# Refractive index

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In optics, the refractive index or index of refraction of a material is a dimensionless number that describes how fast light propagates through the material. It is defined as

Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultraviolet, and infrared light. Because light is an electromagnetic wave, other forms of electromagnetic radiation such as X-rays, microwaves, and radio waves exhibit similar properties.

An optical medium is material through which electromagnetic waves propagate. It is a form of transmission medium. The permittivity and permeability of the medium define how electromagnetic waves propagate in it. The medium has an intrinsic impedance, given by

## Contents

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

where c is the speed of light in vacuum and v is the phase velocity of light in the medium. For example, the refractive index of water is 1.333, meaning that light travels 1.333 times faster in vacuum than in water.

The speed of light in vacuum, commonly denoted c, is a universal physical constant important in many areas of physics. Its exact value is 299,792,458 metres per second. It is exact because by international agreement a metre is defined as the length of the path travelled by light in vacuum during a time interval of 1/299792458 second. According to special relativity, c is the maximum speed at which all conventional matter and hence all known forms of information in the universe can travel. Though this speed is most commonly associated with light, it is in fact the speed at which all massless particles and changes of the associated fields travel in vacuum. Such particles and waves travel at c regardless of the motion of the source or the inertial reference frame of the observer. In the special and general theories of relativity, c interrelates space and time, and also appears in the famous equation of mass–energy equivalence E = mc2.

Vacuum is space devoid of matter. The word stems from the Latin adjective vacuus for "vacant" or "void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Physicists often discuss ideal test results that would occur in a perfect vacuum, which they sometimes simply call "vacuum" or free space, and use the term partial vacuum to refer to an actual imperfect vacuum as one might have in a laboratory or in space. In engineering and applied physics on the other hand, vacuum refers to any space in which the pressure is lower than atmospheric pressure. The Latin term in vacuo is used to describe an object that is surrounded by a vacuum.

The phase velocity of a wave is the rate at which the phase of the wave propagates in space. This is the velocity at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave will appear to travel at the phase velocity. The phase velocity is given in terms of the wavelength λ (lambda) and time period T as

The refractive index determines how much the path of light is bent, or refracted, when entering a material. This is described by Snell's law of refraction, n1 sinθ1 = n2 sinθ2, where θ1 and θ2 are the angles of incidence and refraction, respectively, of a ray crossing the interface between two media with refractive indices n1 and n2. The refractive indices also determine the amount of light that is reflected when reaching the interface, as well as the critical angle for total internal reflection and Brewster's angle. [1]

In physics refraction is the change in direction of a wave passing from one medium to another or from a gradual change in the medium. Refraction of light is the most commonly observed phenomenon, but other waves such as sound waves and water waves also experience refraction. How much a wave is refracted is determined by the change in wave speed and the initial direction of wave propagation relative to the direction of change in speed.

Snell's law is a formula used to describe the relationship between the angles of incidence and refraction, when referring to light or other waves passing through a boundary between two different isotropic media, such as water, glass, or air.

In geometric optics, the angle of incidence is the angle between a ray incident on a surface and the line perpendicular to the surface at the point of incidence, called the normal. The ray can be formed by any wave: optical, acoustic, microwave, X-ray and so on. In the figure below, the line representing a ray makes an angle θ with the normal. The angle of incidence at which light is first totally internally reflected is known as the critical angle. The angle of reflection and angle of refraction are other angles related to beams.

The refractive index can be seen as the factor by which the speed and the wavelength of the radiation are reduced with respect to their vacuum values: the speed of light in a medium is v = c/n, and similarly the wavelength in that medium is λ = λ0/n, where λ0 is the wavelength of that light in vacuum. This implies that vacuum has a refractive index of 1, and that the frequency (f = v/λ) of the wave is not affected by the refractive index. As a result, the energy (E = hf) of the photon, and therefore the perceived color of the refracted light to a human eye which depends on photon energy, is not affected by the refraction or the refractive index of the medium.

In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is thus the inverse of the spatial frequency. Wavelength is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings and is a characteristic of both traveling waves and standing waves, as well as other spatial wave patterns. Wavelength is commonly designated by the Greek letter lambda (λ). The term wavelength is also sometimes applied to modulated waves, and to the sinusoidal envelopes of modulated waves or waves formed by interference of several sinusoids.

Frequency is the number of occurrences of a repeating event per unit of time. It is also referred to as temporal frequency, which emphasizes the contrast to spatial frequency and angular frequency. The period is the duration of time of one cycle in a repeating event, so the period is the reciprocal of the frequency. For example: if a newborn baby's heart beats at a frequency of 120 times a minute, its period—the time interval between beats—is half a second. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio signals (sound), radio waves, and light.

While the refractive index affects wavelength, it depends on photon frequency, color and energy so the resulting difference in the bending angle causes white light to split into its constituent colors. This is called dispersion. It can be observed in prisms and rainbows, and chromatic aberration in lenses. Light propagation in absorbing materials can be described using a complex-valued refractive index. [2] The imaginary part then handles the attenuation, while the real part accounts for refraction.

In optics, dispersion is the phenomenon in which the phase velocity of a wave depends on its frequency.

A rainbow is a meteorological phenomenon that is caused by reflection, refraction and dispersion of light in water droplets resulting in a spectrum of light appearing in the sky. It takes the form of a multicoloured circular arc. Rainbows caused by sunlight always appear in the section of sky directly opposite the sun.

