### 物理代写|光学代写Optics代考|UNITS 24

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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|Metalens with Optical Vortices

Ultrathin and flat metalenses, for diffraction-limited focusing of light, work in the visible spectrum and have been achieved based on metasurfaces [30], which are composed of sub-wavelength-spaced phase shifters at an interface, and allowing for unprecedented control over the properties of light $[31,51]$. Metalenses are less bulky than conventional lenses, and are more practical and less expensive to manufacture [51]. Their features permit potential applications in microscopes, telescopes, cameras, smartphones, and other devices.

The essence of a lens is to pointwise modify the phase of an incident plane wave to form a focus. There are generally three methods to introduce phase compensation mechanisms into a lens design: shaping the surface of a piece of homogenous

material, such as polishing a piece of glass to have a convex surface; use of diffractive structures to engineer the wavefront, as is done with a Fresnel zone plate [21]; using material inhomogeneities to modify the phase change in space, for instance, a gradient index (GRIN) lens [18]. A metalens applies similar phase compensation mechanisms to metamaterials to bring a plane wave to a focus.

Optical metalenses have been designed based on arrays of optical antennas [51], arrays of nanoholes [27], optical masks [15, 26, 43], and nanoslits [48]. Additionally, flat metamaterial-based hyperlenses and superlenses have been used to achieve sub-diffraction focusing $[8,34,39,46]$. These metalenses are designed based on the concept of an optical phase discontinuity, where the design of the metalens is obtained by imposing a spherical phase profile on the metasurface, and the control of wavefront is achieved via a phase shift experienced by the radiation as it scatters off the optically thin arrays of subwavelength-spaced resonators comprising the metasurface. As specified in [30], these techniques do not provide the ability for complete control of the optical wavefront. Recently, a dielectric nanofin array-based metalens has been proposed in [30], followed by a plasmonic nanoparticle-based metalens with a distribution of nanoparticles determined by an evolutionary approach [25]. These metalenses are composed of arrays of nanostructures (i.e., nanofins or nanoparticles) in large number, which potentially leads to manufacturing challenges.

This section shows that it is possible to predict a new type of convex-like metalens using the topology optimization method, which will achieve diffraction-limited focusing for the visible spectrum. Instead of imposing a phase profile on the metasurface, the inverse design method is used to find a geometrical configuration of the metalens with rotational symmetry, by maximizing the focused energy at the focal spot corresponding to a specified diameter and numerical aperture (NA) of the metalens. Topology optimization is utilized to implement an inverse design procedure. In the derived metalens, a titanium dioxide material distribution with concentric-nanoring configuration focuses the plane wave by modifying the phase of the incident wave to form a spherical wavefront, which converges to the desired focal spot. In this viewpoint, a topology optimization problem that maximizes the focused energy at the desired focal spot is equivalent to imposing a phase profile on the metasurface. By maximizing the localized energy deposition in a specified region, it is shown that the method can achieve compact axicon-like designs, with highly confined line-shaped foci.

