## 物理代写|光学代写Optics代考|CSCl031

statistics-lab™ 为您的留学生涯保驾护航 在代写光学Optics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写光学Optics代写方面经验极为丰富，各种代写光学Optics相关的作业也就用不着说。

• 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|General Stress Tensor for Nematic Liquid Crystals

The general theoretical framework for describing the hydrodynamics of liquid crystals has been developed principally by Leslie [16] and Ericksen [17]. Their approaches account for the fact that the stress tensor depends not only on the velocity gradients but also on the orientation and rotation of the director. Accordingly, the stress tensor is given by
$$\sigma_{\alpha \beta}=\alpha_1 n_\gamma n_\delta A_{\gamma \delta} n_\alpha n_\beta+\alpha_2 n_{\alpha \alpha} n_\beta+\alpha_3 n_\beta n_\alpha+\alpha_4 A_{\alpha \beta}+\alpha_5 n_\gamma A_{\gamma \beta}+\alpha_6 n_\beta n_\gamma A_{\gamma \alpha}$$
where the $A_{\alpha \beta}$ s are defined by
$$A_{\alpha \beta}=\frac{1}{2}\left[\frac{\partial v_\beta}{\partial x_\alpha}+\frac{\partial v_\alpha}{\partial x_\beta}\right]$$
Note that all the other terms on the right-hand side of Eq. (3.59) involve the director orientation, except the fourth term, $\alpha_4 A_{\alpha \beta}$.This is the same term as that for an isotropic fluid (cf. Eq. [3.57]), that is, $\alpha_4=2 \eta$.

Therefore, in this formalism, we have six so-called Leslie coefficients, $\alpha_1, \alpha_2, \ldots$, $\alpha_6$, which have the dimension of viscosity coefficients. It was shown by Parodi [18] that
$$\alpha_2+\alpha_3=\alpha_6-\alpha_5$$
and so there are really five independent coefficients.
In the next few sections, we will study exemplary cases of director axis orientation and deformation, and we will show how these Leslie coefficients are related to other commonly used viscosity coefficients.

## 物理代写|光学代写Optics代考|Flows with Fixed Director Axis Orientation

Consider here the simplest case of flows in which the director axis orientation is held fixed. This may be achieved by a strong externally applied magnetic field (see Figure 3.11), where the magnetic field is along the direction $\hat{n}$. Consider the case of shear flow, where the velocity is in the $z$-direction, and the velocity gradient is along the $x$-direction. This process could occur, for example, in liquid crystals confined by two parallel plates in the $y$-z plane.

In terms of the orientation of the director axis, there are three distinct possibilities involving three corresponding viscosity coefficients:

1. $\eta_1: \hat{n}$ is parallel to the velocity gradient, that is, along the $x$-axis $\left(\theta=90^{\circ}, \phi=0^{\circ}\right)$.
2. $\eta_2: \hat{n}$ is parallel to the flow velocity, that is, along the $z$-axis and lies in the shear plane $x-z\left(\theta=0^{\circ}, \phi=0^{\circ}\right)$.
3. $\eta_3: \hat{n}$ is perpendicular to the shear plane, that is, along the $y$-axis $\left(\theta=0^{\circ}, \phi\right.$ $\left.=90^{\circ}\right)$.

These three configurations have been investigated by Miesowicz [19], and the $\eta$ s are known as Miesowicz coefficients. In the original paper, as well as in the treatment by deGennes [3], the definitions of $\eta_1$ and $\eta_3$ are interchanged. In deGennes notation, in terms of $\eta_a, \eta_b$, and $\eta_c$, we have $\eta_a=\eta_1, \eta_b=\eta_2$, and $\eta_c=\eta_3$. The notation used here is attributed to Helfrich [6], which is now the conventional one.

To obtain the relations between $\eta_{1,2,3}$ and the Leslie coefficients $\alpha_{1,2, \ldots, 6}$, one could evaluate the stress tensor $\sigma_{\alpha \beta}$ and the shear rate $A_{\alpha \beta}$ for various director orientations and flow and velocity gradient directions. From these considerations, the following relationships are obtained [3]:

\begin{aligned} &\eta_1=\frac{1}{2}\left(\alpha_4+\alpha_5-\alpha_2\right) \ &\eta_2=\frac{1}{2}\left(\alpha_3+\alpha_4+\alpha_6\right) \ &\eta_3=\frac{1}{2} \alpha_4 \end{aligned}
In the shear plane $x-z$, the general effective viscosity coefficient is actually more correctly expressed in the form [20]
$$\eta_{\mathrm{eff}}=\eta_1+\eta_2 \cos ^2 \theta+\eta_2$$
in order to account for angular velocity gradients. The coefficient $\eta_{1,2}$ is related to the Leslie coefficient $\alpha_1$ by
$$\eta_{1,2}=\alpha_1 .$$

## 物理代写|光学代写Optics代考|General Stress Tensor for Nematic Liquid Crystals

$$\sigma_{\alpha \beta}=\alpha_1 n_\gamma n_\delta A_{\gamma \delta} n_\alpha n_\beta+\alpha_2 n_{\alpha \alpha} n_\beta+\alpha_3 n_\beta n_\alpha+\alpha_4 A_{\alpha \beta}+\alpha_5 n_\gamma A_{\gamma \beta}+\alpha_6 n_\beta n_\gamma A_{\gamma \alpha}$$

$$A_{\alpha \beta}=\frac{1}{2}\left[\frac{\partial v_\beta}{\partial x_\alpha}+\frac{\partial v_\alpha}{\partial x_\beta}\right]$$

