thumb|A [[ray (optics)|ray of light being refracted through a glass slab|alt=refer to caption]]
thumb|170px|Refraction of a light ray|alt=Illustration of the incidence and refraction angles
In optics, the refractive index (also called refraction index or index of refraction), often denoted n, is the ratio of the speed of light in vacuum (c) to the speed of light in a given optical medium (v), . The refractive index determines how much the path of light is bent, or refracted, when entering a material, as described by Snell's law of refraction, , where and are the angle of incidence and angle of refraction, respectively, of a ray crossing the interface between two media with refractive indices and . 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, their intensity (Fresnel equations) and Brewster's angle.
The refractive index, <math>n</math>, 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 , and similarly the wavelength in that medium is , where is the wavelength of that light in vacuum. This implies that vacuum has a refractive index of 1, and assumes that the frequency () 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 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. Consequently, refractive indices for materials reported using a single value for must specify the wavelength used in the measurement.
The concept of refractive index applies across 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. Refraction also occurs in oceans when light passes into the halocline where salinity has impacted the density of the water column.
For lenses (such as eye glasses), a lens made from a high refractive index glass will be thinner, and hence lighter, than a – usually cheaper – conventional lens with a lower refractive index.
Plastics materials tend to have lower refractive indices than glasses, but have significantly less density than glasses. For many years,<!-- 1980? --> the lightest eyeglasses have been fabricated from plastics.
Definition
The relative refractive index of an optical medium 2 with respect to another reference medium 1 () is given by the ratio of speed of light in medium 1 to that in medium 2. This can be expressed as follows:
<math display="block">n_{21}=\frac{v_1}{v_2}.</math>
If the reference medium 1 is vacuum, then the refractive index of medium 2 is considered with respect to vacuum. It is simply represented as 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 in vacuum, , and the phase velocity of light in the medium,
<math display="block">n=\frac{\mathrm{c{v}.</math>
Since is constant, is inversely proportional to :
<math display="block">n\propto\frac{1}{v}.</math>
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. At the same time he changed this value of refractive power into a single number, instead of the traditional ratio of two numbers. Using a ratio had the disadvantage of it being given inconsistent notation: 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: , , and . The symbol gradually prevailed.
Typical values
<!-- the lead section links here -->
left|thumb|alt=Gemstone diamonds|[[Diamonds have a very high refractive index of 2.417.]]
Refractive index also varies with wavelength of the light as given by Cauchy's equation. The most general form of this equation is
<math display="block"> n(\lambda) = A + \frac {B}{\lambda^2} + \frac{C}{\lambda^4} + \cdots,</math>
where is the refractive index, is the wavelength, and , , , etc., are coefficients that can be determined for a material by fitting the equation to measured refractive indices at known wavelengths. The coefficients are usually quoted for as the vacuum wavelength in micrometres.
Usually, it is sufficient to use a two-term form of the equation:
<math display="block"> n(\lambda) = A + \frac{B}{\lambda^2},</math>
where the coefficients and are determined specifically for this form of the equation.
{| class="wikitable plainrowheaders" style="float:right;"
|+ Selected refractive indices at .
For references, see the extended List of refractive indices.
! scope="col" | Material
! scope="col" |
|-
! scope="row" | Vacuum
|
|-
| colspan="2" style="text-align:center;"| Gases at 0 °C and 1 atm
|-
! scope="row" | Air
|
|-
! scope="row" | Helium
|
|-
! scope="row" | Hydrogen
|
|-
! scope="row" | Carbon dioxide
|
|-
| colspan="2" style="text-align:center;"| Liquids at 20 °C
|-
! scope="row" | Water
| 1.333
|-
! scope="row" | Ethanol
| 1.36
|-
! scope="row" | Olive oil
| 1.47
|-
| colspan="2" style="text-align:center;"| Solids
|-
! scope="row" | Ice
| 1.31
|-
! scope="row" | Fused silica (quartz)
| 1.46
|-
! scope="row" | PMMA (acrylic, plexiglas, lucite, perspex)
| 1.49
|-
! scope="row" | Window glass
| 1.52
|-
! scope="row" | Polycarbonate (Lexan™)
| 1.58
|-
! scope="row" | Flint glass (typical)
| 1.69
|-
! scope="row" | Sapphire
| 1.77
|-
! scope="row" | Cubic zirconia
| 2.15
|-
! scope="row" | Diamond
| 2.417
|-
! scope="row" | 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. 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 light refractive indices can be considerably higher. Germanium is transparent in the wavelength region from and has a refractive index of about 4. Topological insulators can 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.{2, \\
\kappa &= \sqrt{\frac{|\underline{\varepsilon}_\mathrm{r}| - \varepsilon_\mathrm{r{2.
\end{align}</math>
where <math>|\underline{\varepsilon}_\mathrm{r}| = \sqrt{\varepsilon_\mathrm{r}^2 + \tilde{\varepsilon}_\mathrm{r}^2}</math> is the complex modulus.
Wave impedance
The wave impedance of a plane electromagnetic wave in a non-conductive medium is given by
<math display="block">\begin{align}
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}
\end{align}</math>
where is the vacuum wave impedance, and are the absolute permeability and permittivity of the medium, is the material's relative permittivity, and is its relative permeability.
