Refractive index
25880
224761665
2008-07-10T08:53:16Z
Jason Patton
4178568
Reverted [[WP:AGF|good faith]] edits by [[Special:Contributions/85.121.35.7|85.121.35.7]]; Thanks, but already have [[List of refractive indices]]. ([[WP:TW|TW]])
The '''refractive index''' (or '''index of [[refraction]]''') of a medium is a measure for how much the speed of light (or other waves such as sound waves) is reduced inside the medium. For example, typical [[glass]] has a refractive index of 1.5, which means that in glass, light travels at <math>1/1.5=0.67</math> times the speed of light in a vacuum. Two common properties of glass and other transparent materials are directly related to their refractive index. First, light rays change direction when they cross the interface from air to the material, an effect that is used in lenses and [[glasses]]. Second, light reflects partially from surfaces that have a refractive index different from that of their surroundings.
Definition:
The refractive index ''n'' of a medium is defined as the ratio of the [[phase velocity]] ''c'' of a [[wave]] phenomenon such as [[light]] or [[sound]] in a reference medium to the phase velocity <math>v_{\mathrm{p}}</math> in the medium itself:
:<math>n = \frac{c}{v_{\mathrm {p}}}</math>
It is most commonly used in the context of [[light]] with [[vacuum]] as a reference medium, although historically other reference media (e.g. [[air]] at a standardized [[pressure]] and [[temperature]]) have been common. It is usually given the symbol ''n''. In the case of light, it equals
:<math> n=\sqrt{\epsilon_r\mu_r}</math>,
where ''ε<sub>r</sub>'' is the material's relative [[permittivity]], and ''μ<sub>r</sub>'' is its relative [[Permeability (electromagnetism)|permeability]]. For most materials, ''μ<sub>r</sub>'' is very close to 1 at optical frequencies, therefore ''n'' is approximately <math>\sqrt{\epsilon_r}</math>. Contrary to a widespread misconception, ''n'' may be less than 1, for example for [[x-rays]].<ref>Tanya M. Sansosti, [http://laser.physics.sunysb.edu/~tanya/report1/ Compound Refractive Lenses for X-Rays]. 2002</ref>. This has practical technical applications, such as effective mirrors for x-rays based on [[total external reflection]].
The [[phase velocity]] is defined as the rate at which the crests of the [[waveform]] propagate; that is, the rate at which the [[phase (waves)|phase]] of the waveform is moving. The ''[[group velocity]]'' is the rate that the ''envelope'' of the waveform is propagating; that is, the rate of variation of the [[amplitude]] of the waveform. Provided the waveform is not distorted significantly during propagation, it is the group velocity that represents the rate that information (and energy) may be transmitted by the wave, for example the velocity at which a pulse of light travels down an [[optical fiber]].
==Speed of light==
[[Image:Snells law.svg|thumb|[[Refraction]] of light at the interface between two media of different refractive indices, with n<sub>2</sub> > n<sub>1</sub>. Since the phase velocity is lower in the second medium (v<sub>2</sub> < v<sub>1</sub>), the angle of refraction θ<sub>2</sub> is less than the angle of incidence θ<sub>1</sub>; that is, the ray in the higher-index medium is closer to the normal.]]
The speed of all electromagnetic radiation in vacuum is the same, approximately 3×10<sup>8</sup> meters per second, and is denoted by [[speed of light|''c'']].
Therefore, if ''v'' is the [[phase velocity]] of radiation of a specific frequency in a specific material, the refractive index is given by
:<math>n =\frac{c}{v}</math>
or inversely
:<math>v =\frac{c}{n}</math>
This number is typically greater than one: the higher the index of the material, the more the light is slowed down (see [[Cherenkov radiation]]). However, at certain frequencies (e.g. near [[absorption (optics)|absorption]] resonances, and for [[X-ray]]s), ''n'' will actually be smaller than one. This does not contradict the [[theory of relativity]], which holds that no [[Signal (electrical engineering)|information-carrying signal]] can ever propagate faster than ''c'', because the [[phase velocity]] is not the same as the [[group velocity]] or the [[signal velocity]].
Sometimes, a "group velocity refractive index", usually called the ''group index'' is defined:
:<math>n_g=\frac{c}{v_g}</math>
where ''v<sub>g</sub>'' is the group velocity. This value should not be confused with ''n'', which is always defined with respect to the phase velocity. The group index can be written in terms of the wavelength dependence of the refractive index as
:<math>n_g = n - \lambda\frac{dn}{d\lambda},</math>
where <math>\lambda</math> is the wavelength in vacuum.
At the microscale, 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]]s) proportional to the [[permittivity]] of the medium. The charges will, in general, oscillate slightly out of [[phase (waves)|phase]] with respect to the driving electric field. The charges thus radiate their own electromagnetic wave that is at the same frequency but with a phase delay. The macroscopic sum of all such contributions in the material is 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 (see [[scattering]]).
