# 电子工程代写|光子简介代写Introduction to Photonics代考|PHYS3112

## 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|Total Reflection

Under conditions of total reflection [Figs. $2.2$ and $2.10$, Eq. (2.10)], the normal component of the wave vector turns imaginary
$$k_{z}^{\mathrm{t}}=k_{0} \sqrt{n_{\mathrm{t}}^{2}-n_{\mathrm{i}}^{2} \sin ^{2} \theta^{\mathrm{i}}}=:-\mathrm{j} \gamma^{\mathrm{t}} .$$

In medium (t), for which we assume $z>0$, the field is then given by
$$E^{\mathrm{t}}=E_{0}^{\mathrm{t}} \mathrm{e}^{-\mathrm{j}(k \cdot \mathbf{x}-\omega t)}=E_{0}^{\mathrm{t}} \mathrm{e}^{-\gamma^{\mathrm{t}} z} \mathrm{e}^{-\mathrm{j}\left(k_{x}^{\mathrm{t}} x-\omega t\right)} .$$
The amplitude of this inhomogeneous, so-called evanescent wave decays exponentially with increasing distance from the interface so that the wave is essentially confined to a layer of thickness $1 / \gamma^{\mathrm{t}}$. This penetration depth is on the order of a wavelength unless the angle of incidence is very close to the critical angle, where it grows quickly and approaches infinity at $\theta^{\mathrm{i}}=\theta_{\text {crit }}$ [Eq. (2.43)].

According to Eq. (2.19), the reflection coefficient for $\sigma$-polarized light under total reflection conditions is
$$r_{\sigma}=\frac{1+\mathrm{j}\left(\gamma^{\mathrm{t}} / k_{z}^{\mathrm{i}}\right)}{1-\mathrm{j}\left(\gamma^{\mathrm{t}} / k_{\mathrm{z}}^{\mathrm{i}}\right)}=: \mathrm{e}^{\mathrm{j} \phi_{\sigma}},$$
while the reflectance is $R=r r^{*}=1$; the reflectance of a metallic mirror, for comparison, is usually less than $0.9$. According to Eq. (2.45), $r_{\sigma}$ is complex and introduces a phase shift of the reflected wave that amounts to
$$\frac{\phi_{\sigma}}{2}=\arctan \frac{\gamma^{\mathrm{t}}}{k_{z}^{\mathrm{i}}}=\arctan \frac{\left(n_{\mathrm{i}}^{2} \sin ^{2} \theta^{\mathrm{i}}-n_{\mathrm{t}}^{2}\right)^{1 / 2}}{n_{\mathrm{i}} \cos \theta^{\mathrm{i}}}$$
(Fig. 2.14). For $\pi$-polarized light, Eq. (2.28) yields
$$r_{\pi}=\frac{1+\mathrm{j}\left(n_{\mathrm{i}} / n_{\mathrm{t}}\right)^{2}\left(\gamma^{\mathrm{t}} / k_{z}^{\mathrm{i}}\right)}{1-\mathrm{j}\left(n_{\mathrm{i}} / n_{\mathrm{t}}\right)^{2}\left(\gamma^{\mathrm{t}} / k_{\mathrm{z}}^{\mathrm{i}}\right)}=: \mathrm{e}^{\mathrm{j} \phi=}$$
with
$$\frac{\phi_{\pi}}{2}=\arctan \frac{n_{\mathrm{i}}^{2}}{n_{\mathrm{t}}^{2}} \frac{\gamma^{\mathrm{t}}}{k_{2}^{\mathrm{i}}}=\arctan \frac{n_{\mathrm{i}}^{2}}{n_{\mathrm{t}}^{2}} \frac{\left(n_{\mathrm{i}}^{2} \sin ^{2} \theta^{\mathrm{i}}-n_{\mathrm{t}}^{2}\right)^{1 / 2}}{n_{\mathrm{i}} \cos \theta^{\mathrm{i}}} .$$

