## 电子工程代写|光子简介代写Introduction to Photonics代考|LEO242

statistics-lab™ 为您的留学生涯保驾护航 在代写光子简介Introduction to Photonics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子简介Introduction to Photonics方面经验极为丰富，各种代写光子简介Introduction to Photonics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

## 电子工程代写|光子简介代写Introduction to Photonics代考|Beam Velocity

As already mentioned, wave packets can be tailored in time and space to form an optical (pulsed) beam (Sect. 3.1.6). The wave vectors of the Fourier components of such a beam are grouped around a central wave vector $\mathbf{k}^{0}$ that defines the direction of the beam: $\mathbf{k}=\mathbf{k}^{0}+\Delta \mathbf{k}$; each wave vector is related to a frequency $\omega=\omega_{0}+\Delta \omega$ according to the dispersion relation $\omega(\mathbf{k})$ that can be expanded as
$$\omega(\mathbf{k})=\omega_{0}+\frac{\partial \omega}{\partial \mathbf{k}} \Delta \mathbf{k}+\ldots$$
The wave packet can be written as three-dimensional integral over $\Delta \mathbf{k}$,
\begin{aligned} \tilde{\mathbf{E}}(\mathbf{x}, t) &=\int \tilde{\mathbf{E}}(\Delta \mathbf{k}) \mathrm{e}^{-\mathrm{j}\left[\left(\mathbf{k}^{0}+\Delta \mathbf{k}\right) \cdot \mathbf{x}-\omega(\mathbf{k}) t\right]} \mathrm{d}^{3} \Delta \mathbf{k} \ &=\mathrm{e}^{-\mathrm{j}\left(\mathbf{k}^{0} \cdot \mathbf{x}-\omega_{0} t\right)} \int \tilde{\mathbf{E}}(\Delta \mathbf{k}) \mathrm{e}^{-\mathrm{j} \Delta \mathbf{k} \cdot[\mathbf{x}-(\partial \omega / \partial \mathbf{k}) t]} \mathrm{d}^{3} \Delta \mathbf{k}, \end{aligned}
where $\tilde{\mathbf{E}}(\Delta \mathbf{k})$ is the amplitude corresponding to the wave vector $\mathbf{k}^{0}+\Delta \mathbf{k}$. The result is a plane carrier wave $\exp \left[-\mathrm{j}\left(\mathbf{k}^{0} \cdot \mathbf{x}-\omega_{0} t\right)\right]$ with a spatial-temporal envelope represented by the integral; the vectorial group velocity $\mathbf{v}{\text {ray }}$ is obtained by choosing a certain value of the envelope phase $\Delta \mathbf{k} \cdot[\mathbf{x}-(\partial \omega / \partial \mathbf{k}) t]=$ const. and extracting $\dot{\mathbf{x}}$ from the temporal derivative $$\mathbf{v}{\text {ray }}=\dot{\mathbf{x}}=\left[\begin{array}{l} \partial \omega / \partial k_{x} \ \partial \omega / \partial k_{y} \ \partial \omega / \partial k_{z} \end{array}\right]=\nabla_{\mathbf{k}} \omega(\mathbf{k})$$

## 电子工程代写|光子简介代写Introduction to Photonics代考|Energy Transport

The energy transport of electromagnetic waves is described by Poynting’s theorem; to derive it, we multiply Eq. (1.13) with $\mathbf{H}$, and (1.14) with $\mathbf{E}$
$$\mathbf{E} \cdot(\nabla \times \mathbf{H})=\mathbf{E} \cdot \frac{\partial}{\partial t}\left(\varepsilon_{0} \mathbf{E}+\mathbf{P}\right)$$

