} \cdot \overline{\mathbf{E}} \mathrm{d} \Omega$$
where $J_{0}$ is the square norm of the electric field in the resonating domain, with dielectric filled in the design domain. The obtained resonator topology is shown in Fig. 3.9a, with incident wave and resonating field respectively shown in Fig. $3.9 \mathrm{c}$ and $\mathrm{d}$, where the incident wave is set to be the uniform plane wave $\mathbf{E}{i}=\left(0,0, e^{-j k{0} x}\right)$ with $k_{0}=20 \pi \mathrm{rad} / \mathrm{m}$. The objective convergent histories for both cases are plotted in Figs. $3.8 \mathrm{~b}$ and $3.9 \mathrm{~b}$, and demonstrate the robustness of the convergence process of the solution procedure. The computationally designed resonators have effectively focused the optical energy in the resonating domain of the resonator, where the optical field has been enhanced effectively; and this can be confirmed by inspecting the field distribution shown in Figs. $3.8 \mathrm{~d}$ and $3.9 \mathrm{~d}$.

To check the optimality of the derived resonator topologies in Figs. $3.8 \mathrm{a}$ and $3.9 \mathrm{a}$, the similar cross comparison method is adopted as that in Sect. 3.2.1. By exchanging the corresponding incident waves, the magnetic field distribution around the resonator in Fig. 3.9a induced by the incident wave in Fig. 3.8c is shown in Fig. 3.10a; and electric field distribution around the resonator in Fig. 3.8a induced by the incident wave in Fig. 3.9c is shown in Fig. 3.10b. The objective values corresponding to Figs. $3.8 \mathrm{~d}$ and $3.10 \mathrm{a}, 3.9 \mathrm{~d}$ and $3.10 \mathrm{~b}$ are listed in Table 3.2. From the comparison of the values in Table $3.2$, the optimality can be confirmed for the derived resonator topologies in Figs. $3.8 \mathrm{a}$ and $3.9 \mathrm{a}$.

## 物理代写|光学代写Optics代考|Beam Splitter

An optical splitter is topologically optimized in the following, in order to demonstrate the robustness of the developed method when applied to max-min-type optimization

problems. For computationally designing the splitters, the computational domain is set up as shown in Fig.3.11, where the optical energy enters the domain from the inlet $\Gamma_{i}$ and output from the two specified outlets $\Gamma_{o 1}$ and $\Gamma_{o 2}$. The computational domain is discretized by $60 \times 60 \times 12$ elements. The incident wave is set to be the $z$-polarized uniform plane wave with a frequency equal to $1 \times 10^{9} \mathrm{~Hz}$.

The design objective of the splitter is to achieve equal and maximized energy levels at the two outlets. Therefore, for the magnetic field case, the design objective is set to maximize

$$\min \left{\int_{\Gamma_{a 1}} \frac{1}{2} \mu_{0} \mu_{r} \mathbf{H} \cdot \overline{\mathbf{H}} \mathrm{d} \Gamma, \int_{\Gamma_{a} 2} \frac{1}{2} \mu_{0} \mu_{r} \mathbf{H} \cdot \overline{\mathbf{H}} \mathrm{d} \Gamma\right}$$
and for the electric field case, the design objective is modified to maximize
$$\min \left{\int_{\Gamma_{e 1}} \frac{1}{2} \varepsilon_{0} \varepsilon_{r} \mathbf{E} \cdot \overline{\mathbf{E}} \mathrm{d} \Gamma, \int_{\Gamma_{c 2}} \frac{1}{2} \varepsilon_{0} \varepsilon_{r} \mathbf{E} \cdot \overline{\mathbf{E}} \mathrm{d} \Gamma\right}$$
The splitter topology is derived as shown in Figs. $3.12$ and $3.13$, where the convergence histories of objective values and field distribution are included. From the field distribution in Figs. $3.12 \mathrm{c}$ and $3.13 \mathrm{c}$, one can confirm by inspection the wave splitting performance of the computationally designed splitters. The splitting and parallelization was achieved in approximately eight wavelengths for both versions.
The optimality of the derived splitter topologies in Figs. 3.12a and $3.13 \mathrm{a}$ is checked with the cross comparison implemented by exchanging the corresponding incident waves. When the splitter in Fig. 3.13a is used for the magnetic wave, the magnetic field is distributed as shown Fig. 3.14a; and when the splitter in Fig. 3.12a is used for the electric field, the electric field is distributed as shown in Fig. 3.14b. The objective values corresponding to Figs. $3.12 \mathrm{c}$ and $3.14 \mathrm{a}, 3.13 \mathrm{c}$ and $3.14 \mathrm{~b}$ are listed in Table $3.3$. From the comparison of the values in Table $3.3$, the optimality for the derived splitter topologies in Figs. 3.12a and 3.13a is confirmed.

## 物理代写|光学代写Optics代考|Cloak for Perfect Conductor

$$J=\frac{1}{J_{0}} \int_{\Omega_{a} } \mathbf{H} {s} \cdot \overline{\mathbf{H}} {s} \mathrm{~d} \Omega$$

Ĵ=1Ĵ0∫Ω○和s⋅和¯s dΩ

## 物理代写|光学代写Optics代考|Dielectric Resonator

0.3乘以入射波长，共振域是中心球域，半径等于0.3乘以入射波长。计算域离散化为50×50×50砖元素。

Ĵ=1Ĵ0∫ΩrH⋅H¯dΩ

Ĵ=1Ĵ0∫Ωr和⋅和¯dΩ

## 物理代写|光学代写Optics代考|Beam Splitter

\min \left{\int_{\Gamma_{a 1}} \frac{1}{2} \mu_{0} \mu_{r} \mathbf{H}\cdot\overline{\mathbf{H}}\ mathrm{d}\Gamma,\int_{\Gamma_{a}2}\frac{1}{2}\mu_{0}\mu_{r}\mathbf{H}\cdot\overline{\mathbf{H} } \mathrm{d}\Gamma\right}\min \left{\int_{\Gamma_{a 1}} \frac{1}{2} \mu_{0} \mu_{r} \mathbf{H}\cdot\overline{\mathbf{H}}\ mathrm{d}\Gamma,\int_{\Gamma_{a}2}\frac{1}{2}\mu_{0}\mu_{r}\mathbf{H}\cdot\overline{\mathbf{H} } \mathrm{d}\Gamma\right}

\min \left{\int_{\Gamma_{e 1}} \frac{1}{2} \varepsilon_{0} \varepsilon_{r}\mathbf{E}\cdot\overline{\mathbf{E}}\ mathrm{d} \Gamma, \int_{\Gamma_{c 2}} \frac{1}{2} \productpsilon_{0}\productpsilon_{r}\mathbf{E}\cdot\overline{\mathbf{E} } \mathrm{d}\Gamma\right}\min \left{\int_{\Gamma_{e 1}} \frac{1}{2} \varepsilon_{0} \varepsilon_{r}\mathbf{E}\cdot\overline{\mathbf{E}}\ mathrm{d} \Gamma, \int_{\Gamma_{c 2}} \frac{1}{2} \productpsilon_{0}\productpsilon_{r}\mathbf{E}\cdot\overline{\mathbf{E} } \mathrm{d}\Gamma\right}

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## MATLAB代写

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