Refractive index

The refractive index of a particular material at a particular frequency is the factor by which electromagnetic radiation of that frequency is slowed down (relative to vacuum) when it travels inside the material. For a non-magnetic material, the square of the refractive index is the dielectric constant ε (sometimes multiplied by ε₀, the permittivity of free space).

The speed of all electromagnetic radiation in vacuum is the same, approximately 3*10⁸ meters per second, and is denoted by c. So if v is the phase velocity of radiation of a specific frequency in a specific material, then the refractive index is given by

n = c/v.

This number is typically bigger than one: the denser the material, the more the light is slowed down. However, at certain frequencies (e.g. near absorption resonances, and for x-rays), n will actually be smaller than one. This does not contradict the theory of relativity, which holds that no information carrying signal can ever propagate faster than c, because the phase velocity is not the same as the group velocity or the signal velocity.

The phase velocity is defined as the rate at which the crests of the waveform propagate; that is, the rate at which the phase of the waveform is moving. The group velocity is the rate that the envelope of the waveform is propagating; that is, the rate of variation of the amplitude of the waveform. It is the group velocity that (almost always) represents the rate that information (and energy) may be transmitted by the wave, for example the velocity at which a pulse of light travels down an optical fibre.

Sometimes, a "group velocity refractive index", usually called the group index is defined:

n_g = v_g / c ,

where v_g is the group velocity. This value should not be confused with n, which is always defined with respect to the phase velocity.

For frequencies corresponding to regions of anomalous dispersion in a material, it is possible for the group velocity v_g to exceed the speed of light in a vacuum, c . While this would seem to indicate that superluminal communication is possible, in reality this is not so; in this case the information (or energy) of the light beam propagates at a rate known as the signal velocity:

v_s = c² / v_g,

which is always less than c if v_g > c.

On an atomic level, the slowing of light as it passes through a material may be considered as a continuous process of absorption and emission of photons as they interact with the atoms of the material. Between each atom, the photons travel at c, as in a vacuum. As they impinge on the atoms, they are absorbed and near-instantly re-emitted, creating a slight delay at each atom which (on a large enough scale) seems to be an overall reduction in the speed of the photons.

The absorption and emission process can be though of as the electric field of a photon creating an oscillating force on the charges of each atom (primarily the electrons). This oscillation of charges itself causes the radiation of an electromagnetic field, which is slightly out-of-phase compared to that of the original photon, leading to a slight retardation of the field and an apparent delay in the photon's travel.

Sometimes the refractive index is defined as a complex number, with the imaginary part of the number representing the absorption of the material. This is particularly useful when analysing the propagation of electromagnetic waves through metals.

If the refractive indices of two materials are known for a given frequency, then one can compute the angle by which radiation of that frequency will be refracted as it moves from the first into the second material from Snell's law.

The refractive index of a material varies with frequency (except in vacuum, where all frequencies travel at the same speed, c). This effect, known as dispersion, lets a prism divide white light into its constituent spectral colors, explains rainbows, and is the cause of chromatic aberration in lenses.

Some representative refractive indices at different wavelengths:

Material n at 589.3 nm
(yellow sodium light) n (x-ray)

helium 1.000036

air 1.0002926

carbon dioxide 1.00045
water ice 1.31

liquid water (20°C) 1.333
glycerine 1.4729

rock salt 1.516

bromine 1.661

glass (typical) 1.5 to 1.9

diamond 2.419

Material	n at 589.3 nm (yellow sodium light)	n (x-ray)
helium	1.000036
air	1.0002926
carbon dioxide	1.00045
water ice	1.31
liquid water (20°C)	1.333
glycerine	1.4729
rock salt	1.516
bromine	1.661
glass (typical)	1.5 to 1.9
diamond	2.419

When light enters a diamond, the high refractive index causes it to suffer multiple total internal reflections, which is the reason for the brilliance of these gemstones.

The refractive index of certain media may be different depending on the polarization and direction of propagation of the light through the medium. This is known as birefringence or anisotropy and is described by the field of crystal optics. In the most general case, the dielectric constant is a rank-2 tensor (a 3 by 3 matrix), which cannot simply be described by refractive indices except for polarizations along principle axes. In magneto-optic (gyro-magnetic) materials, the principle axes are complex (corresponding to elliptical polarizations), and the dielectric tensor is complex-Hermitian (for lossless media); such materials break time-reversal symmetry and are used e.g. to construct optical isolators.

The strong electric field of high intensity light (such as output of a laser) may cause a medium's refractive index to vary as the light passes through it, giving rise to nonlinear optics. If the index varies quadratically with the field (linearly with the intensity), it is called the optical Kerr effect and causes phenomena such as self-focusing and self phase modulation. If the index varies linearly with the field (which is only possible in materials that do not possess inversion symmetry), it is known as the Pockels effect.