### 物理代写|光学代写Optics代考|UNITS 24

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• Statistical Inference 统计推断
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• 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|Electric Field Formulation

In three-dimensional cases, electric field-based descriptions are preferred when the optical performance is evaluated based on values of the electric field. In this case, Maxwell’s equations can be reduced into the electric field-based wave equation
$$\left{\begin{array}{l} \nabla \times\left[\mu_{r}^{-1} \nabla \times\left(\mathbf{E}{s}+\mathbf{E}{i}\right)\right]-k_{0}^{2} \varepsilon_{r}\left(\mathbf{E}{s}+\mathbf{E}{i}\right)=\mathbf{0}, \text { in } \Omega \ \nabla \cdot\left[\varepsilon_{r}\left(\mathbf{E}{s}+\mathbf{E}{i}\right)\right]=0, \text { in } \Omega \end{array}\right.$$
where the scattering field formulation is also used with the electric field $\mathbf{E}$ split into two parts, i.e., the incident wave $\mathbf{E}{i}$ and scattering field $\mathbf{E}{s}$; the second equation is the divergence-free condition of the electric displacement; the incident wave is the wave propagation in free space, satisfying the divergence-free condition $\nabla \cdot \mathbf{E}{i}=0$. The boundary conditions for Eq. $3.14$ include a first-order adsorbing condition, and perfect magnetic and electric conditions, which are respectively expressed as $$\left{\begin{array}{l} \mathbf{n} \times\left(\mu{r}^{-1} \nabla \times \mathbf{E}{s}\right)-j k{0} \sqrt{\mu_{r}^{-1} \varepsilon_{r}} \mathbf{n} \times\left(\mathbf{E}{s} \times \mathbf{n}\right)=\mathbf{0}, \text { on } \Gamma{a} \ \mathbf{n} \times\left[\mu_{r}^{-1} \nabla \times\left(\mathbf{E}{s}+\mathbf{E}{i}\right)\right]=\mathbf{0}, \text { on } \Gamma_{P M C} \ \mathbf{n} \times\left(\mathbf{E}{s}+\mathbf{E}{i}\right)=\mathbf{0}, \text { on } \Gamma_{P E C} \end{array}\right.$$
Being different from the magnetic field-based description case presented in Sect. 3.1.1, the divergence-free condition in Eq. $3.14$ must consider the gradient of the relative permittivity, because the permittivity gradient always arises in the topology optimization procedure. The permittivity gradient could result in the inapplicability of numerical solution methods, e.g., edge element-based finite element method,

which can otherwise fulfil the divergence-free condition of the field in piecewise homogeneous media [28].

To circumvent the above problem, the computational domain $\Omega$ is assumed to be piecewise homogeneous. Under the assumption of piecewise homogeneity, the relative permittivity is a constant distribution in every piecewise domain, i.e.,
$$\varepsilon_{r}\left(\Omega_{n}\right)=\text { Const }(n=1,2 \ldots . N)$$
where $\Omega_{n}$ is a homogeneous piece of the computational domain, satisfying
$$\Omega=\bigcup_{n=1}^{N} \Omega_{n}, \Omega_{p} \bigcap \Omega_{q}=\emptyset(p \neq q, \text { and } p, q=1,2 \ldots N)$$
with $N$ representing the number of homogeneous pieces included in the computational domain. Based on the assumed piecewise homogeneity, the divergence-free condition in Eq. $3.14$ can be transformed into
$$\nabla \cdot \mathbf{E}_{s}=0, \text { in } \Omega$$

## 物理代写|光学代写Optics代考|Adjoint Analysis for Electric Field-Based Topology

The Lagrangian multiplier-based adjoint sensitivity analysis of variational problem in Eq. $3.23$ is implemented as follows. The functional space and trace operators of Eq. $3.14$ are similarly defined as that in Sect. 3.1.2, except that
$$\mathscr{V}{\mathbf{E}}=\left{\mathbf{u} \in \mathscr{H}(\operatorname{curl} ; \Omega) \mid \nabla \cdot \mathbf{u}=0, \text { in } \Omega ; \mathbf{n} \times \mathbf{u}=\mathbf{0}, \text { on } \Gamma{P E C}\right}$$
According to the Kurash-Kuhn-Tucker condition of the PDE constrained optimization problem [22], the adjoint equations can be obtained as