$$\mathbf{H} \cdot(\nabla \times \mathbf{E})=-\mathbf{H} \cdot\left(\mu_{0} \frac{\partial \mathbf{H}}{\partial t}\right)$$
after subtraction and using $2 \mathbf{a} \cdot(\partial \mathbf{a} / \partial t)=\partial(\mathbf{a} \cdot \mathbf{a}) / \partial t$, we obtain the equation
$$\mathbf{E} \cdot(\nabla \times \mathbf{H})-\mathbf{H} \cdot(\nabla \times \mathbf{E})=\frac{\partial}{\partial t}\left(\varepsilon_{0} \frac{\mathbf{E} \cdot \mathbf{E}}{2}+\mu_{0} \frac{\mathbf{H} \cdot \mathbf{H}}{2}\right)+\mathbf{E} \cdot \frac{\partial \mathbf{P}}{\partial t}$$
which, using the identity $\mathbf{b} \cdot(\nabla \times \mathbf{a})-\mathbf{a} \cdot(\nabla \times \mathbf{b})=\nabla \cdot(\mathbf{a} \times \mathbf{b})$, we convert into Poynting’s theorem in its differential form
$$-\nabla \cdot(\mathbf{E} \times \mathbf{H})=\frac{\partial}{\partial t}\left(\varepsilon_{0} \frac{\mathbf{E} \cdot \mathbf{E}}{2}+\mu_{0} \frac{\mathbf{H} \cdot \mathbf{H}}{2}\right)+\mathbf{E} \cdot \frac{\partial \mathbf{P}}{\partial t}$$
For the interpretation of the individual terms, we employ the divergence-theorem
$$\int_{V}(\nabla \cdot \mathbf{u}) \mathrm{d} V=\int_{A} \mathbf{u} \cdot \mathbf{n} \mathrm{d} A$$
where $A$ is the surface of the volume $V, \mathbf{n}$ is the outward pointing unit normal vector of a surface element, and $\mathrm{d} V, \mathrm{~d} A$ are differential volume and surface elements, respectively, to transform Eq. (1.52) into
$$\int_{A}[(\mathbf{E} \times \mathbf{H}) \cdot \mathbf{n}] \mathrm{d} A=-\int_{V}\left[\frac{\partial}{\partial t}\left(\varepsilon_{0} \frac{\mathbf{E} \cdot \mathbf{E}}{2}+\mu_{0} \frac{\mathbf{H} \cdot \mathbf{H}}{2}\right)+\mathbf{E} \cdot \frac{\partial \mathbf{P}}{\partial t}\right] \mathrm{d} V .$$
The terms $\varepsilon_{0} \mathbf{E} \cdot \mathbf{E} / 2$ and $\mu_{0} \mathbf{H} \cdot \mathbf{H} / 2$ represent the electric and magnetic contributions to the vacuum-energy density of the field, while $\mathbf{E} \cdot(\partial \mathbf{P} / \partial t)$ is the power density that is exchanged herween rhe field and rhe medium. Thus, re righr-hand side of Eq. (1.54) is equal to the temporal change of the energy stored in volume $V$. The left-hand side can therefore be interpreted as energy flux through the surface $A$, and the Poynting vector
$$\mathbf{S}=\mathbf{E} \times \mathbf{H}$$
as energy flux density [W $\mathrm{m}^{-2}$ ] of the electromagnetic field.

## 电子工程代写|光子简介代写Introduction to Photonics代考|Beam Velocity

$$\omega(\mathbf{k})=\omega_{0}+\frac{\partial \omega}{\partial \mathbf{k}} \Delta \mathbf{k}+\ldots$$

$$\tilde{\mathbf{E}}(\mathbf{x}, t)=\int \tilde{\mathbf{E}}(\Delta \mathbf{k}) \mathrm{e}^{-\mathrm{j}\left[\left(\mathbf{k}^{0}+\Delta \mathbf{k}\right) \cdot \mathbf{x}-\omega(\mathbf{k}) t\right]} \mathrm{d}^{3} \Delta \mathbf{k} \quad=\mathrm{e}^{-\mathrm{j}\left(\mathbf{k}^{0} \cdot \mathbf{x}-\omega_{0} t\right)} \int \tilde{\mathbf{E}}(\Delta \mathbf{k}) \mathrm{e}^{-\mathrm{j} \Delta \mathbf{k} \cdot[\mathbf{x}-(\partial \omega / \partial \mathbf{k}) t]} \mathrm{d}^{3} \Delta \mathbf{k},$$

$$\text { vray }=\dot{\mathbf{x}}=\left[\partial \omega / \partial k_{x} \partial \omega / \partial k_{y} \partial \omega / \partial k_{z}\right]=\nabla_{\mathbf{k}} \omega(\mathbf{k})$$