In optics, chromatic aberration is a failure of a lens to focus all colors to the same point. It is caused by dispersion: the refractive index of the lens elements varies with the wavelength of light. The refractive index of most transparent materials decreases with increasing wavelength. Since the focal length of a lens depends on the refractive index, this variation in refractive index affects focusing. Chromatic aberration manifests itself as "fringes" of color along boundaries that separate dark and bright parts of the image.

The concept of refractive index applies within the full electromagnetic spectrum, from X-rays to radio waves. It can also be applied to wave phenomena such as sound. In this case the speed of sound is used instead of that of light, and a reference medium other than vacuum must be chosen. [3]

The electromagnetic spectrum is the range of frequencies of electromagnetic radiation and their respective wavelengths and photon energies.

X-rays make up X-radiation, a form of electromagnetic radiation. Most X-rays have a wavelength ranging from 0.01 to 10 nanometers, corresponding to frequencies in the range 30 petahertz to 30 exahertz (3×1016 Hz to 3×1019 Hz) and energies in the range 100 eV to 100 keV. X-ray wavelengths are shorter than those of UV rays and typically longer than those of gamma rays. In many languages, X-radiation is referred to with terms meaning Röntgen radiation, after the German scientist Wilhelm Röntgen who discovered these on November 8, 1895, who usually is credited as its discoverer, and who named it X-radiation to signify an unknown type of radiation. Spelling of X-ray(s) in the English language includes the variants x-ray(s), xray(s), and X ray(s).

Radio waves are a type of electromagnetic radiation with wavelengths in the electromagnetic spectrum longer than infrared light. Radio waves have frequencies as high as 300 gigahertz (GHz) to as low as 30 hertz (Hz). At 300 GHz, the corresponding wavelength is 1 mm, and at 30 Hz is 10,000 km. Like all other electromagnetic waves, radio waves travel at the speed of light. They are generated by electric charges undergoing acceleration, such as time varying electric currents. Naturally occurring radio waves are emitted by lightning and astronomical objects.

## Definition

The refractive index n of an optical medium is defined as the ratio of the speed of light in vacuum, c = 299792458 m/s, and the phase velocity v of light in the medium, [1]

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

The phase velocity is the speed at which the crests or the phase of the wave moves, which may be different from the group velocity, the speed at which the pulse of light or the envelope of the wave moves.

The definition above is sometimes referred to as the absolute refractive index or the absolute index of refraction to distinguish it from definitions where the speed of light in other reference media than vacuum is used. [1] Historically air at a standardized pressure and temperature has been common as a reference medium.

## History

Thomas Young was presumably the person who first used, and invented, the name "index of refraction", in 1807. [4] At the same time he changed this value of refractive power into a single number, instead of the traditional ratio of two numbers. The ratio had the disadvantage of different appearances. Newton, who called it the "proportion of the sines of incidence and refraction", wrote it as a ratio of two numbers, like "529 to 396" (or "nearly 4 to 3"; for water). [5] Hauksbee, who called it the "ratio of refraction", wrote it as a ratio with a fixed numerator, like "10000 to 7451.9" (for urine). [6] Hutton wrote it as a ratio with a fixed denominator, like 1.3358 to 1 (water). [7]

Young did not use a symbol for the index of refraction, in 1807. In the next years, others started using different symbols: n, m, and µ. [8] [9] [10] The symbol n gradually prevailed.

## Typical values

Selected refractive indices at λ=589 nm. For references, see the extended List of refractive indices.
Materialn
Vacuum 1
Gases at 0 °C and 1 atm
Air 1.000293
Helium 1.000036
Hydrogen 1.000132
Carbon dioxide 1.00045
Liquids at 20 °C
Water 1.333
Ethanol 1.36
Olive oil 1.47
Solids
Ice 1.31
Fused silica (quartz)1.46 [11]
PMMA (acrylic, plexiglas, lucite, perspex)1.49
Window glass 1.52 [12]
Polycarbonate (Lexan™)1.58 [13]
Flint glass (typical)1.62
Sapphire 1.77 [14]
Cubic zirconia 2.15
Diamond 2.42
Moissanite 2.65

For visible light most transparent media have refractive indices between 1 and 2. A few examples are given in the adjacent table. These values are measured at the yellow doublet D-line of sodium, with a wavelength of 589 nanometers, as is conventionally done. [15] Gases at atmospheric pressure have refractive indices close to 1 because of their low density. Almost all solids and liquids have refractive indices above 1.3, with aerogel as the clear exception. Aerogel is a very low density solid that can be produced with refractive index in the range from 1.002 to 1.265. [16] Moissanite lies at the other end of the range with a refractive index as high as 2.65. Most plastics have refractive indices in the range from 1.3 to 1.7, but some high-refractive-index polymers can have values as high as 1.76. [17]

For infrared light refractive indices can be considerably higher. Germanium is transparent in the wavelength region from 2 to 14 µm and has a refractive index of about 4. [18] A type of new materials, called topological insulator, was recently found holding higher refractive index of up to 6 in near to mid infrared frequency range. Moreover, topological insulator material are transparent when they have nanoscale thickness. These excellent properties make them a type of significant materials for infrared optics. [19]

### Refractive index below unity

According to the theory of relativity, no information can travel faster than the speed of light in vacuum, but this does not mean that the refractive index cannot be lower than 1. The refractive index measures the phase velocity of light, which does not carry information. [20] The phase velocity is the speed at which the crests of the wave move and can be faster than the speed of light in vacuum, and thereby give a refractive index below 1. This can occur close to resonance frequencies, for absorbing media, in plasmas, and for X-rays. In the X-ray regime the refractive indices are lower than but very close to 1 (exceptions close to some resonance frequencies). [21] As an example, water has a refractive index of 0.99999974 = 1 − 2.6×10−7 for X-ray radiation at a photon energy of 30  keV (0.04 nm wavelength). [21]

