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optics 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, ultra ...
, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the
light Light or visible light is electromagnetic radiation that can be perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 nanometres (nm), corresponding to frequencies of 750–420 te ...
bending ability of that medium. 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, , where ''θ''1 and ''θ''2 are the
angle of incidence Angle of incidence is a measure of deviation of something from "straight on" and may refer to: * Angle of incidence (aerodynamics), angle between a wing chord and the longitudinal axis, as distinct from angle of attack In fluid dynamics, ang ...
and angle of refraction, respectively, of a ray crossing the interface between two media with refractive indices ''n''1 and ''n''2. The refractive indices also determine the amount of light that is
reflected Reflection or reflexion may refer to: Science and technology * Reflection (physics), a common wave phenomenon ** Specular reflection, reflection from a smooth surface *** Mirror image, a reflection in a mirror or in water ** Signal reflection, in ...
when reaching the interface, as well as the critical angle for total internal reflection, their intensity (
Fresnel's equations The Fresnel equations (or Fresnel coefficients) describe the reflection and transmission of light (or electromagnetic radiation in general) when incident on an interface between different optical media. They were deduced by Augustin-Jean Fresne ...
) and Brewster's angle. The refractive index can be seen as the factor by which the speed and the
wavelength In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tr ...
of the radiation are reduced with respect to their vacuum values: the speed of light in a medium is , and similarly the wavelength in that medium is , where ''λ''0 is the wavelength of that light in vacuum. This implies that vacuum has a refractive index of 1, and assumes that the
frequency Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from '' angular frequency''. Frequency is measured in hertz (Hz) which is ...
() of the wave is not affected by the refractive index. The refractive index may vary with wavelength. This causes white light to split into constituent colors when refracted. This is called dispersion. This effect can be observed in prisms and rainbows, and as
chromatic aberration In optics, chromatic aberration (CA), also called chromatic distortion and spherochromatism, 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 ...
in lenses. Light propagation in absorbing materials can be described using a complex-valued refractive index. The imaginary part then handles the attenuation, while the real part accounts for refraction. For most materials the refractive index changes with wavelength by several percent across the visible spectrum. Nevertheless, refractive indices for materials are commonly reported using a single value for ''n'', typically measured at 633 nm. The concept of refractive index applies across the full
electromagnetic spectrum The electromagnetic spectrum is the range of frequencies (the spectrum) of electromagnetic radiation and their respective wavelengths and photon energies. The electromagnetic spectrum covers electromagnetic waves with frequencies ranging from ...
, from
X-ray X-rays (or rarely, ''X-radiation'') are a form of high-energy electromagnetic radiation. In many languages, it is referred to as Röntgen radiation, after the German scientist Wilhelm Conrad Röntgen, who discovered it in 1895 and named it ' ...
s to
radio wave Radio waves are a type of electromagnetic radiation with the longest wavelengths in the electromagnetic spectrum, typically with frequencies of 300 gigahertz ( GHz) and below. At 300 GHz, the corresponding wavelength is 1 mm (sho ...
s. It can also be applied to
wave In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities. Waves can be periodic, in which case those quantities oscillate repeatedly about an equilibrium (r ...
phenomena such as
sound In physics, sound is a vibration that propagates as an acoustic wave, through a transmission medium such as a gas, liquid or solid. In human physiology and psychology, sound is the ''reception'' of such waves and their ''perception'' by ...
. In this case, the speed of sound is used instead of that of light, and a reference medium other than vacuum must be chosen. For
lenses A lens is a transmissive optical device which focuses or disperses a light beam by means of refraction. A simple lens consists of a single piece of transparent material, while a compound lens consists of several simple lenses (''elements ...
(such as eye glasses), a lens made from a high refractive index material will be thinner, and hence lighter, than a conventional lens with a lower refractive index. Such lenses are generally more expensive to manufacture than conventional ones.