If the refractive indices of two materials are known for a given frequency, then one can compute the angle by which radiation of that frequency will be [[refraction|refracted]] as it moves from the first into the second material from [[Snell's law]].
If in a given region the values of refractive indices ''n'' or ''n<sub>g</sub>'' were found to differ from unity (whether homogeneously, or isotropically, or not), then this region was distinct from vacuum in the above sense for lacking
[[Poincaré group|Poincaré symmetry]].
==Negative Refractive Index==
Recent research has also demonstrated the existence of [[negative refractive index]] which can occur if the real parts of both <math>\epsilon_r</math> and <math>\mu_r</math> are ''simultaneously'' negative, although such is a necessary but not sufficient condition. Not thought to occur naturally, this can be achieved with so-called [[metamaterial]]s and offers the possibility of [[superlenses|perfect lenses]] and other exotic phenomena such as a reversal of [[Snell%27s_law|Snell's law]]. [http://www.newscientisttech.com/article/dn10816.html] [http://www.sciencedaily.com/releases/2007/03/070322132145.htm]
==Dispersion and absorption==
[[Image:Dispersion-curve.png|right|thumb|320px|The variation of refractive index vs. wavelength for various glasses.]]
In real materials, the [[polarization (electrostatics)|polarization]] does not respond instantaneously to an applied field. This causes [[dielectric loss]], which can be expressed by a [[permittivity]] that is both [[complex number|complex]] and [[frequency]] dependent. Real materials are not perfect [[Electrical insulation|insulator]]s either, i.e. they have non-zero [[direct current]] [[electrical conductivity|conductivity]]. Taking both aspects into consideration, we can define a complex index of refraction:
:<math>\tilde{n}=n+i\kappa</math>
Here, ''n'' is the refractive index indicating the phase velocity as above, while ''κ'' is called the [[extinction coefficient]], which indicates the amount of [[absorption (optics)|absorption]] loss when the electromagnetic wave propagates through the material. Both ''n'' and ''κ'' are dependent on the frequency ([[wavelength]]). Note that the sign of the complex part is a matter of convention, which is important due to possible confusion between loss and gain. The notation above, which is usually used by physicists, corresponds to waves with time evolution given by <math>e^{-i\omega t}</math>.
The effect that ''n'' varies with [[frequency]] (except in vacuum, where all frequencies travel at the same speed, ''c'') is known as [[dispersion (optics)|dispersion]], and it is what causes a [[Triangular prism (optics)|prism]] to divide white light into its constituent spectral [[color]]s, explains [[rainbow]]s, and is the cause of [[chromatic aberration]] in [[Lens (optics)|lenses]]. In regions of the spectrum where the material does not absorb, the real part of the refractive index tends to increase with frequency. Near absorption peaks, the curve of the refractive index is a complex form given by the [[Kramers–Kronig relation]]s, and can decrease with frequency.
Since the refractive index of a material varies with the frequency (and thus wavelength) of light, it is usual to specify the corresponding vacuum wavelength at which the refractive index is measured. Typically, this is done at various well-defined spectral [[emission line]]s; for example, ''n''<sub>D</sub> is the refractive index at the [[Fraunhofer lines|Fraunhofer]] "D" line, the centre of the yellow [[sodium]] double emission at 589.29 [[nanometre|nm]] wavelength.
The [[Sellmeier equation]] is an empirical formula that works well in describing dispersion, and Sellmeier coefficients are often quoted instead of the refractive index in tables. For some representative refractive indices at different wavelengths, see [[list of indices of refraction]].
As shown above, 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 (κ ≈ 0). However, at higher frequencies (such as visible light), dielectric loss may increase absorption significantly, reducing the material's [[transparency (optics)|transparency]] to these frequencies.
The real and imaginary parts of the complex refractive index are related through use of the [[Kramers–Kronig relation]]s. For example, one can determine a material's full complex refractive index as a function of wavelength from an absorption spectrum of the material.
==Relation to dielectric constant==
The [[dielectric constant]] (which is often dependent on wavelength) is simply the square of the (complex) refractive index. The refractive index is used for optics in [[Fresnel equations]] and [[Snell's law]]; while the dielectric constant is used in [[Maxwell's equations]] and electronics.
Where <math>\tilde\epsilon</math>, <math>\epsilon_1</math>, <math>\epsilon_2</math>, <math>n</math>, and <math>\kappa</math> are functions of wavelength:
:<math>\tilde\epsilon=\epsilon_1+i\epsilon_2= (n+i\kappa)^2 </math>
Conversion between refractive index and dielectric constant is done by:
:<math> \epsilon_1= n^2 - \kappa^2</math>
:<math> \epsilon_2 = 2n\kappa</math>
:<math> n = \sqrt{\frac{\sqrt{\epsilon_1^2+\epsilon_2^2}+\epsilon_1}{2}}</math> <ref>p.49 Frederick Wooten. Optical properties of solids. Academic Press, New York, 1972</ref>
:<math> \kappa = \sqrt{ \frac{ \sqrt{ \epsilon_1^2+ \epsilon_2^2}- \epsilon_1}{2}}</math><ref>p.49 Frederick Wooten. Optical properties of solids. Academic Press, New York, 1972</ref>
==Anisotropy==
[[Image:Calcite.jpg|thumb|A calcite crystal laid upon a paper with some letters showing [[birefringence]]]]
The refractive index of certain media may be different depending on the [[polarization]] and direction of propagation of the light through the medium. This is known as [[birefringence]] or anisotropy and is described by the field of [[crystal optics]]. In the most general case, the ''[[dielectric constant]]'' is a rank-2 [[tensor]] (a 3 by 3 matrix), which cannot simply be described by refractive indices except for polarizations along principal axes.
