2024-03-28 Reflection and Absorption of EM waves

Last time

• Reflection and transmission of EM waves

$$\vec{E}={\vec{E}}_0{e}^{i(\vec{k}-\vec{r}-\omega t)}$$

• EM energy density proportional to $E^2$ and $B^2$
• Fresnel’s Equations
• Plane of incidence and Electric Field Polarization
• The behavior of EM waves being reflected and absorbed depends on the polarization of the electric field

Fresnel’s Equations

Incident wave:

$${\vec{E}}_i={\vec{E}}_{io}{e}^{i(\vec{k}\cdot \vec{r}-\omega t)}$$

with $k=\omega \sqrt{\mu ϵ}$ Refracted wave $k^{\prime }=\omega \sqrt{{\mu }^{\prime }{ϵ}^{\prime }}$ Reflected wave $\equiv$ incident wave

Boundary conditions

$$(ϵ({\vec{E}}_{ic}+{\vec{E}}_{rc})-{ϵ}^{\prime }{\vec{E}}_{t_0})\cdot \vec{n}=0$$

3 more

$$(\vec{k}\times {\vec{E}}_{io}+{\vec{k}}^{\prime \prime }\times {\vec{E}}_{r_0}-{\vec{k}}^{\prime }\times {\vec{E}}_{t_0})\cdot \vec{n}=0$$ $$({\vec{E}}_{i_0}+{\vec{E}}_{r_0}-{\vec{E}}_{to})\times \vec{n}=0$$

…

Fresnel’s Equation for $\perp$ electric field yields one case, while the $\parallel$ case yields another Refer to slides/textbook for the 4 different equations (Perpendicular transmission/reflection)/(parallel transmission/reflection)

Reflection and transmission coefficients are calculated from the boundary conditions and ratio of magnitude of field

For normal incidence $t=\frac{2n}{n^{\prime }+n}$ and $r=\frac{n^{\prime }-n}{n^{\prime }+n}$

intensity of transmission is given by

$$T=t^2={\frac{E_{t_0}}{E_{i_0}}}^2$$

and reflection

$$R=r^2=(E_{r_0/}\dots$$

Inventor of Fresnel lenses!! cool stuff. stepped lens design that allows lighthouses to be more visible at sea

Electromagnetic waves traveling in conducting media

• When the wave travels through a conducting media it loses energy
• Electric and Magnetic field are out of phase
• Look at maxwell’s equations in conducting media. All laws are the same exp. amperes law

$$\vec{\nabla }\times \vec{B}=\mu \sigma \vec{E}+\mu ϵ\frac{\partial \vec{E}}{\partial t}$$

Where $\mu$ is the conductivity in the current density term

• The wave equation can be derived from the amperes law.
• Solutions to this are the following

$$\vec{E}(z,t)=E_0{e}^{-kz}\cos (kz-\omega t+{\delta }_E)\vec{x}$$ $$\vec{B}(z,t)=B_0{e}^{-kz}\cos (kz-\omega t+{\delta }_e+\varphi )\hat{y}$$

for a wave moving in the z direction

In vacuum and nonconducting media the maximum amplitude of the $B$ and $E$ field are aligned, but in conducting media they become out of phase.

The skin depth is given by $d$, where

$$d=\frac{1}{k}$$

The skin depth is the thickness of the layer where the parallel component of the electric field penetrates the field.

The differnece between the elctcif and magnetic field is given by

$$\varphi =\tan \dots$$

fill from ppt

TLDR: A wave travelling in a conducting media is attenuated, dependent on wavelength. The magnetic and electric field also get out of phase.

Frequency dependent dielectric function

huhhhhh not a constant any more uhhhhh

TLDR: microscopically the eletric fields of the waves interacts wiht bound electrons and makes them oscillates time-dependnet dipole moments

the model of the damped harmonic externally driven osciallatior can be used to find the macroscopic polarization

9.19/9.21