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

## 电子工程代写|信号处理与线性系统作业代写Signal Processing and Linear Systems代考|Absorption and Reflection

Plane monochromatic waves
$$\tilde{\mathbf{E}}(\mathbf{x}, t)=\tilde{\mathbf{E}}{0} \mathrm{e}^{-\mathrm{j}(\mathbf{k} \cdot \mathbf{x}-\omega t)}$$ are solutions of the wave equation Eq. (1.20), even if the permittivity $\tilde{\varepsilon}=\tilde{\chi}+1$ is complex. The dispersion relation Eq. (1.27) requires $$\tilde{\mathbf{k}}^{2}=\tilde{\varepsilon}\left(\frac{\omega}{c{0}}\right)^{2}=: \tilde{n}^{2} k_{0}^{2},$$
implying a complex propagation index
$$\tilde{n}=\sqrt{\varepsilon^{\prime}+\mathrm{j} \varepsilon^{\prime \prime}}=: n-\mathrm{j} \kappa \text {; }$$
$n$ and $\kappa$ are obtained from $\tilde{\varepsilon}$ by setting
$$n^{2}-\kappa^{2}=\varepsilon^{\prime}, \quad 2 n \kappa=-\varepsilon^{\prime \prime} .$$
Elimination of $\kappa$ yields
$$4 n^{4}-4 n^{2} \varepsilon^{\prime}-\varepsilon^{\prime \prime 2}=0$$
so that
\begin{aligned} &n=\sqrt{\frac{\left(\varepsilon^{\prime 2}+\varepsilon^{\prime \prime 2}\right)^{1 / 2}+\varepsilon^{\prime}}{2}} \ &\kappa=\sqrt{\frac{\left(\varepsilon^{\prime 2}+\varepsilon^{\prime \prime 2}\right)^{1 / 2}-\varepsilon^{\prime}}{2}} . \end{aligned}
Figure $2.18$ shows the frequency dependence of these two parameters for the $\tilde{\varepsilon}(\omega)$ shown in Fig. 2.17. Equation (2.62) then assumes the form
$$\tilde{\mathbf{E}}(\mathbf{x}, t)=\tilde{\mathbf{E}}{0} \mathrm{e}^{-\kappa k{0} z} \mathrm{e}^{-\mathrm{j}\left(n \mathbf{k}_{0} \cdot \mathbf{x}-\omega t\right)}$$

## 电子工程代写|信号处理与线性系统作业代写Signal Processing and Linear Systems代考|Free Electron Gas Model of Metals

The resonant behavior discussed above is due to the restoring force $a x$ in the equation of motion Eq. (2.51) and is characteristic for bound electrons. Many optical and electronic properties of metals, on the other hand, can be described in good

approximation by modeling the conduction electrons as a free electron gas, i.e., by setting the restoring force in the equation of motion equal to zero
$$\ddot{\mathbf{x}}+\Gamma \dot{\mathbf{x}}=-\frac{e}{m_{e}} \mathbf{E}(t)$$
the complex susceptibility $\tilde{\chi}(\omega)$ according to Eq. $(2.56)$ is then
$$\tilde{\chi}(\omega)=-\left(\frac{n_{\mathrm{e}} e^{2}}{\varepsilon_{0} m_{\mathrm{e}}}\right) \frac{1}{\omega^{2}-\mathrm{j} \omega \Gamma} .$$
The physical source of the damping term in metals are electron collisions that occur within an average collision time $\tau_{\mathrm{e}}$; to establish a relation between $\Gamma$ and $\tau_{\mathrm{e}}$, we expose the electrons to a constant electric field; the stationary velocity of the electrons according to $\mathrm{Eq} .(2.78)$ is
$$\dot{\mathbf{x}}=-\frac{e}{m_{\mathrm{e}} \Gamma} \mathbf{E} .$$
Assuming that the electron velocity is completely randomized by a collision and the average velocity immediately after a collision is consequently equal to zero, the average velocity of the electrons in the static field is also equal to the acceleration of the electrons, $-\left(e / m_{\mathrm{e}}\right) \mathbf{E}(t)$, multiplied with the average time $\tau_{\mathrm{e}}$ between consecutive collisions
$$\dot{\mathbf{x}}=-\frac{e}{m_{\mathrm{e}}} \tau_{\mathrm{e}} \mathbf{E},$$ so that we can set
$$\Gamma=\frac{1}{\tau_{\mathrm{e}}} .$$

## 电子工程代写|信号处理与线性系统作业代写Signal Processing and Linear Systems代考|Absorption and Reflection

$$\tilde{\mathbf{E}}(\mathbf{x}, t)=\tilde{\mathbf{E}} 0 \mathrm{e}^{-\mathrm{j}(\mathbf{k} \cdot \mathbf{x}-\omega t)}$$

$$\tilde{\mathbf{k}}^{2}=\tilde{\varepsilon}\left(\frac{\omega}{c 0}\right)^{2}=: \tilde{n}^{2} k_{0}^{2}$$

$$\tilde{n}=\sqrt{\varepsilon^{\prime}+\mathrm{j} \varepsilon^{\prime \prime}}=: n-\mathrm{j} \kappa ;$$
$n$ 和 $\kappa$ 获得自 $\tilde{\varepsilon}$ 通过设置
$$n^{2}-\kappa^{2}=\varepsilon^{\prime}, \quad 2 n \kappa=-\varepsilon^{\prime \prime} .$$

$$4 n^{4}-4 n^{2} \varepsilon^{\prime}-\varepsilon^{\prime \prime 2}=0$$

$$n=\sqrt{\frac{\left(\varepsilon^{\prime 2}+\varepsilon^{\prime \prime 2}\right)^{1 / 2}+\varepsilon^{\prime}}{2}} \quad \kappa=\sqrt{\frac{\left(\varepsilon^{\prime 2}+\varepsilon^{\prime \prime 2}\right)^{1 / 2}-\varepsilon^{\prime}}{2}} .$$

$$n=\sqrt{\frac{\left(\varepsilon^{\prime 2}+\varepsilon^{\prime \prime 2}\right)^{1 / 2}+\varepsilon^{\prime}}{2}} \quad \kappa=\sqrt{\frac{\left(\varepsilon^{\prime 2}+\varepsilon^{\prime \prime 2}\right)^{1 / 2}-\varepsilon^{\prime}}{2}}$$

$$\tilde{\mathbf{E}}(\mathbf{x}, t)=\tilde{\mathbf{E}} 0 \mathrm{e}^{-\kappa k 0 z} \mathrm{e}^{-\mathrm{j}\left(n \mathbf{k}_{0} \cdot \mathbf{x}-\omega t\right)}$$

## 电子工程代写|信号处理与线性系统作业代写Signal Processing and Linear Systems代考|Free Electron Gas Model of Metals

$$\ddot{\mathbf{x}}+\Gamma \dot{\mathbf{x}}=-\frac{e}{m_{e}} \mathbf{E}(t)$$

$$\tilde{\chi}(\omega)=-\left(\frac{n_{\mathrm{e}} e^{2}}{\varepsilon_{0} m_{\mathrm{e}}}\right) \frac{1}{\omega^{2}-\mathrm{j} \omega \Gamma} .$$

$$\dot{\mathbf{x}}=-\frac{e}{m_{\mathrm{e}} \Gamma} \mathbf{E}$$

$$\dot{\mathbf{x}}=-\frac{e}{m_{\mathrm{e}}} \tau_{\mathrm{e}} \mathbf{E},$$

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