## 电子工程代写|光子简介代写Introduction to Photonics代考|Energy Transport

\begin{aligned} &\mathbf{E} \cdot(\nabla \times \mathbf{H})=\mathbf{E} \cdot \frac{\partial}{\partial t}\left(\varepsilon_{0} \mathbf{E}+\mathbf{P}\right) \ &\mathbf{H} \cdot(\nabla \times \mathbf{E})=-\mathbf{H} \cdot\left(\mu_{0} \frac{\partial \mathbf{H}}{\partial t}\right) \end{aligned}

$$\mathbf{E} \cdot(\nabla \times \mathbf{H})-\mathbf{H} \cdot(\nabla \times \mathbf{E})=\frac{\partial}{\partial t}\left(\varepsilon_{0} \frac{\mathbf{E} \cdot \mathbf{E}}{2}+\mu_{0} \frac{\mathbf{H} \cdot \mathbf{H}}{2}\right)+\mathbf{E} \cdot \frac{\partial \mathbf{P}}{\partial t}$$

$$-\nabla \cdot(\mathbf{E} \times \mathbf{H})=\frac{\partial}{\partial t}\left(\varepsilon_{0} \frac{\mathbf{E} \cdot \mathbf{E}}{2}+\mu_{0} \frac{\mathbf{H} \cdot \mathbf{H}}{2}\right)+\mathbf{E} \cdot \frac{\partial \mathbf{P}}{\partial t}$$

$$\int_{V}(\nabla \cdot \mathbf{u}) \mathrm{d} V=\int_{A} \mathbf{u} \cdot \mathbf{n} \mathrm{d} A$$

$$\int_{A}[(\mathbf{E} \times \mathbf{H}) \cdot \mathbf{n}] \mathrm{d} A=-\int_{V}\left[\frac{\partial}{\partial t}\left(\varepsilon_{0} \frac{\mathbf{E} \cdot \mathbf{E}}{2}+\mu_{0} \frac{\mathbf{H} \cdot \mathbf{H}}{2}\right)+\mathbf{E} \cdot \frac{\partial \mathbf{P}}{\partial t}\right] \mathrm{d} V .$$

$$\mathbf{S}=\mathbf{E} \times \mathbf{H}$$

## 广义线性模型代考

statistics-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。

## 电子工程代写|光子简介代写Introduction to Photonics代考|PHYS3112

statistics-lab™ 为您的留学生涯保驾护航 在代写光子简介Introduction to Photonics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子简介Introduction to Photonics方面经验极为丰富，各种代写光子简介Introduction to Photonics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

## 电子工程代写|光子简介代写Introduction to Photonics代考|Phase Velocity

To determine the phase velocity of the wave function Eq. (1.26), we choose a certain value of the phase
$$\mathbf{k} \cdot \mathbf{x}-\omega t=\text { const., }$$
and calculate the speed at which it propagates through space by taking the spatial derivative
$$\mathbf{k} \cdot \frac{\mathrm{d} \mathbf{x}}{\mathrm{d} t}-\omega=0 .$$
The phase velocity in the direction of the wave vector is then, using Eq. (1.28),
$$v_{\mathrm{ph}}=\left|\frac{\mathrm{d} \mathbf{x}}{\mathrm{d} t}\right|=\frac{\omega}{k}=\frac{c_{0}}{n} .$$
In vacuum, $n=\sqrt{\varepsilon}=1$ and the phase velocity is equal to the vacuum velocity of light, $c_{0}$. In the visible, this is also a good approximation for the phase velocity in gasses at moderate pressure; the propagation index of transparent condensed media ranges between 1 and about 3 , the corresponding phase velocity between $c_{0}$ and $c_{0} / 3$. The phase velocity is the relevant velocity for the description of interference effects (Chap. 4). The propagation of optical pulses is governed by the so-called group velocity, which we discuss in the following.

## 电子工程代写|光子简介代写Introduction to Photonics代考|Group Velocity

According to the Fourier theorem, wave packets can be understood as superpositions of monochromatic waves. Since the permittivity $\varepsilon$ and thus the phase velocity of the individual Fourier components is frequency dependent, the propagation velocity of the wave packet may be difficult to define and evaluate. We will treat this problem in some detail in Sect. 3.2. As a first approach, we superimpose two waves with slightly different frequencies and determine the propagation velocity of the resulting “beating” envelope. The two frequencies $\omega_{0} \pm \Delta \omega$ correspond to two wave numbers $k^{0} \pm \Delta k$; assuming equal amplitudes of the two waves, the total field is given by
\begin{aligned} \mathbf{E}(z, t) &=\operatorname{Re}\left[\mathbf{E}{0} \mathrm{e}^{-\mathrm{j}\left[\left(k^{0}+\Delta k\right) z-\left(\omega{0}+\Delta \omega\right) t\right]}+\mathbf{E}{0} \mathrm{e}^{-\mathrm{j}\left[\left(k^{0}-\Delta k\right) z-\left(\omega{0}-\Delta \omega\right) t\right]}\right] \ &=2 \mathbf{E}{0} \cos (z \Delta k-t \Delta \omega) \cos \left(\omega{0} t-k^{0} z\right) \end{aligned}
which is an amplitude modulated wave with the carrier frequency $\omega_{0}$ and the envelope $\cos (z \Delta k-t \Delta \omega)$ (Fig.1.2). The envelope propagates at the velocity $\mathrm{d} z / \mathrm{d} t=\Delta \omega / \Delta k$, while the phase fronts move at the phase velocity of the carrier,

