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# 物理代写|傅立叶光学代写Fourier optics代考|SK2340 PLANE EM WAVES

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## 物理代写|傅立叶光学代写Fourier optics代考|PLANE EM WAVES

Consider a plane wave propagating along the $z$-direction. The electric and magnetic fields can be written as
\begin{aligned} &\mathbf{E}=E_x \mathrm{e}^{j(k z-w t)} \hat{e}_{\mathrm{x}}+E_y \mathrm{e}^{j(k z-w t)} \hat{e}_y \ &\mathbf{H}=H_x \mathrm{e}^{j(k z-w t)} \hat{e}_x+H_y \mathrm{e}^{j(k z-w t)} \hat{e}_y, \end{aligned}

where the real parts are actually the physical solutions. They can be more generally written as
\begin{aligned} &\mathbf{E}(\mathbf{r}, t)=\mathbf{E}0 \cos (\mathbf{k} \bullet \mathbf{r}-w t) \ &\mathbf{H}(\mathbf{r}, t)=\mathbf{H}_0 \cos (\mathbf{k} \bullet \mathbf{r}-w t), \end{aligned} where $\mathbf{E}_0$ and $\mathbf{H}_0$ have components $\left(E_x, E_y\right),\left(H_x, H_y\right)$, and $\mathbf{k}, \mathbf{r}$ in this case are simply $k=2 \pi / \lambda$ and $z$, respectively. Substituting Eqs. (3.7-1) in to $\nabla \times \mathbf{E}=-\mu \frac{\partial \mathbf{H}}{\partial t}$ gives the following: $$k E_y \hat{e}{\mathbf{x}}-k E_x \hat{e}y=-\mu w\left[H_x \hat{e}{\mathbf{x}}+H_y \hat{e}_y\right]$$
Hence,
\begin{aligned} &H_x=-\frac{1}{\eta} E_y \ &H_y=\frac{1}{\eta} E_x \end{aligned}
where $\eta$ is called the characteristic impedance of the medium. It is given by
$$\eta=\frac{w}{k} \mu=v \mu=\sqrt{\mu / \varepsilon}$$

## 物理代写|傅立叶光学代写Fourier optics代考|Scalar Diffraction Theory

When the wavelength of a wave field is larger than the “aperture” sizes of the diffraction device used to control the wave, the scalar diffraction theory can be used. Even when this is not true, scalar diffraction theory has been found to be quite accurate [Mellin and Nordin, 2001]. Scalar diffraction theory involves the conversion of the wave equation, which is a partial differential equation, into an integral equation. It can be used to analyze most types of diffraction phenomena and imaging systems within its realm of validity. For example, Figure $4.1$ shows the diffraction pattern from a double slit illuminated with a monochromatic plane wave. The resulting wave propagation can be quite accurately described with scalar diffraction theory.

In this chapter, scalar diffraction theory will be first derived for monochromatic waves with a single wavelength. Then, the results will be generalized to nonmonochromatic waves by using Fourier analysis and synthesis in the time direction.

This chapter consists of eight sections. In Section 4.2, the Helmholtz equation is derived. It characterizes the spatial variation of the wave field, by characterizing the time variation as a complex exponential factor. In Section 4.3, the solution of the Helmholtz equation in homogeneous media is obtained in terms of the angular spectrum of plane waves. This formulation also characterizes wave propagation in a homogeneous medium as a linear system. The FFT implementation of the angular spectrum of plane waves is discussed in Section 4.4.

Diffraction can also be treated by starting with the Helmholtz equation and converting it to an integral equation using Green’s theorem. The remaining sections cover this topic. In Section 4.5, the Kirchoff theory of diffraction results in one formulation of this approach. The Rayleigh-Sommerfeld theory of diffraction covered in Sections $4.6$ and $4.7$ is another formulation of the same approach. The Rayleigh-Sommerfeld theory of diffraction for nonmonochromatic waves is treated in Section $4.8$.

## 物理代写|傅立叶光学代写Fourier optics代考|PLANE EM WAVES

. PLANE EM WAVES

\begin{aligned} &\mathbf{E}=E_x \mathrm{e}^{j(k z-w t)} \hat{e}_{\mathrm{x}}+E_y \mathrm{e}^{j(k z-w t)} \hat{e}_y \ &\mathbf{H}=H_x \mathrm{e}^{j(k z-w t)} \hat{e}_x+H_y \mathrm{e}^{j(k z-w t)} \hat{e}_y, \end{aligned}

\begin{aligned} &\mathbf{E}(\mathbf{r}, t)=\mathbf{E}0 \cos (\mathbf{k} \bullet \mathbf{r}-w t) \ &\mathbf{H}(\mathbf{r}, t)=\mathbf{H}_0 \cos (\mathbf{k} \bullet \mathbf{r}-w t), \end{aligned}，其中$\mathbf{E}_0$和$\mathbf{H}_0$有组件$\left(E_x, E_y\right),\left(H_x, H_y\right)$，而$\mathbf{k}, \mathbf{r}$在本例中分别简单地是$k=2 \pi / \lambda$和$z$。替换方程式。(3.7-1)在$\nabla \times \mathbf{E}=-\mu \frac{\partial \mathbf{H}}{\partial t}$给出如下:$$k E_y \hat{e}{\mathbf{x}}-k E_x \hat{e}y=-\mu w\left[H_x \hat{e}{\mathbf{x}}+H_y \hat{e}_y\right]$$

\begin{aligned} &H_x=-\frac{1}{\eta} E_y \ &H_y=\frac{1}{\eta} E_x \end{aligned}

$$\eta=\frac{w}{k} \mu=v \mu=\sqrt{\mu / \varepsilon}$$

## 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。