Light as Electromagnetic Waves

Unifying Light and Electromagnetic Waves

Maxwell Review

• Guass's law

\[\begin{align*} \nabla \vec{E }=&\frac{\rho}{\epsilon_0}\\ \nabla \vec{B}=&0 \end{align*}\]

• Faraday

\[\begin{align*} \nabla \times \vec{E}+\frac{\partial \vec{B }}{\partial t}=0 \end{align*}\]

• Ampere-maxwell

\[\begin{align*} \nabla \times \vec{B }-\mu_0 \epsilon_0\frac{\partial\vec{E }}{\partial t}=\vec{j} \end{align*}\]

• Without change or current (in vacuum (真空))

\[\begin{align*} \nabla^2 \vec{E}=&\frac{1}{c^2}\frac{\partial^2\vec{E}}{\partial t^2}\\ \nabla^2\vec{B}=&\frac{1}{c }\frac{\partial^2\vec{B }}{\partial t^2} \end{align*}\]

Plane Wave

Dimension (量纲)

\[[c]=[\frac{[distance]}{[time]}]\]

\[[\frac{\partial ^2}{\partial t^2}]=\frac{1}{[time]^2}\]

\[[\nabla^2]=[\frac{\partial ^2}{\partial x^2}+\frac{\partial ^2}{\partial y^2}+\frac{\partial ^2}{\partial z^2}]=\frac{1}{[length^2]}\]

Dimension (维度)

• one space dimension (scalar field (标量场))

\[\begin{align*} &c^2\frac{\partial ^2}{\partial x^2}f(x, t)=\frac{\partial ^2}{\partial t^2}f(x, t)\\ &f(x, t)=f(x-ct, t)\\ &f(x,t)=f_0 \cos \left( kx-\omega t + \phi \right), \,k=\frac{2\pi }{\lambda}, \, \omega=\frac{2 \pi }{T} \end{align*}\]

\[\begin{align*} \vec{E }(x,t) =\vec{E }_m \cos (k x-\omega t+\phi)\\ \vec{B }(x,t) =\vec{B }_m \cos (k x-\omega t+\phi) \end{align*}\]

• three space dimension (vector field (矢量场))

\[\psi(\vec{r})=A\cos(\vec{k}\cdot\vec{r}-\omega t+\phi).\]

\[\begin{align*} \vec{E }(\vec{r},t) =\vec{E }_m \cos (\vec{k }\cdot\vec{r }-\omega t+\phi)\\ \vec{B }(\vec{r},t) =\vec{B }_m \cos (\vec{k }\cdot\vec{r }-\omega t+\phi) \end{align*}\]

Wave Fronts(波前)

At any instant a wavefront is a surface of constant phase. For the plane wave such a surface is

\[\begin{align*} \vec{k }\cdot \vec{r }=constant \end{align*}\]

The Propagation of Light Rays(光传播)

The study of the properties of light waves under the approximation that it travels in a straight line is called geometrical optics(几何光学).

Transmission of Light in Matter

• In dielectric materials, the electric field is altered relative permittivity (相对介电常数) (the dielectric constant\(k\)) \(\epsilon_r\)

• In magnetic materials (therefore, not in glass or plastic),the magnetic field is altered by relative permeability (相对渗透性) \(\mu_r\)

• A light wave propagating through any substantive medium travels at a speed

\[\begin{align*} v=\frac{1}{\sqrt{\epsilon_r \mu_r}}\frac{1}{\sqrt{\epsilon_0 \mu_0}}=\frac{c }{n } \end{align*}\]

where the index of refracton \(n=\sqrt{\epsilon_r \mu_r}\).

• The dispersion relation (散射率) becomes

\[\begin{align*} \omega=vk=\frac{ck }{n} \end{align*}\]

hence \(k=nk_0\), where \(k_0\) is the wave number in vacuum.

Reflection(反射) and Refraction(折射)

• Law of reflection: \(\boxed{\theta_1^{\prime}=\theta_1}\).
• Law of refraction (or Snell's law): \(\boxed{n_2\sin\theta_2=n_1\sin\theta_1}\)

Total Internal Reflection

With \(n_glass > n_air\), at a critical (临界) \(\theta_0\) , the refracted ray points directly along the interface, i.e.

\[\begin{align*} n_{glass}\sin\theta_c=n_{air}\sin\left( \frac{\pi}{2} \right)\\ \end{align*}\]

For larger \(\theta > \theta_c\), all the light is reflected.

\[\theta_c = \sin^{-1}\frac{n_{next}}{n_i}\]

for \(n_{next} > n_i\)

Fermat's Principle

Principle of least time: the actual path between two points taken by abeam of light is the one that is traversed in the least time.

Deriving laws of reflection and refraction

1. Reflection

The straight-line path S′BP,which corresponds to \(θ_i=θ_r\), is the shortest possible one.

1. Refraction

\[\begin{align*} &t=\frac{\overrightarrow{SO }}{v_i }+\frac{\overrightarrow{OP }}{v_t }, \text{ Let } \frac{dt }{dx }=0\\ \therefore& n_i\sin \theta_i = n_t \sin \theta_t \end{align*}\]

Hugen's Principle

Huygens’ principle :All points on a wavefront serve as point sources of spherical secondary wavelets. After a time \(t\), the new position of the wavefront will be that of a surface tangent to these secondary wavelets.

wavefront

Huygens’s principle applied to a straight wave front. Each point on the wave front emits a semicircular wavelet that moves a distance \(s = vt\). The new wave front is a line tangent to the wavelets.