$v_{\mathrm{ph}}=\omega_{0} / k^{0}$. For $\Delta \omega \rightarrow 0$, the Taylor expansion
$$\Delta \omega=\frac{\mathrm{d} \omega}{\mathrm{d} k} \Delta k+\ldots,$$
allows us to identify $\Delta \omega / \Delta k$ with $\mathrm{d} \omega / \mathrm{d} k$, so that the group velocity can be defined as
$$v_{\mathrm{g}}:=\frac{\mathrm{d} \omega}{\mathrm{d} k} .$$

## 电子工程代写|光子简介代写Introduction to Photonics代考|Phase Velocity

$$\mathbf{k} \cdot \mathbf{x}-\omega t=\text { const., }$$

$$\mathbf{k} \cdot \frac{\mathrm{d} \mathbf{x}}{\mathrm{d} t}-\omega=0$$

$$v_{\mathrm{ph}}=\left|\frac{\mathrm{d} \mathbf{x}}{\mathrm{d} t}\right|=\frac{\omega}{k}=\frac{c_{0}}{n} .$$

## 电子工程代写|光子简介代写Introduction to Photonics代考|Group Velocity

$$\mathbf{E}(z, t)=\operatorname{Re}\left[\mathbf{E} 0 \mathrm{e}^{-\mathrm{j}\left[\left(k^{0}+\Delta k\right) z-(\omega 0+\Delta \omega) t\right]}+\mathbf{E} 0 \mathrm{e}^{-\mathrm{j}\left[\left(k^{0}-\Delta k\right) z-(\omega 0-\Delta \omega) t\right]}\right] \quad=2 \mathbf{E} 0 \cos (z \Delta k-t \Delta \omega) \cos$$

$v_{\mathrm{ph}}=\omega_{0} / k^{0}$. 为了 $\Delta \omega \rightarrow 0$, 泰勒展开
$$\Delta \omega=\frac{\mathrm{d} \omega}{\mathrm{d} k} \Delta k+\ldots,$$

$$v_{\mathrm{g}}:=\frac{\mathrm{d} \omega}{\mathrm{d} k} .$$

## 广义线性模型代考

statistics-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。

## 电子工程代写|光子简介代写Introduction to Photonics代考|ELEC3405

statistics-lab™ 为您的留学生涯保驾护航 在代写光子简介Introduction to Photonics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子简介Introduction to Photonics方面经验极为丰富，各种代写光子简介Introduction to Photonics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