Definition

The relative refractive index of an optical medium 2 with respect to another ''reference'' medium 1 (n21) is given by the ratio of speed of light in medium 1 to that in medium 2. This can be expressed as follows: :n_=\frac. If the ''reference'' medium 1 is
vacuum A vacuum is a space devoid of matter. The word is derived 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 di ...
, then the refractive index of medium 2 is considered with respect to vacuum. It is simply represented as n2 and is called the absolute refractive index of medium 2. The absolute refractive index ''n'' of an optical medium is defined as the ratio of the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit fo ...
in vacuum, , and the phase velocity ''v'' of light in the medium, :n=\frac. Since ''c'' is constant, ''n'' is inversely proportional to ''v'' : :n\propto\frac. The phase velocity is the speed at which the crests or the phase of the
wave In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities. Waves can be periodic, in which case those quantities oscillate repeatedly about an equilibrium (r ...
moves, which may be different from the group velocity, the speed at which the pulse of light or the
envelope An envelope is a common packaging item, usually made of thin, flat material. It is designed to contain a flat object, such as a letter or card. Traditional envelopes are made from sheets of paper cut to one of three shapes: a rhombus, a ...
of the wave moves. Historically air at a standardized
pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country a ...
and
temperature Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer. Thermometers are calibrated in various temperature scales that historically have relied on ...
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. 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). Hauksbee, who called it the "ratio of refraction", wrote it as a ratio with a fixed numerator, like "10000 to 7451.9" (for urine). Hutton wrote it as a ratio with a fixed denominator, like 1.3358 to 1 (water). Young did not use a symbol for the index of refraction, in 1807. In the later years, others started using different symbols: ''n, m'', and µ. Exponent des Brechungsverhältnisses is index of refraction The symbol ''n'' gradually prevailed.


Typical values

Refractive index also varies with wavelength of the light as given by Cauchy's equation: The most general form of Cauchy's equation is n(\lambda) = A + \frac + \frac + \cdots, where ''n'' is the refractive index, λ is the wavelength, ''A'', ''B'', ''C'', etc., are
coefficient In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves ...
s that can be determined for a material by fitting the equation to measured refractive indices at known wavelengths. The coefficients are usually quoted for λ as the vacuum wavelength in
micrometre The micrometre ( international spelling as used by the International Bureau of Weights and Measures; SI symbol: μm) or micrometer ( American spelling), also commonly known as a micron, is a unit of length in the International System of ...
s. Usually, it is sufficient to use a two-term form of the equation: n(\lambda) = A + \frac, where the coefficients ''A'' and ''B'' are determined specifically for this form of the equation. For
visible light Light or visible light is electromagnetic radiation that can be perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 nanometres (nm), corresponding to frequencies of 750–420 tera ...
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 Sodium is a chemical element with the symbol Na (from Latin ''natrium'') and atomic number 11. It is a soft, silvery-white, highly reactive metal. Sodium is an alkali metal, being in group 1 of the periodic table. Its only stable ...
, with a wavelength of 589 nanometers, as is conventionally done. 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. 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. For
infrared Infrared (IR), sometimes called infrared light, is electromagnetic radiation (EMR) with wavelengths longer than those of visible light. It is therefore invisible to the human eye. IR is generally understood to encompass wavelengths from aroun ...
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. A type of new materials termed "topological insulators", was recently found which have high refractive index of up to 6 in the near to mid infrared frequency range. Moreover, topological insulators are transparent when they have nanoscale thickness. These properties are potentially important for applications in infrared optics.


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 less than 1. The refractive index measures the phase velocity of light, which does not carry
information Information is an abstract concept that refers to that which has the power to inform. At the most fundamental level information pertains to the interpretation of that which may be sensed. Any natural process that is not completely random, ...
. 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 Resonance describes the phenomenon of increased amplitude that occurs when the frequency of an applied Periodic function, periodic force (or a Fourier analysis, Fourier component of it) is equal or close to a natural frequency of the system ...
, for absorbing media, in plasmas, and for
X-ray X-rays (or rarely, ''X-radiation'') are a form of high-energy electromagnetic radiation. In many languages, it is referred to as Röntgen radiation, after the German scientist Wilhelm Conrad Röntgen, who discovered it in 1895 and named it ' ...
s. In the X-ray regime the refractive indices are lower than but very close to 1 (exceptions close to some resonance frequencies). As an example, water has a refractive index of = 1 − for X-ray radiation at a photon energy of (0.04 nm wavelength). 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.


