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Light passing through a composite block

Consider light passing through the composite block composed of two parallel-sided slabs of transparent material of refractive indices n₁ and n₂ as shown in the diagram.

Considering the refractions at the three boundaries, we have:

n₁ = sini/sini₁
₁n₂ = sini₁/sini₂
₂n = sini₂/sini = 1/n₂

and this gives us the important relationship that:

n₁sin i₁ = n₂ sin i₂

This can be extended to cover the passage of light through a multiple block. Notice also that if the block is parallel- sided (as shown in the diagram) then the light emerges travelling in the same direction as its original path but displaced sideways (see below). You should expect this result from the principle of reversibility of light.

Example problem
A film of oil of refractive index 1.40 floats on water.
Calculate the angle of refraction in the water for a beam of light that strikes the oil surface at 35^o (The refractive index of water is 1.33.)

We have: n_asin i₁ = n_w sin q = n_w sin i₂
and so: sin 35 = 1.4 sin q = 1.33 sin i₂ sini₂ = sin35/1.33 = 0.43
Therefore i₂ = 25.5^o

Notice that the oil layer has no effect on the direction of the light in the water.

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