Refraction

Refraction is the bending of light as it passes from one medium into another in which it travels at a different speed. The refractive index $n$ measures that speed, $v = c/n$, and a ray crossing the boundary obliquely turns toward the normal on entering a higher-index medium and away from it on leaving. Typical visible-range indices are 1.0003 for air, 1.333 for water, 1.45–1.9 for optical glasses and 2.42 for diamond; semiconductors in the near infrared reach 3.2–4.3. The direction change is given by Snell's law,

$$n_1 \sin\theta_1 = n_2 \sin\theta_2 ,$$

with angles measured from the surface normal.

Why light bends

Along the boundary the incident and transmitted waves must stay in step, since the fields on both sides are matched at every point of the surface. The frequency cannot change, so the wavelength inside the medium shortens to $\lambda_0/n$: 633 nm light has a wavelength of 417 nm inside N-BK7 ($n = 1.5168$). For the wave crests to remain continuous across an inclined boundary with this shorter spacing, the wavefront has to tilt, and the ray, perpendicular to the wavefront in an isotropic medium, changes direction. Fermat's principle of least time gives the same result: the path light takes makes the optical path length $\int n\,ds$ stationary (usually a minimum), so it spends proportionally less distance in the slower medium.

Worked example

A ray in air strikes a water surface ($n$ = 1.333) at 45°. In the water

$$\sin\theta_2 = \frac{\sin 45^\circ}{1.333} = 0.530,$$

so $\theta_2 = 32.0^\circ$. Going the other way, from water into air, the angle opens up, and beyond the critical angle of 48.6° for water there is no refracted ray at all. At every boundary some light is also returned by reflection, about 2% here at normal incidence and more at steep angles.

Apparent depth

Seen from directly above, an object under water appears raised, because the rays leaving the surface bend away from the normal and seem to come from a shallower point. For near-normal viewing the apparent depth is the real depth divided by $n$: the bottom of a pool 1.00 m deep appears to be at 0.75 m. The same effect shifts the focus of a beam passing through a flat window or a cover glass, which is why microscope objectives are designed for a specified cover-slip thickness.

Dispersion

Because $n$ depends on wavelength, the refraction angle does too. For N-BK7, $n$ is 1.5224 at 486.1 nm and 1.5143 at 656.3 nm, a difference captured by the Abbe number of 64.2. A prism exploits this to spread white light into a spectrum; in a lens it produces chromatic aberration, and in fiber the same wavelength dependence underlies chromatic dispersion.

Where it matters

Every lens, prism and fiber coupler is designed with refraction. Gradual index changes bend rays continuously: gradient-index lenses and graded-index fiber use this deliberately, while temperature gradients in air cause mirages, image shimmer over hot surfaces, and thermal lensing in laser crystals. Near the horizon, atmospheric refraction lifts the apparent position of the Sun by about half a degree, roughly its own diameter.

Measurement

Refractive index is measured from refraction itself. The prism minimum-deviation method, using a prism of the sample, reaches about 1 part in $10^5$ with a precision goniometer, and $10^6$ with the best instruments. Abbe and critical-angle refractometers locate the sharp boundary of total internal reflection against a reference prism and are standard in chemistry and food laboratories. Thin films are measured by ellipsometry or from the interference fringes in their reflectance spectra.

Common questions

What causes refraction?

A change in the phase velocity of light between two media. The frequency is fixed by the source, so the wavelength changes, and matching the wavefronts across an oblique boundary forces a change in direction. At normal incidence the speed and wavelength still change, but the ray does not bend.

Does light bend toward or away from the normal?

Toward the normal when it enters a medium of higher index (air into glass or water), away from it when it enters a lower index (glass into air). In the second case, a large enough angle of incidence leaves no transmitted ray, and all the light undergoes total internal reflection.

Why does a straw look bent in a glass of water?

The light from the submerged part of the straw refracts away from the normal as it leaves the water, so it appears to come from a point higher and displaced toward the surface. Each point below the waterline is shifted by a different amount, and the straw appears kinked at the surface.

References: E. Hecht, Optics, 5th ed. (Pearson, 2017), Ch. 4; M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999), Ch. 1 and 3; B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019), Ch. 1.