Physics
Refraction of light and Snell's law, with a diagram
Pick the two media and the angle of incidence, or drag the ray on the drawing: see at once where the refracted light goes, how much is reflected, and from which angle on the reflection becomes total.
Why light bends from one medium to another
Light travels more slowly in materials than in a vacuum, and the refractive index says by how much: in water, with n = 1.333, it moves at three quarters of its vacuum speed. When a ray crosses the boundary between two media at a slant, the part of the wavefront that enters first slows down first, and the ray changes direction. Snell's law, n₁ sin θ₁ = n₂ sin θ₂, says by how much, with angles measured from the normal to the surface.
Entering a denser medium, as from air into water, the ray bends towards the normal. Leaving for a less dense one it bends away, and beyond a certain angle, the critical angle, sin θ₂ would have to exceed 1: there is no refracted ray and all the light is reflected. That total internal reflection is what makes optical fibres work and diamonds sparkle, their critical angle against air being only 24°.
Even when light gets through, some of it is always reflected. The Fresnel equations say how much: about 4% on glass hit head-on, more and more towards grazing incidence. At Brewster's angle the reflected light is fully polarised, which is the principle behind polarised sunglasses.
Common mistakes
- Measuring angles from the surface instead of the normal: Snell's law wants the angle between the ray and the perpendicular to the surface.
- Looking for an angle of refraction when sin θ₂ comes out above 1: it is not a calculation error, it is total internal reflection.
- Swapping n₁ and n₂: n₁ is the medium the light comes from. Swapping them changes the result entirely and makes the critical angle appear or disappear.
Frequently asked questions
How is the critical angle calculated?
Only going from a denser medium to a less dense one: sin θc = n₂ / n₁. From glass (1.52) to air the critical angle is about 41°, from water to air about 48.6°.
Why is the index of air 1.000293 and not 1?
Because air does slow light, if only very slightly. Most exercises round it to 1, and the difference in the angle is a few hundredths of a degree.
What is Brewster's angle?
The angle of incidence at which the reflected and refracted rays are perpendicular: tan θB = n₂ / n₁. At that angle the reflected light is completely polarised parallel to the surface.
Does the refractive index depend on the colour of light?
Yes, slightly: that is dispersion, which spreads the colours in a prism. The indices offered here are for yellow light at 589 nm, the handbook reference.
How this calculation works
Snell's law: n₁ sin θ₁ = n₂ sin θ₂. Critical angle: θc = arcsin(n₂ / n₁), only if n₁ > n₂. Brewster's angle: θB = arctan(n₂ / n₁). Speed in the medium: v = c / n. Reflected light, from the Fresnel equations for unpolarised light: R = (Rs + Rp) / 2, with Rs = ((n₁ cos θ₁ − n₂ cos θ₂) / (n₁ cos θ₁ + n₂ cos θ₂))² and Rp = ((n₂ cos θ₁ − n₁ cos θ₂) / (n₂ cos θ₁ + n₁ cos θ₂))². Transmitted light: T = 1 − R.
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