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# Refraction

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

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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.

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

$${n_1}\sin {\theta _1} = {n_2}\sin {\theta _2}$$

where <$n_1$> and <$n_2$> 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.

## Calculator for Snell's Law

 Refractive index of medium 1: Refractive index of medium 2: Angle of incidence: calc Output angle: calc

Enter input values with units, where appropriate. After you have modified some values, click a “calc” button to recalculate the field left of it.

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 (<$\theta_1$>): 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.

### Bibliography

  H. Kosaka et al., “Superprism phenomena in photonic crystals”, Phys. Rev. B 58, R10096 (1998), DOI:10.1103/PhysRevB.58.R10096  H. Kosaka et al., “Superprism phenomena in photonic crystals: toward microscale lightwave circuits”, J. Lightwave Technol. 17 (11), 2032 (1999), DOI:10.1109/50.802991  S. Foteinopoulou, E. N. Economou and C. M. Soukoulis, “Refraction in media with a negative refractive index”, Phys. Rev. Lett. 90, 107402 (2003), DOI:10.1103/PhysRevLett.90.107402  E. Cubukcu et al., “Negative refraction by photonic crystals”, Nature 423, 604 (2003), DOI:10.1038/423604b  N. Yu, “Light propagation with phase discontinuities: Generalized laws of reflection and refraction”, Science 334 (6054), 333 (2011), DOI:10.1126/science.1210713  F. Aieta et al., “Out-of-plane reflection and refraction of light by anisotropic optical antenna metasurfaces with phase discontinuities”, Nano Lett. 12, 1702 (2012), DOI:10.1021/nl300204s  S. Larouche and D. R. Smith, “Reconciliation of generalized refraction with diffraction theory”, Opt. Lett. 37 (12), 2391 (2012), DOI:10.1364/OL.37.002391

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