### 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|Topology Optimization Problem

In a two-dimensional case, surface plasmon polaritons are excited by transverse magnetic (magnetic field in the $z$ direction) polarized waves, scattered by metallic nanostructures. For transverse magnetic waves propagating in the $x-y$ plane, the scattered-field formulation is used in order to reduce the dispersion error
$$\nabla \cdot\left[\varepsilon_{r}^{-1} \nabla\left(H_{z s}+H_{z i}\right)\right]+k_{0}^{2} \mu_{r}\left(H_{z s}+H_{z i}\right)=0, \text { in } \Omega$$
where $H_{z}=H_{z s}+H_{z i}$ is the total field, $H_{z s}$ and $H_{z i}$ are the scattered and incident fields, respectively; $\varepsilon_{r}$ and $\mu_{r}$ are the relative permittivity and permeability, respectively; $k_{0}=\omega \sqrt{\varepsilon_{0} \mu_{0}}$ is the free space wave number with $\omega, \varepsilon_{0}$ and $\mu_{0}$ representing the angular frequency, free space permittivity and permeability, respectively; $\Omega$ is the computational domain; the time dependence of the fields is given by the factor $e^{j \omega t}$, with $t$ representing the time. The incident field can be obtained by solving the electromagnetic equations in free space, with boundary conditions representing realistic working conditions.

The boundary conditions of Eq. $4.18$ usually include the first-order absorbing condition, periodic boundary condition and symmetric condition. The first-order absorbing condition is usually used to truncate the field distribution at infinity [46]
$$\varepsilon_{r}^{-1} \nabla H_{s z} \cdot \mathbf{n}+j k_{0} \sqrt{\varepsilon_{r}^{-1} \mu_{r}} H_{s z}=0, \text { on } \Gamma_{a b}$$
where $j$ is the imaginary unit; $\mathbf{n}$ is the unit outward normal vector at the boundary $\partial \Omega$ of the computational domain; $\Gamma_{a b}$ is the absorbing boundary included in $\partial \Omega$. Periodicity of nanostructures plays a crucial role in tuning the optical response; and single nanostructure can be approximated by the periodic case with low volume ratio of the nanostructure. Therefore, the periodic boundary condition for the scattered field, induced by the periodic incident wave, is often imposed on the piecewise pair included in $\partial \Omega$
$$\left.\begin{array}{l} H_{s z}(\mathbf{x}+\mathbf{a})=H_{s z}(\mathbf{x}) e^{-j \mathbf{k} \cdot \mathbf{a}} \ \mathbf{n}(\mathbf{x}+\mathbf{a}) \cdot \nabla H_{s z}(\mathbf{x}+\mathbf{a})=-e^{-j \mathbf{k} \mathbf{a}} \mathbf{n}(\mathbf{x}) \cdot \nabla H_{s z}(\mathbf{x}) \end{array}\right} \text { for } \forall \mathbf{x} \in \Gamma_{p s}, \mathbf{x}+\mathbf{a} \in \Gamma_{p d}$$
where $\Gamma_{p d}$ and $\Gamma_{p s}$ composes one piecewise periodic boundary pair, with $\Gamma_{p d}$ and $\Gamma_{p s}$ respectively being the destination and source boundaries; $\mathbf{k}$ is the wave vector; $\mathbf{a}$ is the lattice vector of the periodic nanostructures. The symmetry of the incident wave and material distribution gives rise to the symmetrical characteristic of the scattered

field. Then the symmetric condition can be used to reduce the computational cost and ensure the computational accuracy effectively
$$\varepsilon_{r}^{-1} \nabla H_{s z} \cdot \mathbf{n}=0 \text {, on } \Gamma_{s m}$$

In this section, the variational problem for computational design is analyzed to obtain the gradient information used to iteratively evolve the design variable. According to the Refs. $[38,41,64]$, the adjoint method is an efficient approach to derive the derivative of the objective in the partial differential equation constrained variational problem. Then, the adjoint Eqs. $4.18$ and $4.21$ are obtained using the Lagrangian multiplier-based adjoint method (see Appendix $4.4$ for more details)
where $\bar{H}{z s} \in \mathscr{H}^{1 *}(\Omega)$ and $\bar{\rho}{f} \in \mathscr{H}^{1 *}(\Omega)$ are the adjoint variables of the state variables $H_{z s} \in \mathscr{H}^{1}(\Omega)$ and $\rho_{f} \in \mathscr{H}^{1}(\Omega)$, respectively; $\mathscr{H}^{1}(\Omega)$ is the first-order Sobolev space, and $\mathscr{H}^{1 *}(\Omega)$ is the dual space of $\mathscr{H}^{1}(\Omega)$; for complex, * represents the conjugate operation. It is valuable to notice that $\bar{H}{z s}^{}$ and $\rho{f}^{}$ are more convenient to be solved than $\bar{H}{z s}$ and $\rho{f}$ in the adjoint Eqs. $4.11$ and $4.12$. Therefore, the adjoint Eqs. $4.11$ and $4.12$ are utilized to solve $\tilde{H}{z s}^{}$ and $\rho{f}^{}$, and $\bar{H}{z s}$ and $\rho{f}$ can be obtained using conjugate operation. The adjoint derivative of the computational design problem is obtained as (see Appendix $4.4$ for more details)
$$\frac{\delta \hat{J}}{\delta \rho}=\operatorname{Re}\left(\frac{\partial A}{\partial \rho}-\bar{\rho}{f}^{*}\right), \text { in } \Omega$$ where $\rho$ is valued in $\mathscr{L}{2}(\Omega)$, the second-order Lebesgue integrable functional space; $\operatorname{Re}(\cdot)$ is the real part of an expression. In Eq. 4.26, only the real part of the adjoint derivative is utilized, because the design variable $\rho$ is the distribution defined on real space.