Driving Law of Refraction

\[\begin{align*} \because \, &\frac{\lambda_1}{\lambda_2}=\frac{v_1}{v_2}\\ & \sin \theta_1=\frac{\lambda_1}{hc}, \, \sin\theta_2=\frac{\lambda_2}{hc}\\ \therefore \, & \frac{\sin \theta_1}{\sin \theta_2}=\frac{\lambda_1}{\lambda_2}=\frac{v_1}{v_2}=\frac{\frac{c }{n_1 }}{\frac{c }{n_2 }}=\frac{n_2 }{n_1}\\ &\Rightarrow n_2\sin\theta_2=n_1\sin\theta_1 \end{align*}\]

The Electromagnetic Approach

Suppose that the incident, reflected, and transmitted (\(\vec{E}_i, \vec{E}_r, \vec{E}_t\), 入射波, 反射波, 透射波) waves can be written as

\[\begin{align*} \vec{E}_i=&\vec{E}_{0i}\cos\left( \vec{k}_i\cdot\vec{r}-\omega_i t \right)\\ \vec{E}_r=&\vec{E}_{0r}\cos\left( \vec{k}_r\cdot\vec{r}-\omega_r t +\phi_r\right)\\ \vec{E}_t=&\vec{E}_{0t}\cos\left( \vec{k}_t\cdot\vec{r}-\omega_t t +\phi_t\right) \end{align*}\]

\(\vec{k}\) is space angular velocity.

So we have \(\vec{E}=\vec{E}_i+\vec{E}_r\) above the interface and \(\vec{E}=\vec{E}_t\) below.
For simplicity, we consider the case that \(\vec{E}_{0i}, \vec{E}_{0r}, \vec{E}_{0t}\) are constant in time.

Boundary Conditions

We draw a narrow closed path \(C\) that runs parallel to the interface inside both media. According to Faraday's induction law

\[\begin{align*} \oint \vec{E}\cdot \mathrm{d} \vec{s}=-\frac{\mathrm{d }\Phi_B }{\mathrm{d}t} \end{align*}\]

The loop can be made so narrow such that there is no flux through \(C\) . Define \(\hat{u}_n\) to be the unit vector normal(法向量) to the interface.
The boundary condition leads to

\[\begin{align*} \hat{u}_n \times (\vec{E}_i+\vec{E}_r)-\hat{u}_n \times \vec{E}_t=0 \end{align*}\]

which is satisfied for all values of time and at any point on the interface. That is

\[\begin{align*} &\hat{u}_n \times \vec{E}_{0i}\cos\left( \vec{k}_i\cdot\vec{r}-\omega_i t \right)\\ +& \hat{u}_n \times \vec{E}_{0r}\cos\left( \vec{k}_r\cdot\vec{r}-\omega_r t +\phi_r\right) \\ =& \hat{u}_n \times \vec{E}_{0t}\cos\left( \vec{k}_t\cdot\vec{r}-\omega_t t +\phi_t\right) \end{align*}\]

This can only be satisfied if \(\omega_i=\omega_r=\omega_t\), which means the charged particles (带电粒子) within the media are undergoing force oscillations (振荡) at the frequency of the incident wave.（共振）

Law of Reflection

for any \(\vec{r}\) terminating on the interface

\[\begin{align*} &\left. \left( \vec{k}_i \cdot \vec{r} \right)\right|_{y=b} = \left. \left( \vec{k}_r \cdot \vec{r} +\phi_r \right)\right|_{y=b} = \left. \left( \vec{k}_t \cdot \vec{r} +\phi_t \right)\right|_{y=b}\\ &\Rightarrow\left[ (\vec{k}_i-\vec{k}_r)\cdot \vec{r} \right]=\phi_r\\ \end{align*}\]

for any pair of \(\vec{r}_1\) and \(\vec{r}_2\) terminating on the interface

\[(\vec{k}_i-\vec{k}_r)\cdot(\vec{r}_1-\vec{r}_2)=0\]

And we also have \(\hat{u}_n\cdot(\vec{r}_1-\vec{r}_2)=0\), so \((\vec{k}_i-\vec{k}_r)\) is parallel to \(\hat{u}_n\),

\[\begin{align*} &n_i\left( \hat{k}_i\times \hat{u}_n \right)=n_r\left( \hat{k}_r\times \hat {u}_n \right)\\ &k_i\sin\theta_i=k_r\sin\theta_r\\ \end{align*}\]

Since the incident and reflected waves are in the same medium

\[k_i = k_r \Rightarrow \theta_i = \theta_r\]

Law of Refraction

Similarly, \((\vec{k}_i-\vec{k}_t)\) is also parallel to \(\hat{u}_n\),

\[\begin{align*} &\vec{k}_i\times \hat{u}_n=\vec{k}_t\times \hat{u}_n\\ &\because\vec{k}=nk_0\hat{k}\\ &\boxed{n_i\left( \hat{k}_i\times \hat{u}_n \right)=n_t\left( \hat{k}_t\times \hat{u}_n \right)} \end{align*}\]

Note that the law of reflection and the law of refraction only rely on the phase relationship (相位关系) that exists among the phases of \(\vec{E}_i\), \(\vec{E}_r\) and \(\vec{E}_t\) at the boundary.