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# Refractive index

The parameter used to describe the interaction of electromagnetic radiation with matter is the complex index of refraction, $\tilde{n}$, which is a combination of a real part and an imaginary part.

$\tilde{n}=n-ik$

Here, n can also be called 'index of refraction' which sometime leads to confusion, and k is called the 'extinction coefficient'. In a dielectric material such as glass, none of the light is absorbed and therefore k = 0.

The refractive index of a material is the factor by which electromagnetic radiation 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 material's dielectric constant ε (sometimes expressed as the relative permittivity εr multiplied by the permittivity of free space, ε0). For a general material it is given by:

$n=\sqrt{\varepsilon\mu}$
where μ is the permeability 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, the refractive index is given by

$n =\frac{c}{v}$

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:

$n_g=\frac{c}{v_g}$,
where vg is the group velocity. This value should not be confused with n, 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 ε and μ are simultaneously negative. Not thought to occur naturally this can be achieved with so called metamaterials and offers the possibility of perfect lenses and other exotic phenomena such as a reversal of Snell's law.

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## Dispersion and Absorption

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, 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.

In general, the refractive index is defined as a complex number with both a real and imaginary part, where the latter indicates the strength of absorption loss at a particular wavelength—thus, the imaginary part is sometimes called the extinction coefficient k. Such losses become particularly significant, for example, in metals at short (e.g. visible) wavelengths, and must be included in any description of the refractive index. The real and imaginary parts of the complex refractive index are related through use of the Kramers-Kronig relations. For example, one can determine a material's full complex refractive index as a function of wavelength from an absorption spectrum of the material.

## 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 Faraday 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.

## Inhomogeneity

If the refractive index of a medium is not constant, but varies gradually with position, the material is known as a gradient-index medium and is described by gradient index optics. Light travelling through such a medium can be bent or focussed, and this effect can be exploited to produce lenses, some optical fibers and other devices. Some common mirages are caused by a spatially-varying refactive index of air.

## Practical applications

The refractive index of a solution of sugar can be used to determine the sugar content. See Brix