### Scattering and absorption

Up until now we have considered only light propagating in free space. Obviously, the subject of optics also concerns what happens inside matter. To understand this we need to examine what happens when a light wave, or a photon is incident on an atom. All the processes of transmission, reflection and refraction, are macroscopic manifestations of scattering effects on the atomic and subatomic levels.

When a photon (or light wave) encounters an atom, there are two possibilities: the atom may scatter
the light, redirecting it without changing its frequency or energy, or it may absorb the light, using the
energy to make a quantum jump into an excited energy state (more precisely, one of its electrons makes
the jump). With absorption it is likely that the excitation energy will rapidly be transferred to atomic
motion, via collisions, thus producing thermal energy before the atom decays back to the lower energy state
re-emitting a photon. Scattering usually occurs only when the frequency of the photon is too
small to cause a transition to any higher state. However, the electric and magnetic fields of the light drives
the electron cloud of the atom into oscillation, causing it to re-radiate in all directions at that same
frequency. We can imagine, on a simplistic level, the atom acting as point source of electromagnetic
spherical waves. This process is elastic--no energy is lost to the atom. Importantly, the amount of
scattering increases for frequencies closer to the resonant frequency for the particular atom. These
resonant frequencies are those corresponding (via *E* = *hν*) to the exact, quantized differences
between energy levels in an atom. Of course, a particular atom has many resonance frequencies,
corresponding to jumps between different energy levels, and each one has a different likelihood of occurring.
The gas atoms in air have resonant frequencies in the UV (Ultra-violet) range; thus air tends to scatter
more blue light than red light. For sunlight coming laterally through the atmosphere, more blue light than
red light is scattered towards the ground, so the sky looks blue! When the sun is low on the horizon,
light passes through a greater thickness of air; the blue is scattered out and we see the left over red light
coming directly along the line of sight to the sun, causing red sunsets.

However, this random scattering only occurs for media such as gases where the atoms are far apart and randomly placed at distances far greater than the wavelength of the light. In dense, homogeneous media, where the spacing of atoms is much less than the wavelength of the light, it turns out that very little light gets scattered in the backwards direction or any direction perpendicular to the propagation of the wave, but most gets propagated in the forward direction. This allows light to propagate through dielectrics. This is somewhat surprising. Why shouldn't the light be scattered equally in all directions? Consider , showing a regular array of very many closely spaced atoms.

The incoming light wave stimulates two atoms A and B. For every atom A, there will be an atom B that is hit by the wave at exactly the same instant, and that is a distance*λ*/2 away. As shown, the spherical waves of the two atoms will cancel out due to destructive interference in the perpendicular direction. In the forward direction, for any point

*P*, there will be a large number of atoms (if

*λ*is much greater than the atomic spacing) for which the distance to

*P*is roughly the same. Thus all the spherical waves from those atoms will arrive at

*P*more or less in-phase, causing constructive interference in the forward direction. For very dense media, light propagates virtually undiminished in the forward direction.

### Fermat's Principle

Before we embark upon an analysis of reflection and refraction from the point of view of scattering light waves, it is worth exploring an alternative explanation for the propagation of light. Fermat's principle is a variational principle which states that:

The path taken by light going between any two points is the one that is traversed in the least time.

Indeed, by considering all possible paths for a light ray and choosing one which takes the least time, it is possible to determine how a light ray will move. Consider a situation where a particle moves from one medium to another.

If the point at which the light crosses the boundary is a distance*x*from the origin, and the speeds in the media are

*v*

_{A}and

*v*

_{B}respectively, then the time taken by the light is:

t = + |

Minimizing time with respect to

*x*:

= + = 0 |

Rearranging this we find:

= |

which is the law of refraction. In general, paths of minimum time are those paths which deviate little from their original value when the path is varied slightly (hence the word 'variational').

There is a good reason why light behaves in this way. After all, you may well ask, how does light know
beforehand which path will take the least time? The answer lies in the variational nature of the principle; as
stated, the path taken is the one for which adjacent paths correspond to very nearly the same time. Consider
two points near the turning point of a smooth graph. Since the gradient is close to zero here a small *x*
difference between the two points will only correspond to a small different *y*. However, for points
located where the gradient is large in magnitude, small differences in *x* can correspond to large differences
in *y*. Imagine for a moment that light takes * all possible * paths between two points. For points
not close to the minimum path, adjacent paths will differ greatly in the amount of time required, thus the
light along these paths will arrive at different times, and hence out-of-phase, interfering destructively.
Light traversing the shortest-time path, however, will have adjacent paths that take nearly the same time, so
light along these paths will arrive with in-phase, causing constructive interference. Thus light from all
but this extremal path cancels out.