An example of a plasma with an index of refraction less than unity is Earth's ionosphere. Since the refractive index of the ionosphere (a plasma), is less than unity, electromagnetic waves propagating through the plasma are bent "away from the normal" (see Geometric optics) allowing the radio wave to be refracted back toward earth, thus enabling long-distance radio communications. See also Radio Propagation and Skywave. [22]

### Negative refractive index

Recent research has also demonstrated the existence of materials with a negative refractive index, which can occur if permittivity and permeability have simultaneous negative values. [23] This can be achieved with periodically constructed metamaterials. The resulting negative refraction (i.e., a reversal of Snell's law) offers the possibility of the superlens and other exotic phenomena. [24]

## Microscopic explanation

At the atomic scale, an electromagnetic wave's phase velocity is slowed in a material because the electric field creates a disturbance in the charges of each atom (primarily the electrons) proportional to the electric susceptibility of the medium. (Similarly, the magnetic field creates a disturbance proportional to the magnetic susceptibility.) As the electromagnetic fields oscillate in the wave, the charges in the material will be "shaken" back and forth at the same frequency. [1] :67 The charges thus radiate their own electromagnetic wave that is at the same frequency, but usually with a phase delay, as the charges may move out of phase with the force driving them (see sinusoidally driven harmonic oscillator). The light wave traveling in the medium is the macroscopic superposition (sum) of all such contributions in the material: the original wave plus the waves radiated by all the moving charges. This wave is typically a wave with the same frequency but shorter wavelength than the original, leading to a slowing of the wave's phase velocity. Most of the radiation from oscillating material charges will modify the incoming wave, changing its velocity. However, some net energy will be radiated in other directions or even at other frequencies (see scattering).

Depending on the relative phase of the original driving wave and the waves radiated by the charge motion, there are several possibilities:

• If the electrons emit a light wave which is 90° out of phase with the light wave shaking them, it will cause the total light wave to travel slower. This is the normal refraction of transparent materials like glass or water, and corresponds to a refractive index which is real and greater than 1. [25]
• If the electrons emit a light wave which is 270° out of phase with the light wave shaking them, it will cause the wave to travel faster. This is called "anomalous refraction", and is observed close to absorption lines (typically in infrared spectra), with X-rays in ordinary materials, and with radio waves in Earth's ionosphere. It corresponds to a permittivity less than 1, which causes the refractive index to be also less than unity and the phase velocity of light greater than the speed of light in vacuum c (note that the signal velocity is still less than c, as discussed above). If the response is sufficiently strong and out-of-phase, the result is a negative value of permittivity and imaginary index of refraction, as observed in metals or plasma. [25]
• If the electrons emit a light wave which is 180° out of phase with the light wave shaking them, it will destructively interfere with the original light to reduce the total light intensity. This is light absorption in opaque materials and corresponds to an imaginary refractive index.
• If the electrons emit a light wave which is in phase with the light wave shaking them, it will amplify the light wave. This is rare, but occurs in lasers due to stimulated emission. It corresponds to an imaginary index of refraction, with the opposite sign to that of absorption.

For most materials at visible-light frequencies, the phase is somewhere between 90° and 180°, corresponding to a combination of both refraction and absorption.

## Dispersion

The refractive index of materials varies with the wavelength (and frequency) of light. [26] This is called dispersion and causes prisms and rainbows to divide white light into its constituent spectral colors. [27] As the refractive index varies with wavelength, so will the refraction angle as light goes from one material to another. Dispersion also causes the focal length of lenses to be wavelength dependent. This is a type of chromatic aberration, which often needs to be corrected for in imaging systems. In regions of the spectrum where the material does not absorb light, the refractive index tends to decrease with increasing wavelength, and thus increase with frequency. This is called "normal dispersion", in contrast to "anomalous dispersion", where the refractive index increases with wavelength. [26] For visible light normal dispersion means that the refractive index is higher for blue light than for red.

For optics in the visual range, the amount of dispersion of a lens material is often quantified by the Abbe number: [27]

${\displaystyle V={\frac {n_{\mathrm {yellow} }-1}{n_{\mathrm {blue} }-n_{\mathrm {red} }}}.}$

For a more accurate description of the wavelength dependence of the refractive index, the Sellmeier equation can be used. [28] It is an empirical formula that works well in describing dispersion. Sellmeier coefficients are often quoted instead of the refractive index in tables.

Because of dispersion, it is usually important to specify the vacuum wavelength of light for which a refractive index is measured. Typically, measurements are done at various well-defined spectral emission lines; for example, nD usually denotes the refractive index at the Fraunhofer "D" line, the centre of the yellow sodium double emission at 589.29 nm wavelength. [15]

## Complex refractive index

When light passes through a medium, some part of it will always be attenuated. This can be conveniently taken into account by defining a complex refractive index,

${\displaystyle {\underline {n}}=n+i\kappa .}$

Here, the real part n is the refractive index and indicates the phase velocity, while the imaginary part κ is called the extinction coefficient — although κ can also refer to the mass attenuation coefficient [29] :3 and indicates the amount of attenuation when the electromagnetic wave propagates through the material. [1] :128

