19th Ave New York, NY 95822, USA

# 物理代写|傅立叶光学代写Fourier optics代考|ECE502 THE KIRCHOFF THEORY OF DIFFRACTION

my-assignmentexpert™提供最专业的一站式服务：Essay代写，Dissertation代写，Assignment代写，Paper代写，Proposal代写，Proposal代写，Literature Review代写，Online Course，Exam代考等等。my-assignmentexpert™专注为留学生提供Essay代写服务，拥有各个专业的博硕教师团队帮您代写，免费修改及辅导，保证成果完成的效率和质量。同时有多家检测平台帐号，包括Turnitin高级账户，检测论文不会留痕，写好后检测修改，放心可靠，经得起任何考验！

## 物理代写|傅立叶光学代写Fourier optics代考|THE KIRCHOFF THEORY OF DIFFRACTION

The propagation of the angular spectrum of plane waves as discussed in Section $4.5$ does characterize diffraction. However, diffraction can also be treated by starting with the Helmholtz equation and converting it to an integral equation using Green’s theorem.

Green’s theorem involves two complex-valued functions $U(\mathbf{r})$ and $G(\mathbf{r})$ (Green’s function). We let $S$ be a closed surface surrounding a volume $V$, as shown in Figure 4.6.

If $U$ and $G$, their first and second partial derivatives respectively, are singlevalued and continuous, without any singular points within and on $S$, Green’s theorem states that
$$\iint_V\left(G \nabla^2 U-U \nabla^2 G\right) \mathrm{d} v=\iint_S\left(G \frac{\partial U}{\partial n}-U \frac{\partial G}{\partial n}\right) \mathrm{d} s$$
where $\partial / \partial n$ indicates a partial derivative in the outward normal direction at each point of $S$. In our case, $U$ corresponds to the wave field.

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

The Green function chosen by Kirchhoff is a spherical wave given by
$$G(\mathbf{r})=\frac{\mathrm{e}^{j k r_{01}}}{r_{01}}$$
where $\mathbf{r}$ is the position vector pointing from $P_0$ to $P_1$, and $r_{01}$ is the corresponding distance, given by
$$r_{01}=\left[\left(x_0-x\right)^2+\left(y_0-y\right)^2+z^2\right]^{1 / 2}$$
$U(\mathbf{r})$ satisfies the Helmholtz equation. As $G(\mathbf{r})$ is an expanding spherical wave, it also satisfies the Helmholtz equation:
$$\nabla^2 G(\mathbf{r})+k^2 G(\mathbf{r})=0$$
The left hand side of Eq. (4.5-1) can now be written as
$$\iiint_V\left[G \nabla^2 U-U \nabla^2 G\right] \mathrm{d} V=\iiint_V k^2[U G-U G] \mathrm{d} v=0$$
Hence, Eq. (4.5-1) becomes
$$\iint_S\left[G \frac{\partial U}{\partial n}-U \frac{\partial G}{\partial n}\right] \mathrm{d} s=0$$
On the surface $S_0$ of Figure 4.7, we have
\begin{aligned} G(\mathbf{r}) &=\frac{\mathrm{e}^{j k R}}{R} \ \frac{\partial G(\mathbf{r})}{\partial n} &=\left(j k-\frac{1}{R}\right) \frac{\mathrm{e}^{j k R}}{R}\left(\frac{\partial U}{\partial n}-j k U\right) \end{aligned}

## 物理代写|傅立叶光学代写傅里叶光学代考|THE KIRCHOFF THEORY OF衍射

$$\iint_V\left(G \nabla^2 U-U \nabla^2 G\right) \mathrm{d} v=\iint_S\left(G \frac{\partial U}{\partial n}-U \frac{\partial G}{\partial n}\right) \mathrm{d} s$$
，其中$\partial / \partial n$表示在$S$的每一点上向外法向的偏导数。在我们的例子中，$U$对应于波场

## 物理代写|傅立叶光学代写傅里叶光学代考|基尔霍夫衍射理论

Kirchhoff选择的Green函数是一个球面波，由
$$G(\mathbf{r})=\frac{\mathrm{e}^{j k r_{01}}}{r_{01}}$$

$$r_{01}=\left[\left(x_0-x\right)^2+\left(y_0-y\right)^2+z^2\right]^{1 / 2}$$
$U(\mathbf{r})$给出，满足Helmholtz方程。由于$G(\mathbf{r})$是一个膨胀的球形波，它也满足Helmholtz方程:
$$\nabla^2 G(\mathbf{r})+k^2 G(\mathbf{r})=0$$
Eq.(4.5-1)的左边现在可以写成
$$\iiint_V\left[G \nabla^2 U-U \nabla^2 G\right] \mathrm{d} V=\iiint_V k^2[U G-U G] \mathrm{d} v=0$$

$$\iint_S\left[G \frac{\partial U}{\partial n}-U \frac{\partial G}{\partial n}\right] \mathrm{d} s=0$$

\begin{aligned} G(\mathbf{r}) &=\frac{\mathrm{e}^{j k R}}{R} \ \frac{\partial G(\mathbf{r})}{\partial n} &=\left(j k-\frac{1}{R}\right) \frac{\mathrm{e}^{j k R}}{R}\left(\frac{\partial U}{\partial n}-j k U\right) \end{aligned}

## 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。