$$\alpha_2+\alpha_3=\alpha_6-\alpha_5$$

## 物理代写|光学代写Optics代考|Flows with Fixed Director Axis Orientation

1. $\eta_1: \hat{n}$ 平行于速度梯度，即沿 $x$-轴 $\left(\theta=90^{\circ}, \phi=0^{\circ}\right)$.
2. $\eta_2: \hat{n}$ 平行于流速，即沿 $z$-轴并且位于剪切平面内 $x-z\left(\theta=0^{\circ}, \phi=0^{\circ}\right)$.
3. $\eta_3: \hat{n}$ 垂直于剪切面，即沿 $y$-轴 $\left(\theta=0^{\circ}, \phi=90^{\circ}\right)$.
Miesowicz [19] 研究了这三种配置，并且 $\eta$ s 称为 Miesowicz 系数。在原始论文以及 deGennes [3] 的处理中，定 义 $\eta_1$ 和 $\eta_3$ 被互换。在 deGennes 表示法中，根据 $\eta_a, \eta_b$ ，和 $\eta_c$ ，我们有 $\eta_a=\eta_1, \eta_b=\eta_2$ ，和 $\eta_c=\eta_3$. 这里 使用的符号归功于 Helfrich [6]，它现在是传统的符号。
获得之间的关系 $\eta_{1,2,3}$ 和莱斯利系数 $\alpha_{1,2, \ldots, 6}$ ，可以评估应力张量 $\sigma_{\alpha \beta}$ 和剪切速率 $A_{\alpha \beta}$ 适用于各种导向器方向以及 流动和速度梯度方向。从这些考虑，得到以下关系[3]:
$$\eta_1=\frac{1}{2}\left(\alpha_4+\alpha_5-\alpha_2\right) \quad \eta_2=\frac{1}{2}\left(\alpha_3+\alpha_4+\alpha_6\right) \eta_3=\frac{1}{2} \alpha_4$$
在剪切平面 $x-z ， 一$ 般有效粘度系数实际上更正确地表示为 [20]
$$\eta_{\text {eff }}=\eta_1+\eta_2 \cos ^2 \theta+\eta_2$$
为了考虑角速度梯度。系数 $\eta_{1,2}$ 与莱斯利系数有关 $\alpha_1$ 经过
$$\eta_{1,2}=\alpha_1$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 物理代写|光学代写Optics代考|PHS2062

statistics-lab™ 为您的留学生涯保驾护航 在代写光学Optics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写光学Optics代写方面经验极为丰富，各种代写光学Optics相关的作业也就用不着说。

• 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|Equilibrium Temperature

The two principal refractive indices $n_{\perp}$ and $n_{|}$of a uniaxial liquid crystal and the anisotropy $n_{|}-n_{\perp}$ have been the subject of intensive studies for their fundamental importance in the understanding of liquid crystal physics and for their vital roles in applied electro-optic devices. Since the dielectric constants $\left(\varepsilon_{\perp}\right.$ and $\left.\varepsilon_{||}\right)$enter directly and linearly into the constitutive equations (Eqs. (3.30a)-(3.30c)), it is theoretically more convenient to discuss the fundamentals of these temperature dependences in terms of the dielectric constants.

From Eq. (3.34) for the local field $\vec{E}^{\mathrm{loc}}$ and Eq. (3.31) for the induced dipole moments, we can express the polarization $\vec{p} \equiv N \vec{d}$ by
$$\vec{P}=N \overrightarrow{\bar{\alpha}}:(\overrightarrow{\overrightarrow{\bar{K}}}: \vec{E})$$
where $\overrightarrow{\bar{\alpha}}$ is the polarizability tensor of the molecule, $N$ is the number of molecules per unit volume, and the parentheses denote averaging over the orientations of all molecules.
The dielectric constant $\overrightarrow{\bar{\varepsilon}}$ (in units of $\varepsilon_0$ ) is therefore given by
$$\overrightarrow{\vec{\varepsilon}}=1+\frac{N}{\varepsilon_0} \overrightarrow{\vec{\alpha}}: \overrightarrow{\vec{K}}$$
and
\begin{aligned} \Delta \varepsilon &=\varepsilon_{|}-\varepsilon_{\perp} \ &=\frac{N}{\varepsilon_0}\left(\langle\overrightarrow{\vec{\alpha}}: \overrightarrow{\vec{K}}\rangle_{|}-\langle\overrightarrow{\vec{\alpha}}: \overrightarrow{\bar{K}}\rangle_{\perp}\right) . \end{aligned}
From these considerations and from observations by deJeu and Bordewijk [13] that
$$\Delta \varepsilon \propto \rho S$$

## 物理代写|光学代写Optics代考|Hydrodynamics of Ordinary Isotropic Fluids

Consider an elementary volume $d V=d x d y d z$ of a fluid moving in space as shown in Figure 3.9. The following parameters are needed to describe its dynamics:
position vector: $\vec{r}$.
velucity: $\vec{v}(\vec{r}, t)$,
density: $\rho(\vec{r}, t)$,
pressure: $p(\vec{r}, t)$, and
forces in general: $\vec{f}(\vec{r}, t)$.
In later chapters where we study laser-induced acoustic (sound, density) waves in liquid crystals, or generally, when one deals with acoustic waves, it is necessary to assume that the density $\rho(\vec{r}, t)$ is a spatially and temporally varying function. In this chapter, however, we “decouple” such density wave excitation from all the processes under consideration and basically limit our attention to the flow and orientational effects of an incompressible fluid. In that case, we have
$$\rho(\vec{r}, t)=\text { constant }$$
For all liquids, in fact for all gas particles or charges in motion, the equation of continuity also holds
$$\nabla \cdot(\rho \vec{v})=-\frac{\partial \rho}{\partial t} .$$
This equation states that the total variation of $\rho \vec{v}$ over the surface of an enclosing volume is equal to the rate of decrease of the density. Since $\partial \rho / \partial t=0$, we thus have, from Eq. (3.51),
$$\nabla \cdot \vec{v}=0$$
The equation of motion describing the acceleration $d \vec{v} / d t$ of the fluid elements is simply Newton’s law:
$$\rho \frac{d \vec{v}}{d t}=\vec{f}$$

