5.5 Reflection, Refraction, and Transmission

The derivation of Snell's Law for electromagnetic waves is identical to that for acoustic waves (see Section 4.3.2). However, in the context of optics, it is expressed in terms of the index of refraction

$\begin{displaymath} n = \frac{c}{v} \end{displaymath}$

(5.35)

instead of phase velocity. Hence, it takes the form

$\begin{displaymath} n_1 \sin \theta_1 = n_2 \sin \theta_2 \end{displaymath}$

(5.36)

There is a critical angle of incidence $\theta_c$ at which $\theta_2 = \frac{\pi}{2}$ and above which total internal reflection occurs. The equation is identical to that for the acoustic case. (See problem 1 of Homework Assignment 7).

The derivations of reflection and refraction coefficients require a deeper grounding in electromagnetic theory than some of us currently have, so we will accept the results without proof. We must define two sets of coefficients,

$\begin{eqnarray*} R_\parallel &=& \frac{E_{ro\parallel}} {E_{o\parallel}} \\ R_... ...{E_{o\parallel}} \\ T_\perp &=& \frac{E_{to\perp}} {E_{o\perp}} \end{eqnarray*}$

for the components of the electric field parallel and perpendicular to the plane of incidence, in which $\vec{k}$ , $\vec{k}_r$ , and $\vec{k}_t$ lie.

For a planar interface between two dielectric materials with permittivities $\epsilon_1$ and $\epsilon_2$ and identical permeabilities $\mu_1 = \mu_2$ ,

$\displaystyle R_\parallel$	$\textstyle =$	$\displaystyle \frac{n_2 \cos \theta_1 - n_1 \cos \theta_2} {n_1 \cos \theta_2 + n_2 \cos \theta_1}$
$\displaystyle R_\perp$	$\textstyle =$	$\displaystyle \frac{n_1 \cos \theta_1 - n_2 \cos \theta_2} {n_1 \cos \theta_1 + n_2 \cos \theta_2}$
$\displaystyle T_\parallel$	$\textstyle =$	$\displaystyle \frac{2 n_1 \cos \theta_1} {n_1 \cos \theta_2 + n_2 \cos \theta_1}$
$\displaystyle T_\perp$	$\textstyle =$	$\displaystyle \frac{2 n_1 \cos \theta_1} {n_1 \cos \theta_1 + n_2 \cos \theta_2}$	(5.37)

There is an angle at which $R_\parallel$ is zero, known as Brewster's angle. $R_\parallel = 0$ gives

$\begin{displaymath} \frac{\cos \theta_1}{\cos \theta_2} = \frac{n_1}{n_2} \end{displaymath}$

which combines with Snell's law to eliminate the indices of refraction,

$\begin{displaymath} \cos \theta_1 \sin \theta_1 = \sin \theta_2 \cos \theta_2 \end{displaymath}$

which is equivalent to

$\begin{displaymath} \sin (2 \theta_1) = \sin (2 \theta_2) \end{displaymath}$

(5.38)

From Eq. 5.38, we either have $\theta_1 = \theta_2$ which corresponds to the trivial case in which

$\begin{displaymath} 2 \theta_1 = \pi - 2 \theta_2 \end{displaymath}$

which we can rearrange as

$\begin{displaymath} \theta_1 + \theta_2 = \frac{\pi}{2} \end{displaymath}$

(5.39)

which transforms Snell's law into

$\begin{displaymath} \tan \theta_b = \frac{n_2}{n_1} \end{displaymath}$

(5.40)

It also appears that $R_\perp$ can be zero, but you will show for homework that it cannot. $T_\parallel$ and $T_\perp$ are never zero, because the cosine function is nonzero over the range of reasonable angles of incidence.

Updated Wed Jan 18 09:51:28 2006