## 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|Optical Tunneling Effect

If the optically thinner medium (refraction index $n_{\mathrm{L}}$ ) is sandwiched between two optically dense media $\left(n_{\mathrm{H}, \mathrm{I}}, n_{\mathrm{H}, 2}\right)$, a light wave can be transmitted through the optically thin medium even under total reflection conditions $\left|\mathbf{k}{|}\right|>n{\mathrm{L}} k_{0}$, provided that the refractive index of the output medium is large enough to support a propagating wave, $n_{\mathrm{H}, 2} k_{0}>\left|\mathbf{k}{|}\right|$; the transmission coefficient decreases roughly exponentially with distance $\propto \mathrm{e}^{-\gamma^{d} d}$ and the direction of the wave vector of the transmitted wave is given by Snell’s law, $n{\mathrm{H}, 2} \sin \theta^{\mathrm{t}}=n_{\mathrm{i}} \sin \theta^{\mathrm{H} . \mathrm{I}}$. This so-called optical tunnel effect is used in various photonic components (for example, high power beam splitters) and is the basis of scanning-tunneling optical microscopy that allows “tapping” the evanescent light scattered from sub-wavelength features of a specimen.

We have introduced the propagation or refractive index $n=\sqrt{\varepsilon}=\sqrt{1+\chi}$ as a function of the susceptibility of the medium, which relates the polarization density to the electric field; the susceptibility itself was treated as a phenomenological property of the medium that was considered as a continuum. We now want to present a simple mechanistic model of the susceptibility that qualitatively cxplains, among other things, the frequency dependence of the refractive index and the absorption coefficient of a medium. The approach of this Drude-Lorentz model is to treat the medium as containing discrete charges (electrons or ions) of a certain mass, held in place by a force that resembles a spring. In this picture, the polarization density of a medium is the vectorial sum over all microscopic dipole moments per unit volume. As we will see, the mass of the oscillating charged particles limits the frequency up to which they can contribute to the polarization; in the visible and near infrared region of the electromagnetic frequency spectrum, only electrons and protons (hydrogen ions) are light enough to contribute.

## 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|Total Reflection

$$k_{z}^{\mathrm{t}}=k_{0} \sqrt{n_{\mathrm{t}}^{2}-n_{\mathrm{i}}^{2} \sin ^{2} \theta^{\mathrm{i}}}=:-\mathrm{j} \gamma^{\mathrm{t}} .$$

$$E^{\mathrm{t}}=E_{0}^{\mathrm{t}} \mathrm{e}^{-\mathrm{j}(k \cdot \mathbf{x}-\omega t)}=E_{0}^{\mathrm{t}} \mathrm{e}^{-\gamma^{\mathrm{t}} z} \mathrm{e}^{-\mathrm{j}\left(k_{x}^{t} x-\omega t\right)} .$$

$$\frac{\phi_{\sigma}}{2}=\arctan \frac{\gamma^{\mathrm{t}}}{k_{z}^{\mathrm{i}}}=\arctan \frac{\left(n_{\mathrm{i}}^{2} \sin ^{2} \theta^{\mathrm{i}}-n_{\mathrm{t}}^{2}\right)^{1 / 2}}{n_{\mathrm{i}} \cos \theta^{\mathrm{i}}}$$
(图 2.14) 。为了 $\pi$-偏振光，方程式。(2.28) 收益率
$$r_{\pi}=\frac{1+\mathrm{j}\left(n_{\mathrm{i}} / n_{\mathrm{t}}\right)^{2}\left(\gamma^{\mathrm{t}} / k_{z}^{\mathrm{i}}\right)}{1-\mathrm{j}\left(n_{\mathrm{i}} / n_{\mathrm{t}}\right)^{2}\left(\gamma^{\mathrm{t}} / k_{\mathrm{z}}^{\mathrm{i}}\right)}=: \mathrm{e}^{\mathrm{j} \phi=}$$

$$\frac{\phi_{\pi}}{2}=\arctan \frac{n_{\mathrm{i}}^{2}}{n_{\mathrm{t}}^{2}} \frac{\gamma^{\mathrm{t}}}{k_{2}^{\mathrm{i}}}=\arctan \frac{n_{\mathrm{i}}^{2}}{n_{\mathrm{t}}^{2}} \frac{\left(n_{\mathrm{i}}^{2} \sin ^{2} \theta^{\mathrm{i}}-n_{\mathrm{t}}^{2}\right)^{1 / 2}}{n_{\mathrm{i}} \cos \theta^{\mathrm{i}}}$$

## 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|Optical Tunneling Effect

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