## 物理代写|光学代写Optics代考|Equilibrium Temperature

$$\vec{P}=N \overrightarrow{\bar{\alpha}}:(\overrightarrow{\overrightarrow{\vec{K}}}: \vec{E})$$

$$\vec{\varepsilon}=1+\frac{N}{\varepsilon_0} \vec{\alpha}: \overrightarrow{\vec{K}}$$

$$\Delta \varepsilon=\varepsilon_{\mid}-\varepsilon_{\perp} \quad=\frac{N}{\varepsilon_0}\left(\langle\overrightarrow{\vec{\alpha}}: \overrightarrow{\vec{K}}\rangle_{\mid}-\langle\overrightarrow{\vec{\alpha}}: \overrightarrow{\bar{K}}\rangle_{\perp}\right)$$

$\Delta \varepsilon \propto \rho S$

## 物理代写|光学代写Optics代考|Hydrodynamics of Ordinary Isotropic Fluids

$$\rho(\vec{r}, t)=\text { constant }$$

$$\nabla \cdot(\rho \vec{v})=-\frac{\partial \rho}{\partial t} .$$

$$\nabla \cdot \vec{v}=0$$

$$\rho \frac{d \vec{v}}{d t}=\vec{f}$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 物理代写|光学代写Optics代考|UNITS24

statistics-lab™ 为您的留学生涯保驾护航 在代写光学Optics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写光学Optics代写方面经验极为丰富，各种代写光学Optics相关的作业也就用不着说。

• 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|DIELECTRIC CONSTANTS AND REFRACTIVE INDICES

Dielectric constants and refractive indices, as well as electrical conductivities of liquid crystals, are physical parameters that characterize the electronic responses of liquid crystals to externally applied fields (electric, magnetic, or optical). Because of the molecular and energy level structures of nematic molecules, these responses are highly dependent on the direction and the frequencies of the field. Accordingly, we shall classify our studies of dielectric permittivity and other electro-optical parameters into two distinctive frequency regimes: (1) dc and low frequency and (2) optical frequency. Where the transition from the regime (1) to (2) occurs, of course, is governed by the dielectric relaxation processes and the dynamical time constant; typically, the Debye relaxation frequencies in nematics are on the order of $10^{10} \mathrm{~Hz}$.

Most nematics (e.g. E7, pentyl cyanobiphenyl [5CB], etc.) are said to possess positive (dielectric) anisotropy $\left(\varepsilon_{||}>\varepsilon_{\perp}\right)$. On the other hand, some nematics, such as MBBA, possess negative anisotropy (i.e. $\varepsilon_{|}<\varepsilon_{\perp}$ ). The controlling factors are the molecular constituents and structures.

In general, $\varepsilon_{|}$and $\varepsilon_{\perp}$ have different dispersion regions, as shown in Figure $3.4$ for 4-methoxy-4′-n-butylazoxy-benzene [7], which possess negative dielectric anisotropy $(\Delta \varepsilon<0)$. Also plotted in Figure $3.4$ is the dispersion of $\varepsilon_{\text {iso }}$, the dielectric constant for the isotropic case. Notice that for frequencies of $10^9 \mathrm{~Hz}$ or less, $\varepsilon_{\perp}>\varepsilon_{|}$. At higher frequencies and in the optical regime, $\varepsilon_{| 1}>\varepsilon_{\perp}$ (i.e. the dielectric anisotropy changes sign).

For some nematic liquid crystals, this changeover in the sign of $\Delta \varepsilon=\varepsilon_{|}-\varepsilon_{\perp}$ occurs at a much lower frequency (cf. Figure $3.5$ for phenylbenzoates [8]). This changeover frequency $f_{\mathrm{co}}$ is lower because of the long three-ringed molecular structure, which is highly resistant to the rotation of molecules around the short axes.

## 物理代写|光学代写Optics代考|Free Energy and Torques by Electric and Magnetic Fields

In this section, we consider the interactions of nematic liquid crystals with applied fields (electric or magnetic); we will limit our discussion to only dielectric and diamagnetic interactions.

For a generally applied (dc, low frequency, or optical) electric field $\vec{E}$, the displacement $\vec{D}$ may be written in the form
$$\vec{D}=\varepsilon_{\perp} \vec{E}+\left(\varepsilon_{|}-\varepsilon_{\perp}\right)(n \cdot \vec{E}) n$$

The electric interaction energy density is therefore
$$\mu_E=-\int_0^E \vec{D} \cdot d \vec{E}=-\frac{1}{2} \varepsilon_{\perp}(\vec{E} \cdot \vec{E})-\frac{\Delta \varepsilon}{2}(n \cdot \vec{E})^2 .$$
Note that the first term on the right-hand side of Eq. (3.24) is independent of the orientation of the director axis. It can therefore be neglected in the director axis deformation energy. Accordingly, the free-energy density term associated with the application of an electric field is given by
$$F_E=-\frac{\Delta \varepsilon}{2}(n \cdot \vec{E})^2$$
in SI units (in cgs units, $F_E=-(\Delta \varepsilon / 8 \pi)(\hat{n} \cdot \vec{E})^2$ ). The molecular torque produced by the electric field is given by
$$\vec{\Gamma}E=\vec{D} \times \vec{E}=\Delta \varepsilon(n \cdot \vec{E})(n \times \vec{E}) .$$ Similar considerations for the magnetic field yield a magnetic energy density term $U_m$ given by $$U_m=-\int_0^M \vec{B} \cdot d \vec{M}=\frac{1}{2 \mu_0} \chi{\perp}^m B^2-\frac{1}{2 \mu_0} \Delta \chi^m(n \cdot \vec{B})^2,$$
a magnetic free-energy density (associated with director axis reorientation) $F_m$ given by
$$F_m=\frac{1}{2 \mu_0} \Delta \chi^m(n \cdot \vec{B})^2,$$
and a magnetic torque density
\begin{aligned} \vec{\Gamma}_m &=\vec{M} \times \vec{H} \ &=\Delta \chi^m(n \cdot \vec{H})(n \cdot \vec{H}) . \end{aligned}

