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Refractive Index

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Definition: a measure of the reduction in the velocity of light in a medium

German: Brechungsindex

Category: general optics

Formula symbol: n

Units: (dimensionless number)

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The refractive index of a transparent optical medium, also called the index of refraction, is the factor by which the phase velocity vph is decreased relative to the velocity of light in vacuum:

refractive index

Here, one assumes linear propagation (i.e. with low optical intensities) of plane waves. Via the phase velocity, the refractive index also determines phenomena such as refraction, reflection and diffraction at optical interfaces.

The wavelength of light in the medium is n times smaller than the vacuum wavelength.

The refractive index can be calculated from the relative permittivity ε and the relative permeability μ of a material:

refractive index

Note that the values of ε and μ at the optical frequency have to be used, which can deviate substantially from those at low frequencies. For usual optical materials, μ is close to unity.

The refractive index of a material depends on the optical frequency or wavelength; this dependency is called chromatic dispersion. Typical refractive index values for glasses and crystals (e.g. laser crystals) in the visible spectral region are in the range from 1.4 to 2.8, and typically the refractive index increases for shorter wavelengths (normal dispersion). This is a consequence of the fact that the visible spectral region, with high transmission of such materials, lies between spectral regions of strong absorbance: the ultraviolet region with photon energies above the bandgap, and the near- or mid-infrared region with vibrational resonances and their overtones.

The refractive index is generally also dependent on the temperature of the material. In many cases, it rises with increasing temperature, but particularly for glasses the opposite is often the case, essentially because the density decreases with temperature.

Further modifications of the refractive index can occur through mechanical stress (photoelastic effect). Of course, changing the chemical composition e.g. by doping a material with some impurities can also affect the refractive index; this is widely used for graded-index lenses and for optical fibers, for example. In case of rare-earth-doped laser crystals, the refractive index change caused by the doping is often quite small due to a low doping concentration.

refractive index of silica

Figure 1: Refractive index (solid lines) and group index (dotted lines) of silica versus wavelength at temperatures of 0 °C (blue), 100 °C (black) and 200 °C (red). The plots are based on data from M. Medhat et al., J. Opt. A: Pure Appl. Opt. 4, 174 (2002).

Compared with glasses, for example, semiconductors exhibit much higher refractive indices in their transparency region. For example, gallium arsenide (GaAs) has a refractive index of ≈ 3.5 at 1 μm. This is caused by the strong absorption at wavelengths below the bandgap wavelength of ≈ 870 nm. Consequences of the high index of refraction are strong Fresnel reflections and a large critical angle for total internal reflection at semiconductor–air interfaces.

The wavelength-dependent refractive index of a transparent optical material can often be described analytically with a Sellmeier formula, which contains several empirically obtained parameters. Extended versions of such equations also describe the temperature dependence; such an equation has been used for Figure 1. A precise knowledge of the wavelength and temperature dependence of the refractive index is important for phase matching of nonlinear frequency conversion in nonlinear crystal materials.

In anisotropic media, the refractive index generally depends on the polarization direction (→ birefringence) and the propagation direction (anisotropy). If a medium has a so-called optical axis, the refractive index for light propagation along this axis does not depend on the polarization direction.

A complex refractive index is sometimes used to quantify not only the phase change per unit length, but also (via its imaginary part) optical gain or propagation losses (e.g. caused by absorption).

There is another type of refractive index, the group index, which quantifies the reduction in the group velocity. Extreme excursions of the refractive index and particularly the group index can occur near sharp resonances, as are observed in certain quantum optics experiments. This can be related to extremely large or small values of the group velocity (slow light).

Even a negative refractive index is possible for certain photonic metamaterials (usually consisting of metal–dielectric composites), which have been demonstrated first in the microwave region, but begin to become a reality also in the optical domain. Negative refractive index values give rise to a range of intriguing phenomena. For example, refraction at the interface between vacuum and such a material works such that the refracted beam is on the same side of the surface normal as the incident beam.

For waveguides, each propagation mode can be assigned an effective refractive index according to its phase velocity.

See also: refraction, velocity of light, Sellmeier formula, Kramers–Kronig relations, group index, effective refractive index, nonlinear index

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Why does the use of a laser crystal with a very broad wavelength tuning range typically lead to a high laser threshold?

(a) because a broad emission bandwith leads to strong ASE

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(c) because the broad bandwidth results from microscopic disorder, which increases scattering losses

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