## 电子工程代写|光子简介代写Introduction to Photonics代考|Complex Wave Functions and Amplitudes

The structure of Eq. (1.20) allows us to factorize its solutions $\mathbf{E}(\mathbf{x}, t)$ into a spatial and a temporal part. For the temporal part, we choose harmonically oscillating functions: not only do they describe the output of a single mode laser very well, they also represent the base for the Fourier decomposition of more general time varying signals. The ansatz
$$\mathbf{E}(\mathbf{x}, t)=\operatorname{Re}\left[\tilde{\mathbf{E}}(\mathbf{x}, \omega) \mathrm{e}^{\mathrm{j} \omega t}\right]=\frac{1}{2}\left[\tilde{\mathbf{E}}(\mathbf{x}, \omega) \mathrm{e}^{\mathrm{j} \omega t}+c . c .\right]$$
where $\omega$ is the angular frequency and c.c. stands for “complex conjugate,” is a solution of Eq. (1.20), if $\tilde{\mathbf{E}}(\mathbf{x}, \omega)$ is a solution of the Helmholtz equation
$$\nabla^{2} \tilde{\mathbf{E}}(\mathbf{x}, \omega)+\frac{\omega^{2} \varepsilon}{c_{0}^{2}} \tilde{\mathbf{E}}(\mathbf{x}, \omega)=\mathbf{0}$$
In cartesian coordinates, each component of $\tilde{E}{i}$ must be a solution of the scalar Helmholtz equation $$\left[\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}}+\frac{\partial^{2}}{\partial z^{2}}+\frac{\omega^{2} \varepsilon}{c{0}^{2}}\right] \tilde{E}_{i}(\mathbf{x}, \omega)=0$$
A particularly simple solution is the harmonically oscillating function
$$\tilde{\mathbf{E}}(\mathbf{x}, \omega)=\tilde{\mathbf{E}}(\mathbf{k}, \omega) \mathrm{e}^{-\mathrm{j} \mathbf{k} \cdot \mathbf{x}}$$
where $\mathbf{k}$ is known as wave vector and its absolute value $k$ as angular wave number ${ }^{2}$ or propagation constant. The complete electric wave function is then
$$\mathbf{E}(\mathbf{x}, t)=\operatorname{Re}[\tilde{\mathbf{E}}(\mathbf{x}, t)],$$
where
$$\tilde{\mathbf{E}}(\mathbf{x}, t)=\overline{\mathbf{E}}(\mathbf{k}, \omega) \mathrm{e}^{-J(\mathbf{k} \cdot \mathbf{x}-\omega t)}$$
is the so-called complex wave function and $\tilde{\mathbf{E}}(\mathbf{k}, \omega)$ the complex amplitude; the imaginary part of the argument of the exponential function is called phase. Inserting Eq. (1.26) into the Helmholtz equation Eq.

## 电子工程代写|光子简介代写Introduction to Photonics代考|Plane Waves

Surfaces of constant phase of Eq. (1.26), $\mathbf{k} \cdot \mathbf{x}-\omega t=$ const., are planes normal to the wave vector $\mathbf{k}$ (Fig. 1.1); these waves therefore are called plane waves; the distance between planes of equal phase are separated by integer multiples of the so-called wavelength
$$\lambda:=\frac{2 \pi}{|\mathbf{k}|} .$$
The number $k / 2 \pi$ is equal to the number of spatial periods per unit length, measured in the direction of $\mathbf{k} ; k$ is therefore also called spatial (angular) frequency. In vacuum,
$$\lambda_{0}=\frac{2 \pi}{k_{0}}=2 \pi \frac{c_{0}}{\omega} ;$$
the vacuum wavelength in the optical region of the electromagnetic spectrum is of the order of $1 \mu \mathrm{m}$. The corresponding temporal oscillation period, $2 \pi / \omega$, is about $3 \times 10^{-15} \mathrm{~s}$, or 3 femtoseconds (fs).

Similar to harmonically oscillating temporal functions that allow “synthesizing” arbitrary temporal functions, plane waves can be used to synthesize arbitrary spatial wave functions via a Fourier integral over all possible wave vectors (Sect. 3.1.6).
In practice, there are different conventions to specify the frequency of a wave: the temporal frequency $\nu=\omega / 2 \pi$, the quantum energy $\hbar \omega$, the spatial vacuum frequency (spectroscopic wave number) $k / 2 \pi=1 / \lambda_{0}$, or the vacuum wave length $\lambda_{0}$. Table $1.1$ summarizes the relations between the different parameters.

## 电子工程代写|光子简介代写Introduction to Photonics代考|Complex Wave Functions and Amplitudes

$$\mathbf{E}(\mathbf{x}, t)=\operatorname{Re}\left[\tilde{\mathbf{E}}(\mathbf{x}, \omega) \mathrm{e}^{\mathrm{j} \omega t}\right]=\frac{1}{2}\left[\tilde{\mathbf{E}}(\mathbf{x}, \omega) \mathrm{e}^{\mathrm{j} \omega t}+c . c .\right]$$

$$\nabla^{2} \tilde{\mathbf{E}}(\mathbf{x}, \omega)+\frac{\omega^{2} \varepsilon}{c_{0}^{2}} \tilde{\mathbf{E}}(\mathbf{x}, \omega)=\mathbf{0}$$

$$\left[\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}}+\frac{\partial^{2}}{\partial z^{2}}+\frac{\omega^{2} \varepsilon}{c 0^{2}}\right] \tilde{E}_{i}(\mathbf{x}, \omega)=0$$