## 物理代写|光学代写Optics代考|Free Energy and Torques by Electric and Magnetic Fields

$$\vec{D}=\varepsilon_{\perp} \vec{E}+\left(\varepsilon_{\mid}-\varepsilon_{\perp}\right)(n \cdot \vec{E}) n$$

$$\mu_E=-\int_0^E \vec{D} \cdot d \vec{E}=-\frac{1}{2} \varepsilon_{\perp}(\vec{E} \cdot \vec{E})-\frac{\Delta \varepsilon}{2}(n \cdot \vec{E})^2 .$$

$$F_E=-\frac{\Delta \varepsilon}{2}(n \cdot \vec{E})^2$$

$$\vec{\Gamma} E=\vec{D} \times \vec{E}=\Delta \varepsilon(n \cdot \vec{E})(n \times \vec{E}) .$$

$$U_m=-\int_0^M \vec{B} \cdot d \vec{M}=\frac{1}{2 \mu_0} \chi \perp^m B^2-\frac{1}{2 \mu_0} \Delta \chi^m(n \cdot \vec{B})^2,$$

$$F_m=\frac{1}{2 \mu_0} \Delta \chi^m(n \cdot \vec{B})^2,$$

$$\vec{\Gamma}_m=\vec{M} \times \vec{H} \quad=\Delta \chi^m(n \cdot \vec{H})(n \cdot \vec{H})$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 物理代写|光学代写Optics代考|CSCI031

statistics-lab™ 为您的留学生涯保驾护航 在代写光学Optics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写光学Optics代写方面经验极为丰富，各种代写光学Optics相关的作业也就用不着说。

• 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|When Must You Wear Laser Safety Glasses

The class of a laser must be printed on the laser housing. It’s usually shown somewhere near the aperture. Never use a laser whose class you do not know. If there is no label telling you the class and the wavelength, don’t use the laser! Also, never use a laser that has been tampered with because this can affect the output beam in unexpected ways. (Some lasers, such as green laser pointers, are actually frequency-doubled infrared lasers. The infrared component is typically prevented from leaving the laser by a filter. If that filter is broken or removed, you could receive a very hazardous eye exposure to an infrared beam without realizing it.) If the warning label on the laser says words like “danger” or “avoid exposure to beam” take it seriously!

Laser glasses should be obtained from a reputable supplier and should conform to national certification requirements. The ability of laser glasses to block radiation is described by the optical density (OD) rating for the wavelength of the laser in use. Higher OD, means more light attenuation. Note that an OD rating that is sufficiently high to be safe at one wavelength does not imply that the glasses are safe for any other laser wavelength. In general, different laser wavelengths require different safety glasses. The OD rating is a logarithmic scale that specifies the transmission of the glasses at the rated wavelength.
$$\mathrm{OD}=\log _{10}\left(\frac{1}{T}\right)$$
where $T$ is the transmittance of the glasses. The transmittance of the glasses is the fraction of incident power that is transmitted through the glasses. For example, if a set of laser glasses has an OD of 5 at $1,064 \mathrm{~nm}$, they will transmit 1/100,000th of the incident power at this wavelength.

## 物理代写|光学代写Optics代考|Good Safety Habits

If you inculcate good habits when working with lasers and associated optics, laser safety will become a natural way of working and not a burden. Many of these habits will also help with lab productivity in general. Here’s a list.
Remember the safety glasses
Keep safety glasses in a particular place. Make a habit of putting them on before turning on the laser! For keyed lasers, it can be a good reminder to store the key near the glasses.
Keep upright
Never put your eyes at the height of the beam. Other than wearing your safety glasses, this is the most important precaution you can take. Similarly, don’t bend down to get a dropped object or lean down over the optic chain in such a way as to bring your eyes near the height of the laser. When you are not upright, your safety glasses may not sit properly, potentially exposing you to a very hazardous situation.
No jewelry
Remove all reflective objects on your body or clothes before working with a laser. This includes watches, smartwatches, rings of all types, brooches, and so on. Hair should be tied back so as not to fall into the beam and so as not to contaminate the optics.
Horizontal beam path
If at all possible, keep the beam path at a single uniform height. Most importantly, avoid any beams that don’t travel horizontally. If you must change the beam height, use periscopes specifically designed for that purpose.
Close the shutter first
Always block the laser before adding or removing an optic. Usually, this is as simple as closing the laser shutter. Align a new optic as best you can without the beam. Then unblock the laser and adjust the alignment as required.