## 物理代写|光学代写Optics代考|Nanostructures for Localized Surface Plasmonic

Localized surface plasmon resonances are the strong interaction between metal nanostructures and visible light through the resonant excitations of collective oscillations of conduction electrons. In localized surface plasmon resonances, the local electromagnetic field near the nanostructure can be many orders of magnitude higher than the incident field, and the incident field around the resonant-peak wavelength is scattered strongly; the enhanced electric field is confined within only a tiny region of the nanometer length scale near the surface of the nanostructures and decays significantly thereafter [79]. Surface enhanced Raman spectroscopy (SERS) is one typical application of localized surface plasmon resonances [65]. In this section, the computational design is carried out for the metallic nanostructures of surface enhanced Raman spectroscopy using the proposed methodology.

In surface enhanced Raman spectroscopy, the strength of localized surface plasmon resonances can be measured by the maximal enhancement factor (EF) defined as $\sup {\mathbf{x} \in \Omega}|\mathbf{E}|^{4} / E{0}^{4}$, where
$$\mathbf{E}=\frac{1}{j \varepsilon_{r} \varepsilon_{0} \omega} \nabla \times\left(0,0, H_{z}\right)$$
is the total electric field and $E_{0}=\sqrt{\mu_{0} / \varepsilon_{0}}$ is the amplitude of the electric wave corresponding to the incident magnetic wave. Then the design objective can be chosen to maximize the enhancement factor
$$J=\left.\frac{1}{f_{e 0}} \frac{|\mathbf{E}|^{4}}{E_{0}^{4}}\right|{\mathbf{x}=\mathbf{x}{0}}=\frac{1}{f_{e 0}} \int_{\Omega} \frac{|\mathbf{E}|^{4}}{E_{0}^{4}} \delta\left(\text { dist }\left(\mathbf{x}, \mathbf{x}_{0}\right)\right) \mathrm{d} \Omega$$ where the enhancement factor is normalized by $f_{e 0}$; and $f_{e 0}$ is the enhancement factor at $\mathbf{x}{0}$, corresponding to the nanostructure with metal material filled the design domain completely; $\mathbf{x}{0}$ is the reasonably chosen enhancement position in $\Omega ; \delta(\cdot)$ is the Dirac function; dist $\left(\mathbf{x}, \mathbf{x}{0}\right.$ ) is the Euclidean distance between the point $\forall \mathbf{x} \in \Omega$ and the specified position $\mathbf{x}{0}$. The enhancement position $\mathbf{x}_{0}$ should be presented at the surface or coupling position of nanostructures, because the maximal enhancement factor must be at the metal surface or coupling position in localized surface plasmon resonances.

## 物理代写|光学代写Optics代考|Topology Optimization Problem

∇⋅[er−1∇(H和s+H和一世)]+ķ02μr(H和s+H和一世)=0, 在 Ω

er−1∇Hs和⋅n+jķ0er−1μrHs和=0, 上 Γ一个b

\left.\begin{array}{l} H_{s z}(\mathbf{x}+\mathbf{a})=H_{s z}(\mathbf{x}) e^{-j \mathbf{k} \cdot \mathbf{a}} \ \mathbf{n}(\mathbf{x}+\mathbf{a}) \cdot \nabla H_{s z}(\mathbf{x}+\mathbf{a})=- e^{-j \mathbf{k} \mathbf{a}} \mathbf{n}(\mathbf{x}) \cdot \nabla H_{s z}(\mathbf{x}) \end{array}\right } \text { for } \forall \mathbf{x} \in \Gamma_{p s}, \mathbf{x}+\mathbf{a} \in \Gamma_{p d}\left.\begin{array}{l} H_{s z}(\mathbf{x}+\mathbf{a})=H_{s z}(\mathbf{x}) e^{-j \mathbf{k} \cdot \mathbf{a}} \ \mathbf{n}(\mathbf{x}+\mathbf{a}) \cdot \nabla H_{s z}(\mathbf{x}+\mathbf{a})=- e^{-j \mathbf{k} \mathbf{a}} \mathbf{n}(\mathbf{x}) \cdot \nabla H_{s z}(\mathbf{x}) \end{array}\right } \text { for } \forall \mathbf{x} \in \Gamma_{p s}, \mathbf{x}+\mathbf{a} \in \Gamma_{p d}

er−1∇Hs和⋅n=0， 上 Γs米

dĴ^dρ=回覆⁡(∂一个∂ρ−ρ¯F∗), 在 Ω在哪里ρ被重视大号2(Ω)，二阶勒贝格可积函数空间；回覆⁡(⋅)是表达式的实部。在等式。4.26，只使用伴随导数的实部，因为设计变量ρ是在真实空间上定义的分布。

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## MATLAB代写

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