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

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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

## 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

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

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

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

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

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

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

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