### 电子工程代写|光子简介代写Introduction to Photonics代考|ELEC3405

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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|Complex Wave Functions and Amplitudes

The structure of Eq. (1.20) allows us to factorize its solutions $\mathbf{E}(\mathbf{x}, t)$ into a spatial and a temporal part. For the temporal part, we choose harmonically oscillating functions: not only do they describe the output of a single mode laser very well, they also represent the base for the Fourier decomposition of more general time varying signals. The ansatz
$$\mathbf{E}(\mathbf{x}, t)=\operatorname{Re}\left[\tilde{\mathbf{E}}(\mathbf{x}, \omega) \mathrm{e}^{\mathrm{j} \omega t}\right]=\frac{1}{2}\left[\tilde{\mathbf{E}}(\mathbf{x}, \omega) \mathrm{e}^{\mathrm{j} \omega t}+c . c .\right]$$
where $\omega$ is the angular frequency and c.c. stands for “complex conjugate,” is a solution of Eq. (1.20), if $\tilde{\mathbf{E}}(\mathbf{x}, \omega)$ is a solution of the Helmholtz equation
$$\nabla^{2} \tilde{\mathbf{E}}(\mathbf{x}, \omega)+\frac{\omega^{2} \varepsilon}{c_{0}^{2}} \tilde{\mathbf{E}}(\mathbf{x}, \omega)=\mathbf{0}$$
In cartesian coordinates, each component of $\tilde{E}{i}$ must be a solution of the scalar Helmholtz equation $$\left[\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}}+\frac{\partial^{2}}{\partial z^{2}}+\frac{\omega^{2} \varepsilon}{c{0}^{2}}\right] \tilde{E}_{i}(\mathbf{x}, \omega)=0$$
A particularly simple solution is the harmonically oscillating function
$$\tilde{\mathbf{E}}(\mathbf{x}, \omega)=\tilde{\mathbf{E}}(\mathbf{k}, \omega) \mathrm{e}^{-\mathrm{j} \mathbf{k} \cdot \mathbf{x}}$$
where $\mathbf{k}$ is known as wave vector and its absolute value $k$ as angular wave number ${ }^{2}$ or propagation constant. The complete electric wave function is then
$$\mathbf{E}(\mathbf{x}, t)=\operatorname{Re}[\tilde{\mathbf{E}}(\mathbf{x}, t)],$$
where
$$\tilde{\mathbf{E}}(\mathbf{x}, t)=\overline{\mathbf{E}}(\mathbf{k}, \omega) \mathrm{e}^{-J(\mathbf{k} \cdot \mathbf{x}-\omega t)}$$
is the so-called complex wave function and $\tilde{\mathbf{E}}(\mathbf{k}, \omega)$ the complex amplitude; the imaginary part of the argument of the exponential function is called phase. Inserting Eq. (1.26) into the Helmholtz equation Eq.

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

Surfaces of constant phase of Eq. (1.26), $\mathbf{k} \cdot \mathbf{x}-\omega t=$ const., are planes normal to the wave vector $\mathbf{k}$ (Fig. 1.1); these waves therefore are called plane waves; the distance between planes of equal phase are separated by integer multiples of the so-called wavelength
$$\lambda:=\frac{2 \pi}{|\mathbf{k}|} .$$
The number $k / 2 \pi$ is equal to the number of spatial periods per unit length, measured in the direction of $\mathbf{k} ; k$ is therefore also called spatial (angular) frequency. In vacuum,
$$\lambda_{0}=\frac{2 \pi}{k_{0}}=2 \pi \frac{c_{0}}{\omega} ;$$
the vacuum wavelength in the optical region of the electromagnetic spectrum is of the order of $1 \mu \mathrm{m}$. The corresponding temporal oscillation period, $2 \pi / \omega$, is about $3 \times 10^{-15} \mathrm{~s}$, or 3 femtoseconds (fs).

Similar to harmonically oscillating temporal functions that allow “synthesizing” arbitrary temporal functions, plane waves can be used to synthesize arbitrary spatial wave functions via a Fourier integral over all possible wave vectors (Sect. 3.1.6).
In practice, there are different conventions to specify the frequency of a wave: the temporal frequency $\nu=\omega / 2 \pi$, the quantum energy $\hbar \omega$, the spatial vacuum frequency (spectroscopic wave number) $k / 2 \pi=1 / \lambda_{0}$, or the vacuum wave length $\lambda_{0}$. Table $1.1$ summarizes the relations between the different parameters.

## 电子工程代写|光子简介代写Introduction to Photonics代考|Complex Wave Functions and Amplitudes

$$\mathbf{E}(\mathbf{x}, t)=\operatorname{Re}\left[\tilde{\mathbf{E}}(\mathbf{x}, \omega) \mathrm{e}^{\mathrm{j} \omega t}\right]=\frac{1}{2}\left[\tilde{\mathbf{E}}(\mathbf{x}, \omega) \mathrm{e}^{\mathrm{j} \omega t}+c . c .\right]$$

$$\nabla^{2} \tilde{\mathbf{E}}(\mathbf{x}, \omega)+\frac{\omega^{2} \varepsilon}{c_{0}^{2}} \tilde{\mathbf{E}}(\mathbf{x}, \omega)=\mathbf{0}$$

$$\left[\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}}+\frac{\partial^{2}}{\partial z^{2}}+\frac{\omega^{2} \varepsilon}{c 0^{2}}\right] \tilde{E}_{i}(\mathbf{x}, \omega)=0$$

$$\tilde{\mathbf{E}}(\mathbf{x}, \omega)=\tilde{\mathbf{E}}(\mathbf{k}, \omega) \mathrm{e}^{-\mathrm{j} \mathbf{k} \cdot \mathbf{x}}$$

$$\mathbf{E}(\mathbf{x}, t)=\operatorname{Re}[\tilde{\mathbf{E}}(\mathbf{x}, t)],$$

$$\tilde{\mathbf{E}}(\mathbf{x}, t)=\overline{\mathbf{E}}(\mathbf{k}, \omega) \mathrm{e}^{-J(\mathbf{k} \cdot \mathbf{x}-\omega t)}$$

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

$$\lambda:=\frac{2 \pi}{|\mathbf{k}|} .$$

$$\lambda_{0}=\frac{2 \pi}{k_{0}}=2 \pi \frac{c_{0}}{\omega} ;$$

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