Total Internal Reflection & Dispersion
- Compute the critical angle for a boundary between two media
- State the two conditions required for total internal reflection
- Explain dispersion in terms of a wavelength-dependent index of refraction
Two conditions, both necessary
Total internal reflection occurs only when light travels from a more optically dense medium to a less dense one — from higher n to lower n — and the angle of incidence exceeds the critical angle. Both are required. Light going from air into glass can never totally internally reflect however steep the angle, because the first condition fails.
Where the critical angle comes from
Snell's law is n₁ sin θ₁ = n₂ sin θ₂. Going into a less dense medium bends the ray away from the normal, so θ₂ exceeds θ₁ and grows faster. At some incidence angle θ₂ reaches 90° — the refracted ray grazes along the boundary. That incidence angle is the critical angle, and beyond it there is no solution for θ₂ at all, so no light escapes: it is all reflected.
Fiber optics and prisms
An optical fiber works by keeping every ray above the critical angle at the core-cladding boundary, so light reflects along the fiber with almost no loss — far better than any mirror, since total internal reflection is genuinely total. A 45-45-90 prism exploits the same effect: glass-to-air has a critical angle near 42°, so a ray meeting the hypotenuse at 45° is entirely reflected. That is why binoculars and periscopes use prisms rather than mirrored surfaces.
Dispersion: n depends on wavelength
The index of refraction is slightly larger for shorter wavelengths, so violet light bends more than red. A prism therefore spreads white light into a spectrum with violet deviated most and red least. The same effect produces a rainbow, where sunlight refracts entering a droplet, reflects internally at the back, and refracts again on the way out — the two refractions separating the colors. And it is a real limitation on lenses: chromatic aberration is dispersion showing up as color fringing, corrected by combining glasses with different dispersions.
Light travels inside glass of index 1.52 toward a boundary with water of index 1.33. Find the critical angle, and state what happens at incidence angles of 55° and 70°.
- 1.n₁ = 1.52 (glass) is greater than n₂ = 1.33 (water), so total internal reflection is possible.
- 2.sin θ_c = n₂/n₁ = 1.33/1.52 = 0.875.
- 3.θ_c = arcsin(0.875) ≈ 61°.
- 4.At 55°, which is below the critical angle, the light refracts into the water (with partial reflection).
- 5.At 70°, above the critical angle, all the light is reflected back into the glass.
The formula is sin θ_c = n₂/n₁ with n₁ the medium the light is in. If your calculation gives a sine greater than 1, you have the media the wrong way round — and that impossibility is itself the correct physical answer that total internal reflection cannot occur in that direction.
Total internal reflection can occur when light travels from:
A prism separates white light into colors because the index of refraction:
Increasing the index contrast between two media makes the critical angle:
Before computing a critical angle, check that n₁ > n₂. If it is not, the correct answer is that total internal reflection cannot occur in that direction — which is worth stating explicitly rather than producing an impossible arcsine.
Answer the 3 checkpoints as you read.
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