折射率(refractive index)

定义：光速在介质中减小的因子。

透明介质的折射率，是相速度vph相对于真空光速减小的因子： 

这里假设平面波是线性传播的（光强比较低）。折射率通过相速度决定界面处的折射，反射和衍射现象。 介质中光的波长等于真空波长的n分之一。 

根据材料的相对介电常数 ε 和相对磁导率 μ可以计算其折射率： 

需要注意的是，这里ε和μ是处于光频率时的值，与其在低频时的值差别很大。普通的光学材料的磁导率约为1。 

材料中折射率与光频率或波长有关的现象称为色散。普通的玻璃和晶体（例如，激光晶体）在可见光区域的折射率范围是1.4-2.8，并且波长越短时，折射率会增加（正常色散）。这是以下现象的一个结果，可见光区域（介质具有很大的透射率），在两个强吸收区域之间：紫外光区域光子能量大于带隙，而近红外或中红外光区域会发生振动共振。 

图1：二氧化硅在温度分别为0 °C（蓝色），100 °C（黑色）和200 °C（红色）时折射率（实线）和群速度（虚线）随波长的变化。 

半导体在透明区域具有更大的折射率。例如，砷化镓的折射率在1微米时约为3.5。这是由于当波长小于带隙波长（约870nm）时，发生强吸收现象。高折射率的结果就是菲涅尔反射很强，并且在半导体-空气界面处的全反射角比较大。 

透明光学材料中折射率与波长有关的现象可以由Sellmeier方程描述，方程包含一些经验得到的参数。该方程的拓展版本可以描述温度特性，图1中即采用了这一方程。

在非线性晶体材料中发生的非线性频率转换中的相位匹配需要知道折射率随温度和波长变化的具体情况。 

在非各向同性介质中，折射率与偏振方向（参阅双折射）和传播方向（非各向同性）有关。如果介质具有光轴，光在该轴上传播时折射率与偏振方向无关。 复折射率不仅可以定量表示单位长度的相位变化，还可以表示（虚部）光学增益或传播损耗（例如，由于吸收产生）。 

还有一种折射率为群折射率，可以量化群速度的减小。在共振情况下折射率会与群折射率差别很大，这在有些量子光学实验中可以看到。在群速度非常大或者非常小时会用到（慢光）。 

有些光子超材料（通常包含金属-介电复合材料）可以实现负折射率，最早是在微波区域实现，现在光学领域也得到了。负折射率会引起很多非常规的现象。例如，在真空与该介质界面处的折射光束与入射光束位于法线的同一侧。 

在波导中，每一个传播模式都对应一个有效折射率，与其相速度相关。

Definition: a measure of the reduction in the velocity of light in a medium

Alternative term: index of refraction

The refractive index n of a transparent optical medium, also called the index of refraction, is the factor by which the phase velocity vph is decreased relative to the velocity of light in vacuum:

Here, one assumes linear propagation (i.e. with low optical intensities) of plane waves. Via the phase velocity, the refractive index also determines phenomena such as refraction, reflection and diffraction at optical interfaces; see the article on Fresnel equations.

The wavelength of light in the medium is n times smaller than the vacuum wavelength.

The refractive index can be calculated from the relative permittivity ε and the relative permeability μ of an optical material:

Note that the values of ε and μ at the optical frequency have to be used, which can deviate substantially from those at low frequencies. For usual optical materials, μ is close to unity.

The refractive index of a material depends on the optical frequency or wavelength; this dependency is called chromatic dispersion. Typical refractive index values for glasses and crystals (e.g. laser crystals) in the visible spectral region are in the range from 1.4 to 2.8, and typically the refractive index increases for shorter wavelengths (normal dispersion). This is a consequence of the fact that the visible spectral region, with high transmission of such materials, lies between spectral regions of strong absorbance: the ultraviolet region with photon energies above the band gap energy, and the near- or mid-infrared region with vibrational resonances and their overtones. Note that the refractive index at one wavelength can be influenced by absorption in any spectral regions, as described by Kramers–Kronig relations.

The wavelength-dependent refractive index of a transparent optical material can often be described analytically with a Sellmeier formula, which contains several empirically obtained parameters. Extended versions of such equations also describe the temperature dependence; such an equation has been used for Figure 1.

Figure 1: Refractive index (solid lines) and group index (dotted lines) of silica versus wavelength at temperatures of 0 °C (blue), 100 °C (black) and 200 °C (red). The plots are based on data from M. Medhat et al., J. Opt. A: Pure Appl. Opt. 4, 174 (2002).

The refractive index is generally also dependent on the temperature of the material (see Figure 1). In many cases, it rises with increasing temperature, but particularly for glasses the opposite is often the case, essentially because the density decreases with temperature.

Further modifications of the refractive index can occur through mechanical stress (photoelastic effect). Of course, changing the chemical composition e.g. by doping a material with some impurities can also affect the refractive index; this is widely used for graded-index lenses and for optical fibers, for example. In case of rare-earth-doped laser crystals, the refractive index change caused by the doping is often quite small due to a low doping concentration.

Compared with glasses, for example, semiconductors exhibit much higher refractive indices in their transparency region. For example, gallium arsenide (GaAs) has a refractive index of ≈ 3.5 at 1 μm. This is caused by the strong absorption at wavelengths below the bandgap wavelength of ≈ 870 nm. Consequences of the high index of refraction are strong Fresnel reflections and a large critical angle for total internal reflection at semiconductor–air interfaces.

A precise knowledge of the wavelength and temperature dependence of the refractive index is important for phase matching of nonlinear frequency conversion in nonlinear crystal materials.

In anisotropic optical materials, the refractive index generally depends on the polarization direction (→ birefringence) and the propagation direction (anisotropy). If a medium has a so-called optical axis, the refractive index for light propagation along this axis does not depend on the polarization direction.

A complex refractive index is sometimes used to quantify not only the phase change per unit length, but also (via its imaginary part) propagation losses (e.g. caused by absorption) or optical gain. For light propagation in z direction, the optical intensity evolves according to

where one can see that the absorption coefficient α can be calculated as:

(Note that different sign conventions are used in the literature.)

There is another type of refractive index, the group index, which quantifies the reduction in the group velocity. Extreme excursions of the refractive index and particularly the group index can occur near sharp resonances, as are observed in certain quantum optics experiments. This can be related to extremely large or small values of the group velocity (slow light).

For waveguides, each propagation mode can be assigned an effective refractive index according to its phase velocity.

Even a negative refractive index is possible for certain photonic metamaterials (usually consisting of metal–dielectric composites), which have been demonstrated first in the microwave region, but begin to become a reality also in the optical domain. Negative refractive index values give rise to a range of intriguing phenomena such as negative refraction [4]. For example, refraction at the interface between vacuum and such a material works such that the refracted beam is on the same side of the surface normal as the incident beam. Such phenomena also occur in certain photonic crystals.

Negative refractive indices also sometimes occur in geometrical optics, because some authors formally assume the sign of the refractive index to be flipped upon reflection on a surface. With that convention, one can apply certain equations for refraction phenomena also to reflecting surfaces.

In many situations, the refractive index contrast between two different transparent media is relevant. Some examples:

Various calculations are simulations can be done in mathematically simplified ways for cases with weak refractive index contrast. For example, the modes of fibers can then be calculated as LP modes, based on a scalar description of the light field amplitudes (i.e., ignoring the vector nature). The criterion for a refractive index contrast to be weak depend on the circumstances, but often it means that the refractive index difference must be far smaller than 1.