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## 物理代写|傅立叶光学代写Fourier optics代考|Fundamentals of Wave Propagation

In this chapter and Chapter 4 , waves are considered in 3-D, in general. However, in some applications such as in integrated optics in which propagation of waves on a surface is often considered, 2-D waves are of interest. For example, see Chapter 19 on dense wavelength division multiplexing. Two-dimensional equations are simpler because one of the space variables, say, $y$ is omitted from the equations. Hence, the results discussed in 3-D in what follows can be easily reduced to the 2-D counterparts.

Electromagnetic (EM) waves will be of main concern. They are generated when a time-varying electric field $\mathbf{E}(\mathbf{r}, t)$ produces a time-varying field $\mathbf{H}(\mathbf{r}, t)$. EM waves propagate through unguided media such as free space or air and in guided media such as an optical fiber or the medium between the earth’s surface and the ionosphere. In this chapter, we will be mainly concerned with unbounded media.
Spherical waves result when a source such as an antenna emits EM energy as shown in Figure 3.1(a). At a far away distance from the source, the spherical wave appears like a plane wave with uniform properties at all points of the wavefront, as seen in Figure 3.1(b). Another example would be an electric dipole directed along the $z$-axis, located at the origin, and oscillating with the circular frequency $w$. It generates electric and magnetic fields with a complicated expression, but far from the origin where the fields look like plane waves. A perfect plane wave does not exist physically, but it is a component that is very useful in modeling all kinds of waves.
Waves propagate in a medium. In the case of optical waves, the optical medium is characterized by a quantity $n$ called the refractive index. It is the ratio of the speed of light in free space to that of the speed of light in the medium. The medium is homogeneous if $n$ is constant, otherwise, it is inhomogeneous. In this chapter, we will assume that the medium is homogeneous.

## 物理代写|傅立叶光学代写Fourier optics代考|WAVES

Nature is rich in a large variety of waves, such as electromagnetic, acoustical, water, and brain waves. A wave can be considered as a disturbance of some kind that can travel with a fixed velocity and is unchanged in form from point to point.

Let $u(x, t)$ denote a 1-D wave in the $x$-direction in a homogeneous medium. If $v$ is its velocity, $u(x, t)$ satisfies
$$u(x, t)=u(x-v t, 0)$$
if it is traveling to the right and
$$u(x, t)=u(x+v t, 0)$$
if it is traveling to the left.
Assuming the wave is traveling to the left, let $s$ be given by
$$s=x+v t$$

## 物理代写|傅立叶光学代写傅里叶光学代考|WAVES

$$u(x, t)=u(x-v t, 0)$$

$$u(x, t)=u(x+v t, 0)$$

$$s=x+v t$$ 给出

## 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。

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## 物理代写|傅立叶光学代写Fourier optics代考|AMPLITUDE AND PHASE SPECTRA

$U_c(f)$ can be written as
$$U_c(f)=U_a(f) \mathrm{e}^{j \phi(f)},$$
where the amplitude (magnitude) spectrum $U_a(f)$ and the phase spectrum $\phi(f)$ of the signal $x(t)$ are defined as
$$\begin{gathered} U_a(f)=\left|U_c(f)\right|=\frac{1}{2}\left[\left|U_1(f)\right|^2+\left|U_0(f)\right|^2\right]^{1 / 2} \ \phi(f)=\tan ^{-1}\left[\frac{\operatorname{Imaginary}\left[U_c(f)\right]}{\text { Real }\left[U_c(f)\right]}\right] \end{gathered}$$
$U_a(f)$ is an even function. With real signals, $\phi(f)$ is an odd function and can be written as
$$\phi(f)=\tan ^{-1}\left[-U_0(f) / U_1(f)\right]$$

## 物理代写|傅立叶光学代写Fourier optics代考|HANKEL TRANSFORMS

Functions having radial symmetry are easier to handle in polar coordinates. This is often the case, for example, in optics where lenses, aperture stops, and so on are often circular in shape.

Let us first consider the Fourier transform in polar coordinates. The rectangular and polar coordinates are shown in Figure 2.4. The transformation to polar coordinates is given by
\begin{aligned} r &=\left[x^2+y^2\right]^{1 / 2} \ \theta &=\tan ^{-1}\left(\frac{y}{x}\right) \ \rho &=\left[f_x^2+f_y^2\right]^{1 / 2} \ \phi &=\tan ^{-1}\left(\frac{f_y}{f_x}\right) \end{aligned}
The FT of $u(x, y)$ is given by
$$U\left(f_x, f_y\right)=\int_{-\infty}^{\infty} u(x, y) \mathrm{e}^{-j 2 \pi\left(f_x x+f_y y\right)} \mathrm{d} x \mathrm{~d} y$$
$f(x, y)$ in polar coordinates is $f(r, \theta) . F\left(f_x, f_y\right)$ in polar coordinates is $F(\rho, \phi)$, given by
$$U(\rho, \phi)=\int_0^{2 \pi} \mathrm{d} \theta \int_0^{\infty} u(r, \theta) \mathrm{e}^{-j 2 \pi r \rho(\cos \theta \cos \phi+\sin \theta \sin \phi)} r \mathrm{~d} r$$

## 物理代写|傅立叶光学代写傅里叶光学代考|振幅和相位谱

.