Negative refractive index

Recent research has also demonstrated the existence of materials with a negative refractive index, which can occur if
permittivity In electromagnetism, the absolute permittivity, often simply called permittivity and denoted by the Greek letter ''ε'' ( epsilon), is a measure of the electric polarizability of a dielectric. A material with high permittivity polarizes more i ...
and permeability have simultaneous negative values. 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 new phenomena to be actively developed by means of metamaterials.


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
electron The electron (, or in nuclear reactions) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary partic ...
s) proportional to the electric susceptibility of the medium. (Similarly, the
magnetic field A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and t ...
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. 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. * 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-ray X-rays (or rarely, ''X-radiation'') are a form of high-energy electromagnetic radiation. In many languages, it is referred to as Röntgen radiation, after the German scientist Wilhelm Conrad Röntgen, who discovered it in 1895 and named it ' ...
s in ordinary materials, and with radio waves in Earth's ionosphere. It corresponds to a
permittivity In electromagnetism, the absolute permittivity, often simply called permittivity and denoted by the Greek letter ''ε'' ( epsilon), is a measure of the electric polarizability of a dielectric. A material with high permittivity polarizes more i ...
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 In electromagnetism, the absolute permittivity, often simply called permittivity and denoted by the Greek letter ''ε'' ( epsilon), is a measure of the electric polarizability of a dielectric. A material with high permittivity polarizes more i ...
and imaginary index of refraction, as observed in metals or plasma. * 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
laser A laser is a device that emits light through a process of optical amplification based on the stimulated emission of electromagnetic radiation. The word "laser" is an acronym for "light amplification by stimulated emission of radiation". The firs ...
s 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 Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from '' angular frequency''. Frequency is measured in hertz (Hz) which is ...
) of light.R. Paschotta, article o
chromatic dispersion
in th

, accessed on 2014-09-08
This is called dispersion and causes prisms and rainbows to divide white light into its constituent spectral
color Color (American English) or colour (British English) is the visual perceptual property deriving from the spectrum of light interacting with the photoreceptor cells of the eyes. Color categories and physical specifications of color are assoc ...
s.Carl R. Nave, page o
Dispersion
i

, Department of Physics and Astronomy, Georgia State University, accessed on 2014-09-08
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 A lens is a transmissive optical device which focuses or disperses a light beam by means of refraction. A simple lens consists of a single piece of transparent material, while a compound lens consists of several simple lenses (''elements ...
to be wavelength dependent. This is a type of
chromatic aberration In optics, chromatic aberration (CA), also called chromatic distortion and spherochromatism, 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 ...
, 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 ''de''crease with increasing wavelength, and thus ''in''crease with frequency. This is called "normal dispersion", in contrast to "anomalous dispersion", where the refractive index ''in''creases with wavelength. 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: :V = \frac. For a more accurate description of the wavelength dependence of the refractive index, the
Sellmeier equation The Sellmeier equation is an empirical relationship between refractive index and wavelength for a particular transparent medium. The equation is used to determine the dispersion of light in the medium. It was first proposed in 1872 by Wolfgan ...
can be used. It is an empirical formula that works well in describing dispersion. ''Sellmeier coefficients'' are often quoted instead of the refractive index in tables.


Principal refractive index wavelength ambiguity

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. Manufacturers of optical glass in general define principal index of refraction at yellow spectral line of helium (587.56 nm) and alternatively at a green spectral line of mercury (546.07 nm), called ''d'' and ''e'' lines respectively. Abbe number is defined for both and denoted ''Vd'' and ''Ve''. The spectral data provided by glass manufacturers is also often more precise for these 2 wavelengths. Both, ''d'' and ''e'' spectral lines are singlets and thus are suitable to perform a very precise measurements, such as spectral goniometric method. In practical applications, measurements of refractive index are performed on various refractometers, such as Abbe refractometer. Measurement accuracy of such typical commercial devices is in the order of 0.0002. Refractometers usually measure refractive index ''nD'', defined for sodium doublet ''D'' (589.29 nm), which is actually a midpoint between 2 adjacent yellow spectral lines of sodium. Yellow spectral lines of helium (''d'') and sodium (''D'') are 1.73 nm apart, which can be considered negligible for typical refractometers, but can cause confusion and lead to errors if accuracy is critical. All 3 typical principle refractive indices definitions can be found depending on application and region, so a proper subscript should be used to avoid ambiguity.


