# Snellius' law of refraction

Refraction and reflection of a light beam when it enters glass. The beam reflected to the top right has the same angle to the perpendicular to the surface as the beam incident from the top left (60 ° in each case). The angle of the beam penetrating the glass (angle of refraction) is only 35 ° in accordance with the law of refraction.

The law of refraction , also Snellius' law of refraction , Snellius' law or Snellius' law describes the change in direction of the direction of propagation of a plane wave when passing into another medium . The cause of the change in direction, called refraction , is the change in the material- dependent phase velocity , which is included in the law of refraction as a refractive index. The best-known phenomenon, which is described by the law of refraction, is the directional deflection of a light beam when passing a media boundary. The law is not limited to optical phenomena, but applies to any wave, especially ultrasonic waves .

The law of refraction is named after the Dutch astronomer and mathematician Willebrord van Roijen Snell , in some languages ​​after the Latinized form "Snellius", who was not the first to find it in 1621, but was the first to publish it.

## The law

Angular dependence in the refraction for the media air, water and glass. Proportionality is a good approximation up to angles of around 50 ° .

The direction of the incident beam and the perpendicular to the interface determine the plane of incidence . The refracted and reflected rays also lie in this plane. The angles are measured towards the perpendicular. The law of refraction is the following relationship between the angle of incidence and the angle of the refracted ray: ${\ displaystyle \ delta _ {1}}$${\ displaystyle \ delta _ {2}}$

${\ displaystyle n_ {1} \ sin \ delta _ {1} = n_ {2} \ sin \ delta _ {2}}$.

It contains and the refractive indices of the respective media. Air has an index of refraction that is very close to . In the transition from air to glass, the law of refraction can therefore be approximated as: ${\ displaystyle n_ {1}}$${\ displaystyle n_ {2}}$${\ displaystyle n = 1}$

${\ displaystyle \ sin \ delta _ {\ mathrm {air}} = n _ {\ mathrm {glass}} \ sin \ delta _ {\ mathrm {glass}}}$.

The refractive index of an optical medium is generally dependent on the wavelength. This dispersion is included in the law of refraction. Different wavelengths are refracted to different degrees. This is used in dispersion prisms to separate the light into colors.

The law of refraction only applies to weakly absorbing media.

## history

Refraction was described by Ptolemy in his work "Optics". Its linear law is only valid for small angles. The law of refraction was correctly stated for the first time in the 10th century by Ibn Sahl . The law was rediscovered but not published by Thomas Harriot in 1601 and by Willebrord van Roijen Snell around 1621 . While Harriot's discovery did not become public until 350 years later, Snellius' contribution was made known by Jacob Golius in 1632. Almost at the same time and presumably independently of Snellius, René Descartes published a similar connection in his Dioptrique in 1637 . His derivation was wrong, however, as he assumed a higher speed of light in the optically denser medium ( Pierre de Fermat first correctly derived it ).

## Derivation

A plane wave that propagates across an interface. Behind the interface, the medium has a higher refractive index and the wave has a shorter wavelength .

The refractive index of a medium indicates by how much the phase velocity and the wavelength are lower or shorter than in a vacuum: ${\ displaystyle n}$ ${\ displaystyle c}$ ${\ displaystyle \ lambda}$

${\ displaystyle n = {\ frac {c} {c _ {\ rm {Medium}}}} = {\ frac {\ lambda} {\ lambda _ {\ rm {Medium}}}} \ ,.}$

From one medium to another, the wavelength changes by the factor ; in the transition to an optically denser medium shown on the right ( ), the wave is compressed. This compression leads to the distraction. ${\ displaystyle n_ {1} / n_ {2}}$${\ displaystyle n_ {2}> n_ {1}}$

Schematic representation for the derivation of the law of refraction

In the 2nd picture the same process is shown schematically. A wavefront is drawn in at two special points between two parallel rays: The wavefront has just reached the interface on one ray (A) and has to cover the distance L 1 (= | BB '|) in medium 1 on the other ray until it touches the interface (at B '). For this, the second beam in medium 1 needs the time : ${\ displaystyle t}$

