# Geometric Optics

### Reflection

#### External Reflection

The first macroscopic manifestation of scattering we must examine is reflection. In the case of a homogenous, isotropic, dense medium we know that a light wave would continue to propagate in the forward direction. However, the argument we presented for this breaks down when there is a discontinuity. In fact, at any interface between media such a discontinuity will cause some part of the wave to propagate in the backwards direction -- this is called reflection. Referring to , which is a modification of , we can see that this must occur because atoms at the very surface cannot 'pair off' with atoms λ/2 away in order to cancel out radiation in the backwards direction, as all atoms deep in the medium can. When light is reflected while moving from a less to a more optically dense medium (air to water, for example) it is called 'external reflection.' Importantly, reflection occurs without color-bias; all wavelengths are reflected equally from a dielectric surface.

Figure %: Unpaired scatterers cause reflection.

#### The Law of Reflection

Consider the diagram . The direction of the reflected wave is determined by the phase difference between the scatterers on the surface. This, in turn, is caused by the angle made by the incident wave and the surface (the angle of incidence, θ i ). If AB is an incoming wavefront and CD is an outgoing wavefront such that the spherical wave emitted from A will be in- phase with the wave just emitted from D (this is true is AB = CD ).

Figure %: Wavefronts reflecting from a surface.
This is the condition for all the surface waves to be in-phase. From the triangles ABD and ACD , which have a common hypotenuse, we can conclude = , where θ r is the angle of the reflected wave. But clearly, BD = AC , so:

 sinθ i = sinθ râá’θ i = θ r

This is called the "law of reflection." To state the law in its full generality, it is also necessary to say that the incident ray and the reflected ray, as well as the ray perpendicular to the surface, all lie in the same plane. Recall also, that the angles of incidence and reflection correspond to the angles between the rays and the normal to the surface (also, to the angles between the wavefronts and the surface, as in the diagram).

Reflection from a smooth surface (such as a mirror) is called specular reflection (any irregularities in the surface are small compared to λ ). When the surface is rough in comparison to λ , diffuse reflection results.

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