$$\tilde{\mathbf{E}}(\mathbf{x}, \omega)=\tilde{\mathbf{E}}(\mathbf{k}, \omega) \mathrm{e}^{-\mathrm{j} \mathbf{k} \cdot \mathbf{x}}$$

$$\mathbf{E}(\mathbf{x}, t)=\operatorname{Re}[\tilde{\mathbf{E}}(\mathbf{x}, t)],$$

$$\tilde{\mathbf{E}}(\mathbf{x}, t)=\overline{\mathbf{E}}(\mathbf{k}, \omega) \mathrm{e}^{-J(\mathbf{k} \cdot \mathbf{x}-\omega t)}$$

## 电子工程代写|光子简介代写Introduction to Photonics代考|Plane Waves

$$\lambda:=\frac{2 \pi}{|\mathbf{k}|} .$$

$$\lambda_{0}=\frac{2 \pi}{k_{0}}=2 \pi \frac{c_{0}}{\omega} ;$$

## 广义线性模型代考

statistics-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。

## 电子工程代写|光子简介代写Introduction to Photonics代考|ECSE423

statistics-lab™ 为您的留学生涯保驾护航 在代写光子简介Introduction to Photonics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子简介Introduction to Photonics方面经验极为丰富，各种代写光子简介Introduction to Photonics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

## 电子工程代写|光子简介代写Introduction to Photonics代考|The Electromagnetic Field

Maxwell’s equations, relating the electric field $\mathbf{E}\left[\mathrm{Vm}^{-1}\right]$ and the magnetic field $\mathbf{H}$ $\left[\mathrm{Am}^{-1}\right]$ in a medium with polarization density $\mathbf{P}\left[\mathrm{Asm}^{-2}\right]$, magnetization density $\mathbf{M}\left[\mathrm{Am}^{-1}\right]$, density of free charges $\rho\left[\mathrm{Asm}^{-3}\right]$, and current density $\mathbf{j}\left[\mathrm{Am}^{-2}\right]$, have the form
\begin{aligned} &\nabla \times \mathbf{E}=-\mu_{0} \frac{\partial \mathbf{H}}{\partial t}-\mu_{0} \frac{\partial \mathbf{M}}{\partial t} \ &\nabla \times \mathbf{H}=\varepsilon_{0} \frac{\partial \mathbf{E}}{\partial t}+\frac{\partial \mathbf{P}}{\partial t}+\mathbf{j} \end{aligned}

\begin{aligned} \nabla \cdot\left(\varepsilon_{0} \mathbf{E}\right) &=-\nabla \cdot \mathbf{P}+\rho \ \nabla \cdot\left(\mu_{0} \mathbf{H}\right) &=-\nabla \cdot\left(\mu_{0} \mathbf{M}\right), \end{aligned}
where $\varepsilon_{0}=8.854 \times 10^{-12} \mathrm{AsV}^{-1} \mathrm{~m}^{-1}$ is the vacuum permittivity and $\mu_{0}=4 \pi 10^{-7} \mathrm{VsA}^{-1} \mathrm{~m}^{-1}$ the magnetic constant (also called vacuum permeability). In cartesian coordinates, the differential operator $\nabla$ is given by
$$\nabla=\left[\begin{array}{l} \partial / \partial x \ \partial / \partial y \ \partial / \partial z \end{array}\right]$$
or
$$\nabla=[\partial / \partial x, \partial / \partial y, \partial / \partial z],$$
depending on the vector operation. $\mathbf{P}$ is the response of the medium to the electric field and, for moderate optical fields, a linear function of $\mathbf{E}$,
$$\mathbf{P}=\varepsilon_{0} \chi \mathbf{E}$$
$\chi$ is the (dimensionless) electric susceptibility and represents the dielectric properties of the medium. It is common to introduce the electric displacement density $\mathbf{D}$ $\left[\mathrm{Asm}^{-2}\right]$
$$\mathbf{D}:=\varepsilon_{0} \mathbf{E}+\mathbf{P}$$
that combines the “vacuum displacement density” $\varepsilon_{0} \mathbf{E}$ with the material polarization density.