${\ displaystyle t = {\ frac {L _ {\ mathrm {1}}} {c _ {\ mathrm {Medium, 1}}}} = L _ {\ mathrm {1}} {\ frac {n_ {1}} { c}} \ ,.}$

Analogously to this, the first ray traverses the distance L 2 (= | AA '|) in the medium 2 during this time . By converting and equating according to c , the distance is shorter than the above compression factor . ${\ displaystyle L_ {2}}$${\ displaystyle n_ {1} / n_ {2}}$${\ displaystyle L_ {1}}$

The same angles and occur between the interface and the two wavefronts as between the perpendicular and the incident or refracted rays. The opposite cathets of these angles are L 1 and L 2 , respectively , the hypotenuse of length | AB '| lying in the interface they have in common. Hence applies ${\ displaystyle \ delta _ {1}}$${\ displaystyle \ delta _ {2}}$

${\ displaystyle \ sin \ delta _ {1} = {\ frac {L_ {1}} {| AB '|}}}$

and

${\ displaystyle \ sin \ delta _ {2} = {\ frac {L_ {2}} {| AB '|}}}$.

By converting and equating to | AB '| results from it

${\ displaystyle {\ frac {\ sin \ delta _ {1}} {\ sin \ delta _ {2}}} = {\ frac {L_ {1}} {L_ {2}}} \ ,,}$

or with the above-mentioned relationship between the refractive index and the distances and${\ displaystyle L_ {1}}$${\ displaystyle L_ {2}}$

${\ displaystyle {\ frac {\ sin \ delta _ {1}} {\ sin \ delta _ {2}}} = {\ frac {L_ {1}} {L_ {2}}} = {\ frac {n_ {2}} {n_ {1}}} \ ,,}$

which is equivalent to the law of refraction.

## Relationship to Fermat's Principle

The law of refraction can also be deduced from Fermat's principle, which states that small changes in the path that light takes between two points P and Q do not change the optical path length . In the case of refraction, a systematic variation would be the displacement of the kink point within the interface, for example from A to B 'in the figure above. In the case of the displacement, which is so small compared to the distance to points P and Q that the angles do not change, the geometric path in medium 1 increases by L 1 , while in medium 2 L 2 is added. Because of the different phase velocities, the phase does not change overall.

## Total reflection

For and sufficiently large is ${\ displaystyle n_ {1}> n_ {2}}$${\ displaystyle \ delta _ {1}}$

${\ displaystyle \ sin \ delta _ {2} = {\ frac {n_ {1}} {n_ {2}}} \ sin \ delta _ {1}> 1}$

and thus cannot be fulfilled by anything (real) . In these cases total reflection occurs, in which the light is completely reflected. ${\ displaystyle \ delta _ {2}}$

Equality applies to the critical angle of total reflection , i.e. ${\ displaystyle \ delta _ {g}}$

${\ displaystyle \ sin \ delta _ {g} = {\ frac {n_ {2}} {n_ {1}}} \ ,.}$

Total reflection is used, for example, in the erecting prisms of binoculars .

## Optical enhancement

Due to the refraction on the surface, objects under water appear shortened in a vertical direction. Straight objects plunging in at an angle seem to have a kink on the surface.

If you look at objects under water from outside the water, they appear compressed in a vertical direction. The bottom of the vessel appears higher than in a picture of the same scene without water. This phenomenon is therefore also called optical enhancement . On a straight rod that dips into the water at an angle, you can see a kink on the surface of the water. Due to the different refractive indices of water and air, a different angle of refraction of the light rays coming from the rod into the eye occurs above and below the water surface at the interface with the glass. The human brain does not take these different angles of refraction into account and extends the rays in a straight line backwards, so that the rod appears flatter under water than the rod above water.

