Reckoning / Light / Refraction
Light · one surface
Refraction and total reflection
Light slows down in glass and water, and a ray that meets the boundary at a slant changes direction. Here a laser is aimed at the middle of the flat face of a half-cylinder. On the round side every ray runs along a radius, so it crosses that face without bending, and the flat face is the only one that counts. Switch the direction to send the light out of the glass instead, then tilt it past the critical angle.
Set up
Measured on the ray
Snell’s law
Angles are measured from the normal, the line square to the surface. The refractive index n of a material is how many times slower light travels in it than in vacuum: n = c/v. Going into a slower material, the ray turns towards the normal. Coming out, it turns away.
The index depends on the colour (wavelength) of the light. Blue light is slowed more than red, and that is all a prism needs to make a spectrum. The table gives the index at three wavelengths the glass trade uses: the blue F line of hydrogen (486.1 nm), the yellow d line of helium (587.6 nm) and the red C line of hydrogen (656.3 nm). The Abbe number V = (nd − 1)/(nF − nC) measures how little a material spreads the colours: the higher it is, the less the colours separate.
| Material | nF | nd | nC | Abbe V | Critical angle | Reflected at 0° | Source |
|---|---|---|---|---|---|---|---|
| Water (20 °C) | 1.3371 | 1.3330 | 1.3312 | 55.8 | 48.6° | 2.0 % | Daimon and Masumura 2007 |
| Fused silica | 1.4631 | 1.4585 | 1.4564 | 67.8 | 43.3° | 3.5 % | Malitson 1965 |
| Crown glass (N-BK7) | 1.5224 | 1.5168 | 1.5143 | 64.2 | 41.2° | 4.2 % | SCHOTT glass data |
| Flint glass (N-F2) | 1.6321 | 1.6201 | 1.6151 | 36.4 | 38.1° | 5.6 % | SCHOTT glass data |
| Diamond | 2.4355 | 2.4175 | 2.4099 | 55.3 | 24.4° | 17.2 % | Peter 1923 |
Computed from each material’s published dispersion formula, the same code the bench runs. Indices are relative to air. Water at 20 °C. Peter’s diamond measurements end at 644 nm, so the bench’s diamond beyond that is a slight extrapolation.
Total internal reflection
Going from glass into air, the refracted ray leans further from the normal than the incoming one. At the critical angle it skims along the surface, and beyond that angle no light gets out at all: the surface becomes a perfect mirror. For crown glass the critical angle is 41.2°, for water 48.6° and for diamond only 24.4°. That small angle traps light inside a cut diamond and helps it sparkle. Optical fibres, the prisms in binoculars and the rain sensor on a windscreen all work by total internal reflection.
How much is reflected
Every surface reflects some light. Fresnel’s equations give the share. It depends on the angle and on the polarisation, the direction in which the light’s electric field vibrates. Head on, crown glass reflects 4.2 % from each surface. Towards grazing incidence it reflects almost everything, which is why a lake mirrors the far shore but lets you see the bottom at your feet.
At Brewster’s angle, tan θB = n2/n1 (56.6° for crown glass), light polarised in the plane of the drawing is not reflected at all. The reflected and refracted rays are then exactly at right angles. Glare off water and roads is therefore mostly polarised horizontally, and polarising sunglasses block that direction. The bench shows the average for unpolarised light.
The law was first written down by Ibn Sahl in Baghdad around 984, in a treatise on burning mirrors and lenses (identified by the historian Roshdi Rashed in 1990). It was found again by Thomas Harriot about 1602 and by Willebrord Snell in 1621, neither of whom published it, and first printed by René Descartes in La Dioptrique (1637). Augustin Fresnel derived the reflection formulas in 1823, from a theory of light as a wave.
These are rays, the geometrical-optics picture: light is treated as lines that bend and split at surfaces. Diffraction and interference, which need the wave picture, are not shown. The bench is a flat cross-section, so a lens here is really a cylinder of glass seen end-on. That is exact for rays in the plane of the drawing. Air is taken as n = 1, and the glass and water indices are given relative to air, so the angles are right. Colours on screen are an approximation of the spectrum.