Complex refractive index

When light passes through a medium, some part of it will always be absorbed. This can be conveniently taken into account by defining a complex refractive index, :\underline = 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 or absorption coefficient—although ''κ'' can also refer to the mass attenuation coefficient—and indicates the amount of attenuation when the electromagnetic wave propagates through the material. That ''κ'' corresponds to absorption can be seen by inserting this refractive index into the expression for electric field of a plane electromagnetic wave traveling in the ''x''-direction. This can be done by relating the complex wave number to the complex refractive index through , with ''λ'' being the vacuum wavelength; this can be inserted into the plane wave expression for a wave travelling in the x direction as: :\mathbf(s, t) = \operatorname\! \left mathbf_0 e^\right = \operatorname\! \left mathbf_0 e^\right = e^ \operatorname\! \left mathbf_0 e^\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, intensity will depend on the depth into the material as :I(x)= I_0 e^ . and thus the absorption coefficient is , and the penetration depth (the distance after which the intensity is reduced by a factor of 1/''e'') is . Both ''n'' and ''κ'' are dependent on the frequency. In most circumstances (light is absorbed) or (light travels forever without loss). In special situations, especially in the gain medium of
laser A laser is a device that emits light through a process of optical amplification based on the stimulated emission of electromagnetic radiation. The word "laser" is an acronym for "light amplification by stimulated emission of radiation". The firs ...
s, it is also possible that , corresponding to an amplification of the light. An alternative convention uses instead of , but where 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
xp(−''iωt'') XP may refer to: Medicine * Xanthelasma palpebrarum, a cholesterol deposit in the eyelid * Xeroderma pigmentosum, a genetic disorder Computing * Windows XP, an operating system * Microsoft Office XP, a version of the software suite * Athlon XP, ...
versus Re
xp(+''iωt'') XP may refer to: Medicine * Xanthelasma palpebrarum, a cholesterol deposit in the eyelid * Xeroderma pigmentosum, a genetic disorder Computing * Windows XP, an operating system * Microsoft Office XP, a version of the software suite * Athlon XP, ...
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 Transparency, transparence or transparent most often refer to: * Transparency (optics), the physical property of allowing the transmission of light through a material They may also refer to: Literal uses * Transparency (photography), a still ...
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'', are typically measured from 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.


X-ray and extreme UV

For
X-ray X-rays (or rarely, ''X-radiation'') are a form of high-energy electromagnetic radiation. In many languages, it is referred to as Röntgen radiation, after the German scientist Wilhelm Conrad Röntgen, who discovered it in 1895 and named it ' ...
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 (or with the alternative convention mentioned above). Far above the atomic resonance frequency delta can be given by : \delta = \frac where r_0 is the classical electron radius, \lambda is the X-ray wavelength, and 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 f = Z + f' + i f'' . It follows that : \delta = \frac (Z + f') n_\text : \beta = \frac f'' n_\text with \delta and \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, :\text = 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.


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 ''n''1 to one with refractive index ''n''2, with an incidence angle to the
surface normal In geometry, a normal is an object such as a line, ray, or vector that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the (infinite) line perpendicular to the tangent line to the curve ...
of ''θ''1, the refraction angle ''θ''2 can be calculated from Snell's law: :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 ''θ''2 fulfilling Snell's law, i.e., :\frac \sin \theta_1 > 1, the light cannot be transmitted and will instead undergo total internal reflection. 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 ''θ''1 must be larger than the critical angle :\theta_\mathrm = \arcsin\!\left(\frac\right)\!.


