Atmospheric refraction: The laws of refraction
 

When a ray of light passes from a transparent substance of one density into another transparent substance of a different density, the ray changes direction. It is said to be refracted and the amount of the deviation from its original direction depends upon the relative densities of the substances.
Let a ray of light, AB, passing through a vacuum, meet the upper boundary PQ of a plane parallel slab of glass at an angle i to the normal BN to the slab (figure 1). The angle NBA is called the angle of incidence. The ray is refracted upon entering the slab so that it leaves the lower boundary RS of the slab at C. Angle MBC, or r , is called the angle of refraction and is less than i . The ray’s path on leaving at the point C is along the line CD, at an angle ZCD to the normal ZC.
Then the first law of refraction states that the incident ray AB, the normal BN and the refracted
ray BC are coplanar.

Figure1. Refraction of a light beam by a slab.

Figure 2. Refraction by a series of slabs.
The second law (Snell’s law) states that
sin i/sin r = n (1)
where n is called the index of refraction of the substance making up the slab. Since r < i, n > 1 .
If YCZ is the normal at C, ∠YCB = ∠MBC = r (PQ being parallel to RS) and hence, since ray paths are reversible, ∠ZCD = i . In other words, the emergent ray CD will be parallel to AB but not collinear with it.
For a number of plane, parallel slabs of indices of refraction n₁, n₂, n₃, . . . , n _j , we can extend relation (1). In figure 2, the ray of light passes from slab 1, of refractive index n₁, into slab 2, of refractive index n₂. Then by Snell’s law,
sin i = n₁ sin r₁.
Also, for the second slab,
sin i = n₂ sin r₂.
Hence, for j slabs, we have
sin i = n₁ sin r₁ = n₂ sin r₂ = n₃ sin r₃ = · · · = n _j sin r _j . (2)

Figure 3. Refraction by the Earth’s atmosphere of light from a star.