## 物理代写|光学代写Optics代考|When Must You Wear Laser Safety Glasses

$$\mathrm{OD}=\log _{10}\left(\frac{1}{T}\right)$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 物理代写|光学代写Optics代考|PHS2062

statistics-lab™ 为您的留学生涯保驾护航 在代写光学Optics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写光学Optics代写方面经验极为丰富，各种代写光学Optics相关的作业也就用不着说。

• 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|The Paraxial Approximation

We’re going to be most interested in the propagation of “beams” of light. ${ }^{2}$ Beams of light propagate mostly in one direction and we choose our axes so that the $z$-axis lies in the direction of propagation. The paraxial approximation is the assumption that all wavefront normals make small angles with the $z$-axis. It is appropriate for beams and any situation where light travels mostly in one direction. We consider the propagation between two planes perpendicular to the $z$-axis as shown in Figure 1.1: a source plane $\mathbf{S}{\mathbf{1}}$ at $z=z{1}$ and a “downstream” field plane $\mathbf{S}$ at some unspecified $z$. As before, we assume that in the source plane, the complex scalar field $\tilde{E}\left(x, y, z_{1}\right)$ is known.

The Helmholtz equation simplifies in the paraxial approximation. Since the light is propagating primarily in the z-direction, it’s useful to separate out the rapid phase accumulation in $z$ due to the wave nature of the light by writing
$$\tilde{E}(x, y, z)=u(x, y, z) e^{-i k z} .$$
The idea is that as long as the light is traveling largely in the $z$-direction, the $u(x, y, z)$ will vary very little over distances on the order of a wavelength. In other words, our wave can be treated as something close to a plane wave but with a complex amplitude $u(x, y, z)$ that varies slowly with position. $u(x, y, z)$ is sometimes known as the complex field amplitude or just the field amplitude. Substituting Eq. (1.22) into Eq. (1.9), we get
$$\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}+\frac{\partial^{2} u}{\partial z^{2}}-2 i k \frac{\partial u}{\partial z}=0 .$$
In the common case where the wave occupies only a small region in the source plane and the phasefronts are fairly flat – typical characteristics of what we might call “beams” then $u$ will vary more slowly in the z-direction than in any other direction, namely
\begin{aligned} &\left|\frac{\partial^{2} u}{\partial z^{2}}\right| \ll\left|\frac{\partial^{2} u}{\partial x^{2}}\right| \ &\left|\frac{\partial^{2} u}{\partial z^{2}}\right| \ll\left|\frac{\partial^{2} u}{\partial y^{2}}\right| \end{aligned}

## 物理代写|光学代写Optics代考|Coherence

Huygens’ principle also allows us to discuss one of the ways in which we classify the statistical properties of light. Light sources are often discussed in terms of their coherence, which comes in two types: temporal coherence and spatial coherence. In a temporally coherent emitter, all of the Huygens’ wavelets are emitting at the same frequency and the phase of each individual emitter remains fixed for a long time (e.g. many nanoseconds for a HeNe laser). This is known as the coherence time. Lasers have high temporal coherence compared to other sources of light. As a result, lasers tend to be very narrow-band emitters, emitting in only a very narrow band of wavelengths around some nominal wavelength. In lasers, the bandwidth of the output light is referred to as the “linewidth.” For example, HeNe lasers, which have fairly narrow linewidths, may emit wavelengths in the band $\lambda=$ $632.816 \pm 0.001 \mathrm{~nm}$. The coherence length of a HeNe is the distance traveled by the beam in the coherence time. For a HeNe, it’s typically a few tens of centimeters but can be tens of meters for carefully designed units.

High temporal coherence does not in itself require that all the Huygens’ emitters have the same phase, only that the phase of each individual emitter should vary slowly. Spatial coherence describes the phase relationship between the different Huygens’ emitters. In a source with high spatial coherence, all the emitters are in phase with one another, or nearly so. For example, Young’s double-slit experiment only yields the expected diffraction pattern when the spatial coherence of the incident light is sufficient that parts of the beam separated by the slit distance have similar phase. Sources with high spatial coherence can be focused to very small spot sizes and can be collimated so that they approximate plane waves. Note that high spatial coherence does not require high temporal coherence even though they usually occur together. As long as the phases of all the emitters stay the same, spatial coherence is preserved whether the overall phase changes are fast or slow, random or not. In Chapter 6, we discuss the properties of etendue and radiance, which are closely related to spatial coherence. Sources with high spatial coherence will have high radiance and low etendue. As a rule, both the temporal and spatial coherence of lasers are the highest of all light sources, which is the main reason they’re so useful.

## 物理代写|光学代写Optics代考|The Paraxial Approximation

$$\tilde{E}(x, y, z)=u(x, y, z) e^{-i k z}$$

$$\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}+\frac{\partial^{2} u}{\partial z^{2}}-2 i k \frac{\partial u}{\partial z}=0 .$$

$$\left|\frac{\partial^{2} u}{\partial z^{2}}\right| \ll\left|\frac{\partial^{2} u}{\partial x^{2}}\right| \quad\left|\frac{\partial^{2} u}{\partial z^{2}}\right| \ll\left|\frac{\partial^{2} u}{\partial y^{2}}\right|$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 物理代写|光学代写Optics代考|UNITS24

statistics-lab™ 为您的留学生涯保驾护航 在代写光学Optics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写光学Optics代写方面经验极为丰富，各种代写光学Optics相关的作业也就用不着说。