Reflectivity

Apart from the transmitted light there is also a
reflected Reflection or reflexion may refer to: Science and technology * Reflection (physics), a common wave phenomenon ** Specular reflection, reflection from a smooth surface *** Mirror image, a reflection in a mirror or in water ** Signal reflection, in ...
part. The reflection angle is equal to the incidence angle, and the amount of light that is reflected is determined by the
reflectivity The reflectance of the surface of a material is its effectiveness in reflecting radiant energy. It is the fraction of incident electromagnetic power that is reflected at the boundary. Reflectance is a component of the response of the electroni ...
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 :R_0 = \left, \frac\^2\!. For common glass in air, ''n''1 = 1 and ''n''2 = 1.5, and thus about 4% of the incident power is reflected. At other incidence angles the reflectivity will also depend on the
polarization Polarization or polarisation may refer to: Mathematics *Polarization of an Abelian variety, in the mathematics of complex manifolds *Polarization of an algebraic form, a technique for expressing a homogeneous polynomial in a simpler fashion by ...
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 :\theta_\mathrm = \arctan\!\left(\frac\right)\!.


Lenses

The focal length of a
lens A lens is a transmissive optical device which focuses or disperses a light beam by means of refraction. A simple lens consists of a single piece of transparent material, while a compound lens consists of several simple lenses (''elements'' ...
is determined by its refractive index ''n'' and the radii of curvature ''R''1 and ''R''2 of its surfaces. The power of a thin lens in air is given by the Lensmaker's formula: :\frac = (n - 1)\!\left(\frac - \frac\right)\!, where ''f'' is the focal length of the lens.


Microscope resolution

The resolution of a good optical
microscope A microscope () is a laboratory instrument used to examine objects that are too small to be seen by the naked eye. Microscopy is the science of investigating small objects and structures using a microscope. Microscopic means being invisibl ...
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 :\mathrm = 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.


Relative permittivity and permeability

The refractive index of electromagnetic radiation equals :n = \sqrt, where ''ε''r is the material's
relative permittivity The relative permittivity (in older texts, dielectric constant) is the permittivity of a material expressed as a ratio with the electric permittivity of a vacuum. A dielectric is an insulating material, and the dielectric constant of an insul ...
, and ''μ''r is its relative permeability. 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 Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. Th ...
and electronics. Most naturally occurring materials are non-magnetic at optical frequencies, that is ''μr'' is very close to 1, therefore ''n'' is approximately . In this particular case, the complex relative permittivity ''ε''r, with real and imaginary parts ''ε''r and ''ɛ̃''r, and the complex refractive index ''n'', with real and imaginary parts ''n'' and ''κ'' (the latter called the "extinction coefficient"), follow the relation :\underline_\mathrm = \varepsilon_\mathrm + i\tilde_\mathrm = \underline^2 = (n + i\kappa)^2, and their components are related by: :\varepsilon_\mathrm = n^2 - \kappa^2, :\tilde_\mathrm = 2n\kappa, and: :n = \sqrt, :\kappa = \sqrt. where , \underline_\mathrm, = \sqrt is the complex modulus.


Wave impedance

The wave impedance of a plane electromagnetic wave in a non-conductive medium is given by :Z = \sqrt = \sqrt = \sqrt\sqrt = Z_0\sqrt = Z_0\frac where Z_0 is the vacuum wave impedance, ''μ'' and ''ϵ'' are the absolute permeability and permittivity of the medium, ''ε''r is the material's
relative permittivity The relative permittivity (in older texts, dielectric constant) is the permittivity of a material expressed as a ratio with the electric permittivity of a vacuum. A dielectric is an insulating material, and the dielectric constant of an insul ...
, and ''μ''r is its relative permeability. In non-magnetic media with \mu_\mathrm=1, :Z = \frac, :n = \frac. 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 R_0 between two media can thus be expressed both by the wave impedances and the refractive indices as :R_0 = \left, \frac\^2\! = \left, \frac\^2\!.


