# Refractive index

*See also list of indices of refraction.*

The **refractive index** of a 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* (sometimes multiplied by , the permittivity of free space). For a general material it is given by

where is the permeability of free space.

The speed of all electromagnetic radiation in vacuum is the same, approximately 3×10^{8} meters per second, and is denoted by .
So if is the phase velocity of radiation of a specific frequency in a specific material, the refractive index is given by

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), 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 , 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:

,

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

At the microscale an electromagnetic wave is slowed in a material because the electric field creates a disturbance in the charges of each atom (primarily the electrons) proportional to the permittivity. This oscillation of charges itself causes the radiation of an electromagnetic wave that is slightly out-of-phase with the original. The sum of the two waves creates a wave with the same frequency but shorter wavelength than the original, leading to a slowing in the wave's travel.

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.

Recent research has also demonstrated the existence of negative refractive index which can occur if and are *simultaneously* negative. Not thought to occur naturally this can be achieved with so called meta materials and offers the possibility of perfect lenses and other exotic phenomena such as a reversal of Snell's law.

## Dispersion and Absorption

The refractive index of a material varies with frequency (except in vacuum, where all frequencies travel at the same speed, ). This effect, known as dispersion, is what causes a prism to divide white light into its constituent spectral colors, explains rainbows, and is the cause of chromatic aberration in lenses. In regions of the spectrum where the material does not absorb, the refractive index increases with frequency. Near absorption peaks, the refractive index decreases with frequency.

The Sellmeier equation is an empirical formula that works well in describing dispersion, and Sellmeier coefficients are often quoted instead of the refractive index in tables. For some representative refractive indices at different wavelengths, see list of indices of refraction.

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 in metals. The real and imaginary parts of the complex refractive index are related by the Kramers-Kronig relations.

## Anisotropy

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 principal axes.

In magneto-optic (gyro-magnetic) and optically active materials, the principal 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.

## Nonlinearity

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.