In magneto-optic (gyro-magnetic) and [[optical activity|optically active]] materials, the principal axes are complex (corresponding to elliptical polarizations), and the dielectric tensor is complex-[[Hermitian]] (for lossless media); such materials break time-reversal symmetry and are used e.g. to construct [[Faraday isolator]]s.
==Nonlinearity==
The strong [[electric field]] of high intensity light (such as output of a [[laser]]) may cause a medium's refractive index to vary as the light passes through it, giving rise to [[nonlinear optics]]. If the index varies quadratically with the field (linearly with the intensity), it is called the [[Kerr effect|optical Kerr effect]] and causes phenomena such as [[self-focusing]] and [[self phase modulation]]. If the index varies linearly with the field (which is only possible in materials that do not possess [[inversion symmetry]]), it is known as the [[Pockels effect]].
==Inhomogeneity==
[[Image:Grin-lens.png|thumb|A gradient-index lens with a parabolic variation of refractive index (''n'') with radial distance (''x''). The lens focuses light in the same way as a conventional lens.]]
If the refractive index of a medium is not constant, but varies gradually with position, the material is known as a gradient-index medium and is described by [[gradient index optics]]. Light travelling through such a medium can be bent or focussed, and this effect can be exploited to produce [[lens (optics)|lenses]], some [[optical fiber]]s and other devices. Some common [[mirage]]s are caused by a spatially-varying refractive index of [[Earth's atmosphere|air]].
==Relation to density==
[[Image:density-nd.GIF|thumb|Relation between the refractive index and the density of silicate and borosilicate glasses (<ref>[http://www.glassproperties.com/refractive_index/ Glassproperties.com]</ref>).]]
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 [[lithium oxide|Li<sub>2</sub>O]] and [[magnesium oxide|MgO]], while the opposite trend is observed with glasses containing [[lead(II) oxide|PbO]] and [[barium oxide|BaO]] as seen in the diagram at the right.
==Momentum Paradox==
The momentum of a refracted ray, ''p'', was calculated by [[Hermann Minkowski]]<ref>Nacht Ges. Wiss. Göttn. Math.-Phys. Kl.53 (1908).</ref> in 1908, where ''E'' is energy of the photon, ''c'' is the speed of light in vacuum and ''n'' is the refractive index of the medium.
:<math>p=\frac{nE}{c}</math>
In 1909 [[Max Abraham]]<ref>Rend. Circ. Matem. Palermo 28, 1 (1909).</ref> proposed
:<math>p=\frac{E}{nc}</math>
[[Rudolf Peierls]] raises this in his "More Surprises in Theoretical Physics" Princeton (1991). Ulf Leonhardt, Chair in Theoretical Physics at the [[University of St Andrews ]] has discussed<ref>{{cite journal|title=Optics: Momentum in an uncertain light|journal=Nature|volume=444|date=14 December 2006|pages=823–824|doi=10.1038/444823a|author=Leonhardt, Ulf}}</ref> this including experiments to resolve.
==Applications==
The refractive index of a material is the most important property of any [[optics|optical]] system that uses [[refraction]]. It is used to calculate the focusing power of lenses, and the dispersive power of prisms.
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 (glasses and gemstones), liquids, and gases. Most commonly it is used to measure the concentration of a [[solute]] in an [[aqueous]] [[solution]]. 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]]).
==See also==
* [[List of refractive indices]]
* [[Optical properties of water and ice]]
* [[Sellmeier equation]]
* [[Total internal reflection]]
* [[Negative refractive index]] or [[Metamaterial]]
* [[Index-matching material]]
* [[Birefringence]]
* [[Calculation of glass properties]], incl. refractive index
* [[Ellipsometry]]
==References==
{{reflist}}
==External links==
*[http://www.tf.uni-kiel.de/matwis/amat/elmat_en/index.html Dielectric materials]
*[http://www.metamaterials.net/ Negative Refractive Index]
*[http://scienceworld.wolfram.com/physics/IndexofRefraction.html Science World]
*[http://RefractiveIndex.INFO/ RefractiveIndex.INFO] refractive index database
[[Category:Optics]]
[[Category:Fundamental physics concepts]]
[[Category:Optical mineralogy]]
[[Category:Glass engineering and science]]
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