Density

In general, the refractive index of a glass increases with its
density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematicall ...
. However, there does not exist an overall linear relationship 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 Lead(II) oxide, also called lead monoxide, is the inorganic compound with the molecular formula Pb O. PbO occurs in two polymorphs: litharge having a tetragonal crystal structure, and massicot having an orthorhombic crystal structure. Modern ap ...
and
BaO Baozi (), Pao-tsih or bao, is a type of yeast-leavened filled bun in various Chinese cuisines. There are many variations in fillings (meat or vegetarian) and preparations, though the buns are most often steamed. They are a variation of '' man ...
as seen in the diagram at the right. Many oils (such as
olive oil Olive oil is a liquid fat obtained from olives (the fruit of ''Olea europaea''; family Oleaceae), a traditional tree crop of the Mediterranean Basin, produced by pressing whole olives and extracting the oil. It is commonly used in cooking: ...
) and
ethanol Ethanol (abbr. EtOH; also called ethyl alcohol, grain alcohol, drinking alcohol, or simply alcohol) is an organic compound. It is an alcohol with the chemical formula . Its formula can be also written as or (an ethyl group linked to a h ...
are examples of liquids that are more refractive, but less dense, than water, contrary to the general correlation between density and refractive index. For air, ''n'' − 1 is proportional to the density of the gas as long as the chemical composition does not change. 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: :n_\mathrm = \frac, where ''v''g 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 :v_\mathrm = v - \lambda\frac, 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 :n_\mathrm = \frac. 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) :v_\mathrm = \mathrm\!\left(n - \lambda_0 \frac\right)^\!, :n_\mathrm = n - \lambda_0 \frac, where ''λ''0 is the wavelength in vacuum.


Other relations

As shown in the
Fizeau experiment The Fizeau experiment was carried out by Hippolyte Fizeau in 1851 to measure the relative speeds of light in moving water. Fizeau used a special interferometer arrangement to measure the effect of movement of a medium upon the speed of light. ...
, 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: :V = \frac + \frac\approx \frac + v\left(1-\frac\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, refractivity is defined as ''N'' = ''n'' – 1, often scaled as either ''N'' = (''n'' – 1) or ''N'' = (''n'' – 1); 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 :A = \frac \frac, where ''ρ'' is the
density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematicall ...
, and ''M'' is the
molar mass In chemistry, the molar mass of a chemical compound is defined as the mass of a sample of that compound divided by the amount of substance which is the number of moles in that sample, measured in moles. The molar mass is a bulk, not molecula ...
.