That κ corresponds to attenuation can be seen by inserting this refractive index into the expression for electric field of a plane electromagnetic wave traveling in the z-direction. We can do this by relating the complex wave number k to the complex refractive index n through k = 2πn/λ0, with λ0 being the vacuum wavelength; this can be inserted into the plane wave expression as

${\displaystyle \mathbf {E} (z,t)=\operatorname {Re} \!\left[\mathbf {E} _{0}e^{i({\underline {k}}z-\omega t)}\right]=\operatorname {Re} \!\left[\mathbf {E} _{0}e^{i(2\pi (n+i\kappa )z/\lambda _{0}-\omega t)}\right]=e^{-2\pi \kappa z/\lambda _{0}}\operatorname {Re} \!\left[\mathbf {E} _{0}e^{i(kz-\omega t)}\right].}$

Here we see that κ gives an exponential decay, as expected from the Beer–Lambert law. Since intensity is proportional to the square of the electric field, it will depend on the depth into the material as exp(−4πκz/λ0), and the attenuation coefficient becomes α = 4πκ/λ0. [1] :128 This also relates it to the penetration depth, the distance after which the intensity is reduced by 1/e, δp = 1/α = λ0/(4πκ).

Both n and κ are dependent on the frequency. In most circumstances κ > 0 (light is absorbed) or κ = 0 (light travels forever without loss). In special situations, especially in the gain medium of lasers, it is also possible that κ < 0, corresponding to an amplification of the light.

An alternative convention uses n = n instead of n = n + , but where κ > 0 still corresponds to loss. Therefore, these two conventions are inconsistent and should not be confused. The difference is related to defining sinusoidal time dependence as Re[exp(−iωt)] versus Re[exp(+iωt)]. See Mathematical descriptions of opacity.

Dielectric loss and non-zero DC conductivity in materials cause absorption. Good dielectric materials such as glass have extremely low DC conductivity, and at low frequencies the dielectric loss is also negligible, resulting in almost no absorption. However, at higher frequencies (such as visible light), dielectric loss may increase absorption significantly, reducing the material's transparency to these frequencies.

The real, n, and imaginary, κ, parts of the complex refractive index are related through the Kramers–Kronig relations. In 1986 A.R. Forouhi and I. Bloomer deduced an equation describing κ as a function of photon energy, E, applicable to amorphous materials. Forouhi and Bloomer then applied the Kramers–Kronig relation to derive the corresponding equation for n as a function of E. The same formalism was applied to crystalline materials by Forouhi and Bloomer in 1988.

The refractive index and extinction coefficient, n and κ, cannot be measured directly. They must be determined indirectly from measurable quantities that depend on them, such as reflectance, R, or transmittance, T, or ellipsometric parameters, ψ and δ. The determination of n and κ from such measured quantities will involve developing a theoretical expression for R or T, or ψ and δ in terms of a valid physical model for n and κ. By fitting the theoretical model to the measured R or T, or ψ and δ using regression analysis, n and κ can be deduced.

For X-ray and extreme ultraviolet radiation the complex refractive index deviates only slightly from unity and usually has a real part smaller than 1. It is therefore normally written as n = 1 − δ + (or n = 1 − δ with the alternative convention mentioned above). [2] Far above the atomic resonance frequency delta can be given by

${\displaystyle \delta ={\frac {r_{0}\lambda ^{2}n_{e}}{2\pi }}}$

where ${\displaystyle r_{0}}$ is the classical electron radius, ${\displaystyle \lambda }$ is the X-ray wavelength, and ${\displaystyle n_{e}}$ is the electron density. One may assume the electron density is simply the number of electrons per atom Z multiplied by the atomic density, but more accurate calculation of the refractive index requires replacing Z with the complex atomic form factor ${\displaystyle f=Z+f'+if''}$. It follows that

${\displaystyle \delta ={\frac {r_{0}\lambda ^{2}}{2\pi }}(Z+f')n_{Atom}}$
${\displaystyle \beta ={\frac {r_{0}\lambda ^{2}}{2\pi }}f''n_{Atom}}$

with ${\displaystyle \delta }$ and ${\displaystyle \beta }$ typically of the order of 10−5 and 10−6.

## Relations to other quantities

### Optical path length

Optical path length (OPL) is the product of the geometric length d of the path light follows through a system, and the index of refraction of the medium through which it propagates, [30]

${\displaystyle {\text{OPL}}=nd.}$

This is an important concept in optics because it determines the phase of the light and governs interference and diffraction of light as it propagates. According to Fermat's principle, light rays can be characterized as those curves that optimize the optical path length. [1] :68–69

### Refraction

When light moves from one medium to another, it changes direction, i.e. it is refracted. If it moves from a medium with refractive index n1 to one with refractive index n2, with an incidence angle to the surface normal of θ1, the refraction angle θ2 can be calculated from Snell's law: [31]

${\displaystyle n_{1}\sin \theta _{1}=n_{2}\sin \theta _{2}.}$

When light enters a material with higher refractive index, the angle of refraction will be smaller than the angle of incidence and the light will be refracted towards the normal of the surface. The higher the refractive index, the closer to the normal direction the light will travel. When passing into a medium with lower refractive index, the light will instead be refracted away from the normal, towards the surface.

