定义：一束光从一个介质射向另一个介质时传播方向会发生改变的现象。

图1：两介质界面处的折射。

当光（例如激光光束）从一个透明各向同性介质中进入另一个，其传播方向会发生改变（见图1）。这种现象称为折射。这是由于在两介质边界处，入射光和透射光需要满足边界条件。

波矢的切向分量相等，不然在边界处两波的相位差与位置有关，然后波前就不连续了。由于波矢的大小与介质的折射率有关，边界条件只会在某些传播方向上满足。正入射的情况例外，这时波矢在界面上没有分量。 

根据以上讨论，可以得到斯涅耳定律： 

其中n1和n2是两个介质的折射率。很显然折射率小的介质对应的光束与法线之间的夹角更大。 

图2是折射现象的动态图。可以看到右侧介质中波长更短（由于光速变慢），右侧介质折射率大。并且在界面处波前并没有发生变化，只是方向发生了变化。这是由于传播角度发生了变化。 

图2：折射现象的动态图。（这里只能截取静态图）波前由灰色线表示。波前方向的变化来自于右侧介质中光速减小。 

补充说明 
如果入射光来自于折射率大的介质，并且入射角度比较大，那么并不是所有的出射角度都满足斯涅耳定律，因为出射角的正弦值最大为1。这种情况下不可能得到透射光，发生了全反射。 

在非各向同性介质中，折射率与光的偏振方向有关。因此，折射角也与偏振有关。 
折射在基础光学和光子学领域经常碰到： 

1. 大多数光学透镜中会用到折射。 
2. 在棱镜中，需要分离开不同波长成分时，需要采用与波长相关的折射效应。 
3. 采用双折射材料也可以得到与偏振有关的角度。这在有些偏振器中会用到。 
4. 有时需要采用与折射引起的光束偏移有关的光力。 

折射率为负值的光子超材料会产生不寻常的负折射效应。这时折射光与入射光位于法线的同一侧。

Definition: the change of the propagation direction when a wave comes from one medium into another one

When light, e.g. a laser beam, propagates from one transparent homogeneous medium into another, its propagation direction will generally change (see Figure 1). This phenomenon is called refraction. It results from the boundary conditions which the incoming and the transmitted wave need to fulfill at the boundary between the two media. Essentially, the tangential components of the wave vectors need to be identical, as otherwise the phase difference between the waves at the boundary would be position-dependent, and the wavefronts could not be continuous. As the magnitude of the wave vector depends on the refractive index of the medium, the said condition can in general only be fulfilled with different propagation directions. An exception is of course the case of normal incidence, where the wave vectors have no component along the surface.

Figure 1: Refraction at an interface between two media.

From the above considerations, one can easily derive Snell's law (the law of Snellius) for the angles:

where n1 and n2 are the refractive indices of the two media. It is apparent that the larger angle against the normal direction must occur in the medium with the smaller refractive index.

Figure 2 is an animated illustration of refraction. One can see that the wavelength is smaller on the right side (as a result of the reduced velocity of light), where the refractive index is larger. Also, the wavefronts are not interrupted at the interface, but only changed in direction. This is possible only with a modified angle of propagation.

Figure 2: An animated illustration of refraction. The wavefronts are shown as gray lines. The change of direction results from the reduced velocity of light in the right medium.

Additional Remarks

If the incident beam comes from the medium with the higher refractive index, and its angle of incidence is large, it may not be possible to fulfill Snell's law with any output angle, since the sine of the output angle can be at most 1. In that case, transmission is not possible – total internal reflection occurs.

For the reflected beam, the angle against the surface normal is always the same as that of the incident beam (θ1): its direction is not affected by refraction.

For non-isotropic media, the refractive index can depend on the polarization direction of the light. Therefore, the refraction angle can be polarization-dependent.

The amplitude transmission and reflection coefficients of the boundary are described by the Fresnel equations.

The phenomenon of refraction is very often encountered in general optics and other fields of photonics:

• Refraction is used in most optical lenses.
• One may exploit wavelength-dependent refraction angles e.g. in prisms for separating different wavelength components.
• With birefringent optical materials, one may also obtain polarization-dependent angles. This is exploited, for example, in some types of polarizers.
• In some cases, light forces related to beam deflections by refraction are relevant.

Unusual phenomena such a negative refraction [3, 4] can occur with photonic metamaterials with a negative refractive index, at certain photonic metasurfaces, or with certain photonic crystals. Here, the refracted beam can be on the same side of the surface normal as the incident beam.

Another curious refraction phenomenon are superprism effects [1, 2], where the refraction angle exhibits an unusually strong dependence of the propagation direction at the output on details like the input direction and the optical wavelength.

For electromagnetic radiation at very high frequencies, such as X-rays, refraction effects are far weaker than for light, since the refractive index is then close to 1.

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