Refraction and Snell's Law

Light changes direction as it crosses the boundary between two transparent media. The change is summarised by Snell's law and quantified by the refractive index.

Concept

Refractive index (absolute). For a medium,

$n = \frac{c}{v}$

where $c$ is the speed of light in vacuum and $v$ is the speed in the medium. Since $v \le c$, we have $n \ge 1$.

Relative refractive index of medium 2 with respect to medium 1:

$n_{21} = \frac{n_2}{n_1} = \frac{v_1}{v_2}$

Snell's law. If $i$ is the angle of incidence and $r$ the angle of refraction (both measured from the normal),

$n_1 \sin i = n_2 \sin r$

The incident ray, refracted ray, and normal lie in the same plane.

Reversibility. A ray retraces its path if reversed: $n_{12}\cdot n_{21}=1$.

Derivation

A short derivation via Fermat's principle: light travelling from $A$ in medium 1 to $B$ in medium 2 chooses the path that minimises optical path length $n_1 \ell_1 + n_2 \ell_2$. Setting the derivative with respect to the crossing point equal to zero gives

$n_1 \sin i = n_2 \sin r$

(See the Huygens-principle derivation in Chapter 10 for a wave-front argument.)

Apparent depth. An object at real depth $h$ in a medium of index $n$, viewed from above (air), appears at depth

$h' = \frac{h}{n}$

This explains why a swimming pool looks shallower than it is.

Worked Example

A ray strikes a glass slab ($n = 1.5$) from air at $i = 45^\circ$. Find the angle of refraction.

$\sin r = \frac{\sin 45^\circ}{1.5} = \frac{0.7071}{1.5} = 0.4714$

$r = \sin^{-1}(0.4714) \approx 28.1^\circ$

The ray bends towards the normal as expected when entering a denser medium.

Common Confusions

• Snell's law uses sines, not tangents.
• Refractive index of vacuum is exactly 1; air is $\approx 1.0003$ and taken as 1 in most problems.
• The bending direction: denser medium $\Rightarrow$ ray bends towards the normal.
• $n$ depends on wavelength (this is dispersion). $n_{\rm violet} > n_{\rm red}$ for ordinary glass.

Key Takeaways

• $n = c/v$, always $\ge 1$.
• Snell: $n_1 \sin i = n_2 \sin r$.
• Reversibility gives $n_{12} = 1/n_{21}$.
• Apparent depth scales by $1/n$.