### Total internal reflection

If there is no angle θ2 fulfilling Snell's law, i.e.,

${\displaystyle {\frac {n_{1}}{n_{2}}}\sin \theta _{1}>1,}$

the light cannot be transmitted and will instead undergo total internal reflection. [32] :49–50 This occurs only when going to a less optically dense material, i.e., one with lower refractive index. To get total internal reflection the angles of incidence θ1 must be larger than the critical angle [33]

${\displaystyle \theta _{\mathrm {c} }=\arcsin \!\left({\frac {n_{2}}{n_{1}}}\right)\!.}$

### Reflectivity

Apart from the transmitted light there is also a reflected part. The reflection angle is equal to the incidence angle, and the amount of light that is reflected is determined by the reflectivity of the surface. The reflectivity can be calculated from the refractive index and the incidence angle with the Fresnel equations, which for normal incidence reduces to [32] :44

${\displaystyle R_{0}=\left|{\frac {n_{1}-n_{2}}{n_{1}+n_{2}}}\right|^{2}\!.}$

For common glass in air, n1 = 1 and n2 = 1.5, and thus about 4% of the incident power is reflected. [34] At other incidence angles the reflectivity will also depend on the polarization of the incoming light. At a certain angle called Brewster's angle, p-polarized light (light with the electric field in the plane of incidence) will be totally transmitted. Brewster's angle can be calculated from the two refractive indices of the interface as [1] :245

${\displaystyle \theta _{\mathrm {B} }=\arctan \!\left({\frac {n_{2}}{n_{1}}}\right)\!.}$

### Lenses

The focal length of a lens is determined by its refractive index n and the radii of curvature R1 and R2 of its surfaces. The power of a thin lens in air is given by the Lensmaker's formula: [35]

${\displaystyle {\frac {1}{f}}=(n-1)\!\left({\frac {1}{R_{1}}}-{\frac {1}{R_{2}}}\right)\!,}$

where f is the focal length of the lens.

### Microscope resolution

The resolution of a good optical microscope is mainly determined by the numerical aperture (NA) of its objective lens. The numerical aperture in turn is determined by the refractive index n of the medium filling the space between the sample and the lens and the half collection angle of light θ according to [36] :6

${\displaystyle \mathrm {NA} =n\sin \theta .}$

For this reason oil immersion is commonly used to obtain high resolution in microscopy. In this technique the objective is dipped into a drop of high refractive index immersion oil on the sample under study. [36] :14

### Relative permittivity and permeability

The refractive index of electromagnetic radiation equals

${\displaystyle n={\sqrt {\varepsilon _{\mathrm {r} }\mu _{\mathrm {r} }}},}$

where εr is the material's relative permittivity, and μr is its relative permeability. [37] :229 The refractive index is used for optics in Fresnel equations and Snell's law; while the relative permittivity and permeability are used in Maxwell's equations and electronics. Most naturally occurring materials are non-magnetic at optical frequencies, that is μr is very close to 1,[ citation needed ] therefore n is approximately εr. In this particular case, the complex relative permittivity εr, with real and imaginary parts εr and ɛ̃r, and the complex refractive index n, with real and imaginary parts n and κ (the latter called the "extinction coefficient"), follow the relation

${\displaystyle {\underline {\varepsilon }}_{\mathrm {r} }=\varepsilon _{\mathrm {r} }+i{\tilde {\varepsilon }}_{\mathrm {r} }={\underline {n}}^{2}=(n+i\kappa )^{2},}$

and their components are related by: [38]

${\displaystyle \varepsilon _{\mathrm {r} }=n^{2}-\kappa ^{2},}$
${\displaystyle {\tilde {\varepsilon }}_{\mathrm {r} }=2n\kappa ,}$

and:

${\displaystyle n={\sqrt {\frac {|{\underline {\varepsilon }}_{\mathrm {r} }|+\varepsilon _{\mathrm {r} }}{2}}},}$
${\displaystyle \kappa ={\sqrt {\frac {|{\underline {\varepsilon }}_{\mathrm {r} }|-\varepsilon _{\mathrm {r} }}{2}}}.}$

where ${\displaystyle |{\underline {\varepsilon }}_{\mathrm {r} }|={\sqrt {\varepsilon _{\mathrm {r} }^{2}+{\tilde {\varepsilon }}_{\mathrm {r} }^{2}}}}$ is the complex modulus.

### Wave impedance

The wave impedance of a plane electromagnetic wave in a non-conductive medium is given by

${\displaystyle Z={\sqrt {\frac {\mu }{\varepsilon }}}={\sqrt {\frac {\mu _{\mathrm {0} }\mu _{\mathrm {r} }}{\varepsilon _{\mathrm {0} }\varepsilon _{\mathrm {r} }}}}={\sqrt {\frac {\mu _{\mathrm {0} }}{\varepsilon _{\mathrm {0} }}}}{\sqrt {\frac {\mu _{\mathrm {r} }}{\varepsilon _{\mathrm {r} }}}}=Z_{0}{\sqrt {\frac {\mu _{\mathrm {r} }}{\varepsilon _{\mathrm {r} }}}}=Z_{0}{\frac {\mu _{\mathrm {r} }}{n}}}$

where ${\displaystyle Z_{0}}$ is the vacuum wave impedance, μ and ϵ are the absolute permeability and permittivity of the medium, εr is the material's relative permittivity, and μr is its relative permeability.

In non-magnetic media with ${\displaystyle \mu _{\mathrm {r} }=1}$,

${\displaystyle Z={\frac {Z_{0}}{n}},}$
${\displaystyle n={\frac {Z_{0}}{Z}}.}$

Thus refractive index in a non-magnetic media is the ratio of the vacuum wave impedance to the wave impedance of the medium.

