### 物理代写|光学工程代写Optical Engineering代考|ES4C5

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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

## 物理代写|光学工程代写Optical Engineering代考|COMPLEX REPRESENTATION OF WAVE

In general, a plane wave is represented by Eq. (1.29), which is rewritten as
$$u(r, t)=\operatorname{Re}{A \exp [\mathrm{i}(k \cdot r-\omega t)]},$$
because
$$\exp (i \alpha)=\cos \dot{\alpha}+\mathrm{i} \sin \alpha,$$
where $\operatorname{Re}{\ldots}$ denotes the real part of complex number. Equation (1.36) is also represented by
\begin{aligned} u(r, t) & =\frac{1}{2}\left{A \exp [\mathrm{i}(k \cdot r-\omega t)]+A^* \exp [-\mathrm{i}(k \cdot r-\omega t)]\right} \ & =\frac{1}{2} A \exp [\mathrm{i}(k \cdot r-\omega t)]+\text { c.c., } \end{aligned}
where $A^*$ denotes the complex conjugate of $A$, and c.c. means the complex conjugate of its former term. In some cases, the symbol of the real part $\operatorname{Re}{\ldots}$ is omitted so that Eq. (1.36) is simply written as
$$u(r, t)=A \exp [\mathrm{i}(k \cdot r-\omega t)]$$

This is called the complex amplitude. It should be noted that a wave that exists physically is represented by a real number and so the complex amplitude is only a mathematical expression. We have introduced the complex amplitude for mathematical convenience. For example, to calculate a sum of waves
$$A=\sum_m A_m \exp \left[\mathrm{i}\left(k_m \cdot r-\omega t\right)\right]=\left[\sum_m A_m \exp \left(\mathrm{i} k_m \cdot r\right)\right] \cdot \exp (-\mathrm{i} \omega t)$$
we can separate a spatial part and a temporal part at first, and then calculate the spatial parts independently, and finally multiply the temporal part $\exp (-\mathrm{i} \omega t)$. The real amplitude is given by the real part of the final result. In many optical eases, only the spatial terms are considered. If necessary, the time-dependent term is multiplied with the final results of spatial calculation. It should be noted that such methods in the complex notation of wave are valid only in the case of linear operations.

## 物理代写|光学工程代写Optical Engineering代考|SCALAR WAVE AND VECTOR WAVE

Until now, we did not consider the direction of the electric field variation $u$. The light is an electromagnetic wave. Assuming the light is propagating to the direction of the $\mathrm{z}$ axis, the variation direction of the electric and magnetic fields are the directions of the $x$ and $y$ axes. This type of wave is called a transverse wave. On the other hand, an acoustic wave is a longitudinal wave, where the direction of variation is in the propagation direction $\mathrm{z}$.

In general, the electric field and the magnetic field are vectors with three components: $E\left(E_x, E_y, E_z\right)$ and $H\left(H_x, H_y, H_z\right)$, respectively. Therefore, the light wave propagation is vertical by nature.

In a homogeneous media, like vacuum, water or glass, the optical properties do not depend on the position and the propagation direction, and hence the components $E_x, E_y, E_z, H_x, H_y, H_z$ satisfy the wave equation independently:
$$\nabla^2 E_x=\frac{1}{v^2} \frac{\partial^2 E_x}{\partial t^2},$$

and so on. Those equations are integrated into the wave equation Eq. (1.11). This wave is called the scalar wave.

Generally, the light wave is considered as a scalar wave, but in an inhomogeneous media or near an aperture or boundary of homogeneous media, the components of electric and magnetic fields are not independent and interact with each other. In such a case, the scalar approximation is not valid and the light wave should be considered as a vector wave.

Next, consider a complex sinusoidal wave as a solution of scalar wave equation,
$$u(r, t)=U(r) \exp (-\mathrm{i} \omega t),$$
where
$$U(r)=A(r) \exp [\mathrm{i} \phi(r)] .$$
Since this equation satisfies the wave equation (1.11), substituting Eq. (1.50) into Eq. (1.11) gives the Helmholtz equation
$$\left(\nabla^2+k^2\right) U=0,$$
where $k$ denotes the wave number (1.14). The Helmholtz equation Eq. (1.52) describes the monochromatic wave propagation in a homogeneous medium.

# 光学工程代考

## 物理代写|光学工程代写Optical Engineering代考|COMPLEX REPRESENTATION OF WAVE

$$u(r, t)=\operatorname{Re} A \exp [\mathrm{i}(k \cdot r-\omega t)]$$

$$\exp (i \alpha)=\cos \dot{\alpha}+\mathrm{i} \sin \alpha$$

Ibegin ${$ aligned $} u(r, t) \&=\backslash f r a c{1}{2} \backslash \operatorname{left}\left{A\right.$ lexp $[\backslash m a t h r m{i}(k \backslash c$ dot $r-$ lomega $t)]+A^{\wedge} * \backslash \operatorname{lexp}[-\backslash m a t h r r$

$$u(r, t)=A \exp [\mathrm{i}(k \cdot r-\omega t)]$$

$$A=\sum_m A_m \exp \left[\mathrm{i}\left(k_m \cdot r-\omega t\right)\right]=\left[\sum_m A_m \exp \left(\mathrm{i} k_m \cdot r\right)\right] \cdot \exp (-\mathrm{i} \omega t)$$

## 物理代写|光学工程代写Optical Engineering代考|SCALAR WAVE AND VECTOR WAVE

$$\nabla^2 E_x=\frac{1}{v^2} \frac{\partial^2 E_x}{\partial t^2},$$

$$u(r, t)=U(r) \exp (-\mathrm{i} \omega t),$$

$$U(r)=A(r) \exp [\mathrm{i} \phi(r)] .$$

$$\left(\nabla^2+k^2\right) U=0,$$

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## MATLAB代写

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