• 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|Maxwell’s Equations

The field of optics describes the behavior of light as it propagates through space and materials. To understand the behavior of light, we start with the fundamental classical physics model describing it: Maxwell’s equations of electrodynamics. Maxwell’s equations show that the electric and magnetic fields can travel as waves. In a source-free region, Maxwell’s equations in linear media are $^{1}$
\begin{aligned} \vec{\nabla} \cdot \vec{E} &=0, \ \vec{\nabla} \cdot \vec{B} &=0, \ \vec{\nabla} \times \vec{E} &=-\frac{\partial \vec{B}}{\partial t}, \ \vec{\nabla} \times \vec{B} &=\mu \epsilon \frac{\partial \vec{E}}{\partial t}, \end{aligned}
where $\vec{E}$ is the electric field, $\vec{B}$ is the magnetic field, $\mu$ is the permeability of the medium, and $\epsilon$ is the permittivity of the medium. If we take the curl of both sides of Eq. (1.3), apply the vector identity $\vec{\nabla} \times \vec{\nabla} \times \vec{E}=\vec{\nabla}(\vec{\nabla} \cdot \vec{E})-\nabla^{2} \vec{E}$ to the left-hand side and exchange the order of the time derivative and the curl on the right-hand side, we get
$$\vec{\nabla}(\vec{\nabla} \cdot \vec{E})-\nabla^{2} \vec{E}=-\frac{\partial(\vec{\nabla} \times \vec{B})}{\partial t} .$$
Then substitute from Eqs. (1.1) and (1.4) to get
$$\nabla^{2} \vec{E}=\mu \epsilon \frac{\partial^{2} \vec{E}}{\partial t^{2}} .$$
This is the wave equation in three dimensions where the wave speed is $v=1 / \sqrt{\mu \epsilon}$. Taking the curl of Eq. (1.4) and performing similar algebra shows that the magnetic field also satisfies the wave equation with the same wave speed. Thus Maxwell’s equations allow for electromagnetic waves. In vacuum, the speed is $v=1 / \sqrt{\mu_{0} \epsilon_{0}} \equiv c$, the speed of light in vacuum. Light is indeed an electromagnetic wave.

We now look for solutions to Eq. (1.6) and its magnetic field counterpart. We actually only need to solve for the electric field because the magnetic field can always be found from $\vec{B}=\frac{1}{c} \hat{k} \times \vec{E}$, where $\hat{k}$ is the direction of travel.

## 物理代写|光学代写Optics代考|Huygens’ Principle

In many cases of interest in optics, Eq. (1.9) is solved to a good approximation by Huygens’ integral. The field is assumed to be known on a “source plane” $S_{1}$ perpendicular to the $z$-axis and is only nonzero in some finite region of that plane. The values of the field on $S_{1}$ serve as a boundary condition for solving Eq. (1.9). The solution is given by Huygens’ integral for the complex scalar field at any desired point $x, y, z$.
$$\tilde{E}(x, y, z)=\frac{i}{\lambda} \iint_{S_{1}} \tilde{E}\left(x^{\prime}, y^{\prime}, z^{\prime}\right) \cos \phi \frac{e^{-i k r}}{r} \mathrm{~d} S^{\prime} .$$
The integration over the source plane $S_{1}$ is performed using the integration variables $x^{\prime}$, $y^{\prime}, z^{\prime}$. The vector $\vec{r}$ joins points in the source plane $S_{1}$ with the point $(x, y, z)$ at which we are calculating the field. The angle between $\vec{r}$ and the z-axis is $\phi$ (see Figure 1.1). The solution represented by Huygens’ integral is satisfying because it encapsulates an intuitive understanding of how light waves behave that was understood long before the formal mathematics was fully worked out.

The intuitive description of Eq. (1.10) is known as Huygens’ principle, due to Christiaan Huygens (1629-1695), a Dutch mathematician and scientist. Under Huygens’ principle, every point in the source is considered to be emitting light with spherical wavefronts propagating outward – the so-called Huygens’ wavelets. These wavefronts are represented by the factor $\cos \phi \frac{e^{–i t}}{r}$. They are emitted preferentially in the direction perpendicular to the source plane due to the presence of $\cos \phi$. The constant $\frac{i}{\lambda}$ out-front contributes $90^{\circ}$ of phase and the $\lambda$ in the denominator serves to keep the units the same on both sides of the equation. The complex scalar field in the source plane $E\left(x, y, z_{1}\right)$ sets the relative amplitudes and phases of these tiny spherical emitters. The field $\tilde{F}(x, y, z)$ is then simply the linear superposition of all the spherical wavefronts emitted from the source.

## 物理代写|光学代写Optics代考|Maxwell’s Equations

$$\vec{\nabla} \cdot \vec{E}=0, \vec{\nabla} \cdot \vec{B} \quad=0, \vec{\nabla} \times \vec{E}=-\frac{\partial \vec{B}}{\partial t}, \vec{\nabla} \times \vec{B} \quad=\mu \epsilon \frac{\partial \vec{E}}{\partial t},$$

$$\vec{\nabla}(\vec{\nabla} \cdot \vec{E})-\nabla^{2} \vec{E}=-\frac{\partial(\vec{\nabla} \times \vec{B})}{\partial t}$$

$$\nabla^{2} \vec{E}=\mu \epsilon \frac{\partial^{2} \vec{E}}{\partial t^{2}}$$

## 物理代写|光学代写Optics代考|Huygens’ Principle

$$\tilde{E}(x, y, z)=\frac{i}{\lambda} \iint_{S_{1}} \tilde{E}\left(x^{\prime}, y^{\prime}, z^{\prime}\right) \cos \phi \frac{e^{-i k r}}{r} \mathrm{~d} S^{\prime} .$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 物理代写|光学代写Optics代考|CSCl 031

statistics-lab™ 为您的留学生涯保驾护航 在代写光学Optics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写光学Optics代写方面经验极为丰富，各种代写光学Optics相关的作业也就用不着说。

• 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|Functionalized and Discotic Liquid Crystals

Through various chemical synthesis techniques as well as nanotechnologies, an entire class of novel or so-called functionalized liquid crystals have emerged [11-13]. Figure $1.14$ shows, for example, the shuttlecock-shaped liquid crystal formed by incorporating fullerene $\mathrm{C} 60$ to various crystals and liquid crystals reported by Sawamura et al. [11]. Others have investigated a special class of liquid crystals comprising disc-like molecules, discotic liquid crystals, that possess interesting and useful semiconducting properties suitable for optoelectronic applications [12, 13].

