Refraction and total internal reflection

A spoon in a glass seems bent and the bottom seems raised. The objects have not changed; the light's route has. Why does a ray turn at a transparent boundary, and when does it fail to emerge at all?

Transparency does not mean light travels through matter as it does through empty space. Interaction with a medium changes propagation speed. Define absolute refractive index as n = c/v, where c is vacuum light speed and v is its speed in the medium. Ordinary transparent glass has n > 1 and v < c. This ratio describes light of a chosen frequency: the index can differ slightly for different colours.

If n = 1,5, then c is about one and a half times the speed in the medium: v = c/1,5. It does not mean the light is brighter by that factor. Both speeds use m/s, so n has no unit. c ≈ 3·10⁸ м/с. Greater n means lower speed for this light; at n = 1, its speed is c. The number describes speed; direction depends on how the wave crosses the boundary.

An oblique wavefront does not enter the new medium all at once. One part already travels at the new speed while another still has the old speed, so the front rotates. In a medium of higher n, the ray bends towards the normal; for lower n, away from it. This is a wave explanation of refraction. If the front is parallel to the boundary, all parts change speed together: at normal incidence the ray stays straight even though speed changes.

The quantitative angle relation is n₁ sin α = n₂ sin β. Subscript 1 refers to the incident medium, 2 to the transmitted medium; α and β are measured from the normal. Rays and normal lie in one plane. If n₁ = n₂, then β = α: no turn. If α = 0, its sine is zero and β is also zero. The law predicts an angle, not just 'towards' or 'away from' the normal.

Predict before calculating. With incident medium and α unchanged, increasing n₂ reduces β because sin β = (n₁/n₂) sin α. The ray should move closer to the normal. Entering a lower-n₂ medium instead increases the angle, while a refracted ray exists. This check catches swapped indices or angles and explains why one large n alone, without the other medium, cannot determine the turn.

Increase the incidence angle as light leaves higher n for lower n. The refracted ray bends progressively farther from the normal, reaching the boundary direction at β = 90°. This defines critical angle αкр, with sin αкр = n₂/n₁. The ratio must be less than one. If n₁ ≤ n₂, no such limiting angle for total internal reflection exists.

What happens at α > αкр? The law would require sin β > 1, which no direction can satisfy: there is no propagating refracted ray. In the transparent-media model, total internal reflection occurs. Both conditions are needed: n₁ > n₂ and incidence above the critical angle. Simply going from glass towards air is insufficient: at small angles, some light emerges. A field exists immediately at the boundary, but that belongs to the wave description, not an ordinary outgoing ray.

An optical fibre's core and cladding have a suitable index relationship. Rays hitting their boundary at suitable angles undergo total internal reflection and are guided along the fibre. Geometry and indices both matter: not every ray is automatically trapped. Frequency stays the same at a stationary boundary; the new speed changes wavelength through v = λν. Refraction should therefore not be explained by a change in source frequency.

Light goes from air, n₁ = 1, into glass, n₂ = 1,5. The incidence angle is 30°. Estimate the refraction angle, given sin 30° = 0,5.

First predict: n₂ = 1,5 exceeds n₁ = 1, so the ray in glass must bend towards the normal and β must be less than 30°.

Rearrange n₁ sin α = n₂ sin β: sin β = n₁ sin α/n₂ = 1 · 0,5/1,5 = 1/3.

An angle whose sine is 1/3 is about 19.5°. This is measured from the normal, just like the given 30°.

19.5° is less than 30°: the calculation agrees with the physical prediction for slower propagation in glass.

β ≈ 19,5° from the normal. Entering glass bends the ray towards the normal, reducing the original 30° angle.

n = c/v compares speeds; the ray turns because different parts of an oblique wavefront cross the boundary at different times.

The refraction law relates both media and angles from the normal; at normal incidence, speed can change without a turn.

Total internal reflection requires higher n to lower n and incidence above the critical angle.

Curved lens boundaries bring rays together to form an image.