$U_c(f)$可以写成
$$U_c(f)=U_a(f) \mathrm{e}^{j \phi(f)},$$
，其中信号$x(t)$的幅(幅)谱$U_a(f)$和相位谱$\phi(f)$被定义为
$$\begin{gathered} U_a(f)=\left|U_c(f)\right|=\frac{1}{2}\left[\left|U_1(f)\right|^2+\left|U_0(f)\right|^2\right]^{1 / 2} \ \phi(f)=\tan ^{-1}\left[\frac{\operatorname{Imaginary}\left[U_c(f)\right]}{\text { Real }\left[U_c(f)\right]}\right] \end{gathered}$$
$U_a(f)$是偶函数。对于实数信号，$\phi(f)$是一个奇函数，可以写成
$$\phi(f)=\tan ^{-1}\left[-U_0(f) / U_1(f)\right]$$

## 物理代写|傅立叶光学代写傅里叶光学代考|HANKEL TRANSFORMS

.

\begin{aligned} r &=\left[x^2+y^2\right]^{1 / 2} \ \theta &=\tan ^{-1}\left(\frac{y}{x}\right) \ \rho &=\left[f_x^2+f_y^2\right]^{1 / 2} \ \phi &=\tan ^{-1}\left(\frac{f_y}{f_x}\right) \end{aligned}
$u(x, y)$的FT由
$$U\left(f_x, f_y\right)=\int_{-\infty}^{\infty} u(x, y) \mathrm{e}^{-j 2 \pi\left(f_x x+f_y y\right)} \mathrm{d} x \mathrm{~d} y$$
$f(x, y)$在极坐标下是$f(r, \theta) . F\left(f_x, f_y\right)$在极坐标下是$F(\rho, \phi)$，由
$$U(\rho, \phi)=\int_0^{2 \pi} \mathrm{d} \theta \int_0^{\infty} u(r, \theta) \mathrm{e}^{-j 2 \pi r \rho(\cos \theta \cos \phi+\sin \theta \sin \phi)} r \mathrm{~d} r$$ 给出

## 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。

Posted on Categories:傅立叶光学, 物理代写

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## 物理代写|傅立叶光学代写Fourier optics代考|Linear Systems and Transforms

Diffraction as well as imaging can often be modeled as linear systems. First of all, a system is an input-output mapping. Thus, given an input, the system generates an output. For example, in a diffraction or imaging problem, the input and output are typically a wave at an input plane and the corresponding diffracted wave at a distance from the input plane.

Optical systems are quite analogous to communication systems. Both types of systems have a primary purpose of collecting and processing information. Speech signals processed by communication systems are 1-D whereas images are 2-D. Onedimensional signals are typically temporal whereas 2-D signals are typically spatial. For example, an optical system utilizing a laser beam has spatial coherence. Then, the signals can be characterized as 2-D or 3-D complex-valued field amplitudes. Spatial coherence is necessary in order to observe diffraction. Illumination such as ordinary daylight does not have spatial coherence. Then, the signals can be characterized as 2-D spatial, real-valued intensities.

Linear time-invariant and space-invariant communication and optical systems are usually analyzed by frequency analysis using the Fourier transform. Nonlinear optical elements such as the photographic film and nonlinear electronic components such as diodes have similar input-output characteristics.

In both types of systems, Fourier techniques can be used for system synthesis as well. An example is two-dimensional filtering. Theoretically optical matched filters, optical image processing techniques are analogous to matched filters and image processing techniques used in communications and signal processing.

## 物理代写|傅立叶光学代写Fourier optics代考|LINEAR SYSTEMS AND SHIFT INVARIANCE

Linearity allows the decomposition of a complex signal into elementary signals often called basis signals. In Fourier analysis, basis signals or functions are sinusoids.
In a linear system, a given input maps into a unique output. However, more than one input may map into the same output. Thus, the mapping may be one-to-one, or many-to-one.

A 2-D system is shown in Figure 2.1, where $u(x, y)$ is the input signal, and $g(x, y)$ is the output signal. Mathematically, the system can be written as
$$g(x, y)=O[u(x, y)]$$
in the continuous-space case. $O[\bullet]$ is an operator, mapping the input to the output. In the discrete-space case, the point $(x, y)$ is sampled as $[\Delta x \bullet m, \Delta y \bullet n]$, where $\Delta x$ and $\Delta y$ are the sampling intervals along the two directions. $[\Delta x \bullet m, \Delta y \bullet n]$ can be simply represented as $[m, n]$, and the system can be written as
$$g[m, n]=O[u[m, n]]$$
Below the continuous-space case is considered. The system is called linear if any linear combination of two inputs $u_1(x, y)$, and $u_2(x, y)$ generates the same combination of their respective outputs $g_1(x, y)$ and $g_2(x, y)$. This is called superposition principle and written as
$$O\left[a_1 u_1\left(t_1, t_2\right)+a_2 u_2(x, y)\right]=a_1 O\left[u_1(x, y)\right]+a_2 O\left[u_2(x, y)\right]$$

where $a_1$ and $a_2$ are scalars. Above $(x, y)$ is replaced by $[m, n]$ in the case of a linear discrete-space system.

## 物理代写|傅立叶光学代写傅里叶光学代考|线性系统和移位不变性

$$g(x, y)=O[u(x, y)]$$
。$O[\bullet]$是一个操作符，将输入映射到输出。在离散空间的情况下，点$(x, y)$被采样为$[\Delta x \bullet m, \Delta y \bullet n]$，其中$\Delta x$和$\Delta y$是沿两个方向的采样间隔。$[\Delta x \bullet m, \Delta y \bullet n]$可以简单地表示为$[m, n]$，系统可以写成
$$g[m, n]=O[u[m, n]]$$

$$O\left[a_1 u_1\left(t_1, t_2\right)+a_2 u_2(x, y)\right]=a_1 O\left[u_1(x, y)\right]+a_2 O\left[u_2(x, y)\right]$$

## 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。