ABSTRACT
If we refract a light ray with momentum p1 at a surface with normal n, and it comes out as a ray with momentum p2 after refraction, p2 and p1 are related by (see Chapter 12)
p2 = p1 − [(p1 ⋅ n) + √ _________________
2 + (p1 ⋅ n)2] n (13.1)
where n1 and n2 are the refractive indices before and after refraction. If we had a refl ection instead of refraction, the refl ected ray would have momentum
p2 = p1 − 2(p1 ⋅ n)n (13.2)
where n is the refractive index of the material in which the refl ection occurs. In any case, we can write
p2 = p1 + σn (13.3)
where σ is a scalar. We now consider a different situation in which we have the general coordi-
nate axes i1i2i3 and obtain the mathematical relations between the angles α1, α2, and α3, which a ray of light makes with the coordinate axes i1, i2, and i3, as well as the angles β1 and β2 that its projection onto the plane i1i2 makes with the axes i1 and i2, as shown in Figure 13.1.
