W9_parallel_resonance.eps

Week 11: Light 375

θ θi

incident reflected

cancelled

l phase shift ofπ

Figure 143: When light is incident on a perfectly reflecting surface, itcreates little antennas/sources
that radiate theoppositefield in the direction of the incident field. These antennas cause the light
to be reflected at the same angle and with the opposite phase from the surface.

currents that oppose and cancel magnetic fields. In the dynamical case this is still true for

good conductors and optical frequencies. An incominglightwave strikes the conductor,

and its electric fieldpolarizesthe surface atoms so that they become little antennae

that oscillate along with the electric and magnetic field of the light. However, the fields

producedflip over(the way a dipole field does) and hence propagate in the leading

direction with theopposite phase, cancelling the forward directed field quite rapidly at

the surface (often within a few layers of atoms).

Since the conductor is good, very little energy is lost to eddy current heating during this

cancellation. The oscillating surface currents must reradiate theirenergy, and the only

direction they can do so that conserves energy and momentum is toreflectthe incident

energy. However, the reflected wave (in order to achieve the cancellation at the surface)

must have theopposite phasefrom the incoming wave. The situation is very much like

the reflection of a wave pulse on a string from a fixed point on the wall– the reflected

wave flips so it is upside down for precisely the same reasons (energyand momentum

conservation).

In an elastic collision with the conductor, the component of the momentum of the light

alongthe surface is unchanged, but the perpedicular component inverts (becomes minus

itself). The only way this can be true is for the light to bounce off of the surface, with

its phase inverted, at an angle of reflectionθr(measured relative to the normal at the

surface at that point) equal to the angle of incidenceθias drawn above.

So that’s it:

θi=θℓ (919)

is the Law of Reflection. The polarization properties of the reflected light will be discussed

later below.

Note well that for this to be strictly true requires that the surface in question be extremely

smooth – “shiny” as it were. Otherwise neighboring rays would be reflected at different

angles because of small differences in the direction of a normal at different point on

a rough surface. Many (even most) surfaces of real materials are indeed rough on a

microscopic scale (compared to the wavelengths of the incoming light) and hence are

diffusely illuminated ty light instead of perfectly reflecting it accordingto this rule.