In general, temperature ranges for the various mesophases of single constituent liquid crystals are quite limited. Therefore, while many fundamental studies are still conducted on such liquid crystalline materials, industrial applications employ mostly mixturēs, composites, or specially doped liquid crystals with large operrating temperature range and tailor-made physical and optical properties.

There are many techniques for modifying the physical properties of a liquid crystal. At the most fundamental level, various chemical groups such as bonds or atoms can be introduced to modify the LC molecule. A good example is the cyanobiphenyl homologous series $n \mathrm{CB}(n=1,2,3 \ldots$ ). As $n$ is increased through synthesis, the viscosities, anisotropies, molecular sizes, and many other parameters are greatly modified. Some of these physical properties can also be modified by substitution. For example, the hydrogen in the 2,3 , and 4 positions of the phenyl ring may be substituted by some fluoro $(\mathrm{F})$ or chloro $(\mathrm{Cl})$ group [14].

Besides these molecular synthesis techniques, there are other ways to dramatically improve the performance characteristics of liquid crystals. In the following sections, we describe three well-developed methods, focusing our discussion on nematic liquid crystals as they exemplify the unique characteristics of liquid crystals widely used in optical and photonic applications.

## 物理代写|光学代写Optics代考|Dye-doped Liquid Crystals

An obvious effect of introducing dye molecule to liquid crystals is to increase the absorption of a particular liquid crystal at some specified wavelength region. In particular, dye molecules with absorption anisotropy, or those that undergo conformation changes such as trans-cis isomorphism or produce photo-charges, are often used for photonic applications [15-17]. For example, dichroic dye molecules that are more absorptive for optical field polarization parallel than perpendicular to its long axis are often used for the guest-host effect as their oblong shape makes them compatible for dispersing in the host nematic liquid crystals without disturbing the order. These dichroic molecules can then be oriented and reoriented by an external field applied to the host NLC to switch the transmission of the cell (cf. Figure 1.17); such dichroic dye-doped liquid crystals have been utilized to demonstrate optical diode action [15] in the transmission of polarized light.

If the dye molecules undergo some physical changes such as trans-cis isomorphism or produce space charges following photon absorption, they could give rise to nonlinear optical effects [16]; others [17] have shown that dye molecules deposited on the cell windows can be optically aligned as an effective means of surface alignment mechanism for I.C. cell fahrication. These and other effects due to the presence of dye molecules or other photosensitive agents in liquid crystals are discussed in more detail in Chapter 8 .

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 物理代写|光学代写Optics代考|PHS 2062

statistics-lab™ 为您的留学生涯保驾护航 在代写光学Optics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写光学Optics代写方面经验极为丰富，各种代写光学Optics相关的作业也就用不着说。

• 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|Lyotropic Liquid Crystals

Lyotropic liquid crystals are obtained when an appropriate concentration of material is dissolved in some solvent. The most common systems are those formed by water and amphiphilic molecules (molecules that possess a hydrophilic part that interacts strongly with water and a hydrophobic part that is water insoluble) such as soaps, detergents, and lipids. Here the most important variable controlling the existence of the liquid crystalline phase is the amount of solvent (or concentration). There are quite a number of phases observed in such water-amphiphilic systems, as the composition and temperature are varied; some appear as spherical micelles, and others possess ordered structures with 1-, 2-, or 3 – $\mathrm{D}$ positional order.

Examples of these kinds of molecules are soaps (Figure 1.8) and various phospholipids like those present in cell membranes. Lyotropic liquid crystals are of interest in biological studies [7].

Polymeric liquid crystals are basically the polymer versions of the monomers discussed in Section 1.1. A good account of polymeric liquid crystals may be found in [9]. There are three common types of polymers, as shown in Figure $1.9 \mathrm{a}-\mathrm{c}$, which are characterized by the degree of flexibility. The vinyl type (Figure $1.9 \mathrm{a}$ ) is the most flexible, the Dupont Kevlar polymer (Figure 1.9b) is semirigid, and the polypeptide chain (Figure $1.9 \mathrm{c}$ ) is the most rigid. Mesogenic (or liquid crystalline) polymers are classified in accordance with the molecular architectural arrangement of the mesogenic monomer. Main-chain polymers are built by linking rigid mesogenic groups in a manner depicted schematically in Figure $1.10$; the link may be a direct bond or some flexible spacer. Liquid crystal side-chain polymers are formed by pendant side attachment of mesogenic monomers to a conventional polymeric chain, as depicted in Figure 1.10b.

## 物理代写|光学代写Optics代考|Thermotropic Liquid Crystals: Smectic, Nematic, Cholesteric

Although the molecular structures of thermotropic liquid crystals are quite complicated, they are often represented as “rigid rods” that interact with one another to form distinctive ordered structures (or phases) as a function of ascending temperature: crystals, smectic, nematic, cholesteric (including blue-phase), and the isotropic liquid phase. In smectic liquid crystals, there are several subclassifications in accordance with the positional and directional arrangement of the molecules.

As explained in greater detail in the following chapters, these mesophases are defined and characterized by many physical parameters such as long- and shortrange order, orientational distribution functions, and so on. Here we continue to use the rigid-rod model and pictorially describe these phases in terms of their molecularr arrangement.

Figure 1.11a depicts the collective arrangement of the rodlike molecules in the nematic phase schematically. These molecules are, however, directionally correlated; they are aligned in a general direction défined by a unit vector $n$, the so-called director axis, which may be regarded as the crystal axis. Nevertheless, the molecules are positionally random and exhibit flow very much like liquids; X-ray diffraction from nematics does not exhibit any diffraction peak.