Find $\mathbf{E}{s a} \in \mathscr{V}{\mathbf{E}}$ such that
$\int_{\Omega} \frac{\partial A}{\partial \mathbf{E}{s}} \cdot \phi+\frac{\partial A}{\partial \nabla \times \mathbf{E}{s}} \cdot(\nabla \times \phi)+\mu_{r}^{-1}\left(\nabla \times \overline{\mathbf{E}}{s a}\right) \cdot(\nabla \times \boldsymbol{\phi})-k{0}^{2} \varepsilon_{r} \overline{\mathbf{E}}{s a} \cdot \boldsymbol{\phi} \mathrm{d} \Omega$ $+\int{\Gamma_{a}} j k_{0} \sqrt{\varepsilon_{r} \mu_{r}^{-1}}\left(\mathbf{n} \times \overline{\mathbf{E}}{s a} \times \mathbf{n}\right) \cdot(\mathbf{n} \times \phi \times \mathbf{n})+\frac{\partial B}{\partial \mathbf{E}{s}} \cdot \phi \mathrm{d} \Gamma$
$+\int_{\Gamma_{P M C}} \frac{\partial B}{\partial \mathbf{E}{s}} \cdot \phi \mathrm{d} \Gamma=0, \forall \phi \in \mathscr{V}{\mathbf{E}}$
$\int_{\Omega} r^{2} \nabla \gamma_{f a} \cdot \nabla \phi+\gamma_{f a} \phi+A_{\gamma_{\epsilon}} \phi-S_{\gamma_{c}} \phi \mathrm{d} \Omega=0, \forall \phi \in \mathscr{H}(\Omega)$,
where $A_{\gamma_{\epsilon}}(\Omega)$ is defined as
$$A_{\gamma_{e}}=\sum_{n=1}^{N} A_{\gamma_{n e}}\left(\Omega_{n}\right)$$
with
$$A_{\gamma_{a e}}\left(\Omega_{n}\right)=\left{\begin{array}{l} \frac{1}{V_{\Omega_{n}}} \int_{\Omega_{n}} \frac{\partial A}{\partial \gamma_{p}} \frac{\partial \gamma_{p}}{\partial \gamma_{e}} \mathrm{~d} \Omega, \forall \mathbf{x} \in \Omega_{n} \ 0, \forall \mathbf{x} \in \Omega \backslash \Omega_{n} \end{array}\right.$$
and $S_{\gamma_{c}}(\Omega)$ is defined to be
$$S_{\gamma_{e}}=\sum_{n=1}^{N} S_{\gamma_{n, e}}\left(\Omega_{n}\right)$$
with
$$S_{\gamma_{s e}}\left(\Omega_{n}\right)=\left{\begin{array}{l} \frac{1}{V_{\Omega_{n}}} \int_{\Omega_{n}} k_{0}^{2} \frac{\partial \varepsilon_{r}}{\partial \gamma_{p}} \frac{\partial \gamma_{p}}{\partial \gamma_{e}}\left(\mathbf{E}{s}+\mathbf{E}{i}\right) \cdot \overline{\mathbf{E}}{s a} \mathrm{~d} \Omega, \quad \forall \mathbf{x} \in \Omega{n} \ 0, \forall \mathbf{x} \in \Omega \backslash \Omega_{n} \end{array}\right.$$
The adjoint derivative of the cost functional can be derived as
$$\delta J=\int_{\Omega} \operatorname{Re}\left(\frac{\partial A}{\partial \gamma}-\bar{\gamma}_{f a}\right) \delta \gamma \mathrm{d} \Omega$$

## 物理代写|光学代写Optics代考|Numerical implementation

In the wave equations and corresponding adjoint equations, a divergence-free condition needs to be satisfied for both the state variable and the adjoint variable. Therefore,

the edge element-based finite element method is utilized to solve the wave equations and adjoint equations, where brick elements are used to discretize the computational domain and simultaneously ensure the divergence-free condition [28]. For the Helmholtz filter, the filter Eq. $3.4$ and its adjoint equation are solved using the standard Galerkin finite element method.