## 电子工程代写|光子简介代写Introduction to Photonics代考|Wave Equation

We can eliminate the magnetic field from Eqs. (1.13) and (1.14) to obtain a single wave equation for the electric field: taking the rotation of Eq. (1.13) and substituting the time derivative of Eq. (1.14), we obtain
$$\nabla \times(\nabla \times \mathbf{E})+\mu_{0} \frac{\partial^{2} \mathbf{D}}{\partial t^{2}}=\mathbf{0} .$$
In isotropic media, the relation between $\mathbf{P}$ and $\mathbf{E}$ is expressed by a scalar susceptibility $\chi$, and $\varepsilon=1+\chi$. From Eq. (1.3) in the form $\nabla \cdot \mathbf{D}=\nabla \cdot \varepsilon \varepsilon_{0} \mathbf{E}=0$ follows, for homogeneous media, $\nabla \cdot \mathbf{E}=0$. With the identity
$$\nabla \times(\nabla \times \mathbf{a})=\nabla(\nabla \cdot \mathbf{a})-\nabla^{2} \mathbf{a},$$
we can formulate Eq. (1.15) as
$$-\nabla^{2} \mathbf{E}+\mu_{0} \frac{\partial^{2} \mathbf{D}}{\partial t^{2}}=\mathbf{0},$$
where the Laplace operator $\nabla^{2}$, in cartesian coordinates, is given by
$$\nabla^{2}=\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}}+\frac{\partial^{2}}{\partial z^{2}} .$$
With
$$c_{0}:=\frac{1}{\sqrt{\varepsilon_{0} \mu_{0}}},$$
Eq. (1.17) assumes the form
$$\nabla^{2} \mathbf{E}(\mathbf{x}, t)-\frac{\varepsilon}{c_{0}^{3}} \frac{\partial^{2} \mathbf{E}(\mathbf{x}, t)}{\partial t^{2}}=\mathbf{0} ;$$
for reasons that will become obvious, $c_{0}=2.998 \times 10^{8} \mathrm{~ms}^{-1}$ is called vacuum speed of light. Equation (1.20) is the wave equation for the electric field in isotropic, linear, and local media.

## 电子工程代写|光子简介代写Introduction to Photonics代考|The Electromagnetic Field

\begin{aligned} \nabla \times \mathbf{E}=-\mu_{0} \frac{\partial \mathbf{H}}{\partial t}-\mu_{0} \frac{\partial \mathbf{M}}{\partial t} \quad \nabla \times \mathbf{H} &=\varepsilon_{0} \frac{\partial \mathbf{E}}{\partial t}+\frac{\partial \mathbf{P}}{\partial t}+\mathbf{j} \ \nabla \cdot\left(\varepsilon_{0} \mathbf{E}\right)=-\nabla \cdot \mathbf{P}+\rho \nabla \cdot\left(\mu_{0} \mathbf{H}\right) &=-\nabla \cdot\left(\mu_{0} \mathbf{M}\right), \end{aligned}

$$\nabla=[\partial / \partial x \partial / \partial y \partial / \partial z]$$

$$\nabla=[\partial / \partial x, \partial / \partial y, \partial / \partial z]$$

$$\mathbf{P}=\varepsilon_{0} \chi \mathbf{E}$$
$\chi$ 是 (无量纲的) 电敏感性，代表介质的介电特性。通常引入电位移密度 $\mathbf{D}\left[\mathrm{Asm}^{-2}\right]$
$$\mathbf{D}:=\varepsilon_{0} \mathbf{E}+\mathbf{P}$$

## 电子工程代写|光子简介代写Introduction to Photonics代考|Wave Equation

$$\nabla \times(\nabla \times \mathbf{E})+\mu_{0} \frac{\partial^{2} \mathbf{D}}{\partial t^{2}}=\mathbf{0} .$$

$$\nabla \times(\nabla \times \mathbf{a})=\nabla(\nabla \cdot \mathbf{a})-\nabla^{2} \mathbf{a},$$

$$-\nabla^{2} \mathbf{E}+\mu_{0} \frac{\partial^{2} \mathbf{D}}{\partial t^{2}}=\mathbf{0}$$

$$\nabla^{2}=\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}}+\frac{\partial^{2}}{\partial z^{2}} .$$

$$c_{0}:=\frac{1}{\sqrt{\varepsilon_{0} \mu_{0}}}$$

$$\nabla^{2} \mathbf{E}(\mathbf{x}, t)-\frac{\varepsilon}{c_{0}^{3}} \frac{\partial^{2} \mathbf{E}(\mathbf{x}, t)}{\partial t^{2}}=\mathbf{0}$$

## 广义线性模型代考

statistics-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。