ABSTRACT
A ray i traveling in a medium of refractive index n1 incident on a surface A with normal n is refracted thereupon into a medium of refractive index n2. The angle α1 that the ray makes w ith the normal before refraction is related to the angle it makes with the normal after refraction, by Snell’s law of refraction:
n1 sin α1 = n2 sin α2R (12.1)
If surface A was a mirror the ray would be refl ected, and it would continue traveling in the medium of refractive index n1. In this case, expression 12.1 still holds if we make n1 = n2 and replace α2R for refraction with α2X for refl ection, obtaining (Figure 12.1)
sin α1 = sin α2X ⇔ α1 = α2X (12.2)
The incident ray i is traveling in a direction defi ned by unit vector i, the normal to the surface is given by unit vector n, and the refracted ray rR travels in a direction defi ned by unit vector rR. In the case of refl ection, the refl ected ray rX travels in a direction defi ned by unit vector rX. As seen in Chapter 10, in the case of refraction, unit vectors i, n, and rR are all in the same plane. In the case of refl ection too, unit vectors i, n, and rX are all in the same plane. This means that the direction of the refracted or refl ected rays can be obtained by a linear combination of the incident direction i and normal n to the surface as (Equation 10.19)
r = λi + µn (12.3)
where r = rR in the case of refraction and r = rX in the case of refl ection. Coeffi cients λ and µ are also different in both cases.