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 Polarization or polarisation may refer to: Mathematics *Polarization of an Abelian variety, in the mathematics of complex manifolds *Polarization of an algebraic form, a technique for expressing a homogeneous polynomial in a simpler fashion by ...
and propagation direction of the light. This is called
birefringence Birefringence is the optical property of a material having a refractive index that depends on the polarization and propagation direction of light. These optically anisotropic materials are said to be birefringent (or birefractive). The birefrin ...
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. Light with linear polarization perpendicular to this axis will experience an ''ordinary'' refractive index ''n''o while light polarized in parallel will experience an ''extraordinary'' refractive index ''n''e. The birefringence of the material is the difference between these indices of refraction, Δ''n'' = ''n''e − ''n''o. Light propagating in the direction of the optical axis will not be affected by the birefringence since the refractive index will be ''n''o 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. This can be used to change the polarization direction of linearly polarized light or to convert between linear, circular, and elliptical polarizations with waveplates. Many
crystal A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macr ...
s are naturally birefringent, but isotropic materials such as
plastic Plastics are a wide range of synthetic or semi-synthetic materials that use polymers as a main ingredient. Their plasticity makes it possible for plastics to be moulded, extruded or pressed into solid objects of various shapes. This adaptab ...
s and
glass Glass is a non-Crystallinity, crystalline, often transparency and translucency, transparent, amorphous solid that has widespread practical, technological, and decorative use in, for example, window panes, tableware, and optics. Glass is most ...
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 polarizer or polariser is an optical filter that lets light waves of a specific polarization pass through while blocking light waves of other polarizations. It can filter a beam of light of undefined or mixed polarization into a beam of well ...
. 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 In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tens ...
(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 the output of a
laser A laser is a device that emits light through a process of optical amplification based on the stimulated emission of electromagnetic radiation. The word "laser" is an acronym for "light amplification by stimulated emission of radiation". The firs ...
) 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 (a nontrivial linear coefficient 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 the position, the material is known as a gradient-index or GRIN medium and is described by gradient index optics. Light traveling through such a medium can be bent or focused, and this effect can be exploited to produce
lenses A lens is a transmissive optical device which focuses or disperses a light beam by means of refraction. A simple lens consists of a single piece of transparent material, while a compound lens consists of several simple lenses (''elements ...
, some
optical fiber An optical fiber, or optical fibre in Commonwealth English, is a flexible, transparency and translucency, transparent fiber made by Drawing (manufacturing), drawing glass (silica) or plastic to a diameter slightly thicker than that of a Hair ...
s, 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. 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. 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. 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 A telescope is a device used to observe distant objects by their emission, absorption, or reflection of electromagnetic radiation. Originally meaning only an optical instrument using lenses, curved mirrors, or a combination of both to obse ...
, 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 , where ''n''G is the refractive index of the prism. This type of device is commonly used in
chemical A chemical substance is a form of matter having constant chemical composition and characteristic properties. Some references add that chemical substance cannot be separated into its constituent elements by physical separation methods, i.e., wit ...
laboratories for identification of substances and for quality control. Handheld variants are used in
agriculture Agriculture or farming is the practice of cultivating plants and livestock. Agriculture was the key development in the rise of sedentary human civilization, whereby farming of domesticated species created food surpluses that enabled peop ...
by, e.g., wine makers to determine sugar content in
grape A grape is a fruit, botanically a berry (botany), berry, of the deciduous woody vines of the flowering plant genus ''Vitis''. Grapes are a non-Climacteric (botany), climacteric type of fruit, generally occurring in clusters. The cultivation of ...
juice, and inline process refractometers are used in, e.g.,
chemical A chemical substance is a form of matter having constant chemical composition and characteristic properties. Some references add that chemical substance cannot be separated into its constituent elements by physical separation methods, i.e., wit ...
and
pharmaceutical industry The pharmaceutical industry discovers, develops, produces, and markets drugs or pharmaceutical drugs for use as medications to be administered to patients (or self-administered), with the aim to cure them, Vaccine, vaccinate them, or alleviate s ...
for process control. In gemology, a different type of refractometer is used to measure the 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.


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 constitute 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 the 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 In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
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 Interferometry is a technique which uses the '' interference'' of superimposed waves to extract information. Interferometry typically uses electromagnetic waves and is an important investigative technique in the fields of astronomy, fiber o ...
. Zernike phase-contrast microscopy introduces a phase shift to the low spatial frequency components of the
image An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensio ...
with a phase-shifting annulus in the
Fourier plane Fourier may refer to: People named Fourier *Joseph Fourier (1768–1830), French mathematician and physicist *Charles Fourier (1772–1837), French utopian socialist thinker * Peter Fourier (1565–1640), French saint in the Roman Catholic Church ...
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 the transverse shift. 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.


Applications

The refractive index is an 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 An optical fiber, or optical fibre in Commonwealth English, is a flexible, transparency and translucency, transparent fiber made by Drawing (manufacturing), drawing glass (silica) or plastic to a diameter slightly thicker than that of a Hair ...
. Since the 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. The 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 An aqueous solution is a solution in which the solvent is water. It is mostly shown in chemical equations by appending (aq) to the relevant chemical formula. For example, a solution of table salt, or sodium chloride (NaCl), in water would ...
. 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 the refractive index. For a solution of sugar, the refractive index can be used to determine the sugar content (see Brix).


See also

* Fermat's principle * Calculation of glass properties * Clausius–Mossotti relation * Ellipsometry *
Index-matching material In optics, an index-matching material is a substance, usually a liquid, cement (adhesive), or gel, which has an index of refraction that closely approximates that of another object (such as a lens, material, fiber-optic, etc.). When two substances ...
* Index ellipsoid * Laser Schlieren Deflectometry * Optical properties of water and ice * Prism-coupling refractometry * Phase-contrast X-ray imaging * Velocity factor


References


External links


NIST calculator for determining the refractive index of air





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