Optical path length are determined by the
optical path length through the thin soap film in a phenomenon called
thin-film interference.
Optical path length (OPL) is the product of the geometric length of the path light follows through a system, and the index of refraction of the medium through which it propagates, \text{OPL} = nd. This is an important concept in optics because it determines the
phase of the light and governs
interference and
diffraction of light as it propagates. According to
Fermat's principle, light rays can be characterized as those curves that
optimize the optical path length. 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 can be seen at the air-water boundary. If there is no angle fulfilling Snell's law, i.e., \frac{n_1}{n_2} \sin \theta_1 > 1, the light cannot be transmitted and will instead undergo
total internal reflection. \theta_\mathrm{c} = \arcsin\!\left(\frac{n_2}{n_1}\right)\!.
Reflectivity Apart from the transmitted light there is also a
reflected part. The reflection angle is equal to the incidence angle, and the amount of light that is reflected is determined by the
reflectivity of the surface. The reflectivity can be calculated from the refractive index and the incidence angle with the
Fresnel equations, which for
normal incidence reduces to At other incidence angles the reflectivity will also depend on the
polarization of the incoming light. At a certain angle called
Brewster's angle,
p-polarized light (light with the electric field in the
plane of incidence) will be totally transmitted. Brewster's angle can be calculated from the two refractive indices of the interface as \frac{1}{f} = (n - 1)\left[\frac{1}{R_1} - \frac{1}{R_2}\right]\ , where is the focal length of the lens.
Microscope resolution The
resolution of a good optical
microscope is mainly determined by the
numerical aperture () of its
objective lens. The numerical aperture in turn is determined by the refractive index of the medium filling the space between the sample and the lens and the half collection angle of light according to Carlsson (2007): A_\mathrm{Num} = 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. The refractive index is used for optics in
Fresnel equations and
Snell's law; while the relative permittivity and permeability are used in
Maxwell's equations and electronics. Most naturally occurring materials are non-magnetic at optical frequencies, that is is very close to 1, therefore is approximately . In this particular case, the complex relative permittivity , with real and imaginary parts and , and the complex refractive index , with real and imaginary parts and (the latter called the "extinction coefficient"), follow the relation \underline{\varepsilon}_\mathrm{r} = \varepsilon_\mathrm{r} + i\tilde{\varepsilon}_\mathrm{r} = \underline{n}^2 = (n + i\kappa)^2, and their components are related by: \begin{align} \varepsilon_\mathrm{r} &= n^2 - \kappa^2\,, \\ \tilde{\varepsilon}_\mathrm{r} &= 2n\kappa\,, \end{align} and: \begin{align} n &= \sqrt{\frac{|\underline{\varepsilon}_\mathrm{r}| + \varepsilon_\mathrm{r}}{2}}, \\ \kappa &= \sqrt{\frac{|\underline{\varepsilon}_\mathrm{r}| - \varepsilon_\mathrm{r}}{2}}. \end{align} where |\underline{\varepsilon}_\mathrm{r}| = \sqrt{\varepsilon_\mathrm{r}^2 + \tilde{\varepsilon}_\mathrm{r}^2} is the
complex modulus.
Wave impedance The wave impedance of a plane electromagnetic wave in a non-conductive medium is given by \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} 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 ), Z = {Z_0 \over n} and n = {Z_0 \over Z}\,. Thus refractive index in a non-magnetic media is the ratio of the vacuum wave impedance to the wave impedance of the medium. The reflectivity between two media can thus be expressed both by the wave impedances and the refractive indices as \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}
Density 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 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: n_\mathrm{g} = \frac{\mathrm{c}}{v_\mathrm{g}}, 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 v_\mathrm{g} = v - \lambda\frac{\mathrm{d}v}{\mathrm{d}\lambda}, 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 n_\mathrm{g} = \frac{n}{1 + \frac{\lambda}{n}\frac{\mathrm{d}n}{\mathrm{d}\lambda}}. When the refractive index of a medium is known as a function of the vacuum wavelength (instead of the wavelength in the medium), the corresponding expressions for the group velocity and index are (for all values of dispersion) \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} 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: \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} 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 A = \frac{M}{\rho} \cdot \frac{n^2 - 1}{n^2 + 2}\ , where is the
density, and is the
molar mass. ==Nonscalar, nonlinear, or nonhomogeneous refraction==