数学代写|拓扑学代写Topology代考|MATH3061

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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|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
$$J=\left.\frac{1}{f_{e 0}} \frac{|\mathbf{E}|^{4}}{E_{0}^{4}}\right|{\mathbf{x}=\mathbf{x}{0}}=\frac{1}{f_{e 0}} \int_{\Omega} \frac{|\mathbf{E}|^{4}}{E_{0}^{4}} \delta\left(\operatorname{dist}\left(\mathbf{x}, \mathbf{x}_{0}\right)\right) \mathrm{d} \Omega$$ 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 $\Omega ; \delta(*)$ 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.

数学代写|拓扑学代写Topology代考|Nanoslits for Extraordinary Optical Transmission

This section presents the inverse design of resonant nanostructures for extraordinary optical transmission of periodic metallic slits, where topology optimization approach is utilized to implement the inverse design procedure and find the geometrical configurations of the nanostructures. By using the inverse design method, the subwavelength-sized resonant nanostructures, localized at the inlet and outlet sides of the periodic metallic slits, are derived with transmission peaks at the prescribed incident wavelengths. The transmissivity is enhanced by effective excitation and guidance of surface plasmon polariton at the inlet side of the slits, coherent resonance of surface plasmon polariton inside the slits and radiation of the photonic energy at the outlet side of the slits.

The transmission peaks of the periodic metallic slits, with inversely designed resonant nanostructures, are raised along with the red shift of the incident wavelength. The position of the transmission peak of periodic metallic slits can be controlled and localized at a desired frequency, by specifying the incident wave with the wavelength

corresponding to the desired frequency preset in the inverse design procedure. By maximizing the minimum transmissivity of the periodic metallic slits with incident wavelengths in a prescribed wavelength range, the extraordinary optical transmission bandwidth can be enlarged, and the sensitivity of transmissivity to wavelength can be decreased equivalently.

Extraordinary optical transmission is the phenomenon of greatly enhanced transmission of light through a subwavelength aperture in an otherwise opaque metallic film which has been patterned with a regularly repeating periodic structure. It was firstly described by Ebbesen et al in 1998 [29]. In extraordinary optical transmission, the regularly repeating structures enable much higher transmissivity to occur, up to several orders of magnitude greater than that predicted by classical aperture theory. The mechanism of extraordinary optical transmission is attributed to the scattering of surface plasmon polaritons $[35,56]$. Extraordinary optical transmission offers one key advantage over a surface plasmonic resonance device, which is a nanometermicrometer scale device, and it is particularly amenable to miniaturization.

数学代写|拓扑学代写Topology代考|Modeling

For optical waves propagating in a plane, transverse magnetic polarized waves can excite the surface plasmon resonances in the cross sections of metal nanostructures with an infinite thickness. Therefore, the incident wave is chosen as a transverse magnetic wave. A cross section of the periodic metallic slits is illuminated in Fig. $4.9$ with a uniform monochromatic transverse magnetic wave propagation.

The computational domain is set to be one period of the metallic slits. Topology optimization approach is utilized to inversely design the nanostructures localized in the bilateral regions of the preset metallic slits. The design domain, where the design variable is defined, is set to be those two bilateral regions. To truncate the infinitive free space, the first-order absorbing boundary condition is imposed on the inlet $\left(\Gamma_{i}\right)$ and outlet $\left(\Gamma_{o}\right)$ boundaries of the computational domain, and the periodic boundary condition is imposed on the left $\left(\Gamma_{p s}\right)$ and right $\left(\Gamma_{p d}\right)$ boundaries of the slit to reduce the computational cost.

Based on the above computational setup, the inverse design problem is to find the geometrical configurations of the bilateral nanostructures for the preset slit to maximize the transmission of the electromagnetic energy. The propagating wave in the metallic slits is time-harmonic transverse magnetic wave governed by the twodimensional Maxwell’s equations. Those equations can be reformulated into the scalar Helmholtz equation together with the boundary conditions: \begin{aligned} &\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 \ &\varepsilon_{r}^{-1} \nabla H_{z s} \cdot \mathbf{n}+j k_{0} \sqrt{\varepsilon_{r}^{-1} \mu_{r}} H_{z s}=0, \text { on } \Gamma_{i} \cup \Gamma_{o} \ &H_{z s}(\mathbf{x}+\mathbf{a})=H_{z s}(\mathbf{x}) e^{-j \mathbf{k} \cdot \mathbf{a}}, \mathbf{n}(\mathbf{x}+\mathbf{a}) \cdot \nabla H_{z s}(\mathbf{x}+\mathbf{a})=-e^{-j \mathbf{k} \cdot \mathbf{a}} \mathbf{n}(\mathbf{x}) \cdot \nabla H_{z s}(\mathbf{x}) \ &\text { for } \forall \mathbf{x} \in \Gamma_{p s}, \mathbf{x}+\mathbf{a} \in \Gamma_{p d} \end{aligned}
where the scattering-field formulation, with $H_{z}=H_{z s}+H_{z i}$, is used to reduce the dispersion error; $H_{z s}$ and $H_{z i}$ are the scattering 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; $\mathbf{k}$ is the wave vector;, the time dependence of the fields is given by the factor $e^{j \text { wot }}$, with $t$ representing the time; $\mathbf{n}$ is the unit outward normal vector at $\partial \Omega ; j=\sqrt{-1}$ is the imaginary unit; $\Gamma_{i}$ and $\Gamma_{o}$ are the inlet and outlet boundaries of the photonic energy, respectively; and $\Gamma_{p s}$ and $\Gamma_{p d}$ are respectively the source and destination boundary of the periodic boundary pair, with a lattice vector a. The incident field $H_{z i}$ is set to be the parallel-plane wave with unit amplitude.

数学代写|拓扑学代写Topology代考|Nanostructures for Localized Surface Plasmonic

Ĵ=1F和0|和|4和04|X=X0=1F和0∫Ω|和|4和04d(距离⁡(X,X0))dΩ其中增强因子归一化为F和0; 和F和0是增强因子X0，对应于金属材料的纳米结构完全填充了设计域；X0是合理选择的增强位置Ω;d(∗)是狄拉克函数；距离(X,X0) 是点之间的欧几里得距离∀X∈Ω和指定的位置X0. 增强位置X0应该呈现在纳米结构的表面或耦合位置，因为在局部表面等离子体共振中，最大增强因子必须在金属表面或耦合位置。

数学代写|拓扑学代写Topology代考|Modeling

∇⋅[er−1∇(H和s+H和一世)]+ķ02μr(H和s+H和一世)=0, 在 Ω er−1∇H和s⋅n+jķ0er−1μrH和s=0, 上 Γ一世∪Γ○ H和s(X+一个)=H和s(X)和−jķ⋅一个,n(X+一个)⋅∇H和s(X+一个)=−和−jķ⋅一个n(X)⋅∇H和s(X)  为了 ∀X∈Γps,X+一个∈Γpd

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MATLAB代写

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