Index of Refraction

For a passive, causal medium, the phase refractive index is ≥ 1 at all frequencies. Values below 1 occur only in two special cases. First, in the immediate vicinity of an absorption resonance (within what is called the anomalous dispersion region), the phase index can dip below 1 over a very narrow frequency band, but the material is highly absorbing there, so light cannot propagate far. Second, in artificially structured metamaterials designed to have a negative effective permittivity and permeability, the effective refractive index is negative; this gives rise to phenomena like negative refraction and superlensing. For all ordinary transparent materials — air, water, glass, plastics — n is comfortably greater than 1.

The phase velocity is the speed at which a single-frequency wave crest moves: v_phase = ω/k = c/n. The group velocity is the speed at which the envelope of a pulse — the "signal" — moves: v_group = dω/dk. For most transparent materials, v_group is slightly less than v_phase, and both are less than c. The refractive index that appears in the formula n = c/v is the phase index, because it is defined via the phase velocity. But when you send a short pulse through a medium, it is the group velocity that determines how long the pulse takes to arrive. The difference between the two is related to the dispersion of the material: v_group = v_phase − λ (dv_phase/dλ).

The refractive index varies with wavelength — a phenomenon called dispersion. In the visible range, for most transparent materials, n decreases as wavelength increases: blue light (short wavelength) has a higher index than red light (long wavelength). This is why a prism spreads white light into a spectrum, with violet bent more than red. The standard way to characterise dispersion is the Abbe number V_d = (n_d − 1)/(n_F − n_C), where n_d, n_F, and n_C are the indices at the helium d line (587.6 nm), hydrogen F line (486.1 nm), and hydrogen C line (656.3 nm). Low-dispersion glasses have high Abbe numbers (crowns, V_d > 50); high-dispersion glasses have low Abbe numbers (flints, V_d < 50).

When light travels from a higher-index medium to a lower-index one (for example, from glass to air), the refracted ray bends away from the normal. As the angle of incidence increases, the angle of refraction increases, until at a critical angle θ_c = arcsin(n₂/n₁) the refracted ray grazes the boundary at 90°. Beyond this critical angle, no light is transmitted — all of it is reflected back into the first medium. This is total internal reflection, and it is the basis of optical fibres, where light is guided along a high-index core surrounded by a lower-index cladding, and of many prisms and retroreflectors. Total internal reflection only occurs when going from a higher to a lower index (n₁ > n₂).

The Brewster angle is the angle of incidence at which light of a particular polarization (p-polarized, with its electric field in the plane of incidence) is perfectly transmitted with no reflection. It is given by θ_B = arctan(n₂/n₁). At this angle, the reflected and refracted rays are perpendicular to each other, and the reflected light is entirely s-polarized. This is why polarizing sunglasses reduce glare from horizontal surfaces like water and roads: light reflected at the Brewster angle is strongly polarized, and the sunglasses' polarizers block it. The Brewster angle is also used in laser optics to make low-loss windows and in ellipsometry to measure the refractive index of thin films.

The classical method is the prism refractometer: a prism of the material is placed on a spectrometer, and the angle of minimum deviation is measured. From this angle and the prism's apex angle, the refractive index is computed. For liquids, the Abbe refractometer is standard, using the critical angle at the interface between the liquid and a high-index prism. For thin films, ellipsometry measures the change in polarization on reflection and infers both n and the film thickness. Interferometric methods (like the Rayleigh interferometer) compare the optical path length in the sample to that in a reference. In all cases, the measurement is made at a specified wavelength and temperature, because n depends on both. Modern instruments can achieve accuracy of a few parts in 10⁵.