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# Refractive Index

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

Alternative term: index of refraction

German: Brechungsindex

Formula symbol: n

Units: (dimensionless number)

Author:

The refractive index n 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: 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; see the article on Fresnel equations.

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 an optical material: 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.

## Wavelength Dependence and Others Dependencies

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 band gap energy, and the near- or mid-infrared region with vibrational resonances and their overtones. Note that the refractive index at one wavelength can be influenced by absorption in any spectral regions, as described by Kramers–Kronig relations.

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. 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).

The refractive index is generally also dependent on the temperature of the material (see Figure 1). 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.

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.

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.

## Anisotropic Media

In anisotropic optical materials, 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.

## Complex Refractive Index

A complex refractive index is sometimes used to quantify not only the phase change per unit length, but also (via its imaginary part) propagation losses (e.g. caused by absorption) or optical gain. For light propagation in z direction, the optical intensity evolves according to where one can see that the absorption coefficient α can be calculated as: (Note that different sign conventions are used in the literature.)

## Group Index and Effective Index

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).

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

## Measuring of Refractive Indices

There are various methods for measuring refractive indices of transparent solid optical materials or liquids. Instruments suitable for such measurements are called refractometers. Some examples:

• The most common type of refractometer, developed by Ernst Abbe in the late 19th century, is based on measuring the critical angle of total internal reflection at an interface between the investigated material and a reference prism with a known higher refractive index. One can use incident light spanning a small range of angles around the direction parallel to that interface, and some of that light will be refracted into the investigated material. Alternatively, one can try out the incidence angle of light coming from the sample material where total internal reflection starts. If the interface is between two solids, one usually requires a liquid with high refractive index for obtaining an optical contact.
• Another possibility is the minimum deviation method. Here, one uses incident light from air which is deflected by a prism made from the test material and measures the minimum possible angular deviation for optimum (symmetric) orientation of the prism. When using a prism with precisely known opening angle, a precise goniometer (angle measuring instruments) and a carefully chosen method for determining the refractive index of the surrounding air, a rather high precision of the order of 10−6 for the refractive index can be achieved.
• There are spectrophotometric methods, where one measures reflectance and transmittance of incident p- and s- polarized light and uses Fresnel equations to calculate the refractive index. Some of them require a relatively large polished area of the sample (assuming that it is solid) for high accuracy measurements. While some methods work at special angles such as Brewster's angle (with vanishing reflectance for p polarization, but not for s polarization), others work at near normal incidence.
• Ellipsometry is another option, suitable for index measurements on thin films. Here, one measures changes of light polarization upon reflection or transmission through an optical interface and calculates refractive indices by comparison with a mathematical model of the configuration.

Some of those methods are goniometric methods, i.e., involving precise angle measurements.

Refractive index measurements are important in a wide range of application areas, e.g. for the identification of gemstones, concentration measurements in liquids and chacterization of optical materials.

From refractive index measurements at different wavelengths (e.g. some standard spectral lines from spectral lamps), one can obtain chromatic dispersion data.

## Negative Refractive Indices

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 such as negative refraction . 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. Such phenomena also occur in certain photonic crystals.

Negative refractive indices also sometimes occur in geometrical optics, because some authors formally assume the sign of the refractive index to be flipped upon reflection on a surface. With that convention, one can apply certain equations for refraction phenomena also to reflecting surfaces.

## Refractive Index Contrast

In many situations, the refractive index contrast between two different transparent media is relevant. Some examples:

• A refractive index contrast in an optical fiber between fiber core and fiber cladding is usually used to guide light along the core.
• Random refractive index variations due to varying chemical compositions e.g. within a block of glass can cause distortions of laser beams or images, because they lead to spatially dependent deflections of light and also to scattering of light.

Various calculations are simulations can be done in mathematically simplified ways for cases with weak refractive index contrast. For example, the modes of fibers can then be calculated as LP modes, based on a scalar description of the light field amplitudes (i.e., ignoring the vector nature). The criterion for a refractive index contrast to be weak depend on the circumstances, but often it means that the refractive index difference must be far smaller than 1.

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### Bibliography

  J. C. Owens, “Optical refractive index of air: dependence on pressure, temperature and composition”, Appl. Opt. 6 (1), 51 (1967), doi:10.1364/AO.6.000051  J. C. Martinez, “A robust photo-interferometric technique to obtain the refractive index and thickness of non-absorbing stand-alone films”, Meas. Sci. Technol. 11, 1138 (2000)  A. Bruner et al., “Temperature-dependent Sellmeier equation for the refractive index of stoichiometric lithium tantalate”, Opt. Lett. 28 (3), 194 (2003), doi:10.1364/OL.28.000194  S. Foteinopoulou, E. N. Economou and C. M. Soukoulis, “Refraction in media with a negative refractive index”, Phys. Rev. Lett. 90, 107402 (2003), doi:10.1103/PhysRevLett.90.107402  E. Cubukcu et al., “Negative refraction by photonic crystals”, Nature 423, 604 (2003), doi:10.1038/423604b  refractiveindex.info, a public refractive index database Share this with your friends and colleagues, e.g. via social media:   These sharing buttons are implemented in a privacy-friendly way!