The reflectivity ${\displaystyle R_{0}}$ between two media can thus be expressed both by the wave impedances and the refractive indices as

${\displaystyle R_{0}=\left|{\frac {n_{1}-n_{2}}{n_{1}+n_{2}}}\right|^{2}\!=\left|{\frac {Z_{2}-Z_{1}}{Z_{2}+Z_{1}}}\right|^{2}\!.}$

### Density

In general, the refractive index of a glass increases with its density. However, there does not exist an overall linear relation between the refractive index and the density for all silicate and borosilicate glasses. A relatively high refractive index and low density can be obtained with glasses containing light metal oxides such as Li2O and MgO, while the opposite trend is observed with glasses containing PbO and BaO as seen in the diagram at the right.

Many oils (such as olive oil) and ethyl alcohol are examples of liquids which are more refractive, but less dense, than water, contrary to the general correlation between density and refractive index.

For gases, n − 1 is proportional to the density of the gas as long as the chemical composition does not change. [40] This means that it is also proportional to the pressure and inversely proportional to the temperature for ideal gases.

### Group index

Sometimes, a "group velocity refractive index", usually called the group index is defined:[ citation needed ]

${\displaystyle n_{\mathrm {g} }={\frac {\mathrm {c} }{v_{\mathrm {g} }}},}$

where vg is the group velocity. This value should not be confused with n, which is always defined with respect to the phase velocity. When the dispersion is small, the group velocity can be linked to the phase velocity by the relation [32] :22

${\displaystyle v_{\mathrm {g} }=v-\lambda {\frac {\mathrm {d} v}{\mathrm {d} \lambda }},}$

where λ is the wavelength in the medium. In this case the group index can thus be written in terms of the wavelength dependence of the refractive index as

${\displaystyle n_{\mathrm {g} }={\frac {n}{1+{\frac {\lambda }{n}}{\frac {\mathrm {d} n}{\mathrm {d} \lambda }}}}.}$

When the refractive index of a medium is known as a function of the vacuum wavelength (instead of the wavelength in the medium), the corresponding expressions for the group velocity and index are (for all values of dispersion) [41]

${\displaystyle v_{\mathrm {g} }=\mathrm {c} \!\left(n-\lambda _{0}{\frac {\mathrm {d} n}{\mathrm {d} \lambda _{0}}}\right)^{-1}\!,}$
${\displaystyle n_{\mathrm {g} }=n-\lambda _{0}{\frac {\mathrm {d} n}{\mathrm {d} \lambda _{0}}},}$

where λ0 is the wavelength in vacuum.

### Momentum (Abraham–Minkowski controversy)

In 1908, Hermann Minkowski calculated the momentum p of a refracted ray as follows: [42]

${\displaystyle p={\frac {nE}{\mathrm {c} }},}$

where E is energy of the photon, c is the speed of light in vacuum and n is the refractive index of the medium. In 1909, Max Abraham proposed the following formula for this calculation: [43]

${\displaystyle p={\frac {E}{n\mathrm {c} }}.}$

A 2010 study suggested that both equations are correct, with the Abraham version being the kinetic momentum and the Minkowski version being the canonical momentum, and claims to explain the contradicting experimental results using this interpretation. [44]

### Other relations

As shown in the Fizeau experiment, when light is transmitted through a moving medium, its speed relative to an observer traveling with speed v in the same direction as the light is:

${\displaystyle V={\frac {\mathrm {c} }{n}}+{\frac {v\left(1-{\frac {1}{n^{2}}}\right)}{1+{\frac {v}{cn}}}}\approx {\frac {\mathrm {c} }{n}}+v\left(1-{\frac {1}{n^{2}}}\right)\ .}$

The refractive index of a substance can be related to its polarizability with the Lorentz–Lorenz equation or to the molar refractivities of its constituents by the Gladstone–Dale relation.

### Refractivity

In atmospheric applications, the refractivity is taken as N = n – 1. Atmospheric refractivity is often expressed as either [45] N = 106(n – 1) [46] [47] or N = 108(n – 1) [48] The multiplication factors are used because the refractive index for air, n deviates from unity by at most a few parts per ten thousand.

Molar refractivity, on the other hand, is a measure of the total polarizability of a mole of a substance and can be calculated from the refractive index as

${\displaystyle A={\frac {M}{\rho }}{\frac {n^{2}-1}{n^{2}+2}},}$

where ρ is the density, and M is the molar mass. [32] :93

## Nonscalar, nonlinear, or nonhomogeneous refraction

So far, we have assumed that refraction is given by linear equations involving a spatially constant, scalar refractive index. These assumptions can break down in different ways, to be described in the following subsections.

### Birefringence

In some materials the refractive index depends on the polarization and propagation direction of the light. [49] This is called birefringence or optical anisotropy.

In the simplest form, uniaxial birefringence, there is only one special direction in the material. This axis is known as the optical axis of the material. [1] :230 Light with linear polarization perpendicular to this axis will experience an ordinary refractive index no while light polarized in parallel will experience an extraordinary refractive index ne. [1] :236 The birefringence of the material is the difference between these indices of refraction, Δn = neno. [1] :237 Light propagating in the direction of the optical axis will not be affected by the birefringence since the refractive index will be no independent of polarization. For other propagation directions the light will split into two linearly polarized beams. For light traveling perpendicularly to the optical axis the beams will have the same direction. [1] :233 This can be used to change the polarization direction of linearly polarized light or to convert between linear, circular and elliptical polarizations with waveplates. [1] :237

Many crystals are naturally birefringent, but isotropic materials such as plastics and glass can also often be made birefringent by introducing a preferred direction through, e.g., an external force or electric field. This effect is called photoelasticity, and can be used to reveal stresses in structures. The birefringent material is placed between crossed polarizers. A change in birefringence alters the polarization and thereby the fraction of light that is transmitted through the second polarizer.

