Snell's Law and Total Internal Reflection

1. Why light bends at an interface

Light travels at different speeds in different media. In vacuum it moves at c = 299,792,458 m/s; in water it slows to about 0.75c, in glass to roughly 0.66c. We describe this slowdown with the refractive index n = c / v — a dimensionless number that is 1.0003 for air, 1.33 for water, and 1.5–1.9 for common glasses.

When a ray crosses the boundary between two media at an angle, one side of the wavefront enters the slower medium before the other side does. That side is "held back", which rotates the direction of travel — exactly the way a marching band bends when one file steps onto muddy ground while the rest is still on pavement. The ray bends towards the normal when entering a denser medium, and away from the normal when leaving it.

2. Deriving Snell's law from Fermat's principle

Fermat's principle states that light travels between two points along the path that takes the least time (more precisely, a stationary time — a local extremum). Consider a ray leaving point A in medium 1 (index n₁), crossing a flat interface, and arriving at point B in medium 2 (index n₂). Let the crossing point be a horizontal distance x from directly below A.

Minimising t(x) means setting dt/dx = 0. Differentiating and recognising that x/√(a²+x²) = sin θ₁ and (d−x)/√(b²+(d−x)²) = sin θ₂ (the sines of the angles measured from the normal), the equation collapses to:

This is remarkable: the messy geometric optimisation reduces to a one-line relationship between the sines of the incidence and refraction angles. It says nothing about why light "chooses" the fastest path — that deeper explanation comes from the wave picture (Huygens' principle) or from quantum electrodynamics, where every path is summed and only paths near the stationary-time path survive destructive interference.

3. The critical angle and total internal reflection

Rearrange Snell's law to solve for the refraction angle when going from a denser medium (n₁) into a rarer one (n₂ < n₁):

Because n₁/n₂ > 1, sin θ₂ grows faster than sin θ₁. At some incidence angle θ_c, sin θ₂ reaches exactly 1 — meaning θ₂ = 90°, and the refracted ray skims along the interface itself. Beyond that angle there is no mathematical solution for θ₂: no light can refract out at all. Every photon is reflected back into the denser medium. This is total internal reflection (TIR).

Unlike ordinary reflection off a mirror, TIR is 100% efficient — no absorption, no partial transmission, provided the surfaces are clean and the second medium's index really is lower. Some example critical angles for light travelling from a denser medium into air (n₂ ≈ 1.0):

Notice diamond's exceptionally small critical angle: light entering a cut diamond struggles to find an exit angle below 24.4° and instead bounces internally many times before escaping — one of the reasons diamonds appear so brilliant when cut with the correct facet angles.

4. Applications: fibers, diamonds, mirages

Optical fibers

A glass fiber has a core with a slightly higher refractive index than its surrounding cladding (typically n_core ≈ 1.46, n_clad ≈ 1.44). Any light ray that enters within the fiber's acceptance cone hits the core-cladding boundary at an angle steeper than the critical angle and undergoes TIR over and over, bouncing down the fiber for kilometres with almost no loss. This is the same physics explored in more depth in our article on fiber optics, including the related concept of numerical aperture.

Gemstone brilliance

Diamond cutters exploit the small critical angle of diamond (24.4°) by choosing facet angles so that light entering through the top always strikes the bottom facets steeper than θ_c and bounces back out towards the viewer, rather than leaking out of the bottom of the stone.

Mirages

Hot air near asphalt is less dense and has a slightly lower refractive index than the cooler air above it. Light from the sky travelling at a very shallow (near-grazing) angle towards the ground can exceed the local critical angle between these thin, continuously-varying air layers and curve back upward — the brain interprets this bent ray as a "reflection" from a non-existent puddle. This is continuous refraction rather than a single sharp interface, but the same critical-angle condition governs when the ray turns back.

5. Beyond the critical angle: the evanescent wave

Total internal reflection isn't perfectly "nothing happens" on the far side of the interface. Maxwell's equations require the field to be continuous across the boundary, so a thin non-propagating evanescent wave exists just beyond the surface, decaying exponentially with distance:

If a second medium of similar index is brought within roughly one wavelength of the interface, this evanescent field can couple energy across the gap and light "tunnels" through — a direct optical analogue of quantum tunnelling, known as frustrated total internal reflection. It's used in beam-splitter cubes and fingerprint scanners.

6. Ray tracing Snell's law in code

A robust refraction routine (used in ray tracers, and in the simulations linked below) needs to detect TIR and fall back to pure reflection when it occurs:

function refract(incident, normal, n1, n2):
  cosI = -dot(normal, incident)
  eta  = n1 / n2
  k = 1 - eta*eta * (1 - cosI*cosI)

  if k < 0:
    // Total internal reflection: no refracted ray exists
    return reflect(incident, normal)

  cosT = sqrt(k)
  return eta * incident + (eta * cosI - cosT) * normal

The key branch is k < 0: it corresponds exactly to sin θ₂ > 1 in the algebra above — the point at which Snell's law runs out of real solutions and reflection becomes the only option.