Although individual molecules of nematic liquid crystal (NLC), cholesteric liquid crystal (CLC), and blue-phase liquid crystal (BPLC) may be polar, i.e. carry a permanent dipole, they tend to self-assemble themselves in such a manner that bulk liquid crystals are centrosymmetric, cf. Figure 1.12; their physical properties are the same in the $+\hat{n}$ and the optically uniaxial $-\hat{n}$ directions.

Cholesteric liquid crystals, often also called chiral nematic liquid crystals, resemble nematic liquid crystals except that the molecules assembled in a helical manner, as depicted in Figure 1.11. This property results from the addition of chiral agents to nematic constituents in the starting mixture. Owing to the spatially (helical) varying refractive index, CLCs possess special optical properties such as photonic bandgaps for transmission of circularly polarized lights. More details on CLC as well as cholesteric BPLCs obtained by increasing the concentration of the chiral constituent $[10]$ in the starting mixture are presented in Chapter 4 .

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 物理代写|光学代写Optics代考|UNITS 24

statistics-lab™ 为您的留学生涯保驾护航 在代写光学Optics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写光学Optics代写方面经验极为丰富，各种代写光学Optics相关的作业也就用不着说。

• 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|MOLECULAR STRUCTURES AND CHEMICAL COMPOSITIONS

With few exceptions, liquid crystals are composed of organic substances with a typical structure, as depicted in Figure 1.1. They are aromatic and, if they contain benzene rings, they are often referred to as benzene derivatives. In general, aromatic liquid crystal molecules such as those shown in Figure $1.1$ comprise a side chain $\mathrm{R}$, two or more aromatic rings $A$ and $A^{\prime}$, connected by linkage groups $X$ and $Y$, and at the other end connected to a terminal group $\mathrm{R}^{\prime}$.

Examples of side chain and terminal groups are alkyl $\left(\mathrm{C}{n} \mathrm{H}{2 n+1}\right)$, alkoxy $\left(\mathrm{C}{n} \mathrm{H}{2 n+1} \mathrm{O}\right)$, and others such as acyloxyl, alkyl carbonate, alkoxy carbonyl, and the nitro and cyano groups. The Xs of the linkage groups are simple bonds or groups such as stilbene $(-\mathrm{CH}==\mathrm{CH}-)$, ester $\left(-|_{\mathrm{C}}^{\mathrm{O}} \mathrm{O}-\right)$, tolane $(-\mathrm{C} \equiv \equiv \mathrm{C}-)$, azoxy $(-\mathrm{N}==\mathrm{N}-)$, Schiff base $(-\mathrm{CH}==\mathrm{N}-)$, acetylene $(-\mathrm{C} \equiv \equiv \mathrm{C}-)$, and diacetylene $(-\mathrm{C} \equiv \equiv \mathrm{C}-\mathrm{C} \equiv \equiv \mathrm{C}-)$. The names of liquid crystals are often fashioned after the linkage group (e.g. Schiff-base liquid crystal). There are quite a number of aromatic rings. These include saturated cyclohexane or unsaturated phenyl, biphenyl, and terphenyl in various combinations.

The majority of liquid crystals are benzene derivatives; the rest include heterocyclics, organometallics, sterols, and some organic salts or fatty acids. Their typical structures are shown in Figures 1.2-1.4. Heterocyclic liquid crystals are similar in structure to benzene derivatives, with one or more of the benzene rings replaced by a pyridine, pyrimidine, or another similar group. Cholesterol derivatives are the most common chemical compounds that exhibit the cholesteric (or chiral nematic) phase of liquid crystals. Organometallic compounds are special in that they contain metallic atoms and possess interesting dynamical and magneto-optical properties.

## 物理代写|光学代写Optics代考|Electronic Optical Transitions and UV Absorption

Since liquid crystal constituent molecules are quite large, their energy level structures are rather complex. In essence, the basic quantum mechanical theory is similar to the one described in Chapter 10 for a multilevel molecule. Generally, the energy levels are referred to as orbitals: $\pi, n$, and $\sigma$ orbitals for the ground and low-lying levels and $\pi^{}, n^{}$, and $\sigma^{}$ for their excited counterparts. Since most liquid crystals are aromatic compounds, containing one or more aromatic rings, the energy levels or orbitals of aromatic rings play a major role. In particular, the $\pi \rightarrow \pi^{}$ transitions in a benzene molecule have been extensively studied. Figure $1.6$ shows three possible $\pi \rightarrow \pi^{*}$ transitions in a benzene molecule.

In general, these transitions correspond to the absorption of light in the near-UV spectral region $(\leq 200 \mathrm{~nm})$ [2]. These results for a benzene molecule can also be used for interpreting the absorption of liquid crystals containing phenyl rings. On the other hand, in a saturated cyclohexane ring or band, usually only $\sigma$ electrons are involved. The $\sigma \rightarrow \sigma^{}$ transitions correspond to absorption of light of shorter wavelength $(\leq 180 \mathrm{~nm})$ in comparison to the $\pi \rightarrow \pi^{}$ transition mentioned previously.
These optical properties are also related to the presence or absence of conjugation (i.e. alternations of single and double bonds, as in the case of a benzene ring). In such conjugated molecules, the $\pi$ electron’s wave function is delocalized along the conjugation length, resulting in absorption of light in a longer wavelength region compared to, for example, that associated with the $\sigma$ electron in compounds that do not possess conjugation. Absorption data and spectral dependence for a variety of molecular constituents, including phenyl rings, biphenyls, terphenyls, tolanes, and diphenyl-diacetylenes, may be found in [2].

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 物理代写|光学代写Optics代考|PHS2062

statistics-lab™ 为您的留学生涯保驾护航 在代写光学Optics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写光学Optics代写方面经验极为丰富，各种代写光学Optics相关的作业也就用不着说。

• 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。

## 有限元方法代写

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。