In the more general case of trirefringent materials described by the field of crystal optics, the dielectric constant is a rank-2 tensor (a 3 by 3 matrix). In this case the propagation of light cannot simply be described by refractive indices except for polarizations along principal axes.

### 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. [1] :502 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. [1] :264 If the index varies linearly with the field (a nontrivial linear coefficient is only possible in materials that do not possess inversion symmetry), it is known as the Pockels effect. [1] :265

### Inhomogeneity

If the refractive index of a medium is not constant, but varies gradually with position, the material is known as a gradient-index or GRIN medium and is described by gradient index optics. [1] :273 Light traveling through such a medium can be bent or focused, and this effect can be exploited to produce lenses, some optical fibers and other devices. Introducing GRIN elements in the design of an optical system can greatly simplify the system, reducing the number of elements by as much as a third while maintaining overall performance. [1] :276 The crystalline lens of the human eye is an example of a GRIN lens with a refractive index varying from about 1.406 in the inner core to approximately 1.386 at the less dense cortex. [1] :203 Some common mirages are caused by a spatially varying refractive index of air.

## Refractive index measurement

### Homogeneous media

The refractive index of liquids or solids can be measured with refractometers. They typically measure some angle of refraction or the critical angle for total internal reflection. The first laboratory refractometers sold commercially were developed by Ernst Abbe in the late 19th century. [50] The same principles are still used today. In this instrument a thin layer of the liquid to be measured is placed between two prisms. Light is shone through the liquid at incidence angles all the way up to 90°, i.e., light rays parallel to the surface. The second prism should have an index of refraction higher than that of the liquid, so that light only enters the prism at angles smaller than the critical angle for total reflection. This angle can then be measured either by looking through a telescope,[ clarification needed ] or with a digital photodetector placed in the focal plane of a lens. The refractive index n of the liquid can then be calculated from the maximum transmission angle θ as n = nG sin θ, where nG is the refractive index of the prism. [51]

This type of devices are commonly used in chemical laboratories for identification of substances and for quality control. Handheld variants are used in agriculture by, e.g., wine makers to determine sugar content in grape juice, and inline process refractometers are used in, e.g., chemical and pharmaceutical industry for process control.

In gemology a different type of refractometer is used to measure index of refraction and birefringence of gemstones. The gem is placed on a high refractive index prism and illuminated from below. A high refractive index contact liquid is used to achieve optical contact between the gem and the prism. At small incidence angles most of the light will be transmitted into the gem, but at high angles total internal reflection will occur in the prism. The critical angle is normally measured by looking through a telescope. [52]

### Refractive index variations

Unstained biological structures appear mostly transparent under Bright-field microscopy as most cellular structures do not attenuate appreciable quantities of light. Nevertheless, the variation in the materials that constitutes these structures also corresponds to a variation in the refractive index. The following techniques convert such variation into measurable amplitude differences:

To measure the spatial variation of refractive index in a sample phase-contrast imaging methods are used. These methods measure the variations in phase of the light wave exiting the sample. The phase is proportional to the optical path length the light ray has traversed, and thus gives a measure of the integral of the refractive index along the ray path. The phase cannot be measured directly at optical or higher frequencies, and therefore needs to be converted into intensity by interference with a reference beam. In the visual spectrum this is done using Zernike phase-contrast microscopy, differential interference contrast microscopy (DIC) or interferometry.

Zernike phase-contrast microscopy introduces a phase shift to the low spatial frequency components of the image with a phase-shifting annulus in the Fourier plane of the sample, so that high-spatial-frequency parts of the image can interfere with the low-frequency reference beam. In DIC the illumination is split up into two beams that are given different polarizations, are phase shifted differently, and are shifted transversely with slightly different amounts. After the specimen, the two parts are made to interfere, giving an image of the derivative of the optical path length in the direction of the difference in transverse shift. [36] In interferometry the illumination is split up into two beams by a partially reflective mirror. One of the beams is let through the sample before they are combined to interfere and give a direct image of the phase shifts. If the optical path length variations are more than a wavelength the image will contain fringes.

There exist several phase-contrast X-ray imaging techniques to determine 2D or 3D spatial distribution of refractive index of samples in the X-ray regime. [53]

## Applications

The refractive index is a very important property of the components of any optical instrument. It determines the focusing power of lenses, the dispersive power of prisms, the reflectivity of lens coatings, and the light-guiding nature of optical fiber. Since refractive index is a fundamental physical property of a substance, it is often used to identify a particular substance, confirm its purity, or measure its concentration. Refractive index is used to measure solids, liquids, and gases. Most commonly it is used to measure the concentration of a solute in an aqueous solution. It can also be used as a useful tool to differentiate between different types of gemstone, due to the unique chatoyance each individual stone displays. A refractometer is the instrument used to measure refractive index. For a solution of sugar, the refractive index can be used to determine the sugar content (see Brix).

## Related Research Articles

The Beer–Lambert law, also known as Beer's law, the Lambert–Beer law, or the Beer–Lambert–Bouguer law relates the attenuation of light to the properties of the material through which the light is travelling. The law is commonly applied to chemical analysis measurements and used in understanding attenuation in physical optics, for photons, neutrons or rarefied gases. In mathematical physics, this law arises as a solution of the BGK equation.