The topology optimization method for three-dimensional optical waves is implemented by a gradient-based iterative procedure, where the gradient information is derived by sensitivity analysis as demonstrated in Sects. 3.1.2 and 3.1.4 respectively corresponding to the variational problems in Eqs. $3.7$ and 3.23. The flowcharts for iteratively solving the variational problems (Eqs.3.7 and 3.23) respectively corresponding to the magnetic field formulation and electric field formulation are respectively shown in Fig. 3.1a and b. The iterative procedure includes the following steps: (a) solve the wave equations with the current design variable; (b) solve the adjoint equations based on the solution of the wave equations; (c) compute the adjoint derivative of the design objective; and (d) update the design variable using the method of moving asymptotes [47].

During the solving procedure, the filter radius $r$ of the Helmholtz filter in Eq.3.4 is set to be the size of the finite elements used to discretize the computational domain; the threshold parameter $\xi$ in Eqs. $3.5$ and $3.21$ is set to be $0.5$; the initial value of the projection parameter $\beta$ is set to be 1 and it is doubled after every fixed number of iterations until the preset maximal value $2^{10}$ is reached (eleven cycles). The above steps are implemented iteratively until the stopping criterion is satisfied, specified to be the change of the objective values in five consecutive iterations satisfying
$$\frac{1}{5} \sum_{i=1}^{4}\left|J_{k-i}-J_{k-i-1}\right| /\left|J_{k}\right| \leq \varepsilon, \beta \geq 2^{10}$$
in the $k$ th iteration, where $J_{k}$ is the objective value computed in the $k$ th iteration; $\varepsilon$ is the tolerance chosen to be $1 \times 10^{-3}$. Because the iteration number is set to be 40 before doubling the projection parameter, the maximal iterative number is set to be $440 .$

In the optimization procedure for magnetic field described optical waves, the magnetic field is interpolated using linear edge elements (Fig. 3.2a); the design variable and filtered design variable is interpolated using linear nodal element (Fig. 3.2b). In the optimization procedure for electric field described optical waves, the electric field is interpolated using linear edge elements (Fig. 3.2a); the design variable and filtered design variable is interpolated using linear nodal elements (Fig. 3.2b); the filtered design variable is converted to piecewise form by interpolating the piecewise design variable using zeroth-order discontinuous elements (Fig. 3.2c), where $\Omega_{n}$ in Eq. $3.20$ is set to be the space taken up by the brick elements.