## Acoustics

Mode conversion of an ultrasonic wave. An incident longitudinal wave ( "P" for English.
Pressure wave , the pressure wave ) is partially as a shear wave ( ), and pressure wave ( reflected) and transmitted ( and ). The nomenclature is as follows: The first letter stands for the wave type of the causative wave (primary wave) and the second letter for the type of secondary waves that have arisen after the mode conversion.${\ displaystyle P_ {i}}$${\ displaystyle PS_ {r}}$${\ displaystyle PP_ {r}}$${\ displaystyle PS_ {t}}$${\ displaystyle PP_ {t}}$

The law of refraction also applies to mechanical waves , i.e. pressure or shear waves . In the context of acoustics and ultrasound technology, Snell's law of refraction is formulated without refractive indices, but with the help of the wave number . The following applies (see adjacent picture for the angle designations): ${\ displaystyle k}$

${\ displaystyle k = k_ {P1} \ sin \ theta _ {P1} = k_ {S1} \ sin \ theta _ {S1} = k_ {P2} \ sin \ theta _ {P2} = k_ {S2} \ sin \ theta _ {S2}.}$

With the definition with , one obtains the law of refraction in the formulation with the phase velocities of the relevant wave types in the relevant medium and thus the same formulation as in optics (if the vacuum speed of light were reduced there). The law in acoustics is derived from the requirement that the continuity equation for mechanical stresses and displacements at the media boundary be fulfilled . The adjacent picture shows an incident longitudinal wave in a solid , which is partially reflected and transmitted at an interface with a second solid. In general, new wave types arise from the incident longitudinal wave (P wave) at the interface, so that two different wave types are reflected and transmitted: P waves and S waves ( shear wave ). Both types of waves propagate in the two media with different phase velocities, so they are also refracted at different angles. These angles can be calculated using the above law if the individual phase velocities and the angle of incidence of the primary wave are known. In the case of media without shear stress (liquids and gases), no shear waves occur, so that the incident P wave would only generate a reflected and a transmitted P wave. ${\ displaystyle k_ {i} = {\ frac {2 \ pi} {\ lambda _ {i}}} = {\ frac {2 \ pi f} {c_ {i}}}}$${\ displaystyle i = \ {S1, S2, P1, P2 \}}$${\ displaystyle c_ {P1}, \, c_ {S1}, \, c_ {P2}, \, c_ {S2}}$${\ displaystyle \ theta _ {P1}}$

## literature

• Eugene Hecht: optics . 4th, revised edition. Oldenbourg Wissenschaftsverlag, Munich a. a. 2005, ISBN 3-486-27359-0 .
• Klaus Hentschel : The law of refraction in the version of Snellius. Reconstruction of his path of discovery and a translation of his Latin manuscript and supplementary documents, Archive for History of Exact Sciences 55.4 (2001): 297-344.

## Individual evidence

1. Torsten Fließbach : Textbook on Theoretical Physics. Volume 2: Electrodynamics. 4th edition. Spectrum, Academic Publishing House, Heidelberg a. a. 2004, ISBN 3-8274-1530-6 (Chapter 36).
2. Lucio Russo: The forgotten revolution or the rebirth of ancient knowledge . Springer-Verlag, 2005 ( limited preview in Google Book Search [accessed January 22, 2017]).
3. Jim Al-Khalili : The House of Wisdom . S. Fischer, 2011, ISBN 978-3-10-000424-6 , pp. 251 f .
4. Harriot . In: Spectrum of Science (Ed.): Lexicon of Physics . 1998 ( Spektrum.de [accessed on January 22, 2017]).
5. a b Klaus Hentschel: The law of refraction in the version of Snellius . In: Arch. Hist. Exact Sci. tape 55 , no. 4 , 2001, p. 297-344 , doi : 10.1007 / s004070000026 .
6. ^ Constantin Carathéodory : Geometric optics . Julius Springer, 1937, p. 6th f . ( limited preview in Google Book search).
7. cf. Wolfgang Demtröder : Experimental Physics. Volume 1: Mechanics and Warmth. 5th, revised and updated edition. Springer, Berlin a. a. 2008, ISBN 978-3-540-79295-6 .
8. Tribikram Kundu (Ed.): Ultrasonic and Electromagnetic NDE for Structure and Material Characterization . CRC Press, Boca Raton FL et al. 2012, ISBN 978-1-4398-3663-7 , pp. 42–56 ( limited preview in Google Book search).