In non-magnetic media (that is, in materials with ), <math>Z = {Z_0 \over n}</math> and <math>n = {Z_0 \over Z}\,.</math>
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 between two media can thus be expressed both by the wave impedances and the refractive indices as
<math display="block">\begin{align}
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\,.
\end{align}</math>
Density
thumb|upright=1.7|alt=A scatter plot showing a strong correlation between glass density and refractive index for different glasses|The relation between the refractive index and the density of [[silicate glass|silicate and borosilicate glasses]]
In general, it is assumed that the refractive index of a glass increases with its density. 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 lithium oxide| and magnesium oxide|, while the opposite trend is observed with glasses containing lead(II) oxide| and barium oxide| as seen in the diagram at the right.
Many oils (such as olive oil) and ethanol 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, 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. For liquids the same observation can be made as for gases, for instance, the refractive index in alkanes increases nearly perfectly linear with the density. On the other hand, for carboxylic acids, the density decreases with increasing number of C-atoms within the homologeous series. The simple explanation of this finding is that it is not density, but the molar concentration of the chromophore that counts. In homologeous series, this is the excitation of the C-H-bonding. August Beer must have intuitively known that when he gave Hans H. Landolt in 1862 the tip to investigate the refractive index of compounds of homologeous series. While Landolt did not find this relationship, since, at this time dispersion theory was in its infancy, he had the idea of molar refractivity which can even be assigned to single atoms. Based on this concept, the refractive indices of organic materials can be calculated.
Bandgap
thumb|A scatter plot of bandgap energy versus optical refractive index for many common IV, III-V, and II-VI semiconducting elements / compounds.
The optical refractive index of a semiconductor tends to increase as the bandgap energy decreases. Many attempts have been made to model this relationship beginning with T. S. Moses in 1949. Empirical models can match experimental data over a wide range of materials and yet fail for important cases like InSb, PbS, and Ge.
This negative correlation between refractive index and bandgap energy, along with a negative correlation between bandgap and temperature, means that many semiconductors exhibit a positive correlation between refractive index and temperature. This is the opposite of most materials, where the refractive index decreases with temperature as a result of a decreasing material density.
Group index
Sometimes, a "group velocity refractive index", usually called the group index is defined:
<math display="block">n_\mathrm{g} = \frac{\mathrm{c{v_\mathrm{g,</math>
where is the group velocity. This value should not be confused with , 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
<math display="block">v_\mathrm{g} = v - \lambda\frac{\mathrm{d}v}{\mathrm{d}\lambda},</math>
where 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
<math display="block">n_\mathrm{g} = \frac{n}{1 + \frac{\lambda}{n}\frac{\mathrm{d}n}{\mathrm{d}\lambda.</math>
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)
<math display="block">\begin{align}
v_\mathrm{g} &= \mathrm{c}\!\left(n - \lambda_0 \frac{\mathrm{d}n}{\mathrm{d}\lambda_0}\right)^{-1}\!, \\
n_\mathrm{g} &= n - \lambda_0 \frac{\mathrm{d}n}{\mathrm{d}\lambda_0},
\end{align}</math>
where is the wavelength in vacuum.
Velocity, momentum, and polarizability
As shown in the Fizeau experiment, when light is transmitted through a moving medium, its speed relative to an observer traveling with speed in the same direction as the light is:
<math display="block">\begin{align}
V &= \frac{\mathrm{c{n} + \frac{v \left(1 - \frac{1}{n^2} \right)}{1 + \frac{v}{c n \\
&\approx \frac{\mathrm{c{n} + v \left(1 - \frac{1}{n^2} \right)\,.
\end{align}</math>
The momentum of photons in a medium of refractive index is a complex and controversial issue with two different values having different physical interpretations.
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 , often rescaled as either or ; the multiplication factors are used because the refractive index for air, 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
<math display="block">A = \frac{M}{\rho} \cdot \frac{n^2 - 1}{n^2 + 2}\ ,</math>
where is the density, and is the molar mass. 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.
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, or with a digital photodetector placed in the focal plane of a lens. The refractive index of the liquid can then be calculated from the maximum transmission angle as , where is the refractive index of the prism.
thumb|alt=A small cylindrical refractometer with a surface for the sample at one end and an eye piece to look into at the other end|A handheld refractometer used to measure the sugar content of fruits
This type of device is 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 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
thumb|alt=Budding yeast cells with dark borders to the upper left and bright borders to lower right|A [[differential interference contrast microscopy image of budding yeast cells]]
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 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 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.
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. 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. It can be used, for example, 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 the refractive index. For a solution of sugar, the refractive index can be used to determine the sugar content (see Brix).
See also
- Calculation of glass properties
- Clausius–Mossotti relation
- Ellipsometry
- Fermat's principle
- Index ellipsoid
- Index-matching material
- Laser Schlieren deflectometry
- Optical properties of water and ice
- Phase-contrast X-ray imaging
- Prism-coupling refractometry
- Velocity factor
Footnotes
References
External links
- NIST calculator for determining the refractive index of air
- Dielectric materials
- Science World
- Filmetrics' online database Free database of refractive index and absorption coefficient information
- RefractiveIndex.INFO Refractive index database featuring online plotting and parameterisation of data
- LUXPOP Thin film and bulk index of refraction and photonics calculations
- The Feynman Lectures on Physics Vol. II Ch. 32: Refractive Index of Dense Materials