The Fresnel equations describe the reflection and transmission of light when incident on an interface between different optical media. They were deduced by Augustin-Jean Fresnel who was the first to understand that light is a transverse wave, even though no one realized that the "vibrations" of the wave were electric and magnetic fields. For the first time, polarization could be understood quantitatively, as Fresnel's equations correctly predicted the differing behaviour of waves of the s and p polarizations incident upon a material interface.

Nonlinear optics (NLO) is the branch of optics that describes the behaviour of light in nonlinear media, that is, media in which the polarization density P responds non-linearly to the electric field E of the light. The non-linearity is typically observed only at very high light intensities (values of atomic electric fields, typically 108 V/m) such as those provided by lasers. Above the Schwinger limit, the vacuum itself is expected to become nonlinear. In nonlinear optics, the superposition principle no longer holds.

In physics, intensity is the power transferred per unit area, where the area is measured on the plane perpendicular to the direction of propagation of the energy. In the SI system, it has units watts per square metre (W/m2). It is used most frequently with waves, in which case the average power transfer over one period of the wave is used. Intensity can be applied to other circumstances where energy is transferred. For example, one could calculate the intensity of the kinetic energy carried by drops of water from a garden sprinkler.

In electromagnetism, absolute permittivity, often simply called permittivity, usually denoted by the Greek letter ε (epsilon), is the measure of capacitance that is encountered when forming an electric field in a particular medium. More specifically, permittivity describes the amount of charge needed to generate one unit of electric flux in a particular medium. Accordingly, a charge will yield more electric flux in a medium with low permittivity than in a medium with high permittivity. Permittivity is the measure of a material's ability to store an electric field in the polarization of the medium.

Crystal optics is the branch of optics that describes the behaviour of light in anisotropic media, that is, media in which light behaves differently depending on which direction the light is propagating. The index of refraction depends on both composition and crystal structure and can be calculated using the Gladstone–Dale relation. Crystals are often naturally anisotropic, and in some media it is possible to induce anisotropy by applying an external electric field.

In the physical sciences, the wavenumber is the spatial frequency of a wave, measured in cycles per unit distance or radians per unit distance. Whereas temporal frequency can be thought of as the number of waves per unit time, wavenumber is the number of waves per unit distance.

In chemistry, absorbance or decadic absorbance is the common logarithm of the ratio of incident to transmitted radiant power through a material, and spectral absorbance or spectral decadic absorbance is the common logarithm of the ratio of incident to transmitted spectral radiant power through a material. Absorbance is dimensionless, and in particular is not a length, though it is a monotonically increasing function of path length, and approaches zero as the path length approaches zero. The use of the term "optical density" for absorbance is discouraged. In physics, a closely related quantity called "optical depth" is used instead of absorbance: the natural logarithm of the ratio of incident to transmitted radiant power through a material. The optical depth equals the absorbance times ln(10).

Transmittance of the surface of a material is its effectiveness in transmitting radiant energy. It is the fraction of incident electromagnetic power that is transmitted through a sample, in contrast to the transmission coefficient, which is the ratio of the transmitted to incident electric field.

In radiometry, irradiance is the radiant flux (power) received by a surface per unit area. The SI unit of irradiance is the watt per square metre. The CGS unit erg per square centimetre per second is often used in astronomy. Irradiance is often called intensity because it has the same physical dimensions, but this term is avoided in radiometry where such usage leads to confusion with radiant intensity.

The Kerr effect, also called the quadratic electro-optic (QEO) effect, is a change in the refractive index of a material in response to an applied electric field. The Kerr effect is distinct from the Pockels effect in that the induced index change is directly proportional to the square of the electric field instead of varying linearly with it. All materials show a Kerr effect, but certain liquids display it more strongly than others. The Kerr effect was discovered in 1875 by John Kerr, a Scottish physicist.

Geometrical optics, or ray optics, describes light propagation in terms of rays. The ray in geometric optics is an abstraction useful for approximating the paths along which light propagates under certain circumstances.

In physical sciences and electrical engineering, dispersion relations describe the effect of dispersion in a medium on the properties of a wave traveling within that medium. A dispersion relation relates the wavelength or wavenumber of a wave to its frequency. From this relation the phase velocity and group velocity of the wave have convenient expressions which then determine the refractive index of the medium. More general than the geometry-dependent and material-dependent dispersion relations, there are the overarching Kramers–Kronig relations that describe the frequency dependence of wave propagation and attenuation.

Self-phase modulation (SPM) is a nonlinear optical effect of light-matter interaction. An ultrashort pulse of light, when travelling in a medium, will induce a varying refractive index of the medium due to the optical Kerr effect. This variation in refractive index will produce a phase shift in the pulse, leading to a change of the pulse's frequency spectrum.

Acousto-optics is a branch of physics that studies the interactions between sound waves and light waves, especially the diffraction of laser light by ultrasound through an ultrasonic grating.

When an electromagnetic wave travels through a medium in which it gets attenuated, it undergoes exponential decay as described by the Beer–Lambert law. However, there are many possible ways to characterize the wave and how quickly it is attenuated. This article describes the mathematical relationships among:

In the fields of nonlinear optics and fluid dynamics, modulational instability or sideband instability is a phenomenon whereby deviations from a periodic waveform are reinforced by nonlinearity, leading to the generation of spectral-sidebands and the eventual breakup of the waveform into a train of pulses.

Surface plasmon polaritons (SPPs) are infrared or visible-frequency electromagnetic waves that travel along a metal–dielectric or metal–air interface. The term "surface plasmon polariton" explains that the wave involves both charge motion in the metal and electromagnetic waves in the air or dielectric ("polariton").

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