## 物理代写|光学代写Optics代考|Analyzing and Solving

The topology optimization problem in Eq. $3.58$ is solved using a gradient-based iterative procedure, where the gradient of the inverse design objective is used to iteratively evolve the design variable. The variational problem is analyzed using the adjoint method [22], with the adjoint derivative of the electric field density at $\mathbf{p}{f}$ derived to be $$\delta J=h \int{-r_{0}}^{r_{0}}-\tilde{\rho}^{a} \delta \rho \mathrm{d} r$$
where $\bar{\rho}^{a}$ is the adjoint variable of $\bar{\rho} ; \delta \rho$ is the first-order variational of $\rho$. The following weak forms of adjoint equations are solved for the derivation of $\tilde{\rho}^{a}$
Find $H_{s}^{a}$ with $\operatorname{Re}\left(H_{s}^{a}\right) \in \mathscr{H}(\Omega)$ and $\operatorname{Im}\left(H_{s}^{a}\right) \in \mathscr{H}(\Omega)$, such that
\begin{aligned} &\int_{\Omega_{c}} \delta\left(\mathbf{p}-\mathbf{p}{f}\right) 2 \mathbf{E}^{} \cdot \frac{\partial \mathbf{E}}{\partial \nabla H{s}} \cdot \nabla \phi+\varepsilon_{r}^{-1} \nabla H_{s}^{a} \cdot \nabla \phi-k_{0}^{2} \mu_{r} H_{s}^{a} \phi \mathrm{d} v \ &+\int_{\Omega_{P M B}} \varepsilon_{r}^{-1}\left(\frac{\partial \mathbf{p}^{\prime}}{\partial \mathbf{p}} \nabla H_{s}^{a}\right) \cdot\left(\frac{\partial \mathbf{p}}{\partial \mathbf{p}} \nabla \phi\right)\left|\frac{\partial \mathbf{p}^{\prime}}{\partial \mathbf{p}}\right|^{-1} \ &-k_{0}^{2} \mu_{r} H_{s}^{a} \phi\left|\frac{\partial \mathbf{p}^{\prime}}{\partial \mathbf{p}}\right| \mathrm{d} v=0, \forall \phi \in \mathscr{H}(\Omega) \end{aligned}
Find $H_{s}^{a}$ with $\bar{\rho}^{a} \in \mathscr{H}\left(\Omega_{d}^{r}\right)$, such that
$$\int_{-r_{0}}^{r_{0}}\left(r_{f}^{2} \frac{\mathrm{d} \bar{\rho}^{a}}{\mathrm{~d} r} \frac{\mathrm{d} \psi}{\mathrm{d} r}+\bar{\rho}^{a} \psi+\frac{\psi}{h} \int_{-h}^{0} \sum_{n=1}^{N} S_{\hat{p}{u}} \mathrm{~d} z\right) \mathrm{d} r=0, \forall \psi \in \mathscr{H}\left(\Omega{d}^{r}\right)$$
where $H_{s}^{a}$ is the adjoint variable of $H_{s} ; \mathscr{H}(\Omega)$ and $\mathscr{H}\left(\Omega_{d}^{r}\right)$ are respectively the first order Hilbert space defined on $\Omega$ and $\Omega_{d}^{r} ; \Omega_{d}^{r}=\left(-r_{0}, r_{0}\right)$ is the projection of $\Omega_{d}$ on the $r$-axis; $\Omega_{P M L s}$ is the union of the PMLs; $\operatorname{Re}()$ and Im (*) are the operators

used to extract the real and imaginary parts of a complex variable; and $S_{\hat{\rho}{n}}$ is defined to be $$S{\hat{\rho}{u}}\left(\Omega{n}\right)=\left{\begin{array}{l} \frac{1}{V_{\Omega_{n}}} \int_{\Omega_{n}} \operatorname{Re}\left(\frac{\partial \varepsilon_{r}^{-1}}{\partial \bar{\rho}} \frac{\partial \bar{\rho}}{\partial \hat{\rho}{n}} \nabla H\right) \cdot \operatorname{Re}\left(\nabla H{s}^{a}\right)- \ \operatorname{Im}\left(\frac{\partial \varepsilon_{r}^{-1}}{\partial \bar{\rho}} \frac{\partial \bar{\rho}}{\partial \hat{\rho}{n}} \nabla H\right) \cdot \operatorname{Im}\left(\nabla H{s}^{a}\right) \mathrm{d} v, \text { in } \Omega_{n} \ 0, \text { in } \Omega_{d} \backslash \Omega_{n} \end{array}\right.$$
After adjoint analysis, the inverse design problem is solved iteratively using a numerical method. In our setup, the equations and corresponding adjoint equations are solved using the finite element method. The design variable is evolved using the method of moving asymptotes [47].