## 物理代写|光学代写Optics代考|Electric Field Formulation

$$\left{ ∇×[μr−1∇×(和s+和一世)]−ķ02er(和s+和一世)=0, 在 Ω ∇⋅[er(和s+和一世)]=0, 在 Ω\正确的。 在H和r和吨H和sC一个吨吨和r一世nGF一世和ldF○r米在l一个吨一世○n一世s一个ls○在s和d在一世吨H吨H和和l和C吨r一世CF一世和ld和spl一世吨一世n吨○吨在○p一个r吨s,一世.和.,吨H和一世nC一世d和n吨在一个在和和一世一个ndsC一个吨吨和r一世nGF一世和ld和s;吨H和s和C○nd和q在一个吨一世○n一世s吨H和d一世在和rG和nC和−Fr和和C○nd一世吨一世○n○F吨H和和l和C吨r一世Cd一世spl一个C和米和n吨;吨H和一世nC一世d和n吨在一个在和一世s吨H和在一个在和pr○p一个G一个吨一世○n一世nFr和和sp一个C和,s一个吨一世sF是一世nG吨H和d一世在和rG和nC和−Fr和和C○nd一世吨一世○n∇⋅和一世=0.吨H和b○在nd一个r是C○nd一世吨一世○nsF○r和q.3.14一世nCl在d和一个F一世rs吨−○rd和r一个ds○rb一世nGC○nd一世吨一世○n,一个ndp和rF和C吨米一个Gn和吨一世C一个nd和l和C吨r一世CC○nd一世吨一世○ns,在H一世CH一个r和r和sp和C吨一世在和l是和Xpr和ss和d一个s\剩下{ n×(μr−1∇×和s)−jķ0μr−1ern×(和s×n)=0, 上 Γ一个 n×[μr−1∇×(和s+和一世)]=0, 上 Γ磷米C n×(和s+和一世)=0, 上 Γ磷和C\正确的。$$

er(Ωn)= 常量 (n=1,2….ñ)

Ω=⋃n=1ñΩn,Ωp⋂Ωq=∅(p≠q, 和 p,q=1,2…ñ)

∇⋅和s=0, 在 Ω

## 物理代写|光学代写Optics代考|Adjoint Analysis for Electric Field-Based Topology

\mathscr{V}{\mathbf{E}}=\left{\mathbf{u} \in \mathscr{H}(\operatorname{curl} ; \Omega) \mid \nabla \cdot \mathbf{u}= 0, \text { 在} \Omega ; \mathbf{n} \times \mathbf{u}=\mathbf{0}, \text {on} \Gamma{PEC}\right}\mathscr{V}{\mathbf{E}}=\left{\mathbf{u} \in \mathscr{H}(\operatorname{curl} ; \Omega) \mid \nabla \cdot \mathbf{u}= 0, \text { 在} \Omega ; \mathbf{n} \times \mathbf{u}=\mathbf{0}, \text {on} \Gamma{PEC}\right}

∫Ω∂一个∂和s⋅φ+∂一个∂∇×和s⋅(∇×φ)+μr−1(∇×和¯s一个)⋅(∇×φ)−ķ02er和¯s一个⋅φdΩ +∫Γ一个jķ0erμr−1(n×和¯s一个×n)⋅(n×φ×n)+∂乙∂和s⋅φdΓ
+∫Γ磷米C∂乙∂和s⋅φdΓ=0,∀φ∈在和
∫Ωr2∇CF一个⋅∇φ+CF一个φ+一个Cεφ−小号CCφdΩ=0,∀φ∈H(Ω),

$$A_{\ gamma_ {ae}}\left(\Omega_{n}\right)=\left{ 1在Ωn∫Ωn∂一个∂Cp∂Cp∂C和 dΩ,∀X∈Ωn 0,∀X∈Ω∖Ωn\正确的。 一个nd小号CC(Ω)一世sd和F一世n和d吨○b和 S_{\gamma_{e}}=\sum_{n=1}^{N} S_{\gamma_{n, e}}\left(\Omega_{n}\right) 在一世吨H S_{\gamma_{se}}\left(\Omega_{n}\right)=\left{ 1在Ωn∫Ωnķ02∂er∂Cp∂Cp∂C和(和s+和一世)⋅和¯s一个 dΩ,∀X∈Ωn 0,∀X∈Ω∖Ωn\正确的。 吨H和一个dj○一世n吨d和r一世在一个吨一世在和○F吨H和C○s吨F在nC吨一世○n一个lC一个nb和d和r一世在和d一个s \delta J=\int_{\Omega} \operatorname{Re}\left(\frac{\partial A}{\partial \gamma}-\bar{\gamma}_{fa}\right) \delta \gamma \数学{d} \欧米茄$$

## 物理代写|光学代写Optics代考|Numerical implementation

15∑一世=14|Ĵķ−一世−Ĵķ−一世−1|/|Ĵķ|≤e,b≥210

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