﻿ Snell's law

# Snell's law

The law of refraction, also Snell law Snell's law or Snell's law describes the change in direction of the propagation direction of a plane wave in the transition to another medium. Cause the refractive said direction change is the change of the material dependent phase velocity, which is included as a refractive index in the law of refraction. The best-known phenomenon which is described by the Snell's law, the direction of deflection of a light beam during the passage of a media line. The law is not limited to optical phenomena, but is valid for any waves, in particular ultrasound waves.

The law of refraction is named after the Dutch astronomer and mathematician Willibrord van Roijen Snell, in some languages ​​are not published by the Latinized form " Snell ", who found it in 1621, but.

## The law

The direction of the incident beam and the perpendicular to the interface to determine the plane of incidence. In this plane are also crushed and the reflected beam. The angles are measured relative to the perpendicular towards. The law of refraction is the following relationship between the angle of incidence and the angle of the refracted beam:

This includes, and the refractive indices of the respective media. Air has a refractive index which is very close to. In the transition from air to glass can therefore be approximated as the law of refraction:

The refractive index of an optical medium is dependent on the wavelength, in general. This dispersion is included in the law of refraction. Different wavelengths are refracted to different degrees. This is utilized in the dispersion of the light by prisms for separating colors.

The law of refraction is only valid for weakly absorbing media.

## History

Refraction was described by Ptolemy in his " Optics". Its linear law is valid only for small angles. The law of refraction is specified correctly for the first time in the 10th century by Ibn Sahl. In 1601 it was rediscovered by Thomas Harriot, but not published. In 1618 it was used by the Dutch Willibrord van Roijen Snell. Almost at the same time, René Descartes published in 1637 in his Dioptrique a similar context, which is not known whether he found it independently ( he cites no sources ). However, his derivation was wrong, because he went from a higher speed of light in optically dense medium ( properly managed it until Pierre de Fermat from ). The law was already Sad Abu al -Ala ibn Sahl 984 and Thomas Harriot (c. 1601) known.

## Derivation

The refractive index of a medium indicates by how much the phase velocity and there are low and the wavelength is shorter than in a vacuum:

From one medium to another, the wavelength changes by the factor shown on the right in the transition to an optically denser medium (), the shaft is so compressed. This compression leads to distraction.

In the second image, the same process is illustrated schematically. Between two parallel rays of a wave front at two specific points drawn: The wave front has on the one beam, the interface just reached (A ) and must be on the other beam to trace the path L1 ( = | BB '| ) in medium 1 can travel up they ( in B ' ) contacts the interface. For this purpose, the second beam needed in the medium 1, the time:

Similarly, at this time going through the first beam, the distance L2 ( = | A ' | ) in the medium 2 Rearranging and equating to c shows that the route is shorter by the above compression factor as.

Occur between the interface and the two wave fronts the same angle and how. Between the solder and the incident and refracted rays The Gegenkatheten this angle are L1 and L2, the hypotenuse lying in the interface of the length | AB '| they have in common. Consequently applies

And

By rearranging and equating to | AB '| due to the fact

Or with the above-mentioned relationship between the refractive index and the routes and

What is the law of refraction equivalent.

## Relation to Fermat's principle

The law of refraction can also be inferred from the Fermat's principle, which states that small changes of the path that the light between two points P and Q does not change the optical path length. In the case of a systematic variation of refraction would be to shift the inflection point within the interface, for example from A to B ' in the above image. The shift is so small compared to the distance to the points P and Q, that not change the angle, the geometric path in the medium increased by 1 L1, while L2 is added in the medium. Because of different phase velocity overall phase does not change.

## Total reflection

For sufficiently large and

And hence satisfiable by any ( real ). In such cases, total reflection occurs, in which the light is totally reflected.

For the critical angle of total reflection does equality, ie

Total reflection is used, for example in the reverse prism binoculars.

## Optical enhancement

Looking from outside the water items that are underwater, they appear compressed in the vertical direction. The bottom of the vessel appears higher than that of an image of the same scene without water. This phenomenon is therefore also called optical enhancement. On a straight rod which dips at an angle into the water, you can see a kink at the water surface. Due to the different refractive indices of water and air a different angle of refraction of the rod coming from the eye light rays produced above and below the water surface at the interface with the glass. The human brain is not taken into account these different angle of refraction, and the beams extend in a straight line backwards, so that the bar appears to be flat under water rod water.

## Acoustics

For mechanical waves, that is, pressure or shear wave, the Snell's law applies. As part of the acoustic or ultrasonic technology Snell's law but is formulated without refractive indices, but with the help of the wave number. It applies ( see picture of the angular terms ):

With the definition we obtain the law of refraction in the formulation of the phase velocities of the respective wave types in the relevant medium and thus the same formulation as in optics (if you would out of there cut the vacuum speed of light). The derivation of the law in acoustics happens on the demand for the fulfillment of the continuity equation for mechanical stresses and displacements at the media boundary. The picture shows an incident longitudinal wave in a solid body, which is partially reflected at an interface to a second solid body and transmits. In general, arise at the interface of the incident longitudinal wave (P- wave) new wave types, so that two different types of waves are reflected and transmitted P - waves and S- waves ( shear wave ). Both types of waves propagate at different phase velocities in the two media, so they are also broken at different angles. These angles can be calculated with the above law, if the individual phase velocities and the angle of incidence of the primary shaft are known. In the case of shear-stress- free media (liquids and gases ) occur no shear waves, so that the incident P-wave only a reflected and produce a transmitted P- wave would.

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