## 物理代写|光学代写Optics代考|Extension

The same methodology can be used to predict lenses with other focal distributions. For example, instead of a spot, an extended beam focus of axicon lenses can be targeted. Typically, axicons are conically shaped lenses that can focus light and create hollow Bessel beams $[35,36,42]$. The inverse design objective for an axiconlike metalens is to maximize the minimal electric field density on the central line of the desired focal beam
$$J(\mathbf{E})=\min {\mathbf{p} f \in I} \int{\Omega_{f}}|\mathbf{E}|^{2} \delta\left(\mathbf{p}-\mathbf{p}{f}\right) \mathrm{d} v$$ where $\mathbf{p}{f} \in \Omega_{f}$ is the point on the central line of the desired focal beam; $I$ is the central line segment of the desired focal beam. For an incident wavelength of $700 \mathrm{~nm}$ and a design domain radius of $9 \mu \mathrm{m}$, axicon-like metalenses are derived in Fig. 3.27ac with zoomed views shown in Fig. $3.27 \mathrm{~d}$, where the NA of the metalens was set to $0.7$ and the length of the desired focal beam was set to be a factor $0=, 4=$, and 8 -fold of the incident wavelength. The normalized electric field energy density in the focal plane localized at the center of the focal beam is plotted in Fig. $3.27 \mathrm{e}$. And the focusing efficiencies of the derived designs in Fig. 3.27a-c are respectively $37.2 \%, 32.8 \%$, and $26.6 \%$, where longer focal beam corresponds to larger FWHM and lower focusing efficiency. The corresponding electric field energy distribution and real part magnetic field distributions are shown in Fig.3.28, where the field phase of the propagating wave after the metalenses are indicated by the black curves in Fig.3.28d-f. In Fig.3.27, the metalens with focal length equal to 0-fold of the incident wavelength, is identical to a convex-like metalens with a focal spot. From the field distributions of the inversely designed metalenses, one can conclude that the focusing efficiency of the derived metalenses decrease along with an increase in focal length, and the parabolic phase distributions are shaped in the zone after the metalenes to achieve a focal beam.

## 物理代写|光学代写Optics代考|Analyzing and Solving

dĴ=H∫−r0r0−ρ~一个dρdr

∫ΩCd(p−pF)2和⋅∂和∂∇Hs⋅∇φ+er−1∇Hs一个⋅∇φ−ķ02μrHs一个φd在 +∫Ω磷米乙er−1(∂p′∂p∇Hs一个)⋅(∂p∂p∇φ)|∂p′∂p|−1 −ķ02μrHs一个φ|∂p′∂p|d在=0,∀φ∈H(Ω)

∫−r0r0(rF2dρ¯一个 drdψdr+ρ¯一个ψ+ψH∫−H0∑n=1ñ小号p^在 d和)dr=0,∀ψ∈H(Ωdr)

## 物理代写|光学代写Optics代考|Extension

Ĵ(和)=分钟pF∈我∫ΩF|和|2d(p−pF)d在在哪里pF∈ΩF是所需焦点光束中心线上的点；我是所需焦点光束的中心线段。对于入射波长700 n米和设计域半径9μ米，轴锥状超透镜在图 3.27ac 中得到，放大视图如图 3.27ac 所示。3.27 d，其中元透镜的 NA 设置为0.7并且所需焦束的长度被设置为一个因素0=,4=, 和 8 倍的入射波长。位于焦束中心的焦平面中的归一化电场能量密度绘制在图 3 中。3.27和. 图 3.27ac 中派生设计的聚焦效率分别为37.2%,32.8%， 和26.6%，其中较长的焦束对应于较大的 FWHM 和较低的聚焦效率。相应的电场能量分布和实部磁场分布如图3.28所示，其中超透镜后传播波的场相位由图3.28df中的黑色曲线表示。在图 3.27 中，焦距等于入射波长 0 倍的元透镜与具有焦斑的凸面元透镜相同。从逆向设计的超透镜的场分布可以得出结论，衍生超透镜的聚焦效率随着焦距的增加而降低，并且在超透镜之后的区域中形成抛物线相位分布以实现聚焦光束。

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## MATLAB代写

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